\\mathcal {F}(\\epsilon )$ , the feasible range of $G$ is $\\mathcal {S}=\\mathcal {S}_1\\bigcup \\mathcal {S}_2$ where $\\mathcal {S}_1=\\lbrace G\\in {Z}^+|1\\le G\\le g_a\\rbrace , \\qquad \\mathcal {S}_2=\\lbrace G\\in {Z}^+| G\\ge g_b \\rbrace ,$ where $g_a$ and $g_b$ are the two roots of the equation $\\xi (G) = \\mathcal {F}(\\epsilon )$ .","Based on Lemma REF , the function $\\xi (G)$ is monotone increasing over $\\mathcal {S}_1$ but monotone decreasing over $\\mathcal {S}_2$ .","In addition, if $g_a < 1$ , $\\mathcal {S}_1$ is empty and the feasibility range of $G$ reduces to $\\mathcal {S}_2$ ."],["Communication Latency"," Recall that the comm-latency of connected mobiles $\\mathsf {T}_{\\textrm {comm}}$ comprises the expected waiting time for offloaded tasks at mobiles, denoted as $\\mathsf {T}_{\\textrm {comm}}^{(a)}$ , and transmission delay, denoted as $\\mathsf {T}_{\\textrm {comm}}^{(b)}$ .","Consider the expected waiting time.","Recalling that the offloading protocol in Section REF , the first task arrival during $L$ slots is delivered to the offloading buffer and the subsequent tasks are forwarded to the local computation unit.","Let $K$ denote the slot index when an offloaded task arrives at the offloading buffer.","It follows that the probability distribution of $K$ follows a conditional geometric distribution, i.e., $\\Pr (K=k)=\\frac{p (1-p)^{k-1}}{1-(1-p)^L}$ , where $k = 1, 2, \\cdots , L$ and the normalization term $1-(1-p)^L$ gives the probability that at least one task arrives during a single frame.","Thereby, the expected waiting time is given as $\\mathsf {T}_{\\textrm {comm}}^{(a)} = \\sum _{k=1}^L(L-k) \\frac{p(1-p)^{k-1}}{1-(1-p)^L}&= \\frac{L}{1-(1-p)^L}-\\frac{1}{p}.$ Next, consider the transmission time for a single task in a frame that spans $L$ slots.","Recall that $L=G\\mathsf {T}_{\\min }$ where $\\mathsf {T}_{\\min }$ is the minimum time for transmitting a task as defined earlier.","Combining $\\mathsf {T}_{\\textrm {comm}}^{(b)} = G\\mathsf {T}_{\\min }$ and $\\mathsf {T}_{\\textrm {comm}}^{(a)}$ in (REF ) gives the following result.","Lemma 3 (Comm-Latency) Given the spreading factor $G$ , the comm-latency of the typical mobile $\\mathsf {T}_{\\textrm {comm}} $ (in slot) is given as $\\mathsf {T}_{\\textrm {comm}}(G)= G{\\mathsf {T}_{\\min }}+ \\frac{G \\mathsf {T}_{\\min }}{1 - (1-p)^{G \\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $\\mathsf {T}_{\\min }$ is the minimum time for transmitting a task using full bandwidth.","Next, consider the minimization of the comm-latency over the spreading factor $G$ .","Using (REF ), it is straightforward to show that the comm-latency $\\mathsf {T}_{\\textrm {comm}}(G)$ is a monotone increasing function of $G$ .","Therefore, minimizing comm-latency is equivalent to minimizing $G$ .","It follows from Proposition REF that the minimum of $G$ , $G^*=\\underset{G\\in {\\mathcal {S}}}{\\min } G$ , is given as $G^* = \\left\\lbrace \\begin{aligned}&g_b, &&\\mathcal {S}_1 = \\emptyset ,\\\\&1, && \\text{otherwise}.\\end{aligned}\\right.$ Substituting $G^*$ into (REF ) gives the minimum comm-latency as shown in the following theorem.","Theorem 1 (Minimum Comm-Latency) By optimizing the spreading factor $G$ , the minimum comm-latency (in slot), denoted as $\\mathsf {T}_{\\textrm {comm}}^* $ , is given as follows.","If $\\mathcal {S}_1$ in (REF ) is non-empty, $\\mathsf {T}_{\\textrm {comm}}^*= \\mathsf {T}_{\\min }+ \\frac{\\mathsf {T}_{\\min } }{1 - (1-p)^{\\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $\\mathsf {T}_{\\min } = \\frac{\\ell }{B \\cdot t_0 \\cdot \\log _2(1+\\theta )}$ .","If $\\mathcal {S}_1$ is empty, $\\mathsf {T}_{\\textrm {comm}}^*= g_b\\mathsf {T}_{\\min }+ \\frac{g_b \\mathsf {T}_{\\min } }{1 - (1-p)^{g_b \\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $g_b$ is specified in Proposition REF .","Consider the second case in Theorem REF .","The comm-latency $\\mathsf {T}_{\\textrm {comm}}^*$ can be approximated in closed-form if $g_b\\mathsf {T}_{\\min }$ is sufficiently large.","For this case, $\\left[1 - (1-p)^{g_b \\mathsf {T}_{\\min }}\\right]\\approx 1$ and thus the function $\\xi (G)$ in (REF ) can be approximated as $\\xi (G) \\approx \\frac{2(1-\\delta )\\ln \\delta ^{-1}}{\\alpha } \\mathcal {B}( \\alpha ) \\left( \\frac{\\lambda _m}{\\lambda _b}\\right) \\left(\\frac{\\theta }{G}\\right)^{\\frac{2}{\\alpha }}.$ It follows from Theorem REF and (REF ) that if $\\mathcal {S}_1$ is empty and $g_b\\mathsf {T}_{\\min }$ is large, $\\mathsf {T}_{\\textrm {comm}}^*\\approx 2 g_b\\mathsf {T}_{\\min } - \\frac{1}{p}, $ where $g_b\\approx \\left[ \\frac{(1-\\delta )\\ln \\delta ^{-1}\\mathcal {B}( \\alpha ) }{\\alpha \\epsilon }\\left( \\frac{\\lambda _m}{\\lambda _b}\\right)\\right]^{\\frac{\\alpha }{2}}\\theta .$ Remark 4 (Sparse Network vs.","Dense Network) The first and second cases in Theorem REF correspond to sparse and dense networks, respectively, as measured by the mobile-to-AP density ratio $\\lambda _m/\\lambda _b$ .","In the first case ($\\mathcal {S}_1\\ne \\emptyset $ ), the network is sufficiently sparse, namely the ratio $\\lambda _m/\\lambda _b$ is sufficiently small, such that the optimal spreading factor $G^*=1$ and the resultant comm-latency is independent of the ratio as shown in the theorem.","In other words, for this case, it is optimal to allocate all bandwidth for increasing the transmission rate instead of reducing it for the purpose of suppressing interference to satisfy the network connectivity constraint.","In contrast, in the second case ($\\mathcal {S}_1= \\emptyset $ ), the network is relatively dense and it is necessary to apply spread spectrum to reduce interference so as to meet the connectivity requirement, corresponding to $G^* > 1$ .","As the result, the minimum comm-latency scales with the density ratio as $\\mathsf {T}_{\\textrm {comm}}^*\\propto \\left(\\frac{\\lambda _m}{\\lambda _b}\\right)^{\\frac{\\alpha }{2}}$ as one can observe from (REF ).","Remark 5 (Effects of Network Parameters) Substituting $\\mathsf {T}_{\\min } = \\frac{\\ell }{B \\cdot t_0\\cdot \\log _2(1+\\theta )}$ into (REF ) gives that for a relatively dense network, the comm-latency scales as $\\boxed{\\mathsf {T}_{\\textrm {comm}}^*\\propto \\frac{\\ell }{B}\\left(\\frac{\\lambda _m}{\\epsilon \\lambda _b}\\right)^{\\frac{\\alpha }{2}} - \\frac{1}{p}.","}$ The scaling laws shows the effects of network parameters including the task size $\\ell $ , bandwidth $B$ , mobile density $\\lambda _m$ and AP density $\\lambda _b$ , and the task-generation probability per slot $p$ ."],["Task-Arrival Rates at APs/CSs","The offloading throughput of the RAN represents the load of the CSN (see Fig.","REF ).","The throughput can be measured by the expected task-arrival rate (in number of tasks per slot) at the typical AP (equivalently the typical CS).","Its scaling law with the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ , is not straightforward due to several factors.","To be specific, the total bandwidth is fixed, the spread factor grows nonlinearly with $\\lambda _m/\\lambda _b$ , and the likelihood of task-generation probability per frame varies with the frame length.","To address this issue, the task arrivals at the typical AP are characterized as follows.","Consider the case of asynchronous offloading.","Based on the model in Section REF , the probability that a mobile generates a task for offloading in each frame is $p_L^*=1 - (1-p)^{L^*}, $ where $L^*$ is the frame length given the optimal spreading factor $G^*$ in (REF ).","The expected task-offloading rate (in number of tasks per slot) for the typical mobile, denoted as $\\beta ^*$ , is given as $\\beta ^* = \\frac{p_L^*}{L^*}$ .","Since $L^* = G^*\\mathsf {T}_{\\min }$ , $\\beta ^*=\\frac{1 - (1-p)^{G^*\\mathsf {T}_{\\min }}}{G^*\\mathsf {T}_{\\min }}.$ where $\\beta ^*=p_L^* \\cdot G^*\\mathsf {T}_{\\min }$ .","Let $\\bar{\\Lambda }^*$ denote the expected task-arrival rate at the typical AP (or CS).","Then $\\bar{\\Lambda }^* = \\bar{N}\\beta ^*$ where $\\bar{N}$ is the expected number of mobiles connected to the AP.","Since $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ , $\\bar{\\Lambda }^* = (1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}\\beta ^*.$ Remark 6 (Effects of Network Parameters) Using (REF ), (REF ) and (REF ), one can infer that $\\bar{\\Lambda }^* \\propto \\left\\lbrace \\begin{aligned}& p\\frac{\\lambda _m}{\\lambda _b}, && \\frac{\\lambda _m}{\\lambda _b}\\rightarrow 0,\\\\&B\\left(\\frac{\\lambda _m}{\\lambda _b}\\right)^{-\\frac{\\alpha }{2}+1}, && \\frac{\\lambda _m}{\\lambda _b}\\rightarrow \\infty .\\end{aligned}\\right.$ The first case corresponds to a sparse network whose performance is not limited by bandwidth and interference.","Then the expected task arrival-rate grows linearly with the task-generation probability per slot, $p$ , and the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ .","For the second case, in a dense network that is bandwidth-and-interference limited, the rate grows linearly the bandwidth $B$ , but decreases with $\\lambda _m/\\lambda _b$ .","The reason for the decrease is the bandwidth for offloading is reduced so that a larger spreading factor is available for suppressing interference to meet the network-coverage requirement.","Consequently, the load for the CSs is lighter for a dense (thus comm-limited) network, reducing comp-latency as shown in the sequel.","Consider tasks arrivals for the case of synchronous offloading.","Unlike the asynchronous counterpart with arrivals spread over each frame, the tasks from mobiles arrive the typical AP at the beginning of each frame.","Thus, it is useful to characterize the expected number of task arrivals per frame, denoted as $\\bar{A}^*$ , which can be written as $\\bar{A}^* = \\bar{N} p_L^*$ .","It follows that $\\bar{A}^* =(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b} p_L^*.","$ Remark 7 (Effects of Network Parameters) In a dense network ($\\lambda _m/\\lambda _b\\rightarrow \\infty $ ), it can be obtained from (REF ), (REF ), and (REF ) that $p_L^* \\approx 1$ .","Then it follows from (REF ) that the expected number of tasks per frame increases linearly with the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ ."],["Computation Latency Analysis: Asynchronous Offloading","This section aims at analyzing the comp-latency of the asynchronous offloading where task arrival and departure are randomly distributed over time.","Given the Markov model of Fig.","REF , we derive the network stability condition in Definition REF and bounds of the average comp-latency."],["Optimal Control of VMs","On one hand, creating a large number of VMs at the typical CS can slow down its computation rate due to the mentioned I/O interference between VMs.","On the other hand, too few VMs can lead to marginal gain from parallel computing.","Therefore, the number of VMs should be optimally controlled based on the number of waiting tasks.","To this end, let $\\mu (m)$ denote the computation rate given $m$ VMs.","Given the computation model in (REF ), it follows from $\\mu (m)=m/T_c(m)$ that: $\\mu (m) = \\frac{m}{T_0}(1+d)^{1-m}.$ By analyzing the derivative of $\\mu (m)$ , one can find that the function is monotone increasing before reaching a global maximum and after that it is monotone decreasing.","Thereby, the value of $m$ that maximizes $\\mu (m)$ , denoted as $m_{\\max }$ , can be found with the integer constraint $m_{\\max } = \\textrm {round}\\left(\\frac{1}{\\ln (1+d)}\\right),$ where $\\textrm {round}(x)$ rounds $x$ to the nearest integer.","The said properties of the function $\\mu (m)$ and the derived $m_{\\max }$ in (REF ) suggest the following optimal VM-control policy.","Proposition 2 (Optimal VM Control) To maximize the computation rate at the typical CS, the optimal VM-control policy is to create $m_{\\max }$ VMs if there are a sufficient number of tasks for computation or otherwise create as many VMs as possible until the buffer is empty.","Consequently, the maximum computation rate, denoted as $\\mu ^*(m)$ , given $m$ tasks at the CS (being computed or in the buffer) is $\\mu ^*(m) = \\left\\lbrace \\begin{aligned}&\\frac{m}{T_0}(1+d)^{1-m}, &&1\\le m \\le m_{\\max }, \\\\&\\frac{m_{\\max }}{T_0}(1+d)^{1-m_{\\max }}, &&m > m_{\\max },\\end{aligned}\\right.$ where $m_{\\max }$ is given in (REF ).","For ease notation, the maximum computation rate, $\\mu ({m_{\\max }})$ , is re-denoted as $\\mu _{\\max }$ hereafter."],["Computation Rates under Network Stability Constraint"," This subsection focuses on analzying the condition for the maximum computation rate of the typical CS to meet the network stability constraint in Definition REF .","The analysis combines the results from queueing theory, stochastic geometry and parallel computing.","The said constraint requires $\\rho $ -fraction of mobiles, or equivalently $\\rho $ -fraction of CSs, to be stable, namely that comp-latency is finite.","According to queuing theory, stabilizing a typical CS requires that the task-arrival rate $\\Lambda $ should be strictly smaller than the maximum departure rate $\\mu ^*(m_{\\max })$ : $\\Lambda <\\mu _{\\max }$ [28].","Note that the former is a RV proportional to the random number of mobiles, $N$ , connected to the typical CS while the latter is a constant.","Then the stability probability $\\mathsf {p}_s$ is given as $\\mathsf {p}_s =\\Pr [\\Lambda <\\mu _{\\max }]=\\Pr \\left[N <\\frac{\\mu _{\\max }}{\\beta ^*}\\right],$ where $\\beta ^*$ is the task-offloading rate given in (REF ).","It follows from the network spatial model that $N$ is a Poisson distributed RV with the mean $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ .","Using the distribution and (REF ) and applying Chernoff bound, we can obtain an upper bound on the maximum computation rate required to meet the stability constraint as shown below.","Proposition 3 (Computation Rates for $\\rho $ -Stability) For the CSN to be $\\rho $ -stable, a sufficient condition for the maximum computation rate of the typical CS is given as $\\mu _{\\max }\\ge \\bar{\\Lambda }^*\\cdot \\exp \\left(W\\left(\\frac{-\\ln (\\rho )}{\\bar{N} e} - \\frac{1}{e}\\right)+1\\right),$ where $W(\\cdot )$ is the Lambert function, the expected mobiles connected to the typical CS $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ , and $\\bar{\\Lambda }^*$ represents the expected arrival rate given in (REF ).","Proof: See Appendix REF .","$\\Box $ The above result shows that to satisfy the network-stability constraint, the maximum computation rate of each CS, $\\mu _{\\max }$ , should be larger than the expected task-arrival rate, $\\bar{\\Lambda }^*$ , scaled by a factor larger than one, namely the exponential term in (REF ).","Moreover, the factor grows as the stability probability $(1-\\rho )$ increases.","Last, it is useful for subsequent analysis to derive the expected arrival rate conditioned on that the typical CS is stable as shown below.","Lemma 4 (Expected Task-Arrival Rates for Stable CSs) Given that the typical CS is stable, the expected task-arrival rate is given as $\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]=\\bar{\\Lambda }^* \\left(1-\\frac{\\Pr (N = \\left\\lfloor R\\right\\rfloor )}{1 - \\rho }\\right),$ where $R=\\frac{\\mu _{\\max }}{\\beta ^*}$ measures the maximum number of mobiles the CS can serve, $\\beta ^*$ is the task-offloading rate per mobile in (REF ), $\\bar{N}$ and $\\bar{\\Lambda }^*$ follow those in Proposition REF , and the Poisson distribution function $\\Pr (N = n )=\\frac{\\bar{N}^ne^{-\\bar{N}}}{n!","}$ .","Proof: See Appendix REF .","$\\Box $"],["Expected Computation Latency","In this subsection, the expected comp-latency, $\\mathsf {T}_{\\textrm {comp}}$ , is analyzed using the Markov chain in Fig.","REF and applying queueing theory.","Exact analysis is intractable due to the fact that the departure rate $\\mu (m)$ in the Markov chain is a non-linear function of state $m$ .","This difficulty is overcome by modifying the Markov chain to give two versions corresponding to a M/M/m and a M/M/1 queues, yielding an upper and a lower bounds on $\\mathsf {T}_{\\textrm {comp}}$ , respectively.","First, consider upper bounding $\\mathsf {T}_{\\textrm {comp}}$ .","To this end, the departure rate $\\mu (m)$ in the Markov chain in Fig.","REF with the following lower bound obtained by fixing all exponents as $(1-m_{\\max })$ : $\\mu ^-(m) = {\\left\\lbrace \\begin{array}{ll}\\frac{m}{T_0}(1+d)^{1-m_{\\max }}, ~~~1\\le m \\le m_{\\max }, \\\\\\frac{m_{\\max }}{T_0}(1+d)^{1-m_{\\max }}, ~~~m > m_{\\max }.\\end{array}\\right.","}$ As a result, the modified Markov chain is a M/M/$m_{\\max }$ queue.","The corresponding waiting time, denoted as $\\mathsf {T}_{\\textrm {comp}}^+$ , upper bounds $\\mathsf {T}_{\\textrm {comp}}$ since it reduces the computation rate.","Applying classic results on M/M/m queues (see e.g., [28]), the waiting time, $\\mathsf {T}_{\\textrm {comp}}^+$ , for task arrival rate $\\Lambda $ is $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda ) &= \\frac{m_{\\max }}{\\mu ^-(m_{\\max })} + \\frac{\\tau \\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^{m_{\\max }}}{m_{\\max }!\\mu ^-(m_{\\max })\\left( 1-\\frac{\\Lambda }{m_{\\max }\\mu ^-(m_{\\max })} \\right)^2} , $ where the coefficient $\\tau $ is given as $\\tau = \\left[\\sum _{m=0}^{m_{\\max }-1} \\frac{1}{m!","}\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^m + \\sum _{m=m_{\\max }}^{\\infty }\\frac{m_{\\max }^{m_{\\max }-m}}{m_{\\max }!","}\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^m\\right]^{-1}.","$ Using (REF ), (REF ) and (REF ), the upper bound is given in the following theorem.","02 Theorem 2.A (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.","The average comp-latency is upper bounded as $\\mathsf {T}_{\\textrm {comp}} \\le \\frac{m_{\\max }}{\\mu _{\\max }} + \\left(\\frac{m_{\\max }}{\\mu _{\\max }}\\right)^2\\cdot \\frac{\\bar{\\Lambda }^*}{(m_{\\max }-1)!\\left( m_{\\max }-1 \\right)^2}\\cdot \\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right),$ where $ R$ follows that in Lemma REF , and $\\bar{\\Lambda }^*$ and $\\mu _{\\max }$ are specified in (REF ) and (REF ), respectively.","Proof: See Appendix REF .","$\\Box $ Note that the positive factor $\\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right)$ accounts for Poisson distribution of mobiles.","Next, a lower bound on $\\mathsf {T}_{\\textrm {comp}}$ is obtained as follows.","One can observe from the Markov chain in Fig.","REF that for states $m \\le m_{\\max }$ , the departure rates are smaller than the maximum, $\\mu _{\\max }$ .","The reason is that for these states, there are not enough tasks for attaining the maximum rate by parallel computing.","Then replacing all departure rates in the said Markov chain with the maximum $\\mu _{\\max }$ leads to a lower bound on $\\mathsf {T}_{\\textrm {comp}}$ .","The resultant Markov chain corresponds to a M/M/1 queue.","Then using the modified Markov chain and the well-known results from M/M/1 queue (see e.g., [28]), the comp-latency for given arrival rate $\\Lambda $ can be lower bounded as $\\mathsf {T}_{\\textrm {comp}}(\\Lambda ) \\ge \\frac{1}{\\mu _{\\max }-\\Lambda }.$ By taking expectation over $\\Lambda $ and applying Jensen's inequality, $\\mathsf {T}_{\\textrm {comp}}= \\mathsf {E}[\\mathsf {T}_{\\textrm {comp}}(\\Lambda )]\\ge \\mathsf {E}\\left[\\left.\\frac{1}{\\mu _{\\max }-\\Lambda }\\right|\\Lambda <\\mu _{\\max }\\right]\\ge \\frac{1}{\\mu _{\\max }-\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]}.$ Using (REF ) and Lemma REF , we obtain the following result.","Theorem 2.B (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.","The average comp-latency is lower bounded as $\\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max } - \\bar{\\Lambda }^*\\cdot \\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right)},$ where $ R$ follows that in Lemma REF , and $\\bar{\\Lambda }^*$ and $\\mu _{\\max }$ are specified in (REF ) and (REF ), respectively.","Remark 8 (Computation-Resource Provisioning) Consider a MEC network provisioned with sufficient computation resources, $\\mu _{\\max }/\\bar{\\Lambda }^*\\gg 1$ .","It follows from Theorem REF $\\nonumber \\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max }}\\left(1 + \\frac{c_1\\bar{\\Lambda }^*}{\\mu _{\\max }}\\right),$ where $c_1$ is a constant.","This lower bound has a similar form as the upper bound in Theorem REF .","From these results, one can infer that the comp-latency for asynchronous offloading can be approximated written in the following form: $\\boxed{\\mathsf {T}_{\\textrm {comp}}\\approx \\frac{c_2}{\\mu _{\\max }}\\left(1 + \\frac{c_3\\bar{\\Lambda }^*}{\\mu _{\\max }}\\right), \\qquad \\frac{\\mu _{\\max }}{\\bar{\\Lambda }^*}\\gg 1,}$ where $\\lbrace c_2, c_3\\rbrace $ are constants.","The result suggests that to contain comp-latency, the provisioning of computation resources for the MEC network must consider two factors.","First of all, the maximum computation rate, $\\mu _{\\max }$ , for each CS must be sufficient large.","At the same time, the computation rate must scale linearly with the total arrival rate such that the computation resource allocated for a single offloaded task, measured by the ratio $\\mu _{\\max }/\\bar{\\Lambda }^*$ , is sufficiently large."],["Energy Efficiency"," Based on the above analytical results so far, the subsection tempts to discuss the energy savings of offloading than local computing.","First, the energy consumption of offloading, denoted by $\\mathsf {E}_{\\mathrm {off}}$ , can be derived via multiplying mobile's transmission power $P$ by the offloading duration $G^* \\mathsf {T}_{\\min }$ .","To satisfy the minimum average signal strength at the boundary of the MEC service zone, $P$ should scale with the radius ${r_0}$ as $P\\propto r_0^{\\alpha }$ .","Recalling $r_0\\propto \\lambda _b^{-\\frac{1}{2}}$ and $\\mathsf {T}_{\\min }\\propto \\ell $ where $\\ell $ is the task size, the resultant energy consumption of $\\mathsf {E}_{\\mathrm {off}}$ is given as $\\mathsf {E}_{\\mathrm {off}}=c_4 \\frac{G^* \\ell }{{\\lambda _b}^{\\frac{\\alpha }{2}}},$ where $c_4$ is a constant depending on the minimum signal strength and $\\mathsf {T}_{\\min }$ .","Next, it is well studied in[6] that the optimal $\\mathsf {E}_{\\textrm {loc}}$ is proportional to ${\\ell }^3$ , and inversely proportional to the square of the deadline requirement which could be set as the total latency $\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}}$ without loss of generality.","Thus, $\\mathsf {E}_{\\textrm {loc}}$ is given as $\\mathsf {E}_{\\textrm {loc}}=c_5 \\frac{{\\ell }^3}{(\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}})^2},$ where $c_5$ is a constant depending on the chip architecture.","where $c_5$ is a constant depending on the chip architecture.","As a result, the condition of energy savings, namely $\\mathsf {E}_{\\mathrm {off}}<\\mathsf {E}_{\\textrm {loc}}$ , is given in terms of $\\ell $ and $\\lambda _b$ as $\\ell ^2 \\lambda _b^{\\frac{\\alpha }{2}}>\\frac{c_4}{c_5}{G^*}\\left(\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}}\\right)^2.$ It is observed that the right-side of (REF ) is dominantly affected by the expected number of mobiles $\\frac{\\lambda _m}{\\lambda _b}$ (see (REF ), (REF ), (REF ) and (REF )).","In other words, given a mobile density $\\lambda _m$ and the task size $\\ell $ , there exists the minimum density of APs $\\lambda _b$ to satisfy the condition in (REF ).","Then under the condition in (REF ), the energy savings due to offloading is given as $\\mathsf {E}_{\\mathrm {loc}}-\\mathsf {E}_{\\mathrm {off}}$ with $\\mathsf {E}_{\\mathrm {off}}$ and $\\mathsf {E}_{\\mathrm {loc}}$ given in (REF ) and (REF ), respectively."],["MEC Network Provisioning and Planning"," Combining the results from the preceding analysis on comm-latency and comp-latency yields some guidelines for the provisioning and planning of a MEC network as discussed below.","Assume that the network is required to support computing for mobiles with density $\\lambda _m$ with targeted expected comm-latency $\\mathsf {T}_{\\textrm {comm}}$ and comp-latency $\\mathsf {T}_{\\textrm {comp}}$ .","The network resources are quantified by the bandwidth $B$ , the density of AP (or CS) $\\lambda _b$ , the maximum computing rate of each CS $\\mu _{\\max }$ .","First, consider the planning of the RAN.","Combining the above results suggest the following guidelines for network provisioning and planning.","As shown in Section REF , under the network-coverage constraint $(1-\\epsilon )$ , the expected task-arrival rate at a AP, representing the RAN offloading throughput, is a quasi-concave function of the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ , with a global maximum.","Therefore, given a mobile density $\\lambda _m$ , the AP density should be chosen for maximizing the RAN offloading throughput.","Next, based on results in Theorem REF and (REF ), sufficient large channel bandwidth $B$ should be provisioned to achieve the targeted $\\mathsf {T}_{\\textrm {comm}}$ for given mobile and AP densities, mobile task-generation rates, and task sizes.","Next, consider the planning of the CSN.","Under the network-stability constraint, the maximum CS computing rate for parallel computing should be planned to be larger than the expected task-arrival rate scaled by a factor larger than one, which is determined by the allowed fraction of unstable CSs (see Proposition REF ).","Then, the maximum computing rate should be further planned to achieve the targeted $\\mathsf {T}_{\\textrm {comp}}$ for computing an offloaded task using Theorems REF and REF .","In the preceding section, the process of asynchronous task arrival at a CS can be approximated using a Markov chain, allowing tractable analysis of comp-latency using theories of M/M/$m$ and M/M/1 queues.","This approach is inapplicable for synchronous offloading and the resultant periodic task arrivals at the CS.","Though tractable analysis in general is difficult, it is possible for two special cases defined as follows.","Definition 3 (Special Cases: Light-Traffic and Heavy-Traffic) A light-traffic case refers to one that the task-arrival rate is much smaller than the computation rate such that the queue at the CS is always empty as observed by a new arriving task.","In contrast, a heavy-traffic case refers to one that the task-arrival rate is close to the computation rate such that there are always at least $m_{\\max }$ tasks in the queue.","The comp-latency for these two special cases are analyzed to give insights into the performance of CSN with underloaded CSs and those with overloaded CSs."],["Expected Computation Latency with Light-Traffic "," First, the dynamics of the task queue at the typical CS is modelled as follows.","Recall that task arrivals are periodical, occurring at the beginning of every frame.","Consider the typical CS.","Let $\\mathcal {Q}_t$ and $\\mathcal {A}_t$ denote be the numbers of existing and arriving tasks at the beginning of frame $t$ , respectively, and $\\mathcal {C}_t$ the number of departing tasks during frame $t$ .","Then the evolution of $\\mathcal {Q}_t$ can be described mathematically as $\\mathcal {Q}_{t+1} = \\max \\left[ \\mathcal {Q}_t + \\mathcal {A}_{t+1} - \\mathcal {C}_t, 0 \\right].$ The general analysis of comp-latency using (REF ) is difficult.","The main difficulty lies in deriving the distribution of $\\mathcal {C}_t$ that depends on the number of VMs that varies continuously in time since the computation time for simultaneous tasks are random and inter-dependent.","To overcome the difficulty, consider the case of light-traffic where a number of offloaded tasks arrives at the typical CS to see an empty queue and an idling server.","Correspondingly, the evolution equation in (REF ) is modified such that given $\\mathcal {A}_{t+1}\\ne 0 $ , $\\mathcal {Q}_t = \\mathcal {C}_t = 0 $ , yielding $\\mathcal {Q}_{t+1} = \\mathcal {A}_{t+1}$ .","Next, given this simple equality, deriving the expected comp-latency reduces to analyzing the latency for computing a random number of $\\mathcal {A}$ tasks at the CS, which arrives at the beginning of an arbitrary frame.","Without loss of generality, the tasks are arranged in an ascending order in terms of computation time and referred to as Task $1, 2, \\cdots , \\mathcal {A}$ .","Moreover, let $\\mathcal {L}_n$ denote the expected computing time for Task $n$ and hence $\\mathcal {L}_1 \\le \\mathcal {L}_2 \\le \\cdots \\le \\mathcal {L}_{\\mathcal {A}}$ .","Then the expected comp-latency $\\mathsf {T}_{\\textrm {comp}}$ can be written in terms of $\\lbrace \\mathcal {L}_n\\rbrace $ as $\\mathsf {T}_{\\textrm {comp}} = \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} {\\mathcal {L}_n} }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right].","$ To obtain bounds on $\\mathsf {T}_{\\textrm {comp}}$ in closed form, a useful result is derived as follows.","Given $m$ VMs, recall that the computation time of a task follows the exponential distribution with the mean being the inverse of the computation rate $\\mu (m)$ in (REF ).","Using the memoryless property of exponential distribution, a useful relation between $\\lbrace \\mathcal {L}_n\\rbrace $ is obtained as $\\mathcal {L}_n =\\mathcal {L}_{n-1}+ \\frac{1}{\\mu (m)}=\\left\\lbrace \\begin{aligned}&\\mathcal {L}_{n-1}+ \\frac{1}{\\mu _{\\max }}, && 1\\le n\\le \\mathcal {A}-m_{\\max }+1,\\\\&\\mathcal {L}_{n-1} +\\frac{1}{\\mu (\\mathcal {A} - n +1)}, && \\text{otherwise},\\end{aligned}\\right.$ with $\\mathcal {L}_0=0$ .","Note that $\\mu (1) \\le \\mu (m) \\le \\mu _{\\max }$ for all $m$ .","Thus, it follows from (REF ) that $\\frac{n}{\\mu _{\\max }} \\le \\mathcal {L}_n \\le \\frac{n}{\\mu (1)}.","$ Substituting (REF ) into (REF ) gives $\\frac{1}{\\mu _{\\max }}\\cdot \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} n }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right]\\le \\mathsf {T}_{\\textrm {comp}} \\le \\frac{1}{\\mu (1)} \\cdot \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} n }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right].$ Recalling the number of arriving tasks $\\mathcal {A}$ follows a Poisson RV with mean $\\bar{A}^*$ of (REF ), bounds on the comp-latency is obtained shown in the following theorem.","Theorem 3 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with light-traffic.","The expected comp-latency can be bounded as $\\nonumber \\frac{1}{2\\mu _{\\max }}\\left(1+\\frac{\\bar{A}^*}{1-e^{-\\bar{A}^*}}\\right)\\le \\mathsf {{T}}_{\\textrm {comp}} \\le \\frac{1}{2\\mu (1)}\\left(1+\\frac{\\bar{A}^*}{1-e^{-\\bar{A}^*}}\\right).$ where $\\bar{A}^*$ is the expected number of arriving tasks to the typical CS per frame given in (REF ).","Remark 9 (Comparison with Asynchronous Offloading) From the results in the above theorem, one can infer that for the current case, the comm-latency can be approximated as $\\boxed{\\mathsf {{T}}_{\\textrm {comp}} \\approx \\left\\lbrace \\begin{aligned}&\\frac{1}{\\mu (1)}, && \\bar{A}^* \\rightarrow 0,\\\\&\\frac{\\bar{A}^*}{2\\mu (m_{\\max })},&& \\bar{A}^* \\gg 1.\\end{aligned}\\right.","}$ Comparing the expression with the counterpart for asynchronous offloading in (REF ), it is unclear which case leads to longer comp-latency.","However, simulation shows that in general, synchronizing offloading tends to incur longer latency by overloading CSs and thereby suffering more from I/O interference in parallel-computing."],["Expected Computation Latency with Heavy-Traffic","This subsection focuses on analyzing the expected comp-latency, $\\mathsf {T}_{\\textrm {comp}}$ , for the case of heavy-traffic as defined in Definition REF .","For this case, with the queue being always non-empty, the equation in (REF ) describing the queue evolution reduces to $\\mathcal {Q}_{t+1} = \\mathcal {Q}_t + \\mathcal {A}_t - \\mathcal {C}_t$ .","The key step in deriving $\\mathsf {T}_{\\textrm {comp}}$ is to apply the said equation to the analysis of the expected queue length.","The technique involves taking expectation of the squares of the two sides of the equation as follows: $\\mathsf {E}\\left[ \\mathcal {Q}^2_{t+1} \\right]= \\mathsf {E}\\left[ \\left(\\mathcal {Q}_t + \\mathcal {A}_t - \\mathcal {C}_t\\right)^2\\right]&=\\mathsf {E}\\left[ \\mathcal {Q}^2_t \\right]+\\mathsf {E}\\left[\\left( \\mathcal {A}_t - \\mathcal {C}_t \\right)^2\\right]+2\\mathsf {E}\\left[{\\mathcal {Q}}_t (\\mathcal {A}_t - \\mathcal {C}_t) \\right].$ Since $\\mathcal {Q}_t$ , $\\mathcal {A}_t$ and $\\mathcal {C}_t$ are independent of each other and $\\mathsf {E}\\left[ \\mathcal {Q}^2_{t+1} \\right]=\\mathsf {E}\\left[ \\mathcal {Q}^2_t\\right]$ given the stable CS, $\\mathsf {E}\\left[ \\mathcal {Q}\\right] = \\frac{\\mathsf {E}\\left[\\mathcal {A}^2\\right] + \\mathsf {E}\\left[\\mathcal {C}^2\\right] - 2\\mathsf {E}\\left[\\mathcal {A}\\right]\\mathsf {E}\\left[\\mathcal {C}\\right]}{2\\left( \\mathsf {E}\\left[\\mathcal {C}\\right] - \\mathsf {E}\\left[\\mathcal {A}\\right] \\right)}, $ where the subscripts $t$ of $\\mathcal {Q}_t$ , $\\mathcal {A}_t$ and $\\mathcal {C}_t$ are omitted to simplify notation.","Given the number of connected mobiles, $N$ , the number of arrival tasks $\\mathcal {A}$ follows a Poisson distribution with the first and second moments being $\\mathsf {E}\\left(\\mathcal {A}\\mid N \\right)={N} p_L^*$ and $\\mathsf {E}\\left(\\mathcal {A}^2\\mid N \\right)=N p_L^*+(Np_L^*)^2$ respectively, where the task-offloading probability $p_L^*$ is given in (REF ).","Next, under the heavy-traffic assumption, the total computation rate of the CS is $\\mu _{\\max }$ .","It follows that the departure process at the typical CS is Poisson distributed where the first and second moments are $\\mathsf {E}[\\mathcal {C}]= \\mu _{\\max }L^*$ and $\\mathsf {E}[\\mathcal {C}^2]= \\mu _{\\max }L^*+[ \\mu _{\\max }L^*]^2$ , respectively.","Substituting the results into (REF ) gives $\\mathsf {E}\\left[ \\mathcal {Q}\\mid N \\right] =\\frac{{N} p_L^*}{\\mu _{\\max }L^*-{N} p_L^*}+\\frac{1}{2}\\cdot \\left(\\mu _{\\max }L^*-{N} p_L^*\\right)+\\frac{1}{2}.","$ In addition, to satisfy the condition for stabilizing the CS, the arrival rate $\\mathsf {E}\\left[\\mathcal {A}\\right]$ should be strictly smaller than the departure rate $\\mathsf {E}[\\mathcal {C}]$ .","This places a constraint on the maximum of $N$ , namely that $N \\le \\left\\lfloor R \\right\\rfloor $ with $ R$ defined in Lemma REF .","Under this constraint, applying Little's theorem obtains the expected comp-latency $\\mathsf {T}_{\\textrm {comp}}$ as $\\mathsf {T}_{\\textrm {comp}}& = \\left\\lbrace \\mathsf {E}\\left[\\left.","\\frac{\\mathsf {E}\\left[ \\mathcal {Q}\\mid N \\right]}{\\mathsf {E}\\left[ \\mathcal {A}\\mid N \\right]}\\right|N \\le \\left\\lfloor R \\right\\rfloor \\right]-\\frac{1}{2}\\right\\rbrace \\cdot L^*.","$ Combining (REF ) and (REF ) yields the main result of this sub-section as shown below.","Theorem 4 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with heavy-traffic.","The expected comp-latency is given as $\\mathsf {T}_{\\textrm {comp}}=\\left\\lbrace \\frac{1}{p_L^*}\\mathsf {E}\\left[\\frac{1}{ R-N}\\right] +\\frac{1}{2}\\left( R + \\frac{1}{p_L^*}\\right) \\mathsf {E}\\left[\\frac{1}{N}\\right] - 1\\right\\rbrace L^*,$ where the constant $ R$ and the distribution of $N$ follow those in Lemma REF .","Remark 10 (Comparison with Asynchronous Offloading) By applying Jensen's inequality, the comp-latency of (REF ) can be lower bounded as $\\mathsf {T}_{\\textrm {comp}}&\\ge \\frac{1}{\\mu _{\\max }-c_4\\bar{\\Lambda }^*}+\\frac{L^*}{2}\\cdot \\frac{\\mu (m_{\\max })}{c_4\\bar{\\Lambda }^*}+\\frac{1}{c_4\\bar{\\Lambda }^*}- L^*,$ where $c_4$ is a constant.","Since for the case of heavy-traffic, the task-arrival rate $c_4\\bar{\\Lambda }^*$ approaches the maximum computation rate $\\mu _{\\max }$ , $\\boxed{\\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max }-c_4\\bar{\\Lambda }^*},\\qquad c_4\\bar{\\Lambda }^*\\rightarrow \\mu _{\\max }.","}$ The above lower bound has the same form as the asynchronous-offloading counterpart in (REF ).","Both diverge as the task-arrival rate approaches the maximum computation rate."],["Simulation Results"," In this section, analytical results on comm-latency and comp-latency are evaluated by simulation.","The simulation parameters have the following default settings unless specified otherwise.","The densities of APs and mobiles are $\\lambda _b = 2 \\times 10^{-2} \\ \\mathrm {m}^{-2}$ and $\\lambda _m = 5 \\times 10^{-2} \\ \\mathrm {m}^{-2}$ , respectively.","The $\\mathsf {SIR}$ threshold is set as $\\theta = 1$ dB and the path-loss exponent is $\\alpha = 3$ .","For the network coverage parameter $\\delta $ is $\\delta = 10^{-2}$ , corresponding to the radius of MEC service zone being $r_0 = 12 \\mathrm {m}$ .","The total bandwidth is $B = 6$ MHz.","The data size per task is fixed as $\\ell = 0.5 \\times 10^{6}$ bits.","The single-task computation time $T_0$ in the parallel-computation model is set as $T_0 = 0.1$ (sec) and the factor arising from I/O interference is $d = 0.2$ .","The task generation probability per slot is $p = 0.2$ .","The parameters $\\epsilon $ and $\\rho $ are both set as $0.05$ .","Figure: Effect of task generating rate.We present the comparisons between Monte Carlo simulations ($10^4$ realizations) and analytical results in all figures.","For each realization, both mobiles and APs are distributed in the plane based on the PPPs.","Each mobile randomly generates an offloading task and transmits it to its corresponding AP.","Then the AP generates VMs and performs the computation upon the task arrivals.","A queue of tasks will appear if the task arrival rate is large than the computation rate (e.g., too many tasks arrived at the AP at the same time).","The VM will be released when the corresponding task is computed.","Fig.","REF compares expected comm-latency and comp-latency for the case of asynchronous offloading.","The effects of mobile density $\\lambda _m$ and task generating rate $p$ are investigated and several observations can be made.","As shown in Fig.","REF (a), the expected comp-latency as a function of the mobile density is observed to exhibit the quasi-concavity described in Remark REF .","In contrast, the expected comm-latency is a monotone increasing function following the scaling law in Remark REF .","These properties lead to the partitioning of the range of mobile density into three network-operation regimes as indicated in Fig.","REF (a).","In particular, the middle range corresponds to comp-limited regime while others are comm-limited.","Next, consider the effect of task-generation rate at mobiles, specified by the task-generation probability $p$ .","Both types of latency are observed to converge to corresponding limits as the rate grows.","Their different scaling laws result in the partitioning of the range of task-generation rate into comm-limited and comp-limited regimes.","Last, one can observe from both figures that the lower bound on comp-latency as derived in (REF ) is tighter than the upper bond therein.","Fig.","REF compares expected comm-latency and comp-latency for the case of synchronous offloading.","The same observations in the case of synchronous offloading also apply in the current case except that the quasi-concavity of the expected comp-latency with respect to mobile density is not shown in the considered range.","Some new observations can be made as follows.","Comparing Fig.","REF and REF shows that synchronizing offloading results in longer comp-latency.","Next, the center of the comp-limited range in Fig.","REF (a) corresponds to the case of heavy-traffic studied in Section REF .","Consequently, the derived upper bound on expected comp-latency for this case is tight.","For other ranges of mobile density, the bounds derived for the case of light traffic are tighter.","Last, Fig.","REF (b) shows that the expected comp-latency is tightly approximated by bounds derived for the light-traffic case when the task-arrival rate is small ($\\le 0.3$ ) and by that for the heavy-traffic case when the rate is large ($> 0.7$ ), validating the results.","Figure: Effect of task generating rate."],["Conclusion Remarks","In this work, we have first studied the network-constrained latency performance of a large-scale MEC network, namely comm-latency and comp-latency under the constraints of RAN connectivity and CSN stability.","To study the tradeoffs between these metrics and constraints and model the cascaded architecture of RAN and CSN, the MEC network has been modelled using stochastic geometry featuring diversified aspects of wireless access and computing.","Based on the model, the average comm-latency and comp-latency have been analyzed by applying the theories of stochastic geometry, queuing and parallel computing.","In particular, their scaling laws have been derived with respect to various network parameters ranging from the densities of mobiles and APs to the computation capabilities of CSs.","The results provide useful guidelines for MEC-network provisioning and planning to avoid either the RAN or CSN being a performance bottleneck.","The current work can be extended in several directions.","In this work, we consider a single type of computation task of which the task size and the average computation time are identical.","However, considering different types of tasks makes the latency analysis more challenging but is of practical relevance.","Next, studying a large-scale hierarchical fog computing network comprising mobiles, edge cloud and central cloud is aligned with recent advancements in edge computing.","Last, considering advanced techniques such as VM migration and cooperative computing to reduce the latency will be another promising direction."],["Proof of Lemma "," The first derivative of $\\xi (G)$ for $G$ given in (REF ) can be derived as: $\\frac{\\partial \\xi (G)}{\\partial G} = -\\frac{\\Delta }{\\alpha }\\left( G^{-\\frac{2 + \\alpha }{\\alpha }}\\left( \\alpha G \\mathsf {T}_{\\min } \\ln (1-p) (1-p)^{G \\mathsf {T}_{\\min }} - 2(1-p)^{G \\mathsf {T}_{\\min }} + 2 \\right) \\right), $ where $\\Delta = \\frac{2(1-\\delta )\\ln \\delta ^{-1}}{\\alpha } \\mathcal {B}( \\alpha ) \\left( \\frac{\\lambda _m}{\\lambda _b}\\right) \\theta ^{\\frac{2}{\\alpha }}$ .","The existence of $g_0$ is easily proved because (REF ) is strictly positive and negative when $G \\rightarrow 0$ and $G \\rightarrow \\infty $ respectively.","Next, the solution of $g_0$ to solve $\\frac{\\partial \\xi (G)}{\\partial G} = 0$ is $g_0 =\\frac{\\alpha W\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right) + 2}{\\alpha \\mathsf {T}_{\\min } \\ln (1-p)}$ , where $W(x)$ is the Lambert function.","Since the value inside the Lambert function is negative, there are two candidates for $g_0$ : one is from the principle branch of Lambert function $W\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right)$ , and the other is the lower branch $W_{-1}\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right)$ .","The principle branch makes $g_0=0$ , but the lower branch satisfies $g_0>0$ , completing the proof."],["Proof of Proposition "," Applying Chernoff bound on $\\mathsf {p}_s$ in (REF ) makes $\\mathsf {p}_s &\\ge 1- \\exp \\left(- \\frac{\\mu _{\\max }}{\\beta ^*} \\ln \\left(\\frac{ \\mu _{\\max } }{\\bar{N} \\beta ^*}\\right) +\\frac{\\mu _{\\max }}{\\beta ^*}-\\bar{N}\\right)\\ge 1 - \\rho ,$ which is equivalent to $\\mu _{\\max }\\ge \\bar{\\Lambda }^*\\cdot \\exp \\left(W\\left(\\frac{-\\ln (\\rho )}{\\bar{N} e} - \\frac{1}{e}\\right)+1\\right)$ as Proposition REF shows."],["Proof of Lemma "," Noting the expect task arrival of stable CSs is proportional to the average number of connected mobiles.","Therefore, we have $\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]= \\beta ^{*} \\cdot \\frac{\\sum _{n=1}^{ R } n \\cdot \\frac{\\bar{N}^{n} e^{-\\bar{N}}}{n!","}}{1- \\rho } =\\bar{N}\\beta ^* \\left(1-\\frac{\\Pr (N = \\left\\lfloor R\\right\\rfloor )}{1 - \\rho }\\right)$ , where $\\Pr (N = \\left\\lfloor R\\right\\rfloor ) = \\frac{\\bar{N}^{\\left\\lfloor R\\right\\rfloor } e^{-\\bar{N}}}{\\left\\lfloor R\\right\\rfloor !","}$ , ending the proof."],["Proof of Theorem "," Substituting $\\tau = 1$ into (REF ) and then applying $\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^{m_{\\max }} \\le \\frac{\\Lambda }{\\mu ^-(m_{\\max })}$ as well as $\\left(1-\\frac{\\Lambda }{m_{\\max }\\mu ^-(m_{\\max })} \\right)^2\\le \\left(1-\\frac{1}{m_{\\max }}\\right)^2$ give an upper bound of $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda )$ as $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda ) \\le \\frac{m_{\\max }}{\\mu ^-(m_{\\max })} + \\frac{\\frac{\\Lambda }{\\mu ^-(m_{\\max })}}{m_{\\max }!\\mu ^-(m_{\\max })\\left( 1-\\frac{1}{m_{\\max }} \\right)^2}.","$ The spatial average on (REF ) based on Lemma REF gives the final result in Theorem REF ."],["Appendix II: Notation Table"," Table: Summary of NotationIn Fig.","REF , the the comp-latency of the Poisson arrival assumption of task-arrival process is compared with that under the Poisson arrival assumption of Assumption REF , showing that the tasks arrival process at AP can be well approximated to the Poisson task arrival assumption.","Fig.","REF represents the effects of AP density, which is shown that the same observations are made in the case of decreasing mobile density.","Figs.","REF and REF show the effects of the computation capability of CS, the computation time $T_0$ and the degradation factor $d$ , on the comp-latency for asynchronous and synchronous offloading cases, respectively.","Both of the comp-latencies increase exponentially when the edge-computing capability worse, i.e., $T_0$ or $d$ becomes larger, which agrees with the intuition.","Finally, we plot the ratio between the comp-latency of asynchronous and synchronous offloading cases in Fig.","REF , which are always less than one because the comp-latency of the asynchronous offloading is always smaller than the synchronous counterpart.","Specifically, in Fig.","REF (a), the ratio first stays constant when mobiles is sparse and then decreases with mobile density grows.","As mobiles becomes denser, the number of offloading tasks at the beginning of each frame increases, resulting in severe I/O interference than the asynchronous counterpart of which the task arrivals are distributed over frame.","On the other hand, in Fig.","REF (b), it is observed that the ratio grows and then converges to a constant when $p$ increases.","In other words, the gap of comp-latency between two offloading cases is becoming smaller when the task arrival becomes heaver, aligning with the discussion in Remark REF ."]],"string":"[\n [\n \"Wireless Networks for Mobile Edge Computing: Spatial Modeling and\\n Latency Analysis (Extended version)\"\n ],\n [\n \"Abstract Next-generation wireless networks will provide users ubiquitous low-latency computing services using devices at the network edge, called mobile edge computing (MEC).\",\n \"The key operation of MEC, mobile computation offloading (MCO), is to offload computation intensive tasks from users.\",\n \"Since each edge device comprises an access point (AP) and a computer server (CS), a MEC network can be decomposed as a radio access network (RAN) cascaded with a CS network (CSN).\",\n \"Based on the architecture, we investigate network constrained latency performance, namely communication latency (comm-latency) and computation latency (comp-latency) under the constraints of RAN coverage and CSN stability.\",\n \"To this end, a spatial random network is modeled featuring random node distribution, parallel computing, non-orthogonal multiple access, and random computation-task generation.\",\n \"Given the model and the said network constraints, we derive the scaling laws of comm-latency and comp-latency with respect to network-load parameters (density of mobiles and their task-generation rates) and network-resource parameters (bandwidth, density of APs/CSs, CS computation rate).\",\n \"Essentially, the analysis involves the interplay of theories of stochastic geometry, queueing, and parallel computing.\",\n \"Combining the derived scaling laws quantifies the tradeoffs between the latencies, network coverage and network stability.\",\n \"The results provide useful guidelines for MEC-network provisioning and planning by avoiding either of the cascaded RAN or CSN being a performance bottleneck.\"\n ],\n [\n \"Introduction\",\n \"One key mission of 5G systems is to provide users ubiquitous computing services (e.g., multimedia processing, gaming and augmented reality) using servers at the network edge, called mobile edge computing (MEC) [1].\",\n \"Compared with cloud computing, MEC can dramatically reduce latency by avoiding transmissions over the backhaul network, among many other advantages such as security and context awareness [2], [3].\",\n \"Most existing work focuses on designing MEC techniques by merging two disciplines: wireless communications and mobile computing.\",\n \"In this work, we explore a different direction, namely the design of large-scale MEC networks with infinite nodes.\",\n \"To this end, a model of MEC network is constructed featuring spatial random distribution of network nodes, wireless transmissions, parallel computing at servers.\",\n \"Based on the model and under network performance constraints, the latencies for communication and computation are analyzed by applying theories of stochastic geometry, queueing, and parallel computing.\",\n \"The results yield useful guidelines for MEC network provisioning and planning.\",\n \"To realize the vision of Internet-of-Things (IoT) and smart cities, MEC is a key enabler providing ubiquitous and low latency access to computing resources.\",\n \"Edge servers in proximity of users are able to process a large volume of data collected from IoT sensors and provide intelligent real-time solutions for various applications, e.g., health care, smart grid, and autonomous driving.\",\n \"Due to its promising potential and the interdisciplinary nature, many new research issues arise in the area of MEC and are widely studied in different fields (see e.g., [3], [4], [5]).\",\n \"In the area of MEC, one research thrust focuses on designing techniques for enabling low-latency and energy-efficient mobile computation offloading (MCO), which offloads computation intensive tasks from mobiles to the edge servers [6], [7], [8], [9], [10], [11], [12], [13].\",\n \"In [6], considering a CPU with a controllable clock, the optimal policy is derived using stochastic-optimization theory for jointly controlling the MCO decision (offload or not) and clock frequency with the objective of minimum mobile energy consumption.\",\n \"A similar design problem is tackled in [7] using a different approach based on Lyapunov optimization theory.\",\n \"Besides MCO, the battery lives of mobile devices can be further lengthened by energy harvesting [8] or wireless power transfer [9].\",\n \"The optimal policies for MEC control are more complex as they need to account for energy randomness [8] or adapt the operation modes (power transfer or offloading) [9].\",\n \"Designing energy-efficient MEC techniques under computation-deadline constraints implicitly attempts to optimize the latency-and-energy tradeoff.\",\n \"The problem of optimizing this tradeoff via computation-task scheduling is formulated explicitly in [10] and [11] and solved using optimization theory.\",\n \"In addition, other design issues for MEC are also investigated in the literature such as optimal program partitioning for partial offloading [12] and data prefetching based on computation prediction [13].\",\n \"Recent research in MEC focuses on designing more complex MEC systems for multiuser MCO [14], [15], [16], [17], [18], [19], [20].\",\n \"One important issue is the joint radio-and-computation resource allocation for minimizing sum mobile energy consumption under their deadline constraints.\",\n \"The problem is challenging due to the multiplicity of parameters and constraints involved in the problem including multi-user channel states, computation capacities of servers and mobiles, and individual deadline and power constraints.\",\n \"A tractable approach for solving the problem is developed in [14] for a single-cell system comprising one edge server for multiple users.\",\n \"Specifically, a so-called offloading priority function is derived that includes all the parameters and used to show a simple threshold based structure of the optimal policy.\",\n \"The problem of joint resource allocation in multi-cell systems is further complicated by the existence of inter-cell interference.\",\n \"An attempt is made in [15] to tackle this problem using optimization theory.\",\n \"In distributed systems without coordination, mobiles make individual offloading decisions.\",\n \"For such systems, it is proposed in [16] that game theory is applied to improve the performance of distributed joint resource allocation in terms of latency and mobile energy consumption.\",\n \"Cooperation between edge servers (or edge clouds) allows their resource pooling and sharing, which helps overcome their limitations in computation capacity.\",\n \"Algorithms for edge-cloud cooperation are designed in [17] based on game theory that enables or disables cooperation so as to maximize the revenues of edge clouds under the constraint of meeting mobiles' computation demands.\",\n \"Compared with the edge cloud, the central cloud has unlimited computation capacity but its long distance from users can incur long latency for offloading.\",\n \"Nevertheless, cooperation between edge and central clouds is desirable when the formers are overloaded.\",\n \"Given such cooperation, queueing theory is applied in [18] to analyze the latency for computation offloading.\",\n \"On the other hand, cooperation between edge clouds can support mobility by migrating computation tasks between servers.\",\n \"Building on the migration technology, a MEC framework for supporting mobility is proposed in [19] to adapt the placements of offloaded tasks in the cloud infrastructure depending on the mobility of the task owners.\",\n \"Besides offloaded tasks, computing services can be also migrated to adapt to mobility but service migration can place a heavy burden on the backhaul network or result in excessive latency.\",\n \"To address this issue, the framework of service duplication by virtualization is proposed in [20].\",\n \"Prior work considers small-scale MEC systems with several users and servers/clouds, allowing the research to focus on designing complex MCO techniques and protocols.\",\n \"On the other hand, it is also important to study a large-scale MEC network with infinite nodes as illustrated in Fig.\",\n \"REF , which is an area not yet explored.\",\n \"From the practical perspective, such studies can yield guidelines and insights useful for operators' provisioning and planning of MEC networks.\",\n \"Figure: A MEC network where mobiles offload computation tasks to computer servers (CSs) by wireless transmission to access points (APs).Figure: The decomposition view of the MEC network.Figure: The spatial model of a MEC network.\"\n ],\n [\n \"Modeling Wireless Networks for Mobile Edge Computing \",\n \"In the past decade, stochastic geometry has been established as a standard tool for modeling and designing wireless networks, creating an active research area [21].\",\n \"A rich set of spatial point processes such as Poisson point process (PPP) and cluster processes have been used to model node locations in a wide range of wireless networks such as cellular networks [22], heterogeneous networks [23], and cognitive radio networks [24].\",\n \"Based on these network models and applying mathematical tools from stochastic geometry, the effects of most key physical-layer techniques on network performance have been investigated ranging from multi-antenna transmissions [25] to multi-cell cooperation [26].\",\n \"Recent advancements in the area can be found in numerous surveys such as [27].\",\n \"Most existing work in this area shares the same theme of how to cope with interference and hostility of wireless channels (e.g., path loss and fading) so as to ensure high coverage and link reliability for radio access networks (RAN) or distributed device-to-device networks.\",\n \"In contrast, the design of large-scale MEC networks in Fig.\",\n \"REF has different objectives, all of which should jointly address two aspects of network performance, namely wireless communication and edge computing.\",\n \"Modeling a MEC network poses new challenges as its architecture is more complex than a traditional RAN and can be decomposed as a RAN cascaded with a computer-server network (CSN) as illustrated in Fig.\",\n \"REF .\",\n \"The power of modeling MEC networks using stochastic geometry lies in allowing network performance to be described by a function of a relatively small set of network parameters.\",\n \"To be specific, as shown in Fig.\",\n \"REF , the process of mobiles is parametrized by mobile density, the RAN by channel bandwidth and access-point (AP) density, and the CSN by CS density and CS computation capacity.\",\n \"Besides the parameters, the performance of a MEC network is measured by numerous metrics.\",\n \"Like small-scale systems (see e.g., [10] and [11]), the link-level performance of the MEC network is measured by latency, which can be divided into latency for offloading in the RAN, called communication latency (comm-latency) and latency for computing at CSs, called computation latency (comp-latency).\",\n \"At the network level, the coverage of the RAN of a MEC network is typically measured by connectivity probability (also called coverage probability [27]), quantifying the fraction of users having reliable links to APs.\",\n \"A similar metric, called stability probability, can be defined for measuring the stability of the CSN, quantifying the fraction of CSs having finite comp-latency.\",\n \"There exist potentially complex relations between these four metrics that are regulated by the said network parameters.\",\n \"Existing results focusing solely on RAN (see e.g., [27]) are insufficient for quantifying these relations.\",\n \"Instead, it calls for developing a more sophisticated analytical approach integrating theories of stochastic geometry, queueing, and parallel computing.\",\n \"Last, it is worth mentioning that comm-latency and comp-latency have been extensively studied in the literature mostly for point-to-point systems using queueing theory (see e.g., [28], [29]).\",\n \"However, studying such latency in large-scale networks is much more challenging due to the existence of interference between randomly distributed nodes.\",\n \"As a result, there exist only limited results on comm-latency in such networks [30], [31], [32].\",\n \"In [30], the comm-latency given retransmission is derived using stochastic geometry for the extreme cases with either static nodes or nodes having high mobility.\",\n \"The analysis is generalized in [31] for finite mobility.\",\n \"Then the approach for comm-latency as proposed in [30] and [31] is further developed in [32] to integrate stochastic geometry and queueing theory.\",\n \"Compared with these studies, the current work considers a different type of network, namely the MEC network, and explores a different research direction, namely the tradeoff between comm-latency and comp-latency under constraints on the mentioned network-level performance metrics.\"\n ],\n [\n \"Contributions\",\n \"This work represents the first attempt on modeling a large-scale MEC network using stochastic geometry.\",\n \"The proposed model has several features admitting tractable analysis of network latency performance.\",\n \"First, the locations of co-located pairs of CS and AP and the mobiles are distributed as two independent homogeneous PPPs.\",\n \"Second, multiple access is enabled by spread spectrum [33], which underpins the technology of code-domain Non-Orthogonal Multiple Access (NOMA) to be deployed in 5G systems for enabling massive access [34].\",\n \"Using the technology, interference is suppressed by a parameter called spreading factor, denoted as $G$ , at the cost of data bandwidth reduction.\",\n \"Third, each mobile randomly generates a computation task in every time slot.\",\n \"Last, each CS computes multiple tasks simultaneously by parallel computing realized via creating a number of virtual machines (VMs), where the so called input/output (I/O) interference in parallel computing is modelled [35].\",\n \"In this work, we propose an approach building on the spatial network model and the joint applications of tools from diversified areas including stochastic geometry, queueing, and parallel computing.\",\n \"Though the network performance analysis relies on well known tools, their applications are far more than straightforward.\",\n \"In fact, new challenges arise from the coupling of communication and edge computing in the MEC network.\",\n \"For example, the simple server model (with memoryless service time) in the traditional queueing theory is now replaced with a more complex MEC server model featuring dynamic virtual machines and their I/O interference.\",\n \"As another example, the random computing-task arrivals are typically modelled as a single stochastic process in conventional computing/queueing systems but the current model has to account for numerous network features ranging from random node distributions to multiple access.\",\n \"The complex network model introduces new technical challenges that call for the development of a systematic framework for studying the MEC network performance and deployment, which forms the theme of this work.\",\n \"The main contributions are summarized below.\",\n \"Modeling a MEC network using stochastic geometry: As mentioned, this work presents a novel model of a large-scale MEC network constructed using stochastic geometry.\",\n \"Given the complexity of the network, the contribution in network modeling lies in proposing a model that is not only sufficiently practical but at the same time allows a tractable approach of analyzing network latency performance, by integrating stochastic geometry, parallel computing, and queuing theory.\",\n \"The results and insights are summarized as follows.\",\n \"Communication latency: The expected comm-latency for an offloaded task, denoted as $\\\\mathsf {T}_{\\\\textrm {comm}}$ , is minimized under a constraint on the network connectivity probability.\",\n \"This is transformed into a constrained optimization problem of the spreading factor $G$ .\",\n \"Solving the problem yields the minimum $\\\\mathsf {T}_{\\\\textrm {comm}}$ .\",\n \"The result shows that when mobiles are sparse, the full bandwidth should be allocated for data transmission so as to minimize $\\\\mathsf {T}_{\\\\textrm {comm}}$ .\",\n \"However, when mobiles are dense, spread spectrum with large $G$ is needed to mitigate interference for satisfying the network-coverage constraint, which increases $\\\\mathsf {T}_{\\\\textrm {comm}}$ .\",\n \"As a result, the minimum $\\\\mathsf {T}_{\\\\textrm {comm}}$ diminishes inversely proportional to the channel bandwidth and as a power function of the allowed fraction of disconnected users with a negative exponent, but grows sub-linearly with the expected number of mobiles per AP (or CS).\",\n \"In addition, $\\\\mathsf {T}_{\\\\textrm {comm}}$ is a monotone increasing function of the task-generation probability per slot that saturates as the probability approaches one.\",\n \"Analysis of RAN offloading throughput: The RAN throughput, which determines the load of the CSN (see Fig.\",\n \"REF ), can be measured by the expected task-arrival rate at a typical AP (or CS).\",\n \"The rate is shown to be a quasi-concave function of the expected number of mobiles per AP, which first increases and then decreases as the ratio grows.\",\n \"In other words, the expected task-arrival rate is low in both sparse and dense networks.\",\n \"The maximum rate is proportional to the bandwidth.\",\n \"Computation latency Analysis: First, to maximize CS computing rates, it is shown that the dynamic number of VMs at each CS should be no more than a derived number to avoid suffering rate loss due to their I/O interference.\",\n \"Then to ensure stable CSN, it is shown that the resultant maximum computing rate should be larger than the task-arrival rate scaled by a factor larger than one, which is determined by the allowed fraction of unstable CSs.\",\n \"Based on the result for parallel computing, tools from stochastic geometry and M/M/m queues are applied to derive bounds on the expected comp-latency for an offloaded task, denoted as $\\\\mathsf {T}_{\\\\textrm {comp}}$ .\",\n \"The bounds show that the latency is inversely proportional to the maximum computing rate and linearly proportional to the total task-arrival rate at the typical CS (or AP).\",\n \"Consequently, $\\\\mathsf {T}_{\\\\textrm {comp}}$ is a quasi-concave function of the expected number of users per CS (or AP) while $\\\\mathsf {T}_{\\\\textrm {comm}}$ is a monotone increasing function.\",\n \"Network provisioning and planning: Combining the above results suggest the following guidelines for network provisioning and planning.\",\n \"Given a mobile density, the AP density should be chosen for maximizing the RAN offloading throughput under the network-coverage constraint.\",\n \"Then sufficient bandwidth should be provisioned to simultaneously achieve the targeted comm-latency for offloading a task.\",\n \"Last, given the mobile and RAN parameters, the CS computation capacities are planned to achieve the targeted comp-latency for a offloaded task as well as enforcing the network-stability constraint.\",\n \"The derived analytical results simplify the calculation in the specific planning process.\"\n ],\n [\n \"Modeling MEC Networks\",\n \"In this section, a mathematical model of the MEC network as illustrated in Fig.\",\n \"REF is presented.\"\n ],\n [\n \"Network Spatial Model\",\n \" APs (and thus their co-located CSs) are randomly distributed in the horizontal plane and are modelled as a homogeneous PPP $\\\\Omega = \\\\lbrace Y\\\\rbrace $ with density $\\\\lambda _b$ , where $Y \\\\in {R}^{2}$ is the coordinate of the corresponding AP.\",\n \"Similarly, mobiles are modelled as another homogeneous PPP $\\\\Phi = \\\\lbrace X\\\\rbrace $ independent of $\\\\Omega $ and having the density $\\\\lambda _m$ .\",\n \"Define a MEC-service zone for each AP, as a disk region centered at $Y$ and having a fixed radius $r_0$ , denoted by $\\\\mathcal {O}(Y, r_0)$ , determined by the maximum uplink transmission power of each mobile (see Fig.\",\n \"REF ).\",\n \"A mobile can access a AP for computing if it is covered by the MEC-service zone of the AP.\",\n \"It is possible that a mobile is within the service ranges of more than one AP.\",\n \"In this case, the mobile randomly selects a single AP to receive the MEC service.\",\n \"As illustrated in Fig.\",\n \"REF , combining the randomly located MEC-service zones, $\\\\cup _{Y \\\\in \\\\Omega } \\\\mathcal {O}(Y, r_0)$ , forms a coverage process.\",\n \"Covered mobiles are referred to as active ones and others inactive since they remain silent.\",\n \"To achieve close-to-full network coverage, let the fraction of inactive mobiles be no more than a small positive number $\\\\delta $ .\",\n \"Then the radius of MEC-service zones, $r_0$ , should be set as $r_0 = \\\\sqrt{\\\\frac{\\\\ln \\\\frac{1}{\\\\delta }}{\\\\pi \\\\lambda _b }}$ [27].\",\n \"Given $r_0$ , the number of mobiles covered by an arbitrary MEC service zone follows a Poisson distribution with mean $\\\\lambda _m \\\\pi r_0^2$ .\",\n \"Consider a typical AP located at the origin.\",\n \"Let $X_0$ denote a typical mobile located in the typical MEC service zone $\\\\mathcal {O}(o, r_0)$ .\",\n \"Without loss of generality, the network performance analysis focuses on the typical mobile.\"\n ],\n [\n \"Model of Mobile Task Generation\",\n \" Time is divided into slots having a unit duration.\",\n \"Consider an arbitrary mobile.\",\n \"A computation task is randomly generated in each slot with probability $p$ , referred to as the task-generation rate0.1cmThe random task generation is an abstracted model allowing tractable analysis, and it is widely used in the literature in the same vein.\",\n \"The statistics of task generation can be empirically measured by counting the number of user service requests, which is shown in [36] and [37] to be bursty and periodical.\",\n \"It is interesting to use a more general task generation model, which is outside the scope of current work..\",\n \"The generated tasks are those favorable offloading in terms of energy efficiency such that offloading can save more energy than the local computing.\",\n \"The analysis on the offloading favorable condition will be given in the sequel.\",\n \"Task generations over two different slots are assumed to be independent.\",\n \"The mobile has a unit buffer to store at most a single task for offloading.\",\n \"A newly generated task is sent for offloading when the buffer is empty or otherwise computed locally.\",\n \"This avoids significant queueing delay that is unacceptable in the considered case of latency-sensitive mobile computation.\",\n \"For simplicity, offloading each task is assumed to require transmission of a fixed amount data.\",\n \"The transmission of a single task occupies a single frame lasting $L$ slots.\",\n \"The mobile checks whether the buffer is empty at the end of every $L$ slots and transmits a stored task to a serving AP.\",\n \"Define the task-offloading probability as the probability that the mobile's buffer is occupied, denoted as $p_L$ .\",\n \"Equivalently, $p_L$ gives the probability that at least one task is generated within one frame: $p_L=1-(1-p)^L.$ Thereby, the task-departure process at a mobile follows a Bernoulli process with parameter $p_L$ provided the radio link is reliable (see discussion in the sequel).\"\n ],\n [\n \"Radio Access Model\",\n \"Consider an uplink channel with the fixed bandwidth of $B$ Hz.\",\n \"The channel is shared by all mobiles for transmitting data containing offloaded tasks to their serving APs.\",\n \"The CDMA (or code-domain NOMA) is applied to enable multiple access.\",\n \"For CDMA based on the spread-spectrum technology, each mobile spreads every transmitted symbol by multiplying it with a pseudo-random (PN) sequence of chips (1s and $-1$ s), which is generated at a much higher rate than the symbols and thereby spreads the signal spectrum [33].\",\n \"The multiple access of mobiles is enabled by assigning unique PN sequences to individual users.\",\n \"A receiver then retrieves the signal sent by the desired transmitter by multiplying the multiuser signal with the corresponding PN sequence.\",\n \"The operation suppresses inference and de-spreads the signal spectrum to yield symbols.\",\n \"Let $G$ denote the spreading factor defined as the ratio between the chip rate and symbol rate, which is equivalent to the number of available PN sequences.\",\n \"The cross-correlation of PN sequences is proportional to $\\\\frac{1}{G}$ and approaches to zero as $G$ increases.\",\n \"As a result, the interference power is reduced by the factor of $G$ [33].0.1cmFor the special case of synchronous multiuser transmissions, orthogonal sequences (e.g., Hadamard sequences) can be used instead of PN sequences to achieve orthogonal access [33].\",\n \"However, the maximum number of simultaneous users is $G$ , making the design unsuitable for massive access.\",\n \"On the other hand, the price for spread spectrum is that the bandwidth available to individual mobiles is reduced by $G$ , namely $\\\\frac{B}{G}$ .\",\n \"Remark 1 (CDMA vs. OFDMA) While CDMA is expected to enable non-orthogonal access in next-generation systems, orthogonal frequency division multiple access (OFDMA) has been widely deployed in existing system.\",\n \"However, OFDMA limits the number of simultaneous users to be no more than the number of orthogonal sub-channels.\",\n \"Compared with OFDMA, CDMA separates different users by PN sequences.\",\n \"The number of possible PN sequences can be up to $2^G -1$ with $G$ being the spreading factor (sequence length).\",\n \"In theory, an equal number of simultaneous users can be supported by CDMA that can be potentially much larger than that by OFDMA.\",\n \"Allowing non-orthogonality via CDMA provides a graceful tradeoff between the system-performance degradation and the number of simultaneous users, facilitating massive access in 5G.\",\n \"The current analysis of comm-latency can be straightforwardly extended to OFDMA by removing interference between scheduled users.\",\n \"For unscheduled users, comm-latency should include scheduling delay and the corresponding analysis is standard (see e.g., [38]).\",\n \"Uplink channels are characterized by path-loss and small-scale Rayleigh fading.\",\n \"Assuming transmission by a mobile with the fixed power $\\\\eta $ , the received signal power at the AP is given by $\\\\eta g_X |Y - X|^{-\\\\alpha }$ , where $\\\\alpha $ is the path-loss exponent, the $\\\\exp (1)$ random variable (RV) $g_X$ represents Rayleigh fading and $|X-Y|$ denotes the Euclidian distance between $X$ and $Y$ .\",\n \"Based on the channel model, the power of interference at the typical AP $Y_0$ , denoted by $I$ , can be derived as follows.\",\n \"Among potential interferers for the typical AP, the fraction of $\\\\delta $ is outside MEC-service zones.\",\n \"Given random task generation discussed earlier, each interferer transmits with probability $p_L$ .\",\n \"Consequently, the active interferers form a PPP given by $\\\\tilde{\\\\Phi }$ with density $(1-\\\\delta )p_L\\\\lambda _m$ resulting from thinning $\\\\Phi $ .\",\n \"It follows that the interference power $I$ can be written as $I = \\\\frac{1}{G}\\\\sum _{X \\\\in \\\\tilde{\\\\Phi }} \\\\eta g_X |X|^{-\\\\alpha }$ , where the factor $\\\\frac{1}{G}$ is due to the spread spectrum.\",\n \"Consider an interference-limited radio-access network where channel noise is negligible.\",\n \"The received $\\\\mathsf {SIR}$ of the typical mobile is thus given as $\\\\mathsf {SIR}_0 = \\\\frac{g_{X_0} |X_0|^{-\\\\alpha }}{\\\\frac{1}{G}\\\\sum _{X \\\\in \\\\tilde{\\\\Phi }} \\\\eta g_X |X|^{-\\\\alpha }}.$ The condition for successful offloading is that $\\\\mathsf {SIR}$ exceeds a fixed threshold $\\\\theta $ depending on the coding rate.\",\n \"Specifically, given $\\\\theta $ , the spectrum efficiency is $\\\\log _2(1+\\\\theta )$ (bits/sec/Hz) [27].\",\n \"It follows that to transmit a task having a size of $\\\\ell $ bits within a frame, the frame length $L$ should satisfy $L =\\\\frac{G\\\\ell }{B \\\\cdot t_0 \\\\cdot \\\\log _2(1+\\\\theta )}$ (in slots) where $t_0$ is the length of a slot (in sec).\",\n \"Define the minimum time for transmitting a task using the full bandwidth $B$ as $\\\\mathsf {T}_{\\\\min } = \\\\frac{\\\\ell }{B \\\\cdot t_0\\\\cdot \\\\log _2(1+\\\\theta )}$ for ease of notation, giving $L = G\\\\mathsf {T}_{\\\\min }$ .\",\n \"Assumption 1 (Slow Fading) We assume that channels vary at a much slower time scale than that for mobile computation.\",\n \"To be specific, the mobile locations and channel coefficients $\\\\lbrace g_X\\\\rbrace $ remain fixed in the considered time window of computation offloading.\",\n \"Remark 2 (Fast Fading) In the presence of sufficiently high mobility, the channel variation can be faster than edge computation, resulting in fast fading.\",\n \"In this case, a mobile facing an unfavorable channel can rely on retransmission to exploit the channel variation for reliable offloading.\",\n \"Nevertheless, this results in retransmission delay and thereby increases comm-latency.\",\n \"It is straightforward to analyze the extra latency in a large-scale network by applying an existing method (see e.g., [30]).\",\n \"By Assumption REF , mobiles' $\\\\mathsf {SIR}$ s remain constant and thereby mobiles can be separated into connected and disconnected mobiles.\",\n \"To be specific, a mobile is connected to an AP if the corresponding SIR is above the threshold $\\\\theta $ or otherwise disconnected.\",\n \"Figure: Markov chain modeling the tasks queueing for computation at the typical CS where Λ\\\\Lambda is the arrival rate and μ(m)\\\\mu (m) is the computation rate given mm waiting tasks.We consider both synchronous and asynchronous multiuser transmissions defined in existing wireless standards such as 3GPP LTE.\",\n \"For synchronous transmissions, the frame boundaries of different users are aligned so as to facilitate protocols such as control signaling and channel feedback.\",\n \"Synchronization incurs network overhead for implementing a common clock as well as increases latency.\",\n \"For asynchronous transmissions, the said constraint on frame boundaries is not applied and thus the transmission of each mobile is independent of those of others.\",\n \"The transmissions modes lead to different task-arrival models for CSs.\",\n \"Specifically, given synchronous transmissions, the offloaded tasks arrive at a CS in batches and periodically as illustrated in Fig.\",\n \"REF (a).\",\n \"The number of arrival tasks in each batch is random depending on the number of connected mobiles in the same MEC-service zone.\",\n \"On the other hand, given asynchronous transmissions, the offloaded tasks arrive at an AP at different time instants as illustrated in Fig.\",\n \"REF (b).\"\n ],\n [\n \"Parallel-Computing Model\",\n \"Upon their arrivals at APs, tasks are assumed to be delivered to CSs without any delay and queue at the CS buffer for computation on the first-come-first-served basis.\",\n \"Moreover, each CS is assumed to be provisioned with large storage modelled as a buffer with infinite capacity.\",\n \"At each CS, parallel computing of multiple tasks is implemented by creating virtual machines (VM) on the same physical machine (PM) [35].\",\n \"VMs are created asynchronously such that a VM can be added or removed at any time instant.\",\n \"It is well known in the literature that simultaneous VMs interfere with each other due to their sharing common computation resources in the PM e.g., CPU, memory, buses for I/O.\",\n \"The effect is called I/O interference that reduces the computation speeds of VMs.\",\n \"The model of I/O interference as proposed in [35] is adopted where the expected computation time for a single taskThe latency caused by creating and releasing VMs is not explicitly considered.\",\n \"It is assumed to be part of computation time., denoted by $T_c$ , is a function of the number of VMs, $m$ : $T_c(m) = T_0 (1+d)^{m-1}, $ where $T_0$ is the expected computation time of a task in the case of a single VM ($m=1$ ) and $d$ is the degradation factor due to I/O interference between VMs.\",\n \"One can observe that $T_c$ is a monotone increasing function of $d$ .\",\n \"For tractability, we assume that the computation time for a task is an $\\\\exp (T_c)$ RV following the common assumption in queueing theory [35].\"\n ],\n [\n \"CS Queuing Model\",\n \"The general approach of analyzing comp-latency relies on the interplay between parallel-computing and queueing theories.\",\n \"In particular, for the case of asynchronous offloading, the task arrival at the typical AP is approximated as a Poisson process for the following reasons.\",\n \"Due to the lack of synchronization between mobiles, the time instants of tasks arrivals are approximately uniform in time.\",\n \"Furthermore, at different time instants, tasks are generated following i.i.d.\",\n \"Bernoulli distributions based on the model in Section REF .\",\n \"It is well known that the superposition of independent arrival process behaves like a Poisson process [39].\",\n \"Assumption 2 For the case of asynchronous offloading, given $N$ connected mobiles and the spreading factor $G$ , the task arrivals at the typical AP are approximated as a Poisson process with the arrival rate of $\\\\Lambda (N, G)= \\\\frac{N p_L }{L} = \\\\frac{N p_L }{G \\\\mathsf {T}_{\\\\min }}$ .\",\n \"The Poisson approximation is shown by simulation to be accurate in Appendix III.\",\n \"Given the Poisson arrival process and exponentially distributed computation time, the random number of tasks queueing at the typical CS can be modelled as a continuous-time Markov chain as illustrated in Fig.\",\n \"REF [28].\",\n \"In the Markov chain, $\\\\Lambda $ denotes the task-arrival rate in Assumption REF and $\\\\mu (k)$ denotes the CS-computation rate (task/slot) given $k$ tasks in the CS.\",\n \"The CS-computation rate is maximized in the sequel by optimizing the number of VMs based on the queue length.\",\n \"Last, the result-downloading phase is not considered for brevity.\",\n \"First, the corresponding latency analysis is similar to that for the offloading phase.\",\n \"Second, the latency for downloading is negligible compared with those for offloading.\",\n \"The reasons are that computation results typically have small sizes compared with offloaded tasks and furthermore downlink transmission rates are typically much higher than uplink rates.\"\n ],\n [\n \"Performance Metrics\",\n \"The network performance is measured by two metrics: comm-latency and comp-latency.\",\n \"The definitions of metrics build on the design constraints for ensuring network connectivity and stability defined as follows.\",\n \"Definition 1 (Network Coverage Constraint) The RAN in Fig.\",\n \"REF is designed to be $\\\\epsilon $ -connected, namely that the portion of mobiles is no less than $(1-\\\\epsilon )$ , where $0< \\\\epsilon \\\\ll 1$ .\",\n \"The fraction of connected mobiles is equivalent to the success probability, a metric widely used for studying the performance of random wireless networks [27].\",\n \"For the MEC network, the success probability is renamed as connectivity probability and defined for the typical mobile as the following function of the spreading factor $G$ : $\\\\mathsf {p}_c(G) = \\\\mathrm {Pr}\\\\left( \\\\mathsf {SIR}_0 \\\\ge \\\\theta \\\\right),$ where $\\\\mathsf {SIR}_0$ is given in (REF ).\",\n \"Then the network coverage constraint can be written as $\\\\mathsf {p}_c(G) \\\\ge (1 - \\\\epsilon )$ .\",\n \"Under the connectivity constraint, most mobiles are connected to APs.\",\n \"Then the comm-latency, denoted as $\\\\mathsf {T}_{\\\\textrm {comm}}$ , is defined as the expected duration required for a connected mobile to offload a task to the connected AP successfully.\",\n \"The latency includes both waiting time at the mobile's buffer and and the transmission time.\",\n \"Next, consider the computation load of the typical AP.\",\n \"Since the number of mobiles connected to the AP is a RV, there exists non-zero probability that the AP is overloaded, resulting in infinite queueing delay.\",\n \"In this case, the connected mobiles are referred to as being unstable.\",\n \"To ensure most mobiles are stable, the following constraint is applied on the network design.\",\n \"Definition 2 (Network Stability Constraint) The CSN in Fig.\",\n \"REF is designed to be $\\\\rho $ -stable, namely that the fraction of stable CSs is no less than $(1-\\\\rho )$ , where $0< \\\\rho \\\\ll 1$ .\",\n \"The fraction $\\\\rho $ is equivalent to the probability that the typical CS is stable, denoted as $\\\\mathsf {p}_s$ .\",\n \"Under the stability constraint, most connected mobiles are stable.\",\n \"Then the comp-latency, denoted by $\\\\mathsf {T}_{\\\\textrm {comp}}$ , is defined for the typical connected mobile as the expected duration from the instant when an offloaded task arrives at the serving CS until the instant when the computation of the task is completed, which includes both queueing delay and actual computation time.\",\n \"Last, given the above definitions, the network is referred to as being communication-limited (comm-limited) if $\\\\mathsf {T}_{\\\\textrm {comm}} \\\\gg \\\\mathsf {T}_{\\\\textrm {comp}}$ and computation-limited (comp-limited) if $\\\\mathsf {T}_{\\\\textrm {comm}} \\\\ll \\\\mathsf {T}_{\\\\textrm {comp}}$ .\"\n ],\n [\n \"Communication Latency Analysis\",\n \"In this section, the comm-latency defined in the preceding section is analyzed building on results from the literature of network modeling using stochastic geometry.\",\n \"Then the latency is minimized by optimizing the spreading factor for CDMA, which regulates the tradeoff between the transmission rates of connected mobiles and network-connectivity performance.\"\n ],\n [\n \"Feasible Range of Spreading Factor\",\n \" As mentioned, the spreading factor $G$ is a key network parameter regulating the tradeoff between network coverage and comm-latency.\",\n \"To facilitate subsequent analysis, under the network constraint in Definition REF , the feasible range of $G$ is derived as follows.\",\n \"The result is useful for minimizing the comm-latency in the next sub-section.\",\n \"To this end, consider the connectivity probability defined in (REF ).\",\n \"Using a similar approach as the well-known one for deriving network success probability using stochastic geometry (see e.g., [22]), we obtain the following result with the proof omitted for brevity.\",\n \"Lemma 1 (Connectivity Probability) Given the spreading factor $G$ , the connectivity probability of a typical mobile is given as $\\\\mathsf {p}_c(G) = \\\\frac{1 - \\\\exp \\\\left(- \\\\xi (G)\\\\right)}{\\\\xi (G)}, $ where $\\\\xi (G)$ is defined as $\\\\xi (G)=\\\\frac{2(1-\\\\delta )\\\\left( 1 - (1-p)^{G \\\\mathsf {T}_{\\\\min }} \\\\right)\\\\ln \\\\delta ^{-1}}{\\\\alpha } \\\\mathcal {B}( \\\\alpha ) \\\\left( \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right)\\\\left( \\\\frac{\\\\theta }{G} \\\\right)^{\\\\frac{2}{\\\\alpha }},$ and $\\\\mathcal {B}( \\\\alpha ) \\\\triangleq \\\\int _{0}^{1}\\\\kappa ^{\\\\frac{2}{\\\\alpha }-1} (1-\\\\kappa )^{-\\\\frac{2}{\\\\alpha }}\\\\mathrm {d} \\\\kappa $ denotes the Beta function.\",\n \"Recall that the network coverage constraint in Definition REF requires that $\\\\mathsf {p}_c(G)\\\\ge (1-\\\\epsilon )$ .\",\n \"Note that $G$ is an important system parameter affecting both the transmission rates and the connectivity probability as elaborated in the following remark.\",\n \"Remark 3 (Transmission Rates vs. Connectivity) The spreading factor $G$ of CDMA controls the tradeoff between mobile transmission rates and network connectivity probability.\",\n \"On one hand, increasing $G$ reduces the bandwidth, $\\\\frac{B}{G}$ , available to each mobile, thereby reducing the transmission rate and increasing comm-latency.\",\n \"As the result, given longer frames with the task-generation rate being fixed, more mobiles are likely to have tasks for offloading at the beginning of each frame, increasing the density of interferers.\",\n \"On the other hand, growing $G$ suppresses interference power by the factor $G$ via spread spectrum.\",\n \"As a result, the connectivity probability grows.\",\n \"Given the two opposite effects, one should expect that in the case of a stringent connectivity constraint, either small or large value for $G$ is preferred but no the moderate ones.\",\n \"Next, the effects of the spreading factor as discussed in Remark REF are quantified by deriving the feasible range of $G$ under the connectivity constraint.\",\n \"Define the Lambert function, $W(x)$ , as the solution for the equation $W(x)e^{W(x)}=x$ .\",\n \"Then using the result in Lemma REF , the coverage constraint $\\\\mathsf {p}_c(G) \\\\ge (1 - \\\\epsilon )$ is equivalent to $\\\\xi (G) \\\\le \\\\mathcal {F}(\\\\epsilon )$ with the function $\\\\mathcal {F}(\\\\epsilon )$ defined as $\\\\mathcal {F}(\\\\epsilon )= W\\\\left( -\\\\frac{e^{-\\\\frac{1}{1-\\\\epsilon }}}{1 - \\\\epsilon } \\\\right) + \\\\frac{1}{1- \\\\epsilon }.$ Notice that $\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\frac{d}{d\\\\epsilon }W\\\\left( -\\\\frac{e^{-\\\\frac{1}{1-\\\\epsilon }}}{1 - \\\\epsilon } \\\\right) = 1$ .\",\n \"Moreover, $W\\\\left( -\\\\frac{e^{-\\\\frac{1}{1-\\\\epsilon }}}{1 - \\\\epsilon } \\\\right) = -1$ at $\\\\epsilon = 0$ .\",\n \"It follows that from these two results that $\\\\mathcal {F}(\\\\epsilon )$ can be approximated as $\\\\mathcal {F}(\\\\epsilon ) \\\\approx 2\\\\epsilon , \\\\qquad \\\\epsilon \\\\ll 1.$ In addition, $\\\\xi (G)$ is maximized at the point of $G=g_0$ of which the existence and uniqueness are proved in Lemma REF .\",\n \"If $\\\\xi (g_0)\\\\le \\\\mathcal {F}(\\\\epsilon )$ , it is straightforward that any $G$ satisfies the condition of (REF ).\",\n \"Otherwise, the feasible range of $G$ satisfying the connectivity is provided in Proposition REF .\",\n \"Lemma 2 (Properties of $\\\\xi (G)$ ) The function $\\\\xi (G)$ in (REF ) attains its maximum at $G = g_0$ with $g_0=\\\\frac{\\\\alpha W\\\\left( -\\\\frac{2}{\\\\alpha }e^{-\\\\frac{2}{\\\\alpha }} \\\\right) + 2}{{\\\\alpha \\\\mathsf {T}_{\\\\min }} \\\\ln (1-p)}.$ Moreover, $\\\\xi (G)$ is monotone increasing in the range $[-\\\\infty , g_0]$ and monotone decreasing in the range $[g_0, \\\\infty ]$ .\",\n \"Proof: See Appendix REF .\",\n \"$\\\\Box $ Proposition 1 (Feasible Range of Spreading Factor) Under the network connectivity constraint, the feasible range of $G$ is $G \\\\ge 1$ if $\\\\xi (g_0)\\\\le \\\\mathcal {F}(\\\\epsilon )$ , where $g_0$ is given in (REF ).\",\n \"If $\\\\xi (g_0) > \\\\mathcal {F}(\\\\epsilon )$ , the feasible range of $G$ is $\\\\mathcal {S}=\\\\mathcal {S}_1\\\\bigcup \\\\mathcal {S}_2$ where $\\\\mathcal {S}_1=\\\\lbrace G\\\\in {Z}^+|1\\\\le G\\\\le g_a\\\\rbrace , \\\\qquad \\\\mathcal {S}_2=\\\\lbrace G\\\\in {Z}^+| G\\\\ge g_b \\\\rbrace ,$ where $g_a$ and $g_b$ are the two roots of the equation $\\\\xi (G) = \\\\mathcal {F}(\\\\epsilon )$ .\",\n \"Based on Lemma REF , the function $\\\\xi (G)$ is monotone increasing over $\\\\mathcal {S}_1$ but monotone decreasing over $\\\\mathcal {S}_2$ .\",\n \"In addition, if $g_a < 1$ , $\\\\mathcal {S}_1$ is empty and the feasibility range of $G$ reduces to $\\\\mathcal {S}_2$ .\"\n ],\n [\n \"Communication Latency\",\n \" Recall that the comm-latency of connected mobiles $\\\\mathsf {T}_{\\\\textrm {comm}}$ comprises the expected waiting time for offloaded tasks at mobiles, denoted as $\\\\mathsf {T}_{\\\\textrm {comm}}^{(a)}$ , and transmission delay, denoted as $\\\\mathsf {T}_{\\\\textrm {comm}}^{(b)}$ .\",\n \"Consider the expected waiting time.\",\n \"Recalling that the offloading protocol in Section REF , the first task arrival during $L$ slots is delivered to the offloading buffer and the subsequent tasks are forwarded to the local computation unit.\",\n \"Let $K$ denote the slot index when an offloaded task arrives at the offloading buffer.\",\n \"It follows that the probability distribution of $K$ follows a conditional geometric distribution, i.e., $\\\\Pr (K=k)=\\\\frac{p (1-p)^{k-1}}{1-(1-p)^L}$ , where $k = 1, 2, \\\\cdots , L$ and the normalization term $1-(1-p)^L$ gives the probability that at least one task arrives during a single frame.\",\n \"Thereby, the expected waiting time is given as $\\\\mathsf {T}_{\\\\textrm {comm}}^{(a)} = \\\\sum _{k=1}^L(L-k) \\\\frac{p(1-p)^{k-1}}{1-(1-p)^L}&= \\\\frac{L}{1-(1-p)^L}-\\\\frac{1}{p}.$ Next, consider the transmission time for a single task in a frame that spans $L$ slots.\",\n \"Recall that $L=G\\\\mathsf {T}_{\\\\min }$ where $\\\\mathsf {T}_{\\\\min }$ is the minimum time for transmitting a task as defined earlier.\",\n \"Combining $\\\\mathsf {T}_{\\\\textrm {comm}}^{(b)} = G\\\\mathsf {T}_{\\\\min }$ and $\\\\mathsf {T}_{\\\\textrm {comm}}^{(a)}$ in (REF ) gives the following result.\",\n \"Lemma 3 (Comm-Latency) Given the spreading factor $G$ , the comm-latency of the typical mobile $\\\\mathsf {T}_{\\\\textrm {comm}} $ (in slot) is given as $\\\\mathsf {T}_{\\\\textrm {comm}}(G)= G{\\\\mathsf {T}_{\\\\min }}+ \\\\frac{G \\\\mathsf {T}_{\\\\min }}{1 - (1-p)^{G \\\\mathsf {T}_{\\\\min }}} - \\\\frac{1}{p}, $ where $\\\\mathsf {T}_{\\\\min }$ is the minimum time for transmitting a task using full bandwidth.\",\n \"Next, consider the minimization of the comm-latency over the spreading factor $G$ .\",\n \"Using (REF ), it is straightforward to show that the comm-latency $\\\\mathsf {T}_{\\\\textrm {comm}}(G)$ is a monotone increasing function of $G$ .\",\n \"Therefore, minimizing comm-latency is equivalent to minimizing $G$ .\",\n \"It follows from Proposition REF that the minimum of $G$ , $G^*=\\\\underset{G\\\\in {\\\\mathcal {S}}}{\\\\min } G$ , is given as $G^* = \\\\left\\\\lbrace \\\\begin{aligned}&g_b, &&\\\\mathcal {S}_1 = \\\\emptyset ,\\\\\\\\&1, && \\\\text{otherwise}.\\\\end{aligned}\\\\right.$ Substituting $G^*$ into (REF ) gives the minimum comm-latency as shown in the following theorem.\",\n \"Theorem 1 (Minimum Comm-Latency) By optimizing the spreading factor $G$ , the minimum comm-latency (in slot), denoted as $\\\\mathsf {T}_{\\\\textrm {comm}}^* $ , is given as follows.\",\n \"If $\\\\mathcal {S}_1$ in (REF ) is non-empty, $\\\\mathsf {T}_{\\\\textrm {comm}}^*= \\\\mathsf {T}_{\\\\min }+ \\\\frac{\\\\mathsf {T}_{\\\\min } }{1 - (1-p)^{\\\\mathsf {T}_{\\\\min }}} - \\\\frac{1}{p}, $ where $\\\\mathsf {T}_{\\\\min } = \\\\frac{\\\\ell }{B \\\\cdot t_0 \\\\cdot \\\\log _2(1+\\\\theta )}$ .\",\n \"If $\\\\mathcal {S}_1$ is empty, $\\\\mathsf {T}_{\\\\textrm {comm}}^*= g_b\\\\mathsf {T}_{\\\\min }+ \\\\frac{g_b \\\\mathsf {T}_{\\\\min } }{1 - (1-p)^{g_b \\\\mathsf {T}_{\\\\min }}} - \\\\frac{1}{p}, $ where $g_b$ is specified in Proposition REF .\",\n \"Consider the second case in Theorem REF .\",\n \"The comm-latency $\\\\mathsf {T}_{\\\\textrm {comm}}^*$ can be approximated in closed-form if $g_b\\\\mathsf {T}_{\\\\min }$ is sufficiently large.\",\n \"For this case, $\\\\left[1 - (1-p)^{g_b \\\\mathsf {T}_{\\\\min }}\\\\right]\\\\approx 1$ and thus the function $\\\\xi (G)$ in (REF ) can be approximated as $\\\\xi (G) \\\\approx \\\\frac{2(1-\\\\delta )\\\\ln \\\\delta ^{-1}}{\\\\alpha } \\\\mathcal {B}( \\\\alpha ) \\\\left( \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right) \\\\left(\\\\frac{\\\\theta }{G}\\\\right)^{\\\\frac{2}{\\\\alpha }}.$ It follows from Theorem REF and (REF ) that if $\\\\mathcal {S}_1$ is empty and $g_b\\\\mathsf {T}_{\\\\min }$ is large, $\\\\mathsf {T}_{\\\\textrm {comm}}^*\\\\approx 2 g_b\\\\mathsf {T}_{\\\\min } - \\\\frac{1}{p}, $ where $g_b\\\\approx \\\\left[ \\\\frac{(1-\\\\delta )\\\\ln \\\\delta ^{-1}\\\\mathcal {B}( \\\\alpha ) }{\\\\alpha \\\\epsilon }\\\\left( \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right)\\\\right]^{\\\\frac{\\\\alpha }{2}}\\\\theta .$ Remark 4 (Sparse Network vs.\",\n \"Dense Network) The first and second cases in Theorem REF correspond to sparse and dense networks, respectively, as measured by the mobile-to-AP density ratio $\\\\lambda _m/\\\\lambda _b$ .\",\n \"In the first case ($\\\\mathcal {S}_1\\\\ne \\\\emptyset $ ), the network is sufficiently sparse, namely the ratio $\\\\lambda _m/\\\\lambda _b$ is sufficiently small, such that the optimal spreading factor $G^*=1$ and the resultant comm-latency is independent of the ratio as shown in the theorem.\",\n \"In other words, for this case, it is optimal to allocate all bandwidth for increasing the transmission rate instead of reducing it for the purpose of suppressing interference to satisfy the network connectivity constraint.\",\n \"In contrast, in the second case ($\\\\mathcal {S}_1= \\\\emptyset $ ), the network is relatively dense and it is necessary to apply spread spectrum to reduce interference so as to meet the connectivity requirement, corresponding to $G^* > 1$ .\",\n \"As the result, the minimum comm-latency scales with the density ratio as $\\\\mathsf {T}_{\\\\textrm {comm}}^*\\\\propto \\\\left(\\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right)^{\\\\frac{\\\\alpha }{2}}$ as one can observe from (REF ).\",\n \"Remark 5 (Effects of Network Parameters) Substituting $\\\\mathsf {T}_{\\\\min } = \\\\frac{\\\\ell }{B \\\\cdot t_0\\\\cdot \\\\log _2(1+\\\\theta )}$ into (REF ) gives that for a relatively dense network, the comm-latency scales as $\\\\boxed{\\\\mathsf {T}_{\\\\textrm {comm}}^*\\\\propto \\\\frac{\\\\ell }{B}\\\\left(\\\\frac{\\\\lambda _m}{\\\\epsilon \\\\lambda _b}\\\\right)^{\\\\frac{\\\\alpha }{2}} - \\\\frac{1}{p}.\",\n \"}$ The scaling laws shows the effects of network parameters including the task size $\\\\ell $ , bandwidth $B$ , mobile density $\\\\lambda _m$ and AP density $\\\\lambda _b$ , and the task-generation probability per slot $p$ .\"\n ],\n [\n \"Task-Arrival Rates at APs/CSs\",\n \"The offloading throughput of the RAN represents the load of the CSN (see Fig.\",\n \"REF ).\",\n \"The throughput can be measured by the expected task-arrival rate (in number of tasks per slot) at the typical AP (equivalently the typical CS).\",\n \"Its scaling law with the expected number of mobiles per AP, $\\\\lambda _m/\\\\lambda _b$ , is not straightforward due to several factors.\",\n \"To be specific, the total bandwidth is fixed, the spread factor grows nonlinearly with $\\\\lambda _m/\\\\lambda _b$ , and the likelihood of task-generation probability per frame varies with the frame length.\",\n \"To address this issue, the task arrivals at the typical AP are characterized as follows.\",\n \"Consider the case of asynchronous offloading.\",\n \"Based on the model in Section REF , the probability that a mobile generates a task for offloading in each frame is $p_L^*=1 - (1-p)^{L^*}, $ where $L^*$ is the frame length given the optimal spreading factor $G^*$ in (REF ).\",\n \"The expected task-offloading rate (in number of tasks per slot) for the typical mobile, denoted as $\\\\beta ^*$ , is given as $\\\\beta ^* = \\\\frac{p_L^*}{L^*}$ .\",\n \"Since $L^* = G^*\\\\mathsf {T}_{\\\\min }$ , $\\\\beta ^*=\\\\frac{1 - (1-p)^{G^*\\\\mathsf {T}_{\\\\min }}}{G^*\\\\mathsf {T}_{\\\\min }}.$ where $\\\\beta ^*=p_L^* \\\\cdot G^*\\\\mathsf {T}_{\\\\min }$ .\",\n \"Let $\\\\bar{\\\\Lambda }^*$ denote the expected task-arrival rate at the typical AP (or CS).\",\n \"Then $\\\\bar{\\\\Lambda }^* = \\\\bar{N}\\\\beta ^*$ where $\\\\bar{N}$ is the expected number of mobiles connected to the AP.\",\n \"Since $\\\\bar{N}=(1-\\\\delta )(1-\\\\epsilon )\\\\frac{\\\\lambda _m}{\\\\lambda _b}$ , $\\\\bar{\\\\Lambda }^* = (1-\\\\delta )(1-\\\\epsilon )\\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\beta ^*.$ Remark 6 (Effects of Network Parameters) Using (REF ), (REF ) and (REF ), one can infer that $\\\\bar{\\\\Lambda }^* \\\\propto \\\\left\\\\lbrace \\\\begin{aligned}& p\\\\frac{\\\\lambda _m}{\\\\lambda _b}, && \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\rightarrow 0,\\\\\\\\&B\\\\left(\\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right)^{-\\\\frac{\\\\alpha }{2}+1}, && \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\rightarrow \\\\infty .\\\\end{aligned}\\\\right.$ The first case corresponds to a sparse network whose performance is not limited by bandwidth and interference.\",\n \"Then the expected task arrival-rate grows linearly with the task-generation probability per slot, $p$ , and the expected number of mobiles per AP, $\\\\lambda _m/\\\\lambda _b$ .\",\n \"For the second case, in a dense network that is bandwidth-and-interference limited, the rate grows linearly the bandwidth $B$ , but decreases with $\\\\lambda _m/\\\\lambda _b$ .\",\n \"The reason for the decrease is the bandwidth for offloading is reduced so that a larger spreading factor is available for suppressing interference to meet the network-coverage requirement.\",\n \"Consequently, the load for the CSs is lighter for a dense (thus comm-limited) network, reducing comp-latency as shown in the sequel.\",\n \"Consider tasks arrivals for the case of synchronous offloading.\",\n \"Unlike the asynchronous counterpart with arrivals spread over each frame, the tasks from mobiles arrive the typical AP at the beginning of each frame.\",\n \"Thus, it is useful to characterize the expected number of task arrivals per frame, denoted as $\\\\bar{A}^*$ , which can be written as $\\\\bar{A}^* = \\\\bar{N} p_L^*$ .\",\n \"It follows that $\\\\bar{A}^* =(1-\\\\delta )(1-\\\\epsilon )\\\\frac{\\\\lambda _m}{\\\\lambda _b} p_L^*.\",\n \"$ Remark 7 (Effects of Network Parameters) In a dense network ($\\\\lambda _m/\\\\lambda _b\\\\rightarrow \\\\infty $ ), it can be obtained from (REF ), (REF ), and (REF ) that $p_L^* \\\\approx 1$ .\",\n \"Then it follows from (REF ) that the expected number of tasks per frame increases linearly with the expected number of mobiles per AP, $\\\\lambda _m/\\\\lambda _b$ .\"\n ],\n [\n \"Computation Latency Analysis: Asynchronous Offloading\",\n \"This section aims at analyzing the comp-latency of the asynchronous offloading where task arrival and departure are randomly distributed over time.\",\n \"Given the Markov model of Fig.\",\n \"REF , we derive the network stability condition in Definition REF and bounds of the average comp-latency.\"\n ],\n [\n \"Optimal Control of VMs\",\n \"On one hand, creating a large number of VMs at the typical CS can slow down its computation rate due to the mentioned I/O interference between VMs.\",\n \"On the other hand, too few VMs can lead to marginal gain from parallel computing.\",\n \"Therefore, the number of VMs should be optimally controlled based on the number of waiting tasks.\",\n \"To this end, let $\\\\mu (m)$ denote the computation rate given $m$ VMs.\",\n \"Given the computation model in (REF ), it follows from $\\\\mu (m)=m/T_c(m)$ that: $\\\\mu (m) = \\\\frac{m}{T_0}(1+d)^{1-m}.$ By analyzing the derivative of $\\\\mu (m)$ , one can find that the function is monotone increasing before reaching a global maximum and after that it is monotone decreasing.\",\n \"Thereby, the value of $m$ that maximizes $\\\\mu (m)$ , denoted as $m_{\\\\max }$ , can be found with the integer constraint $m_{\\\\max } = \\\\textrm {round}\\\\left(\\\\frac{1}{\\\\ln (1+d)}\\\\right),$ where $\\\\textrm {round}(x)$ rounds $x$ to the nearest integer.\",\n \"The said properties of the function $\\\\mu (m)$ and the derived $m_{\\\\max }$ in (REF ) suggest the following optimal VM-control policy.\",\n \"Proposition 2 (Optimal VM Control) To maximize the computation rate at the typical CS, the optimal VM-control policy is to create $m_{\\\\max }$ VMs if there are a sufficient number of tasks for computation or otherwise create as many VMs as possible until the buffer is empty.\",\n \"Consequently, the maximum computation rate, denoted as $\\\\mu ^*(m)$ , given $m$ tasks at the CS (being computed or in the buffer) is $\\\\mu ^*(m) = \\\\left\\\\lbrace \\\\begin{aligned}&\\\\frac{m}{T_0}(1+d)^{1-m}, &&1\\\\le m \\\\le m_{\\\\max }, \\\\\\\\&\\\\frac{m_{\\\\max }}{T_0}(1+d)^{1-m_{\\\\max }}, &&m > m_{\\\\max },\\\\end{aligned}\\\\right.$ where $m_{\\\\max }$ is given in (REF ).\",\n \"For ease notation, the maximum computation rate, $\\\\mu ({m_{\\\\max }})$ , is re-denoted as $\\\\mu _{\\\\max }$ hereafter.\"\n ],\n [\n \"Computation Rates under Network Stability Constraint\",\n \" This subsection focuses on analzying the condition for the maximum computation rate of the typical CS to meet the network stability constraint in Definition REF .\",\n \"The analysis combines the results from queueing theory, stochastic geometry and parallel computing.\",\n \"The said constraint requires $\\\\rho $ -fraction of mobiles, or equivalently $\\\\rho $ -fraction of CSs, to be stable, namely that comp-latency is finite.\",\n \"According to queuing theory, stabilizing a typical CS requires that the task-arrival rate $\\\\Lambda $ should be strictly smaller than the maximum departure rate $\\\\mu ^*(m_{\\\\max })$ : $\\\\Lambda <\\\\mu _{\\\\max }$ [28].\",\n \"Note that the former is a RV proportional to the random number of mobiles, $N$ , connected to the typical CS while the latter is a constant.\",\n \"Then the stability probability $\\\\mathsf {p}_s$ is given as $\\\\mathsf {p}_s =\\\\Pr [\\\\Lambda <\\\\mu _{\\\\max }]=\\\\Pr \\\\left[N <\\\\frac{\\\\mu _{\\\\max }}{\\\\beta ^*}\\\\right],$ where $\\\\beta ^*$ is the task-offloading rate given in (REF ).\",\n \"It follows from the network spatial model that $N$ is a Poisson distributed RV with the mean $\\\\bar{N}=(1-\\\\delta )(1-\\\\epsilon )\\\\frac{\\\\lambda _m}{\\\\lambda _b}$ .\",\n \"Using the distribution and (REF ) and applying Chernoff bound, we can obtain an upper bound on the maximum computation rate required to meet the stability constraint as shown below.\",\n \"Proposition 3 (Computation Rates for $\\\\rho $ -Stability) For the CSN to be $\\\\rho $ -stable, a sufficient condition for the maximum computation rate of the typical CS is given as $\\\\mu _{\\\\max }\\\\ge \\\\bar{\\\\Lambda }^*\\\\cdot \\\\exp \\\\left(W\\\\left(\\\\frac{-\\\\ln (\\\\rho )}{\\\\bar{N} e} - \\\\frac{1}{e}\\\\right)+1\\\\right),$ where $W(\\\\cdot )$ is the Lambert function, the expected mobiles connected to the typical CS $\\\\bar{N}=(1-\\\\delta )(1-\\\\epsilon )\\\\frac{\\\\lambda _m}{\\\\lambda _b}$ , and $\\\\bar{\\\\Lambda }^*$ represents the expected arrival rate given in (REF ).\",\n \"Proof: See Appendix REF .\",\n \"$\\\\Box $ The above result shows that to satisfy the network-stability constraint, the maximum computation rate of each CS, $\\\\mu _{\\\\max }$ , should be larger than the expected task-arrival rate, $\\\\bar{\\\\Lambda }^*$ , scaled by a factor larger than one, namely the exponential term in (REF ).\",\n \"Moreover, the factor grows as the stability probability $(1-\\\\rho )$ increases.\",\n \"Last, it is useful for subsequent analysis to derive the expected arrival rate conditioned on that the typical CS is stable as shown below.\",\n \"Lemma 4 (Expected Task-Arrival Rates for Stable CSs) Given that the typical CS is stable, the expected task-arrival rate is given as $\\\\mathsf {E}[\\\\Lambda |\\\\Lambda <\\\\mu _{\\\\max }]=\\\\bar{\\\\Lambda }^* \\\\left(1-\\\\frac{\\\\Pr (N = \\\\left\\\\lfloor R\\\\right\\\\rfloor )}{1 - \\\\rho }\\\\right),$ where $R=\\\\frac{\\\\mu _{\\\\max }}{\\\\beta ^*}$ measures the maximum number of mobiles the CS can serve, $\\\\beta ^*$ is the task-offloading rate per mobile in (REF ), $\\\\bar{N}$ and $\\\\bar{\\\\Lambda }^*$ follow those in Proposition REF , and the Poisson distribution function $\\\\Pr (N = n )=\\\\frac{\\\\bar{N}^ne^{-\\\\bar{N}}}{n!\",\n \"}$ .\",\n \"Proof: See Appendix REF .\",\n \"$\\\\Box $\"\n ],\n [\n \"Expected Computation Latency\",\n \"In this subsection, the expected comp-latency, $\\\\mathsf {T}_{\\\\textrm {comp}}$ , is analyzed using the Markov chain in Fig.\",\n \"REF and applying queueing theory.\",\n \"Exact analysis is intractable due to the fact that the departure rate $\\\\mu (m)$ in the Markov chain is a non-linear function of state $m$ .\",\n \"This difficulty is overcome by modifying the Markov chain to give two versions corresponding to a M/M/m and a M/M/1 queues, yielding an upper and a lower bounds on $\\\\mathsf {T}_{\\\\textrm {comp}}$ , respectively.\",\n \"First, consider upper bounding $\\\\mathsf {T}_{\\\\textrm {comp}}$ .\",\n \"To this end, the departure rate $\\\\mu (m)$ in the Markov chain in Fig.\",\n \"REF with the following lower bound obtained by fixing all exponents as $(1-m_{\\\\max })$ : $\\\\mu ^-(m) = {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{m}{T_0}(1+d)^{1-m_{\\\\max }}, ~~~1\\\\le m \\\\le m_{\\\\max }, \\\\\\\\\\\\frac{m_{\\\\max }}{T_0}(1+d)^{1-m_{\\\\max }}, ~~~m > m_{\\\\max }.\\\\end{array}\\\\right.\",\n \"}$ As a result, the modified Markov chain is a M/M/$m_{\\\\max }$ queue.\",\n \"The corresponding waiting time, denoted as $\\\\mathsf {T}_{\\\\textrm {comp}}^+$ , upper bounds $\\\\mathsf {T}_{\\\\textrm {comp}}$ since it reduces the computation rate.\",\n \"Applying classic results on M/M/m queues (see e.g., [28]), the waiting time, $\\\\mathsf {T}_{\\\\textrm {comp}}^+$ , for task arrival rate $\\\\Lambda $ is $\\\\mathsf {T}_{\\\\textrm {comp}}^+(\\\\Lambda ) &= \\\\frac{m_{\\\\max }}{\\\\mu ^-(m_{\\\\max })} + \\\\frac{\\\\tau \\\\left(\\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}\\\\right)^{m_{\\\\max }}}{m_{\\\\max }!\\\\mu ^-(m_{\\\\max })\\\\left( 1-\\\\frac{\\\\Lambda }{m_{\\\\max }\\\\mu ^-(m_{\\\\max })} \\\\right)^2} , $ where the coefficient $\\\\tau $ is given as $\\\\tau = \\\\left[\\\\sum _{m=0}^{m_{\\\\max }-1} \\\\frac{1}{m!\",\n \"}\\\\left(\\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}\\\\right)^m + \\\\sum _{m=m_{\\\\max }}^{\\\\infty }\\\\frac{m_{\\\\max }^{m_{\\\\max }-m}}{m_{\\\\max }!\",\n \"}\\\\left(\\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}\\\\right)^m\\\\right]^{-1}.\",\n \"$ Using (REF ), (REF ) and (REF ), the upper bound is given in the following theorem.\",\n \"02 Theorem 2.A (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.\",\n \"The average comp-latency is upper bounded as $\\\\mathsf {T}_{\\\\textrm {comp}} \\\\le \\\\frac{m_{\\\\max }}{\\\\mu _{\\\\max }} + \\\\left(\\\\frac{m_{\\\\max }}{\\\\mu _{\\\\max }}\\\\right)^2\\\\cdot \\\\frac{\\\\bar{\\\\Lambda }^*}{(m_{\\\\max }-1)!\\\\left( m_{\\\\max }-1 \\\\right)^2}\\\\cdot \\\\left(1-\\\\frac{\\\\Pr (N = \\\\lfloor R\\\\rfloor )}{1- \\\\rho }\\\\right),$ where $ R$ follows that in Lemma REF , and $\\\\bar{\\\\Lambda }^*$ and $\\\\mu _{\\\\max }$ are specified in (REF ) and (REF ), respectively.\",\n \"Proof: See Appendix REF .\",\n \"$\\\\Box $ Note that the positive factor $\\\\left(1-\\\\frac{\\\\Pr (N = \\\\lfloor R\\\\rfloor )}{1- \\\\rho }\\\\right)$ accounts for Poisson distribution of mobiles.\",\n \"Next, a lower bound on $\\\\mathsf {T}_{\\\\textrm {comp}}$ is obtained as follows.\",\n \"One can observe from the Markov chain in Fig.\",\n \"REF that for states $m \\\\le m_{\\\\max }$ , the departure rates are smaller than the maximum, $\\\\mu _{\\\\max }$ .\",\n \"The reason is that for these states, there are not enough tasks for attaining the maximum rate by parallel computing.\",\n \"Then replacing all departure rates in the said Markov chain with the maximum $\\\\mu _{\\\\max }$ leads to a lower bound on $\\\\mathsf {T}_{\\\\textrm {comp}}$ .\",\n \"The resultant Markov chain corresponds to a M/M/1 queue.\",\n \"Then using the modified Markov chain and the well-known results from M/M/1 queue (see e.g., [28]), the comp-latency for given arrival rate $\\\\Lambda $ can be lower bounded as $\\\\mathsf {T}_{\\\\textrm {comp}}(\\\\Lambda ) \\\\ge \\\\frac{1}{\\\\mu _{\\\\max }-\\\\Lambda }.$ By taking expectation over $\\\\Lambda $ and applying Jensen's inequality, $\\\\mathsf {T}_{\\\\textrm {comp}}= \\\\mathsf {E}[\\\\mathsf {T}_{\\\\textrm {comp}}(\\\\Lambda )]\\\\ge \\\\mathsf {E}\\\\left[\\\\left.\\\\frac{1}{\\\\mu _{\\\\max }-\\\\Lambda }\\\\right|\\\\Lambda <\\\\mu _{\\\\max }\\\\right]\\\\ge \\\\frac{1}{\\\\mu _{\\\\max }-\\\\mathsf {E}[\\\\Lambda |\\\\Lambda <\\\\mu _{\\\\max }]}.$ Using (REF ) and Lemma REF , we obtain the following result.\",\n \"Theorem 2.B (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.\",\n \"The average comp-latency is lower bounded as $\\\\mathsf {T}_{\\\\textrm {comp}}\\\\ge \\\\frac{1}{\\\\mu _{\\\\max } - \\\\bar{\\\\Lambda }^*\\\\cdot \\\\left(1-\\\\frac{\\\\Pr (N = \\\\lfloor R\\\\rfloor )}{1- \\\\rho }\\\\right)},$ where $ R$ follows that in Lemma REF , and $\\\\bar{\\\\Lambda }^*$ and $\\\\mu _{\\\\max }$ are specified in (REF ) and (REF ), respectively.\",\n \"Remark 8 (Computation-Resource Provisioning) Consider a MEC network provisioned with sufficient computation resources, $\\\\mu _{\\\\max }/\\\\bar{\\\\Lambda }^*\\\\gg 1$ .\",\n \"It follows from Theorem REF $\\\\nonumber \\\\mathsf {T}_{\\\\textrm {comp}}\\\\ge \\\\frac{1}{\\\\mu _{\\\\max }}\\\\left(1 + \\\\frac{c_1\\\\bar{\\\\Lambda }^*}{\\\\mu _{\\\\max }}\\\\right),$ where $c_1$ is a constant.\",\n \"This lower bound has a similar form as the upper bound in Theorem REF .\",\n \"From these results, one can infer that the comp-latency for asynchronous offloading can be approximated written in the following form: $\\\\boxed{\\\\mathsf {T}_{\\\\textrm {comp}}\\\\approx \\\\frac{c_2}{\\\\mu _{\\\\max }}\\\\left(1 + \\\\frac{c_3\\\\bar{\\\\Lambda }^*}{\\\\mu _{\\\\max }}\\\\right), \\\\qquad \\\\frac{\\\\mu _{\\\\max }}{\\\\bar{\\\\Lambda }^*}\\\\gg 1,}$ where $\\\\lbrace c_2, c_3\\\\rbrace $ are constants.\",\n \"The result suggests that to contain comp-latency, the provisioning of computation resources for the MEC network must consider two factors.\",\n \"First of all, the maximum computation rate, $\\\\mu _{\\\\max }$ , for each CS must be sufficient large.\",\n \"At the same time, the computation rate must scale linearly with the total arrival rate such that the computation resource allocated for a single offloaded task, measured by the ratio $\\\\mu _{\\\\max }/\\\\bar{\\\\Lambda }^*$ , is sufficiently large.\"\n ],\n [\n \"Energy Efficiency\",\n \" Based on the above analytical results so far, the subsection tempts to discuss the energy savings of offloading than local computing.\",\n \"First, the energy consumption of offloading, denoted by $\\\\mathsf {E}_{\\\\mathrm {off}}$ , can be derived via multiplying mobile's transmission power $P$ by the offloading duration $G^* \\\\mathsf {T}_{\\\\min }$ .\",\n \"To satisfy the minimum average signal strength at the boundary of the MEC service zone, $P$ should scale with the radius ${r_0}$ as $P\\\\propto r_0^{\\\\alpha }$ .\",\n \"Recalling $r_0\\\\propto \\\\lambda _b^{-\\\\frac{1}{2}}$ and $\\\\mathsf {T}_{\\\\min }\\\\propto \\\\ell $ where $\\\\ell $ is the task size, the resultant energy consumption of $\\\\mathsf {E}_{\\\\mathrm {off}}$ is given as $\\\\mathsf {E}_{\\\\mathrm {off}}=c_4 \\\\frac{G^* \\\\ell }{{\\\\lambda _b}^{\\\\frac{\\\\alpha }{2}}},$ where $c_4$ is a constant depending on the minimum signal strength and $\\\\mathsf {T}_{\\\\min }$ .\",\n \"Next, it is well studied in[6] that the optimal $\\\\mathsf {E}_{\\\\textrm {loc}}$ is proportional to ${\\\\ell }^3$ , and inversely proportional to the square of the deadline requirement which could be set as the total latency $\\\\mathsf {T}_{\\\\textrm {comm}}+\\\\mathsf {T}_{\\\\textrm {comp}}$ without loss of generality.\",\n \"Thus, $\\\\mathsf {E}_{\\\\textrm {loc}}$ is given as $\\\\mathsf {E}_{\\\\textrm {loc}}=c_5 \\\\frac{{\\\\ell }^3}{(\\\\mathsf {T}_{\\\\textrm {comm}}+\\\\mathsf {T}_{\\\\textrm {comp}})^2},$ where $c_5$ is a constant depending on the chip architecture.\",\n \"where $c_5$ is a constant depending on the chip architecture.\",\n \"As a result, the condition of energy savings, namely $\\\\mathsf {E}_{\\\\mathrm {off}}<\\\\mathsf {E}_{\\\\textrm {loc}}$ , is given in terms of $\\\\ell $ and $\\\\lambda _b$ as $\\\\ell ^2 \\\\lambda _b^{\\\\frac{\\\\alpha }{2}}>\\\\frac{c_4}{c_5}{G^*}\\\\left(\\\\mathsf {T}_{\\\\textrm {comm}}+\\\\mathsf {T}_{\\\\textrm {comp}}\\\\right)^2.$ It is observed that the right-side of (REF ) is dominantly affected by the expected number of mobiles $\\\\frac{\\\\lambda _m}{\\\\lambda _b}$ (see (REF ), (REF ), (REF ) and (REF )).\",\n \"In other words, given a mobile density $\\\\lambda _m$ and the task size $\\\\ell $ , there exists the minimum density of APs $\\\\lambda _b$ to satisfy the condition in (REF ).\",\n \"Then under the condition in (REF ), the energy savings due to offloading is given as $\\\\mathsf {E}_{\\\\mathrm {loc}}-\\\\mathsf {E}_{\\\\mathrm {off}}$ with $\\\\mathsf {E}_{\\\\mathrm {off}}$ and $\\\\mathsf {E}_{\\\\mathrm {loc}}$ given in (REF ) and (REF ), respectively.\"\n ],\n [\n \"MEC Network Provisioning and Planning\",\n \" Combining the results from the preceding analysis on comm-latency and comp-latency yields some guidelines for the provisioning and planning of a MEC network as discussed below.\",\n \"Assume that the network is required to support computing for mobiles with density $\\\\lambda _m$ with targeted expected comm-latency $\\\\mathsf {T}_{\\\\textrm {comm}}$ and comp-latency $\\\\mathsf {T}_{\\\\textrm {comp}}$ .\",\n \"The network resources are quantified by the bandwidth $B$ , the density of AP (or CS) $\\\\lambda _b$ , the maximum computing rate of each CS $\\\\mu _{\\\\max }$ .\",\n \"First, consider the planning of the RAN.\",\n \"Combining the above results suggest the following guidelines for network provisioning and planning.\",\n \"As shown in Section REF , under the network-coverage constraint $(1-\\\\epsilon )$ , the expected task-arrival rate at a AP, representing the RAN offloading throughput, is a quasi-concave function of the expected number of mobiles per AP, $\\\\lambda _m/\\\\lambda _b$ , with a global maximum.\",\n \"Therefore, given a mobile density $\\\\lambda _m$ , the AP density should be chosen for maximizing the RAN offloading throughput.\",\n \"Next, based on results in Theorem REF and (REF ), sufficient large channel bandwidth $B$ should be provisioned to achieve the targeted $\\\\mathsf {T}_{\\\\textrm {comm}}$ for given mobile and AP densities, mobile task-generation rates, and task sizes.\",\n \"Next, consider the planning of the CSN.\",\n \"Under the network-stability constraint, the maximum CS computing rate for parallel computing should be planned to be larger than the expected task-arrival rate scaled by a factor larger than one, which is determined by the allowed fraction of unstable CSs (see Proposition REF ).\",\n \"Then, the maximum computing rate should be further planned to achieve the targeted $\\\\mathsf {T}_{\\\\textrm {comp}}$ for computing an offloaded task using Theorems REF and REF .\",\n \"In the preceding section, the process of asynchronous task arrival at a CS can be approximated using a Markov chain, allowing tractable analysis of comp-latency using theories of M/M/$m$ and M/M/1 queues.\",\n \"This approach is inapplicable for synchronous offloading and the resultant periodic task arrivals at the CS.\",\n \"Though tractable analysis in general is difficult, it is possible for two special cases defined as follows.\",\n \"Definition 3 (Special Cases: Light-Traffic and Heavy-Traffic) A light-traffic case refers to one that the task-arrival rate is much smaller than the computation rate such that the queue at the CS is always empty as observed by a new arriving task.\",\n \"In contrast, a heavy-traffic case refers to one that the task-arrival rate is close to the computation rate such that there are always at least $m_{\\\\max }$ tasks in the queue.\",\n \"The comp-latency for these two special cases are analyzed to give insights into the performance of CSN with underloaded CSs and those with overloaded CSs.\"\n ],\n [\n \"Expected Computation Latency with Light-Traffic \",\n \" First, the dynamics of the task queue at the typical CS is modelled as follows.\",\n \"Recall that task arrivals are periodical, occurring at the beginning of every frame.\",\n \"Consider the typical CS.\",\n \"Let $\\\\mathcal {Q}_t$ and $\\\\mathcal {A}_t$ denote be the numbers of existing and arriving tasks at the beginning of frame $t$ , respectively, and $\\\\mathcal {C}_t$ the number of departing tasks during frame $t$ .\",\n \"Then the evolution of $\\\\mathcal {Q}_t$ can be described mathematically as $\\\\mathcal {Q}_{t+1} = \\\\max \\\\left[ \\\\mathcal {Q}_t + \\\\mathcal {A}_{t+1} - \\\\mathcal {C}_t, 0 \\\\right].$ The general analysis of comp-latency using (REF ) is difficult.\",\n \"The main difficulty lies in deriving the distribution of $\\\\mathcal {C}_t$ that depends on the number of VMs that varies continuously in time since the computation time for simultaneous tasks are random and inter-dependent.\",\n \"To overcome the difficulty, consider the case of light-traffic where a number of offloaded tasks arrives at the typical CS to see an empty queue and an idling server.\",\n \"Correspondingly, the evolution equation in (REF ) is modified such that given $\\\\mathcal {A}_{t+1}\\\\ne 0 $ , $\\\\mathcal {Q}_t = \\\\mathcal {C}_t = 0 $ , yielding $\\\\mathcal {Q}_{t+1} = \\\\mathcal {A}_{t+1}$ .\",\n \"Next, given this simple equality, deriving the expected comp-latency reduces to analyzing the latency for computing a random number of $\\\\mathcal {A}$ tasks at the CS, which arrives at the beginning of an arbitrary frame.\",\n \"Without loss of generality, the tasks are arranged in an ascending order in terms of computation time and referred to as Task $1, 2, \\\\cdots , \\\\mathcal {A}$ .\",\n \"Moreover, let $\\\\mathcal {L}_n$ denote the expected computing time for Task $n$ and hence $\\\\mathcal {L}_1 \\\\le \\\\mathcal {L}_2 \\\\le \\\\cdots \\\\le \\\\mathcal {L}_{\\\\mathcal {A}}$ .\",\n \"Then the expected comp-latency $\\\\mathsf {T}_{\\\\textrm {comp}}$ can be written in terms of $\\\\lbrace \\\\mathcal {L}_n\\\\rbrace $ as $\\\\mathsf {T}_{\\\\textrm {comp}} = \\\\mathsf {E}_{\\\\mathcal {A}}\\\\left[\\\\left.\\\\frac{\\\\sum _{n=1}^{{\\\\mathcal {A}}} {\\\\mathcal {L}_n} }{\\\\mathcal {A}}\\\\right|\\\\mathcal {A}>0 \\\\right].\",\n \"$ To obtain bounds on $\\\\mathsf {T}_{\\\\textrm {comp}}$ in closed form, a useful result is derived as follows.\",\n \"Given $m$ VMs, recall that the computation time of a task follows the exponential distribution with the mean being the inverse of the computation rate $\\\\mu (m)$ in (REF ).\",\n \"Using the memoryless property of exponential distribution, a useful relation between $\\\\lbrace \\\\mathcal {L}_n\\\\rbrace $ is obtained as $\\\\mathcal {L}_n =\\\\mathcal {L}_{n-1}+ \\\\frac{1}{\\\\mu (m)}=\\\\left\\\\lbrace \\\\begin{aligned}&\\\\mathcal {L}_{n-1}+ \\\\frac{1}{\\\\mu _{\\\\max }}, && 1\\\\le n\\\\le \\\\mathcal {A}-m_{\\\\max }+1,\\\\\\\\&\\\\mathcal {L}_{n-1} +\\\\frac{1}{\\\\mu (\\\\mathcal {A} - n +1)}, && \\\\text{otherwise},\\\\end{aligned}\\\\right.$ with $\\\\mathcal {L}_0=0$ .\",\n \"Note that $\\\\mu (1) \\\\le \\\\mu (m) \\\\le \\\\mu _{\\\\max }$ for all $m$ .\",\n \"Thus, it follows from (REF ) that $\\\\frac{n}{\\\\mu _{\\\\max }} \\\\le \\\\mathcal {L}_n \\\\le \\\\frac{n}{\\\\mu (1)}.\",\n \"$ Substituting (REF ) into (REF ) gives $\\\\frac{1}{\\\\mu _{\\\\max }}\\\\cdot \\\\mathsf {E}_{\\\\mathcal {A}}\\\\left[\\\\left.\\\\frac{\\\\sum _{n=1}^{{\\\\mathcal {A}}} n }{\\\\mathcal {A}}\\\\right|\\\\mathcal {A}>0 \\\\right]\\\\le \\\\mathsf {T}_{\\\\textrm {comp}} \\\\le \\\\frac{1}{\\\\mu (1)} \\\\cdot \\\\mathsf {E}_{\\\\mathcal {A}}\\\\left[\\\\left.\\\\frac{\\\\sum _{n=1}^{{\\\\mathcal {A}}} n }{\\\\mathcal {A}}\\\\right|\\\\mathcal {A}>0 \\\\right].$ Recalling the number of arriving tasks $\\\\mathcal {A}$ follows a Poisson RV with mean $\\\\bar{A}^*$ of (REF ), bounds on the comp-latency is obtained shown in the following theorem.\",\n \"Theorem 3 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with light-traffic.\",\n \"The expected comp-latency can be bounded as $\\\\nonumber \\\\frac{1}{2\\\\mu _{\\\\max }}\\\\left(1+\\\\frac{\\\\bar{A}^*}{1-e^{-\\\\bar{A}^*}}\\\\right)\\\\le \\\\mathsf {{T}}_{\\\\textrm {comp}} \\\\le \\\\frac{1}{2\\\\mu (1)}\\\\left(1+\\\\frac{\\\\bar{A}^*}{1-e^{-\\\\bar{A}^*}}\\\\right).$ where $\\\\bar{A}^*$ is the expected number of arriving tasks to the typical CS per frame given in (REF ).\",\n \"Remark 9 (Comparison with Asynchronous Offloading) From the results in the above theorem, one can infer that for the current case, the comm-latency can be approximated as $\\\\boxed{\\\\mathsf {{T}}_{\\\\textrm {comp}} \\\\approx \\\\left\\\\lbrace \\\\begin{aligned}&\\\\frac{1}{\\\\mu (1)}, && \\\\bar{A}^* \\\\rightarrow 0,\\\\\\\\&\\\\frac{\\\\bar{A}^*}{2\\\\mu (m_{\\\\max })},&& \\\\bar{A}^* \\\\gg 1.\\\\end{aligned}\\\\right.\",\n \"}$ Comparing the expression with the counterpart for asynchronous offloading in (REF ), it is unclear which case leads to longer comp-latency.\",\n \"However, simulation shows that in general, synchronizing offloading tends to incur longer latency by overloading CSs and thereby suffering more from I/O interference in parallel-computing.\"\n ],\n [\n \"Expected Computation Latency with Heavy-Traffic\",\n \"This subsection focuses on analyzing the expected comp-latency, $\\\\mathsf {T}_{\\\\textrm {comp}}$ , for the case of heavy-traffic as defined in Definition REF .\",\n \"For this case, with the queue being always non-empty, the equation in (REF ) describing the queue evolution reduces to $\\\\mathcal {Q}_{t+1} = \\\\mathcal {Q}_t + \\\\mathcal {A}_t - \\\\mathcal {C}_t$ .\",\n \"The key step in deriving $\\\\mathsf {T}_{\\\\textrm {comp}}$ is to apply the said equation to the analysis of the expected queue length.\",\n \"The technique involves taking expectation of the squares of the two sides of the equation as follows: $\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}^2_{t+1} \\\\right]= \\\\mathsf {E}\\\\left[ \\\\left(\\\\mathcal {Q}_t + \\\\mathcal {A}_t - \\\\mathcal {C}_t\\\\right)^2\\\\right]&=\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}^2_t \\\\right]+\\\\mathsf {E}\\\\left[\\\\left( \\\\mathcal {A}_t - \\\\mathcal {C}_t \\\\right)^2\\\\right]+2\\\\mathsf {E}\\\\left[{\\\\mathcal {Q}}_t (\\\\mathcal {A}_t - \\\\mathcal {C}_t) \\\\right].$ Since $\\\\mathcal {Q}_t$ , $\\\\mathcal {A}_t$ and $\\\\mathcal {C}_t$ are independent of each other and $\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}^2_{t+1} \\\\right]=\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}^2_t\\\\right]$ given the stable CS, $\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}\\\\right] = \\\\frac{\\\\mathsf {E}\\\\left[\\\\mathcal {A}^2\\\\right] + \\\\mathsf {E}\\\\left[\\\\mathcal {C}^2\\\\right] - 2\\\\mathsf {E}\\\\left[\\\\mathcal {A}\\\\right]\\\\mathsf {E}\\\\left[\\\\mathcal {C}\\\\right]}{2\\\\left( \\\\mathsf {E}\\\\left[\\\\mathcal {C}\\\\right] - \\\\mathsf {E}\\\\left[\\\\mathcal {A}\\\\right] \\\\right)}, $ where the subscripts $t$ of $\\\\mathcal {Q}_t$ , $\\\\mathcal {A}_t$ and $\\\\mathcal {C}_t$ are omitted to simplify notation.\",\n \"Given the number of connected mobiles, $N$ , the number of arrival tasks $\\\\mathcal {A}$ follows a Poisson distribution with the first and second moments being $\\\\mathsf {E}\\\\left(\\\\mathcal {A}\\\\mid N \\\\right)={N} p_L^*$ and $\\\\mathsf {E}\\\\left(\\\\mathcal {A}^2\\\\mid N \\\\right)=N p_L^*+(Np_L^*)^2$ respectively, where the task-offloading probability $p_L^*$ is given in (REF ).\",\n \"Next, under the heavy-traffic assumption, the total computation rate of the CS is $\\\\mu _{\\\\max }$ .\",\n \"It follows that the departure process at the typical CS is Poisson distributed where the first and second moments are $\\\\mathsf {E}[\\\\mathcal {C}]= \\\\mu _{\\\\max }L^*$ and $\\\\mathsf {E}[\\\\mathcal {C}^2]= \\\\mu _{\\\\max }L^*+[ \\\\mu _{\\\\max }L^*]^2$ , respectively.\",\n \"Substituting the results into (REF ) gives $\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}\\\\mid N \\\\right] =\\\\frac{{N} p_L^*}{\\\\mu _{\\\\max }L^*-{N} p_L^*}+\\\\frac{1}{2}\\\\cdot \\\\left(\\\\mu _{\\\\max }L^*-{N} p_L^*\\\\right)+\\\\frac{1}{2}.\",\n \"$ In addition, to satisfy the condition for stabilizing the CS, the arrival rate $\\\\mathsf {E}\\\\left[\\\\mathcal {A}\\\\right]$ should be strictly smaller than the departure rate $\\\\mathsf {E}[\\\\mathcal {C}]$ .\",\n \"This places a constraint on the maximum of $N$ , namely that $N \\\\le \\\\left\\\\lfloor R \\\\right\\\\rfloor $ with $ R$ defined in Lemma REF .\",\n \"Under this constraint, applying Little's theorem obtains the expected comp-latency $\\\\mathsf {T}_{\\\\textrm {comp}}$ as $\\\\mathsf {T}_{\\\\textrm {comp}}& = \\\\left\\\\lbrace \\\\mathsf {E}\\\\left[\\\\left.\",\n \"\\\\frac{\\\\mathsf {E}\\\\left[ \\\\mathcal {Q}\\\\mid N \\\\right]}{\\\\mathsf {E}\\\\left[ \\\\mathcal {A}\\\\mid N \\\\right]}\\\\right|N \\\\le \\\\left\\\\lfloor R \\\\right\\\\rfloor \\\\right]-\\\\frac{1}{2}\\\\right\\\\rbrace \\\\cdot L^*.\",\n \"$ Combining (REF ) and (REF ) yields the main result of this sub-section as shown below.\",\n \"Theorem 4 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with heavy-traffic.\",\n \"The expected comp-latency is given as $\\\\mathsf {T}_{\\\\textrm {comp}}=\\\\left\\\\lbrace \\\\frac{1}{p_L^*}\\\\mathsf {E}\\\\left[\\\\frac{1}{ R-N}\\\\right] +\\\\frac{1}{2}\\\\left( R + \\\\frac{1}{p_L^*}\\\\right) \\\\mathsf {E}\\\\left[\\\\frac{1}{N}\\\\right] - 1\\\\right\\\\rbrace L^*,$ where the constant $ R$ and the distribution of $N$ follow those in Lemma REF .\",\n \"Remark 10 (Comparison with Asynchronous Offloading) By applying Jensen's inequality, the comp-latency of (REF ) can be lower bounded as $\\\\mathsf {T}_{\\\\textrm {comp}}&\\\\ge \\\\frac{1}{\\\\mu _{\\\\max }-c_4\\\\bar{\\\\Lambda }^*}+\\\\frac{L^*}{2}\\\\cdot \\\\frac{\\\\mu (m_{\\\\max })}{c_4\\\\bar{\\\\Lambda }^*}+\\\\frac{1}{c_4\\\\bar{\\\\Lambda }^*}- L^*,$ where $c_4$ is a constant.\",\n \"Since for the case of heavy-traffic, the task-arrival rate $c_4\\\\bar{\\\\Lambda }^*$ approaches the maximum computation rate $\\\\mu _{\\\\max }$ , $\\\\boxed{\\\\mathsf {T}_{\\\\textrm {comp}}\\\\ge \\\\frac{1}{\\\\mu _{\\\\max }-c_4\\\\bar{\\\\Lambda }^*},\\\\qquad c_4\\\\bar{\\\\Lambda }^*\\\\rightarrow \\\\mu _{\\\\max }.\",\n \"}$ The above lower bound has the same form as the asynchronous-offloading counterpart in (REF ).\",\n \"Both diverge as the task-arrival rate approaches the maximum computation rate.\"\n ],\n [\n \"Simulation Results\",\n \" In this section, analytical results on comm-latency and comp-latency are evaluated by simulation.\",\n \"The simulation parameters have the following default settings unless specified otherwise.\",\n \"The densities of APs and mobiles are $\\\\lambda _b = 2 \\\\times 10^{-2} \\\\ \\\\mathrm {m}^{-2}$ and $\\\\lambda _m = 5 \\\\times 10^{-2} \\\\ \\\\mathrm {m}^{-2}$ , respectively.\",\n \"The $\\\\mathsf {SIR}$ threshold is set as $\\\\theta = 1$ dB and the path-loss exponent is $\\\\alpha = 3$ .\",\n \"For the network coverage parameter $\\\\delta $ is $\\\\delta = 10^{-2}$ , corresponding to the radius of MEC service zone being $r_0 = 12 \\\\mathrm {m}$ .\",\n \"The total bandwidth is $B = 6$ MHz.\",\n \"The data size per task is fixed as $\\\\ell = 0.5 \\\\times 10^{6}$ bits.\",\n \"The single-task computation time $T_0$ in the parallel-computation model is set as $T_0 = 0.1$ (sec) and the factor arising from I/O interference is $d = 0.2$ .\",\n \"The task generation probability per slot is $p = 0.2$ .\",\n \"The parameters $\\\\epsilon $ and $\\\\rho $ are both set as $0.05$ .\",\n \"Figure: Effect of task generating rate.We present the comparisons between Monte Carlo simulations ($10^4$ realizations) and analytical results in all figures.\",\n \"For each realization, both mobiles and APs are distributed in the plane based on the PPPs.\",\n \"Each mobile randomly generates an offloading task and transmits it to its corresponding AP.\",\n \"Then the AP generates VMs and performs the computation upon the task arrivals.\",\n \"A queue of tasks will appear if the task arrival rate is large than the computation rate (e.g., too many tasks arrived at the AP at the same time).\",\n \"The VM will be released when the corresponding task is computed.\",\n \"Fig.\",\n \"REF compares expected comm-latency and comp-latency for the case of asynchronous offloading.\",\n \"The effects of mobile density $\\\\lambda _m$ and task generating rate $p$ are investigated and several observations can be made.\",\n \"As shown in Fig.\",\n \"REF (a), the expected comp-latency as a function of the mobile density is observed to exhibit the quasi-concavity described in Remark REF .\",\n \"In contrast, the expected comm-latency is a monotone increasing function following the scaling law in Remark REF .\",\n \"These properties lead to the partitioning of the range of mobile density into three network-operation regimes as indicated in Fig.\",\n \"REF (a).\",\n \"In particular, the middle range corresponds to comp-limited regime while others are comm-limited.\",\n \"Next, consider the effect of task-generation rate at mobiles, specified by the task-generation probability $p$ .\",\n \"Both types of latency are observed to converge to corresponding limits as the rate grows.\",\n \"Their different scaling laws result in the partitioning of the range of task-generation rate into comm-limited and comp-limited regimes.\",\n \"Last, one can observe from both figures that the lower bound on comp-latency as derived in (REF ) is tighter than the upper bond therein.\",\n \"Fig.\",\n \"REF compares expected comm-latency and comp-latency for the case of synchronous offloading.\",\n \"The same observations in the case of synchronous offloading also apply in the current case except that the quasi-concavity of the expected comp-latency with respect to mobile density is not shown in the considered range.\",\n \"Some new observations can be made as follows.\",\n \"Comparing Fig.\",\n \"REF and REF shows that synchronizing offloading results in longer comp-latency.\",\n \"Next, the center of the comp-limited range in Fig.\",\n \"REF (a) corresponds to the case of heavy-traffic studied in Section REF .\",\n \"Consequently, the derived upper bound on expected comp-latency for this case is tight.\",\n \"For other ranges of mobile density, the bounds derived for the case of light traffic are tighter.\",\n \"Last, Fig.\",\n \"REF (b) shows that the expected comp-latency is tightly approximated by bounds derived for the light-traffic case when the task-arrival rate is small ($\\\\le 0.3$ ) and by that for the heavy-traffic case when the rate is large ($> 0.7$ ), validating the results.\",\n \"Figure: Effect of task generating rate.\"\n ],\n [\n \"Conclusion Remarks\",\n \"In this work, we have first studied the network-constrained latency performance of a large-scale MEC network, namely comm-latency and comp-latency under the constraints of RAN connectivity and CSN stability.\",\n \"To study the tradeoffs between these metrics and constraints and model the cascaded architecture of RAN and CSN, the MEC network has been modelled using stochastic geometry featuring diversified aspects of wireless access and computing.\",\n \"Based on the model, the average comm-latency and comp-latency have been analyzed by applying the theories of stochastic geometry, queuing and parallel computing.\",\n \"In particular, their scaling laws have been derived with respect to various network parameters ranging from the densities of mobiles and APs to the computation capabilities of CSs.\",\n \"The results provide useful guidelines for MEC-network provisioning and planning to avoid either the RAN or CSN being a performance bottleneck.\",\n \"The current work can be extended in several directions.\",\n \"In this work, we consider a single type of computation task of which the task size and the average computation time are identical.\",\n \"However, considering different types of tasks makes the latency analysis more challenging but is of practical relevance.\",\n \"Next, studying a large-scale hierarchical fog computing network comprising mobiles, edge cloud and central cloud is aligned with recent advancements in edge computing.\",\n \"Last, considering advanced techniques such as VM migration and cooperative computing to reduce the latency will be another promising direction.\"\n ],\n [\n \"Proof of Lemma \",\n \" The first derivative of $\\\\xi (G)$ for $G$ given in (REF ) can be derived as: $\\\\frac{\\\\partial \\\\xi (G)}{\\\\partial G} = -\\\\frac{\\\\Delta }{\\\\alpha }\\\\left( G^{-\\\\frac{2 + \\\\alpha }{\\\\alpha }}\\\\left( \\\\alpha G \\\\mathsf {T}_{\\\\min } \\\\ln (1-p) (1-p)^{G \\\\mathsf {T}_{\\\\min }} - 2(1-p)^{G \\\\mathsf {T}_{\\\\min }} + 2 \\\\right) \\\\right), $ where $\\\\Delta = \\\\frac{2(1-\\\\delta )\\\\ln \\\\delta ^{-1}}{\\\\alpha } \\\\mathcal {B}( \\\\alpha ) \\\\left( \\\\frac{\\\\lambda _m}{\\\\lambda _b}\\\\right) \\\\theta ^{\\\\frac{2}{\\\\alpha }}$ .\",\n \"The existence of $g_0$ is easily proved because (REF ) is strictly positive and negative when $G \\\\rightarrow 0$ and $G \\\\rightarrow \\\\infty $ respectively.\",\n \"Next, the solution of $g_0$ to solve $\\\\frac{\\\\partial \\\\xi (G)}{\\\\partial G} = 0$ is $g_0 =\\\\frac{\\\\alpha W\\\\left( -\\\\frac{2}{\\\\alpha }e^{-\\\\frac{2}{\\\\alpha }} \\\\right) + 2}{\\\\alpha \\\\mathsf {T}_{\\\\min } \\\\ln (1-p)}$ , where $W(x)$ is the Lambert function.\",\n \"Since the value inside the Lambert function is negative, there are two candidates for $g_0$ : one is from the principle branch of Lambert function $W\\\\left( -\\\\frac{2}{\\\\alpha }e^{-\\\\frac{2}{\\\\alpha }} \\\\right)$ , and the other is the lower branch $W_{-1}\\\\left( -\\\\frac{2}{\\\\alpha }e^{-\\\\frac{2}{\\\\alpha }} \\\\right)$ .\",\n \"The principle branch makes $g_0=0$ , but the lower branch satisfies $g_0>0$ , completing the proof.\"\n ],\n [\n \"Proof of Proposition \",\n \" Applying Chernoff bound on $\\\\mathsf {p}_s$ in (REF ) makes $\\\\mathsf {p}_s &\\\\ge 1- \\\\exp \\\\left(- \\\\frac{\\\\mu _{\\\\max }}{\\\\beta ^*} \\\\ln \\\\left(\\\\frac{ \\\\mu _{\\\\max } }{\\\\bar{N} \\\\beta ^*}\\\\right) +\\\\frac{\\\\mu _{\\\\max }}{\\\\beta ^*}-\\\\bar{N}\\\\right)\\\\ge 1 - \\\\rho ,$ which is equivalent to $\\\\mu _{\\\\max }\\\\ge \\\\bar{\\\\Lambda }^*\\\\cdot \\\\exp \\\\left(W\\\\left(\\\\frac{-\\\\ln (\\\\rho )}{\\\\bar{N} e} - \\\\frac{1}{e}\\\\right)+1\\\\right)$ as Proposition REF shows.\"\n ],\n [\n \"Proof of Lemma \",\n \" Noting the expect task arrival of stable CSs is proportional to the average number of connected mobiles.\",\n \"Therefore, we have $\\\\mathsf {E}[\\\\Lambda |\\\\Lambda <\\\\mu _{\\\\max }]= \\\\beta ^{*} \\\\cdot \\\\frac{\\\\sum _{n=1}^{ R } n \\\\cdot \\\\frac{\\\\bar{N}^{n} e^{-\\\\bar{N}}}{n!\",\n \"}}{1- \\\\rho } =\\\\bar{N}\\\\beta ^* \\\\left(1-\\\\frac{\\\\Pr (N = \\\\left\\\\lfloor R\\\\right\\\\rfloor )}{1 - \\\\rho }\\\\right)$ , where $\\\\Pr (N = \\\\left\\\\lfloor R\\\\right\\\\rfloor ) = \\\\frac{\\\\bar{N}^{\\\\left\\\\lfloor R\\\\right\\\\rfloor } e^{-\\\\bar{N}}}{\\\\left\\\\lfloor R\\\\right\\\\rfloor !\",\n \"}$ , ending the proof.\"\n ],\n [\n \"Proof of Theorem \",\n \" Substituting $\\\\tau = 1$ into (REF ) and then applying $\\\\left(\\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}\\\\right)^{m_{\\\\max }} \\\\le \\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}$ as well as $\\\\left(1-\\\\frac{\\\\Lambda }{m_{\\\\max }\\\\mu ^-(m_{\\\\max })} \\\\right)^2\\\\le \\\\left(1-\\\\frac{1}{m_{\\\\max }}\\\\right)^2$ give an upper bound of $\\\\mathsf {T}_{\\\\textrm {comp}}^+(\\\\Lambda )$ as $\\\\mathsf {T}_{\\\\textrm {comp}}^+(\\\\Lambda ) \\\\le \\\\frac{m_{\\\\max }}{\\\\mu ^-(m_{\\\\max })} + \\\\frac{\\\\frac{\\\\Lambda }{\\\\mu ^-(m_{\\\\max })}}{m_{\\\\max }!\\\\mu ^-(m_{\\\\max })\\\\left( 1-\\\\frac{1}{m_{\\\\max }} \\\\right)^2}.\",\n \"$ The spatial average on (REF ) based on Lemma REF gives the final result in Theorem REF .\"\n ],\n [\n \"Appendix II: Notation Table\",\n \" Table: Summary of NotationIn Fig.\",\n \"REF , the the comp-latency of the Poisson arrival assumption of task-arrival process is compared with that under the Poisson arrival assumption of Assumption REF , showing that the tasks arrival process at AP can be well approximated to the Poisson task arrival assumption.\",\n \"Fig.\",\n \"REF represents the effects of AP density, which is shown that the same observations are made in the case of decreasing mobile density.\",\n \"Figs.\",\n \"REF and REF show the effects of the computation capability of CS, the computation time $T_0$ and the degradation factor $d$ , on the comp-latency for asynchronous and synchronous offloading cases, respectively.\",\n \"Both of the comp-latencies increase exponentially when the edge-computing capability worse, i.e., $T_0$ or $d$ becomes larger, which agrees with the intuition.\",\n \"Finally, we plot the ratio between the comp-latency of asynchronous and synchronous offloading cases in Fig.\",\n \"REF , which are always less than one because the comp-latency of the asynchronous offloading is always smaller than the synchronous counterpart.\",\n \"Specifically, in Fig.\",\n \"REF (a), the ratio first stays constant when mobiles is sparse and then decreases with mobile density grows.\",\n \"As mobiles becomes denser, the number of offloading tasks at the beginning of each frame increases, resulting in severe I/O interference than the asynchronous counterpart of which the task arrivals are distributed over frame.\",\n \"On the other hand, in Fig.\",\n \"REF (b), it is observed that the ratio grows and then converges to a constant when $p$ increases.\",\n \"In other words, the gap of comp-latency between two offloading cases is becoming smaller when the task arrival becomes heaver, aligning with the discussion in Remark REF .\"\n ]\n]"},"id":{"kind":"string","value":"1709.01702"}}},{"rowIdx":694,"cells":{"text":{"kind":"list like","value":[["Constraints on Flavored 2d CFT Partition Functions"],["Abstract We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e.","when the partition functions are \"flavored\".","We begin with a new proof of the transformation law for the modular transformation of such partition functions.","Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory.","We improve previous upper bounds on the state with the greatest \"mass-to-charge\" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory.","We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers.","Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise."],["Introduction","Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs).","It is also a special case of crossing symmetry of CFT correlation functions [1], so aside from its intrinsic interest it is useful as a simpler setting in which to explore many conformal bootstrap ideas and techniques [2], [3], [4].","A particularly appealing generalization of the conformal bootstrap equations is to consider correlation functions in the presence of nonlocal operators, since this enlarges the set of CFT data that can be studied.","In general, including nonlocal operators is a difficult problem, since their behavior under conformal transformations may be quite complicated.","However, one case where the problem remains tractable is when we consider modular invariance in the presence of a chemical potential in 2d CFTs.","A chemical potential corresponds to inserting the nonlocal operator $y^{J_0} \\equiv e^{2 \\pi i z J_0}$ into the partition function, $Z( \\tau , z) \\equiv {\\rm tr}\\left( q^{L_0- \\frac{c}{24}} \\bar{q}^{\\bar{L}_0-\\frac{c}{24}} y^{J_0} \\right),$ where $J_0$ is the zero mode of a conserved current and $L_0, \\bar{L}_0$ are Virasoro generators.","The resulting partition function is no longer modular invariant, but nevertheless has a well-defined and theory-independent transformation law [5]: $Z(\\frac{a \\tau +b}{c \\tau +d}, \\frac{c z}{c \\tau + d}) = e^{ \\pi i k \\left( \\frac{ c z^2 }{c \\tau +d} - \\frac{ c \\bar{z}^2}{c \\bar{\\tau }+d}\\right)} Z(\\tau , z) .$ This transformation law was used to constrain the spectrum of charges in general 2d CFTs in [6].","Proofs of (REF ) so far [5], [7], [8], [6] either are fairly complicated and technical or else apply to special cases such as free boson constructions, and so it is not clear what if any general lessons might be learned from them.We emphasize that we do not assume supersymmetry; additional techniques are available to proof the transformation law for the elliptic genus in the case of supersymmetric theories, see e.g.","[9].","However, the very simple form of (REF ) suggests it should have an equally simple derivation.","Moreover, inserting the nonlocal operator $y^{J_0}$ is equivalent to turning on a background gauge field $A^\\mu $ coupled to the conserved current $J^\\mu $ , which suggests that one might be able to prove (REF ) by studying the CFT's effective action for $A_\\mu $ .","We will begin this paper in section 2 by providing such a proof, and its generalization to a non-abelian symmetry current $J^{a \\mu }$ .","Starting in section 3, we perform several analyses of the constraints that follow from (REF ) and its non-abelian generalization using linear programming and semi-definite programming methods.","Our main results are as follows."],["Abelian Bounds","We begin by reproducing, and improving the results of [6], bounding properties of theories with an abelian current.","We place an upper bound on the dimension of the lightest charged state, $ \\begin{split} \\Delta _{*} = \\frac{c}{\\alpha } +\\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.","\\end{split} $ This bound is qualitatively similar to than the bounds in [4], [10] of non-charged states.","We also improve the bound on the smallest “mass-to-charge” ratio in the theory.","These bounds are qualitatively related to the Weak Gravity Conjecture (WGC), though gauge fields in the gravity duals are Chern-Simons fields rather than Maxwell fields.","Provocatively, we find numerical evidence for a bound on the mass-to-charge ratio that scales at large $c$ as $\\sqrt{c}$ , consistent with the bulk gravity expectation.","This is stronger than the bounds in [6], which scale as $c$ .","The improvement again comes from increasing the number of derivatives of the characters used in the analysis.","Then, we discuss the bound on the charge gap $Q_*$ .","Without any further assumptions, the numerical bound of charge gap is always $Q_*=1$ for all $c$ .","We study two examples of $c=2$ and 8 by turning on both a gap in dimension $\\Delta _*$ and in charge $Q_*$ .","There are kinks in the $\\Delta _*$ and $Q_*$ plots which can potentially be associated with full CFTs.","Lastly, we consider in detail spectra that extremize various gaps.","We use the extremal functional method, as well as extra information contained in the charged spectra to study candidate theories at $c=1$ and 8.","At $c=8$ we find that the level 1 $E_{8}$ Sugawara theory saturates both the gap in dimension (as was found in [10]) and in charge.","Using this, we are able to reproduce the full low lying spectrum, including charge assignments.","When the symmetry current $J^a$ is non-abelian, it is more appropriate to consider bounds on the dimensions of different representations in the theory.","We will mainly focus for specificity on the case where the gauge group $G$ is $SU(2)$ and the level $k$ is 1, though our methods easily generalize to any algebra and level; the main advantage of $k=1, G=SU(2)$ is that convergence is fastest here, so our numerical results are most precise.","We first obtain bounds on the gap to all non-vacuum states in non-abelian theories.","As the extended symmetry imposes additional relations on the spectrum, one may have hoped for stronger bounds.","The results, however, are similar to those found in the abelian, or even non-flavored case.","In particular, at large $c$ we find a bound of the form, $ \\begin{split} \\Delta _{*} = \\frac{c}{\\alpha } + \\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.","\\end{split} $ The real power of the modular constraints on the flavored partition function come from the ability to impose constraints independently on different representations.","Taking advantage of this, we search for a bound on the gap to non-vacuum states transforming in the trivial representation, with no constraints imposed for other representations.","At small $c$ , there are interesting “kinks” at values of $c$ where the bounds on the gap in the neutral sector is minimized.","We focus on the case $c=3$ , and use the extremal functional methods to find the low lying degeneracies at this kink.","Reassuringly, we find integer degeneracies.","This numerical spectral information allows us to guess an exact partition function saturating the bound in the neutral sector.","In fact we find multiple partition functions are allowed if we are somewhat liberal in what spins are allowed for states in the theory.The partition functions found are not strictly modular invariant, but invariant under a subgroup generated by $S$ and $T^{n}$ for $n=2$ or $n=4$ .","If all states must have integer or half-integer spins, we find a unique partition function, $ \\begin{split} Z(\\tau , \\bar{\\tau }, z, \\bar{z})&= \\frac{1}{4} \\sum _{a,b,a^{\\prime },b^{\\prime }=0}^1 (-1)^{a b^{\\prime } + a^{\\prime } b} \\left|\\theta \\left[^a_b\\right](\\tau ,\\frac{z}{2})\\right|^4 | \\theta [^{a^{\\prime }}_{b^{\\prime }}](\\tau ,0)|^2\\,.","\\end{split} $ Allowing quarter-integer spins leads to multiple allowed partition functions that maximize the gap.","Perhaps the most interesting aspect of this analysis however is not the specific partition function for this case, but rather that fact that searching for constraints in a representation dependent manner yields structure hidden to a flavor-blind analysis.","This means that the extremal functional analysis allows one to “discover” a larger class of partition functions when flavored information is included than when it is not.","Moreover, uncovering flavored information can potentially split the degeneracy between theories with the same spectrum and therefore the same partition function, allowing us to address the age-old question of whether one can “taste the shape of a drum.” Finally, we continue to refine our representation dependent analysis.","For the case of $SU(2)_{1}$ we prove analytically that the theory either contains all representation, or the partition function splits into a product of the diagonal Sugawara partition function and a neutral, modular invariant partition function.","After this work was completed, the paper [11] appeared on arXiv also considering modular bootstrap constraints on theories with conserved currents, though the analysis there did not use the flavored partition function."],["Partition Function Transformation and Background Gauge Fields","In this section, we will present an argument for the transformation law (REF ) based on the effective action obtained upon integrating out the CFT in the presence of a background gauge field $A_\\mu $ .","Previous treatments have pointed out that in the present context there are two different notions of the partition function that are natural.","One of these is the canonical partition function $Z(\\tau ,z)$ defined by (REF ).","Following [5], we will refer to an alternate definition as the “path integral” $Z_{\\rm PI}(\\tau , z)$ : $Z_{\\rm PI}(\\tau , z) \\sim e^{ \\pi k B(\\tau , z) } Z(\\tau , z) , \\qquad B(\\tau , z) \\equiv \\frac{z^2+\\bar{z}^2}{2{\\rm Im}(\\tau )}.$ Under $z^{\\prime } = \\frac{z}{c \\tau +d}, \\tau ^{\\prime } = \\frac{a \\tau +b}{c \\tau +d}$ , the factor $B$ is easily seen to transform as $B(\\tau ^{\\prime },z^{\\prime }) &=& B(\\tau ,z) - i \\left( \\frac{ c z^2}{c\\tau +d} - \\frac{c \\bar{z}^2}{c \\bar{\\tau }+d} \\right).$ The important point about the extra factor $B(\\tau ,z)$ is that its transformation cancels the transformation of $Z(\\tau ,z)$ , leaving $Z_{\\rm PI}$ invariant.","The basic idea is that $Z_{\\rm PI}$ should be the result of performing a path integral over the torus, and so should be modular invariant.","In free boson constructions, one can explicitly see how this factor is generated by the Legendre transform from the Lagrangian to the Hamiltonian [5].","However, we would like to see how this arises in a general CFT, without making any reference to a specific form of a Lagrangian.","We will begin with the case of an abelian current, and then consider the generalization to a non-abelian symmetry."],["Modular transformation and the ground state energy","First, let us discuss in more detail how to define the “path integral” function $Z(\\tau ,z)$ , what ambiguities are allowed in this definition, and why they do not affect the transformation law (REF ).","In order to be invariant under modular transformations, we will need to define the path integral to be invariant under diffeomorphisms and rigid rescalings $w \\rightarrow \\lambda ^{-1} w$ : $d \\Psi e^{ - S_{\\tau }[\\Psi ]} = d\\Psi ^{\\prime } e^{ - S_{\\tau ^{\\prime }} [\\Psi ^{\\prime }]}.$ Here, $\\Psi $ are all the fields of the CFT.","As we review in appendix , these two symmetries are sufficient to imply that the path integral defined as an integral over this measure, $Z_{\\rm PI}(\\tau , z) &\\equiv & \\int d \\Psi e^{- S_{\\tau }[\\Psi ] -\\frac{i }{2\\pi } \\int _\\tau A_{\\bar{w}} J^{\\bar{w}}}, \\qquad A_{\\bar{w}} = -i \\frac{z}{2 {\\rm Im}(\\tau )},$ is invariant under modular transformations: $Z_{\\rm PI}\\left( \\frac{a \\tau +b}{c \\tau +d}, \\frac{c z}{c\\tau +d}\\right) &=& Z_{\\rm PI}(\\tau , z).$ Different choices of regulators will change $\\log Z_{\\rm PI}$ by local terms.","However, the local terms allowed by diffeomorphism invariance and scale invariance do not affect the transformation law.","For instance, one can shift the effective action by a local term proportional to $\\int _\\tau d^2 x \\sqrt{g} A_\\mu A^\\mu \\sim \\int _\\tau dw d\\bar{w} A_w A_{\\bar{w}} \\sim \\frac{z \\bar{z}}{4 {\\rm Im}(\\tau )}.$ This term arises in the difference between a regulator that preserves the vector current $J^\\mu $ symmetry and one that preserves the axial current $\\epsilon ^{\\mu \\nu }J_\\nu $ .","However, it is easily seen to be both Weyl invariant and diffeomorphism invariant, and is invariant under modular transformations.","So its coefficient is irrelevant for our purposes, and from now on we will neglect such terms without loss of generality.","Now, the next question is how do we relate the “path integral” $Z_{\\rm PI}$ to the partition function $Z$ ?","The key point is that turning on a background field $A_\\mu $ not only turns on a chemical potential, but it can also shift the ground state energy, since at fixed $\\beta $ such a shift affects only the overall normalization of the path integral.","In the example of the free boson, this energy shift is seen explicitly by doing a Legendre transform, but we can see it in full generality by considering the effective action for $A_\\mu $ .","To see the shift, it is sufficient to calculate the ground state energy, so we can take the limit of the torus where $\\tau = i \\frac{\\beta }{2 \\pi } , \\beta \\gg 1 $ .","In this limit, the torus becomes a cylinder, and the effective action is conformally related to that in flat space, where it is universal and known in closed form.","Including the action for a background metric as well, we can write $\\log Z = \\int d^2 x \\sqrt{-g} \\left( \\frac{c}{48 \\pi } R \\Box ^{-1} R + \\frac{k}{8 \\pi } F^{\\mu \\nu } \\Box ^{-1} F_{\\mu \\nu } \\right).$ Because of the inverse Laplacians, the mapping to the cylinder is a bit subtle.","For the metric contribution, it is easiest to work with the Wess-Zumino anomaly action directly, $S_{\\rm WZ} = \\frac{c}{24\\pi } \\int d^2 x\\sqrt{-g} \\left( \\sigma R + (\\partial \\sigma )^2 \\right)$ , and take $\\sigma (w) = w+\\bar{w}$ , which reproduces the standard ground state energy shift $-\\frac{c}{12}$ from the Schwarzian derivative.","By contrast, the gauge field term in (REF ) is invariant under Weyl transformations, and its contribution to the ground state energy just comes from evaluating the non-local term on the cylinder.","To avoid ambiguities associated with the inverse Laplacian, it is clearest to use the fact that the effective action is the generating functional for the $J^\\mu $ correlators, so we know that we can equivalently write the gauge field part of $\\log Z$ as $W_A[A^\\mu ] = \\int \\frac{d^2 x d^2 x^{\\prime }}{(2\\pi i )^2} A_\\mu (x) A_\\nu (x^{\\prime }) \\langle J^\\mu (x) J^\\nu (x^{\\prime }) \\rangle .$ On the plane, $\\langle J^{\\bar{w}}(w) J^{\\bar{w}}(w^{\\prime })\\rangle = \\frac{k}{(w-w^{\\prime })^2}$ .","Mapping to the cylinder and taking $A_{\\bar{w}}$ to be constant, we haveWe performed this integration as follows.","First, shift $w \\rightarrow w+w^{\\prime }$ to eliminate $w^{\\prime }$ and immediately do the $d^2 w^{\\prime }$ integral, producing just a factor of the volume $2 \\pi \\beta $ of the torus.","Passing to $t,\\theta $ coordinates: $W_A[A_\\mu ] = - \\frac{\\beta k A_{\\bar{w}}^2}{2\\pi } \\int _{-\\beta /2}^{\\beta /2} dt \\int _0^{2\\pi } d\\theta \\frac{1}{(e^{\\frac{t+i \\theta }{2}} -e^{\\frac{-t-i \\theta }{2}})^2} .$ If we do the $\\theta $ integral first, this vanishes, except when $t=0$ where it is divergent; the integral over $\\theta $ is proportional to $\\delta (t)$ .","We avoid this subtlety if we do the $t$ integral first, in which case we obtain $W_A[A_\\mu ] =- \\frac{\\beta k A_{\\bar{w}}^2}{2\\pi } \\int _0^{2 \\pi } d \\theta \\frac{\\sinh \\frac{\\beta }{2}}{\\cos \\theta - \\cosh \\frac{\\beta }{2}} = \\beta k A_{\\bar{w}}^2 .$ $W_A[A_\\mu ] \\rightarrow \\frac{k}{(2\\pi i)^2} \\int d^2 w d^2 w^{\\prime } \\frac{A_{\\bar{w}}^2}{\\left( e^{\\frac{w-w^{\\prime }}{2}} - e^{\\frac{w^{\\prime }-w}{2}}\\right)^2} = \\beta k A_{\\bar{w}}^2.$ Combining the above with a symmetric combination from $A_w$ , we put everything together to obtain the ground state energy: $E_0 = -\\lim _{\\beta \\rightarrow \\infty } \\beta ^{-1} \\log Z_{\\rm PI} = - \\frac{c}{12} + \\delta E , \\qquad \\delta E = -k (A_w^2+ A_{\\bar{w}}^2).$ Therefore, the path integral differs from the canonical partition function by an extra factor $e^{-\\beta \\phantom{.}","\\delta E}$ , which in turn produces the factor $- \\pi k B(\\tau ,z)$ in (REF ,REF ).","So at last we see that this factor is universally the contribution to the partition function from the shift in the ground state energy due to the background gauge field.","Summarizing, the canonical partition function $Z(\\tau ,z)$ in (REF ) is defined to have a ground state energy $-\\frac{c}{12}$ .","However, any path integral over the torus using a regulator that preserves diffeomorphisms and rescalings will have a ground state energy equal to $-\\frac{c}{12} - k (A_w^2 + A_{\\bar{w}}^2)$ , plus possible terms that do not affect the modular transformation of $Z(\\tau ,z)$ ."],["Non-abelian current transformation","The generalization to the case of a non-abelian is straightforward, and can be made as follows.","Unlike in the abelian case, the effective action is not quadratic.","However, we can write it formally as the sum over all connected diagrams: $W_A[A^{a \\mu } ] = \\sum _{n=1}^\\infty \\left(\\prod _{i=1}^n \\int \\frac{d^2 w_i}{2\\pi i} A^{a_i}_w \\right) \\langle J^{a_1}(w_1) \\dots J^{a_n}(w_n) \\rangle _{\\rm conn} .$ As before, we want to set $A_w^a, A_{\\bar{w}}^a$ to be constant on the cylinder and integrate over $d^2 w_i$ .","For the part quadratic in $A$ , the computation proceeds just as in the abelian case, we simply have an extra index for the different components of $J^a_w$ .","The background field couples as $\\frac{-i}{2\\pi } \\int _\\tau A_\\mu ^a J^{\\mu a}, \\qquad A_w^a = -i \\frac{\\bar{z}^a}{2 {\\rm Im}(\\tau )}.$ $A$ 's contribution to the ground state energy is $\\delta E \\cong -k((A_w^a)^2+(A_{\\bar{w}}^a)^2),$ which transforms under modular transformations as $-\\beta \\delta E \\rightarrow -\\beta \\delta E - i \\pi k \\left( \\frac{c (z^a)^2}{c\\tau +d} - \\frac{c (\\bar{z}^a)^2}{c \\bar{\\tau }+d} \\right).$ That leaves the contribution from the higher-point functions.","We can always write these in terms of lower-point function by using the recursive formula $J^a(w)J^b(0) \\sim \\frac{k \\delta ^{ab}}{w^2} + \\frac{f^{abc} J^c(0)}{w},$ where $\\sim $ means `up to non-singular terms'.","The $\\frac{k \\delta ^{ab}}{w^2}$ piece manifestly generates disconnected diagrams - it produces the two-point function times the $(n-2)$ -point function - so it does not contribute to the effective action for higher-point correlators.","But, since we multiply the correlator by $A^a_w$ in the effective action, the $f^{abc}$ term also gives no contribution for constant $A_w^a$ : $A_w^a J^a(w) A_w^b J^b(0) \\sim \\frac{k A_w^2 }{ w^2} + A_w^a A_w^b f^{abc} \\frac{J^c(0)}{w} = \\frac{k A_w^2 }{w^2}$ since $A^a A^b f^{abc} =0$ .","Therefore only the two-point functions contribute."],["Basic setup","In all the cases we consider, we will assume the presence of a conserved current $J^a$ in the theory.","In general, it is convenient to separate the stress tensor $T$ of the theory into a Sugawara stress tensor piece and a residual piece: $T^{(0)} \\equiv T - T^{\\rm sug},\\qquad T^{\\rm sug} = \\frac{1/2}{k+\\widetilde{h}_G} \\sum _{a=1}^{|G|} : J^a J^a : ,$ where $\\widetilde{h}_G$ is the dual Coxter number, because the modes of $T^{(0)}$ commute with the modes of $J^a$ .","Furthermore among themselves they form a Virasoro algebra with central charge $c^{(0)} = c- c^{\\rm sug}, \\qquad c^{\\rm sug} =\\frac{k |G|}{k+\\widetilde{h}_G} .$ Similarly, we can separate the Virasoro generators $L_n = L_n^{(0)}+L_n^{\\rm sug}$ and the weights $h = h^{(0)} + h^{\\rm sug}$ into a part that comes from $T^{(0)}$ and a part that comes from $T^{\\rm sug}$ .","For most representations, the distinction between $T$ and $T^{(0)}$ will not make much difference, since the partition function just counts states at each level.","However, for the special cases with shortening conditions, some descendants becomes null and do not contribute to the partition function, and this is easier to see using the modes of $T^{(0)}$ .","The characters of the Kac-moody algebra $\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})={\\rm Tr}_{V_{\\mathbf {\\mu },k}}q^{L_0^{\\rm sug}-\\frac{c^{\\rm sug}}{24}}e^{2\\pi i \\mathbf {z}\\cdot \\mathbf {H}_0}$ are constructed by acting modes of $J^a$ on some highest weight state which has weight $\\mathbf {\\mu }$ .See e.g.","[12] for a standard introduction.","Here, ${\\bf H}_0$ is the vector of Cartan generators of the algebra.","In the case of an abelian symmetry, ${\\bf H_0} = J_0$ and the characters for a generic primary are simply $ q^{ - \\frac{c^{\\rm sug}-1}{24}}e^{2 \\pi i z Q}/\\eta (\\tau )$ .","In the case of a non-abelian symmetry, the characters are more complicated.","Some descendants of such a highest weight state may be null so it is non-trivial to write down its form.","However, for the purpose of the modular bootstrap, the only property of such characters we use is that the characters transform covariantly $\\mathcal {X}_{\\mathbf {\\mu },k}\\left(-\\frac{1}{\\tau },\\frac{\\mathbf {z}}{\\tau }\\right)=e^{\\frac{i \\pi k \\mathbf {z}^2}{\\tau }}\\sum _{\\mathbf {\\mu ^\\prime }}S_{\\mathbf {\\mu \\mu ^\\prime }}^k\\mathcal {X}_{\\mathbf {\\mu ^\\prime },k}(\\tau ,\\mathbf {z}) ,$ where the matrix $S$ depends on the symmetry group and level $k$ .","These characters do not include the modes of $T^{(0)}$ yet.","Since the algebra generated by modes of $J^a$ is completely orthogonal to that generated by modes of $T^{(0)}$ , the character generated by the full extended algebra simply factorizes into a Kac-Moody character and a Virasoro character $\\mathcal {X}_{\\mathbf {\\mu },k,h}(\\tau , \\mathbf {z}) = \\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z}) \\mathcal {X}_{h^{(0)}}(\\tau )~.$ Like the simple Virasoro character, the character is different if the primary saturates the unitarity bound: $\\mathcal {X}_{h^{(0)}} (\\tau ) = \\left\\lbrace \\begin{array}{cc} \\frac{q^{h^{(0)}-\\frac{c^{(0)}-1}{24}}}{\\eta (\\tau )} & h^{(0)} >0 \\\\\\frac{(1-q)q^{-\\frac{c^{(0)}-1}{24}}}{\\eta (\\tau )} & h^{(0)}=0 \\end{array} \\right.",".$ The same goes for the anti-holomorphic part.","The full partition function is $Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}}) =\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}}d_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}} \\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}}) \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\bar{\\tau }) ~.$ In the above equation the $\\bar{\\mu }$ means the representation of the anti-holomorphic part.","When we are dealing with a non-abelian symmetry, it will be convenient to define a matrix $M_{\\mu , \\bar{\\mu }}$ whose components are the coefficients of the contributions to the partition function from the different representations: $ \\begin{split} M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}&= \\sum _{h, \\bar{h}}d_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}} \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\bar{\\tau })\\\\Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}})&= \\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}} M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\,.","\\end{split} $ Modular transformations on the partition function translates into a specific modular transformation of the matrix $M_{\\mu , \\bar{\\mu }}$ .","To see this transformation law, simply separate out the transformation law of $Z$ into its irrep constituents: $ \\begin{split} 0&=Z(-\\frac{1}{\\tau }, -\\frac{1}{\\bar{\\tau }}, \\frac{\\mathbf {z}}{\\tau },\\frac{\\mathbf {\\bar{z}}}{\\bar{\\tau }}) -e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)} Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}}) \\\\&=\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}} M\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}\\left(-\\frac{1}{\\tau },\\frac{\\mathbf {z}}{\\tau }\\right)\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}\\left(-\\frac{1}{\\bar{\\tau }},\\frac{\\bar{\\mathbf {z}}}{\\bar{\\tau }}\\right)-e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)}M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\\\&=\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\left(\\sum _{\\mathbf {\\mu }^{\\prime },\\mathbf {\\bar{\\mu }^{\\prime }}}S_{\\mathbf {\\mu ^\\prime \\mu }}^kM\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mathbf {\\mu ^{\\prime }},\\mathbf {\\bar{\\mu }^{\\prime }}}\\bar{S}_{\\mathbf {\\bar{\\mu }^\\prime \\bar{\\mu }}}^k- M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\right)e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)}\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\,, \\end{split} $ where we have used the transformation rule, (REF ), and the definition (REF ).","Stripping off the characters, the above crossing equation is equivalent to a crossing equation for the matrix $0 = M(\\tau ,\\bar{\\tau })_{\\mu ,\\bar{\\mu }} - S^T_{\\mu ,\\mu ^\\prime }M\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mu ^\\prime ,\\bar{\\mu }^\\prime }\\bar{S}_{\\bar{\\mu }^\\prime ,\\bar{\\mu }}\\,.$ For the constraints on theories with non-abelian currents, equation (REF ) is the form of the constraint that we will use.","For each bootstrap question we will input the symmetry group and level $k$ ."],["Semidefinite Projective Functionals and the Extremal Method","To be self-contained, we will briefly review linear and semidefinite programming methods as applied to the modular bootstrap; for more thorough reviews and some examples of applications, see e.g.","[4], [10], [13], [14], [15], [16], [17], [18], or [19], [20], [21], [22], [23], [24], [25] for reviews and some of the original papers developing methods in the standard bootstrap that we will adopt directly.","The starting point is equation (REF ), which can be written $0 &=& \\sum _{h, \\bar{h}, \\mu ^{\\prime }, \\bar{\\mu }^{\\prime } } d_{\\mu , \\bar{\\mu }, h, \\bar{h} } \\left( F_{\\mu , \\bar{\\mu }, h, \\bar{h}}\\right)_{\\mu ^{\\prime }, \\bar{\\mu }^{\\prime }}(\\tau , \\bar{\\tau }) , \\\\\\left( F_{\\mu , \\bar{\\mu }, h, \\bar{h}}\\right)_{\\mu ^{\\prime }, \\bar{\\mu }^{\\prime }}(\\tau , \\bar{\\tau }) &\\equiv & \\left[ \\delta _{\\mu \\mu ^{\\prime }} \\delta _{\\bar{\\mu } \\bar{\\mu }^{\\prime }} \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\tau ) - S_{\\mu ^{\\prime } \\mu } S_{\\bar{\\mu }^{\\prime }, \\bar{\\mu }} \\mathcal {X}_{h^{(0)}}(-1/\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(-1/\\bar{\\tau }) \\right] .\\nonumber \\\\$ The occupation numbers $d_{\\mu , \\bar{\\mu }, h,\\bar{h}}$ are all non-negative, and include in particular the vacuum $d_{\\rm vac}=1$ .","One is generally interested in proving that there exist states in the theory with various properties, for instance that there exists a state with $\\Delta < \\Delta _{\\rm max}$ for some value of $\\Delta _{\\rm max}$ .","Let us abstractly call a choice of such properties “$P$ ”.","Then, one can prove that there is at least one state in the theory with properties $P$ as long as one can find a linear functional $\\rho $ that maps the characters to real numbers such that it is positive on the vacuum and also positive on all states not satisfying $P$ .","In equations, $\\rho ({\\rm vac})= 1 , \\textrm { and } \\rho (F_{\\mu , \\bar{\\mu }, h, \\bar{h}} ) \\ge 0 \\textrm { unless } (\\mu , \\bar{\\mu }, h, \\bar{h}) \\textrm { satisfies } P.$ The normalization $\\rho ({\\rm vac})=1$ is conventional.","If such a linear functional $\\rho $ exists, then there must be a state in the theory with the properties $P$ , otherwise $\\rho $ acting on equation (REF ) would imply $0 \\ge 1$ .","Usually we will be interested in not just one choice of $P$ but a continuous family of choices $P_s$ parameterized by a continuous variable (or variables) $s$ .","Typically, $s$ will be something like the bound $\\Delta _{\\rm max}$ in the example above, so that as one decreases $s$ the set of states with property $P_s$ grows and therefore the set of linear functionals that are positive on all such states shrinks.","Critical values $s_*$ of $s$ where the set of such linear functionals vanishes are especially interesting: aside from giving the best possible bounds, at these points one can use the “extremal functional method” [19] to determine all of the occupation numbers $d_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ .","The basic idea behind this is that for any $s$ , the space of functions $F_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ spanned by states satisfying $P_s$ is a polytope where $-F_{\\rm vac}$ is inside the polytope for $ss_*$ .","At exactly $s_*$ , $-F_{\\rm vac}$ passes through one of the faces of the polytope, so there is a unique positive semidefinite linear combination of the states satisfying $P_{s_*}$ that cancels the contribution from the vacuum in (REF ).","In practice, we have to work with finite-dimensional projections of the full space of functions $F_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ , but one optimistically expects to converge to a unique solution as the dimensionality of the projected space increases.","We will encounter some exceptions that we will discuss as we come to them."],["Abelian Bounds","In this section, we perform a systematic numerical analysis on the bounds on the gap in dimensions and charges, as well as on the smallest charge-to-mass ratio allowed in a theory with a $U(1)$ current."],["Semi-definite Programming with Continuous Charge $Q$","For the abelian case, for simplicity we will not use the full Kac-Moody characters, but rather just the Virasoro characters $\\chi _h(q)$ : $Z(\\tau , z) =\\sum _{h, \\bar{h}, Q, \\bar{Q}} d_{Q, \\bar{Q}, h , \\bar{h}} y^Q \\bar{y}^{\\bar{Q}} \\chi _h(q) \\chi _{\\bar{h}}(\\bar{q}).$ where $q= e^{2\\pi i \\tau }$ , $\\bar{q} = e^{-2\\pi i \\bar{\\tau }}$ , $y =e^{2\\pi i z}$ , $\\bar{y} = e^{-2\\pi i \\bar{z}}$ .","We will consider left-right symmetric theories with $c = \\bar{c}$ , and for simplicity we set $\\bar{z} = 0$ .","As in [4], we reduce the characters using the $S$ invariant factor $|\\tau |^{\\frac{1}{2}}|\\eta (\\tau )|^2$ .","Furthermore we reduce the partition function as $\\hat{Z}(\\tau , z) \\equiv |e^{\\frac{i\\pi z^2}{2\\tau } }|^2|\\tau |^{\\frac{1}{2}}|\\eta (\\tau )|^2 Z(\\tau , z) ,$ so that $\\hat{Z}(\\tau , z)$ is invariant under $ \\left( \\tau \\mapsto -\\frac{1}{\\tau }, z \\mapsto \\frac{z}{\\tau } \\right)$ .","The characters are reduced into $\\hat{\\chi }_0(q) = e^{\\frac{i\\pi z^2}{2\\tau } } \\tau ^{\\frac{1}{4}}q^{-\\frac{c-1}{24}}(1-q) ,~~~~\\hat{\\chi }_h(q) y^Q = e^{\\frac{i\\pi z^2}{2\\tau } } \\tau ^{\\frac{1}{4}}q^{-\\frac{c-1}{24}}q^h y^Q .$ We consider linear functionals of the form $\\rho \\equiv \\sum _{m+n+k/2 {\\rm ~odd,~} k {\\rm ~even}}\\left.","\\alpha _{m,n,k} \\ \\partial _{t}^m \\, \\partial _{\\bar{t}}^n\\,\\partial _{w}^k\\right|_{t=\\bar{t}=w=0},$ where the change of variable $\\tau = i e^{t}$ and $z =we^{\\frac{t}{2}}$ is made so that $t \\mapsto -t$ and $w^2 \\mapsto -w^2$ under $S$ transformations.In this expression for $\\rho $ , we have not used any $\\bar{z}$ derivatives and so do not use information about the anti-holomorphic charge $\\bar{Q}$ .","This is mainly for simplicity and efficiency; it is in principle straightforward, though more computationally intensive, to use $\\bar{Q}$ information as well.","Later in this section we will in fact perform one analysis where we keep $\\bar{z}$ derivatives to demonstrate this point.","We arrive at a two variable functional $\\rho _l(\\Delta ,Q) &\\equiv \\rho \\left[ \\hat{\\chi }_{\\frac{\\Delta -l}{2}}(\\tau )\\hat{\\bar{\\chi }}_{\\frac{\\Delta +l}{2}}(\\bar{\\tau }) y^Q +\\hat{\\chi }_{\\frac{\\Delta +l}{2}}(\\tau )\\hat{\\bar{\\chi }}_{\\frac{\\Delta -l}{2}}(\\bar{\\tau }) \\right] , \\nonumber \\\\\\rho ({\\rm vac}) &\\equiv \\rho \\left[ \\hat{\\chi }_{0}(\\tau )\\hat{\\bar{\\chi }}_{0}(\\bar{\\tau }) \\right],$ where we assume the spectrum is parity symmetric.","Part of the challenge of the abelian analysis is that we do not assume charge quantization, i.e.","technically we allow the gauge group to be $\\mathbb {R}$ instead of $U(1)$ , which means that we have to deal with not just one but two continuous parameters, $\\Delta $ and $Q$ .","This complicates the application of positive semi-definite approaches, since these are based on constructing positive functionals of the characters and in general the space of such functionals is more complicated for multiple variables than for a single variable.","In particular, for a single variable, positive semi-definite functionals can be written without loss of generality as a sum of squares plus a linear term times a sum of squares.","For multiple variables, such a parameterization is no longer completely general.","One way to deal with this issue is simply to discretize in, say, $Q$ , but we find that such an approach becomes difficult to implement in practice since the discretization needs to become very fine to prevent the numeric search from picking functionals that become negative in between the discretization points.","The approach we take is instead to limit the search space to functionals that are still a sum of squares plus a linear term times sums of squares.","In the limit of very high order polynomials, one might expect that such functionals can approximate the extremal functionals arbitrarily well.","In any case, while such functionals might not give the best possible bounds, they nevertheless produce valid bounds.","Even restricting to polynomial functionals, there remains a practical problem of how to implement the search over such functionals using available software for semi-definite programming analyses.","In appendix , we discuss how to massage this problem into an appropriate form for use with SDPB."],["Bound on Dimension of Lightest Charged State","With the flavored partition function we can bound the dimension $\\Delta _*$ of lightest charged state in any theory with a U(1).","The bound for different $c$ is summerized in Table.","REF .","We extrapolate the bound values to $n_D \\rightarrow \\infty $ using a linear function of $\\frac{1}{n_D}$ similar to what is done in [10] for $c \\le 100$ where the convergence of the bound values is significant.","Then we extrapolate the bounds to $n_D\\rightarrow \\infty $ and $c \\rightarrow \\infty $ by fitting the finite $n_D$ and $c$ results to a linear function of $\\frac{1}{c}$ and $1/n_D$ and extrapolating, as shown in Fig.","REF .","In this test we take $\\tau = \\frac{i \\beta }{2 \\pi }$ with $\\beta $ real to avoid the complication of spin.","It is not understood a priori why the form of this fit works, but empirically it agrees well with the data at large $c$ and $n_D$ .","Table: Bounds on the dimension of lightestcharged state assuming the theory has U(1) symmetry, as a function of cc and the number n D n_D of derivatives used in the bootstrap functionals.Figure: Bounds of the dimension oflightest charged state assuming the theory has U(1) symmetry.","Theextrapolated gaps at n D →∞n_D \\rightarrow \\infty with the trendline.Similarly to the results of [10], extrapolating in $n_{D}$ and then $c$ provides a parametrically stronger bound than the finite $n_{D}$ analysis.","$\\Delta _{*} = \\frac{c}{\\alpha } + \\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.$ This bound (REF ) is similar to the bounds on the non-charged state found in [4], [10], though quantitatively different.","The bound in [4] is parametrically weaker, which is not surprising since that analysis did not use the spins of the characters, and did not perform any extrapolation in the number of derivatives.","The bound in [10] is more analogous since spins and extrapolations were used; €” the result there is very slightly stronger ($\\alpha \\sim 9$ ) than (REF ) for the charged spectrum."],["Bound on Charge-to-Mass Ratio","In this section, we will present results that there must be a state in the theory with a charge-to-mass ratio $r \\equiv \\frac{Q c}{12 \\Delta } = \\frac{Q}{8 G_N m},$ above some critical value $r_*$ , whose value we will determine numerically.The value of $r^*$ increases as the number $n_D$ of derivatives used increases and the numeric accuracy improves, though we emphasize that even for low number of derivatives the values of $r_*$ are a valid bound proving that a state must exist in the theory with $\\frac{Q}{8 G_N m} > r_*$ .","We try to find the linear functional thatWe also impose a dimension cutoff $\\rho (\\Delta ,Q) \\ge 0,~~~ \\Delta >\\frac{100c}{12}$ , because we want the constraint to be a little stronger that not only a state which saturates the ratio bound exists but also the state must have finite dimension.","Different dimension cutoffs do not result in significantly different functionals and bounds.","It just helps the algorithm to find a functional that only is negative at finite $\\Delta $ .","$\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ |Q| \\le Q_{\\Delta } \\equiv \\frac{12 r \\Delta }{c}~.$ We drop the spin index $l$ by only taking functionals of the form $\\rho \\equiv \\sum _{m+k/2 {\\rm ~odd,~} k {\\rm ~even}} (\\partial _t+ \\partial _{\\bar{t}})^m \\partial _w^k ~.$ For the functional to be positive in a bounded region we make a change of variables of the form $Q^2 = \\frac{\\tilde{Q}^2 Q_{\\Delta }^2}{\\tilde{Q}^2 +1 }~.$ $|\\tilde{Q}| \\ge 0$ means $|Q| \\le Q_{\\Delta }$ , inspired by [10].","First, we show in Fig.","REF the bound on $r_*$ as a function of $c$ .","By inspection, one can see that the larger $c$ is, the longer it takes for the bounds to converge.","To see how the bound depends on the number $n_D$ of derivatives in more detail, in Fig.","REF we focus on a specific value of $c$ , $c=10^5$ and show the resulting bound on $r$ as a function of the number $n_D$ of derivatives allows in the functional $\\rho $ .","The best fit as power law suggests that the optimal bound on $r^*$ might be significantly better, i.e.","$(r^{*})^{-1} \\ll 1 $ .","For comparison, the result in [6] was $\\left(\\frac{8 G_N m}{Q}\\right)_* = (r_*)^{-1} < 4 \\sqrt{\\pi } = 7.1$ .","Figure: Bound of mass-to-charge ratio as a function of cc; a trend line ∝c -1/2 \\propto c^{-1/2} is shown for comparison.","The extrapolation (“n D =∞n_D=\\infty ” points) and error bars are computed by performing a fit as a function of n D n_D and extrapolating to n D →∞n_D\\rightarrow \\infty as described in the text.In [6], it was also shown that even with a small number $n_D$ of derivatives, one could obtain a bound on $\\frac{\\Delta }{Q}$ at large $c$ that scaled like $\\sim c $ .","There is an intriguing possibility however that the true bound scales like $c^{1/2}$ , and that this is obscured because it takes more and more derivatives to reach this optimal bound as $c$ increases.","The basic idea for why one might expect a $c^{1/2}$ scaling is that in higher dimensions, the scaling of the WGC limit can easily be read off by demanding that the binding energy from gravity and a Coulomb force cancel each other out.","In the AdS$_3$ case, one can think of the binding energy from gravity as $ \\frac{3\\Delta ^2}{c}$ , whereas from a $k=1$ Chern-Simons gauge field exchange it is $Q^2$ ; demanding equality would set $\\frac{\\Delta }{Q} \\approx \\sqrt{c/3}$ , i.e.","$\\frac{8 G_N m}{Q} \\approx 6.9 c^{-1/2}$ .","One can read off the coefficients by looking at the vacuum conformal block for Virasoro and Kac-Moody algebras in the limit $z\\sim 1$ [26], [27].","For comparison, in Fig.","REF we have also shown a trend line at $c^{-1/2}$ , which becomes further below our best numeric bound with $n_D=29$ derivatives as $c$ increases.","We can try to estimate the optimal bound by taking our result at each $c$ as a function of $n_D$ and extrapolating to $n_D = \\infty $ .","The “$n_D=\\infty $ ” points we show in Fig.","REF fit our results starting at $n_D \\ge 15$ in order to get the extrapolation.","However, there is significant uncertainty in the resulting estimate, as can be gauged by the fact that performing the fit starting at smaller or larger values of $n_D$ gives different answers.","In Fig.","REF we have shown the bound as a function of $n_D$ , where one can explicitly see that the bound is still changing rapidly as a function of $n_D$ even at the upper range of what we have been able to achieve numerically.","In Fig.","REF , the error bars we have shown indicate the range over all the different positive values we obtain if we perform the fit over $n_D \\ge 11, n_D \\ge 13, \\dots n_D \\ge 19$ .","With better numerical accuracy at large values of $c$ , it should be possible to more firmly establish this scaling behavior.","Figure: Upper bound on 8G N m Q\\frac{8 G_N m}{Q} (that is, there existsa state below the bound) as the number n D n_D of derivatives usedin the semidefinite programming analysis increases, for thespecific case c=10 5 c=10^5.","The bound value is still changing rapidlyat n D =41n_D = 41."],["Bound on Lowest Charge","Next we will focus on bounds on the lowest charge $Q_*$ of all charged states in the theory.","We will first consider the charge $Q$ only, and then see how to do better by including information on dimensions and spins.","To determine the upper bound on the gap to the smallest $|Q|$ of all the charged states, we want to find a linear functional satisfying the following conditions: $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge 0 \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge 0 {\\rm ~~~And~~~} |Q| \\ge Q_*$ The resulting bound on $Q_*$ is shown in Fig.","REF .","The result is somewhat surprisingly always just $Q_* = 1$ .","This may be because in some theories with $c\\le 1$ a state of $Q=1$ saturates the bound and theories of larger $c$ can be constructed as a direct product of such theories and other algebras.","Figure: We obtain an upper bound, shown here, on the smallest nonzero charge Q * Q_*; the bound isQ * ≤1Q_* \\le 1 for all cc.In any case, we next turn to including information on dimensions by bounding gaps in both $Q$ and $\\Delta $ simultaneously – $Q_*$ as the lowest charge of charged states and $\\Delta _*$ as the lowest dimension of all non-vacuum states.","To obtain such a bound, the linear functional $\\rho $ should satisfy $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge \\Delta _* \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge \\Delta _* {\\rm ~~~And~~~} |Q| \\ge Q_*$ We take the linear functional to have no spin information.","Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\Delta _* at c=2c=2.","The region near the kink in the left plot is magnified and shown in the right plot.At each individual $c$ , the bound carves out a region in a two-dimensional parameter space.","The exclusion curve of an $c=2$ example is shown in Fig.","REF .","There is a kink at $\\Delta _* \\approx 0.5$ which has a bound $Q_* \\le 1$ .","It would be interesting to identify what if any theory lives at this kink.","Finally, the simutaneous $\\Delta _*$ and $Q_*$ approach can be even more powerful if we turn on spin.","For this case the example we choose is $c=8$ .","In [10] the Sugawara theory $E_8$ lattice shows up as a kink of bounds on lowest dimension of scalar primaries.","We seek a linear functional $\\rho $ satisfying $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho _0(\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge \\Delta _* \\nonumber \\\\\\rho _l(\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge |l| \\nonumber \\\\\\rho _0(\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge \\Delta _* {\\rm ~~~And~~~} |Q| \\ge Q_* \\nonumber \\\\\\rho _l(\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge |l| {\\rm ~~~And~~~} |Q| \\ge Q_*$ for all $l \\in \\mathbb {Z}_{\\ge 0}$ .","Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\Delta _* at c=8c=8.The two parameter plot of $\\Delta _*$ and $Q_*$ is shown in Fig.","REF .","Note that we find that $Q_*$ at $\\Delta _*=0$ is smaller than 1, better than the bound obtained without $Q_*$ informationIt is also possible that by assuming integer spins we throw away the theory that saturates the $Q_{*}=1$ bound.. More interestingly, we see a sharp kink at $\\Delta _{*}=2$ .","In the next subsection we will analyze the extremal functional of this kink and see that this kink is the $E_8$ lattice CFT, and we will obtain the dimension and charge spectrum of the low lying states."],["Extremal Functional Analysis","In this subsection, we will use extremal functional analyses to determine the partition function saturating various bounds."],["Maximal $r_*$ at {{formula:9d4b5483-4fe8-4eda-a64b-878604bb43b5}}","Our first application of extremal function methods will be to the bound on the charge-to-mass ratio $r$ .","Since our bounds have converged for $c=1$ and we can consider the extremal functional $\\rho $ for this case; by design, $\\rho $ is non-negative on the space of states we allow, and so the states in the theory must be at the places where $\\rho $ vanishes.","The functional depends on both $\\Delta $ and $Q$ , so the extremal spectrum contains more data as shown by Fig.","REF .","Figure: Projection functional ρ\\rho of c=1c=1 as a function ofΔ\\Delta and QQ computed at N=29N=29.","The black regions are where ρ≤0\\rho \\le 0; according to our criterion (), ρ\\rho is ≥0\\ge 0 in the allowed region Q≤12r * Δ cQ\\le \\frac{12 r_* \\Delta }{c}, so the extremal spectrum comprises states where ρ=0\\rho =0 in this region.","There is a line of small black dots where ρ=0\\rho =0 along the Q=0Q=0 axis that are difficult to see and so we plot ρ\\rho along this line in Fig.",".The zeros of $\\rho $ occur at points and can be difficult to see in Fig.","REF .","In Fig.","REF , we show $\\rho $ along two particularly relevant lines: the neutral ($Q=0$ ) states, and the states that saturate the mass-to-charge ratio, i.e.","$Q= \\frac{12 r_* \\Delta }{c}$ .","Figure: The functional restricted at Q=0Q=0 is shown on theleft.","The states that saturate the best bound on8G N m Q\\frac{8 G_N m}{Q} is shown on the right."],["Sequential $\\Delta _*$ and {{formula:b25e4990-736d-44d3-b86f-624d64285424}} Approach - Revisiting the {{formula:b4d2cd11-1472-4199-8de0-93f481d9bb5d}} Lattice","In Sec.","REF we found a kink in the simultaneous $\\Delta _*$ and $Q_*$ approach with spin information at $c=8$ .","In order to show that the kink is indeed the level 1 $E_8$ lattice, we can look for extremal flavored partition functions by using a “two-step” approach where we first solve for the spectrum of dimensions and then solve for the spectrum of charges.","The idea is that we can use the extremal functional method on the unflavored partition function, maximizing the gap in dimensions of operators.","This step is just the standard extremal functional method and it will just reproduce previous results [10].","Then, having fixed the weights of the states in the theory, we can impose a gap in the charge of the states in the theory, allowing only the weights $(h,\\bar{h})$ found previously.","For concreteness, we will focus on the case $c=8$ as a representative example.","As shown in [10], the gap in dimensions is maximized at $\\Delta _*=2$ by the $E_8$ theory at $k=1$ (this theory can be described as 8 free bosons on an $E_8$ lattice), and the extremal functional method allows one to independently derive the partition function of this theory.","We have reproduced the extremal functionals $\\rho _\\ell (\\Delta )$ at each spin $\\ell $ in Fig.","REF , which implies the spectrum is ${\\left\\lbrace \\begin{array}{ll}\\Delta = 2,4,6,\\ldots ,& l=0,\\\\\\Delta = l,l+2,l+4,\\ldots ,& l\\ne 0 .\\\\\\end{array}\\right.","}$ Figure: Extremal functional of c=8c=8 theory, at n D =35n_D=35.Moving on to the spectrum of charges, we find that the gap $Q_*$ is maximized at $Q_* = \\frac{1}{\\sqrt{2}}$ .","The corresponding extremal functionals $\\rho _{\\Delta , \\ell }(Q)$ at each dimension $\\Delta $ and spin $\\ell $ are plotted in Fig.","REF .","We note that this is not the flavored partition function that one obtains if one turns on a chemical potential for the charge $J = \\partial \\phi $ in the $E_8$ lattice description; that choice corresponds to the spectrum of charges $\\frac{1}{2}\\mathbb {Z}$ rather than $\\frac{1}{\\sqrt{2}} \\mathbb {Z}$ .","Instead, if one chooses one of the length-2 vectors $\\vec{\\alpha }$ in the $E_8$ lattice, then $J \\equiv V_{\\vec{\\alpha }} \\equiv \\frac{1}{\\sqrt{2}} \\left( e^{\\vec{\\alpha }\\cdot \\vec{\\phi }} + e^{-\\vec{\\alpha } \\cdot \\vec{\\phi }} \\right)$ has $k=1$ , and the lowest charged states include for instance $V_{2\\vec{\\alpha }}$ , with charge $\\frac{1}{\\sqrt{2}}$ .","One can see in Fig.","REF that the extremal functional has zeros at around $0, \\pm \\frac{1}{\\sqrt{2}}$ and $\\pm \\frac{2}{\\sqrt{2}}$ for all $\\Delta , \\ell $ , and we expect that this would continue to be true at $ \\frac{n}{\\sqrt{2}}$ for all $n \\in \\mathbb {Z}$ as the number $n_D$ of derivatives used in the analysis approaches infinity.","Because these dimensions and charges appear to follow such a simple pattern, we will proceed by assuming this pattern continues.","Then, with the allowed weights $\\Delta , \\ell $ and charges $Q$ fixed in the theory, solving the modular bootstrap equation reduces to a linear programming problem, which is much more efficient numerically.Furthermore, since this linear programming analysis fixes the partition function for us to be a particular flavoring of the $E_8$ theory, by uniqueness it will be the correct one, justifying the original Ansatz.","We obtain the occupation numbers indicated in Table REF , where we have flavored separately by both holomorphic and anti-holomorphic charges $Q$ and $\\bar{Q}$ .","We show the occupation numbers of the conserved $ \\ell = 1$ currents assuming the extremal charge spectrum $Q= \\frac{n}{2}$ (right).","In addition, it is straightforward to repeat the analysis assuming $Q = \\frac{n}{\\sqrt{2}}$ (left) for comparison.","In both cases, we obtain a total of 248 currents each in the holomorphic and anti-holomorphic part, but distributed differently among different charges in the two cases.","Table: Occupation numbers from a linear programming analysis.","The left table assumes states at Q∈1 2ℤQ \\in \\frac{1}{\\sqrt{2}} \\mathbb {Z}, whereas the right assumes Q∈1 2ℤQ \\in \\frac{1}{2} \\mathbb {Z}.Figure: Extremal functional of QQ at n D =19n_D=19 when the gap on QQ is maximizedat 1 2\\frac{1}{\\sqrt{2}}."],["Bounds on gaps in operator dimensions","Next we turn to a numeric analysis of gaps in non-abelian theories.","In some cases, the results are somewhat stronger or weaker depending on whether or not we allow for states that saturate the unitarity bound $2 k h \\ge Q^2$ , which we will refer to as “extremal states”, and whether or not we impose gaps in all charge sectors.","We will present results starting with the strongest assumptions first.","In all cases, we will present only the results for $SU(2)$ gauge group at level $k=1$ .","We have analyzed larger gauge groups and higher levels and the results are qualitatively similar, though the rate of numeric convergence is worse; some preliminary results for $SU(2)$ with $k=2$ are shown in appendix .","To begin, we will set the gap in all representations to be the same and restrict to the partition function at $q=\\bar{q}$ ; the resulting gap value will be the lowest dimension of the primary operators.","This “uniform bound” is shown in Fig.","REF .","We have actually done two slightly different analysis, which are compared to each other on the right in Fig.","REF .","These analyses differ in whether or not we allow states in the non-trivial representations with $h^{(0)}= \\bar{h}^{(0)}=0$ , which saturate the unitarity bound in both the holomorphic and anti-holomorphic sectors and which we will refer to as “extremal states;” in the analysis where such states are included, the “gap” for each representation is defined as the smallest $\\Delta ^{(0)}$ among the non-extremal states.","As one can see, the difference between the results of the two analyses is significant at small $c$ but becomes negligible as $c$ approaches $\\infty $ .","Ultimately, this result does not tell us much more than one learns from previous similar analyses without flavored information; all we learn here is that there must be some state in the theory with $\\Delta ^{(0)}$ below some value, which is very similar to the bound on the same quantity from the unflavored modular bootstrap.","Figure: Bound on SU(2) k=1k=1 gaps in Δ\\Delta universal for allrepresentations for 1≤c≤1001 \\le c \\le 100.","Left: Boundsobtained with different values of n D n_D when extremal states arenot allowed.","Dashed lines from blue to red are computed data of3≤n D ≤293 \\le n_D \\le 29.","The solid black line is extrapolated fromdata of 11≤n D ≤29 11 \\le n_D \\le 29 using a function linear in1/n D 1/n_D.","Right: Bounds extrapolated to n D =∞n_D = \\infty forthe analysis without extremal states compared to the result whenextremal states are allowed.","The difference is negligible atlarge cc but significant at small cc.Next, however, we will turn to an analysis that maximizes the bounds separately in different sectors, and this is where we will start to find something qualitatively new compared with what is possible with the unflavored modular bootstrap.","In particular, we will maximize the gap in the trivial representation, and not impose any constraint on the gaps in the other representations.","In equations, our conditions are $\\rho _{\\lambda ,\\bar{\\lambda }}(\\Delta ^{(0)})\\ge 0 {\\rm ~when~}{\\left\\lbrace \\begin{array}{ll}\\Delta ^{(0)} \\ge \\Delta _{*},&\\lambda =(0){\\rm ~and~} \\bar{\\lambda }=(0),\\\\\\Delta ^{(0)} \\ge 0,&\\lambda {\\rm ~or~} \\bar{\\lambda }\\ne (0),\\\\\\end{array}\\right.","}$ in SU(2) $k=1$ , weight $\\lambda $ (or $\\bar{\\lambda }$ ) takes values $(0)$ or $(\\frac{1}{2})$ .","The resulting bound on the neutral sector gap is shown as a function of $c$ in Fig.","REF .By contrast with the previous subsection, here we find that the bound is exactly the same whether or not we allow extremal ($h^{(0)}=\\bar{h}^{(0)}=0$ ) states in the non-trivial representations.","Notably, there is a minimum of about $\\Delta _* =1$ near $c \\equiv c^{(0)}+1 =3$ .","We next turn to a more detailed study of this point.","Figure: Left: The upper bound on the gap in the dimension ofprimaries, Δ * \\Delta _{*}, in the trivial representationobtained at increasing derivative order of the linear functional(from blue to red, up to n D =29n_D = 29).","The black curve is theextrapolated value.","Right: Blown-up plot of the-near minimalΔ * \\Delta _{*} region.","The minimal isexpected at 2.00≤c≤2.042.00\\le c \\le 2.04,Δ * ≈0.995\\Delta _{*} \\approx 0.995."],["Spin-Independent Analysis","Our $q=\\bar{q}$ analysis in subsection (REF ) found a minimum gap bound at $c=3$ .","Using the extremal functional method [19], the dimensions and the degeneracies of states at this point can be extracted, with numerical accuracy being best for the lowest dimension states.","The dimensions of states occur at the zeros of the extremal functional, plotted in Fig.","REF .","Figure: Extremal functional of SU(2) k=1k=1, c=3c=3 spectrum.Furthermore, we find that the occupation numbers of the lowest-energy states of the partition function are uniquely determined to be $M_{(0),(0)}(\\beta ) &\\approx \\chi _{0}(\\beta ) + 28\\chi _{1}(\\beta ) + 76\\chi _{2}(\\beta ) +274 \\chi _{3}(\\beta )+\\ldots \\\\M_{(0),(\\frac{1}{2})}(\\beta ) =M_{(\\frac{1}{2}),(0)}(\\beta ) &\\approx 8\\chi _{0.5}(\\beta ) + 48\\chi _{1.5}(\\beta )+\\ldots \\\\M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\beta ) &\\approx 8\\chi _{0.5}(\\beta ) + 48\\chi _{1.5}(\\beta )+\\ldots $ The subscript on $\\chi _{\\Delta ^{(0)}}$ denotes the non-Sugawara dimension of the state.","At this point, the analysis takes $\\tau \\equiv \\frac{i \\beta }{2\\pi }$ to be pure imaginary, so no information on spins is used: $\\chi _{\\Delta ^{(0)} } = q^{-\\frac{c}{12}} \\left\\lbrace \\begin{array}{cc} \\prod _{n=2}^\\infty (1-q^n)^{-2}, & \\Delta ^{(0)}=0 , \\\\q^{\\Delta ^{(0)}} \\prod _{n=1}^\\infty (1-q^n)^{-2}, & \\Delta ^{(0)} >0 \\end{array} \\right.$ We find that a manifestly modular invariant partition function that reproduces this is $Z(\\beta )_{z=\\bar{z}} &=& \\frac{1}{2 \\eta ^6 } \\left( \\left( \\theta _3^2 - \\theta _4^2 \\right) \\theta _2^4(\\frac{z}{2}) + \\left( \\theta _2^2 + \\theta _4^2 \\right) \\theta _3^4(\\frac{z}{2}) + \\left( \\theta _3^2 - \\theta _2^2 \\right) \\theta _4^4(\\frac{z}{2}) \\right) \\\\&=& \\frac{1}{2 \\eta ^6} \\left( \\theta _2^2 \\theta _2^4(\\frac{z}{2}) + \\theta _3^2 \\theta _3^4(\\frac{z}{2}) + \\theta _4^2 \\theta _4^4(\\frac{z}{2}) + \\left( \\theta _3^2 - \\theta _2^2 - \\theta _4^2 \\right) \\theta _1^4(\\frac{z}{2}) \\right) ,$ where $\\theta _i\\equiv \\theta _i(z=0)$ .","It is straightforward to check that the occupation numbers are non-negative, so that this partition function is unitary, modular invariant, and extremizes the scalar gap.","Therefore (REF ) is the correct partition function by uniqueness."],["Spin-Dependent Analysis","In the previous subsection, we used extremal functional techniques to determine a unique partition function on the subspace $q=\\bar{q}$ when the gap in the scalar sector was maximized for $c=3$ .","We can gain much more information about the theory by relaxing the constraint $q=\\bar{q}$ and varying $q,\\bar{q}$ independently; in particular, the analysis becomes sensitive to the spins $h-\\bar{h}$ of the spectrum.","We could continue to use semi-definite programming methods, but they converge less quickly for independent $q,\\bar{q}$ than they do for $q=\\bar{q}$ .","Instead, we can use the fact that we know the spectrum of dimensions from the $q=\\bar{q}$ analysis, and the fact that spin is quantized.","This allows us to fix the allowed values of $h,\\bar{h}$ to a discrete set, turning the problem into a linear programming problem and thereby making the analysis much more efficient.","There is a remaining ambiguity, however, which is that we have to make a choice about what spins are allowed.","We find that if we allow only integer spins, there is no allowed partition function and in fact we can reduce the bound on the gap somewhat to about 2/3.","If instead we allow fractional spins, then we find a few different possibilities depending on what spins we allow.","We will begin with the conventional case where we allow integer and half-integer total spins, $h-\\bar{h}$ .","The $SU(2)$ Sugawara characters are such that $M_{(0),(0)}$ only has integer spins, $M_{(0),(1/2)}$ and $M_{(1/2),(0)}$ only has quarter spins and $M_{(1/2),(1/2)}$ can have integer and half integer spins.","Then to meet the requirement $M_{(0),(0)}$ and $M_{(1/2),(1/2)}$ can only have spins $\\frac{2n}{4}$ and $M_{(0),(1/2)}$ and $M_{(1/2),0}$ can only have spins $\\frac{2n+1}{4}$ .","Performing the linear programming analysis for such a spectrum (and continuing to maximize the gap in the neutral sector) leads to the following unique set of weights and occupation numbers $d$ :The reader may notice that the numbers at each dimension in eq.","(REF ) do not match the total number of states in the table.","The reason is that without knowledge of spin, there are null states that could not be taken into account in (REF ).","For instance, at level 2, there are a total of 84 states, as compared with 76 in (REF ), because of the 8 conserved currents at spin 1 that consequently have 8 “null” descendants at $\\Delta = 2$ .","At $\\Delta \\ge 3$ , the presence of such null states causes states to get reorganized in increasingly complicated ways and it is easiest to check the number of states is the same by constructing the full partition function.","Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThe non-Sugawara dimensions $\\Delta ^{(0)}$ and spins $\\ell ^{(0)}$ are just $\\Delta ^{(0)} = h^{(0)} + \\bar{h}^{(0)}, \\ell ^{(0)} = h^{(0)}- \\bar{h}^{(0)}$ .","States are evenly divided between $\\ell ^{(0)} = + | \\ell ^{(0)}| $ and $\\ell ^{(0)} = - | \\ell ^{(0)}|$ at each weight, and the occupation numbers for the $(\\mu , \\bar{\\mu }) = (0, \\frac{1}{2})$ representations are the same as for $(\\frac{1}{2}, 0)$ .","To get the full characters one multiplies the non-Sugawara Virasoro characters $\\chi _h(\\tau )$ (i.e.","generated by the modes of $T^{(0)} = T- T^{(\\rm sug)}$ ) by the Weyl characters $\\chi _\\lambda ^{(k)}(\\tau ,z)$ , which in this case areSee eg [12], eqs (14.176) and (15.244).","$\\chi _\\lambda ^{(1)}(\\tau ,z) = \\frac{1}{\\eta } \\sum _{m \\in \\mathbb {Z}+\\lambda } q^{m^2} y^m .$ After some trial and error, we find that the corresponding flavored partition function is reproduced by $Z(\\tau , \\bar{\\tau }, z, \\bar{z}) = \\frac{1}{4|\\eta |^{6}} \\sum _{a,b,a^{\\prime },b^{\\prime }=0}^1 (-1)^{a b^{\\prime } + a^{\\prime } b} \\left|\\theta \\left[^a_b\\right](\\tau ,\\frac{z}{2})\\right|^4 | \\theta [^{a^{\\prime }}_{b^{\\prime }}](\\tau ,0)|^2 ,$ in Jacobi/Erderlyi notation $\\theta _1 = \\theta [{1 \\atop 1}], \\theta _2 = \\theta [{1 \\atop 0}], \\theta _3 = \\theta [{0 \\atop 0}], \\theta _4 = \\theta [ { 0 \\atop 1}]$ .","As this candidate partition function is half integrally modded, it is a little unfamiliar.","A natural guess is that it arises as a $\\mathbb {Z}_{2}$ orbifold of a fully modular invariant theory.There is actually a history of extremal theories arising in such a fashion [28], [29], [30].","Indeed it is possible to project this onto a fully modular invariant partition function.","Taking the unflavored expression for simplicity, $ \\begin{split} Z^{\\rm (inv)}(\\tau , \\bar{\\tau })&=\\frac{1}{2}\\left(Z(\\tau , \\bar{\\tau }, 0,0)+Z(\\tau +1, \\bar{\\tau }+1, 0,0)+Z(-1/(\\tau +1), -1/(\\bar{\\tau }+1), 0,0)\\right) \\end{split} $ Unfortunately, we have not been able to identify a theory corresponding to (REF ).","It is straightforward to check by exhausting the possibilities that the central charge and the number of spin-1 conserved currents (11) is not consistent with this partition function being associated with a pure Sugawara theory for some Lie algebra.","While other choices for the quantization of spin are less conventional, they are still of some interest.For instance [31].","Another possibility we have considered is that the non-Sugawara part of the spin, i.e.","$h^{(0)}-\\bar{h}^{(0)}$ , is an integer or a half-integer.","Because of the contribution to the weight from the Sugawara part of the stress tensor, in this case the states in the $(\\frac{1}{2}, 0)$ and $(0,\\frac{1}{2})$ representations have quarter-integer spins.","Performing the linear programming analysis making this Ansatz for the spins, we find not just a unique solution but in fact a family of solutions given by the following occupation numbers: Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThese partition functions satisfy crossing for all $x$ .","This one-parameter family is shown graphically in Fig.","REF , where we perform the linear programming analysis with the degeneracy $d$ of the $(\\mu , \\bar{\\mu })= (\\frac{1}{2} ,0), (h^{(0)}, \\bar{h}^{(0)}) = (\\frac{1}{4}, \\frac{1}{4})$ chosen by hand and look at how several other degeneracies depend on this choice.","By inspection of the above table, demanding that all occupation numbers be non-negative integers imposes $x\\in \\lbrace 0,1, \\dots , 4\\rbrace $ .","It would be interesting to know if all or any of these partition functions correspond to underlying physical CFTs.","Figure: The linear programming analysis finds a range of possible partition functions if we allow the physical spins to take quarter integer values.","When we fix one of the degeneracies dd by hand, in this case that of the weight (μ,μ ¯,h (0) ,h ¯ (0) )=(1 2,0,1 4,1 4)(\\mu , \\bar{\\mu }, h^{(0)}, \\bar{h}^{(0)})= (\\frac{1}{2}, 0 , \\frac{1}{4}, \\frac{1}{4}), all other degeneracies become uniquely determined, so that we find a one-parameter family of solutions.","The degeneracy on the x-axis here is 4+x4+x in the notation used in the text.The different values of $x$ here correspond to partition functions that have the same spectrum of dimensions $\\Delta = h + \\bar{h}$ , but which can be distinguished by their representation content, i.e.","through the “flavored” partition function.They are similar in this respect to multiple different CFTs at $c=24$ that have the same spectrum but different underlying symmetries [32]."],["Constraints on Representation Content","In this final subsection, we will consider the question of what representations are forced to be present in a theory.","The gravitational AdS dual of any such constraints would imply that even if certain representations were not present among the perturbative degrees of freedom in some theory, they would have to be present non-perturbatively.","The strongest condition one might try to prove is that all theories have all representations present.","This would however be too ambitious since there are simple counter-examples, but one might still try to prove restrictive constraints on which representations can be absent.","We will only be able to take a very modest step in this direction and prove some simple results for $SU(2)$ .","For instance, without referring to numerical methods, we will prove that an $SU(2), k=1$ partition function either has all representations, or else its flavored partition function factorizes into a Sugawara theory partition function times a non-flavored partition function, assuming left-right symmetry.","We begin by proving this $k=1$ result.","The flavored partition function splits into four representations $M(\\tau , \\bar{\\tau }) = \\left(\\begin{array}{cc}M_{(0),(0)}(\\tau , \\bar{\\tau })& M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })\\\\M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })& M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\tau , \\bar{\\tau })\\end{array}\\right) .$ Modular invariance requires $M(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) = SM(\\tau , \\bar{\\tau })S~,$ with $S$ matrix $S = \\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cc}1 & 1 \\\\1 & -1\\end{array}\\right)~.$ Because there are only two (assuming $(\\frac{1}{2},0)$ and $(0,\\frac{1}{2})$ are symmetric) different nontrivial representations, and the modular transformation manifestly forces at least one to be present, we can delete only one of them.","What if we set $M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })=M_{(0),(\\frac{1}{2})}(\\tau ,\\bar{\\tau })=0$ ?","In this case, the $(1,2)$ entry of matrix equation (REF ) is $\\frac{1}{2}\\left( M_{(0),(0)}(\\tau , \\bar{\\tau })-M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\tau , \\bar{\\tau }) \\right) = 0 .$ The two diagonal representations have to be the same and therefore the “non-Sugawara” $\\tau $ -dependence of the flavored partition function is just an overall flavor-independent prefactor $M_{(0),(0)}(\\tau , \\bar{\\tau })$ that factors out.","Similarly, if we set $M_{(\\frac{1}{2}, \\frac{1}{2})}(\\tau ,\\bar{\\tau })=0$ , then the $(2,2)$ entry of (REF ) is $M_{(\\frac{1}{2}),(0)}(\\tau ,\\bar{\\tau }) = \\frac{1}{2} M_{(0),(0)}(\\tau , \\bar{\\tau })$ , so again the non-Sugawara $\\tau $ -dependence factors out completely.","In this case, the residual “Sugawara” matrix is just a symmetric holomorphic plus anti-holomorphic matrix, i.e.","$\\left( \\begin{array}{cc} 2 & 1 \\\\ 1 & 0 \\end{array} \\right) = \\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & 0 \\end{array} \\right) +\\left( \\begin{array}{cc} 1 & 0 \\\\ 1 & 0 \\end{array} \\right)$ .","Beyond $SU(2)$ $k=1$ , similar arguments can be used to somewhat narrow down the possible combinations of representations in any partition function with non-abelian currents.","Speficially, we can prove for SU(2) any $k$ , the partition function factorizes into a Sugawara partition function times a flavor-independent partition function if only diagonal representations are allowed.","We again begin with the transformation rule $M(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) = SM(\\tau , \\bar{\\tau })S~,$ where now the transformation matrix is $S_{(l) (l^\\prime )} = \\sqrt{\\frac{2}{k+2}} \\sin \\left(\\frac{\\pi }{k+2} (l+1)(l^\\prime +1)\\right) .$ Since we allow only diagonal representations, we can write $M_{(l) (r)}(\\tau , \\bar{\\tau }) &= \\delta _{(l) (r)} f_l , \\\\M_{(l) (r)}(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) &= \\delta _{(l) (r)} g_l~,$ for some arbitrary functions $f_l$ and $g_l$ .","The matrix equation can be written as $g_\\alpha S_{\\alpha \\beta }=S_{\\alpha \\beta } f_\\beta .$ For $\\beta = 0$ , $g_\\alpha \\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right) =f_0 \\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right)$ so $g_\\alpha = f_0$ for all $\\alpha $ unless $\\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right) = 0$ .","But the unitary bound $\\alpha 2$ .","The very interesting recent work [33] on a sort of modular invariance for lens spaces in higher dimension is tantalizing from this point of view.","Again, one would face the issue that correlators of currents in $d>2$ are not universal, but one could nevertheless try to obtain a constraint on the partition function in terms of the data in the $\\langle J(x_1) \\dots J(x_n)\\rangle $ correlators.","Somewhat more abstractly, one of the appealing features of understanding the flavored partition function better is that, by turning on background fields, we are exploring constraints beyond the class of those that can be seen by inserting local operators.","There are many such constraints on CFTs that are invisible in the standard bootstrap; the partition function itself can be though of as one such generalization since mapping to the torus (equivalently, inserting twist operators $\\sigma _2$ ) involves imposing new boundary conditions, and adding background fields is another kind of generalization.","It would be very interesting to understand what additional constraints could be obtained by imposing crossing symmetry of correlators in the presence of background fields.","Understanding the transformation law (REF ) as a statement about crossing symmetry for the four-point function $\\langle \\sigma _2 \\sigma _2 \\sigma _2 \\sigma _2\\rangle $ would be a useful warm-up case and could potentially give insight into how to think about more general correlators."],["Acknowledgments","We thank Scott Collier, Shamit Kachru, Jared Kaplan, Emanuel Katz, and Per Kraus for useful conversations.","ED was supported in part by the National Science Foundation under grant NSF-PHY-1316699 and by the Stanford Institute for Theoretical Physics and in part by the Simons Collaboration Grant 488655 on the Non-Perturbative Bootstrap.","ALF and YX were supported in part by the US Department of Energy Office of Science under Award Number DE-SC-0010025, and ALF was supported in part by the Simons Collaboration Grant on the Non-Perturbative Bootstrap."],["Path Integral Modular Transformation","In this appendix, we review how diffeomorphism invariance and rigid rescalings imply the relation $Z_{\\rm PI}\\left( \\frac{a \\tau +b}{c\\tau +d}, \\frac{c z}{c \\tau +d} \\right) = Z_{\\rm PI}(\\tau ,z).$ We begin with invariance of the path integral measure: $d \\Psi e^{ - S_{\\tau _a, \\tau _b}[\\Psi ]} = d\\Psi ^{\\prime } e^{ - S_{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} [\\Psi ^{\\prime }]}.$ Here, $\\Psi $ are all the fields of the CFT, and to keep track of the torus before and after conformal transformations we have introduced $\\tau _a, \\tau _b$ for two of its corners (i.e.","the four corners are at $0,\\tau _a, \\tau _b$ , and $\\tau _a+\\tau _b$ ).","Under rescalings, the operators ${\\cal O}$ and parameters $\\tau _a, \\tau _b$ transform as ${\\cal O}(w,\\bar{w}) \\rightarrow {\\cal O}^{\\prime }(w,\\bar{w}) = \\lambda ^{-h} \\bar{\\lambda }^{-\\bar{h}}{\\cal O}(\\lambda ^{-1} w, \\bar{\\lambda }^{-1} \\bar{w}), \\qquad (\\tau _a, \\tau _b) \\rightarrow (\\tau _a^{\\prime },\\tau _b^{\\prime })= (\\lambda \\tau _a, \\lambda \\tau _b).$ In particular, for a conserved current $J^\\mu $ , we have $\\int _{\\tau _a, \\tau _b} dw d\\bar{w} J^w (w) = \\int _{\\tau _a, \\tau _b} dw d\\bar{w}\\Big ( \\bar{\\lambda } J^{\\prime w}(\\lambda w)\\Big ) = \\lambda ^{-1} \\int _{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} dw d\\bar{w} J^{\\prime w}(w) .$ and consequently $d \\Psi e^{ - S_{\\tau _a, \\tau _b} [\\Psi ] - \\frac{i}{2\\pi } \\int _{\\tau _a, \\tau _b} dw d\\bar{w} A_w J^w} = d \\Psi ^{\\prime } e^{- S_{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b}[\\Psi ^{\\prime }] - \\frac{i}{2\\pi } \\int _{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} dw d\\bar{w} \\lambda ^{-1} A_w J^w }.$ Integrating both sides obtains the relation $(A_w, \\tau _a, \\tau _b) \\cong (\\lambda ^{-1} A_w, \\lambda \\tau _a, \\lambda \\tau _b)$ .","To obtain the transformation under $U: \\tau \\rightarrow \\frac{\\tau }{\\tau +1}$ , we take $(\\tau _a, \\tau _b) = (\\tau , \\tau +1) \\cong (\\tau ,1),$ where the congruence $\\cong $ follows from a large diffeomorphism cutting the torus along the line from 1 to $\\tau +1$ and sewing it back to the line from $\\tau +1$ to $\\tau +2$ .","By inspection of the chemical potential term $\\frac{1}{2\\pi i} \\int _{\\tau ,1} dw d\\bar{w} A_w J^w = 2 \\pi i {\\rm Im}(\\tau ) A_w \\bar{J}_0$ , we read off that $A_w = - i \\frac{\\bar{z}}{2 {\\rm Im}(\\tau )}.$ Finally, we take $\\lambda = (\\tau +1)^{-1}$ , so $(\\tau _a^{\\prime },\\tau _b^{\\prime })=(\\frac{\\tau }{\\tau +1}, 1)$ and $A_w^{\\prime } = (\\tau +1) A_w$ .","Therefore, $\\bar{z}^{\\prime } = 2 i {\\rm Im}(\\tau ^{\\prime }) A_w^{\\prime } = \\frac{\\bar{z}}{\\bar{\\tau }+1}.$ The transformation under $T: \\tau \\rightarrow \\tau +1$ is trivial, since $\\tau $ and $\\tau +1$ are related by a large diffeomorphism without any need for a rescaling, so $\\lambda =1$ , and neither $A_w$ nor $z$ transform.","All other modular transformations are generated from $T$ and $U$ ."],["A “Systematic” Treatment to Multivariate Problems","The bootstrap of flavored partition function introduces another continuous quantum numbers $Q$ in addition to the scaling dimension of $\\Delta $ .","Unlike the unflavored bootstrap where the problem is rigorously converted to a semidefinite programming problem, bootstrap problems with more than one variables do not have a simple and rigorous conversion to semidefinite programming problems.","One can choose to discretize the second variable $Q$ and hope that the bound converges at very small $\\delta Q$ .","However, the bound obtained in this way is not rigorous.","The linear functional can be negative in between discrete $Q$ 's or at large enough $Q$ ."],["Multivariate Positive Definite Functionals","Whether any real positive semidefinite polynomials (PSD) can be written as sum of squares of real polynomials (SOS) is known as the Hilbert's 17th problem.","Hilbert himself proves the special case for univariate polynomials is true.","But for multivariate polynomials it is later proven that PSD is a sum of squares of real rational functions.","We do not like rational functions because we have much less numerical control over them than polynomials.","Although we cannot find a clean SOS representation of multivariate PSD, if we only consider the subset of strictly positive polynomials we can still represent them by SOS in the following cases: Workaround 1: multiply by a common denominator $p(x_1,x_2)$ is positive definite polynomial (PD, also denote as $p(x_1,x_2) > 0$ ) then $p_g(x_1,x_2) = (1+x_1^2+x_2^2)^g p(x_1,x_2)$ is a sum of square of polynomial (SOS) for some $g$ .","[34] Workaround 2: region is bounded For a compact region $\\mathcal {S}$ defined by $f_i(\\vec{x})\\ge 0$ over set of function $f_i$ , any polynomial strictly positive in $\\mathcal {S}$ can be written as the following form $p = \\sum _I s_I(\\vec{x}) f_{i_1} f_{i_2}\\ldots $ where $s_I(\\vec{x})$ are sum of squares.","$I$ denotes some combinations of $f_i$ 's.","[35] The hope is that PD can approximate PSD well enough so that in practise we can still resort to SOS.","Numerically, solvers like SDPB never give nonnegative polynomials with exact zeros, so in practise we never actually encounter any counterexamples.","Another reason to be hopeful is from the proof that PSD can be approximated as closely as desired by SOS [36].","There is a possible loophole – the positive region of the polynomial has to be bounded.","In unflavored case the region that is frequently used is $\\Delta >\\Delta _{\\star }$ , which is a rare special case of unbounded region.","In practise, there is risk of not covering the full space of PD.","Although not rigorously, one can hope that by multiplying the $(1+x_1^2+x_2^2)^g$ factors of higher and higher $g$ we lose less and less."],["Multivariate Problems and SDPB","In this subsection we discuss how to rewrite the semidefinite polynomial programming with 2 variables into a form suitable for SDPB [37] solver.","SDPB solves univariate “Polynomial Matrix Program” (PMP) question stated as follows: $&\\text{maximize } y_0 + \\sum _n b_n y_n \\nonumber \\\\&\\text{such that } M_j^0(x) + \\sum _n y_n M_j^n(x) \\ge 0 \\nonumber \\\\&\\text{for all } x\\ge 0 \\text{ and } 1\\le j \\le J.$ where $M$ matrices are symmetric matrices of polynomials of $x$ .","In SDPB, the PMP question is internally mapped to an SDP question since $ M_j^0(x) + \\sum _n y_n M_j^n(x) \\ge 0$ if and only if $M_j^0(x) + \\sum _n y_n M_j^n(x) =~ {\\rm tr}\\left[ Y_{A} Q_{A}(x) \\right] +x\\, {\\rm tr}\\left[ Y_{B} Q_{B}(x)\\right]$ for some $Y_A, Y_B \\ge 0$ .","This equation is also the (2.8) of 1502.02033.","We are instead trying to solve the problem for two variable cases.","Here for modular bootstrap we are in the special case where the symmetric matrices $M_j$ are one by one, in other words, single polynomials $p_j$ .","For simplicity here we only deal with this one dimensional case.","Generalization to more dimensions and more variables is very easy.","The question is stated as follows: $&\\text{maximize } y_0 + \\sum _n b_n y_n \\nonumber \\\\&\\text{such that } p_j^0(x_1,x_2) + \\sum _n y_n p_j^n(x_1,x_2) \\ge 0 \\nonumber \\\\&\\text{for all } x_1\\ge 0 \\text{ and all $x_2$ and } 1\\le j \\le J.$ Since SDPB only allows one variable to be bounded we cannot add more constraints on the variables.","The $x_1>0$ is needed in SDPB because we usually choose the input $\\Delta \\ge \\Delta _*$ .","The second variable can be the U(1) charge $Q$ , which is not constrained to be positive number.","If one does want to bound the second variable one can make change of variable.","We use the symbol $F_j$ to represent the linear functional $F_j(x_1,x_2) \\equiv p_j^0(x_1,x_2) + \\sum _n p_j^n(x_1,x_2) y_n$ Similar to the univariate case, we assume that $F_j\\ge 0$ is equivalent to finding $Y_{A,j},~Y_{B,j} \\ge 0$ , so that $F_j(x_1,x_2) =~ {\\rm tr}\\left[ Y_{A,j} Q_{A}(x_1,x_2) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} Q_{B}(x_1,x_2)\\right]$ Here we introduce “bilinear basis” $\\vec{q}(X)$ so that $Q(X) = \\vec{q}\\vec{q}^T$ spans the space of polynomials of $X$ .","An easy example of bilinear basis is $\\vec{q}(x) = \\lbrace 1,x,x^2,\\ldots \\rbrace $ .","The bilinear basis of two or more variables can be factored out as a kronecker product of bilinear bases of each single variables $Q_A (x_1,x_2) &= Q_{A1}(x_1) \\otimes Q_2(x_2) \\nonumber \\\\Q_B (x_1,x_2) &= Q_{B1}(x_1) \\otimes Q_2(x_2)$ We define $d_i$ to be the $x_i$ degree of the polynomial $F$ .","the dimensions of the matrices $Q$ are $&{\\rm dim} Q_{A1} = \\delta _{A1} = [d_1/2]+1 \\nonumber \\\\&{\\rm dim} Q_{B1} = \\delta _{B1} = [(d_1-1)/2]+1 \\nonumber \\\\&{\\rm dim} Q_{2} = \\delta _{2} = [d_2/2]+1$ After factoring out $Q_A$ and $Q_B$ the function $F_j$ is written as $F_j(x_1,x_2) =~ {\\rm tr}\\left[ Y_{A,j}\\big ( Q_{A1}(x_1) \\otimes Q_2(x_2)\\big ) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} \\big ( Q_{B1}(x_1) \\otimes Q_2(x_2)\\big ) \\right]$ Since a polynomial is fixed if we know its value at $(d+1)$ different points, we can simply evaluate the above equation at $(d_2+1)$ values of $x_2$ in order to reduce the equation to have only one variable $x_1$We have not investigated what choices of the $x_{2,k}$ s are optimal.","In practice, we have taken them to be $x_{2,k} = 2^{k-1}$ .","$&F_{j,k}(x_1) = F_j(x_1,x_{2,k}) =~ {\\rm tr}\\left[ Y_{A,j}\\big ( Q_{A1}(x_1) \\otimes Q_2(x_{2,k})\\big ) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} \\big ( Q_{B1}(x_1) \\otimes Q_2(x_{2,k})\\big ) \\right]$ The above $(d_2+1)$ equations are equivalent to (REF ).","In the following we omit the $j$ index because the same equation works for all $j$ .","Now the form is already in single variable and is very close to the form of (REF ).","The only difference is the numerical matrices $Q_2(x_{2,k})$ .","Here we can play a trick by shuffling the $(d_2+1)$ equations with linear combination $&\\sum _l \\alpha _{kl} F_l = {\\rm tr}\\left[ Y_{A,j} \\big (Q_{A1}(x_1) \\otimes \\sum _l \\alpha _{kl} Q_2(x_{2,l})\\big ) \\right] + (\\text{B part})$ for some dimension $(d_2+1)$ square matrix $\\alpha _{kl}$ .","In fact, the space of symmetric $Q_2$ matrices is only $(d_2+1= 2\\delta _2 -1)$ dimensional space since it spans the space $(d_2+1)$ dimensional polynomials.","That means we can always find some $\\alpha _{kl}$ which picks up the the orthornormal basis of the polynomial space.","Further we can perform an arbitrary $GL_{\\delta _2}$ transformation on $Q_2$ $Y &\\mapsto G YG^{-1} \\nonumber \\\\\\sum _l \\alpha _{kl} Q_2(x_{2,l}) &\\mapsto G^{-1} \\sum _l \\alpha _{kl}Q_2(x_{2,l}) G$ so that the orthornormal basis maps to the symmetric matrix basis $G^{-1} \\sum _l \\alpha _{kl} Q_2(x_{2,l}) G = E^{r(k) s(k)}$ where $E^{rs} = \\delta _i^r \\delta _j^s + \\delta _j^r \\delta _i^s$ .","Then $\\sum _l \\alpha _{kl} F_l = {\\rm tr}\\left[ Y_A Q_{1A}(x_1) \\otimes E^{r(k)s(k)}\\right] + x_1{\\rm tr}\\left[ Y_B Q_{1B}(x_1) \\otimes E^{r(k)s(k)}\\right]$ Compared to (REF ), we can turn double variable programming of polynomial into single variable programming of symmetric polynomial matrices by substitution $M_j^0 &= \\sum _k \\sum _l E^{r(k) s(k)} \\alpha _{kl}P_j^0(x_1,x_{2,l})\\nonumber \\\\M_j^n &= \\sum _k \\sum _l E^{r(k) s(k)} \\alpha _{kl}P_j^n(x_1,x_{2,l})$ Since $Q_2$ span a $(d_2+1 = 2\\delta _2 -1)$ dimension space it means only the diagonal and next-to-diagonal elements will be nonzero.","If further we only have even powers of $x_2$ , the matrices will be diagonal.","The procedure defined from (REF ) to (REF ) is not the most efficient algorithm to obtain a single-variable matrix basis to input in (REF ).","In practice, the algorithm we actually follow is computationally more straightforward, and is as follows.","We can rewrite (REF ) as $M_j(x_1)_{tu} =Y_{A,j}^{rs,tu}\\, Q_{A1}(x_1)_{rs} + (\\text{B parts}) ~,$ where $r,s,t,u$ are matrix indices.","Given $M_j(x_1)$ , SDPB can optimize this single-variable problem.","Rather than obtaining $M_j(x_1)$ by the procedure outlined above, we can instead start directly from equation (REF ), which says $F_{j,k}(x_1) = Y_{A,j}^{rs,tu}\\, Q_{A1}(x_1)_{rs}\\,Q_2(x_{2,k})_{tu} + (\\text{B parts}) ~.$ Combining the two equations $F_{j,k}(x_1) = M_j(x_1)^{tu}\\, Q_2(x_{2,k})_{tu} ~.$ The point is that both $F_{j,k}(x_1)$ and $Q_2(x_{2,k})_{tu} $ are known, so $M$ can be obtained by solving the above equation, after which it can be fed into SDBP.","Concretely, first flatten the $(t,u)$ indices $ \\alpha := (t_\\alpha ,u_\\alpha )$ to put $M_{tu}$ into $(\\delta _2+1)(\\delta _2+2)/2$ dimensional array ${\\bf M}_\\alpha $ and $Q_{k,tu}$ into $(d_2+1)\\times (\\delta _2+1)(\\delta _2+2)/2$ array $Q_{k,\\alpha }$ .","Then solve the linear equations for ${\\bf M}_\\alpha $ .","To be concrete, in this paper we explicitly choose the flatten map ${\\bf M} := \\big (\\text{diag of $M$},~\\text{next-to-diag},~\\cdots \\big )~.$ A specific solution can be found by taking the SVD decomposition of $Q$ , $Q = U \\, W \\, V^T$ where diagonal matrix $W = \\left( \\ \\widetilde{W} \\quad O \\ \\right)$ has the same rank $(d_2+1)$ as $Q$ .","Then take $B^{T} = V \\, W^\\prime \\, U^T$ where $W^\\prime = \\left(\\begin{array}{c}\\widetilde{W}^{-1} \\\\ O\\end{array} \\right)$ is the pseudo inverse of $W$ .","Contract $B^T$ both sides of (REF ), $B^T {\\bf F} &= V \\, W^\\prime \\, U^T Q {\\bf M}= V \\, W^\\prime \\, U^T U \\, W \\, V^T {\\bf M}= V \\, \\left(\\begin{array}{cc}I_{d_2+1}& O \\\\ O & O\\end{array}\\right) \\, V^T {\\bf M}$ where it's useful to block-decompose the unitary matrix $V$ as $V = \\left(\\begin{array}{cc}V_{11}&V_{12}\\\\ V_{21}&V_{22}\\end{array}\\right)~.$ Continue to simplify the right hand side, $V^TB^T{\\bf F} &= \\left(\\begin{array}{cc}V_{11}&V_{12}\\\\ O & O\\end{array}\\right) {\\bf M} \\nonumber \\\\\\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf F} &= \\left( \\ I_{d_2+1} \\quad (V_{11}^T)^{-1}V_{21}^T \\ \\right){\\bf M} ~.$ From (REF ) we know that there are only $(d_2+1)$ independent elements are unique.","Any dependent element can be removed by a $GL_{\\delta _2}$ transformation defined by REF .","We choose ${\\bf M}$ such that only the first $(d_2+1)$ elements are zero, and the solution can be written as ${\\bf M} = \\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf F}~.$ Since $F_j(x_1,x_2)$ is the linear functional of input polynomials defined by (REF ).","Our single variable input matrices should be substituted in the same way, leading to ${\\bf M}_{j}^n &= \\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf P}_{j}^n \\nonumber \\\\{\\bf M} &\\equiv {\\bf M}_{j}^0 \\, y_0 + \\sum _n {\\bf M}_{j}^n \\, y_n\\nonumber \\\\\\left( {\\bf P}_{j}^n \\right)_k &\\equiv {\\rm Flatten}\\left( p_j^n(x_1,x_{2,k}) \\right)$ Finally, we inverse (REF ) to put flattened array ${\\bf M}_j^n$ 's back into matrix $M_j^n$ 's."],["$k=2, SU(2)$ Analysis","Here we present some preliminary results on our methods applied to the group $SU(2)$ at level $k=2$ .","Our results are qualitatively similar to the $k=1$ case, though with worse numeric accuracy due to the slower convergence.","In Fig.","REF , we show the bound on the gap $\\Delta _*$ to the lightest neutral state in the theory, which is minimized to be $\\Delta _* \\approx 1.344$ at $c \\approx 2.715$ .","Unfortunately, at the point where the bound is minimized, the occupation numbers from our analysis for some of the lowest few states are not particularly close to integers.","It is not clear whether this indicates that such a point is not associated with an underlying CFT or if we simply have not converged to sufficient precision.","The occupation numbers for the lightest neutral state and charged state are shown as a function of $c$ in Fig.","REF .","The lightest neutral state is close to $d=74$ , however the lightest charged state, which is even lighter is relatively far from the nearest integer, $d \\approx 7$ .","Another possibility is that one ought to maximize the gap in not only the neutral sector but also in one or more charged sectors; it would be interesting to pursue this or other conditions further.","Figure: Bound on the gap Δ * \\Delta _* to the lightest neutral state for SU(2)SU(2) at level k=2k=2.","The bound is minimized at c≈2.715c \\approx 2.715.Figure: Occupation numbers for the lightest neutral (left) and charged (right) states from the extremal functional analysis with SU(2)SU(2) at k=2k=2 as a function of cc.","The optimal bound is at c≈2.715c\\approx 2.715, indicated by a vertical line; horizontal lines are shown at integers.s"]],"string":"[\n [\n \"Constraints on Flavored 2d CFT Partition Functions\"\n ],\n [\n \"Abstract We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e.\",\n \"when the partition functions are \\\"flavored\\\".\",\n \"We begin with a new proof of the transformation law for the modular transformation of such partition functions.\",\n \"Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory.\",\n \"We improve previous upper bounds on the state with the greatest \\\"mass-to-charge\\\" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory.\",\n \"We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers.\",\n \"Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise.\"\n ],\n [\n \"Introduction\",\n \"Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs).\",\n \"It is also a special case of crossing symmetry of CFT correlation functions [1], so aside from its intrinsic interest it is useful as a simpler setting in which to explore many conformal bootstrap ideas and techniques [2], [3], [4].\",\n \"A particularly appealing generalization of the conformal bootstrap equations is to consider correlation functions in the presence of nonlocal operators, since this enlarges the set of CFT data that can be studied.\",\n \"In general, including nonlocal operators is a difficult problem, since their behavior under conformal transformations may be quite complicated.\",\n \"However, one case where the problem remains tractable is when we consider modular invariance in the presence of a chemical potential in 2d CFTs.\",\n \"A chemical potential corresponds to inserting the nonlocal operator $y^{J_0} \\\\equiv e^{2 \\\\pi i z J_0}$ into the partition function, $Z( \\\\tau , z) \\\\equiv {\\\\rm tr}\\\\left( q^{L_0- \\\\frac{c}{24}} \\\\bar{q}^{\\\\bar{L}_0-\\\\frac{c}{24}} y^{J_0} \\\\right),$ where $J_0$ is the zero mode of a conserved current and $L_0, \\\\bar{L}_0$ are Virasoro generators.\",\n \"The resulting partition function is no longer modular invariant, but nevertheless has a well-defined and theory-independent transformation law [5]: $Z(\\\\frac{a \\\\tau +b}{c \\\\tau +d}, \\\\frac{c z}{c \\\\tau + d}) = e^{ \\\\pi i k \\\\left( \\\\frac{ c z^2 }{c \\\\tau +d} - \\\\frac{ c \\\\bar{z}^2}{c \\\\bar{\\\\tau }+d}\\\\right)} Z(\\\\tau , z) .$ This transformation law was used to constrain the spectrum of charges in general 2d CFTs in [6].\",\n \"Proofs of (REF ) so far [5], [7], [8], [6] either are fairly complicated and technical or else apply to special cases such as free boson constructions, and so it is not clear what if any general lessons might be learned from them.We emphasize that we do not assume supersymmetry; additional techniques are available to proof the transformation law for the elliptic genus in the case of supersymmetric theories, see e.g.\",\n \"[9].\",\n \"However, the very simple form of (REF ) suggests it should have an equally simple derivation.\",\n \"Moreover, inserting the nonlocal operator $y^{J_0}$ is equivalent to turning on a background gauge field $A^\\\\mu $ coupled to the conserved current $J^\\\\mu $ , which suggests that one might be able to prove (REF ) by studying the CFT's effective action for $A_\\\\mu $ .\",\n \"We will begin this paper in section 2 by providing such a proof, and its generalization to a non-abelian symmetry current $J^{a \\\\mu }$ .\",\n \"Starting in section 3, we perform several analyses of the constraints that follow from (REF ) and its non-abelian generalization using linear programming and semi-definite programming methods.\",\n \"Our main results are as follows.\"\n ],\n [\n \"Abelian Bounds\",\n \"We begin by reproducing, and improving the results of [6], bounding properties of theories with an abelian current.\",\n \"We place an upper bound on the dimension of the lightest charged state, $ \\\\begin{split} \\\\Delta _{*} = \\\\frac{c}{\\\\alpha } +\\\\mathcal {O}(1)\\\\,, \\\\ \\\\ \\\\ \\\\alpha > 8\\\\,.\",\n \"\\\\end{split} $ This bound is qualitatively similar to than the bounds in [4], [10] of non-charged states.\",\n \"We also improve the bound on the smallest “mass-to-charge” ratio in the theory.\",\n \"These bounds are qualitatively related to the Weak Gravity Conjecture (WGC), though gauge fields in the gravity duals are Chern-Simons fields rather than Maxwell fields.\",\n \"Provocatively, we find numerical evidence for a bound on the mass-to-charge ratio that scales at large $c$ as $\\\\sqrt{c}$ , consistent with the bulk gravity expectation.\",\n \"This is stronger than the bounds in [6], which scale as $c$ .\",\n \"The improvement again comes from increasing the number of derivatives of the characters used in the analysis.\",\n \"Then, we discuss the bound on the charge gap $Q_*$ .\",\n \"Without any further assumptions, the numerical bound of charge gap is always $Q_*=1$ for all $c$ .\",\n \"We study two examples of $c=2$ and 8 by turning on both a gap in dimension $\\\\Delta _*$ and in charge $Q_*$ .\",\n \"There are kinks in the $\\\\Delta _*$ and $Q_*$ plots which can potentially be associated with full CFTs.\",\n \"Lastly, we consider in detail spectra that extremize various gaps.\",\n \"We use the extremal functional method, as well as extra information contained in the charged spectra to study candidate theories at $c=1$ and 8.\",\n \"At $c=8$ we find that the level 1 $E_{8}$ Sugawara theory saturates both the gap in dimension (as was found in [10]) and in charge.\",\n \"Using this, we are able to reproduce the full low lying spectrum, including charge assignments.\",\n \"When the symmetry current $J^a$ is non-abelian, it is more appropriate to consider bounds on the dimensions of different representations in the theory.\",\n \"We will mainly focus for specificity on the case where the gauge group $G$ is $SU(2)$ and the level $k$ is 1, though our methods easily generalize to any algebra and level; the main advantage of $k=1, G=SU(2)$ is that convergence is fastest here, so our numerical results are most precise.\",\n \"We first obtain bounds on the gap to all non-vacuum states in non-abelian theories.\",\n \"As the extended symmetry imposes additional relations on the spectrum, one may have hoped for stronger bounds.\",\n \"The results, however, are similar to those found in the abelian, or even non-flavored case.\",\n \"In particular, at large $c$ we find a bound of the form, $ \\\\begin{split} \\\\Delta _{*} = \\\\frac{c}{\\\\alpha } + \\\\mathcal {O}(1)\\\\,, \\\\ \\\\ \\\\ \\\\alpha > 8\\\\,.\",\n \"\\\\end{split} $ The real power of the modular constraints on the flavored partition function come from the ability to impose constraints independently on different representations.\",\n \"Taking advantage of this, we search for a bound on the gap to non-vacuum states transforming in the trivial representation, with no constraints imposed for other representations.\",\n \"At small $c$ , there are interesting “kinks” at values of $c$ where the bounds on the gap in the neutral sector is minimized.\",\n \"We focus on the case $c=3$ , and use the extremal functional methods to find the low lying degeneracies at this kink.\",\n \"Reassuringly, we find integer degeneracies.\",\n \"This numerical spectral information allows us to guess an exact partition function saturating the bound in the neutral sector.\",\n \"In fact we find multiple partition functions are allowed if we are somewhat liberal in what spins are allowed for states in the theory.The partition functions found are not strictly modular invariant, but invariant under a subgroup generated by $S$ and $T^{n}$ for $n=2$ or $n=4$ .\",\n \"If all states must have integer or half-integer spins, we find a unique partition function, $ \\\\begin{split} Z(\\\\tau , \\\\bar{\\\\tau }, z, \\\\bar{z})&= \\\\frac{1}{4} \\\\sum _{a,b,a^{\\\\prime },b^{\\\\prime }=0}^1 (-1)^{a b^{\\\\prime } + a^{\\\\prime } b} \\\\left|\\\\theta \\\\left[^a_b\\\\right](\\\\tau ,\\\\frac{z}{2})\\\\right|^4 | \\\\theta [^{a^{\\\\prime }}_{b^{\\\\prime }}](\\\\tau ,0)|^2\\\\,.\",\n \"\\\\end{split} $ Allowing quarter-integer spins leads to multiple allowed partition functions that maximize the gap.\",\n \"Perhaps the most interesting aspect of this analysis however is not the specific partition function for this case, but rather that fact that searching for constraints in a representation dependent manner yields structure hidden to a flavor-blind analysis.\",\n \"This means that the extremal functional analysis allows one to “discover” a larger class of partition functions when flavored information is included than when it is not.\",\n \"Moreover, uncovering flavored information can potentially split the degeneracy between theories with the same spectrum and therefore the same partition function, allowing us to address the age-old question of whether one can “taste the shape of a drum.” Finally, we continue to refine our representation dependent analysis.\",\n \"For the case of $SU(2)_{1}$ we prove analytically that the theory either contains all representation, or the partition function splits into a product of the diagonal Sugawara partition function and a neutral, modular invariant partition function.\",\n \"After this work was completed, the paper [11] appeared on arXiv also considering modular bootstrap constraints on theories with conserved currents, though the analysis there did not use the flavored partition function.\"\n ],\n [\n \"Partition Function Transformation and Background Gauge Fields\",\n \"In this section, we will present an argument for the transformation law (REF ) based on the effective action obtained upon integrating out the CFT in the presence of a background gauge field $A_\\\\mu $ .\",\n \"Previous treatments have pointed out that in the present context there are two different notions of the partition function that are natural.\",\n \"One of these is the canonical partition function $Z(\\\\tau ,z)$ defined by (REF ).\",\n \"Following [5], we will refer to an alternate definition as the “path integral” $Z_{\\\\rm PI}(\\\\tau , z)$ : $Z_{\\\\rm PI}(\\\\tau , z) \\\\sim e^{ \\\\pi k B(\\\\tau , z) } Z(\\\\tau , z) , \\\\qquad B(\\\\tau , z) \\\\equiv \\\\frac{z^2+\\\\bar{z}^2}{2{\\\\rm Im}(\\\\tau )}.$ Under $z^{\\\\prime } = \\\\frac{z}{c \\\\tau +d}, \\\\tau ^{\\\\prime } = \\\\frac{a \\\\tau +b}{c \\\\tau +d}$ , the factor $B$ is easily seen to transform as $B(\\\\tau ^{\\\\prime },z^{\\\\prime }) &=& B(\\\\tau ,z) - i \\\\left( \\\\frac{ c z^2}{c\\\\tau +d} - \\\\frac{c \\\\bar{z}^2}{c \\\\bar{\\\\tau }+d} \\\\right).$ The important point about the extra factor $B(\\\\tau ,z)$ is that its transformation cancels the transformation of $Z(\\\\tau ,z)$ , leaving $Z_{\\\\rm PI}$ invariant.\",\n \"The basic idea is that $Z_{\\\\rm PI}$ should be the result of performing a path integral over the torus, and so should be modular invariant.\",\n \"In free boson constructions, one can explicitly see how this factor is generated by the Legendre transform from the Lagrangian to the Hamiltonian [5].\",\n \"However, we would like to see how this arises in a general CFT, without making any reference to a specific form of a Lagrangian.\",\n \"We will begin with the case of an abelian current, and then consider the generalization to a non-abelian symmetry.\"\n ],\n [\n \"Modular transformation and the ground state energy\",\n \"First, let us discuss in more detail how to define the “path integral” function $Z(\\\\tau ,z)$ , what ambiguities are allowed in this definition, and why they do not affect the transformation law (REF ).\",\n \"In order to be invariant under modular transformations, we will need to define the path integral to be invariant under diffeomorphisms and rigid rescalings $w \\\\rightarrow \\\\lambda ^{-1} w$ : $d \\\\Psi e^{ - S_{\\\\tau }[\\\\Psi ]} = d\\\\Psi ^{\\\\prime } e^{ - S_{\\\\tau ^{\\\\prime }} [\\\\Psi ^{\\\\prime }]}.$ Here, $\\\\Psi $ are all the fields of the CFT.\",\n \"As we review in appendix , these two symmetries are sufficient to imply that the path integral defined as an integral over this measure, $Z_{\\\\rm PI}(\\\\tau , z) &\\\\equiv & \\\\int d \\\\Psi e^{- S_{\\\\tau }[\\\\Psi ] -\\\\frac{i }{2\\\\pi } \\\\int _\\\\tau A_{\\\\bar{w}} J^{\\\\bar{w}}}, \\\\qquad A_{\\\\bar{w}} = -i \\\\frac{z}{2 {\\\\rm Im}(\\\\tau )},$ is invariant under modular transformations: $Z_{\\\\rm PI}\\\\left( \\\\frac{a \\\\tau +b}{c \\\\tau +d}, \\\\frac{c z}{c\\\\tau +d}\\\\right) &=& Z_{\\\\rm PI}(\\\\tau , z).$ Different choices of regulators will change $\\\\log Z_{\\\\rm PI}$ by local terms.\",\n \"However, the local terms allowed by diffeomorphism invariance and scale invariance do not affect the transformation law.\",\n \"For instance, one can shift the effective action by a local term proportional to $\\\\int _\\\\tau d^2 x \\\\sqrt{g} A_\\\\mu A^\\\\mu \\\\sim \\\\int _\\\\tau dw d\\\\bar{w} A_w A_{\\\\bar{w}} \\\\sim \\\\frac{z \\\\bar{z}}{4 {\\\\rm Im}(\\\\tau )}.$ This term arises in the difference between a regulator that preserves the vector current $J^\\\\mu $ symmetry and one that preserves the axial current $\\\\epsilon ^{\\\\mu \\\\nu }J_\\\\nu $ .\",\n \"However, it is easily seen to be both Weyl invariant and diffeomorphism invariant, and is invariant under modular transformations.\",\n \"So its coefficient is irrelevant for our purposes, and from now on we will neglect such terms without loss of generality.\",\n \"Now, the next question is how do we relate the “path integral” $Z_{\\\\rm PI}$ to the partition function $Z$ ?\",\n \"The key point is that turning on a background field $A_\\\\mu $ not only turns on a chemical potential, but it can also shift the ground state energy, since at fixed $\\\\beta $ such a shift affects only the overall normalization of the path integral.\",\n \"In the example of the free boson, this energy shift is seen explicitly by doing a Legendre transform, but we can see it in full generality by considering the effective action for $A_\\\\mu $ .\",\n \"To see the shift, it is sufficient to calculate the ground state energy, so we can take the limit of the torus where $\\\\tau = i \\\\frac{\\\\beta }{2 \\\\pi } , \\\\beta \\\\gg 1 $ .\",\n \"In this limit, the torus becomes a cylinder, and the effective action is conformally related to that in flat space, where it is universal and known in closed form.\",\n \"Including the action for a background metric as well, we can write $\\\\log Z = \\\\int d^2 x \\\\sqrt{-g} \\\\left( \\\\frac{c}{48 \\\\pi } R \\\\Box ^{-1} R + \\\\frac{k}{8 \\\\pi } F^{\\\\mu \\\\nu } \\\\Box ^{-1} F_{\\\\mu \\\\nu } \\\\right).$ Because of the inverse Laplacians, the mapping to the cylinder is a bit subtle.\",\n \"For the metric contribution, it is easiest to work with the Wess-Zumino anomaly action directly, $S_{\\\\rm WZ} = \\\\frac{c}{24\\\\pi } \\\\int d^2 x\\\\sqrt{-g} \\\\left( \\\\sigma R + (\\\\partial \\\\sigma )^2 \\\\right)$ , and take $\\\\sigma (w) = w+\\\\bar{w}$ , which reproduces the standard ground state energy shift $-\\\\frac{c}{12}$ from the Schwarzian derivative.\",\n \"By contrast, the gauge field term in (REF ) is invariant under Weyl transformations, and its contribution to the ground state energy just comes from evaluating the non-local term on the cylinder.\",\n \"To avoid ambiguities associated with the inverse Laplacian, it is clearest to use the fact that the effective action is the generating functional for the $J^\\\\mu $ correlators, so we know that we can equivalently write the gauge field part of $\\\\log Z$ as $W_A[A^\\\\mu ] = \\\\int \\\\frac{d^2 x d^2 x^{\\\\prime }}{(2\\\\pi i )^2} A_\\\\mu (x) A_\\\\nu (x^{\\\\prime }) \\\\langle J^\\\\mu (x) J^\\\\nu (x^{\\\\prime }) \\\\rangle .$ On the plane, $\\\\langle J^{\\\\bar{w}}(w) J^{\\\\bar{w}}(w^{\\\\prime })\\\\rangle = \\\\frac{k}{(w-w^{\\\\prime })^2}$ .\",\n \"Mapping to the cylinder and taking $A_{\\\\bar{w}}$ to be constant, we haveWe performed this integration as follows.\",\n \"First, shift $w \\\\rightarrow w+w^{\\\\prime }$ to eliminate $w^{\\\\prime }$ and immediately do the $d^2 w^{\\\\prime }$ integral, producing just a factor of the volume $2 \\\\pi \\\\beta $ of the torus.\",\n \"Passing to $t,\\\\theta $ coordinates: $W_A[A_\\\\mu ] = - \\\\frac{\\\\beta k A_{\\\\bar{w}}^2}{2\\\\pi } \\\\int _{-\\\\beta /2}^{\\\\beta /2} dt \\\\int _0^{2\\\\pi } d\\\\theta \\\\frac{1}{(e^{\\\\frac{t+i \\\\theta }{2}} -e^{\\\\frac{-t-i \\\\theta }{2}})^2} .$ If we do the $\\\\theta $ integral first, this vanishes, except when $t=0$ where it is divergent; the integral over $\\\\theta $ is proportional to $\\\\delta (t)$ .\",\n \"We avoid this subtlety if we do the $t$ integral first, in which case we obtain $W_A[A_\\\\mu ] =- \\\\frac{\\\\beta k A_{\\\\bar{w}}^2}{2\\\\pi } \\\\int _0^{2 \\\\pi } d \\\\theta \\\\frac{\\\\sinh \\\\frac{\\\\beta }{2}}{\\\\cos \\\\theta - \\\\cosh \\\\frac{\\\\beta }{2}} = \\\\beta k A_{\\\\bar{w}}^2 .$ $W_A[A_\\\\mu ] \\\\rightarrow \\\\frac{k}{(2\\\\pi i)^2} \\\\int d^2 w d^2 w^{\\\\prime } \\\\frac{A_{\\\\bar{w}}^2}{\\\\left( e^{\\\\frac{w-w^{\\\\prime }}{2}} - e^{\\\\frac{w^{\\\\prime }-w}{2}}\\\\right)^2} = \\\\beta k A_{\\\\bar{w}}^2.$ Combining the above with a symmetric combination from $A_w$ , we put everything together to obtain the ground state energy: $E_0 = -\\\\lim _{\\\\beta \\\\rightarrow \\\\infty } \\\\beta ^{-1} \\\\log Z_{\\\\rm PI} = - \\\\frac{c}{12} + \\\\delta E , \\\\qquad \\\\delta E = -k (A_w^2+ A_{\\\\bar{w}}^2).$ Therefore, the path integral differs from the canonical partition function by an extra factor $e^{-\\\\beta \\\\phantom{.}\",\n \"\\\\delta E}$ , which in turn produces the factor $- \\\\pi k B(\\\\tau ,z)$ in (REF ,REF ).\",\n \"So at last we see that this factor is universally the contribution to the partition function from the shift in the ground state energy due to the background gauge field.\",\n \"Summarizing, the canonical partition function $Z(\\\\tau ,z)$ in (REF ) is defined to have a ground state energy $-\\\\frac{c}{12}$ .\",\n \"However, any path integral over the torus using a regulator that preserves diffeomorphisms and rescalings will have a ground state energy equal to $-\\\\frac{c}{12} - k (A_w^2 + A_{\\\\bar{w}}^2)$ , plus possible terms that do not affect the modular transformation of $Z(\\\\tau ,z)$ .\"\n ],\n [\n \"Non-abelian current transformation\",\n \"The generalization to the case of a non-abelian is straightforward, and can be made as follows.\",\n \"Unlike in the abelian case, the effective action is not quadratic.\",\n \"However, we can write it formally as the sum over all connected diagrams: $W_A[A^{a \\\\mu } ] = \\\\sum _{n=1}^\\\\infty \\\\left(\\\\prod _{i=1}^n \\\\int \\\\frac{d^2 w_i}{2\\\\pi i} A^{a_i}_w \\\\right) \\\\langle J^{a_1}(w_1) \\\\dots J^{a_n}(w_n) \\\\rangle _{\\\\rm conn} .$ As before, we want to set $A_w^a, A_{\\\\bar{w}}^a$ to be constant on the cylinder and integrate over $d^2 w_i$ .\",\n \"For the part quadratic in $A$ , the computation proceeds just as in the abelian case, we simply have an extra index for the different components of $J^a_w$ .\",\n \"The background field couples as $\\\\frac{-i}{2\\\\pi } \\\\int _\\\\tau A_\\\\mu ^a J^{\\\\mu a}, \\\\qquad A_w^a = -i \\\\frac{\\\\bar{z}^a}{2 {\\\\rm Im}(\\\\tau )}.$ $A$ 's contribution to the ground state energy is $\\\\delta E \\\\cong -k((A_w^a)^2+(A_{\\\\bar{w}}^a)^2),$ which transforms under modular transformations as $-\\\\beta \\\\delta E \\\\rightarrow -\\\\beta \\\\delta E - i \\\\pi k \\\\left( \\\\frac{c (z^a)^2}{c\\\\tau +d} - \\\\frac{c (\\\\bar{z}^a)^2}{c \\\\bar{\\\\tau }+d} \\\\right).$ That leaves the contribution from the higher-point functions.\",\n \"We can always write these in terms of lower-point function by using the recursive formula $J^a(w)J^b(0) \\\\sim \\\\frac{k \\\\delta ^{ab}}{w^2} + \\\\frac{f^{abc} J^c(0)}{w},$ where $\\\\sim $ means `up to non-singular terms'.\",\n \"The $\\\\frac{k \\\\delta ^{ab}}{w^2}$ piece manifestly generates disconnected diagrams - it produces the two-point function times the $(n-2)$ -point function - so it does not contribute to the effective action for higher-point correlators.\",\n \"But, since we multiply the correlator by $A^a_w$ in the effective action, the $f^{abc}$ term also gives no contribution for constant $A_w^a$ : $A_w^a J^a(w) A_w^b J^b(0) \\\\sim \\\\frac{k A_w^2 }{ w^2} + A_w^a A_w^b f^{abc} \\\\frac{J^c(0)}{w} = \\\\frac{k A_w^2 }{w^2}$ since $A^a A^b f^{abc} =0$ .\",\n \"Therefore only the two-point functions contribute.\"\n ],\n [\n \"Basic setup\",\n \"In all the cases we consider, we will assume the presence of a conserved current $J^a$ in the theory.\",\n \"In general, it is convenient to separate the stress tensor $T$ of the theory into a Sugawara stress tensor piece and a residual piece: $T^{(0)} \\\\equiv T - T^{\\\\rm sug},\\\\qquad T^{\\\\rm sug} = \\\\frac{1/2}{k+\\\\widetilde{h}_G} \\\\sum _{a=1}^{|G|} : J^a J^a : ,$ where $\\\\widetilde{h}_G$ is the dual Coxter number, because the modes of $T^{(0)}$ commute with the modes of $J^a$ .\",\n \"Furthermore among themselves they form a Virasoro algebra with central charge $c^{(0)} = c- c^{\\\\rm sug}, \\\\qquad c^{\\\\rm sug} =\\\\frac{k |G|}{k+\\\\widetilde{h}_G} .$ Similarly, we can separate the Virasoro generators $L_n = L_n^{(0)}+L_n^{\\\\rm sug}$ and the weights $h = h^{(0)} + h^{\\\\rm sug}$ into a part that comes from $T^{(0)}$ and a part that comes from $T^{\\\\rm sug}$ .\",\n \"For most representations, the distinction between $T$ and $T^{(0)}$ will not make much difference, since the partition function just counts states at each level.\",\n \"However, for the special cases with shortening conditions, some descendants becomes null and do not contribute to the partition function, and this is easier to see using the modes of $T^{(0)}$ .\",\n \"The characters of the Kac-moody algebra $\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}(\\\\tau ,\\\\mathbf {z})={\\\\rm Tr}_{V_{\\\\mathbf {\\\\mu },k}}q^{L_0^{\\\\rm sug}-\\\\frac{c^{\\\\rm sug}}{24}}e^{2\\\\pi i \\\\mathbf {z}\\\\cdot \\\\mathbf {H}_0}$ are constructed by acting modes of $J^a$ on some highest weight state which has weight $\\\\mathbf {\\\\mu }$ .See e.g.\",\n \"[12] for a standard introduction.\",\n \"Here, ${\\\\bf H}_0$ is the vector of Cartan generators of the algebra.\",\n \"In the case of an abelian symmetry, ${\\\\bf H_0} = J_0$ and the characters for a generic primary are simply $ q^{ - \\\\frac{c^{\\\\rm sug}-1}{24}}e^{2 \\\\pi i z Q}/\\\\eta (\\\\tau )$ .\",\n \"In the case of a non-abelian symmetry, the characters are more complicated.\",\n \"Some descendants of such a highest weight state may be null so it is non-trivial to write down its form.\",\n \"However, for the purpose of the modular bootstrap, the only property of such characters we use is that the characters transform covariantly $\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}\\\\left(-\\\\frac{1}{\\\\tau },\\\\frac{\\\\mathbf {z}}{\\\\tau }\\\\right)=e^{\\\\frac{i \\\\pi k \\\\mathbf {z}^2}{\\\\tau }}\\\\sum _{\\\\mathbf {\\\\mu ^\\\\prime }}S_{\\\\mathbf {\\\\mu \\\\mu ^\\\\prime }}^k\\\\mathcal {X}_{\\\\mathbf {\\\\mu ^\\\\prime },k}(\\\\tau ,\\\\mathbf {z}) ,$ where the matrix $S$ depends on the symmetry group and level $k$ .\",\n \"These characters do not include the modes of $T^{(0)}$ yet.\",\n \"Since the algebra generated by modes of $J^a$ is completely orthogonal to that generated by modes of $T^{(0)}$ , the character generated by the full extended algebra simply factorizes into a Kac-Moody character and a Virasoro character $\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k,h}(\\\\tau , \\\\mathbf {z}) = \\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}(\\\\tau ,\\\\mathbf {z}) \\\\mathcal {X}_{h^{(0)}}(\\\\tau )~.$ Like the simple Virasoro character, the character is different if the primary saturates the unitarity bound: $\\\\mathcal {X}_{h^{(0)}} (\\\\tau ) = \\\\left\\\\lbrace \\\\begin{array}{cc} \\\\frac{q^{h^{(0)}-\\\\frac{c^{(0)}-1}{24}}}{\\\\eta (\\\\tau )} & h^{(0)} >0 \\\\\\\\\\\\frac{(1-q)q^{-\\\\frac{c^{(0)}-1}{24}}}{\\\\eta (\\\\tau )} & h^{(0)}=0 \\\\end{array} \\\\right.\",\n \".$ The same goes for the anti-holomorphic part.\",\n \"The full partition function is $Z(\\\\tau , \\\\bar{\\\\tau }, \\\\mathbf {z},\\\\mathbf {\\\\bar{z}}) =\\\\sum _{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }},h,\\\\bar{h}}d_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }},h,\\\\bar{h}} \\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}(\\\\tau ,\\\\mathbf {z})\\\\mathcal {X}_{\\\\mathbf {\\\\bar{\\\\mu }},k}(\\\\bar{\\\\tau },\\\\bar{\\\\mathbf {z}}) \\\\mathcal {X}_{h^{(0)}}(\\\\tau ) \\\\mathcal {X}_{\\\\bar{h}^{(0)}}(\\\\bar{\\\\tau }) ~.$ In the above equation the $\\\\bar{\\\\mu }$ means the representation of the anti-holomorphic part.\",\n \"When we are dealing with a non-abelian symmetry, it will be convenient to define a matrix $M_{\\\\mu , \\\\bar{\\\\mu }}$ whose components are the coefficients of the contributions to the partition function from the different representations: $ \\\\begin{split} M(\\\\tau ,\\\\bar{\\\\tau })_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}&= \\\\sum _{h, \\\\bar{h}}d_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }},h,\\\\bar{h}} \\\\mathcal {X}_{h^{(0)}}(\\\\tau ) \\\\mathcal {X}_{\\\\bar{h}^{(0)}}(\\\\bar{\\\\tau })\\\\\\\\Z(\\\\tau , \\\\bar{\\\\tau }, \\\\mathbf {z},\\\\mathbf {\\\\bar{z}})&= \\\\sum _{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}} M(\\\\tau ,\\\\bar{\\\\tau })_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}(\\\\tau ,\\\\mathbf {z})\\\\mathcal {X}_{\\\\mathbf {\\\\bar{\\\\mu }},k}(\\\\bar{\\\\tau },\\\\bar{\\\\mathbf {z}})\\\\,.\",\n \"\\\\end{split} $ Modular transformations on the partition function translates into a specific modular transformation of the matrix $M_{\\\\mu , \\\\bar{\\\\mu }}$ .\",\n \"To see this transformation law, simply separate out the transformation law of $Z$ into its irrep constituents: $ \\\\begin{split} 0&=Z(-\\\\frac{1}{\\\\tau }, -\\\\frac{1}{\\\\bar{\\\\tau }}, \\\\frac{\\\\mathbf {z}}{\\\\tau },\\\\frac{\\\\mathbf {\\\\bar{z}}}{\\\\bar{\\\\tau }}) -e^{i \\\\pi k \\\\left(\\\\frac{\\\\mathbf {z}^2}{\\\\tau }-\\\\frac{\\\\bar{\\\\mathbf {z}}^2}{\\\\bar{\\\\tau }}\\\\right)} Z(\\\\tau , \\\\bar{\\\\tau }, \\\\mathbf {z},\\\\mathbf {\\\\bar{z}}) \\\\\\\\&=\\\\sum _{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}} M\\\\left(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}\\\\right)_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}\\\\left(-\\\\frac{1}{\\\\tau },\\\\frac{\\\\mathbf {z}}{\\\\tau }\\\\right)\\\\bar{\\\\mathcal {X}}_{\\\\mathbf {\\\\bar{\\\\mu }},k}\\\\left(-\\\\frac{1}{\\\\bar{\\\\tau }},\\\\frac{\\\\bar{\\\\mathbf {z}}}{\\\\bar{\\\\tau }}\\\\right)-e^{i \\\\pi k \\\\left(\\\\frac{\\\\mathbf {z}^2}{\\\\tau }-\\\\frac{\\\\bar{\\\\mathbf {z}}^2}{\\\\bar{\\\\tau }}\\\\right)}M(\\\\tau ,\\\\bar{\\\\tau })_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}\\\\mathcal {X}_{\\\\mathbf {\\\\mu },k}(\\\\tau ,\\\\mathbf {z})\\\\bar{\\\\mathcal {X}}_{\\\\mathbf {\\\\bar{\\\\mu }},k}(\\\\bar{\\\\tau },\\\\bar{\\\\mathbf {z}})\\\\\\\\&=\\\\sum _{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}\\\\left(\\\\sum _{\\\\mathbf {\\\\mu }^{\\\\prime },\\\\mathbf {\\\\bar{\\\\mu }^{\\\\prime }}}S_{\\\\mathbf {\\\\mu ^\\\\prime \\\\mu }}^kM\\\\left(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}\\\\right)_{\\\\mathbf {\\\\mu ^{\\\\prime }},\\\\mathbf {\\\\bar{\\\\mu }^{\\\\prime }}}\\\\bar{S}_{\\\\mathbf {\\\\bar{\\\\mu }^\\\\prime \\\\bar{\\\\mu }}}^k- M(\\\\tau ,\\\\bar{\\\\tau })_{\\\\mathbf {\\\\mu },\\\\mathbf {\\\\bar{\\\\mu }}}\\\\right)e^{i \\\\pi k \\\\left(\\\\frac{\\\\mathbf {z}^2}{\\\\tau }-\\\\frac{\\\\bar{\\\\mathbf {z}}^2}{\\\\bar{\\\\tau }}\\\\right)}\\\\mathcal {X}_{\\\\mathbf {\\\\bar{\\\\mu }},k}(\\\\bar{\\\\tau },\\\\bar{\\\\mathbf {z}})\\\\bar{\\\\mathcal {X}}_{\\\\mathbf {\\\\bar{\\\\mu }},k}(\\\\bar{\\\\tau },\\\\bar{\\\\mathbf {z}})\\\\,, \\\\end{split} $ where we have used the transformation rule, (REF ), and the definition (REF ).\",\n \"Stripping off the characters, the above crossing equation is equivalent to a crossing equation for the matrix $0 = M(\\\\tau ,\\\\bar{\\\\tau })_{\\\\mu ,\\\\bar{\\\\mu }} - S^T_{\\\\mu ,\\\\mu ^\\\\prime }M\\\\left(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}\\\\right)_{\\\\mu ^\\\\prime ,\\\\bar{\\\\mu }^\\\\prime }\\\\bar{S}_{\\\\bar{\\\\mu }^\\\\prime ,\\\\bar{\\\\mu }}\\\\,.$ For the constraints on theories with non-abelian currents, equation (REF ) is the form of the constraint that we will use.\",\n \"For each bootstrap question we will input the symmetry group and level $k$ .\"\n ],\n [\n \"Semidefinite Projective Functionals and the Extremal Method\",\n \"To be self-contained, we will briefly review linear and semidefinite programming methods as applied to the modular bootstrap; for more thorough reviews and some examples of applications, see e.g.\",\n \"[4], [10], [13], [14], [15], [16], [17], [18], or [19], [20], [21], [22], [23], [24], [25] for reviews and some of the original papers developing methods in the standard bootstrap that we will adopt directly.\",\n \"The starting point is equation (REF ), which can be written $0 &=& \\\\sum _{h, \\\\bar{h}, \\\\mu ^{\\\\prime }, \\\\bar{\\\\mu }^{\\\\prime } } d_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h} } \\\\left( F_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}}\\\\right)_{\\\\mu ^{\\\\prime }, \\\\bar{\\\\mu }^{\\\\prime }}(\\\\tau , \\\\bar{\\\\tau }) , \\\\\\\\\\\\left( F_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}}\\\\right)_{\\\\mu ^{\\\\prime }, \\\\bar{\\\\mu }^{\\\\prime }}(\\\\tau , \\\\bar{\\\\tau }) &\\\\equiv & \\\\left[ \\\\delta _{\\\\mu \\\\mu ^{\\\\prime }} \\\\delta _{\\\\bar{\\\\mu } \\\\bar{\\\\mu }^{\\\\prime }} \\\\mathcal {X}_{h^{(0)}}(\\\\tau ) \\\\mathcal {X}_{\\\\bar{h}^{(0)}}(\\\\tau ) - S_{\\\\mu ^{\\\\prime } \\\\mu } S_{\\\\bar{\\\\mu }^{\\\\prime }, \\\\bar{\\\\mu }} \\\\mathcal {X}_{h^{(0)}}(-1/\\\\tau ) \\\\mathcal {X}_{\\\\bar{h}^{(0)}}(-1/\\\\bar{\\\\tau }) \\\\right] .\\\\nonumber \\\\\\\\$ The occupation numbers $d_{\\\\mu , \\\\bar{\\\\mu }, h,\\\\bar{h}}$ are all non-negative, and include in particular the vacuum $d_{\\\\rm vac}=1$ .\",\n \"One is generally interested in proving that there exist states in the theory with various properties, for instance that there exists a state with $\\\\Delta < \\\\Delta _{\\\\rm max}$ for some value of $\\\\Delta _{\\\\rm max}$ .\",\n \"Let us abstractly call a choice of such properties “$P$ ”.\",\n \"Then, one can prove that there is at least one state in the theory with properties $P$ as long as one can find a linear functional $\\\\rho $ that maps the characters to real numbers such that it is positive on the vacuum and also positive on all states not satisfying $P$ .\",\n \"In equations, $\\\\rho ({\\\\rm vac})= 1 , \\\\textrm { and } \\\\rho (F_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}} ) \\\\ge 0 \\\\textrm { unless } (\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}) \\\\textrm { satisfies } P.$ The normalization $\\\\rho ({\\\\rm vac})=1$ is conventional.\",\n \"If such a linear functional $\\\\rho $ exists, then there must be a state in the theory with the properties $P$ , otherwise $\\\\rho $ acting on equation (REF ) would imply $0 \\\\ge 1$ .\",\n \"Usually we will be interested in not just one choice of $P$ but a continuous family of choices $P_s$ parameterized by a continuous variable (or variables) $s$ .\",\n \"Typically, $s$ will be something like the bound $\\\\Delta _{\\\\rm max}$ in the example above, so that as one decreases $s$ the set of states with property $P_s$ grows and therefore the set of linear functionals that are positive on all such states shrinks.\",\n \"Critical values $s_*$ of $s$ where the set of such linear functionals vanishes are especially interesting: aside from giving the best possible bounds, at these points one can use the “extremal functional method” [19] to determine all of the occupation numbers $d_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}}$ .\",\n \"The basic idea behind this is that for any $s$ , the space of functions $F_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}}$ spanned by states satisfying $P_s$ is a polytope where $-F_{\\\\rm vac}$ is inside the polytope for $ss_*$ .\",\n \"At exactly $s_*$ , $-F_{\\\\rm vac}$ passes through one of the faces of the polytope, so there is a unique positive semidefinite linear combination of the states satisfying $P_{s_*}$ that cancels the contribution from the vacuum in (REF ).\",\n \"In practice, we have to work with finite-dimensional projections of the full space of functions $F_{\\\\mu , \\\\bar{\\\\mu }, h, \\\\bar{h}}$ , but one optimistically expects to converge to a unique solution as the dimensionality of the projected space increases.\",\n \"We will encounter some exceptions that we will discuss as we come to them.\"\n ],\n [\n \"Abelian Bounds\",\n \"In this section, we perform a systematic numerical analysis on the bounds on the gap in dimensions and charges, as well as on the smallest charge-to-mass ratio allowed in a theory with a $U(1)$ current.\"\n ],\n [\n \"Semi-definite Programming with Continuous Charge $Q$\",\n \"For the abelian case, for simplicity we will not use the full Kac-Moody characters, but rather just the Virasoro characters $\\\\chi _h(q)$ : $Z(\\\\tau , z) =\\\\sum _{h, \\\\bar{h}, Q, \\\\bar{Q}} d_{Q, \\\\bar{Q}, h , \\\\bar{h}} y^Q \\\\bar{y}^{\\\\bar{Q}} \\\\chi _h(q) \\\\chi _{\\\\bar{h}}(\\\\bar{q}).$ where $q= e^{2\\\\pi i \\\\tau }$ , $\\\\bar{q} = e^{-2\\\\pi i \\\\bar{\\\\tau }}$ , $y =e^{2\\\\pi i z}$ , $\\\\bar{y} = e^{-2\\\\pi i \\\\bar{z}}$ .\",\n \"We will consider left-right symmetric theories with $c = \\\\bar{c}$ , and for simplicity we set $\\\\bar{z} = 0$ .\",\n \"As in [4], we reduce the characters using the $S$ invariant factor $|\\\\tau |^{\\\\frac{1}{2}}|\\\\eta (\\\\tau )|^2$ .\",\n \"Furthermore we reduce the partition function as $\\\\hat{Z}(\\\\tau , z) \\\\equiv |e^{\\\\frac{i\\\\pi z^2}{2\\\\tau } }|^2|\\\\tau |^{\\\\frac{1}{2}}|\\\\eta (\\\\tau )|^2 Z(\\\\tau , z) ,$ so that $\\\\hat{Z}(\\\\tau , z)$ is invariant under $ \\\\left( \\\\tau \\\\mapsto -\\\\frac{1}{\\\\tau }, z \\\\mapsto \\\\frac{z}{\\\\tau } \\\\right)$ .\",\n \"The characters are reduced into $\\\\hat{\\\\chi }_0(q) = e^{\\\\frac{i\\\\pi z^2}{2\\\\tau } } \\\\tau ^{\\\\frac{1}{4}}q^{-\\\\frac{c-1}{24}}(1-q) ,~~~~\\\\hat{\\\\chi }_h(q) y^Q = e^{\\\\frac{i\\\\pi z^2}{2\\\\tau } } \\\\tau ^{\\\\frac{1}{4}}q^{-\\\\frac{c-1}{24}}q^h y^Q .$ We consider linear functionals of the form $\\\\rho \\\\equiv \\\\sum _{m+n+k/2 {\\\\rm ~odd,~} k {\\\\rm ~even}}\\\\left.\",\n \"\\\\alpha _{m,n,k} \\\\ \\\\partial _{t}^m \\\\, \\\\partial _{\\\\bar{t}}^n\\\\,\\\\partial _{w}^k\\\\right|_{t=\\\\bar{t}=w=0},$ where the change of variable $\\\\tau = i e^{t}$ and $z =we^{\\\\frac{t}{2}}$ is made so that $t \\\\mapsto -t$ and $w^2 \\\\mapsto -w^2$ under $S$ transformations.In this expression for $\\\\rho $ , we have not used any $\\\\bar{z}$ derivatives and so do not use information about the anti-holomorphic charge $\\\\bar{Q}$ .\",\n \"This is mainly for simplicity and efficiency; it is in principle straightforward, though more computationally intensive, to use $\\\\bar{Q}$ information as well.\",\n \"Later in this section we will in fact perform one analysis where we keep $\\\\bar{z}$ derivatives to demonstrate this point.\",\n \"We arrive at a two variable functional $\\\\rho _l(\\\\Delta ,Q) &\\\\equiv \\\\rho \\\\left[ \\\\hat{\\\\chi }_{\\\\frac{\\\\Delta -l}{2}}(\\\\tau )\\\\hat{\\\\bar{\\\\chi }}_{\\\\frac{\\\\Delta +l}{2}}(\\\\bar{\\\\tau }) y^Q +\\\\hat{\\\\chi }_{\\\\frac{\\\\Delta +l}{2}}(\\\\tau )\\\\hat{\\\\bar{\\\\chi }}_{\\\\frac{\\\\Delta -l}{2}}(\\\\bar{\\\\tau }) \\\\right] , \\\\nonumber \\\\\\\\\\\\rho ({\\\\rm vac}) &\\\\equiv \\\\rho \\\\left[ \\\\hat{\\\\chi }_{0}(\\\\tau )\\\\hat{\\\\bar{\\\\chi }}_{0}(\\\\bar{\\\\tau }) \\\\right],$ where we assume the spectrum is parity symmetric.\",\n \"Part of the challenge of the abelian analysis is that we do not assume charge quantization, i.e.\",\n \"technically we allow the gauge group to be $\\\\mathbb {R}$ instead of $U(1)$ , which means that we have to deal with not just one but two continuous parameters, $\\\\Delta $ and $Q$ .\",\n \"This complicates the application of positive semi-definite approaches, since these are based on constructing positive functionals of the characters and in general the space of such functionals is more complicated for multiple variables than for a single variable.\",\n \"In particular, for a single variable, positive semi-definite functionals can be written without loss of generality as a sum of squares plus a linear term times a sum of squares.\",\n \"For multiple variables, such a parameterization is no longer completely general.\",\n \"One way to deal with this issue is simply to discretize in, say, $Q$ , but we find that such an approach becomes difficult to implement in practice since the discretization needs to become very fine to prevent the numeric search from picking functionals that become negative in between the discretization points.\",\n \"The approach we take is instead to limit the search space to functionals that are still a sum of squares plus a linear term times sums of squares.\",\n \"In the limit of very high order polynomials, one might expect that such functionals can approximate the extremal functionals arbitrarily well.\",\n \"In any case, while such functionals might not give the best possible bounds, they nevertheless produce valid bounds.\",\n \"Even restricting to polynomial functionals, there remains a practical problem of how to implement the search over such functionals using available software for semi-definite programming analyses.\",\n \"In appendix , we discuss how to massage this problem into an appropriate form for use with SDPB.\"\n ],\n [\n \"Bound on Dimension of Lightest Charged State\",\n \"With the flavored partition function we can bound the dimension $\\\\Delta _*$ of lightest charged state in any theory with a U(1).\",\n \"The bound for different $c$ is summerized in Table.\",\n \"REF .\",\n \"We extrapolate the bound values to $n_D \\\\rightarrow \\\\infty $ using a linear function of $\\\\frac{1}{n_D}$ similar to what is done in [10] for $c \\\\le 100$ where the convergence of the bound values is significant.\",\n \"Then we extrapolate the bounds to $n_D\\\\rightarrow \\\\infty $ and $c \\\\rightarrow \\\\infty $ by fitting the finite $n_D$ and $c$ results to a linear function of $\\\\frac{1}{c}$ and $1/n_D$ and extrapolating, as shown in Fig.\",\n \"REF .\",\n \"In this test we take $\\\\tau = \\\\frac{i \\\\beta }{2 \\\\pi }$ with $\\\\beta $ real to avoid the complication of spin.\",\n \"It is not understood a priori why the form of this fit works, but empirically it agrees well with the data at large $c$ and $n_D$ .\",\n \"Table: Bounds on the dimension of lightestcharged state assuming the theory has U(1) symmetry, as a function of cc and the number n D n_D of derivatives used in the bootstrap functionals.Figure: Bounds of the dimension oflightest charged state assuming the theory has U(1) symmetry.\",\n \"Theextrapolated gaps at n D →∞n_D \\\\rightarrow \\\\infty with the trendline.Similarly to the results of [10], extrapolating in $n_{D}$ and then $c$ provides a parametrically stronger bound than the finite $n_{D}$ analysis.\",\n \"$\\\\Delta _{*} = \\\\frac{c}{\\\\alpha } + \\\\mathcal {O}(1)\\\\,, \\\\ \\\\ \\\\ \\\\alpha > 8\\\\,.$ This bound (REF ) is similar to the bounds on the non-charged state found in [4], [10], though quantitatively different.\",\n \"The bound in [4] is parametrically weaker, which is not surprising since that analysis did not use the spins of the characters, and did not perform any extrapolation in the number of derivatives.\",\n \"The bound in [10] is more analogous since spins and extrapolations were used; €” the result there is very slightly stronger ($\\\\alpha \\\\sim 9$ ) than (REF ) for the charged spectrum.\"\n ],\n [\n \"Bound on Charge-to-Mass Ratio\",\n \"In this section, we will present results that there must be a state in the theory with a charge-to-mass ratio $r \\\\equiv \\\\frac{Q c}{12 \\\\Delta } = \\\\frac{Q}{8 G_N m},$ above some critical value $r_*$ , whose value we will determine numerically.The value of $r^*$ increases as the number $n_D$ of derivatives used increases and the numeric accuracy improves, though we emphasize that even for low number of derivatives the values of $r_*$ are a valid bound proving that a state must exist in the theory with $\\\\frac{Q}{8 G_N m} > r_*$ .\",\n \"We try to find the linear functional thatWe also impose a dimension cutoff $\\\\rho (\\\\Delta ,Q) \\\\ge 0,~~~ \\\\Delta >\\\\frac{100c}{12}$ , because we want the constraint to be a little stronger that not only a state which saturates the ratio bound exists but also the state must have finite dimension.\",\n \"Different dimension cutoffs do not result in significantly different functionals and bounds.\",\n \"It just helps the algorithm to find a functional that only is negative at finite $\\\\Delta $ .\",\n \"$\\\\rho ({\\\\rm vac}) \\\\ge 0,& \\\\nonumber \\\\\\\\\\\\rho (\\\\Delta ,Q) \\\\ge 0,&~~~ |Q| \\\\le Q_{\\\\Delta } \\\\equiv \\\\frac{12 r \\\\Delta }{c}~.$ We drop the spin index $l$ by only taking functionals of the form $\\\\rho \\\\equiv \\\\sum _{m+k/2 {\\\\rm ~odd,~} k {\\\\rm ~even}} (\\\\partial _t+ \\\\partial _{\\\\bar{t}})^m \\\\partial _w^k ~.$ For the functional to be positive in a bounded region we make a change of variables of the form $Q^2 = \\\\frac{\\\\tilde{Q}^2 Q_{\\\\Delta }^2}{\\\\tilde{Q}^2 +1 }~.$ $|\\\\tilde{Q}| \\\\ge 0$ means $|Q| \\\\le Q_{\\\\Delta }$ , inspired by [10].\",\n \"First, we show in Fig.\",\n \"REF the bound on $r_*$ as a function of $c$ .\",\n \"By inspection, one can see that the larger $c$ is, the longer it takes for the bounds to converge.\",\n \"To see how the bound depends on the number $n_D$ of derivatives in more detail, in Fig.\",\n \"REF we focus on a specific value of $c$ , $c=10^5$ and show the resulting bound on $r$ as a function of the number $n_D$ of derivatives allows in the functional $\\\\rho $ .\",\n \"The best fit as power law suggests that the optimal bound on $r^*$ might be significantly better, i.e.\",\n \"$(r^{*})^{-1} \\\\ll 1 $ .\",\n \"For comparison, the result in [6] was $\\\\left(\\\\frac{8 G_N m}{Q}\\\\right)_* = (r_*)^{-1} < 4 \\\\sqrt{\\\\pi } = 7.1$ .\",\n \"Figure: Bound of mass-to-charge ratio as a function of cc; a trend line ∝c -1/2 \\\\propto c^{-1/2} is shown for comparison.\",\n \"The extrapolation (“n D =∞n_D=\\\\infty ” points) and error bars are computed by performing a fit as a function of n D n_D and extrapolating to n D →∞n_D\\\\rightarrow \\\\infty as described in the text.In [6], it was also shown that even with a small number $n_D$ of derivatives, one could obtain a bound on $\\\\frac{\\\\Delta }{Q}$ at large $c$ that scaled like $\\\\sim c $ .\",\n \"There is an intriguing possibility however that the true bound scales like $c^{1/2}$ , and that this is obscured because it takes more and more derivatives to reach this optimal bound as $c$ increases.\",\n \"The basic idea for why one might expect a $c^{1/2}$ scaling is that in higher dimensions, the scaling of the WGC limit can easily be read off by demanding that the binding energy from gravity and a Coulomb force cancel each other out.\",\n \"In the AdS$_3$ case, one can think of the binding energy from gravity as $ \\\\frac{3\\\\Delta ^2}{c}$ , whereas from a $k=1$ Chern-Simons gauge field exchange it is $Q^2$ ; demanding equality would set $\\\\frac{\\\\Delta }{Q} \\\\approx \\\\sqrt{c/3}$ , i.e.\",\n \"$\\\\frac{8 G_N m}{Q} \\\\approx 6.9 c^{-1/2}$ .\",\n \"One can read off the coefficients by looking at the vacuum conformal block for Virasoro and Kac-Moody algebras in the limit $z\\\\sim 1$ [26], [27].\",\n \"For comparison, in Fig.\",\n \"REF we have also shown a trend line at $c^{-1/2}$ , which becomes further below our best numeric bound with $n_D=29$ derivatives as $c$ increases.\",\n \"We can try to estimate the optimal bound by taking our result at each $c$ as a function of $n_D$ and extrapolating to $n_D = \\\\infty $ .\",\n \"The “$n_D=\\\\infty $ ” points we show in Fig.\",\n \"REF fit our results starting at $n_D \\\\ge 15$ in order to get the extrapolation.\",\n \"However, there is significant uncertainty in the resulting estimate, as can be gauged by the fact that performing the fit starting at smaller or larger values of $n_D$ gives different answers.\",\n \"In Fig.\",\n \"REF we have shown the bound as a function of $n_D$ , where one can explicitly see that the bound is still changing rapidly as a function of $n_D$ even at the upper range of what we have been able to achieve numerically.\",\n \"In Fig.\",\n \"REF , the error bars we have shown indicate the range over all the different positive values we obtain if we perform the fit over $n_D \\\\ge 11, n_D \\\\ge 13, \\\\dots n_D \\\\ge 19$ .\",\n \"With better numerical accuracy at large values of $c$ , it should be possible to more firmly establish this scaling behavior.\",\n \"Figure: Upper bound on 8G N m Q\\\\frac{8 G_N m}{Q} (that is, there existsa state below the bound) as the number n D n_D of derivatives usedin the semidefinite programming analysis increases, for thespecific case c=10 5 c=10^5.\",\n \"The bound value is still changing rapidlyat n D =41n_D = 41.\"\n ],\n [\n \"Bound on Lowest Charge\",\n \"Next we will focus on bounds on the lowest charge $Q_*$ of all charged states in the theory.\",\n \"We will first consider the charge $Q$ only, and then see how to do better by including information on dimensions and spins.\",\n \"To determine the upper bound on the gap to the smallest $|Q|$ of all the charged states, we want to find a linear functional satisfying the following conditions: $\\\\rho ({\\\\rm vac}) \\\\ge 0,& \\\\nonumber \\\\\\\\\\\\rho (\\\\Delta ,0) \\\\ge 0,&~~~ \\\\Delta \\\\ge 0 \\\\nonumber \\\\\\\\\\\\rho (\\\\Delta ,Q) \\\\ge 0,&~~~ \\\\Delta \\\\ge 0 {\\\\rm ~~~And~~~} |Q| \\\\ge Q_*$ The resulting bound on $Q_*$ is shown in Fig.\",\n \"REF .\",\n \"The result is somewhat surprisingly always just $Q_* = 1$ .\",\n \"This may be because in some theories with $c\\\\le 1$ a state of $Q=1$ saturates the bound and theories of larger $c$ can be constructed as a direct product of such theories and other algebras.\",\n \"Figure: We obtain an upper bound, shown here, on the smallest nonzero charge Q * Q_*; the bound isQ * ≤1Q_* \\\\le 1 for all cc.In any case, we next turn to including information on dimensions by bounding gaps in both $Q$ and $\\\\Delta $ simultaneously – $Q_*$ as the lowest charge of charged states and $\\\\Delta _*$ as the lowest dimension of all non-vacuum states.\",\n \"To obtain such a bound, the linear functional $\\\\rho $ should satisfy $\\\\rho ({\\\\rm vac}) \\\\ge 0,& \\\\nonumber \\\\\\\\\\\\rho (\\\\Delta ,0) \\\\ge 0,&~~~ \\\\Delta \\\\ge \\\\Delta _* \\\\nonumber \\\\\\\\\\\\rho (\\\\Delta ,Q) \\\\ge 0,&~~~ \\\\Delta \\\\ge \\\\Delta _* {\\\\rm ~~~And~~~} |Q| \\\\ge Q_*$ We take the linear functional to have no spin information.\",\n \"Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\\\Delta _* at c=2c=2.\",\n \"The region near the kink in the left plot is magnified and shown in the right plot.At each individual $c$ , the bound carves out a region in a two-dimensional parameter space.\",\n \"The exclusion curve of an $c=2$ example is shown in Fig.\",\n \"REF .\",\n \"There is a kink at $\\\\Delta _* \\\\approx 0.5$ which has a bound $Q_* \\\\le 1$ .\",\n \"It would be interesting to identify what if any theory lives at this kink.\",\n \"Finally, the simutaneous $\\\\Delta _*$ and $Q_*$ approach can be even more powerful if we turn on spin.\",\n \"For this case the example we choose is $c=8$ .\",\n \"In [10] the Sugawara theory $E_8$ lattice shows up as a kink of bounds on lowest dimension of scalar primaries.\",\n \"We seek a linear functional $\\\\rho $ satisfying $\\\\rho ({\\\\rm vac}) \\\\ge 0,& \\\\nonumber \\\\\\\\\\\\rho _0(\\\\Delta ,0) \\\\ge 0,&~~~ \\\\Delta \\\\ge \\\\Delta _* \\\\nonumber \\\\\\\\\\\\rho _l(\\\\Delta ,0) \\\\ge 0,&~~~ \\\\Delta \\\\ge |l| \\\\nonumber \\\\\\\\\\\\rho _0(\\\\Delta ,Q) \\\\ge 0,&~~~ \\\\Delta \\\\ge \\\\Delta _* {\\\\rm ~~~And~~~} |Q| \\\\ge Q_* \\\\nonumber \\\\\\\\\\\\rho _l(\\\\Delta ,Q) \\\\ge 0,&~~~ \\\\Delta \\\\ge |l| {\\\\rm ~~~And~~~} |Q| \\\\ge Q_*$ for all $l \\\\in \\\\mathbb {Z}_{\\\\ge 0}$ .\",\n \"Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\\\Delta _* at c=8c=8.The two parameter plot of $\\\\Delta _*$ and $Q_*$ is shown in Fig.\",\n \"REF .\",\n \"Note that we find that $Q_*$ at $\\\\Delta _*=0$ is smaller than 1, better than the bound obtained without $Q_*$ informationIt is also possible that by assuming integer spins we throw away the theory that saturates the $Q_{*}=1$ bound.. More interestingly, we see a sharp kink at $\\\\Delta _{*}=2$ .\",\n \"In the next subsection we will analyze the extremal functional of this kink and see that this kink is the $E_8$ lattice CFT, and we will obtain the dimension and charge spectrum of the low lying states.\"\n ],\n [\n \"Extremal Functional Analysis\",\n \"In this subsection, we will use extremal functional analyses to determine the partition function saturating various bounds.\"\n ],\n [\n \"Maximal $r_*$ at {{formula:9d4b5483-4fe8-4eda-a64b-878604bb43b5}}\",\n \"Our first application of extremal function methods will be to the bound on the charge-to-mass ratio $r$ .\",\n \"Since our bounds have converged for $c=1$ and we can consider the extremal functional $\\\\rho $ for this case; by design, $\\\\rho $ is non-negative on the space of states we allow, and so the states in the theory must be at the places where $\\\\rho $ vanishes.\",\n \"The functional depends on both $\\\\Delta $ and $Q$ , so the extremal spectrum contains more data as shown by Fig.\",\n \"REF .\",\n \"Figure: Projection functional ρ\\\\rho of c=1c=1 as a function ofΔ\\\\Delta and QQ computed at N=29N=29.\",\n \"The black regions are where ρ≤0\\\\rho \\\\le 0; according to our criterion (), ρ\\\\rho is ≥0\\\\ge 0 in the allowed region Q≤12r * Δ cQ\\\\le \\\\frac{12 r_* \\\\Delta }{c}, so the extremal spectrum comprises states where ρ=0\\\\rho =0 in this region.\",\n \"There is a line of small black dots where ρ=0\\\\rho =0 along the Q=0Q=0 axis that are difficult to see and so we plot ρ\\\\rho along this line in Fig.\",\n \".The zeros of $\\\\rho $ occur at points and can be difficult to see in Fig.\",\n \"REF .\",\n \"In Fig.\",\n \"REF , we show $\\\\rho $ along two particularly relevant lines: the neutral ($Q=0$ ) states, and the states that saturate the mass-to-charge ratio, i.e.\",\n \"$Q= \\\\frac{12 r_* \\\\Delta }{c}$ .\",\n \"Figure: The functional restricted at Q=0Q=0 is shown on theleft.\",\n \"The states that saturate the best bound on8G N m Q\\\\frac{8 G_N m}{Q} is shown on the right.\"\n ],\n [\n \"Sequential $\\\\Delta _*$ and {{formula:b25e4990-736d-44d3-b86f-624d64285424}} Approach - Revisiting the {{formula:b4d2cd11-1472-4199-8de0-93f481d9bb5d}} Lattice\",\n \"In Sec.\",\n \"REF we found a kink in the simultaneous $\\\\Delta _*$ and $Q_*$ approach with spin information at $c=8$ .\",\n \"In order to show that the kink is indeed the level 1 $E_8$ lattice, we can look for extremal flavored partition functions by using a “two-step” approach where we first solve for the spectrum of dimensions and then solve for the spectrum of charges.\",\n \"The idea is that we can use the extremal functional method on the unflavored partition function, maximizing the gap in dimensions of operators.\",\n \"This step is just the standard extremal functional method and it will just reproduce previous results [10].\",\n \"Then, having fixed the weights of the states in the theory, we can impose a gap in the charge of the states in the theory, allowing only the weights $(h,\\\\bar{h})$ found previously.\",\n \"For concreteness, we will focus on the case $c=8$ as a representative example.\",\n \"As shown in [10], the gap in dimensions is maximized at $\\\\Delta _*=2$ by the $E_8$ theory at $k=1$ (this theory can be described as 8 free bosons on an $E_8$ lattice), and the extremal functional method allows one to independently derive the partition function of this theory.\",\n \"We have reproduced the extremal functionals $\\\\rho _\\\\ell (\\\\Delta )$ at each spin $\\\\ell $ in Fig.\",\n \"REF , which implies the spectrum is ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\Delta = 2,4,6,\\\\ldots ,& l=0,\\\\\\\\\\\\Delta = l,l+2,l+4,\\\\ldots ,& l\\\\ne 0 .\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ Figure: Extremal functional of c=8c=8 theory, at n D =35n_D=35.Moving on to the spectrum of charges, we find that the gap $Q_*$ is maximized at $Q_* = \\\\frac{1}{\\\\sqrt{2}}$ .\",\n \"The corresponding extremal functionals $\\\\rho _{\\\\Delta , \\\\ell }(Q)$ at each dimension $\\\\Delta $ and spin $\\\\ell $ are plotted in Fig.\",\n \"REF .\",\n \"We note that this is not the flavored partition function that one obtains if one turns on a chemical potential for the charge $J = \\\\partial \\\\phi $ in the $E_8$ lattice description; that choice corresponds to the spectrum of charges $\\\\frac{1}{2}\\\\mathbb {Z}$ rather than $\\\\frac{1}{\\\\sqrt{2}} \\\\mathbb {Z}$ .\",\n \"Instead, if one chooses one of the length-2 vectors $\\\\vec{\\\\alpha }$ in the $E_8$ lattice, then $J \\\\equiv V_{\\\\vec{\\\\alpha }} \\\\equiv \\\\frac{1}{\\\\sqrt{2}} \\\\left( e^{\\\\vec{\\\\alpha }\\\\cdot \\\\vec{\\\\phi }} + e^{-\\\\vec{\\\\alpha } \\\\cdot \\\\vec{\\\\phi }} \\\\right)$ has $k=1$ , and the lowest charged states include for instance $V_{2\\\\vec{\\\\alpha }}$ , with charge $\\\\frac{1}{\\\\sqrt{2}}$ .\",\n \"One can see in Fig.\",\n \"REF that the extremal functional has zeros at around $0, \\\\pm \\\\frac{1}{\\\\sqrt{2}}$ and $\\\\pm \\\\frac{2}{\\\\sqrt{2}}$ for all $\\\\Delta , \\\\ell $ , and we expect that this would continue to be true at $ \\\\frac{n}{\\\\sqrt{2}}$ for all $n \\\\in \\\\mathbb {Z}$ as the number $n_D$ of derivatives used in the analysis approaches infinity.\",\n \"Because these dimensions and charges appear to follow such a simple pattern, we will proceed by assuming this pattern continues.\",\n \"Then, with the allowed weights $\\\\Delta , \\\\ell $ and charges $Q$ fixed in the theory, solving the modular bootstrap equation reduces to a linear programming problem, which is much more efficient numerically.Furthermore, since this linear programming analysis fixes the partition function for us to be a particular flavoring of the $E_8$ theory, by uniqueness it will be the correct one, justifying the original Ansatz.\",\n \"We obtain the occupation numbers indicated in Table REF , where we have flavored separately by both holomorphic and anti-holomorphic charges $Q$ and $\\\\bar{Q}$ .\",\n \"We show the occupation numbers of the conserved $ \\\\ell = 1$ currents assuming the extremal charge spectrum $Q= \\\\frac{n}{2}$ (right).\",\n \"In addition, it is straightforward to repeat the analysis assuming $Q = \\\\frac{n}{\\\\sqrt{2}}$ (left) for comparison.\",\n \"In both cases, we obtain a total of 248 currents each in the holomorphic and anti-holomorphic part, but distributed differently among different charges in the two cases.\",\n \"Table: Occupation numbers from a linear programming analysis.\",\n \"The left table assumes states at Q∈1 2ℤQ \\\\in \\\\frac{1}{\\\\sqrt{2}} \\\\mathbb {Z}, whereas the right assumes Q∈1 2ℤQ \\\\in \\\\frac{1}{2} \\\\mathbb {Z}.Figure: Extremal functional of QQ at n D =19n_D=19 when the gap on QQ is maximizedat 1 2\\\\frac{1}{\\\\sqrt{2}}.\"\n ],\n [\n \"Bounds on gaps in operator dimensions\",\n \"Next we turn to a numeric analysis of gaps in non-abelian theories.\",\n \"In some cases, the results are somewhat stronger or weaker depending on whether or not we allow for states that saturate the unitarity bound $2 k h \\\\ge Q^2$ , which we will refer to as “extremal states”, and whether or not we impose gaps in all charge sectors.\",\n \"We will present results starting with the strongest assumptions first.\",\n \"In all cases, we will present only the results for $SU(2)$ gauge group at level $k=1$ .\",\n \"We have analyzed larger gauge groups and higher levels and the results are qualitatively similar, though the rate of numeric convergence is worse; some preliminary results for $SU(2)$ with $k=2$ are shown in appendix .\",\n \"To begin, we will set the gap in all representations to be the same and restrict to the partition function at $q=\\\\bar{q}$ ; the resulting gap value will be the lowest dimension of the primary operators.\",\n \"This “uniform bound” is shown in Fig.\",\n \"REF .\",\n \"We have actually done two slightly different analysis, which are compared to each other on the right in Fig.\",\n \"REF .\",\n \"These analyses differ in whether or not we allow states in the non-trivial representations with $h^{(0)}= \\\\bar{h}^{(0)}=0$ , which saturate the unitarity bound in both the holomorphic and anti-holomorphic sectors and which we will refer to as “extremal states;” in the analysis where such states are included, the “gap” for each representation is defined as the smallest $\\\\Delta ^{(0)}$ among the non-extremal states.\",\n \"As one can see, the difference between the results of the two analyses is significant at small $c$ but becomes negligible as $c$ approaches $\\\\infty $ .\",\n \"Ultimately, this result does not tell us much more than one learns from previous similar analyses without flavored information; all we learn here is that there must be some state in the theory with $\\\\Delta ^{(0)}$ below some value, which is very similar to the bound on the same quantity from the unflavored modular bootstrap.\",\n \"Figure: Bound on SU(2) k=1k=1 gaps in Δ\\\\Delta universal for allrepresentations for 1≤c≤1001 \\\\le c \\\\le 100.\",\n \"Left: Boundsobtained with different values of n D n_D when extremal states arenot allowed.\",\n \"Dashed lines from blue to red are computed data of3≤n D ≤293 \\\\le n_D \\\\le 29.\",\n \"The solid black line is extrapolated fromdata of 11≤n D ≤29 11 \\\\le n_D \\\\le 29 using a function linear in1/n D 1/n_D.\",\n \"Right: Bounds extrapolated to n D =∞n_D = \\\\infty forthe analysis without extremal states compared to the result whenextremal states are allowed.\",\n \"The difference is negligible atlarge cc but significant at small cc.Next, however, we will turn to an analysis that maximizes the bounds separately in different sectors, and this is where we will start to find something qualitatively new compared with what is possible with the unflavored modular bootstrap.\",\n \"In particular, we will maximize the gap in the trivial representation, and not impose any constraint on the gaps in the other representations.\",\n \"In equations, our conditions are $\\\\rho _{\\\\lambda ,\\\\bar{\\\\lambda }}(\\\\Delta ^{(0)})\\\\ge 0 {\\\\rm ~when~}{\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\Delta ^{(0)} \\\\ge \\\\Delta _{*},&\\\\lambda =(0){\\\\rm ~and~} \\\\bar{\\\\lambda }=(0),\\\\\\\\\\\\Delta ^{(0)} \\\\ge 0,&\\\\lambda {\\\\rm ~or~} \\\\bar{\\\\lambda }\\\\ne (0),\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ in SU(2) $k=1$ , weight $\\\\lambda $ (or $\\\\bar{\\\\lambda }$ ) takes values $(0)$ or $(\\\\frac{1}{2})$ .\",\n \"The resulting bound on the neutral sector gap is shown as a function of $c$ in Fig.\",\n \"REF .By contrast with the previous subsection, here we find that the bound is exactly the same whether or not we allow extremal ($h^{(0)}=\\\\bar{h}^{(0)}=0$ ) states in the non-trivial representations.\",\n \"Notably, there is a minimum of about $\\\\Delta _* =1$ near $c \\\\equiv c^{(0)}+1 =3$ .\",\n \"We next turn to a more detailed study of this point.\",\n \"Figure: Left: The upper bound on the gap in the dimension ofprimaries, Δ * \\\\Delta _{*}, in the trivial representationobtained at increasing derivative order of the linear functional(from blue to red, up to n D =29n_D = 29).\",\n \"The black curve is theextrapolated value.\",\n \"Right: Blown-up plot of the-near minimalΔ * \\\\Delta _{*} region.\",\n \"The minimal isexpected at 2.00≤c≤2.042.00\\\\le c \\\\le 2.04,Δ * ≈0.995\\\\Delta _{*} \\\\approx 0.995.\"\n ],\n [\n \"Spin-Independent Analysis\",\n \"Our $q=\\\\bar{q}$ analysis in subsection (REF ) found a minimum gap bound at $c=3$ .\",\n \"Using the extremal functional method [19], the dimensions and the degeneracies of states at this point can be extracted, with numerical accuracy being best for the lowest dimension states.\",\n \"The dimensions of states occur at the zeros of the extremal functional, plotted in Fig.\",\n \"REF .\",\n \"Figure: Extremal functional of SU(2) k=1k=1, c=3c=3 spectrum.Furthermore, we find that the occupation numbers of the lowest-energy states of the partition function are uniquely determined to be $M_{(0),(0)}(\\\\beta ) &\\\\approx \\\\chi _{0}(\\\\beta ) + 28\\\\chi _{1}(\\\\beta ) + 76\\\\chi _{2}(\\\\beta ) +274 \\\\chi _{3}(\\\\beta )+\\\\ldots \\\\\\\\M_{(0),(\\\\frac{1}{2})}(\\\\beta ) =M_{(\\\\frac{1}{2}),(0)}(\\\\beta ) &\\\\approx 8\\\\chi _{0.5}(\\\\beta ) + 48\\\\chi _{1.5}(\\\\beta )+\\\\ldots \\\\\\\\M_{(\\\\frac{1}{2}),(\\\\frac{1}{2})}(\\\\beta ) &\\\\approx 8\\\\chi _{0.5}(\\\\beta ) + 48\\\\chi _{1.5}(\\\\beta )+\\\\ldots $ The subscript on $\\\\chi _{\\\\Delta ^{(0)}}$ denotes the non-Sugawara dimension of the state.\",\n \"At this point, the analysis takes $\\\\tau \\\\equiv \\\\frac{i \\\\beta }{2\\\\pi }$ to be pure imaginary, so no information on spins is used: $\\\\chi _{\\\\Delta ^{(0)} } = q^{-\\\\frac{c}{12}} \\\\left\\\\lbrace \\\\begin{array}{cc} \\\\prod _{n=2}^\\\\infty (1-q^n)^{-2}, & \\\\Delta ^{(0)}=0 , \\\\\\\\q^{\\\\Delta ^{(0)}} \\\\prod _{n=1}^\\\\infty (1-q^n)^{-2}, & \\\\Delta ^{(0)} >0 \\\\end{array} \\\\right.$ We find that a manifestly modular invariant partition function that reproduces this is $Z(\\\\beta )_{z=\\\\bar{z}} &=& \\\\frac{1}{2 \\\\eta ^6 } \\\\left( \\\\left( \\\\theta _3^2 - \\\\theta _4^2 \\\\right) \\\\theta _2^4(\\\\frac{z}{2}) + \\\\left( \\\\theta _2^2 + \\\\theta _4^2 \\\\right) \\\\theta _3^4(\\\\frac{z}{2}) + \\\\left( \\\\theta _3^2 - \\\\theta _2^2 \\\\right) \\\\theta _4^4(\\\\frac{z}{2}) \\\\right) \\\\\\\\&=& \\\\frac{1}{2 \\\\eta ^6} \\\\left( \\\\theta _2^2 \\\\theta _2^4(\\\\frac{z}{2}) + \\\\theta _3^2 \\\\theta _3^4(\\\\frac{z}{2}) + \\\\theta _4^2 \\\\theta _4^4(\\\\frac{z}{2}) + \\\\left( \\\\theta _3^2 - \\\\theta _2^2 - \\\\theta _4^2 \\\\right) \\\\theta _1^4(\\\\frac{z}{2}) \\\\right) ,$ where $\\\\theta _i\\\\equiv \\\\theta _i(z=0)$ .\",\n \"It is straightforward to check that the occupation numbers are non-negative, so that this partition function is unitary, modular invariant, and extremizes the scalar gap.\",\n \"Therefore (REF ) is the correct partition function by uniqueness.\"\n ],\n [\n \"Spin-Dependent Analysis\",\n \"In the previous subsection, we used extremal functional techniques to determine a unique partition function on the subspace $q=\\\\bar{q}$ when the gap in the scalar sector was maximized for $c=3$ .\",\n \"We can gain much more information about the theory by relaxing the constraint $q=\\\\bar{q}$ and varying $q,\\\\bar{q}$ independently; in particular, the analysis becomes sensitive to the spins $h-\\\\bar{h}$ of the spectrum.\",\n \"We could continue to use semi-definite programming methods, but they converge less quickly for independent $q,\\\\bar{q}$ than they do for $q=\\\\bar{q}$ .\",\n \"Instead, we can use the fact that we know the spectrum of dimensions from the $q=\\\\bar{q}$ analysis, and the fact that spin is quantized.\",\n \"This allows us to fix the allowed values of $h,\\\\bar{h}$ to a discrete set, turning the problem into a linear programming problem and thereby making the analysis much more efficient.\",\n \"There is a remaining ambiguity, however, which is that we have to make a choice about what spins are allowed.\",\n \"We find that if we allow only integer spins, there is no allowed partition function and in fact we can reduce the bound on the gap somewhat to about 2/3.\",\n \"If instead we allow fractional spins, then we find a few different possibilities depending on what spins we allow.\",\n \"We will begin with the conventional case where we allow integer and half-integer total spins, $h-\\\\bar{h}$ .\",\n \"The $SU(2)$ Sugawara characters are such that $M_{(0),(0)}$ only has integer spins, $M_{(0),(1/2)}$ and $M_{(1/2),(0)}$ only has quarter spins and $M_{(1/2),(1/2)}$ can have integer and half integer spins.\",\n \"Then to meet the requirement $M_{(0),(0)}$ and $M_{(1/2),(1/2)}$ can only have spins $\\\\frac{2n}{4}$ and $M_{(0),(1/2)}$ and $M_{(1/2),0}$ can only have spins $\\\\frac{2n+1}{4}$ .\",\n \"Performing the linear programming analysis for such a spectrum (and continuing to maximize the gap in the neutral sector) leads to the following unique set of weights and occupation numbers $d$ :The reader may notice that the numbers at each dimension in eq.\",\n \"(REF ) do not match the total number of states in the table.\",\n \"The reason is that without knowledge of spin, there are null states that could not be taken into account in (REF ).\",\n \"For instance, at level 2, there are a total of 84 states, as compared with 76 in (REF ), because of the 8 conserved currents at spin 1 that consequently have 8 “null” descendants at $\\\\Delta = 2$ .\",\n \"At $\\\\Delta \\\\ge 3$ , the presence of such null states causes states to get reorganized in increasingly complicated ways and it is easiest to check the number of states is the same by constructing the full partition function.\",\n \"Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThe non-Sugawara dimensions $\\\\Delta ^{(0)}$ and spins $\\\\ell ^{(0)}$ are just $\\\\Delta ^{(0)} = h^{(0)} + \\\\bar{h}^{(0)}, \\\\ell ^{(0)} = h^{(0)}- \\\\bar{h}^{(0)}$ .\",\n \"States are evenly divided between $\\\\ell ^{(0)} = + | \\\\ell ^{(0)}| $ and $\\\\ell ^{(0)} = - | \\\\ell ^{(0)}|$ at each weight, and the occupation numbers for the $(\\\\mu , \\\\bar{\\\\mu }) = (0, \\\\frac{1}{2})$ representations are the same as for $(\\\\frac{1}{2}, 0)$ .\",\n \"To get the full characters one multiplies the non-Sugawara Virasoro characters $\\\\chi _h(\\\\tau )$ (i.e.\",\n \"generated by the modes of $T^{(0)} = T- T^{(\\\\rm sug)}$ ) by the Weyl characters $\\\\chi _\\\\lambda ^{(k)}(\\\\tau ,z)$ , which in this case areSee eg [12], eqs (14.176) and (15.244).\",\n \"$\\\\chi _\\\\lambda ^{(1)}(\\\\tau ,z) = \\\\frac{1}{\\\\eta } \\\\sum _{m \\\\in \\\\mathbb {Z}+\\\\lambda } q^{m^2} y^m .$ After some trial and error, we find that the corresponding flavored partition function is reproduced by $Z(\\\\tau , \\\\bar{\\\\tau }, z, \\\\bar{z}) = \\\\frac{1}{4|\\\\eta |^{6}} \\\\sum _{a,b,a^{\\\\prime },b^{\\\\prime }=0}^1 (-1)^{a b^{\\\\prime } + a^{\\\\prime } b} \\\\left|\\\\theta \\\\left[^a_b\\\\right](\\\\tau ,\\\\frac{z}{2})\\\\right|^4 | \\\\theta [^{a^{\\\\prime }}_{b^{\\\\prime }}](\\\\tau ,0)|^2 ,$ in Jacobi/Erderlyi notation $\\\\theta _1 = \\\\theta [{1 \\\\atop 1}], \\\\theta _2 = \\\\theta [{1 \\\\atop 0}], \\\\theta _3 = \\\\theta [{0 \\\\atop 0}], \\\\theta _4 = \\\\theta [ { 0 \\\\atop 1}]$ .\",\n \"As this candidate partition function is half integrally modded, it is a little unfamiliar.\",\n \"A natural guess is that it arises as a $\\\\mathbb {Z}_{2}$ orbifold of a fully modular invariant theory.There is actually a history of extremal theories arising in such a fashion [28], [29], [30].\",\n \"Indeed it is possible to project this onto a fully modular invariant partition function.\",\n \"Taking the unflavored expression for simplicity, $ \\\\begin{split} Z^{\\\\rm (inv)}(\\\\tau , \\\\bar{\\\\tau })&=\\\\frac{1}{2}\\\\left(Z(\\\\tau , \\\\bar{\\\\tau }, 0,0)+Z(\\\\tau +1, \\\\bar{\\\\tau }+1, 0,0)+Z(-1/(\\\\tau +1), -1/(\\\\bar{\\\\tau }+1), 0,0)\\\\right) \\\\end{split} $ Unfortunately, we have not been able to identify a theory corresponding to (REF ).\",\n \"It is straightforward to check by exhausting the possibilities that the central charge and the number of spin-1 conserved currents (11) is not consistent with this partition function being associated with a pure Sugawara theory for some Lie algebra.\",\n \"While other choices for the quantization of spin are less conventional, they are still of some interest.For instance [31].\",\n \"Another possibility we have considered is that the non-Sugawara part of the spin, i.e.\",\n \"$h^{(0)}-\\\\bar{h}^{(0)}$ , is an integer or a half-integer.\",\n \"Because of the contribution to the weight from the Sugawara part of the stress tensor, in this case the states in the $(\\\\frac{1}{2}, 0)$ and $(0,\\\\frac{1}{2})$ representations have quarter-integer spins.\",\n \"Performing the linear programming analysis making this Ansatz for the spins, we find not just a unique solution but in fact a family of solutions given by the following occupation numbers: Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThese partition functions satisfy crossing for all $x$ .\",\n \"This one-parameter family is shown graphically in Fig.\",\n \"REF , where we perform the linear programming analysis with the degeneracy $d$ of the $(\\\\mu , \\\\bar{\\\\mu })= (\\\\frac{1}{2} ,0), (h^{(0)}, \\\\bar{h}^{(0)}) = (\\\\frac{1}{4}, \\\\frac{1}{4})$ chosen by hand and look at how several other degeneracies depend on this choice.\",\n \"By inspection of the above table, demanding that all occupation numbers be non-negative integers imposes $x\\\\in \\\\lbrace 0,1, \\\\dots , 4\\\\rbrace $ .\",\n \"It would be interesting to know if all or any of these partition functions correspond to underlying physical CFTs.\",\n \"Figure: The linear programming analysis finds a range of possible partition functions if we allow the physical spins to take quarter integer values.\",\n \"When we fix one of the degeneracies dd by hand, in this case that of the weight (μ,μ ¯,h (0) ,h ¯ (0) )=(1 2,0,1 4,1 4)(\\\\mu , \\\\bar{\\\\mu }, h^{(0)}, \\\\bar{h}^{(0)})= (\\\\frac{1}{2}, 0 , \\\\frac{1}{4}, \\\\frac{1}{4}), all other degeneracies become uniquely determined, so that we find a one-parameter family of solutions.\",\n \"The degeneracy on the x-axis here is 4+x4+x in the notation used in the text.The different values of $x$ here correspond to partition functions that have the same spectrum of dimensions $\\\\Delta = h + \\\\bar{h}$ , but which can be distinguished by their representation content, i.e.\",\n \"through the “flavored” partition function.They are similar in this respect to multiple different CFTs at $c=24$ that have the same spectrum but different underlying symmetries [32].\"\n ],\n [\n \"Constraints on Representation Content\",\n \"In this final subsection, we will consider the question of what representations are forced to be present in a theory.\",\n \"The gravitational AdS dual of any such constraints would imply that even if certain representations were not present among the perturbative degrees of freedom in some theory, they would have to be present non-perturbatively.\",\n \"The strongest condition one might try to prove is that all theories have all representations present.\",\n \"This would however be too ambitious since there are simple counter-examples, but one might still try to prove restrictive constraints on which representations can be absent.\",\n \"We will only be able to take a very modest step in this direction and prove some simple results for $SU(2)$ .\",\n \"For instance, without referring to numerical methods, we will prove that an $SU(2), k=1$ partition function either has all representations, or else its flavored partition function factorizes into a Sugawara theory partition function times a non-flavored partition function, assuming left-right symmetry.\",\n \"We begin by proving this $k=1$ result.\",\n \"The flavored partition function splits into four representations $M(\\\\tau , \\\\bar{\\\\tau }) = \\\\left(\\\\begin{array}{cc}M_{(0),(0)}(\\\\tau , \\\\bar{\\\\tau })& M_{(\\\\frac{1}{2}),(0)}(\\\\tau , \\\\bar{\\\\tau })\\\\\\\\M_{(\\\\frac{1}{2}),(0)}(\\\\tau , \\\\bar{\\\\tau })& M_{(\\\\frac{1}{2}),(\\\\frac{1}{2})}(\\\\tau , \\\\bar{\\\\tau })\\\\end{array}\\\\right) .$ Modular invariance requires $M(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}) = SM(\\\\tau , \\\\bar{\\\\tau })S~,$ with $S$ matrix $S = \\\\frac{1}{\\\\sqrt{2}} \\\\left(\\\\begin{array}{cc}1 & 1 \\\\\\\\1 & -1\\\\end{array}\\\\right)~.$ Because there are only two (assuming $(\\\\frac{1}{2},0)$ and $(0,\\\\frac{1}{2})$ are symmetric) different nontrivial representations, and the modular transformation manifestly forces at least one to be present, we can delete only one of them.\",\n \"What if we set $M_{(\\\\frac{1}{2}),(0)}(\\\\tau , \\\\bar{\\\\tau })=M_{(0),(\\\\frac{1}{2})}(\\\\tau ,\\\\bar{\\\\tau })=0$ ?\",\n \"In this case, the $(1,2)$ entry of matrix equation (REF ) is $\\\\frac{1}{2}\\\\left( M_{(0),(0)}(\\\\tau , \\\\bar{\\\\tau })-M_{(\\\\frac{1}{2}),(\\\\frac{1}{2})}(\\\\tau , \\\\bar{\\\\tau }) \\\\right) = 0 .$ The two diagonal representations have to be the same and therefore the “non-Sugawara” $\\\\tau $ -dependence of the flavored partition function is just an overall flavor-independent prefactor $M_{(0),(0)}(\\\\tau , \\\\bar{\\\\tau })$ that factors out.\",\n \"Similarly, if we set $M_{(\\\\frac{1}{2}, \\\\frac{1}{2})}(\\\\tau ,\\\\bar{\\\\tau })=0$ , then the $(2,2)$ entry of (REF ) is $M_{(\\\\frac{1}{2}),(0)}(\\\\tau ,\\\\bar{\\\\tau }) = \\\\frac{1}{2} M_{(0),(0)}(\\\\tau , \\\\bar{\\\\tau })$ , so again the non-Sugawara $\\\\tau $ -dependence factors out completely.\",\n \"In this case, the residual “Sugawara” matrix is just a symmetric holomorphic plus anti-holomorphic matrix, i.e.\",\n \"$\\\\left( \\\\begin{array}{cc} 2 & 1 \\\\\\\\ 1 & 0 \\\\end{array} \\\\right) = \\\\left( \\\\begin{array}{cc} 1 & 1 \\\\\\\\ 0 & 0 \\\\end{array} \\\\right) +\\\\left( \\\\begin{array}{cc} 1 & 0 \\\\\\\\ 1 & 0 \\\\end{array} \\\\right)$ .\",\n \"Beyond $SU(2)$ $k=1$ , similar arguments can be used to somewhat narrow down the possible combinations of representations in any partition function with non-abelian currents.\",\n \"Speficially, we can prove for SU(2) any $k$ , the partition function factorizes into a Sugawara partition function times a flavor-independent partition function if only diagonal representations are allowed.\",\n \"We again begin with the transformation rule $M(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}) = SM(\\\\tau , \\\\bar{\\\\tau })S~,$ where now the transformation matrix is $S_{(l) (l^\\\\prime )} = \\\\sqrt{\\\\frac{2}{k+2}} \\\\sin \\\\left(\\\\frac{\\\\pi }{k+2} (l+1)(l^\\\\prime +1)\\\\right) .$ Since we allow only diagonal representations, we can write $M_{(l) (r)}(\\\\tau , \\\\bar{\\\\tau }) &= \\\\delta _{(l) (r)} f_l , \\\\\\\\M_{(l) (r)}(-\\\\frac{1}{\\\\tau },-\\\\frac{1}{\\\\bar{\\\\tau }}) &= \\\\delta _{(l) (r)} g_l~,$ for some arbitrary functions $f_l$ and $g_l$ .\",\n \"The matrix equation can be written as $g_\\\\alpha S_{\\\\alpha \\\\beta }=S_{\\\\alpha \\\\beta } f_\\\\beta .$ For $\\\\beta = 0$ , $g_\\\\alpha \\\\sin \\\\left(\\\\frac{\\\\pi }{k+2} (\\\\alpha +1)\\\\right) =f_0 \\\\sin \\\\left(\\\\frac{\\\\pi }{k+2} (\\\\alpha +1)\\\\right)$ so $g_\\\\alpha = f_0$ for all $\\\\alpha $ unless $\\\\sin \\\\left(\\\\frac{\\\\pi }{k+2} (\\\\alpha +1)\\\\right) = 0$ .\",\n \"But the unitary bound $\\\\alpha 2$ .\",\n \"The very interesting recent work [33] on a sort of modular invariance for lens spaces in higher dimension is tantalizing from this point of view.\",\n \"Again, one would face the issue that correlators of currents in $d>2$ are not universal, but one could nevertheless try to obtain a constraint on the partition function in terms of the data in the $\\\\langle J(x_1) \\\\dots J(x_n)\\\\rangle $ correlators.\",\n \"Somewhat more abstractly, one of the appealing features of understanding the flavored partition function better is that, by turning on background fields, we are exploring constraints beyond the class of those that can be seen by inserting local operators.\",\n \"There are many such constraints on CFTs that are invisible in the standard bootstrap; the partition function itself can be though of as one such generalization since mapping to the torus (equivalently, inserting twist operators $\\\\sigma _2$ ) involves imposing new boundary conditions, and adding background fields is another kind of generalization.\",\n \"It would be very interesting to understand what additional constraints could be obtained by imposing crossing symmetry of correlators in the presence of background fields.\",\n \"Understanding the transformation law (REF ) as a statement about crossing symmetry for the four-point function $\\\\langle \\\\sigma _2 \\\\sigma _2 \\\\sigma _2 \\\\sigma _2\\\\rangle $ would be a useful warm-up case and could potentially give insight into how to think about more general correlators.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank Scott Collier, Shamit Kachru, Jared Kaplan, Emanuel Katz, and Per Kraus for useful conversations.\",\n \"ED was supported in part by the National Science Foundation under grant NSF-PHY-1316699 and by the Stanford Institute for Theoretical Physics and in part by the Simons Collaboration Grant 488655 on the Non-Perturbative Bootstrap.\",\n \"ALF and YX were supported in part by the US Department of Energy Office of Science under Award Number DE-SC-0010025, and ALF was supported in part by the Simons Collaboration Grant on the Non-Perturbative Bootstrap.\"\n ],\n [\n \"Path Integral Modular Transformation\",\n \"In this appendix, we review how diffeomorphism invariance and rigid rescalings imply the relation $Z_{\\\\rm PI}\\\\left( \\\\frac{a \\\\tau +b}{c\\\\tau +d}, \\\\frac{c z}{c \\\\tau +d} \\\\right) = Z_{\\\\rm PI}(\\\\tau ,z).$ We begin with invariance of the path integral measure: $d \\\\Psi e^{ - S_{\\\\tau _a, \\\\tau _b}[\\\\Psi ]} = d\\\\Psi ^{\\\\prime } e^{ - S_{\\\\tau ^{\\\\prime }_a, \\\\tau ^{\\\\prime }_b} [\\\\Psi ^{\\\\prime }]}.$ Here, $\\\\Psi $ are all the fields of the CFT, and to keep track of the torus before and after conformal transformations we have introduced $\\\\tau _a, \\\\tau _b$ for two of its corners (i.e.\",\n \"the four corners are at $0,\\\\tau _a, \\\\tau _b$ , and $\\\\tau _a+\\\\tau _b$ ).\",\n \"Under rescalings, the operators ${\\\\cal O}$ and parameters $\\\\tau _a, \\\\tau _b$ transform as ${\\\\cal O}(w,\\\\bar{w}) \\\\rightarrow {\\\\cal O}^{\\\\prime }(w,\\\\bar{w}) = \\\\lambda ^{-h} \\\\bar{\\\\lambda }^{-\\\\bar{h}}{\\\\cal O}(\\\\lambda ^{-1} w, \\\\bar{\\\\lambda }^{-1} \\\\bar{w}), \\\\qquad (\\\\tau _a, \\\\tau _b) \\\\rightarrow (\\\\tau _a^{\\\\prime },\\\\tau _b^{\\\\prime })= (\\\\lambda \\\\tau _a, \\\\lambda \\\\tau _b).$ In particular, for a conserved current $J^\\\\mu $ , we have $\\\\int _{\\\\tau _a, \\\\tau _b} dw d\\\\bar{w} J^w (w) = \\\\int _{\\\\tau _a, \\\\tau _b} dw d\\\\bar{w}\\\\Big ( \\\\bar{\\\\lambda } J^{\\\\prime w}(\\\\lambda w)\\\\Big ) = \\\\lambda ^{-1} \\\\int _{\\\\tau ^{\\\\prime }_a, \\\\tau ^{\\\\prime }_b} dw d\\\\bar{w} J^{\\\\prime w}(w) .$ and consequently $d \\\\Psi e^{ - S_{\\\\tau _a, \\\\tau _b} [\\\\Psi ] - \\\\frac{i}{2\\\\pi } \\\\int _{\\\\tau _a, \\\\tau _b} dw d\\\\bar{w} A_w J^w} = d \\\\Psi ^{\\\\prime } e^{- S_{\\\\tau ^{\\\\prime }_a, \\\\tau ^{\\\\prime }_b}[\\\\Psi ^{\\\\prime }] - \\\\frac{i}{2\\\\pi } \\\\int _{\\\\tau ^{\\\\prime }_a, \\\\tau ^{\\\\prime }_b} dw d\\\\bar{w} \\\\lambda ^{-1} A_w J^w }.$ Integrating both sides obtains the relation $(A_w, \\\\tau _a, \\\\tau _b) \\\\cong (\\\\lambda ^{-1} A_w, \\\\lambda \\\\tau _a, \\\\lambda \\\\tau _b)$ .\",\n \"To obtain the transformation under $U: \\\\tau \\\\rightarrow \\\\frac{\\\\tau }{\\\\tau +1}$ , we take $(\\\\tau _a, \\\\tau _b) = (\\\\tau , \\\\tau +1) \\\\cong (\\\\tau ,1),$ where the congruence $\\\\cong $ follows from a large diffeomorphism cutting the torus along the line from 1 to $\\\\tau +1$ and sewing it back to the line from $\\\\tau +1$ to $\\\\tau +2$ .\",\n \"By inspection of the chemical potential term $\\\\frac{1}{2\\\\pi i} \\\\int _{\\\\tau ,1} dw d\\\\bar{w} A_w J^w = 2 \\\\pi i {\\\\rm Im}(\\\\tau ) A_w \\\\bar{J}_0$ , we read off that $A_w = - i \\\\frac{\\\\bar{z}}{2 {\\\\rm Im}(\\\\tau )}.$ Finally, we take $\\\\lambda = (\\\\tau +1)^{-1}$ , so $(\\\\tau _a^{\\\\prime },\\\\tau _b^{\\\\prime })=(\\\\frac{\\\\tau }{\\\\tau +1}, 1)$ and $A_w^{\\\\prime } = (\\\\tau +1) A_w$ .\",\n \"Therefore, $\\\\bar{z}^{\\\\prime } = 2 i {\\\\rm Im}(\\\\tau ^{\\\\prime }) A_w^{\\\\prime } = \\\\frac{\\\\bar{z}}{\\\\bar{\\\\tau }+1}.$ The transformation under $T: \\\\tau \\\\rightarrow \\\\tau +1$ is trivial, since $\\\\tau $ and $\\\\tau +1$ are related by a large diffeomorphism without any need for a rescaling, so $\\\\lambda =1$ , and neither $A_w$ nor $z$ transform.\",\n \"All other modular transformations are generated from $T$ and $U$ .\"\n ],\n [\n \"A “Systematic” Treatment to Multivariate Problems\",\n \"The bootstrap of flavored partition function introduces another continuous quantum numbers $Q$ in addition to the scaling dimension of $\\\\Delta $ .\",\n \"Unlike the unflavored bootstrap where the problem is rigorously converted to a semidefinite programming problem, bootstrap problems with more than one variables do not have a simple and rigorous conversion to semidefinite programming problems.\",\n \"One can choose to discretize the second variable $Q$ and hope that the bound converges at very small $\\\\delta Q$ .\",\n \"However, the bound obtained in this way is not rigorous.\",\n \"The linear functional can be negative in between discrete $Q$ 's or at large enough $Q$ .\"\n ],\n [\n \"Multivariate Positive Definite Functionals\",\n \"Whether any real positive semidefinite polynomials (PSD) can be written as sum of squares of real polynomials (SOS) is known as the Hilbert's 17th problem.\",\n \"Hilbert himself proves the special case for univariate polynomials is true.\",\n \"But for multivariate polynomials it is later proven that PSD is a sum of squares of real rational functions.\",\n \"We do not like rational functions because we have much less numerical control over them than polynomials.\",\n \"Although we cannot find a clean SOS representation of multivariate PSD, if we only consider the subset of strictly positive polynomials we can still represent them by SOS in the following cases: Workaround 1: multiply by a common denominator $p(x_1,x_2)$ is positive definite polynomial (PD, also denote as $p(x_1,x_2) > 0$ ) then $p_g(x_1,x_2) = (1+x_1^2+x_2^2)^g p(x_1,x_2)$ is a sum of square of polynomial (SOS) for some $g$ .\",\n \"[34] Workaround 2: region is bounded For a compact region $\\\\mathcal {S}$ defined by $f_i(\\\\vec{x})\\\\ge 0$ over set of function $f_i$ , any polynomial strictly positive in $\\\\mathcal {S}$ can be written as the following form $p = \\\\sum _I s_I(\\\\vec{x}) f_{i_1} f_{i_2}\\\\ldots $ where $s_I(\\\\vec{x})$ are sum of squares.\",\n \"$I$ denotes some combinations of $f_i$ 's.\",\n \"[35] The hope is that PD can approximate PSD well enough so that in practise we can still resort to SOS.\",\n \"Numerically, solvers like SDPB never give nonnegative polynomials with exact zeros, so in practise we never actually encounter any counterexamples.\",\n \"Another reason to be hopeful is from the proof that PSD can be approximated as closely as desired by SOS [36].\",\n \"There is a possible loophole – the positive region of the polynomial has to be bounded.\",\n \"In unflavored case the region that is frequently used is $\\\\Delta >\\\\Delta _{\\\\star }$ , which is a rare special case of unbounded region.\",\n \"In practise, there is risk of not covering the full space of PD.\",\n \"Although not rigorously, one can hope that by multiplying the $(1+x_1^2+x_2^2)^g$ factors of higher and higher $g$ we lose less and less.\"\n ],\n [\n \"Multivariate Problems and SDPB\",\n \"In this subsection we discuss how to rewrite the semidefinite polynomial programming with 2 variables into a form suitable for SDPB [37] solver.\",\n \"SDPB solves univariate “Polynomial Matrix Program” (PMP) question stated as follows: $&\\\\text{maximize } y_0 + \\\\sum _n b_n y_n \\\\nonumber \\\\\\\\&\\\\text{such that } M_j^0(x) + \\\\sum _n y_n M_j^n(x) \\\\ge 0 \\\\nonumber \\\\\\\\&\\\\text{for all } x\\\\ge 0 \\\\text{ and } 1\\\\le j \\\\le J.$ where $M$ matrices are symmetric matrices of polynomials of $x$ .\",\n \"In SDPB, the PMP question is internally mapped to an SDP question since $ M_j^0(x) + \\\\sum _n y_n M_j^n(x) \\\\ge 0$ if and only if $M_j^0(x) + \\\\sum _n y_n M_j^n(x) =~ {\\\\rm tr}\\\\left[ Y_{A} Q_{A}(x) \\\\right] +x\\\\, {\\\\rm tr}\\\\left[ Y_{B} Q_{B}(x)\\\\right]$ for some $Y_A, Y_B \\\\ge 0$ .\",\n \"This equation is also the (2.8) of 1502.02033.\",\n \"We are instead trying to solve the problem for two variable cases.\",\n \"Here for modular bootstrap we are in the special case where the symmetric matrices $M_j$ are one by one, in other words, single polynomials $p_j$ .\",\n \"For simplicity here we only deal with this one dimensional case.\",\n \"Generalization to more dimensions and more variables is very easy.\",\n \"The question is stated as follows: $&\\\\text{maximize } y_0 + \\\\sum _n b_n y_n \\\\nonumber \\\\\\\\&\\\\text{such that } p_j^0(x_1,x_2) + \\\\sum _n y_n p_j^n(x_1,x_2) \\\\ge 0 \\\\nonumber \\\\\\\\&\\\\text{for all } x_1\\\\ge 0 \\\\text{ and all $x_2$ and } 1\\\\le j \\\\le J.$ Since SDPB only allows one variable to be bounded we cannot add more constraints on the variables.\",\n \"The $x_1>0$ is needed in SDPB because we usually choose the input $\\\\Delta \\\\ge \\\\Delta _*$ .\",\n \"The second variable can be the U(1) charge $Q$ , which is not constrained to be positive number.\",\n \"If one does want to bound the second variable one can make change of variable.\",\n \"We use the symbol $F_j$ to represent the linear functional $F_j(x_1,x_2) \\\\equiv p_j^0(x_1,x_2) + \\\\sum _n p_j^n(x_1,x_2) y_n$ Similar to the univariate case, we assume that $F_j\\\\ge 0$ is equivalent to finding $Y_{A,j},~Y_{B,j} \\\\ge 0$ , so that $F_j(x_1,x_2) =~ {\\\\rm tr}\\\\left[ Y_{A,j} Q_{A}(x_1,x_2) \\\\right] +x_1 {\\\\rm tr}\\\\left[ Y_{B,j} Q_{B}(x_1,x_2)\\\\right]$ Here we introduce “bilinear basis” $\\\\vec{q}(X)$ so that $Q(X) = \\\\vec{q}\\\\vec{q}^T$ spans the space of polynomials of $X$ .\",\n \"An easy example of bilinear basis is $\\\\vec{q}(x) = \\\\lbrace 1,x,x^2,\\\\ldots \\\\rbrace $ .\",\n \"The bilinear basis of two or more variables can be factored out as a kronecker product of bilinear bases of each single variables $Q_A (x_1,x_2) &= Q_{A1}(x_1) \\\\otimes Q_2(x_2) \\\\nonumber \\\\\\\\Q_B (x_1,x_2) &= Q_{B1}(x_1) \\\\otimes Q_2(x_2)$ We define $d_i$ to be the $x_i$ degree of the polynomial $F$ .\",\n \"the dimensions of the matrices $Q$ are $&{\\\\rm dim} Q_{A1} = \\\\delta _{A1} = [d_1/2]+1 \\\\nonumber \\\\\\\\&{\\\\rm dim} Q_{B1} = \\\\delta _{B1} = [(d_1-1)/2]+1 \\\\nonumber \\\\\\\\&{\\\\rm dim} Q_{2} = \\\\delta _{2} = [d_2/2]+1$ After factoring out $Q_A$ and $Q_B$ the function $F_j$ is written as $F_j(x_1,x_2) =~ {\\\\rm tr}\\\\left[ Y_{A,j}\\\\big ( Q_{A1}(x_1) \\\\otimes Q_2(x_2)\\\\big ) \\\\right] +x_1 {\\\\rm tr}\\\\left[ Y_{B,j} \\\\big ( Q_{B1}(x_1) \\\\otimes Q_2(x_2)\\\\big ) \\\\right]$ Since a polynomial is fixed if we know its value at $(d+1)$ different points, we can simply evaluate the above equation at $(d_2+1)$ values of $x_2$ in order to reduce the equation to have only one variable $x_1$We have not investigated what choices of the $x_{2,k}$ s are optimal.\",\n \"In practice, we have taken them to be $x_{2,k} = 2^{k-1}$ .\",\n \"$&F_{j,k}(x_1) = F_j(x_1,x_{2,k}) =~ {\\\\rm tr}\\\\left[ Y_{A,j}\\\\big ( Q_{A1}(x_1) \\\\otimes Q_2(x_{2,k})\\\\big ) \\\\right] +x_1 {\\\\rm tr}\\\\left[ Y_{B,j} \\\\big ( Q_{B1}(x_1) \\\\otimes Q_2(x_{2,k})\\\\big ) \\\\right]$ The above $(d_2+1)$ equations are equivalent to (REF ).\",\n \"In the following we omit the $j$ index because the same equation works for all $j$ .\",\n \"Now the form is already in single variable and is very close to the form of (REF ).\",\n \"The only difference is the numerical matrices $Q_2(x_{2,k})$ .\",\n \"Here we can play a trick by shuffling the $(d_2+1)$ equations with linear combination $&\\\\sum _l \\\\alpha _{kl} F_l = {\\\\rm tr}\\\\left[ Y_{A,j} \\\\big (Q_{A1}(x_1) \\\\otimes \\\\sum _l \\\\alpha _{kl} Q_2(x_{2,l})\\\\big ) \\\\right] + (\\\\text{B part})$ for some dimension $(d_2+1)$ square matrix $\\\\alpha _{kl}$ .\",\n \"In fact, the space of symmetric $Q_2$ matrices is only $(d_2+1= 2\\\\delta _2 -1)$ dimensional space since it spans the space $(d_2+1)$ dimensional polynomials.\",\n \"That means we can always find some $\\\\alpha _{kl}$ which picks up the the orthornormal basis of the polynomial space.\",\n \"Further we can perform an arbitrary $GL_{\\\\delta _2}$ transformation on $Q_2$ $Y &\\\\mapsto G YG^{-1} \\\\nonumber \\\\\\\\\\\\sum _l \\\\alpha _{kl} Q_2(x_{2,l}) &\\\\mapsto G^{-1} \\\\sum _l \\\\alpha _{kl}Q_2(x_{2,l}) G$ so that the orthornormal basis maps to the symmetric matrix basis $G^{-1} \\\\sum _l \\\\alpha _{kl} Q_2(x_{2,l}) G = E^{r(k) s(k)}$ where $E^{rs} = \\\\delta _i^r \\\\delta _j^s + \\\\delta _j^r \\\\delta _i^s$ .\",\n \"Then $\\\\sum _l \\\\alpha _{kl} F_l = {\\\\rm tr}\\\\left[ Y_A Q_{1A}(x_1) \\\\otimes E^{r(k)s(k)}\\\\right] + x_1{\\\\rm tr}\\\\left[ Y_B Q_{1B}(x_1) \\\\otimes E^{r(k)s(k)}\\\\right]$ Compared to (REF ), we can turn double variable programming of polynomial into single variable programming of symmetric polynomial matrices by substitution $M_j^0 &= \\\\sum _k \\\\sum _l E^{r(k) s(k)} \\\\alpha _{kl}P_j^0(x_1,x_{2,l})\\\\nonumber \\\\\\\\M_j^n &= \\\\sum _k \\\\sum _l E^{r(k) s(k)} \\\\alpha _{kl}P_j^n(x_1,x_{2,l})$ Since $Q_2$ span a $(d_2+1 = 2\\\\delta _2 -1)$ dimension space it means only the diagonal and next-to-diagonal elements will be nonzero.\",\n \"If further we only have even powers of $x_2$ , the matrices will be diagonal.\",\n \"The procedure defined from (REF ) to (REF ) is not the most efficient algorithm to obtain a single-variable matrix basis to input in (REF ).\",\n \"In practice, the algorithm we actually follow is computationally more straightforward, and is as follows.\",\n \"We can rewrite (REF ) as $M_j(x_1)_{tu} =Y_{A,j}^{rs,tu}\\\\, Q_{A1}(x_1)_{rs} + (\\\\text{B parts}) ~,$ where $r,s,t,u$ are matrix indices.\",\n \"Given $M_j(x_1)$ , SDPB can optimize this single-variable problem.\",\n \"Rather than obtaining $M_j(x_1)$ by the procedure outlined above, we can instead start directly from equation (REF ), which says $F_{j,k}(x_1) = Y_{A,j}^{rs,tu}\\\\, Q_{A1}(x_1)_{rs}\\\\,Q_2(x_{2,k})_{tu} + (\\\\text{B parts}) ~.$ Combining the two equations $F_{j,k}(x_1) = M_j(x_1)^{tu}\\\\, Q_2(x_{2,k})_{tu} ~.$ The point is that both $F_{j,k}(x_1)$ and $Q_2(x_{2,k})_{tu} $ are known, so $M$ can be obtained by solving the above equation, after which it can be fed into SDBP.\",\n \"Concretely, first flatten the $(t,u)$ indices $ \\\\alpha := (t_\\\\alpha ,u_\\\\alpha )$ to put $M_{tu}$ into $(\\\\delta _2+1)(\\\\delta _2+2)/2$ dimensional array ${\\\\bf M}_\\\\alpha $ and $Q_{k,tu}$ into $(d_2+1)\\\\times (\\\\delta _2+1)(\\\\delta _2+2)/2$ array $Q_{k,\\\\alpha }$ .\",\n \"Then solve the linear equations for ${\\\\bf M}_\\\\alpha $ .\",\n \"To be concrete, in this paper we explicitly choose the flatten map ${\\\\bf M} := \\\\big (\\\\text{diag of $M$},~\\\\text{next-to-diag},~\\\\cdots \\\\big )~.$ A specific solution can be found by taking the SVD decomposition of $Q$ , $Q = U \\\\, W \\\\, V^T$ where diagonal matrix $W = \\\\left( \\\\ \\\\widetilde{W} \\\\quad O \\\\ \\\\right)$ has the same rank $(d_2+1)$ as $Q$ .\",\n \"Then take $B^{T} = V \\\\, W^\\\\prime \\\\, U^T$ where $W^\\\\prime = \\\\left(\\\\begin{array}{c}\\\\widetilde{W}^{-1} \\\\\\\\ O\\\\end{array} \\\\right)$ is the pseudo inverse of $W$ .\",\n \"Contract $B^T$ both sides of (REF ), $B^T {\\\\bf F} &= V \\\\, W^\\\\prime \\\\, U^T Q {\\\\bf M}= V \\\\, W^\\\\prime \\\\, U^T U \\\\, W \\\\, V^T {\\\\bf M}= V \\\\, \\\\left(\\\\begin{array}{cc}I_{d_2+1}& O \\\\\\\\ O & O\\\\end{array}\\\\right) \\\\, V^T {\\\\bf M}$ where it's useful to block-decompose the unitary matrix $V$ as $V = \\\\left(\\\\begin{array}{cc}V_{11}&V_{12}\\\\\\\\ V_{21}&V_{22}\\\\end{array}\\\\right)~.$ Continue to simplify the right hand side, $V^TB^T{\\\\bf F} &= \\\\left(\\\\begin{array}{cc}V_{11}&V_{12}\\\\\\\\ O & O\\\\end{array}\\\\right) {\\\\bf M} \\\\nonumber \\\\\\\\\\\\big ( (V_{11}^T)^{-1} \\\\ O \\\\big )V^TB^T{\\\\bf F} &= \\\\left( \\\\ I_{d_2+1} \\\\quad (V_{11}^T)^{-1}V_{21}^T \\\\ \\\\right){\\\\bf M} ~.$ From (REF ) we know that there are only $(d_2+1)$ independent elements are unique.\",\n \"Any dependent element can be removed by a $GL_{\\\\delta _2}$ transformation defined by REF .\",\n \"We choose ${\\\\bf M}$ such that only the first $(d_2+1)$ elements are zero, and the solution can be written as ${\\\\bf M} = \\\\big ( (V_{11}^T)^{-1} \\\\ O \\\\big )V^TB^T{\\\\bf F}~.$ Since $F_j(x_1,x_2)$ is the linear functional of input polynomials defined by (REF ).\",\n \"Our single variable input matrices should be substituted in the same way, leading to ${\\\\bf M}_{j}^n &= \\\\big ( (V_{11}^T)^{-1} \\\\ O \\\\big )V^TB^T{\\\\bf P}_{j}^n \\\\nonumber \\\\\\\\{\\\\bf M} &\\\\equiv {\\\\bf M}_{j}^0 \\\\, y_0 + \\\\sum _n {\\\\bf M}_{j}^n \\\\, y_n\\\\nonumber \\\\\\\\\\\\left( {\\\\bf P}_{j}^n \\\\right)_k &\\\\equiv {\\\\rm Flatten}\\\\left( p_j^n(x_1,x_{2,k}) \\\\right)$ Finally, we inverse (REF ) to put flattened array ${\\\\bf M}_j^n$ 's back into matrix $M_j^n$ 's.\"\n ],\n [\n \"$k=2, SU(2)$ Analysis\",\n \"Here we present some preliminary results on our methods applied to the group $SU(2)$ at level $k=2$ .\",\n \"Our results are qualitatively similar to the $k=1$ case, though with worse numeric accuracy due to the slower convergence.\",\n \"In Fig.\",\n \"REF , we show the bound on the gap $\\\\Delta _*$ to the lightest neutral state in the theory, which is minimized to be $\\\\Delta _* \\\\approx 1.344$ at $c \\\\approx 2.715$ .\",\n \"Unfortunately, at the point where the bound is minimized, the occupation numbers from our analysis for some of the lowest few states are not particularly close to integers.\",\n \"It is not clear whether this indicates that such a point is not associated with an underlying CFT or if we simply have not converged to sufficient precision.\",\n \"The occupation numbers for the lightest neutral state and charged state are shown as a function of $c$ in Fig.\",\n \"REF .\",\n \"The lightest neutral state is close to $d=74$ , however the lightest charged state, which is even lighter is relatively far from the nearest integer, $d \\\\approx 7$ .\",\n \"Another possibility is that one ought to maximize the gap in not only the neutral sector but also in one or more charged sectors; it would be interesting to pursue this or other conditions further.\",\n \"Figure: Bound on the gap Δ * \\\\Delta _* to the lightest neutral state for SU(2)SU(2) at level k=2k=2.\",\n \"The bound is minimized at c≈2.715c \\\\approx 2.715.Figure: Occupation numbers for the lightest neutral (left) and charged (right) states from the extremal functional analysis with SU(2)SU(2) at k=2k=2 as a function of cc.\",\n \"The optimal bound is at c≈2.715c\\\\approx 2.715, indicated by a vertical line; horizontal lines are shown at integers.s\"\n ]\n]"},"id":{"kind":"string","value":"1709.01533"}}},{"rowIdx":695,"cells":{"text":{"kind":"list like","value":[["Depression and Self-Harm Risk Assessment in Online Forums"],["Abstract Users suffering from mental health conditions often turn to online resources for support, including specialized online support communities or general communities such as Twitter and Reddit.","In this work, we present a neural framework for supporting and studying users in both types of communities.","We propose methods for identifying posts in support communities that may indicate a risk of self-harm, and demonstrate that our approach outperforms strong previously proposed methods for identifying such posts.","Self-harm is closely related to depression, which makes identifying depressed users on general forums a crucial related task.","We introduce a large-scale general forum dataset (\"RSDD\") consisting of users with self-reported depression diagnoses matched with control users.","We show how our method can be applied to effectively identify depressed users from their use of language alone.","We demonstrate that our method outperforms strong baselines on this general forum dataset."],["Introduction","Mental health remains a major challenge in public health care.","Depression is one of the most common mental disorders and 350 million people are estimated to suffer from depression worldwide [49].","In 2014 an estimated 7% of all U.S. adults had experienced at least one major depressive disorder depression2015.","Suicide and self-harm are major related concerns in public mental health.","Suicide is one of the leading causes of death [7], and each suicide case has major consequences on the physical and emotional well-being of families and on societies in general.","Therefore identifying individuals at risk of self-harm and providing support to prevent it remains an important problem [19].","Social media is often used by people with mental health problems to express their mental issues and seek support.","This makes social media a significant resource for studying language related to depression, suicide, and self-harm, as well as understanding the authors' reasons for making such posts, and identifying individuals at risk of harm [11].","Depression and suicide are closely related given that depression is the psychiatric diagnosis most commonly associated with suicide.","Research has demonstrated that forums are powerful platforms for self-disclosure and social support seeking around mental health concerns [16], [33].","Such support forums are often staffed by moderators who are mental health experts, trained volunteers, or more experienced users whose role is to identify forum posts suggesting that a user is at risk of self-harm and to provide support.","Studies have shown that self expression and social support are beneficial in improving the individual's state of the mind [47], [8] and, thus such communities and interventions are important in suicide prevention.","However, there are often thousands of user posts published in such support forums daily, making it difficult to manually identify individuals at risk of self-harm.","Additionally, users in acute distress need prompt attention, and any delay in responding to these users could have adverse consequences.","Therefore, identifying individuals at risk of self-harm in such support forums is an important challenge.","Identifying signs of depression in general social media, on the other hand, is also a difficult task that has applications for both better understanding the relationship between mental health and language use and for monitoring a specific user's state (e.g., in the context of monitoring a user's response to clinical care).","In this work we propose and evaluate a framework for performing self-harm risk assessment and for identifying depression in online forums.","We present a general neural network architecture for combining posts into a representation of a user's activity that is used to classify the user.","To address the challenge of depression risk assessment over the general forums, we introduce a large-scale novel Reddit dataset that is substantially larger than the existing data and has a much more realistic number of control users.","The dataset contains over 9,000 users with self-reported depression diagnoses matched with over 107,000 control users.","We apply our approach to (1) identify the users with depression on a general forum like Reddit, and to (2) estimate the risk of self-harm indicated by posts in a more specific mental-health support forum.","Our methods perform significantly better on both datasets than strong existing methods, demonstrating that our approach can be used both to identify depressed users and to estimate the risk of self-harm posed by individual posts."],["Related Work","There is a growing body of related work analyzing mental health-related discourse and language usage in social media to better discover and understand mental health related concerns [41], [17], [14], [11], [35], [46], [12], [1], [36], [4].","To investigate NLP methods for identifying depression and PTSD users on Twitter, a shared task [13] at the 2nd Computational Linguistics and Clinical Psychology Workshop (CLPsych 2015) was introduced where the participants evaluated their methods on a dataset of about 1800 Twitter users.","Other work has used data from approximately 900 Reddit.com users to support self-reported diagnosis detection [31].","Previous work identifying depression and other mental health problems, including the methods participating in CLPsych 2015 (e.g.","works by [40] and [39]) heavily rely on utilizing features such as LIWC [38], topic modeling, manual lexicons, or other domain-dependent application-specific features.","Aside from the effort required to design effective features, these approaches usually model the problem with respect to the selected features and ignore other indicators and signals that can improve prediction.","In contrast, our model only relies on text and is not dependent on any external or domain-specific features.","Existing self-reported diagnosis detection datasets contain a limited number of both control users and diagnosed users.","In contrast to this, we construct a new dataset with over 9,000 depressed users matched with a realistic number of control users.","In addition to general studies addressing mental health, related work has also specifically studied suicide and self-harm through social media [23], [45], [20], [18], [15].","Recently, CLPsych 2016 [21] investigated approaches for detecting the self-harm risk of mental health forum posts [34].","Most related work in this area uses variations of linear classifiers with some sort of feature engineering; successful methods have employed: a combination of sparse (bag-of-words) and dense (doc2vec) representation of the target forum posts [26], a stack of feature-rich Random Forest and linear Support Vector Machine (SVM) [32], an RBF SVM classifier utilizing similar sets of features [6], and various contextual and psycholinguistic features [9], [10].","In contrast to the above works, our model does not use any general or domain specific feature engineering; it learns appropriate representations of documents by considering only their textual content.","Our proposed models consist of a shared architecture based on a CNN, a merge layer, model-specific loss functions, and an output layer (as we will describe in §).","While our model shares similarities with CNN-based models in prior work [25], [27], [50], it focuses on learning representations of user's posts and combining the post representations into an overall representation of the user's activity.","In the case of self-harm risk assessment, we experiment with several loss functions to determine whether considering the ordinal nature of self-harm risk labels (i.e., green, amber, red, and crisis) can improve performance.","Evaluation results suggest that the model variant using this loss function is more robust than our other variants."],["Depression dataset construction.","We created a new dataset to support the task of identifying forum users with self-reported depression diagnoses.","The Reddit Self-reported Depression Diagnosis (RSDD) dataset was created by annotating users from a publicly-available Reddit datasethttps://files.pushshift.io/reddit/.","Users to annotate were selected by identifying all users who made a post between January 2006 and October 2016 matching a high-precision diagnosis pattern.e.g., “I was just diagnosed with depression.” Users with fewer than 100 posts made before their diagnosis post were discarded.","Each of the remaining diagnosis posts was then viewed by three layperson annotators to decide whether the user was claiming to have been diagnosed with depression; the most common false positives included hypotheticals (e.g., “if I was diagnosed with depression”), negations (e.g., “it's not like I've been diagnosed with depression”), and quotes (e.g., “my brother announced `I was just diagnosed with depression' ”).","Only users with at least two positive annotations were included in the final group of diagnosed users.","A pool of potential control users was identified by selecting only those users who had (1) never posted in a subreddit related to mental health, and (2) never used a term related to depression or mental health.","These restrictions minimize the likelihood that users with depression are included in the control group.","In order to prevent the diagnosed users from being easily identified by the usage of specific keywords that are never used by the control users, we removed all posts by diagnosed users that met either one of the aforementioned conditions (i.e., that was posted in a mental health subreddit or included a depression term).","For each diagnosed user and potential control user, we calculated the probability that the user would post in each subreddit (while ignoring diagnosed users' posts made to mental health subreddits).","Each diagnosed user was then greedily matched with the 12 control users who had the smallest Hellinger distance between the diagnosed user's and the control user's subreddit post probability distributions, excluding control users with 10% more or fewer posts than the diagnosed user.","This matching approach ensures that diagnosed users are matched with control users who are interested in similar subreddits and have similar activity levels, preventing biases based on the subreddits users are involved in or based on how active the users are on Reddit.","This yielded a dataset containing 9,210 diagnosed users and 107,274 control users.","On average each user in the dataset has 969 posts (median 646).","The mean post length is 148 tokens (median 74).","The Reddit Self-reported Depression Diagnosis (RSDD) dataset differs from prior work creating self-reported diagnoses datasets in several ways: it is an order of magnitude larger, posts were annotated to confirm that they contained claims of a diagnosis, and a realistic number of control users were matched with each diagnosed user.","The lists of terms related to mental health, subreddits related to mental health, high-precision depression diagnosis patterns, and further information are availablehttp://ir.cs.georgetown.edu/data/reddit_depression/.","We note that this dataset has some (inevitable) caveats: (i) the method only captures a subpopulation of depressed people (i.e.","those with self-reported diagnosis), (ii) Reddit users may not be a representative sample of the population as a whole, and (iii) there is no way to verify whether the users with self-reported diagnoses are truthful."],["Self-harm assessment.","For self-harm risk assessment we use data from mental health forum posts from ReachOut.com, which is a successful Australian support forum for young people.","In addition to providing peer-support, ReachOut moderators and trained volunteers monitor and participate in the forum discussions.","The NAACL 2016 Computational Linguistics and Clinical Psychology Workshop [21] released a Triage dataset containing 65,024 forum posts from ReachOut, with annotations for 1,227 posts indicating the author's risk of self-harm [34].","The annotations consist of one of four labels: green (indicating no action is required from ReachOut's moderators), amber (non-urgent attention is required), red (urgent attention is required), and crisis (a risk that requires immediate attention)."],["Ethical concerns.","Social media data are often sensitive, and even more so when the data are related to mental health.","Privacy concerns and the risk to the individuals in the data should always be considered [22], [44], [3].","We note that the risks associated with the data used in this work are minimal.","This assessment is supported by previous work on the ReachOut dataset [34], on Twitter data [13], and on other Reddit data [31].","The RSDD dataset contains only publicly available Reddit posts.","Annotators were shown only anonymized posts and agreed to make no attempts to deanonymize or contact them.","The RSDD dataset will only be made available to researchers who agree to follow ethical guidelines, which include requirements not to contact or attempt to deanonymize any of the users.","Additionally, for the ReachOut forum data that was explicitly related to mental health, the forum's rules require the users to stay anonymous; moderators actively redact any user identifying information.","Figure: The general neural network architecture shared among our user and post classification models.","Each input (e.g., each of a user's posts) is processed by a convolutional network and merged to create a vector representation of the user's activity.","This vector representation is passed through one or more dense layers followed by an output layer that performs classification.","The type of input received, merge operation, and output layer vary with the specific model."],["Methodology","We describe a general neural network architecture for performing text classification over multiple input texts.","We propose models based on this architecture for performing two tasks in the social media and mental health domains that we call self-harm risk classification and detecting depression.","The task of self-harm risk classification is estimating a user's current self-harm risk given the user's post on a mental health support forum and the previous posts in the thread.","The task of detecting depressions in users is identifying Reddit users with self-reported depression diagnoses given the users' post histories (excluding posts containing mental health keywords or posted in subreddits related to mental health).","While both tasks are focused on predicting a user's mental health status, they differ in both the type of classification performed (i.e., estimating severity on a four point scale vs. boolean classification) and in the amount of data available.","Our general architecture is based on a two step process: (1) identifying relevant features in each input text, and (2) combining the features observed in the model's inputs to classify the user.","Figure: The convolutional network component of our architecture.","A convolutional layer takes a series of terms as input (a) and applies ll filters to a kk-term sliding window to derive feature values for each window or region (b); k=2k=2 and l=3l=3 shown here.","A max pooling layer considers non-overlapping region sequences of length nn (b) and keeps the highest feature value for the sequence (c); n=3n=3 shown here."],["Shared Architecture","Our proposed models share a common architecture that takes one or more posts as input, processes the posts using a convolutional layer to identify features present in sliding windows of text, merges the features identified into a vector representation of the user's activity, and uses a series of dense layers to perform classification on the merged vector representation.","The type of merging performed and the output layers are properties of the model variant, which we describe in detail in the following section.","Convolutional networks have commonly been applied to the task of text classification, such as by Kim:2014.","We use categorical cross-entropy as a loss function with both methods, but also experiment with other loss functions when performing severity classification.","First, the model takes one or more posts as input and processes each post with a convolutional network containing a convolutional layer and a pooling layer.","This process is illustrated with a max pooling layer in Figure REF .","The convolutional layer applies filters to a sliding window of $k$ terms (a) and outputs a feature value for each sliding window region and each filter (b).","The same filters are applied to each window; each filter can be viewed as a feature detector and the overall process can be conceptualized as looking for windows of terms that contain specific features.","The features are not specified a priori through feature engineering, but instead are learned automatically when the model is trained.","After identifying the features present in each region (i.e., sliding window), a max pooling layer considers non-overlapping regions of length $n$ and keeps the highest feature value for each region (c).","This step eliminates the regions (i.e., sliding windows) that do not contain useful features, which reduces the size of the convolutional network's output.","The same convolutional network is applied to each input post, meaning that the model learns to look for the same set of features in each.","After each input post has been processed by a convolutional network, the output of each convolutional network is merged to create a representation of the user's activity across all input posts.","This representation is processed by one or more dense layers (i.e., fully connected layers) with dropout [42] before being processed by a final output layer to perform classification.","The type of output layer is dependent on the model variant.","Our shared model architecture is illustrated in Figure REF .","The architecture's hyperparameters (e.g., the sliding window size $k$ , the number of convolutional filters used, and type of pooling) also vary among models and are described in §.","Both the convolutional and dense layers use ReLU activations [37] in all model variants.","Table: The hyperparameters used by each model."],["Depression detection","Our model for depression detection takes a user's posts as input and processes each post with a convolutional network.","Each convolutional network performs average pooling to produce its output.","These post representations are then merged with a second convolutional layer to create a user representation; we found this approach led to more stable performance than using a second average pooling or max pooling layer.","The user representation created by the merge step is then passed to one or more dense layers before being passed to a dense output layer with a softmax activation function to perform classification.","The number of dense layers used is a hyperparameter described in §.","Categorical cross-entropy is used as the model's loss function."],["Self-harm risk assessment","Our model for self-harm risk classification takes two inputs: the target post being classified and the prior posts (if any) in the target post's thread.","The prior posts provide context and are thus useful for estimating the risk of self-harm present in the target post.","The two inputs are both processed by a convolutional network as in user-level classification, but in this case the convolutional network's outputs correspond to a representation of the target post and to a representation of the target post's context (i.e., the prior posts in the thread).","Given that these two outputs represent different aspects, they are merged by concatenating them together.","This merged representation is then passed to one or more dense layers and to an output layer; the type of output layer depends on the loss function used.","There are four self-harm risk assessment model variants in total: Categorical Cross Ent.","uses an output layer with a softmax activation function, and categorical cross-entropy as its loss function.","This mirrors the output layer and loss function used in the user level classification model.","MSE uses an output layer with a linear activation function, and mean squared error as its loss function.","The model's output is thus a single value; to perform classification, this output value is rounded to the nearest integer in the interval $[0, t - 1]$ , where $t$ is the number of target classes.","The final two loss functions perform metric learning rather than performing classification directly.","They learn representations of a user's activity and of the four self-harm risk severity labels; classification is performed by comparing the euclidean distance between a representation of a user's activity (produced by the final layer) and each of the four severity label representations.","Class Metric: Let $d$ be the size of the output layer and $X$ be the layer's $d$ -dimensional output.","Class Metric learns a $d$ -dimensional representation of each class $C_i$ such that $||X - C_i||_2$ is minimized for the correct class $i$ ; this is accomplished with the loss function: $ L_{i,p,n} = \\max \\big (0, ||X_i - C_p||_2 - ||X_i - C_n||_2 + \\alpha \\big ) $ where $C_p$ is the correct (i.e., positive) class for $X_i$ , $C_n$ is a randomly chosen incorrect (i.e., negative) class, and $\\alpha $ is a constant to enforce a minimum margin between classes.","Classification is performed by computing the similarity between $X_i$ and each class $C_j$ .","Class Metric (Ordinal) extends Class Metric to enforce a margin between ordinal classes as a function of the distance between classes.","Given a ranked list of classes such that more similar classes have closer rankings, that is $\\forall i$ $sim(C_i, C_{i \\pm 1}) > sim(C_i, C_{i \\pm 2})$ , we incorporate the class distance into the margin such that more distant incorrect class labels must be further away from the correct class label in the metric space.","The loss function becomes $ L_{i,p,n} = \\max \\big (0, \\;\\; ||X_i - C_p||_2 \\;\\; - \\\\||X_i - C_n||_2 + \\alpha |p-n| \\big )$ where $|p-n|$ causes the margin to scale with the distance between classes $p$ and $n$ ."],["Experiments","In this section, we describe the model hyperparameters used and present our results on the depression detection and self-harm risk assessment tasks.","To facilitate reproducibility we provide our code and will provide the Reddit depression dataset to researchers who sign a data usage agreementhttp://ir.cs.georgetown.edu/data/reddit_depression/."],["Experimental setup.","The hyperparameters used with our models are shown in Table REF .","The severity risk assessment models' hyperparameters were chosen using 10-fold cross validation on the 947 ReachOut training posts, with 15% of each fold used as validation data.","The depression identification model's hyperparameters were chosen using the Reddit validation set.","The depression identification model's second convolutional layer (i.e., the layer used to merge post representations) used filters of length 15, a stride of length 15, and the same number of filters as the first convolutional layer.","All models were trained using stochastic gradient descent with the Adam optimizer [28].","The hyperparameters that varied across models are shown in Table REF .","The convolution size, number of convolutional filters, pooling type, pooling length, and number of dense layers was similar across all post models.","Class balancing was performed with Categorical Cross Ent.","by weighting classes inversely proportional to their frequencies, whereas sampling an equal number of instances for each class worked best with the other methods."],["Addressing limited data.","The post classification models' input consists of skip-thought vectors [29]; each vector used is a 7200-dimensional representation of a sentence.","Thus, the convolutional windows used for post classification are over sentences rather than over terms.","This input representation was chosen to mitigate the effects of the ReachOut dataset's relatively small size.","The skip-thought vectors were generated from the the ReachOut forum dataset by sequentially splitting the posts in the training set into sentences, tokenizing them, and training skip-thoughts using Kiros et al.","'s implementation with the default parameters.","Sentence boundary detection was performed using the Punkt sentence tokenizer [30] available in NLTK [5].","These 2400-dimensional forum post skip-thought vectors were concatenated with the 4800-dimensional book corpus skip-thought vectors available from [29].","Experiments on the training set indicated that using only the ReachOut skip-thought vectors slightly decreased performance, while using only the book corpus skip-thought vectors substantially decreased performance.","As input the post models received the last 20 sentences in each target post and the last 20 sentences in the thread prior to the target post; any prior sentences are ignored."],["Depression detection.","The data used for depression detection was described in §.","As baselines we compare our model against the FastText classifier [24] and MNB and SVM classifiers [48] using features from prior work.","We tune FastText's hyperparameters on the validation set.","Specifically, we consider a maximum n-gram size $\\in [1,2,3,4,5]$ , an embedding size $\\in [50,100, 150]$ , and a learning rate $\\in [0.05, 0.1, 0.25, 0.5]$ as suggested in the documentation.","We consider two sets of features for the MNB and SVM classifiers.","The first set of features is the post content itself represented as sparse bag of words features (BoW baselines).","The second set of features (feature-rich baselines) comprises a large set of features including bag of words features encoded as sparse weighted vectors, external psycholinguistic features captured by LIWChttp://liwc.wpengine.com/ pennebaker2015development, and emotion lexicon features [43].","Since our problem is identifying depression among users, psycholinguistic signals and emotional attributes in the text are potentially important features for the task.","These features (as described in §) have been also previously used by successful methods in the Twitter self-reported diagnosis detection task [13].","Thus, we argue that these are strong baselines for our self-reported diagnosis detection task.","We apply count based and TF-IDF based feature weighting for bag of words features.","We perform standard preprocessing by removing stopwords and lowercasing the input text.During experimentation, we found TF-IDF sparse feature weighting to be superior than other weighting schemes.","Additional features such as LDA topics and $\\chi ^2$ feature selection did not result in any further improvements.","The data is split into training, validation, and testing datasets each containing approximately 3,000 diagnosed users and their matched control users.","The validation set is used for tuning development and hyperparameter tuning of our models and the baselines.","The reported results are on the test set.","The depression detection models' input consisted of raw terms encoded as one-hot vectors.","We used an input layer to learn 50-dimensional representation of the terms.","For each target user, the CNN received up to $n_{post}$ posts containing up to $n_{term}$ terms.","In this section we present results for two values of $n_{post}$ .","The earliest post approach (CNN-E) takes each user's $n_{post}=400$ earliest posts as input.","The random approach (CNN-R) samples $n_{post}=1500$ random posts from each user.","We empirically set $n_{term}=100$ with both approaches.","We later analyze the model's performance as $n_{post}$ and $n_{term}$ vary in §REF and as the post selection strategy varies in §REF .","Table: Performance identifying depressed users on the Reddit test set.The differences between the CNN and baselines are statistically significant (McNemar's test, p<0.05p<0.05).Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 test set.Results for the other methods are from .Differences in performance between the following pairs are statistically significant (McNemar's test, p<0.05p<0.05):Categorical Cross Ent.","and Class Metric, MSE and Categorical Cross Ent., MSE and Class Metric (Ordinal), and Class Metric (Ordinal) and Class Metric.Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 training set using 10-fold cross validation.","Categorical Cross Ent.","performs substantially worse than on the test set, while MSE performs substantially better.","Class Metric (Ordinal) continues to perform well.","The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.","and MSE, Categorical Cross Ent.","and Class Metric, Categorical Cross Ent.","and Class Metric (Ordinal), MSE and Class Metric, and Class Metric and Class Metric (Ordinal).Results.","The results of identifying depressed users for our model and baselines are shown in Table REF .","Our proposed model outperforms the baselines by a large margin in terms of recall and F1 on the diagnosed users (increases of 41% and 16%, respectively), but performs worse in terms of precision.","As described later in the analysis section, the CNN identifies language associated with negative sentiment across a user's posts.","Figure: Sensitivity of the CNN-R model to the parameters n posts n_{posts} (a) and n terms n_{terms} (b) on RSDD's validation set.","F1 increases as n posts n_{posts} does (a), but the rate of increase slows as n posts n_{posts} surpasses 1000.The trend for n terms n_{terms} is less clear (b), but the highest F1 is achieved at n terms =100n_{terms}=100.In Figure (a) the parameter n terms n_{terms} was fixed to 100, and in Figure (b) n posts n_{posts} was fixed to 1500.Figure: Empirical cumulative distribution functions (CDF) of the number of posts per user (a) and the post length (b) in the RSDD dataset."],["Self-harm risk classification.","We train our methods to label the ReachOut posts and compare them against the top methods from CLPsych '16.","We use the same experimental protocol as was used in CLPsych '16; our methods were trained on the 947 training posts and evaluated on the remaining 280 testing posts.","We used 15% of the 947 training posts as validation data.","We report results using the same metrics used in CLPsych, which were: the macro-averaged F1 for the $amber$ , $red$ , and $crisis$ labels (non-green posts); the macro-averaged F1 of $ green $ posts vs. $ amber \\cup red \\cup crisis $ (flagged posts); and the macro-averaged F1 of $ green \\cup amber $ vs. $ red \\cup crisis $ (urgent posts).","The non-green F1 was used as the official CLPsych metric with the intention of placing emphasis on classification performance for the non-green categories (i.e., those that required some response).","The binary flagged meta-class was chosen to measure models' abilities to differentiate between posts that require attention and posts that do not, and the binary urgent meta-class was chosen to measure their abilities to differentiate between posts that require quick responses and posts that do not.","In addition to macro-averaged F1, CLPsych also reported the accuracy for each category.","We additionally report F1 macro-averaged over all classes.","Results.","The results on the self-harm risk assessment task for our models and for the current best-performing methods (briefly explained in §) are shown in Table REF .","We also report a baseline result which is based on a SVM classifier with bigram features.","When measured by non-green F1, the official metric of the CLPsych '16 Triage Task, our proposed models perform up to 19% better than the best existing methods.","Similarly, our models perform up to 11% better when measured with an F1 macro-averaged across all categories (i.e., all column) and up to 5% better with measured accuracy across all categories.","Categorical Cross Ent.","performs best in all of these cases, though the difference between the performance of Categorical Cross Ent.","and Class Metric with an ordinal margin is not statistically significant.","We also evaluate the performance of our methods on the training set using 10-fold cross validation to better observe performance differences (Table REF ).","All model variants perform substantially better on the training set than on the test set.","This is partially explained by the fact that the models were tuned on the training set, but the large difference in some cases (e.g., the increase in the highest non-green F1 from 0.50 to 0.87) suggest there may be qualitative differences between the datasets.","The best-performing method on the test set, Categorical Cross Ent., performs the worst on the training set; worst-performing method on the test set, MSE, performs the best on the training set.","Class Metric (Ordinal) performs well on both the testing and training sets, however, suggesting that it is more robust than the other methods.","Furthermore, there is no statistically significant difference between Class Metric (Ordinal) and the best-performing method on either dataset."],["Posts per user and post length","In this section we consider the effects of the maximum number of posts per user (i.e., $n_{post}$ ) and the maximum post length (i.e., $n_{term}$ ) on the Reddit dataset.","To do so we train the CNN-R model as described in §REF and report F1 on the validation set.","When varying $n_{post}$ we set $n_{term}=100$ , and when varying $n_{term}$ we set $n_{post}=1500$ .","As shown in Figure REF , the best performance of the CNN-R model is reached when it considers 100 terms in posts and up to 1750 posts for each user.","F1 increases as $n_{post}$ increases, up to the maximum tested value of 1750 (Figure REF ).","There is relatively little change in F1 from $n_{post}=1250$ to $n_{post}=1750$ , however, so we use $n_{post}=1500$ in our experiments for efficiency reasons.","As shown in Figure REF , approximately 20% of users have more than 1500 posts.","The effect of the maximum post length is not consistent (Figure REF ), but performance is maximized at $n_{term}=100$ .","As shown in Figure REF , approximately 40% of posts are longer than 100 terms.","Table: Models' performance on RSDD's validation set with different post selection strategiesand values of n post n_{post}.","CNN-E corresponds to the earliest strategy with n post =400n_{post}=400and CNN-R corresponds to the random strategy with n post =1500n_{post}=1500.Table: Example phrases that strongly contributed to a user's depression classification on the RSDD dataset."],["Post selection","For users with more than the maximum number of posts $n_{post}$ , a post selection strategy dictates which posts are used as input to the model.","Table REF shows the effect of the post selection strategy on the Reddit dataset's validation set.","Selecting a user's earliest posts performs the worst regardless of $n_{post}$ 's value, though the differences in F1 are smaller when $n_{post}=400$ .","Randomly selecting posts for each user performs the best across all metrics when $n_{post}=1500$ , with a large increase in precision over selecting users' earliest posts and a small increase over choosing users' latest posts.","Table: Self-harm risk assessment performance on the ReachOut CLPsych '17 test set.All methods perform substantially worse than on the CLPsych '16 test data.The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.","and MSE, and MSE and Class Metric (Ordinal)."],["Phrases contributing to classification","In this section we analyze the language that strongly contributed to the identification of depressed users on the Reddit dataset.","Unfortunately, it is impossible to show entire Reddit posts without compromising users' anonymity; we found that even when a post is paraphrased, enough information remains that it can easily be identified using a Web search engine.","For example, one Reddit post that strongly contributed to the author's classification as a depressed user contained the mention of a specific type of abuse and several comments vaguely related to this type of abuse.","We attempted to paraphrase this post, but found that any paraphrase containing general language related to both the type of abuse and to the user's comments was enough to identify the user.","Thus, to protect the anonymity of the users in our dataset, we do not publish posts in any form.","Rather than publishing posts, we identify key phrases in posts from users who were correctly identified as being depressed.","Phrases from eight self-reported depressed users are shown in Table REF ; to prevent these phrases from being used to identify users, we retain only the top phrase from each user.","These phrases were identified by using the model's convolutional filter weights to identify posts in the validation dataset that are strongly contributing to the model's classification decision, and then using the convolutional filter weights to identify the phrase within each post that most strongly contributed to the post's classification (i.e., had the highest feature values).","In keeping with the design of our dataset, terms related to depression or diagnoses are not present.","Instead, the model identifies phrases that often could be associated with a negative sentiment or outlook.","For example, “my whole” could be part of a negative comment referring to the poster's whole life.","It should be noted that the model makes classification decisions based on the occurrence of phrases across many posts by the same user.","Though one can imagine how the phrases shown here could be used to convey negative sentiment, the presence of a single such phrase is not sufficient to cause the model to classify a user as depressed."],["CLPsych '17 shared task","In this section we report results on the 2017 CLPsych Workshop's self-harm risk classification task.The 2017 test data was released after the initial version of this manuscript had been completed.","An official overview paper for CLPysch '17 is not yet available at the time of writing.","While CLPsych '17 featured the same self-harm risk classification task as CLPsych '16 (§REF ), new test data was used to conduct the evaluation.","This provides an opportunity to further evaluate our model on the task of self-harm risk assessment and to conduct an error analysis.","The methods were configured and evaluated in the same manner as described in §REF .The results in this section differ slightly from the methods' results as reported by CLPsych '17.","Here the methods were trained on only CLPsych '16 training data to match the experimental setup described earlier, whereas the methods were trained on both the CLPsych '16 training and test data in the official results reported by CLPsych '17.","Results are shown in Table REF .","All methods perform substantially worse than they performed on the CLPsych '16 test data as measured by non-green, urgent, and overall F1.","The trends across methods remain similar, however, with Categorical Cross Ent.","performing the best as measured by non-green and overall F1, and with no statistically significant difference between Class Metric (Ordinal) and the best performing method.","Notably, the methods' flagged F1 scores do not see a similar decrease on the CLPsych '17 data.","This suggests that the decreased performance is being caused by an inability to distinguish between the non-green classes (i.e., amber, red, and crisis).","The importance of differentiating between the red and crisis classes increased with the 2017 shared task, because the proportion of crisis labels in the data increased from 0.4% (2016 testing) and 4% (2016 training) to 11% (2017 testing).","The methods rarely classify a post as crisis, however, causing an increase in the number of misclassifications on the 2017 testing data.","For example, Class Metric (Ordinal) classified only four posts from the 2017 test data as crisis, and it classified no posts from the 2016 test data as crisis.","We leave improving the model to better identify crisis posts as future work."],["Conclusion","In this work, we argued for the close connection between social media and mental health, and described a neural network architecture for performing self-harm risk classification and depression detection on social media posts.","We described the construction of the Reddit Self-reported Depression Diagnosis (RSDD) datasethttp://ir.cs.georgetown.edu/data/reddit_depression/, containing over 9,000 users with self-reported depression diagnoses matched with over 107,000 similar control users; the dataset is available under a data usage agreement.","We applied our classification approach to the task of identifying depressed users on this dataset and found that it substantially outperformed strong existing methods in terms of Recall and F1.","While these depression detection results are encouraging, the absolute values of the metrics illustrate that this is a challenging task and worthy of further exploration.","We also applied our classification approach to the task of estimating the self-harm risk posed by posts on the ReachOut.com mental health support forum, and found that it substantially outperformed strong previously-proposed methods.","Our approach and results are significant from several perspectives: they provide a strong approach to identifying posts indicating a risk of self-harm in social media; they demonstrate a means for large scale public mental health studies surrounding the state of depression; and they demonstrate the possibility of sensitive applications in the context of clinical care, where clinicians could be notified if the activities of their patients suggest they are at risk of self-harm.","Furthermore, large-scale datasets such as the one presented in this paper can provide complementary information to existing data on mental health which are generally relatively smaller collections."]],"string":"[\n [\n \"Depression and Self-Harm Risk Assessment in Online Forums\"\n ],\n [\n \"Abstract Users suffering from mental health conditions often turn to online resources for support, including specialized online support communities or general communities such as Twitter and Reddit.\",\n \"In this work, we present a neural framework for supporting and studying users in both types of communities.\",\n \"We propose methods for identifying posts in support communities that may indicate a risk of self-harm, and demonstrate that our approach outperforms strong previously proposed methods for identifying such posts.\",\n \"Self-harm is closely related to depression, which makes identifying depressed users on general forums a crucial related task.\",\n \"We introduce a large-scale general forum dataset (\\\"RSDD\\\") consisting of users with self-reported depression diagnoses matched with control users.\",\n \"We show how our method can be applied to effectively identify depressed users from their use of language alone.\",\n \"We demonstrate that our method outperforms strong baselines on this general forum dataset.\"\n ],\n [\n \"Introduction\",\n \"Mental health remains a major challenge in public health care.\",\n \"Depression is one of the most common mental disorders and 350 million people are estimated to suffer from depression worldwide [49].\",\n \"In 2014 an estimated 7% of all U.S. adults had experienced at least one major depressive disorder depression2015.\",\n \"Suicide and self-harm are major related concerns in public mental health.\",\n \"Suicide is one of the leading causes of death [7], and each suicide case has major consequences on the physical and emotional well-being of families and on societies in general.\",\n \"Therefore identifying individuals at risk of self-harm and providing support to prevent it remains an important problem [19].\",\n \"Social media is often used by people with mental health problems to express their mental issues and seek support.\",\n \"This makes social media a significant resource for studying language related to depression, suicide, and self-harm, as well as understanding the authors' reasons for making such posts, and identifying individuals at risk of harm [11].\",\n \"Depression and suicide are closely related given that depression is the psychiatric diagnosis most commonly associated with suicide.\",\n \"Research has demonstrated that forums are powerful platforms for self-disclosure and social support seeking around mental health concerns [16], [33].\",\n \"Such support forums are often staffed by moderators who are mental health experts, trained volunteers, or more experienced users whose role is to identify forum posts suggesting that a user is at risk of self-harm and to provide support.\",\n \"Studies have shown that self expression and social support are beneficial in improving the individual's state of the mind [47], [8] and, thus such communities and interventions are important in suicide prevention.\",\n \"However, there are often thousands of user posts published in such support forums daily, making it difficult to manually identify individuals at risk of self-harm.\",\n \"Additionally, users in acute distress need prompt attention, and any delay in responding to these users could have adverse consequences.\",\n \"Therefore, identifying individuals at risk of self-harm in such support forums is an important challenge.\",\n \"Identifying signs of depression in general social media, on the other hand, is also a difficult task that has applications for both better understanding the relationship between mental health and language use and for monitoring a specific user's state (e.g., in the context of monitoring a user's response to clinical care).\",\n \"In this work we propose and evaluate a framework for performing self-harm risk assessment and for identifying depression in online forums.\",\n \"We present a general neural network architecture for combining posts into a representation of a user's activity that is used to classify the user.\",\n \"To address the challenge of depression risk assessment over the general forums, we introduce a large-scale novel Reddit dataset that is substantially larger than the existing data and has a much more realistic number of control users.\",\n \"The dataset contains over 9,000 users with self-reported depression diagnoses matched with over 107,000 control users.\",\n \"We apply our approach to (1) identify the users with depression on a general forum like Reddit, and to (2) estimate the risk of self-harm indicated by posts in a more specific mental-health support forum.\",\n \"Our methods perform significantly better on both datasets than strong existing methods, demonstrating that our approach can be used both to identify depressed users and to estimate the risk of self-harm posed by individual posts.\"\n ],\n [\n \"Related Work\",\n \"There is a growing body of related work analyzing mental health-related discourse and language usage in social media to better discover and understand mental health related concerns [41], [17], [14], [11], [35], [46], [12], [1], [36], [4].\",\n \"To investigate NLP methods for identifying depression and PTSD users on Twitter, a shared task [13] at the 2nd Computational Linguistics and Clinical Psychology Workshop (CLPsych 2015) was introduced where the participants evaluated their methods on a dataset of about 1800 Twitter users.\",\n \"Other work has used data from approximately 900 Reddit.com users to support self-reported diagnosis detection [31].\",\n \"Previous work identifying depression and other mental health problems, including the methods participating in CLPsych 2015 (e.g.\",\n \"works by [40] and [39]) heavily rely on utilizing features such as LIWC [38], topic modeling, manual lexicons, or other domain-dependent application-specific features.\",\n \"Aside from the effort required to design effective features, these approaches usually model the problem with respect to the selected features and ignore other indicators and signals that can improve prediction.\",\n \"In contrast, our model only relies on text and is not dependent on any external or domain-specific features.\",\n \"Existing self-reported diagnosis detection datasets contain a limited number of both control users and diagnosed users.\",\n \"In contrast to this, we construct a new dataset with over 9,000 depressed users matched with a realistic number of control users.\",\n \"In addition to general studies addressing mental health, related work has also specifically studied suicide and self-harm through social media [23], [45], [20], [18], [15].\",\n \"Recently, CLPsych 2016 [21] investigated approaches for detecting the self-harm risk of mental health forum posts [34].\",\n \"Most related work in this area uses variations of linear classifiers with some sort of feature engineering; successful methods have employed: a combination of sparse (bag-of-words) and dense (doc2vec) representation of the target forum posts [26], a stack of feature-rich Random Forest and linear Support Vector Machine (SVM) [32], an RBF SVM classifier utilizing similar sets of features [6], and various contextual and psycholinguistic features [9], [10].\",\n \"In contrast to the above works, our model does not use any general or domain specific feature engineering; it learns appropriate representations of documents by considering only their textual content.\",\n \"Our proposed models consist of a shared architecture based on a CNN, a merge layer, model-specific loss functions, and an output layer (as we will describe in §).\",\n \"While our model shares similarities with CNN-based models in prior work [25], [27], [50], it focuses on learning representations of user's posts and combining the post representations into an overall representation of the user's activity.\",\n \"In the case of self-harm risk assessment, we experiment with several loss functions to determine whether considering the ordinal nature of self-harm risk labels (i.e., green, amber, red, and crisis) can improve performance.\",\n \"Evaluation results suggest that the model variant using this loss function is more robust than our other variants.\"\n ],\n [\n \"Depression dataset construction.\",\n \"We created a new dataset to support the task of identifying forum users with self-reported depression diagnoses.\",\n \"The Reddit Self-reported Depression Diagnosis (RSDD) dataset was created by annotating users from a publicly-available Reddit datasethttps://files.pushshift.io/reddit/.\",\n \"Users to annotate were selected by identifying all users who made a post between January 2006 and October 2016 matching a high-precision diagnosis pattern.e.g., “I was just diagnosed with depression.” Users with fewer than 100 posts made before their diagnosis post were discarded.\",\n \"Each of the remaining diagnosis posts was then viewed by three layperson annotators to decide whether the user was claiming to have been diagnosed with depression; the most common false positives included hypotheticals (e.g., “if I was diagnosed with depression”), negations (e.g., “it's not like I've been diagnosed with depression”), and quotes (e.g., “my brother announced `I was just diagnosed with depression' ”).\",\n \"Only users with at least two positive annotations were included in the final group of diagnosed users.\",\n \"A pool of potential control users was identified by selecting only those users who had (1) never posted in a subreddit related to mental health, and (2) never used a term related to depression or mental health.\",\n \"These restrictions minimize the likelihood that users with depression are included in the control group.\",\n \"In order to prevent the diagnosed users from being easily identified by the usage of specific keywords that are never used by the control users, we removed all posts by diagnosed users that met either one of the aforementioned conditions (i.e., that was posted in a mental health subreddit or included a depression term).\",\n \"For each diagnosed user and potential control user, we calculated the probability that the user would post in each subreddit (while ignoring diagnosed users' posts made to mental health subreddits).\",\n \"Each diagnosed user was then greedily matched with the 12 control users who had the smallest Hellinger distance between the diagnosed user's and the control user's subreddit post probability distributions, excluding control users with 10% more or fewer posts than the diagnosed user.\",\n \"This matching approach ensures that diagnosed users are matched with control users who are interested in similar subreddits and have similar activity levels, preventing biases based on the subreddits users are involved in or based on how active the users are on Reddit.\",\n \"This yielded a dataset containing 9,210 diagnosed users and 107,274 control users.\",\n \"On average each user in the dataset has 969 posts (median 646).\",\n \"The mean post length is 148 tokens (median 74).\",\n \"The Reddit Self-reported Depression Diagnosis (RSDD) dataset differs from prior work creating self-reported diagnoses datasets in several ways: it is an order of magnitude larger, posts were annotated to confirm that they contained claims of a diagnosis, and a realistic number of control users were matched with each diagnosed user.\",\n \"The lists of terms related to mental health, subreddits related to mental health, high-precision depression diagnosis patterns, and further information are availablehttp://ir.cs.georgetown.edu/data/reddit_depression/.\",\n \"We note that this dataset has some (inevitable) caveats: (i) the method only captures a subpopulation of depressed people (i.e.\",\n \"those with self-reported diagnosis), (ii) Reddit users may not be a representative sample of the population as a whole, and (iii) there is no way to verify whether the users with self-reported diagnoses are truthful.\"\n ],\n [\n \"Self-harm assessment.\",\n \"For self-harm risk assessment we use data from mental health forum posts from ReachOut.com, which is a successful Australian support forum for young people.\",\n \"In addition to providing peer-support, ReachOut moderators and trained volunteers monitor and participate in the forum discussions.\",\n \"The NAACL 2016 Computational Linguistics and Clinical Psychology Workshop [21] released a Triage dataset containing 65,024 forum posts from ReachOut, with annotations for 1,227 posts indicating the author's risk of self-harm [34].\",\n \"The annotations consist of one of four labels: green (indicating no action is required from ReachOut's moderators), amber (non-urgent attention is required), red (urgent attention is required), and crisis (a risk that requires immediate attention).\"\n ],\n [\n \"Ethical concerns.\",\n \"Social media data are often sensitive, and even more so when the data are related to mental health.\",\n \"Privacy concerns and the risk to the individuals in the data should always be considered [22], [44], [3].\",\n \"We note that the risks associated with the data used in this work are minimal.\",\n \"This assessment is supported by previous work on the ReachOut dataset [34], on Twitter data [13], and on other Reddit data [31].\",\n \"The RSDD dataset contains only publicly available Reddit posts.\",\n \"Annotators were shown only anonymized posts and agreed to make no attempts to deanonymize or contact them.\",\n \"The RSDD dataset will only be made available to researchers who agree to follow ethical guidelines, which include requirements not to contact or attempt to deanonymize any of the users.\",\n \"Additionally, for the ReachOut forum data that was explicitly related to mental health, the forum's rules require the users to stay anonymous; moderators actively redact any user identifying information.\",\n \"Figure: The general neural network architecture shared among our user and post classification models.\",\n \"Each input (e.g., each of a user's posts) is processed by a convolutional network and merged to create a vector representation of the user's activity.\",\n \"This vector representation is passed through one or more dense layers followed by an output layer that performs classification.\",\n \"The type of input received, merge operation, and output layer vary with the specific model.\"\n ],\n [\n \"Methodology\",\n \"We describe a general neural network architecture for performing text classification over multiple input texts.\",\n \"We propose models based on this architecture for performing two tasks in the social media and mental health domains that we call self-harm risk classification and detecting depression.\",\n \"The task of self-harm risk classification is estimating a user's current self-harm risk given the user's post on a mental health support forum and the previous posts in the thread.\",\n \"The task of detecting depressions in users is identifying Reddit users with self-reported depression diagnoses given the users' post histories (excluding posts containing mental health keywords or posted in subreddits related to mental health).\",\n \"While both tasks are focused on predicting a user's mental health status, they differ in both the type of classification performed (i.e., estimating severity on a four point scale vs. boolean classification) and in the amount of data available.\",\n \"Our general architecture is based on a two step process: (1) identifying relevant features in each input text, and (2) combining the features observed in the model's inputs to classify the user.\",\n \"Figure: The convolutional network component of our architecture.\",\n \"A convolutional layer takes a series of terms as input (a) and applies ll filters to a kk-term sliding window to derive feature values for each window or region (b); k=2k=2 and l=3l=3 shown here.\",\n \"A max pooling layer considers non-overlapping region sequences of length nn (b) and keeps the highest feature value for the sequence (c); n=3n=3 shown here.\"\n ],\n [\n \"Shared Architecture\",\n \"Our proposed models share a common architecture that takes one or more posts as input, processes the posts using a convolutional layer to identify features present in sliding windows of text, merges the features identified into a vector representation of the user's activity, and uses a series of dense layers to perform classification on the merged vector representation.\",\n \"The type of merging performed and the output layers are properties of the model variant, which we describe in detail in the following section.\",\n \"Convolutional networks have commonly been applied to the task of text classification, such as by Kim:2014.\",\n \"We use categorical cross-entropy as a loss function with both methods, but also experiment with other loss functions when performing severity classification.\",\n \"First, the model takes one or more posts as input and processes each post with a convolutional network containing a convolutional layer and a pooling layer.\",\n \"This process is illustrated with a max pooling layer in Figure REF .\",\n \"The convolutional layer applies filters to a sliding window of $k$ terms (a) and outputs a feature value for each sliding window region and each filter (b).\",\n \"The same filters are applied to each window; each filter can be viewed as a feature detector and the overall process can be conceptualized as looking for windows of terms that contain specific features.\",\n \"The features are not specified a priori through feature engineering, but instead are learned automatically when the model is trained.\",\n \"After identifying the features present in each region (i.e., sliding window), a max pooling layer considers non-overlapping regions of length $n$ and keeps the highest feature value for each region (c).\",\n \"This step eliminates the regions (i.e., sliding windows) that do not contain useful features, which reduces the size of the convolutional network's output.\",\n \"The same convolutional network is applied to each input post, meaning that the model learns to look for the same set of features in each.\",\n \"After each input post has been processed by a convolutional network, the output of each convolutional network is merged to create a representation of the user's activity across all input posts.\",\n \"This representation is processed by one or more dense layers (i.e., fully connected layers) with dropout [42] before being processed by a final output layer to perform classification.\",\n \"The type of output layer is dependent on the model variant.\",\n \"Our shared model architecture is illustrated in Figure REF .\",\n \"The architecture's hyperparameters (e.g., the sliding window size $k$ , the number of convolutional filters used, and type of pooling) also vary among models and are described in §.\",\n \"Both the convolutional and dense layers use ReLU activations [37] in all model variants.\",\n \"Table: The hyperparameters used by each model.\"\n ],\n [\n \"Depression detection\",\n \"Our model for depression detection takes a user's posts as input and processes each post with a convolutional network.\",\n \"Each convolutional network performs average pooling to produce its output.\",\n \"These post representations are then merged with a second convolutional layer to create a user representation; we found this approach led to more stable performance than using a second average pooling or max pooling layer.\",\n \"The user representation created by the merge step is then passed to one or more dense layers before being passed to a dense output layer with a softmax activation function to perform classification.\",\n \"The number of dense layers used is a hyperparameter described in §.\",\n \"Categorical cross-entropy is used as the model's loss function.\"\n ],\n [\n \"Self-harm risk assessment\",\n \"Our model for self-harm risk classification takes two inputs: the target post being classified and the prior posts (if any) in the target post's thread.\",\n \"The prior posts provide context and are thus useful for estimating the risk of self-harm present in the target post.\",\n \"The two inputs are both processed by a convolutional network as in user-level classification, but in this case the convolutional network's outputs correspond to a representation of the target post and to a representation of the target post's context (i.e., the prior posts in the thread).\",\n \"Given that these two outputs represent different aspects, they are merged by concatenating them together.\",\n \"This merged representation is then passed to one or more dense layers and to an output layer; the type of output layer depends on the loss function used.\",\n \"There are four self-harm risk assessment model variants in total: Categorical Cross Ent.\",\n \"uses an output layer with a softmax activation function, and categorical cross-entropy as its loss function.\",\n \"This mirrors the output layer and loss function used in the user level classification model.\",\n \"MSE uses an output layer with a linear activation function, and mean squared error as its loss function.\",\n \"The model's output is thus a single value; to perform classification, this output value is rounded to the nearest integer in the interval $[0, t - 1]$ , where $t$ is the number of target classes.\",\n \"The final two loss functions perform metric learning rather than performing classification directly.\",\n \"They learn representations of a user's activity and of the four self-harm risk severity labels; classification is performed by comparing the euclidean distance between a representation of a user's activity (produced by the final layer) and each of the four severity label representations.\",\n \"Class Metric: Let $d$ be the size of the output layer and $X$ be the layer's $d$ -dimensional output.\",\n \"Class Metric learns a $d$ -dimensional representation of each class $C_i$ such that $||X - C_i||_2$ is minimized for the correct class $i$ ; this is accomplished with the loss function: $ L_{i,p,n} = \\\\max \\\\big (0, ||X_i - C_p||_2 - ||X_i - C_n||_2 + \\\\alpha \\\\big ) $ where $C_p$ is the correct (i.e., positive) class for $X_i$ , $C_n$ is a randomly chosen incorrect (i.e., negative) class, and $\\\\alpha $ is a constant to enforce a minimum margin between classes.\",\n \"Classification is performed by computing the similarity between $X_i$ and each class $C_j$ .\",\n \"Class Metric (Ordinal) extends Class Metric to enforce a margin between ordinal classes as a function of the distance between classes.\",\n \"Given a ranked list of classes such that more similar classes have closer rankings, that is $\\\\forall i$ $sim(C_i, C_{i \\\\pm 1}) > sim(C_i, C_{i \\\\pm 2})$ , we incorporate the class distance into the margin such that more distant incorrect class labels must be further away from the correct class label in the metric space.\",\n \"The loss function becomes $ L_{i,p,n} = \\\\max \\\\big (0, \\\\;\\\\; ||X_i - C_p||_2 \\\\;\\\\; - \\\\\\\\||X_i - C_n||_2 + \\\\alpha |p-n| \\\\big )$ where $|p-n|$ causes the margin to scale with the distance between classes $p$ and $n$ .\"\n ],\n [\n \"Experiments\",\n \"In this section, we describe the model hyperparameters used and present our results on the depression detection and self-harm risk assessment tasks.\",\n \"To facilitate reproducibility we provide our code and will provide the Reddit depression dataset to researchers who sign a data usage agreementhttp://ir.cs.georgetown.edu/data/reddit_depression/.\"\n ],\n [\n \"Experimental setup.\",\n \"The hyperparameters used with our models are shown in Table REF .\",\n \"The severity risk assessment models' hyperparameters were chosen using 10-fold cross validation on the 947 ReachOut training posts, with 15% of each fold used as validation data.\",\n \"The depression identification model's hyperparameters were chosen using the Reddit validation set.\",\n \"The depression identification model's second convolutional layer (i.e., the layer used to merge post representations) used filters of length 15, a stride of length 15, and the same number of filters as the first convolutional layer.\",\n \"All models were trained using stochastic gradient descent with the Adam optimizer [28].\",\n \"The hyperparameters that varied across models are shown in Table REF .\",\n \"The convolution size, number of convolutional filters, pooling type, pooling length, and number of dense layers was similar across all post models.\",\n \"Class balancing was performed with Categorical Cross Ent.\",\n \"by weighting classes inversely proportional to their frequencies, whereas sampling an equal number of instances for each class worked best with the other methods.\"\n ],\n [\n \"Addressing limited data.\",\n \"The post classification models' input consists of skip-thought vectors [29]; each vector used is a 7200-dimensional representation of a sentence.\",\n \"Thus, the convolutional windows used for post classification are over sentences rather than over terms.\",\n \"This input representation was chosen to mitigate the effects of the ReachOut dataset's relatively small size.\",\n \"The skip-thought vectors were generated from the the ReachOut forum dataset by sequentially splitting the posts in the training set into sentences, tokenizing them, and training skip-thoughts using Kiros et al.\",\n \"'s implementation with the default parameters.\",\n \"Sentence boundary detection was performed using the Punkt sentence tokenizer [30] available in NLTK [5].\",\n \"These 2400-dimensional forum post skip-thought vectors were concatenated with the 4800-dimensional book corpus skip-thought vectors available from [29].\",\n \"Experiments on the training set indicated that using only the ReachOut skip-thought vectors slightly decreased performance, while using only the book corpus skip-thought vectors substantially decreased performance.\",\n \"As input the post models received the last 20 sentences in each target post and the last 20 sentences in the thread prior to the target post; any prior sentences are ignored.\"\n ],\n [\n \"Depression detection.\",\n \"The data used for depression detection was described in §.\",\n \"As baselines we compare our model against the FastText classifier [24] and MNB and SVM classifiers [48] using features from prior work.\",\n \"We tune FastText's hyperparameters on the validation set.\",\n \"Specifically, we consider a maximum n-gram size $\\\\in [1,2,3,4,5]$ , an embedding size $\\\\in [50,100, 150]$ , and a learning rate $\\\\in [0.05, 0.1, 0.25, 0.5]$ as suggested in the documentation.\",\n \"We consider two sets of features for the MNB and SVM classifiers.\",\n \"The first set of features is the post content itself represented as sparse bag of words features (BoW baselines).\",\n \"The second set of features (feature-rich baselines) comprises a large set of features including bag of words features encoded as sparse weighted vectors, external psycholinguistic features captured by LIWChttp://liwc.wpengine.com/ pennebaker2015development, and emotion lexicon features [43].\",\n \"Since our problem is identifying depression among users, psycholinguistic signals and emotional attributes in the text are potentially important features for the task.\",\n \"These features (as described in §) have been also previously used by successful methods in the Twitter self-reported diagnosis detection task [13].\",\n \"Thus, we argue that these are strong baselines for our self-reported diagnosis detection task.\",\n \"We apply count based and TF-IDF based feature weighting for bag of words features.\",\n \"We perform standard preprocessing by removing stopwords and lowercasing the input text.During experimentation, we found TF-IDF sparse feature weighting to be superior than other weighting schemes.\",\n \"Additional features such as LDA topics and $\\\\chi ^2$ feature selection did not result in any further improvements.\",\n \"The data is split into training, validation, and testing datasets each containing approximately 3,000 diagnosed users and their matched control users.\",\n \"The validation set is used for tuning development and hyperparameter tuning of our models and the baselines.\",\n \"The reported results are on the test set.\",\n \"The depression detection models' input consisted of raw terms encoded as one-hot vectors.\",\n \"We used an input layer to learn 50-dimensional representation of the terms.\",\n \"For each target user, the CNN received up to $n_{post}$ posts containing up to $n_{term}$ terms.\",\n \"In this section we present results for two values of $n_{post}$ .\",\n \"The earliest post approach (CNN-E) takes each user's $n_{post}=400$ earliest posts as input.\",\n \"The random approach (CNN-R) samples $n_{post}=1500$ random posts from each user.\",\n \"We empirically set $n_{term}=100$ with both approaches.\",\n \"We later analyze the model's performance as $n_{post}$ and $n_{term}$ vary in §REF and as the post selection strategy varies in §REF .\",\n \"Table: Performance identifying depressed users on the Reddit test set.The differences between the CNN and baselines are statistically significant (McNemar's test, p<0.05p<0.05).Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 test set.Results for the other methods are from .Differences in performance between the following pairs are statistically significant (McNemar's test, p<0.05p<0.05):Categorical Cross Ent.\",\n \"and Class Metric, MSE and Categorical Cross Ent., MSE and Class Metric (Ordinal), and Class Metric (Ordinal) and Class Metric.Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 training set using 10-fold cross validation.\",\n \"Categorical Cross Ent.\",\n \"performs substantially worse than on the test set, while MSE performs substantially better.\",\n \"Class Metric (Ordinal) continues to perform well.\",\n \"The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.\",\n \"and MSE, Categorical Cross Ent.\",\n \"and Class Metric, Categorical Cross Ent.\",\n \"and Class Metric (Ordinal), MSE and Class Metric, and Class Metric and Class Metric (Ordinal).Results.\",\n \"The results of identifying depressed users for our model and baselines are shown in Table REF .\",\n \"Our proposed model outperforms the baselines by a large margin in terms of recall and F1 on the diagnosed users (increases of 41% and 16%, respectively), but performs worse in terms of precision.\",\n \"As described later in the analysis section, the CNN identifies language associated with negative sentiment across a user's posts.\",\n \"Figure: Sensitivity of the CNN-R model to the parameters n posts n_{posts} (a) and n terms n_{terms} (b) on RSDD's validation set.\",\n \"F1 increases as n posts n_{posts} does (a), but the rate of increase slows as n posts n_{posts} surpasses 1000.The trend for n terms n_{terms} is less clear (b), but the highest F1 is achieved at n terms =100n_{terms}=100.In Figure (a) the parameter n terms n_{terms} was fixed to 100, and in Figure (b) n posts n_{posts} was fixed to 1500.Figure: Empirical cumulative distribution functions (CDF) of the number of posts per user (a) and the post length (b) in the RSDD dataset.\"\n ],\n [\n \"Self-harm risk classification.\",\n \"We train our methods to label the ReachOut posts and compare them against the top methods from CLPsych '16.\",\n \"We use the same experimental protocol as was used in CLPsych '16; our methods were trained on the 947 training posts and evaluated on the remaining 280 testing posts.\",\n \"We used 15% of the 947 training posts as validation data.\",\n \"We report results using the same metrics used in CLPsych, which were: the macro-averaged F1 for the $amber$ , $red$ , and $crisis$ labels (non-green posts); the macro-averaged F1 of $ green $ posts vs. $ amber \\\\cup red \\\\cup crisis $ (flagged posts); and the macro-averaged F1 of $ green \\\\cup amber $ vs. $ red \\\\cup crisis $ (urgent posts).\",\n \"The non-green F1 was used as the official CLPsych metric with the intention of placing emphasis on classification performance for the non-green categories (i.e., those that required some response).\",\n \"The binary flagged meta-class was chosen to measure models' abilities to differentiate between posts that require attention and posts that do not, and the binary urgent meta-class was chosen to measure their abilities to differentiate between posts that require quick responses and posts that do not.\",\n \"In addition to macro-averaged F1, CLPsych also reported the accuracy for each category.\",\n \"We additionally report F1 macro-averaged over all classes.\",\n \"Results.\",\n \"The results on the self-harm risk assessment task for our models and for the current best-performing methods (briefly explained in §) are shown in Table REF .\",\n \"We also report a baseline result which is based on a SVM classifier with bigram features.\",\n \"When measured by non-green F1, the official metric of the CLPsych '16 Triage Task, our proposed models perform up to 19% better than the best existing methods.\",\n \"Similarly, our models perform up to 11% better when measured with an F1 macro-averaged across all categories (i.e., all column) and up to 5% better with measured accuracy across all categories.\",\n \"Categorical Cross Ent.\",\n \"performs best in all of these cases, though the difference between the performance of Categorical Cross Ent.\",\n \"and Class Metric with an ordinal margin is not statistically significant.\",\n \"We also evaluate the performance of our methods on the training set using 10-fold cross validation to better observe performance differences (Table REF ).\",\n \"All model variants perform substantially better on the training set than on the test set.\",\n \"This is partially explained by the fact that the models were tuned on the training set, but the large difference in some cases (e.g., the increase in the highest non-green F1 from 0.50 to 0.87) suggest there may be qualitative differences between the datasets.\",\n \"The best-performing method on the test set, Categorical Cross Ent., performs the worst on the training set; worst-performing method on the test set, MSE, performs the best on the training set.\",\n \"Class Metric (Ordinal) performs well on both the testing and training sets, however, suggesting that it is more robust than the other methods.\",\n \"Furthermore, there is no statistically significant difference between Class Metric (Ordinal) and the best-performing method on either dataset.\"\n ],\n [\n \"Posts per user and post length\",\n \"In this section we consider the effects of the maximum number of posts per user (i.e., $n_{post}$ ) and the maximum post length (i.e., $n_{term}$ ) on the Reddit dataset.\",\n \"To do so we train the CNN-R model as described in §REF and report F1 on the validation set.\",\n \"When varying $n_{post}$ we set $n_{term}=100$ , and when varying $n_{term}$ we set $n_{post}=1500$ .\",\n \"As shown in Figure REF , the best performance of the CNN-R model is reached when it considers 100 terms in posts and up to 1750 posts for each user.\",\n \"F1 increases as $n_{post}$ increases, up to the maximum tested value of 1750 (Figure REF ).\",\n \"There is relatively little change in F1 from $n_{post}=1250$ to $n_{post}=1750$ , however, so we use $n_{post}=1500$ in our experiments for efficiency reasons.\",\n \"As shown in Figure REF , approximately 20% of users have more than 1500 posts.\",\n \"The effect of the maximum post length is not consistent (Figure REF ), but performance is maximized at $n_{term}=100$ .\",\n \"As shown in Figure REF , approximately 40% of posts are longer than 100 terms.\",\n \"Table: Models' performance on RSDD's validation set with different post selection strategiesand values of n post n_{post}.\",\n \"CNN-E corresponds to the earliest strategy with n post =400n_{post}=400and CNN-R corresponds to the random strategy with n post =1500n_{post}=1500.Table: Example phrases that strongly contributed to a user's depression classification on the RSDD dataset.\"\n ],\n [\n \"Post selection\",\n \"For users with more than the maximum number of posts $n_{post}$ , a post selection strategy dictates which posts are used as input to the model.\",\n \"Table REF shows the effect of the post selection strategy on the Reddit dataset's validation set.\",\n \"Selecting a user's earliest posts performs the worst regardless of $n_{post}$ 's value, though the differences in F1 are smaller when $n_{post}=400$ .\",\n \"Randomly selecting posts for each user performs the best across all metrics when $n_{post}=1500$ , with a large increase in precision over selecting users' earliest posts and a small increase over choosing users' latest posts.\",\n \"Table: Self-harm risk assessment performance on the ReachOut CLPsych '17 test set.All methods perform substantially worse than on the CLPsych '16 test data.The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.\",\n \"and MSE, and MSE and Class Metric (Ordinal).\"\n ],\n [\n \"Phrases contributing to classification\",\n \"In this section we analyze the language that strongly contributed to the identification of depressed users on the Reddit dataset.\",\n \"Unfortunately, it is impossible to show entire Reddit posts without compromising users' anonymity; we found that even when a post is paraphrased, enough information remains that it can easily be identified using a Web search engine.\",\n \"For example, one Reddit post that strongly contributed to the author's classification as a depressed user contained the mention of a specific type of abuse and several comments vaguely related to this type of abuse.\",\n \"We attempted to paraphrase this post, but found that any paraphrase containing general language related to both the type of abuse and to the user's comments was enough to identify the user.\",\n \"Thus, to protect the anonymity of the users in our dataset, we do not publish posts in any form.\",\n \"Rather than publishing posts, we identify key phrases in posts from users who were correctly identified as being depressed.\",\n \"Phrases from eight self-reported depressed users are shown in Table REF ; to prevent these phrases from being used to identify users, we retain only the top phrase from each user.\",\n \"These phrases were identified by using the model's convolutional filter weights to identify posts in the validation dataset that are strongly contributing to the model's classification decision, and then using the convolutional filter weights to identify the phrase within each post that most strongly contributed to the post's classification (i.e., had the highest feature values).\",\n \"In keeping with the design of our dataset, terms related to depression or diagnoses are not present.\",\n \"Instead, the model identifies phrases that often could be associated with a negative sentiment or outlook.\",\n \"For example, “my whole” could be part of a negative comment referring to the poster's whole life.\",\n \"It should be noted that the model makes classification decisions based on the occurrence of phrases across many posts by the same user.\",\n \"Though one can imagine how the phrases shown here could be used to convey negative sentiment, the presence of a single such phrase is not sufficient to cause the model to classify a user as depressed.\"\n ],\n [\n \"CLPsych '17 shared task\",\n \"In this section we report results on the 2017 CLPsych Workshop's self-harm risk classification task.The 2017 test data was released after the initial version of this manuscript had been completed.\",\n \"An official overview paper for CLPysch '17 is not yet available at the time of writing.\",\n \"While CLPsych '17 featured the same self-harm risk classification task as CLPsych '16 (§REF ), new test data was used to conduct the evaluation.\",\n \"This provides an opportunity to further evaluate our model on the task of self-harm risk assessment and to conduct an error analysis.\",\n \"The methods were configured and evaluated in the same manner as described in §REF .The results in this section differ slightly from the methods' results as reported by CLPsych '17.\",\n \"Here the methods were trained on only CLPsych '16 training data to match the experimental setup described earlier, whereas the methods were trained on both the CLPsych '16 training and test data in the official results reported by CLPsych '17.\",\n \"Results are shown in Table REF .\",\n \"All methods perform substantially worse than they performed on the CLPsych '16 test data as measured by non-green, urgent, and overall F1.\",\n \"The trends across methods remain similar, however, with Categorical Cross Ent.\",\n \"performing the best as measured by non-green and overall F1, and with no statistically significant difference between Class Metric (Ordinal) and the best performing method.\",\n \"Notably, the methods' flagged F1 scores do not see a similar decrease on the CLPsych '17 data.\",\n \"This suggests that the decreased performance is being caused by an inability to distinguish between the non-green classes (i.e., amber, red, and crisis).\",\n \"The importance of differentiating between the red and crisis classes increased with the 2017 shared task, because the proportion of crisis labels in the data increased from 0.4% (2016 testing) and 4% (2016 training) to 11% (2017 testing).\",\n \"The methods rarely classify a post as crisis, however, causing an increase in the number of misclassifications on the 2017 testing data.\",\n \"For example, Class Metric (Ordinal) classified only four posts from the 2017 test data as crisis, and it classified no posts from the 2016 test data as crisis.\",\n \"We leave improving the model to better identify crisis posts as future work.\"\n ],\n [\n \"Conclusion\",\n \"In this work, we argued for the close connection between social media and mental health, and described a neural network architecture for performing self-harm risk classification and depression detection on social media posts.\",\n \"We described the construction of the Reddit Self-reported Depression Diagnosis (RSDD) datasethttp://ir.cs.georgetown.edu/data/reddit_depression/, containing over 9,000 users with self-reported depression diagnoses matched with over 107,000 similar control users; the dataset is available under a data usage agreement.\",\n \"We applied our classification approach to the task of identifying depressed users on this dataset and found that it substantially outperformed strong existing methods in terms of Recall and F1.\",\n \"While these depression detection results are encouraging, the absolute values of the metrics illustrate that this is a challenging task and worthy of further exploration.\",\n \"We also applied our classification approach to the task of estimating the self-harm risk posed by posts on the ReachOut.com mental health support forum, and found that it substantially outperformed strong previously-proposed methods.\",\n \"Our approach and results are significant from several perspectives: they provide a strong approach to identifying posts indicating a risk of self-harm in social media; they demonstrate a means for large scale public mental health studies surrounding the state of depression; and they demonstrate the possibility of sensitive applications in the context of clinical care, where clinicians could be notified if the activities of their patients suggest they are at risk of self-harm.\",\n \"Furthermore, large-scale datasets such as the one presented in this paper can provide complementary information to existing data on mental health which are generally relatively smaller collections.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01848"}}},{"rowIdx":696,"cells":{"text":{"kind":"list like","value":[["Scalar curvature, flat Borromean rings, and the 3-body problem"],["Abstract This paper explains unexpected links between the 3 topics in the title and frames them in a large canvas."],["History","The basic ideas of geometry go back to Euclid and Archimedes, followed later by Gauss, Minkowski and Weyl.","Topology also has ancient roots as evidenced by Alexander the Great cutting the Gordian Knot, and the Popes whose coat of arms became the Borromean rings.","Isaac Newton, standing on the shoulders of Galileo and beyond Papal influence, used Greek ellipses to explain planetary motion and left to posterity the 3-body problem.","The 3 Borromean rings of polished steel at the entrance to the Newton Institute in CambridgeImage reproduced by kind permission of Peter Robinson, Bradshaw Foundation ©John Robinson/Edition Limitée Paris 2007 provided me with inspiration, in August 2017, at the conference celebrating the 60th birthday of Simon Donaldson.","Tantalized by Misha Gromov's lecture on scalar curvature, I decided to combine the two topics and relate them to Newton's planetary orbits.","$\\includegraphics [width=6cm]{ini.jpg}$ Visionary ideas are what we mathematicians grapple with, but illustrative special examples help the understanding.","They also lead us by analogy to more general and realistic examples.","The master whose style I try to emulate is the greatest mathematical physicist since Newton, James Clerk Maxwell.","He understood dynamical stability, with its topological underpinnings, more thoroughly than anyone else and I offer this short note as a tribute to him."],["The Topology and Geometry of Borromean Rings","The linking of circles in 3-space is familiar to us all and led to the formation of chains.","Gauss, Faraday and Maxwell explained electro-magnetism in terms of such links.","The beauty and subtlety of the 3 Borromean rings is that each pair are unlinked but the triple holds together.","This mysterious property had, in medieval times, theological (even Trinitarian) overtones which is why it appealed to Popes, and would-be popes, like the Borromeo family.","Newton might have heard of this mystery but, as a secret unitarian in Trinity College, he would have kept off such dangerous ground.","We all know that geometry and topology are siblings.","Geometry is about measurement and topology is about shape.","This family unity is embodied in the great theorem of Gauss, which starts locally with the area of spherical triangles, or polygons, and concludes globally with the formula for the Euler member of a closed surface as the average of the Gauss or scalar curvature (divided by $2\\pi $ ).","It is natural to ask if there is a similar formula relating the geometry and topology of Borromean rings.","For this to make sense the 3 rings must be geometrical, not just topological.","Each ring should be planar or spherical, with a boundary and the 3 rings should be placed in 3-space so that their mutual topology is Borromean.","Looking at the sculptured steel outside the Newton Institute raises a challenging question, rather like the Rubik Cube or the model sailing boats in wine bottles.","How are they made or dismantled?","If you are the artist commissioned to make the sculpture, what precise instructions do you give the craftsman in the workshop?","For example, suppose you wanted to make the sculpture on the left of the Institute consisting of 3 squares of flat steel, each with a smaller square hole inside it?","The craftsman would need exact measurements and their tolerance : how accurate does he have to be?","This sounds, and is, standard engineering design.","The sculptor, or his mathematical consultant, has to do \"the sums\" and instruct the craftsman, confident that it will give a stable work of art.","Having done this once for the first steel sculpture at the Newton Institute, the design team then have to go through a similar procedure with the square replaced by a rhombus or a triangle, for the two sculptures on the right.","As your sculptures attract attention the team might be flooded with orders, but each client would have different requirements.","Some might want polygons with many sides, perhaps infinitely many, so that polygons become ellipses (with Newtonian approval).","Clearly you will need an efficient algorithm to implement the process.","Does your firm have mathematical consultants of the necessary quality?","If they studied at Trinity or St. John's (the parent Colleges of the Newton Institute) then you might be in luck.","As a Trinity man I feel it incumbent on me to try.","The next sections provide my mathematical report on these problems."],["The Iso-Perimetric Inequality","In the spirit of Maxwell I will consider a simple model based on the actual steel sculptures at Newton's Institute.","My Borromean rings will be modelled by flat regular polygons $C_i $ $(i=1,2,3)$ with $N$ sides ($N\\geqslant 3$ ), having a border of width $\\delta $ .","See the figure for $N=4$ and one polygon $C$ .","$\\includegraphics [width=10cm]{square.pdf}$ The engineering problem is to take 3 such regular $N$ -gons $C_1,C_2,C_3$ of the same width $\\delta $ and then place them in 3-space (or on a 3-sphere) so that they form a Borromean triple.","Clearly the holes in the middle of the $N$ -gons cannot be too small.","We look for an inequality which will give the critical size $d$ , so that a Borromean triple of such (flat bordered) $N$ -gons can be constructed provided $\\delta 3$ of width $\\delta $ which are planar or spherical.","The figure in section 3 shows the planar case for $N=4$ .","The question we ask is (5.1) Can this data be assembled in 3-space ($\\mathbb {R}^3$ or $S^3$ ) by translation and scale changes to form a Borromean triple of $N$ -gons?","The trigonometric calculations in Section 3 show that this is possible provided we have the inequality (5.2) $A_N(r,\\delta ) > r\\delta \\sin (2\\pi /N)$ derived from the equality (3.4).","The \"error\" is the quadratic term (5.3) $\\delta ^2/2.N \\sin (2\\pi /N)$ This answers the questions raised at the end of section 2.","The desired bound $d$ depends on the scale set by the radius $r$ and, for regular $N$ -gons $d=2r$ and we just get the inequality $\\delta /r \\leqslant 2$ .","Embodied in planar steel, as at the Newton Institute, formula (5.2) gives the engineering team the precise measurements needed, for any $N$ , and the tolerance or error.","As the width tends to zero the tolerance also tends to zero so that, in the (conformal) limit, steel is no longer a suitable material and the Borromean rings have to made of silk thread.","For irregular Borromean rings the exact formula (5.3) will be replaced by more general ones, but the leading (topological) term will remain.","While (5.2) is elementary trigonometry, because of the symmetry, its natural context is that of Minkowski's mixed areas, or the Gauss scalar curvature.","Generalizing (5.2) was therefore standard geometry over a century ago.","In more recent times the topology and geometry of Borromean triples has become equally standard though more sophisticated.","Different experts develop different machinery and do not always use the same language.","For applied mathematicians, engineers or for Olympic pole vaulters, the position of the centre of gravity is crucial.","Borromean geometry, based on the stability that comes from topology, may help win gold medals.","Maxwell would have understood it all and he would have explained how a few well-chosen examples are all you need.","Since he is no longer here in person I have tried to stand in for him, like an actor who stands in for Charles Dickens.","3 planets in elliptical orbits may have long term stability, held together by mysterious topology, if the parameters are right.","It seems to me more than likely that some versions of the famous 3-body problem will turn out to be Borromean, perhaps in the rings of Saturn, studied by Maxwell or even in the Red Spot of Jupiter.",".","I hope I have whetted the appetite of the reader, and that the Newton Institute will continue to inspire natural philosophers for decades to come.","But let me conclude with two final sections extending the scope of the discussion."],["Force and Centres","The Borromean rings at the Newton Institute were our starting point.","For artistic reasons they were highly symmetrical with an obvious centre.","But, as we start to deform them, their symmetry and centre get lost.","How do we track the centre?","Newton found the answer by going back to Apollonius and conic sections.","The focus (one of a pair) was the appropriate centre and the eccentricity of the ellipse increased as the focus moved away from the centre.","What Newton realized, and Apollonius did not, was that the focus was the centre of gravity.","In other words, gravitational force changes the geometry, and planetary orbits reflect this changed geometry.","On the time scale of Earth-years the gravitational attraction of the sun does not change and planetary orbits are, with modest variations, stable for millennia.","Mathematicians, from Laplace onwards, wondered about stability in the “long-term”.","Would the solar system survive forever?","To an astronomer or an astrophysicist, this was an idealization too far.","A time-scale may not matter to the pure mathematician but, if it is not commensurate with physical time-scales, it is irrelevant: the “long-term” will never arrive.","Such metaphysical questions acquire more significance in two ways.","On the one hand, if gravity were replaced by electro-magnetism, also based on the inverse-square law, planets could be replaced by electrons travelling in orbits at velocities close to the speed of light.","This dramatic change of scales brings mathematical time-scales down to physical time-scales and long-term stability becomes important for engineers at CERN, as explained to me many years ago by Jürgen Moser.","Another more fundamental change came, a century ago, with Einstein's Theory of General Relativity.","The inverse-square law is just the Newtonian linear approximation but non-linear GR has to incorporate Einsteinian corrections, even for the solar system, and the accuracy of GPS requires such corrections.","If we also want to understand the magnetic fields that generate solar flares and affect our weather, then both kinds of corrections are necessary.","The LIGO triangles designed to detect gravitational waves were testing not just the statics of GR but also its dynamics, which brings in time, so Borromean rings might acquire a cosmic significance.","This may not just be “$\\pi $ in the SKY” since respected astronomers have been searching for “cosmic strings” both in theory and in observation.","The lessons I draw from this global (or cosmic) perspective are the following (6.1) the importance of forces in geometry (6.2) the key role of the centres of force (6.3) the role of centres in dynamical stability (6.4) the importance of effective criteria for stability (6.5) the understanding of different force scales (6.6) the role of topology in qualitative understanding of statics and dynamics (6.7) the role of geometry in quantitative versions of (6.6) with explicit tolerances.","Needless to say most of this is known to most scientists most of the time.","But I suspect that not all is known to all scientists all the time.","Perhaps it is time for all natural philosophers to get under the Newton Apple Tree and admire the Borromean Rings, just as our Druid ancestors admired the Art and Architecture of Stonehenge.","In the next and final section I will descend from Heaven to Earth."],["Down to Earth","The 3 Borromean rings from which we started are the simplest geometric examples in 3-space.","They can be generalized in infinitely many ways and we are free to choose the model that fits our needs and our resources.","There is no single multi-purpose model which will do everything.","I have already indicated that regular polygons can be replaced by convex polygons and that the parameters can all be allowed to vary.","For a bordered polygon with width $\\delta $ we started with 3 polygons $C_i$ having the same width, but we can now allow different widths $\\delta _i$ .","Ellipses appear when the polygon has an infinite number of sides, so we can end up with 3 ellipses of different widths sitting in 3-space.","Topologically they are either Borromean rings or non-Borromean rings, the Papal version of Hamlet's dilemma : To $B$ or not to $B$.","Mathematical techniques now exist to answer Hamlet.","Shakespeare was an actor as well as a playwright, so he would have embraced the new technology.","The new technology discussed at Donaldson 60 would have solved Hamlet's dilemma, depriving the English language of its favourite quotation.","All Hamlet had to do was to find out if the 3 rings could balance stably in the earth's gravitational field.","The nearby apple tree would have shown him that the small convex set (the tolerance), defined by the widths $\\delta _i$ , should contain the centre of gravity.","Fortunately the flat fenland of Cambridge would ensure that the gravitational field had essentially the same strength on both sides of the Institute.","Apples could therefore be substituted by steel rings, though the Borromean property would need a high speed camera to capture any entanglement of 3 falling apples, the dynamics of which were famously observed by the occupant of Woolsthorpe manor.","The engineers would not need to incorporate the relativistic Einsteinian corrections of section 6, since auroras are rarely seen at latitude $52^\\circ $ North.","We thus end up with an effective numerical criterion for the 3 rings to be Borromean or non-Borromean.","This describes all the steel sculptures that might be constructed.","As pointed out in section 3, Minkowski only dealt with convex bodies, which like spheres, have positive scalar curvature, while the Gauss formula allows regions of negative curvature.","But passing to convex hulls swallows up the negative regions.","There are many ways of understanding this difference.","(7.1) A general Morse function $f$ on a closed surface will have maxima, minima and saddle points.","(7.2) The Hessian of $f$ at critical points distinguishes between the 3 cases in (7.1).","It defines local geometries with scalar curvature positive at the maxima and minima, but negative at the saddle points.","(7.3) In the regions of positive curvature the Minkowski theory agrees with Gauss.","(7.4) The Gauss integral formula is about averages and Euler's topological invariant gives the dominant term.","(7.5) An average smooths out local irregularities, and gives good results when the averaging process is tailored to the nature of the irregularities.","(7.6) In all branches of science local irregularities occur and have to be averaged.","Atoms in a molecule, planets in the solar system or galaxies in the cosmos are all “irregularities” in this sense.","(7.7) Analysts have developed very refined techniques of smoothing irregularities which lead to stochastic differential equations.","(7.8) Geometers have developed techniques of stability descending from Minkowski and used in Mirror Symmetry.","(7.9) Physicists have techniques called renormalization which provide powerful tools in quantum field theory.","(7.10) Number theorists since Euler and Riemann have used similar ideas to study the “irregular” distribution of primes, with the Riemann Hypothesis as the ultimate goal.","References There is a vast literature on geometry and dynamics, much too extensive to list here.","The great classic is of course Maxwell's Smith Prize Essay on Saturn's Rings.","For more recent literature I can at least point to some of my own papers which are relevant in various ways.","M.F.Atiyah, Convexity and commuting Hamiltonians, Bull.","Lond.","Math.","Soc.","14 (1981), 1–15.","–, The Non-Existent Complex 6-Sphere, arXiv:1610.09366v2 [math.DG] (2016) –, Geometric Models of Helium, arXiv:1703.02532v1 [physics.gen-ph] (2017) –, Groups of Odd Order and Galois Theory, to appear.","– and J.Berndt, Projective planes, Severi varieties and spheres, Surveys in Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, International Press, Somerville, MA (2003).","–, R.Bott and L.Gårding, Lacunas for hyperbolic differential operators with constant coefficients I., Acta Math.","124 (1970), 109–189.","– and G.Segal, Twisted K-theory and Cohomology, in Inspired by S.S. Chern: A Memorial Volume in Honor of a Great Mathematician (Editor: P. A. Griffiths), Nankai Tracts in Mathematics Vol.","11, World Scientific Publishing Co Inc. (2007).","J.C.Maxwell, On the stability of the motion of Saturn's rings, An essay which obtained the Adams Prize in 1856 in the University of Cambridge, Macmillan (1859) Available online at https://archive.org/details/onstabilityofmot00maxw"]],"string":"[\n [\n \"Scalar curvature, flat Borromean rings, and the 3-body problem\"\n ],\n [\n \"Abstract This paper explains unexpected links between the 3 topics in the title and frames them in a large canvas.\"\n ],\n [\n \"History\",\n \"The basic ideas of geometry go back to Euclid and Archimedes, followed later by Gauss, Minkowski and Weyl.\",\n \"Topology also has ancient roots as evidenced by Alexander the Great cutting the Gordian Knot, and the Popes whose coat of arms became the Borromean rings.\",\n \"Isaac Newton, standing on the shoulders of Galileo and beyond Papal influence, used Greek ellipses to explain planetary motion and left to posterity the 3-body problem.\",\n \"The 3 Borromean rings of polished steel at the entrance to the Newton Institute in CambridgeImage reproduced by kind permission of Peter Robinson, Bradshaw Foundation ©John Robinson/Edition Limitée Paris 2007 provided me with inspiration, in August 2017, at the conference celebrating the 60th birthday of Simon Donaldson.\",\n \"Tantalized by Misha Gromov's lecture on scalar curvature, I decided to combine the two topics and relate them to Newton's planetary orbits.\",\n \"$\\\\includegraphics [width=6cm]{ini.jpg}$ Visionary ideas are what we mathematicians grapple with, but illustrative special examples help the understanding.\",\n \"They also lead us by analogy to more general and realistic examples.\",\n \"The master whose style I try to emulate is the greatest mathematical physicist since Newton, James Clerk Maxwell.\",\n \"He understood dynamical stability, with its topological underpinnings, more thoroughly than anyone else and I offer this short note as a tribute to him.\"\n ],\n [\n \"The Topology and Geometry of Borromean Rings\",\n \"The linking of circles in 3-space is familiar to us all and led to the formation of chains.\",\n \"Gauss, Faraday and Maxwell explained electro-magnetism in terms of such links.\",\n \"The beauty and subtlety of the 3 Borromean rings is that each pair are unlinked but the triple holds together.\",\n \"This mysterious property had, in medieval times, theological (even Trinitarian) overtones which is why it appealed to Popes, and would-be popes, like the Borromeo family.\",\n \"Newton might have heard of this mystery but, as a secret unitarian in Trinity College, he would have kept off such dangerous ground.\",\n \"We all know that geometry and topology are siblings.\",\n \"Geometry is about measurement and topology is about shape.\",\n \"This family unity is embodied in the great theorem of Gauss, which starts locally with the area of spherical triangles, or polygons, and concludes globally with the formula for the Euler member of a closed surface as the average of the Gauss or scalar curvature (divided by $2\\\\pi $ ).\",\n \"It is natural to ask if there is a similar formula relating the geometry and topology of Borromean rings.\",\n \"For this to make sense the 3 rings must be geometrical, not just topological.\",\n \"Each ring should be planar or spherical, with a boundary and the 3 rings should be placed in 3-space so that their mutual topology is Borromean.\",\n \"Looking at the sculptured steel outside the Newton Institute raises a challenging question, rather like the Rubik Cube or the model sailing boats in wine bottles.\",\n \"How are they made or dismantled?\",\n \"If you are the artist commissioned to make the sculpture, what precise instructions do you give the craftsman in the workshop?\",\n \"For example, suppose you wanted to make the sculpture on the left of the Institute consisting of 3 squares of flat steel, each with a smaller square hole inside it?\",\n \"The craftsman would need exact measurements and their tolerance : how accurate does he have to be?\",\n \"This sounds, and is, standard engineering design.\",\n \"The sculptor, or his mathematical consultant, has to do \\\"the sums\\\" and instruct the craftsman, confident that it will give a stable work of art.\",\n \"Having done this once for the first steel sculpture at the Newton Institute, the design team then have to go through a similar procedure with the square replaced by a rhombus or a triangle, for the two sculptures on the right.\",\n \"As your sculptures attract attention the team might be flooded with orders, but each client would have different requirements.\",\n \"Some might want polygons with many sides, perhaps infinitely many, so that polygons become ellipses (with Newtonian approval).\",\n \"Clearly you will need an efficient algorithm to implement the process.\",\n \"Does your firm have mathematical consultants of the necessary quality?\",\n \"If they studied at Trinity or St. John's (the parent Colleges of the Newton Institute) then you might be in luck.\",\n \"As a Trinity man I feel it incumbent on me to try.\",\n \"The next sections provide my mathematical report on these problems.\"\n ],\n [\n \"The Iso-Perimetric Inequality\",\n \"In the spirit of Maxwell I will consider a simple model based on the actual steel sculptures at Newton's Institute.\",\n \"My Borromean rings will be modelled by flat regular polygons $C_i $ $(i=1,2,3)$ with $N$ sides ($N\\\\geqslant 3$ ), having a border of width $\\\\delta $ .\",\n \"See the figure for $N=4$ and one polygon $C$ .\",\n \"$\\\\includegraphics [width=10cm]{square.pdf}$ The engineering problem is to take 3 such regular $N$ -gons $C_1,C_2,C_3$ of the same width $\\\\delta $ and then place them in 3-space (or on a 3-sphere) so that they form a Borromean triple.\",\n \"Clearly the holes in the middle of the $N$ -gons cannot be too small.\",\n \"We look for an inequality which will give the critical size $d$ , so that a Borromean triple of such (flat bordered) $N$ -gons can be constructed provided $\\\\delta 3$ of width $\\\\delta $ which are planar or spherical.\",\n \"The figure in section 3 shows the planar case for $N=4$ .\",\n \"The question we ask is (5.1) Can this data be assembled in 3-space ($\\\\mathbb {R}^3$ or $S^3$ ) by translation and scale changes to form a Borromean triple of $N$ -gons?\",\n \"The trigonometric calculations in Section 3 show that this is possible provided we have the inequality (5.2) $A_N(r,\\\\delta ) > r\\\\delta \\\\sin (2\\\\pi /N)$ derived from the equality (3.4).\",\n \"The \\\"error\\\" is the quadratic term (5.3) $\\\\delta ^2/2.N \\\\sin (2\\\\pi /N)$ This answers the questions raised at the end of section 2.\",\n \"The desired bound $d$ depends on the scale set by the radius $r$ and, for regular $N$ -gons $d=2r$ and we just get the inequality $\\\\delta /r \\\\leqslant 2$ .\",\n \"Embodied in planar steel, as at the Newton Institute, formula (5.2) gives the engineering team the precise measurements needed, for any $N$ , and the tolerance or error.\",\n \"As the width tends to zero the tolerance also tends to zero so that, in the (conformal) limit, steel is no longer a suitable material and the Borromean rings have to made of silk thread.\",\n \"For irregular Borromean rings the exact formula (5.3) will be replaced by more general ones, but the leading (topological) term will remain.\",\n \"While (5.2) is elementary trigonometry, because of the symmetry, its natural context is that of Minkowski's mixed areas, or the Gauss scalar curvature.\",\n \"Generalizing (5.2) was therefore standard geometry over a century ago.\",\n \"In more recent times the topology and geometry of Borromean triples has become equally standard though more sophisticated.\",\n \"Different experts develop different machinery and do not always use the same language.\",\n \"For applied mathematicians, engineers or for Olympic pole vaulters, the position of the centre of gravity is crucial.\",\n \"Borromean geometry, based on the stability that comes from topology, may help win gold medals.\",\n \"Maxwell would have understood it all and he would have explained how a few well-chosen examples are all you need.\",\n \"Since he is no longer here in person I have tried to stand in for him, like an actor who stands in for Charles Dickens.\",\n \"3 planets in elliptical orbits may have long term stability, held together by mysterious topology, if the parameters are right.\",\n \"It seems to me more than likely that some versions of the famous 3-body problem will turn out to be Borromean, perhaps in the rings of Saturn, studied by Maxwell or even in the Red Spot of Jupiter.\",\n \".\",\n \"I hope I have whetted the appetite of the reader, and that the Newton Institute will continue to inspire natural philosophers for decades to come.\",\n \"But let me conclude with two final sections extending the scope of the discussion.\"\n ],\n [\n \"Force and Centres\",\n \"The Borromean rings at the Newton Institute were our starting point.\",\n \"For artistic reasons they were highly symmetrical with an obvious centre.\",\n \"But, as we start to deform them, their symmetry and centre get lost.\",\n \"How do we track the centre?\",\n \"Newton found the answer by going back to Apollonius and conic sections.\",\n \"The focus (one of a pair) was the appropriate centre and the eccentricity of the ellipse increased as the focus moved away from the centre.\",\n \"What Newton realized, and Apollonius did not, was that the focus was the centre of gravity.\",\n \"In other words, gravitational force changes the geometry, and planetary orbits reflect this changed geometry.\",\n \"On the time scale of Earth-years the gravitational attraction of the sun does not change and planetary orbits are, with modest variations, stable for millennia.\",\n \"Mathematicians, from Laplace onwards, wondered about stability in the “long-term”.\",\n \"Would the solar system survive forever?\",\n \"To an astronomer or an astrophysicist, this was an idealization too far.\",\n \"A time-scale may not matter to the pure mathematician but, if it is not commensurate with physical time-scales, it is irrelevant: the “long-term” will never arrive.\",\n \"Such metaphysical questions acquire more significance in two ways.\",\n \"On the one hand, if gravity were replaced by electro-magnetism, also based on the inverse-square law, planets could be replaced by electrons travelling in orbits at velocities close to the speed of light.\",\n \"This dramatic change of scales brings mathematical time-scales down to physical time-scales and long-term stability becomes important for engineers at CERN, as explained to me many years ago by Jürgen Moser.\",\n \"Another more fundamental change came, a century ago, with Einstein's Theory of General Relativity.\",\n \"The inverse-square law is just the Newtonian linear approximation but non-linear GR has to incorporate Einsteinian corrections, even for the solar system, and the accuracy of GPS requires such corrections.\",\n \"If we also want to understand the magnetic fields that generate solar flares and affect our weather, then both kinds of corrections are necessary.\",\n \"The LIGO triangles designed to detect gravitational waves were testing not just the statics of GR but also its dynamics, which brings in time, so Borromean rings might acquire a cosmic significance.\",\n \"This may not just be “$\\\\pi $ in the SKY” since respected astronomers have been searching for “cosmic strings” both in theory and in observation.\",\n \"The lessons I draw from this global (or cosmic) perspective are the following (6.1) the importance of forces in geometry (6.2) the key role of the centres of force (6.3) the role of centres in dynamical stability (6.4) the importance of effective criteria for stability (6.5) the understanding of different force scales (6.6) the role of topology in qualitative understanding of statics and dynamics (6.7) the role of geometry in quantitative versions of (6.6) with explicit tolerances.\",\n \"Needless to say most of this is known to most scientists most of the time.\",\n \"But I suspect that not all is known to all scientists all the time.\",\n \"Perhaps it is time for all natural philosophers to get under the Newton Apple Tree and admire the Borromean Rings, just as our Druid ancestors admired the Art and Architecture of Stonehenge.\",\n \"In the next and final section I will descend from Heaven to Earth.\"\n ],\n [\n \"Down to Earth\",\n \"The 3 Borromean rings from which we started are the simplest geometric examples in 3-space.\",\n \"They can be generalized in infinitely many ways and we are free to choose the model that fits our needs and our resources.\",\n \"There is no single multi-purpose model which will do everything.\",\n \"I have already indicated that regular polygons can be replaced by convex polygons and that the parameters can all be allowed to vary.\",\n \"For a bordered polygon with width $\\\\delta $ we started with 3 polygons $C_i$ having the same width, but we can now allow different widths $\\\\delta _i$ .\",\n \"Ellipses appear when the polygon has an infinite number of sides, so we can end up with 3 ellipses of different widths sitting in 3-space.\",\n \"Topologically they are either Borromean rings or non-Borromean rings, the Papal version of Hamlet's dilemma : To $B$ or not to $B$.\",\n \"Mathematical techniques now exist to answer Hamlet.\",\n \"Shakespeare was an actor as well as a playwright, so he would have embraced the new technology.\",\n \"The new technology discussed at Donaldson 60 would have solved Hamlet's dilemma, depriving the English language of its favourite quotation.\",\n \"All Hamlet had to do was to find out if the 3 rings could balance stably in the earth's gravitational field.\",\n \"The nearby apple tree would have shown him that the small convex set (the tolerance), defined by the widths $\\\\delta _i$ , should contain the centre of gravity.\",\n \"Fortunately the flat fenland of Cambridge would ensure that the gravitational field had essentially the same strength on both sides of the Institute.\",\n \"Apples could therefore be substituted by steel rings, though the Borromean property would need a high speed camera to capture any entanglement of 3 falling apples, the dynamics of which were famously observed by the occupant of Woolsthorpe manor.\",\n \"The engineers would not need to incorporate the relativistic Einsteinian corrections of section 6, since auroras are rarely seen at latitude $52^\\\\circ $ North.\",\n \"We thus end up with an effective numerical criterion for the 3 rings to be Borromean or non-Borromean.\",\n \"This describes all the steel sculptures that might be constructed.\",\n \"As pointed out in section 3, Minkowski only dealt with convex bodies, which like spheres, have positive scalar curvature, while the Gauss formula allows regions of negative curvature.\",\n \"But passing to convex hulls swallows up the negative regions.\",\n \"There are many ways of understanding this difference.\",\n \"(7.1) A general Morse function $f$ on a closed surface will have maxima, minima and saddle points.\",\n \"(7.2) The Hessian of $f$ at critical points distinguishes between the 3 cases in (7.1).\",\n \"It defines local geometries with scalar curvature positive at the maxima and minima, but negative at the saddle points.\",\n \"(7.3) In the regions of positive curvature the Minkowski theory agrees with Gauss.\",\n \"(7.4) The Gauss integral formula is about averages and Euler's topological invariant gives the dominant term.\",\n \"(7.5) An average smooths out local irregularities, and gives good results when the averaging process is tailored to the nature of the irregularities.\",\n \"(7.6) In all branches of science local irregularities occur and have to be averaged.\",\n \"Atoms in a molecule, planets in the solar system or galaxies in the cosmos are all “irregularities” in this sense.\",\n \"(7.7) Analysts have developed very refined techniques of smoothing irregularities which lead to stochastic differential equations.\",\n \"(7.8) Geometers have developed techniques of stability descending from Minkowski and used in Mirror Symmetry.\",\n \"(7.9) Physicists have techniques called renormalization which provide powerful tools in quantum field theory.\",\n \"(7.10) Number theorists since Euler and Riemann have used similar ideas to study the “irregular” distribution of primes, with the Riemann Hypothesis as the ultimate goal.\",\n \"References There is a vast literature on geometry and dynamics, much too extensive to list here.\",\n \"The great classic is of course Maxwell's Smith Prize Essay on Saturn's Rings.\",\n \"For more recent literature I can at least point to some of my own papers which are relevant in various ways.\",\n \"M.F.Atiyah, Convexity and commuting Hamiltonians, Bull.\",\n \"Lond.\",\n \"Math.\",\n \"Soc.\",\n \"14 (1981), 1–15.\",\n \"–, The Non-Existent Complex 6-Sphere, arXiv:1610.09366v2 [math.DG] (2016) –, Geometric Models of Helium, arXiv:1703.02532v1 [physics.gen-ph] (2017) –, Groups of Odd Order and Galois Theory, to appear.\",\n \"– and J.Berndt, Projective planes, Severi varieties and spheres, Surveys in Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, International Press, Somerville, MA (2003).\",\n \"–, R.Bott and L.Gårding, Lacunas for hyperbolic differential operators with constant coefficients I., Acta Math.\",\n \"124 (1970), 109–189.\",\n \"– and G.Segal, Twisted K-theory and Cohomology, in Inspired by S.S. Chern: A Memorial Volume in Honor of a Great Mathematician (Editor: P. A. Griffiths), Nankai Tracts in Mathematics Vol.\",\n \"11, World Scientific Publishing Co Inc. (2007).\",\n \"J.C.Maxwell, On the stability of the motion of Saturn's rings, An essay which obtained the Adams Prize in 1856 in the University of Cambridge, Macmillan (1859) Available online at https://archive.org/details/onstabilityofmot00maxw\"\n ]\n]"},"id":{"kind":"string","value":"1709.01539"}}},{"rowIdx":697,"cells":{"text":{"kind":"list like","value":[["Sequence Prediction with Neural Segmental Models"],["Abstract Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.","Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.","However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.","In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.","We also show that instead of including more features to obtain better accuracy, segmental cascades can be used to speed up training and decoding.","Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.","In the first stage, a frame classifier is trained with manual alignments, and then in the second stage, segmental models are trained with manual alignments and the out- puts of the frame classifier.","However, obtaining manual alignments are time-consuming and expensive.","We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.","We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification.","We present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.","Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks."],[">=latex SEQUENCE PREDICTION WITH NEURAL SEGMENTAL MODELS BY HAO TANG A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science at the TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO Chicago, Illinois September, 2017 Thesis Committee: Karen Livescu (Thesis Advisor) Kevin Gimpel David McAllester Eric Fosler-Lussier James Glass left=0in,right=0in,top=0in,bottom=0in Figure: NO_CAPTION Sequence Prediction with Neural Segmental Models by Hao Tang Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.","Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.","However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.","In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.","Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.","In the first stage, a frame classifier is trained, and in the second stage, segmental models are trained with the outputs of the frame classifier.","Both training stages require manual alignments, and obtaining manual alignments are time-consuming and expensive.","We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.","We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification, and present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.","Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks.","Thesis Supervisor: Karen Livescu Title: Associate Professor First, I would like to thank my advisor, Karen Livescu.","Karen has always been very patient, letting me make mistakes after mistakes, mistakes that she can foresee, mistakes that I have made many times already, mistakes that have no easy fix, so that I can learn from them.","She taught me when to explore novel ideas and when to persist and complete tedious tasks.","I have learned to think critically and argue constructively, while at the same time to be honest and critical to my own work.","There are countless other things I have learned from her including phonetics, spectrogram reading, and English word usage, to name a few.","I can not express how much I appreciate her guidance and mentorship.","I am really fortunate to be her first PhD student.","Next, I want to thank my committee members, Kevin Gimpel, David McAllester, Eric Fosler-Lussier, and Jim Glass.","David's comments and questions always encourage me to view my work in a broader context and in the most general form possible.","I have also greatly benefited from his mathematical rigor through his talks, his lectures, and interactions with him, and have developed the habit of asking whether a mathematical expression is type checked or not.","I am grateful to Eric for his insightful comments.","I have benefited a lot from conversations we had in workshops and conferences.","I enjoyed his humorous and joyful attitude, and him pushing me towards the finish line harder than my advisor does.","I thank Jim for keeping the speech science aspect of my research in mind.","My research is heavily influenced by his prior work, and it is very special to work on something that he has been working on for decades.","I am deeply grateful to Kevin.","He is like my second advisor, caring and helpful in many ways.","I have benefited from his expertise in natural language processing (NLP), and discovered many similar approaches developed in parallel in both the NLP and the speech community.","Besides the technical content, he taught me to be humble yet to be bold when necessary.","I have also enjoyed his humorous comments on my writing before stressful deadlines.","It has been a great pleasure to have his company during my PhD.","I was fortunate to join Mark Hasegawa-Johnson's team in the second Jelinek summer workshop.","I have benefited from Mark's extensive knowledge in both speech science and machine learning.","It is fair to say that I always learn something new when talking to him.","I also thank him for his guidance after the workshop.","It has been a great pleasure to work with the people on the team.","In particular, I have benefited from interactions with Chunxi Liu, Preethi Jyothi, Amit Das, Vimal Manohar, Paul Hager, Tyler Kekona, and Rose Sloan.","People I met during the workshop, including Yi Luan, Yangfeng Ji, Lingpeng Kong, Guoguo Chen, and Trang Tran, also made the overall experience fun and pleasant.","In 2013, I worked as an intern with Shinji Watanabe at the Mitsubishi Electric Research Laboratories (MERL).","I have benefited from the interactions with him and other people at MERL, including John Hershey, Jonathan Le Roux, and Tim Marks.","In particular, I thank Shinji for taking care of me when I had Bell's palsy during the internship.","The fellow interns at MERL, including Niao He and Lingling Tao, also made the internship experience fun and rewarding.","My first research experience related to speech and language processing was acquired from Lin-Shan Lee's guidance in National Taiwan University back in 2009.","Lin-Shan, even with his always busy schedule, has been very caring and helpful.","It would not have been possible for me to pursue a PhD without his help and encouragement.","I thank the people in Toyota Technological Institute at Chicago (TTIC) for making my PhD journey wonderful.","In particular, I have benefited from Julia Chuzhoy's and Madhur Tulsiani's lectures, acquiring the necessary vocabulary to talk to theory people.","I thank Nati Srebro for grilling me during my qualifying exam, forcing me to always keep the precise language and mathematical terms in mind.","I enjoyed the interactions with Greg Shakhnarovich and his humor.","I thank Joseph Keshet for his guidance and mentorship during my first few years of PhD.","I have benefited interactions and collaboration with Liang Lu.","I thank the fellow students, including Behnam Tavakoli, Somaye Hashemifar, Taehwan Kim, Shubham Toshniwal, Shane Settle, Qingming Tang, Bowen Shi, Lifu Tu, Hai Wang, Renyu Zhang, Jian Yao, Jianzhu Ma, Xing Xu, Avleen Bijral, Payman Yadollahpour, Feng Zhao, Zhiyong Wang, Karthik Sridharan, and Peng Jian, for the interactions, cookie breaks, random interruptions, and random questions.","I will never regret choosing a cubicle that invites so many interruptions.","I am grateful to Weiran Wang.","It is fair to say that much of my work would not be possible without his help.","I have also greatly benefited from interactions with past post-docs, including Raman Arora and Herman Kamper.","I thank visiting students, Taiki Kawano and Takayuki Yamabe, from Toyota Technological Institute in Japan, for the fun time we had in Chicago and in Tokyo.","Thanks to Chrissy Novak, Adam Bohlander, and the rest of the staff for making my PhD life easy and smooth.","Lastly, I thank the president of TTIC, Sadaoki Furui, for his guidance, and for pushing me to exercise and to practice my tennis skills with him.","Thanks to my family in Taiwan for their support over the years.","Thanks to Hsin-Yu, for everything.","And thanks to my parents, whom I cannot possibly thank in words.","Segmental models, the subject of this thesis, are a family of models that predict sequences of segments.","Segmental models are designed for a wide range of sequence prediction tasks, such as named entity recognition [118], speech recognition [104], [35], and action recognition [122], [55].","Examples of these tasks are shown in Figure REF .","The input of a sequence prediction task is commonly represented as a sequence of real-valued vectors, and the output is a sequence of discrete labels.","The goal is to find a function that maps the input vectors to the output labels.","Every output label in the output sequence, such as a named entity, a phoneme, or an action, typically has a corresponding chunk of contiguous input vectors, called a segment.","Segments come in varying lengths.","As a consequence, the lengths of the output sequences typically do not match the lengths of the input sequences.","Figure: Examples of sequence prediction tasks.The first example is speech recognition, where the input is a sequence of acousticsignals and the output is a sequence of words.","The second example is American SignLanguage fingerspelling recognition, where the input is a sequence of images andthe output is a sequence of letters.","The third example is named entity recognition,where the input is a sequence of words and the output is the same sequence of wordswith named entities in parentheses.","In the example of speech recognition,the gray arrow connects the word “else” to its corresponding contiguous partin the input sequence.The predominant approach to solving these tasks is to break the varying-length segments into pieces so that the lengths of the input sequences match the lengths of the output sequences.","These smaller pieces are then assembled back into varying-length segments with an additional post-processing step.","Each individual element in the input sequence is referred to as a frame.","Models, such as hidden Markov models [112], [62], linear-chain conditional random fields [43], [99], and recurrent neural networks [40], that map input sequences to output sequences of the same lengths, are called frame-based models [104], [35].","To include a wider context, it is common to use windows of frames instead of individual frames for prediction.","Nevertheless, the number of windowed frames is still the same as the number of output labels.","Because frame-based models are conceptually simple and computationally cheap, they have enjoyed much success and received much attention in the past few decades.","Figure: Frame-based approaches and segment-based approaches for sequence prediction.Model accuracies for prediction tasks are heavily influenced by the set of features used to make prediction, where a feature is a real-value function of the input that correlates well with a label or a subset of labels.","Breaking up varying-length segments is an extremely successful heuristic, but it also makes extracting certain features from segments difficult.","For example, computing durations is difficult for frame-based models.","Variants of frame-based models, such as variable-duration HMMs [28], are proposed in order to overcome this difficulty, but it is impractical to construct a model for every new type of feature.","In contrast, segment-based approaches do not assume anything about the segments, so the users are free to use arbitrary information about the segments during prediction.","For example, given a segment with a precise start time and end time, computing its duration is trivial.","For speech recognition, energy distribution at the left and right boundaries are useful features.","For named entity recognition, we can check whether a phrase (segment) is in the dictionary.","The flexibility makes segment-based models appealing, and many promising results have been reported in the past [147], [104], [35].","A comparison of the two approaches is shown in Figure REF .","More formally, a segment is a tuple of a start time, an end time, and possibly a label.","At a high level, segmental models make predictions by first creating a search space consisting of connected segments, or paths.","Segments are given weights based on the start time, end time, and label.","The weight values reflect how well the labels match the input.","Finding the sequence of segments that best matches the input is equivalent to finding the maximum-weight path.","The prediction process is also referred to as decoding.","Note that the segment weights can be computed by any function as long as it only makes use of the start time, the end time, the label, and the input sequence.","An example is shown in Figure REF .","Figure: An example of a segmental model.","A search space is built based on theinput frames.","Each edge (segment) has a start time, an end time, and a label,which the weight function can make use of.","Once the weights of edgesare computed, decoding is the problem of finding the maximum-weight path.Despite their success and attractive properties, segmental models still fall behind frame-based models on many tasks, such as speech recognition [148].","There are many possible reasons behind this.","The choice of features extracted from the segments might play a role.","However, complex features are typically computationally expensive.","The large number of segments we need to consider prohibits the use of computationally expensive segment features.","The central goal of this thesis is to improve the efficiency of segmental models, allowing us to explore more computationally expensive features and to improve the performance of segmental models.","This thesis studies segmental models along three directions: segment features, inference, and learning.","We introduce discriminative segmental cascades to speed up inference, allowing us to explore computationally expensive segment representations, such as ones computed with neural networks.","We study how segmental models perform when they are trained with various loss functions.","Different loss functions have different training requirements, some of them allowing us to train segmental models without manual alignments.","We also study whether it is possible to train segmental models end to end from random initialization.","We focus on two tasks, phonetic recognition and American Sign Language fingerspelling recognition.","In this section, we briefly describe automatic speech recognition (ASR), a task for which segmental models have been studied extensively.","We describe a few features that were shown to be useful for disambiguating phonemes, the basic sound units that distinguish words.","We then describe why it is hard to incorporate these features in frame-based models, and some of the past efforts to overcome this difficulty.","In brief, human speech production is the process of mapping a sequence of discrete linguistic units, such as words or phonemes, into waveforms, and automatic speech recognition is about inverting this process, mapping speech waveforms to sequences of discrete linguistic units.","On one side of the process, we have speech waveforms, commonly represented as sequences of real-valued vectors called frames.","On the other side of the process, we have the discrete units, namely segments, representing phonemes or words.","A waveform can be represented as a sequence of real-valued vectors in multiple ways.","One common representation is the amount of sinusoids at different frequencies appearing in the signal.","A waveform is first converted into small overlapping chunks.","Typically the size of the chunk is 25ms, and a chunk is created every 10ms.","Each chunk of waveform is represented as a linear combination of different sinusoids at different frequencies.","In particular, a frame in this representation is the vector of coefficients of the linear combination.","This representation is known as the spectrogram.","Examples of spectrograms are shown in Figure REF .","Given a sequence of frames, the goal of ASR is to predict what is being said in the waveform, in the form of a sequence of discrete linguistic units, with words and phonemes being the two most common discrete units.","Phonemes are defined as the sound units that distinguish words [76], and a sequence of phonemes can be converted into words by looking up the pronunciations of words in a lexicon.","In most settings, the set of phonemes and the lexicon are assumed to be given.","A standard evaluation metric for ASR is the Levenshtein distance between the predicted label sequence and the ground-truth sequence.","Levenshstein distance measures the minimum number of edits that needs to be performed to transform the predicted label sequence to the ground-truth sequence; hence it is also called the edit distance.","Formally, the edit distance of two label sequences $y$ and $\\hat{y}$ is defined as $\\text{edit}(\\hat{y}, y) = \\frac{I + D + S}{|y|},$ where $|y|$ is the length of $y$ , $I$ , $D$ , and $S$ are the number of insertion, deletion, and substitution, respectively.","ASR is a difficult task due to the wide range of variabilities in the process of speech production.","Even in the most controlled setting, the same isolated word can hardly be pronounced the same way twice by the same speaker.","Speech production is a highly context-dependent process.","Phonemes are pronounced differently when the neighboring phonemes are different [76].","Similarly, words are pronounced differently in different contexts.","Different speakers pronounce words and phonemes differently due to the differences in their speech organs [80].","Different speaking styles also affect pronunciations.","For example, in conversational speech, words are seldom pronounced in the canonical way presented in the lexicon due to the casual speaking style [83].","All of the above variabilities make ASR difficult.","It is even more difficult when the speech signals are degraded by noise [54].","Since the goal of speech recognition is to predict words based on their pronunciations, much work has been dedicated to finding features, sometimes referred to as acoustic cues, that identify phonemes and differentiate phonemes.","See [47] and citations therein.","Some features are correlated with how humans identify phonemes [93].","In general, there are many such features that are useful for speech recognition.","Each feature alone is probably not enough to identify a phoneme, but a combination of features might be able to [47].","Below we describe two features as examples.","Duration is one of the features that differentiate phonemes.","For example, the duration of the long vowel /iy/The phonemes are written in ARPAbet.","in the word “seat” is typically longer than short vowels /ih/ in the word “sit” [106].","In this case, duration serves as a good feature to differentiate /iy/ from /ih/.","Unvoiced fricatives, such as /f/, /s/, and /sh/, typically have longer duration than voiced fricatives, such /v/, /z/, and /zh/ [136].","Durations have also been shown to help improve recognition results [5].","Perhaps the most salient feature for distinguishing phonemes by looking at the spectrogram is the distribution of energy [76].","For example, fricatives, such as /s/ and /f/, have evenly spread energy across frequency.","Stops, such as /t/ and /p/, have sudden sharp bursts of energy.","Bands of high energy in spectrograms, commonly referred to as formants, are another type of energy patterns useful for identifying vowels [76].","The first and second formants (counting from low frequencies) are commonly used to differentiate vowels [53].","Formants are also useful for speech recognition [57].","Integrating arbitrary segment features into frame-based models has always been considered a difficult task.","Integrating even just the duration requires nontrivial modification to the models [81].","For example, we can expand the label set with durations, but this approach only works for discrete features and generates unnecessarily large search spaces.","Integrating these segment features is also one of the driving forces behind the use and development of segmental models [147].","While hidden Markov models assume that frame labels follow a Markov process, hidden semi-Markov models assume that segments follow a Markov process.","Semi-Markov processes have been used to incorporate duration in frame-based models [116].","Hidden semi-Markov models have been applied to speech recognition with the same reason in mind [81], [117].","The segment-based speech recognizer SUMMIT, is designed with the goal of integrating arbitrary segment features [147].","Segmental models have also been applied to other sequence prediction tasks, such as named entity recognition [118], American Sign Language fingerspelling recognition [71], and action recognition [122], [55].","For named entity recognition, the input is a sequence of words, and the output is a sequence of named entity tags, such as person name, organization name, and location name.","For American Sign Language fingerspelling recognition, the input sequence is a sequence of video frames, and the output is a sequence of letters.","For action unit detection, the input sequence is a sequence of video frames, and the output is a sequence of actions, such as walk, jump, sit, and stand.","In general, these tasks follow a left-to-right (or right-to-left) order in both the input sequence and the output sequence.","A task is said to be linear or monotonic if the task has a left-to-right (or right-to-left) order.","One sequence prediction task that is not monotonic is translation [19].","Depending on the source and the target languages, the order of words in the source language can be different from the order in the target language.","In this thesis, we only focus on segmental models for monotonic sequence prediction tasks.","We have seen why segmental models are favored over frame-based models when incorporating complex segment features is needed.","However, the flexibility of segmental models comes with a price.","Enumerating segments of different lengths and of different labels can be time-consuming.","Consider an input sequence of length 300, and a label set size of 50.","The number of possible segments of length up to 30 is around $30 \\times 50 \\times 300 = 450000$ .","If we make only a single decision at every time point as for frame-based models, we only need to consider $50 \\times 300 = 15000$ different decisions.","The hypothesis space for segmental models is considerably larger, and as a consequence learning and inference for segmental models are significantly slower than for frame-based models.","Many researchers have attacked the problem of large hypothesis spaces either implicitly or explicitly.","The most common approach is to use frame-based models to generate a set of high-confidence segments, and then use segmental models to rescore the segments with segment features [35], [149].","[102] explicitly train a model for pruning segments, while [139] reuse the computation of features whenever possible.","The first approach needs to have two separate models and the second approach depends on the exact form of the features.","We would like to design segmental models that do not depend on other external models, and can handle large hypothesis spaces in a feature-oblivious manner.","Designing and implementing state-of-the-art frame-based models requires a significant amount of engineering effort.","Phonemes are modeled with three-state hidden Markov models [119].","Since phonemes are influenced heavily by nearby phonemes, phoneme labels that include the previous and the next phonemes, referred to as triphones, are introduced [119].","The large number of triphones makes estimating their probabilities difficult, so triphone parameters are shared using with decision trees [143].","These design decisions in model structure makes engineering difficult.","In contrast to frame-based models, the engineering effort in segmental models is put into designing segment features, while the model structure remains the same.","In sum, we aim to design segmental models that perform well on sequence prediction tasks, have a clean and modular mathematical definition, are efficient to train and to decode, and do not depend on other models.","The concept of segmental models was not explicitly defined before the late 1980s, and little work has had segmental models as the primary focus.","[141] were one of the early attempts in the 1970s to build speech recognizers based on segment features.","In the early 1980s, [23] developed a system for recognizing isolated English letters based on segment features.","Isolated digits as segments were considered in [75].","[147] introduced SUMMIT, a segment-based speech recognizer that can handle arbitrary segment features.","While the SUMMIT system was later formulated into a probabilistic framework [34], others were not probabilistic, and have very few parameters to be estimated.","The probabilistic view of segmental models can also be traced back to the 1980s.","As mentioned earlier, [116] introduced semi-Markov processes and the follow-up [117] introduced hidden semi-Markov models to the speech recognition community.","These two studies were mainly motivated by the need to include duration in frame-based models.","[103] formalized segmental models as a probabilistic framework to include arbitrary features, but the authors only used frame-based probabilities with one additional duration feature.","[115] and [29] further proposed segmental hidden Markov models, but the general form stayed mostly the same.","Segmental models proposed in these studies were generative, and were mostly restricted to frame-based Gaussian distributions.","This line of work has been summarized in [104].","Into the 2000s, studies of segmental models shifted from generative to discriminative.","[118] proposed semi-Markov conditional random fields (CRF), the first type of discriminative segmental models.","Semi-Markov CRFs require the use of manual alignments to train the models, making them less applicable to speech recognition.","[149] proposed to use a different training loss for training semi-Markov CRFs, marginalizing over all possible segmentations.","This approach alleviated the need for manual alignments to train segmental models.","Later, [150] used segmental models as second-pass rescoring models for word recognition, showcasing the flexibility of segmental models for incorporating a wide variety of features.","A segmental model is said to be a first-pass model if it exhaustively searches over the entire hypothesis space.","Before [148], all of the segmental models for speech recognition were not first-pass.","The early version of SUMMIT [147] used a heuristic approach to estimate phoneme boundaries, instead of searching over the entire search space.","A more recent version of SUMMIT [34] avoided searching exhaustively by using a frame-based model to generate high confidence hypotheses.","Semi-Markov CRFs are first-pass segmental models, but have only been applied to natural language tasks, where sequences are typically short and label set size is small.","[148] showed that it is feasible to use segmental models as first-pass speech recognizers.","Since then, studies of segmental models have moved on to first-pass recognition.","At the same time, advances in neural networks have allowed us to learn generic feature representations, so many studies have been devoted to finding better segment representations to replace hand-crafted features.","[49] improved upon [148] by using a 3-layer multilayer perceptron to produce phoneme probabilities.","[1] further improved the results by using a deep convolutional neural network, and it was also the first to train segmental models and neural networks end to end.","Along the same line, [85] proposed end-to-end segmental models with segment representations computed with long short-term memory networks.","In this thesis, we make the following contributions to the development of segmental models.","We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.","We develop improved understanding of training approaches and training requirements for segmental models.","We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.","We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.","We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.","Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.","We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.","We develop improved understanding of training approaches and training requirements for segmental models.","We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.","We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.","We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.","Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.","In Chapter 2, we review finite-state transducers (FST), an important tool for describing the hypothesis spaces of segmental models.","For the second part of Chapter 2, we review the essential components in speech recognizers, such as language models, lexicons, and hidden Markov models, represented with FSTs.","In Chapter 3, we formally define segmental models.","The definition is modular and covers many existing segmental models as special cases.","In Chapter 4, we introduce discriminative segmental cascades, which allow us to explore various segment representations while maintaining efficiency.","In Chapter 5, we study segmental models trained in different training conditions, comparing end-to-end training and multi-stage training.","In Chapter 6, we review several variants of segmental models, and describe how they relate to our definition of segmental models.","In Chapter 7, we propose a unified framework encompassing many end-to-end frame-based models and segmental models.","Finally, in Chapter 8, we discuss possible ways to extend segmental models.","This chapter describes finite-state transducers (FST), a useful tool for representing and manipulating the search space of a sequence prediction task.","This chapter also includes a complete example of a standard speech recognizer based on hidden Markov models represented as FSTs.","See [96] for a comprehensive review of FSTs and their algorithms, and [97] for their applications to speech recognition.","We define FSTs based on multigraphs (graphs that allow multiple edges between any two vertices) instead of regular graphs.","A multigraph $G$ is a tuple $(V, E, \\mathrm {tail}, \\mathrm {head})$ , where $V$ is a set of vertices, $E$ is a set of edges, $\\mathrm {tail}: E \\rightarrow V$ is a function that associates an edge to its tail vertex, and $\\mathrm {head}: E \\rightarrow V$ is a function that associates an edge to its head vertex.","Note that $E$ is not a subset of $V \\times V$ but a general set, because we allow many edges to have the same tail and head, and a pair of vertices is not enough to uniquely identify an edge.","An example of a multigraph is shown in Figure REF .","We say that $e_1 \\in E$ and $e_2 \\in E$ are connected if $\\mathrm {head}(e_1) = \\mathrm {tail}(e_2)$ .","In addition, we define a path of length $n$ as a sequence of connected edges $(e_1, e_2, \\dots , e_n)$ where $\\mathrm {head}(e_i) = \\mathrm {tail}(e_{i+1})$ for $i = 1, \\dots , n$ .","A sub-path is a sequence of connected edges within a path.","For example, $(e_3, e_4, e_5)$ is a sub-path of $(e_1, e_2, \\dots , e_7, e_8)$ .","A finite-state transducer is a tuple $(G, \\Sigma , \\Lambda , I, F, i, o, w)$ , where $G$ is a multigraph, $\\Sigma $ is a set of input symbols, $\\Lambda $ is a set of output symbols, $I \\subseteq V$ is a set of initial vertices, $F \\subseteq V$ is a set of final vertices, $i: E \\rightarrow \\Sigma $ is a function that associates an edge to its input symbol, $o: E \\rightarrow \\Lambda $ is a function that associates an edge to its output symbol, and $w: E \\rightarrow \\mathbb {R}$ is a function that puts weights on edges.","An example is shown in Figure REF .","Figure: A multigraph, where V={0,1,2,3}V = \\lbrace 0, 1, 2, 3\\rbrace and E={0,1,2,3,4}E = \\lbrace 0, 1, 2, 3, 4\\rbrace .The functions of tail and head are shown in the table on the right.Figure: An FST based on the multigraph in Figure ,with Σ={𝚊,𝚋,𝚌,𝚍,𝚎}\\Sigma = \\lbrace \\texttt {a}, \\texttt {b}, \\texttt {c}, \\texttt {d}, \\texttt {e}\\rbrace andΛ={𝙰,𝙱,𝙲,𝙳,𝙴}\\Lambda = \\lbrace \\texttt {A}, \\texttt {B}, \\texttt {C}, \\texttt {D}, \\texttt {E}\\rbrace .The initial vertex is shown in bold, and the final vertexis shown with a doubled circle, i.e., I={0}I = \\lbrace 0\\rbrace and F={3}F = \\lbrace 3\\rbrace .We use σ\\sigma :λ\\lambda /w to denote an edgewith input symbol σ\\sigma , output symbol λ\\lambda , and weight ww.The number in parentheses is an identifier of an edge.An FST can be seen as a function that maps strings to strings, where each path in the graph defines an input-output pair.","Specifically, a path $(e_1, e_2, \\dots , e_n)$ associates the input string $i(e_1)i(e_2)\\cdots i(e_n)$ with the output string $o(e_1)o(e_2)\\cdots o(e_n)$ .","For example, the FST shown in Figure REF defines the mapping $\\lbrace (\\texttt {ad}, \\texttt {AD}), (\\texttt {bd}, \\texttt {BD}),(\\texttt {ce}, \\texttt {CE})\\rbrace $ .","There is a special symbol called the empty symbol, denoted $\\epsilon $ .","Any string concatenated with an empty symbol is itself, i.e., $\\sigma _1\\sigma _2\\cdots \\sigma _n\\epsilon = \\epsilon \\sigma _1\\sigma _2\\cdots \\sigma _n= \\sigma _1\\sigma _2\\cdots \\sigma _n$ .","If the input symbol of an edge is the empty symbol, then we can traverse the edge without consuming any input.","If the output symbol of an edge is the empty symbol, then we do not produce any output when traversing the edge.","As an example, the FST in Figure REF defines the mapping $\\lbrace (\\texttt {ad}, \\texttt {AD}), (\\texttt {a}, \\texttt {AF}),(\\texttt {bd}, \\texttt {BD}), (\\texttt {b}, \\texttt {BF}), (\\texttt {ce}, \\texttt {CE})\\rbrace $ .","Figure: An FST with empty symbols.","The edgewith the empty symbol is highlighted in red.We assume there is one unique start vertex and one unique end vertex, i.e., $|I| = |F| = 1$ .","If an FST has more than one initial vertices, we can always create an additional vertex and connect all initial vertices to the additional vertex with empty symbols (and possibly zero weights) without affecting the string mapping of the FST.","For convenience, we also define $\\text{in}(v) = \\lbrace e \\in E : \\mathrm {head}(e) = v\\rbrace $ and $\\text{out}(v) = \\lbrace e \\in E: \\mathrm {tail}(e) = v\\rbrace $ .","We have reviewed FSTs as graphs and string functions.","The third view of FSTs is as state machines.","Take the FST in Figure REF for example.","To get the output AD from the input ad, we first feed the character a and traverse from vertex 0 to vertex 1, and then feed the character d and traverse from vertex 1 to vertex 3.","Similarly, to get BF from b, we first feed the character b and traverse from vertex 0 to vertex 1.","Due to the empty symbol $\\epsilon $ , we get F and traverse from vertex 1 to vertex 3 without feeding any character.","The view of state machines is especially useful when we talk about FST compositions.","Other than just manipulating strings, we are also interested in the weights on the edges, especially when paths are considered.","We introduce two operators $\\otimes $ and $\\oplus $ , where $\\otimes $ is for combining edge weights and $\\oplus $ is for combining path weights.","For example, by traversing the path $(0, 3)$ that produces AD in Figure REF , we collect the weight 0.1 from edge 0 and 0.4 from edge 3, and the weight of the path $(0, 3)$ is defined as $w(0) \\otimes w(3) = 0.1 \\otimes 0.4$ .","Suppose in general $S$ is the set of weights.","It is desirable to have certain properties, such as commutativity and associativity, for the two operators.","For $a \\in S$ , $b \\in S$ , and $c \\in S$ , we say that an operator $*$ is commutative if $a * b = b * a$ , and that an operator $*$ is associative if $(a * b) * c = a * (b * c)$ .","The element $0 \\in S$ is an identity with respect to the operator $*$ if $0 * a = a * 0 = a$ .","The element $0 \\in S$ annihilates $S$ with respect to $*$ if $0 * a = a * 0 = 0$ .","We say that the operator $*$ distributes over the operator $+$ if $a * (b + c) = a * b + a * c$ and $(a + b) * c = a * c + b * c$ .","The tuple $(S, \\oplus , \\otimes )$ is a semiring [95] if $\\oplus $ is associative and commutative with identity $0 \\in S$ , $\\otimes $ is associative with identity $1 \\in S$ , $\\otimes $ distributes over $\\oplus $ , and the element $0 \\in S$ annihilates $S$ with respect to $\\otimes $ .","An example of semiring is $(\\mathbb {R}, +, \\times )$ with 0 being the identity of $+$ , and 1 being the identity of $\\times $ .","Another example is the tropical semiring $(\\mathbb {R} \\cup \\lbrace \\infty \\rbrace , \\min , +)$ with $\\infty $ being the identity of $\\min $ , and 0 being the identity of $+$ .","Yet another example is the log semiring $(\\mathbb {R} \\cup \\lbrace -\\infty \\rbrace , \\text{logadd}, +)$ where $\\text{logadd}(a, b) = \\log (\\exp (a) + \\exp (b))$ , $-\\infty $ is the identity of $\\text{logadd}$ , and 0 is the identity of $+$ .","Defining the operators for combining weights in a general way encourages algorithm reuse.","As we will see in later chapters, FST algorithms tend to look very similar, and typically the only difference is how the weights are combined.","By using general operators, we get a family of algorithms.","In other words, we create algorithms almost for free by choosing a proper semiring.","Given an FST, we are interested in finding a shortest path from the initial vertex to the final vertex.","For example, if the weights are negative log probabilities traversing from one vertex to another, then a shortest path corresponds to one of the most likely paths.","We assume there are multiple shortest paths, but finding one of them is enough.","We also assume the underlying graph is acyclic, meaning that every path can only traverse a vertex at most once.","For acyclic graphs, we have a nice necessary condition for shortest paths—every sub-path within a shortest path is also a shortest path.","The argument for the necessary condition is simple.","If a sub-path is not a shortest path, then we can always substitute the sub-path with another shorter sub-path to create a shorter path overall.","The necessary condition can be viewed as a recursive condition.","Let $\\mathcal {P}(u, v)$ be the set of paths from vertex $u$ to vertex $v$ , and let $w(p)$ be a shorthand of $\\sum _{e \\in p} w(e)$ .","Suppose we want to compute a shortest path from the vertex $u$ to vertex $v$ .","We examine the set of edges $\\text{in}(v) = \\lbrace e \\in E: \\mathrm {head}(e) = v\\rbrace $ leading into vertex $v$ .","If an edge $e \\in \\text{in}(v)$ is part of the shortest path, then by the necessary condition every path from $u$ to $\\mathrm {tail}(e)$ should also be a shortest path.","In other words, $\\min _{p \\in \\mathcal {P}(u, v)} w(p)& = \\min _{e \\in \\text{in}(v)} \\min _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))}[ w(e) + w(p^{\\prime }) ] \\\\& = \\min _{e \\in \\text{in}(v)}\\Bigg [ w(e) + \\min _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))} w(p^{\\prime }) \\Bigg ]$ The recursive condition can be computed efficiently with dynamic programming if we store the shortest distance from vertex $u$ to every other vertex.","Specifically, let $d(v)$ be the shortest distance from $u$ to $v$ , i.e., $d(v) = \\min _{p \\in \\mathcal {P}(u, v)} w(p).$ By the recursive condition, we have $d(v) = \\min _{e \\in \\text{in}(v)} [ w(e) + d(\\mathrm {tail}(e)) ].","$ Before computing the minimization in (REF ), we need to make sure the shortest sub-paths are computed.","Formally, to compute $d(v)$ for vertex $v$ , we need to make sure $d(\\mathrm {tail}(e))$ are computed for $e \\in \\text{in}(v)$ .","In other words, if there is a path from vertex $w$ to vertex $v$ , then $d(w)$ should be computed before $d(v)$ .","We are looking for an order in which $w$ should come before $v$ if there is a path from $w$ to $v$ .","An order in which $w$ comes before $v$ if there is a path from $w$ to $v$ is called a topological order.","To be precise, a topological order is a function $f: V \\rightarrow \\lbrace 1, 2, \\dots , |V|\\rbrace $ such that $f(w) < f(v)$ if there is a path from $w$ to $v$ .","Given an directed acyclic graph, there exist many topological orders.","Any one of them would suffice for our dynamic programming.","One way to find a topological order is to run depth-first search (DFS) on the reversed graph.","The depth-first search algorithm is shown in Algorithm REF .","We maintain a stack $S$ .","We say that a vertex is traversed if it has been put on $S$ , and we use a set $T$ to track the traversed vertices.","When a vertex $v$ is put on stack, we also put a variable indicating whether we have tried to put its neighbors on the stack, where the set of neighbors is defined by $\\text{in}(\\cdot )$ .","We say that a vertex is expanded if we have tried to put its neighbors on the stack.","It is not hard to see that every vertex is put on the stack twice, once before it is expanded and once after.","When a vertex $v$ popped out from the stack is expanded, all vertices that can be traversed from $v$ are also expanded.","Since we put a vertex $v$ in $O$ when $v$ and all vertices traversable from $v$ are expanded, the order $O$ is exactly a topological order.","Algorithm REF is an instance of DFS specifically for computing a topological order by defining the set of neighbors with $\\text{in}(\\cdot )$ .","The algorithm can be modified for other purposes, for example, letting the set of neighbors be $\\text{out}(\\cdot )$ (and changing $\\mathrm {tail}(\\cdot )$ to $\\mathrm {head}(\\cdot )$ accordingly) if we want to search the graph in a forward direction instead of backwards.","Due to the use of a stack, DFS prefers to follow one path until none of the vertices can be expanded; hence the name depth-first search.","Depth-First Search (DFS) Let $r$ be the vertex we start to search.","A list of vertices $O$ is returned in topological order.","$S = \\lbrace (r, \\texttt {false})\\rbrace $ , $T = \\lbrace r\\rbrace $ , $O = ()$ $S$ is not empty pop $(v, z)$ from $S$ $z$ append $v$ to $O$ push $(v, \\texttt {true})$ to $S$ $e \\in \\text{in}(v)$ Expand $v$ .","$\\mathrm {tail}(e) \\notin T$ push $(\\mathrm {tail}(e), \\texttt {false})$ to $S$ $\\mathrm {tail}(e)$ is traversed.","add $v$ to $T$ Shortest-Path Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.","The map $d$ contains the shortest distances to $v$ for all vertices.","Let $O = (o_1, o_2, \\dots , o_n)$ be a topological order by running DFS starting from $t$ .","$i = 1, \\dots , n$ $d(o_i) = \\min _{e \\in \\text{in}(o_i)} [ w(e) + d(\\mathrm {tail}(e)) ]$ $\\pi (o_i) = \\operatornamewithlimits{\\arg \\!\\min }_{e \\in \\text{in}(o_i)} [ w(e) + d(\\mathrm {tail}(e)) ]$ Backtracking Let $s$ be the initial vertex and $t$ be the final vertex.","A shortest path $p$ from $s$ to $t$ .","$v \\leftarrow t$ $p = ()$ $v \\ne s$ add $\\pi (v)$ to $p$ $v \\leftarrow \\mathrm {tail}(\\pi (v))$ reverse $p$ Generalized Shortest-Distance Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.","The map $d$ contains the shortest distances to $v$ for all vertices.","Let $O = (o_1, o_2, \\dots , o_n)$ be a topological order by running DFS starting from $t$ .","$i = 1, \\dots , n$ $d(o_i) = \\bigoplus _{e \\in \\text{in}(o_i)} [ w(e) \\otimes d(\\mathrm {tail}(e)) ]$ Up to this point, we are interested in finding a path that is shortest.","The weight of a path is defined to be the sum of the edge weights, and path weights are combined with the $\\min $ operator.","These operators are the ones used in the tropical semiring, and can be generalized to other semirings, where edge weights are combined with $\\otimes $ and path weights are combined with $\\oplus $ .","To be precise, the weight of a path $p$ is defined as $w(p) = \\bigotimes _{e \\in p} w(e)$ , and the distance can be written as $d(v) & = \\bigoplus _{p \\in \\mathcal {P}(u, v)} w(p)= \\bigoplus _{p \\in \\mathcal {P}(u, v)} \\bigotimes _{e \\in p} w(e)= \\bigoplus _{e \\in \\text{in}(v)} \\bigoplus _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))}[ w(e) \\otimes w(p^{\\prime }) ] \\\\& = \\bigoplus _{e \\in \\text{in}(v)} \\left[ w(e) \\otimes \\bigoplus _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))} w(p^{\\prime }) \\right] \\\\& = \\bigoplus _{e \\in \\text{in}(v)} [w(e) \\otimes d(\\mathrm {tail}(e))].$ The generalized shortest-distance algorithm is shown in Algorithm REF .","See [95] for a general treatment of shortest-distance algorithms.","The recursive condition (REF ) only gives us the distance to the final vertex.","To obtain the path, we can record down the edge that achieves the minimum while computing the distances.","Specifically, let $\\pi (v)$ be the edge that achieves the minimum, i.e., $\\pi (v) = \\operatornamewithlimits{\\arg \\!\\min }_{e \\in \\text{in}(v)} [ w(e) + d(\\mathrm {tail}(e)) ].$ We can iteratively collect the optimal edge with $\\pi $ .","Since the algorithm goes from the final vertex to the initial vertex, it is commonly called backtracking.","The final shortest-path algorithm and the backtracking algorithm are shown in Algorithm REF and REF .","Recall that FSTs can be regarded as functions that map strings to strings.","It is natural to have multiple FSTs with one FST consuming the outputs of another FST, similar to function composition.","Let $T(x, y)$ be the weight of a path in the FST $T$ consuming $x$ as input and producing $y$ as output.","For any two FSTs $T_1$ and $T_2$ , the weight of a path with input $x$ and output $y$ in the composed FST $T_1 \\circ T_2$ is defined as $ (T_1 \\circ T_2)(x, y) = \\bigoplus _{z \\in \\mathcal {Z}(x)} T_1(x, z) \\otimes T_2(z, y),$ where $\\mathcal {Z}(x)$ is the set of output strings that can be produced by feeding $x$ to $T_1$ [2].","The weight is undefined if $\\mathcal {Z}(x)$ is an empty set or there is no path with input $z$ and output $y$ in $T_2$ .","The definition of (REF ) does not provide an explicit structure of the composed FST.","In this thesis, we prefer another approach to composing FSTs which we term structured composition [130].","Formally, the structured composition (or $\\sigma $ -composition) of two FSTs $T_1 = (G_1, \\Sigma _1, \\Lambda _1, I_1, F_1, i_1, o_1, w_1)$ with $G_1 = (V_1, E_1, \\mathrm {tail}_1, \\mathrm {head}_1)$ and $T_2 = (G_2, \\Sigma _2, \\Lambda _2, I_2, F_2, i_2, o_2,w_2)$ with $G_2 = (V_2, E_2, \\mathrm {tail}_2, \\mathrm {head}_2)$ is defined as $T_1 \\circ _\\sigma T_2 = (G, \\Sigma , \\Lambda , I,F, i, o, w)$ where $G = (V, E, \\mathrm {tail}, \\mathrm {head})$ with the following constraints.","$ V & = V_1 \\times V_2 &E & = \\Big \\lbrace \\langle e_1, e_2 \\rangle \\in E_1 \\times E_2 : o_1(e_1) = i_2(e_2) \\Big \\rbrace $ $\\Sigma & = \\Sigma _1 &i(\\langle e_1, e_2 \\rangle ) & = i_1(e_1) \\\\\\Lambda & = \\Lambda _2 &o(\\langle e_1, e_2 \\rangle ) & = o_2(e_2) \\\\I & = I_1 \\times I_2 &\\mathrm {tail}(\\langle e_1, e_2 \\rangle ) & = \\langle \\mathrm {tail}_1(e_1), \\mathrm {tail}_2(e_2) \\rangle \\\\F & = F_1 \\times F_2 &\\mathrm {head}(\\langle e_1, e_2 \\rangle ) & = \\langle \\mathrm {head}_1(e_1), \\mathrm {head}_2(e_2) \\rangle $ The $\\sigma $ -composed graph is defined over pairs of vertices and pairs of edges.","Due to the edge constraints (REF ), the outputs from $T_1$ have to match the inputs to $T_2$ .","An example is shown in Figure REF .","Note that the definition of the weight function is left to the users.","One possible definition is to let $w(\\langle e_1, e_2 \\rangle ) = w_1(e_1) \\otimes w_2(e_2).$ As a consequence of (REF ), the outgoing edges and incoming edges of a vertex in the $\\sigma $ -composed FST can be computed locally with $\\text{out}_1(\\cdot )$ and $\\text{out}_2(\\cdot )$ .","Specifically, $ \\text{out}(\\langle v_1, v_2 \\rangle )= \\Big \\lbrace \\langle e_1, e_2 \\rangle : e_1 \\in \\text{out}_1(v_1), e_2 \\in \\text{out}_2(v_2),o_1(e_1) = i_2(e_2) \\Big \\rbrace .$ Since computing $\\text{out}(\\langle v_1, v_2 \\rangle )$ only requires access to edges connected to $v_1$ and $v_2$ , traversing the composed FST, for example with DFS (Algorithm REF ), only needs to keep track of the vertex pairs.","The edges can be computed on the fly and do not need to be stored in memory.","In general, composing FSTs without explicitly storing the entire graph is commonly known as on-the-fly composition or lazy composition.","Another way to compute structured composition is to view FSTs as state machines.","Take Figure REF for example.","We start from vertex 0 in both $T_1$ and $T_2$ .","To get to vertex 1 in $T_1$ , we can either produce A or produce B as output.","In $T_2$ , to go from vertex 0 to vertex 1, we only have the choice to consume A.","Therefore, in the composed FST $T_1 \\circ _\\sigma T_2$ we have the edge from $(0, 0)$ to $(1, 1)$ with input a and output $\\alpha $ .","By walking through the vertices one by one, we are essentially performing a search on both FSTs while computing (REF ) on the fly.","Figure: An example of structured composition.Vertices in T 1 ∘ σ T 2 T_1 \\circ _\\sigma T_2 are labeled with pairs of vertices from T 1 T_1 and T 2 T_2.We now have all the necessary tools to describe a basic speech recognizer based on FSTs.","Modern speech recognizers are based on a series of string function compositions.","Following [98], we represent the string functions as FSTs, and the compositions are realized with structured composition.","The first FST $U$ takes acoustic frames as inputs and produces a sequence of frame labels.","The second FST $H$ takes in frame labels and converts them into phonemes.","The third FST $L$ takes phonemes as inputs and produces words as outputs.","The fourth FST $G$ takes words as inputs and produces words as outputs.","Finally, the four FSTs are $\\sigma $ -composed into $D = U \\circ _\\sigma H \\circ _\\sigma L \\circ _\\sigma G.$ The path weights in $D$ are typically negative log probabilities, so the maximum-weight path is the most likely path based on the probabilities.","To find the maximum-weight path, we simply negate the weights and find the shortest path.","The tropical semiring is used to combine the negated weights.","In sum, to predict a sequence from a sequence of frames, we create the FST $D$ and find the maximum-weight path in $D$ .","Below we describe the four FSTs $U$ , $H$ , $L$ and $G$ in detail.","The dynamics of speech are commonly modeled by hidden Markov models (HMM).","The generative story of a hidden Markov model is as follows.","We first generate a state $y_1$ based on a prior distribution and generate a frame $x_1$ given $y_1$ .","A state $y_2$ is generated given $y_1$ , and a frame $x_2$ is generated given $y_2$ .","In general, for some $t = 2, \\dots , T$ , the state $y_t$ is generated given $y_{t-1}$ , and $x_t$ is generated given $y_t$ .","For a sequence of $T$ frames $x_1, \\dots , x_T$ , the probability distribution defined by an HMM is $p(x_{1:T}, y_{1:T}) = p(y_1) \\prod _{t=2}^T p(y_t | y_{t-1}) \\prod _{t=1}^T p(x_t | y_t),$ where $x_{1:T}$ is a shorthand for $x_1, \\dots , x_T$ , $x_t \\in \\mathbb {R}^d$ for some $d \\in \\mathbb {N}$ and $y_t \\in \\lbrace 1, \\dots , S\\rbrace $ for some $S \\in \\mathbb {N}$ for $t=1, \\dots , T$ .","An example of a four-frame HMM is shown in Figure REF .","Note that Figure REF is not an FST but a graphical model where vertices are random variables and edges represent conditional dependencies.","The learnable parameters in an HMM are $p(y_1)$ , $p(y_t | y_{t-1})$ , and $p(x_t | y_t)$ for $t = 1, \\dots , T$ .","The probabilities $p(y_t | y_{t-1})$ are commonly known as transition probabilities, and $p(x_t | y_t)$ emission probabilities.","Figure: A hidden Markov model for 4 frames x 1 ,x 2 ,x 3 ,x 4 x_1, x_2, x_3, x_4with hidden labels y 1 ,y 2 ,y 3 ,y 4 y_1, y_2, y_3, y_4.Note that this is a graphical model where vertices are random variablesand edges represent conditional dependencies, not an FST.The observed variables are shaded, and the unobserved are not.The transition probabilities $p(y_t | y_{t-1})$ can be represented as a square matrix $A$ of size $S \\times S$ in which $A_{ij} \\in [0, 1]$ is the probability of transitioning from state $i$ to state $j$ satisfying $\\sum _{j=1}^S A_{ij} = 1$ .","To forbid the transition from state $i$ to state $j$ , we simply let $A_{ij} = 0$ .","It is common to model a phoneme as a 3-state HMM, and parameterize $A$ as $\\begin{pmatrix}a_{11} & a_{12} & 0 \\\\0 & a_{22} & a_{23} \\\\0 & 0 & a_{33}\\end{pmatrix}$ where we are only allowed to either stay in the current state or move to the next state.","The allowed transitions can be conveniently represented as an FST, where vertices are states and edges are transitions associated with transition probabilities.","An example of the allowed state transitions is shown in Figure REF .","Figure: A 3-state FST for the phoneme ih, where loga ij \\log a_{ij} isthe log transition probability from state ii to state jj.To allow transitions from phonemes to phonemes, we connect the FSTs, such as the ones in Figure REF , in parallel.","An example is shown in Figure REF .","An additional start state and an additional end state are added.","A backward edge from the end state to the start state is also added to allow a sequence of phonemes with indefinite length.","The prior distribution $p(y_1)$ entering the phoneme is also added from the start state to each of the phoneme FSTs.","Note that when entering one of the phoneme FSTs, the edge consumes a phoneme state and produces a phoneme, and the phoneme FSTs only consume phoneme states and do not produce any output.","Silences are modeled with five states rather than three, because they are allowed to be longer than other phonemes.","This finishes the construction of the FST $H$ that converts phoneme states to phonemes.","Figure: An FST HH that converts phoneme states to phonemes.The emission probabilities $p(x_t | y_t)$ in HMM are typically modeled by Gaussian mixture models (GMM).","Specifically, $p(x_t | y_t) = \\sum _{i=1}^C \\pi _{i, y_t} p( x_t | \\mu _{i, y_t}, \\sigma ^2_{i, y_t})$ where there are $C$ components for each state in $\\lbrace 1, \\dots , S\\rbrace $ , each component $p(x_t | \\mu _{i, y}, \\sigma ^2_{i, y})$ is a Gaussian distribution with mean $\\mu _{i, y}$ and variance $\\sigma ^2_{i, y}$ , and $\\pi _{i, y}$ is the probability of selecting the $i$ -th component.","For a sequence of $T$ frames, the emission probability for each frame can be independently evaluated.","Since each state has $C$ Gaussian components, there are $S \\times C$ evaluations for each frame.","We can build an FST that has $S \\times C$ edges for every frame.","Each edge has a weight computed from its Gaussian component and has a phone state as the output symbol.","An example is shown in Figure REF .","Note that the FST is constructed by following the generative story of Gaussian mixture models, evaluating a single Gaussian component (selected based on $\\pi $ ) for every frame rather than evaluating the weighted average of all Gaussian components.","Figure: An FST UU for 4 frames with GMM probabilities.For example, if each phoneme has 3 states, and eachstate has 64 Gaussian components, thens∈{𝚊𝚊1,𝚊𝚊2,𝚊𝚊3,⋯,𝚣𝚑1,𝚣𝚑2,𝚣𝚑3}s \\in \\lbrace \\texttt {aa1}, \\texttt {aa2}, \\texttt {aa3}, \\dots , \\texttt {zh1}, \\texttt {zh2}, \\texttt {zh3} \\rbrace ,i∈{1,⋯,C=64}i \\in \\lbrace 1, \\dots , C = 64\\rbrace .If there are LL phonemes (i.e., S=3LS = 3L), then there are 3L×643L \\times 64 (i.e., S×C)S \\times C) edges between each pair of vertices.By $\\sigma $ -composing $U$ and $H$ , we construct an FST where each path in the FST is a sample drawn from the generative story and the weight of each path is the corresponding log probability.","To better see the connection, we can traverse the FST $U$ and $H$ synchronously as the way we compute $U \\circ _\\sigma H$ .","First when we traverse an edge in $U$ , we select one phoneme state and one Gaussian component, collecting the log probability and producing a phoneme state as output.","Then the FST $H$ receives the phoneme state and waits for the next phoneme state, collecting transition probabilities and producing a phoneme when necessary.","This completes the construction of $U$ and $H$ .","The pronunciation dictionary, or lexicon, is a mapping from a word to its pronunciation.","To construct an FST representing the lexicon, we create, for each entry in the lexicon, a path consuming a sequence of phonemes and producing a word.","Similar to the FST $H$ , we have a start vertex and an end vertex that joins the pronunciation paths, and we also have a backward edge that allows indefinite amount of words to be produced.","An example is shown in Figure REF .","Note that the weights on the edges can either be 0 or the probability of a chosen pronunciation.","The FST also allows a word to have multiple pronunciations.","Silences are considered as words and are included in the FST.","The size of the FST can be significantly reduced if we maintain a prefix tree of the phonemes instead of having parallel pronunciations for every word.","Figure: An FST LL representing a pronunciation dictionary.Language models assign probabilities to word sequences.","For any $k \\in \\mathbb {N}$ , a word sequence $w_{1:k} = w_1, \\dots , w_k$ of length $k$ has probability $p(w_{1:k}) = \\prod _{i=1}^k p(w_i | w_{1:i-1}).$ As the sequence gets longer, i.e., as $i$ gets larger, $p(w_i | w_{1:i-1})$ depends on more words, which makes estimating the probabilities difficult.","A simple approach to remedy this problem is to introduce the Markov assumption $p(w_i | w_{1:i-1}) = p(w_i | w_{i-n+1:i-1}).$ A language model that gives a probability for the current word based on the previous $n-1$ words is commonly known as an $n$ -gram language model.","To construct an FST representing a language model, we create vertices that corresponds to history words $w_{i-n+1:i-1}$ .","For each vertex, the outgoing edges produce the next word $w_i$ with the weight $\\log p(w_i | w_{i-n+1:i-1})$ .","A path in this FST consumes a sequence of words and has the probability of the word sequence assigned by the language model.","There is a start vertex representing the state of having no history words.","We assume the utterance starts and ends with silences, so there is only one edge going out from the start state expecting a silence.","An example is shown in Figure REF .","Back-off language models can also be approximated with FSTs [3].","Figure: An FST GG representing a bigram language modelfor the vocabulary {sil, A, cat, runs}.Some edges are dashed to avoid clutter.The output symbols are the same as the input symbols for every edge,hence ignored.","The vertex in red is the silence state.In this chapter, we have reviewed the definition of finite-state transducers (FST) and constructed a basic speech recognizer by $\\sigma $ -composing an HMM emission FST $U$ , an HMM transition FST $H$ , a lexicon $L$ , and a language model $G$ .","Since $H \\circ _\\sigma L \\circ _\\sigma G$ is shared across all utterances, it is typically $\\sigma $ -composed and saved.","Each individual FST and the composed FST can also be determinized and minimized for efficiency, which we do not cover.","Interested readers should refer to [98] for further details.","The problem of prediction in general can be considered as a search problem.","Given an input $x$ , we first construct a set of possible outputs $\\mathcal {Y}(x)$ , referred to as the search space.","For every output hypothesis $y$ in the search space $\\mathcal {Y}(x)$ , we measure how well the output matches the input, assigning a weight to each pair $(x, y)$ .","Prediction can be considered as finding the hypothesis $\\hat{y}$ such that the weight of $(x, \\hat{y})$ is larger than any other pairs.","Extending the above paradigm, the problem of sequence prediction, such as speech recognition as we have seen in Chapter REF , can be considered as a search problem.","Given an input sequence, the search space in this case is a set of sequences of connected segments.","The number of such sequences is exponential in the number of possible segments.","Fortunately, we do not need to store exponentially many such sequences, and can represent the search space compactly as an finite-state transducer (FST).","Consider an FST with vertices corresponding to time points, and edges corresponding to segments.","A path in the FST corresponds to a sequence of connected segments, and the set of all paths defined by the FST is the search space.","The FST is weighted, and the weights assigned to the edges are based on how well the segments match the input.","Prediction can be considered as finding the maximum-weight path, because the maximum-weight path, by definition, is a path that best matches the input.","In this chapter, we formally define segmental models, including search spaces constructed from sequences of input vectors, weight functions that measure how well a segment matches the acoustic signals, and loss functions used for training segmental models.","Let $\\mathcal {X}$ be the input space, the set of all sequences of real-valued vectors, e.g., log mel filter bank features or mel frequency cepstral coefficients (MFCCs).","Specifically, for a sequence of $T$ vectors $x = (x_1, \\dots , x_T) \\in \\mathcal {X}$ , each $x_t \\in \\mathbb {R}^d$ is a $d$ -dimensional vector, also referred to as a frame, for $t \\in \\lbrace 1, \\dots , T\\rbrace $ .","Let $\\mathcal {Y}$ be the output space, the set of all label sequences, where each label in a label sequence comes from a label set $L$ , e.g., a phoneme set in the case of phoneme recognition.","Given any $T$ frames, a segmentation of length $K$ is a sequence of time points $((1 = s_1, t_1), \\dots , (s_K, t_K = T))$ , where $s_k \\le t_k$ and $t_k + 1 = s_{k+1}$ for $k \\in \\lbrace 2, \\dots , K\\rbrace $ .","A segment (typically denoted $e$ in later sections) is a tuple $(\\ell , s, t)$ where $\\ell \\in L$ is its label, $s$ is the start time, and $t$ is the end time.","A segmental model is a tuple $(\\Theta , w)$ where $\\Theta $ is a set of parameters, and $w: \\mathcal {X} \\times E \\rightarrow \\mathbb {R}$ is a weight function parameterized by $\\Theta $ and $E$ is the set of all segment tuples $(\\ell , s, t)$ .","A sequence of segments forms a path.","Specifically, a path of length $K$ is a sequence of segments $(e_1, \\dots , e_K)$ , where $e_k \\in E$ for $k \\in \\lbrace 1, \\dots , K\\rbrace $ .","Let $\\mathcal {P}$ be the set of all paths.","For any path $p$ , we overload $w$ such that $w(x, p) = \\sum _{e \\in p} w(x, e)$ .","We will also abbreviate $w(x, e)$ and $w(x, p)$ as $w(e)$ and $w(p)$ respectively when the context is clear.","The concrete form of the weight function will be defined in later sections.","Given an input $x \\in \\mathcal {X}$ , segmental models aim to solve sequence prediction by reducing it to finding the maximum-weight path $ \\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(x, p).$ The set of paths $\\mathcal {P}$ , also referred to as the search space, can be compactly represented as an FST.","Once we have the search space FST, we can simply invoke Algorithm REF to find the maximum-weight path.","We will describe how the search space FST is constructed in the next section.","A segmental model can be trained by finding a set of parameters that minimizes a loss function.","We emphasize that the model definition is not tied to any loss function, allowing us to study the behavior of segmental models under different loss functions.","We define various loss functions for training segmental models in Section REF , and discuss how the properties of the losses affect the training requirement.","To represent the set of paths $\\mathcal {P}$ as an FST, we place a vertex at every time point and connect vertices based on the set of segments.","Suppose we have $T$ frames.","The set of segments $E$ is an exhaustive enumeration of tuples $(\\ell , s, t)$ for all $\\ell \\in L$ and $1 \\le s \\le t \\le T$ .","We create a set of vertices $V = \\lbrace v_0, v_1, \\dots , v_T\\rbrace $ and a time function $\\tau : V \\rightarrow \\mathbb {N}$ such that $\\tau (v_t) = t$ for $t \\in \\lbrace 0, 1, \\dots , T\\rbrace $ .","For every segment $(\\ell , s, t) \\in E$ , we create an edge $e$ such that $i(e) = o(e) = \\ell $ , $\\mathrm {tail}(e) = v_{s-1}$ , and $\\mathrm {head}(e) = v_t$ .","In other words, for any $e \\in E$ , the corresponding segment $(\\ell , s, t) = (o(e), \\tau (\\mathrm {tail}(e)), \\tau (\\mathrm {head}(e)))$ .","As a result, we will use $w(e)$ and $w((\\ell , s, t))$ interchangeably.","We set $\\Sigma = \\Lambda = L$ , $I = \\lbrace v_0\\rbrace $ , and $F = \\lbrace v_T\\rbrace $ to complete the construction of the FST given any $T$ frames.","The number of segments in the graph is $O(|L|T^2)$ .","To reduce the size of the search space, a maximum duration constraint is typically imposed while creating the search space.","Specifically, we only create segments $(\\ell , s, t)$ with $t - s + 1 \\le D$ , for some maximum duration $D$ .","Adding such constraint makes the number of segments $O(|L|TD)$ .","An example of a search space is shown in Figure REF .","Figure: An example search space for a five-frame input sequencewith a label set LL of size three and maximum segment duration DD of two frames,i.e., T=5T = 5, |L|=3|L| = 3, and D=3D = 3.The three edges between any two nodes are associated with the three labels.Here we detail two types of weight functions based on prior work by ourselves and others [49], [130], [1], [85].","We will compare the two weight functions and in Chapter REF .","The term feature function is often used in the literature to denote the function $\\phi : \\mathcal {X} \\times E \\rightarrow \\mathbb {R}^m$ for some $m$ , when the weight function $w(x, e)$ is of the form $\\theta ^\\top \\phi (x, e)$ for some parameter vector $\\theta \\in \\mathbb {R}^m$ .","In general, the weight function need not be a dot product, but can be any (sub-)differentiable real-valued function.","A weight function is typically a composition of two steps: first the input vectors are converted into an intermediate representation, and second the intermediate representation is converted into segment weights.","The function in the first step that converts input vectors to an intermediate representation are called an encoder.","Specifically, an encoder is a function that takes in $T$ frames $x_1, \\dots , x_T$ and outputs $\\tilde{T}$ feature vectors $h_1, \\dots , h_{\\tilde{T}}$ .","The number of output vectors $\\tilde{T}$ can be different from the number of input vectors $T$ depending on the encoder architecture.","We defer the actual implementation of encoders in the experimental sections.","Here we define weight functions based on $h_1, \\dots , h_{\\tilde{T}}$ .","We use $\\Theta _\\text{enc}$ to denote the parameters for the encoder, and let $\\Theta _\\text{dec}$ be the remaining parameters in the weight function.","Note that $\\Theta = \\Theta _\\text{enc} \\cup \\Theta _\\text{dec}$ .","The following weight function, termed FC weight, is based on a frame classifier and is similar to weight functions used in a variety of prior work [49], [130], [1].","The frame classifier takes in the output $h_1, \\dots , h_{\\tilde{T}}$ from the encoder, and produces a sequence of log probability vectors over the labels $z_i = \\text{logsoftmax}(W h_i + b)$ where $z_i \\in \\mathbb {R}^{|L|}$ and $W$ and $b$ are the parameters, for $i \\in \\lbrace 1, \\dots \\tilde{T}\\rbrace $ .","Based on these posterior vectors, we define several functions that summarize the posteriors over a segment: The average of transformed log probabilities $w_{\\text{avg}}((\\ell , s, t)) = \\frac{1}{t - s + 1} \\sum _{i=s}^{t} (u_{i})_{\\ell },$ where $u_i = W_\\text{avg} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ .","A sample of transformed log probabilities $w_{\\text{spl-}j}((\\ell , s, t)) = (u_{j})_{\\ell }$ at time $j \\in \\lbrace (t - s)/6, (t - s)/2, 5(t - s)/6\\rbrace $ , where $u_i = W_\\text{spl} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ .","The average of transformed log probabilities around the left boundary (start) of the segment $w_{\\text{left}-k}((\\ell , s, t)) = (u_{i-k})_{\\ell }$ and around the right boundary (end) $w_{\\text{right}-k}((\\ell , s, t)) = (u^{\\prime }_{i+k})_{\\ell }$ where $u_i = W_\\text{left} z_i$ and $u^{\\prime }_i = W_\\text{right} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ , and $k = 1, 2, 3$ .","The label-dependent duration weight $w_{\\text{dur}}((\\ell , s, t)) = d_{\\ell , t - s}.$ A label-dependent bias $w_{\\text{bias}}((\\ell , s, t)) = b^{\\prime }_{\\ell }.$ The final FC weight function is the sum of all the above weight functions.","When the FC weight function is used, $\\Theta _\\text{dec}$ is $\\lbrace W_\\text{avg}, W_\\text{spl},W_\\text{left}, W_\\text{right}, d, b^{\\prime }\\rbrace $ .","Note that $\\lbrace W, b\\rbrace $ are considered parameters of the encoders.","The MLP weight function based on a multi-layer perceptron (MLP) is inspired by [130], [74], [85].","Two hidden layers $z^{(1)}_{\\ell , s, t} & = \\text{ReLU}(W_1 [ h_s; h_t; c_\\ell ; d_{\\lfloor \\log _{1.6}(t-s)\\rfloor }] + b_1) \\\\z^{(2)}_{\\ell , s, t} & = \\tanh (W_2 z^{(1)}_{\\ell , s, t} + b_2)$ are computed directly from the outputs $h_1, \\dots , h_{\\tilde{T}}$ of the encoder before computing the final weight, where $c_\\ell $ is a label embedding vector for the label $\\ell $ , $d_k$ is a duration embedding vector for the duration $k$ in log scaleWe use 5 different duration embedding vectors for our experiments.","The base 1.6 is chosen such that $k \\in \\lbrace 0, \\dots , 4\\rbrace $ ., with $k = \\lfloor \\log _{1.6}(t-s+1)\\rfloor $ , and $\\text{ReLU}(x) = \\max (x, 0)$ .","The final weight for the segment is defined as $w((\\ell , s, t)) & = \\theta ^\\top z^{(2)}_{\\ell , s, t}.$ When the MLP weight function is used, $\\Theta _\\text{dec}$ is $\\lbrace W_1, b_1, W_2, b_2, \\theta \\rbrace $ .","Although the MLP weight function is conceptually simple, it is more expensive to compute than the FC weight function.","In [74], [85], an LSTM is created for each segment consuming the outputs of the encoder, followed by an MLP taking the output vectors of the per segment LSTM at the segment boundary.","In order to reduce the computation, we discard the per segment LSTM, and use the vectors produced by the encoder at the segment boundary as input to the MLP.","Recall that a path $p = ((\\ell _1, s_1, t_1), \\dots , (\\ell _K, s_K, t_K))$ consists of a label sequence $y = (\\ell _1, \\dots , \\ell _K)$ and a segmentation $z = ((s_1, t_1), \\dots , (s_K, t_K))$ .","In the following, we will use $(y, z)$ and $p$ interchangeably.","We will also denote the space of all segmentations $\\mathcal {Z}$ .","Training a segmental model aims to find a set of parameters $\\Theta $ that minimizes the expected task loss, for example, the expected edit distance $\\mathbb {E}_{(x, y) \\sim \\mathcal {D}}[\\mathrm {edit}(h_\\Theta (x), y)]$ where $h$ is the inference algorithm (Algorithm REF ) parameterized with $\\Theta $ , $\\mathrm {edit}$ computes the edit distance between two sequences, and the expectation is taken over samples $(x, y) \\in \\mathcal {X} \\times \\mathcal {Y}$ drawn from a distribution $\\mathcal {D}$ .","The expectation can be decomposed into two steps $\\mathbb {E}_{x \\sim \\mathcal {D}(x)} \\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathrm {edit}(h_\\Theta (x), y)],$ first sampling $x$ and then sampling $y$ .","To obtain a good discriminator $h$ , it suffices to optimize the inner expectation $\\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathrm {edit}(h_\\Theta (x), y)],$ for any $x$ drawn from $\\mathcal {D}(x)$ .","However, the edit distance, due to its discrete nature, is difficult to optimize, so instead we minimize the expected loss $ \\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathcal {L}(\\Theta ; x, y)],$ where $\\mathcal {L}$ is a surrogate loss function.","If segmentations are considered in the loss function, then we can minimize $ \\mathbb {E}_{(y, z) \\sim \\mathcal {D}^{\\prime }(y, z | x)}[\\mathcal {L}(\\Theta ; x, y, z)],$ where $\\mathcal {D}^{\\prime }$ is a conditional distribution over $\\mathcal {X} \\times \\mathcal {Y} \\times \\mathcal {Z}$ .","Since the distribution $\\mathcal {D}$ is unknown, we use a tranining set $S = \\lbrace (x_1, y_1), \\dots , (x_n, y_n)\\rbrace $ of size $n$ to approximate the expectation and instead minimize $\\frac{1}{n} \\sum _{i=1}^n \\mathcal {L}(\\Theta ; x_i, y_i).$ If segmentations are considered, we use a tranining set $S = \\lbrace (x_1, y_1, z_1), \\dots , (x_n, y_n, z_n)\\rbrace $ of size $n$ to approximate $\\mathcal {D}^{\\prime }$ and minimize $\\frac{1}{n} \\sum _{i=1}^n \\mathcal {L}(\\Theta ; x_i, y_i, z_i).$ The connection between the surrogate loss $\\mathcal {L}$ and the edit distance depends on the choice of loss.","We will optimize the loss functions with first-order methods, such as stochastic gradient descent.","When listing the function definitions along with reasons for using them, we will also list the (sub-)gradients with respect to the weight $w(e)$ for some edge $e$ .","We assume the weight function $w$ is (sub-)differentiable and the (sub-)gradients with respect to the parameters can be obtained with back-propagation.","Given an utterance $x$ and a ground truth path $p = (y, z)$ , the hinge loss is defined as $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)= \\max _{p^{\\prime } \\in \\mathcal {P}} \\left[ \\mathrm {cost}(p^{\\prime }, p) - w(p) + w(p^{\\prime }) \\right]$ where $\\mathrm {cost}$ is a user-defined cost function.","The connection between the hinge loss and the task loss is through the cost function.","Suppose $\\hat{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(p)$ is the maximum-weight path found by Algorithm REF .","The cost of the inferred path $\\hat{p}$ against the ground truth $p$ can be upper-bounded by the hinge loss: $\\mathrm {cost}(\\hat{p}, p) \\le \\mathrm {cost}(\\hat{p}, p) - w(p) + w(\\hat{p})\\le \\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p).$ When the cost function is the edit distance, minimizing the hinge loss is minimizing an upper bound on the edit distance of the predicted sequence.","The hinge loss itself is difficult to optimize when the cost function is the edit distance.","In practice, the cost function is assumed to be decomposable $\\mathrm {cost}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}(e^{\\prime }, p)$ to allow efficient dynamic programming.","When the cost is decomposable, the hinge loss can be written as $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)& = \\max _{p^{\\prime } \\in \\mathcal {P}} \\left[ \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}(e^{\\prime }, p)- \\sum _{e \\in p} w(e) + \\sum _{e^{\\prime } \\in p^{\\prime }} w(e^{\\prime }) \\right] \\\\& = \\max _{p^{\\prime } \\in \\mathcal {P}} \\sum _{e^{\\prime } \\in p^{\\prime }} \\left[ \\mathrm {cost}(e^{\\prime }, p)+ w(e^{\\prime }) \\right] - \\sum _{e \\in p} w(e),$ and the $\\max $ operator in the first term can be solved with Algorithm REF by adding the costs to the weights for all segments.","A subgradient of the hinge loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)}{\\partial w(e)}= -{1}_{e \\in p} + {1}_{e \\in \\tilde{p}}$ where $\\tilde{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p^{\\prime } \\in \\mathcal {P}} [\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })],$ which is the path that maximizes the first term in the hinge loss, and can be obtained with Algorithm REF with the cost added.","Linear models are called support vector machines (SVM) when trained with the hinge loss, and are referred to as structured SVMs when used for solving structured prediction problems, e.g., sequence prediction in our case.","A hinge loss with cost zero is also called perceptron loss [77].","Segmental models trained with the hinge loss have been studied in [146], [129], [130].","Segmental models can be treated as probabilistic models by defining probability distributions on the set of all paths $\\mathcal {P}$ .","Specifically, the probability of a path $p = (y, z)$ is defined as $P(y, z | x) = P(p | x) = \\frac{1}{Z(x)} \\exp (w(x, p))$ where $Z(x) = \\sum _{p^{\\prime } \\in \\mathcal {P}} \\exp (w(x, p^{\\prime }))$ is the partition function.","Given an input $x$ and a ground truth path $p$ , the log loss is defined as $\\mathcal {L}_{\\text{log}}(\\Theta ; x, p) = -\\log P(p | x).$ Minimizing the log loss is equivalent to maximizing the conditional likelihood.","In addition, the conditional likelihood can be written as $P(y, z | x) & = \\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) = (y, z)}] \\\\& = 1 - \\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}].$ Therefore, maximizing the conditional likelihood is equivalent to minimizing the expected zero-one loss $\\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}],$ where $P(y, z | x)$ is used to approximate $\\mathcal {D}^{\\prime }(y, z | x)$ and the zero-one loss ${1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}$ is used.","The use of the log loss is justified because the expectation above can be seen as an approximation of (REF ).","Segmental models trained with log loss have been referred to as semi-Markov CRFs [118].","Minimizing log loss is equivalent to maximizing mutual information, and is commonly referred to as the MMI criterion [10].","Since the weight for the ground truth path $p$ can be efficiently computed, we are left with the problem of computing the partition function $Z(x)$ .","The partition function can also be computed efficiently with the following dynamic programming algorithm.","Let $\\mathcal {P}(u, v)$ be the set of paths that start at vertex $u$ and end at vertex $v$ .","For any vertex $v$ , define the forward marginal as $\\alpha (v) = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp (w(p^{\\prime })).$ By expanding the edges ending at $v$ , we have $\\alpha (v) &= \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp \\left(\\sum _{e \\in p^{\\prime }} w(e)\\right) \\\\&= \\log \\sum _{e \\in \\text{in}(v)} \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, \\mathrm {tail}(e))}\\exp \\left(w(e) + \\sum _{e^{\\prime } \\in p^{\\prime }} w(e^{\\prime })\\right) \\\\&= \\log \\sum _{e \\in \\text{in}(v)} \\exp \\left(w(e) + \\alpha (\\mathrm {tail}(e))\\right)$ Similarly, the backward marginal at $v$ is defined as $\\beta (v) = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v, v_T)} \\exp (w(p^{\\prime })),$ and has similar recursive structure.","The complete algorithm is shown in Algorithm REF .","Once all entries in $\\alpha $ and $\\beta $ are computed, the log partition function is $\\log Z(x) = \\alpha (v_T) = \\beta (v_0).$ Note that we store all of the entries in log space to maintain numerical stability.","The gradient of the log loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{log}}(\\Theta ; x, p)}{\\partial w(e)}& = -{1}_{e \\in p} + \\frac{1}{Z(x)}\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime })) \\\\& = -{1}_{e \\in p} + \\exp \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e)+ \\beta (\\mathrm {head}(e)) - \\log Z(x) \\Big ],$ which can also be efficiently computed once the marginals are computed.","Note that the form of Algorithm REF is very similar to that of Algorithm REF .","Recall that to get the standard shortest-path algorithm (Algorithm REF ), we simply use the tropical semiring when running Algorithm REF .","If the tropical semiring is replaced by the log semiring, we arrive at Algorithm REF , demonstrating the generality of FST algorithms and semirings.","Computing forward and backward marginals $\\alpha (v_0) = 0$ $\\beta (v_T) = 0$ $\\text{logadd}(a, b) = \\log (\\exp (a) + \\exp (b))$ $v = v_0, v_1, \\dots , v_T$ $\\alpha (v) = \\text{logadd}_{e \\in \\text{in}(v)} \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e) \\Big ]$ $v = v_T, v_{T-1}, \\dots , v_0$ $\\beta (v) = \\text{logadd}_{e \\in \\text{out}(v)} \\Big [ \\beta (\\mathrm {head}(e)) + w(e) \\Big ]$ Log loss can be modified to include a cost function.","Speficically, we define $P^+_{p}(p^{\\prime } | x) = \\frac{1}{Z(x, p)} \\exp (w(x, p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p))$ where $p$ is the ground-truth path and $Z(x, p) = \\sum _{p^{\\prime \\prime } \\in \\mathcal {P}} \\exp (w(x, p^{\\prime \\prime }) + \\mathrm {cost}(p^{\\prime \\prime }, p)).$ Unlike $P$ , the distribution $P^+$ considers both the weights and the costs.","The modified log loss, also known as the boosted MMI criterion [109], is defined as $\\mathcal {L}_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P^+_{p}(p | x).$ Given an input $x$ and a label sequence $y$ , the marginal log loss is defined as $\\mathcal {L}_{\\text{mll}}(\\Theta ; x, y) = -\\log P(y | x) = -\\log \\sum _{z \\in \\mathcal {Z}} P(y, z | x)$ where the segmentation is marginalized compared to log loss; hence the name.","Following the same argument as for the log loss, the marginal distribution can be written as $P(y | x) = 1 - \\mathbb {E}_{y^{\\prime } \\sim P(y^{\\prime } | x)}[{1}_{y \\ne y^{\\prime }}],$ and maximizing the marginal distribution is equivalent to minimizing the expected zero-one loss $\\mathbb {E}_{y^{\\prime } \\sim P(y^{\\prime } | x)}[{1}_{y \\ne y^{\\prime }}],$ where $P(y | x)$ is used to approximate $\\mathcal {D}(y | x)$ .","Note that the zero-one loss ${1}_{y \\ne y^{\\prime }}$ only depends on the label sequence.","While the log loss has a connection to (REF ), the marginal log loss justifies its use by directly approximating (REF ) with the above expected zero-one loss.","Note that both the hinge loss and the log loss depend on the ground-truth path, or more specifically, the ground-truth segmentation.","The marginal log loss marginalizes over the segmentations, so it only depends on the ground-truth label sequence and not the segmentation.","The lack of reliance on the ground-truth segmentation makes the marginal log loss attractive for tasks such as speech recognition, because collecting ground-truth segmentations for phonemes or words is time-consuming and/or expensive.","In addition, the boundaries of phonemes and words tend to be ambiguous, so it can be preferable to leave the decision to the model.","Segmental models trained with the marginal log loss have been referred to as segmental CRFs [149].","To compute the marginal log loss, we can rewrite it as $\\mathcal {L}_{\\text{mll}}(\\Theta ; x, y) & = -\\log \\sum _{z \\in \\mathcal {Z}} P(y, z | x) \\\\& = -\\log \\sum _{z \\in \\mathcal {Z}} \\exp (w(x, (y, z))) + \\log Z(x) \\\\& = -\\underbrace{\\log \\sum _{p^{\\prime }: \\Gamma (p^{\\prime }) = y} \\exp (w(x, p^{\\prime }))}_{\\log Z(x, y)} + \\log Z(x)$ where $\\Gamma $ extracts the label sequence from a path, i.e., for $p^{\\prime } = (y^{\\prime }, z^{\\prime })$ , $\\Gamma (p^{\\prime }) = y^{\\prime }$ .","Since the partition function can be efficiently computed from Algorithm REF , we only need to compute $\\log Z(x, y)$ .","Since the term $\\log Z(x, y)$ is identical to $\\log Z(x)$ except that it involves a constrained search space considering all paths with the same label sequence $y$ , the strategy is to construct the constrained search space with an FST and run Algorithm REF on the FST.","Let $F$ be a chain FST that represents $y$ , with edges $\\lbrace e_1, \\dots , e_{|y|}\\rbrace $ , where $i(e_k) = o(e_k) = y_k$ for all $k \\in \\lbrace 1, \\dots , |y|\\rbrace $ .","Let $G$ be the search space consisting of all paths in $\\mathcal {P}$ .","The term $\\log Z(x, y)$ can be efficiently computed by running Algorithm REF on the $\\sigma $ -composition of $G$ and $F$ , i.e., $G \\circ _\\sigma F$ .","Note that after $\\sigma $ -composing $G$ and $F$ , we only allow paths in $G$ that can produce output sequences that $F$ accepts.","Since $F$ only accepts $y$ , the paths in $G \\circ _\\sigma F$ are all the paths in $G$ that produce $y$ as desired.","Let the forward and backward marginals computed on $G \\circ _\\sigma F$ be $\\alpha ^{\\prime }$ and $\\beta ^{\\prime }$ .","We have $\\log Z(x, y) = \\alpha ^{\\prime }(v_T) = \\beta ^{\\prime }(v_0)$ .","The gradient of the marginal log loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{mll}}(\\Theta ; x, y)}{\\partial w(e)}& = -\\frac{1}{Z(x, y)} \\sum _{\\begin{array}{c}p^{\\prime } \\ni e\\\\ \\Gamma (p^{\\prime }) = y\\end{array}} \\exp (w(p^{\\prime }))+ \\frac{1}{Z(x)}\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime })) \\\\& = - \\exp \\Big [ \\alpha ^{\\prime }(\\mathrm {tail}(e)) + w(e) + \\beta ^{\\prime }(\\mathrm {head}(e)) - \\log Z(x, y) \\Big ] \\\\& \\qquad {} + \\exp \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e) + \\beta (\\mathrm {head}(e)) - \\log Z(x) \\Big ].$ and can be efficiently computed once all of the marginals are computed.","The empirical Bayes risk (EBR) [125] is defined as $\\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p) = \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)].$ The intuition behind EBR is that if $P$ is a good approximation for $\\mathcal {D}^{\\prime }$ , then EBR is a good approximation for $\\mathbb {E}_{p^{\\prime } \\sim \\mathcal {D}^{\\prime }(p^{\\prime } | x)}[\\mathrm {cost}(p^{\\prime }, p)].$ The goal is to find $P$ that achieves a low expected cost.","Empirical Bayes risk is also referred to as minimum phone error (MPE) when the cost is based on phone errors, and is referred to as minimum word error (MWE) when the cost is based on word errors respectively [108].","A boosted version of EBR can be obtained by replacing $P$ with $P^+$ [89].","Since $Z(x)$ can be computed through the forward and backward marginals $\\alpha $ and $\\beta $ , we are left with $\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)$ .","Similar to the forward and backward marginals, we have $\\alpha ^{\\prime \\prime }(v) & = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{in}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p))\\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, \\mathrm {tail}(e))} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{in}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p) + \\alpha ^{\\prime \\prime }(\\mathrm {tail}(e))) \\\\\\beta ^{\\prime \\prime }(v) & = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v, v_T)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{out}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p))\\sum _{p^{\\prime } \\in \\mathcal {P}(\\mathrm {head}(e), v_T)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{out}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p) + \\beta ^{\\prime \\prime }(\\mathrm {head}(e)))$ where $\\mathrm {cost}$ is assumed to be non-negative, and the marginals are stored in log space to maintain numerical stability.","Given the cost-augmented marginals, we have $\\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)] = \\exp (\\alpha ^{\\prime \\prime }(v_T) - \\log Z(x))= \\exp (\\beta ^{\\prime \\prime }(v_0) - \\log Z(x)).$ The gradient of EBR with respect to $w(e)$ of some edge $e$ is $\\frac{\\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p)}{w(e)}& = \\frac{\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)}{Z(x)}- \\frac{\\left[\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)\\right]\\left[\\sum _{p^{\\prime } \\ni e}\\exp (w(p^{\\prime }))\\right]}{(Z(x))^2} \\\\& = \\exp (\\alpha ^{\\prime \\prime }(\\mathrm {tail}(e)) + w(e) + \\log \\mathrm {cost}(e, p) + \\beta ^{\\prime \\prime }(\\mathrm {head}(e)) - \\log Z(x)) \\\\& \\quad {} - \\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p) [\\exp (\\alpha (\\mathrm {tail}(e)) + w(e) + \\beta (\\mathrm {head}(e)) - \\log Z(x))]$ It is also interesting to note that, in general, $\\frac{\\partial }{\\partial w(e)} \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[f(p^{\\prime })]&= \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ni e} f(p^{\\prime })]- \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ni e}]\\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[f(p^{\\prime })] \\\\&= \\text{Cov}(\\mathbb {1}_{p^{\\prime } \\ni e}, f(p^{\\prime })),$ for any function $f: \\mathcal {P} \\rightarrow \\mathbb {R}$ .","The ramp loss [24] is defined as $\\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p) = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - \\max _{p^{\\prime \\prime }}w(p^{\\prime \\prime })].$ It is easy to see that $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)& = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - w(p)] \\\\& \\ge \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - \\max _{p^{\\prime \\prime }}w(p^{\\prime \\prime })]= \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p).$ In addition, for $\\hat{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p^{\\prime }} w(p^{\\prime })$ , $\\mathrm {cost}(\\hat{p}, p) &= \\mathrm {cost}(\\hat{p}, p) + w(\\hat{p}) - w(\\hat{p}) \\\\& \\le \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })] - w(\\hat{p}) \\\\& = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) ] - \\max _{p^{\\prime }} w(p^{\\prime })= \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p).$ Therefore, we have $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p) \\ge \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p)\\ge \\mathrm {cost}(\\hat{p}, p).$ In other words, ramp loss is also an upper bound on the cost of the predicted sequence, but is tighter than hinge loss.","However, optimizing the ramp loss is more complicated.","Ramp loss can be minimized with the concave-convex procedure (CCCP) [145].","See [33] for a detailed implementation of CCCP for solving ramp loss.","Log loss augmented with a cost function, or boosted MMI, can be seen as a soft version of hinge loss.","By the definition of boosted MMI, where $\\mathcal {L}_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\log \\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p)),$ we can see that the second term acts as a soft-max instead of a max function.","To see this, note that $\\log (e^{x_1} + e^{x_2} + \\dots + e^{x_n})& = \\log e^{x_{\\text{max}}} (e^{x_1 - x_{\\text{max}}} + \\dots + e^{x_n - x_{\\text{max}}}) \\\\& = x_{\\text{max}} + \\log (e^{x_1 - x_{\\text{max}}} + \\dots + e^{x_n - x_{\\text{max}}}),$ where $x_{\\text{max}} = \\max _i x_i$ .","The second term in (REF ) is small when the difference between $x_{\\text{max}}$ and other $x_i$ 's is large.","In fact, we can introduce an addition factor $\\epsilon $ to control this effect.","Specifically, $\\epsilon \\log (e^{x_1/\\epsilon } + e^{x_2/\\epsilon } + \\dots + e^{x_n\\epsilon })& = x_{\\text{max}} + \\epsilon \\log (e^{(x_1 - x_{\\text{max})/\\epsilon }} + \\dots + e^{(x_n - x_{\\text{max}})/\\epsilon }).$ When $\\epsilon \\rightarrow 0$ , the second term vanishes and (REF ) acts exactly like a max function.","We can add the additional $\\epsilon $ factor to the cost-augmented log loss and get $\\mathcal {L}^{\\epsilon }_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\epsilon \\log \\sum _{p^{\\prime }} \\exp \\left(\\frac{1}{\\epsilon }(w(p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p))\\right).$ As $\\epsilon \\rightarrow 0$ , $\\mathcal {L}^{\\epsilon }_{\\text{bMMI}} \\rightarrow \\mathcal {L}_{\\text{hinge}}$ .","This connection was first presented in [52].","[32] independently proposed softmax-margin, the equivalence of cost-augmented log loss.","As we have noted when we introduced log loss, $P(p | x) = \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } = p}]= 1 - \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ne p}].$ Therefore, minimizing log loss is equivalent to minimizing EBR with $\\mathrm {cost}(p^{\\prime }, p) = \\mathbb {1}_{p^{\\prime } \\ne p}$ .","Another connection between cost-augmented log loss and EBR is the following.","Let $\\mathcal {L}_{\\lambda , \\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\log \\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p)),$ where we introduce a scaling factor $\\lambda $ for the cost function in cost-augmented log loss.","If we take the derivative of cost-augmented log loss with respect to $\\lambda $ , we have $\\frac{\\partial }{\\partial \\lambda } \\mathcal {L}_{\\lambda , \\text{bMMI}}&= \\frac{\\sum _{p^{\\prime }} [\\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p)) \\mathrm {cost}(p^{\\prime }, p)]}{\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p))} \\\\&= \\mathbb {E}_{p^{\\prime } \\sim P^+_\\lambda (p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)],$ where $P^+_{\\lambda }(p^{\\prime }|x) = \\frac{\\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p))}{\\sum _{p^{\\prime \\prime }} \\exp (w(p^{\\prime \\prime }) + \\lambda \\mathrm {cost}(p^{\\prime \\prime }, p))}$ Therefore, the derivative of cost-augmented log loss with respect to $\\lambda $ is cost-augmented EBR, or sometimes referred to as boosted MPE.","This connection was first presented in [90].","In this sectionThese results were published in [129], we describe experiments comparing segmental models trained with various loss functions and two cost functions.","Instead of using the entire search space, we follow [149], [150] and use a baseline recognizer to generate sparse search spaces, commonly known as lattices.","Segmental models are then applied on these lattices.","The weight function used in these experiments is a linear function $w(x, e) = \\theta ^\\top \\phi (x, e)$ where $\\theta $ is a parameter vector and $\\phi (x, e)$ is a feature vector for some edge $e$ .","The feature function $\\phi $ is task-dependent, and is described in detail later.","Hinge loss and ramp loss are sensitive to the scale of the cost, so the scale of the cost function is tuned.","Specifically, we introduce the parameter $\\mu $ in the hinge loss $\\mathcal {L}_{\\text{hinge}} = \\max _{p^{\\prime }} [\\mu \\cdot \\mathrm {cost}(p^{\\prime }, p) - w(p) + w(p^{\\prime })],$ and parameters $\\mu _1$ and $\\mu _2$ for the ramp loss $\\mathcal {L}_{\\text{ramp}} = \\max _{p^{\\prime }} [\\mu _1 \\cdot \\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })]- \\max _{p^{\\prime }}[\\mu _2 \\cdot \\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })].$ Note the additional cost term in the above ramp loss.","This version of ramp loss is a generalization of the standard ramp loss [20], [33].","One cost function was proposed in [108] in the context of MPE/MWE training, and we refer to it as MPE cost.","The cost of an edge is the duration of the non-overlapping part of a matching ground-truth edge that gives the lowest error, where the error is one if the label is correct and 0.5 otherwise.","Formally, for any hypothesized edge $e^{\\prime }$ , we define $\\mathrm {cost}_{\\text{MPE}}(e^{\\prime }, p) =1 - \\max _{e \\in p} \\left[ \\mathbb {1}_{o(e) = o(e^{\\prime })}\\frac{|e \\cap e^{\\prime }|}{|e|}+ \\frac{1}{2} \\mathbb {1}_{o(e) \\ne o(e^{\\prime })}\\frac{|e \\cap e^{\\prime }|}{|e|} \\right],$ and $\\mathrm {cost}_{\\text{MPE}}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}_{\\text{MPE}}(e^{\\prime }, p),$ where $|e|$ denotes the length of segment $e$ .","The MPE cost only penalizes false negatives and does not account for false positives.","Therefore, we propose an alternative that we refer to as the overlap cost: $\\mathrm {cost}_{\\text{overlap}}(e^{\\prime }, p) = 1 - \\mathbb {1}_{o(e) = o(\\tilde{e})}\\frac{|e \\cap \\tilde{e}|}{|e \\cup \\tilde{e}|},$ where $\\tilde{e} = \\operatornamewithlimits{\\arg \\!\\max }_{e \\in p} |e^{\\prime } \\cap e|$ .","This cost function finds the most overlapping edge in the ground truth and considers any part of the union of the two edges that is not overlapping to be in error.","The cost for the whole path is again $\\mathrm {cost}_{\\text{overlap}}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}_{\\text{overlap}}(e^{\\prime }, p).$ We study the various losses and cost functions on two tasks.","One is a standard speech recognition task, namely TIMIT [31] phonetic recognition.","The second is a sign language recognition task from video, in particular recognition of fingerspelled letter sequences in American Sign Language (ASL).","Both are tasks on which there is prior work using semi-Markov CRFs [148], [71], and both are small enough (in terms of data set size and decoding search space) to run many empirical comparisons in a reasonable amount of time.","For the ASL task, we use the data and experimental setup of [71]: We obtain baseline lattices using their tandem HMM-based system, and we use the same set of segmental feature functions.","We use forced alignments of the ground-truth transcriptions for training.","We train all models from all-zero weights and optimize with Rprop [114] for 20 epochs.","We use $L_1$ and $L_2$ regularization, with parameters tuned over the grid $\\lbrace 0,10^{-6}, 10^{-5}, \\dots , 0.1, 1\\rbrace ^2$ .","For hinge and ramp loss, we use the standard forms without tuning the cost weights (i.e., $\\mu =1$ , $\\mu _1 = 0$ , and $\\mu _2 = 1$ ).","Table: Letter error rates (%) for a baseline tandem HMM and segmental models trained with various lossesand cost functions on the ASL data set.Table: Phone error rates (%) for a baseline HMM and segmental models trained with various lossesand cost functions on the TIMIT data set.For TIMIT, we use the standard 3696-utterance training set and 192-utterance core test set, plus a random 192 utterances from the full test set (excluding the core test set) as a development set.","We collapse the 61 phones in the phone set to 48 for training, and further collapse them to 39 for evaluation [79].","We use lattices generated by a baseline monophone HMM system with 39-dimensional MFCCs.","The resulting lattices have an average density (average number of hypothesized edges per ground truth edge) of 60.1.","The oracle phone error rate is 6.3% for the development set and 7.0% for the core test set.","We use oracle paths (paths with minimum phone error) from the lattices as ground truth for training.","We implement segmental models with various feature functions.","The base features are the acoustic and language model score from the baseline recognizer, and a bias (a feature that is always one).","We also include a set of features based on spectro-temporal receptive fields implemented as follows.","We begin with 40-dimensional log mel filter bank features.","For each segment, we divide it evenly into thirds in both time and frequency, resulting in nine patches for each segment.","For each patch, we have a $3 \\times 13$ receptive field of all ones, and convolve it with the patch.","The resulting $3 \\times 13 \\times 9$ numbers are lexicalized to form the final features for the segmental model.","Specifically, for any feature vector $v$ , we refer to $v \\otimes \\mathbf {1}_\\ell $ as the lexicalized feature vector, where $\\mathbf {1}_\\ell $ is a one-hot vector for the label $\\ell $ .","We optimize the loss functions using AdaGrad [27], using step size 0.1 for 10 epochs.","$L_1$ and $L_2$ regularization parameters are tuned over the grid $\\lbrace 0, 0.001, 0.1, 1\\rbrace \\times \\lbrace 0, 0.1, 1, 100\\rbrace $ .","The results for ASL recognition, averaged over four signers, are shown in Table REF .","The evaluation metric is the letter error rate, which is the percentage of letters that are substituted, inserted, or deleted.","The results for TIMIT are shown in Table REF .","We observe three consistent conclusions: Segmental models perform significantly better than the baseline HMM.","Across losses, overlap cost is better than MPE cost.","Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.","Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.","We suspect a warm start might be able to remedy this.","Segmental models perform significantly better than the baseline HMM.","Across losses, overlap cost is better than MPE cost.","Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.","Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.","We suspect a warm start might be able to remedy this.","For the ASL task, we tuned on a development set the cost weights for ramp loss over the grid $\\lbrace -100,-10,-1,-0.1,-0.01\\rbrace \\times \\lbrace 0.01,0.1,1,10,100\\rbrace $ , using overlap cost.","The test result of tuned ramp slightly improves over hinge loss, confirming that if ramp is tuned carefully, it is able to outperform hinge.","However, even though tuned ramp loss achieves very good results, considering the time spent tuning $\\mu _1$ and $\\mu _2$ , we still favor hinge and log loss.","We also conducted experiments to determine how the results are affected by different levels of noise in the feature functions, using simulated phone detector-based features.","Similarly to [150], we define a detection event as a (time, phone label) pair, and a feature function that is an indicator of whether a phone detection event occurs in the time span of the edge.","If we set a high weight for the phone event that occurs in an edge with the same phone label, then we can exactly recover the oracle path.","This allows us to conduct a series of simulated experiments with different amounts of noise added to the oracle phone events, or gold events.","For all experiments below, we use the same TIMIT setting except that we only use the acoustic and language model score with the simulated phone detector features, with no regularizer and one epoch of AdaGrad.","The ramp cost weights are set to $\\mu _1=0, \\mu _2=1$ .","The cost weight for hinge is tuned over $\\lbrace 0.01, 0.1, 1, 10, 100\\rbrace $ and results are only shown for the best-performing value.","The first set of experiments randomly perturbs the correct phone label of each event to an incorrect label with the corruption probability shown on the $x$ -axis; the event times are not perturbed.","The second set of experiments perturbs the time for each event but not the label.","We add Gaussian noise with mean set to the time at which the event occurs and with several standard deviations shown on the $x$ -axis.","For the third and fourth set of experiments, we randomly include an edge in the lattice as a false positive event, or randomly delete an event from the gold events.","The conclusions are consistent with our previous observation, namely, that hinge is the consistent winner but only by a very small margin, that log loss is very competitive, that non-convex losses are hard to optimize, and that overlap cost is better than MPE cost.","As a byproduct, we note that we could achieve 17.7% given a phone detector with any of the following characteristics: up to 50% phone error rate but perfect time information, up to 5-frame time perturbations (in standard deviation) but perfect labels, 1.8 false positives per gold edge, or 20% false negatives.","Figure: Lattice rescoring with various noise added to a perfectphone detector.From Left to Right: Perturbing phone labels.","Perturbing time.Adding false positive events.","Adding false negative events.In this chapter, we have described a formal framework for discriminative segmental models.","The definition is not tied to any weight function or any loss function, encompassing a large family of segmental models.","We have described two weight functions, one based on frame classifiers, and one based on segmental recurrent neural networks.","We have also described various loss functions, and have drawn connections between surrogate losses and the true loss that we want to minimize.","We have shown how (sub-)gradients of loss functions can be efficiently computed with FST algorithms.","The flexibility to use different loss functions allows us to train segmental models in different training settings.","Preliminary results have shown that segmental models outperform HMMs by a large margin and that overlap cost is better than MPE cost across loss functions.","In terms of loss functions, non-convex losses are more difficult to optimize.","Hinge loss is the best performer while log loss is very competitive.","Our final task loss is the edit distance, but none of the surrogate losses can be optimized when the cost function is the edit distance.","There are two difficulties for optimizing the edit distance: the edit distance is discrete, and it involves a minimization over all alignments to the ground truth.","Using a cost function that considers all alignments to the ground truth without the minimization has been explored in [51], [137], [121].","Other approaches for directly minimizing task losses [69], [67] are also suitable for minimizing the edit distance.","As we have seen in Chapter REF , search spaces defined by segmental models are dense and redundant, in the sense that many segments that have the same label differ only by a few frames (or even just a single frame) and that many segments with the same start time and end time differ only in the labels.","In other words, most of the random segments do not look like the ground truths.","Decoding in multiple passes is an approach that exploits dense search spaces.","Since the search spaces are dense, a simple weight function is typically enough to significantly reduce the size of search spaces.","After the first pass, we may use more complicated weight functions to make predictions or to further reduce the search spaces.","Reducing the size of search spaces is commonly known as pruning.","Pruning has to be done carefully, because it might hurt the final performance if the ground truth segments are pruned.","To measure how pruning affects the search spaces and the quality of predictions, we measure the lowest achievable task loss, the oracle error rate, of the search spaces after pruning.","A simple experiment on the phonetic recognition task shows that the oracle error rate, i.e., the best edit distance we can achieve, on average stays at zero even if we remove 50% of segments at random.","This result shows that the search spaces are indeed dense and redundant.","Since the pruned search space is smaller than the original space, inference and learning using the pruned search space are faster in the second pass.","As a result, we can afford to use computationally expensive weight functions for better prediction in the second pass.","In general, with the same weight function, multi-pass systems have the potential to decode faster than single-pass systems.","In other words, under the same decoding time budget, multi-pass systems can use computationally more expensive weight functions than single-pass systems to obtain better performance.","Since pruning plays an important role, in the following sections, we present and compare several pruning strategies and their consequences.","We then develop a multi-pass system, trading between the size of the search space and the computational complexity of weight functions.","In this section, we describe several pruning approaches to reduce search spaces.","Note that pruning is a form of inference, and we assume a pruning approach has access to search spaces defined by an FST, and a weight function $w : E \\rightarrow \\mathbb {R}$ parameterized by $\\Theta _{\\text{prn}}$ .","We discuss the choice of $w$ and how $\\Theta _{\\text{prn}}$ is obtained in the experimental sections.","A very naive approach, referred to as greedy pruning, is to prune the edges branching out from a vertex based on their edge weights.","Specifically, the set of edges branching out from a vertex $v$ is pruned based on a threshold $\\tau $ .","Let $S_v = \\lbrace e \\in \\text{out}(v): w(e) \\ge \\tau \\rbrace $ be the set of vertices that survive pruning.","We collect the edges $\\bigcup _{v \\in V} S_v$ to form the pruned graph.","There are two nice properties about this approach.","First, the pruning procedure is embarrassingly parallelizable.","Second, if we let $\\tau = \\lambda \\min _{e \\in \\text{out}(v)} w(e)+ (1 - \\lambda ) \\max _{e \\in \\text{out}(v)} w(e)$ where $0 \\le \\lambda \\le 1$ is a parameter that controls the amount of pruning, the graph is guaranteed to be connected.","To see this, note that there is at least one edge surviving for every vertex.","In other words, from the start vertex, there is at least one edge for us to traverse for every vertex, and we eventually arrive at the final vertex.","Beam pruning, or more generally beam search [84], is a widely used search and pruning method.","The motivation behind beam search is to constrain a search algorithm, such as the shortest-path algorithm (Algorithm REF ), with a fixed memory budget.","Due to the memory budget, we cannot afford to remember all the paths we have traversed, so we prune the paths that are less likely to have high weights.","Specifically, the beam search algorithm performs the following steps while visiting vertices in topological order.","We keep an approximate shortest distance $\\hat{d}(u)$ to every vertex $u$ .","Suppose vertex $v$ is the current vertex being visited.","A set of surviving edges $S_v = \\lbrace e \\in \\text{in}(v): \\hat{d}(\\mathrm {tail}(e)) \\ge \\tau \\rbrace $ is computed based on a threshold $\\tau $ , and then the approximate shortest distance for $v$ is updated by setting $\\hat{d}(v) = \\max _{e \\in S_v} w(e) + \\hat{d}(\\mathrm {tail}(e)).$ Note that this equation is identical to the shortest-path recursion, except that the set of incoming edges $\\text{in}(v)$ is pruned before updating.","To reconstruct the graph after beam pruning, we simply collect the surviving edges $\\bigcup _{v \\in V} S_v$ .","We can also let $\\tau = \\lambda \\min _{e \\in \\text{in}(v)} \\hat{d}(\\mathrm {tail}(e))+ (1 - \\lambda ) \\max _{e \\in \\text{in}(v)} \\hat{d}(\\mathrm {tail}(e))$ where $0 \\le \\lambda \\le 1$ is a parameter that controls the amount of pruning.","Note that beam pruning is very similar to greedy pruning.","For greedy pruning, edges are pruned based on edge weights, while for beam pruning, edges are pruned based on estimates of shortest distances.","It is obvious that if an edge is pruned, paths going through the edge are no longer in the search space.","Whenever we prune an edge, we discard the maximum-weight path that passes through the edge.","Obviously if the maximum-weight path passing through the edge survives pruning, the edge survives pruning.","The max-marginal for an edge is defined as the weight of the maximum-weight path passing through the edge.","Max-marginal pruning was first proposed by [124] and later rediscovered by [142].","Formally, the max-marginal of an edge $e$ is defined as $\\gamma (e) = \\max _{p \\ni e} w(p),$ i.e., the weight of the maximum-weight path that passes through $e$ .","The pruning procedure is simple: choose a threshold $\\tau $ and discard edge $e$ if $\\gamma (e) < \\tau $ .","One way to set the threshold [142] is $\\tau _\\lambda = \\lambda \\max _{e \\in E} \\gamma (e) + (1 - \\lambda ) \\frac{1}{|E|}\\sum _{e \\in E} \\gamma (e),$ where $E$ is the set of segments and $0 \\le \\lambda \\le 1$ .","There are two nice properties about max-marginal pruning with the above threshold: There is at least one path surviving; all paths with weights higher than $\\tau _\\lambda $ survive the pruning.","The first property is true because paths that achieve the maximum weight always survive.","The second property can be proved by contradiction.","Suppose path $p$ has weight $s > \\tau _\\lambda $ but is pruned.","Then there exists an edge $e$ in $p$ such that $\\gamma (e) < \\tau _\\lambda $ .","On the other hand, since $p$ passes through $e$ , $\\gamma (e)$ is at least $s$ , i.e., $\\gamma (e) > s > \\tau _\\lambda $ , a contradiction.","Note that we can only guarantee that paths with weights larger than $\\tau _\\lambda $ survive pruning.","It does not imply that all paths that survive pruning have weights larger than $\\tau _\\lambda $ .","To calculate $\\gamma (e)$ for each $e \\in E$ , note that $\\gamma (e) = \\max _{p \\ni e} w(p)= \\Big [ \\max _{p_1 \\in \\mathcal {P}(s, \\mathrm {tail}(e))} w(p_1) \\Big ]+ w(e)+ \\Big [ \\max _{p_2 \\in \\mathcal {P}(\\mathrm {head}(e), t)} w(p_2) \\Big ],$ where $\\mathcal {P}(u, v)$ is the set of paths that start at vertex $u$ and end at vertex $v$ .","We can construct an algorithm like the shortest-path algorithm to calculate the first and third terms as follows.","$d_1(v) & = \\max _{p_1 \\in \\mathcal {P}(s, v)} w(p_1)= \\max _{e \\in \\text{in}(v)} [ d_1(\\mathrm {tail}(e)) + w(e) ] \\\\d_2(v) & = \\max _{p_2 \\in \\mathcal {P}(v, t)} w(p_2)= \\max _{e \\in \\text{out}(v)} [ d_2(\\mathrm {head}(e)) + w(e) ]$ As shown above, the runtime of max-marginal pruning is the same as that of the shortest-path algorithm (Algorithm REF ).","Max-marginal pruning can also be applied to vertices.","Let $\\gamma (v) = \\max _{p \\ni v} w(p)$ be the max-marginal of the vertex $v$ .","We can also obtain a similar recursion $\\gamma (v) = \\max _{p \\ni v} w(p)= \\Big [ \\max _{p_1 \\in \\mathcal {P}(s, v)} w(p_1) \\Big ]+ \\Big [ \\max _{p_2 \\in \\mathcal {P}(v, t)} w(p_2) \\Big ]= d_1(v) + d_2(v),$ and set a threshold based on the convex combination of $\\max _{v \\in V} \\gamma (v)$ and $\\frac{1}{|V|} \\sum _{v \\in V} \\gamma (v)$ .","Similar guarantees also hold for max-marginal vertex pruning.","Recall that the runtime for finding the shortest path is $O(|E|C)$ where $E$ is the set of segments in the search space and $C$ is the computational cost of evaluating the weight function on a single segment.","Suppose we have a fixed time budget.","Since the pruned search space is smaller than the original space, we can afford to use a more computationally expensive weight function on the pruned space for better prediction.","Since additional pruning can be applied to the search space, we end up with a system having multiple rounds of pruning with increasingly expensive weight functions.","Prediction is then done on the pruned space from the final round.","The above approach is also commonly referred to as rescoring, because hypotheses are given new scores (weights) after pruning.","Models are also named after the pass in which they are used.","For example, a first-pass model is one that searches over the entire search space; a second-pass model is a model that rescores the pruned search space produced by a first-pass model.","Based on rescoring, [142] proposed structured prediction cascades, a type of multi-pass system with multiple rounds of pruning.","The complete search space is denoted $H_1$ .","At round $k$ , we have hypothesis space $H_k$ , train a pruning model $\\Theta _k$ on $H_k$ , and prune $H_k$ to get $H_k^{\\prime }$ .","The pruned space $H_k^{\\prime }$ is expanded to $H_{k+1}$ by considering more context for every segment, such as label $n$ -grams.","The process can be repeated for any number of rounds.","Inspired by [142], we propose discriminative segmental cascades, a multi-pass system consisting of segmental models with increasingly expensive weight functions.","In our cascades, all search spaces are represented as FSTs.","We denote the complete search space $H_1$ .","At the $k$ -th pass, based on the search space $H_k$ , we train a segmental model $\\Theta _k$ , and use it to prune the search space to get $H_k^{\\prime }$ .","In our framework, we can decide whether to expand the search space to include more context.","Since the search spaces are represented as FSTs, we propose a new FST operation, termed self-expansion, to efficiently consider neighboring segments for every segment.","If we decide to self-expand, then $H_{k+1} = {H_k^{\\prime }}^\\uparrow $ , where ${H_k^{\\prime }}^\\uparrow $ is the self-expansion of $H_k^{\\prime }$ .","Otherwise, $H_{k+1} = H_k^{\\prime }$ .","Suppose we decide to self-expand.","Then the weight function for the subsequent pass $w_{k+1}$ can make use of a wider context than the weight function of the previous pass.","Including a wider context is one possible approach to increase the expressiveness of the weight function.","Other approaches include using a deeper or more complex neural network, or extracting more sophisticated features such as tracking formants or pitches, both of which can be done without self-expansion.","The process can be repeated many times, forming a segmental cascade.","An example of a segmental cascade is shown in Figure REF .","Segmental models in each of these passes can be trained with the losses described in Section REF .","[142] proposed to train the pruning models with the filtering loss, measuring how well the ground truths are retained after pruning.","We decide to train the segmental models in the cascade with losses in Section REF that directly measure the final performance, making sure that we have a model for prediction after each pass.","This approach also makes it easier to monitor when to stop stacking the cascade.","Figure: An example segmental cascade.","The search space H 1 H_1is the complete space.","The search space H 1 ' H_1^{\\prime }is pruned from H 1 H_1.","The search space H 2 H_2is the self-expansion of H 1 ' H_1^{\\prime }.The additional vertex and edge added to H 1 ' H_1^{\\prime } beforeself-expansion are shown in dashed lines.Given a search space represented as an FST $H$ , we can expand the context of a segment by considering its neighbors.","Suppose we want to incorporate the previous segment given the current segment.","We create a new FST $H^\\uparrow $ that remembers and keeps track of the previous segment (edge) traversed in $H$ .","Vertices in $H^\\uparrow $ correspond to the possible previous segments in $H$ .","Edges in $H^\\uparrow $ are of the form $\\langle e_1, e_2 \\rangle $ , where $e_1$ is the edge just traversed and $e_2$ is the edge to be traversed.","We refer to $H^\\uparrow $ as the self-expansion of $H$ .","Formally, let $H = ((V, E, \\mathrm {tail}, \\mathrm {head}), \\Sigma , \\Lambda , I, F, i, o)$ .","The self-expansion of $H$ is defined as $H^\\uparrow = ((V^\\uparrow , E^\\uparrow , \\mathrm {tail}^\\uparrow , \\mathrm {head}^\\uparrow ),\\Sigma ^\\uparrow , \\Lambda ^\\uparrow , I^\\uparrow , F^\\uparrow , i^\\uparrow , o^\\uparrow )$ where $V^\\uparrow & = E \\cup \\lbrace e_0 \\rbrace & E^\\uparrow & = \\lbrace \\langle e_1, e_2 \\rangle : \\mathrm {head}(e_1) = \\mathrm {tail}(e_2) \\rbrace $ $I^\\uparrow & = \\lbrace e_0 \\rbrace & \\mathrm {tail}^\\uparrow (\\langle e_1, e_2 \\rangle ) & = e_1 \\\\F^\\uparrow & = \\lbrace e \\in E : \\mathrm {head}(e) \\in F \\rbrace & \\mathrm {head}^\\uparrow (\\langle e_1, e_2 \\rangle ) & = e_2 \\\\\\Sigma ^\\uparrow & = \\Sigma & i^\\uparrow (\\langle e_1, e_2 \\rangle ) & = i(e_2) \\\\\\Lambda ^\\uparrow & = \\Lambda & o^\\uparrow (\\langle e_1, e_2 \\rangle ) & = o(e_2)$ A vertex $v_0$ and an $e_0$ is created in $H$ before self-expansion, where $i(e_0) = o(e_0) = \\epsilon $ , $\\mathrm {tail}(e_0) = v_0$ , and $\\mathrm {head}(e_0) = v_1$ assuming we only have a single start state $v_1$ , i.e., $I = \\lbrace v_1\\rbrace $ .","The edge $e_0$ is created as a sentinel, corresponding to a vertex in $H^\\uparrow $ for not having traversed any edge.","The weight function $w^\\uparrow : E^\\uparrow \\rightarrow \\mathbb {R}$ is task-dependent, so we do not define it explicitly here.","However, since $w^\\uparrow $ is defined on $E^\\uparrow $ , the weight function after taking a edge $\\langle e_1, e_2 \\rangle \\in E^\\uparrow $ can make use of all the information about $e_1$ and $e_2$ to compute the weight, including the start times, end times, and labels.","We also observe that $\\text{out}^\\uparrow (e) & = \\lbrace \\langle e, e^{\\prime } \\rangle : e^{\\prime } \\in \\text{out}(\\mathrm {head}(e)) \\rbrace \\\\\\text{in}^\\uparrow (e) & = \\lbrace \\langle e^{\\prime }, e \\rangle : e^{\\prime } \\in \\text{in}(\\mathrm {tail}(e)) \\rbrace $ for $e \\in V^\\uparrow = E$ .","Therefore, similarly to $\\sigma $ -composition, self-expansion can be computed lazily (on the fly) while traversing $H$ .","We conduct experiments on the TIMIT data set in the same setting as in Section REF , except that here we follow [42] and use 40-dimensional log mel filter bank outputs instead of 39-dimensional MFCCs.","The data set is phonetically transcribed, so we have the option of training the encoders with frame-wise cross entropy based on the ground-truth frame labels.","In the following experiments, we explore convolutional neural network (CNN) encoders.","The CNN encoders are inspired by [123] in that they are deep convolutional neural networks with small $3 \\times 3$ and $5 \\times 5$ filters.","For frame classification, the input to the network is a window of 15 frames of 40-dimensional log mel filter outputs centered on the current frame.","The network has five convolutional layers, with 64 filters of size $5 \\times 5$ for the first layer and 128, 128, 256, and 256 filters of size $3 \\times 3$ for layers two to five respectively, each of which is followed by a rectified linear unit (ReLU) activation [101], with max pooling layers after the first and the third ReLU layers.","The output of the final ReLU layer is concatenated with a window of 15 frames of 39-dimensional MFCCs centered on the current frame, and the resulting vector is passed through three fully connected ReLU layers with 4096 units each.","The network is trained with SGD for 35 epochs with step size 0.001, momentum 0.95, weight decay 0.005, and a mini-batch size of 100 frame predictions.","Dropout [128] are applied to the concatenation layer with rate 20% and to the fully connected layers with rate 50%.","This classifier was tuned on the development set and achieves a 22.3% frame error rate (after collapsing to 39 phone labels) on the development set and 23.0% on the test set.","After confirming that the CNN encoder is able to perform well on frame classification, we use the CNN encoder to generate frame posterior probabilities, and train segmental models with the FC weight function and a maximum duration of 30 frames.","Hinge loss is optimized with AdaGrad [27] with step size 0.01 and a mini-batch size of 1 utterance for 70 epochs.","No explicit regularizer is used except early stopping on the development set.","We use a scaled overlap cost for our experiments: $\\mathrm {cost}(p^{\\prime }, p) &= \\sum _{e^{\\prime } \\in p} \\mathrm {cost}(e^{\\prime }, p) \\\\\\mathrm {cost}(e^{\\prime }, p) &= |e^{\\prime } \\cup \\tilde{e}| - |e^{\\prime } \\cap \\tilde{e}| {1}_{o(e^{\\prime }) = o(\\tilde{e})}$ where $p^{\\prime }$ is the hypothesized path, $p$ is the ground-truth path, $|e^{\\prime } \\cup e|$ denotes the union of the two intervals defined by $e$ and $e^{\\prime }$ , $|e^{\\prime } \\cap e|$ denotes the intersection of the two intervals, $o(e)$ is the output label of $e$ , and $\\tilde{e} = \\operatornamewithlimits{\\arg \\!\\max }_{e \\in p} |e^{\\prime } \\cap e|$ .","In words, $\\tilde{e}$ is the edge that overlaps the most with $e^{\\prime }$ .","The cost is non-negative and is only zero if $e^{\\prime } = e$ , and it can be seen as an estimate of the number of incorrectly predicted frames.","The results are shown in Table REF .","Our first-pass segmental model is on par with [1], where a CNN encoder is also used, and it is also on par with a baseline HMM-DNN hybrid system that we trained with Kaldi using the standard recipe for TIMIT [110].","Table: Phoneme error rates (%) on TIMITcomparing a HMM-DNN hybrid system todiscriminative segmental models.To construct a second-pass segmental model, we first evaluate pruning approaches based on oracle error rates and lattice densities.","The oracle error rate is the lowest achievable phone error rate of a given lattice.","The lattice density is defined as the total number of edges divided by the number of ground-truth edges, i.e., the number of hypothesized edges per ground-truth edge.","In general, a dense lattice has a higher chance to contain the ground-truth path than a sparse lattice.","In fact, the entire search space is a dense lattice that always contains the ground-truth path.","A good pruner is expected to produce sparse lattices while maintaining a low oracle error rate.","We compare max-marginal pruning with $\\lambda \\in \\lbrace 0.5, 0.6, 0.7, 0.8\\rbrace $ , greedy pruning with $\\lambda \\in \\lbrace 0.95, 0.9, 0.875, 0.85, 0.825\\rbrace $ , random pruning with probability $\\lbrace 0.92, 0.94, 0.96, 0.98\\rbrace $ , and beam pruning with $\\lambda =0.9$ .","The results are shown in Figure REF .","For reference, the average density of the complete spaces is 11587.67.","We observe that search spaces of segmental models are robust to missing edges—the oracle error rate is only 1.7% when we randomly drop 92% of the edges.","We also observe that max-marginal pruning produces significantly sparser lattices than greedy pruning and random pruning while maintaining low oracle error rates.","Beam pruning performs the worst.","Since greedy pruning works well by just comparing edges with the same tail vertex (i.e., with the same start time), we suspect the bad performance of beam pruning is due to comparing distances at different time points.","Figure: Left: Oracle error rates vs lattice densities for various pruningapproaches.","Right: A re-scaled version of the left figure focusing ongreedy pruning and max-marginal pruning.Based on the above pruning comparison, we decide to use the lattices generated by max-marginal pruning with $\\lambda = 0.8$ .","Let $H_1$ be the entire search space, and $H_1^{\\prime }$ be the pruned search space.","To train a segmental model with a wider segmental context, we first self-expand $H_1^{\\prime }$ into $H_2 = {H_1^{\\prime }}^\\uparrow $ and introduce the following weight functions.","Note that the weight functions can now depend on two segments instead of one.","Instead of re-learning all of the weight functions in the first-pass model, we reuse them in the second pass by having $w_{\\text{lat}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= \\alpha \\cdot w((\\ell _2, s_2, t_2))$ from the first level of the cascade, where $\\alpha $ is a learnable parameter.","We use the weight $w_{\\text{LM}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= \\beta \\cdot \\log p(\\ell _2 | \\ell _1)$ to include the log probability of the bigram $\\ell _1 \\ell _2$ , where $\\beta $ is a learnable parameter.","We define $w_{\\text{left}-k}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )& = u_{s_2 - k, \\ell _1, \\ell _2} \\\\w_{\\text{right}-k}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )& = u_{t_2 + k, \\ell _1, \\ell _2}^{\\prime }$ similarly to the first-order boundary weights, where $u_{t, \\ell , \\ell ^{\\prime }} = \\theta _{\\ell , \\ell ^{\\prime }}^\\top h_t$ and $u_{t, \\ell , \\ell ^{\\prime }}^{\\prime } = {\\theta ^{\\prime }_{\\ell , \\ell ^{\\prime }}}^\\top h_t$ for $t = 1, \\dots , \\tilde{T}$ and some learnable parameters $\\theta , \\theta ^{\\prime }$ .","Since the search space is sparse, we can afford to compute segment features based on a neural network for every segment.","We choose a similar CNN architecture to the one for frame classification.","We pre-train the CNN with a segment classification task.","Here the features at the input layer are the log mel filter outputs from a 15-frame window around the segment's central frame.","Instead of concatenation with 15-frame MFCCs, we concatenate with a segmental feature vector consisting of the average MFCCs of three sub-segments in the ratio of 3-4-3, plus two four-frame averages at both boundaries and length indicators for length 0 to 20 (similar to the segmental feature vectors of [46], [22]).","This CNN is trained on the ground-truth segments in the training set.","Finally, we build an ensemble of such networks with different random seeds and a majority vote.","This ensemble classifier has a 15.0% segment classification error on the test set.","A comparison of our CNN to other segment classifiers is shown in Table REF .","Table: TIMIT segment classification error rates (%).Directly running the CNN segment classifier for every edge in the lattice is, however, still too time-consuming.","We instead compress the best-performing (single) CNN into a shallow fully connected network with one hidden layer of 512 ReLUs by training it to predict the log probability outputs of the deep network, as proposed by [8].","We then use the log probability outputs of the shallow network.","We refer to the result as segment NN weights.","Let $w_{\\text{seg}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= (u_{s_2, t_2})_{\\ell _2}$ where $u_{s, t} = \\theta ^\\top z_{s, t}$ , $z_{s, t}$ is a log probability vector of the labels obtained by passing the segment $x_s, \\dots , x_t$ to the CNN segment classifier, and $\\theta $ is some learnable parameter vector.","To add more context information, we use the same CNN architecture and training setup to learn a bi-phone frame classifier, but with an added 256-unit bottleneck linear layer before the softmax.","Each frame is labeled with its segment label and one additional label from a neighboring segment.","If the current frame is in the first half of the segment, the additional label is the previous phone; if it is in the second half, then the additional label is the next phone.","The learned bottleneck layer outputs are used to define weights (although they do not correspond to log probabilities) with averaging and sampling as for the uni-phone case.","We refer to the resulting weights as bi-phone NN bottleneck weights.","We use the sum of the lattice weight, the bigram LM weight, second-order boundary weights, segment NN weights, bi-phone NN bottleneck weights, length indicators, and lexicalized bias as our final weight function for the second level of the cascade.","Hinge loss is minimized with AdaGrad for 20 epochs with step size 0.01.","Again, no explicit regularizer is used except early stopping based on the PER on the development set.","Results with these additional weights are shown in Table REF .","Adding the second-order weights, bigram LM, and the above NN weights gives a 1.8% absolute improvement over our best first-pass system, demonstrating the value of including such expensive features with wider context.","Table: Phoneme error rates (%)with discriminative segmental cascades.The search space for the first pass is denoted H 1 H_1,and the search space for the second pass H 1 ' ↑ {H_1^{\\prime }}^\\uparrow is denoted H 2 H_2, where H 1 H_1 is pruned to get H 1 ' H_1^{\\prime }.We have shown that segmental cascades can be used to incorporate computationally expensive weight functions while maintaining efficiency by shifting the computationally expensive weights to later passes.","In this section, instead of incorporating additional weights, we shift weights to later passes to improve decoding speed.","Due to their recent success [42], we also explore long short-term memory (LSTM) networks [56] as encoders instead of CNN encoders.","In this section, we describe long short-term memory (LSTM) networks.","We start by defining a single-layer LSTM.","Suppose the input vectors are $x_1, \\dots , x_T$ .","First we apply a linear transformation to the inputs to get $x_t^{\\prime } = \\begin{pmatrix} W_{xu} \\\\ W_{xi} \\\\ W_{xf} \\\\ W_{xo} \\end{pmatrix} x_t,$ which can be done for $t=1, \\dots , T$ in parallel.","Next, assume $h_0 = c_0 = 0$ .","Suppose we have already computed $h_t$ and $c_t$ , and we want to compute $h_{t+1}$ and $c_{t+1}$ .","We apply a linear transformation to get the candidates $u^{\\prime }$ , $i^{\\prime }$ , $f^{\\prime }$ , and $o^{\\prime }$ for the cell update vectors $u$ and the gates $i$ , $f$ and $o$ .","$\\begin{pmatrix}u^{\\prime } \\\\ i^{\\prime } \\\\ f^{\\prime } \\\\ o^{\\prime }\\end{pmatrix}= x_t^{\\prime } + \\begin{pmatrix} W_{hu} \\\\ W_{hi} \\\\ W_{hf} \\\\ W_{ho} \\end{pmatrix} h_t+ \\begin{pmatrix} 0 \\\\ W_{ci} \\\\ W_{cf} \\\\ 0 \\end{pmatrix} c_t.$ The update vectors $u$ , the input gate $i$ , and the forget gate $f$ are then computed from the candidates by applying nonlinear transformations.","$u & = \\tanh (u^{\\prime }) \\\\i & = \\text{logistic}(i^{\\prime }) \\\\f & = \\text{logistic}(f^{\\prime })$ The new cell vector $c_{t+1}$ is computed with the update vector gated by the input gate and the previous cell.","$c_{t+1} & = i \\odot u + f \\odot c_t$ Finally, the output gate $o$ is computed from the updated cell $c_{t+1}$ , and the new hidden vector $h_{t+1}$ is updated with the cell gated by the output gate.","$o & = \\text{logistic}(o^{\\prime } + W_{co} c_{t+1}) \\\\h_{t+1} & = o \\odot \\tanh (c_{t+1})$ The final results after running an LSTM on $x_1, \\dots , x_T$ are $h_1, \\dots , h_T$ and $c_1, \\dots , c_T$ .","We use $h_{1:T} = \\text{LSTM}(x_{1:T})$ to denote the process above.","Since LSTMs have an assigned direction, it is natural to run two LSTMs in opposite directions and combine the hidden vectors.","Formally, $h^f_{1:T} & = \\text{LSTM}(x_{1:T}) \\\\h^b_{T:1} & = \\text{LSTM}(x_{T:1}) \\\\h_t & = W_f h^f_t + W_b h^b_t$ We use $h_{1:T} = \\text{BiLSTM}(x_{1:T})$ as a shorthand for the bidirectional LSTMs.","Note that the parameters of the two LSTMs are not shared.","Another way to increase the complexity of the acoustic encoder is to stack them on top of each other.","For example, if we stack 3 layers of bidirectional LSTMs, we get $h^{(1)}_{1:T} &= \\text{BLSTM}(x_{1:T}) \\\\h^{(2)}_{1:T} &= \\text{BLSTM}(h^{(1)}_{1:T}) \\\\h^{(3)}_{1:T} &= \\text{BLSTM}(h^{(2)}_{1:T})$ We use $h_{1:T} = \\text{DBLSTM}^3(x_{1:T})$ to denote the 3-layer bidirectional LSTMs (where “D” refers to “deep”).","Again the parameters of the LSTMs in each layer and each direction are not shared.","As in Section REF , we first explore LSTMs on a frame classification task, Specifically, the log probability vector of a frame at time $t$ is $z_t = \\text{logsoftmax}(W h_t + b)$ where $h_t$ is the hidden vector at time $t$ produced by an LSTM, and $W$ and $b$ are the parameters.","We then explore segmental cascades with weight functions based on log probabilities produced by LSTMs.","Experiments are conducted on the TIMIT data set.","We compute 41-dimensional log mel filter bank outputs including energy, and concatenate with the delta's and double delta's to form the final 123-dimensional acoustic feature vectors.","Instead of using a 192-utterance development set, we follow the Kaldi recipe [110] and use 400 utterances from the complete test set (disjoint from the core test set) as the validation set.","We report numbers on the core test set, the same test set as in Section REF .","We build a frame classifier by stacking three layers of bidirectional LSTMs.","The cell and hidden vectors have 256 units.","We train the frame classifier with frame-wise cross entropy and optimize with AdaGrad [27] with mini-batch size of 1 utterance and step size 0.01 for 30 epochs.","We choose the best-performing model on the development set (early stopping).","Following [138], [92], we consider dropping half of the frames for any given utterance in order to save time on feeding forward.","Specifically, we only use $x_2, x_4, \\dots , x_{T - 2}, x_T$ to feed forward through the deep LSTM and generate $z_2, z_4, \\dots ,z_{T - 2}, z_T$ (assuming $T$ is even, without loss of generality).","We then copy each even-indexed output to its previous frame, i.e., $z_{i-1} = z_i$ for $i = 2, 4, \\dots , T-2, T$ .","During training, the cross entropy is calculated over all frames and propagated back.","Specifically, let $E_i$ be the cross entropy at frame $i$ , and $E = \\sum _{i=1}^T E_i$ .","The gradient of $z_i$ is the sum of the gradients from the current frame and the copied frame.","Dropping even-indexed frames is similar to dropping odd-indexed frames, except the outputs are copied from $z_i$ to $z_{i+1}$ for $i = 1, 3, \\dots , T-1$ .","When training LSTMs with subsampling, we alternate between dropping even- and odd-numbered frames every other epoch.","The results are shown in Figure REF .","Figure: Development set frame error rates vs. number of epochs (left)and vs. training hours (right).We observe that with subsampling the model converges more slowly than without, in terms of number of epochs.","However, by the end of epoch 30, there is almost no loss in frame error rates when we drop half of the frames.","Considering the more important measure of training time rather than number of epochs, the LSTMs with frame subsampling converge twice as fast as those without subsampling.","For the remaining experiments, we use the log posteriors at each frame of the subsampled LSTM outputs as the inputs to the segmental models.","Our baseline system, denoted $R$ , is a first-pass segmental model with the FC weight function (Section REF ).","The baseline system is trained by optimizing hinge loss with AdaGrad using mini-batch size 1 utterance, step size 0.1, and early stopping for 50 epochs.","We decide to shift the FC weight function to the second pass, and train a segmental model, denoted $A_1$ , with just the label posterior and a label-independent bias.","Specifically, the label posterior weight is defined as $w((\\ell , s, t)) = \\sum _{i=s}^t (z_i)_\\ell ,$ where $z_i = \\text{logsoftmax}(W h_i + b)$ is the log probability vector at time $i$ based on the vector $h_i$ produced by the LSTM.","This reduces the number of features from 24528 to two.","We use hinge loss optimized with AdaGrad with mini-batch size 1 and step size 1.","Since we only have two features, learning converges very quickly, in only three epochs.","We take the model from the third epoch to produce lattices for subsequent passes in the cascade.","Lattices are generated with max-marginal pruning with $\\lambda = 0.85$ .","The resulting lattices have an average oracle error rates of 1.4% and average density of 213.02.","We train a second-pass model, denoted $A_2$ , with the FC weight function except that we add a “lattice score” feature corresponding to the segment score given by the two-feature system.","Hinge loss is optimized with AdaGrad with mini-batch size 1, step size 0.1, and early stopping for 20 epochs.","The learning curve comparing $R$ and $A_1$ followed by $A_2$ is shown in Figure REF .","We observe that the learning time per epoch of the two-feature system $A_1$ is only one-third of the baseline system $R$ .","We also observe that training of $A_2$ converges faster than training $R$ , despite the fact that they use almost identical feature functions.","The baseline system achieves the best result at epoch 49.","In contrast, the two-pass system is done before the baseline even finishes the third epoch.","Figure: Learning curve of the proposed two-pass system compared withthe baseline system.The time gap between the first pass and the second pass is thetime spent on pruning.Following Section REF , given the first-pass baseline, we apply max-marginal pruning to produce lattices for the second-pass baseline with $\\lambda =0.8$ .","The second-pass baseline features are the lattice score from the first-pass baseline, a bigram language model score, first-order length indicators, and a bias.","Hinge loss is optimized with AdaGrad with mini batch size 1, step size 0.01, and early stopping for 20 epochs.","For the proposed system, we produce lattices with max-marginal pruning and $\\lambda =0.3$ for the third-pass system.","We use the same set of features and hyper-parameters as the second-pass baseline for the third pass.","Phone error rates of all passes are shown in Table REF .","First, if we compare the one-pass baseline with the proposed two-pass system, our system is close to the baseline.","Second, we observe a healthy improvement by just adding the bigram language model score to the second-pass baseline.","The improvement for our third-pass system is small but brings our final performance to within 0.4% of the baseline second pass.","Table: Phone error rates (%) of proposed and baseline systems.Next we report on the speedups in training and decoding obtained with our proposed approach.","Table REF shows the real-time factors for decoding with the baseline and proposed systems.","In terms of decoding time alone, we achieve a factor of 2.4 speedup compared to the baseline.","If the time of feeding forward LSTMs is included, then our proposed system is two times faster than the baseline.","Table: Real-time factors for decoding.Table REF shows the times needed to train a system to get to the performance in Table REF .","The speedup mostly comes from the fast convergence of the first pass.","In terms of training the segmental models alone, we achieve an 18.0-fold speedup.","If the time to train the LSTMs is included, then we obtain a 3.4-fold speedup compared to the baseline.","To summarize some of the above results: With a combination of the first-pass two-feature system and edge pruning, we prune 95% of the segments in the first-pass hypothesis space, leading to significant speedup in both decoding and training.","The feed-forward time for our LSTMs is halved through frame subsampling.","In the end, with a single four-core CPU, we achieve 0.31 times real-time decoding including feeding forward, which is 2.2 times faster than the baseline, and 32.5 hours in total to obtain our final model including LSTM training, which is 3.4 times faster than the baseline.","Excluding the LSTMs, the segmental model decoding alone is 2.4 times faster than the baseline, and training the segmental models alone is 18 times faster than the baseline.","Table: Hours for training the system.In this chapter, we have described discriminative segmental cascades, a multi-pass system for segmental models.","Max-marginal pruning lies at the heart of segmental cascades, producing sparse lattices while having low oracle error rates.","After pruning, it becomes feasible to incorporate computationally expensive weight functions, such as higher-order LMs, and weight functions based on neural networks.","We demonstrate that incorporating these weights significantly improves the performance of segmental models.","We also show that decoding and training speed can be significantly improved with segmental cascades by shifting weights to later passes without sacrificing accuracy.","Though we have shown improvement with segmental cascades, how the density and structure of search spaces affect the training of segmental models is still not fully known.","Task-dependent priors, such as phoneme duration, are not fully explored and can potentially be integrated into the pruning procedure.","The fact that beam search works well for frame-based models but fails miserably for segmental models is also worth investigating.","Converting other more sophisticated search algorithms, such as $\\text{A}^*$ search [68], into pruning algorithms is a possibility that might show us signs as to why beam search fails for segmental models.","In the previous chapter, we have seen how a sequence recognizer can be trained in multiple stages: a frame classifier is first trained with frame-wise cross entropy, and a segmental model is trained with a sequence loss, such as the ones defined in Section REF , based on the classifier's outputs.","In other words, the frame classifier is trained independently from the segmental model, and is held fixed when the segmental model is being trained.","A natural question to ask is whether we can further improve the sequence loss by updating the frame classifier while holding the segmental model fixed.","A more general question would be whether we can improve the sequence loss by updating the frame classifier and the segmental model simultaneously.","Optimizing all of the parameters simultaneously against a single loss function is commonly referred to as end-to-end training.","Since the loss function with respect to all the parameters in a neural network is non-convex, end-to-end training might be more difficult than training in multiple stages.","On the other hand, since end-to-end training is jointly optimizing all parameters, it might achieve a lower loss, leading to better performance.","In this chapter, we formally define these training settings, including multi-stage training and end-to-end training, Different training approaches and losses have different training requirements.","We discuss the pros and cons of these training approaches, and conduct experiments to see empirically how well they compare to each other.","Recall that the set of parameters $\\Theta $ consists of $\\Theta _\\text{enc}$ and $\\Theta _\\text{dec}$ , where $\\Theta _\\text{enc}$ is the set of parameters in the encoder and $\\Theta _\\text{dec}$ are the rest of the parameters in the weight function.","The training approach we have seen in the previous chapter is referred to as multi-stage training.","Specifically, the encoder is trained in the first stage with an encoder-specific loss $\\mathcal {L}_\\text{enc}(\\Theta _\\text{enc}),$ for example, the frame-wise cross entropy.","Let $\\hat{\\Theta }_\\text{enc} = \\operatornamewithlimits{\\arg \\!\\min }_{\\Theta _\\text{enc}}\\mathcal {L}_\\text{enc}(\\Theta _\\text{enc})$ .","In the second stage, we solve the optimization problem $ \\min _{\\Theta _\\text{dec}} \\mathcal {L}_\\text{seq}(\\hat{\\Theta }_\\text{enc}, \\Theta _\\text{dec}),$ where $\\mathcal {L}$ is a sequence loss, such as hinge loss, log loss, or marginal log loss, defined in Section REF .","Note that in the second stage the parameters of the encoder are held fixed.","Typically, after the first stage, we can feed the inputs in the entire data set forward, and reuse the vectors while optimizing the sequence loss to save computation.","Another benefit of optimizing (REF ) with $\\hat{\\Theta }_\\text{enc}$ fixed is that the loss function $\\mathcal {L}$ might be convex in $\\Theta _\\text{dec}$ , for example, when hinge loss or log loss is used and the weight function is linear in $\\Theta _{\\text{dec}}$ .","The characteristics of convex functions are well-understood [13], and many algorithms, other than first-order methods, have been developed for optimizing convex functions [12].","In contrast to multi-stage training, we refer to solving $ \\min _{\\Theta _\\text{enc}, \\Theta _\\text{dec}}\\mathcal {L}_\\text{seq}(\\Theta _\\text{enc}, \\Theta _\\text{dec})$ as end-to-end training, where all parameters $\\Theta _\\text{enc}$ and $\\Theta _\\text{dec}$ are optimized jointly.","Since all parameters are optimized jointly, the loss function is in general non-convex.","Though end-to-end training might be harder to solve, it might lead to a lower loss value.","Typically, the solution found by multi-stage training is a reasonably good solution for the sequence loss.","As a result, a safe strategy, which we refer to as end-to-end fine-tuning, may be to use the solution from multi-stage training as an initialization for solving (REF ).","One major difference between multi-stage training and end-to-end training is their resource requirements.","Multi-stage training requires extra labels for optimizing the encoder loss $\\mathcal {L}_\\text{enc}$ , while end-to-end training requires only whatever the sequence loss $\\mathcal {L}_\\text{seq}$ requires.","For experiments, we use the same phoneme recognition setting on the TIMIT data set as in Section REF .","In the sections below, we first compare the performance of CNNs and LSTMs, and then compare two training settings for training segmental models, multi-stage training followed by end-to-end fine-tuning and end-to-end training from random initialization.","We also explore various weight functions paired with various loss functions in the context of end-to-end training from random initialization.","Recall that the multi-stage training approach demonstrated in Section REF consists of two stages: first training a frame classifier, and second training a segmental model based on the output probabilities of the frame classifier.","Instead of a CNN encoder, a 3-layer bidirectional LSTM with hidden vector size of 250 is trained for frame classification.","Parameters are initialized according to [36].","Vanilla SGD is used to minimize frame-wise cross entropy for 20 epochs with step size 0.1 and a mini-batch size of 1 utterance.","Gradients are clipped to norm 5 if the norm is above 5.","The best model (measured by frame error rates on the development set) is chosen among the 20 epochs, and trained for another 20 epochs with step size 0.75 decayed by 0.75 after every epoch.","For phoneme recognition, we use segmental models with the FC weight function, the same setting as in Section REF .","Hinge loss with the overlap cost are minimized with RMSProp [100] for 20 epochs with step size ${10}^{-4}$ and decay 0.9, The frame error rates (FER) and phoneme error rates (PER) compared to the CNN in Section REF are shown in Table REF .","The LSTM performs better than the CNN for both tasks.","We suspect that the LSTM's superior performance is due to the training approaches: the LSTM is trained on entire utterances, while the CNN is trained on batches of 15-frame windows.","This effect was also observed in [58].","For the rest of the experiments, we use 3-layer LSTMs as our encoders.","Table: Frame error rates (FER) and phoneme error rates (PER)on the development setcomparing the CNN and the LSTM encoder.Next, we compare segmental models trained with different losses, including hinge loss, log loss, and marginal log loss.","We use the FC weight function and a maximum segment duration of 30 frames.","All losses are minimized with RMSProp for 20 epochs with step size ${10}^{-4}$ and decay 0.9.","Results are shown in Table REF .","No explicit regularizers are used except early stopping on the development set.","All losses achieve similar results with log loss slightly behind.","Note that although marginal log loss does not require manual alignments, the frame-wise cross entropy used to train the frame classifier still does.","After two-stage training, we use the trained segmental models and frame classifiers as an initialization for end-to-end training.","Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 decayed by 0.75 after each epoch.","Gradients are clipped with norm 5.","Dropout is added to the LSTMs with rate 0.2, and no other regularizers are used except early stopping on the development set.","Results are shown in Table REF .","End-to-end fine-tuning is able to improve the error rates for all losses with marginal log loss being the best performer.","Table: Phoneme error rates (%) comparing different losseswith two-stage training (2s) followed by end-to-end fine-tuning (ft).After end-to-end fine-tuning, the encoders are not constrained to perform well on frame-wise cross entropy.","We evaluate the encoders on the frame classification task to see how much the encoders deviate from the initialization, and whether the intermediate representations trained for frame classification are still preserved.","Results are shown in Table REF .","The encoders fine-tuned with hinge loss and log loss deviate less compared to the one fine-tuned with marginal log loss.","We suspect this is because hinge loss and log loss use the ground-truth alignments in training while marginal log loss does not.","Table: Frame error rates (%) of LSTM encodersafter end-to-end fine-tuning with different losses .After observing the success of end-to-end fine-tuning, we conduct experiments to see whether it is possible to train segmental models end to end from random initialization.","We use the FC weight function and a maximum duration of 30 frames, the same setting as in the multi-stage experiments.","We initialize the parameters according to [36].","Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 and gradient clipping with norm 5.","The best performing model, chosen in the first 20 epochs, is trained for another 20 epochs with vanilla SGD step size 0.75 decayed by 0.75 after each epoch.","Dropout of rate 0.2 is used throughout the training process.","No other regularizers are used except early stopping on the development set.","Results are shown in Table REF .","There is no significant difference in terms of performance between models trained end to end from random initialization and ones trained in multiple stages followed by fine-tuning.","This shows that it is possible to train segmental models end to end from random initialization, and that models trained in multiple stages can serve as a good initialization for end-to-end training.","Table: Phoneme error rates (%) on the development setcomparing multi-stage trainingfollowed by end-to-end fine-tuning (2s+ft) and end-to-end trainingfrom random initialization (e2e).Next, we consider various weight functions for end-to-end training from random initialization.","Note that the initialization scheme in [36] is not suitable the for log-soft-max operation used after the LSTMs for producing log probabilities, because the variances before and after the log-soft-max operation are very different.","Based on this intuition, we define a new weight function, called FC bottleneck (FCB) weight function, removing the log-soft-max layer when using the FC weight function.","More precisely, suppose $h_1, \\dots , h_T$ is the sequence of vectors produced by an LSTM.","Whereas the FC weight function uses the average, samples, and boundaries of $z_i = \\text{logsoftmax}(W h_i + b)$ for $i = 1, \\dots , T$ , the FCB weight function uses the average, samples, and boundaries of $h_i$ 's directly.","The learning curves of segmental models trained end to end with FC weight function and the FCB weight function are shown in Figure REF .","We observe significantly faster convergence with the FCB weight function than with the FC weight function.","Figure: Learning curves of segmental models trained end to endwith the FC weight function and FCB weight functionwith marginal log loss.We do not compare the MLP weight function to the FC and FCB weight functions in the current setting, because it is too time-consuming to train segmental models with the MLP weight function on the entire search space.","To train segmental models with the MLP weight function, we follow [85] and reduce the time resolution by a factor of four using pyramid LSTMs.","Specifically, for an input sequence $x_{1:T}$ , $h^{(1)}_{1:T} &= \\text{BLSTM}(x_{1:T}) \\\\z^{(1)}_{1:\\lfloor T/2 \\rfloor } &= \\text{subsample}(h^{(1)}_{1:T}) \\\\h^{(2)}_{1:\\lfloor T/2 \\rfloor } &= \\text{BLSTM}(z^{(1)}_{1:\\lfloor T/2 \\rfloor }) \\\\z^{(2)}_{1:\\lfloor T/4 \\rfloor } &= \\text{subsample}(h^{(2)}_{1:\\lfloor T/2 \\rfloor }) \\\\h^{(3)}_{1:\\lfloor T/4 \\rfloor } &= \\text{BLSTM}(z^{(2)}_{1:\\lfloor T/4 \\rfloor })$ where $h_2, h_4, \\dots , h_{2(k-1)}, h_{2k} = \\text{subsample}(h_1, h_2, \\dots , h_{2k})$ if we drop a frame for every two frames.","The final output sequence $h^{(3)}$ is then used to compute edge weights.","An encoder with several layers of LSTMs interleaved with subsampling is also known as a pyramid LSTM and has been used in other prior work, e.g., [16].","In addition, we set the maximum duration to 8 frames, effectively $4 \\times 8 = 32$ frames on the original time scale.","The reduced time resolution and maximum duration result in search spaces 16 times smaller than the original ones, making it feasible to compute the MLP weight function on all edges.","With 3-layer pyramid LSTMs, we compare segmental models trained with the three weight functions and three losses.","All models are trained with the same step size scheduling as in the previous experiments.","Results with and without pyramid LSTMs are both shown in Table REF .","First, we observe that using pyramid LSTMs leads to a degradation in performance for segmental models with the FC and FCB weight functions.","However, using pyramid LSTMs has less impact on segmental models trained with marginal log loss compared to the ones trained with hinge loss and log loss.","We suspect this is because hinge loss and log loss rely on the ground-truth alignments for training while marginal log loss does not.","Second, we observe that while segmental models with the FC and FCB weight functions are able to achieve low hinge loss values, the ones with the MLP weight function completely fail to do so.","Finally, the best result on the development set obtained by using the FC weight function is slightly worse than using the MLP weight function.","Table: Phoneme error rates (%) comparing different lossesfor end-to-end training from random initialization.Segmental models trained with marginal log loss do not require ground-truth alignments during training, making annotating data sets easier, cheaper, and less time-consuming.","However, it is harder to diagnose errors made by models trained end to end because the intermediate representations are not interpretable.","Models trained in multiple stages are easier to diagnose by looking at the performance of proxy tasks, such as frame classification in our case.","We compare the time to train segmental models end to end.","Times are measured on a 3GHz 4-core CPU.","Multithreading is only used for matrix operations; all FST algorithms are implemented single-threaded.","Results are shown in Table REF .","Reducing the time resolution with pyramid LSTMs significantly improves the runtime for all losses.","The MLP weight function is about 2.5 times slower than the FC weight function.","Table: Average number of minutes for one epoch of end-to-end training on TIMIT.We are interested in how well segmental models recover the phone boundaries in the end-to-end setting when manual alignments are not used during training.","The task, commonly known as phonetic segmentation, is to align the phonetic transcription to the acoustic frames.","Unlike sequence prediction, the phonetic transcription is known.","We measure the error rates of the boundaries under some tolerance level.","Specifically, suppose the predicted boundary is at time $\\hat{b}$ and the ground truth is at time $b$ .","The predicted boundary $\\hat{b}$ is considered correct if the absolute distance $t = |\\hat{b} - b| \\le \\tau $ for some $\\tau $ .","The error rates under various thresholds for the segmental model with the FC weight function trained with marginal log loss are shown in Table REF .","Models proposed in [70], [113], [144], shown in row 1–3 in Table REF , are specifically trained for phonetic segmentation.","Though the alignment results are behind models trained specifically to align, the segmental model trained with marginal log loss is not supervised with any ground-truth alignments, and its performance is sufficient for many tasks.","Table: Boundary error rates (%) of phonetic segmentation at different tolerance levelson the TIMIT core test set (except the last row)for segmental models trained end to end with marginal log losscompared to past results.We analyze the errors made by the segmental model.","The top 30 most errorful boundary types sorted by error rates are shown in Table REF .","We observe that most of the boundaries in Table REF involve silences, vowels, and semi-vowels.","The ones involving silences often appears at the start or the end of utterances.","Boundaries between two vowels and between a semi-vowel and a vowel are known to be ambiguous [120], [135].","We see few alignment errors between consonants and vowels.","The top 30 most errorful boundary types sorted by error counts are shown in Table REF .","We are also interested in boundary types that have hight error counts but not necessary have high error rates.","Compared to Table REF , we see a very different pattern.","Most of the errors in Table REF involve silences (including voiced and unvoiced closures).","For $0\\text{ms}$ and $10\\text{ms}$ , the segmental model errs at the boundaries between closures and the bursts of stop consonants, such as /b/, /d/, /g/, /p/, /t/, and /g/.","It also errs at boundaries involving liquids, such as /r/ and /l/, across all tolerance levels.","Similarly to semi-vowels, the boundaries of liquids are also known to be ambiguous [120], [135].","The errors for higher tolerance levels, such as 30ms and 40ms, mostly involve silences that appear at the start or end of utterances.","We notice that 10.5% of the silences in the training set are longer than 30 frames.","We suspect the 30-frame maximum duration constraint is too restrictive for silences.","Some phoneme boundaries are inherently ambiguous.","In other words, there are cases where segments do not have clear start times and end times.","We argue that segmental models are still well-motivated and suitable for these tasks, such as phonetic recognition.","First, the search space are stochastic when the path weights are interpreted as probabilities.","In fact, lattices, when first proposed by [120], were used to account for phoneme boundary ambiguity.","Training segmental models with marginal log loss also takes the ambiguity into account by marginalizing over all segmentations.","For decoding, ideally we want to find the label sequence that has the maximum marginal weight $\\operatornamewithlimits{\\arg \\!\\max }_{y} \\sum _{p \\in \\Gamma (y)} w(p), $ where $\\Gamma (y)$ is the set of paths with the label sequence $y$ .","However, the above is not tractable and we typically approximate it by finding the most probable path $\\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(p).$ Another approach to approximate (REF ) is to use beam search.","In summary, segmental models are capable of handling ambiguous segment boundaries if the training and decoding take the ambiguity into account.","We have compared different training approaches for segmental models, including multi-stage training and end-to-end training.","End-to-end training improves over multi-stage training, and end-to-end training from random initialization is on par with end-to-end fine-tuning.","Multi-stage training is useful for diagnosing end-to-end training, because every model found by multi-stage training is a valid model for end-to-end training objectives.","We have shown that segmental models can be trained with marginal log loss from random initialization, without requiring manual alignments for training.","As a byproduct, segmental models trained with marginal log loss perform reasonably well on phonetic segmentation.","Segmentation errors made by end-to-end segmental models are aligned with the studies in acoustic phonetics.","Table: Top 30 boundaries and its preceding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error rates ofthe segmental model trained with marginal log loss.Table: Top 30 boundaries and its preceeding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error counts ofthe segmental model trained with marginal log loss.Automatic speech recognition (ASR) has been posed as a graph search problem since the 1970s [61].","A search space, represented as a graph, is first constructed; edges in the graph are assigned weights based on the input and the edges; to predict, we simply run the shortest-path algorithm on the graph.","The graph search paradigm has been popularized by the use of hidden Markov models (HMM) [112], [62], where the shortest-path algorithm is known as the Viterbi algorithm.","In the probabilistic setting, finding the shortest path can be seen as finding the maximum a posteriori solution, justifying the graph search paradigm.","The probabilistic view has spawned many algorithms, such as segmental k-means [63] and expectation maximization [116], for estimating parameters in the weight function.","Segmental models follow the same graph search paradigm while having a different type of search spaces and different weight functions.","Many variants of segmental models were proposed in the past, such as stochastic segment models [103], [104], semi-Markov HMMs [117], segmental HMMs [115], [29], semi-Markov conditional random fields (CRF) [118], and segmental CRFs [149].","However, the type of search spaces and the shortest-path algorithm stay the same.","In this chapter, we review these variants from the graph search point of view.","The idea of using segmental features for speech recognition can be traced back to the 1970s [141].","The definition of segmental models was not explicit.","Most of the studies still followed the graph search paradigm, though the graph structures were typically constructed from a lexicon with a small vocabulary, and the weights on the edges were typically estimated with heuristics or a small amount of data [75], [23].","In the following sections, we categorize variants of segmental models as either generative or discriminative.","This categorization also aligns well with their rough chronological order.","Hidden semi-Markov Models are arguably the first segmental models applied to speech recognition [81], [117].","Given a sequence of observations $x = (x_1, \\dots , x_T)$ of length $T$ , let $y = (\\ell _1, \\dots , \\ell _K)$ be the label sequence and $z = ((s_1, t_1), \\dots ,(s_K, t_K))$ be the segmentation, where $\\ell _1, \\dots , \\ell _K \\in L$ for some discrete label set $L$ , and $s_1 = 1$ , $t_k = T$ , $s_k \\le t_k$ , $t_{k-1} + 1 = s_k$ , for $k = 2, \\dots , n$ .","Let $e_k = (\\ell _k, s_k, t_k)$ for $k = 1, \\dots , K$ .","The probability of $x_{1:T}$ defined by hidden semi-Markov models is $p(x, y, z) = p(x_{1:T}, e_{1:K})= p(e_1) \\prod _{k=2}^K p(e_k | e_{k-1}) \\prod _{k=1}^K p(x_{s_k:t_k} | e_k).$ The generative story is straightforward: the segments are generated one by one, following a Markov chain, and each segment $e = (\\ell , s, t)$ generates $t - s + 1$ observations.","Hidden Markov models are special cases of hidden semi-Markov models with the constraints $K = T$ and $s_k = t_k$ for $k = 1, \\dots , K$ .","There are many ways to define and parameterize $p(e_k | e_{k-1})$ and $p(x_{s_k:t_k} | e_k)$ .","The most common assumptions are $p(e_k | e_{k-1}) & = p(\\ell _k | \\ell _{k-1}) \\\\p(x_{s_k:t_k} | e_k) & = p(x_{s_k:t_k} | t-s+1, \\ell _k) p(t-s+1 | \\ell _k)$ where $p(\\ell _k | \\ell _{k-1})$ is commonly known as the transition probability, $p(x_{s_k:t_k} | t-s+1, \\ell _k)$ the emission probability, and $p(t-s+1 | \\ell _k)$ the duration probability.","The observations within a segment are typically assumed to be independent, i.e., $p(x_{s_k:t_k} | t-s+1, \\ell _k) =\\prod _{j=s_k}^{t_k} \\mathcal {N}(x_j; \\mu _{\\ell _k}, \\sigma _{\\ell _k}^2),$ where $\\mu _\\ell $ and $\\sigma _\\ell ^2$ are the mean and variance of the Gaussian distribution for the label $\\ell $ .","The single Gaussian case can be easily extended to a mixture of Gaussians.","Continuously variable duration HMMs [81], stochastic segment models [103] and segmental HMMs [115], [29] are all hidden semi-Markov models with the above assumptions.","The difference among them lies in how the emission probability $p(x_{s_k:t_k} | t-s+1, \\ell _k)$ is defined.","The subtle differences are summarized in [30], [104].","The duration probability is typically defined by a Poisson distribution [116] or Gamma distribution [81].","See [104] and the citations therein for other options.","Decoding with hidden semi-Markov models is done by solving $\\operatornamewithlimits{\\arg \\!\\max }_{y, z} p(y, z | x)& = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(y, z | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log \\frac{p(x, y, z)}{p(x)} \\\\& = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(x, y, z),$ where $y$ is the label sequence and $z$ is the segmentation, and is equivalent to finding the maximum-weight path with $w((\\ell , s, t)) = \\log p(t-s+1 | \\ell )+ \\sum _{j=s}^t \\log \\mathcal {N}(x_j; \\mu _\\ell , \\sigma _\\ell ^2).$ The term $p(\\ell ^{\\prime } | \\ell )$ is ignored here for simplicity, but can be included once the search space is self-expanded (Section REF ).","Training hidden semi-Markov models can be done by maximizing the likelihood of the training set.","The likelihood can be maximized using gradient-based methods or expectation maximization (EM) [116].","See [30] for a detailed explanation of the EM algorithm for estimating the parameters of hidden semi-Markov models.","Originally motivated by using rich segmental features [147], the SUMMIT system was defined as a generative model [34].","However, [34] introduced the notion of anti-phones and later [18] introduced near-misses, training the system with a discriminative touch.","It held the state-of-the-art result for speaker-independent phoneme recognition on TIMIT [45] until the rise of deep neural networks [94].","Since maximum mutual information was introduced as a training criterion for HMMs [10], ASR studies have gradually shifted from generative to discriminative modeling.","Parallel to the development of segmental models in the ASR community, [118] proposed semi-Markov CRFs for named entity recognition.","Using the same notation as in the previous section, the probability of a label sequence $y = (\\ell _1, \\dots , \\ell _n)$ and a segmentation $z = ((s_1, t_1), \\dots , (s_K, t_K))$ given an observation sequence $x = (x_1, \\dots , x_T)$ is defined as $p(y, z | x) = \\frac{1}{Z(x)} \\exp \\left( \\sum _{k=1}^K \\theta ^\\top \\phi (x, e_k) \\right)$ where $Z(x) = \\sum _{y^{\\prime }, z^{\\prime }} \\exp (\\sum _{e \\in (y^{\\prime }, z^{\\prime })} \\theta ^\\top \\phi (x, e))$ and $e \\in (y^{\\prime }, z^{\\prime })$ is a shorthand for enumerating $(\\ell _1, s_1, t_1),\\dots ,(\\ell _{K^{\\prime }}, s_{K^{\\prime }}, t_{K^{\\prime }})$ for $y^{\\prime } = (\\ell _1, \\dots , \\ell _{K^{\\prime }})$ and $z^{\\prime } = ((s_1, t_1), \\dots , (s_{K^{\\prime }}, y_{K^{\\prime }}))$ of length $K^{\\prime }$ .","The function $\\phi $ , called the feature function, extracts feature vectors that are intended to correlate well with $y$ and $z$ .","The parameter vector $\\theta $ can be learned by maximizing the likelihood of the training set.","Decoding with semi-Markov CRFs is done by solving $\\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(y, z | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\sum _{e \\in (y, z)} \\theta ^\\top \\phi (x, e),$ and is equivalent to finding the maximum-weight path if we let $w(x, e) = \\theta ^\\top \\phi (x, e)$ .","Similarly, training is done by maximizing the conditional likelihood of the data, and is equivalent to minimizing the negative log likelihood, or log loss.","Heavily influenced by [118], [149] proposed segmental CRFs.","The difference between segmental CRFs and semi-Markov CRFs lies in the training loss.","Instead of optimizing the conditional likelihood $p(y, z | x)$ , segmental CRFs optimize the marginal likelihood $p(y | x) = \\sum _{z} p(y, z | x).$ The connection between the marginal likelihood and marginal log loss is clear once we take the log of the probability distribution.","As we defined in Chapter REF , we distinguish between segmental models that consider the entire search space and ones that do not.","The former are called first-pass segmental models.","Whether a segmental model is first-pass or not is independent of its definition.","For example, semi-Markov CRFs were used as first-pass segmental models in [118]; segmental CRFs were first used as a second-pass model in [149], and were later used as a first-pass model in [148].","In Table REF , we provide a set of highlights of results in the development of segmental models on the TIMIT data set.","[148] was the first to explore discriminative segmental models that search over sequences and segmentations exhaustively, and did not use neural networks.","[49] first used (shallow) neural network-based frame classifiers to define weight functions, and later extended the idea to deep neural networks in [48].","[1] were the first to use deep convolutional neural networks for the weight functions, and were the first to train segmental models end to end.","We have compared different losses and training strategies for segmental models, first in a rescoring framework [129] and then in first-pass segmental models [131].","We also introduced segment-level classifiers and segmental cascades for incorporating them (and other expensive features) into segmental weight functions [130].","[85] introduced an LSTM-based weight function for every segment, and were also the first to use pyramid LSTMs to speed up inference for segmental models.","Table: TIMIT PERs (%) for various segmental modelscompared with HMMs and the current state of the art.The acoustic features are speaker-independent (spk indep) orspeaker-adapted with mean and variance normalization (mvn)or maximum likelihood linear regression (fMLLR) .Some results were obtained with MFCCs and some with log filter bank features.In this chapter, we have reviewed variants of segmental models categorized as either generative or discriminative in rough chronological order.","We review hidden semi-Markov models, a broad class of generative segmental models subsuming stochastic segment models and segmental HMMs.","We then review semi-Markov CRFs, the discriminative counterpart of hidden semi-Markov models.","We have established connections between these special cases and the general segmental models defined in Chapter REF .","Beyond segmental models, this chapter reviews other modern models trained end to end within the graph search paradigm.","We review connectionist temporal classification (CTC) [41], a popular approach for training frame-based LSTMs end to end.","Drawing on the connection between CTC and marginal log loss, we propose a framework, consisting of search spaces (represented as FSTs), weight functions, and training losses, that can encompass many end-to-end models, such as hidden Markov models trained with lattice-free maximum mutual information [111], and LSTMs trained with CTC, as special cases.","We compare end-to-end segmental models and end-to-end frame-based models, including one-state HMMs and two-state HMMs with LSTM encoders, and LSTMs trained with CTC.","Having these end-to-end models within the unified framework allows us to see the effect of each component while holding the other components fixed.","In this section, we review connectionist temporal classification (CTC) [41].","While being conceptually simple, LSTMs trained with CTC were the state of art for phonetic recognition in 2013 [42], and have achieved competitive results on large-vocabular speech recognition [92], [88], [151].","Consider a sequence of acoustic vectors $x_1, \\dots , x_T$ and its corresponding labels $y_1, \\dots , y_n$ , where $y_i \\in L$ for $i = 1, \\dots , n$ and some label set $L$ .","We assume $n < T$ because a phoneme is typically more than a frame long.","Suppose there exists a function $\\mathcal {P}$ that maps $y_1, \\dots , y_n$ to a path $a_1, \\dots , a_T$ , where $a_t \\in L^{\\prime }$ for some other label set $L^{\\prime }$ .","Minimizing $p(a_{1:T} | x_{1:T})$ can be done by simply minimizing the frame-wise cross entropy $p(a_{1:T} | x_{1:T}) = \\prod _{t=1}^T p(a_t | x_t).$ [41] proposed a mapping $\\mathcal {P}$ as follows.","Given a sequence $a_1, \\dots , a_T$ where $a_t \\in L \\cup \\lbrace \\varnothing \\rbrace $ for $t = 1, \\dots , T$ and $\\varnothing $ is the blank symbol.","Let $\\mathcal {B}$ be a function that first replaces duplicate contiguous labels into a single label, and second removes all the blank symbols.","For example, $\\text{\\texttt {k} \\texttt {ae} \\texttt {t}} = \\mathcal {B}(\\text{$\\varnothing $ \\texttt {k} \\texttt {k} $\\varnothing $ \\texttt {ae} \\texttt {ae} \\texttt {t}\\texttt {t} $\\varnothing $}).$ [41] used $\\mathcal {P}(y) = \\mathcal {B}^{-1}(y) = \\lbrace a = (a_1, \\dots , a_T) :a_t \\in L \\cup \\lbrace \\varnothing \\rbrace , \\mathcal {B}(a) = y\\rbrace $ to map a label sequence to a sequence of the same length as the input sequence.","However, this function $\\mathcal {P}$ returns a set of sequences, so the objective is modified to $p(y_{1:n} | x_{1:T}) = \\sum _{a_{1:T} \\in \\mathcal {B}^{-1}(y_{1:n})} p(a_{1:T} | x_{1:T})= \\sum _{a_{1:T} \\in \\mathcal {B}^{-1}(y_{1:n})} \\prod _{t=1}^T p(a_t | x_t),$ marginalizing over all the possible paths.","Given an input sequence $x$ , predicting a label sequence is done by finding $\\operatornamewithlimits{\\arg \\!\\max }_{y} p(y | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y} \\sum _{a \\in \\mathcal {B}^{-1}(y)} p(a | x).$ However, there are currently no algorithms that can solve the above efficiently, so it is typically approximated by finding the greedy best path $\\operatornamewithlimits{\\arg \\!\\max }_{y} p(y | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y} \\max _{a \\in \\mathcal {B}^{-1}(y)} p(a | x)= \\mathcal {B}(\\operatornamewithlimits{\\arg \\!\\max }_{a} p(a|x)).$ In other words, we simply find the label that achieves the highest probability at every time point and remove the duplicates and blank symbols to produce the final decoded sequence.","As a result of (REF ), the search space of CTC has an edge for every label in the label set (including the blank label) at every time step.","Specifically, the search space $G$ includes the edges $\\lbrace e_{\\ell , t}: \\ell \\in L, t \\in \\lbrace 1, \\dots , T\\rbrace \\rbrace $ with $v_{t-1} = \\mathrm {tail}(e_{\\ell , t})$ and $v_t = \\mathrm {head}(e_{\\ell , t})$ .","An example is shown in Figure REF .","The weight of an edge $e_{\\ell , t}$ is the log probability of label $\\ell $ at time $t$ .","By construction, the decision made at every time point is independent of the decision at other time points conditioned on $x_{1:T}$ .","In addition, since the probabilities at every time point sum to one, the partition function $Z(x)$ of the search space, i.e., the sum of all the path probabilities, is always 1.","Recall that the marginal log loss is defined as $\\mathcal {L}_{\\text{mll}} = -\\log \\sum _{p \\in \\mathcal {P}_{G \\circ _\\sigma F_y}} \\exp (w(p))+ \\log \\sum _{p \\in \\mathcal {P}_{G}} \\exp (w(p))$ where $\\mathcal {P}_K$ is the set of paths in the FST $K$ and $F_y$ is the constraint FST constructed from the ground truth label sequence $y$ .","Since $Z(x) = 1$ for CTC, the second term is zero.","By the definition of $\\mathcal {B}$ , we construct the constraint FST $F_y$ such that it consists of the sequences of one or more labels with zero or more blanks in between labels.","For example, for the label sequence “k ae t,” the constraint FST is the regular expression $\\varnothing ^*\\texttt {k}^+\\varnothing ^*\\texttt {ae}^+\\varnothing ^*\\texttt {t}^+\\varnothing ^*$ .","We have $\\mathcal {B}^{-1}(y) = G \\circ _\\sigma F_y$ .","In other words, the sum of path probabilities in $G \\circ _\\sigma F_y$ exactly matches the CTC objective.","Because the first term matches the CTC objective and the second term is zero, marginal log loss becomes the CTC objective for this type of search spaces and the log probability weight function.","Figure: An example of the CTC search space for a five-frame utterancewith a label set of size three (plus one blank).Most mainstream end-to-end speech recognition models can be broadly categorized as either frame-based models or encoder-decoder models.","Frame-based LSTMs trained with CTC, HMMs, and some newer approaches like the auto-segmentation criterion (ASG) [25] fall under the first category, because these models emit one symbol for every frame.","Falling under the second category, encoder-decoder models proposed by [21], [9], [16] generate labels one at a time while conditioning on the input and the labels generated in the past, without an explicit alignment between labels and frames.","Since frame-based models follow the same graph search framework as segmental models, we will focus on discussing the connection between these and segmental models.","Recall that marginal log loss requires a search space $G$ and a constraint FST $F$ to limit the search space to ground-truth labels.","To compute marginal log loss, we first compute the marginals on $G$ for computing the partition function $Z(x)$ , and then compute the marginals on the $\\sigma $ -composed FST $G \\circ _\\sigma F$ for computing $Z(x, y)$ .","CTC, HMMs whtn trained with lattice-free MMI [111], and ASG can all be seen as special cases of this framework.","Comparing CTC to HMMs, the search space is different depending on the HMM topology.","For example, two-state HMMs are used in [111].","Since the transition probabilities and emission probabilities are all locally normalized, the partition function $Z(x)$ is always 1.","The constraint FST representing the ground-truth labels consists simply of sequences of repeating labels.","For example, for the label sequence “k ae t,” the constraint FST for one-state HMMs is the regular expression $\\texttt {k}^+\\texttt {ae}^+\\texttt {t}^+$ .","For two-state HMMs, the constraint FST is the regular expression $\\texttt {k}_1\\texttt {k}_2^*\\texttt {ae}_1\\texttt {ae}_2^*\\texttt {t}_1\\texttt {t}_2^*$ .","With the above construction, marginal log loss applied to HMMs is equivalent to lattice-free MMI [111].","For ASG, the search space is equivalent to that of one-state HMMs.","Instead of assuming conditional independence as in CTC, ASG includes transition probabilities between states.","The constraint FST is identical to that of HMMs, with repeated ground-truth labels.","However, in ASG the weights on the edges are not locally normalized, so the partition function $Z(x)$ is not always 1 and has to be computed.","With the above search space construction, marginal log loss becomes ASG.","Another approach similar to CTC proposed in [38] is called RNN transducers.","The search space of an RNN transducer is the set of alignments from the speech signal to all possible label sequences, so the search space grows exponentially in the number of labels.","The weight function of a path in this approach relies on an RNN, and is not decomposable as a sum of weights of the edges.","RNN transducers are trained with marginal log loss.","By the independence assumption imposed in [38], the partition function $Z(x)$ is still 1, so we do not need to marginalize over the exponentially large space.","During decoding, however, we still have to search over the exponentially large space with, for example, beam search.","In view of this framework, even when using the same loss function, i.e., marginal log loss, segmental models and frame-based models differ in their search space, weight functions, and how the search space is constrained by the ground-truth labels during training.","A summary of special cases is shown in Table REF .","Table: An example instantiation of the components used in marginal log losswith the ground-truth sequence “k ae t”and TT input framesfor segmental models and various end-to-end models.The search space L T L^T consists of sequences of TT labels.The label sets L 1 L_1 and L 2 L_2 contain labels in LL with subscript 1 and 2 respectively.The search space is denoted GG and the constraint FST is denoted F y F_y in the table.We compare various frame-based models trained end to end on the same phonetic recognition task in the same setting as in Section REF .","We use 3-layer 256-unit biderectional LSTMs paired with CTC, one-state HMM, and two-state HMMs.","We do not use transition probabilities for two-state HMMs.","We optimize marginal log loss with the corresponding weight functions and the corresponding constraint FSTs for CTC, one-state HMM, and two-state HMMs.","We run vanilla SGD with step size 0.1 and gradient clipping of norm 5 for 20 epochs.","Starting from the best performing model of the first 20 epochs, we run vanilla SGD for another 20 epochs with step size 0.75 decayed by 0.75 after each epoch.","The dropout rate is 0.2, and no other explicit regularizer is used except early stopping on the development set.","The same experiments are repeated with pyramid LSTMs.","Results comparing with segmental models are shown in Table REF .","When standard LSTMs are used, we fail to minimize the training loss for one-state HMMs and two-state HMMs The only difference between CTC and one-state HMMs is the use of blank symbols, suggesting that blank symbols play an important role in end-to-end frame-based models.","In particular, one-state HMMs and two-state HMMs explicitly marginalize over segmentations in training and are thus sensitive to time, while LSTMs trained with CTC are not.","Besides, the tail states in $L_2$ can be seen as label-dependent blank symbols.","Since two-state HMMs fail to achieve a low training loss, having label-dependent blanks is not helpful in this case.","When pyramid LSTMs are used, all variants of frame-based models achieve low PERs on the development set and improve over ones with standard LSTMs.","Since the time resolution is reduced by four with pyramid LSTMs, we hypothesize that using the pyramid LSTMs introduces a bias similar to a minimum duration constraint, which helps end-to-end training for frame-based models.","The number of minutes per epoch for training segmental models and LSTMs trained with CTC is shown in Table REF , and a scatter plot showing number of minutes per epoch vs.","PER is shown in Figure REF .","Pyramid LSTMs trained with CTC are the best performer while also being the most efficient to train.","The real-time factors for decoding are shown in Table REF .","Due to smaller and simpler search spaces, frame-based LSTMs trained with CTC decode faster than segmental models.","While frame-based LSTMs train and decode faster, segmental models are more flexible, so they might be improved with additional feature functions.","Explicitly hypothesizing segments can serve as a prior, so segmental models might perform better in low-resource settings.","Table: Phoneme error rates (%) for end-to-end training fromrandom initialization comparing CTCand segmental models trained with marginal log loss.Table: Average number of minutes for one epoch of end-to-end training on TIMIT.Table: Real-time factors for decoding comparing CTC and segmental models.Figure: Training minutes per epoch vs.","PER (%) for segmental modelsand LSTMs trained with CTC.For American Sign Language (ASL) fingerspelling experiments, we follow [72] and compare end-to-end models in both signer-dependent setting and signer-independent setting.","The input is a sequence of 128-dimensional histograms of oriented gradients (HoG) feature vectors computed over $128 \\times 128$ hand shape images.","The output is a sequence of English letters, and the label set consists of the 26 English characters plus two labels for initial and final non-signing regions.","The evaluation metric is the edit distance between the predicted letter sequence and the ground-truth letter sequence.","For the signer-dependent setting, the data for each signer is split into ten folds, in which eight are for training, one is for development, and one is for testing.","We report the letter error rate averaged over the 10 experiments.","We train LSTMs with CTC and segmental models with the FCB weight function.","Both models are trained with marginal log loss from random initialization.","The encoder is a one-layer 128-unit bidirectional LSTM.","For CTC, we run vanilla SGD with step sizes in $\\lbrace 0.1, 0.05, 0.025\\rbrace $ and gradient clipping with norm 5 for 200 epochs.","Dropout is used with a rate of 0.2, and no other explicit regularizer is used except early stopping.","Step sizes and early stopping are tuned on the development set for each individual experiment.","For segmental models, we run vanilla SGD with step size 0.1 for 75 epochs.","The maximum segment length is 75 frames.","Other hyperparameters, such as dropout and gradient clipping, are the same as for CTC.","Results are shown in Table REF .","The frame-based LSTMs trained with CTC are better than Tandem HMMs, but are behind the segmental models.","The segmental models trained end to end are on par with the two-stage system in [72], but are behind the discriminative segmental cascades (DSC).","Table: Letter error rates (%) for signer-dependent models averagedover ten folds.For the signer-independent setting, we train on three signers and use the last signer for development and testing.","Specifically, the data of the last signer is split into ten folds, and we use two folds for development and the rest of the eight folds for testing.","Instead of dividing the accumulated error counts by the accumulated label sequence lengths, letter error rates are averaged over the folds.","We compare LSTMs trained with CTC and segmental models with FCB weight function.","The loss function and training procedure stay the same except that we only run 40 epochs of vanilla SGD for segmental models.","Results are shown in Table REF .","In this setting, CTC and segmental models trained end to end perform significantly better than Tandem HMMs and segmental models trained with the FC weight function in two stages.","To compare signer-independent and signer-dependent settings, we average the letter error rates over the same eight folds.","Results are shown in Table REF .","Segmental models perform significantly better than CTC in both settings.","Table: Letter error rates (%) for signer-independent modelsaveraged over eight folds.Table: Letter error rates (%) for signer-dependent and signer-independent models averagedover the same eight folds.We have discussed how other end-to-end frame-based models, such as CTC, HMMs trained with lattice-free MMI, ASG, and RNN transducers are all trained with marginal log loss.","The differences among them lie in the search spaces and the weight functions.","Drawing these connections allows us to generalize search spaces, loss functions, and weight functions to a broad class of models.","From the results of comparing CTC to one-state HMMs and two-state HMMs, we have found that the blank symbol seems to play an important role in training LSTMs end to end.","Using pyramid LSTMs improves both the performance and the runtime of decoding and training for CTC, one-state HMMs, and two-state HMMs, but not for segmental models.","We have also shown that segmental models with regular LSTMs are better than regular LSTMs trained with CTC on both phonetic recognition and ASL fingerspelling recognition.","In this thesis, we have made the following contributions in advancing the study of discriminative segmental models.","We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.","We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.","We obtain improved performance over most earlier work while greatly improving efficiency.","We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.","We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.","We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.","Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.","We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.","Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.","We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.","We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.","We obtain improved performance over most earlier work while greatly improving efficiency.","We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.","We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.","We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.","Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.","We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.","Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.","In this chapter, we discuss some potential future work extending segmental models to large-vocabulary tasks and to unsupervised settings.","Another potential direction for future work is even richer feature functions.","Since first-pass segmental models are computationally demanding, it is very slow to train and decode segmental models with label sets of size in the order of 10,000.","This poses a challenge for tasks with large label sets, such as word recognition.","[150], [88] bypassed this difficulty by using a baseline HMM recognizer to generate word lattices, and used segmental models to rescore these lattices, exploring various word-level features.","However, the performance is constrained by the quality of the baseline recognizer and the quality of the lattices.","First-pass segmental models have previously been successfully applied to word recognition [35], [50].","This previous work treats first-pass segmental models as a drop-in replacement for HMM phoneme recognizers, because both models serve as functions that map acoustic features to phoneme strings.","The phoneme recognizers are then composed with a lexicon and a language model to form a word recognizer, as described in Section REF .","This is also an option for extending our work to word recognition.","Recent work has explored models that directly predict characters, avoiding the need for a lexicon [39], [92] but still allowing for improved performance when constraining the search space with a lexicon (through FST composition) [92].","Segmental models can also be used to predict characters by changing the label set, but might be hampered by the poor alignment of characters to acoustics.","Syllables might be a better option than characters.","Instead of using intermediate discrete representations, such as phonemes or characters, recent advances in computing power have made it feasible to directly predict words [87], [11], [127], [7].","In this case, rather than using a pronunciation dictionary, only a list of words is needed for decoding.","Segmental models can also be used to directly predict words by using the list of words as the label set.","This approach is worth exploring further, although efficiency issues make it nontrivial to train such models [7].","We have presented segmental models for supervised sequence prediction in this thesis.","Our framework can be extended to unsupervised sequence prediction.","The goal in this setting is typically grouping segments of varying length into clusters.","We define unsupervised sequence prediction as finding a function that maps an input sequence $x_1, \\dots , x_T$ into a sequence of segments $(\\ell _1, s_1, t_1), \\dots , (\\ell _n, s_n, t_n)$ where $\\ell _i \\in L$ for $i=1, \\dots , n$ and some label set $L$ , and $s_1 = 1$ , $t_n = T+1$ , $s_i < t_i$ , $t_{i-1} = s_i$ for $i = 1, \\dots , n$ , given a data set $S$ of sequences without labels.","Since we do not have labels for the data samples, the goal in general is to design a loss function $\\mathcal {L}(\\Theta ; x)$ that only depends on the input sequence $x$ , or even more generally a loss function $\\mathcal {L}(\\Theta ; S)$ that depends on the data set $S$ .","The label set $L$ is typically predefined to be just a set of identifiers, and the correspondence between the input sequence and the identifier sequence is learned from a data set.","As a result, a label in $L$ does not have a predefined meaning unless the user assigns a post hoc meaning or the loss function enforces one.","Generative segmental models, such as hidden semi-Markov models, can be naturally extended to the unsupervised setting by maximizing the marginal likelihood of the data.","For example, Bayesian segmental models have been applied to small-vocabulary [64] and larger-vocabulary [65] word discovery.","Viterbi-style training for such models has also been explored [66].","Recently, [134] proposed unsupervised neural HMMs, which can be easily extended to hidden semi-Markov models.","In fact, [26] proposed a neural version of hidden semi-Markov models similar to [134].","All the above approaches aim to maximize the marginal likelihood of the data.","Besides generative models, approaches for training log-linear models in an unsupervised fashion, notably contrastive estimation [126], [107], noise-contrastive estimation [44], and auto-encoders [4], have also been extensively studied.","These approaches are designed for discriminative models, and can be readily applied to our segmental models.","A segmental model trained in an unsupervised fashion can be used for segment clustering.","One major application of segment clustering is for lexical unit discovery.","There is a long history of lexical unit discovery from acoustic signals [14], sometimes called spoken term discovery [105], [60].","Much of the recent work relies on dynamic time warping (DTW) [59], [15] for spoken term discovery, with [78] being out of the few exceptions who took a nonparametric Bayesian approach.","DTW has also been found to help lexical discovery in an HMM system [140].","Other related tasks that we do not cover here include unsupervised word segmentation [37], unsupervised part-of-speech tagging [91], and unsupervised dependency parsing [73].","Though segmental models were not used in the above studies, many ideas can be borrowed and help advance the study of segmental models in unsupervised settings.","The assumption that input sequences can be decomposed into a sequence of segments serves as a strong inductive bias, and it is particularly useful in unsupervised settings when little is assumed about the data.","Segmental models are more flexible than frame-based models when additional assumptions, such as the form of the feature functions, are needed.","Therefore, we believe segmental models have the potential to perform well in these unsupervised settings."]],"string":"[\n [\n \"Sequence Prediction with Neural Segmental Models\"\n ],\n [\n \"Abstract Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.\",\n \"Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.\",\n \"However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.\",\n \"In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.\",\n \"We also show that instead of including more features to obtain better accuracy, segmental cascades can be used to speed up training and decoding.\",\n \"Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.\",\n \"In the first stage, a frame classifier is trained with manual alignments, and then in the second stage, segmental models are trained with manual alignments and the out- puts of the frame classifier.\",\n \"However, obtaining manual alignments are time-consuming and expensive.\",\n \"We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.\",\n \"We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification.\",\n \"We present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.\",\n \"Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks.\"\n ],\n [\n \">=latex SEQUENCE PREDICTION WITH NEURAL SEGMENTAL MODELS BY HAO TANG A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science at the TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO Chicago, Illinois September, 2017 Thesis Committee: Karen Livescu (Thesis Advisor) Kevin Gimpel David McAllester Eric Fosler-Lussier James Glass left=0in,right=0in,top=0in,bottom=0in Figure: NO_CAPTION Sequence Prediction with Neural Segmental Models by Hao Tang Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.\",\n \"Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.\",\n \"However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.\",\n \"In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.\",\n \"Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.\",\n \"In the first stage, a frame classifier is trained, and in the second stage, segmental models are trained with the outputs of the frame classifier.\",\n \"Both training stages require manual alignments, and obtaining manual alignments are time-consuming and expensive.\",\n \"We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.\",\n \"We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification, and present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.\",\n \"Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks.\",\n \"Thesis Supervisor: Karen Livescu Title: Associate Professor First, I would like to thank my advisor, Karen Livescu.\",\n \"Karen has always been very patient, letting me make mistakes after mistakes, mistakes that she can foresee, mistakes that I have made many times already, mistakes that have no easy fix, so that I can learn from them.\",\n \"She taught me when to explore novel ideas and when to persist and complete tedious tasks.\",\n \"I have learned to think critically and argue constructively, while at the same time to be honest and critical to my own work.\",\n \"There are countless other things I have learned from her including phonetics, spectrogram reading, and English word usage, to name a few.\",\n \"I can not express how much I appreciate her guidance and mentorship.\",\n \"I am really fortunate to be her first PhD student.\",\n \"Next, I want to thank my committee members, Kevin Gimpel, David McAllester, Eric Fosler-Lussier, and Jim Glass.\",\n \"David's comments and questions always encourage me to view my work in a broader context and in the most general form possible.\",\n \"I have also greatly benefited from his mathematical rigor through his talks, his lectures, and interactions with him, and have developed the habit of asking whether a mathematical expression is type checked or not.\",\n \"I am grateful to Eric for his insightful comments.\",\n \"I have benefited a lot from conversations we had in workshops and conferences.\",\n \"I enjoyed his humorous and joyful attitude, and him pushing me towards the finish line harder than my advisor does.\",\n \"I thank Jim for keeping the speech science aspect of my research in mind.\",\n \"My research is heavily influenced by his prior work, and it is very special to work on something that he has been working on for decades.\",\n \"I am deeply grateful to Kevin.\",\n \"He is like my second advisor, caring and helpful in many ways.\",\n \"I have benefited from his expertise in natural language processing (NLP), and discovered many similar approaches developed in parallel in both the NLP and the speech community.\",\n \"Besides the technical content, he taught me to be humble yet to be bold when necessary.\",\n \"I have also enjoyed his humorous comments on my writing before stressful deadlines.\",\n \"It has been a great pleasure to have his company during my PhD.\",\n \"I was fortunate to join Mark Hasegawa-Johnson's team in the second Jelinek summer workshop.\",\n \"I have benefited from Mark's extensive knowledge in both speech science and machine learning.\",\n \"It is fair to say that I always learn something new when talking to him.\",\n \"I also thank him for his guidance after the workshop.\",\n \"It has been a great pleasure to work with the people on the team.\",\n \"In particular, I have benefited from interactions with Chunxi Liu, Preethi Jyothi, Amit Das, Vimal Manohar, Paul Hager, Tyler Kekona, and Rose Sloan.\",\n \"People I met during the workshop, including Yi Luan, Yangfeng Ji, Lingpeng Kong, Guoguo Chen, and Trang Tran, also made the overall experience fun and pleasant.\",\n \"In 2013, I worked as an intern with Shinji Watanabe at the Mitsubishi Electric Research Laboratories (MERL).\",\n \"I have benefited from the interactions with him and other people at MERL, including John Hershey, Jonathan Le Roux, and Tim Marks.\",\n \"In particular, I thank Shinji for taking care of me when I had Bell's palsy during the internship.\",\n \"The fellow interns at MERL, including Niao He and Lingling Tao, also made the internship experience fun and rewarding.\",\n \"My first research experience related to speech and language processing was acquired from Lin-Shan Lee's guidance in National Taiwan University back in 2009.\",\n \"Lin-Shan, even with his always busy schedule, has been very caring and helpful.\",\n \"It would not have been possible for me to pursue a PhD without his help and encouragement.\",\n \"I thank the people in Toyota Technological Institute at Chicago (TTIC) for making my PhD journey wonderful.\",\n \"In particular, I have benefited from Julia Chuzhoy's and Madhur Tulsiani's lectures, acquiring the necessary vocabulary to talk to theory people.\",\n \"I thank Nati Srebro for grilling me during my qualifying exam, forcing me to always keep the precise language and mathematical terms in mind.\",\n \"I enjoyed the interactions with Greg Shakhnarovich and his humor.\",\n \"I thank Joseph Keshet for his guidance and mentorship during my first few years of PhD.\",\n \"I have benefited interactions and collaboration with Liang Lu.\",\n \"I thank the fellow students, including Behnam Tavakoli, Somaye Hashemifar, Taehwan Kim, Shubham Toshniwal, Shane Settle, Qingming Tang, Bowen Shi, Lifu Tu, Hai Wang, Renyu Zhang, Jian Yao, Jianzhu Ma, Xing Xu, Avleen Bijral, Payman Yadollahpour, Feng Zhao, Zhiyong Wang, Karthik Sridharan, and Peng Jian, for the interactions, cookie breaks, random interruptions, and random questions.\",\n \"I will never regret choosing a cubicle that invites so many interruptions.\",\n \"I am grateful to Weiran Wang.\",\n \"It is fair to say that much of my work would not be possible without his help.\",\n \"I have also greatly benefited from interactions with past post-docs, including Raman Arora and Herman Kamper.\",\n \"I thank visiting students, Taiki Kawano and Takayuki Yamabe, from Toyota Technological Institute in Japan, for the fun time we had in Chicago and in Tokyo.\",\n \"Thanks to Chrissy Novak, Adam Bohlander, and the rest of the staff for making my PhD life easy and smooth.\",\n \"Lastly, I thank the president of TTIC, Sadaoki Furui, for his guidance, and for pushing me to exercise and to practice my tennis skills with him.\",\n \"Thanks to my family in Taiwan for their support over the years.\",\n \"Thanks to Hsin-Yu, for everything.\",\n \"And thanks to my parents, whom I cannot possibly thank in words.\",\n \"Segmental models, the subject of this thesis, are a family of models that predict sequences of segments.\",\n \"Segmental models are designed for a wide range of sequence prediction tasks, such as named entity recognition [118], speech recognition [104], [35], and action recognition [122], [55].\",\n \"Examples of these tasks are shown in Figure REF .\",\n \"The input of a sequence prediction task is commonly represented as a sequence of real-valued vectors, and the output is a sequence of discrete labels.\",\n \"The goal is to find a function that maps the input vectors to the output labels.\",\n \"Every output label in the output sequence, such as a named entity, a phoneme, or an action, typically has a corresponding chunk of contiguous input vectors, called a segment.\",\n \"Segments come in varying lengths.\",\n \"As a consequence, the lengths of the output sequences typically do not match the lengths of the input sequences.\",\n \"Figure: Examples of sequence prediction tasks.The first example is speech recognition, where the input is a sequence of acousticsignals and the output is a sequence of words.\",\n \"The second example is American SignLanguage fingerspelling recognition, where the input is a sequence of images andthe output is a sequence of letters.\",\n \"The third example is named entity recognition,where the input is a sequence of words and the output is the same sequence of wordswith named entities in parentheses.\",\n \"In the example of speech recognition,the gray arrow connects the word “else” to its corresponding contiguous partin the input sequence.The predominant approach to solving these tasks is to break the varying-length segments into pieces so that the lengths of the input sequences match the lengths of the output sequences.\",\n \"These smaller pieces are then assembled back into varying-length segments with an additional post-processing step.\",\n \"Each individual element in the input sequence is referred to as a frame.\",\n \"Models, such as hidden Markov models [112], [62], linear-chain conditional random fields [43], [99], and recurrent neural networks [40], that map input sequences to output sequences of the same lengths, are called frame-based models [104], [35].\",\n \"To include a wider context, it is common to use windows of frames instead of individual frames for prediction.\",\n \"Nevertheless, the number of windowed frames is still the same as the number of output labels.\",\n \"Because frame-based models are conceptually simple and computationally cheap, they have enjoyed much success and received much attention in the past few decades.\",\n \"Figure: Frame-based approaches and segment-based approaches for sequence prediction.Model accuracies for prediction tasks are heavily influenced by the set of features used to make prediction, where a feature is a real-value function of the input that correlates well with a label or a subset of labels.\",\n \"Breaking up varying-length segments is an extremely successful heuristic, but it also makes extracting certain features from segments difficult.\",\n \"For example, computing durations is difficult for frame-based models.\",\n \"Variants of frame-based models, such as variable-duration HMMs [28], are proposed in order to overcome this difficulty, but it is impractical to construct a model for every new type of feature.\",\n \"In contrast, segment-based approaches do not assume anything about the segments, so the users are free to use arbitrary information about the segments during prediction.\",\n \"For example, given a segment with a precise start time and end time, computing its duration is trivial.\",\n \"For speech recognition, energy distribution at the left and right boundaries are useful features.\",\n \"For named entity recognition, we can check whether a phrase (segment) is in the dictionary.\",\n \"The flexibility makes segment-based models appealing, and many promising results have been reported in the past [147], [104], [35].\",\n \"A comparison of the two approaches is shown in Figure REF .\",\n \"More formally, a segment is a tuple of a start time, an end time, and possibly a label.\",\n \"At a high level, segmental models make predictions by first creating a search space consisting of connected segments, or paths.\",\n \"Segments are given weights based on the start time, end time, and label.\",\n \"The weight values reflect how well the labels match the input.\",\n \"Finding the sequence of segments that best matches the input is equivalent to finding the maximum-weight path.\",\n \"The prediction process is also referred to as decoding.\",\n \"Note that the segment weights can be computed by any function as long as it only makes use of the start time, the end time, the label, and the input sequence.\",\n \"An example is shown in Figure REF .\",\n \"Figure: An example of a segmental model.\",\n \"A search space is built based on theinput frames.\",\n \"Each edge (segment) has a start time, an end time, and a label,which the weight function can make use of.\",\n \"Once the weights of edgesare computed, decoding is the problem of finding the maximum-weight path.Despite their success and attractive properties, segmental models still fall behind frame-based models on many tasks, such as speech recognition [148].\",\n \"There are many possible reasons behind this.\",\n \"The choice of features extracted from the segments might play a role.\",\n \"However, complex features are typically computationally expensive.\",\n \"The large number of segments we need to consider prohibits the use of computationally expensive segment features.\",\n \"The central goal of this thesis is to improve the efficiency of segmental models, allowing us to explore more computationally expensive features and to improve the performance of segmental models.\",\n \"This thesis studies segmental models along three directions: segment features, inference, and learning.\",\n \"We introduce discriminative segmental cascades to speed up inference, allowing us to explore computationally expensive segment representations, such as ones computed with neural networks.\",\n \"We study how segmental models perform when they are trained with various loss functions.\",\n \"Different loss functions have different training requirements, some of them allowing us to train segmental models without manual alignments.\",\n \"We also study whether it is possible to train segmental models end to end from random initialization.\",\n \"We focus on two tasks, phonetic recognition and American Sign Language fingerspelling recognition.\",\n \"In this section, we briefly describe automatic speech recognition (ASR), a task for which segmental models have been studied extensively.\",\n \"We describe a few features that were shown to be useful for disambiguating phonemes, the basic sound units that distinguish words.\",\n \"We then describe why it is hard to incorporate these features in frame-based models, and some of the past efforts to overcome this difficulty.\",\n \"In brief, human speech production is the process of mapping a sequence of discrete linguistic units, such as words or phonemes, into waveforms, and automatic speech recognition is about inverting this process, mapping speech waveforms to sequences of discrete linguistic units.\",\n \"On one side of the process, we have speech waveforms, commonly represented as sequences of real-valued vectors called frames.\",\n \"On the other side of the process, we have the discrete units, namely segments, representing phonemes or words.\",\n \"A waveform can be represented as a sequence of real-valued vectors in multiple ways.\",\n \"One common representation is the amount of sinusoids at different frequencies appearing in the signal.\",\n \"A waveform is first converted into small overlapping chunks.\",\n \"Typically the size of the chunk is 25ms, and a chunk is created every 10ms.\",\n \"Each chunk of waveform is represented as a linear combination of different sinusoids at different frequencies.\",\n \"In particular, a frame in this representation is the vector of coefficients of the linear combination.\",\n \"This representation is known as the spectrogram.\",\n \"Examples of spectrograms are shown in Figure REF .\",\n \"Given a sequence of frames, the goal of ASR is to predict what is being said in the waveform, in the form of a sequence of discrete linguistic units, with words and phonemes being the two most common discrete units.\",\n \"Phonemes are defined as the sound units that distinguish words [76], and a sequence of phonemes can be converted into words by looking up the pronunciations of words in a lexicon.\",\n \"In most settings, the set of phonemes and the lexicon are assumed to be given.\",\n \"A standard evaluation metric for ASR is the Levenshtein distance between the predicted label sequence and the ground-truth sequence.\",\n \"Levenshstein distance measures the minimum number of edits that needs to be performed to transform the predicted label sequence to the ground-truth sequence; hence it is also called the edit distance.\",\n \"Formally, the edit distance of two label sequences $y$ and $\\\\hat{y}$ is defined as $\\\\text{edit}(\\\\hat{y}, y) = \\\\frac{I + D + S}{|y|},$ where $|y|$ is the length of $y$ , $I$ , $D$ , and $S$ are the number of insertion, deletion, and substitution, respectively.\",\n \"ASR is a difficult task due to the wide range of variabilities in the process of speech production.\",\n \"Even in the most controlled setting, the same isolated word can hardly be pronounced the same way twice by the same speaker.\",\n \"Speech production is a highly context-dependent process.\",\n \"Phonemes are pronounced differently when the neighboring phonemes are different [76].\",\n \"Similarly, words are pronounced differently in different contexts.\",\n \"Different speakers pronounce words and phonemes differently due to the differences in their speech organs [80].\",\n \"Different speaking styles also affect pronunciations.\",\n \"For example, in conversational speech, words are seldom pronounced in the canonical way presented in the lexicon due to the casual speaking style [83].\",\n \"All of the above variabilities make ASR difficult.\",\n \"It is even more difficult when the speech signals are degraded by noise [54].\",\n \"Since the goal of speech recognition is to predict words based on their pronunciations, much work has been dedicated to finding features, sometimes referred to as acoustic cues, that identify phonemes and differentiate phonemes.\",\n \"See [47] and citations therein.\",\n \"Some features are correlated with how humans identify phonemes [93].\",\n \"In general, there are many such features that are useful for speech recognition.\",\n \"Each feature alone is probably not enough to identify a phoneme, but a combination of features might be able to [47].\",\n \"Below we describe two features as examples.\",\n \"Duration is one of the features that differentiate phonemes.\",\n \"For example, the duration of the long vowel /iy/The phonemes are written in ARPAbet.\",\n \"in the word “seat” is typically longer than short vowels /ih/ in the word “sit” [106].\",\n \"In this case, duration serves as a good feature to differentiate /iy/ from /ih/.\",\n \"Unvoiced fricatives, such as /f/, /s/, and /sh/, typically have longer duration than voiced fricatives, such /v/, /z/, and /zh/ [136].\",\n \"Durations have also been shown to help improve recognition results [5].\",\n \"Perhaps the most salient feature for distinguishing phonemes by looking at the spectrogram is the distribution of energy [76].\",\n \"For example, fricatives, such as /s/ and /f/, have evenly spread energy across frequency.\",\n \"Stops, such as /t/ and /p/, have sudden sharp bursts of energy.\",\n \"Bands of high energy in spectrograms, commonly referred to as formants, are another type of energy patterns useful for identifying vowels [76].\",\n \"The first and second formants (counting from low frequencies) are commonly used to differentiate vowels [53].\",\n \"Formants are also useful for speech recognition [57].\",\n \"Integrating arbitrary segment features into frame-based models has always been considered a difficult task.\",\n \"Integrating even just the duration requires nontrivial modification to the models [81].\",\n \"For example, we can expand the label set with durations, but this approach only works for discrete features and generates unnecessarily large search spaces.\",\n \"Integrating these segment features is also one of the driving forces behind the use and development of segmental models [147].\",\n \"While hidden Markov models assume that frame labels follow a Markov process, hidden semi-Markov models assume that segments follow a Markov process.\",\n \"Semi-Markov processes have been used to incorporate duration in frame-based models [116].\",\n \"Hidden semi-Markov models have been applied to speech recognition with the same reason in mind [81], [117].\",\n \"The segment-based speech recognizer SUMMIT, is designed with the goal of integrating arbitrary segment features [147].\",\n \"Segmental models have also been applied to other sequence prediction tasks, such as named entity recognition [118], American Sign Language fingerspelling recognition [71], and action recognition [122], [55].\",\n \"For named entity recognition, the input is a sequence of words, and the output is a sequence of named entity tags, such as person name, organization name, and location name.\",\n \"For American Sign Language fingerspelling recognition, the input sequence is a sequence of video frames, and the output is a sequence of letters.\",\n \"For action unit detection, the input sequence is a sequence of video frames, and the output is a sequence of actions, such as walk, jump, sit, and stand.\",\n \"In general, these tasks follow a left-to-right (or right-to-left) order in both the input sequence and the output sequence.\",\n \"A task is said to be linear or monotonic if the task has a left-to-right (or right-to-left) order.\",\n \"One sequence prediction task that is not monotonic is translation [19].\",\n \"Depending on the source and the target languages, the order of words in the source language can be different from the order in the target language.\",\n \"In this thesis, we only focus on segmental models for monotonic sequence prediction tasks.\",\n \"We have seen why segmental models are favored over frame-based models when incorporating complex segment features is needed.\",\n \"However, the flexibility of segmental models comes with a price.\",\n \"Enumerating segments of different lengths and of different labels can be time-consuming.\",\n \"Consider an input sequence of length 300, and a label set size of 50.\",\n \"The number of possible segments of length up to 30 is around $30 \\\\times 50 \\\\times 300 = 450000$ .\",\n \"If we make only a single decision at every time point as for frame-based models, we only need to consider $50 \\\\times 300 = 15000$ different decisions.\",\n \"The hypothesis space for segmental models is considerably larger, and as a consequence learning and inference for segmental models are significantly slower than for frame-based models.\",\n \"Many researchers have attacked the problem of large hypothesis spaces either implicitly or explicitly.\",\n \"The most common approach is to use frame-based models to generate a set of high-confidence segments, and then use segmental models to rescore the segments with segment features [35], [149].\",\n \"[102] explicitly train a model for pruning segments, while [139] reuse the computation of features whenever possible.\",\n \"The first approach needs to have two separate models and the second approach depends on the exact form of the features.\",\n \"We would like to design segmental models that do not depend on other external models, and can handle large hypothesis spaces in a feature-oblivious manner.\",\n \"Designing and implementing state-of-the-art frame-based models requires a significant amount of engineering effort.\",\n \"Phonemes are modeled with three-state hidden Markov models [119].\",\n \"Since phonemes are influenced heavily by nearby phonemes, phoneme labels that include the previous and the next phonemes, referred to as triphones, are introduced [119].\",\n \"The large number of triphones makes estimating their probabilities difficult, so triphone parameters are shared using with decision trees [143].\",\n \"These design decisions in model structure makes engineering difficult.\",\n \"In contrast to frame-based models, the engineering effort in segmental models is put into designing segment features, while the model structure remains the same.\",\n \"In sum, we aim to design segmental models that perform well on sequence prediction tasks, have a clean and modular mathematical definition, are efficient to train and to decode, and do not depend on other models.\",\n \"The concept of segmental models was not explicitly defined before the late 1980s, and little work has had segmental models as the primary focus.\",\n \"[141] were one of the early attempts in the 1970s to build speech recognizers based on segment features.\",\n \"In the early 1980s, [23] developed a system for recognizing isolated English letters based on segment features.\",\n \"Isolated digits as segments were considered in [75].\",\n \"[147] introduced SUMMIT, a segment-based speech recognizer that can handle arbitrary segment features.\",\n \"While the SUMMIT system was later formulated into a probabilistic framework [34], others were not probabilistic, and have very few parameters to be estimated.\",\n \"The probabilistic view of segmental models can also be traced back to the 1980s.\",\n \"As mentioned earlier, [116] introduced semi-Markov processes and the follow-up [117] introduced hidden semi-Markov models to the speech recognition community.\",\n \"These two studies were mainly motivated by the need to include duration in frame-based models.\",\n \"[103] formalized segmental models as a probabilistic framework to include arbitrary features, but the authors only used frame-based probabilities with one additional duration feature.\",\n \"[115] and [29] further proposed segmental hidden Markov models, but the general form stayed mostly the same.\",\n \"Segmental models proposed in these studies were generative, and were mostly restricted to frame-based Gaussian distributions.\",\n \"This line of work has been summarized in [104].\",\n \"Into the 2000s, studies of segmental models shifted from generative to discriminative.\",\n \"[118] proposed semi-Markov conditional random fields (CRF), the first type of discriminative segmental models.\",\n \"Semi-Markov CRFs require the use of manual alignments to train the models, making them less applicable to speech recognition.\",\n \"[149] proposed to use a different training loss for training semi-Markov CRFs, marginalizing over all possible segmentations.\",\n \"This approach alleviated the need for manual alignments to train segmental models.\",\n \"Later, [150] used segmental models as second-pass rescoring models for word recognition, showcasing the flexibility of segmental models for incorporating a wide variety of features.\",\n \"A segmental model is said to be a first-pass model if it exhaustively searches over the entire hypothesis space.\",\n \"Before [148], all of the segmental models for speech recognition were not first-pass.\",\n \"The early version of SUMMIT [147] used a heuristic approach to estimate phoneme boundaries, instead of searching over the entire search space.\",\n \"A more recent version of SUMMIT [34] avoided searching exhaustively by using a frame-based model to generate high confidence hypotheses.\",\n \"Semi-Markov CRFs are first-pass segmental models, but have only been applied to natural language tasks, where sequences are typically short and label set size is small.\",\n \"[148] showed that it is feasible to use segmental models as first-pass speech recognizers.\",\n \"Since then, studies of segmental models have moved on to first-pass recognition.\",\n \"At the same time, advances in neural networks have allowed us to learn generic feature representations, so many studies have been devoted to finding better segment representations to replace hand-crafted features.\",\n \"[49] improved upon [148] by using a 3-layer multilayer perceptron to produce phoneme probabilities.\",\n \"[1] further improved the results by using a deep convolutional neural network, and it was also the first to train segmental models and neural networks end to end.\",\n \"Along the same line, [85] proposed end-to-end segmental models with segment representations computed with long short-term memory networks.\",\n \"In this thesis, we make the following contributions to the development of segmental models.\",\n \"We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.\",\n \"We develop improved understanding of training approaches and training requirements for segmental models.\",\n \"We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.\",\n \"We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.\",\n \"We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.\",\n \"Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.\",\n \"We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.\",\n \"We develop improved understanding of training approaches and training requirements for segmental models.\",\n \"We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.\",\n \"We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.\",\n \"We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.\",\n \"Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.\",\n \"In Chapter 2, we review finite-state transducers (FST), an important tool for describing the hypothesis spaces of segmental models.\",\n \"For the second part of Chapter 2, we review the essential components in speech recognizers, such as language models, lexicons, and hidden Markov models, represented with FSTs.\",\n \"In Chapter 3, we formally define segmental models.\",\n \"The definition is modular and covers many existing segmental models as special cases.\",\n \"In Chapter 4, we introduce discriminative segmental cascades, which allow us to explore various segment representations while maintaining efficiency.\",\n \"In Chapter 5, we study segmental models trained in different training conditions, comparing end-to-end training and multi-stage training.\",\n \"In Chapter 6, we review several variants of segmental models, and describe how they relate to our definition of segmental models.\",\n \"In Chapter 7, we propose a unified framework encompassing many end-to-end frame-based models and segmental models.\",\n \"Finally, in Chapter 8, we discuss possible ways to extend segmental models.\",\n \"This chapter describes finite-state transducers (FST), a useful tool for representing and manipulating the search space of a sequence prediction task.\",\n \"This chapter also includes a complete example of a standard speech recognizer based on hidden Markov models represented as FSTs.\",\n \"See [96] for a comprehensive review of FSTs and their algorithms, and [97] for their applications to speech recognition.\",\n \"We define FSTs based on multigraphs (graphs that allow multiple edges between any two vertices) instead of regular graphs.\",\n \"A multigraph $G$ is a tuple $(V, E, \\\\mathrm {tail}, \\\\mathrm {head})$ , where $V$ is a set of vertices, $E$ is a set of edges, $\\\\mathrm {tail}: E \\\\rightarrow V$ is a function that associates an edge to its tail vertex, and $\\\\mathrm {head}: E \\\\rightarrow V$ is a function that associates an edge to its head vertex.\",\n \"Note that $E$ is not a subset of $V \\\\times V$ but a general set, because we allow many edges to have the same tail and head, and a pair of vertices is not enough to uniquely identify an edge.\",\n \"An example of a multigraph is shown in Figure REF .\",\n \"We say that $e_1 \\\\in E$ and $e_2 \\\\in E$ are connected if $\\\\mathrm {head}(e_1) = \\\\mathrm {tail}(e_2)$ .\",\n \"In addition, we define a path of length $n$ as a sequence of connected edges $(e_1, e_2, \\\\dots , e_n)$ where $\\\\mathrm {head}(e_i) = \\\\mathrm {tail}(e_{i+1})$ for $i = 1, \\\\dots , n$ .\",\n \"A sub-path is a sequence of connected edges within a path.\",\n \"For example, $(e_3, e_4, e_5)$ is a sub-path of $(e_1, e_2, \\\\dots , e_7, e_8)$ .\",\n \"A finite-state transducer is a tuple $(G, \\\\Sigma , \\\\Lambda , I, F, i, o, w)$ , where $G$ is a multigraph, $\\\\Sigma $ is a set of input symbols, $\\\\Lambda $ is a set of output symbols, $I \\\\subseteq V$ is a set of initial vertices, $F \\\\subseteq V$ is a set of final vertices, $i: E \\\\rightarrow \\\\Sigma $ is a function that associates an edge to its input symbol, $o: E \\\\rightarrow \\\\Lambda $ is a function that associates an edge to its output symbol, and $w: E \\\\rightarrow \\\\mathbb {R}$ is a function that puts weights on edges.\",\n \"An example is shown in Figure REF .\",\n \"Figure: A multigraph, where V={0,1,2,3}V = \\\\lbrace 0, 1, 2, 3\\\\rbrace and E={0,1,2,3,4}E = \\\\lbrace 0, 1, 2, 3, 4\\\\rbrace .The functions of tail and head are shown in the table on the right.Figure: An FST based on the multigraph in Figure ,with Σ={𝚊,𝚋,𝚌,𝚍,𝚎}\\\\Sigma = \\\\lbrace \\\\texttt {a}, \\\\texttt {b}, \\\\texttt {c}, \\\\texttt {d}, \\\\texttt {e}\\\\rbrace andΛ={𝙰,𝙱,𝙲,𝙳,𝙴}\\\\Lambda = \\\\lbrace \\\\texttt {A}, \\\\texttt {B}, \\\\texttt {C}, \\\\texttt {D}, \\\\texttt {E}\\\\rbrace .The initial vertex is shown in bold, and the final vertexis shown with a doubled circle, i.e., I={0}I = \\\\lbrace 0\\\\rbrace and F={3}F = \\\\lbrace 3\\\\rbrace .We use σ\\\\sigma :λ\\\\lambda /w to denote an edgewith input symbol σ\\\\sigma , output symbol λ\\\\lambda , and weight ww.The number in parentheses is an identifier of an edge.An FST can be seen as a function that maps strings to strings, where each path in the graph defines an input-output pair.\",\n \"Specifically, a path $(e_1, e_2, \\\\dots , e_n)$ associates the input string $i(e_1)i(e_2)\\\\cdots i(e_n)$ with the output string $o(e_1)o(e_2)\\\\cdots o(e_n)$ .\",\n \"For example, the FST shown in Figure REF defines the mapping $\\\\lbrace (\\\\texttt {ad}, \\\\texttt {AD}), (\\\\texttt {bd}, \\\\texttt {BD}),(\\\\texttt {ce}, \\\\texttt {CE})\\\\rbrace $ .\",\n \"There is a special symbol called the empty symbol, denoted $\\\\epsilon $ .\",\n \"Any string concatenated with an empty symbol is itself, i.e., $\\\\sigma _1\\\\sigma _2\\\\cdots \\\\sigma _n\\\\epsilon = \\\\epsilon \\\\sigma _1\\\\sigma _2\\\\cdots \\\\sigma _n= \\\\sigma _1\\\\sigma _2\\\\cdots \\\\sigma _n$ .\",\n \"If the input symbol of an edge is the empty symbol, then we can traverse the edge without consuming any input.\",\n \"If the output symbol of an edge is the empty symbol, then we do not produce any output when traversing the edge.\",\n \"As an example, the FST in Figure REF defines the mapping $\\\\lbrace (\\\\texttt {ad}, \\\\texttt {AD}), (\\\\texttt {a}, \\\\texttt {AF}),(\\\\texttt {bd}, \\\\texttt {BD}), (\\\\texttt {b}, \\\\texttt {BF}), (\\\\texttt {ce}, \\\\texttt {CE})\\\\rbrace $ .\",\n \"Figure: An FST with empty symbols.\",\n \"The edgewith the empty symbol is highlighted in red.We assume there is one unique start vertex and one unique end vertex, i.e., $|I| = |F| = 1$ .\",\n \"If an FST has more than one initial vertices, we can always create an additional vertex and connect all initial vertices to the additional vertex with empty symbols (and possibly zero weights) without affecting the string mapping of the FST.\",\n \"For convenience, we also define $\\\\text{in}(v) = \\\\lbrace e \\\\in E : \\\\mathrm {head}(e) = v\\\\rbrace $ and $\\\\text{out}(v) = \\\\lbrace e \\\\in E: \\\\mathrm {tail}(e) = v\\\\rbrace $ .\",\n \"We have reviewed FSTs as graphs and string functions.\",\n \"The third view of FSTs is as state machines.\",\n \"Take the FST in Figure REF for example.\",\n \"To get the output AD from the input ad, we first feed the character a and traverse from vertex 0 to vertex 1, and then feed the character d and traverse from vertex 1 to vertex 3.\",\n \"Similarly, to get BF from b, we first feed the character b and traverse from vertex 0 to vertex 1.\",\n \"Due to the empty symbol $\\\\epsilon $ , we get F and traverse from vertex 1 to vertex 3 without feeding any character.\",\n \"The view of state machines is especially useful when we talk about FST compositions.\",\n \"Other than just manipulating strings, we are also interested in the weights on the edges, especially when paths are considered.\",\n \"We introduce two operators $\\\\otimes $ and $\\\\oplus $ , where $\\\\otimes $ is for combining edge weights and $\\\\oplus $ is for combining path weights.\",\n \"For example, by traversing the path $(0, 3)$ that produces AD in Figure REF , we collect the weight 0.1 from edge 0 and 0.4 from edge 3, and the weight of the path $(0, 3)$ is defined as $w(0) \\\\otimes w(3) = 0.1 \\\\otimes 0.4$ .\",\n \"Suppose in general $S$ is the set of weights.\",\n \"It is desirable to have certain properties, such as commutativity and associativity, for the two operators.\",\n \"For $a \\\\in S$ , $b \\\\in S$ , and $c \\\\in S$ , we say that an operator $*$ is commutative if $a * b = b * a$ , and that an operator $*$ is associative if $(a * b) * c = a * (b * c)$ .\",\n \"The element $0 \\\\in S$ is an identity with respect to the operator $*$ if $0 * a = a * 0 = a$ .\",\n \"The element $0 \\\\in S$ annihilates $S$ with respect to $*$ if $0 * a = a * 0 = 0$ .\",\n \"We say that the operator $*$ distributes over the operator $+$ if $a * (b + c) = a * b + a * c$ and $(a + b) * c = a * c + b * c$ .\",\n \"The tuple $(S, \\\\oplus , \\\\otimes )$ is a semiring [95] if $\\\\oplus $ is associative and commutative with identity $0 \\\\in S$ , $\\\\otimes $ is associative with identity $1 \\\\in S$ , $\\\\otimes $ distributes over $\\\\oplus $ , and the element $0 \\\\in S$ annihilates $S$ with respect to $\\\\otimes $ .\",\n \"An example of semiring is $(\\\\mathbb {R}, +, \\\\times )$ with 0 being the identity of $+$ , and 1 being the identity of $\\\\times $ .\",\n \"Another example is the tropical semiring $(\\\\mathbb {R} \\\\cup \\\\lbrace \\\\infty \\\\rbrace , \\\\min , +)$ with $\\\\infty $ being the identity of $\\\\min $ , and 0 being the identity of $+$ .\",\n \"Yet another example is the log semiring $(\\\\mathbb {R} \\\\cup \\\\lbrace -\\\\infty \\\\rbrace , \\\\text{logadd}, +)$ where $\\\\text{logadd}(a, b) = \\\\log (\\\\exp (a) + \\\\exp (b))$ , $-\\\\infty $ is the identity of $\\\\text{logadd}$ , and 0 is the identity of $+$ .\",\n \"Defining the operators for combining weights in a general way encourages algorithm reuse.\",\n \"As we will see in later chapters, FST algorithms tend to look very similar, and typically the only difference is how the weights are combined.\",\n \"By using general operators, we get a family of algorithms.\",\n \"In other words, we create algorithms almost for free by choosing a proper semiring.\",\n \"Given an FST, we are interested in finding a shortest path from the initial vertex to the final vertex.\",\n \"For example, if the weights are negative log probabilities traversing from one vertex to another, then a shortest path corresponds to one of the most likely paths.\",\n \"We assume there are multiple shortest paths, but finding one of them is enough.\",\n \"We also assume the underlying graph is acyclic, meaning that every path can only traverse a vertex at most once.\",\n \"For acyclic graphs, we have a nice necessary condition for shortest paths—every sub-path within a shortest path is also a shortest path.\",\n \"The argument for the necessary condition is simple.\",\n \"If a sub-path is not a shortest path, then we can always substitute the sub-path with another shorter sub-path to create a shorter path overall.\",\n \"The necessary condition can be viewed as a recursive condition.\",\n \"Let $\\\\mathcal {P}(u, v)$ be the set of paths from vertex $u$ to vertex $v$ , and let $w(p)$ be a shorthand of $\\\\sum _{e \\\\in p} w(e)$ .\",\n \"Suppose we want to compute a shortest path from the vertex $u$ to vertex $v$ .\",\n \"We examine the set of edges $\\\\text{in}(v) = \\\\lbrace e \\\\in E: \\\\mathrm {head}(e) = v\\\\rbrace $ leading into vertex $v$ .\",\n \"If an edge $e \\\\in \\\\text{in}(v)$ is part of the shortest path, then by the necessary condition every path from $u$ to $\\\\mathrm {tail}(e)$ should also be a shortest path.\",\n \"In other words, $\\\\min _{p \\\\in \\\\mathcal {P}(u, v)} w(p)& = \\\\min _{e \\\\in \\\\text{in}(v)} \\\\min _{p^{\\\\prime } \\\\in \\\\mathcal {P}(u, \\\\mathrm {tail}(e))}[ w(e) + w(p^{\\\\prime }) ] \\\\\\\\& = \\\\min _{e \\\\in \\\\text{in}(v)}\\\\Bigg [ w(e) + \\\\min _{p^{\\\\prime } \\\\in \\\\mathcal {P}(u, \\\\mathrm {tail}(e))} w(p^{\\\\prime }) \\\\Bigg ]$ The recursive condition can be computed efficiently with dynamic programming if we store the shortest distance from vertex $u$ to every other vertex.\",\n \"Specifically, let $d(v)$ be the shortest distance from $u$ to $v$ , i.e., $d(v) = \\\\min _{p \\\\in \\\\mathcal {P}(u, v)} w(p).$ By the recursive condition, we have $d(v) = \\\\min _{e \\\\in \\\\text{in}(v)} [ w(e) + d(\\\\mathrm {tail}(e)) ].\",\n \"$ Before computing the minimization in (REF ), we need to make sure the shortest sub-paths are computed.\",\n \"Formally, to compute $d(v)$ for vertex $v$ , we need to make sure $d(\\\\mathrm {tail}(e))$ are computed for $e \\\\in \\\\text{in}(v)$ .\",\n \"In other words, if there is a path from vertex $w$ to vertex $v$ , then $d(w)$ should be computed before $d(v)$ .\",\n \"We are looking for an order in which $w$ should come before $v$ if there is a path from $w$ to $v$ .\",\n \"An order in which $w$ comes before $v$ if there is a path from $w$ to $v$ is called a topological order.\",\n \"To be precise, a topological order is a function $f: V \\\\rightarrow \\\\lbrace 1, 2, \\\\dots , |V|\\\\rbrace $ such that $f(w) < f(v)$ if there is a path from $w$ to $v$ .\",\n \"Given an directed acyclic graph, there exist many topological orders.\",\n \"Any one of them would suffice for our dynamic programming.\",\n \"One way to find a topological order is to run depth-first search (DFS) on the reversed graph.\",\n \"The depth-first search algorithm is shown in Algorithm REF .\",\n \"We maintain a stack $S$ .\",\n \"We say that a vertex is traversed if it has been put on $S$ , and we use a set $T$ to track the traversed vertices.\",\n \"When a vertex $v$ is put on stack, we also put a variable indicating whether we have tried to put its neighbors on the stack, where the set of neighbors is defined by $\\\\text{in}(\\\\cdot )$ .\",\n \"We say that a vertex is expanded if we have tried to put its neighbors on the stack.\",\n \"It is not hard to see that every vertex is put on the stack twice, once before it is expanded and once after.\",\n \"When a vertex $v$ popped out from the stack is expanded, all vertices that can be traversed from $v$ are also expanded.\",\n \"Since we put a vertex $v$ in $O$ when $v$ and all vertices traversable from $v$ are expanded, the order $O$ is exactly a topological order.\",\n \"Algorithm REF is an instance of DFS specifically for computing a topological order by defining the set of neighbors with $\\\\text{in}(\\\\cdot )$ .\",\n \"The algorithm can be modified for other purposes, for example, letting the set of neighbors be $\\\\text{out}(\\\\cdot )$ (and changing $\\\\mathrm {tail}(\\\\cdot )$ to $\\\\mathrm {head}(\\\\cdot )$ accordingly) if we want to search the graph in a forward direction instead of backwards.\",\n \"Due to the use of a stack, DFS prefers to follow one path until none of the vertices can be expanded; hence the name depth-first search.\",\n \"Depth-First Search (DFS) Let $r$ be the vertex we start to search.\",\n \"A list of vertices $O$ is returned in topological order.\",\n \"$S = \\\\lbrace (r, \\\\texttt {false})\\\\rbrace $ , $T = \\\\lbrace r\\\\rbrace $ , $O = ()$ $S$ is not empty pop $(v, z)$ from $S$ $z$ append $v$ to $O$ push $(v, \\\\texttt {true})$ to $S$ $e \\\\in \\\\text{in}(v)$ Expand $v$ .\",\n \"$\\\\mathrm {tail}(e) \\\\notin T$ push $(\\\\mathrm {tail}(e), \\\\texttt {false})$ to $S$ $\\\\mathrm {tail}(e)$ is traversed.\",\n \"add $v$ to $T$ Shortest-Path Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.\",\n \"The map $d$ contains the shortest distances to $v$ for all vertices.\",\n \"Let $O = (o_1, o_2, \\\\dots , o_n)$ be a topological order by running DFS starting from $t$ .\",\n \"$i = 1, \\\\dots , n$ $d(o_i) = \\\\min _{e \\\\in \\\\text{in}(o_i)} [ w(e) + d(\\\\mathrm {tail}(e)) ]$ $\\\\pi (o_i) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\min }_{e \\\\in \\\\text{in}(o_i)} [ w(e) + d(\\\\mathrm {tail}(e)) ]$ Backtracking Let $s$ be the initial vertex and $t$ be the final vertex.\",\n \"A shortest path $p$ from $s$ to $t$ .\",\n \"$v \\\\leftarrow t$ $p = ()$ $v \\\\ne s$ add $\\\\pi (v)$ to $p$ $v \\\\leftarrow \\\\mathrm {tail}(\\\\pi (v))$ reverse $p$ Generalized Shortest-Distance Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.\",\n \"The map $d$ contains the shortest distances to $v$ for all vertices.\",\n \"Let $O = (o_1, o_2, \\\\dots , o_n)$ be a topological order by running DFS starting from $t$ .\",\n \"$i = 1, \\\\dots , n$ $d(o_i) = \\\\bigoplus _{e \\\\in \\\\text{in}(o_i)} [ w(e) \\\\otimes d(\\\\mathrm {tail}(e)) ]$ Up to this point, we are interested in finding a path that is shortest.\",\n \"The weight of a path is defined to be the sum of the edge weights, and path weights are combined with the $\\\\min $ operator.\",\n \"These operators are the ones used in the tropical semiring, and can be generalized to other semirings, where edge weights are combined with $\\\\otimes $ and path weights are combined with $\\\\oplus $ .\",\n \"To be precise, the weight of a path $p$ is defined as $w(p) = \\\\bigotimes _{e \\\\in p} w(e)$ , and the distance can be written as $d(v) & = \\\\bigoplus _{p \\\\in \\\\mathcal {P}(u, v)} w(p)= \\\\bigoplus _{p \\\\in \\\\mathcal {P}(u, v)} \\\\bigotimes _{e \\\\in p} w(e)= \\\\bigoplus _{e \\\\in \\\\text{in}(v)} \\\\bigoplus _{p^{\\\\prime } \\\\in \\\\mathcal {P}(u, \\\\mathrm {tail}(e))}[ w(e) \\\\otimes w(p^{\\\\prime }) ] \\\\\\\\& = \\\\bigoplus _{e \\\\in \\\\text{in}(v)} \\\\left[ w(e) \\\\otimes \\\\bigoplus _{p^{\\\\prime } \\\\in \\\\mathcal {P}(u, \\\\mathrm {tail}(e))} w(p^{\\\\prime }) \\\\right] \\\\\\\\& = \\\\bigoplus _{e \\\\in \\\\text{in}(v)} [w(e) \\\\otimes d(\\\\mathrm {tail}(e))].$ The generalized shortest-distance algorithm is shown in Algorithm REF .\",\n \"See [95] for a general treatment of shortest-distance algorithms.\",\n \"The recursive condition (REF ) only gives us the distance to the final vertex.\",\n \"To obtain the path, we can record down the edge that achieves the minimum while computing the distances.\",\n \"Specifically, let $\\\\pi (v)$ be the edge that achieves the minimum, i.e., $\\\\pi (v) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\min }_{e \\\\in \\\\text{in}(v)} [ w(e) + d(\\\\mathrm {tail}(e)) ].$ We can iteratively collect the optimal edge with $\\\\pi $ .\",\n \"Since the algorithm goes from the final vertex to the initial vertex, it is commonly called backtracking.\",\n \"The final shortest-path algorithm and the backtracking algorithm are shown in Algorithm REF and REF .\",\n \"Recall that FSTs can be regarded as functions that map strings to strings.\",\n \"It is natural to have multiple FSTs with one FST consuming the outputs of another FST, similar to function composition.\",\n \"Let $T(x, y)$ be the weight of a path in the FST $T$ consuming $x$ as input and producing $y$ as output.\",\n \"For any two FSTs $T_1$ and $T_2$ , the weight of a path with input $x$ and output $y$ in the composed FST $T_1 \\\\circ T_2$ is defined as $ (T_1 \\\\circ T_2)(x, y) = \\\\bigoplus _{z \\\\in \\\\mathcal {Z}(x)} T_1(x, z) \\\\otimes T_2(z, y),$ where $\\\\mathcal {Z}(x)$ is the set of output strings that can be produced by feeding $x$ to $T_1$ [2].\",\n \"The weight is undefined if $\\\\mathcal {Z}(x)$ is an empty set or there is no path with input $z$ and output $y$ in $T_2$ .\",\n \"The definition of (REF ) does not provide an explicit structure of the composed FST.\",\n \"In this thesis, we prefer another approach to composing FSTs which we term structured composition [130].\",\n \"Formally, the structured composition (or $\\\\sigma $ -composition) of two FSTs $T_1 = (G_1, \\\\Sigma _1, \\\\Lambda _1, I_1, F_1, i_1, o_1, w_1)$ with $G_1 = (V_1, E_1, \\\\mathrm {tail}_1, \\\\mathrm {head}_1)$ and $T_2 = (G_2, \\\\Sigma _2, \\\\Lambda _2, I_2, F_2, i_2, o_2,w_2)$ with $G_2 = (V_2, E_2, \\\\mathrm {tail}_2, \\\\mathrm {head}_2)$ is defined as $T_1 \\\\circ _\\\\sigma T_2 = (G, \\\\Sigma , \\\\Lambda , I,F, i, o, w)$ where $G = (V, E, \\\\mathrm {tail}, \\\\mathrm {head})$ with the following constraints.\",\n \"$ V & = V_1 \\\\times V_2 &E & = \\\\Big \\\\lbrace \\\\langle e_1, e_2 \\\\rangle \\\\in E_1 \\\\times E_2 : o_1(e_1) = i_2(e_2) \\\\Big \\\\rbrace $ $\\\\Sigma & = \\\\Sigma _1 &i(\\\\langle e_1, e_2 \\\\rangle ) & = i_1(e_1) \\\\\\\\\\\\Lambda & = \\\\Lambda _2 &o(\\\\langle e_1, e_2 \\\\rangle ) & = o_2(e_2) \\\\\\\\I & = I_1 \\\\times I_2 &\\\\mathrm {tail}(\\\\langle e_1, e_2 \\\\rangle ) & = \\\\langle \\\\mathrm {tail}_1(e_1), \\\\mathrm {tail}_2(e_2) \\\\rangle \\\\\\\\F & = F_1 \\\\times F_2 &\\\\mathrm {head}(\\\\langle e_1, e_2 \\\\rangle ) & = \\\\langle \\\\mathrm {head}_1(e_1), \\\\mathrm {head}_2(e_2) \\\\rangle $ The $\\\\sigma $ -composed graph is defined over pairs of vertices and pairs of edges.\",\n \"Due to the edge constraints (REF ), the outputs from $T_1$ have to match the inputs to $T_2$ .\",\n \"An example is shown in Figure REF .\",\n \"Note that the definition of the weight function is left to the users.\",\n \"One possible definition is to let $w(\\\\langle e_1, e_2 \\\\rangle ) = w_1(e_1) \\\\otimes w_2(e_2).$ As a consequence of (REF ), the outgoing edges and incoming edges of a vertex in the $\\\\sigma $ -composed FST can be computed locally with $\\\\text{out}_1(\\\\cdot )$ and $\\\\text{out}_2(\\\\cdot )$ .\",\n \"Specifically, $ \\\\text{out}(\\\\langle v_1, v_2 \\\\rangle )= \\\\Big \\\\lbrace \\\\langle e_1, e_2 \\\\rangle : e_1 \\\\in \\\\text{out}_1(v_1), e_2 \\\\in \\\\text{out}_2(v_2),o_1(e_1) = i_2(e_2) \\\\Big \\\\rbrace .$ Since computing $\\\\text{out}(\\\\langle v_1, v_2 \\\\rangle )$ only requires access to edges connected to $v_1$ and $v_2$ , traversing the composed FST, for example with DFS (Algorithm REF ), only needs to keep track of the vertex pairs.\",\n \"The edges can be computed on the fly and do not need to be stored in memory.\",\n \"In general, composing FSTs without explicitly storing the entire graph is commonly known as on-the-fly composition or lazy composition.\",\n \"Another way to compute structured composition is to view FSTs as state machines.\",\n \"Take Figure REF for example.\",\n \"We start from vertex 0 in both $T_1$ and $T_2$ .\",\n \"To get to vertex 1 in $T_1$ , we can either produce A or produce B as output.\",\n \"In $T_2$ , to go from vertex 0 to vertex 1, we only have the choice to consume A.\",\n \"Therefore, in the composed FST $T_1 \\\\circ _\\\\sigma T_2$ we have the edge from $(0, 0)$ to $(1, 1)$ with input a and output $\\\\alpha $ .\",\n \"By walking through the vertices one by one, we are essentially performing a search on both FSTs while computing (REF ) on the fly.\",\n \"Figure: An example of structured composition.Vertices in T 1 ∘ σ T 2 T_1 \\\\circ _\\\\sigma T_2 are labeled with pairs of vertices from T 1 T_1 and T 2 T_2.We now have all the necessary tools to describe a basic speech recognizer based on FSTs.\",\n \"Modern speech recognizers are based on a series of string function compositions.\",\n \"Following [98], we represent the string functions as FSTs, and the compositions are realized with structured composition.\",\n \"The first FST $U$ takes acoustic frames as inputs and produces a sequence of frame labels.\",\n \"The second FST $H$ takes in frame labels and converts them into phonemes.\",\n \"The third FST $L$ takes phonemes as inputs and produces words as outputs.\",\n \"The fourth FST $G$ takes words as inputs and produces words as outputs.\",\n \"Finally, the four FSTs are $\\\\sigma $ -composed into $D = U \\\\circ _\\\\sigma H \\\\circ _\\\\sigma L \\\\circ _\\\\sigma G.$ The path weights in $D$ are typically negative log probabilities, so the maximum-weight path is the most likely path based on the probabilities.\",\n \"To find the maximum-weight path, we simply negate the weights and find the shortest path.\",\n \"The tropical semiring is used to combine the negated weights.\",\n \"In sum, to predict a sequence from a sequence of frames, we create the FST $D$ and find the maximum-weight path in $D$ .\",\n \"Below we describe the four FSTs $U$ , $H$ , $L$ and $G$ in detail.\",\n \"The dynamics of speech are commonly modeled by hidden Markov models (HMM).\",\n \"The generative story of a hidden Markov model is as follows.\",\n \"We first generate a state $y_1$ based on a prior distribution and generate a frame $x_1$ given $y_1$ .\",\n \"A state $y_2$ is generated given $y_1$ , and a frame $x_2$ is generated given $y_2$ .\",\n \"In general, for some $t = 2, \\\\dots , T$ , the state $y_t$ is generated given $y_{t-1}$ , and $x_t$ is generated given $y_t$ .\",\n \"For a sequence of $T$ frames $x_1, \\\\dots , x_T$ , the probability distribution defined by an HMM is $p(x_{1:T}, y_{1:T}) = p(y_1) \\\\prod _{t=2}^T p(y_t | y_{t-1}) \\\\prod _{t=1}^T p(x_t | y_t),$ where $x_{1:T}$ is a shorthand for $x_1, \\\\dots , x_T$ , $x_t \\\\in \\\\mathbb {R}^d$ for some $d \\\\in \\\\mathbb {N}$ and $y_t \\\\in \\\\lbrace 1, \\\\dots , S\\\\rbrace $ for some $S \\\\in \\\\mathbb {N}$ for $t=1, \\\\dots , T$ .\",\n \"An example of a four-frame HMM is shown in Figure REF .\",\n \"Note that Figure REF is not an FST but a graphical model where vertices are random variables and edges represent conditional dependencies.\",\n \"The learnable parameters in an HMM are $p(y_1)$ , $p(y_t | y_{t-1})$ , and $p(x_t | y_t)$ for $t = 1, \\\\dots , T$ .\",\n \"The probabilities $p(y_t | y_{t-1})$ are commonly known as transition probabilities, and $p(x_t | y_t)$ emission probabilities.\",\n \"Figure: A hidden Markov model for 4 frames x 1 ,x 2 ,x 3 ,x 4 x_1, x_2, x_3, x_4with hidden labels y 1 ,y 2 ,y 3 ,y 4 y_1, y_2, y_3, y_4.Note that this is a graphical model where vertices are random variablesand edges represent conditional dependencies, not an FST.The observed variables are shaded, and the unobserved are not.The transition probabilities $p(y_t | y_{t-1})$ can be represented as a square matrix $A$ of size $S \\\\times S$ in which $A_{ij} \\\\in [0, 1]$ is the probability of transitioning from state $i$ to state $j$ satisfying $\\\\sum _{j=1}^S A_{ij} = 1$ .\",\n \"To forbid the transition from state $i$ to state $j$ , we simply let $A_{ij} = 0$ .\",\n \"It is common to model a phoneme as a 3-state HMM, and parameterize $A$ as $\\\\begin{pmatrix}a_{11} & a_{12} & 0 \\\\\\\\0 & a_{22} & a_{23} \\\\\\\\0 & 0 & a_{33}\\\\end{pmatrix}$ where we are only allowed to either stay in the current state or move to the next state.\",\n \"The allowed transitions can be conveniently represented as an FST, where vertices are states and edges are transitions associated with transition probabilities.\",\n \"An example of the allowed state transitions is shown in Figure REF .\",\n \"Figure: A 3-state FST for the phoneme ih, where loga ij \\\\log a_{ij} isthe log transition probability from state ii to state jj.To allow transitions from phonemes to phonemes, we connect the FSTs, such as the ones in Figure REF , in parallel.\",\n \"An example is shown in Figure REF .\",\n \"An additional start state and an additional end state are added.\",\n \"A backward edge from the end state to the start state is also added to allow a sequence of phonemes with indefinite length.\",\n \"The prior distribution $p(y_1)$ entering the phoneme is also added from the start state to each of the phoneme FSTs.\",\n \"Note that when entering one of the phoneme FSTs, the edge consumes a phoneme state and produces a phoneme, and the phoneme FSTs only consume phoneme states and do not produce any output.\",\n \"Silences are modeled with five states rather than three, because they are allowed to be longer than other phonemes.\",\n \"This finishes the construction of the FST $H$ that converts phoneme states to phonemes.\",\n \"Figure: An FST HH that converts phoneme states to phonemes.The emission probabilities $p(x_t | y_t)$ in HMM are typically modeled by Gaussian mixture models (GMM).\",\n \"Specifically, $p(x_t | y_t) = \\\\sum _{i=1}^C \\\\pi _{i, y_t} p( x_t | \\\\mu _{i, y_t}, \\\\sigma ^2_{i, y_t})$ where there are $C$ components for each state in $\\\\lbrace 1, \\\\dots , S\\\\rbrace $ , each component $p(x_t | \\\\mu _{i, y}, \\\\sigma ^2_{i, y})$ is a Gaussian distribution with mean $\\\\mu _{i, y}$ and variance $\\\\sigma ^2_{i, y}$ , and $\\\\pi _{i, y}$ is the probability of selecting the $i$ -th component.\",\n \"For a sequence of $T$ frames, the emission probability for each frame can be independently evaluated.\",\n \"Since each state has $C$ Gaussian components, there are $S \\\\times C$ evaluations for each frame.\",\n \"We can build an FST that has $S \\\\times C$ edges for every frame.\",\n \"Each edge has a weight computed from its Gaussian component and has a phone state as the output symbol.\",\n \"An example is shown in Figure REF .\",\n \"Note that the FST is constructed by following the generative story of Gaussian mixture models, evaluating a single Gaussian component (selected based on $\\\\pi $ ) for every frame rather than evaluating the weighted average of all Gaussian components.\",\n \"Figure: An FST UU for 4 frames with GMM probabilities.For example, if each phoneme has 3 states, and eachstate has 64 Gaussian components, thens∈{𝚊𝚊1,𝚊𝚊2,𝚊𝚊3,⋯,𝚣𝚑1,𝚣𝚑2,𝚣𝚑3}s \\\\in \\\\lbrace \\\\texttt {aa1}, \\\\texttt {aa2}, \\\\texttt {aa3}, \\\\dots , \\\\texttt {zh1}, \\\\texttt {zh2}, \\\\texttt {zh3} \\\\rbrace ,i∈{1,⋯,C=64}i \\\\in \\\\lbrace 1, \\\\dots , C = 64\\\\rbrace .If there are LL phonemes (i.e., S=3LS = 3L), then there are 3L×643L \\\\times 64 (i.e., S×C)S \\\\times C) edges between each pair of vertices.By $\\\\sigma $ -composing $U$ and $H$ , we construct an FST where each path in the FST is a sample drawn from the generative story and the weight of each path is the corresponding log probability.\",\n \"To better see the connection, we can traverse the FST $U$ and $H$ synchronously as the way we compute $U \\\\circ _\\\\sigma H$ .\",\n \"First when we traverse an edge in $U$ , we select one phoneme state and one Gaussian component, collecting the log probability and producing a phoneme state as output.\",\n \"Then the FST $H$ receives the phoneme state and waits for the next phoneme state, collecting transition probabilities and producing a phoneme when necessary.\",\n \"This completes the construction of $U$ and $H$ .\",\n \"The pronunciation dictionary, or lexicon, is a mapping from a word to its pronunciation.\",\n \"To construct an FST representing the lexicon, we create, for each entry in the lexicon, a path consuming a sequence of phonemes and producing a word.\",\n \"Similar to the FST $H$ , we have a start vertex and an end vertex that joins the pronunciation paths, and we also have a backward edge that allows indefinite amount of words to be produced.\",\n \"An example is shown in Figure REF .\",\n \"Note that the weights on the edges can either be 0 or the probability of a chosen pronunciation.\",\n \"The FST also allows a word to have multiple pronunciations.\",\n \"Silences are considered as words and are included in the FST.\",\n \"The size of the FST can be significantly reduced if we maintain a prefix tree of the phonemes instead of having parallel pronunciations for every word.\",\n \"Figure: An FST LL representing a pronunciation dictionary.Language models assign probabilities to word sequences.\",\n \"For any $k \\\\in \\\\mathbb {N}$ , a word sequence $w_{1:k} = w_1, \\\\dots , w_k$ of length $k$ has probability $p(w_{1:k}) = \\\\prod _{i=1}^k p(w_i | w_{1:i-1}).$ As the sequence gets longer, i.e., as $i$ gets larger, $p(w_i | w_{1:i-1})$ depends on more words, which makes estimating the probabilities difficult.\",\n \"A simple approach to remedy this problem is to introduce the Markov assumption $p(w_i | w_{1:i-1}) = p(w_i | w_{i-n+1:i-1}).$ A language model that gives a probability for the current word based on the previous $n-1$ words is commonly known as an $n$ -gram language model.\",\n \"To construct an FST representing a language model, we create vertices that corresponds to history words $w_{i-n+1:i-1}$ .\",\n \"For each vertex, the outgoing edges produce the next word $w_i$ with the weight $\\\\log p(w_i | w_{i-n+1:i-1})$ .\",\n \"A path in this FST consumes a sequence of words and has the probability of the word sequence assigned by the language model.\",\n \"There is a start vertex representing the state of having no history words.\",\n \"We assume the utterance starts and ends with silences, so there is only one edge going out from the start state expecting a silence.\",\n \"An example is shown in Figure REF .\",\n \"Back-off language models can also be approximated with FSTs [3].\",\n \"Figure: An FST GG representing a bigram language modelfor the vocabulary {sil, A, cat, runs}.Some edges are dashed to avoid clutter.The output symbols are the same as the input symbols for every edge,hence ignored.\",\n \"The vertex in red is the silence state.In this chapter, we have reviewed the definition of finite-state transducers (FST) and constructed a basic speech recognizer by $\\\\sigma $ -composing an HMM emission FST $U$ , an HMM transition FST $H$ , a lexicon $L$ , and a language model $G$ .\",\n \"Since $H \\\\circ _\\\\sigma L \\\\circ _\\\\sigma G$ is shared across all utterances, it is typically $\\\\sigma $ -composed and saved.\",\n \"Each individual FST and the composed FST can also be determinized and minimized for efficiency, which we do not cover.\",\n \"Interested readers should refer to [98] for further details.\",\n \"The problem of prediction in general can be considered as a search problem.\",\n \"Given an input $x$ , we first construct a set of possible outputs $\\\\mathcal {Y}(x)$ , referred to as the search space.\",\n \"For every output hypothesis $y$ in the search space $\\\\mathcal {Y}(x)$ , we measure how well the output matches the input, assigning a weight to each pair $(x, y)$ .\",\n \"Prediction can be considered as finding the hypothesis $\\\\hat{y}$ such that the weight of $(x, \\\\hat{y})$ is larger than any other pairs.\",\n \"Extending the above paradigm, the problem of sequence prediction, such as speech recognition as we have seen in Chapter REF , can be considered as a search problem.\",\n \"Given an input sequence, the search space in this case is a set of sequences of connected segments.\",\n \"The number of such sequences is exponential in the number of possible segments.\",\n \"Fortunately, we do not need to store exponentially many such sequences, and can represent the search space compactly as an finite-state transducer (FST).\",\n \"Consider an FST with vertices corresponding to time points, and edges corresponding to segments.\",\n \"A path in the FST corresponds to a sequence of connected segments, and the set of all paths defined by the FST is the search space.\",\n \"The FST is weighted, and the weights assigned to the edges are based on how well the segments match the input.\",\n \"Prediction can be considered as finding the maximum-weight path, because the maximum-weight path, by definition, is a path that best matches the input.\",\n \"In this chapter, we formally define segmental models, including search spaces constructed from sequences of input vectors, weight functions that measure how well a segment matches the acoustic signals, and loss functions used for training segmental models.\",\n \"Let $\\\\mathcal {X}$ be the input space, the set of all sequences of real-valued vectors, e.g., log mel filter bank features or mel frequency cepstral coefficients (MFCCs).\",\n \"Specifically, for a sequence of $T$ vectors $x = (x_1, \\\\dots , x_T) \\\\in \\\\mathcal {X}$ , each $x_t \\\\in \\\\mathbb {R}^d$ is a $d$ -dimensional vector, also referred to as a frame, for $t \\\\in \\\\lbrace 1, \\\\dots , T\\\\rbrace $ .\",\n \"Let $\\\\mathcal {Y}$ be the output space, the set of all label sequences, where each label in a label sequence comes from a label set $L$ , e.g., a phoneme set in the case of phoneme recognition.\",\n \"Given any $T$ frames, a segmentation of length $K$ is a sequence of time points $((1 = s_1, t_1), \\\\dots , (s_K, t_K = T))$ , where $s_k \\\\le t_k$ and $t_k + 1 = s_{k+1}$ for $k \\\\in \\\\lbrace 2, \\\\dots , K\\\\rbrace $ .\",\n \"A segment (typically denoted $e$ in later sections) is a tuple $(\\\\ell , s, t)$ where $\\\\ell \\\\in L$ is its label, $s$ is the start time, and $t$ is the end time.\",\n \"A segmental model is a tuple $(\\\\Theta , w)$ where $\\\\Theta $ is a set of parameters, and $w: \\\\mathcal {X} \\\\times E \\\\rightarrow \\\\mathbb {R}$ is a weight function parameterized by $\\\\Theta $ and $E$ is the set of all segment tuples $(\\\\ell , s, t)$ .\",\n \"A sequence of segments forms a path.\",\n \"Specifically, a path of length $K$ is a sequence of segments $(e_1, \\\\dots , e_K)$ , where $e_k \\\\in E$ for $k \\\\in \\\\lbrace 1, \\\\dots , K\\\\rbrace $ .\",\n \"Let $\\\\mathcal {P}$ be the set of all paths.\",\n \"For any path $p$ , we overload $w$ such that $w(x, p) = \\\\sum _{e \\\\in p} w(x, e)$ .\",\n \"We will also abbreviate $w(x, e)$ and $w(x, p)$ as $w(e)$ and $w(p)$ respectively when the context is clear.\",\n \"The concrete form of the weight function will be defined in later sections.\",\n \"Given an input $x \\\\in \\\\mathcal {X}$ , segmental models aim to solve sequence prediction by reducing it to finding the maximum-weight path $ \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{p \\\\in \\\\mathcal {P}} w(x, p).$ The set of paths $\\\\mathcal {P}$ , also referred to as the search space, can be compactly represented as an FST.\",\n \"Once we have the search space FST, we can simply invoke Algorithm REF to find the maximum-weight path.\",\n \"We will describe how the search space FST is constructed in the next section.\",\n \"A segmental model can be trained by finding a set of parameters that minimizes a loss function.\",\n \"We emphasize that the model definition is not tied to any loss function, allowing us to study the behavior of segmental models under different loss functions.\",\n \"We define various loss functions for training segmental models in Section REF , and discuss how the properties of the losses affect the training requirement.\",\n \"To represent the set of paths $\\\\mathcal {P}$ as an FST, we place a vertex at every time point and connect vertices based on the set of segments.\",\n \"Suppose we have $T$ frames.\",\n \"The set of segments $E$ is an exhaustive enumeration of tuples $(\\\\ell , s, t)$ for all $\\\\ell \\\\in L$ and $1 \\\\le s \\\\le t \\\\le T$ .\",\n \"We create a set of vertices $V = \\\\lbrace v_0, v_1, \\\\dots , v_T\\\\rbrace $ and a time function $\\\\tau : V \\\\rightarrow \\\\mathbb {N}$ such that $\\\\tau (v_t) = t$ for $t \\\\in \\\\lbrace 0, 1, \\\\dots , T\\\\rbrace $ .\",\n \"For every segment $(\\\\ell , s, t) \\\\in E$ , we create an edge $e$ such that $i(e) = o(e) = \\\\ell $ , $\\\\mathrm {tail}(e) = v_{s-1}$ , and $\\\\mathrm {head}(e) = v_t$ .\",\n \"In other words, for any $e \\\\in E$ , the corresponding segment $(\\\\ell , s, t) = (o(e), \\\\tau (\\\\mathrm {tail}(e)), \\\\tau (\\\\mathrm {head}(e)))$ .\",\n \"As a result, we will use $w(e)$ and $w((\\\\ell , s, t))$ interchangeably.\",\n \"We set $\\\\Sigma = \\\\Lambda = L$ , $I = \\\\lbrace v_0\\\\rbrace $ , and $F = \\\\lbrace v_T\\\\rbrace $ to complete the construction of the FST given any $T$ frames.\",\n \"The number of segments in the graph is $O(|L|T^2)$ .\",\n \"To reduce the size of the search space, a maximum duration constraint is typically imposed while creating the search space.\",\n \"Specifically, we only create segments $(\\\\ell , s, t)$ with $t - s + 1 \\\\le D$ , for some maximum duration $D$ .\",\n \"Adding such constraint makes the number of segments $O(|L|TD)$ .\",\n \"An example of a search space is shown in Figure REF .\",\n \"Figure: An example search space for a five-frame input sequencewith a label set LL of size three and maximum segment duration DD of two frames,i.e., T=5T = 5, |L|=3|L| = 3, and D=3D = 3.The three edges between any two nodes are associated with the three labels.Here we detail two types of weight functions based on prior work by ourselves and others [49], [130], [1], [85].\",\n \"We will compare the two weight functions and in Chapter REF .\",\n \"The term feature function is often used in the literature to denote the function $\\\\phi : \\\\mathcal {X} \\\\times E \\\\rightarrow \\\\mathbb {R}^m$ for some $m$ , when the weight function $w(x, e)$ is of the form $\\\\theta ^\\\\top \\\\phi (x, e)$ for some parameter vector $\\\\theta \\\\in \\\\mathbb {R}^m$ .\",\n \"In general, the weight function need not be a dot product, but can be any (sub-)differentiable real-valued function.\",\n \"A weight function is typically a composition of two steps: first the input vectors are converted into an intermediate representation, and second the intermediate representation is converted into segment weights.\",\n \"The function in the first step that converts input vectors to an intermediate representation are called an encoder.\",\n \"Specifically, an encoder is a function that takes in $T$ frames $x_1, \\\\dots , x_T$ and outputs $\\\\tilde{T}$ feature vectors $h_1, \\\\dots , h_{\\\\tilde{T}}$ .\",\n \"The number of output vectors $\\\\tilde{T}$ can be different from the number of input vectors $T$ depending on the encoder architecture.\",\n \"We defer the actual implementation of encoders in the experimental sections.\",\n \"Here we define weight functions based on $h_1, \\\\dots , h_{\\\\tilde{T}}$ .\",\n \"We use $\\\\Theta _\\\\text{enc}$ to denote the parameters for the encoder, and let $\\\\Theta _\\\\text{dec}$ be the remaining parameters in the weight function.\",\n \"Note that $\\\\Theta = \\\\Theta _\\\\text{enc} \\\\cup \\\\Theta _\\\\text{dec}$ .\",\n \"The following weight function, termed FC weight, is based on a frame classifier and is similar to weight functions used in a variety of prior work [49], [130], [1].\",\n \"The frame classifier takes in the output $h_1, \\\\dots , h_{\\\\tilde{T}}$ from the encoder, and produces a sequence of log probability vectors over the labels $z_i = \\\\text{logsoftmax}(W h_i + b)$ where $z_i \\\\in \\\\mathbb {R}^{|L|}$ and $W$ and $b$ are the parameters, for $i \\\\in \\\\lbrace 1, \\\\dots \\\\tilde{T}\\\\rbrace $ .\",\n \"Based on these posterior vectors, we define several functions that summarize the posteriors over a segment: The average of transformed log probabilities $w_{\\\\text{avg}}((\\\\ell , s, t)) = \\\\frac{1}{t - s + 1} \\\\sum _{i=s}^{t} (u_{i})_{\\\\ell },$ where $u_i = W_\\\\text{avg} z_i$ for $i \\\\in \\\\lbrace 1, \\\\dots , \\\\tilde{T}\\\\rbrace $ .\",\n \"A sample of transformed log probabilities $w_{\\\\text{spl-}j}((\\\\ell , s, t)) = (u_{j})_{\\\\ell }$ at time $j \\\\in \\\\lbrace (t - s)/6, (t - s)/2, 5(t - s)/6\\\\rbrace $ , where $u_i = W_\\\\text{spl} z_i$ for $i \\\\in \\\\lbrace 1, \\\\dots , \\\\tilde{T}\\\\rbrace $ .\",\n \"The average of transformed log probabilities around the left boundary (start) of the segment $w_{\\\\text{left}-k}((\\\\ell , s, t)) = (u_{i-k})_{\\\\ell }$ and around the right boundary (end) $w_{\\\\text{right}-k}((\\\\ell , s, t)) = (u^{\\\\prime }_{i+k})_{\\\\ell }$ where $u_i = W_\\\\text{left} z_i$ and $u^{\\\\prime }_i = W_\\\\text{right} z_i$ for $i \\\\in \\\\lbrace 1, \\\\dots , \\\\tilde{T}\\\\rbrace $ , and $k = 1, 2, 3$ .\",\n \"The label-dependent duration weight $w_{\\\\text{dur}}((\\\\ell , s, t)) = d_{\\\\ell , t - s}.$ A label-dependent bias $w_{\\\\text{bias}}((\\\\ell , s, t)) = b^{\\\\prime }_{\\\\ell }.$ The final FC weight function is the sum of all the above weight functions.\",\n \"When the FC weight function is used, $\\\\Theta _\\\\text{dec}$ is $\\\\lbrace W_\\\\text{avg}, W_\\\\text{spl},W_\\\\text{left}, W_\\\\text{right}, d, b^{\\\\prime }\\\\rbrace $ .\",\n \"Note that $\\\\lbrace W, b\\\\rbrace $ are considered parameters of the encoders.\",\n \"The MLP weight function based on a multi-layer perceptron (MLP) is inspired by [130], [74], [85].\",\n \"Two hidden layers $z^{(1)}_{\\\\ell , s, t} & = \\\\text{ReLU}(W_1 [ h_s; h_t; c_\\\\ell ; d_{\\\\lfloor \\\\log _{1.6}(t-s)\\\\rfloor }] + b_1) \\\\\\\\z^{(2)}_{\\\\ell , s, t} & = \\\\tanh (W_2 z^{(1)}_{\\\\ell , s, t} + b_2)$ are computed directly from the outputs $h_1, \\\\dots , h_{\\\\tilde{T}}$ of the encoder before computing the final weight, where $c_\\\\ell $ is a label embedding vector for the label $\\\\ell $ , $d_k$ is a duration embedding vector for the duration $k$ in log scaleWe use 5 different duration embedding vectors for our experiments.\",\n \"The base 1.6 is chosen such that $k \\\\in \\\\lbrace 0, \\\\dots , 4\\\\rbrace $ ., with $k = \\\\lfloor \\\\log _{1.6}(t-s+1)\\\\rfloor $ , and $\\\\text{ReLU}(x) = \\\\max (x, 0)$ .\",\n \"The final weight for the segment is defined as $w((\\\\ell , s, t)) & = \\\\theta ^\\\\top z^{(2)}_{\\\\ell , s, t}.$ When the MLP weight function is used, $\\\\Theta _\\\\text{dec}$ is $\\\\lbrace W_1, b_1, W_2, b_2, \\\\theta \\\\rbrace $ .\",\n \"Although the MLP weight function is conceptually simple, it is more expensive to compute than the FC weight function.\",\n \"In [74], [85], an LSTM is created for each segment consuming the outputs of the encoder, followed by an MLP taking the output vectors of the per segment LSTM at the segment boundary.\",\n \"In order to reduce the computation, we discard the per segment LSTM, and use the vectors produced by the encoder at the segment boundary as input to the MLP.\",\n \"Recall that a path $p = ((\\\\ell _1, s_1, t_1), \\\\dots , (\\\\ell _K, s_K, t_K))$ consists of a label sequence $y = (\\\\ell _1, \\\\dots , \\\\ell _K)$ and a segmentation $z = ((s_1, t_1), \\\\dots , (s_K, t_K))$ .\",\n \"In the following, we will use $(y, z)$ and $p$ interchangeably.\",\n \"We will also denote the space of all segmentations $\\\\mathcal {Z}$ .\",\n \"Training a segmental model aims to find a set of parameters $\\\\Theta $ that minimizes the expected task loss, for example, the expected edit distance $\\\\mathbb {E}_{(x, y) \\\\sim \\\\mathcal {D}}[\\\\mathrm {edit}(h_\\\\Theta (x), y)]$ where $h$ is the inference algorithm (Algorithm REF ) parameterized with $\\\\Theta $ , $\\\\mathrm {edit}$ computes the edit distance between two sequences, and the expectation is taken over samples $(x, y) \\\\in \\\\mathcal {X} \\\\times \\\\mathcal {Y}$ drawn from a distribution $\\\\mathcal {D}$ .\",\n \"The expectation can be decomposed into two steps $\\\\mathbb {E}_{x \\\\sim \\\\mathcal {D}(x)} \\\\mathbb {E}_{y \\\\sim \\\\mathcal {D}(y|x)}[\\\\mathrm {edit}(h_\\\\Theta (x), y)],$ first sampling $x$ and then sampling $y$ .\",\n \"To obtain a good discriminator $h$ , it suffices to optimize the inner expectation $\\\\mathbb {E}_{y \\\\sim \\\\mathcal {D}(y|x)}[\\\\mathrm {edit}(h_\\\\Theta (x), y)],$ for any $x$ drawn from $\\\\mathcal {D}(x)$ .\",\n \"However, the edit distance, due to its discrete nature, is difficult to optimize, so instead we minimize the expected loss $ \\\\mathbb {E}_{y \\\\sim \\\\mathcal {D}(y|x)}[\\\\mathcal {L}(\\\\Theta ; x, y)],$ where $\\\\mathcal {L}$ is a surrogate loss function.\",\n \"If segmentations are considered in the loss function, then we can minimize $ \\\\mathbb {E}_{(y, z) \\\\sim \\\\mathcal {D}^{\\\\prime }(y, z | x)}[\\\\mathcal {L}(\\\\Theta ; x, y, z)],$ where $\\\\mathcal {D}^{\\\\prime }$ is a conditional distribution over $\\\\mathcal {X} \\\\times \\\\mathcal {Y} \\\\times \\\\mathcal {Z}$ .\",\n \"Since the distribution $\\\\mathcal {D}$ is unknown, we use a tranining set $S = \\\\lbrace (x_1, y_1), \\\\dots , (x_n, y_n)\\\\rbrace $ of size $n$ to approximate the expectation and instead minimize $\\\\frac{1}{n} \\\\sum _{i=1}^n \\\\mathcal {L}(\\\\Theta ; x_i, y_i).$ If segmentations are considered, we use a tranining set $S = \\\\lbrace (x_1, y_1, z_1), \\\\dots , (x_n, y_n, z_n)\\\\rbrace $ of size $n$ to approximate $\\\\mathcal {D}^{\\\\prime }$ and minimize $\\\\frac{1}{n} \\\\sum _{i=1}^n \\\\mathcal {L}(\\\\Theta ; x_i, y_i, z_i).$ The connection between the surrogate loss $\\\\mathcal {L}$ and the edit distance depends on the choice of loss.\",\n \"We will optimize the loss functions with first-order methods, such as stochastic gradient descent.\",\n \"When listing the function definitions along with reasons for using them, we will also list the (sub-)gradients with respect to the weight $w(e)$ for some edge $e$ .\",\n \"We assume the weight function $w$ is (sub-)differentiable and the (sub-)gradients with respect to the parameters can be obtained with back-propagation.\",\n \"Given an utterance $x$ and a ground truth path $p = (y, z)$ , the hinge loss is defined as $\\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p)= \\\\max _{p^{\\\\prime } \\\\in \\\\mathcal {P}} \\\\left[ \\\\mathrm {cost}(p^{\\\\prime }, p) - w(p) + w(p^{\\\\prime }) \\\\right]$ where $\\\\mathrm {cost}$ is a user-defined cost function.\",\n \"The connection between the hinge loss and the task loss is through the cost function.\",\n \"Suppose $\\\\hat{p} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{p \\\\in \\\\mathcal {P}} w(p)$ is the maximum-weight path found by Algorithm REF .\",\n \"The cost of the inferred path $\\\\hat{p}$ against the ground truth $p$ can be upper-bounded by the hinge loss: $\\\\mathrm {cost}(\\\\hat{p}, p) \\\\le \\\\mathrm {cost}(\\\\hat{p}, p) - w(p) + w(\\\\hat{p})\\\\le \\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p).$ When the cost function is the edit distance, minimizing the hinge loss is minimizing an upper bound on the edit distance of the predicted sequence.\",\n \"The hinge loss itself is difficult to optimize when the cost function is the edit distance.\",\n \"In practice, the cost function is assumed to be decomposable $\\\\mathrm {cost}(p^{\\\\prime }, p) = \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} \\\\mathrm {cost}(e^{\\\\prime }, p)$ to allow efficient dynamic programming.\",\n \"When the cost is decomposable, the hinge loss can be written as $\\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p)& = \\\\max _{p^{\\\\prime } \\\\in \\\\mathcal {P}} \\\\left[ \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} \\\\mathrm {cost}(e^{\\\\prime }, p)- \\\\sum _{e \\\\in p} w(e) + \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} w(e^{\\\\prime }) \\\\right] \\\\\\\\& = \\\\max _{p^{\\\\prime } \\\\in \\\\mathcal {P}} \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} \\\\left[ \\\\mathrm {cost}(e^{\\\\prime }, p)+ w(e^{\\\\prime }) \\\\right] - \\\\sum _{e \\\\in p} w(e),$ and the $\\\\max $ operator in the first term can be solved with Algorithm REF by adding the costs to the weights for all segments.\",\n \"A subgradient of the hinge loss with respect to $w(e)$ is $\\\\frac{\\\\partial \\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p)}{\\\\partial w(e)}= -{1}_{e \\\\in p} + {1}_{e \\\\in \\\\tilde{p}}$ where $\\\\tilde{p} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{p^{\\\\prime } \\\\in \\\\mathcal {P}} [\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime })],$ which is the path that maximizes the first term in the hinge loss, and can be obtained with Algorithm REF with the cost added.\",\n \"Linear models are called support vector machines (SVM) when trained with the hinge loss, and are referred to as structured SVMs when used for solving structured prediction problems, e.g., sequence prediction in our case.\",\n \"A hinge loss with cost zero is also called perceptron loss [77].\",\n \"Segmental models trained with the hinge loss have been studied in [146], [129], [130].\",\n \"Segmental models can be treated as probabilistic models by defining probability distributions on the set of all paths $\\\\mathcal {P}$ .\",\n \"Specifically, the probability of a path $p = (y, z)$ is defined as $P(y, z | x) = P(p | x) = \\\\frac{1}{Z(x)} \\\\exp (w(x, p))$ where $Z(x) = \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}} \\\\exp (w(x, p^{\\\\prime }))$ is the partition function.\",\n \"Given an input $x$ and a ground truth path $p$ , the log loss is defined as $\\\\mathcal {L}_{\\\\text{log}}(\\\\Theta ; x, p) = -\\\\log P(p | x).$ Minimizing the log loss is equivalent to maximizing the conditional likelihood.\",\n \"In addition, the conditional likelihood can be written as $P(y, z | x) & = \\\\mathbb {E}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\sim P(y^{\\\\prime }, z^{\\\\prime } | x)}[{1}_{(y^{\\\\prime }, z^{\\\\prime }) = (y, z)}] \\\\\\\\& = 1 - \\\\mathbb {E}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\sim P(y^{\\\\prime }, z^{\\\\prime } | x)}[{1}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\ne (y, z)}].$ Therefore, maximizing the conditional likelihood is equivalent to minimizing the expected zero-one loss $\\\\mathbb {E}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\sim P(y^{\\\\prime }, z^{\\\\prime } | x)}[{1}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\ne (y, z)}],$ where $P(y, z | x)$ is used to approximate $\\\\mathcal {D}^{\\\\prime }(y, z | x)$ and the zero-one loss ${1}_{(y^{\\\\prime }, z^{\\\\prime }) \\\\ne (y, z)}$ is used.\",\n \"The use of the log loss is justified because the expectation above can be seen as an approximation of (REF ).\",\n \"Segmental models trained with log loss have been referred to as semi-Markov CRFs [118].\",\n \"Minimizing log loss is equivalent to maximizing mutual information, and is commonly referred to as the MMI criterion [10].\",\n \"Since the weight for the ground truth path $p$ can be efficiently computed, we are left with the problem of computing the partition function $Z(x)$ .\",\n \"The partition function can also be computed efficiently with the following dynamic programming algorithm.\",\n \"Let $\\\\mathcal {P}(u, v)$ be the set of paths that start at vertex $u$ and end at vertex $v$ .\",\n \"For any vertex $v$ , define the forward marginal as $\\\\alpha (v) = \\\\log \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v_0, v)} \\\\exp (w(p^{\\\\prime })).$ By expanding the edges ending at $v$ , we have $\\\\alpha (v) &= \\\\log \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v_0, v)} \\\\exp \\\\left(\\\\sum _{e \\\\in p^{\\\\prime }} w(e)\\\\right) \\\\\\\\&= \\\\log \\\\sum _{e \\\\in \\\\text{in}(v)} \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v_0, \\\\mathrm {tail}(e))}\\\\exp \\\\left(w(e) + \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} w(e^{\\\\prime })\\\\right) \\\\\\\\&= \\\\log \\\\sum _{e \\\\in \\\\text{in}(v)} \\\\exp \\\\left(w(e) + \\\\alpha (\\\\mathrm {tail}(e))\\\\right)$ Similarly, the backward marginal at $v$ is defined as $\\\\beta (v) = \\\\log \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v, v_T)} \\\\exp (w(p^{\\\\prime })),$ and has similar recursive structure.\",\n \"The complete algorithm is shown in Algorithm REF .\",\n \"Once all entries in $\\\\alpha $ and $\\\\beta $ are computed, the log partition function is $\\\\log Z(x) = \\\\alpha (v_T) = \\\\beta (v_0).$ Note that we store all of the entries in log space to maintain numerical stability.\",\n \"The gradient of the log loss with respect to $w(e)$ is $\\\\frac{\\\\partial \\\\mathcal {L}_{\\\\text{log}}(\\\\Theta ; x, p)}{\\\\partial w(e)}& = -{1}_{e \\\\in p} + \\\\frac{1}{Z(x)}\\\\sum _{p^{\\\\prime } \\\\ni e} \\\\exp (w(p^{\\\\prime })) \\\\\\\\& = -{1}_{e \\\\in p} + \\\\exp \\\\Big [ \\\\alpha (\\\\mathrm {tail}(e)) + w(e)+ \\\\beta (\\\\mathrm {head}(e)) - \\\\log Z(x) \\\\Big ],$ which can also be efficiently computed once the marginals are computed.\",\n \"Note that the form of Algorithm REF is very similar to that of Algorithm REF .\",\n \"Recall that to get the standard shortest-path algorithm (Algorithm REF ), we simply use the tropical semiring when running Algorithm REF .\",\n \"If the tropical semiring is replaced by the log semiring, we arrive at Algorithm REF , demonstrating the generality of FST algorithms and semirings.\",\n \"Computing forward and backward marginals $\\\\alpha (v_0) = 0$ $\\\\beta (v_T) = 0$ $\\\\text{logadd}(a, b) = \\\\log (\\\\exp (a) + \\\\exp (b))$ $v = v_0, v_1, \\\\dots , v_T$ $\\\\alpha (v) = \\\\text{logadd}_{e \\\\in \\\\text{in}(v)} \\\\Big [ \\\\alpha (\\\\mathrm {tail}(e)) + w(e) \\\\Big ]$ $v = v_T, v_{T-1}, \\\\dots , v_0$ $\\\\beta (v) = \\\\text{logadd}_{e \\\\in \\\\text{out}(v)} \\\\Big [ \\\\beta (\\\\mathrm {head}(e)) + w(e) \\\\Big ]$ Log loss can be modified to include a cost function.\",\n \"Speficically, we define $P^+_{p}(p^{\\\\prime } | x) = \\\\frac{1}{Z(x, p)} \\\\exp (w(x, p^{\\\\prime }) + \\\\mathrm {cost}(p^{\\\\prime }, p))$ where $p$ is the ground-truth path and $Z(x, p) = \\\\sum _{p^{\\\\prime \\\\prime } \\\\in \\\\mathcal {P}} \\\\exp (w(x, p^{\\\\prime \\\\prime }) + \\\\mathrm {cost}(p^{\\\\prime \\\\prime }, p)).$ Unlike $P$ , the distribution $P^+$ considers both the weights and the costs.\",\n \"The modified log loss, also known as the boosted MMI criterion [109], is defined as $\\\\mathcal {L}_{\\\\text{bMMI}}(\\\\Theta ; x, p) = -\\\\log P^+_{p}(p | x).$ Given an input $x$ and a label sequence $y$ , the marginal log loss is defined as $\\\\mathcal {L}_{\\\\text{mll}}(\\\\Theta ; x, y) = -\\\\log P(y | x) = -\\\\log \\\\sum _{z \\\\in \\\\mathcal {Z}} P(y, z | x)$ where the segmentation is marginalized compared to log loss; hence the name.\",\n \"Following the same argument as for the log loss, the marginal distribution can be written as $P(y | x) = 1 - \\\\mathbb {E}_{y^{\\\\prime } \\\\sim P(y^{\\\\prime } | x)}[{1}_{y \\\\ne y^{\\\\prime }}],$ and maximizing the marginal distribution is equivalent to minimizing the expected zero-one loss $\\\\mathbb {E}_{y^{\\\\prime } \\\\sim P(y^{\\\\prime } | x)}[{1}_{y \\\\ne y^{\\\\prime }}],$ where $P(y | x)$ is used to approximate $\\\\mathcal {D}(y | x)$ .\",\n \"Note that the zero-one loss ${1}_{y \\\\ne y^{\\\\prime }}$ only depends on the label sequence.\",\n \"While the log loss has a connection to (REF ), the marginal log loss justifies its use by directly approximating (REF ) with the above expected zero-one loss.\",\n \"Note that both the hinge loss and the log loss depend on the ground-truth path, or more specifically, the ground-truth segmentation.\",\n \"The marginal log loss marginalizes over the segmentations, so it only depends on the ground-truth label sequence and not the segmentation.\",\n \"The lack of reliance on the ground-truth segmentation makes the marginal log loss attractive for tasks such as speech recognition, because collecting ground-truth segmentations for phonemes or words is time-consuming and/or expensive.\",\n \"In addition, the boundaries of phonemes and words tend to be ambiguous, so it can be preferable to leave the decision to the model.\",\n \"Segmental models trained with the marginal log loss have been referred to as segmental CRFs [149].\",\n \"To compute the marginal log loss, we can rewrite it as $\\\\mathcal {L}_{\\\\text{mll}}(\\\\Theta ; x, y) & = -\\\\log \\\\sum _{z \\\\in \\\\mathcal {Z}} P(y, z | x) \\\\\\\\& = -\\\\log \\\\sum _{z \\\\in \\\\mathcal {Z}} \\\\exp (w(x, (y, z))) + \\\\log Z(x) \\\\\\\\& = -\\\\underbrace{\\\\log \\\\sum _{p^{\\\\prime }: \\\\Gamma (p^{\\\\prime }) = y} \\\\exp (w(x, p^{\\\\prime }))}_{\\\\log Z(x, y)} + \\\\log Z(x)$ where $\\\\Gamma $ extracts the label sequence from a path, i.e., for $p^{\\\\prime } = (y^{\\\\prime }, z^{\\\\prime })$ , $\\\\Gamma (p^{\\\\prime }) = y^{\\\\prime }$ .\",\n \"Since the partition function can be efficiently computed from Algorithm REF , we only need to compute $\\\\log Z(x, y)$ .\",\n \"Since the term $\\\\log Z(x, y)$ is identical to $\\\\log Z(x)$ except that it involves a constrained search space considering all paths with the same label sequence $y$ , the strategy is to construct the constrained search space with an FST and run Algorithm REF on the FST.\",\n \"Let $F$ be a chain FST that represents $y$ , with edges $\\\\lbrace e_1, \\\\dots , e_{|y|}\\\\rbrace $ , where $i(e_k) = o(e_k) = y_k$ for all $k \\\\in \\\\lbrace 1, \\\\dots , |y|\\\\rbrace $ .\",\n \"Let $G$ be the search space consisting of all paths in $\\\\mathcal {P}$ .\",\n \"The term $\\\\log Z(x, y)$ can be efficiently computed by running Algorithm REF on the $\\\\sigma $ -composition of $G$ and $F$ , i.e., $G \\\\circ _\\\\sigma F$ .\",\n \"Note that after $\\\\sigma $ -composing $G$ and $F$ , we only allow paths in $G$ that can produce output sequences that $F$ accepts.\",\n \"Since $F$ only accepts $y$ , the paths in $G \\\\circ _\\\\sigma F$ are all the paths in $G$ that produce $y$ as desired.\",\n \"Let the forward and backward marginals computed on $G \\\\circ _\\\\sigma F$ be $\\\\alpha ^{\\\\prime }$ and $\\\\beta ^{\\\\prime }$ .\",\n \"We have $\\\\log Z(x, y) = \\\\alpha ^{\\\\prime }(v_T) = \\\\beta ^{\\\\prime }(v_0)$ .\",\n \"The gradient of the marginal log loss with respect to $w(e)$ is $\\\\frac{\\\\partial \\\\mathcal {L}_{\\\\text{mll}}(\\\\Theta ; x, y)}{\\\\partial w(e)}& = -\\\\frac{1}{Z(x, y)} \\\\sum _{\\\\begin{array}{c}p^{\\\\prime } \\\\ni e\\\\\\\\ \\\\Gamma (p^{\\\\prime }) = y\\\\end{array}} \\\\exp (w(p^{\\\\prime }))+ \\\\frac{1}{Z(x)}\\\\sum _{p^{\\\\prime } \\\\ni e} \\\\exp (w(p^{\\\\prime })) \\\\\\\\& = - \\\\exp \\\\Big [ \\\\alpha ^{\\\\prime }(\\\\mathrm {tail}(e)) + w(e) + \\\\beta ^{\\\\prime }(\\\\mathrm {head}(e)) - \\\\log Z(x, y) \\\\Big ] \\\\\\\\& \\\\qquad {} + \\\\exp \\\\Big [ \\\\alpha (\\\\mathrm {tail}(e)) + w(e) + \\\\beta (\\\\mathrm {head}(e)) - \\\\log Z(x) \\\\Big ].$ and can be efficiently computed once all of the marginals are computed.\",\n \"The empirical Bayes risk (EBR) [125] is defined as $\\\\mathcal {L}_{\\\\text{ebr}}(\\\\Theta ; x, p) = \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathrm {cost}(p^{\\\\prime }, p)].$ The intuition behind EBR is that if $P$ is a good approximation for $\\\\mathcal {D}^{\\\\prime }$ , then EBR is a good approximation for $\\\\mathbb {E}_{p^{\\\\prime } \\\\sim \\\\mathcal {D}^{\\\\prime }(p^{\\\\prime } | x)}[\\\\mathrm {cost}(p^{\\\\prime }, p)].$ The goal is to find $P$ that achieves a low expected cost.\",\n \"Empirical Bayes risk is also referred to as minimum phone error (MPE) when the cost is based on phone errors, and is referred to as minimum word error (MWE) when the cost is based on word errors respectively [108].\",\n \"A boosted version of EBR can be obtained by replacing $P$ with $P^+$ [89].\",\n \"Since $Z(x)$ can be computed through the forward and backward marginals $\\\\alpha $ and $\\\\beta $ , we are left with $\\\\sum _{p^{\\\\prime }} \\\\exp (w(p^{\\\\prime }))\\\\mathrm {cost}(p^{\\\\prime }, p)$ .\",\n \"Similar to the forward and backward marginals, we have $\\\\alpha ^{\\\\prime \\\\prime }(v) & = \\\\log \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v_0, v)} \\\\exp (w(p^{\\\\prime }) + \\\\log \\\\mathrm {cost}(p^{\\\\prime }, p)) \\\\\\\\& = \\\\log \\\\sum _{e^{\\\\prime } \\\\in \\\\text{in}(v)} \\\\exp (w(e^{\\\\prime }) + \\\\log \\\\mathrm {cost}(e^{\\\\prime }, p))\\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v_0, \\\\mathrm {tail}(e))} \\\\exp (w(p^{\\\\prime }) + \\\\log \\\\mathrm {cost}(p^{\\\\prime }, p)) \\\\\\\\& = \\\\log \\\\sum _{e^{\\\\prime } \\\\in \\\\text{in}(v)} \\\\exp (w(e^{\\\\prime }) + \\\\log \\\\mathrm {cost}(e^{\\\\prime }, p) + \\\\alpha ^{\\\\prime \\\\prime }(\\\\mathrm {tail}(e))) \\\\\\\\\\\\beta ^{\\\\prime \\\\prime }(v) & = \\\\log \\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(v, v_T)} \\\\exp (w(p^{\\\\prime }) + \\\\log \\\\mathrm {cost}(p^{\\\\prime }, p)) \\\\\\\\& = \\\\log \\\\sum _{e^{\\\\prime } \\\\in \\\\text{out}(v)} \\\\exp (w(e^{\\\\prime }) + \\\\log \\\\mathrm {cost}(e^{\\\\prime }, p))\\\\sum _{p^{\\\\prime } \\\\in \\\\mathcal {P}(\\\\mathrm {head}(e), v_T)} \\\\exp (w(p^{\\\\prime }) + \\\\log \\\\mathrm {cost}(p^{\\\\prime }, p)) \\\\\\\\& = \\\\log \\\\sum _{e^{\\\\prime } \\\\in \\\\text{out}(v)} \\\\exp (w(e^{\\\\prime }) + \\\\log \\\\mathrm {cost}(e^{\\\\prime }, p) + \\\\beta ^{\\\\prime \\\\prime }(\\\\mathrm {head}(e)))$ where $\\\\mathrm {cost}$ is assumed to be non-negative, and the marginals are stored in log space to maintain numerical stability.\",\n \"Given the cost-augmented marginals, we have $\\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathrm {cost}(p^{\\\\prime }, p)] = \\\\exp (\\\\alpha ^{\\\\prime \\\\prime }(v_T) - \\\\log Z(x))= \\\\exp (\\\\beta ^{\\\\prime \\\\prime }(v_0) - \\\\log Z(x)).$ The gradient of EBR with respect to $w(e)$ of some edge $e$ is $\\\\frac{\\\\mathcal {L}_{\\\\text{ebr}}(\\\\Theta ; x, p)}{w(e)}& = \\\\frac{\\\\sum _{p^{\\\\prime } \\\\ni e} \\\\exp (w(p^{\\\\prime }))\\\\mathrm {cost}(p^{\\\\prime }, p)}{Z(x)}- \\\\frac{\\\\left[\\\\sum _{p^{\\\\prime }} \\\\exp (w(p^{\\\\prime }))\\\\mathrm {cost}(p^{\\\\prime }, p)\\\\right]\\\\left[\\\\sum _{p^{\\\\prime } \\\\ni e}\\\\exp (w(p^{\\\\prime }))\\\\right]}{(Z(x))^2} \\\\\\\\& = \\\\exp (\\\\alpha ^{\\\\prime \\\\prime }(\\\\mathrm {tail}(e)) + w(e) + \\\\log \\\\mathrm {cost}(e, p) + \\\\beta ^{\\\\prime \\\\prime }(\\\\mathrm {head}(e)) - \\\\log Z(x)) \\\\\\\\& \\\\quad {} - \\\\mathcal {L}_{\\\\text{ebr}}(\\\\Theta ; x, p) [\\\\exp (\\\\alpha (\\\\mathrm {tail}(e)) + w(e) + \\\\beta (\\\\mathrm {head}(e)) - \\\\log Z(x))]$ It is also interesting to note that, in general, $\\\\frac{\\\\partial }{\\\\partial w(e)} \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[f(p^{\\\\prime })]&= \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathbb {1}_{p^{\\\\prime } \\\\ni e} f(p^{\\\\prime })]- \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathbb {1}_{p^{\\\\prime } \\\\ni e}]\\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[f(p^{\\\\prime })] \\\\\\\\&= \\\\text{Cov}(\\\\mathbb {1}_{p^{\\\\prime } \\\\ni e}, f(p^{\\\\prime })),$ for any function $f: \\\\mathcal {P} \\\\rightarrow \\\\mathbb {R}$ .\",\n \"The ramp loss [24] is defined as $\\\\mathcal {L}_{\\\\text{ramp}}(\\\\Theta ; x, p) = \\\\max _{p^{\\\\prime }}[\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime }) - \\\\max _{p^{\\\\prime \\\\prime }}w(p^{\\\\prime \\\\prime })].$ It is easy to see that $\\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p)& = \\\\max _{p^{\\\\prime }}[\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime }) - w(p)] \\\\\\\\& \\\\ge \\\\max _{p^{\\\\prime }}[\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime }) - \\\\max _{p^{\\\\prime \\\\prime }}w(p^{\\\\prime \\\\prime })]= \\\\mathcal {L}_{\\\\text{ramp}}(\\\\Theta ; x, p).$ In addition, for $\\\\hat{p} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{p^{\\\\prime }} w(p^{\\\\prime })$ , $\\\\mathrm {cost}(\\\\hat{p}, p) &= \\\\mathrm {cost}(\\\\hat{p}, p) + w(\\\\hat{p}) - w(\\\\hat{p}) \\\\\\\\& \\\\le \\\\max _{p^{\\\\prime }}[\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime })] - w(\\\\hat{p}) \\\\\\\\& = \\\\max _{p^{\\\\prime }}[\\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime }) ] - \\\\max _{p^{\\\\prime }} w(p^{\\\\prime })= \\\\mathcal {L}_{\\\\text{ramp}}(\\\\Theta ; x, p).$ Therefore, we have $\\\\mathcal {L}_{\\\\text{hinge}}(\\\\Theta ; x, p) \\\\ge \\\\mathcal {L}_{\\\\text{ramp}}(\\\\Theta ; x, p)\\\\ge \\\\mathrm {cost}(\\\\hat{p}, p).$ In other words, ramp loss is also an upper bound on the cost of the predicted sequence, but is tighter than hinge loss.\",\n \"However, optimizing the ramp loss is more complicated.\",\n \"Ramp loss can be minimized with the concave-convex procedure (CCCP) [145].\",\n \"See [33] for a detailed implementation of CCCP for solving ramp loss.\",\n \"Log loss augmented with a cost function, or boosted MMI, can be seen as a soft version of hinge loss.\",\n \"By the definition of boosted MMI, where $\\\\mathcal {L}_{\\\\text{bMMI}}(\\\\Theta ; x, p) = -\\\\log P(p | x)= -w(p) + \\\\log \\\\sum _{p^{\\\\prime }} \\\\exp (w(p^{\\\\prime }) + \\\\mathrm {cost}(p^{\\\\prime }, p)),$ we can see that the second term acts as a soft-max instead of a max function.\",\n \"To see this, note that $\\\\log (e^{x_1} + e^{x_2} + \\\\dots + e^{x_n})& = \\\\log e^{x_{\\\\text{max}}} (e^{x_1 - x_{\\\\text{max}}} + \\\\dots + e^{x_n - x_{\\\\text{max}}}) \\\\\\\\& = x_{\\\\text{max}} + \\\\log (e^{x_1 - x_{\\\\text{max}}} + \\\\dots + e^{x_n - x_{\\\\text{max}}}),$ where $x_{\\\\text{max}} = \\\\max _i x_i$ .\",\n \"The second term in (REF ) is small when the difference between $x_{\\\\text{max}}$ and other $x_i$ 's is large.\",\n \"In fact, we can introduce an addition factor $\\\\epsilon $ to control this effect.\",\n \"Specifically, $\\\\epsilon \\\\log (e^{x_1/\\\\epsilon } + e^{x_2/\\\\epsilon } + \\\\dots + e^{x_n\\\\epsilon })& = x_{\\\\text{max}} + \\\\epsilon \\\\log (e^{(x_1 - x_{\\\\text{max})/\\\\epsilon }} + \\\\dots + e^{(x_n - x_{\\\\text{max}})/\\\\epsilon }).$ When $\\\\epsilon \\\\rightarrow 0$ , the second term vanishes and (REF ) acts exactly like a max function.\",\n \"We can add the additional $\\\\epsilon $ factor to the cost-augmented log loss and get $\\\\mathcal {L}^{\\\\epsilon }_{\\\\text{bMMI}}(\\\\Theta ; x, p) = -\\\\log P(p | x)= -w(p) + \\\\epsilon \\\\log \\\\sum _{p^{\\\\prime }} \\\\exp \\\\left(\\\\frac{1}{\\\\epsilon }(w(p^{\\\\prime }) + \\\\mathrm {cost}(p^{\\\\prime }, p))\\\\right).$ As $\\\\epsilon \\\\rightarrow 0$ , $\\\\mathcal {L}^{\\\\epsilon }_{\\\\text{bMMI}} \\\\rightarrow \\\\mathcal {L}_{\\\\text{hinge}}$ .\",\n \"This connection was first presented in [52].\",\n \"[32] independently proposed softmax-margin, the equivalence of cost-augmented log loss.\",\n \"As we have noted when we introduced log loss, $P(p | x) = \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathbb {1}_{p^{\\\\prime } = p}]= 1 - \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P(p^{\\\\prime }|x)}[\\\\mathbb {1}_{p^{\\\\prime } \\\\ne p}].$ Therefore, minimizing log loss is equivalent to minimizing EBR with $\\\\mathrm {cost}(p^{\\\\prime }, p) = \\\\mathbb {1}_{p^{\\\\prime } \\\\ne p}$ .\",\n \"Another connection between cost-augmented log loss and EBR is the following.\",\n \"Let $\\\\mathcal {L}_{\\\\lambda , \\\\text{bMMI}}(\\\\Theta ; x, p) = -\\\\log P(p | x)= -w(p) + \\\\log \\\\sum _{p^{\\\\prime }} \\\\exp (w(p^{\\\\prime }) + \\\\lambda \\\\mathrm {cost}(p^{\\\\prime }, p)),$ where we introduce a scaling factor $\\\\lambda $ for the cost function in cost-augmented log loss.\",\n \"If we take the derivative of cost-augmented log loss with respect to $\\\\lambda $ , we have $\\\\frac{\\\\partial }{\\\\partial \\\\lambda } \\\\mathcal {L}_{\\\\lambda , \\\\text{bMMI}}&= \\\\frac{\\\\sum _{p^{\\\\prime }} [\\\\exp (w(p^{\\\\prime }) + \\\\lambda \\\\mathrm {cost}(p^{\\\\prime }, p)) \\\\mathrm {cost}(p^{\\\\prime }, p)]}{\\\\sum _{p^{\\\\prime }} \\\\exp (w(p^{\\\\prime }) + \\\\lambda \\\\mathrm {cost}(p^{\\\\prime }, p))} \\\\\\\\&= \\\\mathbb {E}_{p^{\\\\prime } \\\\sim P^+_\\\\lambda (p^{\\\\prime }|x)}[\\\\mathrm {cost}(p^{\\\\prime }, p)],$ where $P^+_{\\\\lambda }(p^{\\\\prime }|x) = \\\\frac{\\\\exp (w(p^{\\\\prime }) + \\\\lambda \\\\mathrm {cost}(p^{\\\\prime }, p))}{\\\\sum _{p^{\\\\prime \\\\prime }} \\\\exp (w(p^{\\\\prime \\\\prime }) + \\\\lambda \\\\mathrm {cost}(p^{\\\\prime \\\\prime }, p))}$ Therefore, the derivative of cost-augmented log loss with respect to $\\\\lambda $ is cost-augmented EBR, or sometimes referred to as boosted MPE.\",\n \"This connection was first presented in [90].\",\n \"In this sectionThese results were published in [129], we describe experiments comparing segmental models trained with various loss functions and two cost functions.\",\n \"Instead of using the entire search space, we follow [149], [150] and use a baseline recognizer to generate sparse search spaces, commonly known as lattices.\",\n \"Segmental models are then applied on these lattices.\",\n \"The weight function used in these experiments is a linear function $w(x, e) = \\\\theta ^\\\\top \\\\phi (x, e)$ where $\\\\theta $ is a parameter vector and $\\\\phi (x, e)$ is a feature vector for some edge $e$ .\",\n \"The feature function $\\\\phi $ is task-dependent, and is described in detail later.\",\n \"Hinge loss and ramp loss are sensitive to the scale of the cost, so the scale of the cost function is tuned.\",\n \"Specifically, we introduce the parameter $\\\\mu $ in the hinge loss $\\\\mathcal {L}_{\\\\text{hinge}} = \\\\max _{p^{\\\\prime }} [\\\\mu \\\\cdot \\\\mathrm {cost}(p^{\\\\prime }, p) - w(p) + w(p^{\\\\prime })],$ and parameters $\\\\mu _1$ and $\\\\mu _2$ for the ramp loss $\\\\mathcal {L}_{\\\\text{ramp}} = \\\\max _{p^{\\\\prime }} [\\\\mu _1 \\\\cdot \\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime })]- \\\\max _{p^{\\\\prime }}[\\\\mu _2 \\\\cdot \\\\mathrm {cost}(p^{\\\\prime }, p) + w(p^{\\\\prime })].$ Note the additional cost term in the above ramp loss.\",\n \"This version of ramp loss is a generalization of the standard ramp loss [20], [33].\",\n \"One cost function was proposed in [108] in the context of MPE/MWE training, and we refer to it as MPE cost.\",\n \"The cost of an edge is the duration of the non-overlapping part of a matching ground-truth edge that gives the lowest error, where the error is one if the label is correct and 0.5 otherwise.\",\n \"Formally, for any hypothesized edge $e^{\\\\prime }$ , we define $\\\\mathrm {cost}_{\\\\text{MPE}}(e^{\\\\prime }, p) =1 - \\\\max _{e \\\\in p} \\\\left[ \\\\mathbb {1}_{o(e) = o(e^{\\\\prime })}\\\\frac{|e \\\\cap e^{\\\\prime }|}{|e|}+ \\\\frac{1}{2} \\\\mathbb {1}_{o(e) \\\\ne o(e^{\\\\prime })}\\\\frac{|e \\\\cap e^{\\\\prime }|}{|e|} \\\\right],$ and $\\\\mathrm {cost}_{\\\\text{MPE}}(p^{\\\\prime }, p) = \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} \\\\mathrm {cost}_{\\\\text{MPE}}(e^{\\\\prime }, p),$ where $|e|$ denotes the length of segment $e$ .\",\n \"The MPE cost only penalizes false negatives and does not account for false positives.\",\n \"Therefore, we propose an alternative that we refer to as the overlap cost: $\\\\mathrm {cost}_{\\\\text{overlap}}(e^{\\\\prime }, p) = 1 - \\\\mathbb {1}_{o(e) = o(\\\\tilde{e})}\\\\frac{|e \\\\cap \\\\tilde{e}|}{|e \\\\cup \\\\tilde{e}|},$ where $\\\\tilde{e} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{e \\\\in p} |e^{\\\\prime } \\\\cap e|$ .\",\n \"This cost function finds the most overlapping edge in the ground truth and considers any part of the union of the two edges that is not overlapping to be in error.\",\n \"The cost for the whole path is again $\\\\mathrm {cost}_{\\\\text{overlap}}(p^{\\\\prime }, p) = \\\\sum _{e^{\\\\prime } \\\\in p^{\\\\prime }} \\\\mathrm {cost}_{\\\\text{overlap}}(e^{\\\\prime }, p).$ We study the various losses and cost functions on two tasks.\",\n \"One is a standard speech recognition task, namely TIMIT [31] phonetic recognition.\",\n \"The second is a sign language recognition task from video, in particular recognition of fingerspelled letter sequences in American Sign Language (ASL).\",\n \"Both are tasks on which there is prior work using semi-Markov CRFs [148], [71], and both are small enough (in terms of data set size and decoding search space) to run many empirical comparisons in a reasonable amount of time.\",\n \"For the ASL task, we use the data and experimental setup of [71]: We obtain baseline lattices using their tandem HMM-based system, and we use the same set of segmental feature functions.\",\n \"We use forced alignments of the ground-truth transcriptions for training.\",\n \"We train all models from all-zero weights and optimize with Rprop [114] for 20 epochs.\",\n \"We use $L_1$ and $L_2$ regularization, with parameters tuned over the grid $\\\\lbrace 0,10^{-6}, 10^{-5}, \\\\dots , 0.1, 1\\\\rbrace ^2$ .\",\n \"For hinge and ramp loss, we use the standard forms without tuning the cost weights (i.e., $\\\\mu =1$ , $\\\\mu _1 = 0$ , and $\\\\mu _2 = 1$ ).\",\n \"Table: Letter error rates (%) for a baseline tandem HMM and segmental models trained with various lossesand cost functions on the ASL data set.Table: Phone error rates (%) for a baseline HMM and segmental models trained with various lossesand cost functions on the TIMIT data set.For TIMIT, we use the standard 3696-utterance training set and 192-utterance core test set, plus a random 192 utterances from the full test set (excluding the core test set) as a development set.\",\n \"We collapse the 61 phones in the phone set to 48 for training, and further collapse them to 39 for evaluation [79].\",\n \"We use lattices generated by a baseline monophone HMM system with 39-dimensional MFCCs.\",\n \"The resulting lattices have an average density (average number of hypothesized edges per ground truth edge) of 60.1.\",\n \"The oracle phone error rate is 6.3% for the development set and 7.0% for the core test set.\",\n \"We use oracle paths (paths with minimum phone error) from the lattices as ground truth for training.\",\n \"We implement segmental models with various feature functions.\",\n \"The base features are the acoustic and language model score from the baseline recognizer, and a bias (a feature that is always one).\",\n \"We also include a set of features based on spectro-temporal receptive fields implemented as follows.\",\n \"We begin with 40-dimensional log mel filter bank features.\",\n \"For each segment, we divide it evenly into thirds in both time and frequency, resulting in nine patches for each segment.\",\n \"For each patch, we have a $3 \\\\times 13$ receptive field of all ones, and convolve it with the patch.\",\n \"The resulting $3 \\\\times 13 \\\\times 9$ numbers are lexicalized to form the final features for the segmental model.\",\n \"Specifically, for any feature vector $v$ , we refer to $v \\\\otimes \\\\mathbf {1}_\\\\ell $ as the lexicalized feature vector, where $\\\\mathbf {1}_\\\\ell $ is a one-hot vector for the label $\\\\ell $ .\",\n \"We optimize the loss functions using AdaGrad [27], using step size 0.1 for 10 epochs.\",\n \"$L_1$ and $L_2$ regularization parameters are tuned over the grid $\\\\lbrace 0, 0.001, 0.1, 1\\\\rbrace \\\\times \\\\lbrace 0, 0.1, 1, 100\\\\rbrace $ .\",\n \"The results for ASL recognition, averaged over four signers, are shown in Table REF .\",\n \"The evaluation metric is the letter error rate, which is the percentage of letters that are substituted, inserted, or deleted.\",\n \"The results for TIMIT are shown in Table REF .\",\n \"We observe three consistent conclusions: Segmental models perform significantly better than the baseline HMM.\",\n \"Across losses, overlap cost is better than MPE cost.\",\n \"Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.\",\n \"Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.\",\n \"We suspect a warm start might be able to remedy this.\",\n \"Segmental models perform significantly better than the baseline HMM.\",\n \"Across losses, overlap cost is better than MPE cost.\",\n \"Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.\",\n \"Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.\",\n \"We suspect a warm start might be able to remedy this.\",\n \"For the ASL task, we tuned on a development set the cost weights for ramp loss over the grid $\\\\lbrace -100,-10,-1,-0.1,-0.01\\\\rbrace \\\\times \\\\lbrace 0.01,0.1,1,10,100\\\\rbrace $ , using overlap cost.\",\n \"The test result of tuned ramp slightly improves over hinge loss, confirming that if ramp is tuned carefully, it is able to outperform hinge.\",\n \"However, even though tuned ramp loss achieves very good results, considering the time spent tuning $\\\\mu _1$ and $\\\\mu _2$ , we still favor hinge and log loss.\",\n \"We also conducted experiments to determine how the results are affected by different levels of noise in the feature functions, using simulated phone detector-based features.\",\n \"Similarly to [150], we define a detection event as a (time, phone label) pair, and a feature function that is an indicator of whether a phone detection event occurs in the time span of the edge.\",\n \"If we set a high weight for the phone event that occurs in an edge with the same phone label, then we can exactly recover the oracle path.\",\n \"This allows us to conduct a series of simulated experiments with different amounts of noise added to the oracle phone events, or gold events.\",\n \"For all experiments below, we use the same TIMIT setting except that we only use the acoustic and language model score with the simulated phone detector features, with no regularizer and one epoch of AdaGrad.\",\n \"The ramp cost weights are set to $\\\\mu _1=0, \\\\mu _2=1$ .\",\n \"The cost weight for hinge is tuned over $\\\\lbrace 0.01, 0.1, 1, 10, 100\\\\rbrace $ and results are only shown for the best-performing value.\",\n \"The first set of experiments randomly perturbs the correct phone label of each event to an incorrect label with the corruption probability shown on the $x$ -axis; the event times are not perturbed.\",\n \"The second set of experiments perturbs the time for each event but not the label.\",\n \"We add Gaussian noise with mean set to the time at which the event occurs and with several standard deviations shown on the $x$ -axis.\",\n \"For the third and fourth set of experiments, we randomly include an edge in the lattice as a false positive event, or randomly delete an event from the gold events.\",\n \"The conclusions are consistent with our previous observation, namely, that hinge is the consistent winner but only by a very small margin, that log loss is very competitive, that non-convex losses are hard to optimize, and that overlap cost is better than MPE cost.\",\n \"As a byproduct, we note that we could achieve 17.7% given a phone detector with any of the following characteristics: up to 50% phone error rate but perfect time information, up to 5-frame time perturbations (in standard deviation) but perfect labels, 1.8 false positives per gold edge, or 20% false negatives.\",\n \"Figure: Lattice rescoring with various noise added to a perfectphone detector.From Left to Right: Perturbing phone labels.\",\n \"Perturbing time.Adding false positive events.\",\n \"Adding false negative events.In this chapter, we have described a formal framework for discriminative segmental models.\",\n \"The definition is not tied to any weight function or any loss function, encompassing a large family of segmental models.\",\n \"We have described two weight functions, one based on frame classifiers, and one based on segmental recurrent neural networks.\",\n \"We have also described various loss functions, and have drawn connections between surrogate losses and the true loss that we want to minimize.\",\n \"We have shown how (sub-)gradients of loss functions can be efficiently computed with FST algorithms.\",\n \"The flexibility to use different loss functions allows us to train segmental models in different training settings.\",\n \"Preliminary results have shown that segmental models outperform HMMs by a large margin and that overlap cost is better than MPE cost across loss functions.\",\n \"In terms of loss functions, non-convex losses are more difficult to optimize.\",\n \"Hinge loss is the best performer while log loss is very competitive.\",\n \"Our final task loss is the edit distance, but none of the surrogate losses can be optimized when the cost function is the edit distance.\",\n \"There are two difficulties for optimizing the edit distance: the edit distance is discrete, and it involves a minimization over all alignments to the ground truth.\",\n \"Using a cost function that considers all alignments to the ground truth without the minimization has been explored in [51], [137], [121].\",\n \"Other approaches for directly minimizing task losses [69], [67] are also suitable for minimizing the edit distance.\",\n \"As we have seen in Chapter REF , search spaces defined by segmental models are dense and redundant, in the sense that many segments that have the same label differ only by a few frames (or even just a single frame) and that many segments with the same start time and end time differ only in the labels.\",\n \"In other words, most of the random segments do not look like the ground truths.\",\n \"Decoding in multiple passes is an approach that exploits dense search spaces.\",\n \"Since the search spaces are dense, a simple weight function is typically enough to significantly reduce the size of search spaces.\",\n \"After the first pass, we may use more complicated weight functions to make predictions or to further reduce the search spaces.\",\n \"Reducing the size of search spaces is commonly known as pruning.\",\n \"Pruning has to be done carefully, because it might hurt the final performance if the ground truth segments are pruned.\",\n \"To measure how pruning affects the search spaces and the quality of predictions, we measure the lowest achievable task loss, the oracle error rate, of the search spaces after pruning.\",\n \"A simple experiment on the phonetic recognition task shows that the oracle error rate, i.e., the best edit distance we can achieve, on average stays at zero even if we remove 50% of segments at random.\",\n \"This result shows that the search spaces are indeed dense and redundant.\",\n \"Since the pruned search space is smaller than the original space, inference and learning using the pruned search space are faster in the second pass.\",\n \"As a result, we can afford to use computationally expensive weight functions for better prediction in the second pass.\",\n \"In general, with the same weight function, multi-pass systems have the potential to decode faster than single-pass systems.\",\n \"In other words, under the same decoding time budget, multi-pass systems can use computationally more expensive weight functions than single-pass systems to obtain better performance.\",\n \"Since pruning plays an important role, in the following sections, we present and compare several pruning strategies and their consequences.\",\n \"We then develop a multi-pass system, trading between the size of the search space and the computational complexity of weight functions.\",\n \"In this section, we describe several pruning approaches to reduce search spaces.\",\n \"Note that pruning is a form of inference, and we assume a pruning approach has access to search spaces defined by an FST, and a weight function $w : E \\\\rightarrow \\\\mathbb {R}$ parameterized by $\\\\Theta _{\\\\text{prn}}$ .\",\n \"We discuss the choice of $w$ and how $\\\\Theta _{\\\\text{prn}}$ is obtained in the experimental sections.\",\n \"A very naive approach, referred to as greedy pruning, is to prune the edges branching out from a vertex based on their edge weights.\",\n \"Specifically, the set of edges branching out from a vertex $v$ is pruned based on a threshold $\\\\tau $ .\",\n \"Let $S_v = \\\\lbrace e \\\\in \\\\text{out}(v): w(e) \\\\ge \\\\tau \\\\rbrace $ be the set of vertices that survive pruning.\",\n \"We collect the edges $\\\\bigcup _{v \\\\in V} S_v$ to form the pruned graph.\",\n \"There are two nice properties about this approach.\",\n \"First, the pruning procedure is embarrassingly parallelizable.\",\n \"Second, if we let $\\\\tau = \\\\lambda \\\\min _{e \\\\in \\\\text{out}(v)} w(e)+ (1 - \\\\lambda ) \\\\max _{e \\\\in \\\\text{out}(v)} w(e)$ where $0 \\\\le \\\\lambda \\\\le 1$ is a parameter that controls the amount of pruning, the graph is guaranteed to be connected.\",\n \"To see this, note that there is at least one edge surviving for every vertex.\",\n \"In other words, from the start vertex, there is at least one edge for us to traverse for every vertex, and we eventually arrive at the final vertex.\",\n \"Beam pruning, or more generally beam search [84], is a widely used search and pruning method.\",\n \"The motivation behind beam search is to constrain a search algorithm, such as the shortest-path algorithm (Algorithm REF ), with a fixed memory budget.\",\n \"Due to the memory budget, we cannot afford to remember all the paths we have traversed, so we prune the paths that are less likely to have high weights.\",\n \"Specifically, the beam search algorithm performs the following steps while visiting vertices in topological order.\",\n \"We keep an approximate shortest distance $\\\\hat{d}(u)$ to every vertex $u$ .\",\n \"Suppose vertex $v$ is the current vertex being visited.\",\n \"A set of surviving edges $S_v = \\\\lbrace e \\\\in \\\\text{in}(v): \\\\hat{d}(\\\\mathrm {tail}(e)) \\\\ge \\\\tau \\\\rbrace $ is computed based on a threshold $\\\\tau $ , and then the approximate shortest distance for $v$ is updated by setting $\\\\hat{d}(v) = \\\\max _{e \\\\in S_v} w(e) + \\\\hat{d}(\\\\mathrm {tail}(e)).$ Note that this equation is identical to the shortest-path recursion, except that the set of incoming edges $\\\\text{in}(v)$ is pruned before updating.\",\n \"To reconstruct the graph after beam pruning, we simply collect the surviving edges $\\\\bigcup _{v \\\\in V} S_v$ .\",\n \"We can also let $\\\\tau = \\\\lambda \\\\min _{e \\\\in \\\\text{in}(v)} \\\\hat{d}(\\\\mathrm {tail}(e))+ (1 - \\\\lambda ) \\\\max _{e \\\\in \\\\text{in}(v)} \\\\hat{d}(\\\\mathrm {tail}(e))$ where $0 \\\\le \\\\lambda \\\\le 1$ is a parameter that controls the amount of pruning.\",\n \"Note that beam pruning is very similar to greedy pruning.\",\n \"For greedy pruning, edges are pruned based on edge weights, while for beam pruning, edges are pruned based on estimates of shortest distances.\",\n \"It is obvious that if an edge is pruned, paths going through the edge are no longer in the search space.\",\n \"Whenever we prune an edge, we discard the maximum-weight path that passes through the edge.\",\n \"Obviously if the maximum-weight path passing through the edge survives pruning, the edge survives pruning.\",\n \"The max-marginal for an edge is defined as the weight of the maximum-weight path passing through the edge.\",\n \"Max-marginal pruning was first proposed by [124] and later rediscovered by [142].\",\n \"Formally, the max-marginal of an edge $e$ is defined as $\\\\gamma (e) = \\\\max _{p \\\\ni e} w(p),$ i.e., the weight of the maximum-weight path that passes through $e$ .\",\n \"The pruning procedure is simple: choose a threshold $\\\\tau $ and discard edge $e$ if $\\\\gamma (e) < \\\\tau $ .\",\n \"One way to set the threshold [142] is $\\\\tau _\\\\lambda = \\\\lambda \\\\max _{e \\\\in E} \\\\gamma (e) + (1 - \\\\lambda ) \\\\frac{1}{|E|}\\\\sum _{e \\\\in E} \\\\gamma (e),$ where $E$ is the set of segments and $0 \\\\le \\\\lambda \\\\le 1$ .\",\n \"There are two nice properties about max-marginal pruning with the above threshold: There is at least one path surviving; all paths with weights higher than $\\\\tau _\\\\lambda $ survive the pruning.\",\n \"The first property is true because paths that achieve the maximum weight always survive.\",\n \"The second property can be proved by contradiction.\",\n \"Suppose path $p$ has weight $s > \\\\tau _\\\\lambda $ but is pruned.\",\n \"Then there exists an edge $e$ in $p$ such that $\\\\gamma (e) < \\\\tau _\\\\lambda $ .\",\n \"On the other hand, since $p$ passes through $e$ , $\\\\gamma (e)$ is at least $s$ , i.e., $\\\\gamma (e) > s > \\\\tau _\\\\lambda $ , a contradiction.\",\n \"Note that we can only guarantee that paths with weights larger than $\\\\tau _\\\\lambda $ survive pruning.\",\n \"It does not imply that all paths that survive pruning have weights larger than $\\\\tau _\\\\lambda $ .\",\n \"To calculate $\\\\gamma (e)$ for each $e \\\\in E$ , note that $\\\\gamma (e) = \\\\max _{p \\\\ni e} w(p)= \\\\Big [ \\\\max _{p_1 \\\\in \\\\mathcal {P}(s, \\\\mathrm {tail}(e))} w(p_1) \\\\Big ]+ w(e)+ \\\\Big [ \\\\max _{p_2 \\\\in \\\\mathcal {P}(\\\\mathrm {head}(e), t)} w(p_2) \\\\Big ],$ where $\\\\mathcal {P}(u, v)$ is the set of paths that start at vertex $u$ and end at vertex $v$ .\",\n \"We can construct an algorithm like the shortest-path algorithm to calculate the first and third terms as follows.\",\n \"$d_1(v) & = \\\\max _{p_1 \\\\in \\\\mathcal {P}(s, v)} w(p_1)= \\\\max _{e \\\\in \\\\text{in}(v)} [ d_1(\\\\mathrm {tail}(e)) + w(e) ] \\\\\\\\d_2(v) & = \\\\max _{p_2 \\\\in \\\\mathcal {P}(v, t)} w(p_2)= \\\\max _{e \\\\in \\\\text{out}(v)} [ d_2(\\\\mathrm {head}(e)) + w(e) ]$ As shown above, the runtime of max-marginal pruning is the same as that of the shortest-path algorithm (Algorithm REF ).\",\n \"Max-marginal pruning can also be applied to vertices.\",\n \"Let $\\\\gamma (v) = \\\\max _{p \\\\ni v} w(p)$ be the max-marginal of the vertex $v$ .\",\n \"We can also obtain a similar recursion $\\\\gamma (v) = \\\\max _{p \\\\ni v} w(p)= \\\\Big [ \\\\max _{p_1 \\\\in \\\\mathcal {P}(s, v)} w(p_1) \\\\Big ]+ \\\\Big [ \\\\max _{p_2 \\\\in \\\\mathcal {P}(v, t)} w(p_2) \\\\Big ]= d_1(v) + d_2(v),$ and set a threshold based on the convex combination of $\\\\max _{v \\\\in V} \\\\gamma (v)$ and $\\\\frac{1}{|V|} \\\\sum _{v \\\\in V} \\\\gamma (v)$ .\",\n \"Similar guarantees also hold for max-marginal vertex pruning.\",\n \"Recall that the runtime for finding the shortest path is $O(|E|C)$ where $E$ is the set of segments in the search space and $C$ is the computational cost of evaluating the weight function on a single segment.\",\n \"Suppose we have a fixed time budget.\",\n \"Since the pruned search space is smaller than the original space, we can afford to use a more computationally expensive weight function on the pruned space for better prediction.\",\n \"Since additional pruning can be applied to the search space, we end up with a system having multiple rounds of pruning with increasingly expensive weight functions.\",\n \"Prediction is then done on the pruned space from the final round.\",\n \"The above approach is also commonly referred to as rescoring, because hypotheses are given new scores (weights) after pruning.\",\n \"Models are also named after the pass in which they are used.\",\n \"For example, a first-pass model is one that searches over the entire search space; a second-pass model is a model that rescores the pruned search space produced by a first-pass model.\",\n \"Based on rescoring, [142] proposed structured prediction cascades, a type of multi-pass system with multiple rounds of pruning.\",\n \"The complete search space is denoted $H_1$ .\",\n \"At round $k$ , we have hypothesis space $H_k$ , train a pruning model $\\\\Theta _k$ on $H_k$ , and prune $H_k$ to get $H_k^{\\\\prime }$ .\",\n \"The pruned space $H_k^{\\\\prime }$ is expanded to $H_{k+1}$ by considering more context for every segment, such as label $n$ -grams.\",\n \"The process can be repeated for any number of rounds.\",\n \"Inspired by [142], we propose discriminative segmental cascades, a multi-pass system consisting of segmental models with increasingly expensive weight functions.\",\n \"In our cascades, all search spaces are represented as FSTs.\",\n \"We denote the complete search space $H_1$ .\",\n \"At the $k$ -th pass, based on the search space $H_k$ , we train a segmental model $\\\\Theta _k$ , and use it to prune the search space to get $H_k^{\\\\prime }$ .\",\n \"In our framework, we can decide whether to expand the search space to include more context.\",\n \"Since the search spaces are represented as FSTs, we propose a new FST operation, termed self-expansion, to efficiently consider neighboring segments for every segment.\",\n \"If we decide to self-expand, then $H_{k+1} = {H_k^{\\\\prime }}^\\\\uparrow $ , where ${H_k^{\\\\prime }}^\\\\uparrow $ is the self-expansion of $H_k^{\\\\prime }$ .\",\n \"Otherwise, $H_{k+1} = H_k^{\\\\prime }$ .\",\n \"Suppose we decide to self-expand.\",\n \"Then the weight function for the subsequent pass $w_{k+1}$ can make use of a wider context than the weight function of the previous pass.\",\n \"Including a wider context is one possible approach to increase the expressiveness of the weight function.\",\n \"Other approaches include using a deeper or more complex neural network, or extracting more sophisticated features such as tracking formants or pitches, both of which can be done without self-expansion.\",\n \"The process can be repeated many times, forming a segmental cascade.\",\n \"An example of a segmental cascade is shown in Figure REF .\",\n \"Segmental models in each of these passes can be trained with the losses described in Section REF .\",\n \"[142] proposed to train the pruning models with the filtering loss, measuring how well the ground truths are retained after pruning.\",\n \"We decide to train the segmental models in the cascade with losses in Section REF that directly measure the final performance, making sure that we have a model for prediction after each pass.\",\n \"This approach also makes it easier to monitor when to stop stacking the cascade.\",\n \"Figure: An example segmental cascade.\",\n \"The search space H 1 H_1is the complete space.\",\n \"The search space H 1 ' H_1^{\\\\prime }is pruned from H 1 H_1.\",\n \"The search space H 2 H_2is the self-expansion of H 1 ' H_1^{\\\\prime }.The additional vertex and edge added to H 1 ' H_1^{\\\\prime } beforeself-expansion are shown in dashed lines.Given a search space represented as an FST $H$ , we can expand the context of a segment by considering its neighbors.\",\n \"Suppose we want to incorporate the previous segment given the current segment.\",\n \"We create a new FST $H^\\\\uparrow $ that remembers and keeps track of the previous segment (edge) traversed in $H$ .\",\n \"Vertices in $H^\\\\uparrow $ correspond to the possible previous segments in $H$ .\",\n \"Edges in $H^\\\\uparrow $ are of the form $\\\\langle e_1, e_2 \\\\rangle $ , where $e_1$ is the edge just traversed and $e_2$ is the edge to be traversed.\",\n \"We refer to $H^\\\\uparrow $ as the self-expansion of $H$ .\",\n \"Formally, let $H = ((V, E, \\\\mathrm {tail}, \\\\mathrm {head}), \\\\Sigma , \\\\Lambda , I, F, i, o)$ .\",\n \"The self-expansion of $H$ is defined as $H^\\\\uparrow = ((V^\\\\uparrow , E^\\\\uparrow , \\\\mathrm {tail}^\\\\uparrow , \\\\mathrm {head}^\\\\uparrow ),\\\\Sigma ^\\\\uparrow , \\\\Lambda ^\\\\uparrow , I^\\\\uparrow , F^\\\\uparrow , i^\\\\uparrow , o^\\\\uparrow )$ where $V^\\\\uparrow & = E \\\\cup \\\\lbrace e_0 \\\\rbrace & E^\\\\uparrow & = \\\\lbrace \\\\langle e_1, e_2 \\\\rangle : \\\\mathrm {head}(e_1) = \\\\mathrm {tail}(e_2) \\\\rbrace $ $I^\\\\uparrow & = \\\\lbrace e_0 \\\\rbrace & \\\\mathrm {tail}^\\\\uparrow (\\\\langle e_1, e_2 \\\\rangle ) & = e_1 \\\\\\\\F^\\\\uparrow & = \\\\lbrace e \\\\in E : \\\\mathrm {head}(e) \\\\in F \\\\rbrace & \\\\mathrm {head}^\\\\uparrow (\\\\langle e_1, e_2 \\\\rangle ) & = e_2 \\\\\\\\\\\\Sigma ^\\\\uparrow & = \\\\Sigma & i^\\\\uparrow (\\\\langle e_1, e_2 \\\\rangle ) & = i(e_2) \\\\\\\\\\\\Lambda ^\\\\uparrow & = \\\\Lambda & o^\\\\uparrow (\\\\langle e_1, e_2 \\\\rangle ) & = o(e_2)$ A vertex $v_0$ and an $e_0$ is created in $H$ before self-expansion, where $i(e_0) = o(e_0) = \\\\epsilon $ , $\\\\mathrm {tail}(e_0) = v_0$ , and $\\\\mathrm {head}(e_0) = v_1$ assuming we only have a single start state $v_1$ , i.e., $I = \\\\lbrace v_1\\\\rbrace $ .\",\n \"The edge $e_0$ is created as a sentinel, corresponding to a vertex in $H^\\\\uparrow $ for not having traversed any edge.\",\n \"The weight function $w^\\\\uparrow : E^\\\\uparrow \\\\rightarrow \\\\mathbb {R}$ is task-dependent, so we do not define it explicitly here.\",\n \"However, since $w^\\\\uparrow $ is defined on $E^\\\\uparrow $ , the weight function after taking a edge $\\\\langle e_1, e_2 \\\\rangle \\\\in E^\\\\uparrow $ can make use of all the information about $e_1$ and $e_2$ to compute the weight, including the start times, end times, and labels.\",\n \"We also observe that $\\\\text{out}^\\\\uparrow (e) & = \\\\lbrace \\\\langle e, e^{\\\\prime } \\\\rangle : e^{\\\\prime } \\\\in \\\\text{out}(\\\\mathrm {head}(e)) \\\\rbrace \\\\\\\\\\\\text{in}^\\\\uparrow (e) & = \\\\lbrace \\\\langle e^{\\\\prime }, e \\\\rangle : e^{\\\\prime } \\\\in \\\\text{in}(\\\\mathrm {tail}(e)) \\\\rbrace $ for $e \\\\in V^\\\\uparrow = E$ .\",\n \"Therefore, similarly to $\\\\sigma $ -composition, self-expansion can be computed lazily (on the fly) while traversing $H$ .\",\n \"We conduct experiments on the TIMIT data set in the same setting as in Section REF , except that here we follow [42] and use 40-dimensional log mel filter bank outputs instead of 39-dimensional MFCCs.\",\n \"The data set is phonetically transcribed, so we have the option of training the encoders with frame-wise cross entropy based on the ground-truth frame labels.\",\n \"In the following experiments, we explore convolutional neural network (CNN) encoders.\",\n \"The CNN encoders are inspired by [123] in that they are deep convolutional neural networks with small $3 \\\\times 3$ and $5 \\\\times 5$ filters.\",\n \"For frame classification, the input to the network is a window of 15 frames of 40-dimensional log mel filter outputs centered on the current frame.\",\n \"The network has five convolutional layers, with 64 filters of size $5 \\\\times 5$ for the first layer and 128, 128, 256, and 256 filters of size $3 \\\\times 3$ for layers two to five respectively, each of which is followed by a rectified linear unit (ReLU) activation [101], with max pooling layers after the first and the third ReLU layers.\",\n \"The output of the final ReLU layer is concatenated with a window of 15 frames of 39-dimensional MFCCs centered on the current frame, and the resulting vector is passed through three fully connected ReLU layers with 4096 units each.\",\n \"The network is trained with SGD for 35 epochs with step size 0.001, momentum 0.95, weight decay 0.005, and a mini-batch size of 100 frame predictions.\",\n \"Dropout [128] are applied to the concatenation layer with rate 20% and to the fully connected layers with rate 50%.\",\n \"This classifier was tuned on the development set and achieves a 22.3% frame error rate (after collapsing to 39 phone labels) on the development set and 23.0% on the test set.\",\n \"After confirming that the CNN encoder is able to perform well on frame classification, we use the CNN encoder to generate frame posterior probabilities, and train segmental models with the FC weight function and a maximum duration of 30 frames.\",\n \"Hinge loss is optimized with AdaGrad [27] with step size 0.01 and a mini-batch size of 1 utterance for 70 epochs.\",\n \"No explicit regularizer is used except early stopping on the development set.\",\n \"We use a scaled overlap cost for our experiments: $\\\\mathrm {cost}(p^{\\\\prime }, p) &= \\\\sum _{e^{\\\\prime } \\\\in p} \\\\mathrm {cost}(e^{\\\\prime }, p) \\\\\\\\\\\\mathrm {cost}(e^{\\\\prime }, p) &= |e^{\\\\prime } \\\\cup \\\\tilde{e}| - |e^{\\\\prime } \\\\cap \\\\tilde{e}| {1}_{o(e^{\\\\prime }) = o(\\\\tilde{e})}$ where $p^{\\\\prime }$ is the hypothesized path, $p$ is the ground-truth path, $|e^{\\\\prime } \\\\cup e|$ denotes the union of the two intervals defined by $e$ and $e^{\\\\prime }$ , $|e^{\\\\prime } \\\\cap e|$ denotes the intersection of the two intervals, $o(e)$ is the output label of $e$ , and $\\\\tilde{e} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{e \\\\in p} |e^{\\\\prime } \\\\cap e|$ .\",\n \"In words, $\\\\tilde{e}$ is the edge that overlaps the most with $e^{\\\\prime }$ .\",\n \"The cost is non-negative and is only zero if $e^{\\\\prime } = e$ , and it can be seen as an estimate of the number of incorrectly predicted frames.\",\n \"The results are shown in Table REF .\",\n \"Our first-pass segmental model is on par with [1], where a CNN encoder is also used, and it is also on par with a baseline HMM-DNN hybrid system that we trained with Kaldi using the standard recipe for TIMIT [110].\",\n \"Table: Phoneme error rates (%) on TIMITcomparing a HMM-DNN hybrid system todiscriminative segmental models.To construct a second-pass segmental model, we first evaluate pruning approaches based on oracle error rates and lattice densities.\",\n \"The oracle error rate is the lowest achievable phone error rate of a given lattice.\",\n \"The lattice density is defined as the total number of edges divided by the number of ground-truth edges, i.e., the number of hypothesized edges per ground-truth edge.\",\n \"In general, a dense lattice has a higher chance to contain the ground-truth path than a sparse lattice.\",\n \"In fact, the entire search space is a dense lattice that always contains the ground-truth path.\",\n \"A good pruner is expected to produce sparse lattices while maintaining a low oracle error rate.\",\n \"We compare max-marginal pruning with $\\\\lambda \\\\in \\\\lbrace 0.5, 0.6, 0.7, 0.8\\\\rbrace $ , greedy pruning with $\\\\lambda \\\\in \\\\lbrace 0.95, 0.9, 0.875, 0.85, 0.825\\\\rbrace $ , random pruning with probability $\\\\lbrace 0.92, 0.94, 0.96, 0.98\\\\rbrace $ , and beam pruning with $\\\\lambda =0.9$ .\",\n \"The results are shown in Figure REF .\",\n \"For reference, the average density of the complete spaces is 11587.67.\",\n \"We observe that search spaces of segmental models are robust to missing edges—the oracle error rate is only 1.7% when we randomly drop 92% of the edges.\",\n \"We also observe that max-marginal pruning produces significantly sparser lattices than greedy pruning and random pruning while maintaining low oracle error rates.\",\n \"Beam pruning performs the worst.\",\n \"Since greedy pruning works well by just comparing edges with the same tail vertex (i.e., with the same start time), we suspect the bad performance of beam pruning is due to comparing distances at different time points.\",\n \"Figure: Left: Oracle error rates vs lattice densities for various pruningapproaches.\",\n \"Right: A re-scaled version of the left figure focusing ongreedy pruning and max-marginal pruning.Based on the above pruning comparison, we decide to use the lattices generated by max-marginal pruning with $\\\\lambda = 0.8$ .\",\n \"Let $H_1$ be the entire search space, and $H_1^{\\\\prime }$ be the pruned search space.\",\n \"To train a segmental model with a wider segmental context, we first self-expand $H_1^{\\\\prime }$ into $H_2 = {H_1^{\\\\prime }}^\\\\uparrow $ and introduce the following weight functions.\",\n \"Note that the weight functions can now depend on two segments instead of one.\",\n \"Instead of re-learning all of the weight functions in the first-pass model, we reuse them in the second pass by having $w_{\\\\text{lat}}(\\\\langle (\\\\ell _1, s_1, t_1), (\\\\ell _2, s_2, t_2) \\\\rangle )= \\\\alpha \\\\cdot w((\\\\ell _2, s_2, t_2))$ from the first level of the cascade, where $\\\\alpha $ is a learnable parameter.\",\n \"We use the weight $w_{\\\\text{LM}}(\\\\langle (\\\\ell _1, s_1, t_1), (\\\\ell _2, s_2, t_2) \\\\rangle )= \\\\beta \\\\cdot \\\\log p(\\\\ell _2 | \\\\ell _1)$ to include the log probability of the bigram $\\\\ell _1 \\\\ell _2$ , where $\\\\beta $ is a learnable parameter.\",\n \"We define $w_{\\\\text{left}-k}(\\\\langle (\\\\ell _1, s_1, t_1), (\\\\ell _2, s_2, t_2) \\\\rangle )& = u_{s_2 - k, \\\\ell _1, \\\\ell _2} \\\\\\\\w_{\\\\text{right}-k}(\\\\langle (\\\\ell _1, s_1, t_1), (\\\\ell _2, s_2, t_2) \\\\rangle )& = u_{t_2 + k, \\\\ell _1, \\\\ell _2}^{\\\\prime }$ similarly to the first-order boundary weights, where $u_{t, \\\\ell , \\\\ell ^{\\\\prime }} = \\\\theta _{\\\\ell , \\\\ell ^{\\\\prime }}^\\\\top h_t$ and $u_{t, \\\\ell , \\\\ell ^{\\\\prime }}^{\\\\prime } = {\\\\theta ^{\\\\prime }_{\\\\ell , \\\\ell ^{\\\\prime }}}^\\\\top h_t$ for $t = 1, \\\\dots , \\\\tilde{T}$ and some learnable parameters $\\\\theta , \\\\theta ^{\\\\prime }$ .\",\n \"Since the search space is sparse, we can afford to compute segment features based on a neural network for every segment.\",\n \"We choose a similar CNN architecture to the one for frame classification.\",\n \"We pre-train the CNN with a segment classification task.\",\n \"Here the features at the input layer are the log mel filter outputs from a 15-frame window around the segment's central frame.\",\n \"Instead of concatenation with 15-frame MFCCs, we concatenate with a segmental feature vector consisting of the average MFCCs of three sub-segments in the ratio of 3-4-3, plus two four-frame averages at both boundaries and length indicators for length 0 to 20 (similar to the segmental feature vectors of [46], [22]).\",\n \"This CNN is trained on the ground-truth segments in the training set.\",\n \"Finally, we build an ensemble of such networks with different random seeds and a majority vote.\",\n \"This ensemble classifier has a 15.0% segment classification error on the test set.\",\n \"A comparison of our CNN to other segment classifiers is shown in Table REF .\",\n \"Table: TIMIT segment classification error rates (%).Directly running the CNN segment classifier for every edge in the lattice is, however, still too time-consuming.\",\n \"We instead compress the best-performing (single) CNN into a shallow fully connected network with one hidden layer of 512 ReLUs by training it to predict the log probability outputs of the deep network, as proposed by [8].\",\n \"We then use the log probability outputs of the shallow network.\",\n \"We refer to the result as segment NN weights.\",\n \"Let $w_{\\\\text{seg}}(\\\\langle (\\\\ell _1, s_1, t_1), (\\\\ell _2, s_2, t_2) \\\\rangle )= (u_{s_2, t_2})_{\\\\ell _2}$ where $u_{s, t} = \\\\theta ^\\\\top z_{s, t}$ , $z_{s, t}$ is a log probability vector of the labels obtained by passing the segment $x_s, \\\\dots , x_t$ to the CNN segment classifier, and $\\\\theta $ is some learnable parameter vector.\",\n \"To add more context information, we use the same CNN architecture and training setup to learn a bi-phone frame classifier, but with an added 256-unit bottleneck linear layer before the softmax.\",\n \"Each frame is labeled with its segment label and one additional label from a neighboring segment.\",\n \"If the current frame is in the first half of the segment, the additional label is the previous phone; if it is in the second half, then the additional label is the next phone.\",\n \"The learned bottleneck layer outputs are used to define weights (although they do not correspond to log probabilities) with averaging and sampling as for the uni-phone case.\",\n \"We refer to the resulting weights as bi-phone NN bottleneck weights.\",\n \"We use the sum of the lattice weight, the bigram LM weight, second-order boundary weights, segment NN weights, bi-phone NN bottleneck weights, length indicators, and lexicalized bias as our final weight function for the second level of the cascade.\",\n \"Hinge loss is minimized with AdaGrad for 20 epochs with step size 0.01.\",\n \"Again, no explicit regularizer is used except early stopping based on the PER on the development set.\",\n \"Results with these additional weights are shown in Table REF .\",\n \"Adding the second-order weights, bigram LM, and the above NN weights gives a 1.8% absolute improvement over our best first-pass system, demonstrating the value of including such expensive features with wider context.\",\n \"Table: Phoneme error rates (%)with discriminative segmental cascades.The search space for the first pass is denoted H 1 H_1,and the search space for the second pass H 1 ' ↑ {H_1^{\\\\prime }}^\\\\uparrow is denoted H 2 H_2, where H 1 H_1 is pruned to get H 1 ' H_1^{\\\\prime }.We have shown that segmental cascades can be used to incorporate computationally expensive weight functions while maintaining efficiency by shifting the computationally expensive weights to later passes.\",\n \"In this section, instead of incorporating additional weights, we shift weights to later passes to improve decoding speed.\",\n \"Due to their recent success [42], we also explore long short-term memory (LSTM) networks [56] as encoders instead of CNN encoders.\",\n \"In this section, we describe long short-term memory (LSTM) networks.\",\n \"We start by defining a single-layer LSTM.\",\n \"Suppose the input vectors are $x_1, \\\\dots , x_T$ .\",\n \"First we apply a linear transformation to the inputs to get $x_t^{\\\\prime } = \\\\begin{pmatrix} W_{xu} \\\\\\\\ W_{xi} \\\\\\\\ W_{xf} \\\\\\\\ W_{xo} \\\\end{pmatrix} x_t,$ which can be done for $t=1, \\\\dots , T$ in parallel.\",\n \"Next, assume $h_0 = c_0 = 0$ .\",\n \"Suppose we have already computed $h_t$ and $c_t$ , and we want to compute $h_{t+1}$ and $c_{t+1}$ .\",\n \"We apply a linear transformation to get the candidates $u^{\\\\prime }$ , $i^{\\\\prime }$ , $f^{\\\\prime }$ , and $o^{\\\\prime }$ for the cell update vectors $u$ and the gates $i$ , $f$ and $o$ .\",\n \"$\\\\begin{pmatrix}u^{\\\\prime } \\\\\\\\ i^{\\\\prime } \\\\\\\\ f^{\\\\prime } \\\\\\\\ o^{\\\\prime }\\\\end{pmatrix}= x_t^{\\\\prime } + \\\\begin{pmatrix} W_{hu} \\\\\\\\ W_{hi} \\\\\\\\ W_{hf} \\\\\\\\ W_{ho} \\\\end{pmatrix} h_t+ \\\\begin{pmatrix} 0 \\\\\\\\ W_{ci} \\\\\\\\ W_{cf} \\\\\\\\ 0 \\\\end{pmatrix} c_t.$ The update vectors $u$ , the input gate $i$ , and the forget gate $f$ are then computed from the candidates by applying nonlinear transformations.\",\n \"$u & = \\\\tanh (u^{\\\\prime }) \\\\\\\\i & = \\\\text{logistic}(i^{\\\\prime }) \\\\\\\\f & = \\\\text{logistic}(f^{\\\\prime })$ The new cell vector $c_{t+1}$ is computed with the update vector gated by the input gate and the previous cell.\",\n \"$c_{t+1} & = i \\\\odot u + f \\\\odot c_t$ Finally, the output gate $o$ is computed from the updated cell $c_{t+1}$ , and the new hidden vector $h_{t+1}$ is updated with the cell gated by the output gate.\",\n \"$o & = \\\\text{logistic}(o^{\\\\prime } + W_{co} c_{t+1}) \\\\\\\\h_{t+1} & = o \\\\odot \\\\tanh (c_{t+1})$ The final results after running an LSTM on $x_1, \\\\dots , x_T$ are $h_1, \\\\dots , h_T$ and $c_1, \\\\dots , c_T$ .\",\n \"We use $h_{1:T} = \\\\text{LSTM}(x_{1:T})$ to denote the process above.\",\n \"Since LSTMs have an assigned direction, it is natural to run two LSTMs in opposite directions and combine the hidden vectors.\",\n \"Formally, $h^f_{1:T} & = \\\\text{LSTM}(x_{1:T}) \\\\\\\\h^b_{T:1} & = \\\\text{LSTM}(x_{T:1}) \\\\\\\\h_t & = W_f h^f_t + W_b h^b_t$ We use $h_{1:T} = \\\\text{BiLSTM}(x_{1:T})$ as a shorthand for the bidirectional LSTMs.\",\n \"Note that the parameters of the two LSTMs are not shared.\",\n \"Another way to increase the complexity of the acoustic encoder is to stack them on top of each other.\",\n \"For example, if we stack 3 layers of bidirectional LSTMs, we get $h^{(1)}_{1:T} &= \\\\text{BLSTM}(x_{1:T}) \\\\\\\\h^{(2)}_{1:T} &= \\\\text{BLSTM}(h^{(1)}_{1:T}) \\\\\\\\h^{(3)}_{1:T} &= \\\\text{BLSTM}(h^{(2)}_{1:T})$ We use $h_{1:T} = \\\\text{DBLSTM}^3(x_{1:T})$ to denote the 3-layer bidirectional LSTMs (where “D” refers to “deep”).\",\n \"Again the parameters of the LSTMs in each layer and each direction are not shared.\",\n \"As in Section REF , we first explore LSTMs on a frame classification task, Specifically, the log probability vector of a frame at time $t$ is $z_t = \\\\text{logsoftmax}(W h_t + b)$ where $h_t$ is the hidden vector at time $t$ produced by an LSTM, and $W$ and $b$ are the parameters.\",\n \"We then explore segmental cascades with weight functions based on log probabilities produced by LSTMs.\",\n \"Experiments are conducted on the TIMIT data set.\",\n \"We compute 41-dimensional log mel filter bank outputs including energy, and concatenate with the delta's and double delta's to form the final 123-dimensional acoustic feature vectors.\",\n \"Instead of using a 192-utterance development set, we follow the Kaldi recipe [110] and use 400 utterances from the complete test set (disjoint from the core test set) as the validation set.\",\n \"We report numbers on the core test set, the same test set as in Section REF .\",\n \"We build a frame classifier by stacking three layers of bidirectional LSTMs.\",\n \"The cell and hidden vectors have 256 units.\",\n \"We train the frame classifier with frame-wise cross entropy and optimize with AdaGrad [27] with mini-batch size of 1 utterance and step size 0.01 for 30 epochs.\",\n \"We choose the best-performing model on the development set (early stopping).\",\n \"Following [138], [92], we consider dropping half of the frames for any given utterance in order to save time on feeding forward.\",\n \"Specifically, we only use $x_2, x_4, \\\\dots , x_{T - 2}, x_T$ to feed forward through the deep LSTM and generate $z_2, z_4, \\\\dots ,z_{T - 2}, z_T$ (assuming $T$ is even, without loss of generality).\",\n \"We then copy each even-indexed output to its previous frame, i.e., $z_{i-1} = z_i$ for $i = 2, 4, \\\\dots , T-2, T$ .\",\n \"During training, the cross entropy is calculated over all frames and propagated back.\",\n \"Specifically, let $E_i$ be the cross entropy at frame $i$ , and $E = \\\\sum _{i=1}^T E_i$ .\",\n \"The gradient of $z_i$ is the sum of the gradients from the current frame and the copied frame.\",\n \"Dropping even-indexed frames is similar to dropping odd-indexed frames, except the outputs are copied from $z_i$ to $z_{i+1}$ for $i = 1, 3, \\\\dots , T-1$ .\",\n \"When training LSTMs with subsampling, we alternate between dropping even- and odd-numbered frames every other epoch.\",\n \"The results are shown in Figure REF .\",\n \"Figure: Development set frame error rates vs. number of epochs (left)and vs. training hours (right).We observe that with subsampling the model converges more slowly than without, in terms of number of epochs.\",\n \"However, by the end of epoch 30, there is almost no loss in frame error rates when we drop half of the frames.\",\n \"Considering the more important measure of training time rather than number of epochs, the LSTMs with frame subsampling converge twice as fast as those without subsampling.\",\n \"For the remaining experiments, we use the log posteriors at each frame of the subsampled LSTM outputs as the inputs to the segmental models.\",\n \"Our baseline system, denoted $R$ , is a first-pass segmental model with the FC weight function (Section REF ).\",\n \"The baseline system is trained by optimizing hinge loss with AdaGrad using mini-batch size 1 utterance, step size 0.1, and early stopping for 50 epochs.\",\n \"We decide to shift the FC weight function to the second pass, and train a segmental model, denoted $A_1$ , with just the label posterior and a label-independent bias.\",\n \"Specifically, the label posterior weight is defined as $w((\\\\ell , s, t)) = \\\\sum _{i=s}^t (z_i)_\\\\ell ,$ where $z_i = \\\\text{logsoftmax}(W h_i + b)$ is the log probability vector at time $i$ based on the vector $h_i$ produced by the LSTM.\",\n \"This reduces the number of features from 24528 to two.\",\n \"We use hinge loss optimized with AdaGrad with mini-batch size 1 and step size 1.\",\n \"Since we only have two features, learning converges very quickly, in only three epochs.\",\n \"We take the model from the third epoch to produce lattices for subsequent passes in the cascade.\",\n \"Lattices are generated with max-marginal pruning with $\\\\lambda = 0.85$ .\",\n \"The resulting lattices have an average oracle error rates of 1.4% and average density of 213.02.\",\n \"We train a second-pass model, denoted $A_2$ , with the FC weight function except that we add a “lattice score” feature corresponding to the segment score given by the two-feature system.\",\n \"Hinge loss is optimized with AdaGrad with mini-batch size 1, step size 0.1, and early stopping for 20 epochs.\",\n \"The learning curve comparing $R$ and $A_1$ followed by $A_2$ is shown in Figure REF .\",\n \"We observe that the learning time per epoch of the two-feature system $A_1$ is only one-third of the baseline system $R$ .\",\n \"We also observe that training of $A_2$ converges faster than training $R$ , despite the fact that they use almost identical feature functions.\",\n \"The baseline system achieves the best result at epoch 49.\",\n \"In contrast, the two-pass system is done before the baseline even finishes the third epoch.\",\n \"Figure: Learning curve of the proposed two-pass system compared withthe baseline system.The time gap between the first pass and the second pass is thetime spent on pruning.Following Section REF , given the first-pass baseline, we apply max-marginal pruning to produce lattices for the second-pass baseline with $\\\\lambda =0.8$ .\",\n \"The second-pass baseline features are the lattice score from the first-pass baseline, a bigram language model score, first-order length indicators, and a bias.\",\n \"Hinge loss is optimized with AdaGrad with mini batch size 1, step size 0.01, and early stopping for 20 epochs.\",\n \"For the proposed system, we produce lattices with max-marginal pruning and $\\\\lambda =0.3$ for the third-pass system.\",\n \"We use the same set of features and hyper-parameters as the second-pass baseline for the third pass.\",\n \"Phone error rates of all passes are shown in Table REF .\",\n \"First, if we compare the one-pass baseline with the proposed two-pass system, our system is close to the baseline.\",\n \"Second, we observe a healthy improvement by just adding the bigram language model score to the second-pass baseline.\",\n \"The improvement for our third-pass system is small but brings our final performance to within 0.4% of the baseline second pass.\",\n \"Table: Phone error rates (%) of proposed and baseline systems.Next we report on the speedups in training and decoding obtained with our proposed approach.\",\n \"Table REF shows the real-time factors for decoding with the baseline and proposed systems.\",\n \"In terms of decoding time alone, we achieve a factor of 2.4 speedup compared to the baseline.\",\n \"If the time of feeding forward LSTMs is included, then our proposed system is two times faster than the baseline.\",\n \"Table: Real-time factors for decoding.Table REF shows the times needed to train a system to get to the performance in Table REF .\",\n \"The speedup mostly comes from the fast convergence of the first pass.\",\n \"In terms of training the segmental models alone, we achieve an 18.0-fold speedup.\",\n \"If the time to train the LSTMs is included, then we obtain a 3.4-fold speedup compared to the baseline.\",\n \"To summarize some of the above results: With a combination of the first-pass two-feature system and edge pruning, we prune 95% of the segments in the first-pass hypothesis space, leading to significant speedup in both decoding and training.\",\n \"The feed-forward time for our LSTMs is halved through frame subsampling.\",\n \"In the end, with a single four-core CPU, we achieve 0.31 times real-time decoding including feeding forward, which is 2.2 times faster than the baseline, and 32.5 hours in total to obtain our final model including LSTM training, which is 3.4 times faster than the baseline.\",\n \"Excluding the LSTMs, the segmental model decoding alone is 2.4 times faster than the baseline, and training the segmental models alone is 18 times faster than the baseline.\",\n \"Table: Hours for training the system.In this chapter, we have described discriminative segmental cascades, a multi-pass system for segmental models.\",\n \"Max-marginal pruning lies at the heart of segmental cascades, producing sparse lattices while having low oracle error rates.\",\n \"After pruning, it becomes feasible to incorporate computationally expensive weight functions, such as higher-order LMs, and weight functions based on neural networks.\",\n \"We demonstrate that incorporating these weights significantly improves the performance of segmental models.\",\n \"We also show that decoding and training speed can be significantly improved with segmental cascades by shifting weights to later passes without sacrificing accuracy.\",\n \"Though we have shown improvement with segmental cascades, how the density and structure of search spaces affect the training of segmental models is still not fully known.\",\n \"Task-dependent priors, such as phoneme duration, are not fully explored and can potentially be integrated into the pruning procedure.\",\n \"The fact that beam search works well for frame-based models but fails miserably for segmental models is also worth investigating.\",\n \"Converting other more sophisticated search algorithms, such as $\\\\text{A}^*$ search [68], into pruning algorithms is a possibility that might show us signs as to why beam search fails for segmental models.\",\n \"In the previous chapter, we have seen how a sequence recognizer can be trained in multiple stages: a frame classifier is first trained with frame-wise cross entropy, and a segmental model is trained with a sequence loss, such as the ones defined in Section REF , based on the classifier's outputs.\",\n \"In other words, the frame classifier is trained independently from the segmental model, and is held fixed when the segmental model is being trained.\",\n \"A natural question to ask is whether we can further improve the sequence loss by updating the frame classifier while holding the segmental model fixed.\",\n \"A more general question would be whether we can improve the sequence loss by updating the frame classifier and the segmental model simultaneously.\",\n \"Optimizing all of the parameters simultaneously against a single loss function is commonly referred to as end-to-end training.\",\n \"Since the loss function with respect to all the parameters in a neural network is non-convex, end-to-end training might be more difficult than training in multiple stages.\",\n \"On the other hand, since end-to-end training is jointly optimizing all parameters, it might achieve a lower loss, leading to better performance.\",\n \"In this chapter, we formally define these training settings, including multi-stage training and end-to-end training, Different training approaches and losses have different training requirements.\",\n \"We discuss the pros and cons of these training approaches, and conduct experiments to see empirically how well they compare to each other.\",\n \"Recall that the set of parameters $\\\\Theta $ consists of $\\\\Theta _\\\\text{enc}$ and $\\\\Theta _\\\\text{dec}$ , where $\\\\Theta _\\\\text{enc}$ is the set of parameters in the encoder and $\\\\Theta _\\\\text{dec}$ are the rest of the parameters in the weight function.\",\n \"The training approach we have seen in the previous chapter is referred to as multi-stage training.\",\n \"Specifically, the encoder is trained in the first stage with an encoder-specific loss $\\\\mathcal {L}_\\\\text{enc}(\\\\Theta _\\\\text{enc}),$ for example, the frame-wise cross entropy.\",\n \"Let $\\\\hat{\\\\Theta }_\\\\text{enc} = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\min }_{\\\\Theta _\\\\text{enc}}\\\\mathcal {L}_\\\\text{enc}(\\\\Theta _\\\\text{enc})$ .\",\n \"In the second stage, we solve the optimization problem $ \\\\min _{\\\\Theta _\\\\text{dec}} \\\\mathcal {L}_\\\\text{seq}(\\\\hat{\\\\Theta }_\\\\text{enc}, \\\\Theta _\\\\text{dec}),$ where $\\\\mathcal {L}$ is a sequence loss, such as hinge loss, log loss, or marginal log loss, defined in Section REF .\",\n \"Note that in the second stage the parameters of the encoder are held fixed.\",\n \"Typically, after the first stage, we can feed the inputs in the entire data set forward, and reuse the vectors while optimizing the sequence loss to save computation.\",\n \"Another benefit of optimizing (REF ) with $\\\\hat{\\\\Theta }_\\\\text{enc}$ fixed is that the loss function $\\\\mathcal {L}$ might be convex in $\\\\Theta _\\\\text{dec}$ , for example, when hinge loss or log loss is used and the weight function is linear in $\\\\Theta _{\\\\text{dec}}$ .\",\n \"The characteristics of convex functions are well-understood [13], and many algorithms, other than first-order methods, have been developed for optimizing convex functions [12].\",\n \"In contrast to multi-stage training, we refer to solving $ \\\\min _{\\\\Theta _\\\\text{enc}, \\\\Theta _\\\\text{dec}}\\\\mathcal {L}_\\\\text{seq}(\\\\Theta _\\\\text{enc}, \\\\Theta _\\\\text{dec})$ as end-to-end training, where all parameters $\\\\Theta _\\\\text{enc}$ and $\\\\Theta _\\\\text{dec}$ are optimized jointly.\",\n \"Since all parameters are optimized jointly, the loss function is in general non-convex.\",\n \"Though end-to-end training might be harder to solve, it might lead to a lower loss value.\",\n \"Typically, the solution found by multi-stage training is a reasonably good solution for the sequence loss.\",\n \"As a result, a safe strategy, which we refer to as end-to-end fine-tuning, may be to use the solution from multi-stage training as an initialization for solving (REF ).\",\n \"One major difference between multi-stage training and end-to-end training is their resource requirements.\",\n \"Multi-stage training requires extra labels for optimizing the encoder loss $\\\\mathcal {L}_\\\\text{enc}$ , while end-to-end training requires only whatever the sequence loss $\\\\mathcal {L}_\\\\text{seq}$ requires.\",\n \"For experiments, we use the same phoneme recognition setting on the TIMIT data set as in Section REF .\",\n \"In the sections below, we first compare the performance of CNNs and LSTMs, and then compare two training settings for training segmental models, multi-stage training followed by end-to-end fine-tuning and end-to-end training from random initialization.\",\n \"We also explore various weight functions paired with various loss functions in the context of end-to-end training from random initialization.\",\n \"Recall that the multi-stage training approach demonstrated in Section REF consists of two stages: first training a frame classifier, and second training a segmental model based on the output probabilities of the frame classifier.\",\n \"Instead of a CNN encoder, a 3-layer bidirectional LSTM with hidden vector size of 250 is trained for frame classification.\",\n \"Parameters are initialized according to [36].\",\n \"Vanilla SGD is used to minimize frame-wise cross entropy for 20 epochs with step size 0.1 and a mini-batch size of 1 utterance.\",\n \"Gradients are clipped to norm 5 if the norm is above 5.\",\n \"The best model (measured by frame error rates on the development set) is chosen among the 20 epochs, and trained for another 20 epochs with step size 0.75 decayed by 0.75 after every epoch.\",\n \"For phoneme recognition, we use segmental models with the FC weight function, the same setting as in Section REF .\",\n \"Hinge loss with the overlap cost are minimized with RMSProp [100] for 20 epochs with step size ${10}^{-4}$ and decay 0.9, The frame error rates (FER) and phoneme error rates (PER) compared to the CNN in Section REF are shown in Table REF .\",\n \"The LSTM performs better than the CNN for both tasks.\",\n \"We suspect that the LSTM's superior performance is due to the training approaches: the LSTM is trained on entire utterances, while the CNN is trained on batches of 15-frame windows.\",\n \"This effect was also observed in [58].\",\n \"For the rest of the experiments, we use 3-layer LSTMs as our encoders.\",\n \"Table: Frame error rates (FER) and phoneme error rates (PER)on the development setcomparing the CNN and the LSTM encoder.Next, we compare segmental models trained with different losses, including hinge loss, log loss, and marginal log loss.\",\n \"We use the FC weight function and a maximum segment duration of 30 frames.\",\n \"All losses are minimized with RMSProp for 20 epochs with step size ${10}^{-4}$ and decay 0.9.\",\n \"Results are shown in Table REF .\",\n \"No explicit regularizers are used except early stopping on the development set.\",\n \"All losses achieve similar results with log loss slightly behind.\",\n \"Note that although marginal log loss does not require manual alignments, the frame-wise cross entropy used to train the frame classifier still does.\",\n \"After two-stage training, we use the trained segmental models and frame classifiers as an initialization for end-to-end training.\",\n \"Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 decayed by 0.75 after each epoch.\",\n \"Gradients are clipped with norm 5.\",\n \"Dropout is added to the LSTMs with rate 0.2, and no other regularizers are used except early stopping on the development set.\",\n \"Results are shown in Table REF .\",\n \"End-to-end fine-tuning is able to improve the error rates for all losses with marginal log loss being the best performer.\",\n \"Table: Phoneme error rates (%) comparing different losseswith two-stage training (2s) followed by end-to-end fine-tuning (ft).After end-to-end fine-tuning, the encoders are not constrained to perform well on frame-wise cross entropy.\",\n \"We evaluate the encoders on the frame classification task to see how much the encoders deviate from the initialization, and whether the intermediate representations trained for frame classification are still preserved.\",\n \"Results are shown in Table REF .\",\n \"The encoders fine-tuned with hinge loss and log loss deviate less compared to the one fine-tuned with marginal log loss.\",\n \"We suspect this is because hinge loss and log loss use the ground-truth alignments in training while marginal log loss does not.\",\n \"Table: Frame error rates (%) of LSTM encodersafter end-to-end fine-tuning with different losses .After observing the success of end-to-end fine-tuning, we conduct experiments to see whether it is possible to train segmental models end to end from random initialization.\",\n \"We use the FC weight function and a maximum duration of 30 frames, the same setting as in the multi-stage experiments.\",\n \"We initialize the parameters according to [36].\",\n \"Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 and gradient clipping with norm 5.\",\n \"The best performing model, chosen in the first 20 epochs, is trained for another 20 epochs with vanilla SGD step size 0.75 decayed by 0.75 after each epoch.\",\n \"Dropout of rate 0.2 is used throughout the training process.\",\n \"No other regularizers are used except early stopping on the development set.\",\n \"Results are shown in Table REF .\",\n \"There is no significant difference in terms of performance between models trained end to end from random initialization and ones trained in multiple stages followed by fine-tuning.\",\n \"This shows that it is possible to train segmental models end to end from random initialization, and that models trained in multiple stages can serve as a good initialization for end-to-end training.\",\n \"Table: Phoneme error rates (%) on the development setcomparing multi-stage trainingfollowed by end-to-end fine-tuning (2s+ft) and end-to-end trainingfrom random initialization (e2e).Next, we consider various weight functions for end-to-end training from random initialization.\",\n \"Note that the initialization scheme in [36] is not suitable the for log-soft-max operation used after the LSTMs for producing log probabilities, because the variances before and after the log-soft-max operation are very different.\",\n \"Based on this intuition, we define a new weight function, called FC bottleneck (FCB) weight function, removing the log-soft-max layer when using the FC weight function.\",\n \"More precisely, suppose $h_1, \\\\dots , h_T$ is the sequence of vectors produced by an LSTM.\",\n \"Whereas the FC weight function uses the average, samples, and boundaries of $z_i = \\\\text{logsoftmax}(W h_i + b)$ for $i = 1, \\\\dots , T$ , the FCB weight function uses the average, samples, and boundaries of $h_i$ 's directly.\",\n \"The learning curves of segmental models trained end to end with FC weight function and the FCB weight function are shown in Figure REF .\",\n \"We observe significantly faster convergence with the FCB weight function than with the FC weight function.\",\n \"Figure: Learning curves of segmental models trained end to endwith the FC weight function and FCB weight functionwith marginal log loss.We do not compare the MLP weight function to the FC and FCB weight functions in the current setting, because it is too time-consuming to train segmental models with the MLP weight function on the entire search space.\",\n \"To train segmental models with the MLP weight function, we follow [85] and reduce the time resolution by a factor of four using pyramid LSTMs.\",\n \"Specifically, for an input sequence $x_{1:T}$ , $h^{(1)}_{1:T} &= \\\\text{BLSTM}(x_{1:T}) \\\\\\\\z^{(1)}_{1:\\\\lfloor T/2 \\\\rfloor } &= \\\\text{subsample}(h^{(1)}_{1:T}) \\\\\\\\h^{(2)}_{1:\\\\lfloor T/2 \\\\rfloor } &= \\\\text{BLSTM}(z^{(1)}_{1:\\\\lfloor T/2 \\\\rfloor }) \\\\\\\\z^{(2)}_{1:\\\\lfloor T/4 \\\\rfloor } &= \\\\text{subsample}(h^{(2)}_{1:\\\\lfloor T/2 \\\\rfloor }) \\\\\\\\h^{(3)}_{1:\\\\lfloor T/4 \\\\rfloor } &= \\\\text{BLSTM}(z^{(2)}_{1:\\\\lfloor T/4 \\\\rfloor })$ where $h_2, h_4, \\\\dots , h_{2(k-1)}, h_{2k} = \\\\text{subsample}(h_1, h_2, \\\\dots , h_{2k})$ if we drop a frame for every two frames.\",\n \"The final output sequence $h^{(3)}$ is then used to compute edge weights.\",\n \"An encoder with several layers of LSTMs interleaved with subsampling is also known as a pyramid LSTM and has been used in other prior work, e.g., [16].\",\n \"In addition, we set the maximum duration to 8 frames, effectively $4 \\\\times 8 = 32$ frames on the original time scale.\",\n \"The reduced time resolution and maximum duration result in search spaces 16 times smaller than the original ones, making it feasible to compute the MLP weight function on all edges.\",\n \"With 3-layer pyramid LSTMs, we compare segmental models trained with the three weight functions and three losses.\",\n \"All models are trained with the same step size scheduling as in the previous experiments.\",\n \"Results with and without pyramid LSTMs are both shown in Table REF .\",\n \"First, we observe that using pyramid LSTMs leads to a degradation in performance for segmental models with the FC and FCB weight functions.\",\n \"However, using pyramid LSTMs has less impact on segmental models trained with marginal log loss compared to the ones trained with hinge loss and log loss.\",\n \"We suspect this is because hinge loss and log loss rely on the ground-truth alignments for training while marginal log loss does not.\",\n \"Second, we observe that while segmental models with the FC and FCB weight functions are able to achieve low hinge loss values, the ones with the MLP weight function completely fail to do so.\",\n \"Finally, the best result on the development set obtained by using the FC weight function is slightly worse than using the MLP weight function.\",\n \"Table: Phoneme error rates (%) comparing different lossesfor end-to-end training from random initialization.Segmental models trained with marginal log loss do not require ground-truth alignments during training, making annotating data sets easier, cheaper, and less time-consuming.\",\n \"However, it is harder to diagnose errors made by models trained end to end because the intermediate representations are not interpretable.\",\n \"Models trained in multiple stages are easier to diagnose by looking at the performance of proxy tasks, such as frame classification in our case.\",\n \"We compare the time to train segmental models end to end.\",\n \"Times are measured on a 3GHz 4-core CPU.\",\n \"Multithreading is only used for matrix operations; all FST algorithms are implemented single-threaded.\",\n \"Results are shown in Table REF .\",\n \"Reducing the time resolution with pyramid LSTMs significantly improves the runtime for all losses.\",\n \"The MLP weight function is about 2.5 times slower than the FC weight function.\",\n \"Table: Average number of minutes for one epoch of end-to-end training on TIMIT.We are interested in how well segmental models recover the phone boundaries in the end-to-end setting when manual alignments are not used during training.\",\n \"The task, commonly known as phonetic segmentation, is to align the phonetic transcription to the acoustic frames.\",\n \"Unlike sequence prediction, the phonetic transcription is known.\",\n \"We measure the error rates of the boundaries under some tolerance level.\",\n \"Specifically, suppose the predicted boundary is at time $\\\\hat{b}$ and the ground truth is at time $b$ .\",\n \"The predicted boundary $\\\\hat{b}$ is considered correct if the absolute distance $t = |\\\\hat{b} - b| \\\\le \\\\tau $ for some $\\\\tau $ .\",\n \"The error rates under various thresholds for the segmental model with the FC weight function trained with marginal log loss are shown in Table REF .\",\n \"Models proposed in [70], [113], [144], shown in row 1–3 in Table REF , are specifically trained for phonetic segmentation.\",\n \"Though the alignment results are behind models trained specifically to align, the segmental model trained with marginal log loss is not supervised with any ground-truth alignments, and its performance is sufficient for many tasks.\",\n \"Table: Boundary error rates (%) of phonetic segmentation at different tolerance levelson the TIMIT core test set (except the last row)for segmental models trained end to end with marginal log losscompared to past results.We analyze the errors made by the segmental model.\",\n \"The top 30 most errorful boundary types sorted by error rates are shown in Table REF .\",\n \"We observe that most of the boundaries in Table REF involve silences, vowels, and semi-vowels.\",\n \"The ones involving silences often appears at the start or the end of utterances.\",\n \"Boundaries between two vowels and between a semi-vowel and a vowel are known to be ambiguous [120], [135].\",\n \"We see few alignment errors between consonants and vowels.\",\n \"The top 30 most errorful boundary types sorted by error counts are shown in Table REF .\",\n \"We are also interested in boundary types that have hight error counts but not necessary have high error rates.\",\n \"Compared to Table REF , we see a very different pattern.\",\n \"Most of the errors in Table REF involve silences (including voiced and unvoiced closures).\",\n \"For $0\\\\text{ms}$ and $10\\\\text{ms}$ , the segmental model errs at the boundaries between closures and the bursts of stop consonants, such as /b/, /d/, /g/, /p/, /t/, and /g/.\",\n \"It also errs at boundaries involving liquids, such as /r/ and /l/, across all tolerance levels.\",\n \"Similarly to semi-vowels, the boundaries of liquids are also known to be ambiguous [120], [135].\",\n \"The errors for higher tolerance levels, such as 30ms and 40ms, mostly involve silences that appear at the start or end of utterances.\",\n \"We notice that 10.5% of the silences in the training set are longer than 30 frames.\",\n \"We suspect the 30-frame maximum duration constraint is too restrictive for silences.\",\n \"Some phoneme boundaries are inherently ambiguous.\",\n \"In other words, there are cases where segments do not have clear start times and end times.\",\n \"We argue that segmental models are still well-motivated and suitable for these tasks, such as phonetic recognition.\",\n \"First, the search space are stochastic when the path weights are interpreted as probabilities.\",\n \"In fact, lattices, when first proposed by [120], were used to account for phoneme boundary ambiguity.\",\n \"Training segmental models with marginal log loss also takes the ambiguity into account by marginalizing over all segmentations.\",\n \"For decoding, ideally we want to find the label sequence that has the maximum marginal weight $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y} \\\\sum _{p \\\\in \\\\Gamma (y)} w(p), $ where $\\\\Gamma (y)$ is the set of paths with the label sequence $y$ .\",\n \"However, the above is not tractable and we typically approximate it by finding the most probable path $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{p \\\\in \\\\mathcal {P}} w(p).$ Another approach to approximate (REF ) is to use beam search.\",\n \"In summary, segmental models are capable of handling ambiguous segment boundaries if the training and decoding take the ambiguity into account.\",\n \"We have compared different training approaches for segmental models, including multi-stage training and end-to-end training.\",\n \"End-to-end training improves over multi-stage training, and end-to-end training from random initialization is on par with end-to-end fine-tuning.\",\n \"Multi-stage training is useful for diagnosing end-to-end training, because every model found by multi-stage training is a valid model for end-to-end training objectives.\",\n \"We have shown that segmental models can be trained with marginal log loss from random initialization, without requiring manual alignments for training.\",\n \"As a byproduct, segmental models trained with marginal log loss perform reasonably well on phonetic segmentation.\",\n \"Segmentation errors made by end-to-end segmental models are aligned with the studies in acoustic phonetics.\",\n \"Table: Top 30 boundaries and its preceding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error rates ofthe segmental model trained with marginal log loss.Table: Top 30 boundaries and its preceeding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error counts ofthe segmental model trained with marginal log loss.Automatic speech recognition (ASR) has been posed as a graph search problem since the 1970s [61].\",\n \"A search space, represented as a graph, is first constructed; edges in the graph are assigned weights based on the input and the edges; to predict, we simply run the shortest-path algorithm on the graph.\",\n \"The graph search paradigm has been popularized by the use of hidden Markov models (HMM) [112], [62], where the shortest-path algorithm is known as the Viterbi algorithm.\",\n \"In the probabilistic setting, finding the shortest path can be seen as finding the maximum a posteriori solution, justifying the graph search paradigm.\",\n \"The probabilistic view has spawned many algorithms, such as segmental k-means [63] and expectation maximization [116], for estimating parameters in the weight function.\",\n \"Segmental models follow the same graph search paradigm while having a different type of search spaces and different weight functions.\",\n \"Many variants of segmental models were proposed in the past, such as stochastic segment models [103], [104], semi-Markov HMMs [117], segmental HMMs [115], [29], semi-Markov conditional random fields (CRF) [118], and segmental CRFs [149].\",\n \"However, the type of search spaces and the shortest-path algorithm stay the same.\",\n \"In this chapter, we review these variants from the graph search point of view.\",\n \"The idea of using segmental features for speech recognition can be traced back to the 1970s [141].\",\n \"The definition of segmental models was not explicit.\",\n \"Most of the studies still followed the graph search paradigm, though the graph structures were typically constructed from a lexicon with a small vocabulary, and the weights on the edges were typically estimated with heuristics or a small amount of data [75], [23].\",\n \"In the following sections, we categorize variants of segmental models as either generative or discriminative.\",\n \"This categorization also aligns well with their rough chronological order.\",\n \"Hidden semi-Markov Models are arguably the first segmental models applied to speech recognition [81], [117].\",\n \"Given a sequence of observations $x = (x_1, \\\\dots , x_T)$ of length $T$ , let $y = (\\\\ell _1, \\\\dots , \\\\ell _K)$ be the label sequence and $z = ((s_1, t_1), \\\\dots ,(s_K, t_K))$ be the segmentation, where $\\\\ell _1, \\\\dots , \\\\ell _K \\\\in L$ for some discrete label set $L$ , and $s_1 = 1$ , $t_k = T$ , $s_k \\\\le t_k$ , $t_{k-1} + 1 = s_k$ , for $k = 2, \\\\dots , n$ .\",\n \"Let $e_k = (\\\\ell _k, s_k, t_k)$ for $k = 1, \\\\dots , K$ .\",\n \"The probability of $x_{1:T}$ defined by hidden semi-Markov models is $p(x, y, z) = p(x_{1:T}, e_{1:K})= p(e_1) \\\\prod _{k=2}^K p(e_k | e_{k-1}) \\\\prod _{k=1}^K p(x_{s_k:t_k} | e_k).$ The generative story is straightforward: the segments are generated one by one, following a Markov chain, and each segment $e = (\\\\ell , s, t)$ generates $t - s + 1$ observations.\",\n \"Hidden Markov models are special cases of hidden semi-Markov models with the constraints $K = T$ and $s_k = t_k$ for $k = 1, \\\\dots , K$ .\",\n \"There are many ways to define and parameterize $p(e_k | e_{k-1})$ and $p(x_{s_k:t_k} | e_k)$ .\",\n \"The most common assumptions are $p(e_k | e_{k-1}) & = p(\\\\ell _k | \\\\ell _{k-1}) \\\\\\\\p(x_{s_k:t_k} | e_k) & = p(x_{s_k:t_k} | t-s+1, \\\\ell _k) p(t-s+1 | \\\\ell _k)$ where $p(\\\\ell _k | \\\\ell _{k-1})$ is commonly known as the transition probability, $p(x_{s_k:t_k} | t-s+1, \\\\ell _k)$ the emission probability, and $p(t-s+1 | \\\\ell _k)$ the duration probability.\",\n \"The observations within a segment are typically assumed to be independent, i.e., $p(x_{s_k:t_k} | t-s+1, \\\\ell _k) =\\\\prod _{j=s_k}^{t_k} \\\\mathcal {N}(x_j; \\\\mu _{\\\\ell _k}, \\\\sigma _{\\\\ell _k}^2),$ where $\\\\mu _\\\\ell $ and $\\\\sigma _\\\\ell ^2$ are the mean and variance of the Gaussian distribution for the label $\\\\ell $ .\",\n \"The single Gaussian case can be easily extended to a mixture of Gaussians.\",\n \"Continuously variable duration HMMs [81], stochastic segment models [103] and segmental HMMs [115], [29] are all hidden semi-Markov models with the above assumptions.\",\n \"The difference among them lies in how the emission probability $p(x_{s_k:t_k} | t-s+1, \\\\ell _k)$ is defined.\",\n \"The subtle differences are summarized in [30], [104].\",\n \"The duration probability is typically defined by a Poisson distribution [116] or Gamma distribution [81].\",\n \"See [104] and the citations therein for other options.\",\n \"Decoding with hidden semi-Markov models is done by solving $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} p(y, z | x)& = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} \\\\log p(y, z | x) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} \\\\log \\\\frac{p(x, y, z)}{p(x)} \\\\\\\\& = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} \\\\log p(x, y, z),$ where $y$ is the label sequence and $z$ is the segmentation, and is equivalent to finding the maximum-weight path with $w((\\\\ell , s, t)) = \\\\log p(t-s+1 | \\\\ell )+ \\\\sum _{j=s}^t \\\\log \\\\mathcal {N}(x_j; \\\\mu _\\\\ell , \\\\sigma _\\\\ell ^2).$ The term $p(\\\\ell ^{\\\\prime } | \\\\ell )$ is ignored here for simplicity, but can be included once the search space is self-expanded (Section REF ).\",\n \"Training hidden semi-Markov models can be done by maximizing the likelihood of the training set.\",\n \"The likelihood can be maximized using gradient-based methods or expectation maximization (EM) [116].\",\n \"See [30] for a detailed explanation of the EM algorithm for estimating the parameters of hidden semi-Markov models.\",\n \"Originally motivated by using rich segmental features [147], the SUMMIT system was defined as a generative model [34].\",\n \"However, [34] introduced the notion of anti-phones and later [18] introduced near-misses, training the system with a discriminative touch.\",\n \"It held the state-of-the-art result for speaker-independent phoneme recognition on TIMIT [45] until the rise of deep neural networks [94].\",\n \"Since maximum mutual information was introduced as a training criterion for HMMs [10], ASR studies have gradually shifted from generative to discriminative modeling.\",\n \"Parallel to the development of segmental models in the ASR community, [118] proposed semi-Markov CRFs for named entity recognition.\",\n \"Using the same notation as in the previous section, the probability of a label sequence $y = (\\\\ell _1, \\\\dots , \\\\ell _n)$ and a segmentation $z = ((s_1, t_1), \\\\dots , (s_K, t_K))$ given an observation sequence $x = (x_1, \\\\dots , x_T)$ is defined as $p(y, z | x) = \\\\frac{1}{Z(x)} \\\\exp \\\\left( \\\\sum _{k=1}^K \\\\theta ^\\\\top \\\\phi (x, e_k) \\\\right)$ where $Z(x) = \\\\sum _{y^{\\\\prime }, z^{\\\\prime }} \\\\exp (\\\\sum _{e \\\\in (y^{\\\\prime }, z^{\\\\prime })} \\\\theta ^\\\\top \\\\phi (x, e))$ and $e \\\\in (y^{\\\\prime }, z^{\\\\prime })$ is a shorthand for enumerating $(\\\\ell _1, s_1, t_1),\\\\dots ,(\\\\ell _{K^{\\\\prime }}, s_{K^{\\\\prime }}, t_{K^{\\\\prime }})$ for $y^{\\\\prime } = (\\\\ell _1, \\\\dots , \\\\ell _{K^{\\\\prime }})$ and $z^{\\\\prime } = ((s_1, t_1), \\\\dots , (s_{K^{\\\\prime }}, y_{K^{\\\\prime }}))$ of length $K^{\\\\prime }$ .\",\n \"The function $\\\\phi $ , called the feature function, extracts feature vectors that are intended to correlate well with $y$ and $z$ .\",\n \"The parameter vector $\\\\theta $ can be learned by maximizing the likelihood of the training set.\",\n \"Decoding with semi-Markov CRFs is done by solving $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} \\\\log p(y, z | x) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y, z} \\\\sum _{e \\\\in (y, z)} \\\\theta ^\\\\top \\\\phi (x, e),$ and is equivalent to finding the maximum-weight path if we let $w(x, e) = \\\\theta ^\\\\top \\\\phi (x, e)$ .\",\n \"Similarly, training is done by maximizing the conditional likelihood of the data, and is equivalent to minimizing the negative log likelihood, or log loss.\",\n \"Heavily influenced by [118], [149] proposed segmental CRFs.\",\n \"The difference between segmental CRFs and semi-Markov CRFs lies in the training loss.\",\n \"Instead of optimizing the conditional likelihood $p(y, z | x)$ , segmental CRFs optimize the marginal likelihood $p(y | x) = \\\\sum _{z} p(y, z | x).$ The connection between the marginal likelihood and marginal log loss is clear once we take the log of the probability distribution.\",\n \"As we defined in Chapter REF , we distinguish between segmental models that consider the entire search space and ones that do not.\",\n \"The former are called first-pass segmental models.\",\n \"Whether a segmental model is first-pass or not is independent of its definition.\",\n \"For example, semi-Markov CRFs were used as first-pass segmental models in [118]; segmental CRFs were first used as a second-pass model in [149], and were later used as a first-pass model in [148].\",\n \"In Table REF , we provide a set of highlights of results in the development of segmental models on the TIMIT data set.\",\n \"[148] was the first to explore discriminative segmental models that search over sequences and segmentations exhaustively, and did not use neural networks.\",\n \"[49] first used (shallow) neural network-based frame classifiers to define weight functions, and later extended the idea to deep neural networks in [48].\",\n \"[1] were the first to use deep convolutional neural networks for the weight functions, and were the first to train segmental models end to end.\",\n \"We have compared different losses and training strategies for segmental models, first in a rescoring framework [129] and then in first-pass segmental models [131].\",\n \"We also introduced segment-level classifiers and segmental cascades for incorporating them (and other expensive features) into segmental weight functions [130].\",\n \"[85] introduced an LSTM-based weight function for every segment, and were also the first to use pyramid LSTMs to speed up inference for segmental models.\",\n \"Table: TIMIT PERs (%) for various segmental modelscompared with HMMs and the current state of the art.The acoustic features are speaker-independent (spk indep) orspeaker-adapted with mean and variance normalization (mvn)or maximum likelihood linear regression (fMLLR) .Some results were obtained with MFCCs and some with log filter bank features.In this chapter, we have reviewed variants of segmental models categorized as either generative or discriminative in rough chronological order.\",\n \"We review hidden semi-Markov models, a broad class of generative segmental models subsuming stochastic segment models and segmental HMMs.\",\n \"We then review semi-Markov CRFs, the discriminative counterpart of hidden semi-Markov models.\",\n \"We have established connections between these special cases and the general segmental models defined in Chapter REF .\",\n \"Beyond segmental models, this chapter reviews other modern models trained end to end within the graph search paradigm.\",\n \"We review connectionist temporal classification (CTC) [41], a popular approach for training frame-based LSTMs end to end.\",\n \"Drawing on the connection between CTC and marginal log loss, we propose a framework, consisting of search spaces (represented as FSTs), weight functions, and training losses, that can encompass many end-to-end models, such as hidden Markov models trained with lattice-free maximum mutual information [111], and LSTMs trained with CTC, as special cases.\",\n \"We compare end-to-end segmental models and end-to-end frame-based models, including one-state HMMs and two-state HMMs with LSTM encoders, and LSTMs trained with CTC.\",\n \"Having these end-to-end models within the unified framework allows us to see the effect of each component while holding the other components fixed.\",\n \"In this section, we review connectionist temporal classification (CTC) [41].\",\n \"While being conceptually simple, LSTMs trained with CTC were the state of art for phonetic recognition in 2013 [42], and have achieved competitive results on large-vocabular speech recognition [92], [88], [151].\",\n \"Consider a sequence of acoustic vectors $x_1, \\\\dots , x_T$ and its corresponding labels $y_1, \\\\dots , y_n$ , where $y_i \\\\in L$ for $i = 1, \\\\dots , n$ and some label set $L$ .\",\n \"We assume $n < T$ because a phoneme is typically more than a frame long.\",\n \"Suppose there exists a function $\\\\mathcal {P}$ that maps $y_1, \\\\dots , y_n$ to a path $a_1, \\\\dots , a_T$ , where $a_t \\\\in L^{\\\\prime }$ for some other label set $L^{\\\\prime }$ .\",\n \"Minimizing $p(a_{1:T} | x_{1:T})$ can be done by simply minimizing the frame-wise cross entropy $p(a_{1:T} | x_{1:T}) = \\\\prod _{t=1}^T p(a_t | x_t).$ [41] proposed a mapping $\\\\mathcal {P}$ as follows.\",\n \"Given a sequence $a_1, \\\\dots , a_T$ where $a_t \\\\in L \\\\cup \\\\lbrace \\\\varnothing \\\\rbrace $ for $t = 1, \\\\dots , T$ and $\\\\varnothing $ is the blank symbol.\",\n \"Let $\\\\mathcal {B}$ be a function that first replaces duplicate contiguous labels into a single label, and second removes all the blank symbols.\",\n \"For example, $\\\\text{\\\\texttt {k} \\\\texttt {ae} \\\\texttt {t}} = \\\\mathcal {B}(\\\\text{$\\\\varnothing $ \\\\texttt {k} \\\\texttt {k} $\\\\varnothing $ \\\\texttt {ae} \\\\texttt {ae} \\\\texttt {t}\\\\texttt {t} $\\\\varnothing $}).$ [41] used $\\\\mathcal {P}(y) = \\\\mathcal {B}^{-1}(y) = \\\\lbrace a = (a_1, \\\\dots , a_T) :a_t \\\\in L \\\\cup \\\\lbrace \\\\varnothing \\\\rbrace , \\\\mathcal {B}(a) = y\\\\rbrace $ to map a label sequence to a sequence of the same length as the input sequence.\",\n \"However, this function $\\\\mathcal {P}$ returns a set of sequences, so the objective is modified to $p(y_{1:n} | x_{1:T}) = \\\\sum _{a_{1:T} \\\\in \\\\mathcal {B}^{-1}(y_{1:n})} p(a_{1:T} | x_{1:T})= \\\\sum _{a_{1:T} \\\\in \\\\mathcal {B}^{-1}(y_{1:n})} \\\\prod _{t=1}^T p(a_t | x_t),$ marginalizing over all the possible paths.\",\n \"Given an input sequence $x$ , predicting a label sequence is done by finding $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y} p(y | x) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y} \\\\sum _{a \\\\in \\\\mathcal {B}^{-1}(y)} p(a | x).$ However, there are currently no algorithms that can solve the above efficiently, so it is typically approximated by finding the greedy best path $\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y} p(y | x) = \\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{y} \\\\max _{a \\\\in \\\\mathcal {B}^{-1}(y)} p(a | x)= \\\\mathcal {B}(\\\\operatornamewithlimits{\\\\arg \\\\!\\\\max }_{a} p(a|x)).$ In other words, we simply find the label that achieves the highest probability at every time point and remove the duplicates and blank symbols to produce the final decoded sequence.\",\n \"As a result of (REF ), the search space of CTC has an edge for every label in the label set (including the blank label) at every time step.\",\n \"Specifically, the search space $G$ includes the edges $\\\\lbrace e_{\\\\ell , t}: \\\\ell \\\\in L, t \\\\in \\\\lbrace 1, \\\\dots , T\\\\rbrace \\\\rbrace $ with $v_{t-1} = \\\\mathrm {tail}(e_{\\\\ell , t})$ and $v_t = \\\\mathrm {head}(e_{\\\\ell , t})$ .\",\n \"An example is shown in Figure REF .\",\n \"The weight of an edge $e_{\\\\ell , t}$ is the log probability of label $\\\\ell $ at time $t$ .\",\n \"By construction, the decision made at every time point is independent of the decision at other time points conditioned on $x_{1:T}$ .\",\n \"In addition, since the probabilities at every time point sum to one, the partition function $Z(x)$ of the search space, i.e., the sum of all the path probabilities, is always 1.\",\n \"Recall that the marginal log loss is defined as $\\\\mathcal {L}_{\\\\text{mll}} = -\\\\log \\\\sum _{p \\\\in \\\\mathcal {P}_{G \\\\circ _\\\\sigma F_y}} \\\\exp (w(p))+ \\\\log \\\\sum _{p \\\\in \\\\mathcal {P}_{G}} \\\\exp (w(p))$ where $\\\\mathcal {P}_K$ is the set of paths in the FST $K$ and $F_y$ is the constraint FST constructed from the ground truth label sequence $y$ .\",\n \"Since $Z(x) = 1$ for CTC, the second term is zero.\",\n \"By the definition of $\\\\mathcal {B}$ , we construct the constraint FST $F_y$ such that it consists of the sequences of one or more labels with zero or more blanks in between labels.\",\n \"For example, for the label sequence “k ae t,” the constraint FST is the regular expression $\\\\varnothing ^*\\\\texttt {k}^+\\\\varnothing ^*\\\\texttt {ae}^+\\\\varnothing ^*\\\\texttt {t}^+\\\\varnothing ^*$ .\",\n \"We have $\\\\mathcal {B}^{-1}(y) = G \\\\circ _\\\\sigma F_y$ .\",\n \"In other words, the sum of path probabilities in $G \\\\circ _\\\\sigma F_y$ exactly matches the CTC objective.\",\n \"Because the first term matches the CTC objective and the second term is zero, marginal log loss becomes the CTC objective for this type of search spaces and the log probability weight function.\",\n \"Figure: An example of the CTC search space for a five-frame utterancewith a label set of size three (plus one blank).Most mainstream end-to-end speech recognition models can be broadly categorized as either frame-based models or encoder-decoder models.\",\n \"Frame-based LSTMs trained with CTC, HMMs, and some newer approaches like the auto-segmentation criterion (ASG) [25] fall under the first category, because these models emit one symbol for every frame.\",\n \"Falling under the second category, encoder-decoder models proposed by [21], [9], [16] generate labels one at a time while conditioning on the input and the labels generated in the past, without an explicit alignment between labels and frames.\",\n \"Since frame-based models follow the same graph search framework as segmental models, we will focus on discussing the connection between these and segmental models.\",\n \"Recall that marginal log loss requires a search space $G$ and a constraint FST $F$ to limit the search space to ground-truth labels.\",\n \"To compute marginal log loss, we first compute the marginals on $G$ for computing the partition function $Z(x)$ , and then compute the marginals on the $\\\\sigma $ -composed FST $G \\\\circ _\\\\sigma F$ for computing $Z(x, y)$ .\",\n \"CTC, HMMs whtn trained with lattice-free MMI [111], and ASG can all be seen as special cases of this framework.\",\n \"Comparing CTC to HMMs, the search space is different depending on the HMM topology.\",\n \"For example, two-state HMMs are used in [111].\",\n \"Since the transition probabilities and emission probabilities are all locally normalized, the partition function $Z(x)$ is always 1.\",\n \"The constraint FST representing the ground-truth labels consists simply of sequences of repeating labels.\",\n \"For example, for the label sequence “k ae t,” the constraint FST for one-state HMMs is the regular expression $\\\\texttt {k}^+\\\\texttt {ae}^+\\\\texttt {t}^+$ .\",\n \"For two-state HMMs, the constraint FST is the regular expression $\\\\texttt {k}_1\\\\texttt {k}_2^*\\\\texttt {ae}_1\\\\texttt {ae}_2^*\\\\texttt {t}_1\\\\texttt {t}_2^*$ .\",\n \"With the above construction, marginal log loss applied to HMMs is equivalent to lattice-free MMI [111].\",\n \"For ASG, the search space is equivalent to that of one-state HMMs.\",\n \"Instead of assuming conditional independence as in CTC, ASG includes transition probabilities between states.\",\n \"The constraint FST is identical to that of HMMs, with repeated ground-truth labels.\",\n \"However, in ASG the weights on the edges are not locally normalized, so the partition function $Z(x)$ is not always 1 and has to be computed.\",\n \"With the above search space construction, marginal log loss becomes ASG.\",\n \"Another approach similar to CTC proposed in [38] is called RNN transducers.\",\n \"The search space of an RNN transducer is the set of alignments from the speech signal to all possible label sequences, so the search space grows exponentially in the number of labels.\",\n \"The weight function of a path in this approach relies on an RNN, and is not decomposable as a sum of weights of the edges.\",\n \"RNN transducers are trained with marginal log loss.\",\n \"By the independence assumption imposed in [38], the partition function $Z(x)$ is still 1, so we do not need to marginalize over the exponentially large space.\",\n \"During decoding, however, we still have to search over the exponentially large space with, for example, beam search.\",\n \"In view of this framework, even when using the same loss function, i.e., marginal log loss, segmental models and frame-based models differ in their search space, weight functions, and how the search space is constrained by the ground-truth labels during training.\",\n \"A summary of special cases is shown in Table REF .\",\n \"Table: An example instantiation of the components used in marginal log losswith the ground-truth sequence “k ae t”and TT input framesfor segmental models and various end-to-end models.The search space L T L^T consists of sequences of TT labels.The label sets L 1 L_1 and L 2 L_2 contain labels in LL with subscript 1 and 2 respectively.The search space is denoted GG and the constraint FST is denoted F y F_y in the table.We compare various frame-based models trained end to end on the same phonetic recognition task in the same setting as in Section REF .\",\n \"We use 3-layer 256-unit biderectional LSTMs paired with CTC, one-state HMM, and two-state HMMs.\",\n \"We do not use transition probabilities for two-state HMMs.\",\n \"We optimize marginal log loss with the corresponding weight functions and the corresponding constraint FSTs for CTC, one-state HMM, and two-state HMMs.\",\n \"We run vanilla SGD with step size 0.1 and gradient clipping of norm 5 for 20 epochs.\",\n \"Starting from the best performing model of the first 20 epochs, we run vanilla SGD for another 20 epochs with step size 0.75 decayed by 0.75 after each epoch.\",\n \"The dropout rate is 0.2, and no other explicit regularizer is used except early stopping on the development set.\",\n \"The same experiments are repeated with pyramid LSTMs.\",\n \"Results comparing with segmental models are shown in Table REF .\",\n \"When standard LSTMs are used, we fail to minimize the training loss for one-state HMMs and two-state HMMs The only difference between CTC and one-state HMMs is the use of blank symbols, suggesting that blank symbols play an important role in end-to-end frame-based models.\",\n \"In particular, one-state HMMs and two-state HMMs explicitly marginalize over segmentations in training and are thus sensitive to time, while LSTMs trained with CTC are not.\",\n \"Besides, the tail states in $L_2$ can be seen as label-dependent blank symbols.\",\n \"Since two-state HMMs fail to achieve a low training loss, having label-dependent blanks is not helpful in this case.\",\n \"When pyramid LSTMs are used, all variants of frame-based models achieve low PERs on the development set and improve over ones with standard LSTMs.\",\n \"Since the time resolution is reduced by four with pyramid LSTMs, we hypothesize that using the pyramid LSTMs introduces a bias similar to a minimum duration constraint, which helps end-to-end training for frame-based models.\",\n \"The number of minutes per epoch for training segmental models and LSTMs trained with CTC is shown in Table REF , and a scatter plot showing number of minutes per epoch vs.\",\n \"PER is shown in Figure REF .\",\n \"Pyramid LSTMs trained with CTC are the best performer while also being the most efficient to train.\",\n \"The real-time factors for decoding are shown in Table REF .\",\n \"Due to smaller and simpler search spaces, frame-based LSTMs trained with CTC decode faster than segmental models.\",\n \"While frame-based LSTMs train and decode faster, segmental models are more flexible, so they might be improved with additional feature functions.\",\n \"Explicitly hypothesizing segments can serve as a prior, so segmental models might perform better in low-resource settings.\",\n \"Table: Phoneme error rates (%) for end-to-end training fromrandom initialization comparing CTCand segmental models trained with marginal log loss.Table: Average number of minutes for one epoch of end-to-end training on TIMIT.Table: Real-time factors for decoding comparing CTC and segmental models.Figure: Training minutes per epoch vs.\",\n \"PER (%) for segmental modelsand LSTMs trained with CTC.For American Sign Language (ASL) fingerspelling experiments, we follow [72] and compare end-to-end models in both signer-dependent setting and signer-independent setting.\",\n \"The input is a sequence of 128-dimensional histograms of oriented gradients (HoG) feature vectors computed over $128 \\\\times 128$ hand shape images.\",\n \"The output is a sequence of English letters, and the label set consists of the 26 English characters plus two labels for initial and final non-signing regions.\",\n \"The evaluation metric is the edit distance between the predicted letter sequence and the ground-truth letter sequence.\",\n \"For the signer-dependent setting, the data for each signer is split into ten folds, in which eight are for training, one is for development, and one is for testing.\",\n \"We report the letter error rate averaged over the 10 experiments.\",\n \"We train LSTMs with CTC and segmental models with the FCB weight function.\",\n \"Both models are trained with marginal log loss from random initialization.\",\n \"The encoder is a one-layer 128-unit bidirectional LSTM.\",\n \"For CTC, we run vanilla SGD with step sizes in $\\\\lbrace 0.1, 0.05, 0.025\\\\rbrace $ and gradient clipping with norm 5 for 200 epochs.\",\n \"Dropout is used with a rate of 0.2, and no other explicit regularizer is used except early stopping.\",\n \"Step sizes and early stopping are tuned on the development set for each individual experiment.\",\n \"For segmental models, we run vanilla SGD with step size 0.1 for 75 epochs.\",\n \"The maximum segment length is 75 frames.\",\n \"Other hyperparameters, such as dropout and gradient clipping, are the same as for CTC.\",\n \"Results are shown in Table REF .\",\n \"The frame-based LSTMs trained with CTC are better than Tandem HMMs, but are behind the segmental models.\",\n \"The segmental models trained end to end are on par with the two-stage system in [72], but are behind the discriminative segmental cascades (DSC).\",\n \"Table: Letter error rates (%) for signer-dependent models averagedover ten folds.For the signer-independent setting, we train on three signers and use the last signer for development and testing.\",\n \"Specifically, the data of the last signer is split into ten folds, and we use two folds for development and the rest of the eight folds for testing.\",\n \"Instead of dividing the accumulated error counts by the accumulated label sequence lengths, letter error rates are averaged over the folds.\",\n \"We compare LSTMs trained with CTC and segmental models with FCB weight function.\",\n \"The loss function and training procedure stay the same except that we only run 40 epochs of vanilla SGD for segmental models.\",\n \"Results are shown in Table REF .\",\n \"In this setting, CTC and segmental models trained end to end perform significantly better than Tandem HMMs and segmental models trained with the FC weight function in two stages.\",\n \"To compare signer-independent and signer-dependent settings, we average the letter error rates over the same eight folds.\",\n \"Results are shown in Table REF .\",\n \"Segmental models perform significantly better than CTC in both settings.\",\n \"Table: Letter error rates (%) for signer-independent modelsaveraged over eight folds.Table: Letter error rates (%) for signer-dependent and signer-independent models averagedover the same eight folds.We have discussed how other end-to-end frame-based models, such as CTC, HMMs trained with lattice-free MMI, ASG, and RNN transducers are all trained with marginal log loss.\",\n \"The differences among them lie in the search spaces and the weight functions.\",\n \"Drawing these connections allows us to generalize search spaces, loss functions, and weight functions to a broad class of models.\",\n \"From the results of comparing CTC to one-state HMMs and two-state HMMs, we have found that the blank symbol seems to play an important role in training LSTMs end to end.\",\n \"Using pyramid LSTMs improves both the performance and the runtime of decoding and training for CTC, one-state HMMs, and two-state HMMs, but not for segmental models.\",\n \"We have also shown that segmental models with regular LSTMs are better than regular LSTMs trained with CTC on both phonetic recognition and ASL fingerspelling recognition.\",\n \"In this thesis, we have made the following contributions in advancing the study of discriminative segmental models.\",\n \"We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.\",\n \"We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.\",\n \"We obtain improved performance over most earlier work while greatly improving efficiency.\",\n \"We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.\",\n \"We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.\",\n \"We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.\",\n \"Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.\",\n \"We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.\",\n \"Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.\",\n \"We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.\",\n \"We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.\",\n \"We obtain improved performance over most earlier work while greatly improving efficiency.\",\n \"We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.\",\n \"We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.\",\n \"We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.\",\n \"Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.\",\n \"We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.\",\n \"Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.\",\n \"In this chapter, we discuss some potential future work extending segmental models to large-vocabulary tasks and to unsupervised settings.\",\n \"Another potential direction for future work is even richer feature functions.\",\n \"Since first-pass segmental models are computationally demanding, it is very slow to train and decode segmental models with label sets of size in the order of 10,000.\",\n \"This poses a challenge for tasks with large label sets, such as word recognition.\",\n \"[150], [88] bypassed this difficulty by using a baseline HMM recognizer to generate word lattices, and used segmental models to rescore these lattices, exploring various word-level features.\",\n \"However, the performance is constrained by the quality of the baseline recognizer and the quality of the lattices.\",\n \"First-pass segmental models have previously been successfully applied to word recognition [35], [50].\",\n \"This previous work treats first-pass segmental models as a drop-in replacement for HMM phoneme recognizers, because both models serve as functions that map acoustic features to phoneme strings.\",\n \"The phoneme recognizers are then composed with a lexicon and a language model to form a word recognizer, as described in Section REF .\",\n \"This is also an option for extending our work to word recognition.\",\n \"Recent work has explored models that directly predict characters, avoiding the need for a lexicon [39], [92] but still allowing for improved performance when constraining the search space with a lexicon (through FST composition) [92].\",\n \"Segmental models can also be used to predict characters by changing the label set, but might be hampered by the poor alignment of characters to acoustics.\",\n \"Syllables might be a better option than characters.\",\n \"Instead of using intermediate discrete representations, such as phonemes or characters, recent advances in computing power have made it feasible to directly predict words [87], [11], [127], [7].\",\n \"In this case, rather than using a pronunciation dictionary, only a list of words is needed for decoding.\",\n \"Segmental models can also be used to directly predict words by using the list of words as the label set.\",\n \"This approach is worth exploring further, although efficiency issues make it nontrivial to train such models [7].\",\n \"We have presented segmental models for supervised sequence prediction in this thesis.\",\n \"Our framework can be extended to unsupervised sequence prediction.\",\n \"The goal in this setting is typically grouping segments of varying length into clusters.\",\n \"We define unsupervised sequence prediction as finding a function that maps an input sequence $x_1, \\\\dots , x_T$ into a sequence of segments $(\\\\ell _1, s_1, t_1), \\\\dots , (\\\\ell _n, s_n, t_n)$ where $\\\\ell _i \\\\in L$ for $i=1, \\\\dots , n$ and some label set $L$ , and $s_1 = 1$ , $t_n = T+1$ , $s_i < t_i$ , $t_{i-1} = s_i$ for $i = 1, \\\\dots , n$ , given a data set $S$ of sequences without labels.\",\n \"Since we do not have labels for the data samples, the goal in general is to design a loss function $\\\\mathcal {L}(\\\\Theta ; x)$ that only depends on the input sequence $x$ , or even more generally a loss function $\\\\mathcal {L}(\\\\Theta ; S)$ that depends on the data set $S$ .\",\n \"The label set $L$ is typically predefined to be just a set of identifiers, and the correspondence between the input sequence and the identifier sequence is learned from a data set.\",\n \"As a result, a label in $L$ does not have a predefined meaning unless the user assigns a post hoc meaning or the loss function enforces one.\",\n \"Generative segmental models, such as hidden semi-Markov models, can be naturally extended to the unsupervised setting by maximizing the marginal likelihood of the data.\",\n \"For example, Bayesian segmental models have been applied to small-vocabulary [64] and larger-vocabulary [65] word discovery.\",\n \"Viterbi-style training for such models has also been explored [66].\",\n \"Recently, [134] proposed unsupervised neural HMMs, which can be easily extended to hidden semi-Markov models.\",\n \"In fact, [26] proposed a neural version of hidden semi-Markov models similar to [134].\",\n \"All the above approaches aim to maximize the marginal likelihood of the data.\",\n \"Besides generative models, approaches for training log-linear models in an unsupervised fashion, notably contrastive estimation [126], [107], noise-contrastive estimation [44], and auto-encoders [4], have also been extensively studied.\",\n \"These approaches are designed for discriminative models, and can be readily applied to our segmental models.\",\n \"A segmental model trained in an unsupervised fashion can be used for segment clustering.\",\n \"One major application of segment clustering is for lexical unit discovery.\",\n \"There is a long history of lexical unit discovery from acoustic signals [14], sometimes called spoken term discovery [105], [60].\",\n \"Much of the recent work relies on dynamic time warping (DTW) [59], [15] for spoken term discovery, with [78] being out of the few exceptions who took a nonparametric Bayesian approach.\",\n \"DTW has also been found to help lexical discovery in an HMM system [140].\",\n \"Other related tasks that we do not cover here include unsupervised word segmentation [37], unsupervised part-of-speech tagging [91], and unsupervised dependency parsing [73].\",\n \"Though segmental models were not used in the above studies, many ideas can be borrowed and help advance the study of segmental models in unsupervised settings.\",\n \"The assumption that input sequences can be decomposed into a sequence of segments serves as a strong inductive bias, and it is particularly useful in unsupervised settings when little is assumed about the data.\",\n \"Segmental models are more flexible than frame-based models when additional assumptions, such as the form of the feature functions, are needed.\",\n \"Therefore, we believe segmental models have the potential to perform well in these unsupervised settings.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01572"}}},{"rowIdx":698,"cells":{"text":{"kind":"list like","value":[["Weak solutions for a thermoelectric problem with power-type boundary\n effects"],["Abstract This paper deals with thermoelectric problems including the Peltier and Seebeck effects.","The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects.","To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3).","They are essentially approximated solutions based on the Rothe method.","It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero.","The proof of the existence of time discretized solutions relies on fixed point and compactness arguments.","In this study, we establish quantitative estimates to clarify the smallness conditions."],["Introduction","The study of the heat equation with constant coefficients is a simplification from both mathematical and engineering points of view.","From the real world point of view, constant coefficients are not appropriate because the density and the thermal conductivity both depend on the temperature itself, and often also on the spatial variable.","The concern of discontinuous leading coefficient is being a long matter of study in the mathematical literature, as long as the works [22], [25].","The complete concern is achieved by the doubly quasilinear parabolic equation [1], [3], [5].","It is well known that the determination of estimates is the crucial key in the theory of partial differential equations (PDE), which involve the so-called universal bounds.","With their abstract form, these bounds are only qualitative and they do not have any practical use on the real world applications.","In their majority, if the proof of estimates should be remade step by step, the expression of the qualitative bounds would be truly cumbersome, or even impossible if the contradiction argument is applied.","Also regularity estimates have being a subject of study in the last decades [11], [12], [15], but these ones only occur by admitting data smoothness.","With this in mind, our main objective is to find quantitative estimates, i.e.","their involved constants have an explicit expression, that are useful on the real applications.","In particular, the quantitative estimates clarify the smallness conditions on the data when a fixed point argument is used.","For the two-dimensional space situation, a first attempt on the finding smallness conditions that assure the existence and regularity results for some thermoelectric problems is presented in [9], [10], where some domain dependent constants were kept abstract.","Indeed, the central existence result of a weak solution for a class of elliptic systems on divergence form, which is only 2D valid, is provided under some higher regularity ($W^{1,p}$ regularity, with $p > 2$ ).","The Gehring-type higher integrability technique makes the smallness conditions quite bizarre.","Here, we establish more elegant smallness conditions and they are extended to the $n$ -dimensional space situation, by finding weak solutions.","The present model also extends the thermal effects, of the previous works [9], [10], to the unsteady state.","Existence of solutions for parabolic-elliptic systems with nonlinear no-flux boundary conditions is not a new idea if taking constant coefficients into account [4].","Application of elliptic PDE system in divergence form with Dirichlet boundary conditions in doubly-connected domain of the plane are given in [7] to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour effects.","In [8], the authors deal with a traditional RLC circuit in which a thermistor has been inserted, representing the microwave heating process with temperature-induced modulations on the electric field.","In particular, the existence of a solution to a coupled system of three differential equations (an ODE, an elliptic equation and a nonlinear parabolic PDE) and appropriate initial and boundary conditions is proved.","A one-dimensional thermal analysis for the performance of thermoelectric cooler is conducted in [13] under the influence of the Thomson effect, the Joule heating, the Fourier heat conduction, and the radiation and convection heat transfer.","Simulation studies have been performed to investigate the thermal balance affected by anode shorting in an aluminum reduction cell [6].","The method of discretization in time, whose basic idea (coming from the implicit Euler formula) was investigated by Rothe, is a very well-known effective technique for both theoretical and numerical analysis, [14], [17], [24] and [21], [26], respectively (see also the pioneering work [1] of Alt and Luckhaus).","This paper is organized as follows.","The thermoelectric (TE) model is introduced in Section .","After discussing the physical model, the main result with respect to this model is formulated with a detailed description of the relevant constants.","In Section , one abstract model related to the problem under consideration is introduced to simplify the proofs of the existence results of time-discretized solutions (Section ), and their corresponding steady-state solutions (Section ).","Indeed, the analysis of the problem is structured via two different approaches to exemplify alternative assumptions on the data smallness, namely the existence results of time-discretized solutions (Subsections REF and REF ), and their corresponding steady-state solutions (Subsections REF and REF )."],["The thermoelectric model","Let $[0, T] \\subset {\\mathbb {R}}$ be the time interval with $ T >0$ being an arbitrary (but preassigned) time.","Let $\\Omega $ be a bounded domain (that is, connected open set) in $\\mathbb {R}^n$ ($n\\ge 2$ ).","Its boundary is constituted by two disjoint open $(n-1)$ -dimensional sets $\\partial \\Omega =\\overline{\\Gamma _\\mathrm {N}}\\cup \\overline{\\Gamma }$ .","We consider $\\Gamma _{\\rm N}$ over which the Neumann boundary condition is taken into account, and $\\Gamma $ over which the radiative effects may occur.","Each one, $\\Gamma _{\\rm N}$ and $\\Gamma $ , may be alternatively of zero $(n-1)$ -Lebesgue measure.","Set $Q_T=\\Omega \\times ]0,T[$ and $\\Sigma _T=\\Gamma \\times ]0,T[$ .","The electrical current density $\\bf j$ and the energy flux density ${\\bf J}={\\bf q}+\\phi {\\bf j}$ , with $\\bf q$ being the heat flux vector, are given by the constitutive relations (see [9] and the references therein) ${\\bf q}&= &- k (\\cdot ,\\theta ) \\nabla \\theta -\\Pi (\\cdot ,\\theta ) \\sigma (\\cdot ,\\theta ) \\nabla \\phi ;\\\\{\\bf j}&=& -\\alpha _{\\rm S}(\\cdot ,\\theta ) \\sigma (\\cdot ,\\theta ) \\nabla \\theta -\\sigma (\\cdot ,\\theta ) \\nabla \\phi .$ Here, $\\theta $ denotes the absolute temperature, $\\phi $ is the electric potential, $\\alpha _{\\rm S}$ represents the Seebeck coefficient, and the Peltier coefficient $\\Pi (\\theta )=\\theta \\alpha _{\\rm s}(\\theta )$ is due to the first Kelvin relation.","The electrical conductivity $\\sigma $ , and the thermal conductivity $k=k_{\\rm T}+\\Pi \\alpha _{\\rm s}\\sigma $ , with $k_T$ denotes the purely conductive contribution, are, respectively, the known positive coefficients of Ohm and Fourier laws.","The Seebeck coefficient $\\alpha _{\\rm S}$ has a constant sign corresponding to the Hall effect.","With positive sign ($\\alpha _{\\rm S}>0$ ), there are as examples: the alkali metals Li, Rb and Cs [2], and the noble metals Ag and Au [2] or [20].","With negative sign ($\\alpha _{\\rm S}<0$ ), there are as examples: the alkali metals Na and K [20], the transition metals Fe and Ni [2], and the semiconductor Pb [2].","We refer to [10], and the references therein, for more examples and their increase and decrease behaviors.","Although heat generation starts instantaneously when the current begins to flow, it takes time before the heat transfer process is initiated to allow the transient conditions to disappear.","Thus, the electrical current density $\\bf j$ and the energy flux density ${\\bf J}$ satisfy $&&\\left\\lbrace \\begin{array}{ll}\\nabla \\cdot {\\bf j}=0 &\\mbox{ in }\\Omega \\\\-{\\bf j}\\cdot {\\bf n}=g &\\mbox{ on }\\Gamma _{\\rm N}\\\\{\\bf j}\\cdot {\\bf n}=0 &\\mbox{ on }\\Gamma \\end{array}\\right.\\\\ &&\\left\\lbrace \\begin{array}{ll}\\rho (\\cdot ,\\theta ) c_\\mathrm {v}(\\cdot , \\theta )\\partial _t \\theta -\\nabla \\cdot {\\bf J}=0 &\\mbox{ in } Q_T\\\\{\\bf J}\\cdot {\\bf n}=0 &\\mbox{ on }\\Gamma _{\\rm N}\\times ]0,T[\\\\-{\\bf J}\\cdot {\\bf n}=\\gamma (\\cdot ,\\theta ) |\\theta |^{\\ell -2}\\theta -h &\\mbox{ on }\\Sigma _T,\\end{array}\\right.$ for $\\ell \\ge 2$ .","Here, $\\rho $ denotes the density, $c_\\mathrm {v}$ denotes the heat capacity (at constant volume), $\\bf n$ is the unit outward normal to the boundary $\\partial \\Omega $ , and $g$ denotes the surface current source, The boundary operators, $\\gamma $ and $h$ , are temperature dependent functions that express, respectively, the radiative convection depending on the wavelength, and the external heat sources.","For $\\ell =5$ , the Stefan-Boltzmann radiation law says that $\\gamma (T)=\\sigma _{\\rm SB}\\epsilon (T)$ and $h(T)=\\sigma _{\\rm SB} \\alpha (T) \\theta _\\mathrm {e}^{\\ell -1}$ , where $\\sigma _{\\rm SB}=\\, 5.67\\times 10^{-8}$ W m$^{-2} $ K$^{-4}$ is the Stefan-Boltzmann constant for blackbodies, and $ \\theta _\\mathrm {e}$ denotes an external temperature.","The parameters, the emissivity $\\epsilon $ and the absorptivity $\\alpha $ , both depend on the space variable and the temperature function $\\theta $ .","If $\\ell =2$ , the boundary condition corresponds to the Newton law of cooling with heat transfer coefficient $\\gamma =h/ \\theta _\\mathrm {e}^{\\ell -1}$ .","In the framework of Sobolev and Lebesgue functional spaces, we use the following spaces of test functions: $V &=& \\left\\lbrace \\begin{array}{l}V(\\Omega ) = \\left\\lbrace v\\in H^{1}(\\Omega ):\\ \\int _{\\Omega } v \\mathrm {dx}=0 \\right\\rbrace \\\\V(\\partial \\Omega ) = \\left\\lbrace v\\in H^{1}(\\Omega ):\\ \\int _{\\partial \\Omega } v \\mathrm {ds}=0 \\right\\rbrace \\end{array}\\right.","\\\\V_{\\ell }(\\Omega ) &=&\\left\\lbrace v\\in H^{1}(\\Omega ) :\\ v|_{\\Gamma }\\in L^{\\ell }(\\Gamma ) \\right\\rbrace ;\\\\V_{\\ell }(Q_T) &=&\\left\\lbrace v\\in L^2(0,T; H^{1}(\\Omega )) :\\ v|_{\\Sigma _T }\\in L^{\\ell }(\\Sigma _T) \\right\\rbrace ,$ with their usual norms, $\\ell >1$ .","Notice that $V_{\\ell }(\\Omega )\\equiv H^{1}(\\Omega ) $ if $\\ell \\le 2_*$ , where $ 2_*$ is the critical trace exponent, i.e.","$2_*=2(n-1)/(n-2)$ if $n>2$ and $2_*>1$ is arbitrary if $n=2$ .","In the presence of the previous considerations, the temperature-potential pair does not be expectable to be regular nor even bounded.","The thermoelectric problem is formulated as follows.","(TE) Find the temperature-potential pair $(\\theta ,\\phi )$ such that if it verifies the variational problem: $\\int _0^T\\langle \\rho (\\cdot , \\theta ) c_\\mathrm {v}(\\cdot , \\theta )\\partial _t\\theta , v\\rangle \\mathrm {dt} +\\int _{Q_T} k (\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} +\\nonumber \\\\ +\\int _{Q_T} \\sigma (\\cdot ,\\theta )\\left(T_\\mathcal {M}(\\phi )\\alpha _{\\rm S}(\\cdot ,\\theta )\\nabla \\theta +(\\Pi (\\cdot ,\\theta )+T_\\mathcal {M}(\\phi ))\\nabla \\phi \\right) \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\nonumber \\\\+\\int _{\\Sigma _T} \\gamma (\\cdot ,\\theta )|\\theta |^{\\ell -2} \\theta v\\mathrm {ds} \\mathrm {dt}= \\int _{\\Sigma _T} h(\\cdot ,\\theta ) v\\mathrm {ds} \\mathrm {dt}; \\\\\\int _\\Omega \\sigma (\\cdot ,\\theta )\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+\\int _\\Omega \\sigma (\\cdot ,\\theta ) \\alpha _{\\rm S}(\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\ \\mbox{ a.e.","in } ]0,T[ , $ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ , where $\\langle \\cdot ,\\cdot \\rangle $ accounts for the duality product, and $T_\\mathcal {M}$ is the $\\mathcal {M}$ -truncation function defined by $T_\\mathcal {M}(z)=\\max (-\\mathcal {M},\\min (\\mathcal {M},z))$ .","We assume the following.","(H1) The density and the heat capacity $\\rho , c_\\mathrm {v}: \\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions, i.e.","measurable with respect to $x\\in \\Omega $ and continuous with respect to $e\\in \\mathbb {R}$ .","Furthermore, they verify $\\exists b^\\#,b_\\#>0: \\quad b_\\#\\le \\rho (x,e)c_\\mathrm {v}(x,e)\\le b^\\# ,\\quad \\mbox{for a.e. }","x\\in \\Omega ,\\quad \\forall e \\in \\mathbb {R} .","$ (H2) The thermal and electrical conductivities $k,\\sigma :\\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions.","Furthermore, they verify $\\exists k^\\#,k_\\#>0:&&k_\\#\\le k(x,e)\\le k^\\#;\\\\\\exists \\sigma ^\\#, \\sigma _\\#>0:&&\\sigma _\\#\\le \\sigma (x,e)\\le \\sigma ^\\#\\quad \\mbox{for a.e. }","x\\in \\Omega ,\\quad \\forall e \\in \\mathbb {R}.","$ (H3) The Seebeck and Peltier coefficients $\\alpha _{\\rm S},\\Pi :\\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions such that $\\exists \\alpha ^\\#>0: &&|\\alpha _{\\rm S}(x,e)|\\le \\alpha ^\\#; \\\\\\exists \\Pi ^\\#>0: &&|\\Pi (x,e)|\\le \\Pi ^\\#,\\quad \\mbox{for a.e. }","x\\in \\Omega ,\\quad \\forall e\\in \\mathbb {R}.$ (H4) The boundary function $h$ belongs to $L^{\\ell ^{\\prime }}(\\Sigma _T)$ .","(H5) The boundary function $g$ belongs to $ L^{2}(\\Gamma _{\\rm N})$ .","(H6) The boundary operator $\\gamma $ is a Carathéodory function from $\\Sigma _T\\times \\mathbb {R}$ into $\\mathbb {R}$ such that $\\exists \\gamma _\\#, \\gamma ^\\#>0:\\quad \\gamma _\\#\\le \\gamma (x,t,e)\\le \\gamma ^\\#; \\quad \\mbox{for a.e. }","(x,t)\\in \\Sigma _T,\\quad \\forall e\\in \\mathbb {R}.","$ Moreover, $\\gamma $ is strongly monotone: $\\left(\\gamma ( u )| u |^{\\ell -2} u- \\gamma ( v )| v |^{\\ell -2} v \\right) ( u- v)\\ge \\gamma _\\# |u - v|^\\ell .$ Let us state our main existence theorem.","Theorem 2.1 Let (H1)-(H6) be fulfilled.","The thermoelectric problem (TE) admits a solution $(\\theta ,\\phi )\\in V_\\ell (Q_T)\\times L^2(0,T; V)$ , for $\\mathcal {M}$ being such that $\\mathcal {M}\\alpha ^\\#\\sigma ^\\#< k_\\#,$ and one of the following hypothesis is assured: there holds $4( k_\\# - \\mathcal {M}\\alpha ^\\#\\sigma ^\\# ) \\sigma _\\# > (\\sigma ^\\#)^2 ( \\Pi ^\\#+\\mathcal {M} +\\alpha ^\\#)^2 ;$ there holds $4( k_\\# - \\mathcal {M}\\alpha ^\\#\\sigma ^\\# ) >\\sigma ^\\# ( \\Pi ^\\#+\\mathcal {M} +\\alpha ^\\#)^2 ;$ there holds $k_\\#> \\sigma ^\\# \\alpha ^\\# ( 2 \\Pi ^\\#+ 3 \\mathcal {M}) .$"],["Existence of approximated solutions","The thermoelectric problem provides the abstract initial boundary value problem $b(\\theta )\\partial _t \\theta -\\nabla \\cdot \\left( a ( \\theta ,\\phi )\\nabla \\theta \\right)= \\nabla \\cdot ( \\sigma ( \\theta ) F (\\theta ,\\phi )\\nabla \\phi ) && \\\\-\\nabla \\cdot (\\sigma ( \\theta )\\nabla \\phi )= \\nabla \\cdot \\left( \\sigma ( \\theta ) \\alpha _\\mathrm {S} (\\theta ) \\nabla \\theta \\right) && \\mbox{ in }Q_T ; \\\\\\left( a ( \\theta ,\\phi )\\nabla \\theta + \\sigma ( \\theta ) F ( \\theta ,\\phi )\\nabla \\phi \\right)\\cdot \\mathbf {n} =\\left(h - \\gamma (\\theta ) |\\theta |^{\\ell -2} \\theta \\right)\\chi _{\\Gamma } && \\\\\\left( \\sigma ( \\theta )\\nabla \\phi + \\sigma ( \\theta ) \\alpha _\\mathrm {S} (\\theta )\\nabla \\theta \\right)\\cdot {\\bf n} =g \\chi _{\\Gamma _\\mathrm {N}}&&\\mbox{ on } \\partial \\Omega \\times ]0,T[ .","$ This abstract problem is formulated in the form that the coefficients are correlated with the leading coefficient $\\sigma $ .","We emphasize that this interrelation must be clear.","Let us assume the hypothesis set.","(H) The operators $ a,F $ and $b,\\sigma , \\alpha _\\mathrm {S}$ are Carathéodory functions from $\\Omega \\times \\mathbb {R}^2$ and $\\Omega \\times \\mathbb {R}$ , respectively, into $\\mathbb {R}$ , which enjoy the following properties.","There exist positive constants $F^\\#, a_\\#,a^\\#, b_\\#,b^\\#$ such that $| F(x,e,d )|\\le F^\\#; && \\\\a_\\#\\le a(x,e,d )\\le a^\\#; && \\\\b_\\#\\le b (x,e )\\le b^\\# &&\\mbox{for a.e. }","x\\in \\Omega ,\\quad \\forall e,d \\in \\mathbb {R} , $ and $\\sigma _\\#,\\sigma ^\\#, \\alpha ^\\#$ verifying (), (REF ), respectively.","Definition 3.1 We say that $(\\theta ,\\phi )$ is a weak solution to (REF )-() if it solves the variational problem $\\int _0^T\\langle b(\\cdot , \\theta )\\partial _t\\theta , v\\rangle \\mathrm {dt} +\\int _{Q_T} a (\\cdot ,\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma (\\cdot ,\\theta )|\\theta |^{\\ell -2} \\theta v\\mathrm {ds} \\mathrm {dt}=\\nonumber \\\\ = - \\int _{Q_T}\\sigma ( \\theta ) F (\\cdot ,\\theta ,\\phi )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\int _{\\Sigma _T} h(\\cdot ,\\theta ) v\\mathrm {ds} \\mathrm {dt}; \\\\\\int _\\Omega \\sigma (\\cdot ,\\theta )\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\cdot , \\theta ) \\alpha _\\mathrm {S} (\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla w\\mathrm {dx} =\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\mbox{ a.e.","in } ]0,T[ ,$ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ .","We define an auxiliary operator.","Denote by $B$ the operator from $H^1(\\Omega )$ into $L^2(\\Omega )$ defined by $B(v)=\\int _0^v b(\\cdot ,z)\\mathrm {dz},$ for all $v\\in H^1(\\Omega )$ .","Different approaches in the finding of solutions according to Definition REF provide different smallness conditions (REF ), (REF ) or (REF ).","We emphasize that the difference between these smallness conditions has its importance in the real-world applications.","Theorem 3.1 Let (H) and (H4)-(H6) be fulfilled.","If there exists $\\varepsilon >0$ such that one the following relation holds, that is, either $a_\\# >\\varepsilon \\sigma ^\\#(F^\\#+\\alpha ^\\#)/2\\quad \\mbox{ and } \\quad \\varepsilon \\sigma _\\# >\\sigma ^\\#(F^\\#+\\alpha ^\\#)/2,$ or $a_\\# >\\varepsilon \\sqrt{\\sigma ^\\#}(F^\\#+\\alpha ^\\#)/2 \\quad \\mbox{ and } \\quad \\varepsilon >\\sqrt{\\sigma ^\\#}(F^\\#+\\alpha ^\\#)/2, $ then the variational problem (REF )-() admits a sequence of approximate solutions $\\lbrace (\\theta _M,\\phi _M)\\rbrace _{M\\in \\mathbb {N}}$ in the sense established in Section REF .","The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx} =\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _{\\Gamma } h_{m} v \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (\\theta ^m)\\nabla \\phi ^m\\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\theta ^m) \\alpha _\\mathrm {S}(\\theta ^m)\\nabla \\theta ^{m} \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ where $\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\in \\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).","We call $\\phi ^m$ the corresponding solution to the time independent temperature $\\theta ^{m}$ .","Theorem 3.2 Let (H) and (H4)-(H6) be fulfilled.","If there holds $a_\\#>2\\sigma ^\\# \\alpha ^\\# F^\\# ,$ then the variational problem (REF )-() admits a sequence of approximate solutions $\\lbrace (\\theta _M,\\phi _M)\\rbrace _{M\\in \\mathbb {N}}$ in the sense established in Section REF .","The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx} =\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _\\Gamma h_{m} v \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (\\theta ^{m-1})\\nabla \\phi ^m\\cdot \\nabla w\\mathrm {dx}= - \\int _\\Omega \\sigma (\\theta ^{m-1}) \\alpha _\\mathrm {S}(\\theta ^{m-1})\\nabla \\theta ^{m-1} \\cdot \\nabla w \\mathrm {dx} +\\nonumber \\\\ + \\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ where $\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\in \\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).","We call $\\phi ^m$ the corresponding solution to the time independent temperature $\\theta ^{m-1}$ ."],["Steady-state solvability","In this section, we prove the existence of solutions to the recurrent sequence of time-discretized problems (REF )-() and (REF )-() in Subsections REF and REF , respectively.","Since $m\\in \\mathbb {N}$ is fixed and $\\theta ^{m-1}\\in V_{\\ell }(\\Omega )$ is given, for the sake of simplicity, we set $f= B(\\theta ^{m-1})$ and $H=h_m$ , and we omit the index to the unknown pair, i.e.","we simply write $(\\theta ,\\phi )$ .","Denoting by $K_{2}$ the continuity constant of the trace embedding $H^{1}(\\Omega )\\hookrightarrow L^2(\\Gamma )$ , with $2_*=2(n-1)/(n-2)$ if $n>2$ , and any $2_*>2$ if $n=2$ , and by $P_2$ the Poincaré constant correspondent to the space exponent 2, the constant $K_2(P_2+1)$ obeys $\\Vert v\\Vert _{2,\\Gamma }\\le K_2 \\left( \\Vert v\\Vert _{2,\\Omega }+ \\Vert \\nabla v\\Vert _{2,\\Omega }\\right)\\le K_2(P_2+1) \\Vert \\nabla v\\Vert _{2,\\Omega },\\quad \\forall v\\in H^1(\\Omega ).$ Let us introduce [1], [16] $\\Psi (s) := B(s)s-\\int _0^s B(r)\\mathrm {dr} = \\int _0^s(B(s) - B(r) )\\mathrm {dr}.$ We state the main properties of the auxiliary operators $B$ and $\\Psi $ , the ones that we will use later.","For completeness sake, we sketch the proof of the property (REF ).","Lemma 4.1 There holds $\\int _\\Omega (B(u)-B(v) ) u \\mathrm {dx} \\ge \\int _\\Omega \\Psi (u)\\mathrm {dx} - \\int _\\Omega \\Psi (v)\\mathrm {dx} .$ In particular, if the assumption () is fulfilled then there holds $\\int _\\Omega \\Psi (u)\\mathrm {dx}\\le \\int _\\Omega B(u) u \\mathrm {dx} \\le b^\\# \\Vert u\\Vert _{2,\\Omega }^2 .$ Under the assumption () the operator $B$ verifies $(B(u)-B(v),u-v)\\ge b_\\# \\Vert u-v\\Vert _{2,\\Omega }^2 .$ Let us write the decomposition $(B(u)-B(v) ) u = B(u)u-B(v) v - B(v)(u-v).$ Thanks to the mean value theorem for definite integrals, there exists $c$ between $u$ and $v$ such that $\\int _{v} ^{u} B(r)\\mathrm {dr}= B(c)(u-v).$ Since $-B$ is a decreasing function, we obtain $\\int _\\Omega (B(u)-B(v) ) u \\mathrm {dx} \\ge \\int _\\Omega (B(u)u-B(v) v )\\mathrm {dx} -\\int _\\Omega \\int _{v} ^{u} B(r)\\mathrm {dr} \\mathrm {dx},$ which concludes the proof by definition of $\\Psi $ .","Finally, we recall the following remarkable lemma [1].","Lemma 4.2 Suppose $u_m$ weakly converge to $u$ in $L^p(0,T;W^{1,p}(\\Omega ))$ , $p>1$ , with the estimates $\\int _\\Omega \\Psi (u_m(t)) \\mathrm {dx}\\le C\\quad \\mbox{for } 00$ $\\int _0^{T-z}\\int _\\Omega (B(u_m(t+z))-B(u_m(t)) ) (u_m(t+z)-u_m(t)) \\mathrm {dx} \\mathrm {dt} \\le Cz ,$ with $C$ being positive constants.","Then, $B(u_m) \\rightarrow B(u)$ in $L^1(Q_T)$ and $\\Psi (u_m) \\rightarrow \\Psi (u) $ almost everywhere in $Q_T$ ."],["Fixed point argument (solvability to (","Let $\\ell \\ge 2$ , and define an operator $\\mathcal {T}$ from $\\mathbf {V}_{\\ell } =V_\\ell (\\Omega )\\times V$ into itself such that $(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ is the unique solution of Proposition REF .","Proposition 4.1 Let $\\mathbf {u} =(u_1,u_2)\\in \\mathbf {V}_{\\ell }$ , and $u=u_1$ .","Then, there exists a unique solution $(\\theta ,\\phi )\\in \\mathbf {V}_{\\ell }$ to the Neumann-power type elliptic problem $\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta v\\mathrm {dx}+\\int _\\Omega a( \\mathbf {u} )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}+\\int _\\Omega \\sigma (u) F( \\mathbf {u}) \\nabla \\phi \\cdot \\nabla v\\mathrm {dx} +\\nonumber \\\\+\\int _\\Gamma \\gamma (u) |\\theta |^{\\ell -2} \\theta v \\mathrm {ds} =\\frac{1}{\\tau }\\int _\\Omega fv \\mathrm {dx} +\\int _\\Gamma Hv \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (u)\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (u) \\alpha _\\mathrm {S}(u) \\nabla \\theta \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ and $w\\in V$ .","In addition, the following estimate $\\frac{b_\\#}{2\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2 + (L_{1})_\\#\\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2 +\\frac{(L_{2})_\\# }{2}\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\frac{ 1 }{ 2\\tau b_\\# }\\Vert f\\Vert _{2,\\Omega }^2+ \\nonumber \\\\ +\\frac{1}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}} \\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+\\frac{(K_2)^2(P_2+1)^2}{2 (L_{2})_\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N} } ^2 := \\mathcal {R}( \\Vert f\\Vert _{2,\\Omega }^2 ,\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} ) \\quad $ holds true, if provided by one the following definition $&&\\left\\lbrace \\begin{array}l(L_{1})_\\# = a_\\#- \\varepsilon \\sigma ^\\# \\left( F^\\# + \\alpha ^\\#\\right) /2\\\\(L_{2})_\\# = \\sigma _\\# - \\sigma ^\\# \\left( F^\\# + \\alpha ^\\#\\right) /(2\\varepsilon )\\end{array}\\right.","\\\\ &&\\left\\lbrace \\begin{array}l(L_{1})_\\# = a_\\#- \\varepsilon \\sqrt{\\sigma ^\\# }( F^\\#+\\alpha ^\\#) /2\\\\(L_{2})_\\# = \\sigma _\\#\\left( 1- \\sqrt{\\sigma ^\\# }( F^\\#+\\alpha ^\\#) /(2\\varepsilon ) \\right)\\end{array}\\right.",".$ The existence of a solution to the variational system (REF )-() relies on the direct application of the Browder-Minty Theorem [18].","Indeed, the form $\\mathcal {F}: \\mathbf {V}_{\\ell }\\rightarrow \\mathbb {R}$ defined by $\\mathcal {F}(v,w)=\\frac{1}{\\tau }\\int _\\Omega fv\\mathrm {dx}+\\int _\\Gamma Hv \\mathrm {ds} +\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds}$ is continuous and linear, and the form $\\mathcal {L}: \\mathbf {V}_{\\ell }\\times \\mathbf {V}_{\\ell }\\rightarrow \\mathbb {R}$ defined by $\\mathcal {L} \\left( (\\theta ,\\phi ), (v,w)\\right)=\\frac{1}{\\tau }\\int _\\Omega b(u) \\theta v\\mathrm {dx}+\\int _\\Omega \\left(\\mathsf {L}( \\mathbf {u} )\\nabla \\left[\\begin{array}{c}\\theta \\\\\\phi \\end{array} \\right]\\right)\\cdot \\nabla \\left[\\begin{array}{c} v\\\\w\\end{array} \\right]\\mathrm {dx},$ is continuous and bilinear, with $\\mathsf {L}$ being the $(2\\times 2)$ -matrix $\\mathsf {L} ( \\mathbf {u} )=\\left[\\begin{array}{cc}a( \\mathbf {u} )& \\sigma (u) F( \\mathbf {u} ) \\\\\\sigma (u) \\alpha _\\mathrm {S} (u)& \\sigma (u)\\end{array}\\right] .$ Moreover, $\\mathcal {L}$ is coercive: $\\sum _{i,j=1}^{ 2}\\sum _{l=1}^n \\left( L_{i,j} ( \\mathbf {u})\\xi _{j,l}\\right)\\xi _{l,i}\\ge (L_{1})_\\#|\\xi _1|^2+ (L_{2})_\\#|\\xi _2|^2,$ with $(L_{1})_\\#$ and $(L_{2})_\\#$ being the positive constants defined in (REF ) or (), taking the assumptions (REF ) and (REF ) into account.","The difference of the definitions is consequence of the different application of the Young inequality $2AB\\le \\varepsilon A^2+B^2/\\varepsilon $ ($\\varepsilon ,A,B>0$ ), see Remark REF .","Namely, with $A=| \\xi _{1}| $ and $B=| \\xi _{2}|$ , for (REF ).","That is, $ \\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right) \\le \\sigma ^\\# \\left( F^\\#+\\alpha ^\\# \\right)\\left( \\frac{\\varepsilon }{2}A^2+\\frac{1}{2\\varepsilon } B^2\\right).$ $A=| \\xi _{1}| $ and $B=\\sqrt{\\sigma (u)}| \\xi _{2}| $ , for ().","That is, $ \\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right)\\le \\sqrt{\\sigma ^\\# }\\left( F^\\#+\\alpha ^\\# \\right)\\left( \\frac{\\varepsilon }{2}A^2+\\frac{ 1}{2\\varepsilon }B^2\\right).$ Finally, observing that the function $e\\in \\mathbb {R}\\mapsto \\gamma (u)|e|^{\\ell -2}e $ is monotonically increasing, we conclude the existence of the required solution.","In order to obtain (REF ), we take $v=\\theta $ and $w=\\phi $ as test functions in (REF ) and (), respectively.","Summing the obtained relations, and applying (), (REF ), the coercivity (REF ) of $\\mathsf {L}$ , and the Hölder inequality, we find $\\frac{ b_\\# }{\\tau } \\Vert \\theta \\Vert _{2,\\Omega }^2+ (L_{1})_\\# \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+(L_{2})_\\#\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\gamma _\\# \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\nonumber \\\\ \\le \\frac{ 1 }{\\tau } \\Vert f\\Vert _{2,\\Omega }\\Vert \\theta \\Vert _{2,\\Omega }+\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }\\Vert \\theta \\Vert _{\\ell ,\\Gamma }+\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}\\Vert \\phi \\Vert _{2,\\Gamma _\\mathrm {N}}.$ We successively apply (REF ) and the Young inequality to obtain $\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma } \\Vert \\theta \\Vert _{\\ell ,\\Gamma }+\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N} }\\Vert \\phi \\Vert _{2,\\Gamma _\\mathrm {N}} \\le \\nonumber \\\\\\le \\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} }\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+\\frac{\\gamma _\\#}{\\ell }\\Vert \\theta \\Vert _{\\ell ,\\Gamma }^\\ell +\\frac{K_2^2(P_2+1)^2}{ 2(L_{2})_\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2+ \\frac{(L_{2})_\\#}{2}\\Vert \\nabla \\phi \\Vert _{2,\\Omega } ^2.","$ Inserting (REF ) into (REF ), we deduce (REF ).","Remark 4.1 Even $\\varepsilon >0$ may be an arbitrary (but fixed) number, we may differently define $(L_1)_\\#$ and $(L_2)_\\#$ .","Indeed, the Young inequality $2AB\\le \\varepsilon A^2+B^2/\\varepsilon $ ($\\varepsilon ,A,B>0$ ) may be applied to obtain $\\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right) \\le \\sigma ^\\# \\left( F^\\#\\left( \\frac{\\varepsilon _1}{2}| \\xi _{1}|^2+\\frac{1}{2\\varepsilon _1} | \\xi _{2}|^2\\right)+\\right.","\\\\ \\left.","+\\alpha ^\\#\\left( \\frac{\\varepsilon _2}{2}| \\xi _{1}|^2+\\frac{1}{2\\varepsilon _2} | \\xi _{2}|^2\\right) \\right).$ Next, let us determine whose radius make possible that the operator $\\mathcal {T}$ maps a closed ball into itself.","Proposition 4.2 For $R=\\max \\lbrace R_1,R_2\\rbrace $ with $R_1$ and $R_2$ being defined in (REF ) and (REF ), respectively, the operator $\\mathcal {T}$ verifies $\\mathcal {T}(K)\\subset K$ , with $K=\\left\\lbrace (v,w)\\in \\mathbf {V}_\\ell : \\ \\Vert \\nabla w\\Vert _{2,\\Omega }+\\Vert \\nabla v\\Vert _{2,\\Omega }+ \\Vert v\\Vert _{\\ell ,\\Gamma } \\le R\\right\\rbrace .$ Let $\\mathbf {u}\\in \\mathbf {V}_{\\ell }$ , $u=u_1$ and $(\\theta ,\\phi )$ be the unique solution of Proposition REF , i.e.","$(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ .","In order to prove that $(\\theta ,\\phi )\\in K$ we consider two different cases: (1) if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le 1$ ; and (2) if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } >1$ , if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le 1$ , then there holds $\\Vert \\nabla \\phi \\Vert _{2,\\Omega }+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }+ \\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le \\sqrt{2}\\left(\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+ \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2\\right)^{1/2}+1,$ by applying the elementary inequality $(a+b)^2\\le 2(a^2+b^2)$ for every $a,b\\ge 0$ .","By using (REF ), we may take $R_1=\\left(\\frac{2\\mathcal {R}}{\\min \\left\\lbrace (L_{1})_\\#, (L_{2})_\\#/2\\right\\rbrace } \\right)^{1/2}+1 .","$ if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } > 1$ , then using $\\ell \\ge 2$ there holds $\\Vert \\nabla \\phi \\Vert _{2,\\Omega }+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }+ \\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le \\sqrt{2}\\left( 2(\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2) + \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\right)^{1/2},$ by applying the elementary inequality $(a+b)^2\\le 2(a^2+b^2)$ for every $a,b\\ge 0$ .","By using (REF ), we may take $R_2^2=\\left(\\frac{2 }{\\min \\left\\lbrace (L_{1})_\\#, (L_{2})_\\#/2\\right\\rbrace } +\\frac{\\ell ^{\\prime }}{\\gamma _\\#}\\right)\\mathcal {R} .","$ Then, the proof is complete by taking $R$ such that is the maximum of $R_1$ and $R_2$ defined in (REF ) and (REF ), respectively.","Proposition 4.3 The operator $\\mathcal {T}$ is continuous.","Let $\\lbrace \\mathbf {u}^m\\rbrace _{m\\in \\mathbb {N}}$ be a sequence such that weakly converges to $\\mathbf {u}=(u,u_2)$ in $ \\mathbf {V}_\\ell $ , and $(\\theta _m,\\phi _m)=\\mathcal {T}(\\mathbf {u}^m)$ for each $m\\in \\mathbb {N}$ .","Proposition REF guarantees that $(\\theta _m,\\phi _m)$ solves, for each $m\\in \\mathbb {N}$ , the variational system (REF )$_m$ -()$_m$ , with $\\mathbf {u}$ replaced by $\\mathbf {u}^m$ .","The uniform boundedness ensured by Proposition REF guarantees the existence of a limit $(\\theta ,\\phi )\\in \\mathbf {V}_\\ell $ , for at least a subsequence of $(\\theta _m,\\phi _m)$ still denoted by $(\\theta _m,\\phi _m)$ , such that $\\theta _m\\rightharpoonup \\theta \\quad \\mbox{in } V_\\ell (\\Omega )\\quad \\mbox{and}\\quad \\phi _m\\rightharpoonup \\phi \\quad \\mbox{in } V \\quad (\\mbox{as } m\\rightarrow +\\infty ) .$ The Rellich-Kondrachov theorem guarantees the strong convergences $u^m\\rightarrow u &\\mbox{ and }& u_2^m\\rightarrow u_2 \\quad \\mbox{in } L^2(\\Omega ); \\\\\\theta _m\\rightarrow \\theta &\\mbox{ and }& \\phi _m\\rightarrow \\phi \\quad \\mbox{in } L^2(\\Omega ); \\\\u^m\\rightarrow u &\\mbox{ and }& \\theta _m\\rightarrow \\theta \\quad \\mbox{in } L^2(\\Gamma ).$ To show that $(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ , it remains to pass to the limit in the system (REF )$_m$ -()$_m$ as $m$ tends to infinity.","Applying the Krasnoselski theorem to the Nemytskii operators $b$ , $a$ , $\\sigma $ , we have $b(u^m) v\\rightarrow b(u) v &\\mbox{ in }& L^2(\\Omega );\\\\a(\\mathbf {u}^m) \\nabla v\\rightarrow a(\\mathbf {u}) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(\\Omega ); \\\\\\sigma (u^m)\\nabla v\\rightarrow \\sigma ( u)\\nabla v &\\mbox{ in }& \\mathbf {L}^2(\\Omega ),$ for all $v\\in H^1(\\Omega )$ , making use of the Lebesgue dominated convergence theorem and the assumptions () and ()-().","Also the terms $\\sigma (u^m)F(\\mathbf {u}^m) \\nabla v$ and $ \\sigma (u^m) \\alpha _\\mathrm {S} (u^m) \\nabla v$ pass to the limit making recourse to the assumptions (REF ) and (REF ), respectively.","Similarly, the boundary term $\\gamma (u^m)v$ converges to $\\gamma (u) v$ in $L^{\\ell ^{\\prime }}(\\Gamma )$ , for all $v\\in L^{\\ell ^{\\prime }}(\\Gamma )$ , due to (REF ).","Observe that $ \\theta _m$ strongly converges to $\\theta $ in $L^p(\\Gamma )$ , for all $10$ , and finally letting $\\delta \\rightarrow 0^+$ we arrive to $\\int _{\\Gamma }\\gamma (u)(\\Lambda - |\\theta |^{\\ell -2}\\theta )\\varphi \\mathrm {ds} \\ge 0, \\quad \\forall \\varphi \\in \\mathcal {D}(\\Gamma ),$ which implies that $\\Lambda = |\\theta |^{\\ell -2}\\theta $ .","Thus, we are in the condition of concluding that $(\\theta ,\\phi )$ is the required limit solution, i.e.","it solves the limit system (REF )-().","Thanks to Propositions REF , REF and REF , there exists at least one fixed point of $\\mathcal {T}$ , that is $(\\theta ,\\phi )=\\mathcal {T}(\\theta ,\\phi )$ , which concludes the solvability to (REF )-()."],["Fixed point argument (solvability to (","Let $\\ell \\ge 2$ , and define an operator $\\mathcal {T}$ from $V_{\\ell }(\\Omega )$ into itself such that $\\theta =\\mathcal {T}(u)$ is the unique solution of Proposition REF .","Denote by the well defined continuous operator such that $\\mathcal {F}( u)=\\phi $ .","The existence of a unique weak auxiliary solution $\\phi $ to the variational equality () is standard and it can be stated as follows.","Proposition 4.4 Let $u\\in H^1(\\Omega )$ .","Under the assumptions (), (REF ) and (H5), the Neumann problem $\\int _\\Omega \\sigma (u) \\nabla \\phi \\cdot \\nabla w\\mathrm {dx}=\\int _\\Omega \\sigma (u) \\alpha _\\mathrm {S} (u)\\nabla u\\cdot \\nabla w \\mathrm {dx}+\\int _{\\Gamma _\\mathrm {N}} gw \\mathrm {ds},\\qquad \\forall w\\in V,$ admits a unique solution $\\phi \\in V$ .","Moreover, the estimate $\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega }\\le \\sqrt{\\sigma ^\\#}\\alpha ^\\# \\Vert \\nabla u\\Vert _{2,\\Omega }+\\frac{K_2 (P_2+1) }{\\sqrt{\\sigma _\\#}}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}$ holds true.","Let us establish the quantitative estimate (REF ).","We take $w=\\phi $ as a test function in (REF ), and we compute by applying the Hölder inequality and (REF ) $\\Vert \\sqrt{\\sigma (u)}\\nabla \\phi \\Vert _{2,\\Omega } ^2 \\le \\left( \\alpha ^\\#\\Vert \\sqrt{\\sigma (u)}\\nabla u\\Vert _{2,\\Omega }+\\frac{ K_2(P_2+1)}{\\sqrt{\\sigma _\\#}} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}\\right)\\Vert \\sqrt{\\sigma (u)}\\nabla \\phi \\Vert _{2,\\Omega } .$ Then, (REF ) arises.","Proposition 4.5 Let $\\mathbf {u} =(u,\\phi ) \\in \\left( H^1(\\Omega ) \\right)^2$ .","Under the assumptions (), (REF ) and (REF )-(), there exists a unique solution $\\theta \\in V_\\ell (\\Omega )$ to the power type elliptic problem $\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta v\\mathrm {dx}+\\int _\\Omega a (\\mathbf {u} )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}+\\int _\\Omega \\sigma (u) F ( \\mathbf {u} ) \\nabla \\phi \\cdot \\nabla v\\mathrm {dx} +\\nonumber \\\\+\\int _\\Gamma \\gamma (u) |\\theta |^{\\ell -2}\\theta v \\mathrm {ds} =\\frac{1}{\\tau }\\int _\\Omega fv \\mathrm {dx} +\\int _\\Gamma Hv \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ .","If $\\phi \\in V$ satisfies (REF ), then the following estimate $\\frac{b_\\#}{2\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2 +\\frac{ a_\\#}{2} \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\frac{ 1 }{ 2\\tau b_\\# }\\Vert f\\Vert _{2,\\Omega }^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}}\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+ \\nonumber \\\\ +\\frac{(F^\\#)^2\\sigma ^\\#}{ a_\\#}\\left( \\sigma ^\\#(\\alpha ^\\# )^2\\Vert \\nabla u \\Vert _{2,\\Omega }^2 +\\frac{K_2 ^2(P_2+1)^2 }{\\sigma _\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2\\right) .","$ holds true.","Taking $v=\\theta $ as a test function in (REF ), and applying (), (), (REF ), (REF ), and the Hölder inequality, we find $\\frac{b_\\#}{\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2+ a_\\# \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+\\gamma _\\# \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\\\ \\le \\frac{ 1 }{\\tau } \\Vert f\\Vert _{2,\\Omega }\\Vert \\theta \\Vert _{2,\\Omega }+F^\\#\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega } \\Vert \\sqrt{\\sigma (u)} \\nabla \\theta \\Vert _{2,\\Omega }+\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }\\Vert \\theta \\Vert _{\\ell ,\\Gamma } .$ Applying (REF ) and the Young inequality, we compute $\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega }\\Vert \\sqrt{\\sigma (u)} \\nabla \\theta \\Vert _{2,\\Omega }\\le \\frac{a_\\#}{2} \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2 + \\\\+\\frac{\\sigma ^\\# }{a_\\#} \\left(\\sigma ^\\# (\\alpha ^\\#)^2 \\Vert \\nabla u\\Vert _{2,\\Omega } ^2+\\frac{K_2^2(P_2+1) ^2}{\\sigma _\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 \\right).$ Then, arguing as in (REF ), we deduce (REF ).","Next, let us determine whose radius make possible that the operator $\\mathcal {T}$ maps a closed ball into itself.","Proposition 4.6 Let (REF ) be fulfilled.","For $\\tau \\le a_\\# / b_\\#$ and $R>0$ being defined as in (REF ), the operator $\\mathcal {T}$ verifies $\\mathcal {T}(\\overline{B_R})\\subset \\overline{B_R}$ , with $B_R$ denoting the open ball of $H^1(\\Omega )$ with radius $R$ .","Let $u\\in H^1(\\Omega )$ and $\\theta =\\mathcal {T}(u)$ be the unique solution according to Proposition REF .","Considering (REF ) and $\\min \\left\\lbrace b_\\#/\\tau ,a_\\#\\right\\rbrace =a_\\#$ , the proof is complete by defining $R$ such that $R\\left(\\sqrt{ a_\\# }-2\\frac{F^\\# \\sigma ^\\#\\alpha ^\\# }{ \\sqrt{a_\\#}} \\right) &=&\\left(\\frac{ 1 }{\\tau b_\\# }\\Vert f \\Vert _{2,\\Omega }^2 +\\frac{2}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}} \\Vert H \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} \\right)^{1/2}+\\nonumber \\\\&&+ 2F^\\# K_2(P_2+1) \\sqrt{\\frac{\\sigma ^\\#}{ a_\\#\\sigma _\\#}}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}} , \\quad $ taking the assumption (REF ) be account.","Proposition 4.7 Let $\\lbrace u_m\\rbrace _{m\\in \\mathbb {N}}$ be a sequence such that weakly converges to $u$ in $ H^1(\\Omega )$ , then the solution $(\\theta _m,\\phi _m)$ according to Propositions REF and REF weakly converges in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ , and its limit is a solution according to Propositions REF and REF Let $(\\theta _m,\\phi _m)$ be the solution according to Propositions REF and REF and corresponding to $u_m$ for each $m\\in \\mathbb {N}$ .","The estimates (REF ) and (REF ) guarantee that the sequence $(\\theta _m,\\phi _m)$ is uniformly bounded in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ .","Thus, we can extract a subsequence of $(\\theta _m,\\phi _m)$ still denoted by $(\\theta _m,\\phi _m)$ , weakly convergent to $(\\theta ,\\phi )$ in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ .","Similar arguments in the proof of Proposition REF the weak limit $(\\theta ,\\phi )$ solves the variational system consisting of (REF ) and (REF ), which concludes the proof of Proposition REF .","Thanks to Propositions REF , REF , REF and REF , there exists at least one fixed point of $\\mathcal {T}:u\\mapsto (u, \\mathcal {F}(u))\\mapsto \\theta ,$ that is $\\theta =\\mathcal {T}(\\theta )$ and $\\phi =\\mathcal {F}(\\theta )$ , which concludes the solvability to (REF )-()."],["Time discretization technique","In this section, we apply the method of discretization in time [17], [23], [24].","We decompose the time interval $I=[0,T]$ into $M$ subintervals $I_{m,M}$ of size $\\tau $ such that $M=T/\\tau \\in \\mathbb {N}$ , i.e.","$I_{m,M}=[(m-1)T/M,mT/M]$ for $m\\in \\lbrace 1,\\cdot \\cdot \\cdot ,M\\rbrace $ .","We set $t_{m,M}=mT/M$ .","Thus, the problem (REF ) is approximated by the following recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx}=\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _\\Gamma h(t_{m,M}) v \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ , and the problem () is approximated by either () or () for all $w\\in V$ , corresponding to the two different approaches.","The existence of weak solutions pair $(\\theta ^m,\\phi ^m) \\in V_{\\ell }(\\Omega )\\times V$ to the above systems of elliptic problems is established in Section with $H =h(t_{m,M})$ .","Since $\\theta ^0\\in L^2(\\Omega )$ is known, we determine $(\\theta ^{1},\\phi ^1)$ as the unique solution of the Neumann-power type elliptic problems (REF )-() or (REF )-(), and we inductively proceed.","Denote by $\\lbrace {\\theta }_M\\rbrace _{M\\in \\mathbb {N}}$ , $\\lbrace {\\phi }_M\\rbrace _{M\\in \\mathbb {N}}$ and $\\lbrace Z_M\\rbrace _{M\\in \\mathbb {N}}$ the sequences of the (piecewise constant in time) functions, $ {\\theta }_M : [0,T] \\rightarrow V_\\ell (\\Omega )$ , $ {\\phi }_M : ]0,T] \\rightarrow V$ and $ Z_M:[0,T]\\rightarrow L^2(\\Omega )$ , defined by, respectively, a.e.","in $\\Omega $ ${\\theta }_M (t) &:=& \\left\\lbrace \\begin{array}{ll}\\theta ^{0}&\\mbox{ for }t=0\\\\\\theta ^{m}&\\mbox{ for } t\\in ]t_{m-1,M},t_{m,M}]\\end{array}\\right.","\\\\{\\phi }_M (t) &:=& \\phi ^m \\quad \\mbox{ for all }t\\in ]t_{m-1,M},t_{m,M} ],$ in accordance with one of the two variational formulations () and (), while $Z_M (t):=\\left\\lbrace \\begin{array}{ll}B(\\theta ^{0} ) &\\mbox{ for }t=0\\\\Z^{m}&\\mbox{ for } t\\in ]t_{m-1,M},t_{m,M}]\\end{array}\\right.","\\mbox{ in }\\Omega ,$ with the discrete derivative with respect to $t$ at the time $t=t_{m,M}$ : $Z^{m}:=\\frac{B(\\theta ^{m})-B(\\theta ^{m-1}) }{\\tau }.$ While $\\theta _M$ is the Rothe function obtained from $\\theta ^m$ by piecewise constant interpolation with respect to time $t$ , the Rothe function, obtained from $\\theta ^m$ by piecewise linear interpolation with respect to time $t$ , $\\Theta _M$ is $\\Theta _M(\\cdot ,t)=\\theta ^{m-1}+(t-t_{m-1,M}) \\frac{\\theta ^m- \\theta ^{m-1}}{ \\tau }.$ For our purposes, we introduce the following definition.","Definition 5.1 We say that $\\lbrace \\widetilde{B}_M= \\widetilde{B}(\\theta _M)\\rbrace _{M\\in \\mathbb {N}}$ is the Rothe sequence (affine on each time interval) if $\\widetilde{B}(\\cdot ,\\theta _M (t)) = B(\\cdot ,\\theta ^{m-1}) + \\frac{ t-t_{m-1,M}}{\\tau }\\left( B(\\cdot ,\\theta ^m) - B(\\cdot ,\\theta ^{m-1}) \\right)$ in $\\Omega $ , for all $t\\in I_{m, M}$ , for all $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ .","Denoting $h_M(t)=h(t_{m,M})$ for $t\\in ]t_{m-1,M},t_{m,M}]$ and $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ , the triple $ ({\\theta }_M ,\\phi _M,{Z}_M )$ solve $\\int _0^T\\int _\\Omega {Z}_M v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} a ( {\\theta }_M , {\\phi }_M )\\nabla {\\theta }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma ( {\\theta }_M )| {\\theta }_M |^{\\ell -2} {\\theta }_M v \\mathrm {ds}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\sigma ( \\theta _M ) F( \\theta _M , \\phi _M ) \\nabla {\\phi }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}=\\int _{\\Sigma _T} h_M v \\mathrm {ds}\\mathrm {dt} .\\qquad $"],["Proof of Theorem ","Here, $ {\\phi }_M $ solves $\\int _\\Omega \\sigma (\\theta _M)\\nabla \\phi _M\\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\theta _M) \\alpha _\\mathrm {S}(\\theta _M)\\nabla \\theta _M \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ for $w\\in V$ and a.e.","in $]0,T[$ .","We begin by establishing the uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ .","Proposition 5.1 Let $ {\\theta }_M $ and $ {\\phi }_M $ be the (piecewise constant in time) functions defined in (REF )-().","Then the following estimate holds: $\\max _{1\\le m\\le M} \\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}+( L_1)_\\# \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ^2+\\frac{(L_{2})_\\# }{2}\\Vert \\nabla \\phi _M\\Vert _{2,Q_T} ^2 +\\nonumber \\\\+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } ^\\ell \\le b^\\# \\Vert \\theta ^0\\Vert _{2,\\Omega } ^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} } \\Vert h \\Vert _{\\ell ^{\\prime },\\Sigma _T }^{\\ell ^{\\prime }} +T \\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 .$ Let $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ be arbitrary.","Choosing $v=\\theta ^m\\in V_{\\ell }(\\Omega )$ and $w=\\phi ^m \\in V$ as test functions in (REF )-(), we sum the obtained relations, and arguing as in (REF )-(REF ), we have $\\frac{1}{\\tau }\\int _\\Omega (B(\\theta ^m)-B(\\theta ^{m-1}) )\\theta ^m\\mathrm {dx} +( L_1)_\\# \\Vert \\nabla \\theta ^{m} \\Vert _{2,\\Omega }^2 +\\frac{(L_{2})_\\#}{2} \\Vert \\nabla \\phi ^{m} \\Vert _{2,\\Omega }^2 +\\nonumber \\\\+\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds}\\le \\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} )+\\frac{1}{\\ell }\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds}, $ with $\\mathcal {R}$ being the increasing continuous function defined in (REF ).","By (REF ), we have $\\sum _{i=1}^m\\int _\\Omega (B(\\theta ^i)-B(\\theta ^{i-1}) )\\theta ^i\\mathrm {dx} \\ge \\int _\\Omega (\\Psi (\\theta ^m) - \\Psi (\\theta ^{0}) )\\mathrm {dx} .$ Therefore, summing over $i=1,\\cdots , m$ into (REF ), multiplying by $\\tau $ , and inserting the previous inequality, we obtain $\\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}& +&\\tau \\sum _{i=1}^m \\left( ( L_1)_\\# \\Vert \\nabla \\theta ^{i} \\Vert _{2,\\Omega }^2+\\frac{1}{\\ell ^{\\prime }}\\int _\\Gamma \\gamma (\\theta ^{i})|\\theta ^{i}|^{\\ell } \\mathrm {ds} +\\frac{(L_{2})_\\#}{2} \\Vert \\nabla \\phi ^{i} \\Vert _{2,\\Omega }^2 \\right) \\nonumber \\\\ &\\le &\\int _\\Omega \\Psi ( \\theta ^0 )\\mathrm {dx} +\\tau \\sum _{i=1}^m\\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} ) .","$ Therefore, we find the uniform estimate (REF ) by taking the maximum over $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ in the previous estimate and applying Lemma REF provided by ().","A direct application of Proposition REF ensures the following proposition.","Proposition 5.2 There exist $\\theta ,\\phi : Q_T\\rightarrow \\mathbb {R}$ and subsequences of $(\\theta _M,\\phi _M)$ , still labelled by $(\\theta _M,\\phi _M)$ , such that ${\\theta }_M \\rightharpoonup \\theta &\\mbox{ in }& V_\\ell (Q_T); \\\\\\phi _M \\rightharpoonup \\phi & \\mbox{ in }& L^2(0,T;V),$ as $M$ tends to infinity.","Moreover, there exists $Z: Q_T\\rightarrow \\mathbb {R}$ such that $\\partial _t\\widetilde{B}(\\theta _M)\\rightharpoonup Z \\quad \\mbox{in } L^{\\ell ^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime }),$ as $M$ tends to infinity.","Considering the uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ that are established in Proposition REF , we extract subsequences, still denoted by $ {\\theta }_M $ and $ {\\phi }_M $ , weakly convergent in $ V_\\ell (Q_T )$ and $ L^2(0,T;V)$ , respectively, to $\\theta $ and $\\phi $ .","Let $p=\\max \\lbrace \\ell ,2\\rbrace =\\ell $ .","Note that $L^p(0,T;V_\\ell (\\Omega ))\\hookrightarrow V_\\ell (Q_T)$ .","By definition of norm, we find $\\Vert \\partial _t\\widetilde{B}(\\theta _M)\\Vert _{L^{p^{\\prime }}(0,T;(V_\\ell (\\Omega ))^{\\prime })} = \\sum _{m=1}^M\\int _{(m-1)\\tau }^{m\\tau }\\sup _{v\\in L^p(0,T;V_\\ell (\\Omega )) \\atop \\Vert v\\Vert \\le 1} \\langle Z^m,v\\rangle \\mathrm {dt}\\le C,$ with $C>0$ being a constant independent on $M$ , by estimating in (REF ) the term involving $Z^m$ by means of the rest terms using the uniform estimates established in (REF ).","Hence, we can extract a subsequence, still denoted by $\\partial _t\\widetilde{B}(\\theta _M)$ , weakly convergent to $Z$ in $L^{p^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime })$ .","In the following proposition, we state the strong convergence of $B(\\theta _M)$ and of $\\theta _M$ .","Proposition 5.3 Under ()-(REF ) and (REF )-(), the solution $\\theta ^m$ of (REF ) satisfies $b_\\#\\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{2,\\Omega } ^2\\le \\int _\\Omega (B(\\theta ^{m })-B(\\theta ^{m-1 })) (\\theta ^{m }-\\theta ^{m-1 }) \\mathrm {dx} \\le \\nonumber \\\\\\le C\\left(\\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{\\ell ,\\Gamma } \\tau ^{1/\\ell } + \\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{2,\\Omega }\\sqrt{\\tau } \\right),$ with $C$ being a positive constant.","Moreover, for a subsequence, there hold $B( {\\theta }_M) \\rightarrow B(\\theta ) &\\mbox{ in }& L^1(Q_T); \\\\\\theta _M \\rightarrow \\theta & \\mbox{ a.e.","in }& Q_T,$ as $M$ tends to infinity.","Let $k\\in \\mathbb {N}$ .","Let us sum up (REF ) for $m=j+1,\\cdots ,j+k$ and multiply by $\\tau $ , obtaining $\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) v \\mathrm {dx} \\le \\mathcal {I }_\\Gamma ^j +\\mathcal {I}_\\Omega ^j ,$ where $\\mathcal {I}_\\Gamma ^j&:=& \\tau \\sum _{m=j+1}^{j+k} \\int _\\Gamma | (\\gamma (\\theta ^m)|\\theta ^{m}|^{\\ell -2} \\theta ^{m} -h(t_{m,M}) )v| \\mathrm {dx} \\\\&\\le & \\int _{j\\tau }^{(j+k)\\tau }\\Vert \\gamma (\\theta _M)|\\theta _M|^{\\ell -2} \\theta _M -h \\Vert _{\\ell ^{\\prime },\\Gamma } \\Vert v \\Vert _{\\ell ,\\Gamma }\\mathrm {dt} ; \\\\\\mathcal {I}_\\Omega ^j &:=& \\tau \\sum _{m=j+1}^{j+k}\\int _\\Omega |( a ( \\theta ^m,\\phi ^m )\\nabla \\theta ^{m} +F ( \\theta ^m,\\phi ^m) \\nabla \\phi ^m )v| \\mathrm {dx} \\\\&\\le & \\int _{j\\tau }^{(j+k)\\tau }\\Vert a ( \\theta _M,\\phi _M )\\nabla \\theta _M +F ( \\theta _M,\\phi _M) \\nabla \\phi _M \\Vert _{2,\\Omega }\\Vert v\\Vert _{2,\\Omega } \\mathrm {dt} .$ Here, we used the Hölder inequality and the definition of $ \\theta _M$ and of $\\phi _M$ .","Let us compute $\\mathcal {I }_\\Gamma ^j $ and $\\mathcal {I}_\\Omega ^j$ by applying the estimate (REF ).","Using the assumption (REF ) and after the Hölder inequality, we deduce $\\mathcal {I}_\\Gamma ^j\\le \\Vert v \\Vert _{\\ell ,\\Gamma } \\int _{j\\tau }^{(j+k)\\tau }\\left(\\gamma ^\\#\\Vert \\theta _M\\Vert _{\\ell ,\\Gamma }^{\\ell -1}+\\Vert h \\Vert _{\\ell ^{\\prime },\\Gamma } \\right)\\mathrm {dt}\\le \\Vert v \\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell }.$ Using the assumptions (), (REF ) and (), and after the Hölder inequality, we deduce $\\mathcal {I}_\\Omega ^j \\le \\Vert v\\Vert _{2,\\Omega } \\int _{j\\tau }^{(j+k)\\tau }\\left(a^\\#\\Vert \\nabla \\theta _M\\Vert _{2,\\Omega } + \\sigma ^\\#F ^\\#\\Vert \\nabla \\phi _M \\Vert _{2,\\Omega } \\right) \\mathrm {dt} \\le \\Vert v\\Vert _{2,\\Omega }C\\sqrt{k\\tau }.$ Hence, we find $\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) v \\mathrm {dx} \\le \\Vert v \\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell } + \\Vert v\\Vert _{2,\\Omega }C\\sqrt{k\\tau } .$ In particular, the estimate (REF ) follows by taking $j=m-1$ , $k=1$ and $v=\\theta ^{m }-\\theta ^{m-1 }$ , and applying Lemma REF .","To prove the convergences, we will apply Lemma REF .","Considering the weak convergence of $\\theta _M$ established in Proposition REF and the estimate (REF ), in order to apply Lemma REF it remains to prove that the condition (REF ) is fulfilled.","Let $0T/z$ , which means $\\tau k+1$ deducing $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\\\le \\sum _{j=1 }^{M-k} \\int _{(j-1)\\tau }^{(j+k)\\tau }\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) (\\theta ^{j+k }-\\theta ^{j })\\mathrm {dx} .$ Taking $v=\\theta ^{j+k }-\\theta ^j$ in (REF ) and then summing up for $j=1,\\cdots , M-k$ , we find $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\\\le (k+\\tau )\\sum _{j=1 }^{M-k}\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) (\\theta ^{j+k }-\\theta ^{j })\\mathrm {dx} \\le \\\\ \\le \\sum _{j=1 }^{M-k} \\int _{(j-1)\\tau }^{j\\tau +k }\\left( \\Vert \\theta ^{j+k }-\\theta ^{j }\\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell } +\\Vert \\theta ^{j+k }-\\theta ^{j }\\Vert _{2,\\Omega }C\\sqrt{k\\tau } \\right) \\mathrm {dt} .$ Arguing as in (REF ) and (REF ), we conclude $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\ \\le C\\left( (k\\tau )^{1/\\ell }(k\\tau +\\tau )^{ 1/\\ell ^{\\prime }} + (k\\tau )^{1/2} (k\\tau +\\tau )^{1/2}\\right)=C\\left( 2^{ 1/\\ell ^{\\prime }}+ 2^{ 1/2}\\right) z.$ Thus, all hypothesis of Lemma REF are fulfilled.","Therefore, Lemma REF assures that $B(\\theta _M)$ strongly converges to $B(\\theta )$ in $L^1(Q_T)$ .","Consequently, up to a subsequence, $B(\\theta _M)$ converges to $B(\\theta )$ a.e.","in $Q_T$ .","Since $B$ is strictly monotone, $\\theta _M$ converges to $\\theta $ a.e.","in $Q_T$ (see, for instance, [19]).","Now we are able to identify the limit $Z$ .","Proposition 5.4 The limit $Z$ satisfies $Z=\\partial _t (B(\\theta )) \\quad \\mbox{ in } L^{\\ell ^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime }).$ For a fixed $t$ , there exists $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ such that $t\\in ]t_{m-1,M},t_{m,M}]$ .","From definition REF we have $\\int ^t_0 {Z}_M(\\varsigma )\\mathrm {d\\varsigma }=\\sum _{j=1}^{m-1}\\int _{(j-1)\\tau }^{j\\tau }\\frac{B(\\theta ^{j})(\\varsigma )-B(\\theta ^{j-1})(\\varsigma )}{ \\tau }\\mathrm {d\\varsigma } + \\\\+\\int _{(m-1)\\tau }^{t}\\frac{ B(\\theta ^m) (\\varsigma )- B(\\theta ^{m-1})(\\varsigma ) }{\\tau }\\mathrm {d\\varsigma } =\\\\=B(\\theta ^{m-1})- B(\\theta ^{0}) + \\frac{ t-(m-1)\\tau }{\\tau }\\left( B(\\theta ^m) - B(\\theta ^{m-1}) \\right) = \\widetilde{B}(\\theta _M(t)) - B(\\theta ^{0})$ in $\\Omega $ .","By the Riesz theorem, the bounded linear functional $v\\in L^2(\\Omega )\\mapsto \\int ^t_0 ({Z}_M(\\varsigma ),v)\\mathrm {d\\varsigma }$ is (uniquely) representable by the element $\\widetilde{B}(\\theta _M(t)) - B(\\theta ^{0}) $ from $L^2(\\Omega ).$ Using the corresponding definitions we compute $\\int _0^T\\Vert \\widetilde{B}(\\theta _M(t)) - B(\\theta _M) \\Vert _{2,\\Omega }^2\\mathrm {dt} =\\sum _{m=1}^M \\Vert Z^m \\Vert _{2,\\Omega }^2\\int _{(m-1)\\tau }^{m\\tau } (t-m\\tau )^2\\mathrm {dt}= \\\\=\\frac{\\tau ^3}{3}\\sum _{m=1}^M \\Vert Z^m \\Vert _{2,\\Omega }^2 =\\frac{\\tau }{3}\\sum _{m=1}^M\\Vert B(\\theta ^m) - B(\\theta ^{m-1}) \\Vert _{2,\\Omega }^2\\le \\\\\\le (b^\\#)^2 \\frac{\\tau }{3}\\sum _{m=1}^M\\Vert \\theta ^m - \\theta ^{m-1} \\Vert _{2,\\Omega }^2.$ Applying (REF ) and Proposition REF we have $\\int _0^T\\Vert \\widetilde{B}(\\theta _M(t)) - B(\\theta _M) \\Vert _{2,\\Omega }^2\\mathrm {dt}\\le C\\left( \\tau ^{1/\\ell +1/\\ell ^{\\prime }} +\\tau ^{1/2+1/2} \\right) = C\\tau ,$ and consequently $\\widetilde{B}(\\theta _M)$ converges to $ B(\\theta ) $ .","By the uniqueness of limit, we deduce $\\int ^t_0 Z(\\varsigma )\\mathrm {d\\varsigma }=B(\\theta ) - B(\\theta ^{0}),$ which concludes the proof.","We emphasize that the above convergences are sufficient to identify the limit $\\phi $ as stated in the following proposition, but they are not sufficient to identify the temperature $\\theta $ as a solution, because on the one hand the apparent nonlinearity of the coefficients destroy the weak convergence, on the other hand, the weak-weak convergence does not imply weak convergence.","Corollary 5.1 Let $(\\theta ,\\phi )$ be in accordance with Propositions REF and REF , then they verify ().","Let $(\\theta _M,\\phi _M)$ solve (REF )-(REF ).","Applying Propositions REF and REF , and the Krasnoselski theorem to the Nemytskii operators $\\sigma $ and $\\alpha _\\mathrm {S}$ , we have $\\sigma ( \\theta _M )\\nabla \\phi _M\\rightharpoonup \\sigma ( \\theta )\\nabla \\phi &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\alpha _\\mathrm {S}( \\theta _M ) \\nabla \\theta _M\\rightharpoonup \\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla \\theta &\\mbox{ in }& \\mathbf {L}^2(Q_T) \\quad \\mbox{ as } M\\rightarrow +\\infty .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\theta ,\\phi )$ verifies ()."],["Proof of Theorem ","This proof follows mutatis mutandis the structure of the proof of Theorem REF (cf.","Subsection REF ).","We only sketch its main steps.","The uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ are as follows.","The quantitative estimate (REF ) reads $\\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}+ a_\\# \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } ^\\ell \\le \\nonumber \\\\\\le b^\\# \\Vert \\theta ^0\\Vert _{2,\\Omega } ^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} } \\Vert h \\Vert _{\\ell ^{\\prime },\\Sigma _T }^{\\ell ^{\\prime }} +T \\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 ,$ by using the argument to estimate (REF ).","Namely, (REF ) reads $\\frac{1}{\\tau }\\int _\\Omega (B(\\theta ^m)-B(\\theta ^{m-1}) )\\theta ^m\\mathrm {dx} +a_\\# \\Vert \\nabla \\theta ^{m} \\Vert _{2,\\Omega }^2+\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds} \\le \\\\\\le \\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} )+\\frac{1}{\\ell }\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds},$ where $\\mathcal {R}$ is the increasing continuous function defined in (REF ), with $(L_2)_\\#=a_\\#\\sigma _\\#/ (2 F^\\#\\sigma ^\\#)$ .","In addition, summing the quantitative estimate $\\sqrt{\\sigma _\\#} \\Vert \\nabla \\phi ^{m} \\Vert _{2,\\Omega } \\le \\sqrt{\\sigma _\\#}a^\\# \\Vert \\nabla \\theta ^{m-1} \\Vert _{2,\\Omega }+\\frac{K_2(P_2+1)}{ \\sqrt{\\sigma _\\#} } \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}},$ it results in $\\Vert \\nabla \\phi _M\\Vert _{2,Q_T} \\le a^\\# \\Vert \\nabla \\theta _M \\Vert _{2,Q_T}+ T\\frac{K_2(P_2+1)}{ \\sigma _\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}} .$ For subsequences of $ {\\theta }_M $ and $ {\\phi }_M $ , the weak convergences hold according to Propositions REF and REF , which guarantee the required result."],["Existence of solutions to the TE problem","The objective is the passage to the limit in the abstract boundary value problems introduced in Section as the time step goes to zero ($M\\rightarrow +\\infty $ ), with the coefficients being defined by $b(\\cdot , v) &=& \\rho (\\cdot ,v)c_\\mathrm {v}(\\cdot , v);\\\\a(\\cdot , v,w)&=&k(\\cdot , v) + T_\\mathcal {M}(w)\\alpha _\\mathrm {S}(\\cdot , v)\\sigma (\\cdot , v);\\\\F(\\cdot , v,w)&=&\\Pi (\\cdot , v)+T_\\mathcal {M}(w),$ where $T_\\mathcal {M}$ is the $\\mathcal {M}$ -truncation function defined by $T_\\mathcal {M}(z)=\\max (-\\mathcal {M},\\min (\\mathcal {M},z))$ .","By the definition of truncated functions, we choose $a_\\# &=& k_\\#-\\mathcal {M}\\alpha ^\\#\\sigma ^\\#;\\\\F^\\# &=& \\Pi ^\\#+\\mathcal {M}.$ taking (REF ) into account.","Under these choices, the assumptions (REF ), (REF ) and (REF ) imply (REF ), (REF ) and (REF ), respectively.","Let us foccus the present proof in accordance with the approximated solutions that are established in Theorem REF .","Analogous argument is valid for the approximated solutions that are established in Theorem REF .","Let us redefine the electrical current density as $\\mathbf {j}( \\theta , \\phi )= \\sigma ( \\theta )\\nabla \\phi +\\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla \\theta .$ Analogously for $\\mathbf {j}_M= \\mathbf {j}(\\theta _M ,\\phi _M)$ or simply $\\mathbf {j}_M$ and $\\mathbf {j}$ whenever the meaning is not ambiguous.","Let $ (\\theta _M, {\\phi }_M )$ solve (REF )-(REF ), which may be rewritten as $\\int _0^T\\int _\\Omega {Z}_M v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} k ( {\\theta }_M )\\nabla {\\theta }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma ( {\\theta }_M )| {\\theta }_M |^{\\ell -2} {\\theta }_M v \\mathrm {ds}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\sigma ( \\theta _M ) \\Pi ( \\theta _M ) \\nabla {\\phi }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\phi _M \\mathbf {j}( \\theta _M , \\phi _M ) \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} =\\nonumber \\\\=\\int _{\\Sigma _T} h_M v \\mathrm {ds}\\mathrm {dt} ;\\qquad \\\\\\int _\\Omega \\mathbf {j}( \\theta _M , \\phi _M)\\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\qquad $ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ .","There exist $\\theta ,\\phi : Q_T\\rightarrow \\mathbb {R}$ and subsequences of $(\\theta _M,\\phi _M)$ , still labelled by $(\\theta _M,\\phi _M)$ , weakly convergent in accordance with Proposition REF .","By Corollary REF , $(\\theta ,\\phi )$ verify the electric equality $\\\\\\int _\\Omega \\mathbf {j}( \\theta , \\phi )\\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds},$ for every $w\\in V$ .","We emphasize that the weak convergence of $\\mathbf {j}_M$ to $ \\mathbf {j}$ in $ \\mathbf {L}^2(Q_T)$ , taking (REF )-() into account, is not sufficient to pass to the limit the term $\\phi _M \\mathbf {j}( \\theta _M , \\phi _M ) $ .","Moreover, the non smoothness of the coefficients destroy the possibility of obtaining strong convergences of $\\nabla \\theta _M$ and of $\\phi _M$ .","Thanks to Proposition REF , we have a.e.","pointwise convergence for a subsequence of $\\theta _M$ , which we still denote by $\\theta _M$ .","Considering the assumptions (REF )-(), the Nemytskii operators are continuous due to the Krasnoselski theorem, and applying the Lebesgue dominated convergence theorem, we obtain $k( \\theta _M )\\nabla v\\rightarrow k(\\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\alpha _\\mathrm {S}( \\theta _M ) \\nabla v\\rightarrow \\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\Pi ( \\theta _M ) \\nabla v\\rightarrow \\sigma ( \\theta ) \\Pi ( \\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T).$ Applying (REF ) to the following estimates $\\Vert T_\\mathcal {M}(\\phi _M) \\nabla \\theta _M\\Vert _{2,Q_T}\\le \\mathcal {M} \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ; \\\\\\Vert \\sigma ( \\theta _M ) T_\\mathcal {M}( \\phi _M ) \\nabla \\phi _M\\Vert _{2,Q_T}\\le \\sigma ^\\# \\mathcal {M} \\Vert \\nabla \\phi _M\\Vert _{2,Q_T} ;\\\\\\Vert \\gamma ( \\theta _M ) | \\theta _M|^{\\ell -2} \\theta _M\\Vert _{\\ell ^{\\prime },\\Sigma _T } \\le \\gamma ^\\# \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } .$ there exist $\\Lambda _1 , \\Lambda _2 \\in \\mathbf {L}^2(Q_T)$ and $\\Lambda _3\\in L^{\\ell ^{\\prime }}(\\Sigma _T ) $ such that $T_\\mathcal {M}( \\phi _M )\\nabla \\theta _M\\rightharpoonup \\Lambda _1 &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) T_\\mathcal {M}( \\phi _M ) \\nabla \\phi _M\\rightharpoonup \\Lambda _2 &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\gamma ( \\theta _M ) | \\theta _M|^{\\ell -2} \\theta _M\\rightharpoonup \\Lambda _3 &\\mbox{ in }& L^{\\ell ^{\\prime }}(\\Sigma _T ) .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\theta ,\\phi )$ verifies $\\int _0^T\\int _\\Omega Z v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} k (\\theta )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+ \\nonumber \\\\ +\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\sigma ( \\theta ) \\alpha _\\mathrm {S}(\\theta ) \\Lambda _1 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 v \\mathrm {ds}\\mathrm {dt}=\\int _{\\Sigma _T} h v \\mathrm {ds}\\mathrm {dt} ,\\quad \\forall v \\in V_\\ell (Q_T) .","$ To identify the temperature $\\theta $ as a solution, we need to identify $\\Lambda _1 , \\Lambda _2 \\in \\mathbf {L}^2(Q_T)$ and $\\Lambda _3\\in L^{\\ell ^{\\prime }}(\\Sigma _T ) $ .","To prove that $\\Lambda _1= T_\\mathcal {M}( \\phi )\\nabla \\theta $ , let us consider the Green formula $\\int _{Q_T} T_\\mathcal {M}( \\phi _M )\\nabla \\theta _M \\cdot \\mathbf {v}\\mathrm {dx}\\mathrm {dt}=-\\int _{Q_T[|\\phi _M|<\\mathcal {M}]} \\theta _M \\nabla \\phi _M \\cdot \\mathbf {v}\\mathrm {dx}\\mathrm {dt},$ for every $\\mathbf {v}\\in \\mathbf {L}^p(0,T; \\mathbf {W}^{1,p}(\\Omega ))$ such that $\\nabla \\cdot \\mathbf {v}=0$ in $Q_T$ and $\\mathbf {v}\\cdot \\mathbf {n}=0$ in $\\partial \\Omega \\times ]0,T [$ .","Next we choose the exponent $p>1$ to ensure the meaning of the involved terms.","By $\\theta _M\\in L^\\infty (0,T; L^2(\\Omega ))\\cap L^2(0,T; H^1(\\Omega ))$ and $ H^1(\\Omega )\\hookrightarrow L^{2^*}(\\Omega )$ with $2^*$ being the critical Sobolev exponent, i.e.","$2^*=2n/(n-2)$ if $n>2$ and any $2^*>1$ if $n=2$ , making recourse to the interpolation with exponents being $\\frac{\\beta }{2}= \\frac{1-\\beta }{2}+\\frac{\\beta }{2^*}=\\frac{1}{q}, \\qquad (0<\\beta <1),$ then $\\theta _M$ converges to $\\theta $ in $L^q(Q_T)$ for every $q<2(n+2)/n$ .","In particular, we take $p>n+2$ such that $\\frac{1}{2}=\\frac{1}{p}+\\frac{1}{q} >\\frac{1}{p}+\\frac{n}{2(n+2)}.$ Consequently, we have that $\\theta _M\\mathbf {v}$ converges to $\\theta \\mathbf {v}$ in $\\mathbf {L}^2 (Q_T)$ .","Therefore, the uniqueness of the weak limit implies that $\\Lambda _1= T_\\mathcal {M}( \\phi )\\nabla \\theta $ .","In particular, we find $\\int _{Q_T} a(\\theta _M,\\phi _M)\\nabla \\theta _M\\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}\\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} \\int _{Q_T} a(\\theta ,\\phi ) \\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt},$ and consequently (REF ) reads $\\int _0^T\\int _\\Omega Z v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} a(\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+ \\nonumber \\\\+\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 v \\mathrm {ds}\\mathrm {dt}=\\int _{\\Sigma _T} h v \\mathrm {ds}\\mathrm {dt} ,\\quad \\forall v \\in V_\\ell (Q_T) .","\\qquad $ Now, we are in the conditions to identify the limits $\\Lambda _2$ and $\\Lambda _3$ by making recourse to the Minty argument as follows.","We rephrase (REF ) as $\\mathcal {J}_M -\\mathcal {J}_1^M-\\mathcal {J}_2^M -\\mathcal {J}_3^M -\\mathcal {J}_4^M -\\mathcal {J}_5^M + \\\\+\\int _{\\Sigma _T} \\left(\\gamma (\\theta _M) |\\theta _{M}|^{\\ell -2} \\theta _{M} -\\gamma (v) |v|^{\\ell -2}v\\right)(\\theta _{M} -v)\\mathrm {ds}\\mathrm {dt} \\ge \\\\ \\ge (L_1)_\\# \\int _{Q_T}|\\nabla (\\theta _M-v) |^2\\mathrm {dx} \\mathrm {dt}+(L_2)_\\#\\int _{Q_T}|\\nabla (\\phi _M - \\phi ) |^2\\mathrm {dx}\\mathrm {dt}\\ge 0,$ where $\\mathcal {J}_M &:=& \\int _{Q_T}\\left( a(\\theta _M,\\phi _M) |\\nabla \\theta _M|^2 +\\sigma (\\theta _M) F (\\theta _M, \\phi _M) \\nabla \\phi _M\\cdot \\nabla \\theta _M\\right)\\mathrm {dx}\\mathrm {dt} + \\\\ &&+\\int _{Q_T}\\left(\\sigma (\\theta _M) |\\nabla \\phi _M|^2 + \\sigma (\\theta _M) \\alpha _\\mathrm {S} (\\theta _M) \\nabla \\theta _M\\cdot \\nabla \\phi _M\\right)\\mathrm {dx} \\mathrm {dt}+ \\\\ &&+\\int _{\\Sigma _T}\\gamma (\\theta _M) |\\theta _M|^\\ell \\mathrm {ds}\\mathrm {dt}; \\\\\\mathcal {J}_1^M &:=& 2\\int _{Q_T} a(\\theta _M,\\phi _M)\\nabla \\theta _M\\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}-\\int _{Q_T} a(\\theta _M,\\phi _M)|\\nabla v|^2\\mathrm {dx}\\mathrm {dt}; \\\\\\mathcal {J}_2^M &:=& 2\\int _{Q_T}\\sigma (\\theta _M)\\nabla \\phi _M\\cdot \\nabla \\phi \\mathrm {dx}\\mathrm {dt} -\\int _{Q_T}\\sigma (\\theta _M) |\\nabla \\phi |^2\\mathrm {dx}\\mathrm {dt} ;\\\\\\mathcal {J}_3^M &:=&\\int _{Q_T} \\sigma (\\theta _M) \\Pi (\\theta _M) \\Big (\\nabla \\phi _M \\cdot \\nabla v +\\nabla \\phi \\cdot \\nabla ( \\theta _M-v) \\Big ) \\mathrm {dx} \\mathrm {dt} ; \\\\\\mathcal {J}_4^M &:=&\\int _{Q_T} \\sigma (\\theta _M) T_\\mathcal {M} (\\phi _M) \\Big (\\nabla \\phi _M \\cdot \\nabla v +\\nabla \\phi \\cdot \\nabla ( \\theta _M-v) \\Big ) \\mathrm {dx} \\mathrm {dt} ; \\\\\\mathcal {J}_5^M &:=&\\int _{Q_T} \\sigma (\\theta _M) \\alpha _\\mathrm {S} (\\theta _M)\\Big ( \\nabla \\theta _M \\cdot \\nabla \\phi +\\nabla v\\cdot \\nabla (\\phi _M - \\phi ) \\Big ) \\mathrm {dx} \\mathrm {dt}.$ Considering the convergences $\\mathcal {J}_1^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} & 2\\int _{Q_T} a(\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} -\\int _{Q_T} a(\\theta ,\\phi ) |\\nabla v|^2\\mathrm {dx} \\mathrm {dt}:= \\mathcal {J}_1;\\\\\\mathcal {J}_2^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) |\\nabla \\phi |^2\\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_2; \\\\\\mathcal {J}_3^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) \\Pi (\\theta ) \\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_3;\\\\\\mathcal {J}_4^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt} +\\int _{Q_T}\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\cdot \\nabla ( \\theta -v)\\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_4;\\\\\\mathcal {J}_5^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) \\alpha _\\mathrm {S} (\\theta )\\nabla \\theta \\cdot \\nabla \\phi \\mathrm {dx}\\mathrm {dt}:= \\mathcal {J}_5,$ we deduce $\\lim _{ M\\rightarrow \\infty }\\mathcal {J}_M\\ge \\int _{\\Sigma _T}\\Lambda _3 v\\mathrm {ds} \\mathrm {dt}+\\int _{\\Sigma _T}\\gamma (v)|v|^{\\ell -2} v(\\theta -v)\\mathrm {ds}\\mathrm {dt}+ \\\\ + \\mathcal {J}_1+\\mathcal {J}_2+ \\mathcal {J}_3 +\\mathcal {J}_4 +\\mathcal {J}_5.$ We continue the Minty argument by taking in (REF ) and (), respectively, the test function $v=\\theta _M$ and the test function $w=\\phi _M$ , in (REF ) the test function $v=\\theta $ , and in (REF ) the test function $w=\\phi $ , we deduce $\\lim _{M\\rightarrow \\infty }\\mathcal {J}_M = -\\int _{Q_T} Z\\theta \\mathrm {dx} \\mathrm {dt}+\\int _{\\Sigma _T} h\\theta \\mathrm {ds} \\mathrm {dt}+\\int _0^T \\int _{\\Gamma _\\mathrm {N}} g \\theta \\mathrm {ds}\\mathrm {dt}=\\\\=\\int _{Q_T} a( \\theta ,\\phi ) |\\nabla \\theta |^2\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx} \\mathrm {dt} + \\\\ +\\int _{Q_T} \\Lambda _2\\cdot \\nabla \\theta \\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 \\theta \\mathrm {ds}\\mathrm {dt}+\\int _{Q_T} \\mathbf {j}( \\theta , \\phi ) \\cdot \\nabla \\phi \\mathrm {dx} \\mathrm {dt}.$ Gathering the above two relations, we find $\\int _{Q_T} a( \\theta ,\\phi )|\\nabla (\\theta - v)|^2\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\Big (\\Lambda _2 -\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\Big )\\cdot \\nabla ( \\theta -v)\\mathrm {dx} \\mathrm {dt} + \\nonumber \\\\+\\int _{\\Sigma _T} \\left(\\Lambda _3 -\\gamma (v)|v|^{\\ell -2}v \\right) (\\theta -v) \\mathrm {ds}\\mathrm {dt} \\ge 0.$ Next, taking $v=\\theta -\\delta \\varphi $ , with $\\varphi \\in \\mathcal {D}(Q_T)$ , and after dividing by $\\delta >0$ , we arrive to $\\delta \\int _{Q_T} a( \\theta ,\\phi )|\\nabla \\varphi |^2\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\Big (\\Lambda _2 -\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\Big )\\cdot \\nabla \\varphi \\mathrm {dx} \\mathrm {dt} \\ge 0.$ Finally letting $\\delta \\rightarrow 0^+$ then we obtain $\\Lambda _2 =\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi $ .","We conclude the Minty argument by taking $v=\\theta -\\delta \\varphi $ , with $\\varphi \\in \\mathcal {D}(\\Sigma _T)$ , in (REF ).","After dividing by $\\delta >0$ , and finally letting $\\delta \\rightarrow 0^+$ we arrive to $\\int _{\\Sigma _T}(\\Lambda _3 -\\gamma (\\theta ) |\\theta |^{\\ell -2}\\theta )\\varphi \\mathrm {ds}\\mathrm {dt} \\ge 0, \\quad \\forall \\varphi \\in \\mathcal {D}(\\Sigma _T),$ which implies that $\\Lambda _3 =\\gamma (\\theta )|\\theta |^{\\ell -2}\\theta $ .","Therefore, the weak formulation (REF ) yields concluding the proof of Theorem REF ."]],"string":"[\n [\n \"Weak solutions for a thermoelectric problem with power-type boundary\\n effects\"\n ],\n [\n \"Abstract This paper deals with thermoelectric problems including the Peltier and Seebeck effects.\",\n \"The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects.\",\n \"To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3).\",\n \"They are essentially approximated solutions based on the Rothe method.\",\n \"It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero.\",\n \"The proof of the existence of time discretized solutions relies on fixed point and compactness arguments.\",\n \"In this study, we establish quantitative estimates to clarify the smallness conditions.\"\n ],\n [\n \"Introduction\",\n \"The study of the heat equation with constant coefficients is a simplification from both mathematical and engineering points of view.\",\n \"From the real world point of view, constant coefficients are not appropriate because the density and the thermal conductivity both depend on the temperature itself, and often also on the spatial variable.\",\n \"The concern of discontinuous leading coefficient is being a long matter of study in the mathematical literature, as long as the works [22], [25].\",\n \"The complete concern is achieved by the doubly quasilinear parabolic equation [1], [3], [5].\",\n \"It is well known that the determination of estimates is the crucial key in the theory of partial differential equations (PDE), which involve the so-called universal bounds.\",\n \"With their abstract form, these bounds are only qualitative and they do not have any practical use on the real world applications.\",\n \"In their majority, if the proof of estimates should be remade step by step, the expression of the qualitative bounds would be truly cumbersome, or even impossible if the contradiction argument is applied.\",\n \"Also regularity estimates have being a subject of study in the last decades [11], [12], [15], but these ones only occur by admitting data smoothness.\",\n \"With this in mind, our main objective is to find quantitative estimates, i.e.\",\n \"their involved constants have an explicit expression, that are useful on the real applications.\",\n \"In particular, the quantitative estimates clarify the smallness conditions on the data when a fixed point argument is used.\",\n \"For the two-dimensional space situation, a first attempt on the finding smallness conditions that assure the existence and regularity results for some thermoelectric problems is presented in [9], [10], where some domain dependent constants were kept abstract.\",\n \"Indeed, the central existence result of a weak solution for a class of elliptic systems on divergence form, which is only 2D valid, is provided under some higher regularity ($W^{1,p}$ regularity, with $p > 2$ ).\",\n \"The Gehring-type higher integrability technique makes the smallness conditions quite bizarre.\",\n \"Here, we establish more elegant smallness conditions and they are extended to the $n$ -dimensional space situation, by finding weak solutions.\",\n \"The present model also extends the thermal effects, of the previous works [9], [10], to the unsteady state.\",\n \"Existence of solutions for parabolic-elliptic systems with nonlinear no-flux boundary conditions is not a new idea if taking constant coefficients into account [4].\",\n \"Application of elliptic PDE system in divergence form with Dirichlet boundary conditions in doubly-connected domain of the plane are given in [7] to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour effects.\",\n \"In [8], the authors deal with a traditional RLC circuit in which a thermistor has been inserted, representing the microwave heating process with temperature-induced modulations on the electric field.\",\n \"In particular, the existence of a solution to a coupled system of three differential equations (an ODE, an elliptic equation and a nonlinear parabolic PDE) and appropriate initial and boundary conditions is proved.\",\n \"A one-dimensional thermal analysis for the performance of thermoelectric cooler is conducted in [13] under the influence of the Thomson effect, the Joule heating, the Fourier heat conduction, and the radiation and convection heat transfer.\",\n \"Simulation studies have been performed to investigate the thermal balance affected by anode shorting in an aluminum reduction cell [6].\",\n \"The method of discretization in time, whose basic idea (coming from the implicit Euler formula) was investigated by Rothe, is a very well-known effective technique for both theoretical and numerical analysis, [14], [17], [24] and [21], [26], respectively (see also the pioneering work [1] of Alt and Luckhaus).\",\n \"This paper is organized as follows.\",\n \"The thermoelectric (TE) model is introduced in Section .\",\n \"After discussing the physical model, the main result with respect to this model is formulated with a detailed description of the relevant constants.\",\n \"In Section , one abstract model related to the problem under consideration is introduced to simplify the proofs of the existence results of time-discretized solutions (Section ), and their corresponding steady-state solutions (Section ).\",\n \"Indeed, the analysis of the problem is structured via two different approaches to exemplify alternative assumptions on the data smallness, namely the existence results of time-discretized solutions (Subsections REF and REF ), and their corresponding steady-state solutions (Subsections REF and REF ).\"\n ],\n [\n \"The thermoelectric model\",\n \"Let $[0, T] \\\\subset {\\\\mathbb {R}}$ be the time interval with $ T >0$ being an arbitrary (but preassigned) time.\",\n \"Let $\\\\Omega $ be a bounded domain (that is, connected open set) in $\\\\mathbb {R}^n$ ($n\\\\ge 2$ ).\",\n \"Its boundary is constituted by two disjoint open $(n-1)$ -dimensional sets $\\\\partial \\\\Omega =\\\\overline{\\\\Gamma _\\\\mathrm {N}}\\\\cup \\\\overline{\\\\Gamma }$ .\",\n \"We consider $\\\\Gamma _{\\\\rm N}$ over which the Neumann boundary condition is taken into account, and $\\\\Gamma $ over which the radiative effects may occur.\",\n \"Each one, $\\\\Gamma _{\\\\rm N}$ and $\\\\Gamma $ , may be alternatively of zero $(n-1)$ -Lebesgue measure.\",\n \"Set $Q_T=\\\\Omega \\\\times ]0,T[$ and $\\\\Sigma _T=\\\\Gamma \\\\times ]0,T[$ .\",\n \"The electrical current density $\\\\bf j$ and the energy flux density ${\\\\bf J}={\\\\bf q}+\\\\phi {\\\\bf j}$ , with $\\\\bf q$ being the heat flux vector, are given by the constitutive relations (see [9] and the references therein) ${\\\\bf q}&= &- k (\\\\cdot ,\\\\theta ) \\\\nabla \\\\theta -\\\\Pi (\\\\cdot ,\\\\theta ) \\\\sigma (\\\\cdot ,\\\\theta ) \\\\nabla \\\\phi ;\\\\\\\\{\\\\bf j}&=& -\\\\alpha _{\\\\rm S}(\\\\cdot ,\\\\theta ) \\\\sigma (\\\\cdot ,\\\\theta ) \\\\nabla \\\\theta -\\\\sigma (\\\\cdot ,\\\\theta ) \\\\nabla \\\\phi .$ Here, $\\\\theta $ denotes the absolute temperature, $\\\\phi $ is the electric potential, $\\\\alpha _{\\\\rm S}$ represents the Seebeck coefficient, and the Peltier coefficient $\\\\Pi (\\\\theta )=\\\\theta \\\\alpha _{\\\\rm s}(\\\\theta )$ is due to the first Kelvin relation.\",\n \"The electrical conductivity $\\\\sigma $ , and the thermal conductivity $k=k_{\\\\rm T}+\\\\Pi \\\\alpha _{\\\\rm s}\\\\sigma $ , with $k_T$ denotes the purely conductive contribution, are, respectively, the known positive coefficients of Ohm and Fourier laws.\",\n \"The Seebeck coefficient $\\\\alpha _{\\\\rm S}$ has a constant sign corresponding to the Hall effect.\",\n \"With positive sign ($\\\\alpha _{\\\\rm S}>0$ ), there are as examples: the alkali metals Li, Rb and Cs [2], and the noble metals Ag and Au [2] or [20].\",\n \"With negative sign ($\\\\alpha _{\\\\rm S}<0$ ), there are as examples: the alkali metals Na and K [20], the transition metals Fe and Ni [2], and the semiconductor Pb [2].\",\n \"We refer to [10], and the references therein, for more examples and their increase and decrease behaviors.\",\n \"Although heat generation starts instantaneously when the current begins to flow, it takes time before the heat transfer process is initiated to allow the transient conditions to disappear.\",\n \"Thus, the electrical current density $\\\\bf j$ and the energy flux density ${\\\\bf J}$ satisfy $&&\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\nabla \\\\cdot {\\\\bf j}=0 &\\\\mbox{ in }\\\\Omega \\\\\\\\-{\\\\bf j}\\\\cdot {\\\\bf n}=g &\\\\mbox{ on }\\\\Gamma _{\\\\rm N}\\\\\\\\{\\\\bf j}\\\\cdot {\\\\bf n}=0 &\\\\mbox{ on }\\\\Gamma \\\\end{array}\\\\right.\\\\\\\\ &&\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\rho (\\\\cdot ,\\\\theta ) c_\\\\mathrm {v}(\\\\cdot , \\\\theta )\\\\partial _t \\\\theta -\\\\nabla \\\\cdot {\\\\bf J}=0 &\\\\mbox{ in } Q_T\\\\\\\\{\\\\bf J}\\\\cdot {\\\\bf n}=0 &\\\\mbox{ on }\\\\Gamma _{\\\\rm N}\\\\times ]0,T[\\\\\\\\-{\\\\bf J}\\\\cdot {\\\\bf n}=\\\\gamma (\\\\cdot ,\\\\theta ) |\\\\theta |^{\\\\ell -2}\\\\theta -h &\\\\mbox{ on }\\\\Sigma _T,\\\\end{array}\\\\right.$ for $\\\\ell \\\\ge 2$ .\",\n \"Here, $\\\\rho $ denotes the density, $c_\\\\mathrm {v}$ denotes the heat capacity (at constant volume), $\\\\bf n$ is the unit outward normal to the boundary $\\\\partial \\\\Omega $ , and $g$ denotes the surface current source, The boundary operators, $\\\\gamma $ and $h$ , are temperature dependent functions that express, respectively, the radiative convection depending on the wavelength, and the external heat sources.\",\n \"For $\\\\ell =5$ , the Stefan-Boltzmann radiation law says that $\\\\gamma (T)=\\\\sigma _{\\\\rm SB}\\\\epsilon (T)$ and $h(T)=\\\\sigma _{\\\\rm SB} \\\\alpha (T) \\\\theta _\\\\mathrm {e}^{\\\\ell -1}$ , where $\\\\sigma _{\\\\rm SB}=\\\\, 5.67\\\\times 10^{-8}$ W m$^{-2} $ K$^{-4}$ is the Stefan-Boltzmann constant for blackbodies, and $ \\\\theta _\\\\mathrm {e}$ denotes an external temperature.\",\n \"The parameters, the emissivity $\\\\epsilon $ and the absorptivity $\\\\alpha $ , both depend on the space variable and the temperature function $\\\\theta $ .\",\n \"If $\\\\ell =2$ , the boundary condition corresponds to the Newton law of cooling with heat transfer coefficient $\\\\gamma =h/ \\\\theta _\\\\mathrm {e}^{\\\\ell -1}$ .\",\n \"In the framework of Sobolev and Lebesgue functional spaces, we use the following spaces of test functions: $V &=& \\\\left\\\\lbrace \\\\begin{array}{l}V(\\\\Omega ) = \\\\left\\\\lbrace v\\\\in H^{1}(\\\\Omega ):\\\\ \\\\int _{\\\\Omega } v \\\\mathrm {dx}=0 \\\\right\\\\rbrace \\\\\\\\V(\\\\partial \\\\Omega ) = \\\\left\\\\lbrace v\\\\in H^{1}(\\\\Omega ):\\\\ \\\\int _{\\\\partial \\\\Omega } v \\\\mathrm {ds}=0 \\\\right\\\\rbrace \\\\end{array}\\\\right.\",\n \"\\\\\\\\V_{\\\\ell }(\\\\Omega ) &=&\\\\left\\\\lbrace v\\\\in H^{1}(\\\\Omega ) :\\\\ v|_{\\\\Gamma }\\\\in L^{\\\\ell }(\\\\Gamma ) \\\\right\\\\rbrace ;\\\\\\\\V_{\\\\ell }(Q_T) &=&\\\\left\\\\lbrace v\\\\in L^2(0,T; H^{1}(\\\\Omega )) :\\\\ v|_{\\\\Sigma _T }\\\\in L^{\\\\ell }(\\\\Sigma _T) \\\\right\\\\rbrace ,$ with their usual norms, $\\\\ell >1$ .\",\n \"Notice that $V_{\\\\ell }(\\\\Omega )\\\\equiv H^{1}(\\\\Omega ) $ if $\\\\ell \\\\le 2_*$ , where $ 2_*$ is the critical trace exponent, i.e.\",\n \"$2_*=2(n-1)/(n-2)$ if $n>2$ and $2_*>1$ is arbitrary if $n=2$ .\",\n \"In the presence of the previous considerations, the temperature-potential pair does not be expectable to be regular nor even bounded.\",\n \"The thermoelectric problem is formulated as follows.\",\n \"(TE) Find the temperature-potential pair $(\\\\theta ,\\\\phi )$ such that if it verifies the variational problem: $\\\\int _0^T\\\\langle \\\\rho (\\\\cdot , \\\\theta ) c_\\\\mathrm {v}(\\\\cdot , \\\\theta )\\\\partial _t\\\\theta , v\\\\rangle \\\\mathrm {dt} +\\\\int _{Q_T} k (\\\\cdot ,\\\\theta )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\nonumber \\\\\\\\ +\\\\int _{Q_T} \\\\sigma (\\\\cdot ,\\\\theta )\\\\left(T_\\\\mathcal {M}(\\\\phi )\\\\alpha _{\\\\rm S}(\\\\cdot ,\\\\theta )\\\\nabla \\\\theta +(\\\\Pi (\\\\cdot ,\\\\theta )+T_\\\\mathcal {M}(\\\\phi ))\\\\nabla \\\\phi \\\\right) \\\\cdot \\\\nabla v\\\\mathrm {dx} \\\\mathrm {dt}+\\\\nonumber \\\\\\\\+\\\\int _{\\\\Sigma _T} \\\\gamma (\\\\cdot ,\\\\theta )|\\\\theta |^{\\\\ell -2} \\\\theta v\\\\mathrm {ds} \\\\mathrm {dt}= \\\\int _{\\\\Sigma _T} h(\\\\cdot ,\\\\theta ) v\\\\mathrm {ds} \\\\mathrm {dt}; \\\\\\\\\\\\int _\\\\Omega \\\\sigma (\\\\cdot ,\\\\theta )\\\\nabla \\\\phi \\\\cdot \\\\nabla w\\\\mathrm {dx}+\\\\int _\\\\Omega \\\\sigma (\\\\cdot ,\\\\theta ) \\\\alpha _{\\\\rm S}(\\\\cdot ,\\\\theta )\\\\nabla \\\\theta \\\\cdot \\\\nabla w\\\\mathrm {dx}=\\\\int _{\\\\Gamma _{\\\\rm N}}g w\\\\mathrm {ds}, \\\\ \\\\mbox{ a.e.\",\n \"in } ]0,T[ , $ for every $v\\\\in V_{\\\\ell }(Q_T)$ and $w\\\\in V$ , where $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle $ accounts for the duality product, and $T_\\\\mathcal {M}$ is the $\\\\mathcal {M}$ -truncation function defined by $T_\\\\mathcal {M}(z)=\\\\max (-\\\\mathcal {M},\\\\min (\\\\mathcal {M},z))$ .\",\n \"We assume the following.\",\n \"(H1) The density and the heat capacity $\\\\rho , c_\\\\mathrm {v}: \\\\Omega \\\\times \\\\mathbb {R}\\\\rightarrow \\\\mathbb {R}$ are Carathéodory functions, i.e.\",\n \"measurable with respect to $x\\\\in \\\\Omega $ and continuous with respect to $e\\\\in \\\\mathbb {R}$ .\",\n \"Furthermore, they verify $\\\\exists b^\\\\#,b_\\\\#>0: \\\\quad b_\\\\#\\\\le \\\\rho (x,e)c_\\\\mathrm {v}(x,e)\\\\le b^\\\\# ,\\\\quad \\\\mbox{for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\quad \\\\forall e \\\\in \\\\mathbb {R} .\",\n \"$ (H2) The thermal and electrical conductivities $k,\\\\sigma :\\\\Omega \\\\times \\\\mathbb {R}\\\\rightarrow \\\\mathbb {R}$ are Carathéodory functions.\",\n \"Furthermore, they verify $\\\\exists k^\\\\#,k_\\\\#>0:&&k_\\\\#\\\\le k(x,e)\\\\le k^\\\\#;\\\\\\\\\\\\exists \\\\sigma ^\\\\#, \\\\sigma _\\\\#>0:&&\\\\sigma _\\\\#\\\\le \\\\sigma (x,e)\\\\le \\\\sigma ^\\\\#\\\\quad \\\\mbox{for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\quad \\\\forall e \\\\in \\\\mathbb {R}.\",\n \"$ (H3) The Seebeck and Peltier coefficients $\\\\alpha _{\\\\rm S},\\\\Pi :\\\\Omega \\\\times \\\\mathbb {R}\\\\rightarrow \\\\mathbb {R}$ are Carathéodory functions such that $\\\\exists \\\\alpha ^\\\\#>0: &&|\\\\alpha _{\\\\rm S}(x,e)|\\\\le \\\\alpha ^\\\\#; \\\\\\\\\\\\exists \\\\Pi ^\\\\#>0: &&|\\\\Pi (x,e)|\\\\le \\\\Pi ^\\\\#,\\\\quad \\\\mbox{for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\quad \\\\forall e\\\\in \\\\mathbb {R}.$ (H4) The boundary function $h$ belongs to $L^{\\\\ell ^{\\\\prime }}(\\\\Sigma _T)$ .\",\n \"(H5) The boundary function $g$ belongs to $ L^{2}(\\\\Gamma _{\\\\rm N})$ .\",\n \"(H6) The boundary operator $\\\\gamma $ is a Carathéodory function from $\\\\Sigma _T\\\\times \\\\mathbb {R}$ into $\\\\mathbb {R}$ such that $\\\\exists \\\\gamma _\\\\#, \\\\gamma ^\\\\#>0:\\\\quad \\\\gamma _\\\\#\\\\le \\\\gamma (x,t,e)\\\\le \\\\gamma ^\\\\#; \\\\quad \\\\mbox{for a.e. }\",\n \"(x,t)\\\\in \\\\Sigma _T,\\\\quad \\\\forall e\\\\in \\\\mathbb {R}.\",\n \"$ Moreover, $\\\\gamma $ is strongly monotone: $\\\\left(\\\\gamma ( u )| u |^{\\\\ell -2} u- \\\\gamma ( v )| v |^{\\\\ell -2} v \\\\right) ( u- v)\\\\ge \\\\gamma _\\\\# |u - v|^\\\\ell .$ Let us state our main existence theorem.\",\n \"Theorem 2.1 Let (H1)-(H6) be fulfilled.\",\n \"The thermoelectric problem (TE) admits a solution $(\\\\theta ,\\\\phi )\\\\in V_\\\\ell (Q_T)\\\\times L^2(0,T; V)$ , for $\\\\mathcal {M}$ being such that $\\\\mathcal {M}\\\\alpha ^\\\\#\\\\sigma ^\\\\#< k_\\\\#,$ and one of the following hypothesis is assured: there holds $4( k_\\\\# - \\\\mathcal {M}\\\\alpha ^\\\\#\\\\sigma ^\\\\# ) \\\\sigma _\\\\# > (\\\\sigma ^\\\\#)^2 ( \\\\Pi ^\\\\#+\\\\mathcal {M} +\\\\alpha ^\\\\#)^2 ;$ there holds $4( k_\\\\# - \\\\mathcal {M}\\\\alpha ^\\\\#\\\\sigma ^\\\\# ) >\\\\sigma ^\\\\# ( \\\\Pi ^\\\\#+\\\\mathcal {M} +\\\\alpha ^\\\\#)^2 ;$ there holds $k_\\\\#> \\\\sigma ^\\\\# \\\\alpha ^\\\\# ( 2 \\\\Pi ^\\\\#+ 3 \\\\mathcal {M}) .$\"\n ],\n [\n \"Existence of approximated solutions\",\n \"The thermoelectric problem provides the abstract initial boundary value problem $b(\\\\theta )\\\\partial _t \\\\theta -\\\\nabla \\\\cdot \\\\left( a ( \\\\theta ,\\\\phi )\\\\nabla \\\\theta \\\\right)= \\\\nabla \\\\cdot ( \\\\sigma ( \\\\theta ) F (\\\\theta ,\\\\phi )\\\\nabla \\\\phi ) && \\\\\\\\-\\\\nabla \\\\cdot (\\\\sigma ( \\\\theta )\\\\nabla \\\\phi )= \\\\nabla \\\\cdot \\\\left( \\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S} (\\\\theta ) \\\\nabla \\\\theta \\\\right) && \\\\mbox{ in }Q_T ; \\\\\\\\\\\\left( a ( \\\\theta ,\\\\phi )\\\\nabla \\\\theta + \\\\sigma ( \\\\theta ) F ( \\\\theta ,\\\\phi )\\\\nabla \\\\phi \\\\right)\\\\cdot \\\\mathbf {n} =\\\\left(h - \\\\gamma (\\\\theta ) |\\\\theta |^{\\\\ell -2} \\\\theta \\\\right)\\\\chi _{\\\\Gamma } && \\\\\\\\\\\\left( \\\\sigma ( \\\\theta )\\\\nabla \\\\phi + \\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S} (\\\\theta )\\\\nabla \\\\theta \\\\right)\\\\cdot {\\\\bf n} =g \\\\chi _{\\\\Gamma _\\\\mathrm {N}}&&\\\\mbox{ on } \\\\partial \\\\Omega \\\\times ]0,T[ .\",\n \"$ This abstract problem is formulated in the form that the coefficients are correlated with the leading coefficient $\\\\sigma $ .\",\n \"We emphasize that this interrelation must be clear.\",\n \"Let us assume the hypothesis set.\",\n \"(H) The operators $ a,F $ and $b,\\\\sigma , \\\\alpha _\\\\mathrm {S}$ are Carathéodory functions from $\\\\Omega \\\\times \\\\mathbb {R}^2$ and $\\\\Omega \\\\times \\\\mathbb {R}$ , respectively, into $\\\\mathbb {R}$ , which enjoy the following properties.\",\n \"There exist positive constants $F^\\\\#, a_\\\\#,a^\\\\#, b_\\\\#,b^\\\\#$ such that $| F(x,e,d )|\\\\le F^\\\\#; && \\\\\\\\a_\\\\#\\\\le a(x,e,d )\\\\le a^\\\\#; && \\\\\\\\b_\\\\#\\\\le b (x,e )\\\\le b^\\\\# &&\\\\mbox{for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\quad \\\\forall e,d \\\\in \\\\mathbb {R} , $ and $\\\\sigma _\\\\#,\\\\sigma ^\\\\#, \\\\alpha ^\\\\#$ verifying (), (REF ), respectively.\",\n \"Definition 3.1 We say that $(\\\\theta ,\\\\phi )$ is a weak solution to (REF )-() if it solves the variational problem $\\\\int _0^T\\\\langle b(\\\\cdot , \\\\theta )\\\\partial _t\\\\theta , v\\\\rangle \\\\mathrm {dt} +\\\\int _{Q_T} a (\\\\cdot ,\\\\theta ,\\\\phi )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\gamma (\\\\cdot ,\\\\theta )|\\\\theta |^{\\\\ell -2} \\\\theta v\\\\mathrm {ds} \\\\mathrm {dt}=\\\\nonumber \\\\\\\\ = - \\\\int _{Q_T}\\\\sigma ( \\\\theta ) F (\\\\cdot ,\\\\theta ,\\\\phi )\\\\nabla \\\\phi \\\\cdot \\\\nabla v\\\\mathrm {dx} \\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} h(\\\\cdot ,\\\\theta ) v\\\\mathrm {ds} \\\\mathrm {dt}; \\\\\\\\\\\\int _\\\\Omega \\\\sigma (\\\\cdot ,\\\\theta )\\\\nabla \\\\phi \\\\cdot \\\\nabla w\\\\mathrm {dx}+ \\\\int _\\\\Omega \\\\sigma (\\\\cdot , \\\\theta ) \\\\alpha _\\\\mathrm {S} (\\\\cdot ,\\\\theta )\\\\nabla \\\\theta \\\\cdot \\\\nabla w\\\\mathrm {dx} =\\\\int _{\\\\Gamma _{\\\\rm N}}g w\\\\mathrm {ds}, \\\\mbox{ a.e.\",\n \"in } ]0,T[ ,$ for every $v\\\\in V_{\\\\ell }(Q_T)$ and $w\\\\in V$ .\",\n \"We define an auxiliary operator.\",\n \"Denote by $B$ the operator from $H^1(\\\\Omega )$ into $L^2(\\\\Omega )$ defined by $B(v)=\\\\int _0^v b(\\\\cdot ,z)\\\\mathrm {dz},$ for all $v\\\\in H^1(\\\\Omega )$ .\",\n \"Different approaches in the finding of solutions according to Definition REF provide different smallness conditions (REF ), (REF ) or (REF ).\",\n \"We emphasize that the difference between these smallness conditions has its importance in the real-world applications.\",\n \"Theorem 3.1 Let (H) and (H4)-(H6) be fulfilled.\",\n \"If there exists $\\\\varepsilon >0$ such that one the following relation holds, that is, either $a_\\\\# >\\\\varepsilon \\\\sigma ^\\\\#(F^\\\\#+\\\\alpha ^\\\\#)/2\\\\quad \\\\mbox{ and } \\\\quad \\\\varepsilon \\\\sigma _\\\\# >\\\\sigma ^\\\\#(F^\\\\#+\\\\alpha ^\\\\#)/2,$ or $a_\\\\# >\\\\varepsilon \\\\sqrt{\\\\sigma ^\\\\#}(F^\\\\#+\\\\alpha ^\\\\#)/2 \\\\quad \\\\mbox{ and } \\\\quad \\\\varepsilon >\\\\sqrt{\\\\sigma ^\\\\#}(F^\\\\#+\\\\alpha ^\\\\#)/2, $ then the variational problem (REF )-() admits a sequence of approximate solutions $\\\\lbrace (\\\\theta _M,\\\\phi _M)\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ in the sense established in Section REF .\",\n \"The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^m) v\\\\mathrm {dx}+\\\\int _\\\\Omega a (\\\\theta ^m,\\\\phi ^m )\\\\nabla \\\\theta ^{m}\\\\cdot \\\\nabla v\\\\mathrm {dx}+\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^m)|\\\\theta ^m|^{\\\\ell -2} \\\\theta ^{m} v \\\\mathrm {ds} +\\\\nonumber \\\\\\\\ +\\\\int _\\\\Omega \\\\sigma (\\\\theta ^m) F( \\\\theta ^m,\\\\phi ^m ) \\\\nabla \\\\phi ^m \\\\cdot \\\\nabla v\\\\mathrm {dx} =\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^{m-1} )v\\\\mathrm {dx}+\\\\int _{\\\\Gamma } h_{m} v \\\\mathrm {ds} ; \\\\\\\\\\\\int _\\\\Omega \\\\sigma (\\\\theta ^m)\\\\nabla \\\\phi ^m\\\\cdot \\\\nabla w\\\\mathrm {dx}+ \\\\int _\\\\Omega \\\\sigma (\\\\theta ^m) \\\\alpha _\\\\mathrm {S}(\\\\theta ^m)\\\\nabla \\\\theta ^{m} \\\\cdot \\\\nabla w \\\\mathrm {dx}=\\\\int _{\\\\Gamma _\\\\mathrm {N}} g w \\\\mathrm {ds} ,$ where $\\\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\\\in \\\\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).\",\n \"We call $\\\\phi ^m$ the corresponding solution to the time independent temperature $\\\\theta ^{m}$ .\",\n \"Theorem 3.2 Let (H) and (H4)-(H6) be fulfilled.\",\n \"If there holds $a_\\\\#>2\\\\sigma ^\\\\# \\\\alpha ^\\\\# F^\\\\# ,$ then the variational problem (REF )-() admits a sequence of approximate solutions $\\\\lbrace (\\\\theta _M,\\\\phi _M)\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ in the sense established in Section REF .\",\n \"The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^m) v\\\\mathrm {dx}+\\\\int _\\\\Omega a (\\\\theta ^m,\\\\phi ^m )\\\\nabla \\\\theta ^{m}\\\\cdot \\\\nabla v\\\\mathrm {dx}+\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^m)|\\\\theta ^m|^{\\\\ell -2} \\\\theta ^{m} v \\\\mathrm {ds} +\\\\nonumber \\\\\\\\ +\\\\int _\\\\Omega \\\\sigma (\\\\theta ^m) F( \\\\theta ^m,\\\\phi ^m ) \\\\nabla \\\\phi ^m \\\\cdot \\\\nabla v\\\\mathrm {dx} =\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^{m-1} )v\\\\mathrm {dx}+\\\\int _\\\\Gamma h_{m} v \\\\mathrm {ds} ; \\\\\\\\\\\\int _\\\\Omega \\\\sigma (\\\\theta ^{m-1})\\\\nabla \\\\phi ^m\\\\cdot \\\\nabla w\\\\mathrm {dx}= - \\\\int _\\\\Omega \\\\sigma (\\\\theta ^{m-1}) \\\\alpha _\\\\mathrm {S}(\\\\theta ^{m-1})\\\\nabla \\\\theta ^{m-1} \\\\cdot \\\\nabla w \\\\mathrm {dx} +\\\\nonumber \\\\\\\\ + \\\\int _{\\\\Gamma _\\\\mathrm {N}} g w \\\\mathrm {ds} ,$ where $\\\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\\\in \\\\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).\",\n \"We call $\\\\phi ^m$ the corresponding solution to the time independent temperature $\\\\theta ^{m-1}$ .\"\n ],\n [\n \"Steady-state solvability\",\n \"In this section, we prove the existence of solutions to the recurrent sequence of time-discretized problems (REF )-() and (REF )-() in Subsections REF and REF , respectively.\",\n \"Since $m\\\\in \\\\mathbb {N}$ is fixed and $\\\\theta ^{m-1}\\\\in V_{\\\\ell }(\\\\Omega )$ is given, for the sake of simplicity, we set $f= B(\\\\theta ^{m-1})$ and $H=h_m$ , and we omit the index to the unknown pair, i.e.\",\n \"we simply write $(\\\\theta ,\\\\phi )$ .\",\n \"Denoting by $K_{2}$ the continuity constant of the trace embedding $H^{1}(\\\\Omega )\\\\hookrightarrow L^2(\\\\Gamma )$ , with $2_*=2(n-1)/(n-2)$ if $n>2$ , and any $2_*>2$ if $n=2$ , and by $P_2$ the Poincaré constant correspondent to the space exponent 2, the constant $K_2(P_2+1)$ obeys $\\\\Vert v\\\\Vert _{2,\\\\Gamma }\\\\le K_2 \\\\left( \\\\Vert v\\\\Vert _{2,\\\\Omega }+ \\\\Vert \\\\nabla v\\\\Vert _{2,\\\\Omega }\\\\right)\\\\le K_2(P_2+1) \\\\Vert \\\\nabla v\\\\Vert _{2,\\\\Omega },\\\\quad \\\\forall v\\\\in H^1(\\\\Omega ).$ Let us introduce [1], [16] $\\\\Psi (s) := B(s)s-\\\\int _0^s B(r)\\\\mathrm {dr} = \\\\int _0^s(B(s) - B(r) )\\\\mathrm {dr}.$ We state the main properties of the auxiliary operators $B$ and $\\\\Psi $ , the ones that we will use later.\",\n \"For completeness sake, we sketch the proof of the property (REF ).\",\n \"Lemma 4.1 There holds $\\\\int _\\\\Omega (B(u)-B(v) ) u \\\\mathrm {dx} \\\\ge \\\\int _\\\\Omega \\\\Psi (u)\\\\mathrm {dx} - \\\\int _\\\\Omega \\\\Psi (v)\\\\mathrm {dx} .$ In particular, if the assumption () is fulfilled then there holds $\\\\int _\\\\Omega \\\\Psi (u)\\\\mathrm {dx}\\\\le \\\\int _\\\\Omega B(u) u \\\\mathrm {dx} \\\\le b^\\\\# \\\\Vert u\\\\Vert _{2,\\\\Omega }^2 .$ Under the assumption () the operator $B$ verifies $(B(u)-B(v),u-v)\\\\ge b_\\\\# \\\\Vert u-v\\\\Vert _{2,\\\\Omega }^2 .$ Let us write the decomposition $(B(u)-B(v) ) u = B(u)u-B(v) v - B(v)(u-v).$ Thanks to the mean value theorem for definite integrals, there exists $c$ between $u$ and $v$ such that $\\\\int _{v} ^{u} B(r)\\\\mathrm {dr}= B(c)(u-v).$ Since $-B$ is a decreasing function, we obtain $\\\\int _\\\\Omega (B(u)-B(v) ) u \\\\mathrm {dx} \\\\ge \\\\int _\\\\Omega (B(u)u-B(v) v )\\\\mathrm {dx} -\\\\int _\\\\Omega \\\\int _{v} ^{u} B(r)\\\\mathrm {dr} \\\\mathrm {dx},$ which concludes the proof by definition of $\\\\Psi $ .\",\n \"Finally, we recall the following remarkable lemma [1].\",\n \"Lemma 4.2 Suppose $u_m$ weakly converge to $u$ in $L^p(0,T;W^{1,p}(\\\\Omega ))$ , $p>1$ , with the estimates $\\\\int _\\\\Omega \\\\Psi (u_m(t)) \\\\mathrm {dx}\\\\le C\\\\quad \\\\mbox{for } 00$ $\\\\int _0^{T-z}\\\\int _\\\\Omega (B(u_m(t+z))-B(u_m(t)) ) (u_m(t+z)-u_m(t)) \\\\mathrm {dx} \\\\mathrm {dt} \\\\le Cz ,$ with $C$ being positive constants.\",\n \"Then, $B(u_m) \\\\rightarrow B(u)$ in $L^1(Q_T)$ and $\\\\Psi (u_m) \\\\rightarrow \\\\Psi (u) $ almost everywhere in $Q_T$ .\"\n ],\n [\n \"Fixed point argument (solvability to (\",\n \"Let $\\\\ell \\\\ge 2$ , and define an operator $\\\\mathcal {T}$ from $\\\\mathbf {V}_{\\\\ell } =V_\\\\ell (\\\\Omega )\\\\times V$ into itself such that $(\\\\theta ,\\\\phi )=\\\\mathcal {T}(\\\\mathbf {u})$ is the unique solution of Proposition REF .\",\n \"Proposition 4.1 Let $\\\\mathbf {u} =(u_1,u_2)\\\\in \\\\mathbf {V}_{\\\\ell }$ , and $u=u_1$ .\",\n \"Then, there exists a unique solution $(\\\\theta ,\\\\phi )\\\\in \\\\mathbf {V}_{\\\\ell }$ to the Neumann-power type elliptic problem $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega b(u)\\\\theta v\\\\mathrm {dx}+\\\\int _\\\\Omega a( \\\\mathbf {u} )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}+\\\\int _\\\\Omega \\\\sigma (u) F( \\\\mathbf {u}) \\\\nabla \\\\phi \\\\cdot \\\\nabla v\\\\mathrm {dx} +\\\\nonumber \\\\\\\\+\\\\int _\\\\Gamma \\\\gamma (u) |\\\\theta |^{\\\\ell -2} \\\\theta v \\\\mathrm {ds} =\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega fv \\\\mathrm {dx} +\\\\int _\\\\Gamma Hv \\\\mathrm {ds} ; \\\\\\\\\\\\int _\\\\Omega \\\\sigma (u)\\\\nabla \\\\phi \\\\cdot \\\\nabla w\\\\mathrm {dx}+ \\\\int _\\\\Omega \\\\sigma (u) \\\\alpha _\\\\mathrm {S}(u) \\\\nabla \\\\theta \\\\cdot \\\\nabla w \\\\mathrm {dx}=\\\\int _{\\\\Gamma _\\\\mathrm {N}} g w \\\\mathrm {ds} ,$ for all $v\\\\in V_{\\\\ell }(\\\\Omega )$ and $w\\\\in V$ .\",\n \"In addition, the following estimate $\\\\frac{b_\\\\#}{2\\\\tau }\\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }^2 + (L_{1})_\\\\#\\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2 +\\\\frac{(L_{2})_\\\\# }{2}\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }^2+\\\\frac{\\\\gamma _\\\\#}{\\\\ell ^{\\\\prime }} \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } ^\\\\ell \\\\le \\\\frac{ 1 }{ 2\\\\tau b_\\\\# }\\\\Vert f\\\\Vert _{2,\\\\Omega }^2+ \\\\nonumber \\\\\\\\ +\\\\frac{1}{\\\\ell ^{\\\\prime }\\\\gamma _\\\\#^{1/(\\\\ell -1)}} \\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }}+\\\\frac{(K_2)^2(P_2+1)^2}{2 (L_{2})_\\\\#}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N} } ^2 := \\\\mathcal {R}( \\\\Vert f\\\\Vert _{2,\\\\Omega }^2 ,\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }} ) \\\\quad $ holds true, if provided by one the following definition $&&\\\\left\\\\lbrace \\\\begin{array}l(L_{1})_\\\\# = a_\\\\#- \\\\varepsilon \\\\sigma ^\\\\# \\\\left( F^\\\\# + \\\\alpha ^\\\\#\\\\right) /2\\\\\\\\(L_{2})_\\\\# = \\\\sigma _\\\\# - \\\\sigma ^\\\\# \\\\left( F^\\\\# + \\\\alpha ^\\\\#\\\\right) /(2\\\\varepsilon )\\\\end{array}\\\\right.\",\n \"\\\\\\\\ &&\\\\left\\\\lbrace \\\\begin{array}l(L_{1})_\\\\# = a_\\\\#- \\\\varepsilon \\\\sqrt{\\\\sigma ^\\\\# }( F^\\\\#+\\\\alpha ^\\\\#) /2\\\\\\\\(L_{2})_\\\\# = \\\\sigma _\\\\#\\\\left( 1- \\\\sqrt{\\\\sigma ^\\\\# }( F^\\\\#+\\\\alpha ^\\\\#) /(2\\\\varepsilon ) \\\\right)\\\\end{array}\\\\right.\",\n \".$ The existence of a solution to the variational system (REF )-() relies on the direct application of the Browder-Minty Theorem [18].\",\n \"Indeed, the form $\\\\mathcal {F}: \\\\mathbf {V}_{\\\\ell }\\\\rightarrow \\\\mathbb {R}$ defined by $\\\\mathcal {F}(v,w)=\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega fv\\\\mathrm {dx}+\\\\int _\\\\Gamma Hv \\\\mathrm {ds} +\\\\int _{\\\\Gamma _\\\\mathrm {N}} g w \\\\mathrm {ds}$ is continuous and linear, and the form $\\\\mathcal {L}: \\\\mathbf {V}_{\\\\ell }\\\\times \\\\mathbf {V}_{\\\\ell }\\\\rightarrow \\\\mathbb {R}$ defined by $\\\\mathcal {L} \\\\left( (\\\\theta ,\\\\phi ), (v,w)\\\\right)=\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega b(u) \\\\theta v\\\\mathrm {dx}+\\\\int _\\\\Omega \\\\left(\\\\mathsf {L}( \\\\mathbf {u} )\\\\nabla \\\\left[\\\\begin{array}{c}\\\\theta \\\\\\\\\\\\phi \\\\end{array} \\\\right]\\\\right)\\\\cdot \\\\nabla \\\\left[\\\\begin{array}{c} v\\\\\\\\w\\\\end{array} \\\\right]\\\\mathrm {dx},$ is continuous and bilinear, with $\\\\mathsf {L}$ being the $(2\\\\times 2)$ -matrix $\\\\mathsf {L} ( \\\\mathbf {u} )=\\\\left[\\\\begin{array}{cc}a( \\\\mathbf {u} )& \\\\sigma (u) F( \\\\mathbf {u} ) \\\\\\\\\\\\sigma (u) \\\\alpha _\\\\mathrm {S} (u)& \\\\sigma (u)\\\\end{array}\\\\right] .$ Moreover, $\\\\mathcal {L}$ is coercive: $\\\\sum _{i,j=1}^{ 2}\\\\sum _{l=1}^n \\\\left( L_{i,j} ( \\\\mathbf {u})\\\\xi _{j,l}\\\\right)\\\\xi _{l,i}\\\\ge (L_{1})_\\\\#|\\\\xi _1|^2+ (L_{2})_\\\\#|\\\\xi _2|^2,$ with $(L_{1})_\\\\#$ and $(L_{2})_\\\\#$ being the positive constants defined in (REF ) or (), taking the assumptions (REF ) and (REF ) into account.\",\n \"The difference of the definitions is consequence of the different application of the Young inequality $2AB\\\\le \\\\varepsilon A^2+B^2/\\\\varepsilon $ ($\\\\varepsilon ,A,B>0$ ), see Remark REF .\",\n \"Namely, with $A=| \\\\xi _{1}| $ and $B=| \\\\xi _{2}|$ , for (REF ).\",\n \"That is, $ \\\\sum _{l=1}^n \\\\left(\\\\sigma (u) F( \\\\mathbf {u} ) \\\\xi _{2,l}\\\\xi _{l,1}+\\\\sigma (u) \\\\alpha _\\\\mathrm {S} (u) \\\\xi _{1,l}\\\\xi _{l,2} \\\\right) \\\\le \\\\sigma ^\\\\# \\\\left( F^\\\\#+\\\\alpha ^\\\\# \\\\right)\\\\left( \\\\frac{\\\\varepsilon }{2}A^2+\\\\frac{1}{2\\\\varepsilon } B^2\\\\right).$ $A=| \\\\xi _{1}| $ and $B=\\\\sqrt{\\\\sigma (u)}| \\\\xi _{2}| $ , for ().\",\n \"That is, $ \\\\sum _{l=1}^n \\\\left(\\\\sigma (u) F( \\\\mathbf {u} ) \\\\xi _{2,l}\\\\xi _{l,1}+\\\\sigma (u) \\\\alpha _\\\\mathrm {S} (u) \\\\xi _{1,l}\\\\xi _{l,2} \\\\right)\\\\le \\\\sqrt{\\\\sigma ^\\\\# }\\\\left( F^\\\\#+\\\\alpha ^\\\\# \\\\right)\\\\left( \\\\frac{\\\\varepsilon }{2}A^2+\\\\frac{ 1}{2\\\\varepsilon }B^2\\\\right).$ Finally, observing that the function $e\\\\in \\\\mathbb {R}\\\\mapsto \\\\gamma (u)|e|^{\\\\ell -2}e $ is monotonically increasing, we conclude the existence of the required solution.\",\n \"In order to obtain (REF ), we take $v=\\\\theta $ and $w=\\\\phi $ as test functions in (REF ) and (), respectively.\",\n \"Summing the obtained relations, and applying (), (REF ), the coercivity (REF ) of $\\\\mathsf {L}$ , and the Hölder inequality, we find $\\\\frac{ b_\\\\# }{\\\\tau } \\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }^2+ (L_{1})_\\\\# \\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2+(L_{2})_\\\\#\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }^2+\\\\gamma _\\\\# \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } ^\\\\ell \\\\le \\\\nonumber \\\\\\\\ \\\\le \\\\frac{ 1 }{\\\\tau } \\\\Vert f\\\\Vert _{2,\\\\Omega }\\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }+\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma }+\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}\\\\Vert \\\\phi \\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}.$ We successively apply (REF ) and the Young inequality to obtain $\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma } \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma }+\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N} }\\\\Vert \\\\phi \\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}} \\\\le \\\\nonumber \\\\\\\\\\\\le \\\\frac{1}{\\\\ell ^{\\\\prime }\\\\gamma _{\\\\#}^{1/(\\\\ell -1)} }\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }}+\\\\frac{\\\\gamma _\\\\#}{\\\\ell }\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma }^\\\\ell +\\\\frac{K_2^2(P_2+1)^2}{ 2(L_{2})_\\\\#}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}^2+ \\\\frac{(L_{2})_\\\\#}{2}\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega } ^2.\",\n \"$ Inserting (REF ) into (REF ), we deduce (REF ).\",\n \"Remark 4.1 Even $\\\\varepsilon >0$ may be an arbitrary (but fixed) number, we may differently define $(L_1)_\\\\#$ and $(L_2)_\\\\#$ .\",\n \"Indeed, the Young inequality $2AB\\\\le \\\\varepsilon A^2+B^2/\\\\varepsilon $ ($\\\\varepsilon ,A,B>0$ ) may be applied to obtain $\\\\sum _{l=1}^n \\\\left(\\\\sigma (u) F( \\\\mathbf {u} ) \\\\xi _{2,l}\\\\xi _{l,1}+\\\\sigma (u) \\\\alpha _\\\\mathrm {S} (u) \\\\xi _{1,l}\\\\xi _{l,2} \\\\right) \\\\le \\\\sigma ^\\\\# \\\\left( F^\\\\#\\\\left( \\\\frac{\\\\varepsilon _1}{2}| \\\\xi _{1}|^2+\\\\frac{1}{2\\\\varepsilon _1} | \\\\xi _{2}|^2\\\\right)+\\\\right.\",\n \"\\\\\\\\ \\\\left.\",\n \"+\\\\alpha ^\\\\#\\\\left( \\\\frac{\\\\varepsilon _2}{2}| \\\\xi _{1}|^2+\\\\frac{1}{2\\\\varepsilon _2} | \\\\xi _{2}|^2\\\\right) \\\\right).$ Next, let us determine whose radius make possible that the operator $\\\\mathcal {T}$ maps a closed ball into itself.\",\n \"Proposition 4.2 For $R=\\\\max \\\\lbrace R_1,R_2\\\\rbrace $ with $R_1$ and $R_2$ being defined in (REF ) and (REF ), respectively, the operator $\\\\mathcal {T}$ verifies $\\\\mathcal {T}(K)\\\\subset K$ , with $K=\\\\left\\\\lbrace (v,w)\\\\in \\\\mathbf {V}_\\\\ell : \\\\ \\\\Vert \\\\nabla w\\\\Vert _{2,\\\\Omega }+\\\\Vert \\\\nabla v\\\\Vert _{2,\\\\Omega }+ \\\\Vert v\\\\Vert _{\\\\ell ,\\\\Gamma } \\\\le R\\\\right\\\\rbrace .$ Let $\\\\mathbf {u}\\\\in \\\\mathbf {V}_{\\\\ell }$ , $u=u_1$ and $(\\\\theta ,\\\\phi )$ be the unique solution of Proposition REF , i.e.\",\n \"$(\\\\theta ,\\\\phi )=\\\\mathcal {T}(\\\\mathbf {u})$ .\",\n \"In order to prove that $(\\\\theta ,\\\\phi )\\\\in K$ we consider two different cases: (1) if $\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } \\\\le 1$ ; and (2) if $\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } >1$ , if $\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } \\\\le 1$ , then there holds $\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }+\\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }+ \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } \\\\le \\\\sqrt{2}\\\\left(\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }^2+ \\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2\\\\right)^{1/2}+1,$ by applying the elementary inequality $(a+b)^2\\\\le 2(a^2+b^2)$ for every $a,b\\\\ge 0$ .\",\n \"By using (REF ), we may take $R_1=\\\\left(\\\\frac{2\\\\mathcal {R}}{\\\\min \\\\left\\\\lbrace (L_{1})_\\\\#, (L_{2})_\\\\#/2\\\\right\\\\rbrace } \\\\right)^{1/2}+1 .\",\n \"$ if $\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } > 1$ , then using $\\\\ell \\\\ge 2$ there holds $\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }+\\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }+ \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } \\\\le \\\\sqrt{2}\\\\left( 2(\\\\Vert \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }^2+\\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2) + \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } ^\\\\ell \\\\right)^{1/2},$ by applying the elementary inequality $(a+b)^2\\\\le 2(a^2+b^2)$ for every $a,b\\\\ge 0$ .\",\n \"By using (REF ), we may take $R_2^2=\\\\left(\\\\frac{2 }{\\\\min \\\\left\\\\lbrace (L_{1})_\\\\#, (L_{2})_\\\\#/2\\\\right\\\\rbrace } +\\\\frac{\\\\ell ^{\\\\prime }}{\\\\gamma _\\\\#}\\\\right)\\\\mathcal {R} .\",\n \"$ Then, the proof is complete by taking $R$ such that is the maximum of $R_1$ and $R_2$ defined in (REF ) and (REF ), respectively.\",\n \"Proposition 4.3 The operator $\\\\mathcal {T}$ is continuous.\",\n \"Let $\\\\lbrace \\\\mathbf {u}^m\\\\rbrace _{m\\\\in \\\\mathbb {N}}$ be a sequence such that weakly converges to $\\\\mathbf {u}=(u,u_2)$ in $ \\\\mathbf {V}_\\\\ell $ , and $(\\\\theta _m,\\\\phi _m)=\\\\mathcal {T}(\\\\mathbf {u}^m)$ for each $m\\\\in \\\\mathbb {N}$ .\",\n \"Proposition REF guarantees that $(\\\\theta _m,\\\\phi _m)$ solves, for each $m\\\\in \\\\mathbb {N}$ , the variational system (REF )$_m$ -()$_m$ , with $\\\\mathbf {u}$ replaced by $\\\\mathbf {u}^m$ .\",\n \"The uniform boundedness ensured by Proposition REF guarantees the existence of a limit $(\\\\theta ,\\\\phi )\\\\in \\\\mathbf {V}_\\\\ell $ , for at least a subsequence of $(\\\\theta _m,\\\\phi _m)$ still denoted by $(\\\\theta _m,\\\\phi _m)$ , such that $\\\\theta _m\\\\rightharpoonup \\\\theta \\\\quad \\\\mbox{in } V_\\\\ell (\\\\Omega )\\\\quad \\\\mbox{and}\\\\quad \\\\phi _m\\\\rightharpoonup \\\\phi \\\\quad \\\\mbox{in } V \\\\quad (\\\\mbox{as } m\\\\rightarrow +\\\\infty ) .$ The Rellich-Kondrachov theorem guarantees the strong convergences $u^m\\\\rightarrow u &\\\\mbox{ and }& u_2^m\\\\rightarrow u_2 \\\\quad \\\\mbox{in } L^2(\\\\Omega ); \\\\\\\\\\\\theta _m\\\\rightarrow \\\\theta &\\\\mbox{ and }& \\\\phi _m\\\\rightarrow \\\\phi \\\\quad \\\\mbox{in } L^2(\\\\Omega ); \\\\\\\\u^m\\\\rightarrow u &\\\\mbox{ and }& \\\\theta _m\\\\rightarrow \\\\theta \\\\quad \\\\mbox{in } L^2(\\\\Gamma ).$ To show that $(\\\\theta ,\\\\phi )=\\\\mathcal {T}(\\\\mathbf {u})$ , it remains to pass to the limit in the system (REF )$_m$ -()$_m$ as $m$ tends to infinity.\",\n \"Applying the Krasnoselski theorem to the Nemytskii operators $b$ , $a$ , $\\\\sigma $ , we have $b(u^m) v\\\\rightarrow b(u) v &\\\\mbox{ in }& L^2(\\\\Omega );\\\\\\\\a(\\\\mathbf {u}^m) \\\\nabla v\\\\rightarrow a(\\\\mathbf {u}) \\\\nabla v &\\\\mbox{ in }& \\\\mathbf {L}^2(\\\\Omega ); \\\\\\\\\\\\sigma (u^m)\\\\nabla v\\\\rightarrow \\\\sigma ( u)\\\\nabla v &\\\\mbox{ in }& \\\\mathbf {L}^2(\\\\Omega ),$ for all $v\\\\in H^1(\\\\Omega )$ , making use of the Lebesgue dominated convergence theorem and the assumptions () and ()-().\",\n \"Also the terms $\\\\sigma (u^m)F(\\\\mathbf {u}^m) \\\\nabla v$ and $ \\\\sigma (u^m) \\\\alpha _\\\\mathrm {S} (u^m) \\\\nabla v$ pass to the limit making recourse to the assumptions (REF ) and (REF ), respectively.\",\n \"Similarly, the boundary term $\\\\gamma (u^m)v$ converges to $\\\\gamma (u) v$ in $L^{\\\\ell ^{\\\\prime }}(\\\\Gamma )$ , for all $v\\\\in L^{\\\\ell ^{\\\\prime }}(\\\\Gamma )$ , due to (REF ).\",\n \"Observe that $ \\\\theta _m$ strongly converges to $\\\\theta $ in $L^p(\\\\Gamma )$ , for all $10$ , and finally letting $\\\\delta \\\\rightarrow 0^+$ we arrive to $\\\\int _{\\\\Gamma }\\\\gamma (u)(\\\\Lambda - |\\\\theta |^{\\\\ell -2}\\\\theta )\\\\varphi \\\\mathrm {ds} \\\\ge 0, \\\\quad \\\\forall \\\\varphi \\\\in \\\\mathcal {D}(\\\\Gamma ),$ which implies that $\\\\Lambda = |\\\\theta |^{\\\\ell -2}\\\\theta $ .\",\n \"Thus, we are in the condition of concluding that $(\\\\theta ,\\\\phi )$ is the required limit solution, i.e.\",\n \"it solves the limit system (REF )-().\",\n \"Thanks to Propositions REF , REF and REF , there exists at least one fixed point of $\\\\mathcal {T}$ , that is $(\\\\theta ,\\\\phi )=\\\\mathcal {T}(\\\\theta ,\\\\phi )$ , which concludes the solvability to (REF )-().\"\n ],\n [\n \"Fixed point argument (solvability to (\",\n \"Let $\\\\ell \\\\ge 2$ , and define an operator $\\\\mathcal {T}$ from $V_{\\\\ell }(\\\\Omega )$ into itself such that $\\\\theta =\\\\mathcal {T}(u)$ is the unique solution of Proposition REF .\",\n \"Denote by the well defined continuous operator such that $\\\\mathcal {F}( u)=\\\\phi $ .\",\n \"The existence of a unique weak auxiliary solution $\\\\phi $ to the variational equality () is standard and it can be stated as follows.\",\n \"Proposition 4.4 Let $u\\\\in H^1(\\\\Omega )$ .\",\n \"Under the assumptions (), (REF ) and (H5), the Neumann problem $\\\\int _\\\\Omega \\\\sigma (u) \\\\nabla \\\\phi \\\\cdot \\\\nabla w\\\\mathrm {dx}=\\\\int _\\\\Omega \\\\sigma (u) \\\\alpha _\\\\mathrm {S} (u)\\\\nabla u\\\\cdot \\\\nabla w \\\\mathrm {dx}+\\\\int _{\\\\Gamma _\\\\mathrm {N}} gw \\\\mathrm {ds},\\\\qquad \\\\forall w\\\\in V,$ admits a unique solution $\\\\phi \\\\in V$ .\",\n \"Moreover, the estimate $\\\\Vert \\\\sqrt{\\\\sigma (u)} \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }\\\\le \\\\sqrt{\\\\sigma ^\\\\#}\\\\alpha ^\\\\# \\\\Vert \\\\nabla u\\\\Vert _{2,\\\\Omega }+\\\\frac{K_2 (P_2+1) }{\\\\sqrt{\\\\sigma _\\\\#}}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}$ holds true.\",\n \"Let us establish the quantitative estimate (REF ).\",\n \"We take $w=\\\\phi $ as a test function in (REF ), and we compute by applying the Hölder inequality and (REF ) $\\\\Vert \\\\sqrt{\\\\sigma (u)}\\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega } ^2 \\\\le \\\\left( \\\\alpha ^\\\\#\\\\Vert \\\\sqrt{\\\\sigma (u)}\\\\nabla u\\\\Vert _{2,\\\\Omega }+\\\\frac{ K_2(P_2+1)}{\\\\sqrt{\\\\sigma _\\\\#}} \\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}\\\\right)\\\\Vert \\\\sqrt{\\\\sigma (u)}\\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega } .$ Then, (REF ) arises.\",\n \"Proposition 4.5 Let $\\\\mathbf {u} =(u,\\\\phi ) \\\\in \\\\left( H^1(\\\\Omega ) \\\\right)^2$ .\",\n \"Under the assumptions (), (REF ) and (REF )-(), there exists a unique solution $\\\\theta \\\\in V_\\\\ell (\\\\Omega )$ to the power type elliptic problem $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega b(u)\\\\theta v\\\\mathrm {dx}+\\\\int _\\\\Omega a (\\\\mathbf {u} )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}+\\\\int _\\\\Omega \\\\sigma (u) F ( \\\\mathbf {u} ) \\\\nabla \\\\phi \\\\cdot \\\\nabla v\\\\mathrm {dx} +\\\\nonumber \\\\\\\\+\\\\int _\\\\Gamma \\\\gamma (u) |\\\\theta |^{\\\\ell -2}\\\\theta v \\\\mathrm {ds} =\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega fv \\\\mathrm {dx} +\\\\int _\\\\Gamma Hv \\\\mathrm {ds} ,$ for all $v\\\\in V_{\\\\ell }(\\\\Omega )$ .\",\n \"If $\\\\phi \\\\in V$ satisfies (REF ), then the following estimate $\\\\frac{b_\\\\#}{2\\\\tau }\\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }^2 +\\\\frac{ a_\\\\#}{2} \\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2+\\\\frac{\\\\gamma _\\\\#}{\\\\ell ^{\\\\prime }} \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } ^\\\\ell \\\\le \\\\frac{ 1 }{ 2\\\\tau b_\\\\# }\\\\Vert f\\\\Vert _{2,\\\\Omega }^2 +\\\\frac{1}{\\\\ell ^{\\\\prime }\\\\gamma _\\\\#^{1/(\\\\ell -1)}}\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }}+ \\\\nonumber \\\\\\\\ +\\\\frac{(F^\\\\#)^2\\\\sigma ^\\\\#}{ a_\\\\#}\\\\left( \\\\sigma ^\\\\#(\\\\alpha ^\\\\# )^2\\\\Vert \\\\nabla u \\\\Vert _{2,\\\\Omega }^2 +\\\\frac{K_2 ^2(P_2+1)^2 }{\\\\sigma _\\\\#}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}^2\\\\right) .\",\n \"$ holds true.\",\n \"Taking $v=\\\\theta $ as a test function in (REF ), and applying (), (), (REF ), (REF ), and the Hölder inequality, we find $\\\\frac{b_\\\\#}{\\\\tau }\\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }^2+ a_\\\\# \\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2+\\\\gamma _\\\\# \\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } ^\\\\ell \\\\le \\\\\\\\ \\\\le \\\\frac{ 1 }{\\\\tau } \\\\Vert f\\\\Vert _{2,\\\\Omega }\\\\Vert \\\\theta \\\\Vert _{2,\\\\Omega }+F^\\\\#\\\\Vert \\\\sqrt{\\\\sigma (u)} \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega } \\\\Vert \\\\sqrt{\\\\sigma (u)} \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }+\\\\Vert H\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }\\\\Vert \\\\theta \\\\Vert _{\\\\ell ,\\\\Gamma } .$ Applying (REF ) and the Young inequality, we compute $\\\\Vert \\\\sqrt{\\\\sigma (u)} \\\\nabla \\\\phi \\\\Vert _{2,\\\\Omega }\\\\Vert \\\\sqrt{\\\\sigma (u)} \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }\\\\le \\\\frac{a_\\\\#}{2} \\\\Vert \\\\nabla \\\\theta \\\\Vert _{2,\\\\Omega }^2 + \\\\\\\\+\\\\frac{\\\\sigma ^\\\\# }{a_\\\\#} \\\\left(\\\\sigma ^\\\\# (\\\\alpha ^\\\\#)^2 \\\\Vert \\\\nabla u\\\\Vert _{2,\\\\Omega } ^2+\\\\frac{K_2^2(P_2+1) ^2}{\\\\sigma _\\\\#}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}^2 \\\\right).$ Then, arguing as in (REF ), we deduce (REF ).\",\n \"Next, let us determine whose radius make possible that the operator $\\\\mathcal {T}$ maps a closed ball into itself.\",\n \"Proposition 4.6 Let (REF ) be fulfilled.\",\n \"For $\\\\tau \\\\le a_\\\\# / b_\\\\#$ and $R>0$ being defined as in (REF ), the operator $\\\\mathcal {T}$ verifies $\\\\mathcal {T}(\\\\overline{B_R})\\\\subset \\\\overline{B_R}$ , with $B_R$ denoting the open ball of $H^1(\\\\Omega )$ with radius $R$ .\",\n \"Let $u\\\\in H^1(\\\\Omega )$ and $\\\\theta =\\\\mathcal {T}(u)$ be the unique solution according to Proposition REF .\",\n \"Considering (REF ) and $\\\\min \\\\left\\\\lbrace b_\\\\#/\\\\tau ,a_\\\\#\\\\right\\\\rbrace =a_\\\\#$ , the proof is complete by defining $R$ such that $R\\\\left(\\\\sqrt{ a_\\\\# }-2\\\\frac{F^\\\\# \\\\sigma ^\\\\#\\\\alpha ^\\\\# }{ \\\\sqrt{a_\\\\#}} \\\\right) &=&\\\\left(\\\\frac{ 1 }{\\\\tau b_\\\\# }\\\\Vert f \\\\Vert _{2,\\\\Omega }^2 +\\\\frac{2}{\\\\ell ^{\\\\prime }\\\\gamma _\\\\#^{1/(\\\\ell -1)}} \\\\Vert H \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }} \\\\right)^{1/2}+\\\\nonumber \\\\\\\\&&+ 2F^\\\\# K_2(P_2+1) \\\\sqrt{\\\\frac{\\\\sigma ^\\\\#}{ a_\\\\#\\\\sigma _\\\\#}}\\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}} , \\\\quad $ taking the assumption (REF ) be account.\",\n \"Proposition 4.7 Let $\\\\lbrace u_m\\\\rbrace _{m\\\\in \\\\mathbb {N}}$ be a sequence such that weakly converges to $u$ in $ H^1(\\\\Omega )$ , then the solution $(\\\\theta _m,\\\\phi _m)$ according to Propositions REF and REF weakly converges in $V_\\\\ell (\\\\Omega )\\\\times H^1(\\\\Omega )$ , and its limit is a solution according to Propositions REF and REF Let $(\\\\theta _m,\\\\phi _m)$ be the solution according to Propositions REF and REF and corresponding to $u_m$ for each $m\\\\in \\\\mathbb {N}$ .\",\n \"The estimates (REF ) and (REF ) guarantee that the sequence $(\\\\theta _m,\\\\phi _m)$ is uniformly bounded in $V_\\\\ell (\\\\Omega )\\\\times H^1(\\\\Omega )$ .\",\n \"Thus, we can extract a subsequence of $(\\\\theta _m,\\\\phi _m)$ still denoted by $(\\\\theta _m,\\\\phi _m)$ , weakly convergent to $(\\\\theta ,\\\\phi )$ in $V_\\\\ell (\\\\Omega )\\\\times H^1(\\\\Omega )$ .\",\n \"Similar arguments in the proof of Proposition REF the weak limit $(\\\\theta ,\\\\phi )$ solves the variational system consisting of (REF ) and (REF ), which concludes the proof of Proposition REF .\",\n \"Thanks to Propositions REF , REF , REF and REF , there exists at least one fixed point of $\\\\mathcal {T}:u\\\\mapsto (u, \\\\mathcal {F}(u))\\\\mapsto \\\\theta ,$ that is $\\\\theta =\\\\mathcal {T}(\\\\theta )$ and $\\\\phi =\\\\mathcal {F}(\\\\theta )$ , which concludes the solvability to (REF )-().\"\n ],\n [\n \"Time discretization technique\",\n \"In this section, we apply the method of discretization in time [17], [23], [24].\",\n \"We decompose the time interval $I=[0,T]$ into $M$ subintervals $I_{m,M}$ of size $\\\\tau $ such that $M=T/\\\\tau \\\\in \\\\mathbb {N}$ , i.e.\",\n \"$I_{m,M}=[(m-1)T/M,mT/M]$ for $m\\\\in \\\\lbrace 1,\\\\cdot \\\\cdot \\\\cdot ,M\\\\rbrace $ .\",\n \"We set $t_{m,M}=mT/M$ .\",\n \"Thus, the problem (REF ) is approximated by the following recurrent sequence of time-discretized problems $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^m) v\\\\mathrm {dx}+\\\\int _\\\\Omega a (\\\\theta ^m,\\\\phi ^m )\\\\nabla \\\\theta ^{m}\\\\cdot \\\\nabla v\\\\mathrm {dx}+\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^m)|\\\\theta ^m|^{\\\\ell -2} \\\\theta ^{m} v \\\\mathrm {ds} +\\\\nonumber \\\\\\\\ +\\\\int _\\\\Omega \\\\sigma (\\\\theta ^m) F( \\\\theta ^m,\\\\phi ^m ) \\\\nabla \\\\phi ^m \\\\cdot \\\\nabla v\\\\mathrm {dx}=\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega B (\\\\theta ^{m-1} )v\\\\mathrm {dx}+\\\\int _\\\\Gamma h(t_{m,M}) v \\\\mathrm {ds} ,$ for all $v\\\\in V_{\\\\ell }(\\\\Omega )$ , and the problem () is approximated by either () or () for all $w\\\\in V$ , corresponding to the two different approaches.\",\n \"The existence of weak solutions pair $(\\\\theta ^m,\\\\phi ^m) \\\\in V_{\\\\ell }(\\\\Omega )\\\\times V$ to the above systems of elliptic problems is established in Section with $H =h(t_{m,M})$ .\",\n \"Since $\\\\theta ^0\\\\in L^2(\\\\Omega )$ is known, we determine $(\\\\theta ^{1},\\\\phi ^1)$ as the unique solution of the Neumann-power type elliptic problems (REF )-() or (REF )-(), and we inductively proceed.\",\n \"Denote by $\\\\lbrace {\\\\theta }_M\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ , $\\\\lbrace {\\\\phi }_M\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ and $\\\\lbrace Z_M\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ the sequences of the (piecewise constant in time) functions, $ {\\\\theta }_M : [0,T] \\\\rightarrow V_\\\\ell (\\\\Omega )$ , $ {\\\\phi }_M : ]0,T] \\\\rightarrow V$ and $ Z_M:[0,T]\\\\rightarrow L^2(\\\\Omega )$ , defined by, respectively, a.e.\",\n \"in $\\\\Omega $ ${\\\\theta }_M (t) &:=& \\\\left\\\\lbrace \\\\begin{array}{ll}\\\\theta ^{0}&\\\\mbox{ for }t=0\\\\\\\\\\\\theta ^{m}&\\\\mbox{ for } t\\\\in ]t_{m-1,M},t_{m,M}]\\\\end{array}\\\\right.\",\n \"\\\\\\\\{\\\\phi }_M (t) &:=& \\\\phi ^m \\\\quad \\\\mbox{ for all }t\\\\in ]t_{m-1,M},t_{m,M} ],$ in accordance with one of the two variational formulations () and (), while $Z_M (t):=\\\\left\\\\lbrace \\\\begin{array}{ll}B(\\\\theta ^{0} ) &\\\\mbox{ for }t=0\\\\\\\\Z^{m}&\\\\mbox{ for } t\\\\in ]t_{m-1,M},t_{m,M}]\\\\end{array}\\\\right.\",\n \"\\\\mbox{ in }\\\\Omega ,$ with the discrete derivative with respect to $t$ at the time $t=t_{m,M}$ : $Z^{m}:=\\\\frac{B(\\\\theta ^{m})-B(\\\\theta ^{m-1}) }{\\\\tau }.$ While $\\\\theta _M$ is the Rothe function obtained from $\\\\theta ^m$ by piecewise constant interpolation with respect to time $t$ , the Rothe function, obtained from $\\\\theta ^m$ by piecewise linear interpolation with respect to time $t$ , $\\\\Theta _M$ is $\\\\Theta _M(\\\\cdot ,t)=\\\\theta ^{m-1}+(t-t_{m-1,M}) \\\\frac{\\\\theta ^m- \\\\theta ^{m-1}}{ \\\\tau }.$ For our purposes, we introduce the following definition.\",\n \"Definition 5.1 We say that $\\\\lbrace \\\\widetilde{B}_M= \\\\widetilde{B}(\\\\theta _M)\\\\rbrace _{M\\\\in \\\\mathbb {N}}$ is the Rothe sequence (affine on each time interval) if $\\\\widetilde{B}(\\\\cdot ,\\\\theta _M (t)) = B(\\\\cdot ,\\\\theta ^{m-1}) + \\\\frac{ t-t_{m-1,M}}{\\\\tau }\\\\left( B(\\\\cdot ,\\\\theta ^m) - B(\\\\cdot ,\\\\theta ^{m-1}) \\\\right)$ in $\\\\Omega $ , for all $t\\\\in I_{m, M}$ , for all $m\\\\in \\\\lbrace 1,\\\\cdots ,M\\\\rbrace $ .\",\n \"Denoting $h_M(t)=h(t_{m,M})$ for $t\\\\in ]t_{m-1,M},t_{m,M}]$ and $m\\\\in \\\\lbrace 1,\\\\cdots ,M\\\\rbrace $ , the triple $ ({\\\\theta }_M ,\\\\phi _M,{Z}_M )$ solve $\\\\int _0^T\\\\int _\\\\Omega {Z}_M v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\int _{Q_T} a ( {\\\\theta }_M , {\\\\phi }_M )\\\\nabla {\\\\theta }_M \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\gamma ( {\\\\theta }_M )| {\\\\theta }_M |^{\\\\ell -2} {\\\\theta }_M v \\\\mathrm {ds}\\\\mathrm {dt} +\\\\nonumber \\\\\\\\+\\\\int _{Q_T} \\\\sigma ( \\\\theta _M ) F( \\\\theta _M , \\\\phi _M ) \\\\nabla {\\\\phi }_M \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}=\\\\int _{\\\\Sigma _T} h_M v \\\\mathrm {ds}\\\\mathrm {dt} .\\\\qquad $\"\n ],\n [\n \"Proof of Theorem \",\n \"Here, $ {\\\\phi }_M $ solves $\\\\int _\\\\Omega \\\\sigma (\\\\theta _M)\\\\nabla \\\\phi _M\\\\cdot \\\\nabla w\\\\mathrm {dx}+ \\\\int _\\\\Omega \\\\sigma (\\\\theta _M) \\\\alpha _\\\\mathrm {S}(\\\\theta _M)\\\\nabla \\\\theta _M \\\\cdot \\\\nabla w \\\\mathrm {dx}=\\\\int _{\\\\Gamma _\\\\mathrm {N}} g w \\\\mathrm {ds} ,$ for $w\\\\in V$ and a.e.\",\n \"in $]0,T[$ .\",\n \"We begin by establishing the uniform estimates to $ {\\\\theta }_M $ and $ {\\\\phi }_M $ .\",\n \"Proposition 5.1 Let $ {\\\\theta }_M $ and $ {\\\\phi }_M $ be the (piecewise constant in time) functions defined in (REF )-().\",\n \"Then the following estimate holds: $\\\\max _{1\\\\le m\\\\le M} \\\\int _\\\\Omega \\\\Psi ( \\\\theta ^m )\\\\mathrm {dx}+( L_1)_\\\\# \\\\Vert \\\\nabla \\\\theta _M\\\\Vert _{2,Q_T} ^2+\\\\frac{(L_{2})_\\\\# }{2}\\\\Vert \\\\nabla \\\\phi _M\\\\Vert _{2,Q_T} ^2 +\\\\nonumber \\\\\\\\+\\\\frac{\\\\gamma _\\\\#}{\\\\ell ^{\\\\prime }} \\\\Vert \\\\theta _M\\\\Vert _{\\\\ell ,\\\\Sigma _T } ^\\\\ell \\\\le b^\\\\# \\\\Vert \\\\theta ^0\\\\Vert _{2,\\\\Omega } ^2 +\\\\frac{1}{\\\\ell ^{\\\\prime }\\\\gamma _{\\\\#}^{1/(\\\\ell -1)} } \\\\Vert h \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Sigma _T }^{\\\\ell ^{\\\\prime }} +T \\\\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\\\#} \\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}^2 .$ Let $m\\\\in \\\\lbrace 1,\\\\cdots ,M\\\\rbrace $ be arbitrary.\",\n \"Choosing $v=\\\\theta ^m\\\\in V_{\\\\ell }(\\\\Omega )$ and $w=\\\\phi ^m \\\\in V$ as test functions in (REF )-(), we sum the obtained relations, and arguing as in (REF )-(REF ), we have $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega (B(\\\\theta ^m)-B(\\\\theta ^{m-1}) )\\\\theta ^m\\\\mathrm {dx} +( L_1)_\\\\# \\\\Vert \\\\nabla \\\\theta ^{m} \\\\Vert _{2,\\\\Omega }^2 +\\\\frac{(L_{2})_\\\\#}{2} \\\\Vert \\\\nabla \\\\phi ^{m} \\\\Vert _{2,\\\\Omega }^2 +\\\\nonumber \\\\\\\\+\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^{m})|\\\\theta ^{m}|^{\\\\ell } \\\\mathrm {ds}\\\\le \\\\mathcal {R}(0, \\\\Vert h(t_{m,M}) \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }} )+\\\\frac{1}{\\\\ell }\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^{m})|\\\\theta ^{m}|^{\\\\ell } \\\\mathrm {ds}, $ with $\\\\mathcal {R}$ being the increasing continuous function defined in (REF ).\",\n \"By (REF ), we have $\\\\sum _{i=1}^m\\\\int _\\\\Omega (B(\\\\theta ^i)-B(\\\\theta ^{i-1}) )\\\\theta ^i\\\\mathrm {dx} \\\\ge \\\\int _\\\\Omega (\\\\Psi (\\\\theta ^m) - \\\\Psi (\\\\theta ^{0}) )\\\\mathrm {dx} .$ Therefore, summing over $i=1,\\\\cdots , m$ into (REF ), multiplying by $\\\\tau $ , and inserting the previous inequality, we obtain $\\\\int _\\\\Omega \\\\Psi ( \\\\theta ^m )\\\\mathrm {dx}& +&\\\\tau \\\\sum _{i=1}^m \\\\left( ( L_1)_\\\\# \\\\Vert \\\\nabla \\\\theta ^{i} \\\\Vert _{2,\\\\Omega }^2+\\\\frac{1}{\\\\ell ^{\\\\prime }}\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^{i})|\\\\theta ^{i}|^{\\\\ell } \\\\mathrm {ds} +\\\\frac{(L_{2})_\\\\#}{2} \\\\Vert \\\\nabla \\\\phi ^{i} \\\\Vert _{2,\\\\Omega }^2 \\\\right) \\\\nonumber \\\\\\\\ &\\\\le &\\\\int _\\\\Omega \\\\Psi ( \\\\theta ^0 )\\\\mathrm {dx} +\\\\tau \\\\sum _{i=1}^m\\\\mathcal {R}(0, \\\\Vert h(t_{m,M}) \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }} ) .\",\n \"$ Therefore, we find the uniform estimate (REF ) by taking the maximum over $m\\\\in \\\\lbrace 1,\\\\cdots ,M\\\\rbrace $ in the previous estimate and applying Lemma REF provided by ().\",\n \"A direct application of Proposition REF ensures the following proposition.\",\n \"Proposition 5.2 There exist $\\\\theta ,\\\\phi : Q_T\\\\rightarrow \\\\mathbb {R}$ and subsequences of $(\\\\theta _M,\\\\phi _M)$ , still labelled by $(\\\\theta _M,\\\\phi _M)$ , such that ${\\\\theta }_M \\\\rightharpoonup \\\\theta &\\\\mbox{ in }& V_\\\\ell (Q_T); \\\\\\\\\\\\phi _M \\\\rightharpoonup \\\\phi & \\\\mbox{ in }& L^2(0,T;V),$ as $M$ tends to infinity.\",\n \"Moreover, there exists $Z: Q_T\\\\rightarrow \\\\mathbb {R}$ such that $\\\\partial _t\\\\widetilde{B}(\\\\theta _M)\\\\rightharpoonup Z \\\\quad \\\\mbox{in } L^{\\\\ell ^{\\\\prime } }(0,T;(V_\\\\ell (\\\\Omega ))^{\\\\prime }),$ as $M$ tends to infinity.\",\n \"Considering the uniform estimates to $ {\\\\theta }_M $ and $ {\\\\phi }_M $ that are established in Proposition REF , we extract subsequences, still denoted by $ {\\\\theta }_M $ and $ {\\\\phi }_M $ , weakly convergent in $ V_\\\\ell (Q_T )$ and $ L^2(0,T;V)$ , respectively, to $\\\\theta $ and $\\\\phi $ .\",\n \"Let $p=\\\\max \\\\lbrace \\\\ell ,2\\\\rbrace =\\\\ell $ .\",\n \"Note that $L^p(0,T;V_\\\\ell (\\\\Omega ))\\\\hookrightarrow V_\\\\ell (Q_T)$ .\",\n \"By definition of norm, we find $\\\\Vert \\\\partial _t\\\\widetilde{B}(\\\\theta _M)\\\\Vert _{L^{p^{\\\\prime }}(0,T;(V_\\\\ell (\\\\Omega ))^{\\\\prime })} = \\\\sum _{m=1}^M\\\\int _{(m-1)\\\\tau }^{m\\\\tau }\\\\sup _{v\\\\in L^p(0,T;V_\\\\ell (\\\\Omega )) \\\\atop \\\\Vert v\\\\Vert \\\\le 1} \\\\langle Z^m,v\\\\rangle \\\\mathrm {dt}\\\\le C,$ with $C>0$ being a constant independent on $M$ , by estimating in (REF ) the term involving $Z^m$ by means of the rest terms using the uniform estimates established in (REF ).\",\n \"Hence, we can extract a subsequence, still denoted by $\\\\partial _t\\\\widetilde{B}(\\\\theta _M)$ , weakly convergent to $Z$ in $L^{p^{\\\\prime } }(0,T;(V_\\\\ell (\\\\Omega ))^{\\\\prime })$ .\",\n \"In the following proposition, we state the strong convergence of $B(\\\\theta _M)$ and of $\\\\theta _M$ .\",\n \"Proposition 5.3 Under ()-(REF ) and (REF )-(), the solution $\\\\theta ^m$ of (REF ) satisfies $b_\\\\#\\\\Vert \\\\theta ^{m }-\\\\theta ^{m-1 }\\\\Vert _{2,\\\\Omega } ^2\\\\le \\\\int _\\\\Omega (B(\\\\theta ^{m })-B(\\\\theta ^{m-1 })) (\\\\theta ^{m }-\\\\theta ^{m-1 }) \\\\mathrm {dx} \\\\le \\\\nonumber \\\\\\\\\\\\le C\\\\left(\\\\Vert \\\\theta ^{m }-\\\\theta ^{m-1 }\\\\Vert _{\\\\ell ,\\\\Gamma } \\\\tau ^{1/\\\\ell } + \\\\Vert \\\\theta ^{m }-\\\\theta ^{m-1 }\\\\Vert _{2,\\\\Omega }\\\\sqrt{\\\\tau } \\\\right),$ with $C$ being a positive constant.\",\n \"Moreover, for a subsequence, there hold $B( {\\\\theta }_M) \\\\rightarrow B(\\\\theta ) &\\\\mbox{ in }& L^1(Q_T); \\\\\\\\\\\\theta _M \\\\rightarrow \\\\theta & \\\\mbox{ a.e.\",\n \"in }& Q_T,$ as $M$ tends to infinity.\",\n \"Let $k\\\\in \\\\mathbb {N}$ .\",\n \"Let us sum up (REF ) for $m=j+1,\\\\cdots ,j+k$ and multiply by $\\\\tau $ , obtaining $\\\\int _\\\\Omega (B(\\\\theta ^{j+k })-B(\\\\theta ^{j })) v \\\\mathrm {dx} \\\\le \\\\mathcal {I }_\\\\Gamma ^j +\\\\mathcal {I}_\\\\Omega ^j ,$ where $\\\\mathcal {I}_\\\\Gamma ^j&:=& \\\\tau \\\\sum _{m=j+1}^{j+k} \\\\int _\\\\Gamma | (\\\\gamma (\\\\theta ^m)|\\\\theta ^{m}|^{\\\\ell -2} \\\\theta ^{m} -h(t_{m,M}) )v| \\\\mathrm {dx} \\\\\\\\&\\\\le & \\\\int _{j\\\\tau }^{(j+k)\\\\tau }\\\\Vert \\\\gamma (\\\\theta _M)|\\\\theta _M|^{\\\\ell -2} \\\\theta _M -h \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma } \\\\Vert v \\\\Vert _{\\\\ell ,\\\\Gamma }\\\\mathrm {dt} ; \\\\\\\\\\\\mathcal {I}_\\\\Omega ^j &:=& \\\\tau \\\\sum _{m=j+1}^{j+k}\\\\int _\\\\Omega |( a ( \\\\theta ^m,\\\\phi ^m )\\\\nabla \\\\theta ^{m} +F ( \\\\theta ^m,\\\\phi ^m) \\\\nabla \\\\phi ^m )v| \\\\mathrm {dx} \\\\\\\\&\\\\le & \\\\int _{j\\\\tau }^{(j+k)\\\\tau }\\\\Vert a ( \\\\theta _M,\\\\phi _M )\\\\nabla \\\\theta _M +F ( \\\\theta _M,\\\\phi _M) \\\\nabla \\\\phi _M \\\\Vert _{2,\\\\Omega }\\\\Vert v\\\\Vert _{2,\\\\Omega } \\\\mathrm {dt} .$ Here, we used the Hölder inequality and the definition of $ \\\\theta _M$ and of $\\\\phi _M$ .\",\n \"Let us compute $\\\\mathcal {I }_\\\\Gamma ^j $ and $\\\\mathcal {I}_\\\\Omega ^j$ by applying the estimate (REF ).\",\n \"Using the assumption (REF ) and after the Hölder inequality, we deduce $\\\\mathcal {I}_\\\\Gamma ^j\\\\le \\\\Vert v \\\\Vert _{\\\\ell ,\\\\Gamma } \\\\int _{j\\\\tau }^{(j+k)\\\\tau }\\\\left(\\\\gamma ^\\\\#\\\\Vert \\\\theta _M\\\\Vert _{\\\\ell ,\\\\Gamma }^{\\\\ell -1}+\\\\Vert h \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma } \\\\right)\\\\mathrm {dt}\\\\le \\\\Vert v \\\\Vert _{\\\\ell ,\\\\Gamma } C (k\\\\tau )^{1/\\\\ell }.$ Using the assumptions (), (REF ) and (), and after the Hölder inequality, we deduce $\\\\mathcal {I}_\\\\Omega ^j \\\\le \\\\Vert v\\\\Vert _{2,\\\\Omega } \\\\int _{j\\\\tau }^{(j+k)\\\\tau }\\\\left(a^\\\\#\\\\Vert \\\\nabla \\\\theta _M\\\\Vert _{2,\\\\Omega } + \\\\sigma ^\\\\#F ^\\\\#\\\\Vert \\\\nabla \\\\phi _M \\\\Vert _{2,\\\\Omega } \\\\right) \\\\mathrm {dt} \\\\le \\\\Vert v\\\\Vert _{2,\\\\Omega }C\\\\sqrt{k\\\\tau }.$ Hence, we find $\\\\int _\\\\Omega (B(\\\\theta ^{j+k })-B(\\\\theta ^{j })) v \\\\mathrm {dx} \\\\le \\\\Vert v \\\\Vert _{\\\\ell ,\\\\Gamma } C (k\\\\tau )^{1/\\\\ell } + \\\\Vert v\\\\Vert _{2,\\\\Omega }C\\\\sqrt{k\\\\tau } .$ In particular, the estimate (REF ) follows by taking $j=m-1$ , $k=1$ and $v=\\\\theta ^{m }-\\\\theta ^{m-1 }$ , and applying Lemma REF .\",\n \"To prove the convergences, we will apply Lemma REF .\",\n \"Considering the weak convergence of $\\\\theta _M$ established in Proposition REF and the estimate (REF ), in order to apply Lemma REF it remains to prove that the condition (REF ) is fulfilled.\",\n \"Let $0T/z$ , which means $\\\\tau k+1$ deducing $\\\\int _0^{T-z}\\\\int _\\\\Omega (B(\\\\theta _M(t+z))-B(\\\\theta _M(t))) (\\\\theta _M(t+z)-\\\\theta _M(t))\\\\mathrm {dx} \\\\mathrm {dt}\\\\le \\\\\\\\\\\\le \\\\sum _{j=1 }^{M-k} \\\\int _{(j-1)\\\\tau }^{(j+k)\\\\tau }\\\\int _\\\\Omega (B(\\\\theta ^{j+k })-B(\\\\theta ^{j })) (\\\\theta ^{j+k }-\\\\theta ^{j })\\\\mathrm {dx} .$ Taking $v=\\\\theta ^{j+k }-\\\\theta ^j$ in (REF ) and then summing up for $j=1,\\\\cdots , M-k$ , we find $\\\\int _0^{T-z}\\\\int _\\\\Omega (B(\\\\theta _M(t+z))-B(\\\\theta _M(t))) (\\\\theta _M(t+z)-\\\\theta _M(t))\\\\mathrm {dx} \\\\mathrm {dt}\\\\le \\\\\\\\\\\\le (k+\\\\tau )\\\\sum _{j=1 }^{M-k}\\\\int _\\\\Omega (B(\\\\theta ^{j+k })-B(\\\\theta ^{j })) (\\\\theta ^{j+k }-\\\\theta ^{j })\\\\mathrm {dx} \\\\le \\\\\\\\ \\\\le \\\\sum _{j=1 }^{M-k} \\\\int _{(j-1)\\\\tau }^{j\\\\tau +k }\\\\left( \\\\Vert \\\\theta ^{j+k }-\\\\theta ^{j }\\\\Vert _{\\\\ell ,\\\\Gamma } C (k\\\\tau )^{1/\\\\ell } +\\\\Vert \\\\theta ^{j+k }-\\\\theta ^{j }\\\\Vert _{2,\\\\Omega }C\\\\sqrt{k\\\\tau } \\\\right) \\\\mathrm {dt} .$ Arguing as in (REF ) and (REF ), we conclude $\\\\int _0^{T-z}\\\\int _\\\\Omega (B(\\\\theta _M(t+z))-B(\\\\theta _M(t))) (\\\\theta _M(t+z)-\\\\theta _M(t))\\\\mathrm {dx} \\\\mathrm {dt}\\\\le \\\\\\\\ \\\\le C\\\\left( (k\\\\tau )^{1/\\\\ell }(k\\\\tau +\\\\tau )^{ 1/\\\\ell ^{\\\\prime }} + (k\\\\tau )^{1/2} (k\\\\tau +\\\\tau )^{1/2}\\\\right)=C\\\\left( 2^{ 1/\\\\ell ^{\\\\prime }}+ 2^{ 1/2}\\\\right) z.$ Thus, all hypothesis of Lemma REF are fulfilled.\",\n \"Therefore, Lemma REF assures that $B(\\\\theta _M)$ strongly converges to $B(\\\\theta )$ in $L^1(Q_T)$ .\",\n \"Consequently, up to a subsequence, $B(\\\\theta _M)$ converges to $B(\\\\theta )$ a.e.\",\n \"in $Q_T$ .\",\n \"Since $B$ is strictly monotone, $\\\\theta _M$ converges to $\\\\theta $ a.e.\",\n \"in $Q_T$ (see, for instance, [19]).\",\n \"Now we are able to identify the limit $Z$ .\",\n \"Proposition 5.4 The limit $Z$ satisfies $Z=\\\\partial _t (B(\\\\theta )) \\\\quad \\\\mbox{ in } L^{\\\\ell ^{\\\\prime } }(0,T;(V_\\\\ell (\\\\Omega ))^{\\\\prime }).$ For a fixed $t$ , there exists $m\\\\in \\\\lbrace 1,\\\\cdots ,M\\\\rbrace $ such that $t\\\\in ]t_{m-1,M},t_{m,M}]$ .\",\n \"From definition REF we have $\\\\int ^t_0 {Z}_M(\\\\varsigma )\\\\mathrm {d\\\\varsigma }=\\\\sum _{j=1}^{m-1}\\\\int _{(j-1)\\\\tau }^{j\\\\tau }\\\\frac{B(\\\\theta ^{j})(\\\\varsigma )-B(\\\\theta ^{j-1})(\\\\varsigma )}{ \\\\tau }\\\\mathrm {d\\\\varsigma } + \\\\\\\\+\\\\int _{(m-1)\\\\tau }^{t}\\\\frac{ B(\\\\theta ^m) (\\\\varsigma )- B(\\\\theta ^{m-1})(\\\\varsigma ) }{\\\\tau }\\\\mathrm {d\\\\varsigma } =\\\\\\\\=B(\\\\theta ^{m-1})- B(\\\\theta ^{0}) + \\\\frac{ t-(m-1)\\\\tau }{\\\\tau }\\\\left( B(\\\\theta ^m) - B(\\\\theta ^{m-1}) \\\\right) = \\\\widetilde{B}(\\\\theta _M(t)) - B(\\\\theta ^{0})$ in $\\\\Omega $ .\",\n \"By the Riesz theorem, the bounded linear functional $v\\\\in L^2(\\\\Omega )\\\\mapsto \\\\int ^t_0 ({Z}_M(\\\\varsigma ),v)\\\\mathrm {d\\\\varsigma }$ is (uniquely) representable by the element $\\\\widetilde{B}(\\\\theta _M(t)) - B(\\\\theta ^{0}) $ from $L^2(\\\\Omega ).$ Using the corresponding definitions we compute $\\\\int _0^T\\\\Vert \\\\widetilde{B}(\\\\theta _M(t)) - B(\\\\theta _M) \\\\Vert _{2,\\\\Omega }^2\\\\mathrm {dt} =\\\\sum _{m=1}^M \\\\Vert Z^m \\\\Vert _{2,\\\\Omega }^2\\\\int _{(m-1)\\\\tau }^{m\\\\tau } (t-m\\\\tau )^2\\\\mathrm {dt}= \\\\\\\\=\\\\frac{\\\\tau ^3}{3}\\\\sum _{m=1}^M \\\\Vert Z^m \\\\Vert _{2,\\\\Omega }^2 =\\\\frac{\\\\tau }{3}\\\\sum _{m=1}^M\\\\Vert B(\\\\theta ^m) - B(\\\\theta ^{m-1}) \\\\Vert _{2,\\\\Omega }^2\\\\le \\\\\\\\\\\\le (b^\\\\#)^2 \\\\frac{\\\\tau }{3}\\\\sum _{m=1}^M\\\\Vert \\\\theta ^m - \\\\theta ^{m-1} \\\\Vert _{2,\\\\Omega }^2.$ Applying (REF ) and Proposition REF we have $\\\\int _0^T\\\\Vert \\\\widetilde{B}(\\\\theta _M(t)) - B(\\\\theta _M) \\\\Vert _{2,\\\\Omega }^2\\\\mathrm {dt}\\\\le C\\\\left( \\\\tau ^{1/\\\\ell +1/\\\\ell ^{\\\\prime }} +\\\\tau ^{1/2+1/2} \\\\right) = C\\\\tau ,$ and consequently $\\\\widetilde{B}(\\\\theta _M)$ converges to $ B(\\\\theta ) $ .\",\n \"By the uniqueness of limit, we deduce $\\\\int ^t_0 Z(\\\\varsigma )\\\\mathrm {d\\\\varsigma }=B(\\\\theta ) - B(\\\\theta ^{0}),$ which concludes the proof.\",\n \"We emphasize that the above convergences are sufficient to identify the limit $\\\\phi $ as stated in the following proposition, but they are not sufficient to identify the temperature $\\\\theta $ as a solution, because on the one hand the apparent nonlinearity of the coefficients destroy the weak convergence, on the other hand, the weak-weak convergence does not imply weak convergence.\",\n \"Corollary 5.1 Let $(\\\\theta ,\\\\phi )$ be in accordance with Propositions REF and REF , then they verify ().\",\n \"Let $(\\\\theta _M,\\\\phi _M)$ solve (REF )-(REF ).\",\n \"Applying Propositions REF and REF , and the Krasnoselski theorem to the Nemytskii operators $\\\\sigma $ and $\\\\alpha _\\\\mathrm {S}$ , we have $\\\\sigma ( \\\\theta _M )\\\\nabla \\\\phi _M\\\\rightharpoonup \\\\sigma ( \\\\theta )\\\\nabla \\\\phi &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T); \\\\\\\\\\\\sigma ( \\\\theta _M ) \\\\alpha _\\\\mathrm {S}( \\\\theta _M ) \\\\nabla \\\\theta _M\\\\rightharpoonup \\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S}( \\\\theta ) \\\\nabla \\\\theta &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T) \\\\quad \\\\mbox{ as } M\\\\rightarrow +\\\\infty .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\\\theta ,\\\\phi )$ verifies ().\"\n ],\n [\n \"Proof of Theorem \",\n \"This proof follows mutatis mutandis the structure of the proof of Theorem REF (cf.\",\n \"Subsection REF ).\",\n \"We only sketch its main steps.\",\n \"The uniform estimates to $ {\\\\theta }_M $ and $ {\\\\phi }_M $ are as follows.\",\n \"The quantitative estimate (REF ) reads $\\\\int _\\\\Omega \\\\Psi ( \\\\theta ^m )\\\\mathrm {dx}+ a_\\\\# \\\\Vert \\\\nabla \\\\theta _M\\\\Vert _{2,Q_T} ^2+\\\\frac{\\\\gamma _\\\\#}{\\\\ell ^{\\\\prime }} \\\\Vert \\\\theta _M\\\\Vert _{\\\\ell ,\\\\Sigma _T } ^\\\\ell \\\\le \\\\nonumber \\\\\\\\\\\\le b^\\\\# \\\\Vert \\\\theta ^0\\\\Vert _{2,\\\\Omega } ^2 +\\\\frac{1}{\\\\ell ^{\\\\prime }\\\\gamma _{\\\\#}^{1/(\\\\ell -1)} } \\\\Vert h \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Sigma _T }^{\\\\ell ^{\\\\prime }} +T \\\\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\\\#} \\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}}^2 ,$ by using the argument to estimate (REF ).\",\n \"Namely, (REF ) reads $\\\\frac{1}{\\\\tau }\\\\int _\\\\Omega (B(\\\\theta ^m)-B(\\\\theta ^{m-1}) )\\\\theta ^m\\\\mathrm {dx} +a_\\\\# \\\\Vert \\\\nabla \\\\theta ^{m} \\\\Vert _{2,\\\\Omega }^2+\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^{m})|\\\\theta ^{m}|^{\\\\ell } \\\\mathrm {ds} \\\\le \\\\\\\\\\\\le \\\\mathcal {R}(0, \\\\Vert h(t_{m,M}) \\\\Vert _{\\\\ell ^{\\\\prime },\\\\Gamma }^{\\\\ell ^{\\\\prime }} )+\\\\frac{1}{\\\\ell }\\\\int _\\\\Gamma \\\\gamma (\\\\theta ^{m})|\\\\theta ^{m}|^{\\\\ell } \\\\mathrm {ds},$ where $\\\\mathcal {R}$ is the increasing continuous function defined in (REF ), with $(L_2)_\\\\#=a_\\\\#\\\\sigma _\\\\#/ (2 F^\\\\#\\\\sigma ^\\\\#)$ .\",\n \"In addition, summing the quantitative estimate $\\\\sqrt{\\\\sigma _\\\\#} \\\\Vert \\\\nabla \\\\phi ^{m} \\\\Vert _{2,\\\\Omega } \\\\le \\\\sqrt{\\\\sigma _\\\\#}a^\\\\# \\\\Vert \\\\nabla \\\\theta ^{m-1} \\\\Vert _{2,\\\\Omega }+\\\\frac{K_2(P_2+1)}{ \\\\sqrt{\\\\sigma _\\\\#} } \\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}},$ it results in $\\\\Vert \\\\nabla \\\\phi _M\\\\Vert _{2,Q_T} \\\\le a^\\\\# \\\\Vert \\\\nabla \\\\theta _M \\\\Vert _{2,Q_T}+ T\\\\frac{K_2(P_2+1)}{ \\\\sigma _\\\\#} \\\\Vert g\\\\Vert _{2,\\\\Gamma _\\\\mathrm {N}} .$ For subsequences of $ {\\\\theta }_M $ and $ {\\\\phi }_M $ , the weak convergences hold according to Propositions REF and REF , which guarantee the required result.\"\n ],\n [\n \"Existence of solutions to the TE problem\",\n \"The objective is the passage to the limit in the abstract boundary value problems introduced in Section as the time step goes to zero ($M\\\\rightarrow +\\\\infty $ ), with the coefficients being defined by $b(\\\\cdot , v) &=& \\\\rho (\\\\cdot ,v)c_\\\\mathrm {v}(\\\\cdot , v);\\\\\\\\a(\\\\cdot , v,w)&=&k(\\\\cdot , v) + T_\\\\mathcal {M}(w)\\\\alpha _\\\\mathrm {S}(\\\\cdot , v)\\\\sigma (\\\\cdot , v);\\\\\\\\F(\\\\cdot , v,w)&=&\\\\Pi (\\\\cdot , v)+T_\\\\mathcal {M}(w),$ where $T_\\\\mathcal {M}$ is the $\\\\mathcal {M}$ -truncation function defined by $T_\\\\mathcal {M}(z)=\\\\max (-\\\\mathcal {M},\\\\min (\\\\mathcal {M},z))$ .\",\n \"By the definition of truncated functions, we choose $a_\\\\# &=& k_\\\\#-\\\\mathcal {M}\\\\alpha ^\\\\#\\\\sigma ^\\\\#;\\\\\\\\F^\\\\# &=& \\\\Pi ^\\\\#+\\\\mathcal {M}.$ taking (REF ) into account.\",\n \"Under these choices, the assumptions (REF ), (REF ) and (REF ) imply (REF ), (REF ) and (REF ), respectively.\",\n \"Let us foccus the present proof in accordance with the approximated solutions that are established in Theorem REF .\",\n \"Analogous argument is valid for the approximated solutions that are established in Theorem REF .\",\n \"Let us redefine the electrical current density as $\\\\mathbf {j}( \\\\theta , \\\\phi )= \\\\sigma ( \\\\theta )\\\\nabla \\\\phi +\\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S}( \\\\theta ) \\\\nabla \\\\theta .$ Analogously for $\\\\mathbf {j}_M= \\\\mathbf {j}(\\\\theta _M ,\\\\phi _M)$ or simply $\\\\mathbf {j}_M$ and $\\\\mathbf {j}$ whenever the meaning is not ambiguous.\",\n \"Let $ (\\\\theta _M, {\\\\phi }_M )$ solve (REF )-(REF ), which may be rewritten as $\\\\int _0^T\\\\int _\\\\Omega {Z}_M v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\int _{Q_T} k ( {\\\\theta }_M )\\\\nabla {\\\\theta }_M \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\gamma ( {\\\\theta }_M )| {\\\\theta }_M |^{\\\\ell -2} {\\\\theta }_M v \\\\mathrm {ds}\\\\mathrm {dt} +\\\\nonumber \\\\\\\\+\\\\int _{Q_T} \\\\sigma ( \\\\theta _M ) \\\\Pi ( \\\\theta _M ) \\\\nabla {\\\\phi }_M \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{Q_T} \\\\phi _M \\\\mathbf {j}( \\\\theta _M , \\\\phi _M ) \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt} =\\\\nonumber \\\\\\\\=\\\\int _{\\\\Sigma _T} h_M v \\\\mathrm {ds}\\\\mathrm {dt} ;\\\\qquad \\\\\\\\\\\\int _\\\\Omega \\\\mathbf {j}( \\\\theta _M , \\\\phi _M)\\\\cdot \\\\nabla w\\\\mathrm {dx}=\\\\int _{\\\\Gamma _{\\\\rm N}}g w\\\\mathrm {ds}, \\\\qquad $ for every $v\\\\in V_{\\\\ell }(Q_T)$ and $w\\\\in V$ .\",\n \"There exist $\\\\theta ,\\\\phi : Q_T\\\\rightarrow \\\\mathbb {R}$ and subsequences of $(\\\\theta _M,\\\\phi _M)$ , still labelled by $(\\\\theta _M,\\\\phi _M)$ , weakly convergent in accordance with Proposition REF .\",\n \"By Corollary REF , $(\\\\theta ,\\\\phi )$ verify the electric equality $\\\\\\\\\\\\int _\\\\Omega \\\\mathbf {j}( \\\\theta , \\\\phi )\\\\cdot \\\\nabla w\\\\mathrm {dx}=\\\\int _{\\\\Gamma _{\\\\rm N}}g w\\\\mathrm {ds},$ for every $w\\\\in V$ .\",\n \"We emphasize that the weak convergence of $\\\\mathbf {j}_M$ to $ \\\\mathbf {j}$ in $ \\\\mathbf {L}^2(Q_T)$ , taking (REF )-() into account, is not sufficient to pass to the limit the term $\\\\phi _M \\\\mathbf {j}( \\\\theta _M , \\\\phi _M ) $ .\",\n \"Moreover, the non smoothness of the coefficients destroy the possibility of obtaining strong convergences of $\\\\nabla \\\\theta _M$ and of $\\\\phi _M$ .\",\n \"Thanks to Proposition REF , we have a.e.\",\n \"pointwise convergence for a subsequence of $\\\\theta _M$ , which we still denote by $\\\\theta _M$ .\",\n \"Considering the assumptions (REF )-(), the Nemytskii operators are continuous due to the Krasnoselski theorem, and applying the Lebesgue dominated convergence theorem, we obtain $k( \\\\theta _M )\\\\nabla v\\\\rightarrow k(\\\\theta ) \\\\nabla v &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T); \\\\\\\\\\\\sigma ( \\\\theta _M ) \\\\alpha _\\\\mathrm {S}( \\\\theta _M ) \\\\nabla v\\\\rightarrow \\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S}( \\\\theta ) \\\\nabla v &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T); \\\\\\\\\\\\sigma ( \\\\theta _M ) \\\\Pi ( \\\\theta _M ) \\\\nabla v\\\\rightarrow \\\\sigma ( \\\\theta ) \\\\Pi ( \\\\theta ) \\\\nabla v &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T).$ Applying (REF ) to the following estimates $\\\\Vert T_\\\\mathcal {M}(\\\\phi _M) \\\\nabla \\\\theta _M\\\\Vert _{2,Q_T}\\\\le \\\\mathcal {M} \\\\Vert \\\\nabla \\\\theta _M\\\\Vert _{2,Q_T} ; \\\\\\\\\\\\Vert \\\\sigma ( \\\\theta _M ) T_\\\\mathcal {M}( \\\\phi _M ) \\\\nabla \\\\phi _M\\\\Vert _{2,Q_T}\\\\le \\\\sigma ^\\\\# \\\\mathcal {M} \\\\Vert \\\\nabla \\\\phi _M\\\\Vert _{2,Q_T} ;\\\\\\\\\\\\Vert \\\\gamma ( \\\\theta _M ) | \\\\theta _M|^{\\\\ell -2} \\\\theta _M\\\\Vert _{\\\\ell ^{\\\\prime },\\\\Sigma _T } \\\\le \\\\gamma ^\\\\# \\\\Vert \\\\theta _M\\\\Vert _{\\\\ell ,\\\\Sigma _T } .$ there exist $\\\\Lambda _1 , \\\\Lambda _2 \\\\in \\\\mathbf {L}^2(Q_T)$ and $\\\\Lambda _3\\\\in L^{\\\\ell ^{\\\\prime }}(\\\\Sigma _T ) $ such that $T_\\\\mathcal {M}( \\\\phi _M )\\\\nabla \\\\theta _M\\\\rightharpoonup \\\\Lambda _1 &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T); \\\\\\\\\\\\sigma ( \\\\theta _M ) T_\\\\mathcal {M}( \\\\phi _M ) \\\\nabla \\\\phi _M\\\\rightharpoonup \\\\Lambda _2 &\\\\mbox{ in }& \\\\mathbf {L}^2(Q_T); \\\\\\\\\\\\gamma ( \\\\theta _M ) | \\\\theta _M|^{\\\\ell -2} \\\\theta _M\\\\rightharpoonup \\\\Lambda _3 &\\\\mbox{ in }& L^{\\\\ell ^{\\\\prime }}(\\\\Sigma _T ) .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\\\theta ,\\\\phi )$ verifies $\\\\int _0^T\\\\int _\\\\Omega Z v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\int _{Q_T} k (\\\\theta )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+ \\\\nonumber \\\\\\\\ +\\\\int _{Q_T} \\\\sigma (\\\\theta ) \\\\Pi (\\\\theta )\\\\nabla \\\\phi \\\\cdot \\\\nabla v\\\\mathrm {dx} \\\\mathrm {dt}+\\\\int _{Q_T} \\\\sigma ( \\\\theta ) \\\\alpha _\\\\mathrm {S}(\\\\theta ) \\\\Lambda _1 \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\nonumber \\\\\\\\+\\\\int _{Q_T} \\\\Lambda _2 \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\Lambda _3 v \\\\mathrm {ds}\\\\mathrm {dt}=\\\\int _{\\\\Sigma _T} h v \\\\mathrm {ds}\\\\mathrm {dt} ,\\\\quad \\\\forall v \\\\in V_\\\\ell (Q_T) .\",\n \"$ To identify the temperature $\\\\theta $ as a solution, we need to identify $\\\\Lambda _1 , \\\\Lambda _2 \\\\in \\\\mathbf {L}^2(Q_T)$ and $\\\\Lambda _3\\\\in L^{\\\\ell ^{\\\\prime }}(\\\\Sigma _T ) $ .\",\n \"To prove that $\\\\Lambda _1= T_\\\\mathcal {M}( \\\\phi )\\\\nabla \\\\theta $ , let us consider the Green formula $\\\\int _{Q_T} T_\\\\mathcal {M}( \\\\phi _M )\\\\nabla \\\\theta _M \\\\cdot \\\\mathbf {v}\\\\mathrm {dx}\\\\mathrm {dt}=-\\\\int _{Q_T[|\\\\phi _M|<\\\\mathcal {M}]} \\\\theta _M \\\\nabla \\\\phi _M \\\\cdot \\\\mathbf {v}\\\\mathrm {dx}\\\\mathrm {dt},$ for every $\\\\mathbf {v}\\\\in \\\\mathbf {L}^p(0,T; \\\\mathbf {W}^{1,p}(\\\\Omega ))$ such that $\\\\nabla \\\\cdot \\\\mathbf {v}=0$ in $Q_T$ and $\\\\mathbf {v}\\\\cdot \\\\mathbf {n}=0$ in $\\\\partial \\\\Omega \\\\times ]0,T [$ .\",\n \"Next we choose the exponent $p>1$ to ensure the meaning of the involved terms.\",\n \"By $\\\\theta _M\\\\in L^\\\\infty (0,T; L^2(\\\\Omega ))\\\\cap L^2(0,T; H^1(\\\\Omega ))$ and $ H^1(\\\\Omega )\\\\hookrightarrow L^{2^*}(\\\\Omega )$ with $2^*$ being the critical Sobolev exponent, i.e.\",\n \"$2^*=2n/(n-2)$ if $n>2$ and any $2^*>1$ if $n=2$ , making recourse to the interpolation with exponents being $\\\\frac{\\\\beta }{2}= \\\\frac{1-\\\\beta }{2}+\\\\frac{\\\\beta }{2^*}=\\\\frac{1}{q}, \\\\qquad (0<\\\\beta <1),$ then $\\\\theta _M$ converges to $\\\\theta $ in $L^q(Q_T)$ for every $q<2(n+2)/n$ .\",\n \"In particular, we take $p>n+2$ such that $\\\\frac{1}{2}=\\\\frac{1}{p}+\\\\frac{1}{q} >\\\\frac{1}{p}+\\\\frac{n}{2(n+2)}.$ Consequently, we have that $\\\\theta _M\\\\mathbf {v}$ converges to $\\\\theta \\\\mathbf {v}$ in $\\\\mathbf {L}^2 (Q_T)$ .\",\n \"Therefore, the uniqueness of the weak limit implies that $\\\\Lambda _1= T_\\\\mathcal {M}( \\\\phi )\\\\nabla \\\\theta $ .\",\n \"In particular, we find $\\\\int _{Q_T} a(\\\\theta _M,\\\\phi _M)\\\\nabla \\\\theta _M\\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}\\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} \\\\int _{Q_T} a(\\\\theta ,\\\\phi ) \\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt},$ and consequently (REF ) reads $\\\\int _0^T\\\\int _\\\\Omega Z v\\\\mathrm {dx}\\\\mathrm {dt} +\\\\int _{Q_T} a(\\\\theta ,\\\\phi )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{Q_T} \\\\sigma (\\\\theta ) \\\\Pi (\\\\theta )\\\\nabla \\\\phi \\\\cdot \\\\nabla v\\\\mathrm {dx} \\\\mathrm {dt}+ \\\\nonumber \\\\\\\\+\\\\int _{Q_T} \\\\Lambda _2 \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\Lambda _3 v \\\\mathrm {ds}\\\\mathrm {dt}=\\\\int _{\\\\Sigma _T} h v \\\\mathrm {ds}\\\\mathrm {dt} ,\\\\quad \\\\forall v \\\\in V_\\\\ell (Q_T) .\",\n \"\\\\qquad $ Now, we are in the conditions to identify the limits $\\\\Lambda _2$ and $\\\\Lambda _3$ by making recourse to the Minty argument as follows.\",\n \"We rephrase (REF ) as $\\\\mathcal {J}_M -\\\\mathcal {J}_1^M-\\\\mathcal {J}_2^M -\\\\mathcal {J}_3^M -\\\\mathcal {J}_4^M -\\\\mathcal {J}_5^M + \\\\\\\\+\\\\int _{\\\\Sigma _T} \\\\left(\\\\gamma (\\\\theta _M) |\\\\theta _{M}|^{\\\\ell -2} \\\\theta _{M} -\\\\gamma (v) |v|^{\\\\ell -2}v\\\\right)(\\\\theta _{M} -v)\\\\mathrm {ds}\\\\mathrm {dt} \\\\ge \\\\\\\\ \\\\ge (L_1)_\\\\# \\\\int _{Q_T}|\\\\nabla (\\\\theta _M-v) |^2\\\\mathrm {dx} \\\\mathrm {dt}+(L_2)_\\\\#\\\\int _{Q_T}|\\\\nabla (\\\\phi _M - \\\\phi ) |^2\\\\mathrm {dx}\\\\mathrm {dt}\\\\ge 0,$ where $\\\\mathcal {J}_M &:=& \\\\int _{Q_T}\\\\left( a(\\\\theta _M,\\\\phi _M) |\\\\nabla \\\\theta _M|^2 +\\\\sigma (\\\\theta _M) F (\\\\theta _M, \\\\phi _M) \\\\nabla \\\\phi _M\\\\cdot \\\\nabla \\\\theta _M\\\\right)\\\\mathrm {dx}\\\\mathrm {dt} + \\\\\\\\ &&+\\\\int _{Q_T}\\\\left(\\\\sigma (\\\\theta _M) |\\\\nabla \\\\phi _M|^2 + \\\\sigma (\\\\theta _M) \\\\alpha _\\\\mathrm {S} (\\\\theta _M) \\\\nabla \\\\theta _M\\\\cdot \\\\nabla \\\\phi _M\\\\right)\\\\mathrm {dx} \\\\mathrm {dt}+ \\\\\\\\ &&+\\\\int _{\\\\Sigma _T}\\\\gamma (\\\\theta _M) |\\\\theta _M|^\\\\ell \\\\mathrm {ds}\\\\mathrm {dt}; \\\\\\\\\\\\mathcal {J}_1^M &:=& 2\\\\int _{Q_T} a(\\\\theta _M,\\\\phi _M)\\\\nabla \\\\theta _M\\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt}-\\\\int _{Q_T} a(\\\\theta _M,\\\\phi _M)|\\\\nabla v|^2\\\\mathrm {dx}\\\\mathrm {dt}; \\\\\\\\\\\\mathcal {J}_2^M &:=& 2\\\\int _{Q_T}\\\\sigma (\\\\theta _M)\\\\nabla \\\\phi _M\\\\cdot \\\\nabla \\\\phi \\\\mathrm {dx}\\\\mathrm {dt} -\\\\int _{Q_T}\\\\sigma (\\\\theta _M) |\\\\nabla \\\\phi |^2\\\\mathrm {dx}\\\\mathrm {dt} ;\\\\\\\\\\\\mathcal {J}_3^M &:=&\\\\int _{Q_T} \\\\sigma (\\\\theta _M) \\\\Pi (\\\\theta _M) \\\\Big (\\\\nabla \\\\phi _M \\\\cdot \\\\nabla v +\\\\nabla \\\\phi \\\\cdot \\\\nabla ( \\\\theta _M-v) \\\\Big ) \\\\mathrm {dx} \\\\mathrm {dt} ; \\\\\\\\\\\\mathcal {J}_4^M &:=&\\\\int _{Q_T} \\\\sigma (\\\\theta _M) T_\\\\mathcal {M} (\\\\phi _M) \\\\Big (\\\\nabla \\\\phi _M \\\\cdot \\\\nabla v +\\\\nabla \\\\phi \\\\cdot \\\\nabla ( \\\\theta _M-v) \\\\Big ) \\\\mathrm {dx} \\\\mathrm {dt} ; \\\\\\\\\\\\mathcal {J}_5^M &:=&\\\\int _{Q_T} \\\\sigma (\\\\theta _M) \\\\alpha _\\\\mathrm {S} (\\\\theta _M)\\\\Big ( \\\\nabla \\\\theta _M \\\\cdot \\\\nabla \\\\phi +\\\\nabla v\\\\cdot \\\\nabla (\\\\phi _M - \\\\phi ) \\\\Big ) \\\\mathrm {dx} \\\\mathrm {dt}.$ Considering the convergences $\\\\mathcal {J}_1^M & \\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} & 2\\\\int _{Q_T} a(\\\\theta ,\\\\phi )\\\\nabla \\\\theta \\\\cdot \\\\nabla v\\\\mathrm {dx}\\\\mathrm {dt} -\\\\int _{Q_T} a(\\\\theta ,\\\\phi ) |\\\\nabla v|^2\\\\mathrm {dx} \\\\mathrm {dt}:= \\\\mathcal {J}_1;\\\\\\\\\\\\mathcal {J}_2^M & \\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} &\\\\int _{Q_T}\\\\sigma (\\\\theta ) |\\\\nabla \\\\phi |^2\\\\mathrm {dx} \\\\mathrm {dt} := \\\\mathcal {J}_2; \\\\\\\\\\\\mathcal {J}_3^M & \\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} &\\\\int _{Q_T}\\\\sigma (\\\\theta ) \\\\Pi (\\\\theta ) \\\\nabla \\\\phi \\\\cdot \\\\nabla \\\\theta \\\\mathrm {dx} \\\\mathrm {dt} := \\\\mathcal {J}_3;\\\\\\\\\\\\mathcal {J}_4^M & \\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} &\\\\int _{Q_T} \\\\Lambda _2 \\\\cdot \\\\nabla v\\\\mathrm {dx} \\\\mathrm {dt} +\\\\int _{Q_T}\\\\sigma (\\\\theta )T_\\\\mathcal {M}(\\\\phi )\\\\nabla \\\\phi \\\\cdot \\\\nabla ( \\\\theta -v)\\\\mathrm {dx} \\\\mathrm {dt} := \\\\mathcal {J}_4;\\\\\\\\\\\\mathcal {J}_5^M & \\\\mathrel {\\\\mathop {\\\\hspace{0.0pt}\\\\longrightarrow }\\\\limits _{M\\\\rightarrow \\\\infty }} &\\\\int _{Q_T}\\\\sigma (\\\\theta ) \\\\alpha _\\\\mathrm {S} (\\\\theta )\\\\nabla \\\\theta \\\\cdot \\\\nabla \\\\phi \\\\mathrm {dx}\\\\mathrm {dt}:= \\\\mathcal {J}_5,$ we deduce $\\\\lim _{ M\\\\rightarrow \\\\infty }\\\\mathcal {J}_M\\\\ge \\\\int _{\\\\Sigma _T}\\\\Lambda _3 v\\\\mathrm {ds} \\\\mathrm {dt}+\\\\int _{\\\\Sigma _T}\\\\gamma (v)|v|^{\\\\ell -2} v(\\\\theta -v)\\\\mathrm {ds}\\\\mathrm {dt}+ \\\\\\\\ + \\\\mathcal {J}_1+\\\\mathcal {J}_2+ \\\\mathcal {J}_3 +\\\\mathcal {J}_4 +\\\\mathcal {J}_5.$ We continue the Minty argument by taking in (REF ) and (), respectively, the test function $v=\\\\theta _M$ and the test function $w=\\\\phi _M$ , in (REF ) the test function $v=\\\\theta $ , and in (REF ) the test function $w=\\\\phi $ , we deduce $\\\\lim _{M\\\\rightarrow \\\\infty }\\\\mathcal {J}_M = -\\\\int _{Q_T} Z\\\\theta \\\\mathrm {dx} \\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} h\\\\theta \\\\mathrm {ds} \\\\mathrm {dt}+\\\\int _0^T \\\\int _{\\\\Gamma _\\\\mathrm {N}} g \\\\theta \\\\mathrm {ds}\\\\mathrm {dt}=\\\\\\\\=\\\\int _{Q_T} a( \\\\theta ,\\\\phi ) |\\\\nabla \\\\theta |^2\\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{Q_T} \\\\sigma (\\\\theta ) \\\\Pi (\\\\theta )\\\\nabla \\\\phi \\\\cdot \\\\nabla \\\\theta \\\\mathrm {dx} \\\\mathrm {dt} + \\\\\\\\ +\\\\int _{Q_T} \\\\Lambda _2\\\\cdot \\\\nabla \\\\theta \\\\mathrm {dx}\\\\mathrm {dt}+\\\\int _{\\\\Sigma _T} \\\\Lambda _3 \\\\theta \\\\mathrm {ds}\\\\mathrm {dt}+\\\\int _{Q_T} \\\\mathbf {j}( \\\\theta , \\\\phi ) \\\\cdot \\\\nabla \\\\phi \\\\mathrm {dx} \\\\mathrm {dt}.$ Gathering the above two relations, we find $\\\\int _{Q_T} a( \\\\theta ,\\\\phi )|\\\\nabla (\\\\theta - v)|^2\\\\mathrm {dx} \\\\mathrm {dt}+\\\\int _{Q_T} \\\\Big (\\\\Lambda _2 -\\\\sigma (\\\\theta )T_\\\\mathcal {M}(\\\\phi )\\\\nabla \\\\phi \\\\Big )\\\\cdot \\\\nabla ( \\\\theta -v)\\\\mathrm {dx} \\\\mathrm {dt} + \\\\nonumber \\\\\\\\+\\\\int _{\\\\Sigma _T} \\\\left(\\\\Lambda _3 -\\\\gamma (v)|v|^{\\\\ell -2}v \\\\right) (\\\\theta -v) \\\\mathrm {ds}\\\\mathrm {dt} \\\\ge 0.$ Next, taking $v=\\\\theta -\\\\delta \\\\varphi $ , with $\\\\varphi \\\\in \\\\mathcal {D}(Q_T)$ , and after dividing by $\\\\delta >0$ , we arrive to $\\\\delta \\\\int _{Q_T} a( \\\\theta ,\\\\phi )|\\\\nabla \\\\varphi |^2\\\\mathrm {dx} \\\\mathrm {dt}+\\\\int _{Q_T} \\\\Big (\\\\Lambda _2 -\\\\sigma (\\\\theta )T_\\\\mathcal {M}(\\\\phi )\\\\nabla \\\\phi \\\\Big )\\\\cdot \\\\nabla \\\\varphi \\\\mathrm {dx} \\\\mathrm {dt} \\\\ge 0.$ Finally letting $\\\\delta \\\\rightarrow 0^+$ then we obtain $\\\\Lambda _2 =\\\\sigma (\\\\theta )T_\\\\mathcal {M}(\\\\phi )\\\\nabla \\\\phi $ .\",\n \"We conclude the Minty argument by taking $v=\\\\theta -\\\\delta \\\\varphi $ , with $\\\\varphi \\\\in \\\\mathcal {D}(\\\\Sigma _T)$ , in (REF ).\",\n \"After dividing by $\\\\delta >0$ , and finally letting $\\\\delta \\\\rightarrow 0^+$ we arrive to $\\\\int _{\\\\Sigma _T}(\\\\Lambda _3 -\\\\gamma (\\\\theta ) |\\\\theta |^{\\\\ell -2}\\\\theta )\\\\varphi \\\\mathrm {ds}\\\\mathrm {dt} \\\\ge 0, \\\\quad \\\\forall \\\\varphi \\\\in \\\\mathcal {D}(\\\\Sigma _T),$ which implies that $\\\\Lambda _3 =\\\\gamma (\\\\theta )|\\\\theta |^{\\\\ell -2}\\\\theta $ .\",\n \"Therefore, the weak formulation (REF ) yields concluding the proof of Theorem REF .\"\n ]\n]"},"id":{"kind":"string","value":"1709.01871"}}},{"rowIdx":699,"cells":{"text":{"kind":"list like","value":[["Nehari's theorem for convex domain Hankel and Toeplitz operators in\n several variables"],["Abstract We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables.","A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows.","Let $\\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\\Xi$, consider the Hankel operator $$\\Gamma_f (g)(x)=\\int_{\\Xi} f(x+y) g(y) \\, dy, \\quad x \\in\\Xi.$$ Then $\\Gamma_f$ extends to a bounded operator on $L^2(\\Xi)$ if and only if there is a bounded function $b$ on $\\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2\\Xi$.","This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix.","In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks."],["Introduction","For an open connected set $\\Xi \\subset {\\mathbb {R}}^d$ , $d \\ge 1$ , let $\\Omega = \\Xi + \\Xi = \\lbrace x + y \\, : \\, x\\in \\Xi , \\, y\\in \\Xi \\rbrace ,$ and consider a distribution $f$ defined on $\\Omega $ .","The associated general domain Hankel operator $\\Gamma _f = \\Gamma _{f, \\Xi }$ is the (densely defined) operator $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ , given by $\\Gamma _f (g)(x)=\\int _{\\Xi } f(x+y) g(y) \\, dy, \\quad x \\in \\Xi ,$ where $dy$ is the Lebesgue measure on ${\\mathbb {R}}^d$ .","The case $\\Xi = {\\mathbb {R}}_+ = (0, \\infty )$ for $d=1$ corresponds to the class of usual Hankel operators; when represented in the appropriate basis of $L^2({\\mathbb {R}}_+)$ , the operator $\\Gamma _{f, {\\mathbb {R}}_+}$ is realized as an infinite Hankel matrix $\\lbrace a_{n+m}\\rbrace _{n,m =0}^\\infty $ [31].","Nehari's theorem characterizes the bounded Hankel operators $\\Gamma _f \\colon L^2({\\mathbb {R}}_+) \\rightarrow L^2({\\mathbb {R}}_+)$ .","For a function $g$ on ${\\mathbb {R}}^d$ , we let $\\hat{g} = \\mathcal {F}g$ denote its Fourier transform, $\\hat{g}(\\xi ) = \\mathcal {F}g(\\xi ) = \\int _{\\mathbb {R}^d} g(x) e^{-2\\pi i x \\cdot \\xi } \\, dx, \\quad \\xi \\in \\mathbb {R}^d.$ Theorem (Nehari [25]) Suppose that $f$ is a distribution in ${\\mathbb {R}}_+$ , $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}_+)$ .","Then $\\Gamma _f \\colon L^2(\\mathbb {R}_+) \\rightarrow L^2(\\mathbb {R}_+)$ is bounded if and only if there exists a function $b\\in L^\\infty ({\\mathbb {R}})$ such that $\\hat{b}|_{{\\mathbb {R}}_+}=f$ .","Moreover, it is possible to choose $b$ so that $ \\Vert \\Gamma _f\\Vert = \\Vert b\\Vert _{L^\\infty }.$ Nehari's theorem is canonical in operator theory.","The two most common proofs proceed either by factorization in the single variable Hardy space or by making use of the commutant lifting theorem.","For $d > 1$ , the operators $\\Gamma _{f, {\\mathbb {R}}_+^d}$ , $\\Xi = {\\mathbb {R}}_+^d$ , correspond to (small) Hankel operators on the product domain multi-variable Hardy space $H^2_d$ .","In this case, the analogue of Nehari's theorem remains true, apart from (REF ), but it is significantly more difficult to prove.","It was established by Ferguson and Lacey ($d=2$ ) and Lacey and Terwilleger ($d > 2$ ) [18], [23].","A precise statement is given in Theorem REF .","The main purpose of this article is to prove Nehari's theorem when $\\Xi \\subset {\\mathbb {R}}^d$ is a simple convex polytope.","t1altTheorem REF Let $\\Xi $ be a simple convex polytope, and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ where $\\Omega =2\\Xi $ .","Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ When $d = 1$ , the only open connected sets $\\Xi \\subset {\\mathbb {R}}$ are the intervals $\\Xi = I$ .","In this case, Theorem REF is due to Rochberg [35], who called the corresponding operators $\\Gamma _{f, I}$ Hankel/Toeplitz operators on the Paley–Wiener space.","They have also been called Wiener–Hopf operators on a finite interval [30].","These operators have inspired a wealth of theory in the single variable setting – see Section REF , where we shall interpret Theorem REF in the context of Paley–Wiener spaces.","Even for $d=1$ , our proof of Theorem REF appears to be new.","However, in several variables our proof relies on the Nehari theorem of Ferguson–Lacey–Terwilleger, and can therefore not be used to give a new proof of their results.","We shall also consider general domain Toeplitz operators $\\Theta _f=\\Theta _{f, \\Xi } \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ .","In this context, $f$ is a distribution defined on $\\Omega = \\Xi - \\Xi $ , and $\\Theta _f$ is densely defined via $\\Theta _f(g)(x) = \\int _{\\Xi } f(x-y) g(y) \\, dy, \\quad x \\in \\Xi .$ If $\\Xi $ after a translation is invariant under the reflection $x \\mapsto -x$ , then the classes of Hankel operators $\\Gamma _{f, \\Xi }$ and Toeplitz operators $\\Theta _{\\widetilde{f}, \\Xi }$ are essentially the same, and Theorem REF immediately yields a boundedness result.","This reasoning is applicable to the cube $\\Xi = (0,1)^d$ , for example.","t2Corollary REF Let $\\Xi $ be a simple convex polytope such that for some $z \\in {\\mathbb {R}}^d$ it holds that $\\Xi +z = -\\Xi - z$ .","Let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =\\Xi - \\Xi = 2\\Xi + 2z$ .","Then $\\Theta _f$ is bounded if and only if there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .","There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Theta _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ On the other hand, when $\\Xi $ is a proper convex unbounded set, containing an open cone say, it is clear that the boundedness characterizations of $\\Theta _{f, \\Xi }$ and $\\Gamma _{f,\\Xi }$ may be completely different; plainly explained by the fact that $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ in the Toeplitz case, while $\\Omega = \\Xi + \\Xi = 2\\Xi \\subsetneq {\\mathbb {R}}^d$ for Hankel operators.","In this setting, identifying the boundedness of $\\Theta _f$ carries none of the subtleties of Nehari-type theorems.","In Theorem REF we obtain the expected boundedness result for a class of “cone-like” domains $\\Xi $ .","Rather than giving a precise statement here, let us record the following corollary of Theorem REF .","cor:exCorollary REF Let $\\Xi \\subset {\\mathbb {R}}^d$ be any open connected domain such that $(1,\\infty )^d \\subset \\Xi \\subset (0,\\infty )^d,$ and let $f \\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ .","Then $\\Theta _{f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if $f$ is a tempered distribution and $\\Vert \\hat{f}\\Vert _{L^\\infty ({\\mathbb {R}}^d)} < \\infty $ , and in this case $\\Vert \\Theta _f\\Vert = \\Vert \\hat{f} \\Vert _{L^\\infty }.$ In the final part of the paper we shall give an application of Theorem REF to matrix completion theory, essentially obtained by discretizing Corollary REF when $\\Xi $ is a cube.","To avoid introducing further notation, we shall only state the result in words for now.","Recall that a Toeplitz matrix is one whose diagonals are constant.","An $N \\times N$ $d$ -multilevel block Toeplitz matrix is an $N \\times N$ Toeplitz matrix whose entries are $N\\times N$ $(d-1)$ -multilevel block Toeplitz matrices.","Here $N$ could be finite or infinite.","A 1-multilevel block Toeplitz matrix is simply an ordinary Toeplitz matrix.","A 2-multilevel block Toeplitz matrix is what is usually considered a block Toeplitz matrix where each block itself is Toeplitz.","thm:blocktoepTheorem REF Every finite $N \\times N$ $d$ -multilevel block Toeplitz matrix can be extended to an infinite $d$ -multilevel block Toeplitz matrix bounded on $\\ell ^2$ , with a constant which only depends on the dimension $d$ .","For scalar Toeplitz matrices ($d=1$ ) this result is well-known [5], [26], [36], [37], although not as firmly cemented in the literature as the Nehari theorem itself; see [28] for a proof based on Parrot's lemma and a discussion of the result's history.","For $d=1$ , the converse deduction of Theorem REF starting from Theorem REF can be found in [13].","The paper is laid out as follows.","In Section we will give a more formal background and introduce necessary notation.","We will also discuss the relationship between $\\Gamma _{f, \\Xi }$ , Paley–Wiener spaces, and co-invariant subspaces of the Hardy spaces.","In Section we will prove approximation results for distribution symbols with respect to Hankel and Toeplitz operators, allowing us to reduce to smooth symbols.","Section briefly outlines what we need to know about convex sets and polytopes.","In Section we prove Theorem REF , our Nehari theorem for Hankel operators.","We also indicate how the proof extends to certain unbounded polyhedral domains.","In Section our main result on Toeplitz operators is shown, Theorem REF .","Finally, Section gives the proof of Theorem REF ."],["Hankel operators on multi-variable Hardy spaces","Let us begin by placing Hankel operators $\\Gamma _f$ into the context of classical Hankel operators on Hardy spaces.","As before, for $g \\in L^2(\\mathbb {R}^d)$ , let $\\hat{g} = \\mathcal {F}g$ denote its Fourier transform, $\\hat{g}(\\xi ) = \\mathcal {F}g(\\xi ) = \\int _{\\mathbb {R}^d} g(x) e^{-2\\pi i x \\cdot \\xi } \\, dx, \\quad \\xi \\in \\mathbb {R}^d.$ For the inverse transform we write $\\mathcal {F}^{-1}(g)=\\check{g}$ .","The product domain Hardy space $H^2_d$ is the proper subspace of $L^2(\\mathbb {R}^d)$ of functions whose Fourier transforms are supported in the cone $\\overline{{\\mathbb {R}}_+^d}$ , ${\\mathbb {R}}_+ = (0,\\infty )$ , $ H^2_d = \\left\\lbrace G \\in L^2(\\mathbb {R}^d) \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{\\mathbb {R}_+^d}\\right\\rbrace .$ We let $P_d \\colon L^2(\\mathbb {R}^d) \\rightarrow H^2_d$ denote the orthogonal projection, and let $J \\colon L^2(\\mathbb {R}^d) \\rightarrow L^2(\\mathbb {R}^d)$ be the involution defined by $JG(x) = G(-x)$ , $x \\in \\mathbb {R}$ .","Consider $\\Gamma _f = \\Gamma _{f, \\Xi }$ for $\\Xi = \\mathbb {R}_+^d$ with $f \\in L^2(\\mathbb {R}_+^d)$ .","For a dense set of $g, h \\in L^2(\\mathbb {R}_+^d)$ we have that $ \\langle \\Gamma _f g, h \\rangle _{L^2(\\mathbb {R}_+^d)} = \\langle \\check{f} J\\check{g}, \\check{h}\\rangle _{H^2_d}.$ It follows that the (possibly unbounded) operator $\\Gamma _f \\colon L^2({\\mathbb {R}}_+^d) \\rightarrow L^2({\\mathbb {R}}_+^d)$ is unitarily equivalent to the small Hankel operator $Z_{\\check{f}} \\colon H^2_d \\rightarrow H^2_d$ , $Z_{\\check{f}} G = P_d(\\check{f} \\cdot JG).$ Note that any $b$ such that $\\hat{b}|_{\\mathbb {R}_+^d} = f$ generates the same Hankel operator as $\\check{f}$ , $Z_b = Z_{\\check{f}}$ .","To justify the above computation easily we assumed that $f \\in L^2({\\mathbb {R}}_+^d)$ .","An approximation argument is needed to consider general symbols $f$ , which may only be distributions in ${\\mathbb {R}}_+^d$ .","We provide this later in Proposition REF .","We can then read off the boundedness of $\\Gamma _f$ from the boundedness of the corresponding Hankel operator on $H^2_d$ .","When $d=1$ and $\\Xi =\\Omega ={\\mathbb {R}}_+$ , the analogue of Theorem REF is exactly the classical Nehari theorem.","In higher dimensions the corresponding theorem is due to Ferguson–Lacey–Terwilleger [18], [23].","In our notation, their results read as follows.","Theorem 2.1 Suppose $\\Xi =\\Omega ={\\mathbb {R}}_+^d$ and that $f$ is a distribution in ${\\mathbb {R}}^d_+$ , $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d_+)$ .","Then $\\Gamma _f \\colon L^2(\\mathbb {R}^d_+) \\rightarrow L^2(\\mathbb {R}^d_+)$ is bounded if and only if there exists a function $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}_+^d}=f$ .","Moreover, there exists a constant $c > 0$ , depending on $d$ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty }\\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ For $d > 1$ it is not possible to take $c=1$ in (REF ), see for example [29]."],["Hankel operators on bounded domains","We now discuss bounded domains $\\Xi $ , the setting of our main result.","The only convex bounded domains in ${\\mathbb {R}}$ are the intervals $I\\subset {\\mathbb {R}}$ .","Translations, dilations, and reflections carry the operator $\\Theta _{f,I}$ onto $\\Gamma _{\\tilde{f},J}$ , where $J \\subset \\mathbb {R}$ is any other interval and $\\tilde{f}$ arises from transforming $f$ appropriately.","In one variable it thus suffices to consider operators $\\Gamma _{f,(0,1)}$ where $\\Xi = (0,1)$ .","Rochberg [35] called these operators Hankel operators on the Paley-Wiener space and proved Theorem REF in the one-dimensional case.","In the same article [35], it is posed as an open problem to characterize the bounded Hankel operators $\\Gamma _{f,\\Xi }$ when $\\Xi $ is a disc in $\\mathbb {R}^2$ .","We are not able to settle this question, but Theorem REF does provide the answer when $\\Xi = (0,1)^d$ is a cube in ${\\mathbb {R}}^d$ .","As we will see, the Hankel operators $\\Gamma _{f, (0,1)^d}$ constitute a natural generalization of the Hankel operators on the Paley-Wiener space.","On a technical level, the reason that we are able to prove Theorem REF when $\\Xi $ is a simple convex polytope, but not when $\\Xi $ is a ball, is that we rely on Theorem REF .","In applying Theorem REF to our situation, the corners of the boundary of $\\Xi $ are actually of help rather than hindrance.","We consider the case of a ball to be an interesting open problem for which we do not dare to make a firm conjecture.","In view of Fefferman's disproof of the disc conjecture [17], Nehari theorems might turn out to be quite different for balls and polytopes."],["Toeplitz operators","When $d=1$ and $\\Xi ={\\mathbb {R}}_+$ , $\\Omega = {\\mathbb {R}}$ , the operators $\\Theta _f$ are known as Wiener-Hopf operators [11].","Analogously with Hankel operators, these can be shown to be unitarily equivalent to Toeplitz matrix operators on $\\ell ^2({\\mathbb {N}})$ .","In this case the boundedness characterization is easy to both state and prove, $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }.$ In Theorem REF we extend (REF ) to Toeplitz operators $\\Theta _{f, \\Xi }$ for a class of “cone-like” domains $\\Xi \\subset {\\mathbb {R}}^d$ , for which $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ ."],["Truncated correlation operators","For open connected sets $\\Xi , \\Upsilon \\subset {\\mathbb {R}}^d$ it is also convenient to introduce the more general “truncated correlation operators” $\\Psi _{f, \\Upsilon , \\Xi } \\colon L^2(\\Upsilon )\\rightarrow L^2(\\Xi )$ , defined by $\\Psi _{f}(g)({x})=\\int _{\\Upsilon }f({x}+{y}) g({y}) ~d{y},\\quad {x}\\in \\Xi ,$ where $f$ lives on $\\Omega =\\Xi +\\Upsilon $ .","This class of operators includes both general domain Hankel and Toeplitz operators, by letting $\\Upsilon = \\Xi $ and $\\Upsilon = -\\Xi $ , respectively.","For our purposes, general truncated correlation operators will only appear in intermediate steps toward proving the main results, but they also carry independent interest.","They were introduced in [1], where their finite rank structure was investigated.","In [2] it was shown that they have a fundamental connection with frequency estimation on general domains, motivating the practical need for understanding such operators not only on domains of simple geometrical structure.","In [3] it is explained how one may infer certain results for the integral operators $\\Psi _f$ from their discretized matrix counterparts.","We warn the reader that in naming the operators $\\Gamma _f$ , $\\Theta _f$ , and $\\Psi _f$ we have slightly departed from previous work, reserving the term (general domain) Hankel operator for truncated correlation operators of the form $\\Psi _{f,\\Xi ,\\Xi }$ ."],["Hankel operators on multi-variable Paley–Wiener spaces","Another viewpoint is offered through co-invariant subspaces of the Hardy spaces $H^2_d$ .","For a domain $\\Xi \\subset {\\mathbb {R}}^d$ , let $\\operatorname{PW}_\\Xi $ denote the subspace of $L^2({\\mathbb {R}}^d)$ of functions with Fourier transforms supported in $\\Xi $ , $\\operatorname{PW}_\\Xi = \\lbrace G \\in L^2({\\mathbb {R}}^d) \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{\\Xi } \\rbrace .$ In the classical case $\\Xi = (0,1) \\subset {\\mathbb {R}}$ , note that $\\operatorname{PW}_{(0,1)} = H^2_1 \\ominus \\lbrace G \\in H^2_1 \\, : \\, \\operatorname{supp}\\hat{G} \\subset [1,\\infty )\\rbrace = H^2_1 \\ominus \\theta H^2_1,$ where $\\theta (x) = e^{i2\\pi x}, \\quad x \\in \\mathbb {R}.$ Hence $\\operatorname{PW}_{(0,1)}$ is the ortho-complement (in $H^2_1$ ) of $\\theta H^2_1$ , the shift-invariant subspace of $H^2_1$ with inner factor $\\theta $ .","This space is usually denoted $K_\\theta $ , $\\operatorname{PW}_{(0,1)} = K_\\theta := (\\theta H^2_1)^\\perp .$ By a calculation similar to (REF ) we see that $\\Gamma _{f, (0,1)}$ is unitarily equivalent to the compression of the Hankel operator $Z_{\\check{f}}$ to $\\operatorname{PW}_{(0,1)}$ , $\\Gamma _{f, (0,1)} \\simeq P_{\\operatorname{PW}_{(0,1)}} Z_{\\check{f}}|_{\\operatorname{PW}_{(0,1)}},$ where $P_{\\operatorname{PW}_{(0,1)}} \\colon H^2_1 \\rightarrow \\operatorname{PW}_{(0,1)}$ denotes the orthogonal projection onto $\\operatorname{PW}_{(0,1)}$ .","Such truncated Toeplitz and Hankel operators are now very well studied on general $K_\\theta $ -spaces [6], [7], [9], [10], [14], [20], [27], [30], [36].","In the case of the cube $\\Xi = (0,1)^d \\subset {\\mathbb {R}}^d$ , $d > 1$ , the Hankel operator $\\Gamma _{f,\\Xi }$ may, just as for $d=1$ , be understood as the compression of a Hankel operator to a co-invariant subspace of $H^2_d$ .","Namely, $\\operatorname{PW}_{(0,1)^d} = \\lbrace G \\in H_d^2 \\, : \\, \\operatorname{supp}\\hat{G} \\subset [0,1]^d\\rbrace = \\lbrace G \\in H_d^2 \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{{\\mathbb {R}}_+^d} \\setminus (0,1)^d\\rbrace ^\\perp .$ If $G \\in H_d^2 \\cap L^\\infty ({\\mathbb {R}}^d)$ , it is clear that $G\\operatorname{PW}_{(0,1)^d}^\\perp \\subset \\operatorname{PW}_{(0,1)^d}^\\perp $ , since $\\mathcal {F}(GH)(\\xi ) = \\int _{{\\mathbb {R}}_+^d} \\hat{G}(y) \\hat{H}(\\xi -y) \\, dy = 0, \\quad H \\in \\operatorname{PW}_{(0,1)^d}^\\perp , \\; \\xi \\in [0,1]^d.$ Hence $\\operatorname{PW}_{(0,1)^d}^\\perp \\subset H^2_d$ is an invariant subspace (under multiplication by bounded holomorphic functions), and as before we have that $\\Gamma _{f, (0,1)^d} \\simeq P_{\\operatorname{PW}_{(0,1)^d}} Z_{\\check{f}}|_{\\operatorname{PW}_{(0,1)^d}},$ where $P_{\\operatorname{PW}_{(0,1)^d}} \\colon H^2_d \\rightarrow \\operatorname{PW}_{(0,1)^d}$ denotes the orthogonal projection onto $\\operatorname{PW}_{(0,1)^d}$ .","Finally, let us briefly discuss the viewpoint of weak factorization.","The Hardy space $H^1_d$ is defined as the closure of $\\mathcal {F}^{-1}(C_c^\\infty ({\\mathbb {R}}_+^d))$ in $L^1({\\mathbb {R}}^d)$ .","Similarly, we define $\\operatorname{PW}^1_\\Xi $ as the closure of $\\mathcal {F}^{-1}(C_c^\\infty (\\Xi ))$ in $L^1({\\mathbb {R}}^d)$ .","As is well known, see for example [24], Theorem REF is equivalent to the fact that $H^1_d$ is the projective tensor product of two copies of $H^2_d$ , $ H^1_d = H^2_d \\odot H^2_d,$ with equivalence of norms.","Here the projective tensor product norm on $X \\odot X$ , $X$ a Banach space of functions, is given by $\\Vert G\\Vert _{X \\odot X} = \\inf \\left\\lbrace \\sum _j \\Vert G_j\\Vert _{X} \\Vert H_j\\Vert _{X} \\, : \\, G = \\sum _j G_j H_j, \\; G_j, H_j \\in X \\right\\rbrace ,$ $X \\odot X$ being defined as the completion of finite sums $\\sum _j G_j H_j$ in this norm.","The reason that Theorem REF is equivalent to (REF ) is the following: by (REF ), $\\Gamma _{f, {\\mathbb {R}}_+^d}$ is bounded if and only if $|\\langle \\check{f}, GH \\rangle _{H^2_d} | \\le C\\Vert G\\Vert _{H^2_d} \\Vert H\\Vert _{H^2_d},$ which means precisely that $\\check{f}$ induces a bounded functional on $H^2_d \\odot H^2_d$ , $\\check{f} \\in (H^2_d \\odot H^2_d)^*$ .","On the other hand, the existence of $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}_+^d} = f|_{{\\mathbb {R}}_+^d}$ , so that $\\langle \\check{f}, GH \\rangle _{H^2_d} = \\langle b, GH \\rangle _{H^2_d}$ , $G,H \\in H^2_d$ , means, by the Hahn–Banach theorem, precisely that $\\check{f} \\in (H_d^1)^*$ .","Theorem REF yields a similar weak factorization theorem for Paley–Wiener spaces.","We postpone the proof to Section , mentioning only that corresponding weak factorization for $K_\\theta $ -spaces plays an important role in [6] and [9].","cor:factCorollary REF Let $\\Xi $ be a simple convex polytope, and let $\\Omega = 2\\Xi $ .","Then $PW_{\\Omega }^1 = \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ The norms of these Banach spaces are equivalent."],["Brief historical overview","Z. Nehari published his famous theorem in 1957 [25], inspiring the search for analogous statements in other contexts; positive results are themselves often referred to as Nehari theorems.","The most natural inquiries are perhaps those related to Hankel operators on Hardy spaces of several variables.","Nehari's theorem for the Hardy space of the unit ball was proven by Coifman, Rochberg and Weiss in 1976 [15], but this setting is rather different from the one considered in this paper.","For the product domain Hardy space $H^2_d$ , Hankel operators can be defined by either projecting on $H^2_d$ or on the larger space $L^2({\\mathbb {R}}^d) \\ominus H^2_d$ .","The first option leads to the “small” Hankel operators considered in Section REF , while the second type of operator is commonly referred to as a “big” Hankel operator.","In the notation of Section REF , a small Hankel operator is an operator $\\Psi _{f,{\\mathbb {R}}_+^d,{\\mathbb {R}}_+^d}=\\Gamma _{f,{\\mathbb {R}}_+^d}$ , whereas big Hankel operators are of the form $\\Psi _{f,{\\mathbb {R}}_+^d,{\\mathbb {R}}^d\\setminus \\overline{{\\mathbb {R}}_+^d}}$ .","When transferred to operators on the Hardy space of the polydisc, small Hankel operators correspond, in the standard basis, to infinite matrices with a certain block Hankel structure (cf.","Section ).","The big Hankel operators were extensively studied by Cotlar and Sadosky.","In particular, boundedness of the big Hankel operators was characterized in terms of certain $\\operatorname{BMO}$ type estimates in [16].","Small Hankel operators were investigated by Janson and Peetre [22] in 1988.","They introduced “generalized Hankel and Toeplitz operators” as particular cases of a more general class of pseudo-differential operators called paracommutators.","In their terminology, an operator of the form $\\Psi _{f,\\Xi ,\\Upsilon }$ is a generalized Hankel operator if $\\Xi $ and $\\Upsilon $ are open cones and $\\overline{\\Xi } \\cap (-\\overline{\\Upsilon })=\\lbrace 0\\rbrace $ , whereas it is called Toeplitz if $\\Xi \\cap (-\\Upsilon ) \\ne \\emptyset .$ Hence the general domain Hankel operators $\\Gamma _{f,\\Xi }$ are generalized Hankel operators a lá Janson–Peetre whenever $\\Xi $ is a cone with mild restrictions, while $\\Theta _{f,\\Xi }$ is a generalized Toeplitz operator a lá Janson–Peetre for every open cone $\\Xi $ .","In the Toeplitz case, a full boundedness characterization is given in [22].","In the Hankel case, only sufficient conditions for boundedness and Schatten class membership are provided, in terms of $\\operatorname{BMO}$ and Besov spaces, respectively.","As previously mentioned, R. Rochberg considered Hankel operators for bounded domains in 1987 [35], studying the case of a finite interval in one dimension.","Furthermore, he posed as an open problem to understand the case when $\\Xi \\subset {\\mathbb {R}}^2$ is a disc.","In this latter setting, L. Peng [32] characterized when $\\Gamma _{f, \\Xi }$ belongs to the Schatten class $S_p$ , for $1 \\le p \\le 2$ , in terms of certain Besov spaces adapted to the disc.","L. Peng also carried out a similar study [33] for the case of the multidimensional cube, $\\Xi = (-1,1)^d$ , describing membership in $S_p$ for all $p$ , $0 < p < \\infty $ , as well as giving a sufficient condition for boundedness.","Since then it seems that the field did not see progress until the results of Ferguson–Lacey–Terwilleger [18], [23] settled the issue of boundedness of small Hankel operators."],["Distribution symbols","Let $\\Xi ,\\Upsilon \\subset {\\mathbb {R}}^d$ be any open connected sets and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ be a distribution on $\\Omega $ , $\\Omega =\\Xi +\\Upsilon $ .","We follow the notation of [21] in our use of distributions.","We then define the truncated correlation operator $\\Psi _f$ as an operator $\\Psi _{f, \\Upsilon , \\Xi }:C^\\infty _c(\\Upsilon )\\rightarrow C^\\infty (\\Xi )$ by the formula $\\Psi _f(\\varphi )(x)=({f,T_x\\varphi }), \\quad x \\in \\Xi ,$ where $(f,\\varphi )$ denotes the action of $f$ on $\\varphi $We reserve the notation $\\langle f,\\varphi \\rangle $ for scalar products which are anti-linear in the second entry.","and $T_x\\varphi (\\cdot )=\\varphi (\\cdot -x).$ Since $T_x\\varphi $ is compactly supported in $\\Omega $ for $x \\in \\Xi $ , it follows that $\\Psi _f(\\varphi )$ this is well-defined and smooth in $\\Xi $ (see e.g.","[21]).","Since $C^\\infty _c(\\Upsilon )$ is dense in $L^2(\\Upsilon )$ , $\\Psi _f$ gives rise to a densely defined operator on the latter space which extends to a bounded operator $\\Psi _f \\colon L^2(\\Upsilon ) \\rightarrow L^2(\\Xi )$ if and only if $\\Vert \\Psi _f\\Vert =\\sup \\left\\lbrace \\frac{\\Vert \\Psi _f(\\varphi )\\Vert _{L^2(\\Xi )}}{\\Vert \\varphi \\Vert _{L^2(\\Upsilon )}}:~\\varphi \\in C_c^\\infty (\\Upsilon ), ~\\varphi \\ne 0\\right\\rbrace <\\infty .$ It is clear that $\\Psi _f(\\varphi )(x) = \\int f(x+y)\\varphi (y) \\, dy$ whenever $f \\in L^1_{\\textrm {loc}}(\\Omega )$ .","By slight abuse of notation, we write the action of $\\Psi _f$ in this way even when $f$ is not locally integrable.","The central question in this paper is the following: for which domains $\\Upsilon $ and $\\Xi $ is the boundedness of $\\Psi _f$ equivalent to the existence of a function $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega }=f$ ?","Some care must be taken in interpreting this question.","For example, the prototypical example of a bounded Hankel operator is the Carleman operator $\\Gamma _{1/x, {\\mathbb {R}}_+} = \\Psi _{1/x, {\\mathbb {R}}_+, {\\mathbb {R}}_+}.$ The symbol $f(x) = \\frac{1}{x}\\chi _{{\\mathbb {R}}_+}(x)$ is in this case not a tempered distribution on $\\mathbb {R}$ (so the meaning of $\\check{f}$ is unclear) – it is, however, the restriction of the tempered distribution $\\operatorname{p.v.","}\\frac{1}{x}$ to ${\\mathbb {R}}_+$ .","An example with a delta function makes it clear that it is not necessary for $f$ to be locally integrable in $\\Omega $ either.","We first record the answer to our question in the trivial direction.","Proposition 3.1 Consider any connected open domains $\\Xi ,$ $\\Upsilon \\subset {\\mathbb {R}}^d$ , with associated domain $\\Omega =\\Upsilon +\\Xi $ .","Let $b\\in L^\\infty ({\\mathbb {R}}^d)$ be given and suppose $f=\\hat{b}|_{\\Omega }$ .","Then $\\Psi _f \\colon L^2(\\Upsilon ) \\rightarrow L^2(\\Xi )$ is bounded and $\\Vert \\Psi _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ For $\\varphi \\in C^\\infty _c(\\Upsilon )$ we have that $ \\Psi _{f}(\\varphi )={\\mathcal {F}}M_b J{\\mathcal {F}}^{-1} \\varphi |_{\\Xi },$ where $M_b$ is the operator of multiplication by $b$ .","The statement is obvious from here.","Next we establish two technical results on the approximation of distribution symbols by smooth compactly supported functions, Propositions REF and REF .","They will help us to overcome the technical issues mentioned earlier, in particular allowing us to deduce Theorem REF from the corresponding statements in [18], [23].","Given open connected domains $\\Xi ,$ $\\Upsilon \\subset {\\mathbb {R}}^d$ , let $\\left(\\Upsilon _n\\right)_{n=1}^\\infty $ be an increasing sequence of connected open subdomains $\\Upsilon _n \\subset \\Upsilon $ such that $\\operatorname{dist}(\\Upsilon _n,\\partial \\Upsilon )>1/n, \\quad \\cup _{n=1}^\\infty \\Upsilon _n=\\Upsilon .$ Note that $\\Omega _n = \\Upsilon _n + \\Xi $ is also increasing and satisfies $\\operatorname{dist}(\\Omega _n,\\partial \\Omega )>1/n, \\quad \\cup _{n=1}^\\infty \\Omega _n = \\Omega .$ Let $\\psi \\in C^\\infty _c({\\mathbb {R}}^d)$ be a fixed non-negative function with compact support in the ball $B(0,1/2)$ such that $\\int _{{\\mathbb {R}}^d} \\psi (x) \\, dx =1$ .","For $n\\ge 1$ let $\\psi _n(x)={n^d}\\psi (nx),$ so that $(\\psi _n)_{n=1}^\\infty $ is an approximation of the identity.","Since $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ and $\\operatorname{supp}\\psi _n \\subset B(0,1/2n)$ , the convolution $f*\\psi _n$ is well-defined as a function in $C^\\infty (\\Omega _{2n})$ .","Let $\\rho _n$ be a smooth cut-off function which is 1 in a neighborhood of $\\overline{\\Omega _n}$ but zero in a neighborhood of $\\Omega _{2n}^c$ , and note that $\\rho _n(f*\\psi _n)$ then naturally defines a function in $C^\\infty ({\\mathbb {R}}^n)$ .","Finally, for a non-negative function $\\eta \\in C^\\infty _c({\\mathbb {R}}^d)$ with $\\Vert \\eta \\Vert _{L^2} = 1$ , let $\\omega = \\eta * \\tilde{\\eta }$ , where $\\tilde{\\eta }(x) = \\eta (-x)$ .","Then $\\omega \\in C^\\infty _c({\\mathbb {R}}^d)$ and $\\omega (0)= \\Vert \\hat{\\omega }\\Vert _{L^1} = 1.$ Let $\\omega _n(x)=\\omega (x/n)$ .","We introduce $f_n=\\omega _n\\rho _n (f*\\psi _n)$ as an approximant of $f$ , where the role of $\\omega _n$ is to enforce compact support in case $\\Omega $ is unbounded.","By construction, $f_n \\in C_c^\\infty (\\Omega )$ and it is straightforward to check that $f_n \\rightarrow f$ in ${\\mathcal {D}}^{\\prime }(\\Omega )$ .","As for $\\Psi _{f_n, \\Upsilon _n, \\Xi }$ , we have the following result.","Proposition 3.2 Let $\\Xi ,$ $\\Upsilon $ be connected open domains, $\\Omega =\\Upsilon +\\Xi $ , and suppose $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ .","For $n \\ge 1$ , let $\\Omega _n=\\Upsilon _n+\\Xi $ and $f_n$ be constructed as above.","Then $\\Vert \\Psi _{f_n,\\Upsilon _n,\\Xi }\\Vert \\le \\Vert \\Psi _{f,\\Upsilon ,\\Xi }\\Vert .$ First note that $\\omega _n(x)=\\int _{{\\mathbb {R}}^d} n^d \\hat{\\omega }(n \\xi )e^{2\\pi i x \\cdot \\xi } \\, d\\xi ,$ the integrand on the right having $L^1$ -norm equal to $\\Vert \\hat{\\omega }\\Vert _{L^1({\\mathbb {R}}^d)}$ .","Letting $g_n=\\rho _n (f*\\psi _n)$ , we have for $\\varphi \\in C^\\infty _c(\\Upsilon _n)$ and $x \\in \\Xi $ that $\\Psi _{f_n}(\\varphi )(x)=\\int _{\\Upsilon _n} \\int _{{\\mathbb {R}}^d} n^d\\hat{\\omega }(n\\xi )e^{2\\pi i (x+y)\\cdot \\xi }\\,d\\xi \\, g_n(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}^d} n^d \\hat{\\omega }(n \\xi ){e^{2\\pi i \\xi \\cdot x}}\\Psi _{g_n}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y\\cdot \\xi }\\varphi (y)$ .","Since $\\Vert \\varphi _\\xi \\Vert _{L^2} = \\Vert \\varphi \\Vert _{L^2}$ it follows by the triangle inequality (for $L^2$ -valued Bochner integrals) that $\\Vert \\Psi _{f_n, \\Upsilon _n, \\Xi }\\Vert \\le \\Vert \\hat{\\omega }\\Vert _{L^1}\\Vert \\Psi _{g_n, \\Upsilon _n, \\Xi }\\Vert = \\Vert \\Psi _{g_n, \\Upsilon _n, \\Xi }\\Vert .$ This reduces our task to proving that the operators $\\Psi _{g_n, \\Upsilon _n, \\Xi } = \\Psi _{\\rho _n (f*\\psi _n), \\Upsilon _n, \\Xi } = \\Psi _{f*\\psi _n, \\Upsilon _n, \\Xi }$ are uniformly bounded in $n$ .","We have for $\\varphi \\in C^\\infty _c(\\Upsilon _n)$ and $x \\in \\Xi $ that $\\Psi _{f*\\psi _n}(\\varphi )(x) &=\\int _{{\\mathbb {R}}^d} \\int _{{\\mathbb {R}}^d} f((x+y)-z) \\psi _n(z) \\, dz~\\varphi (y) \\, dy \\\\&= \\int _{{\\mathbb {R}}^d} f(x+z) \\int _{{\\mathbb {R}}^d} \\psi _n(y-z)\\varphi (y)\\, dy \\,dz=\\Psi _f(\\widetilde{\\psi }_n * \\varphi )(x),$ where $\\widetilde{\\psi }_n(x) = \\psi _n(-x)$ .","Since $\\Vert \\widetilde{\\psi }_n * \\varphi \\Vert _{L^2(\\Upsilon )} \\le \\Vert \\psi _n\\Vert _{L^1} \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)} = \\Vert \\psi \\Vert _{L^1} \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)} = \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)},$ this completes the proof.","Suppose that $\\Gamma _{f, {\\mathbb {R}}_+^d} = \\Psi _{f, \\Xi , \\Upsilon }$ is bounded, where $\\Xi = \\Upsilon = {\\mathbb {R}}_+^d$ .","In this case, we let $\\Upsilon _n = (2/n, \\infty )^d$ .","By Proposition REF we then have that $\\Vert \\Gamma _{f_n, \\Upsilon _n}\\Vert \\le \\Vert \\Psi _{f_n, \\Upsilon _n,\\Xi }\\Vert \\le \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert , \\quad n \\ge 1.$ Since $\\Upsilon _n = z_n + {\\mathbb {R}}_+^d$ , $z_n = (2/n, \\ldots , 2/n)$ , we have that $\\Gamma _{f_n, \\Upsilon _n}(g)(x) = \\Gamma _{\\tilde{f}_n, {\\mathbb {R}}_+^d} (\\tilde{g})(x-z_n),$ where $\\tilde{f}_n(x) = f_n(x+2z_n)$ and $\\tilde{g}_n(x) = g(x+z_n)$ .","Since $\\tilde{f}_n \\in L^2({\\mathbb {R}}_+^d)$ , the computation that lead to (REF ) is justified, and we conclude from [18], [23] that there is $b_n \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}_n|_{2\\Upsilon _n} = f_n|_{2\\Upsilon _n}, \\quad \\Vert b_n\\Vert _{L^\\infty } \\le C \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert .$ By Alaoglu's theorem it follows that there is a weak-star convergent subsequence $(b_{n_k})_{k=1}^\\infty $ with limit $b \\in L^\\infty $ having norm less than $C \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert $ .","It remains to prove that $f=\\hat{b}|_{{\\mathbb {R}}_+^d}$ , i.e.","that $(f,\\varphi )=(b,\\hat{\\varphi })$ holds for all $\\varphi \\in C^\\infty _c({\\mathbb {R}}_+^d)$ .","However, this is clear from the construction; since $\\hat{\\varphi }\\in L^1$ we have that $(b,\\hat{\\varphi })=\\lim _{k\\rightarrow \\infty }(b_{n_k},\\hat{\\varphi })=\\lim _{k\\rightarrow \\infty }(f_{n_k},{\\varphi }) =(f,{\\varphi }).","$ In Section we will consider Toeplitz operators $\\Theta _{f, \\Xi }$ for which $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ .","In this case $f * \\psi _n$ is a smooth function defined in all of ${\\mathbb {R}}^d$ , and there is no need to multiply with $\\rho _n$ or to introduce the subdomains $\\Upsilon _n$ .","In this case we simply let $f_n = \\omega _n (f * \\psi _n).$ Clearly, $f_n \\rightarrow f$ in ${\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ and we have, with the exact same proof as for Proposition REF , the following approximation result.","Proposition 3.3 Let $\\Xi ,$ $\\Upsilon $ be connected open domains for which $\\Omega =\\Upsilon +\\Xi = {\\mathbb {R}}^d$ , and suppose $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ .","For $n \\ge 1$ , let $f_n$ be constructed as above.","Then $\\Vert \\Psi _{f_n,\\Upsilon ,\\Xi }\\Vert \\le \\Vert \\Psi _{f,\\Upsilon ,\\Xi }\\Vert .$"],["On convex sets and polytopes","We recall some basic properties of convex sets.","Given an unbounded convex set $\\Omega \\subset {\\mathbb {R}}^d$ which is either open or closed, its characteristic cone, also known as its recession cone, is the closed set $\\operatorname{cc}_{\\Omega }=\\lbrace x\\in {\\mathbb {R}}^d \\, : \\, \\Omega +x{\\mathbb {R}}_+\\subset \\Omega \\rbrace .$ The support function $h_{\\Omega }:{\\mathbb {R}}^d\\rightarrow (-\\infty ,\\infty ]$ is defined by $h_{\\Omega }({\\theta })=\\sup _{{x}\\in \\Omega } {x}\\cdot {\\theta }.$ We refer to [21] for the basic properties of $h_\\Omega $ .","The barrier cone of $\\Omega $ is the set $ \\operatorname{bc}_{\\Omega }=\\lbrace \\theta \\in {\\mathbb {R}}^d \\, : \\, h_{\\Omega }({\\theta })<\\infty \\rbrace .$ The characteristic cone $\\operatorname{cc}_{\\Omega }$ coincides with the polar cone of the barrier cone $\\operatorname{bc}_{\\Omega }$ , that is, $\\operatorname{cc}_{\\Omega } = \\lbrace x \\in {\\mathbb {R}}^d \\, : \\, x \\cdot y \\le 0, \\; \\: \\forall y\\in \\operatorname{bc}_{\\Omega }\\rbrace .$ To give a complete reference for this claim, first note that for closed convex sets $\\Omega $ , $\\operatorname{cc}_{\\Omega }$ coincides with the asymptotic cone of $\\Omega $ , giving (REF ) by [4].","When $\\Omega $ instead is open and convex we have that $\\Omega $ is equal to its relative interior $\\operatorname{ri}(\\Omega )$ , and since $\\operatorname{cc}_{\\operatorname{ri}(\\Omega )} = \\operatorname{cc}_{\\overline{\\Omega }}$ [8], it follows that $\\operatorname{cc}_{\\Omega } = \\operatorname{cc}_{\\overline{\\Omega }}$ in this case.","We next recall some standard terminology and facts of polytopes, referring to for example [12].","By an open halfspace in ${\\mathbb {R}}^d$ we mean a set $H_\\nu ^r=\\lbrace x \\in {\\mathbb {R}}^d \\, : \\, x\\cdot \\nu >r\\rbrace ,$ where $\\nu \\in {\\mathbb {R}}^d$ is a non-zero vector and $r\\in {\\mathbb {R}}$ .","A closed half-space is the closure of such a set.","A finite intersection of half-spaces is called a polyhedral set.","A convex polytope is a bounded polyhedral set.","A closed convex polytope is the convex hull of a finite set of points.","The minimal set of such points coincides with the extreme points of the polytope, that is, its vertices.","If the minimal number of defining hyperspaces of a convex polytope is $d+1$ (equivalently, if it has precisely $d+1$ vertices), the polytope is called a simplex.","For a non-closed polytope we define its vertices (and its edges and facets) as those of its closure.","The boundary of a polytope set is made up of a finite amount of facets (i.e.","$d-1$ dimensional faces), see Corollary 7.4 and Theorem 8.1 of [12].","For a polytope $\\Pi $ with vertex $x_j$ , we denote by $\\partial _{\\operatorname{far},x_j}\\Pi $ the part of its boundary made up of all facets not containing $x_j$ .","A vertex of a polytope will be called simple if it is contained in precisely $d$ of its edges.","We say that a polytope is simple if all of its vertices are simple, which coincides with the standard terminology.","Equivalently, this means that each vertex is contained in precisely $d$ of its facets (cf.","[12]).","By an affine linear transformation we mean a map of the form $A(x)=x_0+L(x)$ where $L$ is a linear map, and we call $x_0$ the origin of such a map.","The following simple lemma gives a third characterization of simple vertices.","Lemma 4.1 Let $\\lbrace x_j\\rbrace _{j=1}^J$ be the vertices of a closed polytope $\\Pi $ .","Then the vertex $x_j$ is simple if and only if it is the origin of an invertible affine transformation $A_j$ such that $\\Pi $ locally coincides with $A_j(\\overline{{\\mathbb {R}}_+^d})$ around $x_j$ , i.e.","for any neighborhood $U$ of $x_j$ such that $\\overline{U} \\cap \\partial _{\\operatorname{far}, x_j}\\Pi = \\emptyset $ we have that $A_j^{-1}(\\Pi \\cap U) = \\overline{{\\mathbb {R}}_+^d} \\cap A_j^{-1}(U).$ By compactness it is easy to construct a partition of unity adapted to the vertices of $\\Pi $ .","Lemma 4.2 Given a polytope $\\Pi $ with vertices $\\lbrace x_j\\rbrace _{j=1}^J$ there exist functions $\\lbrace \\mu _j\\rbrace _{j=1}^J$ such that $\\mu _j \\in C_c^\\infty ({\\mathbb {R}}^d)$ , $\\sum _{j=1}^J \\mu _j(x) = 1$ for $x \\in \\overline{\\Pi }$ , and $\\operatorname{supp}\\mu _j\\cap \\partial _{\\operatorname{far}, x_j}\\Pi =\\emptyset $ ."],["General domain Hankel operators","We now consider general domain Hankel operators $\\Gamma _{f,\\Xi }$ for convex domains $\\Xi $ .","Observe that in this case $\\Omega =\\Xi +\\Xi = 2\\Xi $ .","We begin with a proposition that links the bounded Hankel operators with weak factorization.","Proposition 5.1 Let $\\Xi $ be an open convex domain.","Then $X = \\left\\lbrace \\Gamma _{f, \\Xi } \\, : \\, \\Vert \\Gamma _{f, \\Xi } \\Vert < \\infty \\right\\rbrace $ is a closed subspace of the space of bounded linear operators on $L^2(\\Xi )$ .","As a Banach space, it is isometrically isomorphic to the dual space $(\\operatorname{PW}_\\Xi \\odot \\operatorname{PW}_\\Xi )^*$ .","More precisely, bounded functionals $\\mu $ on the projective tensor product correspond to distributions $f$ on $\\Omega = 2\\Xi $ , $( f, g ) = \\mu (\\mathcal {F}^{-1}g), \\quad g \\in C_c^\\infty (\\Omega ),$ for which $\\Vert \\Gamma _{f, \\Xi }\\Vert = \\Vert \\mu \\Vert .$ The main fact to be proved is that $\\mathcal {F}^{-1}(C_c^\\infty (\\Omega )) \\subset \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ Since $C_c^\\infty (\\Xi )$ is dense in $L^2(\\Xi )$ , it then follows that $\\mathcal {F}^{-1}(C_c^\\infty (\\Omega ))$ is dense in the product $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ .","We will actually show a little more than the claim.","Namely, every $g \\in C_c^\\infty (\\Omega )$ can be written $g = \\sum _{k} g_k * h_k, \\quad g_k, h_k \\in L^2(\\Xi ),$ in such a way that the corresponding map $g \\mapsto \\sum _k \\Vert g_k\\Vert _{L^2(\\Xi )} \\Vert h_k\\Vert _{L^2(\\Xi )}$ is continuous from $C_c^\\infty (\\Omega )$ , equipped with the usual test function topology, to $\\mathbb {R}$ .","By employing a partition of unity in which each member is compactly supported in a cube, it is sufficient to prove the claim when $\\Xi = (0, 1/2)^d$ .","For this we employ Fourier series.","Let $\\lambda (t) = 1/2 - |t-1/2|$ , $t \\in [0,1]$ , and let $\\Lambda (x) = \\prod _{i=1}^d \\lambda (x_i), \\quad x \\in (0,1)^d.$ Note that $\\lambda $ is in the Wiener algebra $A([0,1])$ , the space of functions on $[0,1]$ with absolutely convergent Fourier series, equipped with pointwise multiplication.","Therefore $\\Lambda $ is in the Wiener algebra $A([0,1]^d)$ , since $\\Lambda $ is a tensor power of $\\lambda $ .","Since $g \\in C_c^\\infty ((0,1)^d)$ and $\\Lambda $ is non-zero on compact subsets of $(0,1)^d$ it follows by Wiener's lemma [19] that $g/\\Lambda \\in A([0,1]^d)$ (to apply Wiener's lemma, first modify $\\Lambda $ to be nonzero outside the support of $g$ ).","Expanding $g/\\Lambda $ in a Fourier series, $(g/\\Lambda )(x) = \\sum _{k \\in {\\mathbb {Z}}^d} a_k e^{i2\\pi k \\cdot x}, \\quad \\sum _{k \\in {\\mathbb {Z}}^d} |a_k| < \\infty , \\quad x \\in [0,1]^d,$ let $h_k(x) = e^{i2\\pi k \\cdot x} \\chi _{(0,1/2)^d}(x)$ , $g_k = a_k h_k$ .","Then a computation shows that $(g_k * h_k)(x) = a_k e^{i2\\pi k \\cdot x} \\Lambda (x), \\quad x \\in (0,1)^d,$ so that $g = \\sum _{k \\in {\\mathbb {Z}}^d} g_k * h_k, \\quad \\sum _{k \\in {\\mathbb {Z}}^d} \\Vert g_k\\Vert _{L^2((0,1/2)^d)} \\Vert h_k\\Vert _{L^2((0,1/2)^d)} < \\infty .$ An inspection of the argument shows that the $g \\mapsto g/\\Lambda $ is continuous from $C_c^\\infty ((0,1)^d)$ to $A([0,1]^d)$ , and therefore $g \\mapsto \\sum _k \\Vert g_k\\Vert _{L^2((0,1/2)^d)} \\Vert h_k\\Vert _{L^2((0,1/2)^d)}$ is continuous on $C_c^\\infty ((0,1)^d)$ as promised.","Suppose now that $\\mu \\in (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ .","We have just demonstrated that $( f, g ) = \\mu (\\mathcal {F}^{-1}g)$ , $g \\in C_c^\\infty (\\Omega )$ , defines a distribution on $\\Omega $ .","Hence we may consider the Hankel operator $\\Gamma _{f, \\Xi }$ .","For $g, h \\in C_c^\\infty (\\Xi )$ we have that $ \\langle \\Gamma _{f}g, h \\rangle _{L^2(\\Xi )} = ( f, g * \\bar{h} ) = \\mu (\\mathcal {F}^{-1}g \\cdot \\mathcal {F}^{-1}\\bar{h}).$ Since $\\mu $ is a bounded functional on $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ we conclude that $|\\langle \\Gamma _{f}g, h \\rangle _{L^2(\\Xi )}| \\le \\Vert \\mu \\Vert \\Vert \\mathcal {F}^{-1} g\\Vert _{\\operatorname{PW}_\\Xi } \\Vert \\mathcal {F}^{-1}\\bar{h}\\Vert _{\\operatorname{PW}_\\Xi } = \\Vert \\mu \\Vert \\Vert g\\Vert _{L^2(\\Xi )} \\Vert h\\Vert _{L^2(\\Xi )},$ that is, $\\Gamma _{f, \\Xi }$ is bounded, and in fact $\\Vert \\Gamma _f\\Vert = \\Vert \\mu \\Vert $ .","Conversely, if $f$ is a distribution on $\\Omega $ such that $\\Gamma _{f, \\Xi }$ is bounded, it is clear that $f$ induces a bounded functional $\\mu $ on $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ by (REF ).","This proves that $X$ is isometrically isomorphic to the Banach space $(\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ , which also entails that $X$ is closed, completing the proof.","In the remainder of this section we assume that $\\Xi $ is a convex polytope.","Next we prove Theorem REF under the additional assumption that $f$ is supported around one simple vertex of $\\Omega $ .","Proposition 5.2 Let $\\Xi \\subset {\\mathbb {R}}^d$ be an open convex polytope, $x$ a simple vertex of $\\Omega =2\\Xi $ , and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ be such that $\\operatorname{supp}f \\cap \\partial _{\\operatorname{far},x} \\Omega = \\emptyset $ .","If $\\Gamma _f$ is bounded as an operator on $L^2(\\Xi )$ , then there exists a $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ As in Lemma REF , let $A$ be an affine transformation with origin $x$ such that $A({\\mathbb {R}}_+^d)$ coincides with $\\Xi $ in a neighborhood of $x$ .","It is straightforward to verify that it suffices to prove the proposition for $\\Gamma _{f\\circ A, A^{-1}(\\Xi )}$ .","Since $A^{-1}(\\Xi )$ is also a convex polytope, we may hence assume that $x=0$ and that $\\Omega $ coincides with ${\\mathbb {R}}_+^d$ in a neighborhood $U$ of $\\operatorname{supp}f$ , $\\overline{U} \\cap \\partial _{far,0} \\Omega = \\emptyset $ .","In particular, since $\\Omega $ is a convex polytope, we have that $\\Omega \\subset \\mathbb {R}_+^d$ .","Since $\\operatorname{supp}f \\subset U \\cap \\overline{\\Omega }$ and $\\overline{U} \\cap \\partial _{far,0} \\Omega = \\emptyset $ , we can extend $f$ to a distribution on all of ${\\mathbb {R}}_+^d$ by letting it be zero outside $\\Omega $ .","Our strategy is to show that the operator $\\Gamma _{f,{\\mathbb {R}}_+^d}$ is bounded and to then apply Theorem REF .","For $n\\in {\\mathbb {N}}^d$ let $C_n$ denote the cube $(n_1,n_1+1)\\times \\ldots \\times (n_d,n_d+1)$ .","For a set $X \\subset {\\mathbb {R}}_+^d$ , let $P_X \\colon L^2({\\mathbb {R}}_+^d) \\rightarrow L^2({\\mathbb {R}}_+^d)$ denote the orthogonal projection of $L^2({\\mathbb {R}}_+^d)$ onto $L^2(X)$ , and let $r>0$ be such that $2\\sqrt{d}r<\\operatorname{dist}(U \\cap \\overline{\\Omega },\\partial _{far,0}\\Omega ).$ By considering test functions $g \\in C_c^\\infty ({\\mathbb {R}}_+^d)$ such that $\\operatorname{supp}g \\cap \\overline{rC_m} \\subset rC_m$ for every $m$ , we give meaning to the equality $\\Gamma _{f,{\\mathbb {R}}_+^d}=\\left(\\sum _{n\\in {\\mathbb {N}}^d}P_{rC_n}\\right)\\Gamma _{f,{\\mathbb {R}}_+^d} \\left(\\sum _{m\\in {\\mathbb {N}}^d}P_{rC_m}\\right)=\\sum _{m,n\\in {\\mathbb {N}}^d}P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m},$ a term $P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m}$ being non-zero only if $(rC_m+rC_n)\\cap \\operatorname{supp}f \\ne \\emptyset .$ Hence there are only finitely many non-zero terms in the decomposition.","Since $\\Vert P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m}\\Vert =\\Vert \\Psi _{f,rC_m,rC_n}\\Vert ,$ recalling the definition of $\\Psi _f$ from Section , it therefore suffices to prove that $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert $ is bounded whenever (REF ) holds.","If $rC_m,rC_n\\subset \\Xi $ there is nothing to prove since $\\Gamma _{f,\\Xi }$ is bounded by hypothesis.","For the other terms, note that (REF ), $\\operatorname{supp}f \\subset U \\cap \\overline{\\Omega }$ , and the choice of $r$ implies that $rC_m+rC_n\\subset \\Omega ,$ since $2\\sqrt{d} r$ is the diameter of $rC_m+rC_n$ .","For any $z\\in {\\mathbb {R}}^d$ , $x \\in rC_n$ , and $g \\in C_c^\\infty (rC_m)$ we have that $\\Psi _{f,rC_m,rC_n}(g)(x)=\\int _{rC_m} f(x+y)g(y) \\, dy=\\int _{rC_m+z} f(x+(y-z))g(y-z)dy,$ and hence $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert =\\Vert \\Psi _{f,rC_m+z,rC_n-z}\\Vert .$ In particular, for $z=r(n-m)/2$ we obtain that $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert =\\Vert \\Psi _{f,rC_{\\frac{m+n}{2}},rC_{\\frac{m+n}{2}}}\\Vert .$ However, $2rC_{\\frac{m+n}{2}}=rC_m+rC_n$ so by (REF ) we conclude that $rC_{\\frac{m+n}{2}}\\subset \\Xi $ .","The desired boundedness now follows as it did in the first case considered.","We have just demonstrated that $\\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert < \\infty $ .","By Theorem REF there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}^d_+} = f$ .","This in particular implies that $\\hat{b}|_{\\Omega } = f$ when we return to the initial interpretation of $f$ as a distribution on $\\Omega $ .","Theorem 5.3 Let $\\Xi $ be a simple convex polytope, and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =2\\Xi $ .","Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ Assume that $\\Gamma _f$ is bounded.","Let $\\lbrace x_j\\rbrace _{j=1}^J$ be the vertices of $\\Omega $ , and let $\\lbrace \\mu _j\\rbrace _{j=1}^J$ be partition of unity as in Lemma REF .","For $\\varphi \\in C_c^\\infty (\\Xi )$ and $x \\in \\Xi $ we have that $\\Gamma _{\\mu _j f}(\\varphi )(x)=\\int _{\\Xi } \\int _{{\\mathbb {R}}^d} \\hat{\\mu }_j(\\xi )e^{2\\pi i (x+y)\\cdot \\xi }\\,d\\xi \\, f(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}^d} \\hat{\\mu }_j(\\xi ){e^{2\\pi i \\xi \\cdot x}}\\Gamma _{f}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y\\cdot \\xi }\\varphi (y)$ .","Hence, $\\Gamma _{\\mu _j f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded, $\\Vert \\Gamma _{\\mu _j f}\\Vert \\le \\Vert \\hat{\\mu }_j\\Vert _{L^1} \\Vert \\Gamma _f\\Vert .$ Therefore, by Proposition REF there are functions $b_j \\in L^\\infty $ such that $\\mu _jf=\\hat{b}_j|_\\Omega $ .","Thus $f=\\hat{b}|_\\Omega $ , where $b = \\sum _{j=1}^J b_j \\in L^\\infty $ .","Conversely, if $f=\\hat{b}|_\\Omega $ , where $b \\in L^\\infty $ , then $\\Gamma _f$ is bounded by Proposition REF .","The constant $c$ now arises from abstract reasoning.","Consider the Banach space $X = \\left\\lbrace \\Gamma _{f, \\Xi } \\, : \\, \\Vert \\Gamma _{f, \\Xi } \\Vert < \\infty \\right\\rbrace $ of Proposition REF .","We have just shown that $b \\mapsto \\Gamma _{\\hat{b} |_{\\Omega }, \\Xi }$ is a map of $L^\\infty $ onto $X$ .","The open mapping theorem hence guarantees the existence of $c$ .","We immediately obtain the corresponding result for Toeplitz operators, when $\\Xi $ is a simple convex polytope which, possibly after a translation, is symmetric under $x \\mapsto -x$ .","Corollary 5.4 Let $\\Xi $ be a simple convex polytope such that for some $z \\in {\\mathbb {R}}^d$ it holds that $\\Xi +z = -\\Xi - z$ .","Let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =\\Xi - \\Xi = 2\\Xi + 2z$ .","Then $\\Theta _f$ is bounded if and only if there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .","There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Theta _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ In this case $\\Theta _{f} g = \\Gamma _{\\tilde{f}} \\tilde{g}$ , where $\\tilde{f}(x) = f(x + 2z)$ , $x \\in 2\\Xi $ , and $\\tilde{g}(x) = g(-x-2z)$ , $x \\in \\Xi $ .","Hence the result follows from Theorem REF .","We also deduce the weak factorization result for $\\operatorname{PW}_\\Omega ^1$ , see Section REF .","Corollary 5.5 Let $\\Xi $ be a simple convex polytope, and let $\\Omega = 2\\Xi $ .","Then $PW_{\\Omega }^1 = \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ The norms of these Banach spaces are equivalent.","By Cauchy-Schwarz, the inclusion $I \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow PW_{\\Omega }^1$ is bounded.","Since $I$ has dense range by Proposition REF , the adjoint $I^* \\colon (\\operatorname{PW}^1_\\Omega )^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ has empty kernel.","Suppose $\\mu \\in (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ .","Note that $CG(x) = \\overline{G(-x)}$ defines an anti-linear isometric involution $C \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ .","This induces an anti-linear isometric involution $D \\colon (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ , $D\\mu (G) = \\overline{\\mu (CG)}, \\quad G \\in \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ According to Proposition REF , $( f, g ) = \\mu (\\mathcal {F}^{-1}g)$ , $g \\in C_c^\\infty (\\Omega )$ , defines a distribution on $\\Omega $ such that $\\Vert \\Gamma _{f,\\Xi }\\Vert = \\Vert \\mu \\Vert $ .","By Theorem REF , there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .","Since $\\operatorname{PW}_\\Omega ^1 \\subset L^1({\\mathbb {R}}^d)$ , we can interpret $b$ as an element of $(\\operatorname{PW}_\\Omega ^1)^*$ , $b(G) = \\langle G, b \\rangle _{L^2({\\mathbb {R}}^d)}$ .","Then, recalling that $JG(x) = G(-x)$ , we have that $DI^* b (G) = (b, JG) = ( f, \\mathcal {F}^{-1} JG ) = ( f, \\mathcal {F} G ) = \\mu (G), \\quad G \\in \\mathcal {F}^{-1}(C_c^\\infty (\\Omega )),$ that is, $DI^*b = \\mu $ , or $I^*b = D\\mu $ .","Since $D$ is an involution, it follows that $I^*$ is onto.","In other words, $I^* \\colon (\\operatorname{PW}^1_\\Omega )^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ is a Banach space isomorphism, and therefore the inclusion $I \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow \\operatorname{PW}^1_\\Omega $ is as well.","Hence, $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } = \\operatorname{PW}^1_\\Omega ,$ and the norms of these two Banach spaces are equivalent, by the open mapping theorem.","The method used to prove Theorem REF extends to many unbounded polyhedral sets.","Instead of pursuing a general statement, let us consider the example of a strip in ${\\mathbb {R}}^2$ , $ \\Xi = {\\mathbb {R}}_+ \\times (0,1).$ This is an interesting addition to Theorem REF , since $\\Xi $ does not have a simple vertex at infinity.","In fact, $\\partial \\Xi $ may be considered to have a cusp point there.","Proposition 5.6 Let $\\Xi $ be the strip defined in (REF ), and let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega = 2\\Xi $ .","Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ Let $\\nu _1, \\nu _2 \\in C_c^\\infty ({\\mathbb {R}})$ be functions such that $\\nu _1(t) + \\nu _2(t) = 1$ for $t \\in [0,2]$ , $\\nu _1$ vanishes in a neighborhood of 2, and $\\nu _2$ vanishes in a neighborhood of 0.","Let $\\mu _j(x) = \\nu _j(x_2), \\quad j=1,2, \\; x = (x_1, x_2) \\in \\Omega .$ Then for $\\varphi \\in C_c^\\infty (\\Xi )$ and $x = (x_1, x_2) \\in \\Xi $ we have that $\\Gamma _{\\mu _j f}(\\varphi )(x)=\\int _{\\Xi } \\int _{{\\mathbb {R}}} \\hat{\\nu }_j(\\xi )e^{2\\pi i (x_2+y_2) \\xi }\\,d\\xi \\, f(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}} \\hat{\\nu }_j(\\xi ){e^{2\\pi i x_2\\xi }}\\Gamma _{f}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y_2 \\xi }\\varphi (y)$ , $y = (y_1, y_2) \\in \\Xi $ .","Hence, as before we see that $ \\Vert \\Gamma _{\\mu _j f, \\, \\Xi }\\Vert \\le \\Vert \\hat{\\nu }_j\\Vert _{L^1} \\Vert \\Gamma _{f, \\, \\Xi } \\Vert , \\quad j=1,2.$ As in Proposition REF and Theorem REF it is sufficient to see that $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ and $\\Gamma _{\\mu _2 f} \\colon L^2({\\mathbb {R}}_+ \\times (-\\infty , 1) ) \\rightarrow L^2({\\mathbb {R}}_+ \\times (-\\infty , 1))$ define bounded operators, and by symmetry it is sufficient to consider the first of the two.","For $n \\in {\\mathbb {N}}$ , let $S_n$ denote the strip ${\\mathbb {R}}_+ \\times (n, n+1)$ , and let $r > 0$ be such that $2r < \\operatorname{dist}([0,2] \\cap \\operatorname{supp}\\nu _1, 2).$ We decompose $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ according to strips instead of cubes, $\\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} = \\sum _{m,n \\in {\\mathbb {N}}} P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m}.$ There are only a finite number of non-zero terms in this decomposition, and for any such term we by our choice of $r$ that $ rS_m + rS_n \\subset \\Omega .$ For $n,m$ corresponding to a non-zero term, we have that $\\Vert P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m} \\Vert = \\Vert \\Psi _{\\mu _1f, rS_m, rS_n}\\Vert = \\Vert \\Psi _{\\mu _1f, rS_m + z, rS_n - z}\\Vert = \\Vert \\Psi _{\\mu _1f, rS_{\\frac{m+n}{2}}, rS_{\\frac{m+n}{2}}}\\Vert ,$ where $z = (0, r(n-m)/2)$ .","Since $rS_{\\frac{m+n}{2}} \\subset \\Xi $ by (REF ) and $\\Gamma _{\\mu _1 f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded by (REF ), we conclude that each non-zero term $P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m}$ is bounded.","Hence $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ is bounded, finishing the proof."],["General domain Toeplitz operators","In this section we consider general domain Toeplitz operators on open convex domains $\\tilde{\\Xi }\\subset {\\mathbb {R}}^d$ such that both $\\operatorname{cc}_{\\tilde{\\Xi }}$ and $\\operatorname{bc}_{\\tilde{\\Xi }}$ have non-empty interior (as in the classical case $\\tilde{\\Xi }={\\mathbb {R}}_+$ ).","This forces $\\tilde{\\Xi }$ to be unbounded and, as we shall soon see, it also entails that $\\tilde{\\Omega }=\\tilde{\\Xi }-\\tilde{\\Xi }= {\\mathbb {R}}^d$ .","We shall also consider more general open connected sets $\\Xi $ such that there are points $x_0$ and $x_1$ for which $x_1+\\tilde{\\Xi }\\subset \\Xi \\subset x_0+\\tilde{\\Xi },$ and prove that $\\Vert \\Theta _{f, \\Xi }\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ under this hypothesis.","This allows for domains $\\Xi $ with very irregular boundaries, in sharp contrast to Theorem REF .","The corresponding class of operators $\\Theta _{f,\\Xi }$ partially extends the class of generalized Toeplitz operators considered in [22], see Section REF .","The next theorem can also be recovered by verifying the hypotheses of and keeping track of the constants in the proof of [22].","However, for completeness we prefer to give our own concrete proof.","Theorem 6.1 Let $\\Xi $ be a set as above.","Then $\\Xi -\\Xi ={\\mathbb {R}}^d$ and, for $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ , we have that $\\Theta _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if $f\\in {\\mathcal {F}}^{-1}(L^\\infty )$ .","Moreover, $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ .","Fix $z\\in {\\mathbb {R}}^d$ and set $|z|=R$ .","Pick a vector $e\\in \\operatorname{int}(\\operatorname{cc}_{{\\tilde{\\Xi }}})$ with distance greater than $R$ to the complement of $\\operatorname{cc}_{\\tilde{\\Xi }}$ , which is possible since $\\operatorname{cc}_{{\\tilde{\\Xi }}}$ is a cone with non-empty interior.","Then $e+z\\in \\operatorname{cc}_{{\\tilde{\\Xi }}}$ , so for any $x \\in \\tilde{\\Xi }$ we have that $x_1+x+e+z\\in x_1+\\tilde{\\Xi }\\subset \\Xi $ .","Similarly, $x_1+x+e\\in \\Xi $ .","Since $z$ is the difference of these two vectors, the first claim follows.","Suppose that we have proven the theorem for all $f \\in C_c^\\infty ({\\mathbb {R}}^d)$ .","If $f$ is a general symbol for which $\\Theta _f$ is bounded, consider the sequence of functions $f_n \\in C_c^\\infty ({\\mathbb {R}}^d)$ from Proposition REF .","Then $\\hat{f}_n$ has, by Alaoglu's theorem, a subsequence $\\hat{f}_{n_k}$ which converges weak-star in $L^\\infty $ to some element $g$ .","Since $f_n$ converges to $f$ in distribution, it must be that $g = \\hat{f}$ .","Hence $f\\in {\\mathcal {F}}^{-1}(L^\\infty )$ and, by Propositions REF and REF , we have that $\\Vert \\hat{f} \\Vert _{L^\\infty } \\le \\varlimsup _{k \\rightarrow \\infty } \\Vert \\hat{f}_{n_k} \\Vert _{L^\\infty } = \\varlimsup _{k \\rightarrow \\infty } \\Vert \\Theta _{f_{n_k}}\\Vert \\le \\Vert \\Theta _{f}\\Vert \\le \\Vert \\hat{f} \\Vert _{L^\\infty }.$ This proves the theorem for general symbols.","Hence we assume that $f \\in C_c^\\infty ({\\mathbb {R}}^d)$ .","Fix $\\xi \\in {\\mathbb {R}}^d$ , pick any vector $\\nu $ in $\\operatorname{int}(\\operatorname{bc}_{\\tilde{\\Xi }})$ , and consider for $\\varepsilon > 0$ the function $E_{\\varepsilon }(x)=e^{\\varepsilon x\\cdot \\nu +2\\pi i x\\cdot \\xi }\\chi _{\\Xi }(x), \\quad x \\in \\Xi .$ By [1] this function is in $L^2(x_0+\\tilde{\\Xi })$ ,The set $\\operatorname{bc}_{\\tilde{\\Xi }}$ was denoted $\\Theta $ in [1].","and hence $E_\\varepsilon \\in L^2(\\Xi )$ .","We use $E_\\varepsilon $ as a test function: $\\Vert \\Theta _f\\Vert \\ge \\left|\\frac{\\langle \\Theta _f E_{\\varepsilon },E_{\\varepsilon }\\rangle }{\\Vert E_\\varepsilon \\Vert ^2}\\right| &= \\left|\\frac{1}{\\Vert E_\\varepsilon \\Vert ^2}\\int \\int f(x-y)e^{\\varepsilon (x+y)\\cdot \\nu }e^{2\\pi i (y-x)\\cdot \\xi }\\chi _{\\Xi }(y)\\chi _{\\Xi }(x) \\, dy \\, dx\\right| \\\\&=\\left|\\frac{1}{\\Vert E_\\varepsilon \\Vert ^2}\\int f(z)e^{-2\\pi i z\\cdot \\xi }\\int e^{\\varepsilon (z+2y) \\cdot \\nu }\\chi _{\\Xi }(z+y)\\chi _{\\Xi }(y) \\, dy \\,dz\\right|$ Hence it follows that $\\Vert \\Theta _f\\Vert \\ge |\\hat{f}(\\xi )|$ upon showing that $\\lim _{\\varepsilon \\rightarrow 0^+} \\frac{e^{\\varepsilon z \\cdot \\nu }}{\\Vert E_\\varepsilon \\Vert ^2} \\int e^{2\\varepsilon y \\cdot \\nu }\\chi _{\\Xi }(z+y)\\chi _{\\Xi }(y) \\, dy = 1$ uniformly on compacts in $z$ .","Since $\\xi $ is arbitrary this establishes that $\\Vert \\Theta _f\\Vert \\ge \\Vert \\hat{f}\\Vert _{L^\\infty }$ and by Proposition REF we then conclude that $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ .","Fix $R>0$ and suppose that $z \\in {\\mathbb {R}}^d$ with $|z|1,$ each multi-sequence $a=(a_n)_{n\\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d},$ generates a multi-level block Toeplitz matrix $T_a$ .","As an operator on $\\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace ^d)$ it is given by the formula $T_a(v)(m)=\\sum _{n\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}a_{m-n}v_n, \\quad v \\in \\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace ^d), \\; m \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d.$ To understand this matrix, consider the $d$ -level block Toeplitz matrix $T_a$ as an ordinary $N \\times N$ -Toeplitz matrix with entries which are $(d-1)$ -level block Toeplitz matrices, $T_a = \\lbrace A_{i-j}\\rbrace _{i,j \\in \\lbrace 0,\\ldots ,N-1\\rbrace }, \\quad A_i = \\lbrace a_{(i,m-n)}\\rbrace _{m,n \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^{d-1}}.$ For instance, a multi-level block Toeplitz matrix for $d=2$ is an $N\\times N$ Toeplitz matrix whose entries are $N \\times N$ Toeplitz matrices.","Again, we allow for the possibility that $N=\\infty $ .","We now provide the multi-level block Toeplitz matrix analogue of the Toeplitz matrix completion theorem.","Theorem 7.1 There exists a constant $C_d > 0$ such that any finite multi-sequence $a$ can be extended to an infinite multi-sequence $\\tilde{a}$ on ${\\mathbb {Z}}^d$ for which $T_{\\tilde{a}} \\colon \\ell ^2({\\mathbb {N}}^d) \\rightarrow \\ell ^2({\\mathbb {N}}^d)$ is bounded with norm $\\Vert T_{\\tilde{a}}\\Vert \\le C_d\\Vert T_a\\Vert .$ Let $f=\\sum _{n\\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d} a_n\\delta _n,$ where $\\delta _n$ is the Dirac delta function at $n$ , $\\delta _n(\\varphi )=\\varphi (n), \\quad \\varphi \\in C_c^\\infty ({\\mathbb {R}}^d).$ Set $\\Xi =(0,N)^d$ and consider $\\Theta _f=\\Theta _{f,\\Xi }$ .","Given $g\\in C^\\infty _c(\\Xi )$ , a short calculation shows that $\\Theta _f(g)(x)=\\sum _{n\\in {\\mathbb {Z}}^d\\cap (x-\\Xi )} a_n g(x-n), \\quad x \\in (0, N)^d.$ With $x=m+r$ , where $m \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d$ and $r\\in [0,1)^d$ , this can be rewritten $\\Theta _f(g)(m+r)=\\sum _{k\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d} a_{m-k}g(r+k).$ In other words, with $g_r=\\lbrace g(r+n)\\rbrace _{n\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}$ , we have that $\\Theta _f(g)(m+r)=T_a(g_r)(m).$ Hence $\\sum _{m\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}|\\Theta _f(g)(m+r)|^2=\\Vert T_a(g_r)\\Vert ^2\\le \\Vert T_a\\Vert ^2\\Vert g_r\\Vert ^2=\\Vert T_a\\Vert ^2\\sum _{m\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}|g(m+r)|^2.$ Integrating both sides over $r \\in (0,1)^d$ gives us that $\\Vert \\Theta _f(g)\\Vert ^2\\le \\Vert T_a\\Vert ^2\\Vert g\\Vert ^2$ .","In other words, $\\Theta _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded and $\\Vert \\Theta _f\\Vert \\le \\Vert T_a\\Vert .$ Noting that the constant $c$ in Corollary REF is invariant under homotheties, we find that there exists a distribution $\\tilde{f} = \\hat{b} \\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ , coinciding with $f$ on $(-N,N)^d$ , such that $\\Vert \\Theta _{\\tilde{f},{\\mathbb {R}}^d}\\Vert \\le C_d\\Vert T_a\\Vert ,$ where $C_d$ only depends on the dimension $d$ .","Of course, $\\Theta _{\\tilde{f},{\\mathbb {R}}^d} \\colon L^2({\\mathbb {R}}^d) \\rightarrow L^2({\\mathbb {R}}^d)$ is nothing but the operator of convolution with $\\tilde{f}$ .","Now pick any function $\\varphi \\in C^\\infty _c((-1/2,1/2)^d)$ with $\\int |\\varphi |^2 dx=1$ and consider the isometry $I \\colon \\ell ^2({\\mathbb {N}}^d)\\rightarrow L^2({\\mathbb {R}}^d)$ given by $I v(x)=\\sum _{n\\in {\\mathbb {N}}^d}v_n \\varphi (x-n), \\quad v \\in \\ell ^2({\\mathbb {N}}^d), \\; x \\in {\\mathbb {R}}^d.$ Then $I^* g(n) = \\int _{{\\mathbb {R}}^d} g(x) \\overline{\\varphi (x-n)} \\, dx, \\quad g \\in L^2({\\mathbb {R}}^d), \\; n \\in {\\mathbb {N}}^d.$ It follows that $I^*\\Theta _{\\tilde{f}} I v (m) = \\sum _{n \\in {\\mathbb {N}}^d} \\tilde{a}_{m-n} v_n, \\quad v \\in \\ell ^2({\\mathbb {N}}^d), \\; m \\in {\\mathbb {N}}^d,$ where $\\tilde{a}_n = \\int _{{\\mathbb {R}}^d} \\int _{{\\mathbb {R}}^d} f(x-y + n) \\varphi (y) \\overline{\\varphi (x)} \\, dy \\, dx, \\quad n \\in {\\mathbb {Z}}^d.$ That is, $I^*\\Theta _{\\tilde{f}}I = T_{\\tilde{a}}$ .","It is clear by construction that $\\tilde{a}$ is an extension of $a$ , $\\tilde{a}_n = a_n \\int _{{\\mathbb {R}}^d} |\\varphi (y)|^2 \\, dy = a_n, \\quad n \\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d.$ This finishes the proof."]],"string":"[\n [\n \"Nehari's theorem for convex domain Hankel and Toeplitz operators in\\n several variables\"\n ],\n [\n \"Abstract We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables.\",\n \"A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows.\",\n \"Let $\\\\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\\\\Xi$, consider the Hankel operator $$\\\\Gamma_f (g)(x)=\\\\int_{\\\\Xi} f(x+y) g(y) \\\\, dy, \\\\quad x \\\\in\\\\Xi.$$ Then $\\\\Gamma_f$ extends to a bounded operator on $L^2(\\\\Xi)$ if and only if there is a bounded function $b$ on $\\\\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2\\\\Xi$.\",\n \"This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix.\",\n \"In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.\"\n ],\n [\n \"Introduction\",\n \"For an open connected set $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ , $d \\\\ge 1$ , let $\\\\Omega = \\\\Xi + \\\\Xi = \\\\lbrace x + y \\\\, : \\\\, x\\\\in \\\\Xi , \\\\, y\\\\in \\\\Xi \\\\rbrace ,$ and consider a distribution $f$ defined on $\\\\Omega $ .\",\n \"The associated general domain Hankel operator $\\\\Gamma _f = \\\\Gamma _{f, \\\\Xi }$ is the (densely defined) operator $\\\\Gamma _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ , given by $\\\\Gamma _f (g)(x)=\\\\int _{\\\\Xi } f(x+y) g(y) \\\\, dy, \\\\quad x \\\\in \\\\Xi ,$ where $dy$ is the Lebesgue measure on ${\\\\mathbb {R}}^d$ .\",\n \"The case $\\\\Xi = {\\\\mathbb {R}}_+ = (0, \\\\infty )$ for $d=1$ corresponds to the class of usual Hankel operators; when represented in the appropriate basis of $L^2({\\\\mathbb {R}}_+)$ , the operator $\\\\Gamma _{f, {\\\\mathbb {R}}_+}$ is realized as an infinite Hankel matrix $\\\\lbrace a_{n+m}\\\\rbrace _{n,m =0}^\\\\infty $ [31].\",\n \"Nehari's theorem characterizes the bounded Hankel operators $\\\\Gamma _f \\\\colon L^2({\\\\mathbb {R}}_+) \\\\rightarrow L^2({\\\\mathbb {R}}_+)$ .\",\n \"For a function $g$ on ${\\\\mathbb {R}}^d$ , we let $\\\\hat{g} = \\\\mathcal {F}g$ denote its Fourier transform, $\\\\hat{g}(\\\\xi ) = \\\\mathcal {F}g(\\\\xi ) = \\\\int _{\\\\mathbb {R}^d} g(x) e^{-2\\\\pi i x \\\\cdot \\\\xi } \\\\, dx, \\\\quad \\\\xi \\\\in \\\\mathbb {R}^d.$ Theorem (Nehari [25]) Suppose that $f$ is a distribution in ${\\\\mathbb {R}}_+$ , $f\\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}_+)$ .\",\n \"Then $\\\\Gamma _f \\\\colon L^2(\\\\mathbb {R}_+) \\\\rightarrow L^2(\\\\mathbb {R}_+)$ is bounded if and only if there exists a function $b\\\\in L^\\\\infty ({\\\\mathbb {R}})$ such that $\\\\hat{b}|_{{\\\\mathbb {R}}_+}=f$ .\",\n \"Moreover, it is possible to choose $b$ so that $ \\\\Vert \\\\Gamma _f\\\\Vert = \\\\Vert b\\\\Vert _{L^\\\\infty }.$ Nehari's theorem is canonical in operator theory.\",\n \"The two most common proofs proceed either by factorization in the single variable Hardy space or by making use of the commutant lifting theorem.\",\n \"For $d > 1$ , the operators $\\\\Gamma _{f, {\\\\mathbb {R}}_+^d}$ , $\\\\Xi = {\\\\mathbb {R}}_+^d$ , correspond to (small) Hankel operators on the product domain multi-variable Hardy space $H^2_d$ .\",\n \"In this case, the analogue of Nehari's theorem remains true, apart from (REF ), but it is significantly more difficult to prove.\",\n \"It was established by Ferguson and Lacey ($d=2$ ) and Lacey and Terwilleger ($d > 2$ ) [18], [23].\",\n \"A precise statement is given in Theorem REF .\",\n \"The main purpose of this article is to prove Nehari's theorem when $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ is a simple convex polytope.\",\n \"t1altTheorem REF Let $\\\\Xi $ be a simple convex polytope, and let $f\\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ where $\\\\Omega =2\\\\Xi $ .\",\n \"Then $\\\\Gamma _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded if and only if there is a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b} |_{\\\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\\\Xi $ , such that $b$ can be chosen to satisfy $c\\\\Vert b\\\\Vert _{L^\\\\infty } \\\\le \\\\Vert \\\\Gamma _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ When $d = 1$ , the only open connected sets $\\\\Xi \\\\subset {\\\\mathbb {R}}$ are the intervals $\\\\Xi = I$ .\",\n \"In this case, Theorem REF is due to Rochberg [35], who called the corresponding operators $\\\\Gamma _{f, I}$ Hankel/Toeplitz operators on the Paley–Wiener space.\",\n \"They have also been called Wiener–Hopf operators on a finite interval [30].\",\n \"These operators have inspired a wealth of theory in the single variable setting – see Section REF , where we shall interpret Theorem REF in the context of Paley–Wiener spaces.\",\n \"Even for $d=1$ , our proof of Theorem REF appears to be new.\",\n \"However, in several variables our proof relies on the Nehari theorem of Ferguson–Lacey–Terwilleger, and can therefore not be used to give a new proof of their results.\",\n \"We shall also consider general domain Toeplitz operators $\\\\Theta _f=\\\\Theta _{f, \\\\Xi } \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ .\",\n \"In this context, $f$ is a distribution defined on $\\\\Omega = \\\\Xi - \\\\Xi $ , and $\\\\Theta _f$ is densely defined via $\\\\Theta _f(g)(x) = \\\\int _{\\\\Xi } f(x-y) g(y) \\\\, dy, \\\\quad x \\\\in \\\\Xi .$ If $\\\\Xi $ after a translation is invariant under the reflection $x \\\\mapsto -x$ , then the classes of Hankel operators $\\\\Gamma _{f, \\\\Xi }$ and Toeplitz operators $\\\\Theta _{\\\\widetilde{f}, \\\\Xi }$ are essentially the same, and Theorem REF immediately yields a boundedness result.\",\n \"This reasoning is applicable to the cube $\\\\Xi = (0,1)^d$ , for example.\",\n \"t2Corollary REF Let $\\\\Xi $ be a simple convex polytope such that for some $z \\\\in {\\\\mathbb {R}}^d$ it holds that $\\\\Xi +z = -\\\\Xi - z$ .\",\n \"Let $f \\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ , $\\\\Omega =\\\\Xi - \\\\Xi = 2\\\\Xi + 2z$ .\",\n \"Then $\\\\Theta _f$ is bounded if and only if there exists a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{\\\\Omega } = f$ .\",\n \"There exists a constant $c > 0$ , depending on $\\\\Xi $ , such that $b$ can be chosen to satisfy $c\\\\Vert b\\\\Vert _{L^\\\\infty } \\\\le \\\\Vert \\\\Theta _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ On the other hand, when $\\\\Xi $ is a proper convex unbounded set, containing an open cone say, it is clear that the boundedness characterizations of $\\\\Theta _{f, \\\\Xi }$ and $\\\\Gamma _{f,\\\\Xi }$ may be completely different; plainly explained by the fact that $\\\\Omega = \\\\Xi - \\\\Xi = {\\\\mathbb {R}}^d$ in the Toeplitz case, while $\\\\Omega = \\\\Xi + \\\\Xi = 2\\\\Xi \\\\subsetneq {\\\\mathbb {R}}^d$ for Hankel operators.\",\n \"In this setting, identifying the boundedness of $\\\\Theta _f$ carries none of the subtleties of Nehari-type theorems.\",\n \"In Theorem REF we obtain the expected boundedness result for a class of “cone-like” domains $\\\\Xi $ .\",\n \"Rather than giving a precise statement here, let us record the following corollary of Theorem REF .\",\n \"cor:exCorollary REF Let $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ be any open connected domain such that $(1,\\\\infty )^d \\\\subset \\\\Xi \\\\subset (0,\\\\infty )^d,$ and let $f \\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d)$ .\",\n \"Then $\\\\Theta _{f} \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded if and only if $f$ is a tempered distribution and $\\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty ({\\\\mathbb {R}}^d)} < \\\\infty $ , and in this case $\\\\Vert \\\\Theta _f\\\\Vert = \\\\Vert \\\\hat{f} \\\\Vert _{L^\\\\infty }.$ In the final part of the paper we shall give an application of Theorem REF to matrix completion theory, essentially obtained by discretizing Corollary REF when $\\\\Xi $ is a cube.\",\n \"To avoid introducing further notation, we shall only state the result in words for now.\",\n \"Recall that a Toeplitz matrix is one whose diagonals are constant.\",\n \"An $N \\\\times N$ $d$ -multilevel block Toeplitz matrix is an $N \\\\times N$ Toeplitz matrix whose entries are $N\\\\times N$ $(d-1)$ -multilevel block Toeplitz matrices.\",\n \"Here $N$ could be finite or infinite.\",\n \"A 1-multilevel block Toeplitz matrix is simply an ordinary Toeplitz matrix.\",\n \"A 2-multilevel block Toeplitz matrix is what is usually considered a block Toeplitz matrix where each block itself is Toeplitz.\",\n \"thm:blocktoepTheorem REF Every finite $N \\\\times N$ $d$ -multilevel block Toeplitz matrix can be extended to an infinite $d$ -multilevel block Toeplitz matrix bounded on $\\\\ell ^2$ , with a constant which only depends on the dimension $d$ .\",\n \"For scalar Toeplitz matrices ($d=1$ ) this result is well-known [5], [26], [36], [37], although not as firmly cemented in the literature as the Nehari theorem itself; see [28] for a proof based on Parrot's lemma and a discussion of the result's history.\",\n \"For $d=1$ , the converse deduction of Theorem REF starting from Theorem REF can be found in [13].\",\n \"The paper is laid out as follows.\",\n \"In Section we will give a more formal background and introduce necessary notation.\",\n \"We will also discuss the relationship between $\\\\Gamma _{f, \\\\Xi }$ , Paley–Wiener spaces, and co-invariant subspaces of the Hardy spaces.\",\n \"In Section we will prove approximation results for distribution symbols with respect to Hankel and Toeplitz operators, allowing us to reduce to smooth symbols.\",\n \"Section briefly outlines what we need to know about convex sets and polytopes.\",\n \"In Section we prove Theorem REF , our Nehari theorem for Hankel operators.\",\n \"We also indicate how the proof extends to certain unbounded polyhedral domains.\",\n \"In Section our main result on Toeplitz operators is shown, Theorem REF .\",\n \"Finally, Section gives the proof of Theorem REF .\"\n ],\n [\n \"Hankel operators on multi-variable Hardy spaces\",\n \"Let us begin by placing Hankel operators $\\\\Gamma _f$ into the context of classical Hankel operators on Hardy spaces.\",\n \"As before, for $g \\\\in L^2(\\\\mathbb {R}^d)$ , let $\\\\hat{g} = \\\\mathcal {F}g$ denote its Fourier transform, $\\\\hat{g}(\\\\xi ) = \\\\mathcal {F}g(\\\\xi ) = \\\\int _{\\\\mathbb {R}^d} g(x) e^{-2\\\\pi i x \\\\cdot \\\\xi } \\\\, dx, \\\\quad \\\\xi \\\\in \\\\mathbb {R}^d.$ For the inverse transform we write $\\\\mathcal {F}^{-1}(g)=\\\\check{g}$ .\",\n \"The product domain Hardy space $H^2_d$ is the proper subspace of $L^2(\\\\mathbb {R}^d)$ of functions whose Fourier transforms are supported in the cone $\\\\overline{{\\\\mathbb {R}}_+^d}$ , ${\\\\mathbb {R}}_+ = (0,\\\\infty )$ , $ H^2_d = \\\\left\\\\lbrace G \\\\in L^2(\\\\mathbb {R}^d) \\\\, : \\\\, \\\\operatorname{supp}\\\\hat{G} \\\\subset \\\\overline{\\\\mathbb {R}_+^d}\\\\right\\\\rbrace .$ We let $P_d \\\\colon L^2(\\\\mathbb {R}^d) \\\\rightarrow H^2_d$ denote the orthogonal projection, and let $J \\\\colon L^2(\\\\mathbb {R}^d) \\\\rightarrow L^2(\\\\mathbb {R}^d)$ be the involution defined by $JG(x) = G(-x)$ , $x \\\\in \\\\mathbb {R}$ .\",\n \"Consider $\\\\Gamma _f = \\\\Gamma _{f, \\\\Xi }$ for $\\\\Xi = \\\\mathbb {R}_+^d$ with $f \\\\in L^2(\\\\mathbb {R}_+^d)$ .\",\n \"For a dense set of $g, h \\\\in L^2(\\\\mathbb {R}_+^d)$ we have that $ \\\\langle \\\\Gamma _f g, h \\\\rangle _{L^2(\\\\mathbb {R}_+^d)} = \\\\langle \\\\check{f} J\\\\check{g}, \\\\check{h}\\\\rangle _{H^2_d}.$ It follows that the (possibly unbounded) operator $\\\\Gamma _f \\\\colon L^2({\\\\mathbb {R}}_+^d) \\\\rightarrow L^2({\\\\mathbb {R}}_+^d)$ is unitarily equivalent to the small Hankel operator $Z_{\\\\check{f}} \\\\colon H^2_d \\\\rightarrow H^2_d$ , $Z_{\\\\check{f}} G = P_d(\\\\check{f} \\\\cdot JG).$ Note that any $b$ such that $\\\\hat{b}|_{\\\\mathbb {R}_+^d} = f$ generates the same Hankel operator as $\\\\check{f}$ , $Z_b = Z_{\\\\check{f}}$ .\",\n \"To justify the above computation easily we assumed that $f \\\\in L^2({\\\\mathbb {R}}_+^d)$ .\",\n \"An approximation argument is needed to consider general symbols $f$ , which may only be distributions in ${\\\\mathbb {R}}_+^d$ .\",\n \"We provide this later in Proposition REF .\",\n \"We can then read off the boundedness of $\\\\Gamma _f$ from the boundedness of the corresponding Hankel operator on $H^2_d$ .\",\n \"When $d=1$ and $\\\\Xi =\\\\Omega ={\\\\mathbb {R}}_+$ , the analogue of Theorem REF is exactly the classical Nehari theorem.\",\n \"In higher dimensions the corresponding theorem is due to Ferguson–Lacey–Terwilleger [18], [23].\",\n \"In our notation, their results read as follows.\",\n \"Theorem 2.1 Suppose $\\\\Xi =\\\\Omega ={\\\\mathbb {R}}_+^d$ and that $f$ is a distribution in ${\\\\mathbb {R}}^d_+$ , $f\\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d_+)$ .\",\n \"Then $\\\\Gamma _f \\\\colon L^2(\\\\mathbb {R}^d_+) \\\\rightarrow L^2(\\\\mathbb {R}^d_+)$ is bounded if and only if there exists a function $b\\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{{\\\\mathbb {R}}_+^d}=f$ .\",\n \"Moreover, there exists a constant $c > 0$ , depending on $d$ , such that $b$ can be chosen to satisfy $c\\\\Vert b\\\\Vert _{L^\\\\infty }\\\\le \\\\Vert \\\\Gamma _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ For $d > 1$ it is not possible to take $c=1$ in (REF ), see for example [29].\"\n ],\n [\n \"Hankel operators on bounded domains\",\n \"We now discuss bounded domains $\\\\Xi $ , the setting of our main result.\",\n \"The only convex bounded domains in ${\\\\mathbb {R}}$ are the intervals $I\\\\subset {\\\\mathbb {R}}$ .\",\n \"Translations, dilations, and reflections carry the operator $\\\\Theta _{f,I}$ onto $\\\\Gamma _{\\\\tilde{f},J}$ , where $J \\\\subset \\\\mathbb {R}$ is any other interval and $\\\\tilde{f}$ arises from transforming $f$ appropriately.\",\n \"In one variable it thus suffices to consider operators $\\\\Gamma _{f,(0,1)}$ where $\\\\Xi = (0,1)$ .\",\n \"Rochberg [35] called these operators Hankel operators on the Paley-Wiener space and proved Theorem REF in the one-dimensional case.\",\n \"In the same article [35], it is posed as an open problem to characterize the bounded Hankel operators $\\\\Gamma _{f,\\\\Xi }$ when $\\\\Xi $ is a disc in $\\\\mathbb {R}^2$ .\",\n \"We are not able to settle this question, but Theorem REF does provide the answer when $\\\\Xi = (0,1)^d$ is a cube in ${\\\\mathbb {R}}^d$ .\",\n \"As we will see, the Hankel operators $\\\\Gamma _{f, (0,1)^d}$ constitute a natural generalization of the Hankel operators on the Paley-Wiener space.\",\n \"On a technical level, the reason that we are able to prove Theorem REF when $\\\\Xi $ is a simple convex polytope, but not when $\\\\Xi $ is a ball, is that we rely on Theorem REF .\",\n \"In applying Theorem REF to our situation, the corners of the boundary of $\\\\Xi $ are actually of help rather than hindrance.\",\n \"We consider the case of a ball to be an interesting open problem for which we do not dare to make a firm conjecture.\",\n \"In view of Fefferman's disproof of the disc conjecture [17], Nehari theorems might turn out to be quite different for balls and polytopes.\"\n ],\n [\n \"Toeplitz operators\",\n \"When $d=1$ and $\\\\Xi ={\\\\mathbb {R}}_+$ , $\\\\Omega = {\\\\mathbb {R}}$ , the operators $\\\\Theta _f$ are known as Wiener-Hopf operators [11].\",\n \"Analogously with Hankel operators, these can be shown to be unitarily equivalent to Toeplitz matrix operators on $\\\\ell ^2({\\\\mathbb {N}})$ .\",\n \"In this case the boundedness characterization is easy to both state and prove, $\\\\Vert \\\\Theta _f\\\\Vert =\\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty }.$ In Theorem REF we extend (REF ) to Toeplitz operators $\\\\Theta _{f, \\\\Xi }$ for a class of “cone-like” domains $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ , for which $\\\\Omega = \\\\Xi - \\\\Xi = {\\\\mathbb {R}}^d$ .\"\n ],\n [\n \"Truncated correlation operators\",\n \"For open connected sets $\\\\Xi , \\\\Upsilon \\\\subset {\\\\mathbb {R}}^d$ it is also convenient to introduce the more general “truncated correlation operators” $\\\\Psi _{f, \\\\Upsilon , \\\\Xi } \\\\colon L^2(\\\\Upsilon )\\\\rightarrow L^2(\\\\Xi )$ , defined by $\\\\Psi _{f}(g)({x})=\\\\int _{\\\\Upsilon }f({x}+{y}) g({y}) ~d{y},\\\\quad {x}\\\\in \\\\Xi ,$ where $f$ lives on $\\\\Omega =\\\\Xi +\\\\Upsilon $ .\",\n \"This class of operators includes both general domain Hankel and Toeplitz operators, by letting $\\\\Upsilon = \\\\Xi $ and $\\\\Upsilon = -\\\\Xi $ , respectively.\",\n \"For our purposes, general truncated correlation operators will only appear in intermediate steps toward proving the main results, but they also carry independent interest.\",\n \"They were introduced in [1], where their finite rank structure was investigated.\",\n \"In [2] it was shown that they have a fundamental connection with frequency estimation on general domains, motivating the practical need for understanding such operators not only on domains of simple geometrical structure.\",\n \"In [3] it is explained how one may infer certain results for the integral operators $\\\\Psi _f$ from their discretized matrix counterparts.\",\n \"We warn the reader that in naming the operators $\\\\Gamma _f$ , $\\\\Theta _f$ , and $\\\\Psi _f$ we have slightly departed from previous work, reserving the term (general domain) Hankel operator for truncated correlation operators of the form $\\\\Psi _{f,\\\\Xi ,\\\\Xi }$ .\"\n ],\n [\n \"Hankel operators on multi-variable Paley–Wiener spaces\",\n \"Another viewpoint is offered through co-invariant subspaces of the Hardy spaces $H^2_d$ .\",\n \"For a domain $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ , let $\\\\operatorname{PW}_\\\\Xi $ denote the subspace of $L^2({\\\\mathbb {R}}^d)$ of functions with Fourier transforms supported in $\\\\Xi $ , $\\\\operatorname{PW}_\\\\Xi = \\\\lbrace G \\\\in L^2({\\\\mathbb {R}}^d) \\\\, : \\\\, \\\\operatorname{supp}\\\\hat{G} \\\\subset \\\\overline{\\\\Xi } \\\\rbrace .$ In the classical case $\\\\Xi = (0,1) \\\\subset {\\\\mathbb {R}}$ , note that $\\\\operatorname{PW}_{(0,1)} = H^2_1 \\\\ominus \\\\lbrace G \\\\in H^2_1 \\\\, : \\\\, \\\\operatorname{supp}\\\\hat{G} \\\\subset [1,\\\\infty )\\\\rbrace = H^2_1 \\\\ominus \\\\theta H^2_1,$ where $\\\\theta (x) = e^{i2\\\\pi x}, \\\\quad x \\\\in \\\\mathbb {R}.$ Hence $\\\\operatorname{PW}_{(0,1)}$ is the ortho-complement (in $H^2_1$ ) of $\\\\theta H^2_1$ , the shift-invariant subspace of $H^2_1$ with inner factor $\\\\theta $ .\",\n \"This space is usually denoted $K_\\\\theta $ , $\\\\operatorname{PW}_{(0,1)} = K_\\\\theta := (\\\\theta H^2_1)^\\\\perp .$ By a calculation similar to (REF ) we see that $\\\\Gamma _{f, (0,1)}$ is unitarily equivalent to the compression of the Hankel operator $Z_{\\\\check{f}}$ to $\\\\operatorname{PW}_{(0,1)}$ , $\\\\Gamma _{f, (0,1)} \\\\simeq P_{\\\\operatorname{PW}_{(0,1)}} Z_{\\\\check{f}}|_{\\\\operatorname{PW}_{(0,1)}},$ where $P_{\\\\operatorname{PW}_{(0,1)}} \\\\colon H^2_1 \\\\rightarrow \\\\operatorname{PW}_{(0,1)}$ denotes the orthogonal projection onto $\\\\operatorname{PW}_{(0,1)}$ .\",\n \"Such truncated Toeplitz and Hankel operators are now very well studied on general $K_\\\\theta $ -spaces [6], [7], [9], [10], [14], [20], [27], [30], [36].\",\n \"In the case of the cube $\\\\Xi = (0,1)^d \\\\subset {\\\\mathbb {R}}^d$ , $d > 1$ , the Hankel operator $\\\\Gamma _{f,\\\\Xi }$ may, just as for $d=1$ , be understood as the compression of a Hankel operator to a co-invariant subspace of $H^2_d$ .\",\n \"Namely, $\\\\operatorname{PW}_{(0,1)^d} = \\\\lbrace G \\\\in H_d^2 \\\\, : \\\\, \\\\operatorname{supp}\\\\hat{G} \\\\subset [0,1]^d\\\\rbrace = \\\\lbrace G \\\\in H_d^2 \\\\, : \\\\, \\\\operatorname{supp}\\\\hat{G} \\\\subset \\\\overline{{\\\\mathbb {R}}_+^d} \\\\setminus (0,1)^d\\\\rbrace ^\\\\perp .$ If $G \\\\in H_d^2 \\\\cap L^\\\\infty ({\\\\mathbb {R}}^d)$ , it is clear that $G\\\\operatorname{PW}_{(0,1)^d}^\\\\perp \\\\subset \\\\operatorname{PW}_{(0,1)^d}^\\\\perp $ , since $\\\\mathcal {F}(GH)(\\\\xi ) = \\\\int _{{\\\\mathbb {R}}_+^d} \\\\hat{G}(y) \\\\hat{H}(\\\\xi -y) \\\\, dy = 0, \\\\quad H \\\\in \\\\operatorname{PW}_{(0,1)^d}^\\\\perp , \\\\; \\\\xi \\\\in [0,1]^d.$ Hence $\\\\operatorname{PW}_{(0,1)^d}^\\\\perp \\\\subset H^2_d$ is an invariant subspace (under multiplication by bounded holomorphic functions), and as before we have that $\\\\Gamma _{f, (0,1)^d} \\\\simeq P_{\\\\operatorname{PW}_{(0,1)^d}} Z_{\\\\check{f}}|_{\\\\operatorname{PW}_{(0,1)^d}},$ where $P_{\\\\operatorname{PW}_{(0,1)^d}} \\\\colon H^2_d \\\\rightarrow \\\\operatorname{PW}_{(0,1)^d}$ denotes the orthogonal projection onto $\\\\operatorname{PW}_{(0,1)^d}$ .\",\n \"Finally, let us briefly discuss the viewpoint of weak factorization.\",\n \"The Hardy space $H^1_d$ is defined as the closure of $\\\\mathcal {F}^{-1}(C_c^\\\\infty ({\\\\mathbb {R}}_+^d))$ in $L^1({\\\\mathbb {R}}^d)$ .\",\n \"Similarly, we define $\\\\operatorname{PW}^1_\\\\Xi $ as the closure of $\\\\mathcal {F}^{-1}(C_c^\\\\infty (\\\\Xi ))$ in $L^1({\\\\mathbb {R}}^d)$ .\",\n \"As is well known, see for example [24], Theorem REF is equivalent to the fact that $H^1_d$ is the projective tensor product of two copies of $H^2_d$ , $ H^1_d = H^2_d \\\\odot H^2_d,$ with equivalence of norms.\",\n \"Here the projective tensor product norm on $X \\\\odot X$ , $X$ a Banach space of functions, is given by $\\\\Vert G\\\\Vert _{X \\\\odot X} = \\\\inf \\\\left\\\\lbrace \\\\sum _j \\\\Vert G_j\\\\Vert _{X} \\\\Vert H_j\\\\Vert _{X} \\\\, : \\\\, G = \\\\sum _j G_j H_j, \\\\; G_j, H_j \\\\in X \\\\right\\\\rbrace ,$ $X \\\\odot X$ being defined as the completion of finite sums $\\\\sum _j G_j H_j$ in this norm.\",\n \"The reason that Theorem REF is equivalent to (REF ) is the following: by (REF ), $\\\\Gamma _{f, {\\\\mathbb {R}}_+^d}$ is bounded if and only if $|\\\\langle \\\\check{f}, GH \\\\rangle _{H^2_d} | \\\\le C\\\\Vert G\\\\Vert _{H^2_d} \\\\Vert H\\\\Vert _{H^2_d},$ which means precisely that $\\\\check{f}$ induces a bounded functional on $H^2_d \\\\odot H^2_d$ , $\\\\check{f} \\\\in (H^2_d \\\\odot H^2_d)^*$ .\",\n \"On the other hand, the existence of $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{{\\\\mathbb {R}}_+^d} = f|_{{\\\\mathbb {R}}_+^d}$ , so that $\\\\langle \\\\check{f}, GH \\\\rangle _{H^2_d} = \\\\langle b, GH \\\\rangle _{H^2_d}$ , $G,H \\\\in H^2_d$ , means, by the Hahn–Banach theorem, precisely that $\\\\check{f} \\\\in (H_d^1)^*$ .\",\n \"Theorem REF yields a similar weak factorization theorem for Paley–Wiener spaces.\",\n \"We postpone the proof to Section , mentioning only that corresponding weak factorization for $K_\\\\theta $ -spaces plays an important role in [6] and [9].\",\n \"cor:factCorollary REF Let $\\\\Xi $ be a simple convex polytope, and let $\\\\Omega = 2\\\\Xi $ .\",\n \"Then $PW_{\\\\Omega }^1 = \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }.$ The norms of these Banach spaces are equivalent.\"\n ],\n [\n \"Brief historical overview\",\n \"Z. Nehari published his famous theorem in 1957 [25], inspiring the search for analogous statements in other contexts; positive results are themselves often referred to as Nehari theorems.\",\n \"The most natural inquiries are perhaps those related to Hankel operators on Hardy spaces of several variables.\",\n \"Nehari's theorem for the Hardy space of the unit ball was proven by Coifman, Rochberg and Weiss in 1976 [15], but this setting is rather different from the one considered in this paper.\",\n \"For the product domain Hardy space $H^2_d$ , Hankel operators can be defined by either projecting on $H^2_d$ or on the larger space $L^2({\\\\mathbb {R}}^d) \\\\ominus H^2_d$ .\",\n \"The first option leads to the “small” Hankel operators considered in Section REF , while the second type of operator is commonly referred to as a “big” Hankel operator.\",\n \"In the notation of Section REF , a small Hankel operator is an operator $\\\\Psi _{f,{\\\\mathbb {R}}_+^d,{\\\\mathbb {R}}_+^d}=\\\\Gamma _{f,{\\\\mathbb {R}}_+^d}$ , whereas big Hankel operators are of the form $\\\\Psi _{f,{\\\\mathbb {R}}_+^d,{\\\\mathbb {R}}^d\\\\setminus \\\\overline{{\\\\mathbb {R}}_+^d}}$ .\",\n \"When transferred to operators on the Hardy space of the polydisc, small Hankel operators correspond, in the standard basis, to infinite matrices with a certain block Hankel structure (cf.\",\n \"Section ).\",\n \"The big Hankel operators were extensively studied by Cotlar and Sadosky.\",\n \"In particular, boundedness of the big Hankel operators was characterized in terms of certain $\\\\operatorname{BMO}$ type estimates in [16].\",\n \"Small Hankel operators were investigated by Janson and Peetre [22] in 1988.\",\n \"They introduced “generalized Hankel and Toeplitz operators” as particular cases of a more general class of pseudo-differential operators called paracommutators.\",\n \"In their terminology, an operator of the form $\\\\Psi _{f,\\\\Xi ,\\\\Upsilon }$ is a generalized Hankel operator if $\\\\Xi $ and $\\\\Upsilon $ are open cones and $\\\\overline{\\\\Xi } \\\\cap (-\\\\overline{\\\\Upsilon })=\\\\lbrace 0\\\\rbrace $ , whereas it is called Toeplitz if $\\\\Xi \\\\cap (-\\\\Upsilon ) \\\\ne \\\\emptyset .$ Hence the general domain Hankel operators $\\\\Gamma _{f,\\\\Xi }$ are generalized Hankel operators a lá Janson–Peetre whenever $\\\\Xi $ is a cone with mild restrictions, while $\\\\Theta _{f,\\\\Xi }$ is a generalized Toeplitz operator a lá Janson–Peetre for every open cone $\\\\Xi $ .\",\n \"In the Toeplitz case, a full boundedness characterization is given in [22].\",\n \"In the Hankel case, only sufficient conditions for boundedness and Schatten class membership are provided, in terms of $\\\\operatorname{BMO}$ and Besov spaces, respectively.\",\n \"As previously mentioned, R. Rochberg considered Hankel operators for bounded domains in 1987 [35], studying the case of a finite interval in one dimension.\",\n \"Furthermore, he posed as an open problem to understand the case when $\\\\Xi \\\\subset {\\\\mathbb {R}}^2$ is a disc.\",\n \"In this latter setting, L. Peng [32] characterized when $\\\\Gamma _{f, \\\\Xi }$ belongs to the Schatten class $S_p$ , for $1 \\\\le p \\\\le 2$ , in terms of certain Besov spaces adapted to the disc.\",\n \"L. Peng also carried out a similar study [33] for the case of the multidimensional cube, $\\\\Xi = (-1,1)^d$ , describing membership in $S_p$ for all $p$ , $0 < p < \\\\infty $ , as well as giving a sufficient condition for boundedness.\",\n \"Since then it seems that the field did not see progress until the results of Ferguson–Lacey–Terwilleger [18], [23] settled the issue of boundedness of small Hankel operators.\"\n ],\n [\n \"Distribution symbols\",\n \"Let $\\\\Xi ,\\\\Upsilon \\\\subset {\\\\mathbb {R}}^d$ be any open connected sets and let $f\\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ be a distribution on $\\\\Omega $ , $\\\\Omega =\\\\Xi +\\\\Upsilon $ .\",\n \"We follow the notation of [21] in our use of distributions.\",\n \"We then define the truncated correlation operator $\\\\Psi _f$ as an operator $\\\\Psi _{f, \\\\Upsilon , \\\\Xi }:C^\\\\infty _c(\\\\Upsilon )\\\\rightarrow C^\\\\infty (\\\\Xi )$ by the formula $\\\\Psi _f(\\\\varphi )(x)=({f,T_x\\\\varphi }), \\\\quad x \\\\in \\\\Xi ,$ where $(f,\\\\varphi )$ denotes the action of $f$ on $\\\\varphi $We reserve the notation $\\\\langle f,\\\\varphi \\\\rangle $ for scalar products which are anti-linear in the second entry.\",\n \"and $T_x\\\\varphi (\\\\cdot )=\\\\varphi (\\\\cdot -x).$ Since $T_x\\\\varphi $ is compactly supported in $\\\\Omega $ for $x \\\\in \\\\Xi $ , it follows that $\\\\Psi _f(\\\\varphi )$ this is well-defined and smooth in $\\\\Xi $ (see e.g.\",\n \"[21]).\",\n \"Since $C^\\\\infty _c(\\\\Upsilon )$ is dense in $L^2(\\\\Upsilon )$ , $\\\\Psi _f$ gives rise to a densely defined operator on the latter space which extends to a bounded operator $\\\\Psi _f \\\\colon L^2(\\\\Upsilon ) \\\\rightarrow L^2(\\\\Xi )$ if and only if $\\\\Vert \\\\Psi _f\\\\Vert =\\\\sup \\\\left\\\\lbrace \\\\frac{\\\\Vert \\\\Psi _f(\\\\varphi )\\\\Vert _{L^2(\\\\Xi )}}{\\\\Vert \\\\varphi \\\\Vert _{L^2(\\\\Upsilon )}}:~\\\\varphi \\\\in C_c^\\\\infty (\\\\Upsilon ), ~\\\\varphi \\\\ne 0\\\\right\\\\rbrace <\\\\infty .$ It is clear that $\\\\Psi _f(\\\\varphi )(x) = \\\\int f(x+y)\\\\varphi (y) \\\\, dy$ whenever $f \\\\in L^1_{\\\\textrm {loc}}(\\\\Omega )$ .\",\n \"By slight abuse of notation, we write the action of $\\\\Psi _f$ in this way even when $f$ is not locally integrable.\",\n \"The central question in this paper is the following: for which domains $\\\\Upsilon $ and $\\\\Xi $ is the boundedness of $\\\\Psi _f$ equivalent to the existence of a function $b\\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{\\\\Omega }=f$ ?\",\n \"Some care must be taken in interpreting this question.\",\n \"For example, the prototypical example of a bounded Hankel operator is the Carleman operator $\\\\Gamma _{1/x, {\\\\mathbb {R}}_+} = \\\\Psi _{1/x, {\\\\mathbb {R}}_+, {\\\\mathbb {R}}_+}.$ The symbol $f(x) = \\\\frac{1}{x}\\\\chi _{{\\\\mathbb {R}}_+}(x)$ is in this case not a tempered distribution on $\\\\mathbb {R}$ (so the meaning of $\\\\check{f}$ is unclear) – it is, however, the restriction of the tempered distribution $\\\\operatorname{p.v.\",\n \"}\\\\frac{1}{x}$ to ${\\\\mathbb {R}}_+$ .\",\n \"An example with a delta function makes it clear that it is not necessary for $f$ to be locally integrable in $\\\\Omega $ either.\",\n \"We first record the answer to our question in the trivial direction.\",\n \"Proposition 3.1 Consider any connected open domains $\\\\Xi ,$ $\\\\Upsilon \\\\subset {\\\\mathbb {R}}^d$ , with associated domain $\\\\Omega =\\\\Upsilon +\\\\Xi $ .\",\n \"Let $b\\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ be given and suppose $f=\\\\hat{b}|_{\\\\Omega }$ .\",\n \"Then $\\\\Psi _f \\\\colon L^2(\\\\Upsilon ) \\\\rightarrow L^2(\\\\Xi )$ is bounded and $\\\\Vert \\\\Psi _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ For $\\\\varphi \\\\in C^\\\\infty _c(\\\\Upsilon )$ we have that $ \\\\Psi _{f}(\\\\varphi )={\\\\mathcal {F}}M_b J{\\\\mathcal {F}}^{-1} \\\\varphi |_{\\\\Xi },$ where $M_b$ is the operator of multiplication by $b$ .\",\n \"The statement is obvious from here.\",\n \"Next we establish two technical results on the approximation of distribution symbols by smooth compactly supported functions, Propositions REF and REF .\",\n \"They will help us to overcome the technical issues mentioned earlier, in particular allowing us to deduce Theorem REF from the corresponding statements in [18], [23].\",\n \"Given open connected domains $\\\\Xi ,$ $\\\\Upsilon \\\\subset {\\\\mathbb {R}}^d$ , let $\\\\left(\\\\Upsilon _n\\\\right)_{n=1}^\\\\infty $ be an increasing sequence of connected open subdomains $\\\\Upsilon _n \\\\subset \\\\Upsilon $ such that $\\\\operatorname{dist}(\\\\Upsilon _n,\\\\partial \\\\Upsilon )>1/n, \\\\quad \\\\cup _{n=1}^\\\\infty \\\\Upsilon _n=\\\\Upsilon .$ Note that $\\\\Omega _n = \\\\Upsilon _n + \\\\Xi $ is also increasing and satisfies $\\\\operatorname{dist}(\\\\Omega _n,\\\\partial \\\\Omega )>1/n, \\\\quad \\\\cup _{n=1}^\\\\infty \\\\Omega _n = \\\\Omega .$ Let $\\\\psi \\\\in C^\\\\infty _c({\\\\mathbb {R}}^d)$ be a fixed non-negative function with compact support in the ball $B(0,1/2)$ such that $\\\\int _{{\\\\mathbb {R}}^d} \\\\psi (x) \\\\, dx =1$ .\",\n \"For $n\\\\ge 1$ let $\\\\psi _n(x)={n^d}\\\\psi (nx),$ so that $(\\\\psi _n)_{n=1}^\\\\infty $ is an approximation of the identity.\",\n \"Since $f \\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ and $\\\\operatorname{supp}\\\\psi _n \\\\subset B(0,1/2n)$ , the convolution $f*\\\\psi _n$ is well-defined as a function in $C^\\\\infty (\\\\Omega _{2n})$ .\",\n \"Let $\\\\rho _n$ be a smooth cut-off function which is 1 in a neighborhood of $\\\\overline{\\\\Omega _n}$ but zero in a neighborhood of $\\\\Omega _{2n}^c$ , and note that $\\\\rho _n(f*\\\\psi _n)$ then naturally defines a function in $C^\\\\infty ({\\\\mathbb {R}}^n)$ .\",\n \"Finally, for a non-negative function $\\\\eta \\\\in C^\\\\infty _c({\\\\mathbb {R}}^d)$ with $\\\\Vert \\\\eta \\\\Vert _{L^2} = 1$ , let $\\\\omega = \\\\eta * \\\\tilde{\\\\eta }$ , where $\\\\tilde{\\\\eta }(x) = \\\\eta (-x)$ .\",\n \"Then $\\\\omega \\\\in C^\\\\infty _c({\\\\mathbb {R}}^d)$ and $\\\\omega (0)= \\\\Vert \\\\hat{\\\\omega }\\\\Vert _{L^1} = 1.$ Let $\\\\omega _n(x)=\\\\omega (x/n)$ .\",\n \"We introduce $f_n=\\\\omega _n\\\\rho _n (f*\\\\psi _n)$ as an approximant of $f$ , where the role of $\\\\omega _n$ is to enforce compact support in case $\\\\Omega $ is unbounded.\",\n \"By construction, $f_n \\\\in C_c^\\\\infty (\\\\Omega )$ and it is straightforward to check that $f_n \\\\rightarrow f$ in ${\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ .\",\n \"As for $\\\\Psi _{f_n, \\\\Upsilon _n, \\\\Xi }$ , we have the following result.\",\n \"Proposition 3.2 Let $\\\\Xi ,$ $\\\\Upsilon $ be connected open domains, $\\\\Omega =\\\\Upsilon +\\\\Xi $ , and suppose $f\\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ .\",\n \"For $n \\\\ge 1$ , let $\\\\Omega _n=\\\\Upsilon _n+\\\\Xi $ and $f_n$ be constructed as above.\",\n \"Then $\\\\Vert \\\\Psi _{f_n,\\\\Upsilon _n,\\\\Xi }\\\\Vert \\\\le \\\\Vert \\\\Psi _{f,\\\\Upsilon ,\\\\Xi }\\\\Vert .$ First note that $\\\\omega _n(x)=\\\\int _{{\\\\mathbb {R}}^d} n^d \\\\hat{\\\\omega }(n \\\\xi )e^{2\\\\pi i x \\\\cdot \\\\xi } \\\\, d\\\\xi ,$ the integrand on the right having $L^1$ -norm equal to $\\\\Vert \\\\hat{\\\\omega }\\\\Vert _{L^1({\\\\mathbb {R}}^d)}$ .\",\n \"Letting $g_n=\\\\rho _n (f*\\\\psi _n)$ , we have for $\\\\varphi \\\\in C^\\\\infty _c(\\\\Upsilon _n)$ and $x \\\\in \\\\Xi $ that $\\\\Psi _{f_n}(\\\\varphi )(x)=\\\\int _{\\\\Upsilon _n} \\\\int _{{\\\\mathbb {R}}^d} n^d\\\\hat{\\\\omega }(n\\\\xi )e^{2\\\\pi i (x+y)\\\\cdot \\\\xi }\\\\,d\\\\xi \\\\, g_n(x+y)\\\\varphi (y) \\\\, dy= \\\\int _{\\\\mathbb {R}^d} n^d \\\\hat{\\\\omega }(n \\\\xi ){e^{2\\\\pi i \\\\xi \\\\cdot x}}\\\\Psi _{g_n}(\\\\varphi _\\\\xi )(x) \\\\, d\\\\xi ,$ where $\\\\varphi _\\\\xi (y) = e^{2\\\\pi i y\\\\cdot \\\\xi }\\\\varphi (y)$ .\",\n \"Since $\\\\Vert \\\\varphi _\\\\xi \\\\Vert _{L^2} = \\\\Vert \\\\varphi \\\\Vert _{L^2}$ it follows by the triangle inequality (for $L^2$ -valued Bochner integrals) that $\\\\Vert \\\\Psi _{f_n, \\\\Upsilon _n, \\\\Xi }\\\\Vert \\\\le \\\\Vert \\\\hat{\\\\omega }\\\\Vert _{L^1}\\\\Vert \\\\Psi _{g_n, \\\\Upsilon _n, \\\\Xi }\\\\Vert = \\\\Vert \\\\Psi _{g_n, \\\\Upsilon _n, \\\\Xi }\\\\Vert .$ This reduces our task to proving that the operators $\\\\Psi _{g_n, \\\\Upsilon _n, \\\\Xi } = \\\\Psi _{\\\\rho _n (f*\\\\psi _n), \\\\Upsilon _n, \\\\Xi } = \\\\Psi _{f*\\\\psi _n, \\\\Upsilon _n, \\\\Xi }$ are uniformly bounded in $n$ .\",\n \"We have for $\\\\varphi \\\\in C^\\\\infty _c(\\\\Upsilon _n)$ and $x \\\\in \\\\Xi $ that $\\\\Psi _{f*\\\\psi _n}(\\\\varphi )(x) &=\\\\int _{{\\\\mathbb {R}}^d} \\\\int _{{\\\\mathbb {R}}^d} f((x+y)-z) \\\\psi _n(z) \\\\, dz~\\\\varphi (y) \\\\, dy \\\\\\\\&= \\\\int _{{\\\\mathbb {R}}^d} f(x+z) \\\\int _{{\\\\mathbb {R}}^d} \\\\psi _n(y-z)\\\\varphi (y)\\\\, dy \\\\,dz=\\\\Psi _f(\\\\widetilde{\\\\psi }_n * \\\\varphi )(x),$ where $\\\\widetilde{\\\\psi }_n(x) = \\\\psi _n(-x)$ .\",\n \"Since $\\\\Vert \\\\widetilde{\\\\psi }_n * \\\\varphi \\\\Vert _{L^2(\\\\Upsilon )} \\\\le \\\\Vert \\\\psi _n\\\\Vert _{L^1} \\\\Vert \\\\varphi \\\\Vert _{L^2(\\\\Upsilon _n)} = \\\\Vert \\\\psi \\\\Vert _{L^1} \\\\Vert \\\\varphi \\\\Vert _{L^2(\\\\Upsilon _n)} = \\\\Vert \\\\varphi \\\\Vert _{L^2(\\\\Upsilon _n)},$ this completes the proof.\",\n \"Suppose that $\\\\Gamma _{f, {\\\\mathbb {R}}_+^d} = \\\\Psi _{f, \\\\Xi , \\\\Upsilon }$ is bounded, where $\\\\Xi = \\\\Upsilon = {\\\\mathbb {R}}_+^d$ .\",\n \"In this case, we let $\\\\Upsilon _n = (2/n, \\\\infty )^d$ .\",\n \"By Proposition REF we then have that $\\\\Vert \\\\Gamma _{f_n, \\\\Upsilon _n}\\\\Vert \\\\le \\\\Vert \\\\Psi _{f_n, \\\\Upsilon _n,\\\\Xi }\\\\Vert \\\\le \\\\Vert \\\\Gamma _{f, {\\\\mathbb {R}}_+^d}\\\\Vert , \\\\quad n \\\\ge 1.$ Since $\\\\Upsilon _n = z_n + {\\\\mathbb {R}}_+^d$ , $z_n = (2/n, \\\\ldots , 2/n)$ , we have that $\\\\Gamma _{f_n, \\\\Upsilon _n}(g)(x) = \\\\Gamma _{\\\\tilde{f}_n, {\\\\mathbb {R}}_+^d} (\\\\tilde{g})(x-z_n),$ where $\\\\tilde{f}_n(x) = f_n(x+2z_n)$ and $\\\\tilde{g}_n(x) = g(x+z_n)$ .\",\n \"Since $\\\\tilde{f}_n \\\\in L^2({\\\\mathbb {R}}_+^d)$ , the computation that lead to (REF ) is justified, and we conclude from [18], [23] that there is $b_n \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}_n|_{2\\\\Upsilon _n} = f_n|_{2\\\\Upsilon _n}, \\\\quad \\\\Vert b_n\\\\Vert _{L^\\\\infty } \\\\le C \\\\Vert \\\\Gamma _{f, {\\\\mathbb {R}}_+^d}\\\\Vert .$ By Alaoglu's theorem it follows that there is a weak-star convergent subsequence $(b_{n_k})_{k=1}^\\\\infty $ with limit $b \\\\in L^\\\\infty $ having norm less than $C \\\\Vert \\\\Gamma _{f, {\\\\mathbb {R}}_+^d}\\\\Vert $ .\",\n \"It remains to prove that $f=\\\\hat{b}|_{{\\\\mathbb {R}}_+^d}$ , i.e.\",\n \"that $(f,\\\\varphi )=(b,\\\\hat{\\\\varphi })$ holds for all $\\\\varphi \\\\in C^\\\\infty _c({\\\\mathbb {R}}_+^d)$ .\",\n \"However, this is clear from the construction; since $\\\\hat{\\\\varphi }\\\\in L^1$ we have that $(b,\\\\hat{\\\\varphi })=\\\\lim _{k\\\\rightarrow \\\\infty }(b_{n_k},\\\\hat{\\\\varphi })=\\\\lim _{k\\\\rightarrow \\\\infty }(f_{n_k},{\\\\varphi }) =(f,{\\\\varphi }).\",\n \"$ In Section we will consider Toeplitz operators $\\\\Theta _{f, \\\\Xi }$ for which $\\\\Omega = \\\\Xi - \\\\Xi = {\\\\mathbb {R}}^d$ .\",\n \"In this case $f * \\\\psi _n$ is a smooth function defined in all of ${\\\\mathbb {R}}^d$ , and there is no need to multiply with $\\\\rho _n$ or to introduce the subdomains $\\\\Upsilon _n$ .\",\n \"In this case we simply let $f_n = \\\\omega _n (f * \\\\psi _n).$ Clearly, $f_n \\\\rightarrow f$ in ${\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d)$ and we have, with the exact same proof as for Proposition REF , the following approximation result.\",\n \"Proposition 3.3 Let $\\\\Xi ,$ $\\\\Upsilon $ be connected open domains for which $\\\\Omega =\\\\Upsilon +\\\\Xi = {\\\\mathbb {R}}^d$ , and suppose $f\\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d)$ .\",\n \"For $n \\\\ge 1$ , let $f_n$ be constructed as above.\",\n \"Then $\\\\Vert \\\\Psi _{f_n,\\\\Upsilon ,\\\\Xi }\\\\Vert \\\\le \\\\Vert \\\\Psi _{f,\\\\Upsilon ,\\\\Xi }\\\\Vert .$\"\n ],\n [\n \"On convex sets and polytopes\",\n \"We recall some basic properties of convex sets.\",\n \"Given an unbounded convex set $\\\\Omega \\\\subset {\\\\mathbb {R}}^d$ which is either open or closed, its characteristic cone, also known as its recession cone, is the closed set $\\\\operatorname{cc}_{\\\\Omega }=\\\\lbrace x\\\\in {\\\\mathbb {R}}^d \\\\, : \\\\, \\\\Omega +x{\\\\mathbb {R}}_+\\\\subset \\\\Omega \\\\rbrace .$ The support function $h_{\\\\Omega }:{\\\\mathbb {R}}^d\\\\rightarrow (-\\\\infty ,\\\\infty ]$ is defined by $h_{\\\\Omega }({\\\\theta })=\\\\sup _{{x}\\\\in \\\\Omega } {x}\\\\cdot {\\\\theta }.$ We refer to [21] for the basic properties of $h_\\\\Omega $ .\",\n \"The barrier cone of $\\\\Omega $ is the set $ \\\\operatorname{bc}_{\\\\Omega }=\\\\lbrace \\\\theta \\\\in {\\\\mathbb {R}}^d \\\\, : \\\\, h_{\\\\Omega }({\\\\theta })<\\\\infty \\\\rbrace .$ The characteristic cone $\\\\operatorname{cc}_{\\\\Omega }$ coincides with the polar cone of the barrier cone $\\\\operatorname{bc}_{\\\\Omega }$ , that is, $\\\\operatorname{cc}_{\\\\Omega } = \\\\lbrace x \\\\in {\\\\mathbb {R}}^d \\\\, : \\\\, x \\\\cdot y \\\\le 0, \\\\; \\\\: \\\\forall y\\\\in \\\\operatorname{bc}_{\\\\Omega }\\\\rbrace .$ To give a complete reference for this claim, first note that for closed convex sets $\\\\Omega $ , $\\\\operatorname{cc}_{\\\\Omega }$ coincides with the asymptotic cone of $\\\\Omega $ , giving (REF ) by [4].\",\n \"When $\\\\Omega $ instead is open and convex we have that $\\\\Omega $ is equal to its relative interior $\\\\operatorname{ri}(\\\\Omega )$ , and since $\\\\operatorname{cc}_{\\\\operatorname{ri}(\\\\Omega )} = \\\\operatorname{cc}_{\\\\overline{\\\\Omega }}$ [8], it follows that $\\\\operatorname{cc}_{\\\\Omega } = \\\\operatorname{cc}_{\\\\overline{\\\\Omega }}$ in this case.\",\n \"We next recall some standard terminology and facts of polytopes, referring to for example [12].\",\n \"By an open halfspace in ${\\\\mathbb {R}}^d$ we mean a set $H_\\\\nu ^r=\\\\lbrace x \\\\in {\\\\mathbb {R}}^d \\\\, : \\\\, x\\\\cdot \\\\nu >r\\\\rbrace ,$ where $\\\\nu \\\\in {\\\\mathbb {R}}^d$ is a non-zero vector and $r\\\\in {\\\\mathbb {R}}$ .\",\n \"A closed half-space is the closure of such a set.\",\n \"A finite intersection of half-spaces is called a polyhedral set.\",\n \"A convex polytope is a bounded polyhedral set.\",\n \"A closed convex polytope is the convex hull of a finite set of points.\",\n \"The minimal set of such points coincides with the extreme points of the polytope, that is, its vertices.\",\n \"If the minimal number of defining hyperspaces of a convex polytope is $d+1$ (equivalently, if it has precisely $d+1$ vertices), the polytope is called a simplex.\",\n \"For a non-closed polytope we define its vertices (and its edges and facets) as those of its closure.\",\n \"The boundary of a polytope set is made up of a finite amount of facets (i.e.\",\n \"$d-1$ dimensional faces), see Corollary 7.4 and Theorem 8.1 of [12].\",\n \"For a polytope $\\\\Pi $ with vertex $x_j$ , we denote by $\\\\partial _{\\\\operatorname{far},x_j}\\\\Pi $ the part of its boundary made up of all facets not containing $x_j$ .\",\n \"A vertex of a polytope will be called simple if it is contained in precisely $d$ of its edges.\",\n \"We say that a polytope is simple if all of its vertices are simple, which coincides with the standard terminology.\",\n \"Equivalently, this means that each vertex is contained in precisely $d$ of its facets (cf.\",\n \"[12]).\",\n \"By an affine linear transformation we mean a map of the form $A(x)=x_0+L(x)$ where $L$ is a linear map, and we call $x_0$ the origin of such a map.\",\n \"The following simple lemma gives a third characterization of simple vertices.\",\n \"Lemma 4.1 Let $\\\\lbrace x_j\\\\rbrace _{j=1}^J$ be the vertices of a closed polytope $\\\\Pi $ .\",\n \"Then the vertex $x_j$ is simple if and only if it is the origin of an invertible affine transformation $A_j$ such that $\\\\Pi $ locally coincides with $A_j(\\\\overline{{\\\\mathbb {R}}_+^d})$ around $x_j$ , i.e.\",\n \"for any neighborhood $U$ of $x_j$ such that $\\\\overline{U} \\\\cap \\\\partial _{\\\\operatorname{far}, x_j}\\\\Pi = \\\\emptyset $ we have that $A_j^{-1}(\\\\Pi \\\\cap U) = \\\\overline{{\\\\mathbb {R}}_+^d} \\\\cap A_j^{-1}(U).$ By compactness it is easy to construct a partition of unity adapted to the vertices of $\\\\Pi $ .\",\n \"Lemma 4.2 Given a polytope $\\\\Pi $ with vertices $\\\\lbrace x_j\\\\rbrace _{j=1}^J$ there exist functions $\\\\lbrace \\\\mu _j\\\\rbrace _{j=1}^J$ such that $\\\\mu _j \\\\in C_c^\\\\infty ({\\\\mathbb {R}}^d)$ , $\\\\sum _{j=1}^J \\\\mu _j(x) = 1$ for $x \\\\in \\\\overline{\\\\Pi }$ , and $\\\\operatorname{supp}\\\\mu _j\\\\cap \\\\partial _{\\\\operatorname{far}, x_j}\\\\Pi =\\\\emptyset $ .\"\n ],\n [\n \"General domain Hankel operators\",\n \"We now consider general domain Hankel operators $\\\\Gamma _{f,\\\\Xi }$ for convex domains $\\\\Xi $ .\",\n \"Observe that in this case $\\\\Omega =\\\\Xi +\\\\Xi = 2\\\\Xi $ .\",\n \"We begin with a proposition that links the bounded Hankel operators with weak factorization.\",\n \"Proposition 5.1 Let $\\\\Xi $ be an open convex domain.\",\n \"Then $X = \\\\left\\\\lbrace \\\\Gamma _{f, \\\\Xi } \\\\, : \\\\, \\\\Vert \\\\Gamma _{f, \\\\Xi } \\\\Vert < \\\\infty \\\\right\\\\rbrace $ is a closed subspace of the space of bounded linear operators on $L^2(\\\\Xi )$ .\",\n \"As a Banach space, it is isometrically isomorphic to the dual space $(\\\\operatorname{PW}_\\\\Xi \\\\odot \\\\operatorname{PW}_\\\\Xi )^*$ .\",\n \"More precisely, bounded functionals $\\\\mu $ on the projective tensor product correspond to distributions $f$ on $\\\\Omega = 2\\\\Xi $ , $( f, g ) = \\\\mu (\\\\mathcal {F}^{-1}g), \\\\quad g \\\\in C_c^\\\\infty (\\\\Omega ),$ for which $\\\\Vert \\\\Gamma _{f, \\\\Xi }\\\\Vert = \\\\Vert \\\\mu \\\\Vert .$ The main fact to be proved is that $\\\\mathcal {F}^{-1}(C_c^\\\\infty (\\\\Omega )) \\\\subset \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }.$ Since $C_c^\\\\infty (\\\\Xi )$ is dense in $L^2(\\\\Xi )$ , it then follows that $\\\\mathcal {F}^{-1}(C_c^\\\\infty (\\\\Omega ))$ is dense in the product $\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }$ .\",\n \"We will actually show a little more than the claim.\",\n \"Namely, every $g \\\\in C_c^\\\\infty (\\\\Omega )$ can be written $g = \\\\sum _{k} g_k * h_k, \\\\quad g_k, h_k \\\\in L^2(\\\\Xi ),$ in such a way that the corresponding map $g \\\\mapsto \\\\sum _k \\\\Vert g_k\\\\Vert _{L^2(\\\\Xi )} \\\\Vert h_k\\\\Vert _{L^2(\\\\Xi )}$ is continuous from $C_c^\\\\infty (\\\\Omega )$ , equipped with the usual test function topology, to $\\\\mathbb {R}$ .\",\n \"By employing a partition of unity in which each member is compactly supported in a cube, it is sufficient to prove the claim when $\\\\Xi = (0, 1/2)^d$ .\",\n \"For this we employ Fourier series.\",\n \"Let $\\\\lambda (t) = 1/2 - |t-1/2|$ , $t \\\\in [0,1]$ , and let $\\\\Lambda (x) = \\\\prod _{i=1}^d \\\\lambda (x_i), \\\\quad x \\\\in (0,1)^d.$ Note that $\\\\lambda $ is in the Wiener algebra $A([0,1])$ , the space of functions on $[0,1]$ with absolutely convergent Fourier series, equipped with pointwise multiplication.\",\n \"Therefore $\\\\Lambda $ is in the Wiener algebra $A([0,1]^d)$ , since $\\\\Lambda $ is a tensor power of $\\\\lambda $ .\",\n \"Since $g \\\\in C_c^\\\\infty ((0,1)^d)$ and $\\\\Lambda $ is non-zero on compact subsets of $(0,1)^d$ it follows by Wiener's lemma [19] that $g/\\\\Lambda \\\\in A([0,1]^d)$ (to apply Wiener's lemma, first modify $\\\\Lambda $ to be nonzero outside the support of $g$ ).\",\n \"Expanding $g/\\\\Lambda $ in a Fourier series, $(g/\\\\Lambda )(x) = \\\\sum _{k \\\\in {\\\\mathbb {Z}}^d} a_k e^{i2\\\\pi k \\\\cdot x}, \\\\quad \\\\sum _{k \\\\in {\\\\mathbb {Z}}^d} |a_k| < \\\\infty , \\\\quad x \\\\in [0,1]^d,$ let $h_k(x) = e^{i2\\\\pi k \\\\cdot x} \\\\chi _{(0,1/2)^d}(x)$ , $g_k = a_k h_k$ .\",\n \"Then a computation shows that $(g_k * h_k)(x) = a_k e^{i2\\\\pi k \\\\cdot x} \\\\Lambda (x), \\\\quad x \\\\in (0,1)^d,$ so that $g = \\\\sum _{k \\\\in {\\\\mathbb {Z}}^d} g_k * h_k, \\\\quad \\\\sum _{k \\\\in {\\\\mathbb {Z}}^d} \\\\Vert g_k\\\\Vert _{L^2((0,1/2)^d)} \\\\Vert h_k\\\\Vert _{L^2((0,1/2)^d)} < \\\\infty .$ An inspection of the argument shows that the $g \\\\mapsto g/\\\\Lambda $ is continuous from $C_c^\\\\infty ((0,1)^d)$ to $A([0,1]^d)$ , and therefore $g \\\\mapsto \\\\sum _k \\\\Vert g_k\\\\Vert _{L^2((0,1/2)^d)} \\\\Vert h_k\\\\Vert _{L^2((0,1/2)^d)}$ is continuous on $C_c^\\\\infty ((0,1)^d)$ as promised.\",\n \"Suppose now that $\\\\mu \\\\in (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ .\",\n \"We have just demonstrated that $( f, g ) = \\\\mu (\\\\mathcal {F}^{-1}g)$ , $g \\\\in C_c^\\\\infty (\\\\Omega )$ , defines a distribution on $\\\\Omega $ .\",\n \"Hence we may consider the Hankel operator $\\\\Gamma _{f, \\\\Xi }$ .\",\n \"For $g, h \\\\in C_c^\\\\infty (\\\\Xi )$ we have that $ \\\\langle \\\\Gamma _{f}g, h \\\\rangle _{L^2(\\\\Xi )} = ( f, g * \\\\bar{h} ) = \\\\mu (\\\\mathcal {F}^{-1}g \\\\cdot \\\\mathcal {F}^{-1}\\\\bar{h}).$ Since $\\\\mu $ is a bounded functional on $\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }$ we conclude that $|\\\\langle \\\\Gamma _{f}g, h \\\\rangle _{L^2(\\\\Xi )}| \\\\le \\\\Vert \\\\mu \\\\Vert \\\\Vert \\\\mathcal {F}^{-1} g\\\\Vert _{\\\\operatorname{PW}_\\\\Xi } \\\\Vert \\\\mathcal {F}^{-1}\\\\bar{h}\\\\Vert _{\\\\operatorname{PW}_\\\\Xi } = \\\\Vert \\\\mu \\\\Vert \\\\Vert g\\\\Vert _{L^2(\\\\Xi )} \\\\Vert h\\\\Vert _{L^2(\\\\Xi )},$ that is, $\\\\Gamma _{f, \\\\Xi }$ is bounded, and in fact $\\\\Vert \\\\Gamma _f\\\\Vert = \\\\Vert \\\\mu \\\\Vert $ .\",\n \"Conversely, if $f$ is a distribution on $\\\\Omega $ such that $\\\\Gamma _{f, \\\\Xi }$ is bounded, it is clear that $f$ induces a bounded functional $\\\\mu $ on $\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }$ by (REF ).\",\n \"This proves that $X$ is isometrically isomorphic to the Banach space $(\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ , which also entails that $X$ is closed, completing the proof.\",\n \"In the remainder of this section we assume that $\\\\Xi $ is a convex polytope.\",\n \"Next we prove Theorem REF under the additional assumption that $f$ is supported around one simple vertex of $\\\\Omega $ .\",\n \"Proposition 5.2 Let $\\\\Xi \\\\subset {\\\\mathbb {R}}^d$ be an open convex polytope, $x$ a simple vertex of $\\\\Omega =2\\\\Xi $ , and let $f\\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ be such that $\\\\operatorname{supp}f \\\\cap \\\\partial _{\\\\operatorname{far},x} \\\\Omega = \\\\emptyset $ .\",\n \"If $\\\\Gamma _f$ is bounded as an operator on $L^2(\\\\Xi )$ , then there exists a $b\\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b} |_{\\\\Omega }=f.$ As in Lemma REF , let $A$ be an affine transformation with origin $x$ such that $A({\\\\mathbb {R}}_+^d)$ coincides with $\\\\Xi $ in a neighborhood of $x$ .\",\n \"It is straightforward to verify that it suffices to prove the proposition for $\\\\Gamma _{f\\\\circ A, A^{-1}(\\\\Xi )}$ .\",\n \"Since $A^{-1}(\\\\Xi )$ is also a convex polytope, we may hence assume that $x=0$ and that $\\\\Omega $ coincides with ${\\\\mathbb {R}}_+^d$ in a neighborhood $U$ of $\\\\operatorname{supp}f$ , $\\\\overline{U} \\\\cap \\\\partial _{far,0} \\\\Omega = \\\\emptyset $ .\",\n \"In particular, since $\\\\Omega $ is a convex polytope, we have that $\\\\Omega \\\\subset \\\\mathbb {R}_+^d$ .\",\n \"Since $\\\\operatorname{supp}f \\\\subset U \\\\cap \\\\overline{\\\\Omega }$ and $\\\\overline{U} \\\\cap \\\\partial _{far,0} \\\\Omega = \\\\emptyset $ , we can extend $f$ to a distribution on all of ${\\\\mathbb {R}}_+^d$ by letting it be zero outside $\\\\Omega $ .\",\n \"Our strategy is to show that the operator $\\\\Gamma _{f,{\\\\mathbb {R}}_+^d}$ is bounded and to then apply Theorem REF .\",\n \"For $n\\\\in {\\\\mathbb {N}}^d$ let $C_n$ denote the cube $(n_1,n_1+1)\\\\times \\\\ldots \\\\times (n_d,n_d+1)$ .\",\n \"For a set $X \\\\subset {\\\\mathbb {R}}_+^d$ , let $P_X \\\\colon L^2({\\\\mathbb {R}}_+^d) \\\\rightarrow L^2({\\\\mathbb {R}}_+^d)$ denote the orthogonal projection of $L^2({\\\\mathbb {R}}_+^d)$ onto $L^2(X)$ , and let $r>0$ be such that $2\\\\sqrt{d}r<\\\\operatorname{dist}(U \\\\cap \\\\overline{\\\\Omega },\\\\partial _{far,0}\\\\Omega ).$ By considering test functions $g \\\\in C_c^\\\\infty ({\\\\mathbb {R}}_+^d)$ such that $\\\\operatorname{supp}g \\\\cap \\\\overline{rC_m} \\\\subset rC_m$ for every $m$ , we give meaning to the equality $\\\\Gamma _{f,{\\\\mathbb {R}}_+^d}=\\\\left(\\\\sum _{n\\\\in {\\\\mathbb {N}}^d}P_{rC_n}\\\\right)\\\\Gamma _{f,{\\\\mathbb {R}}_+^d} \\\\left(\\\\sum _{m\\\\in {\\\\mathbb {N}}^d}P_{rC_m}\\\\right)=\\\\sum _{m,n\\\\in {\\\\mathbb {N}}^d}P_{rC_n}\\\\Gamma _{f,{\\\\mathbb {R}}_+^d} P_{rC_m},$ a term $P_{rC_n}\\\\Gamma _{f,{\\\\mathbb {R}}_+^d} P_{rC_m}$ being non-zero only if $(rC_m+rC_n)\\\\cap \\\\operatorname{supp}f \\\\ne \\\\emptyset .$ Hence there are only finitely many non-zero terms in the decomposition.\",\n \"Since $\\\\Vert P_{rC_n}\\\\Gamma _{f,{\\\\mathbb {R}}_+^d} P_{rC_m}\\\\Vert =\\\\Vert \\\\Psi _{f,rC_m,rC_n}\\\\Vert ,$ recalling the definition of $\\\\Psi _f$ from Section , it therefore suffices to prove that $\\\\Vert \\\\Psi _{f,rC_m,rC_n}\\\\Vert $ is bounded whenever (REF ) holds.\",\n \"If $rC_m,rC_n\\\\subset \\\\Xi $ there is nothing to prove since $\\\\Gamma _{f,\\\\Xi }$ is bounded by hypothesis.\",\n \"For the other terms, note that (REF ), $\\\\operatorname{supp}f \\\\subset U \\\\cap \\\\overline{\\\\Omega }$ , and the choice of $r$ implies that $rC_m+rC_n\\\\subset \\\\Omega ,$ since $2\\\\sqrt{d} r$ is the diameter of $rC_m+rC_n$ .\",\n \"For any $z\\\\in {\\\\mathbb {R}}^d$ , $x \\\\in rC_n$ , and $g \\\\in C_c^\\\\infty (rC_m)$ we have that $\\\\Psi _{f,rC_m,rC_n}(g)(x)=\\\\int _{rC_m} f(x+y)g(y) \\\\, dy=\\\\int _{rC_m+z} f(x+(y-z))g(y-z)dy,$ and hence $\\\\Vert \\\\Psi _{f,rC_m,rC_n}\\\\Vert =\\\\Vert \\\\Psi _{f,rC_m+z,rC_n-z}\\\\Vert .$ In particular, for $z=r(n-m)/2$ we obtain that $\\\\Vert \\\\Psi _{f,rC_m,rC_n}\\\\Vert =\\\\Vert \\\\Psi _{f,rC_{\\\\frac{m+n}{2}},rC_{\\\\frac{m+n}{2}}}\\\\Vert .$ However, $2rC_{\\\\frac{m+n}{2}}=rC_m+rC_n$ so by (REF ) we conclude that $rC_{\\\\frac{m+n}{2}}\\\\subset \\\\Xi $ .\",\n \"The desired boundedness now follows as it did in the first case considered.\",\n \"We have just demonstrated that $\\\\Vert \\\\Gamma _{f, {\\\\mathbb {R}}_+^d}\\\\Vert < \\\\infty $ .\",\n \"By Theorem REF there exists a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{{\\\\mathbb {R}}^d_+} = f$ .\",\n \"This in particular implies that $\\\\hat{b}|_{\\\\Omega } = f$ when we return to the initial interpretation of $f$ as a distribution on $\\\\Omega $ .\",\n \"Theorem 5.3 Let $\\\\Xi $ be a simple convex polytope, and let $f\\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ , $\\\\Omega =2\\\\Xi $ .\",\n \"Then $\\\\Gamma _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded if and only if there is a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b} |_{\\\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\\\Xi $ , such that $b$ can be chosen to satisfy $c\\\\Vert b\\\\Vert _{L^\\\\infty } \\\\le \\\\Vert \\\\Gamma _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ Assume that $\\\\Gamma _f$ is bounded.\",\n \"Let $\\\\lbrace x_j\\\\rbrace _{j=1}^J$ be the vertices of $\\\\Omega $ , and let $\\\\lbrace \\\\mu _j\\\\rbrace _{j=1}^J$ be partition of unity as in Lemma REF .\",\n \"For $\\\\varphi \\\\in C_c^\\\\infty (\\\\Xi )$ and $x \\\\in \\\\Xi $ we have that $\\\\Gamma _{\\\\mu _j f}(\\\\varphi )(x)=\\\\int _{\\\\Xi } \\\\int _{{\\\\mathbb {R}}^d} \\\\hat{\\\\mu }_j(\\\\xi )e^{2\\\\pi i (x+y)\\\\cdot \\\\xi }\\\\,d\\\\xi \\\\, f(x+y)\\\\varphi (y) \\\\, dy= \\\\int _{\\\\mathbb {R}^d} \\\\hat{\\\\mu }_j(\\\\xi ){e^{2\\\\pi i \\\\xi \\\\cdot x}}\\\\Gamma _{f}(\\\\varphi _\\\\xi )(x) \\\\, d\\\\xi ,$ where $\\\\varphi _\\\\xi (y) = e^{2\\\\pi i y\\\\cdot \\\\xi }\\\\varphi (y)$ .\",\n \"Hence, $\\\\Gamma _{\\\\mu _j f} \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded, $\\\\Vert \\\\Gamma _{\\\\mu _j f}\\\\Vert \\\\le \\\\Vert \\\\hat{\\\\mu }_j\\\\Vert _{L^1} \\\\Vert \\\\Gamma _f\\\\Vert .$ Therefore, by Proposition REF there are functions $b_j \\\\in L^\\\\infty $ such that $\\\\mu _jf=\\\\hat{b}_j|_\\\\Omega $ .\",\n \"Thus $f=\\\\hat{b}|_\\\\Omega $ , where $b = \\\\sum _{j=1}^J b_j \\\\in L^\\\\infty $ .\",\n \"Conversely, if $f=\\\\hat{b}|_\\\\Omega $ , where $b \\\\in L^\\\\infty $ , then $\\\\Gamma _f$ is bounded by Proposition REF .\",\n \"The constant $c$ now arises from abstract reasoning.\",\n \"Consider the Banach space $X = \\\\left\\\\lbrace \\\\Gamma _{f, \\\\Xi } \\\\, : \\\\, \\\\Vert \\\\Gamma _{f, \\\\Xi } \\\\Vert < \\\\infty \\\\right\\\\rbrace $ of Proposition REF .\",\n \"We have just shown that $b \\\\mapsto \\\\Gamma _{\\\\hat{b} |_{\\\\Omega }, \\\\Xi }$ is a map of $L^\\\\infty $ onto $X$ .\",\n \"The open mapping theorem hence guarantees the existence of $c$ .\",\n \"We immediately obtain the corresponding result for Toeplitz operators, when $\\\\Xi $ is a simple convex polytope which, possibly after a translation, is symmetric under $x \\\\mapsto -x$ .\",\n \"Corollary 5.4 Let $\\\\Xi $ be a simple convex polytope such that for some $z \\\\in {\\\\mathbb {R}}^d$ it holds that $\\\\Xi +z = -\\\\Xi - z$ .\",\n \"Let $f \\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ , $\\\\Omega =\\\\Xi - \\\\Xi = 2\\\\Xi + 2z$ .\",\n \"Then $\\\\Theta _f$ is bounded if and only if there exists a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{\\\\Omega } = f$ .\",\n \"There exists a constant $c > 0$ , depending on $\\\\Xi $ , such that $b$ can be chosen to satisfy $c\\\\Vert b\\\\Vert _{L^\\\\infty } \\\\le \\\\Vert \\\\Theta _f\\\\Vert \\\\le \\\\Vert b\\\\Vert _{L^\\\\infty }.$ In this case $\\\\Theta _{f} g = \\\\Gamma _{\\\\tilde{f}} \\\\tilde{g}$ , where $\\\\tilde{f}(x) = f(x + 2z)$ , $x \\\\in 2\\\\Xi $ , and $\\\\tilde{g}(x) = g(-x-2z)$ , $x \\\\in \\\\Xi $ .\",\n \"Hence the result follows from Theorem REF .\",\n \"We also deduce the weak factorization result for $\\\\operatorname{PW}_\\\\Omega ^1$ , see Section REF .\",\n \"Corollary 5.5 Let $\\\\Xi $ be a simple convex polytope, and let $\\\\Omega = 2\\\\Xi $ .\",\n \"Then $PW_{\\\\Omega }^1 = \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }.$ The norms of these Banach spaces are equivalent.\",\n \"By Cauchy-Schwarz, the inclusion $I \\\\colon \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi } \\\\rightarrow PW_{\\\\Omega }^1$ is bounded.\",\n \"Since $I$ has dense range by Proposition REF , the adjoint $I^* \\\\colon (\\\\operatorname{PW}^1_\\\\Omega )^* \\\\rightarrow (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ has empty kernel.\",\n \"Suppose $\\\\mu \\\\in (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ .\",\n \"Note that $CG(x) = \\\\overline{G(-x)}$ defines an anti-linear isometric involution $C \\\\colon \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi } \\\\rightarrow \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }$ .\",\n \"This induces an anti-linear isometric involution $D \\\\colon (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^* \\\\rightarrow (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ , $D\\\\mu (G) = \\\\overline{\\\\mu (CG)}, \\\\quad G \\\\in \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi }.$ According to Proposition REF , $( f, g ) = \\\\mu (\\\\mathcal {F}^{-1}g)$ , $g \\\\in C_c^\\\\infty (\\\\Omega )$ , defines a distribution on $\\\\Omega $ such that $\\\\Vert \\\\Gamma _{f,\\\\Xi }\\\\Vert = \\\\Vert \\\\mu \\\\Vert $ .\",\n \"By Theorem REF , there is a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b}|_{\\\\Omega } = f$ .\",\n \"Since $\\\\operatorname{PW}_\\\\Omega ^1 \\\\subset L^1({\\\\mathbb {R}}^d)$ , we can interpret $b$ as an element of $(\\\\operatorname{PW}_\\\\Omega ^1)^*$ , $b(G) = \\\\langle G, b \\\\rangle _{L^2({\\\\mathbb {R}}^d)}$ .\",\n \"Then, recalling that $JG(x) = G(-x)$ , we have that $DI^* b (G) = (b, JG) = ( f, \\\\mathcal {F}^{-1} JG ) = ( f, \\\\mathcal {F} G ) = \\\\mu (G), \\\\quad G \\\\in \\\\mathcal {F}^{-1}(C_c^\\\\infty (\\\\Omega )),$ that is, $DI^*b = \\\\mu $ , or $I^*b = D\\\\mu $ .\",\n \"Since $D$ is an involution, it follows that $I^*$ is onto.\",\n \"In other words, $I^* \\\\colon (\\\\operatorname{PW}^1_\\\\Omega )^* \\\\rightarrow (\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi })^*$ is a Banach space isomorphism, and therefore the inclusion $I \\\\colon \\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi } \\\\rightarrow \\\\operatorname{PW}^1_\\\\Omega $ is as well.\",\n \"Hence, $\\\\operatorname{PW}_{\\\\Xi }\\\\odot \\\\operatorname{PW}_{\\\\Xi } = \\\\operatorname{PW}^1_\\\\Omega ,$ and the norms of these two Banach spaces are equivalent, by the open mapping theorem.\",\n \"The method used to prove Theorem REF extends to many unbounded polyhedral sets.\",\n \"Instead of pursuing a general statement, let us consider the example of a strip in ${\\\\mathbb {R}}^2$ , $ \\\\Xi = {\\\\mathbb {R}}_+ \\\\times (0,1).$ This is an interesting addition to Theorem REF , since $\\\\Xi $ does not have a simple vertex at infinity.\",\n \"In fact, $\\\\partial \\\\Xi $ may be considered to have a cusp point there.\",\n \"Proposition 5.6 Let $\\\\Xi $ be the strip defined in (REF ), and let $f \\\\in {\\\\mathcal {D}}^{\\\\prime }(\\\\Omega )$ , $\\\\Omega = 2\\\\Xi $ .\",\n \"Then $\\\\Gamma _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded if and only if there is a function $b \\\\in L^\\\\infty ({\\\\mathbb {R}}^d)$ such that $\\\\hat{b} |_{\\\\Omega }=f.$ Let $\\\\nu _1, \\\\nu _2 \\\\in C_c^\\\\infty ({\\\\mathbb {R}})$ be functions such that $\\\\nu _1(t) + \\\\nu _2(t) = 1$ for $t \\\\in [0,2]$ , $\\\\nu _1$ vanishes in a neighborhood of 2, and $\\\\nu _2$ vanishes in a neighborhood of 0.\",\n \"Let $\\\\mu _j(x) = \\\\nu _j(x_2), \\\\quad j=1,2, \\\\; x = (x_1, x_2) \\\\in \\\\Omega .$ Then for $\\\\varphi \\\\in C_c^\\\\infty (\\\\Xi )$ and $x = (x_1, x_2) \\\\in \\\\Xi $ we have that $\\\\Gamma _{\\\\mu _j f}(\\\\varphi )(x)=\\\\int _{\\\\Xi } \\\\int _{{\\\\mathbb {R}}} \\\\hat{\\\\nu }_j(\\\\xi )e^{2\\\\pi i (x_2+y_2) \\\\xi }\\\\,d\\\\xi \\\\, f(x+y)\\\\varphi (y) \\\\, dy= \\\\int _{\\\\mathbb {R}} \\\\hat{\\\\nu }_j(\\\\xi ){e^{2\\\\pi i x_2\\\\xi }}\\\\Gamma _{f}(\\\\varphi _\\\\xi )(x) \\\\, d\\\\xi ,$ where $\\\\varphi _\\\\xi (y) = e^{2\\\\pi i y_2 \\\\xi }\\\\varphi (y)$ , $y = (y_1, y_2) \\\\in \\\\Xi $ .\",\n \"Hence, as before we see that $ \\\\Vert \\\\Gamma _{\\\\mu _j f, \\\\, \\\\Xi }\\\\Vert \\\\le \\\\Vert \\\\hat{\\\\nu }_j\\\\Vert _{L^1} \\\\Vert \\\\Gamma _{f, \\\\, \\\\Xi } \\\\Vert , \\\\quad j=1,2.$ As in Proposition REF and Theorem REF it is sufficient to see that $\\\\Gamma _{\\\\mu _1 f} \\\\colon L^2({\\\\mathbb {R}}_+^2) \\\\rightarrow L^2({\\\\mathbb {R}}_+^2)$ and $\\\\Gamma _{\\\\mu _2 f} \\\\colon L^2({\\\\mathbb {R}}_+ \\\\times (-\\\\infty , 1) ) \\\\rightarrow L^2({\\\\mathbb {R}}_+ \\\\times (-\\\\infty , 1))$ define bounded operators, and by symmetry it is sufficient to consider the first of the two.\",\n \"For $n \\\\in {\\\\mathbb {N}}$ , let $S_n$ denote the strip ${\\\\mathbb {R}}_+ \\\\times (n, n+1)$ , and let $r > 0$ be such that $2r < \\\\operatorname{dist}([0,2] \\\\cap \\\\operatorname{supp}\\\\nu _1, 2).$ We decompose $\\\\Gamma _{\\\\mu _1 f} \\\\colon L^2({\\\\mathbb {R}}_+^2) \\\\rightarrow L^2({\\\\mathbb {R}}_+^2)$ according to strips instead of cubes, $\\\\Gamma _{\\\\mu _1 f, \\\\, {\\\\mathbb {R}}_+^2} = \\\\sum _{m,n \\\\in {\\\\mathbb {N}}} P_{rS_n} \\\\Gamma _{\\\\mu _1 f, \\\\, {\\\\mathbb {R}}_+^2} P_{r S_m}.$ There are only a finite number of non-zero terms in this decomposition, and for any such term we by our choice of $r$ that $ rS_m + rS_n \\\\subset \\\\Omega .$ For $n,m$ corresponding to a non-zero term, we have that $\\\\Vert P_{rS_n} \\\\Gamma _{\\\\mu _1 f, \\\\, {\\\\mathbb {R}}_+^2} P_{r S_m} \\\\Vert = \\\\Vert \\\\Psi _{\\\\mu _1f, rS_m, rS_n}\\\\Vert = \\\\Vert \\\\Psi _{\\\\mu _1f, rS_m + z, rS_n - z}\\\\Vert = \\\\Vert \\\\Psi _{\\\\mu _1f, rS_{\\\\frac{m+n}{2}}, rS_{\\\\frac{m+n}{2}}}\\\\Vert ,$ where $z = (0, r(n-m)/2)$ .\",\n \"Since $rS_{\\\\frac{m+n}{2}} \\\\subset \\\\Xi $ by (REF ) and $\\\\Gamma _{\\\\mu _1 f} \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded by (REF ), we conclude that each non-zero term $P_{rS_n} \\\\Gamma _{\\\\mu _1 f, \\\\, {\\\\mathbb {R}}_+^2} P_{r S_m}$ is bounded.\",\n \"Hence $\\\\Gamma _{\\\\mu _1 f} \\\\colon L^2({\\\\mathbb {R}}_+^2) \\\\rightarrow L^2({\\\\mathbb {R}}_+^2)$ is bounded, finishing the proof.\"\n ],\n [\n \"General domain Toeplitz operators\",\n \"In this section we consider general domain Toeplitz operators on open convex domains $\\\\tilde{\\\\Xi }\\\\subset {\\\\mathbb {R}}^d$ such that both $\\\\operatorname{cc}_{\\\\tilde{\\\\Xi }}$ and $\\\\operatorname{bc}_{\\\\tilde{\\\\Xi }}$ have non-empty interior (as in the classical case $\\\\tilde{\\\\Xi }={\\\\mathbb {R}}_+$ ).\",\n \"This forces $\\\\tilde{\\\\Xi }$ to be unbounded and, as we shall soon see, it also entails that $\\\\tilde{\\\\Omega }=\\\\tilde{\\\\Xi }-\\\\tilde{\\\\Xi }= {\\\\mathbb {R}}^d$ .\",\n \"We shall also consider more general open connected sets $\\\\Xi $ such that there are points $x_0$ and $x_1$ for which $x_1+\\\\tilde{\\\\Xi }\\\\subset \\\\Xi \\\\subset x_0+\\\\tilde{\\\\Xi },$ and prove that $\\\\Vert \\\\Theta _{f, \\\\Xi }\\\\Vert =\\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty }$ under this hypothesis.\",\n \"This allows for domains $\\\\Xi $ with very irregular boundaries, in sharp contrast to Theorem REF .\",\n \"The corresponding class of operators $\\\\Theta _{f,\\\\Xi }$ partially extends the class of generalized Toeplitz operators considered in [22], see Section REF .\",\n \"The next theorem can also be recovered by verifying the hypotheses of and keeping track of the constants in the proof of [22].\",\n \"However, for completeness we prefer to give our own concrete proof.\",\n \"Theorem 6.1 Let $\\\\Xi $ be a set as above.\",\n \"Then $\\\\Xi -\\\\Xi ={\\\\mathbb {R}}^d$ and, for $f\\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d)$ , we have that $\\\\Theta _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded if and only if $f\\\\in {\\\\mathcal {F}}^{-1}(L^\\\\infty )$ .\",\n \"Moreover, $\\\\Vert \\\\Theta _f\\\\Vert =\\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty }$ .\",\n \"Fix $z\\\\in {\\\\mathbb {R}}^d$ and set $|z|=R$ .\",\n \"Pick a vector $e\\\\in \\\\operatorname{int}(\\\\operatorname{cc}_{{\\\\tilde{\\\\Xi }}})$ with distance greater than $R$ to the complement of $\\\\operatorname{cc}_{\\\\tilde{\\\\Xi }}$ , which is possible since $\\\\operatorname{cc}_{{\\\\tilde{\\\\Xi }}}$ is a cone with non-empty interior.\",\n \"Then $e+z\\\\in \\\\operatorname{cc}_{{\\\\tilde{\\\\Xi }}}$ , so for any $x \\\\in \\\\tilde{\\\\Xi }$ we have that $x_1+x+e+z\\\\in x_1+\\\\tilde{\\\\Xi }\\\\subset \\\\Xi $ .\",\n \"Similarly, $x_1+x+e\\\\in \\\\Xi $ .\",\n \"Since $z$ is the difference of these two vectors, the first claim follows.\",\n \"Suppose that we have proven the theorem for all $f \\\\in C_c^\\\\infty ({\\\\mathbb {R}}^d)$ .\",\n \"If $f$ is a general symbol for which $\\\\Theta _f$ is bounded, consider the sequence of functions $f_n \\\\in C_c^\\\\infty ({\\\\mathbb {R}}^d)$ from Proposition REF .\",\n \"Then $\\\\hat{f}_n$ has, by Alaoglu's theorem, a subsequence $\\\\hat{f}_{n_k}$ which converges weak-star in $L^\\\\infty $ to some element $g$ .\",\n \"Since $f_n$ converges to $f$ in distribution, it must be that $g = \\\\hat{f}$ .\",\n \"Hence $f\\\\in {\\\\mathcal {F}}^{-1}(L^\\\\infty )$ and, by Propositions REF and REF , we have that $\\\\Vert \\\\hat{f} \\\\Vert _{L^\\\\infty } \\\\le \\\\varlimsup _{k \\\\rightarrow \\\\infty } \\\\Vert \\\\hat{f}_{n_k} \\\\Vert _{L^\\\\infty } = \\\\varlimsup _{k \\\\rightarrow \\\\infty } \\\\Vert \\\\Theta _{f_{n_k}}\\\\Vert \\\\le \\\\Vert \\\\Theta _{f}\\\\Vert \\\\le \\\\Vert \\\\hat{f} \\\\Vert _{L^\\\\infty }.$ This proves the theorem for general symbols.\",\n \"Hence we assume that $f \\\\in C_c^\\\\infty ({\\\\mathbb {R}}^d)$ .\",\n \"Fix $\\\\xi \\\\in {\\\\mathbb {R}}^d$ , pick any vector $\\\\nu $ in $\\\\operatorname{int}(\\\\operatorname{bc}_{\\\\tilde{\\\\Xi }})$ , and consider for $\\\\varepsilon > 0$ the function $E_{\\\\varepsilon }(x)=e^{\\\\varepsilon x\\\\cdot \\\\nu +2\\\\pi i x\\\\cdot \\\\xi }\\\\chi _{\\\\Xi }(x), \\\\quad x \\\\in \\\\Xi .$ By [1] this function is in $L^2(x_0+\\\\tilde{\\\\Xi })$ ,The set $\\\\operatorname{bc}_{\\\\tilde{\\\\Xi }}$ was denoted $\\\\Theta $ in [1].\",\n \"and hence $E_\\\\varepsilon \\\\in L^2(\\\\Xi )$ .\",\n \"We use $E_\\\\varepsilon $ as a test function: $\\\\Vert \\\\Theta _f\\\\Vert \\\\ge \\\\left|\\\\frac{\\\\langle \\\\Theta _f E_{\\\\varepsilon },E_{\\\\varepsilon }\\\\rangle }{\\\\Vert E_\\\\varepsilon \\\\Vert ^2}\\\\right| &= \\\\left|\\\\frac{1}{\\\\Vert E_\\\\varepsilon \\\\Vert ^2}\\\\int \\\\int f(x-y)e^{\\\\varepsilon (x+y)\\\\cdot \\\\nu }e^{2\\\\pi i (y-x)\\\\cdot \\\\xi }\\\\chi _{\\\\Xi }(y)\\\\chi _{\\\\Xi }(x) \\\\, dy \\\\, dx\\\\right| \\\\\\\\&=\\\\left|\\\\frac{1}{\\\\Vert E_\\\\varepsilon \\\\Vert ^2}\\\\int f(z)e^{-2\\\\pi i z\\\\cdot \\\\xi }\\\\int e^{\\\\varepsilon (z+2y) \\\\cdot \\\\nu }\\\\chi _{\\\\Xi }(z+y)\\\\chi _{\\\\Xi }(y) \\\\, dy \\\\,dz\\\\right|$ Hence it follows that $\\\\Vert \\\\Theta _f\\\\Vert \\\\ge |\\\\hat{f}(\\\\xi )|$ upon showing that $\\\\lim _{\\\\varepsilon \\\\rightarrow 0^+} \\\\frac{e^{\\\\varepsilon z \\\\cdot \\\\nu }}{\\\\Vert E_\\\\varepsilon \\\\Vert ^2} \\\\int e^{2\\\\varepsilon y \\\\cdot \\\\nu }\\\\chi _{\\\\Xi }(z+y)\\\\chi _{\\\\Xi }(y) \\\\, dy = 1$ uniformly on compacts in $z$ .\",\n \"Since $\\\\xi $ is arbitrary this establishes that $\\\\Vert \\\\Theta _f\\\\Vert \\\\ge \\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty }$ and by Proposition REF we then conclude that $\\\\Vert \\\\Theta _f\\\\Vert =\\\\Vert \\\\hat{f}\\\\Vert _{L^\\\\infty }$ .\",\n \"Fix $R>0$ and suppose that $z \\\\in {\\\\mathbb {R}}^d$ with $|z|1,$ each multi-sequence $a=(a_n)_{n\\\\in \\\\lbrace -N+1,\\\\ldots ,N-1\\\\rbrace ^d},$ generates a multi-level block Toeplitz matrix $T_a$ .\",\n \"As an operator on $\\\\ell ^2(\\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d)$ it is given by the formula $T_a(v)(m)=\\\\sum _{n\\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d}a_{m-n}v_n, \\\\quad v \\\\in \\\\ell ^2(\\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d), \\\\; m \\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d.$ To understand this matrix, consider the $d$ -level block Toeplitz matrix $T_a$ as an ordinary $N \\\\times N$ -Toeplitz matrix with entries which are $(d-1)$ -level block Toeplitz matrices, $T_a = \\\\lbrace A_{i-j}\\\\rbrace _{i,j \\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace }, \\\\quad A_i = \\\\lbrace a_{(i,m-n)}\\\\rbrace _{m,n \\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^{d-1}}.$ For instance, a multi-level block Toeplitz matrix for $d=2$ is an $N\\\\times N$ Toeplitz matrix whose entries are $N \\\\times N$ Toeplitz matrices.\",\n \"Again, we allow for the possibility that $N=\\\\infty $ .\",\n \"We now provide the multi-level block Toeplitz matrix analogue of the Toeplitz matrix completion theorem.\",\n \"Theorem 7.1 There exists a constant $C_d > 0$ such that any finite multi-sequence $a$ can be extended to an infinite multi-sequence $\\\\tilde{a}$ on ${\\\\mathbb {Z}}^d$ for which $T_{\\\\tilde{a}} \\\\colon \\\\ell ^2({\\\\mathbb {N}}^d) \\\\rightarrow \\\\ell ^2({\\\\mathbb {N}}^d)$ is bounded with norm $\\\\Vert T_{\\\\tilde{a}}\\\\Vert \\\\le C_d\\\\Vert T_a\\\\Vert .$ Let $f=\\\\sum _{n\\\\in \\\\lbrace -N+1,\\\\ldots ,N-1\\\\rbrace ^d} a_n\\\\delta _n,$ where $\\\\delta _n$ is the Dirac delta function at $n$ , $\\\\delta _n(\\\\varphi )=\\\\varphi (n), \\\\quad \\\\varphi \\\\in C_c^\\\\infty ({\\\\mathbb {R}}^d).$ Set $\\\\Xi =(0,N)^d$ and consider $\\\\Theta _f=\\\\Theta _{f,\\\\Xi }$ .\",\n \"Given $g\\\\in C^\\\\infty _c(\\\\Xi )$ , a short calculation shows that $\\\\Theta _f(g)(x)=\\\\sum _{n\\\\in {\\\\mathbb {Z}}^d\\\\cap (x-\\\\Xi )} a_n g(x-n), \\\\quad x \\\\in (0, N)^d.$ With $x=m+r$ , where $m \\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d$ and $r\\\\in [0,1)^d$ , this can be rewritten $\\\\Theta _f(g)(m+r)=\\\\sum _{k\\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d} a_{m-k}g(r+k).$ In other words, with $g_r=\\\\lbrace g(r+n)\\\\rbrace _{n\\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d}$ , we have that $\\\\Theta _f(g)(m+r)=T_a(g_r)(m).$ Hence $\\\\sum _{m\\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d}|\\\\Theta _f(g)(m+r)|^2=\\\\Vert T_a(g_r)\\\\Vert ^2\\\\le \\\\Vert T_a\\\\Vert ^2\\\\Vert g_r\\\\Vert ^2=\\\\Vert T_a\\\\Vert ^2\\\\sum _{m\\\\in \\\\lbrace 0,\\\\ldots ,N-1\\\\rbrace ^d}|g(m+r)|^2.$ Integrating both sides over $r \\\\in (0,1)^d$ gives us that $\\\\Vert \\\\Theta _f(g)\\\\Vert ^2\\\\le \\\\Vert T_a\\\\Vert ^2\\\\Vert g\\\\Vert ^2$ .\",\n \"In other words, $\\\\Theta _f \\\\colon L^2(\\\\Xi ) \\\\rightarrow L^2(\\\\Xi )$ is bounded and $\\\\Vert \\\\Theta _f\\\\Vert \\\\le \\\\Vert T_a\\\\Vert .$ Noting that the constant $c$ in Corollary REF is invariant under homotheties, we find that there exists a distribution $\\\\tilde{f} = \\\\hat{b} \\\\in {\\\\mathcal {D}}^{\\\\prime }({\\\\mathbb {R}}^d)$ , coinciding with $f$ on $(-N,N)^d$ , such that $\\\\Vert \\\\Theta _{\\\\tilde{f},{\\\\mathbb {R}}^d}\\\\Vert \\\\le C_d\\\\Vert T_a\\\\Vert ,$ where $C_d$ only depends on the dimension $d$ .\",\n \"Of course, $\\\\Theta _{\\\\tilde{f},{\\\\mathbb {R}}^d} \\\\colon L^2({\\\\mathbb {R}}^d) \\\\rightarrow L^2({\\\\mathbb {R}}^d)$ is nothing but the operator of convolution with $\\\\tilde{f}$ .\",\n \"Now pick any function $\\\\varphi \\\\in C^\\\\infty _c((-1/2,1/2)^d)$ with $\\\\int |\\\\varphi |^2 dx=1$ and consider the isometry $I \\\\colon \\\\ell ^2({\\\\mathbb {N}}^d)\\\\rightarrow L^2({\\\\mathbb {R}}^d)$ given by $I v(x)=\\\\sum _{n\\\\in {\\\\mathbb {N}}^d}v_n \\\\varphi (x-n), \\\\quad v \\\\in \\\\ell ^2({\\\\mathbb {N}}^d), \\\\; x \\\\in {\\\\mathbb {R}}^d.$ Then $I^* g(n) = \\\\int _{{\\\\mathbb {R}}^d} g(x) \\\\overline{\\\\varphi (x-n)} \\\\, dx, \\\\quad g \\\\in L^2({\\\\mathbb {R}}^d), \\\\; n \\\\in {\\\\mathbb {N}}^d.$ It follows that $I^*\\\\Theta _{\\\\tilde{f}} I v (m) = \\\\sum _{n \\\\in {\\\\mathbb {N}}^d} \\\\tilde{a}_{m-n} v_n, \\\\quad v \\\\in \\\\ell ^2({\\\\mathbb {N}}^d), \\\\; m \\\\in {\\\\mathbb {N}}^d,$ where $\\\\tilde{a}_n = \\\\int _{{\\\\mathbb {R}}^d} \\\\int _{{\\\\mathbb {R}}^d} f(x-y + n) \\\\varphi (y) \\\\overline{\\\\varphi (x)} \\\\, dy \\\\, dx, \\\\quad n \\\\in {\\\\mathbb {Z}}^d.$ That is, $I^*\\\\Theta _{\\\\tilde{f}}I = T_{\\\\tilde{a}}$ .\",\n \"It is clear by construction that $\\\\tilde{a}$ is an extension of $a$ , $\\\\tilde{a}_n = a_n \\\\int _{{\\\\mathbb {R}}^d} |\\\\varphi (y)|^2 \\\\, dy = a_n, \\\\quad n \\\\in \\\\lbrace -N+1,\\\\ldots ,N-1\\\\rbrace ^d.$ This finishes the proof.\"\n 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[
"Nonequilibrium quasistationary spin disordered state in the\n Kitaev-Heisenberg magnet $\\alpha$-RuCl$_3$"
],
[
"Abstract Excitation by light pulses enables the manipulation of phases of quantum condensed matter.",
"Here, we photoexcite high-energy holon-doublon pairs as a way to alter the magnetic free energy landscape of the Kitaev-Heisenberg magnet $\\alpha$-RuCl$_3$, with the aim to dynamically stabilize a proximate spin liquid phase.",
"The holon-doublon pair recombination through multimagnon emission is tracked through the time-evolution of the magnetic linear dichroism originating from the competing zigzag spin ordered ground state.",
"A small holon-doublon density suffices to reach a spin disordered state.",
"The phase transition is described within a dynamic Ginzburg-Landau framework, corroborating the quasistationary nature of the transient spin disordered phase.",
"Our work provides insight into the coupling between the electronic and magnetic degrees of freedom in $\\alpha$-RuCl$_3$ and suggests a new route to reach a proximate spin liquid phase in Kitaev-Heisenberg magnets."
],
[
"Introduction",
"Light can be utilized as a tool to manipulate and engineer novel phases in quantum materials [1].",
"In particular, excitation via intense light pulses has been used to create nonequilibrium states of matter nonexistent at thermal equilibrium, such as transient superconductivity in underdoped cuprates [2], metastable ferroelectricity in SrTiO$_3$ [3], [4], and unconventional charge-density wave order in LaTe$_3$ [5].",
"The light pulses excite a transient population of quasiparticles or collective excitations, which acts as a dynamic parameter to alter the material's free-energy landscape.",
"For sufficiently strong excitation densities, a nonequilibrium phase transition can eventually occur [5], [6], [7].",
"By the same token, intense pulsed light holds promise to manipulate the spin state of frustrated magnets [8], [9].",
"These materials, in fact, can host exotic and elusive phases, such as spin liquids (SL).",
"Whereas SLs harbors rich many-body phenomena resulting from spin frustration and possible spin fractionalization [10], [11], these phases often compete with a magnetically ordered ground state, which is typically energetically favoured.",
"Pulsed light excitation can then provide a mechanism to tip the energetic balance away from the magnetically ordered ground state towards a nonequilibrium proximate spin liquid phase.",
"We explore this concept for the Kitaev-Heisenberg frustrated magnet, a type of Mott insulator with a layered honeycomb structure and strong spin-orbit coupling [12], [13], [14].",
"For these materials the large spin-orbit interaction leads to a sizeable bond directional spin exchange, whereas the symmetric Heisenberg exchange cancels out by virtue of the edge-sharing octahedra geometry, making them promising candidates for Kitaev physics [12], [13], [14].",
"Still, the remaining Heisenberg interaction, present due to small structural distortions away from the ideal honeycomb structure, [15] is an adversary to spin liquid formation, and generally favors a spin-ordered ground state [13], [16], [17].",
"By modulating spin entropy through finite temperature effects [18], [19] or by adding external magnetic fields [20], one can however stabilize proximate or field-induced spin liquid phases at thermal equilibrium.",
"These spin liquid realizations show emergent behavior expected for the pure Kitaev spin liquid,[10] most notably, fractionalized particle statistics [19] and quantized conduction phenomena.",
"[20] A case in point is $\\alpha $ -RuCl$_{3}$ .",
"This honeycomb Mott insulator has nearly-ideal $j_{\\rm eff}$ $=$ $\\tfrac{1}{2}$ isospins in highly symmetric octahedra, [21], [22] making it possibly the most promising Kitaev spin liquid host studied to date [18], [19], [20].",
"The ($B$ ,$T$ )-plane in Fig.",
"REF provides the equilibrium phase diagram as a function of magnetic field and temperature.",
"Below $T_{\\rm N}$ $\\approx $ 7K, the isospins couple in a zigzag fashion, consistent with the types of magnetic order captured by the Kitaev-Heisenberg model.",
"[13] Strong short-range spin correlations persist between $T_{\\rm N}$ and the crossover temperature $T_{\\rm H}$ $\\approx $ 100 K, hinting at the formation of a proximate spin liquid (pSL) phase within this intermediate temperature regime.",
"[18], [23], [24] Above 100 K thermal fluctuations bring the system into a conventional paramagnetic phase.",
"An additional tuning parameter is provided by an in-plane magnetic field.",
"A field of $B_{\\rm c}$ $\\approx $ 7 T is sufficient to destabilize the zigzag order.",
"For fields between 7 - 8 T a much-debated field-induced SL is then stabilized, [20] whereas for higher fields a quantum disordered state with partial field alignment of the effective moments forms.",
"[15], [25], [26] Figure: A nonequilibrium dimension to α\\alpha -RuCl 3 _{3}'s magnetic phase diagram.The (BB,TT)-plane sketches the equilibrium magnetic phase diagram.",
"Photoexcited holon-doublon pairs n γ n_{\\gamma } form a new nonequilibrium parameter.For small (red) to intermediate (magenta) quenches the system stays inside the zigzag ordered phase.",
"Above a critical density n γ, crit n_{\\gamma ,\\rm crit} a nonequilibrium proximate spin liquid state may be induced (light blue arrow).In this work, we report on the observation of a transient long-lived spin disordered state in the Kitaev-Heisenberg magnet $\\alpha $ -RuCl$_{3}$ induced by pulsed light excitation.",
"Holon-doublon pairs are created by photoexcitation above the Mott gap, and provide a new nonequilibrium dimension to $\\alpha $ -RuCl$_{3}$ 's magnetic free energy landscape and resulting phase diagram, as illustrated in Fig.",
"REF .",
"The subsequent holon-doublon pair recombination through multimagnon emission leads to a decrease of the zigzag magnetic order.",
"This is tracked through the magnetic linear dichroism (MLD) response of the system.",
"For a sufficiently large holon-doublon density the MLD rotation vanishes, implying that the zigzag ground state is fully suppressed and that a long-lived transient spin-disordered phase is induced.",
"The disordering dynamics of the zigzag order parameter is captured by a time-dependent Ginzburg-Landau model, corroborating the nonequilibrium quasistationary nature of the transient phase.",
"Our work provides insight into the coupling between high-energy electronic and low-energy magnetic degrees of freedom in $\\alpha $ -RuCl$_{3}$ and suggests a new route to reach a proximate spin-liquid phase in honeycomb Mott insulators with residual interactions beyond the bond-directional Kitaev exchange.",
"The photoinduced change in reflected polarization rotation from $\\alpha $ -RuCl$_{3}$ was measured as a function of temperature and photoexcitation density.",
"The sample is excited above the $\\Delta _{\\rm MH}$ $\\sim $ $1.0$ eV Mott-Hubbard gap [27] with a photon energy of $\\hbar \\omega $ $\\approx $ $1.55$ eV.",
"The probe light has $2.42$ eV photon energy.",
"Under zero-field conditions, two contributions to the total optical polarization rotation $\\theta _{\\rm tot}$ can be distinguished: $\\theta _{\\rm tot}= \\theta _{\\rm LD} + \\theta _{\\rm MLD}(\\vec{L}^2).$ The first term $\\theta _{\\rm LD}$ , linear dichroism, originates from the monoclinic distortion of RuCl$_{3}$ , [15] and will only show a negligible temperature dependence over the relevant temperature range.",
"[28] The second term $\\theta _{\\rm MLD}$ , magnetic linear dichroism (MLD), [29], [30] is proportional to the square of the zigzag antiferromagnetic order parameter $\\vec{L}=\\vec{M_{\\uparrow }}-\\vec{M_{\\downarrow }}$ , where $\\vec{M_{\\uparrow }}$ and $\\vec{M_{\\downarrow }}$ give the sublattice magnetizations.",
"As such, the MLD rotation provides an optical probe of the zigzag spin order in $\\alpha $ -RuCl$_{3}$ .",
"Figure: Temperature dependent transient polarization rotation and critical slowing down.",
"a) Photoinduced change in polarization rotation -Δθ MLD (t)-\\Delta \\theta _{\\rm MLD}(t) for various temperatures below and above T N T_{\\rm N} ≈\\approx 7 K. The signal below T N T_{\\rm N} is dominated by the proper stacking phase.",
"Above T N T_{\\rm N} a small signal with opposite rotational sign is observed, originating from the stacking-fault phase.",
"b) Integrated change in rotation (black spheres) and τ decay \\tau _{\\rm decay}[ps] (red circles) as a function of temperature.",
"A critical slowing down of the disordering is observed upon approaching the phase transition.Figure REF a displays the photoinduced change in polarization rotation $-\\Delta \\theta _{\\rm MLD}(t)$ for various bath temperatures.",
"A low-excitation fluence $F$ $\\sim $ $1.7$ $\\mu $ J/cm$^{2}$ was used, corresponding to a photoexcitation density of $n_{\\gamma }$ $\\approx $ $0.8$ $\\cdot $ $10^{17}$ cm$^{-3}$ (Ref. footnoteS1).",
"For temperatures below $T_{\\rm N}$ $\\approx $ 7 K an initial fast demagnetization on the tens of ps timescale is observed, after which the signal recovers on the ns-timescale.",
"Above $T_{\\rm N}$ a small amplitude response is observed with an opposite rotational sense, originating from a fraction of unavoidable stacking-fault-phase contributions at the sample surface.",
"[27], [32] In Fig.",
"REF b the integrated change in rotation $\\Delta \\theta _{\\rm max}$ is plotted versus temperature.",
"The integrated rotation change shows a pronounced increase, followed by a rapid reduction upon approaching $T_{\\rm N}$ $\\approx $ 7 K. This behavior is qualitatively rationalized by considering that the photoexcitation will have the largest transient effect where the derivative of the zigzag order parameter with respect to temperature is the largest [33].",
"Concomitantly, we observe a critical slowing down of the disordering upon approaching the phase transition [34], [35].",
"This behavior is well captured by a $\\tau _{\\rm decay}$ $\\propto $ $\\vert 1-T/T_{\\rm N}\\vert ^{-\\nu z}$ power law with critical exponent $\\nu z$ $=$ $-2.1$ , compatible with the universality class of the 2D Ising model-A dynamics, applicable to $\\alpha $ -RuCl$_{3}$ [33], [34], [36].",
"Figure REF a shows the transient rotation traces $\\theta _{\\rm MLD}(t)$ for various initial photoexcitation densities $n_{\\gamma }$ (sphere symbols).",
"The photoexcitation dependence of the maximum MLD change, $\\Delta \\theta _{\\rm MLD,max}$ , is depicted in Fig.",
"REF b. Qualitatively, two excitation regimes can be distinguished.",
"For lower excitation densities ($n_{\\gamma }$ $<$ $n_{\\rm \\gamma ,crit}$ $\\approx $ 3 $\\cdot $ $10^{17}$ cm$^{-3}$ ), the spin system partially disorders, followed by a subsequent recovery.",
"In this regime the disordering time slows down with increasing photoexcitation density.",
"For the high excitation densities ($n_{\\gamma }$ $>$ $n_{\\rm \\gamma , crit}$ $\\approx $ 3 $\\cdot $ $10^{17}$ cm$^{-3}$ ) a faster disordering time is observed and the change $\\Delta \\theta _{\\rm MLD,max}$ saturates (Fig.",
"REF b), implying that the photoexcited system resides in a $L$ = 0 state for multiple 100s of ps.",
"Referring to the magnetic phase diagram (Fig.",
"REF and Refs.",
"johnson2015,kasahara2018majorana), this means that for quench strengths above $n_{\\rm \\gamma ,crit}$ the zigzag order can be fully suppressed, leaving the system in a spin disordered state.",
"The disordering mechanism and the nature of the long-lived transient state is corroborated below.",
"Figure: Nonequilibrium magnetic phase transition and holon-doublon pair recombination by multimagnon emission.",
"a) Density-dependent θ MLD (t)\\theta _{\\rm MLD}(t) for different excitation densities n γ n_{\\gamma }, as indicated with spheres.",
"The modelled rotation θ(t)\\theta (t) is indicated with thick lines.",
"b) Maximum change in the magnetic linear dichroism (MLD) rotation Δθ MLD (t)\\Delta \\theta _{\\rm MLD}(t) as a function of photon density n γ n_{\\gamma }.",
"Above the critical density n γ, crit n_{\\rm \\gamma ,crit} ≈\\approx 3 ·\\cdot 10 17 10^{17} cm -3 ^{-3} the maximum change in MLD-rotation saturates.c) The honeycomb lattice, consisting of Ru-sites (dark-blue sites) and chloride ligand ions (red sites).",
"The lower processshows the photogeneration of a holon-doublon pair.",
"The upper process shows the subsequent multimagnon emission by holon-doublon recombination.The inherently strong charge-spin coupling of Mott insulators leads to an efficient nonlinear demagnetization mechanism upon photoexcitation above the Mott-Hubbard gap.",
"[37], [38] In order to illustrate this mechanism, first consider the photoexcitation process corresponding to the lowest $t_{2g}^{5}$ $t_{2g}^{5}$ $\\rightarrow $ $t_{2g}^{4}$ $t_{2g}^{6}$ hopping-type excitation across the Mott-Hubbard gap, as illustrated by the lower hopping process in Fig.",
"REF c. Within a quasiparticle picture, this intermediate excited state corresponds to a spinless holon ($t_{2g}^{4}$ ) and doublon ($t_{2g}^{6}$ ), by which effectively two magnetic moments are removed from the zigzag lattice.",
"The mere creation of these quasiparticles at the used low densities of 4 - 85 ppm photons/Ru$^{3+}$ -site however does not suffice to explain the magnitude and timescale of the zigzag disordering.",
"[31] Instead, once created, the dominant decay mechanism of the holon-doublon pairs is recombination through multimagnon emission (upper hopping process Fig.",
"REF c).",
"[37] An order of magnitude estimate for the released amount of magnons per decayed $hd$ -pair is provided by $\\Delta _{\\rm MH}/W$ $\\sim $ 25 (Refs.",
"lenarcic2013), with $W$ $\\approx $ $4.0$ meV being the bandwidth of the low-energy spin wave branch in the zigzag phase.",
"[39] As such, this quasiparticle recombination provides an efficient electronic demagnetization mechanism.",
"In order to further delineate the excitation mechanism and resulting magnetization dynamics, we model the time-domain data within a dynamic Ginzburg-Landau (GL) model.",
"[34], [36] The holon-doublon density, representing the nonequilibrium dimension in Fig.",
"REF , comes in as a new dynamical variable here.",
"We first consider the modified free energy for the antiferromagnetic order parameter $L$ and the holon-doublon-pair density $n$ : $\\mathcal {F}(n,L) = \\frac{a_1}{2}(n-n_{\\rm c,eq}) L^2 + \\frac{a_2}{4}L^4 + \\tilde{\\mathcal {F}}(n),$ with $\\tilde{\\mathcal {F}}(n) = a_3 n + \\frac{a_4}{2} n^2 +\\frac{a_5}{3}n^3 ,$ where $a_i$ , $i=1, \\dots , 5$ are phenomenological parameters.",
"The terms with even powers in the zigzag order parameter $L$ are the standard symmetry-allowed terms in the Landau free energy expansion for an antiferromagnet.",
"[36], [40] Notice that odd powers of $L$ are ruled out by the inversion symmetry of RuCl$_{3}$ .",
"[15] The initial value of the holon-doublon pair density $n(0)$ , or quench strength, is taken proportional to the experimental photoexcitation densities $n_{\\gamma }$ , i.e., $n(0)$ $\\propto $ $n_{\\gamma }$ , where each photon creates one $hd$ -pair.",
"The first term in $\\mathcal {F}(n,L)$ , coupling the $hd$ -pair density $n$ to the order parameter $L$ , leads to a destabilization of the magnetic order for a sufficiently strong excitation of $hd$ -pairs, thus reproducing the process of annihilation of $hd$ -pairs into magnons.",
"[37], [41] The parameter $n_{\\rm c,eq}$ is introduced as the critical $hd$ -pair density at equilibrium.",
"The functional $\\tilde{\\mathcal {F}}(n)$ , independent of the order parameter $L$ , describes the excess energy of the $hd$ -density and its relaxation in the absence of magnetization.",
"It therefore accounts for decay mechanisms other than the nonradiative multimagnon emission discussed above, such as nonradiative phonon emission, spontaneous decay under radiative emission,[42], and possible $hd$ -pair diffusion out of the probe volume.",
"The form of $\\tilde{\\mathcal {F}}(n)$ is chosen as a third-order polynomial, although its exact form is not crucial for the analysis.",
"The time evolution of the $hd$ -pair density $n$ and magnetic order parameter $L$ is described by the coupled equations of motion: $\\frac{d L}{dt}=-\\frac{\\delta \\mathcal {F}}{\\delta L}, \\qquad \\frac{d n}{dt}=-\\frac{\\delta \\mathcal {F}}{\\delta n}.$ In order to relate Eqs.",
"(REF ) to the experimentally measured rotation $\\theta _{\\rm MLD}$ , we rewrite the equations in terms of the polarization rotation $\\theta $ $=$ $L^2/2$ , to finally obtain: $\\frac{d \\theta }{d t} &=-2a_1 (n-n_{\\rm c,eq})\\theta -4a_2\\theta ^2, \\\\\\frac{d n}{d t} & = -a_1\\theta -\\frac{\\delta }{\\delta n}\\tilde{\\mathcal {F}}(n)$ By using Eq.",
"(REF )a, the trajectories $n(t),\\theta (t)$ can be modelled for different initial quench strengths $n(0)$ , taken proportional to the experimental $n_{\\gamma }$ densities.",
"The curves for $\\theta (t)$ are superimposed on the experimental $\\theta _{\\rm MLD}(t)$ in Fig.",
"REF a.",
"The model captures the dependence of the demagnetization time on the excitation density and the position of $t_{\\rm max}$ , i.e., the time at which $\\Delta \\theta _{\\rm max}$ is reached.",
"For the higher excitation densities the magnetic order vanishes, reproducing the long-lived transient $L$ $=$ 0 state.",
"The inclusion of the $n^3$ -term in $\\tilde{\\mathcal {F}}(n)$ ensures that the GL-description does not overestimate the lifetime of the $L$ $=$ 0 state.",
"[43] The density-dependent rotation transients are well captured considering the minimal amount of parameters needed in the nonequilibrium GL-description.",
"The $hd$ -pair density-dependent free-energy landscape $\\mathcal {F}(n,L)$ is shown in Fig.",
"REF .",
"For low densities, the free energy retains its double-well profile, whereas for higher densities a single well forms.",
"[35] Representative trajectories $n(t),L(t)$ for different excitation densities are drawn into the free-energy landscape, with colors corresponding to the conceptual trajectories of Fig.",
"REF .",
"The quench $n(0)$ brings the system into a high-energy state, after which $n$ and $L$ relax along the minimal energy trajectory.",
"For a small quench (red trajectory) the zigzag order parameter $L(t)$ stays finite and eventually recovers.",
"For the intermediate densities (magenta trajectory) the $n(t),L(t)$ coordinates approach the $L$ $=$ 0 line.",
"For the higher excitation densities (light blue trajectory) the $hd$ -pairs have sufficient excess energy to let $n(t),L(t)$ follow a trajectory along the $L$ $=$ 0 line, i.e., full spin disordering is reached.",
"We emphasize that, for strong quenches, the excitation density $n(t)$ still varies in time, even though $L(t)$ takes the quasistationary value $L(t)$ $=$ 0.",
"A sufficiently strong photoexcitation quench thus provides a mechanism to dynamically stabilize a nonequilibrium quasistationary spin disorded state in $\\alpha $ -RuCl$_{3}$ .",
"Figure: Free energy landscape and nonequilibrium quasistationary spin disordered state.",
"Free energy landscape F(n,L)F(n,L) as a function of the zigzag order parameter LL and hdhd-pair density nn (cf.",
"the phase diagram in Fig. ).",
"The initial quench n(0)n(0) brings the system to a high energy state, after which the system relaxes.",
"For small quenches (red trajectory), the order parameter LL stays finite under the relaxation of the density nn.",
"For intermediate quenches (magenta trajectory) the system approaches the LL =0 line.",
"For strong quenches (light blue trajectory, highest energies not shown) the system relaxes along the LL =0 line, implying that the system is described as a nonequilibrium quasistationary spin disordered state.The maximum lifetime of the nonequilibrium quasistationary spin disordered state is dictated by the recombination rate of the $hd$ -pairs.",
"The time evolution of $n(t)$ obtained from the nonequilibrium GL model provides us with an estimate of a few nanoseconds for the recombination timescale of the $hd$ -pairs.",
"[43] This timescale is expected to grow exponentially with the number of magnons needed to traverse the Mott gap, i.e., $\\tau $ $\\sim $ $e^{\\Delta _{\\rm MH}/W}$ .",
"[37] Considering the weak exchange-interaction scale $W$ in $\\alpha $ -RuCl$_{3}$ , one may indeed expect significantly longer recombination times compared to materials with an order of magnitude stronger exchange, such as Nd$_2$ CuO$_4$ and Sr$_2$ IrO$_4$ , where $hd$ -pair lifetimes on the order of $0.1$ ps have been reported.",
"[38], [44] A large ratio between the Mott gap and the exchange interaction energy thus is the key element to ensure a long lifetime of the nonequilibrium quasistationary state.",
"The microscopic nature of the transient long-lived spin disordered state currently remains elusive.",
"The used low excitation densities by far do not provide sufficient energy to drive the material into a conventional paramagnetic state, nor to change the dominant interactions in the system.",
"Considering the phase diagram, it therefore seems plausible that the system is driven into a transient proximate spin liquid phase, reminiscent of the thermodynamic state just above $T_{\\rm N}$ .",
"Energy-resolved ultrafast techniques may provide more insight into the microscopic properties of the induced phase.",
"[17], [19], [45] Furthermore, exact diagonalization [46] and nonequilibrium dynamical mean-field theory [47] methods may elucidate the role of $hd$ -excitations in the Kitaev-Heisenberg model and the resulting phase diagram.",
"We have unveiled a pulsed light excitation driven mechanism allowing to trap a Kitaev-Heisenberg magnet into a quasistationary spin disordered state.",
"Photoexcitation above the Mott-gap generates a transient density of holon-doublon quasiparticle pairs.",
"The subsequent recombination of these quasiparticles through efficient multimagnon emission provides a way to dynamically destabilize the competing zigzag ordered ground state and thereby keeps the system in an out-of-equilibrium spin disordered state, up until the transient electronic quasiparticle density gets depleted.",
"Our work provides insight into the coupling between electronic and magnetic degrees of freedom in $\\alpha $ -RuCl$_{3}$ , and suggest a new way to reach a proximate spin liquid phase in Kitaev- Heisenberg magnets.",
"High-quality $\\alpha $ -RuCl$_{3}$ crystals were prepared by vacuum sublimation.",
"[26] Different samples of the batch were characterized by SQUID magnetometry, showing a sharp phase transition at $T_{\\rm N}$ $\\approx $ 7K.",
"This bulk technique can only provide a first indication of sample quality for an optics study.",
"Cleaving or polishing of RuCl$_{3}$ samples introduces strain, which leads to stacking faults.",
"For the optics study, we therefore refrained from any sample treatment, and used an as-grown RuCl$_{3}$ sample with a shiny $\\sim $ $1.5$ $\\times $ $1.5$ mm$^2$ surface area.",
"The temperature dependence shown in Fig.",
"REF b shows a clear phase transition at $T_{\\rm N}$ $\\approx $ 7K.",
"The $\\alpha $ -RuCl$_{3}$ sample is mounted in a bath cryostat.",
"The time-resolved magneto-optical experiment was performed using 800 nm pump pulses with a temporal with of 40 fs, and probe pulses of 512 nm with a temporal width of 250 fs.",
"The pump and probe beam were focused down to a radius of $r_{\\rm pump}$ $\\approx $ 39 $\\mu $ m and $r_{\\rm probe}$ $\\approx $ 25 $\\mu $ m, respectively.",
"The repetition rate of the amplified laser system was set to $f$ $=$ 30 kHz in order to ensure that the system can relax back to the ground state between consecutive pulses.",
"The change in polarization rotation of the reflected probe pulse is measured via a standard polarization bridge scheme.",
"The optical conductivity reported in Ref.",
"sandilands2016 and the structural properties reported in Ref.",
"johnson2015 allows us to calculate the photoexcitation densities, as outlined in more detail in the Supplementary Material.",
"[31] The authors thank A. Rosch (Cologne, DE) and Z. Lenarčič (Berkeley, USA) for fruitful discussions.",
"This project was partially financed by the Deutsche Forschungsgemeinschaft (DFG) through Project No.",
"277146847 - Collaborative Research Center 1238: Control and Dynamics of Quantum Materials (Subprojects No.",
"B05 and No.",
"C04) and through project INST 216/783-1 FUGG.",
"S.D.",
"acknowledges support by the European Research Council (ERC) under the Horizon 2020 research and innovation program, Grant Agreement No.",
"647434 (DOQS)."
]
] | 2005.14189 |
[
[
"Influence of grain growth on the thermal structure of protoplanetary\n discs"
],
[
"Abstract The thermal structure of a protoplanetary disc is regulated by the opacity that dust grains provide.",
"However, previous works have often considered simplified prescriptions for the dust opacity in hydrodynamical disc simulations, e.g.",
"by considering only a single particle size.",
"In the present work we perform 2D hydrodynamical simulations of protoplanetary discs where the opacity is self-consistently calculated for the dust population, taking into account the particle size, composition and abundance.",
"We first compare simulations using single grain sizes to two different multi-grain size distributions at different levels of turbulence strengths, parameterized through the $\\alpha$-viscosity, and different gas surface densities.",
"Assuming a single dust size leads to inaccurate calculations of the thermal structure of discs, because the grain size dominating the opacity increases with orbital radius.",
"Overall the two grain size distributions, one limited by fragmentation only and the other determined from a more complete fragmentation-coagulation equilibrium, give similar results for the thermal structure.",
"We find that both grain size distributions give less steep opacity gradients that result in less steep aspect ratio gradients, in comparison to discs with only micrometer sized dust.",
"Moreover, in the discs with a grain size distribution, the innermost outward migration region is removed and planets embedded is such discs experience lower migration rates.",
"We also investigate the dependency of the water iceline position on the alpha-viscosity, the initial gas surface density at 1 AU and the dust-to-gas ratio and find $r_{ice} \\propto \\alpha^{0.61} \\Sigma_{g,0}^{0.8} f_{DG}^{0.37}$ independently of the distribution used.",
"The inclusion of the feedback loop between grain growth, opacities and disc thermodynamics allows for more self-consistent simulations of accretion discs and planet formation."
],
[
"Introduction",
"Protoplanetary discs surround young stars for the first few million years after their formation and they are the birthplaces of planetary systems.",
"The position of the iceline within the discs influences the formation and growth of planets.",
"Planetesimal formation has been found to be enhanced or even initiated there because of water vapor that is diffused outwards from the hot, inner disc and recondenses after the iceline [92].",
"This recondensation increases the abundances of icy pebbles, which have better sticking properties compared to dry aggregates [100], [104], [45], causing a pile-up near the iceline and triggering the streaming instability [44], [96], [39].",
"An increase in the dust surface density after the iceline can also aid in the growth of gas giant planet cores [99].",
"The location and the evolution of the iceline location can be defining for the innermost boundary of gas giant formation and along with other parameters, such as the disc's mass, it can also determine what kind of planets will be created [56] and their masses [74], [73].",
"In addition to that, the location of the iceline transition affects the composition of exoplanetary atmospheres [66], [31], [65] .",
"The location of the iceline is determined by the local temperature in the disc [48], [94], [86].",
"The thermal structure of the discs is thus decisive for planetesimal and planet formation.",
"It is, though, greatly affected by the dust content of the protoplanetary disc and the opacity that the dust grains provide.",
"This complex interplay is caused by the influence between gas and dust.",
"The relative velocity for each pair of grains is determined by the aerodynamic properties of the grains, namely the Stokes number, and the local properties of the gas, such as the temperature or the volume density [81].",
"The variety in the relative velocities results in different collisional outcomes between grain sizes, such as coagulation or fragmentation [25], [111], [15], [14].",
"As a result, the dust content of the protoplanetary disc is described by a distribution of grain sizes, with number densities that are not necessarily equally distributed between all existing sizes.",
"Each grain size population has a different opacity, therefore having a distribution instead of a single grain size means that the disc's total opacity will be affected and as a consequence it will affect the resulting structure of the disc.",
"As stated in [13], the opacity, defines the observational characteristics of a protoplanetary disc by influencing the dust thermal continuum emission and the excitation conditions for the gas lines.",
"Additionally, since opacity regulates the amount of light that can be absorbed by the disc, it determines its thermal structure.",
"These reasons make opacity a very important factor of the structure and evolution of a protoplanetary disc.",
"The interplay between opacity and the thermal structure creates a feedback loop that we include in hydrodynamical simulations of equilibrium discs (Fig.",
"REF ).",
"Even though the goal of theoretical models is to simulate protoplanetary discs as realistically as possible, typically they only include specific parts of the feedback loop, contrary to this work.",
"In the following paragraphs we will introduce what work has been done in parts of the feedback loop.",
"Figure: Graphical illustration of the feedback loop.",
"The thermal and density structure of aprotoplanetary disc is determined by this loop: temperature and gas surface density affect the relative velocities for grains of different sizes.",
"Through the relative velocities we find the outcomes of collisions between grains, therefore a grain size distribution is created.",
"The grains are then vertically distributed according to their sizes and the turbulence strength.",
"The spatial distribution of the grain sizes determines the opacity of the disc, which then affects its cooling rate and the stellar heating.",
"This way the temperature and density of the disc change and subsequently its whole structure.Relative velocities and grain size distribution.",
"Early on, [93] worked on dust growth within the context of planet formation and on the time evolution equation for grain size distributions (often called Smoluchowski equation, [98]).",
"A lot of work was also done on dust dynamics and how they would affect collisional outcomes and, as a consequence, coagulation and fragmentation of dust particles [107], [108], [77], [25].",
"A grain size distribution has been widely assumed to follow a power-law derived from the equilibrium between coagulation and fragmentation, inspired by the work of [36] on the number density distribution of objects in the asteroid belt.",
"The number density, thus, can be approximated as $n(s) \\propto s^\\xi $ , where s is the grain size and $\\xi $ a constant.",
"Several attempts were made in order to define this constant, mainly through analytical calculations combined with observational data for the interstellar medium grains [67], but also through experimental studies [33].",
"It was shown by [102] that the $\\xi $ constant is independent of the specific parameters of the collisional outcome model, as long as it is self-similar, which in this case means that the outcome of impacts between dust grains depends on the masses of two colliding particles only through their ratio.",
"However, such a description of a grain size distribution with only one power law is a simplification, since it only takes into account the coagulation/fragmentation equilibrium.",
"More recently, the work on grain size distributions has been aided by laboratory experiments of dust collisions [23], [46].",
"Such experiments determine what the collisional outcomes are between particles of equal or different size, for different relative velocities.",
"They also help in creating models to simulate such collisions accurately and they can be used in the effort of understanding which processes are relevant within the context of planetesimal formation in protoplanetary discs [112].",
"If additional effects are also taken into account, such as cratering or different regimes due to size-dependent relative velocities, then the size distribution is described by broken power laws [15].",
"The studies that were discussed above focused on the local distribution of grains in a protoplanetary disc patch due to fragmentation and growth by coagulation, and typically assume that the gas disc does not evolve in time and the dust has no effect on the gas.",
"Opacity.",
"As a first step, some work has been done on opacity alone within the context of protoplanetary discs [70], [38], [32].",
"The goal of those works is to create a simple opacity model that can describe as realistically as possible the dust opacity and can be then used in disc simulations [18] or help in the interpretation of disc observations [12].",
"Alongside the theoretical models, several observations of the dust emission have been performed in order to connect opacity with the particle sizes present in the protoplanetary discs [78], [2], [3], [91], [63], [89], [90], [103].",
"Disc structure and grain size distribution.",
"Several works in the recent years aimed to couple the dust and gas components of protoplanetary discs in simulations and in most of the cases such models include a grain size distribution.",
"However, the models that will be discussed here simulate the gas component of a protoplanetary disc and how the dust component is affected by the gas, but the solids do not influence the gas.",
"Even without the back-reactions of dust on gas, modeling grain size distributions can be computationally challenging, given the long list of effects and parameters to be taken into account, especially using N-body like techniques to treat dust particles.",
"As a consequence, some of the first attempts on this kind of models were made using the Monte-Carlo method [84], [83] and the goal was to examine how the internal structure of dust affects the collisional evolution of the particles and the disc structure.",
"The Monte-Carlo method has been also used in [112], [111], while in their work the experimental collisional outcomes from [46] were implemented and the effect of the porosity and settling of the dust grains on the collisional outcomes was tested.",
"[25] and [11] numerically solve the Smoluchowski equation for the coagulation/fragmentation equilibrium in vertically isothermal steady-state gas discs, while [80]studied the effects of the dust grain porosity on the dust evolution in a similar disc setup.",
"In the works discussed above the feedback of the dust on the gas disc structure and especially its thermal part, is not taken into account.",
"Disc structure, opacity and cooling rate.",
"The category of models that was described above neglected the effects of opacity, even though the dust opacity regulates the cooling rate of the disc, which affects the disc structure.",
"In recent years, some studies tried to fill this gap by including the effect of dust opacity in disc simulations.",
"[79] performed 1+1D In the 1+1D approach, the vertical structure of each annulus is solved independently and then all of the annuli are used to construct the radial and vertical structure of the disc.",
"simulation focusing on the effect that water-ice opacity has on the location of the iceline.",
"In the aforementioned study the wavelength-dependent opacities of water-ice and silicates are directly used when calculating the radiative transfer.",
"In [18], [22], [20] the [10] opacity profile is followed (in [18] constant opacity discs were also modeled) and 2D simulations (radial and vertical direction, assuming axisymmetry) are performed using the NIRVANA and the FARGOCA code adding radial heat diffusion and stellar irradiation.",
"The effect of the water-ice to silicates ratio on the resulting thermal disc structures has also been studied recently [19] using the FARGOCA code and the opacity module from the RADMC-3D code to calculate the mean opacities (as in the present work), but the opacity differences for the water-to-silicate fractions considered are then translated into differences in the [10] opacity model.",
"The [10] opacity model gives approximate values for the frequency averaged opacities within specific temperature regimes (e.g.",
"ice grains, evaporation of ice grains, metal grains, etc.)",
"assuming micrometer sized particles.",
"The fixed opacity profile then gives the cooling rate and the stellar heating, therefore it defines the disc structure.",
"Even though including the opacity feedback in disc simulations is an important improvement, the aforementioned studies did not include the effect of grain growth and fragmentation, and thus only employed opacities derived for single grain sizes.",
"In addition to this, all of these studies assumed a uniform dust-to-gas ratio in the vertical direction of the disc which is in contrast to our approach in this work (see section REF ).",
"Disc structure, grain size distribution and opacity.",
"[95] coupled the dust and gas evolution in 1D simulations, while they also took into consideration the grain opacity.",
"For the mean opacity calculations they followed the approach of [49], which is similar to the [10] opacity model approach.",
"Moreover, the size distribution follows the [67] power-law.",
"It was found that since grains determine the opacity, their evolution will subsequently change the opacity and therefore affect the structure and evolution of a protoplanetary disc.",
"Prior to this study, [72] and [71] included the dust component evolution in accretion discs and used the results to perform grain opacity calculations.",
"In [101] the coagulation/fragmentation equilibrium is included in order to investigate how the dust emission is affected by the grain size distribution and its corresponding opacity.",
"In this work the size distribution follows the [67] power-law and opacity was calculated using Mie theory.",
"However, in the studies discussed above the back-reaction of the opacity onto the disc structure was not taken into account.",
"In the previous paragraphs some examples were given of the work that has been done in the context of grain growth within protoplanetary discs.",
"Nevertheless, previous models were based on several simplifications, most important of which was that they neglected parts of the feedback loop (Fig.",
"REF ) that defines protoplanetary disc structures [15] or used simplified assumptions for the opacity [18].",
"The few attempts that have been made to include the dust feedback on the gas of the disc, were 1D simulations or assumed an isothermal vertical structure for the gas, in contrast to the 2D hydrodynamical models that will be presented here.",
"Secondly, the opacity is either not included in the actual simulations or the opacities were included only for single fixed grain sizes.",
"The motivation for this project is to approach a more realistic model for disc structures and their evolution and more specifically to simulate the whole feedback loop including a detailed opacity module.",
"We will consider how grain dynamics and more specifically how grain size distributions affect the opacity and as a consequence the thermal structure of the disc in order to simulate the whole feedback loop.",
"As far as the grain size distribution is concerned, two models were used for the simulations of this project.",
"A simple power-law model following [67], hereafter MRN distribution and also a more complex model following [15], hereafter BOD distribution.",
"Moreover, an opacity module was included in the 2D hydrodynamical disc simulations in order to more accurately calculate the opacity of the dust grain distribution and account for the back-reactions of dust to gas.",
"In this opacity module, the Rosseland and Planck mean opacities as a function of temperature are used and they are calculated via Mie theory.",
"The simulations were run until the disc reached thermal equilibrium.",
"Such simulations offer us the opportunity to discuss the implications of the resulting disc structures to planet formation and could also serve as the basis to compare with observations (e.g.",
"ALMA images) in future work.",
"The structure of this paper is as follows: In Sect.",
"we describe the energy equations used in the hydrodynamical simulations, the new opacity module and the two grain size distributions that we included in the code.",
"Also, we mention the input parameters of the simulations and how the disc is set up.",
"Then, in Sect.",
"we compare the discs utilizing the two different grain size distributions for one set of simulation with a nominal turbulence parameter and initial gas surface density.",
"In Sect.",
", we discuss how the simulations change when we vary the turbulence parameter and how it changes when the surface density is different.",
"The implications of the results are discussed in Sect.",
"and a summary follows in Sect.",
"."
],
[
"Methods",
"In this section we discuss the different methods used in our work.",
"We review the hydrodynamical equations in section REF , the opacities and how they are calculated in section REF .",
"In section REF we discuss the grain growth mechanism and in section REF we discuss the effects of vertical grain settling.",
"We then finally describe our simulation setup in section REF ."
],
[
"Hydrodynamical simulations",
"Calculations with mean opacities derived from single grain sizes were first introduced into the FARGOCA code by [61] and [22], who performed 2D and 3D radiation hydrodynamical simulations of discs and planet-disc interactions.",
"The FARGOCA code solves the continuity and the Navier-Stokes equations and uses the flux-limited diffusion approach to radiative transfer.",
"More specifically, the time evolution of the energy profile of the protoplanetary disc is determined by $ \\frac{\\partial E_R}{\\partial t} + \\nabla \\cdot {\\bf F} = \\rho \\kappa _P [B(T) - cE_R] $ $ \\frac{\\partial \\epsilon }{\\partial t} + \\nabla \\cdot ({\\bf u}\\cdot \\nabla )\\epsilon = -P\\nabla \\cdot {\\bf u} -\\rho \\kappa _P[B(T)-cE_R]+ Q^+ + S~.",
"$ The radiative energy density $E_R$ is thus independent from the thermal energy density $\\epsilon $ .",
"In the expressions above the blackbody radiation energy is $B(T) = 4\\sigma T^4$ , where $\\sigma $ is the Stefan-Boltzmann constant, $\\rho $ is the gas density, $\\kappa _P$ the Planck mean opacity (further specified in Sec.",
"REF ), u the velocity, P is the thermal pressure, $Q^+$ is the viscous dissipation or heating function and $S$ is the stellar heating component [62], [35], [28].",
"In our simulations we use the flux-limited diffusion (FLD) for the radiation flux ${\\bf F}$ as described in [62] ${\\bf F} = -\\frac{\\lambda c}{\\rho \\kappa _R }\\nabla E_R~.",
"$ In the flux-limited diffusion equation, $c$ is the speed of light, $\\alpha _R$ is the radiation constant, $\\kappa _R$ is the Rosseland mean opacity and $\\lambda $ the flux-limiter of [57].",
"More details on the energy equations can be found in [17].",
"The opacities that were introduced in the above equations will be discussed in the following section.",
"The stellar heating density received by a grid cell of width $\\Delta $ r is defined as [35]: $S = F_{\\star }e^{-\\tau }\\frac{1-e^{-\\rho \\kappa _{\\star }\\Delta r}}{\\Delta r}~,$ with $F_{\\star } = R_{\\star }^2\\sigma T_{\\star }^4/r^2$ being the stellar flux, $R_{\\star }$ the stellar radius, $T_{\\star }$ the stellar surface temperature, $\\tau $ the radially integrated optical depth (up to each grid cell) and $\\kappa _{\\star }$ the stellar opacity (further specified in Sec.",
"REF )."
],
[
"Opacity-Temperature module",
"In the energy equations (Eqs.",
"REF and REF ) of the hydrodynamical simulation, we use the mean opacities that are averaged over all wavelengths.",
"If we use the Planck black body radiation energy density distribution $B_{\\lambda }(\\lambda , T)$ as a weighting function we can define the Planck mean opacity as $\\kappa _P = \\frac{\\int _0^{\\infty }\\kappa _{\\lambda , ns}(T,\\rho ) B_{\\lambda }(\\lambda , T)d\\lambda }{\\int _0^{\\infty }B_{\\lambda }(\\lambda ,T) d\\lambda }~.",
"$ Since the mean free path of thermal radiation in the disc is small compared to the disc's scale height, the radiation field can be considered isotropic, blackbody emission.",
"The Rosseland mean opacity uses the temperature derivative of the Planck distribution as a weighting function and is defined as $\\kappa _R^{-1} = \\frac{\\int _0^{\\infty }\\kappa _{\\lambda ,s}^{-1}(T,\\rho ) (\\partial B_{\\lambda } (\\lambda , T)/\\partial T)d\\lambda }{\\int _0^{\\infty }(\\partial B_{\\lambda } (\\lambda , T)/\\partial T)d\\lambda }~.",
"$ It should be noted that scattering processes are neglected (subscript ns) when calculating the wavelength dependent opacities $\\kappa _{\\lambda }$ for the Planck mean, but are included in the Rosseland mean opacity (subscript s).",
"We also consider the stellar radiation and define the stellar or optical opacity as $ \\kappa _{\\star } = \\frac{\\int _0^{\\infty }\\kappa _{\\lambda ,ns}(T,\\rho ) B_{\\lambda }(\\lambda ,T_{\\star })d\\lambda }{\\int _0^{\\infty }B_{\\lambda }(\\lambda ,T_{\\star })d\\lambda }~.",
"$ The stellar opacity is then the Planck mean opacity taking into consideration the stellar radiation temperature instead of the local disc temperature.",
"We calculate the mean Rosseland, Planck and stellar opacities as a function of temperature using the RADMC-3D http://www.ita.uni-heidelberg.de/~dullemond/software/radmc-3d/ code.",
"Note that dust opacities are independent of the gas density, as opposed to gas opacities.",
"The latter are not considered in this work as opacities in the disc are dominated by the dust component and the high temperature needed for dust evaporation are not reached within our simulations.",
"The code utilizes Mie-scattering theory and the optical constants for water-ice [105] and silicates [54], [37] in order to calculate the wavelength-dependent opacities, which are then averaged over all wavelengths.",
"The main input parameters are the size of the grains and the dust-to-gas ratio of the disc.",
"We can also choose the dust grain species, silicates, water ice and carbon or the fraction between those in the dust mixture.",
"In this work we include a mixture of 50% water-ice and 50% silicates and the dust-to-gas ratio for the calculation of the opacities is 1%.",
"Finally, the Rosseland, Planck and stellar mean opacities [17] are calculated.",
"In Fig.",
"REF it is illustrated how each mean opacity scales with temperature for six different grain sizes, from 0.1 $\\mu $ m to 1 cm.",
"The wavelength dependent opacities and subsequently the mean opacities depend on the size parameter $x=\\frac{2\\pi s}{\\lambda }$ , but also on the refractive index of the given grain species, which is also itself dependent on wavelength [76].",
"By Wien's law the wavelength is inversely proportional to the temperature.",
"Using the size parameter we find that the regime changes at approximately x = 1 and more specifically at x $\\ll $ 1 we have the Rayleigh scattering, whereas at x $\\gg $ 1 we have the geometric optics regime [24].",
"Consequently, if the size of the particle is a lot smaller than the wavelength of the incident radiation, absorption dominates over scattering and the wavelength dependent opacities become independent of grain size.",
"In the case of the larger grain sizes, or when x $\\gg $ 1, the opacities become independent of wavelength (and consequently temperature), but depend on the grain size.",
"Most of the regions though lie somewhere in between, which means that calculating the opacity depends on both the grain size with its individual refractive index and the given wavelength or temperature.",
"The Rosseland mean opacities (Fig.",
"REF ) for the largest particles of the set (100 $\\mu $ m, 1 mm and 1 cm) are almost flat, except for the transition around the iceline at 170 K $\\pm $ 10 K. At this temperature, ice sublimates and the opacity is then only determined by silicates.",
"For those large particle sizes, the size parameter is greater than 1, therefore we are in the geometric optics regime and the Rosseland mean opacity is independent of temperature.",
"However, we note that for a grain size of 100$\\mu $ m and temperature below 20K the opacity depends on the temperature.",
"In this region the regime has changed and the opacity is determined by Rayleigh scattering.",
"Equally, the size parameter is well below 1.",
"The same trend can be seen for the smaller particles, namely 0.1,1 and 10 $\\mu $ m before the iceline.",
"The opacity of the 10 $\\mu $ m grain sizes goes into the geometric optics regime after the iceline and tends to become independent of temperature.",
"In the region after the iceline for the smallest particles (0.1 and 1 $\\mu $ m) the size parameter is closer to 1, so the opacities are also influenced by the refractive indices.",
"The Planck opacities have a weaker dependency on temperature compared to the Rosseland mean opacities.",
"The stellar opacities depend only on the stellar temperature and grain sizes, but not on the disc temperature, except for the transition at the water ice line, when water rich particles evaporate.",
"Both the Planck and stellar opacities are calculated using only the absorption coefficient, which does not have a strong dependency on wavelength and consequently temperature, as opposed to the extinction coefficient.",
"The Planck mean opacities are calculated taking into account the temperature of the disc, while the stellar opacities, use the temperature of the star, which is constant.",
"Using RADMC-3D a number of files is created with the mean opacity values as a function of temperature.",
"These files are then used in the hydrodynamical code (FARGOCA).",
"The opacity calculations from these files are interpolated and in this way we get in the code the appropriate opacity values given the temperature of the grid cell.",
"The reason why the interpolation is done instead of directly calculating opacity using Mie theory is because the computational time would be very long.",
"We include the direct opacity-temperature calculations for at least 10 grain sizes, from 0.1 $\\mu $ m to 1 mm or 2 cm.",
"We then create size bins and as a simplification, each grain size within a bin shares the same opacity-temperature calculations (corresponding to the logarithmic mean size of that bin).",
"We note here that these bins are different and for computational reasons wider than the bins used for the calculations of the vertically integrated dust surface densities (see Sec.",
"REF ).",
"As a comparison we will also use the frequency averaged [10] opacity law.",
"The opacity in this case, depends on the local temperature and density.",
"There are several transitions in this opacity regime caused by the processes which dominate each temperature region, such as the evaporation of ices interior to the iceline, which is also present in the prescription we are using for the discs with the grain size distributions.",
"However, the greatest difference between the two opacity regimes is that the [10] opacity law is based on micrometer-sized dust and does not take the opacity provided by all of the dust sizes present in the disc into account.",
"The [10] law also considers the gas opacities, but these are relevant for high temperatures that will not be reached in the simulations presented here."
],
[
"Grain size distributions",
"The collision between two dust grains can result in various outcomes.",
"The possible outcomes are coagulation, fragmentation, cratering and bouncing .",
"The outcome of a collision is determined by the relative velocities of the colliding bodies and their mass ratio [106], [25].",
"The relative velocities between grains are determined by the mass of the particles, but they are also greatly affected by the local temperature and the gas scale height.",
"Dust dynamics involve not only collisions between grains, but also with the molecules of the protoplanetary disc's gas.",
"These collisions with the gas cause a lag to the dust particles that leads to relative velocities between themselves.",
"Figure: Vertically integrated dust surface density distribution per logarithmic bin of grain size as a function of grain size for the two distributions used here, after (BOD) and (MRN), at 10 AU, for the simulation with α=5×10 -3 \\alpha = 5\\times 10^{-3} (see Fig.",
"for the distributions over all orbital distances).",
"The dust to gas ratio is 1% and the gas surface density is 1000g/cm 2 1000~g/cm^2 at 1 AU in both cases.",
"For both of the distributions we additionally used u f =1m/su_f = 1~m/s and ρ s =1.6g/cm 3 \\rho _s = 1.6~g/cm^3.",
"In the BOD, we can distinguish three regions.",
"Small particles follow Brownian motion (BM), then as they grow they follow the turbulent gas motions (T1) and finally they are affected by the turbulence, but get decoupled from the gas as their stopping times are much larger than the turn-over time of the eddies (T2).",
"The barrier s BT s_{BT} is the grain size limit for Brownian motion, while the barrier s 12 s_{12} separates the two turbulent regimes.",
"The bump near the end of the distribution is caused by cratering, since large particles only lose part of their mass this way, while small particles can only coagulate and form larger grains.",
"This causes the distribution to be top-heavy.",
"The MRN distribution is a single power-law (Eq.).",
"Both grain size distributions have a maximum value determined by the same fragmentation limit (Eq.",
"), but they are not the same for the BOD and the MRN distribution because of the self-consistently calculated temperature.We compare in this work two different grain size distribution models.",
"These models provide the vertically integrated surface density of dust as a function of the grain size.",
"The first and simple model (hereafter MRN) is inspired by the groundwork on dust distributions [36], [67], [102].",
"At a given distance to the star, the equilibrium between fragmentation and coagulation results in a steady-state size distribution, where the number density of the particles can be written as $n(m)dm \\propto m^{-\\xi }dm$ or $n(s)ds \\propto s^{2-3\\xi }ds~,$ with m the particle mass, s the particle size and $\\xi $ a constant.",
"The mass of a specific size, within a size bin $[s_i-ds^{\\prime },s_i+ds^{\\prime \\prime }]$ is $M_{s_i} = \\int _{s_i-ds^{\\prime }}^{s_i+ds^{\\prime \\prime }} m\\cdot n(s) ds \\propto \\left[\\frac{s^{6-3\\xi }}{6-3\\xi }\\right]_{s_i-ds^{\\prime }}^{s_i+ds^{\\prime \\prime }}~,$ assuming 5-3$\\xi \\ne $ -1.",
"The grain sizes for this project are distributed over a logarithmic grid, so $s_i-ds^{\\prime }$ is $\\sqrt{s_i\\cdot s_{i-1}}$ and $s_i+ds^{\\prime \\prime }$ is $\\sqrt{s_i\\cdot s_{i+1}}$ .",
"The vertically integrated surface density of each grain size bin is then $\\Sigma _{d,s_i} \\propto f_{DG}\\Sigma _g \\left[\\left(\\sqrt{s_i\\cdot s_{i+1}}\\right)^{6-3\\xi }-\\left(\\sqrt{s_i\\cdot s_{i-1}}\\right)^{6-3\\xi }\\right]~,$ where $f_{DG}$ is the dust-to-gas ratio and $\\Sigma _g$ is the gas surface density.",
"We use a grain size grid, such as $s_{i+1} = c\\cdot s_i$ and the assumption that $\\xi $ =11/6 [36], [109], so then the expression for the unnormalized vertically integrated surface density for each grain size bin can be simplified to $\\Sigma _{d,s_i} \\propto s_i^{1/2}f_{DG}\\Sigma _g~.$ The contributions from each grain size are then summed up.",
"In order to get the normalized surface density values we divide each contribution by the aforementioned sum $ \\tilde{\\Sigma }_{d,s_i} =\\frac{s_i^{1/2}f_{DG}\\Sigma _g}{\\sum _i s_i^{1/2}}.$ The second and more complex model [15] takes into account fragmentation, coagulation and also cratering, where only part of the mass of the target body is excavated after the collision with a small impactor.",
"The input parameters for this model are the dust and gas surface densities ($\\Sigma _{d,0}$ and $\\Sigma _{g,0}$ ), the local disc temperature ($T$ ), the alpha turbulence parameter ($\\alpha $ ), the volume density of the particles ($\\rho _s$ ) and finally the fragmentation velocity ($u_f$ ), which is the critical velocity above which all collisions lead to either fragmentation or cratering.",
"The logarithmic grid for the sizes of both distributions is defined as $ s_{i+1} = 1.12s_i$ , while the smallest grain size is 0.025 $\\mu $ m. As mentioned in Sec.",
"REF this grain size grid is finer than the size grid we use to determine the opacities.",
"Considering that different particle sizes lead to different collision outcomes, this recipe takes into account the relative velocities that particles of different sizes will develop in order to create different regimes for each size.",
"These regimes are created according to size boundaries, within which different power-laws apply for the fit to the size distribution.",
"It should be mentioned that these size boundaries are defined by the corresponding relative velocities of the dust grains.",
"The smallest particles of the distribution follow Brownian motion, which means their motions are affected by collisions with the gas molecules, there is no preferred direction and they do not have angular momentum.",
"The next regime regards larger particles that start to get affected by turbulent mixing.",
"It was also found [81] that when particles have stopping times approximately equal or larger compared to the turn-over time of the smallest eddy of the gas, they start to decouple from the gas, so they follow a different regime.",
"Finally, the distribution has an upper end or a fragmentation barrier above which particles can no longer grow and only fragmentation occurs.",
"Between two size boundaries, the distribution is described by a power-law $n(m)\\cdot m \\cdot s = s_i ^{\\delta _i}$ of different powers $\\delta _i$ , depending on the regime (Brownian motion or turbulent mixing).",
"Within each one of the regimes, the power-law indices are different if the grains are affected by settling, given their sizes and the disc parameters (see Sec.",
"REF for a discussion on the vertical distribution of grains).",
"The powers for each regime are found in Table REF and using these we can create a first fit $f(s_i)$ .",
"It is necessary then to include a bump caused by cratering and the cut-off effects of the distribution that cause an increase in the fit for large enough particles.",
"This boundary effect is caused by the fact that large particles near the upper boundary of the grain sizes grid do not have larger particles to collide with, but the mass transfer from one size bin to the other needs to be constant to keep a steady-state grain size distribution.",
"Therefore the number density is increased.",
"Similarly, erosion by small impactors slows down the growth of large particles and an increase in the number density is needed to keep the flux constant.",
"More details for this recipe can be found in [15].",
"Finally the fit is normalized according to the total dust surface density at the given location (Fig.",
"REF , also see Sect.",
"5.2 in [15]) as in the first model.",
"This fit represents the vertically integrated dust surface densities per logarithmic bin of grain size, N(s)$\\cdot $ m$\\cdot \\Delta \\log $ s, where $N(s) = \\int _0^{z_{max}} n(s)~dz$ is the vertically integrated number density.",
"Table: Power-law exponents for each regime in the grain size distribution .",
"The distribution in each regime is n(m)·m·s∝s i δ i n(m) \\cdot m \\cdot s \\propto s_i^{\\delta _i}.The maximum grain size or in other words the fragmentation barrier is defined in the simulations with both of the grain size distributions as $ s_{max} \\simeq \\frac{2\\Sigma _g}{\\pi \\alpha \\rho _s}\\frac{u_f^2}{c_s^2}~, $ with $\\rho _s=1.6~g/cm^2$ the density of each particle, $u_f= 1~m/s$ the fragmentation threshold velocity, $k_B$ the Boltzmann constant, $m_p$ the proton mass and $\\mu $ = 2.3 the mean molecular weight in proton masses.",
"The sound speed is given by $ c_s = \\sqrt{\\frac{k_BT}{\\mu m_p}}~.",
"$ The threshold velocity $u_f \\sim 1~m/s$ corresponds to the threshold after which collisions between silicates always lead to fragmentation [87].",
"However, it has also been experimentally found that water-ice shows a higher threshold velocity, $u_f \\sim 10~m/s$ [45].",
"We choose to use only the lower fragmentation threshold in the here presented work, but the composition dependency will be studied in future work.",
"Because of Eq.",
"REF , which applies to both distributions, and the different regime boundaries in BOD, which depend on the local disc parameters (see Fig.",
"REF and Table REF ), there is not a global size distribution, but rather a self-consistent spatial distribution of grain sizes both radially and vertically (see also Sec.",
"REF ).",
"It is noteworthy that even though we consider the coagulation/fragmentation equilibrium and the effects of cratering and settling, we neglect in the following work the drift of grains and the effect of bouncing.",
"However, in the simulations presented here we find that the fragmentation barrier is always smaller than the drift barrier.",
"This means that the particles have already fragmented and replenished the smaller pieces before they would have the chance to experience drift.",
"The small particles are less affected by radial drift [106] and since they coagulate, an equilibrium forms that drives the grain size distribution.",
"The fragmentation barrier decreases with increasing $\\alpha $ -viscosity parameter, which is expected, since an increased $\\alpha $ leads to increased turbulent relative velocities.",
"The maximum possible grain size also decreases when the fragmentation threshold velocity decreases.",
"Drift is an important effect acting on dust grains in protoplanetary discs, but it is a reasonable simplification to neglect it for the chosen parameters of our simulations.",
"In future work where, for example, the fragmentation threshold velocities are increased or the composition dependency is included, drift is an effect that needs to be taken into consideration."
],
[
"Vertical distribution of grains",
"The grains of a given size are vertically distributed according to the following $\\rho _d = \\rho _{d,0}\\exp \\left(-\\frac{z^2}{2H_d^2}\\right)~,$ with $ \\rho _{d,0} = \\frac{\\Sigma _d}{\\sqrt{2\\pi }H_d } $ the dust density at midplane and the dust scale height derived by [40] $ H_d = H_g\\sqrt{\\frac{\\alpha }{\\alpha + St}}~,$ where $H_d$ and $H_g$ is the dust and gas scale height respectively.",
"The Stokes number of the particles in the Epstein regime is $\\text{\\it St}=\\tau _f\\Omega _K = \\sqrt{\\frac{\\pi }{8}} \\frac{\\rho _s s}{\\rho _g H_g}~, $ with $\\tau _f$ the stopping time of the particles, $\\Omega _K$ the Keplerian velocity and $\\rho _g$ the gas volume density.",
"At midplane, where $\\rho _g = \\rho _{g,0}$ (resembling Eq.",
"REF ), the Stokes number is $St = \\frac{\\pi }{2} \\frac{\\rho _s s}{\\Sigma _g}~.",
"$ The vertically integrated dust surface densities $\\Sigma _d$ as a function of orbital distance are determined by the grain growth and fragmentation equilibrium prescriptions that were introduced in Sec.",
"REF .",
"The BOD grain size distribution has already taken into account the effect of settling (to calculate the distribution itself), depending on the grain sizes and the disc parameters (see Sect.",
"REF and Table REF ).",
"We then distribute in our model the grains vertically according to their sizes and how much they are expected to be affected, in a fashion consistent with the assumptions made in BOD (Eqs.",
"REF -REF ).",
"However, it has been shown that small particles can get trapped in lower altitudes by the concentration of larger grains due to settling [58].",
"This effect is not taken into account here as it is beyond the scope of this work, but could be an improvement in future work.",
"We use the volume density of dust within a grid cell to find the opacity through $ \\bar{\\kappa } = \\sum _i \\left(\\frac{\\rho _{d,i}}{\\rho _g}100\\right)~\\kappa _i~,$ where $\\kappa _i$ is the opacity of each grain size i, as shown in Fig.",
"REF , and $\\rho _{d,i}/\\rho _g$ the dust-to-gas volume density ratio for a given grain size i.",
"In the case of single grain sizes summing is not needed.",
"The dust-to-gas term for the volume densities includes the settling effect (Eq.",
"REF ).",
"In this expression we multiply the volume density dust-to-gas ratios by 100 to account for the fact that in the calculations of $\\kappa _i$ (with the module from RADMC-3D) a dust-to-gas ratio of 1% was assumed.",
"This way we multiply this $\\kappa _i$ with the appropriate factor depending on the volume density dust-to-gas ratio.",
"As an example for the effect of settling we show in Fig.",
"REF the dust density as a function of height at 3AU for 5 different grain sizes.",
"In this simulation, the $\\alpha $ value is $10^{-4}$ , the initial gas surface density at 1 AU is $\\Sigma _0 = 1000~g/cm^2$ and the grain size distribution used is the BOD.",
"As a reference, we also plot the gas density to indicate the different volume density dust-to-gas ratios depending on the grain size and the volume density dust-to-gas ratio.",
"Figure: Dust density as a function of height for grains of five representative grain sizes within a disc with the BOD, α=10 -4 \\alpha =10^{-4} and Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2, at 3 AU.",
"The black line shows the gas density of the disc.",
"The dashed gray line shows the dust-to-gas ratio as a function of height for the whole grain size distribution.",
"The dust-to gas ratio of the smallest particles of the sample remain constant, but as grains get larger they get more affected by settling.",
"Consequently, the largest grain (1 cm) is mostly concentrated at the midplane.Considering the $\\alpha $ prescription for viscosity of [97] the turbulence must be $\\alpha \\le St$ for settling to become important [6].",
"We see from Eq.",
"REF that the larger the Stokes number of a particle, the more it will be affected by settling for a given $\\alpha $ .",
"Additionally, the lower the $\\alpha $ value is, the more effective settling will be for even smaller dust particles.",
"For this reason we choose to show an example in Fig.",
"REF of a simulation with $\\alpha =10^{-4}$ which is the lowest $\\alpha $ value we used in our simulations.",
"Not only the size, but the location within the disc matters, because the Stokes number for a given grain size depends on the gas density which decreases with the increasing orbital distance.",
"As a consequence, the same particles experience less settling the closer they are to the inner boundary of the disc.",
"The smallest particles shown in Figure REF , namely 1 and 10 $\\mu $ m are not affected by settling, despite the low turbulence strength.",
"Their dust-to-gas ratio remains the same at all heights, so they are well coupled to the gas.",
"Then, the larger the particle, the more effective settling is.",
"The 100 $\\mu $ m sized dust particles are already affected by settling, but beyond this grain size the difference is even larger.",
"The cm-sized dust, which is nearly the maximum grain size in this simulation, is almost constrained at the midplane.",
"The dust-to-gas ratio (dashed gray line), $\\rho _d/\\rho _g$ , is well below 1% above z=0.05 AU, but reaches 4% at midplane.",
"The main difference between various grain sizes is their different opacities as a function of temperature.",
"However, as illustrated in Fig.",
"REF , settling is another important effect, with a distinct efficiency depending on the grain size.",
"Test simulations (presented in Appendix ) show that a significant settling changes significantly the thermal structure of the disk.",
"Indeed, without it a constant dust-to-gas ratio leads to overestimated opacities above midplane, hence more \"puffed-up\" inner discs with higher temperatures, which cause a shadowing of the outermost region and prevent it from reaching an equilibrium state.",
"Thus, settling is an important effect which needs to be taken into account in models in order to accurately study the thermal structures of protoplanetary discs."
],
[
"Simulations setup",
"The stellar mass used in the simulations is M$_\\star $ = 1M$_\\odot $ , the temperature is 4370K and the radius is R$_\\star $ = 1.5R$_\\odot $ .",
"The total dust-to-gas ratio is $f_{DG} = 1\\%$ .",
"Viscosity in the simulations follows an $\\alpha $ prescription [97], where $ \\nu = \\alpha \\frac{c_s^2}{\\Omega _K}~.",
"$ Recent observations of protoplanetary discs find $\\alpha $ values from $10^{-4}$ to $10^{-2}$ or even $10^{-1}$ [51], [4], [88], [5], but such large $\\alpha $ would cause discs to rapidly expand to great extend [47] in contrast to observations.",
"We use in this work five sets of simulations with $\\alpha =10^{-2}, 5\\times 10^{-3}, 10^{-3}, 5\\times 10^{-4}~\\text{and}~10^{-4}$ in order to test the effect of turbulence on the thermal structure of the disc.",
"We choose these values in order to include a simulation with $\\alpha = 5\\times 10^{-3}$ so that one can directly compare with the work in [20] and a simulation with $\\alpha = 10^{-3}$ to allow comparison with the discs in [18].",
"In the simulations with $\\alpha =10^{-2}~\\text{and}~5\\times 10^{-3}$ the grid cells are 480$\\times $ 70 (radial-vertical direction) and the disc extends from 2 to 50 AU, while in the simulations with $10^{-3}, 5\\times 10^{-4}~\\text{and}~10^{-4}$ the grid cells are 150$\\times $ 35 and the disc extends from 0.1 to 3.1 AU.",
"The gas surface density follows a profile $\\Sigma _{g} = \\Sigma _{g,0}\\cdot (r/AU)^{-p}~,$ with $p = 1/2$ and we test two different initial surface densities, $\\Sigma _{g,0} = 100~\\text{and}~1000~g/cm^2$ for every $\\alpha $ value that was mentioned above.",
"We run more combinations of different initial gas surface densities and total dust-to-gas ratios, however these are mainly used in order to produce a fitting for the iceline position as a function of the three parameters, $\\alpha $ , $\\Sigma _{g,0}$ and $f_{DG}$ (see Sect.",
"REF , Appendix and ).",
"Since we simulate equilibrium discs, the surface density profile does not evolve significantly during the simulation, because the thermal equilibrium is reached faster than the viscous evolution equilibrium.",
"At the top of the disc we manually set T=3 K, the temperature of the interstellar medium, so that the disc can be cooled by the upper boundary (as described in [17]).",
"The simulations run for some hundreds of orbits (typically 200-1000 orbits) until they reach thermal equilibrium.",
"Nevertheless, some of the simulations might show signs of convection [16], which means that they will remain unstable regardless of integration time.",
"At first, we perform simulations with single grain sizes in order to see the difference in the disc structures between them.",
"Dust grains affect the hydrodynamical simulation through the opacity in each grid cell.",
"Every simulation has a different grain size and the opacity values for this specific size are used (see Fig.",
"REF ).",
"The simulations of single sizes offer the chance to examine the extend to which different grain sizes affect the disc's evolution and equilibrium structure and predict how much grain growth or a grain size distribution will change the outcome.",
"In the next step we also consider settling and how it affects large grains.",
"For these simulations we only use single grain sizes and the dust surface density is assumed to be $\\Sigma _{d} = f_{DG}\\Sigma _{g}$ or specifically $\\Sigma _{d} = 0.01\\Sigma _{g}$ as before.",
"The difference between discs without and with settling is further discussed in Appendix .",
"Figure: Aspect ratio (left plot) and midplane temperature (right plot) as a function of orbital distance in AU, for discs with 5 different single grain sizes from 0.1 μ\\mu m to 1 mm (see Fig.",
"for the opacities of those 5 grain sizes).All of the simulations include viscous heating and stellar irradiation, have α=5×10 -3 \\alpha = 5 \\times 10^{-3} as the turbulence parameter in viscosity, Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2 as the initial gas surface density and the dust-to-gas ratio is f DG =1%f_{DG} = 1\\%.",
"In this set of simulations we also consider settling, so that we can compare with the simulations that include the grain size distributions.",
"The gray areas in the plot indicate the water iceline transition.",
"Overplotted with dashed lines are the discs with the MRN distribution in reddish pink and the BOD distribution in dark blue.",
"The simulations with the distributions shows influence from small particles at the inner part of the disc, while the outer parts are mostly affected by larger particles, around 0.1 mm (see Figs.",
").",
"The small differences in the aspect ratio and temperature of the discs with the distributions stem from the different slope of the vertically integrated dust surface densities of the two distributions (see Figs.",
"and ).We, then, include the two grain size distribution models that were discussed in Sect.",
"REF .",
"The distributions are self-consistently calculated in the code using as input parameters in each time-step and grid cell, the gas surface density (for both distributions) and the temperature (only for the BOD).",
"The upper size boundary for the MRN-power law model can be either fixed or follow the same fragmentation barrier formula as the second, more complex model [15], but in this work we use the latter."
],
[
"Comparison between the two grain size distributions",
"In this section we compare the simulations utilizing the two different grain size distributions (MRN and BOD).",
"At first, we focus only on the case of $\\alpha =5\\times 10^{-3}$ , $\\Sigma _{g,0} =1000~g/cm^2$ and $f_{DG} =1\\%$ .",
"In Fig.",
"REF we can see the aspect ratio as a function of orbital distance from the star for simulations of different grain sizes along with the two grain size distributions.",
"The gray areas correspond to the iceline transition (T = 170 $\\pm $ 10 K).",
"In this area the change in opacity is responsible for the bumps in the aspect ratios.",
"We first focus on the simulation with 0.1 $\\mu $ m, which roughly corresponds to an unevolved dust population as found in the interstellar medium.",
"The simulation with particles of 0.1 $\\mu $ m results in an increasing aspect ratio as a function of orbital distance up to 6 AU, where it reaches a maximum and then decreases up to approximately 15 AU.",
"Using 1 $\\mu $ m-sized particles we see a similar disc structure.",
"The aspect ratio is almost constant for the first few AU and has a small increase around 6 AU.",
"Then it converges with the simulation of 0.1 $\\mu $ m up to the minimum around 15 AU.",
"If the grain size increases to 10 $\\mu $ m, then the aspect ratio also increases with distance, features a bump closer to 7.5 AU and decreases with the same slope as the previous two simulations.",
"The larger particles have distinct profiles.",
"Specifically, the aspect ratio of the simulation with particles of 0.1 mm is a monotonically increasing function of orbital distance with a small bump at 3.5 AU.",
"The same can be seen for the simulation with the largest particles, namely 1 mm, but in this case a bump is not visible at any parts of the aspect ratio profile, since the iceline transition does not exist within the boundaries of the simulation.",
"The gradients in the aspect ratios for the inner region of the discs are determined by the opacity of the disc.",
"We can compare the opacity gradients in Fig.",
"REF with the aspect ratio gradients keeping in mind that the temperature decreases as we move further away from the star.",
"Depending on the temperature at each orbital distance of the disc, we see that the opacity gradients are responsible for the dips and bumps in the aspect ratio profiles of the discs.",
"Figure: Aspect ratio (left) and midplane temperature (right) as a function of orbital distance for the discs with the BOD, the MRN distribution and a disc that utilizes the opacities.",
"All of the simulations have α=5×10 -3 \\alpha = 5 \\times 10^{-3}, Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2 and f DG =1%f_{DG} = 1\\%.We also show the aspect ratio of the simulations with 1 and 100 μ\\mu m for reference.",
"The opacities are based on micrometer sized particles resulting in comparable aspect ratios.",
"The main differences with the discs including the full grain size distributions are the steeper radial temperature (and thus aspect ratio) gradient for the disc with the opacities and the reversed slope within 4 AU.",
"These influence the migration speed and direction of planets embedded in those protoplanetary discs (see Figs.",
", , ).",
"The gray areas correspond to the water iceline transition.Figure: Mean Rosseland opacity values for the disc that includes the BOD distribution (top) and the MRN (bottom).",
"These opacities correspond to the discs with the grain size distributions which were presented in Figs.",
"and .",
"The highest opacity values are found at the iceline transition (gray band in the right plots).",
"Moving outwards, the disc gets colder and the opacity of the disc decreases.",
"The light blue line is the location where the vertically integrated optical depth reaches unity (τ\\tau =1), so it divides the disc in the optically thick lower region and the optically thin upper region.",
"Above this line the opacities decrease due to the efficient cooling of the disc.",
"The uppermost layers show increased opacities caused by the direct stellar heating, which increases the temperature.",
"The same gradients can be seen in the temperature plots (right).The outer region of the discs is dominated by stellar irradiation, which causes the flaring of the discs.",
"The simulation with the BOD shows influence from the smaller particles at the inner parts of the disc, but moving outwards the aspect ratio gets affected by larger particles, around 100 $\\mu $ m. A more detailed analysis of the grain sizes that contribute the most to the opacity of the disc will be done in Sect.",
"REF .",
"The simulation with the MRN distribution shows similar aspect ratio gradients.",
"For this case, with high $\\alpha $ , there is only a minimal difference between the dust surface densities, which leads to similar opacities in total.",
"Both discs are affected by grains of similar sizes and for this reason the aspect ratios are almost the same there.",
"In Fig.",
"REF we show the midplane temperature as a function of orbital distance for the same set of simulations.",
"We can see that the changes in the temperature gradients correspond to the changes in the aspect ratio gradients.",
"The gray horizontal line is again the iceline transition.",
"The simulation with the millimeter-sized particles does not reach the iceline temperature within the extend of the simulated disc and thus does not feature the aspect ratio bump.",
"We can compare in Fig.",
"REF the resulting aspect ratios and midplane temperatures of the simulations with the BOD and MRN distributions with a simulation utilizing the opacities from [10].",
"This opacity profile is based on micrometer sized particles, hence it is expected that it resembles the simulation with only micrometer sized particles.",
"It should be pointed out that the [10] prescription includes the gas opacities as well, but these are relevant for high temperatures that are not reached within the extend of the discs here.",
"We notice in Fig.",
"REF that the gradient after the iceline of the simulation with the [10] prescription or only micrometer sized particles is much steeper than the corresponding gradient in the simulations of the full distributions.",
"This is an important difference as the gradients in the aspect ratio affect the migration speed of planets that could be embedded in such a protoplanetary disc (see Sect.",
"REF ).",
"In conclusion, we find that including either the BOD or the MRN distribution leads to comparable results.",
"The differences between the two grain size distributions tested in this work for different values of the turbulence parameter $\\alpha $ and different surface densities will be discussed in Sect.",
".",
"Prior to that, a more extended discussion on the dust surface densities, dominant grain sizes, opacities and temperatures follows in the next paragraphs."
],
[
"Opacities and temperature",
"The opacity and temperature within the disc for the BOD and MRN distribution is plotted in Fig.",
"REF as a function of orbital distance on the x-axis and height on the y-axis.",
"The total opacity of the disc is determined by accounting for the contribution of each grain size according to Eq.",
"REF .",
"The highest opacity values in the figures correspond to the iceline as it can be also seen in the temperature plot (gray band).",
"Almost every particle size has its highest opacity at the iceline, consequently the total opacity of the disc is the highest at the iceline, as it can be seen already in Fig.",
"REF .",
"It was already briefly mentioned in Sect.",
"REF (detailed discussion in Sect.",
"REF ) that the dominant grain sizes at the inner disc are small, therefore we see the same pattern in the opacity of the disc around the iceline as the opacities of the small particles, with the bump around the transition.",
"For this reason we are tracing the iceline at the opacity plot.",
"The total opacity is scaled down in the simulation with the MRN distribution compared to the one with the BOD, which is what we also find in the aspect ratio.",
"Since the opacities have the same pattern and since the bumps in the aspect ratio are caused by opacity transitions it is expected to find there the same gradient.",
"Viscous dissipation is the dominant source of heating for the inner parts of the disc, while stellar irradiation becomes important at larger orbital distances and more importantly for the upper layers of the disc [42], [17].",
"Since the upper layers are heated up, the opacities are also higher there.",
"If we move vertically up, the iceline moves inwards as viscous heating becomes weaker.",
"The specific radius and height at which stellar heating begins to affect the structure of the disc is, among other parameters, influenced by the strength of turbulence.",
"The dependence of the disc's thermal structure on the turbulence strength is discussed in Sect.",
"and more opacity plots as a function of orbital distance and height can be found in Appendix .",
"In Fig.",
"REF the $\\tau $ = 1 line is overplotted (light blue line).",
"The optical depth $\\tau $ is defined as $\\tau = \\int _{z_{max}}^0 \\kappa _R \\rho _g dz~, $ therefore it increases as the height z is decreasing.",
"The $\\tau =1$ line marks the difference between the optically thick and the optically thin medium.",
"When $\\tau \\ge 1$ , then the disc is optically thick or in other words, the mean free path of the photons is much smaller than the length scale over which temperature changes.",
"In optically thin parts of the disc, photons can \"freely\" travel out of the disc.",
"The $\\tau =1$ line thus marks the region of the disc where cooling becomes efficient.",
"A $\\tau =1$ line close to midplane corresponds to lower opacities, which results in a cooler disc.",
"Even though the regions above this line are optically thin and cool down very efficiently, the uppermost layers are directly heated by stellar irradiation and we also see an increase in the opacity.",
"The transition from the optically thin part of the disc to the optically thick (moving from the top layers towards the midplane) is also where the boundary for observations would be if these were integrated over all wavelengths.",
"Mid-infrared wavelength observations of the optically thick disc probe the temperature of the dust \"photosphere\", the effective surface layer of the disc.",
"Observations are in general carried out at various wavelengths, thus probing different grain sizes and different information for the disc [1].",
"The optical depth relevant for such observations might differ for individual grain sizes."
],
[
"Dust surface densities",
"The vertically integrated dust surface densities per orbital distance and grain size are presented in Appendix .",
"The maximum grain sizes in both of the distributions depend on the local temperature and gas surface density, which change with time.",
"Additionally, all of the boundary sizes of the BOD distribution depend on the local gas surface density.",
"We stress here the loop that is created; the dust surface densities play a major role in determining the opacity of the disc (Eq.",
"REF ), which then influences the cooling rate and the stellar heating and thus changes the temperature.",
"The shift in the temperature leads to a new fragmentation barrier (and regime boundaries for the BOD), hence the dust surface densities change and so forth.",
"Given the fact that this loop exists between the dust and the gas, it is important to consider the self-consistent calculations of the dust surface densities in the simulations."
],
[
"Dominant grain sizes",
"Depending on the local gas disc parameters, the grain size which plays the role of the dominant opacity source will change.",
"We find the individual contribution of each grain size to the total opacity of the disc through Eq.",
"REF .",
"For each set of particles of the same size we calculate its contribution, $\\frac{\\rho _{d,i}}{\\rho _g}100\\kappa _i$ and we present it as a function of orbital distance in Fig.",
"REF for the nominal case of $\\alpha =5\\times 10^{-3},~ \\Sigma _{g,0}=1000~g/cm^2$ .",
"In order for a grain size to dominate the opacity it needs a combination of high dust-to-gas volume density ratio (for this specific particle size) and high opacity at the given part of the disc.",
"Figure: Aspect ratio as a function of orbital distance for the discs with high α\\alpha -viscosity values (top left), namely α=10 -2 \\alpha = 10^{-2}, 5×10 -3 5\\times 10^{-3} and low α\\alpha -viscosity values (bottom left), namely α=10 -3 ,5×10 -4 \\alpha = 10^{-3},~5\\times 10^{-4} and 5×10 -4 5\\times 10^{-4}, for the two grain size distributions (BOD in dark blue, MRN in reddish pink).",
"The iceline moves inwards as α\\alpha decreases, due to reduced viscous heating.",
"The wiggles that can be seen in parts of the low α\\alpha discs are caused by convection.",
"The right plots show the temperature as a function of orbital distance for the discs with high α\\alpha -viscosity values (top), namely α=10 -2 \\alpha = 10^{-2}, 5×10 -3 5\\times 10^{-3} and for low α\\alpha -viscosity values (bottom), namely α=10 -3 ,5×10 -4 \\alpha = 10^{-3},~5\\times 10^{-4} and 10 -4 10^{-4}.We can see in the same figure, plotted as a black solid line, that the inner disc with either the BOD or the MRN distribution is influenced by very small particles, around 3 $\\mu $ m. In other words, those small particles have the maximum contribution to the total opacity or in other words their opacity dominates the opacity of the disc.",
"The dominant grain sizes in the disc with the BOD grain size distribution feature a jump at around 20 AU.",
"This jump is caused by the dip in the distribution in the transition between the first turbulence regime and the second (see Fig.",
"REF ), after which particles are large enough to get completely decoupled from the gas.",
"After this jump the dominant grain size is near 200 $\\mu $ m. The dominant grain size in the disc with the MRN distribution smoothly increases in the inner regions and then remains constant at around 90 $\\mu $ m exterior to 20 AU.",
"At the inner, hot parts of the disc, the grains around the micrometer size have surface densities which are around an order of magnitude lower than the larger grains (see Appendix ).",
"However, they have the highest opacity by several orders of magnitude (see Fig.",
"REF ) at these high temperatures.",
"This results in them being the dominant particles at that region.",
"Farther out, the temperature of the disc decreases and as consequence, larger particles carry the highest opacity.",
"The decreased temperature means that the opacities of the larger particles get comparable with those of the smaller particles.",
"At the same time, the difference between the very small and the largest grain sizes slightly increases, aiding the dominance of the largest grains.",
"In Fig.",
"REF the grain sizes below the dark red line give 25% of the contribution to the total opacity, thus the grain sizes above this line give 75% contribution.",
"The line above this divides the grain sizes in two groups, each contributing 50% to the opacity of the disc.",
"In the same way the uppermost line has grain sizes with 75% of the contribution beneath it and sizes with 25% contribution above it.",
"These lines show the general trend of the contribution to the opacity, which mainly comes from the small grains in the inner disc and by the large grains in the outer disc.",
"For a given grain size, the contribution to the opacity shows similar patterns as the dust surface densities.",
"The maximum opacities per grain size are seen at approximately 6.5 AU.",
"This location corresponds to the iceline and it is expected to have the highest opacity contribution by almost all of the grain sizes (Fig.",
"REF ).",
"In conclusion, what defines the grain size with the maximum contribution to the total opacity is the combination of the dust surface densities and the opacity of each grain size at a specific orbital distance (which is determined by the local temperature).",
"Once more it is evident that the self-consistent calculations within the feedback loop (Fig.",
"REF ) are crucial to the disc structure evolution.",
"Decreasing the $\\alpha $ -viscosity values is expected to affect the discs in two ways.",
"First of all, the viscous heating decreases, therefore the discs will cool down and their aspect ratio will be lower (see Fig.",
"REF ).",
"Secondly, the lower turbulence means that particles will face less destructive collisions and thus they will be able to grow to larger sizes.",
"The larger grains have in general lower opacities, which means that the discs experience an additional cooling because of the change in the opacities.",
"The general trend that we show in Fig.",
"REF is that the aspect ratio indeed decreases as the turbulence parameter decreases.",
"The location of the iceline also moves further in.",
"The aspect ratios and corresponding midplane temperatures in the low $\\alpha $ models (bottom plots in Fig.",
"REF ) show some wiggles due to convection.",
"Convection is caused by the vertical temperature gradient which depends on the opacity gradient in the vertical direction and is present in the optically thick regions of the disc [16].",
"This also implies that as the grains are more vertically diffused in the higher $\\alpha $ case and the vertical temperature gradients are less steep, the effect should be less strong.",
"Indeed, convection is also present in the high $\\alpha $ simulations, but only at the inner, hotter regions of the disc.",
"All regions that are affected by convection experience some sort of instability, so that reaching a steady state is very hard, if not impossible.",
"The vertically integrated dust surface densities as a function of orbital distance and grain size for the simulations with the two grain size distributions and the rest of the $\\alpha $ values ($10^{-2}, 10^{-3}, 5 \\times 10^{-4}, 10^{-4}$ ) are presented in Fig.",
"REF in the appendix.",
"In Sect.",
"we presented the vertically integrated dust surface densities for the nominal value of $\\alpha = 5 \\times 10^{-3}$ .",
"We also plotted the 50% contribution line for the same $\\alpha $ value in Fig.",
"REF .",
"This line divides the grain sizes into two groups that contribute equally to the total opacity at the given orbital distance.",
"In Fig.",
"REF we present the comparison of the 50% contribution lines as a function of orbital distance and $\\alpha $ -viscosity.",
"We plot also, in thicker lines of the same color, the maximum grain size in each disc so that the two groups of grain sizes with equal contribution to the opacity of the disc can be seen.",
"The lower boundary of this plot corresponds to the minimum grain size (constant in all of the simulations).",
"Figure: 50% contribution lines as a function of orbital distance for the discs with the high α\\alpha values at the top and the low α\\alpha values at the bottom.",
"The group of thicker lines at the top correspond to the maximum grain sizes of each disc, so that the 50% contribution lines below divide the grain sizes into two groups, which contribute equally to the total opacity of the disc (see the red lines in Fig.",
").The difference between the two grain size distributions is small.",
"As α\\alpha -viscosity decreases, the maximum grain size increases and more influence to the total opacity comes from larger grains.We find that the position of this 50% contribution line is similar almost independently of the grain size distribution utilized in the model.",
"Similarly the position of the 50% contribution line within the grain size range (vertical axis in Figs.",
"REF & REF ) does not change significantly as $\\alpha $ decreases (see Fig.",
"REF ), but at the same time the maximum grain size increases.",
"The very large particles ($\\ge $ 100 $\\mu $ m) have significantly lower opacities and thus each order of magnitude added in grain size only adds a small contribution to the total opacity of the disc.",
"If we decrease the gas surface density then the total dust surface density also scales down.",
"The reduced surface density results in a colder disc because of two effects.",
"On one hand the viscous heating decreases and on the other hand the radiative cooling increases, as it is inversely proportional to the disc's density.",
"This is what we find in the simulations with the lowest initial gas surface density we used, namely $\\Sigma _{g,0} = 100 g/cm^2$ (Fig.",
"REF ).",
"The discs with lower surface density are much colder, therefore have significantly lower aspect ratio.",
"In this case the difference in the aspect ratio of the discs with the two distributions almost vanishes completely, therefore the position of the iceline is also practically the same for the two discs.",
"This is explained by the fact that the dust surface densities of the two distributions are comparable, leading to contributions to the opacity from similar grain sizes (see Fig.REF ).",
"On the contrary, if we increase the dust-to-gas ratio the total dust surface density is by definition enhanced.",
"This means that the viscous heating is higher and the cooling rate decreased, resulting in hotter discs with higher aspect ratios.",
"In addition to that, the opacity increases because of the enhanced total dust surface densities and as a consequence the optically thick region of the discs extends to higher heights compared to the discs with lower dust-to-gas ratio (see example of a disc with $f_{DG}=3\\%$ in Appendix ).",
"Figure: Aspect ratio as a function of orbital distance for two different surface densities (Σ g,0 =100g/cm 2 \\Sigma _{g,0} = 100~g/cm^2 and Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2).",
"The aspect ratio decreases when the gas surface density decreases (while keeping the α\\alpha -viscosity constant at the nominal value of 5×10 -3 5\\times 10^{-3}) due to reduced viscous heating and increased cooling, which is inversely proportional to the density of the disc.",
"As a result the position of the water ice line moves inwards, close to 1 AU.",
"This inward movement of the water ice line due to a reduced gas surface density follows the trend of previous disc evolution simulations that show an inward movement of the water ice line as the disc evolves and the gas surface density reduces (e.g.",
", , ).",
"The low gas surface density results in almost the same aspect ratio profile for the discs with the two distributions.The opacities as a function of orbital distance and height for the discs with the two distributions and the rest of the $\\alpha $ values ($10^{-2}, 10^{-3}, 5 \\times 10^{-4}, 10^{-4}$ ) can be found in Fig.",
"REF in the appendix.",
"The turbulence strength affects the viscous heating and the orbital distance and height at which stellar heating takes over.",
"This direct influence to the thermal structure of the disc leads to different opacities at each position in the disc, depending on the $\\alpha $ -viscosity value.",
"A decrease in the gas surface density, as mentioned, decreases the viscous heating and increases the cooling rate, but the lower temperature decreases the opacity of the disc and cooling is enhanced.",
"A comparison between the opacities in discs with all of the $\\alpha $ values can be found in Fig.",
"REF in the appendix.",
"We also show in Appendix the opacity of the disc with the lowest initial gas surface density $\\Sigma _{g,0}=100~g/cm^2$ (with $\\alpha =10^{-2}$ , $f_{DG}=1\\%$ ) and the opacity of the disc with the highest dust-to-gas ratio, $f_{DG}=3\\%$ (with $\\alpha =10^{-2}$ again and $\\Sigma _{g,0}=1000~g/cm^2$ )."
],
[
"Iceline",
"The location of the iceline can be theoretically calculated by considering the viscous and stellar radiation heating and the partial pressure of water vapor [86], [34], [27].",
"But it is also well predicted by setting a single sublimation temperature [48], [69] as we do here as well.",
"Nevertheless, the location where this sublimation temperature is reached depends on several parameters, as described by the feedback loop (Fig.",
"REF ).",
"Planet formation studies indicate that the iceline in protoplanetary disc models should be outside of 1 AU, the Earth’s orbit.",
"Otherwise, mechanisms like blocking the inward flow of pebbles from the outer disc by growing planets need to be invoked to keep the planets in the inner system dry [73].",
"If we take into consideration the observations of the composition of the bodies in the asteroid belt, the iceline at the time of formation of the asteroids would be at $\\sim $ 2.7 AU.",
"But if icy planetesimals were formed at such small orbital distances and contributed to the formation of the terrestrial planets, we would observe larger amounts of water on Earth than what we observe today.",
"The composition of asteroids from the inner region of the asteroid belt suggests that at their time of formation they should have been interior to the iceline.",
"Evolving disc models indicate that at the time of the formation of terrestrial planets the iceline has already moved towards 1 AU [20].",
"In [79] it is suggested that in order to reach a better conclusion about the location of the iceline a grain size distribution is required, rather than uniform dust grain sizes.",
"In [43] the decoupling of dust particles from gas is discussed as a potential influence to the thermal structure of the disc.",
"[60] argue that in order to move the iceline outside 3 AU one possible solution would be to increase the opacity used in their model.",
"Here we do not include the opacity of a single grain size, but use an evolving grain size distribution that regulates the opacity of the disc more realistically, because we take into account their individual contributions.",
"All of these suggested effects have been therefore taken into account in the here presented work where we study the influence of $\\alpha $ -viscosity, initial gas surface density and total dust-to-gas ratio on the position of the iceline.",
"In Appendix we present the simulations which were used and the procedure that was followed to do the fitting of the iceline position as a function of those three disc parameters.",
"We find that the location of the iceline is independent of the grain size distribution which was utilized in the disc and it follows $r_{ice}=9.2\\cdot \\left(\\frac{\\alpha }{0.01}\\right)^{0.61}\\cdot \\left(\\frac{\\Sigma _{g,0}}{1000~g/cm^2}\\right)^{0.8}\\cdot \\left(\\frac{f_{DG}}{0.01}\\right)^{0.37}~\\text{AU.",
"}$ Figure: Iceline position as a function of α\\alpha turbulence and initial gas surface density at 1AU with a constant f DG =1%f_{DG}=1\\% (Eq.",
").",
"The iceline transition is defined as T=170±10KT=170\\pm 10~K.",
"The black lines mark r ice =0.5,1,2.7and5AUr_{ice} = 0.5, 1, 2.7\\text{ and }5~AU.",
"Higher viscosity or gas surface density leads to hotter discs, with the iceline located at greater distances from the star.",
"The same applies to higher total dust-to-gas ratio.",
"The gray dashed lines mark r ice =0.5,1,2.7and5AUr_{ice} = 0.5, 1, 2.7\\text{ and }5~AU for a disc with f DG =3%f_{DG}=3\\%.In order to investigate the theoretical background of the above power law fit we can start, as in [17], by considering the heating and cooling balance, $Q^+ = Q^-$ , which means $2\\sigma T_{eff}^4 = \\frac{9}{4}\\Sigma _g\\nu \\Omega _K^2~.",
"$ Given that the midplane temperature can be expressed as $T_{mid} = \\left(\\frac{3\\tau _d}{4}\\right)^{1/4} T_{eff}$ , the above equation becomes $\\frac{8\\sigma }{3\\tau _d}T_{mid}^4 = \\frac{9}{4}\\Sigma _g\\nu \\Omega _K^2~.$ We also substitute viscosity with the $\\alpha $ prescription (Eq.",
"REF and Eq.",
"REF for the sound speed) and express the vertical optical depth as $\\tau _d = \\frac{1}{2}\\Sigma _g f_{DG} \\kappa $ , so we get $T_{mid}^3 = \\left(\\frac{27}{64} \\frac{k_B}{\\sigma \\mu m_H}\\right) \\Sigma _g^2 ~f_{DG}~\\kappa ~ \\alpha ~ \\Omega _K~.",
"$ The surface density profile follows $\\Sigma _g = \\Sigma _{g,0} \\left(\\frac{r}{AU}\\right)^{-1/2}$ and $\\Omega _K = \\sqrt{\\frac{GM_*}{r^3}}$ , so we obtain $T_{mid}^3 \\propto \\Sigma _{g,0}^2 ~f_{DG}~\\kappa ~\\alpha ~r^{-5/2}~.$ We can then solve for the position of the iceline $r=r_{ice}$ where $T_{mid}=T_{ice}$ $ r_{ice} \\propto \\Sigma _{g,0}^{4/5} ~ f_{DG}^{2/5} ~\\kappa ^{2/5}~\\alpha ^{2/5}~.",
"$ We thus find that the power-law indices for the dependencies on $\\Sigma _{g,0}$ and $f_{DG}$ are very similar to what we find in our fitting (Eq.",
"REF ).",
"Comparing Eqs.",
"REF and REF suggests that at the iceline $\\kappa \\propto \\alpha ^{1/2}$ , but is almost independent of $\\Sigma _{g,0}$ and $ f_{DG}$ .",
"The reason for this dependency has no easy analytical explanation, but it appears to be the outcome of the feedback between the disc structure and the dust evolution.",
"This further illustrates, as we also discuss in Sec.",
"REF , that we cannot rely on single grain size opacities to accurately describe the disc structure.",
"The position of the iceline as a function of the $\\alpha $ -turbulence parameter and the initial gas surface density $\\Sigma _{g,0}$ from our fitting formula is presented in Fig.",
"REF , for discs with constant $f_{DG}=1\\%$ .",
"The iceline transition is defined as the location where $T=(170\\pm 10)~K$ .",
"Increasing values of either one of the three parameters, $\\alpha $ , $\\Sigma _{g,0}$ , $f_{DG}$ , leads to hotter discs so that the iceline moves closer to the star (see Sect.",
").",
"In the models with any of the grain size distributions, the iceline is located outside 1 AU for $\\alpha \\ge 2.6\\times 10^{-4}$ and exterior to 2.7 AU only when $\\alpha \\ge 1.4 \\times 10^{-3}$ for a disc with $\\Sigma _{g,0} = 1000~g/cm^2$ and $f_{DG}=1\\%$ .",
"However, this also depends on the disc's surface density and total dust-to-gas ratio.",
"If the surface density reduces in time, the disc becomes cooler and the ice line moves inwards, even for the high viscosity cases.",
"For example, for $\\Sigma _{g,0} = 100~g/cm^2$ the iceline is located outside 1 AU for $\\alpha \\ge 5.4 \\times 10^{-3}$ and exterior to 2.7 AU only when $\\alpha \\ge 2.8 \\times 10^{-2}$ (again with a constant $f_{DG}=1\\%$ ).",
"In contrast, if the total dust-to-gas ratio is increased to $3\\%$ , then we find the iceline outside 1 AU even for $\\alpha \\ge 1.4 \\times 10^{-4}$ and outside 2.7 AU for $\\alpha \\ge 6.9 \\times 10^{-4}$ .",
"We can conclude that utilizing Mie theory for the opacities of the grains and taking into account a distribution of grain sizes helps in keeping the iceline sufficiently far out from the star, especially for high $\\alpha $ and $\\Sigma _{g,0}$ values.",
"In general, the location of the iceline might also depend on the composition of the grains, which will be examined in future work.",
"In contrast to [79], who suggest that grain size distributions might help to keep the ice line at larger distances compared to single grain size discs, we actually find the opposite.",
"Including a distribution in a disc simulation results in a similar position of the iceline to the discs with the smallest grain sizes (0.1 and 1 $\\mu $ m).",
"At low viscosities the opacity is dominated by larger grains and thus the disc becomes colder.",
"Unrealistic single grain opacities (typically of micrometer size particles) result in discs that are too hot.",
"This implies that potentially other heating source are needed to keep the iceline at large orbital distances, especially if viscous heating is low [75]."
],
[
"Planet migration",
"The protoplanetary disc's structure also affects planet migration.",
"Very roughly said, if the aspect ratio increases with orbital distance then planets migrate inwards (Type-I migration, [85]).",
"On the contrary, if the aspect ratio is a decreasing function of orbital distance, planets will migrate outwards [18], [22], [20], if the viscosity is large enough [9].",
"We will focus here on the results with $\\alpha = 5\\times 10^{-3}$ , a viscosity large enough to trigger outward migration by the entropy driven corotation torque.",
"For the discs with the grain size distributions, we see an aspect ratio which is a decreasing function of orbital distance beyond the iceline, therefore in those disc regions planets could migrate outwards.",
"Interior to the iceline the aspect ratio is an increasing function of orbital distance which means that planets embedded in this region of the disc would only migrate inwards.",
"The minima in the aspect ratio are locations where planets could get trapped and if (as it is more likely) more than one planet existed, they could get into resonance and remain at those fixed orbital distances until the local parameters of the discs changed sufficiently to force them to migrate again [50], [30], [29], [53], [52].",
"Figure: Torque acting on planets with different masses for the disc utilizing the BOD distribution for the nominal viscosity of α=5×10 -3 \\alpha = 5\\times 10^{-3}.",
"The black line encircles the regions of outward migration and corresponds to the region of the disc where the aspect ratio decreases as a function of the orbital distance.",
"The temperature at the same region shows the steepest gradient.Figure: Same as Fig.",
"for the disc with the MRN distribution.",
"The difference to the BOD distribution is small regarding the size of the region of outward migration, however, the torque is weaker for the MRN distribution.",
"This could lead to different migration and growth behavior of planets forming in the outer disc.Figure: Same as Fig.",
"for the disc with the opacity profile.",
"In addition to the BOD and MRN grain size distribution the disc simulated with the opacity law shows an inner region of outward migration, which is caused by another bump in the H/r profile in the inner disc (Fig.5).Simulations with single particle sizes larger than 100 $\\mu $ m show a monotonically increasing aspect ratio with orbital distance (Fig.",
"REF ), implying that planets in these colder discs would migrate inwards.",
"The speed of the migration scales with the inverse of the square of the aspect ratio.",
"If the aspect ratio decreases, then the migration is faster [8].",
"Therefore, generally speaking, migration would be faster in the single grain size models with large particles and the exact migration speed of planets depends on their exact position in the protoplanetary disc.",
"To predict more precisely the migration of planets in the discs presented here, we include the migration maps for two discs with the distributions (Figs.",
"REF and REF ).",
"Migration rates are derived from the torque formula of [85].",
"For comparison we also show the migration map of the disc with the [10] opacities (Fig.",
"REF ), as these were the opacities used in [20] in Fig.18.",
"The black solid line in Figs.",
"REF , REF , REF encircles the regions of outward migration.",
"Planets with masses less than approximately 10 $M_{\\oplus }$ always migrate inwards in the simulations with the distributions, whereas for the simulation using the [10], pure micrometer opacities, inward migration is the only possibility for planets less than 6 $M_{\\oplus }$ .",
"The innermost region of outward migration in Fig.REF corresponds to the area where the aspect ratio decreases as the orbital distance increases (Fig.",
"REF ).",
"We can also see that this area has a steeper temperature gradient (Fig.",
"REF ) both interior and exterior to the water iceline transition.",
"The increased torques and consequently migration speeds at the outer region of outward migration between 5 and 20 AU in the [10] disc simulations are caused by the steep increase of opacity for temperatures larger than 170 K, which is not the case for the BOD and MRN distribution, as these distributions are not dominated by micrometer grains, in contrast to the [10] opacity model.",
"This illustrates that the grain sizes which dominate or contribute the most to the opacity of the disc have significant implications, not only for the disc structure itself, but also indirectly for the planets embedded in that disc.",
"In addition, this has important effects for the formation of planetary systems, because the migration rates determine how close planets can migrate towards each other, which sets the stability of the planetary system [68], [26]."
],
[
"Implications for planet formation and protoplanetary disc simulations",
"Our here presented simulations are the first step towards more self-consistent protoplanetary disc structure and evolution simulations as well as planet formation simulations.",
"Planet formation in the pebble assisted core accretion scenario rely crucially on the pebble sizes and distributions [82], [59], [55], as well as on the disc structure to calculate the planet migration rates as the planets grow [21].",
"The here presented model opens the avenue to simulations with self-consistent disc structures and pebble sizes, which can then be accreted onto planets.",
"This can increase the accuracy of future planet formation simulations by pebble accretion.",
"The here used FARGOCA code also allows for 3D hydrodynamical simulations with embedded planets.",
"A combination with the presented model of thermal structures calculated from full grain size distributions allows a very detailed comparison with observations, which are more advanced that the mostly used 2D isothermal simulations followed by 3D radiative transfer (e.g.",
"[110]).",
"This could potentially change our interpretation of observed protoplanetary discs featuring substructures potentially caused by planets."
],
[
"Summary",
"We perform 2D hydrodynamical simulations including the whole feedback loop shown in Fig.",
"REF .",
"Specifically, we include and test two full grain size distributions and mean opacities (calculated via Mie theory) and study their influence on the disc structure.",
"The particles have a minimal size of 0.025 $\\mu $ m and the upper boundary is regulated by the fragmentation barrier (Eq.",
"REF ).",
"We test five different $\\alpha $ -viscosity values ($10^{-2}, 5\\times 10^{-3}, 10^{-3}, 5\\times 10 ^{-4}, 10^{-4}$ ), five values of initial gas surface density $\\Sigma _{g,0}$ (100, 250, 500, 750 and 1000 $g/cm^2$ ) and five values of dust-to-gas ratio $f_{DG}$ (1%, 1.5%, 2%, 2.5%, 3%).",
"We also perform simulations with only single grain sizes and with the [10] opacity law for comparison and in order to understand to greater extend the influence of grains of different sizes on the thermal structures of protoplanetary discs.",
"The dust component in protoplanetary discs is believed to follow a size distribution, regulated by a coagulation-fragmentation equilibrium [25], [15].",
"We utilize and compare two different grain size distributions.",
"The first and simple model (MRN) [36], [67], [102] results from the equilibrium between fragmentation and coagulation, whereas the second and more complex model [15] takes into account fragmentation, coagulation and also cratering and adjusts the dust surface densities according to the grain sizes and how they compare to the size of the gas molecules and the gas turbulent eddies.",
"The dust surface densities are calculated as dictated by the aforementioned grain size distributions and the dust grains are also vertically distributed according to their sizes taking into account the effect of settling.",
"We also have a spatial distribution radially, since the size distribution depends on the local disc parameters and changes self-consistently.",
"In conclusion, a whole loop of growth, fragmentation, and settling of the resulting grains for each vertical slice of the disk is modeled in our simulations and updated at every timestep according to the local disc parameters.",
"We show disc structures calculated with the full grain size distributions and single grain sizes in Figs.",
"REF and REF .",
"Additionally we show that the grain sizes which dominate or contribute the most to the opacity of the disc are not the same at all orbital distances of the disc (Figs.",
"REF and REF ).",
"As a consequence, the opacity prescriptions which assume a single dust size lead to inaccurate calculations of the thermal structures of the discs.",
"It is also important to stress that the dust surface densities, or in other words the distribution of mass among the grain sizes, play a major role in determining the disc opacity (Eq.",
"REF ), which in turn influences the cooling rate and the stellar heating and changes the temperature and surface density of the gas.",
"This shift in the local disc parameters leads to a new fragmentation barrier (and regime boundaries for the BOD), therefore the dust surface densities change and so on.",
"For this reason it is important to include the self-consistent calculations of the dust surface densities in the simulations.",
"The two grain size distributions show minimal differences in the dust surface densities (Fig.",
"REF ).",
"The reason for this is that both of the grain size distributions we have used in the discs, feature the same fragmentation barrier.",
"Therefore the grain size range in the discs with either one of the distributions is similar.",
"Any difference between the discs with the BOD distribution and the discs with the MRN distribution comes mainly from the difference in the surface densities as a function of grain size (see Figs.",
"REF and REF ), which is usually smaller than an order of magnitude.",
"The dominant grain sizes (Sec.",
"REF ) might not be the same, because of the small differences in the dust surface densities per grain size, but the total opacity of the disc is similar independently of the grain size distribution.",
"With this accurate prescription we investigate the dependency of the iceline position on the $\\alpha $ -viscosity, the initial gas surface density and the dust-to-gas ratio, where we see the effect of the feedback loop and find $r_{ice} \\propto \\alpha ^{0.61}\\Sigma _{g,0}^{0.8}f_{DG}^{0.37}$ (Eq.",
"REF , Fig.",
"REF ) independently of the grain size distribution utilized in the disc model.",
"Specifically, for high gas surface density ($\\Sigma _{g,0}=1000~g/cm^2$ ) the position of the iceline is exterior to 1 AU for $\\alpha \\ge 2.64\\times 10^{-4}$ and exterior to 2.7 AU only when $\\alpha \\ge 1.35 \\times 10^{-3}$ .",
"For higher values than the nominal $f_{DG}=1\\%$ we find that the iceline moves closer to the star as it is expected by the enhanced dust surface densities and the consequent hotter discs.",
"However, for the nominal $f_{DG}=1\\%$ , lowering the gas surface density results in colder discs and the iceline is below 2.7 AU, even for the high viscosity models.",
"The changes in the aspect ratio gradient as a function of orbital distance affect the regions where outward migration is possible for planets that could be embedded in the disc (Figs.",
"REF , REF , REF ).",
"Utilizing an $\\alpha $ -viscosity of $5\\times 10^{-3}$ , $\\Sigma _{g,0} = 1000~g/cm^2$ and $f_{DG}=1\\%$ we find that the regions where outward migration could be possible in the discs with the two distributions are similar to the one present in a disc with the [10] opacities, that feature only micrometer sized grains, at around 5-15 AU for planets with masses greater than 10 M$_{\\oplus }$ .",
"However, the region is more extended for the disc with the BOD distribution (up to 20 AU) and the disc with the [10] opacities has one more outward migration regions, near the inner boundary (2-3 AU), which is not present in the discs with the grain size distributions.",
"We can hence conclude that given the complexity and computational expense of the BOD distribution and the fact that it does not take into account radial drift or bouncing of the dust particles it is not necessary to prefer it over a simple MRN-like power-law distribution.",
"As the iceline can be the starting point for planetesimal formation [44], [96], [39] it is important to have as realistic models as possible, therefore include the feedback loop of grain growth and thermodynamics in hydrodynamical models (Fig.",
"REF ).",
"Given also the fact that dust in protoplanetary discs follows a size distribution regulated by a coagulation-fragmentation equilibrium, the opacity prescription of a single grain size is not able to accurately calculate the thermal structures of discs.",
"The here presented model has some limitations that we wish to further investigate in future work.",
"Both of the distributions tested here neglect radial drift [25], [14] and bouncing [112], [64] which can be detrimental for grain growth and in general affect the dust dynamics and subsequently the disc's thermal structure.",
"Also the onset of convection in some regions and for a subset of $\\alpha $ and $\\Sigma _{g,0}$ values might change the vertical distribution of the grains beyond settling and turbulent stirring by viscosity, as taken into account here.",
"A very important future step is to model accretion discs instead of equilibrium discs and in this way we will be able to also study different evolutionary steps of the protoplanetary disc.",
"The particle composition and abundances are also determinant for dust dynamics and opacities, so it is important to consider a population that is as realistic as possible and use more accurate fragmentation velocities depending on our dust composition.",
"Similarly, we are assuming a dust population consisting of 50% silicates and 50% water-ice, but we can relax this assumption and test different fractions (as done for example in [19]).",
"It is hence evident that the prescription that we used and presented for this work opens up new avenues for protoplanetary disc simulations and planet formation.",
"The inclusion of the feedback loop of grain growth and disc thermodynamics leads to more self consistent simulations of protoplanetary accretion discs and planet formation simulations in the pebble accretion scenario.",
"Eventually, such models target a more precise comparison of protoplanetary disc observations to simulations that allow us to move away from simple 2D isothermal models with post-processing of radiation transfer.",
"B.B.",
"and S.S. thank the European Research Council (ERC Starting Grant 757448-PAMDORA) for their financial support.",
"M.L.",
"thanks the Knut and Alice Wallenberg Foundation (grant 2017.0287, PI A. Johansen).",
"S.S is a Fellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD).",
"We would like to thank the anonymous referee for the valuable comments and suggestions that helped us improve the manuscript."
],
[
"Dust surface densities for different $\\alpha $ -viscosity values",
"We present here the vertically integrated dust surface densities and the opacities for the simulations using the rest of the $\\alpha $ values and the two grain size distributions.",
"The maximum grain sizes of the BOD and MRN discs are approximately the same, because they follow the same fragmentation barrier formula (Eq.",
"REF ).",
"The vertically integrated dust surface densities are around one order of magnitude lower in the discs with the BOD grain size distribution compared to the discs with the MRN distribution for small particles (Fig.",
"REF ).",
"On the other hand they are always comparable for the largest particles in the discs.",
"This is already evident by the shape of the two grain size distributions (Fig.",
"REF ).",
"As $\\alpha $ decreases, the surface densities of the smallest particles in the discs with the BOD distribution are several orders of magnitude lower than these of the largest particles.",
"The gradients are smoother in the discs with the MRN distribution as $\\alpha $ decreases.",
"The fragmentation barrier depends on the initial gas surface density so when the latter decreases, the maximum grain size gets smaller (Eq.",
"REF ).",
"In Fig.",
"REF we show the opacities as a function of orbital distance and height for a selection of the simulated discs for this work.",
"The $\\tau $ =1 line is located at similar heights in all of the discs with high $\\alpha $ -viscosity (around 2 AU at the outer edge).",
"The same applies to the low $\\alpha $ -viscosity discs where the $\\tau $ =1 line is always around 0.15 AU at the outer edge.",
"As expected the optically thick region is extended towards higher altitudes with higher total dust-to-gas ratio and contained near the midplane for low gas surface densities.",
"Above the $\\tau $ =1 line opacity always decreases as cooling is more efficient.",
"However, the uppermost layers show increased opacities because of the stellar irradiation that directly heats them up.",
"Figure: MRN, α=5×10 -4 \\alpha =5\\times 10^{-4}, Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2, f DG =1%f_{DG}=1\\%Figure: Dust surface densities as a function of orbital distance and grain size for the different α\\alpha values used here, and for additional simulations with the lowest gas surface densities and with the highest dust-to-gas ratio that we have tried.",
"When the turbulence is reduced, the maximum grain size increases (since less destructive collisions are expected).",
"In addition to that, the reduced α\\alpha -viscosity allows the discs to become cooler, so the opacity of the larger grains (∼\\sim mm) becomes comparable or larger than that of the smaller (∼μ\\sim \\mu m) grains.",
"For α=10 -4 \\alpha =10^{-4} we find that the grains grow up to a few centimeters in the discs with either one of the distributions.",
"The dashed lines divide the grain sizes into two groups which contribute equally to the total opacity of the disc (see Figs.",
"and ).Figure: MRN, α=5×10 -4 \\alpha =5\\times 10^{-4}, Σ g,0 =1000g/cm 2 \\Sigma _{g,0} = 1000~g/cm^2, f DG =1%f_{DG}=1\\%Figure: Mean Rosseland opacities as a function of orbital distance and height for the different α\\alpha values, the lowest gas surface densities and the highest dust-to-gas ratio.",
"Due to the larger grain sizes for the MRN distribution (Fig.",
"), the opacities for the MRN distribution are also generally lower compared to the BOD distribution.",
"The light blue line corresponds to optical depth τ=1\\tau =1 integrated vertically starting from infinity towards midplane, so it divides the optically thin (above) and thick region (below)."
],
[
"Iceline position as a function of $\\alpha $ -turbulence, initial gas surface density and dust-to-gas ratio",
"In Sec.",
"REF we present the position of the iceline as a function of the $\\alpha $ -viscosity parameter, the initial gas surface density, $\\Sigma _{g,0}$ and the total dust-to-gas ratio $f_{DG}$ .",
"In order to do this fitting, we used five simulations for each parameter (see Table REF ).",
"In Fig.",
"REF we show the individual fitting over each one of the parameters.",
"The fit to the three parameters writes $f = C \\cdot \\left(\\frac{\\alpha }{0.01}\\right)^{p_1}\\cdot \\left(\\frac{\\Sigma _{g,0}}{1000~g/cm^2}\\right)^{p_2}\\left(\\frac{f_{DG}}{0.01}\\right)^{p_3}~,$ with C=9.20$\\pm $ 0.05 AU, $p_1$ = 0.61$\\pm $ 0.03, $p_2$ = 0.77$\\pm $ 0.03, $p_3$ = 0.37$\\pm $ 0.01.",
"The resulting fit is the same regardless of the grain size distribution used in the disc (solid line in Fig.",
"REF ).",
"Table: Parameters used in the simulations performed for the fitting of the iceline position to α\\alpha -viscosity, initial gas surface density and total dust-to-gas ratio for the BOD and the MRN distribution."
],
[
"The effect of settling",
"We implement in our work the effect of settling for the grains in the disc as described in Sect.",
"REF , in order to vertically distribute the grains according to their sizes and the local disc parameters.",
"This implies that the disc structure can be affected both by a change in the grain size due to the different opacities that each size provides (Fig.",
"REF ) and by a change in the settling efficiency of the given grain size.",
"In order to test if both of these effects are significant factors that define the disc structure, we run one simulation where the disc only contains millimeter grains and compare with a disc which also contains only millimeter grains but does not take settling into account.",
"Thus in this latter case, the millimeter grains are vertically distributed according to a constant dust-to-gas ratio throughout the whole disc.",
"Additionally, we run a simulation where the opacities of the grains correspond to millimeter grains, but we assume that the grains are vertically distributed as micrometer grains (so we assume $s= 1\\mu m$ in the equations describing settling, Eq.",
"REF - REF ).",
"We choose $\\alpha =10^{-4}$ for which the settling of millimeter grains will be very effective.",
"However, micrometer dust grains are not expected to be affected by settling even at this low $\\alpha $ -viscosity.",
"The models also have an initial gas surface density of $\\Sigma _{g,0} = 1000~g/cm^2$ and total dust-to-gas ratio $f_{DG} = 1\\%$ .",
"In Figure REF we present the aspect ratio of these three disc models.",
"The aspect ratio as a function of orbital distance for the disc where grains are vertically distributed as micrometer sized dust resembles the one of a disc where the dust-to-gas ratio is constant all over the disc.",
"This is expected because micrometer sized dust is not significantly affected by settling even at the low $\\alpha $ -viscosity of $10^{-4}$ (see Fig.",
"REF ).",
"However, we find that the aspect ratio is lower in the inner regions of discs when the millimeter grains are allowed to settle with their corresponding properties.",
"When settling is included, the millimeter grains are mainly concentrated near the midplane (see also Fig.",
"REF ), while at higher altitudes the opacity diminishes.",
"Without settling or with reduced efficiency of settling the opacity is similar at all altitudes, which leads to a reduced cooling rate and higher aspect ratio in the inner regions.",
"Without settling of the millimeter grains according to their size properties, the outer regions are not sufficiently heated.",
"Due to the increased aspect ratio in the inner disc, stellar irradiation to the outer disc is diminished, creating a shadow that cools down the outer region.",
"At the same time the millimeter grains have very low opacity so they cannot absorb the stellar heating efficiently in the outer disc.",
"The disc at the outermost radii might keep cooling down until it reaches the temperature of the surroundings [41].",
"Hence including settling is very important to avoid such complications and inconsistencies in the disc structures.",
"Different grain sizes lead to different disc structures even without any settling implemented.",
"The distinctive structures of discs with different grain sizes comes mainly by their individual opacities (Fig.",
"REF ).",
"However, without settling the disc opacity above midplane is overestimated so the discs are hotter in the inner regions and thus do not allow the stellar irradiation to heat the outer regions.",
"In order to consistently take into account the influence of a grain size distribution to the resulting disc structures, it is important to include settling.",
"Figure: Comparison of the aspect ratio as a function of orbital distance for discs with mm grains with settling, without settling and with the settling of μm\\mu m grains.",
"The disc structure is almost the same when no settling is implemented and when the opacities correspond to mm grains, but the grains are distributed as micrometer sized dust (Eq.",
"- ).",
"When settling is included the millimeter grains are concentrated near the midplane, leaving the upper layers with diminished opacities and enhanced cooling rate.",
"Without efficient settling the inner discs get hotter.",
"However, this creates a shadow that prevents the efficient heating of the outermost regions from stellar irradiation."
]
] | 2005.14097 |
[
[
"Blockchain is Watching You: Profiling and Deanonymizing Ethereum Users"
],
[
"Abstract Ethereum is the largest public blockchain by usage.",
"It applies an account-based model, which is inferior to Bitcoin's unspent transaction output model from a privacy perspective.",
"Due to its privacy shortcomings, recently several privacy-enhancing overlays have been deployed on Ethereum, such as non-custodial, trustless coin mixers and confidential transactions.",
"In our privacy analysis of Ethereum's account-based model, we describe several patterns that characterize only a limited set of users and successfully apply these quasi-identifiers in address deanonymization tasks.",
"Using Ethereum Name Service identifiers as ground truth information, we quantitatively compare algorithms in recent branch of machine learning, the so-called graph representation learning, as well as time-of-day activity and transaction fee based user profiling techniques.",
"As an application, we rigorously assess the privacy guarantees of the Tornado Cash coin mixer by discovering strong heuristics to link the mixing parties.",
"To the best of our knowledge, we are the first to propose and implement Ethereum user profiling techniques based on quasi-identifiers.",
"Finally, we describe a malicious value-fingerprinting attack, a variant of the Danaan-gift attack, applicable for the confidential transaction overlays on Ethereum.",
"By incorporating user activity statistics from our data set, we estimate the success probability of such an attack."
],
[
"Introduction",
"The narrative around cryptocurrency privacy provisions has dramatically changed since the inception of Bitcoin [38].",
"Initially many, especially criminals, thought Bitcoin and other cryptocurrencies provide privacy to hide their illicit business activities [15].",
"The first extensive study about Bitcoin's privacy provisions was done by Meiklejohn et al [35], in which they provide several powerful heuristics allowing one to cluster Bitcoin addresses.",
"The revelation of Bitcoin's privacy shortcomings spurred the creation and implementation of many privacy-enhancing overlays for Bitcoin [60], [10], [51], [69].",
"As of today, several Bitcoin wallets, e.g.",
"Wasabi and Samourai wallets, provide privacy-enhancing solutions to their users.",
"Previous work has focused on assessing the privacy guarantees provided by several UTXO-based (unspent transaction output) cryptocurrencies, such as Bitcoin [3], [35], Monero [14], [37], [8] or Zcash [6], [7], [8], [27], [58].",
"However, perhaps surprisingly, until today there were no similar empirical studies on account-based cryptocurrency privacy provisions.",
"Therefore in this work, we put forth the problem of studying the privacy guarantees of Ethereum's account-based model.",
"Assessing and understanding the privacy guarantees of cryptocurrencies is essential as the lack of financial privacy is detrimental to most cryptocurrency use cases.",
"Furthermore, there are state-sponsored companies and other entities, e.g.",
"Chainalysis [42], performing large-scale deanonymization tasks on cryptocurrency users.",
"In contrast to the UTXO-model, many cryptocurrencies that provide smart contract functionalities operate with accounts.",
"In an account-based cryptocurrency, users store their assets in accounts rather than in UTXOs.",
"Already in the Bitcoin white paper, Nakamoto suggested that “a new key pair should be used for each transaction to keep them from being linked to a common owner” [38].",
"Despite this suggestion, account-based cryptocurrency users tend to use only a handful of addresses for their activities.",
"In an account-based cryptocurrency, native transactions can only move funds between a single sender and a single receiver, hence in a payment transaction, the change remains at the sender account.",
"Thus, a subsequent transaction necessarily uses the same address again to spend the remaining change amount.",
"Therefore, the account-based model essentially relies on address-reuse on the protocol level.",
"This behavior practically renders the account-based cryptocurrencies inferior to UTXO-based currencies from a privacy perspective.",
"Previously, several works had identified the privacy shortcomings of the account-based model, specifically in Ethereum.",
"Those works had proposed trustless coin mixers [34], [53], [55] and confidential transactions [66], [11], [13].",
"However, until recently, none of these schemes has been deployed on Ethereum.",
"Even today, Ethereum's privacy-enhancing overlays are still in a nascent, immature phase especially in comparison with Bitcoin's well-established coin mixer scene [10], [60], [69], [26], [51].",
"Our contributions: We identify and apply several quasi-identifiers stemming from address reuse (time-of-day activity, transaction fee, transaction graph), which allow us to profile and deanonymize Ethereum users.",
"In the cryptocurrency domain, we are the first to quantitatively assess the performance of a recent area of machine learning in graphs, the so-called node embedding algorithms.",
"We establish several heuristics to decrease the privacy guarantees of non-custodial mixers on Ethereum.",
"We describe a version of the malicious value fingerprinting attack, also known as Danaan-gift attack [7], applicable in Ethereum.",
"We collect and analyze a wide source of Etherum related data, including Ethereum name service (ENS), Etherscan blockchain explorer, Tornado Cash mixer contracts, and Twitter.",
"We release the collected data as well as our source code for further researchhttps://github.com/ferencberes/ethereum-privacy.",
"The rest of the paper is organized as follows.",
"In Section , we review related work.",
"In Section , a brief background is given on the inner workings of Ethereum along with the general idea behind node embedding.",
"In Section , we describe our collected data.",
"In Section , we overview the literature on evaluating deanonymization methods and propose our metrics.",
"Our main methods to pair Ethereum addresses that belong to the same user and link Tornado deposits and withdrawals are detailed in Section and .",
"A variant of the Danaan-gift attack is described in Section .",
"Finally, we conclude our paper in Section ."
],
[
"Related Work",
"First results on Ethereum deanonymization [30] attempted to directly apply both on-chain and peer-to-peer (P2P) Bitcoin deanonymization techniques.",
"The starting point of our work is the recognition that common deanonymization methods for Bitcoin are not applicable to Ethereum due to differences in Ethereum's P2P stack and account-based model.",
"The relevant body of more recent literature takes two different approaches.",
"The first analyzes Ethereum smart contracts with unsupervised clustering techniques [43].",
"Kiffer et al.",
"[28] assert a large degree of code reuse which might be problematic in case of vulnerable and buggy contracts.",
"The second branch of literature assesses Ethereum addresses.",
"A crude and initial analysis had been made by Payette et al., who clusters the Ethereum address space into only four different groups [45].",
"More interestingly Friedhelm Victor proposes address clustering techniques based on participation in certain airdrops and ICOs [61].",
"These techniques are indeed powerful, however, they do not generalize well as it assumes participation in certain on-chain events.",
"Our techniques are more general and are applicable to all Ethereum addresses.",
"Victor et al.",
"gave a comprehensive measurement study of Ethereum's ERC-20 token networks, which further facilitates the deanonymization of ERC-20 token holders [62].",
"A completely different and unique approach is taken by [32], which uses stylometry to deanonymize smart contract authors and their respective accounts.",
"The work had been used to identify scams on Ethereum."
],
[
"Background",
"In this section we provide some background on cryptocurrency privacy-enhancing technologies as well as node embedding algorithms.",
"Elementary preliminaries on Ethereum and its applied gas mechanism are included in Appendix ."
],
[
"Non-custodial mixers",
"Coin mixing is a prevalent technique to enhance transaction privacy of cryptocurrency users.",
"Coin mixers may be custodial or non-custodial.",
"In case of custodial mixing, users send their “tainted” coins to a trusted party, who in return sends back “clean” coins after some timeout.",
"This solution is not satisfactory as the user does not retain ownership of her coins during the course of mixing.",
"Hence, the trusted mixing party might just steal funds, as it already happened with custodial mixers [35].",
"Figure: Schematic depiction of non-custodial mixers on EthereumMotivated by these drawbacks, recently several non-custodial mixers have been proposed in the literature [34], [65], [53], [55].",
"The recurring theme of non-custodial mixers is to replace the trusted mixing party with a publicly verifiable transparent smart contract or with secure multi-party computation (MPC).",
"Non-custodial mixing is a two-step procedure.",
"First, users deposit equal amounts of ether or other tokens into a mixer contract from an address $\\mathcal {A}$ , see Figure REF .",
"After some user-defined time interval, they can withdraw their deposited coins with a withdraw transaction to a fresh address $\\mathcal {B}$ .",
"In the withdraw transaction, users can prove to the mixer contract that they deposited without revealing which deposit transaction was issued by them by using one of several available cryptographic techniques, including ring signatures [34], verifiable shuffles [53], threshold signatures [55], and zkSNARKs [65]."
],
[
"Ethereum Name Service",
"Ethereum Name Service (ENS) is a distributed, open, and extensible naming system based on the Ethereum blockchain.",
"In spirit it is similar to the well-known Domain Name Service (DNS).",
"However, in ENS the registry is implemented in Ethereum smart contractsSee: https://docs.ens.domains, hence it is resistant to DoS attacks and data tampering.",
"Like DNS, ENS operates on a system of dot-separated hierarchical names called domains, with the owner of a domain having full control over subdomains.",
"ENS maps human-readable names like alice.eth to machine-readable identifiers such as Ethereum addresses.",
"Therefore, ENS provides a more user-friendly way of transferring assets on Ethereum, where users can use ENS names (alice.eth) as recipient addresses instead of the error-prone hexadecimal Ethereum addresses."
],
[
"Node embeddings",
"Node embedding methods form a class of network representation learning methods that map graph nodes to vectors in a low-dimensional vector space.",
"They are designed to represent vertices with similar neighborhood structure by vectors that are close in the vector space.",
"Intuitively, addresses that interact with the same set of addresses in the Ethereum transaction graph should be close in the embedded space.",
"Perhaps the best methods are Laplacian eigenmaps [5] and graph factorization [1].",
"Research in node embedding has recently been catalyzed by Word2Vec [36], an embedding method for natural language processing.",
"Several node embedding methods have been proposed recently [46], [57], [24], [48] and applied successfully for multi-label classification and link prediction in a variety of real-world networks from diverse domains.",
"In this work, we use these techniques on the Ethereum transaction graph to link addresses owned by the same user.",
"To the best of our knowledge, we are the first to apply node embedding for Ethereum user profiling.",
"Figure: Fraction of ENS names (collected from Twitter) that interacted with the given service topics.",
"Popular services within the categories are shown in Figure .Figure: Most popular services within the Defi, Exchange, Stablecoins and Gaming categories in our data collection."
],
[
"Data collection",
"We collected addresses presumably related to regular users and not automatic (trader or exchange) bots from the following publicly available data sources.",
"Twitter: By using the Twitter APIUsing the Twitter Search and People API endpoints, we collected tweets containing the following keywords {'@ensdomains','.eth','ENS name','ENS address', 'ethereum', '#ethereum'} as well as profiles with an ENS name in their displayed profile name or description.",
"We also searched for ENS names in the name and description of every tweeter in our data.",
"Twitter data collection lasted from 2019-11-15 until 2020-03-05., we were able to collect 890 ENS names included in Twitter profile names or descriptions, and discover the connected Ethereum addresses, see Figure REF .",
"Humanity DAO:See: https://www.humanitydao.org/humansA human registry of Ethereum users, which can include a Twitter handle in addition to the Ethereum address.",
"Tornado Cash mixer contracts: We collected all Ethereum addresses that issued or received transactions from Tornado Cash mixers up to 2020-04-04.",
"Table REF shows the total number of addresses collected from each data source as well as addresses with at least 5 sent transactions.",
"We note that there are overlaps between the three address groups, see the last row of Table REF .",
"By using the Etherscan blockchain explorer API, we collected 1,155,188 transactions sent or received by the addresses in our collection.",
"The final transaction graph contains 159,339 addresses.",
"The transactions span from 2015-07-30 until 2020-04-04.",
"Figure REF shows the average number of transactions sent and received in the three data sources.",
"Addresses collected from Twitter and Humanity DAO have similar behavior, while Tornado accounts have fewer transactions since Tornado Cash has only recently been launched.",
"Finally, using the Etherscan Label Word Cloud, we manually collected service category labels (e.g.",
"exchange, gambling, stablecoins) related to popular addresses in our data set.",
"We summarize the fraction of ENS names in our collection that interacted with the given services in Figure REF .",
"We observed that the publicly revealed ENS names already expose sensitive activities such as gambling and adult services.",
"Therefore, users should avoid sensitive activities on addresses easily linkable to their public identities, such as ENS name or their Twitter handle.",
"Table: Number of Ethereum addresses collected from three different sources.",
"* ^*Tornado ground truth pairs are only heuristically identified, see Section .",
"Due to overlaps between the data sources, the total number of investigated addresses is less than the sum of the records in the top three rows.Figure: Unique address count of ENS names collected from Twitter.",
"Most of the ENS names in our collection are linked to a single Ethereum address, while some entities use multiple accounts.",
"In Section , we use ENS names with exactly two unique addresses (green) to measure the performance of different profiling techniques.Figure: Average number of transactions sent or received by the addresses of each data source.",
"Tornado accounts have less transactions as the service has only recently been launched."
],
[
"Evaluation measures",
"In this paper, we propose deanonymization methods for pairing Etherum accounts of the same user (Section ), Tornado deposits and withdrawals (Section ), and fingerprinting accounts (Section ).",
"To establish an appropriate measure for evaluating our methods, we face the diversity and complexity of estimates of the adversary’s success to breach privacy.",
"In the literature, the adversary's output takes the form of a posterior probability distribution, see the survey [63].",
"The simplest metrics consider the success rate of a deanonymizing adversary.",
"Metrics such as accuracy, coverage, fraction of correctly identified nodes [4], [41], [39] are applicable only when the attack has the potential to exactly identify a significant part of the network.",
"Exact identification is an overly ambitious goal in our experiments, which aim to use very limited public information to rank candidate pairs and quantify the leaked information as risk for a potential systematic deanonymization attack.",
"For this reason, we quantify non-exact matches, since even though our deanonymizing tools might not exactly find a mixing address, they can radically reduce the anonymity set, which is still harmful to privacy.",
"We want to quantify the information leaked from network structure, time-of-day activity, and gas price usage to assess the implications for the future privacy [40] of the account owners.",
"In our first two deanonymization experiments, our algorithms will return a ranked list of candidate pairs for each account in our testing set.",
"Based on the ranked list, we propose a simple metric, the average rank of the target in the output.",
"Recent results consider deanonymization as a classification task and use AUC for evaluation [33].",
"In our experiments, we will compute AUC by the following claim: Lemma 5.1 Consider a set of accounts $a$ , each with a set of candidate pairs $c(a)$ such that exactly one in $c(a)$ is the correct pair of $a$ .",
"Let an algorithm return a ranked list of all sets $c(a)$ .",
"The AUC of this algorithm is equal to the average of $r(a)/|c(a)|$ over all $a$ , where $r(a)$ is the rank of the correct pair of $a$ in the output.",
"Follows since AUC is the probability that a randomly selected correct record pair is ranked higher than another incorrect one [25].",
"Finally, we consider evaluation by variants of entropy, which quantify privacy loss by the number of bits of additional information needed to identify a node.",
"Defining entropy is difficult in our case for two reasons.",
"First, our algorithms provide a ranked list and not a probability distribution.",
"Second, for the Tornado Cash mixer deanonymization, the anonymity set size is dynamic, as users can freely deposit anytime they wish, hence increasing the anonymity set size.",
"In the literature, entropy based evaluation considers the a priori knowledge without a deanonymization method and the a posteriori knowledge after applying one [54].",
"Several papers compute the entropy of the a posteriori knowledge [54], [16], [40], however they assume that the deanonymizer outputs a probability distribution of the candidate records [40].",
"The information the attacker has learned with the attack can be expressed as the difference of the a priori and a posteriori entropy.",
"We call this difference the entropy gain, denoted as gain$(n, p)$ where $n$ and $p$ are the anonymity set size and probability distribution, respectively.",
"The a priori entropy of the target record is typically the base-2 logarithm of the a priori anonymity set size.",
"The problem with varying a priori anonymity set size is that while correctly selecting ten candidate users from a pool of a million is a great achievement, the same entropy of $\\log _2(10)$ is achieved without deanonymization if the initial pool size, for example in a low-utilization mixer, is only 10.",
"We note that in [16], the authors also divide the entropy gain to normalize the value.",
"Next, we describe a new method to infer the a posteriori distribution given varying a priori knowledge and appropriately normalize with respect to the a priori entropy.",
"More precisely, first we give a heuristic argument that the a priori anonymity set size has little effect on the entropy gain, and hence we can compare and average across different measurements.",
"In the formula below, given an a priori anonymity set size $2n$ vs. $n$ , we compare the entropy gain of the same distribution $p$ , gain$(2n,p)-{}$ gain$(n,p)$ .",
"In the formula below, $p_i$ denotes the probability $p([(i-1)/(2n),i/(2n)])$ .",
"$\\nonumber \\text{gain}(2n,p) &=& \\log _2(2n)+ \\sum _{i=1}^{2n} p_i \\log _2(p_i);\\\\\\nonumber \\text{gain}(n,p) &=& \\log _2(n)+ \\sum _{i=1}^{n} (p_{2i-1} + p_{2i}) \\log _2(p_{2i-1} + p_{2i}).$ Since $\\log _2(2n)-\\log _2(n) = 1 = \\sum _i p_i$ , we may group the terms to obtain the difference in the entropy gain as the sum for $1\\le i \\le n$ of $p_{2i-1} \\log _2\\left(\\frac{2p_{2i-1}}{p_{2i-1}+ p_{2i}}\\right) + p_{2i} \\log _2\\left(\\frac{2p_{2i}}{p_{2i-1}+ p_{2i}}\\right),$ which can be bounded from above by using $\\log x <x-1$ as $\\frac{(p_{2i-1} - p_{2i})^2}{p_{2i-1} + p_{2i}}.$ If the probability distribution is smooth with little density changes in a neighborhood, the above value is very small.",
"For example, the value is small if $p_i$ is monotonic in $i$ , which at least approximately holds in our experiments.",
"Based on the above argument, we may infer an empirical probability distribution of the candidates ranked by an algorithm.",
"For each a priori size $n$ and rank $r$ for the ground truth pair of a target record, we define the distribution $P(n,r)$ to be uniform in $[(r-1)/n, r/n]$ , and 0 elsewhere, in accordance with formula (REF ).",
"The empirical probability distribution of an algorithm will be the average of $P(n,r)$ over all the output of the algorithm.",
"In the discussion, we will use the entropy gain of the above empirical probability distribution to quantify the deanonymization power of our algorithms."
],
[
"Linking Ethereum accounts of the same user",
"In this section, we introduce our approach to identify pairs of Ethereum accounts that belong to the same user.",
"In our measurements, we investigate three quasi-identifiers of the account owner: the active time of the day, the gas price selection, and the location in the Ethereum transaction graph.",
"We evaluate our methods by using the set of address pairs in our collection that belong to the same name in the Ethereum Name Service (ENS), see Figure REF .",
"We consider 129 ENS names with exactly two Ethereum addresses to avoid the possible validation bias caused by ENS names with more than two addresses.",
"We also note that Ethereum addresses connected to multiple ENS names were excluded from our experiments."
],
[
"Time-of-day transaction activity",
"Ethereum transaction timestamps reveal the daily activity patterns of the account owner, see Figure REF .",
"In the top row of Figure REF , we show time-of-day profiles for two ENS names that are active in different time zones.",
"Given the set of timestamps, an account is represented by the vector including the mean, median and standard deviation, as well as the time-of-day activity histogram divided into $b_{\\text{hour}}$ bins."
],
[
"Gas price distribution",
"Ethereum transactions also contain the gas price, which is usually automatically set by wallet softwares.",
"Users rarely change this setting manually.",
"Most wallet user interfaces offer three levels of gas prices, slow, average, and fast where the fast gas price guarantees almost immediate inclusion in the blockchain.",
"The changes in daily Ethereum traffic volume sometimes cause temporary network congestion, which affect user gas prices.",
"Hence we normalized the gas price by the daily network average.",
"In Figure REF , the two peaks of the normalized gas price around $0.5$ and 1 correspond to the slow and average gas price options.",
"On the other hand, users only occasionally charge more than three times the daily average gas price.",
"The combination of these gas price levels forms the so-called gas price profile for each Ethereum user.",
"Given the normalized gas prices of the transactions sent, an account is represented by the vector including the mean, median and standard deviation, as well as the normalized gas price histogram divided into $b_{\\text{gas}}$ bins.",
"Figure: Time-of-day and normalized gas price profiles for two ENS names with a pair of addresses each.",
"Both the time-of-day and gas price selection are similar in case of matmeth.eth addresses (red, green) while the addresses of kinchase.eth (blue, orange) have different gas price profiles.",
"Addresses are denoted by different colors."
],
[
"Transaction graph analysis",
"The set of addresses used in interactions characterize a user.",
"Users with multiple accounts might interact with the same addresses or services from most of them.",
"Furthermore, as users move funds between their personal addresses, they may unintentionally reveal their address clusters.",
"Our deanonymization experiments are conducted on a transaction graph with nodes as Ethereum addresses and edges as transactions.",
"From the libraryhttps://github.com/benedekrozemberczki/karateclub of Rozenberczki et al.",
"[49], we selected twelve node embedding methods [31], [68], [48], [47], [44], [12], [46], [56], [5], [17], [50] (see Section REF ) to discover address pairs that might belong to the same user.",
"To the best of our knowledge, we are the first to apply node embedding for Ethereum user profiling.",
"To apply the selected library [49], certain preprocessing steps are required.",
"First, we considered transactions as undirected edges and removed loops and multi-edges.",
"We excluded nodes outside the largest connected component.",
"Due to running time considerations, we also removed nodes with degree one.",
"The resulting graph has 16,704 nodes and 132,231 edges.",
"We generated 128-dimensional representations for the addresses.",
"In order to compare with timestamp and gas price representations, we assign the overall average of the network embedding vectors to the removed nodes."
],
[
"Evaluation",
"Based on timestamp, gas price distributions or network embedding, we generate Euclidean feature vectors for 3321 Ethereum addresses with each having at least five transactions sent, see Table REF .",
"Given a target address, we order the remaining addresses by their Euclidean distance from the target.",
"In the evaluation, we use 129 address pairs that belong to the same ENS name.",
"The accuracy metrics of Section for identifying accounts of the same user by using only time-of-day activity or normalized gas price is given in Figures REF –REF .",
"While time-of-day representation works best with $b_{\\text{hour}}=4$ to 6 (six to four hour long bins), normalized gas price representation performs weaker and the related histogram gives only very small improvement with $b_{\\text{gas}}=50$ over mean, median and standard deviation.",
"The performance of the twelve different node embedding algorithms is shown in Figures REF –REF based on ten independent experiments.",
"The two best performing methods are Diff2Vec [50] and Role2Vec [2].",
"Note that these algorithms capture different aspects of the same graph as Diff2Vec is a neighbourhood preserving and Role2Vec is a structural node embedding.",
"We achieved best Ethereum address linking performance by combining these two methods by the harmonic average of their rank.",
"In Figure REF , we show the fraction of pairs where the rank of the ground truth pair is not more than a given value.",
"Surprisingly, Diff2Vec and Role2Vec find the corresponding ENS address pairs within 100 closest representations by almost $20\\%$ more likely than time-of-day activity and gas price statistics.",
"Our combination based approach further improves the performance.",
"Our results show that the proposed profiling techniques link Ethereum addresses of the same user significantly better than random guessing.",
"More precisely, the combination of Diff2Vec and Role2Vec yield $1.6$ bits of additional information on account owners, see Figure REF .",
"In other words, we can reduce the anonymity set of a particular address by a factor of $2^{1.6}\\approx 3.0314$ .",
"Figure: Average rank at different granularity for daily activity (top) and normalized gas price (bottom).",
"Dashed lines show performance with only mean, median and standard deviation used.",
"Note that the maximum rank is 3321, the total number of Ethereum addresses considered in this experiment.Figure: AUC at different granularity for daily activity (top) and normalized gas price (bottom).",
"Dashed lines show performance with only mean, median and standard deviation used.Figure: Entropy gain at different granularity for daily activity (top) and normalized gas price (bottom).",
"Dashed lines show performance with only mean, median and standard deviation used.Figure: Average rank for node embedding methods.",
"Vertical lines show standard deviation in 10 independent experiments.",
"Reciprocal rank combination of Diff2Vec and Role2Vec gives the best performance.",
"Note that the maximum rank is 3321, the total number of Ethereum addresses considered in this experiment.Figure: AUC for node embedding methods.",
"Vertical lines show standard deviation in 10 independent experiments.",
"Reciprocal rank combination of Diff2Vec and Role2Vec gives the best performance.Figure: Entropy gain for node embedding methods.",
"Vertical lines show standard deviation in 10 independent experiments.",
"Reciprocal rank combination of Diff2Vec and Role2Vec gives the best performance.Figure: Fraction of ENS address pairs correctly identified within a given maximum rank, for different embedding methods."
],
[
"Deanonymizing trustless mixing services on Ethereum",
"As the Ethereum community realises the consequences of the lack of privacy on Ethereum, more and more emphasis is put on increasing transaction privacy [34], [53], [55].",
"Hence, privacy-enhancing tools became crucially important gadgets in the Ethereum ecosystem.",
"Without doubt, the most popular is Tornado Cash (TC), a non-custodial zkSNARK-based mixer.",
"It allows its users to enhance their anonymity by hiding their identity among a set of participating users.",
"In this section, we provide techniques and heuristics to decrease the anonymity achieved in a TC mixer.",
"The Tornado Cash (TC) Mixers are sets of trustless Ethereum smart contracts allowing Ethereum users to enhance their anonymity.",
"A TC mixer contract holds equal amounts of funds (ether or other ERC-20 tokens) from a set of depositors.",
"One mixer contract typically holds one type of asset.",
"In case of the TC mixer, anonymity is achieved by applying zkSNARKs [23].",
"Each depositor inserts a hash value in a Merkle-tree.",
"Later, at withdraw time, each legitimate withdrawer can prove unlinkably with a zero-knowledge proof that they know the pre-image of a previously inserted hash leaf in the Merkle-tree.",
"Subsequently, users can withdraw their asset from the mixer whenever they consider that the size of the anonymity set is satisfactory.",
"Cryptocurrency mixers typically provide $k$ -anonymity (also known as plausible deniability) to their users [52].",
"Generally speaking, a $k$ -anonymized dataset has the property that each record is indistinguishable from at least $k-1$ others.",
"Specifically, if a mixer contract holds $n$ deposits out of which $n-k$ had already been withdrawn, then the next withdrawer will be indistinguishable among at least those $k$ users who have not withdrawn from the mixer yet.",
"Hence each withdrawer can enhance their transaction privacy and make their identity indistinguishable among at least $k$ addresses.",
"We call the set containing the $k$ indistinguishable addresses the anonymity set of the user.",
"Figure: The number of total deposits in each TC mixer over time.",
"This is an upper bound for the achievable anonymity set size when a withdraw transaction is executed.",
"The popularity of the 0.10.1ETH mixer is superior compared to higher value mixers.In Figure REF , we show the changes in the anonymity set size over time for four TC mixer contracts ($0.1$ ETH, 1 ETH, 10 ETH, 100 ETH) respectively.",
"Since TC was launched in December 2019, hundreds of deposits were placed in the mixers as more and more user interacted with this service.",
"In general, we observe orders of magnitude lower activity for the 100ETH mixer, thus it does not provide as much anonymity as mixers with lower values ($0.1$ ETH, 1ETH, 10ETH)."
],
[
"Heuristics for linking mixer deposits and withdraws",
"Unfortunately, careless usage easily reveals links between deposits and withdraws and also impact the anonymity of other users, since if a deposit can be linked to a withdraw, it will no longer belong to the anonymity set.",
"Next, we list three usage patterns that can be used to link deposits and withdraws.",
"The simplest careless usage is applying the same address for deposit and withdraw transactions as well: Heuristic 1.",
"If there is an address from where a deposit and also a withdraw has been made, then we consider these deposits and withdraws linked.",
"The next heuristic is based on salient gas price settings.",
"Most wallet softwares, e.g.",
"Metamask or My Ether Wallet, automatically sets gas prices as multiples of Gwei ($10^{9}$ wei, i.e.",
"giga wei).",
"However, one can observe gas prices whose last 9 digits are non-zero, hence those gas prices are likely set by the transaction issuer manually.",
"These custom-set gas prices can be used to link deposits and withdraw transactions.",
"For instance, one might observe the deposit transactionDepositor:$\\mathit {0x074a3e9451fe3fb47be47786cf2dc4e84e797a6f}$ at block height $9,418,956$ with $5.130909091$ Gwei gas price.",
"Later on, there is a withdraw transactionWithdrawer:$\\mathit {0x0f2437ff38e032596f2226873038230dcb22c485}$ at block height $9,419,096$ in the Ethereum blockchain with exactly the same custom-set gas price.",
"This deposit and withdraw pair can be linked.",
"Heuristic 2.",
"If there is a deposit-withdraw pair with unique and manually set gas prices, then we consider them as linked.",
"Frequently, users reveal links between their deposit and withdraw addresses if they sent transactions from one of their addresses to another address owned by them.",
"We conjecture that users falsely expect that withdraw addresses are clean, therefore they can send transactions from any address to their clean withdraw addresses.",
"However, if the withdraw address can be linked to one of their deposit addresses, then they effectively lose all privacy guarantee accomplished by the fresh withdraw address.",
"Express differently, if users run out of clean funds at their fresh addresses, they might feel tempted to move \"dirty\" assets to their \"clean\" addresses.",
"Again, such a transaction links \"clean\" and \"dirty\" addresses which is captured by the following heuristic.",
"Heuristic 3.",
"Let $d$ be a deposit and $w$ a withdraw address in a TC mixer.",
"If there is a transaction between $d$ and $w$ (or vice versa), we consider the addresses linked.",
"One could easily generalize Heuristic 3 by requiring transactions to be sent from not only a depositor address $d$ , but rather from any address in the cluster of addresses containing $d$ .",
"However, we leave the implementation of this generalization for future work.",
"Table: Number of all withdraws and deanonymized withdraws using the corresponding heuristics in each mixer contract.Applying Heuristics 1–3, we found 218, 110, 60, and 7 withdraws linked in the four mixer contracts ($0.1$ ETH, 1 ETH, 10 ETH, 100 ETH) respectively up to 2020 April 4th, see Table REF .",
"We note that withdraws identified by Heuristic 2 can also overlap with other withdraws identified by Heuristic 1 or 3.",
"Hence the number of total linked withdraws are less than the sum of all withdraws individually identified by each heuristic."
],
[
"Elapsed time between deposits and withdraws, withdraw address reuse",
"In Figure REF , we observe that most users of the linked deposit-withdraw pairs leave their deposit for less than a day in the mixer contract.",
"This user behavior can be exploited for deanonymization by assuming that the vast majority of the deposits are always withdrawn after one or two days.",
"Figure: Withdrawal address reuse in the 0.10.1 ETH mixer contract.",
"Many users withdraw multiple deposits to the same address, which eases deanonymization and reduces the privacy properties of the mixer.Even worse, in Figure REF , we observe several addresses receiving multiple withdrawals from the $0.1$ ETH mixer contract.",
"For instance, there are 83 addresses that have withdrawn 2 times and 27 addresses with 3 withdrawals each.",
"This phenomenon causes privacy risk not just for the owner of these addresses but also reduces the privacy properties of the mixer.",
"Note that proper usage always requires a withdraw to a fresh address.",
"Figure: Average rank of the deposit address in the candidate list of our algorithms for the three different ground truth sets described in Section .Figure: Number of withdraw addresses in the 0.10.1ETH mixer contract such that the corresponding deposit is identified within the given rank in the candidate list of each deanonymization technique, separate for the three ground truth sets described in Section .Figure: Entropy gain of our best deanonymization methods for the three different ground truth sets described in Section ."
],
[
"Deanonymization performance",
"Next we measure how well the techniques of Section identify the linked withdraw-deposit address pairs.",
"We build ground truth by using Heuristics 2–3 of Section REF .",
"We omit withdraw-deposit pairs identified by Heuristic 1 from the ground truth as in that case both withdraw and deposit addresses are the same.",
"Such a “pair” is trivially identified.",
"We define three different ground truth sets, one when the deposit is within the past day of the withdraw, another when within the past week, and the unfiltered full set, see Fig.",
"REF .",
"Experiments on the unfiltered full set is labeled past in Figures REF -REF .",
"Note that our ground truth sets are compiled by using Heuristics 2–3, and hence are correct up to our best knowledge on the data.",
"Since in Heuristic 2 we used gas prices and in Heuristic 3 an edge between the two addresses, in this section, we show gas price only as reference, and omit the edges used by Heuristic 3 for the network analysis algorithms.",
"As we will see, gas price distribution performs weakly for finding the account pairs identified by the Heuristics despite that Heuristic 2 is based on gas price, adding the edges between accounts identified by Heuristic 3 would yield overly strong deanonymization results since the same information is used for deanonymization and testing.",
"Figure REF shows that an address with withdraw within a day or week has significantly smaller anonymity set size, on average, since we only search for the corresponding deposit in a smaller set.",
"For example, for the $0.1$ ETH mixer the original average anonymity set size of 400 could be reduced to almost 12 by assuming that the deposit occurred within one day of the withdraw.",
"We note that in Figure REF and all other measurements over the filtered ground truth sets, we do not discount for the withdraw addresses that are not included in the filtered set.",
"For example, as seen in Figure REF , for 80 0.1-Ether withdraw transactions, we list candidate deposits, but for the remaining 20, we make no deanonymization attempt.",
"To normalize the results by considering these withdraws, we have to assume that the corresponding deposit is not in the 80-element candidate set but in the remaining 320, thus giving an average rank contribution of 160 for 20% of the data.",
"Hence average rank for 0.1-Ether withdraws with deposit within a week have an additional correction of 32 for average rank; by similar calculations, the correction for transactions within a day is 63.",
"Daily activity and Diff2Vec have similar performance while their concatenated feature vectors proved to be the best address representation; for the smaller ground truth sets, they identify related deposit addresses within the 20 and 5 closest representations on average.",
"Withdraw linking performance is further improved by concatenating the two models.",
"Entropy gain is shown in Figure REF and the number of withdraws linked to deposits within a given rank of the output for the best methods are in Figure REF .",
"In Figure REF , we show the withdraw linking performance over time.",
"As the number of active deposits increases, it becomes harder to link withdraws to any of the past deposits.",
"However withdraws that follow the deposit after a few days are still much easier to deanonymize.",
"Figure: Change of average rank in time, cumulated from the beginning of our data, for the 0.10.1 ETH Tornado mixer by using our best deanonymization methods.",
"Results are showed separately for the three ground truth sets described in Section ."
],
[
"Maintaining privacy",
"We believe if users were using the technology in a sound way or a privacy-focused wallet software would have helped them and abstracted away potential privacy leaks, then TC mixers could possibly achieve higher degrees of anonymity."
],
[
"Randomized mixing intervals",
"Mixing participants decrease largely their gained anonymity by withdrawing funds after short time intervals, cf.",
"Figure REF and REF .",
"These heuristics can be defeated by randomized mixing intervals.",
"Randomized mixing intervals cannot be enforced by the mixing contract itself, since withdrawals are unlinkable to the deposits.",
"Therefore, this should be accomplished by the user wallet software."
],
[
"Fresh withdraw addresses",
"Currently, many users apply the same withdraw addresses across several withdraws, see Figure REF .",
"This greatly decreases the complexity of linking deposits and withdraws.",
"Therefore users must use fresh withdraw addresses for each of their withdraws.",
"This issue could have been easily fixed on the user interface level."
],
[
"Mixer usage and user behaviors",
"Mixers mainly attempt to break the link between sets of transaction graphs associated with Ethereum accounts.",
"As such, users need to ensure that their on-chain behaviors are unlinkable between uses of the TC mixers.",
"Therefore, to ensure maximal privacy, users should use the TC mixers after every transaction.",
"However, this decreases the user experience and ability to use applications on Ethereum."
],
[
"Danaan-gift attack in Ethereum",
"The Danaan-gift attack, also known as malicious value fingerprinting, was introduced in [7].",
"In a value fingerprinting attack, an adversary sends a cryptocurrency transaction with a crafted amount to add a fingerprint to the receiver's account balance.",
"Although value fingerprinting was originally introduced in the context of Zcash, we notice that these attacks are applicable to Ethereum as well.",
"Most wallet software denominates gas prices in multiples of gwei ($10^9$ wei where $1\\mathit {ETH} =10^{18}\\mathit {wei}$ ), hence transaction fees overwhelmingly (in $98,1\\%$ ) do not change the last 9 digits of an account balance.",
"Albeit, users might set transaction fees manually, potentially changing their own fingerprint (in $1.9\\%$ ).",
"The last 9 digits of an account balance have no economic significance (1 gwei$\\approx 0.0000003\\$) $ but could be used as a fingerprint by an adversary.",
"Table: Balance fingerprinting statistics for Ethereum users.",
"In each cutoff, we only consider addresses that did not issue more transactions than the cutoff value.",
"We observe that vast majority of fingerprinting transactions were sent by addresses that send numerous transactions.",
"Fingerprinting an address with few sent transactions is obviously easier than an address with many issued transactions.",
"Fingerprint survival probabilities were calculated as in Equation .Figure: Ether account balance fingerprints.",
"Many Ethereum accounts have an integer account balance.",
"This allows an attacker to fingerprint the last 9 digits of an account balance.",
"Account balance fingerprints distribution has a 4.014.01 bit entropy and 6.446.44 bit entropy gain.Figure: Danaan-gift attack in confidential transaction layers.",
"An adversary can fingerprint (2) an unsuspecting user's account balance after she deposited assets (1) in a confidential asset pool, e.g.",
"AZTEC.",
"Adversary can track the user when she leaves (3) the confidential pool.First, we measure the fraction of ether transfer transactions that modify the account fingerprint ($43,7\\%$ ).",
"For the sake of robustness of the measurements, we chose fingerprints with the last eight digits.",
"As seen in Figure REF , account balances are mostly integer values.",
"However, the rest of the fingerprint values modulo $[group-separator={,}]{100000000}$ are moderately uniformly distributed.",
"The entropy of the account balance fingerprints is $4.01$ with a $6.44$ entropy gain.",
"These results suggest that account balances might be easily fingerprinted.",
"In the sequel, we estimate the average fingerprint survival probability.",
"Let $F$ denote the event that a fingerprint of an address remains unchanged.",
"To approximate the event probability $\\Pr (F)$ , let $p$ denote the probability that a transaction modified the fingerprint and let $x$ denote the number of transactions sent or received by the given address in our dataset.",
"By assuming that each transaction is independent from all others, the fingerprint survival probability of this address is $(1-p)^{x}$ .",
"We observe that the distribution of the number $x$ of transactions sent and received by an address follow power-law distribution $\\sim x^{-k}$ with $k=1.91$ .",
"The average survival probability of all addresses can hence be approximated by the following integral, where we group by $x$ , the number of transactions of an address: $ \\Pr (F)=\\int _{1}^{\\infty } x^{-k}(1-p)^{x} dx,$ which can be computed in a closed formula.",
"The numerical values are summarized in Table REF .",
"As the number of transactions sent follow a power-law distribution, the average value is skewed by the tail of the distribution.",
"Therefore it makes sense to calculate the average survival probability for several cutoffs of the tail, see Table REF .",
"Namely, in each cutoff we only consider addresses in our data set that sent less number of transactions than the cutoff value.",
"One can observe how fingerprint survival probability increases among users with a small number of transactions.",
"For example, an adversary could successfully fingerprint $21.83\\%$ of the addresses that send not more than 50 transactions.",
"This result is comparable to the $16.6\\%$ fingerprint survival probability observed in Zcash [7]."
],
[
"Danaan-gift attack for confidential transaction overlays",
"A future application of Danaan-gift attacks in Ethereum might be linking confidential transactions in privacy-enhancing overlays, like the AZTEC protocol [66].",
"In a confidential transaction overlay, users can convert public amounts to confidential notes.",
"Subsequently, they can send confidential notes to intended recipients by splitting and or joining their confidential notes.",
"The amount of confidential notes is hidden, yet publicly verifiable due to range proofs.",
"Users can also convert their confidential tokens back to public amounts.",
"In this scenario, an adversary can fingerprint unsuspecting users inside a confidential transaction overlay, see Figure REF .",
"When a user deposits a public amount to the confidential asset pool, an adversary could fingerprint her account balance by sending her a confidential transaction with a fingerprinting amount.",
"Subsequently, the user might issue several confidential transactions in this privacy-enhanced overlay.",
"If the victim's balance fingerprint survives during the course of issued confidential transactions, the adversary can identify the user withdrawing funds from the confidential asset pool by inspecting the fingerprint on the withdrawn amount.",
"Thus the fingerprinting adversary can assess how much the unsuspecting user paid in the confidential asset pool."
],
[
"Future directions",
"We expect that in the near future more potent and powerful deanonymization tools and techniques will emerge.",
"In this work, we solely applied on-chain data for deanonymizing Ethereum users.",
"Subsequent tools will likely use a combination of on-chain and off-chain data.",
"Therefore we deem the following directions would be extremely valuable for future work for the broader cryptocurrency research community."
],
[
"Further quasi-identifiers",
"In this work we identified several quasi-identifiers of Ethereum accounts, such as time-of-day activity, gas price profile and position in the Ethereum transaction graph.",
"However, we forecast that many more quasi-identifiers can be used for further profiling and deanonymizing Ethereum users.",
"One such potential quasi-identifier is wallet fingerprints.",
"One could establish which wallet a certain user employs by assessing how transaction gas prices are calculated.",
"Different wallet softwares use different methods to compute suggested gas prices [64]."
],
[
"Network-level privacy",
"Assessing Ethereum's privacy provisions entirely can only be established if one considers the full life-cycle of a transaction.",
"Specifically, one also needs to understand how much privacy is lost when users interact with full nodes or wallet providers.",
"As the history of Bitcoin and other cryptocurrencies showed, full nodes and wallet providers can deanonymize regular users and light clients already on the network layer [6], [8], [19], [20], [59], [37].",
"An attacker could establish many well-connected nodes in the peer-to-peer layer to log the timing information of transactions.",
"Due to the symmetry of broadcast, the adversary could infer the origin of the transaction [20], [6].",
"Yet, there are solely measurement studies on Ethereum's P2P network structure [29], [21].",
"Therefore, it would be worthwhile to conduct a study on Ethereum's P2P network, but from a privacy point of view.",
"Fortunately, several proposals had been made to enhance network-level privacy for cryptocurrencies [9], [18].",
"Additionally, in Ethereum, special nodes called relayers gain more and more popularity.",
"Relayers allow senders to issue feeless transactions, i.e.",
"users can send transactions from addresses that do not hold ether yet.",
"Such relayer nodes can also easily deanonymize their users.",
"This is especially problematic in case of non-custodial mixers, like Tornado Cash."
],
[
"Wallet and Browser Privacy",
"It has been shown how online trackers and cookies can aid the deanonymization of cryptocurrency users even when their coins were mixed through the use of a mixer [22].",
"Many users of the Ethereum blockchain make use of a tool called MetaMask, a browser extension available in most desktop browsers.",
"As such, for future research, it would be fascinating to analyze how the use of this extension affects the privacy of Ethereum users, even with the use of mixers.",
"It may be possible to use the techniques presented in [22] to deanonymize users.",
"Furthermore, as many Ethereum users also make use of mobile wallets, it may be useful to investigate how mobile phones can affect cryptocurrency users' privacy and assess the privacy guarantees of these mobile wallet providers [7]."
],
[
"Privacy of UTXO-based cryptocurrencies",
"We note that the deanonymizing power of quasi-identifiers (e.g.",
"temporal activity, wallet fingerprints etc.)",
"is also applicable to UTXO-based cryptocurrencies.",
"Even though in that case deanonymization is slightly more involved as one need to apply our techniques not to individual addresses but rather to clusters of UTXOs.",
"We do foresee that more potent agencies can and will engage in such deanonymization campaigns.",
"We believe that in practice, due to the aforementioned quasi-identifiers, also Bitcoin non-custodial mixers provide drastically less privacy and fungibility than what currently the community expects from those privacy-enhancing technologies."
],
[
"Conclusion",
"In this paper, we studied how graph representation learning, time-of-day activity and gas price based profiling can be used to link Ethereum addresses owned by the same user.",
"The Ethereum Name Service (ENS) relations in our data set provided ground truth information to quantitatively compare and analyze the performance of these quasi-identifiers.",
"Our results showed that recent node embedding methods had superior performance compared to user activity based profiling techniques.",
"Recently, several privacy-enhancing overlays have been deployed on Ethereum, such as Tornado Cash mixers.",
"By our measurements, their decreased usability and immature user behavior prevent them from reaching their highest attainable privacy guarantees.",
"Evaluation on heuristically linked mixing participants showed that profiling techniques, especially novel node embedding algorithms, can significantly reduce the anonymity set sizes of the mixing parties.",
"Finally, we investigated an active attack scenario for Ethereum confidential transactions by repurposing the Danaan-gift attack, originally introduced for Zcash.",
"The estimated success probability of the attack demonstrates that users should be concerned and warned about these attacks against transaction confidentiality.",
"We release the collected data as well as our source code to facilitate further researchhttps://github.com/ferencberes/ethereum-privacy."
],
[
"Acknowledgements",
"We thank Daniel A. Nagy, David Hai Gootvilig, Domokos M. Kelen and Kobi Gurkan for conversations and useful suggestions.",
"Support from Project 2018-1.2.1-NKP-00008: Exploring the Mathematical Foundations of Artificial Intelligence and the “Big Data—–Momentum” grant of the Hungarian Academy of Sciences."
],
[
"Ethereum basics",
"Ethereum is a cryptocurrency built on top of a blockchain [67].",
"There are two types of accounts in Ethereum: externally owned accounts (EOAs) and contract accounts, also known as smart contracts.",
"The global state of the system consists of the state of all different accounts.",
"EOAs are controlled by an asymmetric cryptographic key pair, while smart contracts are controlled by their code stored in persistent, immutable storage.",
"EOAs can issue transactions, which might alter the global state.",
"Transactions can either create a new contract account or call existing accounts.",
"Accounts have balances in ether, the native currency of Ethereum, and are denominated in wei where $1ETH=10^{18}wei$ .",
"Calls to EOAs can transfer Ether to the callee, while contract calls execute the code associated with the smart contract.",
"The contract execution might alter the storage of the account, moreover can call to other accounts - these are called internal transactions.",
"Contract code is executed in the Ethereum Virtual Machine (EVM)."
],
[
"Gas mechanism",
"A crucial aspect of the EVM is the gas mechanism.",
"To every EVM opcode, there is a gas amount assigned, which is deemed to price the computational complexity of that opcode.",
"For instance, adding two elements on top of the stack consumes only 3 gas, but storing a non-zero stack element in the persistent storage burns [group-separator=,]20000 gas.",
"The base gas fee for every transaction is [group-separator=,]21000 gas, which is not paid for internal transactions.",
"Therefore, whenever one executes a smart contract code in the EVM, the execution consumes a certain amount of gas.",
"At each transaction, the sender needs to define the maximum number of gas, called gas limit, they allow their transaction to consume.",
"Usually, due to the dynamic nature of the state, one does not know statically how much gas would her transaction burn.",
"If a transaction does not consume all the gas assigned to it, then surplus gas is refunded to the caller, however, if a transaction runs out of gas, then all state changes are reverted and assigned gas is taken from the caller.",
"As of now, gas can only be purchased by Ethereum's native currency, ether, at a dynamically changing price, called gas price.",
"Miners are naturally incentivised to insert transactions with higher gas prices into their blocks to increase their collected transaction fees."
]
] | 2005.14051 |
[
[
"Optimal estimates for hyperbolic harmonic mappings in Hardy space"
],
[
"Abstract Assume that $p\\in(1,\\infty]$ and $u=P_{h}[\\phi]$, where $\\phi\\in L^{p}(\\mathbb{S}^{n-1},\\mathbb{R}^{n})$.",
"Then for any $x\\in \\mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\\leq \\frac{\\mathbf{C}_{q}^{\\frac{1}{q}}(x)}{(1-|x|^2)^{\\frac{n-1 }{p}}} \\|\\phi\\|_{L^{p}} \\quad\\text{and}\\quad |u(x)|\\leq \\frac{\\mathbf{C}_{q}^{\\frac{1}{q}} }{(1-|x|^2)^{\\frac{n-1 }{p}}} \\|\\phi\\|_{L^{p} } $$ for some function $\\mathbf{C}_{q}(x)$ and constant $\\mathbf{C}_{q}$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$.",
"This result generalize and extend some known result from harmonic mapping theory ([5, Theorems 1.1 and 1.2] and [1, Proposition 6.16])."
],
[
"Introduction",
"For $n\\ge 2$ , let $\\mathbb {R}^{n}$ denote the $n$ -dimensional Euclidean space.",
"We use $\\mathbb {B}^{n}$ and $\\mathbb {S}^{n-1}$ to denote the unit ball $\\lbrace x\\in \\mathbb {R}^{n}:|x|<1\\rbrace $ and the unit sphere $\\lbrace x\\in \\mathbb {R}^{n}:|x|=1\\rbrace $ , respectively.",
"In particular, let $\\overline{\\mathbb {B}}^{n}=\\mathbb {B}^{n} \\cup \\mathbb {S}^{n-1}$ , $\\mathbb {R}^{2}=\\mathbb {C}$ and $\\mathbb {B}^{2}=\\mathbb {D}$ .",
"A mapping $u=(u_1,\\cdots ,u_n)\\in C^{2}(\\mathbb {B}^{n}, \\mathbb {R}^{n})$ is said to be hyperbolic harmonic if $\\Delta _{h}u=(\\Delta _{h}u_{1}, \\cdots ,\\Delta _{h}u_{n})=0,$ that is, for each $j\\in \\lbrace 1,\\cdots , n\\rbrace $ , $u_j$ satisfies the hyperbolic Laplace equation $\\Delta _{h}u_{j} (x)=(1-|x|^2)^2\\Delta u_{j}(x)+2(n-2)(1-|x|^2)\\sum _{i=1}^{n} x_{i} \\frac{\\partial u_{j}}{\\partial x_{i}}(x)=0,$ where $\\Delta $ denotes the usual Laplacian in $\\mathbb {R}^{n}$ .",
"For convenience, in the rest of this paper, we call $\\Delta _{h}$ the hyperbolic Laplacian operator.",
"Obviously, when $n=2$ , $\\Delta _{h}u=(1-|x|^2)^2\\Delta u$ , and thus the class of hyperbolic harmonic mappings coincides with the usual class of harmonic mappings in $\\mathbb {D}$ .",
"However, when $n\\ge 3$ , it is easily seen that the only mappings annihilated by both $\\Delta _{h}$ and $\\Delta $ are the constant mappings (cf.",
"[9]).",
"In this paper, we focus our investigations on the case when $n\\ge 3$ ."
],
[
"Hardy space for hyperbolic harmonic mappings",
"A measurable mapping $f: \\mathbb {B}^{n}\\rightarrow \\mathbb { R}^{n}$ belongs to the Hardy space $\\mathcal {H}^{p}(\\mathbb {B}^{n}, \\mathbb { R}^{n})$ with $p\\in (0,\\infty ]$ , if $M_{p}(r,f)$ exists for all $r\\in (0,1)$ and $||f||_{\\mathcal {H}^p}<\\infty $ , where $||f||_{\\mathcal {H}^p}=\\sup _{0<r<1} \\big \\lbrace M_{p}(r,f)\\big \\rbrace $ and $\\;\\;M_{p}(r,f)={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\;\\left( \\int _{\\mathbb {S}^{n-1}} |f(r\\xi )|^{p}d\\sigma (\\xi )\\right)^{\\frac{1}{p}} , & \\text{ if } p\\in (0,\\infty ),\\\\\\displaystyle \\;\\sup _{\\xi \\in \\mathbb {S}^{n-1}} \\big \\lbrace |f(r\\xi )|\\big \\rbrace , \\;\\;\\;\\;& \\text{ if } p=\\infty .\\end{array}\\right.",
"}$ Here and hereafter, $d \\sigma $ always denotes the normalized surface measure on $\\mathbb {S}^{n-1}$ so that $\\sigma (\\mathbb {S}^{n-1})=1$ .",
"If $\\phi \\in L^{1}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ , we define the invariant Poisson integral or Poisson-Szegö integral of $\\phi $ in $\\mathbb {B}^{n}$ by $P_{h}[\\phi ](x)=\\int _{\\mathbb {S}} P_{h}(x,\\zeta )\\phi (\\zeta )d\\sigma (\\zeta )$ (cf.",
"[9] or [10]), where $P_{h}(x,\\zeta )=\\left(\\frac{1-|x|^2}{|x-\\zeta |^{2}}\\right)^{n-1}$ is the Poisson-Szegö kernel with respective to $\\Delta _{h}$ satisfying $\\int _{\\mathbb {S}} P_{h}(x,\\zeta ) d\\sigma (\\zeta )=1$ (cf.",
"[10]).",
"Similarly, if $\\mu $ is a finite signed Borel measure on $\\mathbb {S}^{n-1}$ , then invariant Poisson integral of $\\mu $ will be denoted by $P_{h}[\\mu ]$ , that is, $P_{h}[\\mu ](x)=\\int _{\\mathbb {S}} P_{h}(x,\\zeta ) d\\mu (\\zeta ).$ Furthermore, both $P_{h}[\\phi ]$ and $P_{h}[\\mu ]$ are hyperbolic harmonic in $\\mathbb {B}^{n}$ (cf.",
"[2], [10]).",
"It is known that if $u$ is a hyperbolic harmonic mapping and $u\\in \\mathcal {H}^p(\\mathbb {B}^n, \\mathbb {R}^{n})$ with $p\\in (1,\\infty ]$ , then $u$ has the following integral representation (cf.",
"[10]) $u(x)= P_{h}[\\phi ](x),$ where $\\phi \\in L^{p}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ is the boundary value of $u$ and $\\Vert \\phi \\Vert _{L^p}= \\Vert u\\Vert _{\\mathcal {H}^p}.$ If $\\Delta _{h}u=0$ and $u\\in \\mathcal {H}^1(\\mathbb {B}^n, \\mathbb {R})$ , then $u$ has the representation $u =P_{h}[\\mu ]$ , where $\\mu $ is a signed Borel measure in $\\mathbb {B}^n$ .",
"Further, the similar arguments as [1] show that $\\Vert u\\Vert _{\\mathcal {H}^1}=||\\mu ||$ , where $||\\mu ||$ is the total variation of $\\mu $ on $\\mathbb {S}^{n-1}$ .",
"Let $H^{p}(\\mathbb {D},\\mathbb {C})$ denote the Hardy space which consists of analytic functions from $\\mathbb {D}$ into $\\mathbb {C}$ , while the analogous space of harmonic mappings from $\\mathbb {B}^{n}$ into $\\mathbb {R}^{n}$ is denoted by $h^{p}(\\mathbb {B}^{n}, \\mathbb { R}^{n})$ .",
"In [7], Pavlović obtained a growth estimate for functions in $H^{p}(\\mathbb {D},\\mathbb {C})$ : If $f\\in H^{p}(\\mathbb {D},\\mathbb {C})$ with $p\\in (0,\\infty ]$ , then for any $x\\in \\mathbb {D}$ , $|f(z)|\\le (1-|z|^{2})^{-\\frac{1}{p}}||f||_{H^{p}}.$ For the harmonic case, by [1], we see that for any $f\\in h^{p}(\\mathbb {B}^{n}, \\mathbb { R}^{n})$ with $p\\in [1,\\infty )$ , $|f(x)|\\le \\left( \\frac{1+|x|}{(1-|x|)^{n-1}}\\right)^{ \\frac{1}{p}}||f||_{h^{p}}$ in $\\mathbb {B}^{n}$ .",
"Further, if $p=2$ , then $h^{2}(\\mathbb {B}^{n}, \\mathbb { R}^{n})$ is a Hilbert space.",
"In this case, we have the following slightly better point estimate (cf.",
"[1]) $|f(x)|\\le \\sqrt{ \\frac{1+|x|^{2}}{(1-|x|^{2})^{n-1}}} \\;||f||_{h^{2}}.$ In the recent paper [5], Kalaj and Marković studied the pointwise estimates of mappings in $h^{p}(\\mathbb {B}^{n}, \\mathbb { R}^{n})$ , where $p\\in (1,\\infty ]$ .",
"They obtained a sharp function $\\mathcal {C}_{p}(x)$ and a sharp constant $\\mathcal {C}_{p}$ for the following two inequalities $|u(x)|\\le \\frac{\\mathcal {C}_{p}(x)}{(1-|x|^{2})^{\\frac{n-1}{p}}}||u||_{h^{p}}\\quad \\text{and}\\quad |u(x)|\\le \\frac{\\mathcal {C}_{p}}{(1-|x|^{2})^{\\frac{n-1}{p}}}||u||_{h^{p}}.$ In this paper, we will establish a counterpart of [5] in the setting of hyperbolic harmonic mappings in $\\mathcal {H}^p(\\mathbb {B}^{n},\\mathbb {R}^{n})$ .",
"In order to state our main results, we need to introduce some notations.",
"For any $a\\in \\mathbb {R}$ and $k\\in \\mathbb {N}=\\lbrace 0,1,2,\\ldots \\rbrace $ , let $(a)_{k}$ denote the factorial function with $(a)_{0}=1$ and $(a)_{k}=a(a+1)\\ldots (a+k-1)$ .",
"For $x\\in \\mathbb {R}$ , we define the Gauss hypergeometric function or hypergeometric series by $\\;_{2}F_{1}(a,b;c;x)=\\sum _{k=0}^{\\infty }\\frac{(a)_{k}(b)_{k}}{k!",
"(c)_{k}}x^{k},$ where $a,b\\in \\mathbb {R}$ and $c$ is neither zero nor a negative integer (cf.",
"[8]).",
"If $c-a-b>0$ , then the series $\\;_{2}F_{1}(a,b;c;x)$ converges absolutely for all $|x|\\le 1$ (cf.",
"[8]).",
"If $c>b>0$ and $|x|<1$ , then (cf.",
"[8]) $\\;_{2}F_{1}(a,b;c;x)= \\frac{\\Gamma (c)}{\\Gamma (b)\\Gamma (c-b)}\\int _{0}^{1}\\frac{t^{b-1}(1-t)^{c-b-1}}{(1-tx)^{a}}dt,$ where $\\Gamma $ is the Gamma function.",
"It is easy to check the following formula $\\frac{d}{dx}\\;_{2}F_{1}(a,b;c;x)= \\frac{ab}{c}\\;_{2}F_{1}(a+1,b+1;c+1;x).$ If $2b$ is neither zero nor a negative integer, and if both $|x|<1$ and $|4x(1+x)^{-2}|<1$ , then we have the following quadratic transformation (cf.",
"[8]) $\\;_{2}F_{1}\\left(a,b;2b;\\frac{4x}{(1+x)^{2}}\\right)= (1+x)^{2a}\\;_{2}F_{1}\\left(a,a-b+\\frac{1}{2};b+\\frac{1}{2};x^{2}\\right).$ The following are our results.",
"Note that a different form of growth estimate for hyperbolic harmonic mappings in $\\mathcal {H}^{p}(\\mathbb {B}^{n},\\mathbb {R}^{n})$ was proved in [3].",
"Theorem 1.1 Let $p\\in (1,\\infty ]$ and $q$ be its conjugate.",
"If $u=P_{h}[\\phi ]$ and $\\phi \\in L^{p}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ , then for any $x\\in \\mathbb {B}^{n}$ , we have the following sharp inequality: $|u(x)|\\le \\frac{\\mathbf {C}_{q}^{\\frac{1}{q}}(x)}{(1-|x|^2)^{\\frac{n-1 }{p}}}\\Vert \\phi \\Vert _{L^{p}},$ where $\\mathbf {C}_{q}(x)=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};|x|^{2}\\right).$ Theorem 1.2 Let $p\\in (1,\\infty ]$ and $q$ be its conjugate.",
"If $u=P_{h}[\\phi ]$ and $\\phi \\in L^{p}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ , then for any $x\\in \\mathbb {B}^{n}$ , we have the following sharp inequality: $|u(x)|\\le \\frac{\\mathbf {C}_{q}^{\\frac{1}{q}} }{(1-|x|^2)^{\\frac{n-1 }{p}}}\\Vert \\phi \\Vert _{L^{p}},$ where $\\mathbf {C}_{q}=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};1\\right).$ Remark 1.1 If $u=P_{h}[\\phi ]$ and $\\phi \\in L^{\\infty }(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ , then by (REF ) and (REF ), it is easy to see that for any $x\\in \\mathbb {B}^{n}$ , $|u(x)|\\le \\int _{\\mathbb {S}^{n-1}}P_{h}(x,\\zeta )d\\sigma (\\zeta )\\cdot \\Vert \\phi \\Vert _{L^{\\infty }}= \\Vert \\phi \\Vert _{L^{\\infty }}.$ By letting $\\phi \\equiv C$ on $\\mathbb {S}^{n-1}$ , we see that the sharpness of (REF ) follows, where $C$ is a constant.",
"In this case, $p=\\infty $ , $q=1$ and $\\mathbf {C}_{1}(x)\\equiv \\mathbf {C}_{1}\\equiv 1$ .",
"If $u=P_{h}[\\phi ]$ and $\\phi \\in L^{1}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ , then by (REF ) and (REF ), we obtain $|u(x)|\\le \\max _{\\zeta \\in \\mathbb {S}^{n-1}}P_{h}(x,\\zeta )\\cdot \\Vert \\phi \\Vert _{L^{1}}= \\frac{(1+|x|)^{2(n-1)} }{( 1-|x|^{2} )^{n-1}} \\Vert \\phi \\Vert _{L^{1}}$ in $\\mathbb {B}^{n}$ .",
"In the following, we show the sharpness of (REF ).",
"For any $x_{0}=|x_{0}|\\eta _{0} \\in \\mathbb {B}^{n}$ and $i\\in \\mathbb {Z}^{+}$ , define $\\phi _{i}(\\zeta )=\\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{ || \\chi _{\\Omega _{i}} ||_{L^{1}}}$ on $\\mathbb {S}^{n-1}$ and $u_{i} =P_{h}[\\phi _{i}] $ in $\\mathbb {B}^{n}$ , where $\\Omega _{i}=\\lbrace \\zeta \\in \\mathbb {S}^{n-1}:|\\zeta -\\eta _{0}|\\le \\frac{1}{i}\\rbrace $ and $\\chi $ is the indicator function.",
"Then for any $i\\in \\mathbb {Z}^{+}$ and $x\\in \\mathbb {B}^{n}$ , $ ||\\phi _{i}||_{L^{1} }=1$ and $ u_{i}(x)= \\int _{\\mathbb {S}^{n-1}}P_{h}(x,\\zeta )\\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{ || \\chi _{\\Omega _{i}} ||_{L^{1}}}d\\sigma (\\zeta ).$ For fixed $x\\in \\mathbb {B}^{n}$ , by the definition of $\\chi _{\\Omega _{i}}$ , we see that $\\lim _{i\\rightarrow \\infty }\\big | P_{h}(x,\\zeta ) -P_{h}(x,\\eta _{0}) \\big |\\cdot \\chi _{\\Omega _{i}}(\\zeta )=0.$ Then for any $\\varepsilon >0$ , there exists a positive integer $m_{1}=m_{1}(\\varepsilon ,x,\\eta _{0})$ such that for any $i\\ge m_{1} $ , $\\big | P_{h}(x,\\zeta ) -P_{h}(x,\\eta _{0}) \\big |\\cdot \\chi _{\\Omega _{i}}(\\zeta )<\\varepsilon ,$ where the notation $m_{1}=m_{1}(\\varepsilon ,x, \\eta _{0})$ means that the constant $m_{1}$ depends only on the quantities $\\varepsilon $ , $x$ and $\\eta _{0}$ .",
"Since $\\int _{\\mathbb {S}^{n-1}}\\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{||\\chi _{\\Omega _{i}}||_{L^{1}}}d\\sigma (\\zeta )=1$ , then for any $i\\ge m_{1} $ , $&\\;\\;&\\left| \\int _{\\mathbb {S}^{n-1}}P_{h}(x,\\zeta )\\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{ || \\chi _{\\Omega _{i}} ||_{L^{1}}}d\\sigma (\\zeta )- P_{h}(x,\\eta _{0}) \\right|\\\\&\\le &\\int _{\\mathbb {S}^{n-1}}\\left|\\big ( P_{h}(x,\\zeta ) -P_{h}(x,\\eta _{0})\\big )\\cdot \\chi _{\\Omega _{i}}(\\zeta )\\right|\\cdot \\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{ || \\chi _{\\Omega _{i}} ||_{L^{1}}}d\\sigma (\\zeta )\\le \\varepsilon .$ This, together with (REF ), means $ \\lim _{i\\rightarrow \\infty } u_{i}(x)=\\lim _{i\\rightarrow \\infty } \\int _{\\mathbb {S}^{n-1}}P_{h}(x,\\zeta )\\frac{ \\chi _{\\Omega _{i}}(\\zeta )}{ || \\chi _{\\Omega _{i}} ||_{L^{1}}}d\\sigma (\\zeta )= P_{h}(x,\\eta _{0}).$ By replacing $x$ with $x_{0}$ in (REF ), we obtain $\\lim _{i\\rightarrow \\infty } u_{i}(x_{0})= P_{h}(x_{0},\\eta _{0})= \\frac{(1+|x_{0}|)^{2(n-1)} }{( 1-|x_{0}|^{2} )^{n-1}} \\lim _{i\\rightarrow \\infty }\\Vert \\phi _{i}\\Vert _{L^{1}},$ and so, (REF ) is sharp."
],
[
"Proofs of the main results",
"The aim of this section is to prove Theorems REF and REF when $p\\in (1,\\infty )$ .",
"The cases $p=1$ and $p=\\infty $ are already considered in Remark REF .",
"Before the proofs of them, we need some preparation which consists of three lemmas.",
"The first one reads as follows.",
"Lemma 2.1 For any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}$ , we have the following sharp inequality $|u(x)|\\le \\frac{1}{(1-|x|^2)^{\\frac{(n-1)(q-1)}{q}}}\\left( \\int _{\\mathbb {S}^{n-1}} |x-\\eta |^{2(n-1)(q-1)} d\\sigma (\\eta )\\right)^{\\frac{1}{q}}\\Vert \\phi \\Vert _{L^{p}}.$ Let $p$ be the conjugate of $q$ , $ \\phi \\in L^{p}(\\mathbb {S}^{n-1},\\mathbb {R}^{n})$ and $u=P_{h}[\\phi ]$ in $\\mathbb {B}^{n}$ , where $p\\in (1,\\infty )$ .",
"For any $x\\in \\mathbb {B}^{n}$ , by (REF ) and Hölder's inequality, we have $|u(x)|\\le \\left(\\int _{\\mathbb {S}^{n-1}}P^{ q}_{h}(x,\\zeta )d\\sigma (\\zeta )\\right)^{\\frac{1}{q}}\\Vert \\phi \\Vert _{L^{p}}.$ Next we show the sharpness of (REF ).",
"For any $x\\in \\mathbb {B}^{n}$ , define $\\phi _{*}(\\zeta )= P_{h}^{q/p}(x,\\zeta )$ on $\\mathbb {S}^{n-1}$ and $u_{*} = P_{h}[\\phi _{*}] $ in $\\mathbb {B}^{n}$ .",
"Then we have $u_{*}(x)= \\left(\\int _{\\mathbb {S}^{n-1}} P^{q}_{h}(x,\\zeta ) d\\sigma (\\zeta )\\right)^{\\frac{1}{q}}\\Vert \\phi _{*}\\Vert _{L^{p}},$ which means that (REF ) is sharp for any $x\\in \\mathbb {B}^{n}$ .",
"In the following, we will calculate the integral above.",
"For any $\\eta \\in \\mathbb {S}^{n-1}$ and $x\\in \\mathbb {B}^{n}$ , let $\\zeta =T_{x}(\\eta )$ , where $T_{x}(\\eta )=x-(1-|x|^{2})\\frac{\\eta -x }{|\\eta -x|^{2}}.$ Then $\\zeta =T_{x}(\\eta )$ is a transformation from $\\mathbb {S}^{n-1}$ onto $\\mathbb {S}^{n-1}$ (cf.",
"[4]) satisfying $|x-\\zeta |=\\frac{1-|x|^{2}}{|\\eta -x|}\\quad \\text{and}\\quad d\\sigma (\\zeta )=\\frac{(1-|x|^2)^{n-1}}{|\\eta -x|^{2n-2}}d\\sigma (\\eta )$ (cf.",
"[6]).",
"Combining (REF ) and (REF ), we get $P_{h}^{q}(x,\\zeta )d\\sigma (\\zeta )=\\frac{(1-|x|^2)^{(n-1)q}}{|x-\\zeta |^{2(n-1)q}}d\\sigma (\\zeta )= \\frac{|x-\\eta |^{2(n-1)(q-1)} }{(1-|x|^2)^{(n-1)(q-1)}} d\\sigma (\\eta ).$ Therefore, for any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}$ , $\\int _{\\mathbb {S}^{n-1}}P_{h}^{q}(x,\\zeta )d\\sigma (\\zeta )=\\frac{1}{(1-|x|^2)^{(n-1)(q-1)}}\\int _{\\mathbb {S}^{n-1}} |x-\\eta |^{2(n-1)(q-1)} d\\sigma (\\eta ).$ From this and (REF ), the lemma follows.",
"For any $q\\in (1,\\infty )$ and $x\\in \\overline{\\mathbb {B}}^{n}$ , let $C_{q}(x)= \\int _{\\mathbb {S}^{n-1}} |x-\\eta |^{2(n-1)(q-1)} d\\sigma (\\eta ).$ For $C_{q}(x)$ , we have the following result.",
"Lemma 2.2 For any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}$ , we have $C_{q}(x)=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};|x|^{2}\\right).$ Let $A$ be an unitary transformation in $\\mathbb {R}^{n}$ .",
"For any $x\\in \\mathbb {B}^{n}$ , by replacing $\\eta $ with $A\\eta $ in (REF ), we get $C_{q}(Ax)= \\int _{\\mathbb {S}^{n-1}} |Ax-A\\eta |^{2(n-1)(q-1)} d\\sigma (A\\eta )= \\int _{\\mathbb {S}^{n-1}} |x-\\eta |^{2(n-1)(q-1)} d\\sigma (\\eta )=C_{q}(x).$ Now, we choose a suitable $A$ such that $Ax=|x|e_{n}$ , where $e_{n}=(0,\\ldots ,0,1)\\in \\mathbb {S}^{n-1}$ .",
"Then we have $C_{q}(x)=C_{q}(|x|e_{n}).$ Hence, to prove the lemma is sufficient to estimate the quality $C_{q}(\\rho e_{n})$ , where $\\rho =|x|\\in [0,1)$ .",
"Using spherical coordinates transformation (cf.",
"[4]), we get $& &C_{q}(\\rho e_{n})\\\\\\nonumber &= &\\frac{1}{\\omega _{n-1}}\\int _{0}^{\\pi } (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)} \\sin ^{n-2}\\theta _{1} d\\theta _{1}\\\\\\nonumber &&\\times \\int _{0}^{\\pi } \\sin ^{n-3}\\theta _{2}d\\theta _{2}\\cdots \\int _{0}^{\\pi } \\sin \\theta _{n-2}d\\theta _{n-2} \\int _{0}^{2\\pi }d\\theta _{n-1}\\\\\\nonumber &= &\\frac{\\omega _{n-2}}{\\omega _{n-1}}\\int _{0}^{\\pi } (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)} \\sin ^{n-2}\\theta _{1} d\\theta _{1}\\\\\\nonumber &=&(1+\\rho )^{2(n-1)(q-1)}\\frac{\\omega _{n-2}}{\\omega _{n-1}}\\int _{0}^{\\pi } \\left(1- \\frac{2\\rho (1+\\cos \\theta _{1})}{(1+\\rho )^{2}}\\right)^{(n-1)(q-1)} \\sin ^{n-2}\\theta _{1} d\\theta _{1},$ where $\\omega _{n-1}=\\frac{2\\pi ^{n/2}}{\\Gamma (n/2)}$ denotes the $(n-1)$ -dimensional Lebesgue measure on $\\mathbb {S}^{n-1}$ and $\\frac{\\omega _{n-2}}{\\omega _{n-1}}=\\frac{1}{ \\int _{0}^{\\pi } \\sin ^{n-2}\\theta _{1}d\\theta _{1}}=\\frac{\\Gamma (\\frac{n}{2})}{\\sqrt{\\pi }\\Gamma (\\frac{n-1}{2})}.$ For any $\\rho \\in [0,1)$ and $\\theta _{1}\\in (0,\\pi )$ , let $s=\\frac{4\\rho }{(1+\\rho )^{2}}\\in [0,1)\\quad \\text{and}\\quad t=\\frac{1+\\cos \\theta _{1}}{2}\\in (0,1).$ By elementary calculations, we have that $\\sin \\theta _{1}=2t^{\\frac{1}{2}}(1-t)^{\\frac{1}{2}}\\quad \\text{and}\\quad d\\theta _{1}=-\\frac{2}{\\sin \\theta _{1}} dt.$ Then (REF ), (REF ), (REF ) and (REF ) imply that $&&\\int _{0}^{\\pi } \\left(1- \\frac{2\\rho (1+\\cos \\theta _{1})}{(1+\\rho )^{2}}\\right)^{(n-1)(q-1)} \\sin ^{n-2}\\theta _{1} d\\theta _{1}\\\\\\nonumber &=&2^{n-2}\\int _{0}^{1} (1-st)^{(n-q)(q-1)}t^{\\frac{n-3}{2}}(1-t)^{\\frac{n-3}{2}}dt\\\\\\nonumber &=&2^{n-2}\\frac{\\big (\\Gamma (\\frac{n-1}{2}) \\big )^{2}}{\\Gamma (n-1)}\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n-1}{2};n-1;\\frac{4\\rho }{(1+\\rho )^2} \\right)\\\\\\nonumber &=&2^{n-2}\\frac{\\big (\\Gamma (\\frac{n-1}{2}) \\big )^{2}}{\\Gamma (n-1)}(1+\\rho )^{-2(n-1)(q-1)}\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};\\rho ^{2}\\right).$ Since $\\sqrt{\\pi }\\Gamma (n-1)=2^{n-2}\\Gamma (\\frac{n-1}{2})\\Gamma (\\frac{n}{2})$ (cf.",
"[8]), then by (REF ), (REF ), (REF ) and (REF ), we get $C_{q}(x)=C_{q}(|x| e_{n})=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};|x|^{2}\\right),$ as required.",
"Lemma 2.3 For any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}$ , we have $\\sup _{x\\in \\mathbb {B}^{n}}C_{q}(x)=C_{q}(e_{n})=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};1\\right).$ For any $q\\in (1,\\infty )$ , since $\\frac{n}{2}+(n-1)(q-1)-(\\frac{n}{2}+q-nq)>0$ , then the series $\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};\\rho ^{2}\\right)$ is both absolutely convergent and continuous with respect to $\\rho $ in $[0,1]$ (cf.",
"[8]).",
"In the following, we divide the proof into two cases according to the value of $q$ .",
"Case 2.1 $q\\in (1,1+\\frac{1}{n-1})$ .",
"In this case, we let $\\varphi (\\rho )=\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};\\rho \\right) $ in $[0.1]$ .",
"By Lemma REF , we see that it suffices to prove $\\max _{\\rho \\in [0,1]}\\varphi (\\rho )=\\varphi (1)$ .",
"Using the formula (REF ), we get $\\begin{split}&\\varphi ^{\\prime }(\\rho )\\\\=& -\\frac{2(n-1)(q-1)}{n}\\left(\\frac{n}{2}+q-nq\\right)\\;_{2}F_{1}\\left(1-(n-1)(q-1),\\frac{n}{2}+q-nq+1;\\frac{n}{2}+1;\\rho \\right)\\\\=& -\\frac{2(n-1)(q-1)}{n}\\left(\\frac{n}{2}+q-nq\\right)\\;_{2}F_{1}\\left(\\frac{n}{2}+q-nq+1 ,1-(n-1)(q-1);\\frac{n}{2}+1;\\rho \\right).\\end{split}$ By the assumption $q\\in (1,1+\\frac{1}{n-1})$ , we know that $\\frac{n}{2}+1>1-(n-1)(q-1)>0.$ Then it follows from (REF ) that $&&\\;_{2}F_{1}\\left(\\frac{n}{2}+q-nq+1 ,1-(n-1)(q-1);\\frac{n}{2}+1;\\rho \\right)\\\\&=&\\frac{\\Gamma (\\frac{n}{2}+1)}{\\Gamma \\big (1-(n-1)(q-1)\\big )\\Gamma \\big (\\frac{n}{2}+(n-1)(q-1)\\big )}\\int _{0}^{1}\\frac{t^{-(n-1)(q-1)}(1-t)^{\\frac{n}{2}+(n-1)(q-1)-1}}{(1-t\\rho )^{\\frac{n}{2}+q-nq+1}}dt\\\\&>&0.$ Further, because $\\frac{n}{2}+q-nq<0$ , we obtain from these equalities that $\\varphi ^{\\prime }(\\rho )\\ge 0$ in $(0,1)$ .",
"Hence, $\\max _{\\rho \\in [0,1]}\\varphi (\\rho )=\\varphi (1).$ This, together with (REF ) and Lemma REF , shows that $\\sup _{x\\in \\mathbb {B}^{n}}C_{q}(x)=C_{q}(e_{n})$ for any $q\\in (1,1+\\frac{1}{n-1})$ , which is what we need.",
"Case 2.2 $q\\in [1+\\frac{1}{n-1},\\infty )$ .",
"For any $\\rho \\in (0,1)$ , it follows from (REF ) that $& &\\frac{\\omega _{n-1}}{\\omega _{n-2}}\\cdot \\frac{d}{d\\rho }C_{q}(\\rho e_{n})\\\\\\nonumber &= &2(n-1)(q-1)\\int _{0}^{\\pi } \\sin ^{n-2}\\theta _{1} (\\rho -\\cos \\theta _{1}) (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)-1} d\\theta _{1}\\\\&= &2(n-1)(q-1)\\rho \\int _{0}^{\\pi } \\sin ^{n-2}\\theta _{1} (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)-1} d\\theta _{1}\\\\&&-2(n-1)(q-1)\\int _{0}^{\\pi } \\cos \\theta _{1} \\sin ^{n-2}\\theta _{1} (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)-1} d\\theta _{1}.$ For the last integral above, basic calculations yield that $\\begin{split}& \\int _{0}^{\\pi } \\cos \\theta _{1} \\sin ^{n-2}\\theta _{1} (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{(n-1)(q-1)-1} d\\theta _{1}\\\\\\nonumber =& -\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}\\sin \\theta _{1} \\cos ^{n-2}\\theta _{1} (1+\\rho ^{2}+ 2\\rho \\sin \\theta _{1})^{(n-1)(q-1)-1} d\\theta _{1}\\\\=& \\int _{0}^{\\frac{\\pi }{2}}\\sin \\theta _{1} \\cos ^{n-2}\\theta _{1}\\big ( (1+\\rho ^{2}- 2\\rho \\sin \\theta _{1})^{(n-1)(q-1)-1}- (1+\\rho ^{2}+ 2\\rho \\sin \\theta _{1})^{(n-1)(q-1)-1}\\big ) d\\theta _{1}\\\\\\le &\\;0.\\end{split}$ Then we obtain $\\frac{d}{d\\rho }C_{q}(\\rho e_{n})\\ge 0$ , which means that $\\max _{\\rho \\in [0,1]}C_{q}(\\rho e_{n})=C_{q}(e_{n})$ .",
"This, together with (REF ), shows that $\\sup _{x\\in \\mathbb {B}^{n}}C_{q}(x)=\\max _{\\rho \\in [0,1]}C_{q}(\\rho e_{n})=C_{q}(e_{n})$ for any $q\\in [1+\\frac{1}{n-1},\\infty )$ .",
"The proof of the lemma is complete."
],
[
"Proofs of Theorems ",
"For any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}$ , by Lemmas REF $\\sim $REF , we see that $|u(x)|\\le \\frac{ C^{\\frac{1}{q}}_{q}(x)}{(1-|x|^2)^{\\frac{(n-1)(q-1)}{q}}}\\Vert \\phi \\Vert _{L^{p}},$ where $C_{q}(x)&=&\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};|x|^{2}\\right)\\\\&\\le &\\;_{2}F_{1}\\left(-(n-1)(q-1),\\frac{n}{2}+q-nq;\\frac{n}{2};1\\right).$ The above inequalities are sharp.",
"$\\Box $ Remark 2.1 In the case $n=3$ , we can find a very explicit sharp point estimate.",
"For any $q\\in (1,\\infty )$ and $\\rho \\in (0,1)$ , it follows from (REF ) and (REF ) that $C_{q}(\\rho e_{3})&= &\\frac{1}{2}\\int _{0}^{\\pi } (1+\\rho ^{2}- 2\\rho \\cos \\theta _{1})^{2(q-1)} \\sin \\theta _{1} d\\theta _{1}\\\\&= &\\frac{1}{2}\\int _{-1}^{1} (1+\\rho ^{2}- 2\\rho x)^{2(q-1)} dx\\\\&= &\\frac{(1+\\rho )^{4q-2}-(1-\\rho )^{4q-2}}{4(2q-1)\\rho }.$ If $\\rho =0$ , we obtain from (REF ) that $ C_{q}(0)=1$ .",
"Therefore, for any $q\\in (1,\\infty )$ and $x\\in \\mathbb {B}^{n}\\backslash \\lbrace 0\\rbrace $ , by Lemma REF and (REF )$\\sim $ (REF ), we have the following sharp inequality $|u(x)|\\le \\frac{1}{(1-|x|^2)^{\\frac{2q-2}{q}}}\\left(\\frac{(1+|x|)^{4q-2}-(1-|x|)^{4q-2}}{4(2q-1)|x|}\\right)^{\\frac{1}{q}}\\Vert \\phi \\Vert _{L^{p}}.$ For $q\\in (1,\\infty )$ and $x=0$ , we have $|u(x)|\\le \\Vert \\phi \\Vert _{L^{p}}$ .",
"Funding.",
"The first author is partially supported by NSFS of China (No.",
"11571216, 11671127 and 11801166), NSF of Hunan Province (No.",
"2018JJ3327), China Scholarship Council and the construct program of the key discipline in Hunan Province."
]
] | 2005.14046 |
[
[
"Valley pseudospin relaxation of charged excitons in monolayer MoTe$_2$"
],
[
"Abstract Zeeman effect induced by the magnetic field introduces a splitting between the two valleys at K$^+$ and K$^-$ points of the Brillouin zone in monolayer semiconducting transition metal dichalcogenides.",
"In consequence, the photoluminescence signal exhibits a field dependent degree of circular polarization.",
"We present a comprehensive study of this effect in the case of a trion in monolayer MoTe$_2$, showing that although time integrated data allows us to deduce a g-factor of the trion state, such an analysis cannot be substantiated by the timescales revealed in the time-resolved experiments."
],
[
"Introduction",
"Valley degree of freedom is one of the central points in the physics of two-dimensional crystals with honeycomb lattice, such as, for example graphene[1] and ultrathin layers of transition metal dichalcogenides [2].",
"This has been first theoretically recognized [3] and then largely studied in experiments, predominantly focused on the optical properties of monolayers of semiconductor transition metal dichalcogenides (S-TMDs).",
"Distinctly, the individual valleys at $K^+$ and $K^-$ points of the Brillouin zone in S-TMD monolayers can be accessed by using light with selected $\\sigma ^+$ or $\\sigma ^-$ circular polarization [4], [5], [6] Up to date, a wide variety of valley-related phenomena has been demonstrated experimentally, including valley Zeeman effect, optical orientation of the valley pseudospin, or valley coherence [7], [8], [9].",
"While most of the research has been so far focused on the neutral exciton (X) due its simple electronic structure, the valley degree of freedom as an integral element of all excitonic complexes, Is an interesting property to look over also in charged excitons (CXs).",
"Similarly to the case of spin degree of freedom, different relaxation mechanisms for X and CX complexes may result in significantly different valley relaxation time — a quantity of paramount importance for any future applications.",
"In this work we focus on the CX valley relaxation in monolayer MoTe$_2$ .",
"The benefit from choosing this particular is the long bright exciton PL decay time in comparison with other monolayer semiconductor TMDs [10].",
"Although the difference in the decay constants is not qualitative, a longer decay time facilitates investigation of the valley effects.",
"The main motivation behind our study is to exploit the field-induced polarization of the negative trion in order to assess the value of the hole g-factor in monolayer MoTe$_2$ .",
"The idea of such a measurement is based on the notion that the difference in trion population between the two valleys is dictated by the Boltzmann factor, which in turn depends on the value of the Zeeman splitting.",
"In our approach the valley polarization is studied by means of time-integrated photoluminescence spectroscopy in the magnetic field up to 10 T. Such an experiment is conducted for a range of excitation intensities, which allows us to address the issue of laser-induced heating of the sample.",
"This kind of time-integrated investigation is additionally confronted with time-resolved data, in order to substantiate a core assumption of thermal equilibrium between trions in the two valleys."
],
[
"Photoluminescence of monolayer MoTe$_2$",
"The experiments reported in this work were carried out on monolayer MoTe$_2$ .",
"The flakes were mechanically exfoliated from bulk crystals with 2H structure purchased from HQ Graphene and deposited on a piece of chemically cleaned and oxygen-plasma ashed Si substrate covered with a 90-nm-thick SiO$_2$ layer.",
"In order to obtain large, high quality monolayers, a two-stage, tape- and polydimethylsiloxane-based exfoliation technique was used.",
"After the exfoliation, the flakes of interest were first identified by their characteristic optical contrast and then subjected to Raman scattering and atomic force microscopy measurements to unambiguously confirm their monolayer thickness.",
"The optical measurements were performed in a photoluminescence (PL) setup with a spatial resolution of about 1 $\\mu $ m. The sample was excited non-resonantly using a femtosecond Ti:sapphire laser with repetition rate of 76 MHz.",
"The PL signal was detected either with an InGaAs CCD camera or a streak camera with an S1 cathode for time-resolved measurements.",
"The temperature of the sample was controlled using a bath cryostat with a variable-temperature insert.",
"The cryostat was equipped with a superconductive magnet producing magnetic field up to 10 T in the Faraday configuration.",
"Figure: (a) PL spectra of a MoTe 2 _2 monolayer taken at different temperatures, but for fixed power and energy of the pulsed laser excitation, equal to, respectively, 0.720.72 mW and 1.71.7 eV.",
"Solid lines represent multi-peak Gaussian fits to the measured spectra.",
"(b) Temporal decay profiles of the CX PL acquired under the same conditions as the PL spectra in (a).",
"Solid lines are Gaussian-convolved mono-exponential fits to the experimental data.",
"The instrument response profile determined using the laser light backscattered from the sample surface is depicted for reference.",
"(c) The lifetime (left axis) and PL intensity (right axis) of the CX optical transition plotted as a function of the temperature.",
"Red solid curve represents the fit of the lifetime dependence with a simple theoretical model described in the text.A representative PL spectrum of a monolayer MoTe$_2$ at $T=10$ K is presented in Fig.",
"REF (a).",
"In accordance with earlier studies [11], [12], the spectrum consists of two emission peaks, corresponding to the neutral exciton (X) and the charged exciton (CX).",
"We corroborate this assignment with time-resolved measurements shown in Fig.",
"REF (b).",
"As in all monolayer S-TMDs, the neutral exciton exhibits an ultra-fast decay on the order of single picoseconds, while the trion decays by an order of magnitude slower[13], [14].",
"The experimentally measured lifetime of the neutral exciton of $\\tau \\approx 3.3$ ps only slightly exceeds the jitter of our setup ($\\sigma = 2.2$ ps), consistently with the measurements reported in Ref.",
"[10].",
"On the other hand, the decay of the charged exciton yields a decay constant of $\\tau \\approx 19$ ps, easily resolved by the streak camera.",
"At longer times the CX decay profile deviates from the single exponential shape, possibly due to population conversion from the dark states, but in further considerations we will only focus on the short-lived component.",
"Figure: (a) Circular-polarization-resolved PL spectra of a MoTe 2 _2 monolayer measured at T=10T=10 K under different magnetic fields applied perpendicularly to the monolayer plane.",
"(b) Values of Zeeman splittings of the X and CX optical transitions determined as a function of magnetic field.",
"Solid lines represent the linear fits to the X and CX data, which correspond to spectral gg-factors of g X =-4.6g_\\mathrm {X}=-4.6 and g CX =-3.8g_\\mathrm {CX}=-3.8, respectively.The properties discussed above correspond to the case of low-temperature limit.",
"Upon increasing the temperature, the spectrum undergoes two distinctive changes, as shown in Fig.",
"REF .",
"First, both X and CX transitions are gradually red-shifted due to the temperature dependence of the MoTe$_2$ bandgap [11].",
"Secondly, the overall intensity of both emission lines diminishes.",
"The quenching occurs at a different rate for X and CX emission, which leads to temperature-induced changes in their relative intensity.",
"It is still under debate whether these changes reflect the mass action law [15] or rather the increasing role of phonon-trion scattering, acting as a non-radiative decay channel for the trions [14].",
"In order to shed more light on this issue we have performed a series of time-resolved measurements at different temperatures.",
"The results of three of such measurements are presented in Fig.",
"REF (b).",
"The data clearly shows an increase of the decay rate upon increasing the temperature.",
"As shown in Fig.",
"REF (c), with the exception of the lowest temperatures, the quenching of the PL intensity and the decrease of the CX decay time exhibit the same temperature dependence.",
"The deviation for low temperatures might be related to the biexponential character of the decay profile, which is more pronounced in that temperature range.",
"The dependence presented in Fig.",
"REF (c) can be fit by a simple model of two competing decay channels: a temperature-independent radiative recombination and a thermally activated phonon-assisted decay: $\\frac{1}{\\tau } = \\frac{1}{\\tau _0} + \\frac{1}{\\tau _\\mathrm {ph}} \\exp \\left(-\\frac{E_\\mathrm {ph}}{kT}\\right).$ The solid line in Fig.",
"REF (c) presents the result of such a fitting with $E_\\mathrm {ph} = 14$ meV.",
"We note that a similar dependence was observed in monolayer MoSe$_2$ in Ref.",
"[14] with $E^\\mathrm {(MoSe_2)}_\\mathrm {ph} = 33$ meV.",
"We tentatively associate the difference between these two values with the difference in the $\\mathrm {A_{1g}}$ phonon energies in the two materials, even though there is only partial quantitative agreement in this respect ($E^\\mathrm {(MoSe_2)}_\\mathrm {A_{1g}} = 30$ meV vs $E^\\mathrm {(MoTe_2)}_\\mathrm {A_{1g}} = 21$ meV [16]) ."
],
[
"Influence of the magnetic field on the charged exciton valley psuedospin relaxation ",
"In order to study the valley-related phenomena we employed a circularly-polarized detection scheme in the PL experiment.",
"Under such conditions we independently access excitons in the K$^+$ or K$^-$ valley by detecting, respectively, in the $\\sigma +$ or $\\sigma -$ polarization [5].",
"In the absence of magnetic field two valleys are degenerate, yielding identical PL signal (see Fig.",
"REF (a)).",
"Upon application of magnetic field in the Faraday geometry, the valley degeneracy is lifted, as evidenced by an increased difference between the PL spectra recorded in both polarizations.",
"The splitting of both the X and CX line grows linearly with magnetic field with a $g$ factor of $-4.6$ and $-3.8$ respectively.",
"These values are consistent with earlier measurements carried out in high magnetic fields [17].",
"As shown in Fig.",
"REF (a), the Zeeman splitting of the X and CX lines is associated with a difference in the overall PL intensity from the two valleys.",
"The dominance of the lower energy component clearly indicates that the effect is due to relaxation of the exciton populations from the K$^-$ to the K$^+$ valley.",
"A systematic analysis of this effect is presented in Fig.",
"REF .",
"Figure: (a-c) Circular-polarization-resolved PL spectra of a MoTe 2 _2 monolayer measured under B=10B=10 T at various (indicated) powers of pulsed-laser excitation and at different bath temperatures: 10 K (a), 30 K (b), and 50 K (c).",
"For the sake of clarity, the spectra acquired at different powers are vertically displaced.",
"Moreover, they are also normalized by indicated factors, in order for the σ + \\sigma ^+-polarized component of the CX transition to remain equally strong at each power/temperature.",
"(d) Exciation power dependence of the ratio of CX PL intensities measured in the σ - \\sigma ^- and σ + \\sigma ^+ polarizations at various temperatures.A comparison between pairs of PL spectra in each of Figs.",
"REF (a-c) evidences that the degree of the relaxation is lower for higher excitation power used in the experiment.",
"Similarly, a comparison across the panels REF (a-c) reveals that the degree of the relaxation is lower for higher temperatures.",
"The combined effect of these two factors is shown in Fig.",
"REF (d).",
"The simplest interpretation of the dependence on the excitation power is the notion that the actual temperature at the laser spot is increased by stronger excitation.",
"In this view the degree of the relaxation at a given magnetic field is governed only by the temperature, but the temperature of the sample is accurately read only for vanishing excitation power.",
"For the sake of deeper analysis of the data in Fig.",
"REF (d) we tentatively assume a perfect thermal equilibrium between trion populations in the two valleys.",
"The validity of this assumption will be challenged in the next section, but it allows us to link the experimentally measured ratio of the PL signals from both valleys to the intrinsic Zeeman splitting.",
"In particular, the Boltzmann distribution dictates that $\\frac{I_+}{I_-} = \\exp \\left( - \\frac{g_\\mathrm {CX,ini}\\, \\mu _\\mathrm {B} B}{kT_\\mathrm {eff}} \\right), $ where $I_\\pm $ denotes the population of trions in the K$^\\pm $ valley, $T_\\mathrm {eff}$ is the effective temperature at the measured spot, and $g_\\mathrm {CX,ini}$ is a g-factor of the CX state.",
"We note that $T_\\mathrm {eff}$ is the temperature of the exciton system, which in the case of optical excitation can be higher than the lattice temperature.",
"We emphasize that the relevant g-factor of the CX state in Eq.",
"REF is different than the previously discussed spectral g-factor $g_\\mathrm {CX}\\approx -3.8$ , as the latter one is modified by the g-factor of the final state for the recombination.",
"The g-factor of the CX may be rather considered as a g-factor of a minority carrier (i.e., the hole in the case of a negatively charged trion), since the two majority carriers form a singlet pair with no magnetic response.",
"The equation (REF ) can be transformed to extract the g-factor: $\\frac{1}{g_\\mathrm {CX,ini}} \\frac{T_\\mathrm {eff}}{T} = \\frac{\\mu _\\mathrm {B} B}{kT} / \\log \\frac{I_-}{I_+}.",
"$ The aim of introducing here the bath temperature $T$ is to make both sides of the equation dimensionless.",
"Figure: (a-b) Pulsed-laser excitation power dependence of the ratio of effective valley pseudospin temperature T eff T_\\mathrm {eff} and bath temperature TT divided by the initial state gg-factor determined for CX (a) and X (b) optical transitions.",
"The plotted quantity (T eff /T)/g ini (T_\\mathrm {eff}/T)/g_\\mathrm {ini} was obtained based on the ratio RR of the σ ± \\sigma ^\\pm -polarized intensities of a given transition (measured at B=10B=10 T at different temperatures TT) using the following expression μ B B/k B Tln(1/R)\\mu _BB/k_BT\\ln (1/R), where μ B \\mu _B stands for the Bohr magneton, while k B k_B represents the Boltzmann constant.",
"The light-red lines depicted in each panel correspond to the values of (T eff /T)/g ini (T_\\mathrm {eff}/T)/g_\\mathrm {ini} obtained at low excitation-power limit, which, for appropriately large bath temperatures (where T eff ≃TT_\\mathrm {eff}\\simeq T), are determined by 1/g ini 1/g_\\mathrm {ini}, thus allowing to extract g ini g_\\mathrm {ini} for both CX and X transitions, yielding -8.5±1.5-8.5\\pm 1.5 and -4.0±0.5-4.0\\pm 0.5, respectively.Figure REF (a) shows the results of application of the formula (REF ) to the data from Fig.",
"REF (d).",
"In such a presentation, each data series corresponding to different cryostat temperature tends to saturate at the same level for low excitation intensity.",
"Such a behavior is expected, as for the lowest excitation powers the ${T_\\mathrm {eff}}/{T}$ factor in Eq.",
"REF should converge to 1.",
"The remaining value should correspond to the inverse of the trion state g-factor: ${1}/{g_\\mathrm {CX,ini}}$ .",
"Based on our experimental data we estimate its value as $g_\\mathrm {CX,ini} = -8.5 \\pm 1.5$ , which is marked in Fig.",
"REF (a) with a light-red bar.",
"The precision of our estimation is limited, as there is no clear-cut criterion for the saturation.",
"Additionally, reaching the sufficiently low excitation power for the lowest bath temperature was not feasible due to progressively increasing accumulation time required in the PL experiment.",
"On the other hand, the obtained value of $-8.5$ is consistent with the expectation for the valence band g-factor based on a simple addition of carrier spin, orbit, and valley contributions to the Zeeman effect [18].",
"Moreover, by repeating the same procedure for the neutral exciton PL signal we obtain an estimation of $g_\\mathrm {X,ini} = -4.0 \\pm 0.5$ (see Fig.",
"REF (b)), which is close to the actual neutral exciton g-factor of $-4.6$ ."
],
[
"Relaxation dynamics of the charged exciton valley pseudospin",
"A crucial element of the analysis presented in the previous section was a premise of thermal equilibrium between the CX populations in the two valleys.",
"In order to verify this assumption we performed time-resolved PL measurements in two circular polarizations.",
"At first sight, the results of such an experiment (see Fig.",
"REF (b)) resemble the decay shown earlier in Fig.",
"REF (b).",
"However, a closer inspection reveals that the dynamics of the signal in the two circular polarization is not identical.",
"In order to evidence these differences, instead of analyzing each polarization separately, we calculate at each point of time the instantaneous degree of polarization.",
"Figures REF (c-e) present such data series for three different experimental conditions.",
"Figure: (a) Scheme of transitions between the two Zeeman-split CX valley-pseudospin states, which were included in the rate equation model of the CX pseudospin relaxation.",
"The Zeeman splitting Δ Z (B)\\Delta _\\mathrm {Z}(B) is calculated using trion state g-factor of 8.58.5.",
"(b) Time-resolved CX PL decay profiles measured at T=1.7T=1.7 K, P=0.72P=0.72 mW, B=10B=10 T in two different circular polarizations of detection (as indicated).",
"(c-e) Temporal profiles of CX PL circular-polarization degree measured under different magnetic fields and various bath temperatures TT and/or excitation powers PP: T=1.7T=1.7 K, P=0.72P=0.72 mW (c), T=10T=10 K, P=0.36P=0.36 mW (d), and T=10T=10 K, P=0.72P=0.72 mW (e).",
"Solid curves in panels (b-e) represent fits to the experimental data with the rate-equation model described in the text.Our analysis of the measured transients is based on a rate-equation model presented schematically in Fig.",
"REF (a).",
"The model assumes a single relaxation time $\\tau _\\mathrm {flip}$ for the valley pseudo-spin.",
"The rate of inverse transition is dictated by the Boltzmann distribution with an effective temperature $T_\\mathrm {eff}$ .",
"The resulting thermalization of the valley pseudo-spin is accompanied by radiative recombination from each valley.",
"The initial populations of each valley are considered as free parameters to abstract from more complex processes occurring during an initial ultra-fast relaxation of the hot photocreated carriers.",
"As shown with solid lines in Figs.",
"REF (c-e), we were able to fit each of the data series using a single effective temperature and relaxation time $\\tau _\\mathrm {flip}$ .",
"Their values are shown on top of each panel in Figs.",
"REF (c-e).",
"A comparison of panel (c) with (e) and panel (d) with (e) proves that both fitted parameters depend on, respectively, the bath temperature and the excitation power.",
"Particularly important is the comparison between the fitted valley relaxation time and the previously measured CX decay time.",
"Clearly, in all the cases shown in Figs.",
"REF (c-e) the $\\tau _\\mathrm {flip}$ is significantly longer than the low-temperature CX recombination time of $\\tau _\\mathrm {CX} \\approx 20$ ps.",
"It entails that the bulk of the PL is emitted before the system reaches the equilibrium between the valleys.",
"From this point of view, the lowering of the excitation power does not improve the situation, giving no warranty that in the low power limit discussed in Section the polarization of the time-integrated PL signal is indeed governed by the g-factor of the initial state.",
"Moreover, our data indicates that the disparity between the valley relaxation time and the CX recombination time is more severe at lower temperatures.",
"It leads to a counter-intuitive conclusion that elevated temperatures can provide a better estimation of the initial state g-factor, despite a worse signal-to-noise ratio due to reduced difference between the PL signals in the two polarizations.",
"As evidenced by this discussion, the timescales revealed by the time-resolved measurements cannot explain why the data in Fig.",
"REF converge at plausible values of the g-factors.",
"This paradox can be perhaps explained by the variation of the initial polarization degree.",
"As seen in Fig.",
"REF (c-e), at $t=0$ the valley polarization does not start at 0, but at some finite field-dependent value.",
"In our analysis we have treated these values as free parameters, as they originate from the interplay between little-known ultrafast cooling processes of the photo-excited carriers and the polarization of the residual carriers.",
"It is possible that upon lowering the excitation power these starting values are getting closer to the saturation values, effectively reducing the impact of slow rate of consecutive valley pseudospin relaxation.",
"Unfortunately, at this point we are unable to directly verify such a scenario due to limited sensitivity of the streak camera measurements."
],
[
"Summary",
"In our work we have analyzed the CX valley polarization using both time-integrated and time-resolved detection of the PL signal.",
"In either case we observed a strong dependence of the measured polarization on the bath temperature and the excitation power.",
"The experimental data was described using simple models accounting for both these factors, which allowed us to extract parameters such as the g-factor of a CX state or a relevant valley pseudospin relaxation rate.",
"The result of the time-resolved experiment clearly shown that the premise of the equilibrium between the CX populations in two valleys is of no merit.",
"Yet, we show that analysis of the time-integrated data based on this assumption still leads to realistic value of the CX g-factor.",
"We conclude that while such an approach inherently underestimates the Zeeman splitting, the introduced error can be mitigated by avoiding too low temperatures."
],
[
"Acknowledgments",
"The work was supported by the ATOMOPTO project carried out within the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund, and National Science Center, Poland under project no.",
"DEC-2015/17/B/ST3/01219."
]
] | 2005.14095 |
[
[
"Equilibrium Validation in Models for Pattern Formation Based on Sobolev\n Embeddings"
],
[
"Abstract In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all.",
"Therefore one usually concentrates on finding individual branches of equilibrium solutions.",
"On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable.",
"On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists.",
"In a series of recent papers, we have aimed for a third option.",
"Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions.",
"In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method.",
"Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations."
],
[
"Introduction",
"The goal of this paper is to present the theoretical underpinnings for computer-assisted branch validation using functional analytic techniques including the constructive implicit function theorem and Neumann series methods, such that pointwise estimates result in solution branch validation.",
"While the individual proof techniques presented here are not novel, we present this approach in a modular way such that it is flexible, adaptable, and as computationally feasible as possible in more than one space dimension.",
"In particular, we apply this methodology in the case of the Ohta–Kawasaki model for diblock copolymers [24].",
"Diblock copolymers are formed by the chemical reaction of two linear polymers (known as blocks) which contain different monomers.",
"Whenever the blocks are thermodynamically incompatible, the blocks are forced to separate after the reaction, but since the blocks are covalently bonded they cannot separate on a macroscopic scale.",
"The competition between these long-range and short-range forces causes microphase separation, resulting in pattern formation on a mesoscopic scale.",
"We study the Ohta-Kawasaki equation in the case of homogeneous Neumann boundary conditions on rectilinear domains $\\Omega $ in dimensions one, two, and three, which is given by $w_t &=& -\\Delta ( \\Delta w + \\lambda f(w) ) - \\lambda \\sigma (w - \\mu )\\quad \\mbox{ in } \\Omega \\;, \\\\[1.5ex]& & \\frac{ \\partial w }{\\partial \\nu } =\\frac{\\partial (\\Delta w)}{\\partial \\nu }= 0\\quad \\mbox{ on } \\partial \\Omega \\; .$ The notation $\\nu $ denotes the unit outward normal on the boundary of $\\Omega $ — corresponding to homogeneous Neumann boundary conditions.",
"The quantity $w(t,x)$ is the local average density of the two blocks.",
"The parameter $\\mu $ is the space average of $w$ , meaning it is a measure of the relative total proportion of the two polymers, which we tersely refer to as the mass of the system.",
"The equation obeys a mass conservation, implying that $\\mu $ is time-invariant.",
"A large value of parameter $\\lambda $ corresponds to a large short-range repulsion, while a large value of the parameter $\\sigma $ corresponds to large long-range elasticity forces.",
"We refer the reader to [16] for a detailed description of how $\\lambda $ and $\\sigma $ are defined.",
"The nonlinear function $f: {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ is often assumed to be $f(u) = u-u^3$ , but the results in this paper still apply as long as $f$ is a $C^2$ -function.",
"Finally, note that the second boundary condition is necessary since this is a fourth order equation.",
"In this paper, we focus on equilibrium solutions $w = w(x)$ .",
"For notational convenience, we reformulate our equation slightly.",
"For a solution $w$ of the diblock copolymer equation, we define $u = w - \\mu $ .",
"Since the space average of $w$ is $\\mu $ , the average of the shifted function $u$ is zero.",
"Therefore the equilibrium equation becomes $ -\\Delta ( \\Delta u + \\lambda f(u+\\mu ) ) - \\lambda \\sigma u &=& 0 \\quad \\mbox{ in } \\Omega \\;, \\nonumber \\\\[1.5ex]\\frac{ \\partial u }{\\partial \\nu }= \\frac{ \\partial (\\Delta u) }{\\partial \\nu }&=& 0 \\quad \\mbox{ on } \\partial \\Omega \\;, \\\\[1.5ex]\\int _\\Omega u \\;dx &=& 0 \\nonumber \\;.$ We will use this version of the equation for the rest of the paper.",
"We focus on solutions to this equation as we vary any of the three parameters: the degree of short-range repulsion $\\lambda $ , the mass $\\mu $ , and the degree of long-range elasticity $\\sigma $ .",
"Our main goal is to establish bounds that make it possible to use a functional analytic approach to rigorous validation using the point of view of the constructive implicit function theorem which we have already developed in previous work [30], [35], [36], [37].",
"Our bounds are developed mostly using theoretical techniques, but in the case of Sobolev embeddings, the bounds themselves are developed using computer-assisted means.",
"This method is designed for validated continuation of branches of solutions which depend on a parameter, in the spirit of the numerical method of pseudo-arclength continuation, such as seen in the software packages AUTO [13] and Matcont [12].",
"Successive application of this theorem allows us to validate branches of equilibrium solutions by giving precise bounds on both the branch approximation error and isolation.",
"This is much more powerful than only validating individual solutions along a branch, since it allows us to guarantee that a set of solutions lie along the same connected branch component.",
"In order to establish what is new in this paper, we give a brief discussion of previous results.",
"A number of papers have previously considered numerical computation of bifurcation diagrams for the Ohta-Kawasaki and Cahn-Hilliard equations, such as for example [5], [6], [7], [8], [11], [16], [19], [20].",
"There are also several decades of results on computer validation for dynamical systems and differential equations solutions which combine fixed point arguments and interval arithmetic; see for example [2], [10], [14], [25], [26], [27], [29], [35], [36].",
"A constructive implicit function theorem was formulated in the work of Chierchia [4].",
"Our approach follows most closely the work of Plum [23], [25], [26], [27], in which functional analytic approaches are given for establishing needed apriori bounds.",
"Such methods have also been applied by Yamamoto [41], [42].",
"In our previous work on the constructive implicit function theorem, our goal has been to give a systematic procedure for adapting these works to the context of parameter continuation.",
"There are several papers that have already considered rigorous validation of parameter-dependent solutions for the Ohta-Kawasaki model [3], [9], [18], [30], [32], [33], [34], [35], [36], [37].",
"Many of these papers also include methods of bounding the terms in a generalized Fourier series, and the estimates on the tail.",
"However, it was necessary to make quite substantial ad hoc calculations in order to establish needed bounds before it is possible to proceed with numerical validation.",
"Our goal in the current paper is to establish a set of flexible bounds on the size of the inverse of the derivative, the required truncation dimension, Lipschitz bounds on the equations with respect to all parameters, as well as constructive Sobolev embedding constant bounds for comparison to the $L^\\infty $ -norm, meaning that equilibrium verifications along branch segments can be done without having to resort to ad hoc calculations which crucially depend on the specific nonlinearity.",
"More precisely, we obtain the following: The approach of this paper derives general estimates that work in one, two, and three space dimensions, and under the natural homogeneous Neumann boundary conditions.",
"This is in contrast to [3] and [36], which only considered the case of one-dimensional domains, or to [33], [34], which considered the three-dimensional case only under periodic boundary conditions and symmetry constraints.",
"Our approach uses the natural functional analytic setting for the diblock copolymer evolution equation, which is based on the Sobolev space of twice weakly differentiable functions.",
"This is in contrast to [33], [34], which seek the equilibria in spaces of analytic functions.",
"As part of our approach, we obtain accurate upper bounds for the operator norm of the inverse of the diblock copolymer Fréchet derivative.",
"For this estimate, we use the natural Sobolev norms of the underlying problem.",
"In contrast to [17], [39], [40] our method is based on Neumann series.",
"Throughout this paper, we focus on the theoretical underpinnings which allow one to apply the constructive implicit function theorem [30].",
"Due to space constraints, we leave the practical application of these results to path-following with slanted boxes as in [30], as well as extensions to pseudo-arclength continuation, for future work.",
"Nevertheless, while this paper is focussed only on the Ohta-Kawasaki model, the general approach can be used for other parabolic partial differential equations as well.",
"The remainder of this paper is organized as follows.",
"In Section , we introduce the necessary functional analytic framework, while Section is devoted to finding bounds on the operator norm of the inverse of the linearized operator.",
"After that, Section establishes Lipschitz bounds on the diblock copolymer operator for continuation with respect to any of the three parameters $\\lambda $ , $\\sigma $ , and $\\mu $ , before in Section we give a brief numerical illustration of how this method rigorously establishes a variety of equilibrium branch pieces for the Ohta-Kawasaki model in multiple dimensions.",
"Finally, in Section we wrap up with conclusions and future plans."
],
[
"Basic definitions and setup",
"In this section, we establish notation and crucial auxiliary bounds.",
"In Section REF we recall the constructive implicit function theorem, before in Section REF we define the function spaces that will be used in our computer-assisted proofs.",
"These spaces are particularly adapted for the use with Fourier series expansions to represent functions with Neumann boundary conditions and zero average.",
"In Section REF , we collect a set of Sobolev embedding results giving precise rigorous bounds on the similarity constants for passing between equivalent norms on these function spaces.",
"Finally, in Section REF we introduce the necessary finite-dimensional spaces and associated projection operators that are used in our computer-assisted proofs."
],
[
"The constructive implicit function theorem",
"In this section we state a constructive implicit function theorem that makes it possible to validate a branch of solutions changing with respect to a parameter.",
"This theorem appears in [30], where we demonstrated the validation of solutions for the lattice Allen-Cahn equation.",
"The theorem is based on previous work of Plum [27] and Wanner [36].",
"To put this in context, our overarching goal is to find a connected curve of values $(\\alpha ,x)$ in the zero set for a specific nonlinear operator ${\\mathcal {G}}(\\alpha ,x)$ .",
"In this paper, the zero set consists of the equilibria of the Ohta-Kawasaki equation.",
"Starting at a point for which the operator ${\\mathcal {G}}$ is close to zero, we use the theorem as the iterative step in a validated continuation.",
"That is, we iteratively validate small portions along the solution curve, each time using the constructive implicit function theorem which is stated below.",
"We also validate that these portions combine to create a piece of a single connected solution curve, and show that it is isolated from any other branch of the solution curve.",
"Rather than getting bogged down in the details of the iterative process, we first concentrate on the single iterative step and the estimates needed in order to perform it.",
"Specifically, we consider solutions to the equation $ {\\mathcal {G}}(\\alpha ,x) = 0 \\; ,$ where ${\\mathcal {G}}: {\\mathcal {P}}\\times {\\mathcal {X}}\\rightarrow {\\mathcal {Y}}$ is a Fréchet differentiable nonlinear operator between two Banach spaces ${\\mathcal {X}}$ and ${\\mathcal {Y}}$ , and the parameter $\\alpha $ is taken from a Banach space ${\\mathcal {P}}$ .",
"The norms on these Banach spaces are denoted by $\\Vert \\cdot \\Vert _{\\mathcal {P}}$ , $\\Vert \\cdot \\Vert _{\\mathcal {X}}$ , and $\\Vert \\cdot \\Vert _{\\mathcal {Y}}$ , respectively.",
"One possible choice of ${\\mathcal {G}}$ would be to directly use the nonlinear operator associated with (REF ), but this is not a numerically viable option for validation of a branch of solutions.",
"Instead we will introduce an extended system which gives a validated version of pseudo-arclength continuation.",
"The system contains not only the Ohta-Kawasaki model equilibrium equation, but is in a way designed to optimize the needed number of validation steps.",
"In order to present the constructive implicit function theorem in detail, we begin by making the following hypotheses.",
"For the classical implicit function theorem, the existence of constants satisfying the hypotheses given below is sufficient.",
"In contrast, since we wish to use a computer assisted proof to validate existence of equilibria with specified error bounds, we require explicit values for each of the constants in (H1)–(H4).",
"(H1) Unlike the traditional implicit function theorem, we assume only an approximate solution to the equation.",
"That is, assume that we are given a pair $(\\alpha ^*,x^*) \\in {\\mathcal {P}}\\times {\\mathcal {X}}$ which is an approximate solution of the nonlinear problem (REF ).",
"More precisely, the residual of the nonlinear operator ${\\mathcal {G}}$ at the pair $(\\alpha ^*,x^*)$ is small, i.e., there exists a constant $\\varrho > 0$ such that $\\left\\Vert {\\mathcal {G}}(\\alpha ^*,x^*) \\right\\Vert _{\\mathcal {Y}}\\le \\varrho \\; .$ (H2) Assume that the operator $D_x{\\mathcal {G}}(\\alpha ^*,x^*)$ is invertible and not very close to being singular.",
"That is, the Fréchet derivative $D_x{\\mathcal {G}}(\\alpha ^*,x^*) \\in {\\mathcal {L}}({\\mathcal {X}},{\\mathcal {Y}})$ , where ${\\mathcal {L}}({\\mathcal {X}},{\\mathcal {Y}})$ denotes the Banach space of all bounded linear operators from ${\\mathcal {X}}$ into ${\\mathcal {Y}}$ , is one-to-one and onto, and its inverse $D_x{\\mathcal {G}}(\\alpha ^*,x^*)^{-1} : {\\mathcal {Y}}\\rightarrow {\\mathcal {X}}$ is bounded and satisfies $\\left\\Vert D_x{\\mathcal {G}}(\\alpha ^*,x^*)^{-1} \\right\\Vert _{{\\mathcal {L}}({\\mathcal {Y}},{\\mathcal {X}})} \\le K\\; ,$ where $\\Vert \\cdot \\Vert _{{\\mathcal {L}}({\\mathcal {Y}},{\\mathcal {X}})}$ denotes the operator norm in ${\\mathcal {L}}({\\mathcal {Y}},{\\mathcal {X}})$ .",
"(H3) For $(\\alpha ,x)$ close to $(\\alpha ^*,x^*)$ , the Fréchet derivative $D_x{\\mathcal {G}}(\\alpha ,x)$ is locally Lipschitz continuous in the following sense.",
"There exist positive real constants $L_1$ , $L_2$ , $\\ell _x$ , and $\\ell _\\alpha \\ge 0$ such that for all pairs $(\\alpha ,x)\\in {\\mathcal {P}}\\times {\\mathcal {X}}$ with $\\Vert x - x^* \\Vert _{\\mathcal {X}}\\le \\ell _x$ and $\\Vert \\alpha - \\alpha ^*\\Vert _{\\mathcal {P}}\\le \\ell _\\alpha $ we have $\\left\\Vert D_x{\\mathcal {G}}(\\alpha ,x) -D_x{\\mathcal {G}}(\\alpha ^*,x^*) \\right\\Vert _{{\\mathcal {L}}({\\mathcal {X}},{\\mathcal {Y}})} \\le L_1 \\left\\Vert x - x^* \\right\\Vert _{\\mathcal {X}}+L_2 \\left\\Vert \\alpha - \\alpha ^* \\right\\Vert _{\\mathcal {P}}\\; .$ To verify this condition, as well as the next one, we will give specific Lipschitz bounds on the Ohta-Kawasaki operator.",
"We will then show the precise way to combine these bounds in order to get the constants $L_k$ .",
"(H4) For $\\alpha $ close to $\\alpha ^*$ , the Fréchet derivative $D_\\alpha {\\mathcal {G}}(\\alpha ,x^*)$ satisfies a Lipschitz-type bound.",
"More precisely, there exist positive real constants $L_3$ and $L_4$ , such that for all $\\alpha \\in {\\mathcal {P}}$ with $\\Vert \\alpha - \\alpha ^*\\Vert _{\\mathcal {P}}\\le \\ell _\\alpha $ one has $\\left\\Vert D_\\alpha {\\mathcal {G}}(\\alpha ,x^*) \\right\\Vert _{{\\mathcal {L}}({\\mathcal {P}},{\\mathcal {Y}})} \\le L_3 + L_4 \\left\\Vert \\alpha - \\alpha ^* \\right\\Vert _{\\mathcal {P}}\\; ,$ where $\\ell _\\alpha $ is the constant that was chosen in (H3).",
"Keeping these hypotheses in mind, the constructive implicit function theorem can then be stated as follows.",
"Theorem 2.1 (Constructive Implicit Function Theorem) Let ${\\mathcal {P}}$ , ${\\mathcal {X}}$ , and ${\\mathcal {Y}}$ be Banach spaces, suppose that the nonlinear operator ${\\mathcal {G}}: {\\mathcal {P}}\\times {\\mathcal {X}}\\rightarrow {\\mathcal {Y}}$ is Fréchet differentiable, and assume that the pair $(\\alpha ^*,x^*) \\in {\\mathcal {P}}\\times {\\mathcal {X}}$ satisfies hypotheses (H1), (H2), (H3), and (H4).",
"Finally, suppose that $ 4 K^2 \\varrho L_1 < 1\\qquad \\mbox{ and }\\qquad 2 K \\varrho < \\ell _x \\; .$ Then there exist pairs of constants $(\\delta _\\alpha ,\\delta _x)$ with $0 \\le \\delta _\\alpha \\le \\ell _\\alpha $ and $0 < \\delta _x \\le \\ell _x$ , as well as $ 2 K L_1 \\delta _x + 2 K L_2 \\delta _\\alpha \\le 1\\qquad \\mbox{ and }\\qquad 2 K \\varrho + 2 K L_3 \\delta _\\alpha + 2 K L_4 \\delta _\\alpha ^2\\le \\delta _x \\; ,$ and for each such pair the following holds.",
"For every $\\alpha \\in {\\mathcal {P}}$ with $\\Vert \\alpha - \\alpha ^*\\Vert _{\\mathcal {P}}\\le \\delta _\\alpha $ there exists a uniquely determined element $x(\\alpha ) \\in {\\mathcal {X}}$ with $\\Vert x(\\alpha ) - x^* \\Vert _{\\mathcal {X}}\\le \\delta _x$ such that ${\\mathcal {G}}(\\alpha , x(\\alpha )) = 0$ .",
"In other words, if we define ${\\mathcal {B}}_\\delta ^{\\mathcal {X}}= \\left\\lbrace \\xi \\in {\\mathcal {X}}\\; : \\;\\left\\Vert \\xi - x^* \\right\\Vert _{\\mathcal {X}}\\le \\delta \\right\\rbrace \\quad \\mbox{ and }\\quad {\\mathcal {B}}_\\delta ^{\\mathcal {P}}= \\left\\lbrace p \\in {\\mathcal {P}}\\; : \\;\\left\\Vert p - \\alpha ^* \\right\\Vert _{\\mathcal {P}}\\le \\delta \\right\\rbrace \\; ,$ then all solutions of the nonlinear problem ${\\mathcal {G}}(\\alpha ,x)=0$ in the set ${\\mathcal {B}}_{\\delta _\\alpha }^{\\mathcal {P}}\\times {\\mathcal {B}}_{\\delta _x}^{\\mathcal {X}}$ lie on the graph of the function $\\alpha \\mapsto x(\\alpha )$ .",
"In addition, the following two statements are satisfied.",
"For all pairs $(\\alpha ,x) \\in {\\mathcal {B}}_{\\delta _\\alpha }^{\\mathcal {P}}\\times {\\mathcal {B}}_{\\delta _x}^{\\mathcal {X}}$ the Fréchet derivative $D_x{\\mathcal {G}}(\\alpha ,x)\\in {\\mathcal {L}}({\\mathcal {X}},{\\mathcal {Y}})$ is a bounded invertible linear operator, whose inverse is in ${\\mathcal {L}}({\\mathcal {Y}},{\\mathcal {X}})$ .",
"If the mapping ${\\mathcal {G}}: {\\mathcal {P}}\\times {\\mathcal {X}}\\rightarrow {\\mathcal {Y}}$ is $k$ -times continuously Fréchet differentiable, then so is the solution function $\\alpha \\mapsto x(\\alpha )$ .",
"Throughout the remainder of this paper, we concentrate on finding computationally accessible versions of hypotheses (H2), (H3), and (H4) for the Ohta-Kawasaki model."
],
[
"Function spaces",
"Throughout this paper, we let $\\Omega = (0,1)^d$ denote the unit cube in dimension $d = 1,2,3$ , and define the constants $c_0 = 1\\quad \\mbox{ and }\\quad c_\\ell = \\sqrt{2}\\quad \\mbox{ for }\\quad \\ell \\in {\\mathbb {N}}\\; .$ If $k \\in {\\mathbb {N}}_0^d$ denotes an arbitrary multi-index of the form $k = (k_1,\\dots ,k_d)$ , then let $c_k = c_{k_1} \\cdot \\ldots \\cdot c_{k_d} \\; .$ If we then define $\\varphi _k(x) = c_k \\prod _{i=1}^d \\cos ( k_i \\pi x_i )\\quad \\mbox{ for all }\\quad x = (x_1,\\ldots ,x_d) \\in \\Omega \\; ,$ then the function collection $\\lbrace \\varphi _k \\rbrace _{k \\in {\\mathbb {N}}_0^d}$ forms a complete orthonormal basis for the space $L^2(\\Omega )$ .",
"Any measurable and square-integrable function $u : \\Omega \\rightarrow {\\mathbb {R}}$ can be written in terms of its Fourier cosine series $ u(x) = \\sum _{k \\in {\\mathbb {N}}_0^d} \\alpha _k \\varphi _k(x) \\; ,$ where $\\alpha _k \\in {\\mathbb {R}}$ are the Fourier coefficients of $u$ .",
"Finally, we define $|k| = (k_1^2 + \\dots + k_d^2)^{1/2}\\quad \\mbox{ and }\\quad |k|_\\infty = \\max ( k_1, \\dots , k_d ) \\; .$ Each function $\\varphi _k(x)$ is an eigenfunction of the negative Laplacian.",
"The corresponding eigenvalue is given by $\\kappa _k$ , defined via the equation $-\\Delta \\varphi _k(x) = \\kappa _k \\varphi _k(x)\\qquad \\mbox{ with }\\qquad \\kappa _k = \\pi ^2 \\left(k_1^2 + k_2^2 + \\dots + k_d^2\\right) =\\pi ^2 |k|^2 \\; .$ A straightforward direct computation shows that each $\\varphi _k(x)$ satisfies the homogeneous Neumann boundary condition $\\partial \\varphi _k /\\partial \\nu = 0$ .",
"In addition, as a result of being an eigenfunction of $-\\Delta $ , each function $\\varphi _k(x)$ also satisfies the second boundary condition in (REF ), since the identity $\\partial (\\Delta \\varphi _k) /\\partial \\nu = -\\kappa _k \\partial \\varphi _k /\\partial \\nu = 0$ holds.",
"Therefore any finite Fourier series as above automatically satisfies both boundary conditions of the diblock copolymer equation.",
"Based on our construction, the family $\\lbrace \\varphi _k \\rbrace _{k \\in {\\mathbb {N}}_0^d}$ is a complete orthonormal basis for the space $L^2(\\Omega )$ .",
"Thus, if $u$ is given as in (REF ) one can easily see that $\\Vert u \\Vert _{L^2} = \\left( \\sum _{k \\in {\\mathbb {N}}_0^d} \\alpha _k^2 \\right)^{1/2} \\; .$ For our application to the diblock copolymer model, we need to work with suitable subspaces of the Sobolev spaces $H^k(\\Omega ) = W^{k,2}(\\Omega )$ , see for example [1].",
"These subspaces have to reflect the required homogeneous Neumann boundary conditions and they can be introduced as follows.",
"For $\\ell \\in {\\mathbb {N}}$ consider the space ${\\mathcal {H}}^\\ell =\\left\\lbrace u = \\sum _{k \\in {\\mathbb {N}}_0^d} \\alpha _k \\varphi _k : \\Vert u \\Vert _{{\\mathcal {H}}^\\ell }< \\infty \\right\\rbrace \\;,$ where $\\Vert u\\Vert _{{\\mathcal {H}}^\\ell } = \\left( \\sum _{k \\in {\\mathbb {N}}_0^d} \\left( 1 +\\kappa _k^\\ell \\right) \\alpha _k^2 \\right)^{1/2} \\; .$ One can easily verify that this is equivalent to the definition $\\Vert u\\Vert ^2_{{\\mathcal {H}}^\\ell } =\\Vert u \\Vert _{L^2}^2 + \\left\\Vert (-\\Delta )^{\\ell /2} u \\right\\Vert ^2_{L^2} \\;,$ where $\\Vert \\cdot \\Vert _{L^2}$ denotes the standard $L^2(\\Omega )$ -norm on the domain $\\Omega $ as mentioned above, and the fractional Laplacian for odd $\\ell $ is defined using the spectral definition.",
"We note that we have incorporated the boundary conditions of (REF ) into our definition of the spaces ${\\mathcal {H}}^\\ell $ .",
"For example, ${\\mathcal {H}}^1 &=& H^1(\\Omega ), \\\\{\\mathcal {H}}^2 &=& \\left\\lbrace u \\in H^2(\\Omega ): \\frac{\\partial u}{\\partial \\nu } =0 \\right\\rbrace \\, \\quad \\mbox{, and } \\\\{\\mathcal {H}}^4 &=& \\left\\lbrace u \\in H^4(\\Omega ): \\frac{\\partial u}{\\partial \\nu } = \\frac{\\partial \\Delta u}{\\partial \\nu } =0 \\right\\rbrace \\, ,$ where the boundary conditions in the second and third equations are considered in the sense of the trace operator.",
"The first identity follows as a special case from the results in [15], [22], the second identity has been established in [21], and also the third identity can be verified as in [21].",
"For the sake of simplicity we further define ${\\mathcal {H}}^0 = L^2(\\Omega )$ .",
"While the spaces ${\\mathcal {H}}^\\ell $ incorporate the boundary conditions of (REF ), recall that we have reformulated the diblock copolymer equation in such a way that solutions satisfy the integral constraint $\\int _\\Omega u \\; dx = 0$ , since the case of nonzero average has been absorbed into the placement of the parameter $\\mu $ .",
"In order to treat this additional constraint, we therefore need to restrict the spaces ${\\mathcal {H}}^\\ell $ further.",
"Consider now an arbitrary integer $\\ell \\in {\\mathbb {Z}}$ and define the space $ \\overline{{\\mathcal {H}}}^\\ell =\\left\\lbrace u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k\\; : \\; \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell } < \\infty \\right\\rbrace \\;,$ where we use the modified norm $ \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell } =\\left( \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\kappa _k^\\ell \\alpha _k^2\\right)^{1/2} \\;.$ Notice that for $\\ell = 0$ this definition reduces to the subspace of $L^2(\\Omega )$ of all functions with average zero equipped with its standard norm, since we removed the constant basis function from the Fourier series.",
"For $\\ell > 0$ one can easily see that $\\overline{{\\mathcal {H}}}^\\ell \\subset {\\mathcal {H}}^\\ell $ , and that the new norm is equivalent to our norm on ${\\mathcal {H}}^\\ell $ .",
"We still need to shed some light on the new definition (REF ) for negative integers $\\ell < 0$ .",
"In this case, the series in (REF ) is interpreted formally, i.e., the element $u \\in \\overline{{\\mathcal {H}}}^\\ell $ for $\\ell < 0$ is identified with the sequence of its Fourier coefficients.",
"Moreover, one can easily see that in this case $u$ acts as a bounded linear functional on $\\overline{{\\mathcal {H}}}^{-\\ell }$ .",
"In fact, for all $\\ell < 0$ the space $\\overline{{\\mathcal {H}}}^\\ell $ can be considered as a subspace of the negative exponent Sobolev space $H^\\ell (\\Omega ) =W^{\\ell ,2}(\\Omega )$ , see again [1].",
"Finally, for every $\\ell \\in {\\mathbb {Z}}$ the space $\\overline{{\\mathcal {H}}}^\\ell $ is a Hilbert space with inner product $(u,v)_{\\overline{{\\mathcal {H}}}^\\ell } =\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\kappa _k^\\ell \\alpha _k \\beta _k \\;,$ where $u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k\\in \\overline{{\\mathcal {H}}}^\\ell \\qquad \\mbox{ and }\\qquad v = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\beta _k \\varphi _k\\in \\overline{{\\mathcal {H}}}^\\ell \\;.$ The above spaces form the functional analytic backbone of this paper, and they allow us to reformulate the equilibrium problem for (REF ) as a zero finding problem.",
"Note first, however, that the functions $\\varphi _k$ can also be used to obtain an orthonormal basis in $\\overline{{\\mathcal {H}}}^\\ell $ .",
"In fact, we only have to drop the constant function $\\varphi _0$ and apply the following rescaling.",
"Lemma 2.2 The set $\\left\\lbrace \\kappa _k^{-\\ell /2} \\varphi _k(x) \\right\\rbrace _{k \\in {\\mathbb {N}}_0^d,\\; |k|>0}$ forms a complete orthonormal set for the Hilbert space $\\overline{{\\mathcal {H}}}^\\ell $ .",
"We close this section by briefly showing how the diblock copolymer equilibrium problem can be stated as a zero set problem in our functional analytic setting.",
"For this, consider the operator $F : {\\mathbb {R}}^3 \\times X \\rightarrow Y \\; ,\\qquad \\mbox{ with }\\qquad X = \\overline{{\\mathcal {H}}}^2\\quad \\mbox{ and }\\quad Y = \\overline{{\\mathcal {H}}}^{-2} \\; ,$ which is defined as $ F(\\lambda ,\\sigma ,\\mu , u) =-\\Delta \\left( \\Delta u + \\lambda f(u + \\mu ) \\right)- \\lambda \\sigma u \\; .$ The problem is now formulated weakly, and in particular, the second boundary condition $\\partial (\\Delta u) / \\partial \\nu = 0$ is no longer explicitly stated in this weak formulation.",
"Note, however, that the first boundary condition $ \\partial u / \\partial \\nu = 0$ has been incorporated into the space $X = \\overline{{\\mathcal {H}}}^2$ .",
"The fact that $f$ is $C^2$ is sufficient to guarantee that the function $F$ maps $X$ to $Y$ , since we only consider domains up to dimension three.",
"Then for fixed parameters, an equilibrium solution $u$ to the diblock copolymer equation (REF ) is a function which satisfies the identity $F(\\lambda ,\\sigma ,\\mu , u) = 0$ .",
"Moreover, the Fréchet derivative of the operator $F$ with respect to $u$ at this equilibrium is given by $ D_u F(\\lambda , \\sigma , \\mu , u) [v] =- \\Delta \\left( \\Delta v + \\lambda f^{\\prime }(u + \\mu )v \\right)- \\lambda \\sigma v \\; .$ In our formulation, the boundary and integral conditions which are part of (REF ) have been incorporated into the choice of the domain $X = \\overline{{\\mathcal {H}}}^2$ of the nonlinear operator $F$ ."
],
[
"Constructive Sobolev embedding and Banach algebra constants",
"For classical Sobolev embedding theorems, it is sufficient to write statements such as “the Sobolev space ${\\mathcal {H}}^2$ can be continuously embedded into $L^\\infty (\\Omega )$ ,” without worrying about the specific constants needed to do so.",
"However, for the purpose of computer-assisted proofs, such statements are insufficient.",
"Instead we need specific numerical bounds to compare the norms of a function or product of functions when considered in different spaces.",
"Parallel to the name constructive implicit function theorem, we refer to the bounds on the constants as constructive Sobolev embedding constants.",
"In addition, we will need a constructive Banach algebra estimate on the relationship between $\\Vert u v \\Vert _{{\\mathcal {H}}^2}$ and the product $\\Vert u\\Vert _{{\\mathcal {H}}^2} \\Vert v\\Vert _{{\\mathcal {H}}^2}$ .",
"In particular, we require the exact values of $C_m$ , $\\overline{C}_m$ , and $C_b$ in one, two, and three dimensions given in the following equations: $\\Vert u \\Vert _\\infty &\\le & C_m \\; \\Vert u \\Vert _{{\\mathcal {H}}^2} \\;,\\qquad \\qquad \\mbox{ for all } u \\in {\\mathcal {H}}^2 \\;,\\nonumber \\\\[1ex]\\Vert u \\Vert _\\infty &\\le & \\overline{C}_m \\; \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2} \\;,\\qquad \\qquad \\mbox{ for all } u \\in \\overline{{\\mathcal {H}}}^2 \\;,\\\\[1.5ex]\\Vert u v \\Vert _{{\\mathcal {H}}^2} &\\le & C_b \\; \\Vert u \\Vert _{{\\mathcal {H}}^2} \\Vert v \\Vert _{{\\mathcal {H}}^2}\\;, \\quad \\;\\;\\;\\mbox{ for all } u,v \\in {\\mathcal {H}}^2 \\nonumber \\;.$ The values of $C_m$ and $C_b$ in dimensions 1, 2, and 3 were established in [38] using rigorous computational techniques.",
"The values of $\\overline{C}_m$ can be obtained by adapting the approach in this paper, as outlined in the next lemma.",
"Table REF summarizes the values of all necessary constants.",
"Table: These values are rigorous upper bounds for the embeddingconstants in ().Lemma 2.3 (Sobolev embedding for the zero mass case) For all functions $u \\in \\overline{{\\mathcal {H}}}^2$ we have the estimate $ \\Vert u \\Vert _\\infty \\quad \\le \\quad \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2} \\cdot \\left( \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0}c_k^2 \\kappa _k^{-2} \\right)^{1/2} \\quad \\le \\quad \\overline{C}_m \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2} \\; ,$ where the value of the constant $\\overline{C}_m$ is given in Table REF .",
"Suppose that $u \\in \\overline{{\\mathcal {H}}}^2$ is given by $u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k$ .",
"According to the definition of the functions $\\varphi _k$ we have $\\Vert \\varphi _k \\Vert _\\infty = c_k$ , which immediately implies for all $x \\in \\Omega $ the estimate $|u(x)| & \\le & \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\left| \\alpha _k \\right|\\, \\left| \\varphi _k(x) \\right|\\; \\le \\;\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\left| \\alpha _k \\right| c_k\\; = \\;\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\left| \\alpha _k \\right|\\kappa _k \\cdot \\frac{c_k}{\\kappa _k} \\\\[2ex]& \\le & \\left( \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k^2\\kappa _k^2 \\right)^{1/2} \\cdot \\left( \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} c_k^2 \\kappa _k^{-2} \\right)^{1/2} \\; ,$ and together with (REF ) this immediately establishes the first estimate in (REF ).",
"In order to complete the proof one only has to find a rigorous upper bound on the second factor in the last line of the above estimate.",
"For this, one can first use the proof of [38] to establish the tail bound $\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k| \\ge N} c_k^2 \\kappa _k^{-2}\\; \\le \\;\\frac{2^d}{\\pi ^4} \\cdot \\gamma _d(N) \\; ,$ where $\\gamma _d(N)$ is explicitly defined in [38].",
"This in turn yields the estimate $\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} c_k^2 \\kappa _k^{-2}\\quad \\le \\quad \\sum _{k \\in {\\mathbb {N}}_0^d, \\; 0<|k| < N} c_k^2 \\kappa _k^{-2} \\; + \\;\\frac{2^d}{\\pi ^4} \\cdot \\gamma _d(N) \\; .$ Evaluating the finite sum and the tail bound using interval arithmetic and $N = 1000$ then furnishes the constant in Table REF .",
"The next lemma derives explicit bounds for the norm equivalence of the norms on the Hilbert spaces $\\overline{{\\mathcal {H}}}^2$ and on ${\\mathcal {H}}^2$ , which contain functions of zero and nonzero average, respectively.",
"Lemma 2.4 (Norm equivalence between zero and nonzero mass) For all $u \\in \\overline{{\\mathcal {H}}}^2$ we have $\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2} \\le \\Vert u \\Vert _{{\\mathcal {H}}^2} \\le \\frac{\\sqrt{1 + \\pi ^4}}{\\pi ^2} \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2} \\;.$ The first inequality is clear from the definitions of the two norms in the last section, since $\\kappa _k^2 \\le 1 + \\kappa _k^2$ .",
"For the second inequality, note that for $|k|>0$ one has the inequality $\\kappa _k = \\pi ^2 |k|^2 \\ge \\pi ^2$ , and therefore $1+ \\kappa _k^2 =\\kappa _k^2 \\left( 1 + \\frac{1}{\\kappa _k^2} \\right) \\le \\kappa _k^2 \\left(1 + \\frac{1}{\\pi ^4} \\right) =\\kappa _k^2 \\, \\frac{1 + \\pi ^4}{\\pi ^4} \\;.$ This in turn implies $\\Vert u \\Vert _{{\\mathcal {H}}^2}^2 =\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} (1 + \\kappa _k^2) \\alpha _k^2 \\le \\frac{1 + \\pi ^4}{\\pi ^4} \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0}\\kappa _k^2 \\alpha _k^2 =\\frac{1 + \\pi ^4}{\\pi ^4} \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^2}^2 \\; ,$ which completes the proof of the lemma.",
"Note that from the above lemma one could conclude $\\overline{C}_m \\le (\\sqrt{1 + \\pi ^4}/\\pi ^2) C_m$ , but the results given in Lemma REF are around an order of magnitude better.",
"Our specific norm choice on the spaces $\\overline{{\\mathcal {H}}}^\\ell $ has some convenient implications for its relation to the Laplacian operator $\\Delta $ .",
"Clearly for any function $u \\in \\overline{{\\mathcal {H}}}^\\ell $ we have both $\\Delta u \\in \\overline{{\\mathcal {H}}}^{\\ell -2}$ and $\\Delta ^{-1} u \\in \\overline{{\\mathcal {H}}}^{\\ell +2}$ .",
"Furthermore, if $u$ is of the form $u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k \\;,\\qquad \\mbox{ then }\\qquad -\\Delta u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0}\\kappa _k \\alpha _k \\varphi _k \\;,$ and we obtain the representation for $-\\Delta ^{-1}u$ if we replace $\\kappa _k$ in the last sum by $\\kappa _k^{-1}$ .",
"This immediately yields $\\Vert \\Delta u \\Vert _{\\overline{{\\mathcal {H}}}^{\\ell -2}}^2 &=&\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\kappa _k^{\\ell -2}\\kappa _k^2 \\alpha _k^2 \\;, \\\\[1.5ex]\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell }^2 &=&\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\kappa _k^\\ell \\alpha _k^2 \\;, \\\\[1.5ex]\\Vert \\Delta ^{-1} u \\Vert _{\\overline{{\\mathcal {H}}}^{\\ell +2}}^2 &=&\\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\kappa _k^{\\ell +2}\\kappa _k^{-2} \\alpha _k^2 \\; ,$ and altogether we have verified the following lemma.",
"Lemma 2.5 (The Laplacian is an isometry) For every $\\ell \\in {\\mathbb {Z}}$ the Laplacian operator $\\Delta $ is an isometry from $\\overline{{\\mathcal {H}}}^\\ell $ to $\\overline{{\\mathcal {H}}}^{\\ell -2}$ , i.e., we have $\\Vert \\Delta ^{-1} u \\Vert _{\\overline{{\\mathcal {H}}}^{\\ell +2}} =\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell } =\\Vert \\Delta u \\Vert _{\\overline{{\\mathcal {H}}}^{\\ell -2}} \\; .$ To close this section we present a final result which relates the standard norm in the Hilbert space $\\overline{{\\mathcal {H}}}^\\ell $ to the norm in $\\overline{{\\mathcal {H}}}^m$ if $\\ell \\le m$ .",
"This inequality will turn out to be useful later on.",
"Lemma 2.6 (Relating the norms in $\\overline{{\\mathcal {H}}}^\\ell $ and $\\overline{{\\mathcal {H}}}^m$ ) For all $u \\in \\overline{{\\mathcal {H}}}^m$ and all $\\ell \\le m$ we have the estimate $\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell } \\; \\le \\;\\frac{1}{\\pi ^{m-\\ell }} \\, \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^m}\\;.$ Furthermore, note that in the special case $\\ell = 0 \\le m$ we have $\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^0} = \\Vert u \\Vert _{L^2}$ .",
"Suppose that $u \\in \\overline{{\\mathcal {H}}}^m$ is given by $u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k$ .",
"Then we have $\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell }^2 \\; = \\; \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0}\\frac{\\kappa _k^m \\alpha _k^2}{\\kappa _k^{m-\\ell }} \\; \\le \\;\\frac{1}{\\pi ^{2(m-\\ell )}} \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0}\\kappa _k^m \\alpha _k^2 \\; = \\;\\frac{1}{\\pi ^{2(m-\\ell )}} \\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^m}^2 \\; ,$ since for all $|k| > 0$ one has $\\kappa _k \\ge \\pi ^2$ ."
],
[
"Projection operators",
"In order to establish computer-assisted existence proofs for equilibrium solutions of (REF ) one needs to work with suitable finite-dimensional approximations.",
"In our framework, we use truncated cosine series, and this is formalized in the current section through the introduction of suitable projection operators.",
"For this, let $N \\in {\\mathbb {N}}$ denote a positive integer, and consider $u \\in {\\mathcal {H}}^\\ell $ for $\\ell \\in {\\mathbb {N}}_0$ , or alternatively $u \\in \\overline{{\\mathcal {H}}}^\\ell $ for $\\ell \\in {\\mathbb {Z}}$ , of the form $u = \\sum _{k \\in {\\mathbb {N}}_0^d} \\alpha _k \\varphi _k$ , where in the latter case $\\alpha _0 = 0$ .",
"Then we define the projection $ P_N u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|_\\infty < N} \\alpha _k \\varphi _k \\; .$ Note that in this definition we use the $\\infty $ -norm of the multi-index $k$ , since this simplifies the implementation of our method.",
"The so-defined operator $P_N$ is a bounded linear operator on ${\\mathcal {H}}^\\ell $ with induced operator norm $\\Vert P_N \\Vert = 1$ , and one can easily see that it leaves the space $\\overline{{\\mathcal {H}}}^\\ell $ invariant if $\\ell \\in {\\mathbb {Z}}$ .",
"Furthermore, it is straightforward to show that for any $N \\in {\\mathbb {N}}$ we have $\\dim P_N {{\\mathcal {H}}}^\\ell = N^d\\qquad \\mbox{ and }\\qquad \\dim P_N \\overline{{\\mathcal {H}}}^\\ell = N^d - 1 \\;.$ For all $\\ell \\in {\\mathbb {N}}_0$ we would like to point out that $(I - P_1) {{\\mathcal {H}}}^\\ell = \\overline{{\\mathcal {H}}}^\\ell $ .",
"Since this is an especially useful operator, we introduce the abbreviation $\\overline{P} = I - P_1 \\; .$ The operator $\\overline{P}$ satisfies the following useful identity.",
"Lemma 2.7 For arbitrary $u \\in {\\mathcal {H}}^0$ and $v \\in \\overline{{\\mathcal {H}}}^0$ we have the equality $\\left( \\overline{P} u, v \\right)_{L^2} =(u,v)_{L^2} \\; .$ This result can be established via direct calculation.",
"Note that $\\left( \\overline{P} u, v \\right)_{L^2} &=&(u - \\alpha _0 \\varphi _0, v)_{L^2} =(u,v)_{L^2} - \\alpha _0 (\\varphi _0,v)_{L^2} \\\\&=& (u,v)_{L^2} - \\alpha _0 \\int _\\Omega v(x) \\; dx =(u,v)_{L^2} - 0 \\; ,$ where for the last step we used the fact that $v \\in \\overline{{\\mathcal {H}}}^0$ .",
"We close this section by deriving a norm bound for the infinite cosine series part that is discarded by the projection $P_N$ in terms of a higher-regularity norm.",
"More precisely, we have the following.",
"Lemma 2.8 (Projection tail estimates) Consider two integers $\\ell \\le m$ and let the function $u \\in \\overline{{\\mathcal {H}}}^m$ be arbitrary.",
"Then the projection tail $(I - P_N)u$ satisfies $\\Vert (I - P_N) u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell } \\; \\le \\;\\frac{1}{\\pi ^{m - \\ell } N^{m - \\ell }} \\,\\Vert (I - P_N) u \\Vert _{\\overline{{\\mathcal {H}}}^m} \\; \\le \\;\\frac{1}{\\pi ^{m - \\ell } N^{m - \\ell }} \\,\\Vert u \\Vert _{\\overline{{\\mathcal {H}}}^m} \\;.$ Suppose that $u \\in \\overline{{\\mathcal {H}}}^m$ is given by $u = \\sum _{k \\in {\\mathbb {N}}_0^d, \\; |k|>0} \\alpha _k \\varphi _k$ .",
"Then we have $\\Vert (I-P_N) u \\Vert _{\\overline{{\\mathcal {H}}}^\\ell }^2 & = &\\sum _{k \\in {\\mathbb {N}}_0^d, \\;|k|_\\infty \\ge N} \\kappa _k^\\ell \\alpha _k^2 \\; = \\;\\sum _{k \\in {\\mathbb {N}}_0^d, \\;|k|_\\infty \\ge N} \\frac{\\kappa _k^m \\alpha _k^2}{\\kappa _k^{m-\\ell }} \\\\[1.5ex]& \\le & \\sum _{k \\in {\\mathbb {N}}_0^d, \\;|k|_\\infty \\ge N}\\frac{\\kappa _k^m \\alpha _k^2}{(\\pi ^2 N^2)^{m-\\ell }} \\; = \\;\\frac{1}{(\\pi ^2 N^2)^{m-\\ell }} \\;\\Vert (I-P_N) u \\Vert _{\\overline{{\\mathcal {H}}}^m}^2 \\; ,$ since the estimate $|k|_\\infty \\ge N$ yields $|k| \\ge N$ ."
],
[
"Derivative inverse estimate",
"This section is devoted to establishing derivative inverse bound in hypothesis (H2), which is required for Theorem REF , the constructive implicit function theorem.",
"More precisely, our goal in the following is to derive a constant $K$ such that $\\left\\Vert (D_u F)^{-1} \\right\\Vert _{{\\mathcal {L}}(Y,X)} \\le K \\; ,$ i.e., we need to find a bound on the operator norm of the inverse of the Fréchet derivative of $F$ with respect to $u$ .",
"We divide the derivation of this estimate into four parts.",
"In Section REF we give an outline of our approach, introduce necessary definitions and auxiliary results, and present the main result of this section.",
"This result will be verified in the following three sections.",
"First, we discuss the finite-dimensional projection of $D_u F$ in Section REF .",
"Using this finite-dimensional operator, we then construct an approximative inverse to the Fréchet derivative in Section REF , before everything is assembled to provide the desired estimate in the final Section REF ."
],
[
"General outline and auxiliary results",
"For convenience of notation in the subsequent discussion, for fixed parameters and $u$ we abbreviate the Fréchet derivative of $F$ by $ L v = D_u F(\\lambda , \\sigma , \\mu , u) [v] \\; , \\quad L \\in {\\mathcal {L}}( X, Y ) \\; , \\quad \\mbox{ with }\\quad X = \\overline{{\\mathcal {H}}}^{2} \\; , \\quad Y = \\overline{{\\mathcal {H}}}^{-2} \\; .$ Standard results imply that $L$ is a bounded linear operator $L \\in {\\mathcal {L}}(\\overline{{\\mathcal {H}}}^{2},\\overline{{\\mathcal {H}}}^{-2})$ , which explicitly is given by $L v = -\\Delta ( \\Delta v + \\lambda \\, f^{\\prime }(u + \\mu ) v ) -\\lambda \\sigma v \\; .$ More precisely, note that since the nonlinearity $f$ is twice continuously differentiable, and in view of Sobolev's imbedding recalled in (REF ), the function $f^{\\prime }(u + \\mu )$ is continuous on $\\overline{\\Omega }$ , which makes the product $\\lambda f^{\\prime }(u + \\mu ) v$ an $L^2(\\Omega )$ -function, and therefore $- \\Delta (\\lambda f^{\\prime }(u + \\mu ) v) \\in \\overline{{\\mathcal {H}}}^{-2}$ .",
"We will also use the abbreviation $ q(x) = \\lambda f^{\\prime }(u(x) + \\mu ) \\;.$ As mentioned earlier, the constructive implicit function theorem crucially relies on being able to find a bound $K$ such that $\\Vert L^{-1}\\Vert \\le K$ .",
"Our goal is to do so by using a finite-dimensional approximation for $L$ , since that can be analyzed via rigorous computational means.",
"Our finite-dimensional approximation for $L$ is given as follows.",
"For fixed $N \\in {\\mathbb {N}}$ define the finite-dimensional spaces $X_N = P_N X\\qquad \\mbox{ and }\\qquad Y_N = P_N Y \\; ,$ where the projection operator is given in (REF ).",
"Define $L_N: X_N \\rightarrow Y_N$ by $ L_N = \\left.",
"P_N L \\right|_{X_N} \\; .$ Let $K_N$ be a bound on the inverse of the finite-dimensional operator $L_N$ , i.e., suppose that $ \\left\\Vert L_N^{-1} \\right\\Vert _{{\\mathcal {L}}(Y_N,X_N)} \\le K_N \\; ,$ where the spaces $X_N$ and $Y_N$ are equipped with the norms of $X$ and $Y$ , respectively.",
"We will discuss further details on appropriate coordinate systems and the actual computation of both $L_N$ and $K_N$ in Section REF .",
"Our main result for this section is as follows.",
"Theorem 3.1 (Derivative inverse estimate) Assume there is a constant $\\tau > 0$ and an integer $N \\in {\\mathbb {N}}$ such that $\\frac{1}{\\pi ^2 N^2} \\sqrt{ K_N^2 \\, \\Vert q \\Vert _\\infty ^2 +C_b^2 \\, \\frac{1+\\pi ^4}{\\pi ^4} \\, \\Vert q \\Vert _{{\\mathcal {H}}^2}^2}\\; \\le \\; \\tau \\; < \\; 1 \\; ,$ where $K_N$ and $q$ are defined in (REF ) and (REF ), respectively.",
"Then the derivative operator $L$ in (REF ) satisfies $\\left\\Vert L^{-1} \\right\\Vert _{{\\mathcal {L}}(X,Y)} \\le \\frac{\\max ( K_N, 1) }{1-\\tau } \\; .$ Before we begin to prove this main theorem, we state a necessary result which is based on a Neumann series argument to derive bounds on the operator norm of an inverse of an operator.",
"This is a standard functional-analytic technique, which we state here for the reader's convenience.",
"A proof can be found in [30].",
"Proposition 1 (Neumann series inverse estimate) Let ${\\mathcal {A}}\\in {\\mathcal {L}}(X,Y)$ be an arbitrary bounded linear operator between two Banach spaces, and let ${\\mathcal {B}}\\in {\\mathcal {L}}(Y,X)$ be one-to-one.",
"Assume that there exist positive constants $\\varrho _1$ and $\\varrho _2$ such that $\\Vert I - {\\mathcal {B}}{\\mathcal {A}}\\Vert _{{\\mathcal {L}}(X,X)} \\le \\varrho _1 < 1\\qquad \\mbox{ and }\\qquad \\Vert {\\mathcal {B}}\\Vert _{{\\mathcal {L}}(Y,X)} \\le \\varrho _2 \\;.$ Then ${\\mathcal {A}}$ is one-to-one and onto, and $\\Vert {\\mathcal {A}}^{-1}\\Vert _{{\\mathcal {L}}(Y,X)} \\le \\frac{\\varrho _2}{1-\\varrho _1} \\;.$ In subsequent discussions, we will refer to ${\\mathcal {B}}$ as an approximate inverse.",
"We are now ready to proceed with the proof of the main result of the section, Theorem REF .",
"For this, we fix all parameters, as well as $u \\in \\overline{{\\mathcal {H}}}^2$ .",
"Our goal is to prove that $L$ is one-to-one, onto, and has an inverse whose operator norm is bounded by the value $K = \\max (K_N,1)/(1-\\tau )$ ."
],
[
"Finite-dimensional projections of the linearization",
"In this section, we consider $L_N$ , the finite dimensional projection of the operator $L$ .",
"The linear map $L_N$ is tractable using rigorous computational methods, since calculating a finite-dimensional inverse is something that can be done using numerical linear algebra.",
"To derive $L_N$ in more detail, we recall the definitions of the following projection spaces, all of which are Hilbert spaces: $\\begin{array}{rclcrclcrcl}X & = & \\overline{{\\mathcal {H}}}^2 \\;, & \\quad &X_N & = & P_N X \\; , & \\quad &X_\\infty & = & (I- P_N) X \\; , \\\\[1ex]Y & = & \\overline{{\\mathcal {H}}}^{-2} \\;, & \\quad &Y_N & = & P_N Y \\; , & \\quad &Y_\\infty & = & (I- P_N) Y \\;.\\end{array}$ Recall that in (REF ) we defined $L_N: X_N \\rightarrow Y_N$ via $L_N = \\left.",
"P_N L \\right|_{X_N}$ .",
"In order to work with this operator in a straightforward computational manner, we need to find its matrix representation.",
"Since both $X_N$ and $Y_N$ have the basis $\\varphi _k$ for all $k \\in {\\mathbb {N}}_0^d$ with $0 < |k|_\\infty < N$ , one obtains such a matrix $B = (b_{k,\\ell }) \\in {\\mathbb {R}}^{(N^d-1) \\times (N^d-1)}$ via the definition $b_{k,\\ell } = (L \\varphi _\\ell ,\\varphi _k)_{L^2} =(L_N \\varphi _\\ell ,\\varphi _k)_{L^2} \\; ,$ where $k,\\ell \\in {\\mathbb {N}}_0^d$ satisfy $0 < |k|_\\infty < N$ and $0 < |\\ell |_\\infty < N$ .",
"The above matrix representation characterizes $L_N$ on the algebraic level in the following sense.",
"If we consider a function $v_N \\in X_N$ , introduce the representations $v_N = \\sum _{k \\in {\\mathbb {N}}_0^d, \\, 0 < |k|_\\infty < N}\\alpha _k \\varphi _k(x)\\qquad \\mbox{ and }\\qquad L_N v_N = \\sum _{k \\in {\\mathbb {N}}_0^d, \\, 0 < |k|_\\infty < N}\\beta _k \\varphi _k(x) \\; ,$ and if we collect the numbers $\\alpha _k$ and $\\beta _k$ in vectors $\\alpha $ and $\\beta $ in the straightforward way, then we have $\\beta = B \\alpha \\; .$ This natural algebraic representation has one drawback.",
"We would like to use the regular Euclidean norm on real vector spaces, as well as the induced matrix norm, to study the ${\\mathcal {L}}(X_N,Y_N)$ -norm of $L_N$ .",
"To achieve this, we recall Lemma REF which shows that the collection $\\lbrace \\kappa _k^{-1} \\varphi _k(x) \\rbrace $ with $k$ as above is an orthonormal basis in $X_N \\subset X$ , and $\\lbrace \\kappa _k \\varphi _k(x) \\rbrace $ is an orthonormal basis in $Y_N \\subset Y$ .",
"Thus, we need to use the representations $v_N = \\sum _{k \\in {\\mathbb {N}}_0^d, \\, 0 < |k|_\\infty < N}\\tilde{\\alpha }_k \\kappa _k^{-1} \\varphi _k(x)\\qquad \\mbox{ and }\\qquad L_N v_N = \\sum _{k \\in {\\mathbb {N}}_0^d, \\, 0 < |k|_\\infty < N}\\tilde{\\beta }_k \\kappa _k \\varphi _k(x)$ instead of the ones given above.",
"In order to pass back and forth between these two representations we define the diagonal matrix $D = \\left( \\begin{array}{cccc}\\kappa _1 & 0 & \\cdots & 0 \\\\0 & \\kappa _2 & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\cdots & 0 & \\kappa _{N-1}\\end{array} \\right) \\; .$ One can easily see that on the level of vectors we have $\\alpha = D^{-1} \\tilde{\\alpha }\\quad \\mbox{ and }\\quad \\beta = D \\tilde{\\beta } \\; ,\\quad \\mbox{ and therefore }\\quad \\tilde{\\beta } = D^{-1} B D^{-1} \\tilde{\\alpha } \\; .$ In view of Lemma REF one then obtains $\\left\\Vert L_N \\right\\Vert _{{\\mathcal {L}}(X_N,Y_N)} =\\Vert \\tilde{B} \\Vert _2\\qquad \\mbox{ with }\\qquad \\tilde{B} = D^{-1} B D^{-1} \\; ,$ where $\\Vert \\cdot \\Vert _2$ denotes the regular induced 2-norm of a matrix.",
"Moreover, one can verify that we also have the identity $ \\left\\Vert L_N^{-1} \\right\\Vert _{{\\mathcal {L}}(Y_N,X_N)} =\\left\\Vert \\tilde{B}^{-1} \\right\\Vert _{L^2} \\; .$ In other words, using this formula, we can use interval arithmetic to establish a rigorous upper bound on the norm of this finite-dimensional inverse.",
"So far our considerations applied to any bounded linear operator between the spaces $X$ and $Y$ .",
"Specifically for the linearization of the diblock copolymer equation we can derive an explicit formula for the matrix entries $b_{k,\\ell }$ .",
"Recall that $\\varphi _k$ as defined in (REF ) is an eigenfunction for the negative Laplacian $-\\Delta $ with eigenvalue $\\kappa _k$ .",
"Therefore, for all multi-indices $k,\\ell \\in {\\mathbb {N}}_0^d$ with $0 < |k|_\\infty < N$ and $0 < |\\ell |_\\infty < N$ one obtains $b_{k,\\ell } & = & (L \\varphi _\\ell ,\\varphi _k)_{L^2} \\; = \\;(-\\kappa _k^2 - \\lambda \\sigma ) (\\varphi _k,\\varphi _\\ell )_{L^2} -(\\Delta (\\lambda f^{\\prime }(u+\\mu ) \\varphi _\\ell ),\\varphi _k)_{L^2}\\nonumber \\\\[1ex]& = & (-\\kappa _k^2 - \\lambda \\sigma ) \\delta _{k,\\ell } -(\\Delta (q \\varphi _\\ell ),\\varphi _k)_{L^2} \\nonumber \\\\[1ex]& = & (-\\kappa _k^2 - \\lambda \\sigma ) \\delta _{k,\\ell } -(q \\varphi _\\ell ,\\Delta \\varphi _k)_{L^2} \\nonumber \\\\[1ex]& = & -\\left( \\kappa _k^2 + \\lambda \\sigma \\right) \\delta _{k,\\ell } +\\kappa _k \\left( q \\varphi _\\ell , \\varphi _k \\right)_{L^2} \\;.$ The above formula explicitly gives the entries of the matrix $B$ .",
"For our computer-assisted proof, we are however interested in the scaled matrix $\\tilde{B} = D^{-1} B D^{-1}$ .",
"One can immediately verify that its entries $\\tilde{b}_{k,\\ell }$ are given by $ \\tilde{b}_{k,\\ell } \\; = \\;-\\left(1 + \\frac{\\lambda \\sigma }{\\kappa _k^2} \\right) \\delta _{k,\\ell } +\\frac{1}{\\kappa _\\ell } (q \\varphi _\\ell , \\varphi _k)_{L^2}\\quad \\mbox{ with }\\quad q(x) = \\lambda f^{\\prime }(\\mu + u(x)) \\; .$ In view of (REF ), this formula will allow us to bound the operator norm of the inverse of the finite-dimensional projection $L_N$ using techniques from interval arithmetic."
],
[
"Construction of an approximative inverse",
"The crucial part in the derivation of our norm bound for the inverse of $L$ is the application of Proposition REF .",
"For this, we need to construct an approximative inverse of this operator.",
"Since this construction has to be explicit, we will approach it in two steps.",
"The first has already been accomplished in the last section, where we considered a finite-dimensional projection of $L$ , which can easily be inverted numerically.",
"In this section, we complement this finite-dimensional part with a consideration of the infinite-dimensional complementary space.",
"For this, we refer the reader again to the definition of the matrix representation $B$ in (REF ).",
"As $N \\rightarrow \\infty $ , this representation leads to better and better approximations of the operator $L$ .",
"Note in particular that the entry $b_{k,\\ell }$ is the sum of two terms.",
"The first of these is a diagonal matrix, and its entries clearly dominate the second term in (REF ).",
"We therefore use the inverse of the first term in order to complement the inverse of $L_N$ .",
"To describe this procedure in more detail, suppose that the function $v \\in Y$ is given by $v = \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty > 0}\\alpha _k \\varphi _k(x)= v_N + v_{\\infty } \\in Y_N \\oplus Y_\\infty \\; ,$ where we define $Y_N = P_N Y\\qquad \\mbox{ and }\\qquad Y_\\infty = \\left( I - P_N \\right) Y \\; .$ Using this representation the approximative inverse $S \\in {\\mathcal {L}}(Y,X)$ of $L \\in {\\mathcal {L}}(X,Y)$ is defined via the formula $S v = L_N^{-1} v_N -\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N} \\frac{\\alpha _k}{\\kappa _k^2 + \\lambda \\sigma } \\, \\varphi _k \\; .$ In addition, consider the operator $T = S|_{Y_\\infty }$ , i.e., let $T \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N} \\alpha _k \\varphi _k =-\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N} \\frac{\\alpha _k}{\\kappa _k^2 + \\lambda \\sigma } \\varphi _k \\; .$ One can easily see that $T : Y_\\infty \\rightarrow X_\\infty = (I-P_N)X$ is one-to-one and onto, and in fact we have the identity $T^{-1} \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N} \\alpha _k \\varphi _k =-\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N} \\left( \\kappa _k^2 +\\lambda \\sigma \\right) \\alpha _k \\varphi _k \\; ,$ which can be rewritten in the form $ T^{-1} v_\\infty = -\\left( \\Delta ^2 v_\\infty +\\lambda \\sigma v_\\infty \\right) \\; .$ Also, from the definition of $S$ we get the alternative representation $ Sv = L_N^{-1} v_N + T v_\\infty \\; .$ To close this section, we now derive a bound on the operator norm of $S$ , since this will be needed in the application of Proposition REF .",
"As a first step, we show that $\\Vert T v_\\infty \\Vert _X \\le \\Vert v_\\infty \\Vert _Y$ for all $y_\\infty \\in Y_\\infty $ , which follows readily from $\\left\\Vert T \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\alpha _k \\varphi _k \\right\\Vert _{X}^2 & = &\\left\\Vert \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\frac{\\alpha _k}{\\kappa _k^2 + \\lambda \\sigma } \\varphi _k\\right\\Vert _{\\overline{{\\mathcal {H}}}^2}^2 \\\\[2ex]& = &\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\frac{\\alpha _k^2 \\kappa _k^2}{(\\kappa _k^2 +\\lambda \\sigma )^2} \\\\[2ex]& \\le &\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\frac{\\alpha _k^2 \\kappa _k^2}{(\\kappa _k^2)^2}\\; = \\;\\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\kappa _k^{-2} \\alpha _k^2 \\\\[2ex]& = &\\left\\Vert \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\alpha _k \\varphi _k \\right\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}^2\\; = \\;\\left\\Vert \\sum _{k \\in {\\mathbb {N}}_0^d, \\, |k|_\\infty \\ge N}\\alpha _k \\varphi _k \\right\\Vert _Y^2.$ This estimate in turn implies for all $v = v_N + v_\\infty \\in Y_N \\oplus Y_\\infty $ the estimate $\\Vert S v \\Vert _X^2 & = &\\Vert L_N^{-1} v_N \\Vert _X^2 + \\Vert T v_\\infty \\Vert _X^2 \\\\[2ex]& \\le &\\underbrace{\\Vert L_N^{-1} \\Vert _{{\\mathcal {L}}(Y_N,X_N)}^2}_{\\le K_N^2}\\Vert v_N \\Vert _Y^2 + \\Vert v_\\infty \\Vert _Y^2\\; \\le \\; \\max (K_N,1)^2 \\Vert v \\Vert _Y^2 \\; ,$ where we used the definition of $K_N$ from (REF ).",
"Altogether, we have shown that $ \\Vert S\\Vert _{{\\mathcal {L}}(Y,X)} \\le \\max (K_N,1) \\; .$ In other words, the operator norm of the approximate inverse $S$ given in (REF ) can be bounded in terms of the inverse bound for the finite-dimensional projection given in (REF ).",
"Furthermore, it follows directly from the definition of $S$ that this operator is one-to-one."
],
[
"Assembling the final inverse estimate",
"In the last section we addressed two crucial aspects of Proposition REF .",
"On the one hand, we provided an explicit construction for the approximative inverse $S \\in {\\mathcal {L}}(Y,X)$ of the Fréchet derivative $L$ defined in (REF ).",
"On the other hand, we derived an upper bound on the operator norm of $S$ , which can be computed using the finite-dimensional projection $L_N$ of $L$ .",
"This in turn provides the constant $\\varrho _2$ in Proposition REF .",
"In this final subsection, we focus on the constant $\\varrho _1$ , i.e., we derive an upper bound on the norm $\\Vert I - S L \\Vert _{{\\mathcal {L}}(X,X)}$ , and show how this bound can be made smaller than one.",
"Altogether, this will complete the proof of the estimate for the constant $K$ in the constructive implicit function theorem, which was given in Theorem REF .",
"Before we begin, recall the abbreviation $q(x) = \\lambda f^{\\prime }(u(x) + \\mu )$ .",
"From our definitions of the operators $L \\in {\\mathcal {L}}(X,Y)$ , $S \\in {\\mathcal {L}}(Y,X)$ , $L_N \\in {\\mathcal {L}}(X_N,Y_N)$ , and $T \\in {\\mathcal {L}}(Y_\\infty ,X_\\infty )$ , as well as the projection $P_N$ , and using the additive representation $v = v_N + v_\\infty \\in Y_N \\oplus Y_\\infty $ , we have the identity $Lv = \\left( L_N v_N - P_N \\Delta (q v_\\infty ) \\right) +\\left( T^{-1} v_\\infty - \\left( I-P_N \\right)\\Delta (q v) \\right) \\; ,$ which will be derived in detail in the following calculation.",
"Notice that the first parentheses contain only terms in the finite-dimensional space $Y_N$ , while the second parentheses contain terms in $Y_\\infty $ .",
"With this in mind, we have $Lv & = &-\\Delta \\left( \\Delta v + q v \\right) - \\lambda \\sigma v \\\\[1ex]& = & -\\Delta ^2 v_N - \\Delta ^2 v_\\infty - P_N \\Delta (q v_N) -(I - P_N) \\Delta (q v_N) \\\\[0.5ex]& & \\qquad - \\Delta (q v_\\infty ) -\\lambda \\sigma v_N - \\lambda \\sigma v_\\infty \\\\[1ex]& = & \\left( -\\Delta ^2 v_N - P_N \\Delta (q v_N) -\\lambda \\sigma v_N \\right) - \\left( \\Delta ^2 v_\\infty +\\lambda \\sigma v_\\infty \\right) \\\\[0.5ex]& & \\qquad - (I - P_N) \\Delta (q v_N) -\\Delta (q v_\\infty ) \\\\[1ex]& = & L_N v_N + T^{-1} v_\\infty - (I - P_N) \\Delta (q v_N) -P_N \\Delta (q v_\\infty ) - (I - P_N) \\Delta (q v_\\infty ) \\\\[1ex]& = & L_N v_N + T^{-1} v_\\infty - P_N \\Delta (q v_\\infty ) -(I - P_N) \\Delta (q v) \\; .$ The first two lines follow just from the definitions, projections, and rearrangements of terms.",
"The third line is a consequence of (REF ) and (REF ).",
"Finally, the fourth and fifth lines involve only rearrangements using the projection operator.",
"Using the above representation (REF ) of the operator $L$ which is split along the subspaces $Y_N$ and $Y_\\infty $ , we can now derive an expression for $I - SL \\in {\\mathcal {L}}(X,X)$ .",
"More precisely, we have $(I - SL)v = L_N^{-1} P_N \\Delta (q v_\\infty ) +T (I - P_N) \\Delta (q v) \\; ,$ and this will be verified in detail below.",
"Notice that in this representation, the first term of the right-hand side lies in the finite-dimensional space $X_N$ , while the second term is contained in the complement $X_\\infty $ .",
"The identity in (REF ) now follows from (REF ) and $SL v & = & L_N^{-1} \\left( L_N v_N - P_N \\Delta (q v_\\infty ) \\right) +T \\left( T^{-1} v_\\infty - (I - P_N) \\Delta (qv) \\right) \\\\[1ex]& = & v_N - L_N^{-1} P_N \\Delta (q v_\\infty ) +v_\\infty - T (I - P_N) \\Delta (qv) \\\\[1ex]& = & I v - L_N^{-1} P_N \\Delta (q v_\\infty ) -T (I - P_N) \\Delta (qv) \\; .$ After these preparation, we can now show that the operator norm of $I - SL$ can be expected to be small for sufficiently large $N$ .",
"This will provide an estimate for the constant $\\varrho _1$ in Proposition REF , and conclude the proof of Theorem REF .",
"In order to show that $\\Vert I - SL \\Vert _{{\\mathcal {L}}(X,X)}$ is indeed small, we separately bound the two terms in (REF ) as $\\begin{array}{rclcrcl}\\displaystyle \\left\\Vert L_N^{-1} P_N \\Delta (q v_\\infty ) \\right\\Vert _X& \\le & \\displaystyle A \\Vert v\\Vert _X & \\quad \\mbox{ with }\\quad &\\displaystyle A & := & \\displaystyle \\frac{K_N \\Vert q\\Vert _\\infty }{\\pi ^2 N^2}\\; , \\\\[3ex]\\displaystyle \\left\\Vert T (I-P_N) \\Delta (q v) \\right\\Vert _X & \\le &\\displaystyle B \\Vert v\\Vert _X & \\quad \\mbox{ with }\\quad &\\displaystyle B & := & \\displaystyle \\frac{C_b \\sqrt{1+\\pi ^4} \\,\\Vert q\\Vert _{{\\mathcal {H}}^2}}{\\pi ^4 N^2} \\; .\\end{array}$ The first of these inequalities is established in the following calculation, which makes liberal use of Sobolev embeddings and other established inequalities: $\\left\\Vert L_N^{-1} P_N \\Delta (q v_\\infty ) \\right\\Vert _X & \\le &\\left\\Vert L_N^{-1} \\right\\Vert _{{\\mathcal {L}}(Y_N,X_N)} \\Vert P_N\\Delta (q v_\\infty ) \\Vert _Y \\\\[1.5ex]& \\le & K_N \\left\\Vert P_N \\Delta (q v_\\infty )\\right\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}\\; \\le \\; K_N \\left\\Vert \\Delta (q v_\\infty )\\right\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1.5ex]& \\le &K_N \\Vert q v_\\infty \\Vert _{{{\\mathcal {H}}}^{0}}\\; \\le \\;K_N \\Vert q\\Vert _{\\infty } \\left\\Vert (I-P_N) v\\right\\Vert _{\\overline{{\\mathcal {H}}}^0} \\\\[1.5ex]& \\le & K_N \\Vert q\\Vert _{\\infty } \\,\\frac{\\Vert v \\Vert _{\\overline{{\\mathcal {H}}}^{2}}}{\\pi ^2 N^2}\\; = \\;\\frac{K_N \\Vert q\\Vert _\\infty }{\\pi ^2 N^2} \\, \\Vert v\\Vert _X\\; = \\;A \\Vert v\\Vert _X \\; ,$ where for the last inequality we used Lemma REF .",
"The second estimate, the one involving the constant $B$ , is verified as follows, again with help from our previously derived inequalities, in particular the fact that $\\Vert T \\Vert _{{\\mathcal {L}}(Y_\\infty ,X_\\infty )} \\le 1$ and Lemmas REF and REF : $\\left\\Vert T (I-P_N) \\Delta (q v) \\right\\Vert _X & \\le &\\left\\Vert (I-P_N) \\Delta (q v) \\right\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}\\; \\le \\;\\frac{\\Vert \\Delta (q v)\\Vert _{\\overline{{\\mathcal {H}}}^{0}} }{\\pi ^2 N^2} \\\\[1.5ex]& = &\\frac{\\left\\Vert \\overline{P} (q v)\\right\\Vert _{\\overline{{\\mathcal {H}}}^{2}}}{\\pi ^2 N^2}\\; \\le \\;\\frac{\\Vert q v \\Vert _{{\\mathcal {H}}^{2}} }{\\pi ^2 N^2}\\; \\le \\;\\frac{C_b \\Vert q \\Vert _{{\\mathcal {H}}^{2}} \\Vert v \\Vert _{{\\mathcal {H}}^{2}} }{\\pi ^2 N^2} \\\\[1.5ex]& \\le &\\frac{C_b \\Vert q\\Vert _{{\\mathcal {H}}^{2}}}{\\pi ^2 N^2} \\cdot \\frac{\\sqrt{1+\\pi ^4}}{\\pi ^2} \\cdot \\Vert v \\Vert _{\\overline{{\\mathcal {H}}}^{2}}\\; = \\; B \\Vert v\\Vert _X \\; .$ Now that we have established these two inequalities, the proof of Theorem REF can easily be completed using an application of Proposition REF .",
"Specifically, the inequalities which involve the constants $A$ ands $B$ combined with (REF ) imply that $\\Vert I - S L \\Vert _{{\\mathcal {L}}(X,X)} \\; \\le \\;\\sqrt{A^2 + B^2} \\; = \\;\\frac{1}{\\pi ^2 N^2} \\, \\sqrt{K_N^2 \\Vert q\\Vert _\\infty ^2 +C_b^2 \\, \\frac{1+\\pi ^4}{\\pi ^4} \\, \\Vert q\\Vert _{{\\mathcal {H}}^2}^2} \\; .$ We also know from (REF ) that $\\Vert S \\Vert _X \\le \\max (K_N,1)$ .",
"Therefore, we can directly apply Proposition REF with the constants $\\varrho _1 = \\sqrt{A^2 + B^2} \\le \\tau < 1$ and $\\varrho _2 = \\max (K_N,1)$ , and this immediately implies that the operator $L \\in {\\mathcal {L}}(X,Y)$ is one-to-one, onto, and the norm of its inverse operator is bounded via $\\left\\Vert L^{-1} \\right\\Vert _{{\\mathcal {L}}(Y,X)} \\; \\le \\;\\frac{\\varrho _2}{1-\\varrho _1} \\; = \\;\\frac{\\max (K_N,1)}{1-\\tau } \\; .$ This completes the proof of Theorem REF ."
],
[
"Lipschitz estimates",
"In this section, our goal is to establish the Lipschitz constants needed in hypotheses (H3) and (H4) required for Theorem REF , the constructive implicit function theorem.",
"Namely, we need to establish Lipschitz bounds for the derivatives of $F$ with respect to both $u$ and with respect to the continuation parameter.",
"We are considering single-parameter continuation, meaning that we have three separate situations to discuss, corresponding to the three different parameters $\\lambda $ , $\\sigma $ , and $\\mu $ .",
"Specifically, for $p$ being one of these three parameters, for a fixed parameter-function pair $(p^*,u^*) \\in {\\mathbb {R}}\\times X$ , and for fixed values of $d_p$ and $d_u$ , we assume that $|p - p^*| \\le d_p$ , and $\\Vert u - u^* \\Vert _X \\le d_u$ .",
"Furthermore, by a slight abuse of notation we drop the parameters different from $p$ from the argument list of $F$ in (REF ).",
"Our goal in the current section is to obtain tight and easily computable bounds on the constants $M_1$ through $M_4$ in the following two formulas: $ \\begin{array}{rcl}\\displaystyle \\Vert D_u F ( p, u) - D_u F ( p, u) \\Vert _{{\\mathcal {L}}(X,Y)} & \\le &\\displaystyle M_1 \\; \\Vert u - u^* \\Vert _X + M_2 \\; |p - p^*| \\; , \\\\[1ex]\\displaystyle \\Vert D_p F ( p, u) - D_p F ( p, u) \\Vert _{{\\mathcal {L}}({\\mathbb {R}},Y)} & \\le &\\displaystyle M_3 \\; \\Vert u - u^* \\Vert _X + M_4 \\; |p - p^*| \\; .\\end{array}$ These bounds will be determined using standard Sobolev embedding theorems and the constants from the previous section, for each of the three parameters $\\lambda $ , $\\sigma $ , and $\\mu $ .",
"Notice that throughout this section, we always assume $\\lambda > 0$ and $\\sigma \\ge 0$ , while the mass $\\mu $ could be a real number of either sign."
],
[
"Variation of the short-range repulsion",
"We now state the Lipschitz estimates for the constructive implicit function theorem in the case where $\\lambda $ , the short-range repulsion term, varies and the remaining parameters $\\mu $ and $\\sigma $ are held fixed.",
"Lemma 4.1 (Lipschitz constants for variation of $\\lambda $ ) Let $\\lambda ^* \\in {\\mathbb {R}}$ and $u^* \\in \\overline{{\\mathcal {H}}}^2$ be arbitrary, and consider fixed positive constants $d_{\\lambda }$ and $d_u$ .",
"Finally let $\\lambda $ and $u$ be such that $|\\lambda -\\lambda ^*| \\le d_{\\lambda }\\quad \\mbox{ and }\\quad \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2 } \\le d_u \\;.$ Then the Lipschitz constants in (REF ) can be chosen as $\\begin{array}{rclcrcl}\\displaystyle M_1 &=& \\displaystyle \\frac{\\overline{C}_m f_{\\max }^{(2)} (\\lambda ^* +d_\\lambda )}{\\pi ^2} \\; , & \\qquad &\\displaystyle M_2 &=& \\displaystyle \\frac{\\Vert f^{\\prime }(u^*+ \\mu ) \\Vert _\\infty }{\\pi ^2} +\\frac{\\sigma }{\\pi ^4} \\; , \\\\[2ex]\\displaystyle M_3 &=& \\displaystyle \\frac{f^{(1)}_{\\max }}{\\pi ^2} +\\frac{\\sigma }{\\pi ^4} \\; , & \\qquad &\\displaystyle M_4 &=& 0 \\; ,\\end{array}$ where $f_{\\max }^{(1)}$ and $f_{\\max }^{(2)}$ are defined as $ f^{(p)}_{\\max } =\\max _{|\\varrho | \\le \\Vert u^*\\Vert _\\infty + \\overline{C}_m d_u}|f^{(p)} (\\varrho + \\mu )| \\;.$ These are well-defined since $f$ is a $C^2$ -function.",
"For our choice of constants $d_\\lambda $ , $d_u$ , reference parameter $\\lambda ^* \\in {\\mathbb {R}}$ and function $u^* \\in \\overline{{\\mathcal {H}}}^2$ , and for arbitrary $v \\in \\overline{{\\mathcal {H}}}^2$ , assume that $|\\lambda - \\lambda ^*| \\le d_\\lambda $ and $\\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^2}\\le d_u$ .",
"We start by deriving expressions for both $M_1$ and $M_2$ .",
"Notice that we have $& & \\hspace*{-56.9055pt} \\Vert D_uF(\\lambda ,u)[v] -D_uF(\\lambda ^*,u^*)[v] \\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\Vert \\Delta ( \\lambda f^{\\prime }(u+\\mu ) v - \\lambda ^* f^{\\prime }(u^*+\\mu )v )\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} + \\sigma \\, |\\lambda - \\lambda ^*|\\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\Vert \\overline{P} ( \\lambda f^{\\prime }(u+\\mu ) v - \\lambda ^* f^{\\prime }(u^*+\\mu )v )\\Vert _{\\overline{{\\mathcal {H}}}^0} + \\sigma \\, |\\lambda - \\lambda ^*| \\, \\frac{1}{\\pi ^4} \\,\\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^2} \\\\[1ex]&\\le & \\Vert \\lambda f^{\\prime }(u+\\mu ) v - \\lambda ^* f^{\\prime }(u^*+\\mu )v \\Vert _{L^2} +\\frac{\\sigma }{\\pi ^4}\\,|\\lambda - \\lambda ^*| \\, \\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^2} \\\\[1ex]&\\le & \\Vert \\lambda f^{\\prime }(u+\\mu ) - \\lambda ^* f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\, \\Vert v\\Vert _{L^2} +\\frac{\\sigma }{\\pi ^4}\\,|\\lambda - \\lambda ^*| \\, \\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^2} \\\\[1ex]&\\le & \\left( \\frac{1}{\\pi ^2} \\Vert \\lambda f^{\\prime }(u+\\mu ) - \\lambda ^* f^{\\prime }(u^*+\\mu ) \\Vert _\\infty + |\\lambda - \\lambda ^*| \\, \\frac{\\sigma }{\\pi ^4}\\right) \\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^2} \\; .$ The first estimate follows straightforwardly from the definition of the Fréchet derivative (REF ), while the second one uses the fact that the Laplacian is an isometry (cf.",
"Lemma REF ) and the Banach scale estimate between $\\overline{{\\mathcal {H}}}^{-2}$ and $\\overline{{\\mathcal {H}}}^{2}$ (cf.",
"Lemma REF ).",
"The third estimate follows from $\\Vert \\overline{P}\\Vert = 1$ , as well as the fact that $\\overline{{\\mathcal {H}}}^0$ and $L^2(\\Omega )$ are equipped with the same norm.",
"Finally, the fourth estimate is straightforward, and the factor $1/\\pi ^2$ in the fifth estimate follows from $v \\in \\overline{{\\mathcal {H}}}^{2} \\subset \\overline{{\\mathcal {H}}}^0$ and the estimate in Lemma REF .",
"The above estimate shows that the operator norm of the difference of the two Fréchet derivatives is bounded by the expression in parentheses.",
"The first of these two terms will now be estimated further.",
"For this, note first that $& & \\Vert \\lambda f^{\\prime }(u+\\mu ) - \\lambda ^* f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\\\[1ex]& & \\qquad \\qquad \\le \\;|\\lambda | \\, \\Vert f^{\\prime }(u+\\mu ) - f^{\\prime }(u^*+\\mu ) \\Vert _\\infty +| \\lambda -\\lambda ^*| \\, \\Vert f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\; .$ For fixed $x \\in \\Omega $ , we know from the mean value theorem that there exists a number $\\xi (x)$ between $u(x)$ and $u^*(x)$ such that $| f^{\\prime }(u(x)+\\mu ) - f^{\\prime }(u^*(x)+\\mu ) | \\le |f^{\\prime \\prime }(\\xi (x)+\\mu )| \\; |u(x)-u^*(x)| \\;.$ Since $\\xi (x)$ is contained between $u(x)$ and $u^*(x)$ for all $x \\in \\Omega $ , the function $\\xi $ is bounded.",
"Combining this fact with the definition of $\\overline{C}_m$ in (REF ) we get $\\Vert \\xi \\Vert _\\infty \\le \\Vert u^*\\Vert _\\infty + \\Vert u - u^* \\Vert _\\infty \\le \\Vert u^*\\Vert _\\infty + \\overline{C}_m \\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} \\le \\Vert u^*\\Vert _\\infty + \\overline{C}_m d_u \\;,$ and therefore $& & \\hspace*{-56.9055pt}\\Vert \\lambda f^{\\prime }(u+\\mu ) - \\lambda ^* f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\\\[1ex]& \\le &|\\lambda | \\, f^{(2)}_{\\max }\\, \\Vert u - u^*\\Vert _\\infty +|\\lambda -\\lambda ^*|\\, \\Vert f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\\\[1ex]& \\le & |\\lambda | \\, f^{(2)}_{\\max }\\, \\overline{C}_m \\,\\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} +|\\lambda -\\lambda ^*|\\, \\Vert f^{\\prime }(u^*+\\mu ) \\Vert _\\infty \\;,$ where $f^{(2)}_{\\max }$ is defined in (REF ).",
"Incorporating this into the previous estimate, we see that $& & \\hspace*{-34.14322pt}\\Vert D_uF(\\lambda ,u) - D_uF(\\lambda ^*,u^*) \\Vert _{{\\mathcal {L}}(\\overline{{\\mathcal {H}}}^2,\\overline{{\\mathcal {H}}}^{-2})} \\\\[1ex]&\\le & \\left( \\frac{\\overline{C}_m \\, f^{(2)}_{\\max } \\,(\\lambda ^* + d_{\\lambda })}{\\pi ^2} \\right) \\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} +\\left( \\frac{\\Vert f^{\\prime }(u^* + \\mu )\\Vert _\\infty }{\\pi ^2} +\\frac{\\sigma }{\\pi ^4}\\right) |\\lambda - \\lambda ^*| \\; .$ This equation directly gives the values of the Lipschitz constants $M_1$ and $M_2$ given in the statement of the lemma.",
"We now turn our attention to the remaining constants $M_3$ and $M_4$ .",
"The Fréchet derivative of $F$ with respect to $\\lambda $ is given by $D_\\lambda F(\\lambda ,u) = - \\Delta f(u+ \\mu ) - \\sigma u \\;.$ Using almost identical steps as the calculation of $M_1$ and $M_2$ , we get $& & \\hspace*{-56.9055pt}\\Vert D_\\lambda F(\\lambda ,u) - D_\\lambda F(\\lambda ^*,u^*)\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]& \\le & \\Vert \\Delta (f(u+\\mu )-f(u^*+\\mu ))\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} +|\\sigma | \\, \\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\Vert f(u+\\mu )-f(u^*+\\mu )\\Vert _{L^2} + \\frac{\\sigma }{\\pi ^4} \\,\\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^{2}} \\\\[1ex]&\\le & f^{(1)}_{\\max } \\, \\Vert u - u^*\\Vert _{L^2} + \\frac{\\sigma }{\\pi ^4} \\,\\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^{2}} \\\\[1ex]&\\le & \\left( \\frac{f^{(1)}_{\\max }}{\\pi ^2} + \\frac{\\sigma }{\\pi ^4} \\right)\\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^{2}} \\;.$ Notice that in estimating the norm of this difference of Fréchet derivatives we use the standard identification of ${\\mathcal {L}}({\\mathbb {R}},\\overline{{\\mathcal {H}}}^{-2})$ with $\\overline{{\\mathcal {H}}}^{-2}$ .",
"Furthermore, in the above inequalities, we have made liberal use of the constructive Sobolev embedding results from the previous section.",
"This gives the constants $M_3$ and $M_4$ given in the statement of the lemma."
],
[
"Variation of the long-range elasticity",
"We now establish Lipschitz constants for the case when the parameter $\\sigma $ varies and both $\\lambda $ and $\\mu $ are held fixed.",
"Lemma 4.2 (Lipschitz constants for variation of $\\sigma $ ) Let $\\sigma ^* \\in {\\mathbb {R}}$ and $u^* \\in \\overline{{\\mathcal {H}}}^2$ be arbitrary, and consider fixed positive constants $d_{\\sigma }$ and $d_u$ .",
"Finally let $\\sigma $ and $u$ be such that $|\\sigma -\\sigma ^*| \\le d_{\\sigma }\\quad \\mbox{ and }\\quad \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2 } \\le d_u \\;.$ Then the Lipschitz constants in (REF ) can be chosen as $M_1 \\; = \\; \\frac{\\lambda \\, f^{(2)}_{\\max } \\, \\overline{C}_m}{\\pi ^2} \\; ,\\qquad M_2 \\; = \\; M_3 \\; = \\; \\frac{\\lambda }{\\pi ^4} \\; ,\\qquad M_4 \\; = \\; 0\\; ,$ where the value of $f^{(2)}_{\\max }$ is defined in (REF ).",
"We start by computing the constants $M_1$ and $M_2$ .",
"Holding $\\mu $ and $\\lambda > 0$ fixed in the equation for $D_uF$ , we are able to follow very similar arguments as in the $\\lambda $ -varying case, including the use of the Sobolev embedding formulas and the mean value theorem.",
"The resulting estimate is given by $& & \\hspace*{-56.9055pt} \\Vert D_uF(\\sigma ,u)[v] - D_uF(\\sigma ^*,u^*)[v]\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]& \\le & \\Vert \\Delta (\\lambda (f^{\\prime }(u + \\mu ) - f^{\\prime }(u^*+\\mu )) v) \\Vert _{\\overline{{\\mathcal {H}}}^{-2}} +\\lambda \\, |\\sigma -\\sigma ^*| \\, \\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]& \\le & \\lambda \\, \\Vert f^{\\prime }(u+\\mu ) - f^{\\prime }(u^*+\\mu )\\Vert _\\infty \\, \\Vert v \\Vert _{L^2} +\\lambda \\, |\\sigma -\\sigma ^*| \\, \\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}\\\\[1ex]& \\le &\\left( \\frac{ \\lambda \\, f^{(2)}_{\\max } \\, \\overline{C}_m}{\\pi ^2} \\right)\\Vert u-u^* \\Vert _{\\overline{{\\mathcal {H}}}^2} \\, \\Vert v \\Vert _{\\overline{{\\mathcal {H}}}^2} +\\left(\\frac{\\lambda }{\\pi ^4} \\right) \\, |\\sigma -\\sigma ^*| \\,\\Vert v\\Vert _{\\overline{{\\mathcal {H}}}^{2}}\\;.$ This establishes constants $M_1$ and $M_2$ given in the lemma.",
"We now turn our attention to the constants $M_3$ and $M_4$ .",
"The derivative of $F$ with respect to $\\sigma $ is given by $D_\\sigma F(\\sigma ,u) = - \\lambda u \\;.$ Therefore, once again Lemma REF , we get $\\Vert D_\\sigma F(\\sigma ,u) - D_\\sigma F(\\sigma ^*,u^*)\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}\\; \\le \\;\\lambda \\, \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}\\; \\le \\;\\frac{\\lambda }{\\pi ^4} \\, \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2 }\\; ,$ which gives the constants $M_3$ and $M_4$ stated in the lemma."
],
[
"Varying the relative proportion of the two polymers",
"In this final subsection we now consider the third parameter variation, namely that of $\\mu $ .",
"Lemma 4.3 (Lipschitz constants for variation of $\\mu $ ) Let $\\mu ^* \\in {\\mathbb {R}}$ and $u^* \\in \\overline{{\\mathcal {H}}}^2$ be arbitrary, and consider fixed positive constants $d_{\\mu }$ and $d_u$ .",
"Finally let $\\mu $ and $u$ be such that $|\\mu -\\mu ^*| \\le d_{\\mu }\\quad \\mbox{ and }\\quad \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2 } \\le d_u \\;.$ Then the Lipschitz constants in (REF ) can be chosen as $M_1 \\; = \\; \\frac{\\lambda \\, f^{(2)}_{\\max ,\\mu } \\, \\overline{C}_m}{\\pi ^2} \\; ,\\qquad M_2 \\; = \\; M_3 \\; = \\; \\frac{\\lambda \\, f^{(2)}_{\\max ,\\mu }}{\\pi ^2} \\; ,\\qquad M_4 \\; = \\; \\lambda \\; f^{(2)}_{\\max ,\\mu } \\; ,$ where the constant $f^{(2)}_{\\max , \\mu }$ is defined as $f^{(2)}_{\\max , \\mu } = \\max _{ |\\varrho | \\le \\Vert u^* + \\mu ^*\\Vert _{\\infty } +\\overline{C}_m d_u + d_\\mu } |f^{\\prime \\prime }(\\varrho )| \\; .$ Using a similar format to the last two proofs, we consider $\\lambda > 0$ and $\\sigma \\ge 0$ to be fixed constants and only allow $\\mu $ to vary.",
"The we have $& & \\hspace*{-56.9055pt} \\Vert D_uF(\\mu ,u)[v] - D_uF(\\mu ^*,u^*)[v]\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\Vert \\Delta ( \\lambda ( f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*))v)\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\lambda \\Vert f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*)\\Vert _{\\infty } \\,\\Vert v \\Vert _{L^2} \\\\[1ex]&\\le & \\frac{\\lambda }{\\pi ^2} \\Vert f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*)\\Vert _{\\infty } \\,\\Vert v \\Vert _{\\overline{{\\mathcal {H}}}^{2}} \\; .$ As in the previous calculations, we use the mean value theorem to bound the value of the maximum norm $\\Vert f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*)\\Vert _{\\infty }$ .",
"To do so, note that if a real value $\\varrho $ is between the two numbers $u^*(x)+\\mu $ and $u(x) + \\mu ^*$ for some $x \\in \\Omega $ , then one has $|\\varrho | &\\le & \\Vert u + \\mu ^*\\Vert _{\\infty } + |\\mu - \\mu ^*| \\\\[1ex]&\\le & \\Vert u^*+\\mu ^*\\Vert _\\infty + \\Vert u - u^*\\Vert _\\infty + |\\mu - \\mu ^*| \\\\[1ex]&\\le & \\Vert u^*+\\mu ^*\\Vert _\\infty + \\overline{C}_m \\Vert u - u^*\\Vert _{\\overline{{\\mathcal {H}}}^2}+ |\\mu - \\mu ^*| \\le \\Vert u^*+\\mu ^*\\Vert _\\infty + \\overline{C}_m d_u + d_\\mu \\; .$ Thus, by the mean value theorem, followed by the use of our Sobolev embedding results, one further obtains $\\Vert f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*)\\Vert _\\infty &\\le &f^{(2)}_{\\max , \\mu } \\, \\Vert (u + \\mu ) - (u^* + \\mu ^*) \\Vert _\\infty \\\\[1ex]&\\le & f^{(2)}_{\\max , \\mu } \\, \\left( \\overline{C}_m\\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} + |\\mu -\\mu ^*| \\right) \\; ,$ and combining this with our previous estimate we finally deduce $\\Vert D_uF(\\mu ,u) - D_uF(\\mu ^*,u^*) \\Vert _{{\\mathcal {L}}(\\overline{{\\mathcal {H}}}^{-2},\\overline{{\\mathcal {H}}}^{2})}\\; \\le \\;\\frac{\\lambda \\; f^{(2)}_{\\max ,\\mu } }{\\pi ^2}\\left( \\overline{C}_m \\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} +|\\mu -\\mu ^*| \\right) \\; .$ This gives the constants $M_1$ and $M_2$ .",
"We now look at the bounds for $M_3$ and $M_4$ .",
"The derivative of $F$ with respect to $\\mu $ is given by $D_\\mu F(\\mu ,u) = -\\Delta (\\lambda f^{\\prime }(u + \\mu )) \\;.$ By similar reasoning as before, we then get $\\Vert D_\\mu F(\\mu ,u) - D_\\mu F(\\mu ^*, u^*)\\Vert _{\\overline{{\\mathcal {H}}}^{-2}}&=& \\lambda \\, \\Vert \\Delta (f^{\\prime }(u + \\mu ) -f^{\\prime }(u^*+\\mu ^*))\\Vert _{\\overline{{\\mathcal {H}}}^{-2}} \\\\[1ex]&\\le & \\lambda \\, \\Vert f^{\\prime }(u + \\mu ) - f^{\\prime }(u^* + \\mu ^*)\\Vert _{L^2} \\\\[1ex]&\\le & \\lambda \\, f_{\\max ,\\mu }^{(2)} \\; \\Vert (u + \\mu ) -(u^* + \\mu ^*)\\Vert _{L^2} \\\\[1ex]&\\le & \\lambda \\, f_{\\max ,\\mu }^{(2)} \\, \\left( \\frac{1}{\\pi ^2}\\Vert u-u^*\\Vert _{\\overline{{\\mathcal {H}}}^2} + |\\mu -\\mu ^*| \\right) \\; .$ This gives the constants $M_3$ and $M_4$ and completes the proof of the lemma.",
"With the above lemma we have completed the discussion of all of the Lipschitz constant bounds for all three equation parameters."
],
[
"Illustrative examples",
"In this section, we present some examples of validated equilibrium solutions in order to illustrate the power of our theoretical validation method.",
"In particular, the theoretical methods developed above can be used to produce a validated region in parameter cross phase space.",
"We emphasize that this section is only intended to present proof of concept.",
"We have not made any attempt to optimize our results or to add computational methods to speed up the code.",
"For example, the interval arithmetic package INTLAB [28] that we have used is not written in parallel, and we have not attempted to parallelize any of our algorithms.",
"As another example, in the past we have found that careful preconditioning can speed up the computation time significantly.",
"Rather than add any of these techniques at this stage, we have chosen to reserve numerical considerations for a future paper, in which we will also address additional questions such as how to use these methods iteratively to validate branches of solutions.",
"Figure: Ten sample validated one-dimensional equilibrium solutions.For all solutions we choose λ=150\\lambda = 150 and σ=6\\sigma = 6.Three of the solutions have total mass μ=0\\mu =0, three are formass μ=0.1\\mu = 0.1, three for μ=0.3\\mu = 0.3, and finally onefor μ=0.5\\mu = 0.5.Figure: There is a tradeoff between high-dimensional calculations andoptimal results.",
"The top left figure shows how the bound of KKvaries with the dimension of the truncated approximation matrixused to calculate K N K_N.",
"These calculations are for dimension one,but a similar effect occurs in higher dimensions as well.",
"The topright figure shows the corresponding estimate for δ x \\delta _x, andthe bottom panel shows the estimate for δ α \\delta _\\alpha ,where α\\alpha is each of the three parameters.",
"The size ofthe validated interval grows larger as the truncation dimensiongrows, but with diminishing returns on the computationalinvestment.Table: A sample of the one-dimensional solution validationparameters for three typical solutions.",
"In each case, weuse σ=6\\sigma = 6 and λ=150\\lambda = 150.",
"If we had chosen alarger value of NN, we could significantly improve the results.Under the hypotheses of Theorem REF , the constructive implicit function theorem, for each $\\delta _\\alpha $ and $\\delta _x$ satisfying both parts of (REF ), we are guaranteed that the solution is uniquely contained in the corresponding $(\\delta _\\alpha ,\\delta _x)$ -box, where $\\alpha $ is the chosen of the three parameters.",
"In fact, if we fix $\\delta _\\alpha $ small enough, then there are a range of values of $\\delta _x$ bounded below by the quadratic second equation and above by the linear first equation.",
"We can view the region bounded by the lower limit of $\\delta _x$ as an accuracy region, within which the equilibrium is guaranteed to lie; and the region bounded by the upper limit of $\\delta _x$ is a uniqueness region, which contains the accuracy region, within which the solution is guaranteed to be unique.",
"If $\\delta _\\alpha $ is chosen to be the point for which the line and curve in (REF ) intersect, then this is the largest possible value of $\\delta _\\alpha $ for which the theorem holds, and the accuracy and uniqueness regions coincide.",
"In our calculations we have validated using this maximal interval in parameter space, and we have done the calculation of the interval size for each of the three parameters.",
"Figure: Six of the seventeen validated two-dimensional equilibrium solutions.For all seventeen solutions we use σ=6\\sigma = 6.",
"Five of these solutionsare for λ=75\\lambda = 75 and μ=0\\mu = 0 (top left).",
"The rest of them useλ=150\\lambda = 150 and μ=0\\mu = 0 (top middle and top right), μ=0.1\\mu =0.1(bottom left), μ=0.3\\mu =0.3 (bottom middle), and μ=0.5\\mu = 0.5 (bottomright).Table: A sample of the two-dimensional validation parametersfor a couple of typical solutions.",
"In all cases, we useσ=6\\sigma = 6.",
"Again as in the previous table, we couldimprove results by choosing a larger value of NN, butin this case since NN is only the linear dimension, thedimension of the calculation varies with N 2 N^2.We have validated ten different equilibrium solutions in one dimension, shown in Figure REF .",
"Some examples of the associated validation parameters are presented in Table REF .",
"Ideally, we are able to validate the largest possible $(\\delta _\\alpha ,\\delta _x)$ -box in which we can guarantee that the solution exists.",
"However, there is a tradeoff between computational cost and optimal bounds.",
"The most computationally costly part of our estimates is the calculation of $K_N$ , the bound on the inverse of the linearization of the truncated system.",
"As depicted in Figure REF , the bounds on $K$ , and correspondingly on $\\delta _x$ and $\\delta _\\alpha $ , depend significantly on the value of $N$ that is chosen for the truncation dimension.",
"Since our goal is to use these validations iteratively for path following, we will not be able to refine our calculations each time.",
"Therefore as a rule of thumb for a starting point, we used the equation in Theorem REF to guess that we would have a successful validation for $N \\approx C\\Vert q\\Vert ^{1/2}_{H^2}$ , where $C$ is a fixed order one constant.",
"In our calculations for the ten solutions, this results in a dimension that varies.",
"For these calculations we chose $N$ values ranging between 50 and 200.",
"The values of $M_i$ become progressively larger as you go from $\\lambda $ to $\\sigma $ to $\\mu $ .",
"This means that the corresponding values of $\\delta _\\alpha $ are worse (i.e., smaller), respectively, often by one or two orders of magnitude.",
"However, the values of $\\delta _x$ for the three cases are of the same order.",
"While we could increase $N$ to improve the estimates, Figure REF shows that there are diminishing returns on computational investment, and eventually at some $N$ , we could not have done much better even with a significantly larger value of $N$ .",
"Figure: A three-dimensional validated solution for the parametervalues λ=75\\lambda = 75, σ=6\\sigma = 6, and μ=0\\mu = 0.Table: Validation parameters for a three-dimensional sample solution.In two dimensions, we have validated seventeen different solutions for varying parameter values.",
"A representative sample are given in Figure REF , with some sample validation parameters presented in Table REF .",
"Again here, there is a tradeoff between computational speed and optimal results, but with all of the computations being significantly longer due to the increased dimension; if the function $u$ is encoded by a Fourier coefficient array of size $N \\times N$ , then the derivative matrix is of size $(N^2 - 1)^2$ , where the $-1$ is due to the fact that we have removed the constant term.",
"As in one dimension, the resulting $\\delta _\\alpha $ values vary significantly, but the $\\delta _x$ values do not.",
"Figure REF and Table REF show the details of a solution which is validated in three dimensions, with much the same observed behavior.",
"Three-dimensional result validation requires a much larger computational effort, since if the function $u$ is given by a Fourier coefficient array of size $N \\times N \\times N$ , then the derivative matrices with inverse being approximated are of size $(N^3-1)^2$ ."
],
[
"Conclusions",
"As outlined in more detail in the introduction, in this paper we presented the theoretical foundations for validating branch segments of equilibrium solutions for the diblock copolymer model.",
"Our approach is based on using the natural Sobolev norms which are used in the study of the underlying evolution equation, and they have been derived in all three relevant physical dimensions.",
"As a side result, we obtained a method based on Neumann series to determine rigorous upper bounds on the inverse Fréchet derivative of the diblock copolymer operator which are of interest in their own right, as they are connected to the pseudo-spectrum of this non-self-adjoint operator, see [31].",
"Moreover, we have demonstrated briefly in the last section how these results can be used to obtain computer-assisted proofs for selected diblock copolymer equilibrium solutions.",
"While the present paper is a first step towards a complete path-following framework for the diblock copolymer model in dimensions up to three, there are still a number of issues that have to be addressed.",
"On the theoretical side, one has to develop a pseudo-arclength continuation method with associated linking conditions which operates in an automatic fashion.",
"This can be done by using the constructive implicit function theorem as a tool, similar to the applications to slanted box continuation and limit point resolution which were presented in [30].",
"In addition, the bottleneck in the current validation step is the estimation of the norm bound for the inverse.",
"Especially in two, and even more so in three dimensions, one has to implement path-following in such a way that the estimate does not have to be validated at every step.",
"This can be accomplished via perturbation arguments, and further speedups are possible by using the sparseness of the involved matrices.",
"However, all of these issues are nontrivial and lie beyond the scope of the current paper — they will therefore be presented elsewhere."
],
[
"Acknowledgments",
"We thank the referee for helpful comments, which improved the quality of this paper.",
"E.S.",
"and T.W.",
"were partially supported by NSF grant DMS-1407087.",
"E.S.",
"was partially supported by NSF grant DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester.",
"In addition, T.W.",
"and E.S.",
"were partially supported by the Simons Foundation under Awards 581334 and 636383, respectively."
]
] | 2005.14224 |
[
[
"A moment matching method for option pricing under stochastic interest\n rates"
],
[
"Abstract In this paper we present a simple, but new, approximation methodology for pricing a call option in a Black \\& Scholes market characterized by stochastic interest rates.",
"The method, based on a straightforward Gaussian moment matching technique applied to a conditional Black \\& Scholes formula, is quite general and it applies to various models, whether affine or not.",
"To check its accuracy and computational time, we implement it for the CIR interest rate model correlated with the underlying, using the Monte Carlo simulations as a benchmark.",
"The method's performance turns out to be quite remarkable, even when compared with analogous results obtained by the affine approximation technique presented in Grzelak and Oosterlee (2011) and by the expansion formula introduced in Kim and Kunimoto (1999), as we show in the last section."
],
[
"Introduction",
"Since the appearance of the seminal Black & Scholes/Merton option pricing fundamental formula, there has been an intensive effort to incorporate in the market model additional stochastic factors, such as the volatility and/or the interest rates, the latter already discussed by Merton himself in [15].",
"Along the years, a huge field of research developed, leading to a very rich literature on stochastic volatility models, while fewer papers aimed at the inclusion of a dynamic term structure into the valuation of derivatives, e.g.",
"[17], [1], [20], [19], [11], [7], [18].",
"Nowadays, the improvement in the performances of option pricing formulas obtained by adding these risk factors is widely recognized in the empirical literature (see e.g [2], [3]), indeed in ([12]) the author remarked that even including solely stochastic interest rates in the model does affect the pricing formula, especially for longer-dated options, in a noticeable manner.",
"Of course this generalization implies a higher degree of mathematical complexity and the search for efficient pricing techniques, able to provide accurate answers in a short computational time (as opposed to Monte Carlo methods) has been relentless, even more so in modern quantitative finance where a huge amount of data allows to consider strategies that call for on real-time model calibration.",
"Hence, computational efficiency has become one of the primary concerns of risk managers and this requirement essentially restricted the choice of models to the affine class (see [7]).",
"Indeed, when the interest rates are modeled in a Gaussian processes framework, as in the very popular Hull-White / Vasicek models, even analytical prices can be obtained.",
"These models are appropriate for modeling periods that admit positive probability of negative rates, such as the current one, but this feature becomes a drawback in usual periods of positive rates.",
"The most popular model used to avoid this drawback is the Cox-Ingersoll-Ross (CIR) one, which guarantees the rate's strict positivity under Feller's condition.",
"Its popularity comes from the fact that falls into the so called affine models, that can exploit a very efficient and fast Fourier transform technique to price the bonds.",
"Unfortunately, the affinity of the model is lost when the interest rate is coupled, with correlation, with a risky asset's dynamics, making the search for efficient approximations of risk-neutral pricing formulas very challenging.",
"Here we present a simple, but new, approximation methodology for pricing a European call option in a market model given by a linear diffusion dynamics for the underlying (a Black & Scholes (BS) framework) coupled with a stochastic short term risk-free rate.",
"The problem is a classical one and the novelty lies on the fact that we propose a quite straightforward moment matching (MM) technique, easy to implement and leading to very efficient approximations.",
"In building our procedure a few issues have to be addressed and we first provide, by appropriate conditioning, a representation formula for the claim's price in terms of the BS formula, then we exploit a Gaussian approximation by properly matching the first two moments of the involved random variables, that allows to use the properties of the Normal cumulative distribution function (c.d.f.)",
"(see Lemma (REF )).",
"When applying the method to the affine models, we also employ a change-of-numeraire technique (introducing the $T$ -forward measure as in [5]) to partially disentangle the contributions due to the underlying and to the interest rate, exploiting the explicit expressions of the bond's price in an affine framework.",
"To keep computations as simple as possible, any time a quantity is computable, it is stored and treated as a constant in the sequel.",
"This leads to an efficient mixed use of the risk free probability and the $T$ -forward measure to evaluate the separate quantities.",
"The paper is organized as follows.",
"In Section 2 we derive a representation formula for the call option's price in Black & Scholes market with stochastic risk-free short rates, while in Section 3 the Moment Matching method is fully described.",
"Finally, in Section 4 we restrict to the affine models and we apply our technique with a CIR interest rate.",
"In the same Section, we briefly introduce other two techniques, the affine approximation, inspired by Grzelak and Oosterlee [9] and the expansion method proposed by Kim and Kunimoto [11] alternative to prices obtained by Monte Carlo simulations.",
"Hence we run a numerical study comparing those methods with ours, using Monte Carlo evaluation as a benchmark."
],
[
"The price of a European call in the BS model with stochastic rates",
"The underlying problem we are concerned with is the pricing of a European call option, whose payoff is given by the function $f(x)=( \\mathrm {e}^x- \\mathrm {e}^\\kappa )^+$ for some $\\kappa \\in \\mathbb {R}$ , when stochastic interest rates come into play.",
"Thus, given a finite time interval $[0,T]$ and a complete probability space $(\\Omega , {\\mathcal {F}}, Q)$ , endowed of a filtration $\\lbrace {\\mathcal {F}}_t\\rbrace _{\\lbrace t\\in [0,T]\\rbrace }$ satisfying the “usual hypotheses\" (see [16]), the market model is defined by the log-price of a risky asset and a risk-free interest $(X_t,r_t)$ , whose joint dynamic for any initial condition $(t,x,r) \\in [0,T] \\times \\mathbb {R} \\times \\mathbb {R}$ and $\\forall s \\in [t,T]$ is given by $\\!\\!",
"{\\left\\lbrace \\begin{array}{ll}X_s =X_t +\\!",
"\\int _t^s(r_v\\!-\\frac{\\sigma ^2}{2})dv+ \\sigma \\Big [\\rho (B_s^1\\!-B^1_t)+\\sqrt{1\\!-\\!\\rho ^2}(B^2_s-B_t^2)\\Big ], \\,\\,\\, X_t=x\\\\r_s = r_t + \\int _t^s\\mu (v, r_v )dv+ \\int _t^s\\eta (v,r_v)dB^1_v,\\quad r_t=r,\\end{array}\\right.",
"}$ where $(B^1, B^2)$ is a two dimensional standard Brownian motion and $\\rho \\in (-1,1)$ .",
"Moreover, we assume that the deterministic functions $\\mu (\\cdot ,\\cdot )$ and $\\eta (\\cdot ,\\cdot )$ are in a class that ensures the existence and uniqueness of a strong solution of (REF ) (see e.g.",
"[10]) and that $Q$ is some risk neutral probability selected by the market.",
"Under these assumptions, the pair $(X_t, r_t)$ is Markovian, whence the arbitrage-free option's price is a deterministic function of the state variables, given by $u(t, x,r, T;\\rho )=\\mathbb {E}(\\mathrm {e}^{-\\int _t^Tr_s}(\\mathrm {e}^{X_T(\\rho )}-\\mathrm {e}^{\\kappa })^+ ds|X_t=x,r_t=r),$ provided that the coefficients $\\mu $ and $\\eta $ are chosen to guarantee the exponential integrability of $X_T$ and $\\int _0^T |r_s| ds$ .",
"Here we wrote $X_T(\\rho )$ , to stress the prices' dependence on the correlation parameter.",
"If $u(t, x,r,T;\\rho )$ is regular enough in $t,x,r$ , Feymann-Kac's formula implies that it is a classical solution of the following two-dimensional parabolic problem ${\\left\\lbrace \\begin{array}{ll}\\frac{\\partial u}{\\partial t}+{{\\mathcal {L}}^{\\rho }}u=0\\\\u(T,x,r,T;\\rho )=(\\mathrm {e}^x-\\mathrm {e}^{\\kappa })^+,\\end{array}\\right.",
"}$ where ${\\mathcal {L}}^{\\rho }= {\\mathcal {L}}^{\\mathbf {0}}+{\\mathcal {A}}$ , with ${\\mathcal {L}}^{\\mathbf {0}}&:=&\\biggl (\\frac{\\sigma ^2}{2}\\frac{\\partial ^2}{\\partial x^2}+(r-\\frac{\\sigma ^2}{2})\\frac{\\partial }{\\partial x}-r\\biggr )+\\biggl (\\frac{\\eta ^2(t,r)}{2}\\frac{\\partial ^2}{\\partial r^2}+\\mu (t,r)\\frac{\\partial }{\\partial r}\\biggr )\\\\{\\mathcal {A}}&:=&\\rho \\sigma \\eta (t,r)\\frac{\\partial ^2}{\\partial x\\partial r}.$ In what follows to keep notation easy, we take $t=0$ and we omit the dependence on $t$ in the pricing function.",
"The general case may be readily obtained substituting $T$ in the final formulas with the time to maturity $T-t$ .",
"By conditioning internally with respect to ${\\mathcal {F}}^1_T=\\sigma (\\lbrace B^1_s:0\\le s\\le T\\rbrace )$ , we have $u(x,r, T;\\rho )=\\mathbb {E}\\left(\\mathrm {e}^{-\\int _0^T r_s ds} (\\mathrm {e}^{X_T(\\rho )}-\\mathrm {e}^\\kappa )^+\\right) = \\mathbb {E}\\left(\\mathrm {e}^{-\\int _0^T r_s ds}\\mathbb {E}\\big ((\\mathrm {e}^{X_T(\\rho )}-\\mathrm {e}^\\kappa )^+|{\\mathcal {F}}^1_T\\big )\\right).$ But $X_T(\\rho )\\big |{\\mathcal {F}}^1_T\\sim N(M_T, \\Sigma _T)$ , where $M_T=x + \\int _0^T(r_s- \\frac{\\sigma ^2}{2})ds + \\sigma \\rho B_T^1,\\quad \\text{and}\\quad \\Sigma ^2_T = \\sigma (1-\\rho ^2)T.$ so we obtain $\\begin{aligned}\\mathbb {E}\\left((\\mathrm {e}^{X_T(\\rho )}-\\mathrm {e}^\\kappa )^+|{\\mathcal {F}}^1_T\\right) =& \\mathrm {e}^{M_T+\\frac{1}{2} \\Sigma _T^2} {\\mathcal {N}}\\left(\\frac{M_T-\\kappa +\\Sigma _T^2}{\\Sigma _T} \\right) - \\mathrm {e}^{\\kappa } {\\mathcal {N}}\\left(\\frac{M_T-\\kappa }{\\Sigma } \\right)\\\\= &\\mathrm {e}^{x + \\int _0^T(r_s- \\frac{\\sigma ^2}{2})ds + \\sigma \\rho B_T^1 +\\frac{1}{2} \\sigma ^2 (1-\\rho ^2)T} {\\mathcal {N}}(d_1(\\rho )) - \\mathrm {e}^{\\kappa } {\\mathcal {N}}(d_2(\\rho )),\\end{aligned}$ where we define $d_1(\\rho ) & = & \\frac{x-\\kappa + \\int _0^T r_s ds + \\sigma \\rho B^1_T + \\frac{\\sigma ^2}{2} T - \\sigma ^2 \\rho ^2 T }{\\sigma \\sqrt{1-\\rho ^2} \\sqrt{T}} \\\\d_2(\\rho ) & = & \\frac{x-\\kappa + \\int _0^T r_s ds + \\sigma \\rho B^1_T - \\frac{\\sigma ^2}{2} T}{\\sigma \\sqrt{1-\\rho ^2} \\sqrt{T}}$ and ${\\mathcal {N}}$ denotes the cumulative distribution function of the standard Gaussian.",
"It is convenient to introduce the following notations $\\Lambda _T=\\int _0^Tr_sds,\\;\\;\\;\\beta (T,\\rho )=\\frac{\\rho }{(1-\\rho ^2)^{1/2}\\sqrt{T}},\\;\\,\\gamma (T,\\rho )=\\frac{1}{\\sigma (1-\\rho ^2)^{1/2}\\sqrt{T}}$ $\\alpha _1(x,T,\\rho )=\\frac{x-\\kappa +\\frac{\\sigma ^2}{2}T-\\sigma ^2\\rho ^2T}{\\sigma (1-\\rho ^2)^{1/2}\\sqrt{T}},\\;\\;\\;\\alpha _2(x,T,\\rho )=\\frac{x-\\kappa -\\frac{\\sigma ^2}{2}T}{\\sigma (1-\\rho ^2)^{1/2}\\sqrt{T}},$ so that $d_i(x,T,\\rho )=\\alpha _i(x,T,\\rho )+\\beta (T,\\rho )B_T^1+\\gamma (T,\\rho )\\Lambda _T, \\quad i=1,2.$ Setting $S_T=\\mathrm {e}^{-\\Lambda _T}$ , we can finally write $ u(x,r,T;\\rho )= \\mathrm {e}^x \\mathrm {e}^{- \\frac{\\sigma ^2 \\rho ^2}{2} T}\\mathbb {E}\\left(\\mathrm {e}^{\\sigma \\rho B_T^1 } {\\mathcal {N}}\\big (d_1(\\rho )\\big )\\right) - \\mathrm {e}^{\\kappa }\\mathbb {E}\\left(S_T {\\mathcal {N}}\\big (d_2(\\rho )\\big )\\right).$ In the forthcoming section we shall introduce the moment matching approximation procedure."
],
[
"Option price approximation by moment matching",
"The main idea of this section is to replace the r.v.",
"'s $d_i(\\rho )$ , $i=1,2$ , defined by (REF ), with Gaussian r.v.",
"'s $D_i(\\rho )$ matching the first and second moments of $d_i(\\rho )$ .",
"We define $D_i(\\rho ):= \\alpha _i(x,T,\\rho )+\\hat{\\beta }(T,\\rho )B_T^1+\\gamma (T,\\rho )\\mathbb {E}(\\Lambda _T) ,\\;\\;\\;i=1,2,$ consequently $\\mathbb {E}\\big (D_i(\\rho )\\big )= \\alpha _i(x,T,\\rho )+ \\gamma (T,\\rho )\\mathbb {E}(\\Lambda _T)= \\mathbb {E}\\big (d_i(\\rho )\\big )$ and the new coefficient $\\hat{\\beta }>0$ is fixed such that $\\begin{aligned}\\mathrm {var}\\big (D_i(\\rho )\\big )=&T\\hat{\\beta }^2(T,\\rho )=\\mathrm {var}(d_1(\\rho ))=\\mathrm {var}(d_2(\\rho ))\\\\=& \\beta ^2(T,\\rho ) T+ \\gamma ^2(T,\\rho )\\mathrm {var}(\\Lambda _T) + 2 \\beta (T,\\rho ) \\gamma (T,\\rho ) \\mathbb {E}(B_T^1 \\Lambda _T)\\end{aligned}$ with $ \\mathrm {var}(\\Lambda _T) &= &\\mathbb {E}\\left( \\Big (\\int _0^T r_s ds\\Big )^2\\right)- \\Big [\\mathbb {E}\\left(\\int _0^T r_s ds\\right)\\Big ]^2, \\\\\\mathbb {E}(B_T^1 \\Lambda _T)&= &\\mathbb {E}\\left(B_T^1 \\int _0^T r_s ds\\right).$ The moment matching method with Gaussian r.v's may be motivated by looking at the empirical distributional properties of the random variables $d_i$ in some well-known rate models: see as examples Figs (REF ), (REF ) and (REF ).",
"We finally introduce a call price approximation $ \\begin{aligned}u^{appr}(x,r,T;\\rho ):= &\\mathrm {e}^x \\mathrm {e}^{- \\frac{1}{2} \\sigma ^2 \\rho ^2 T}\\mathbb {E}\\left(\\mathrm {e}^{\\sigma \\rho B_T^1 } {\\mathcal {N}}\\big (D_1(\\rho )\\big )\\right) - \\mathrm {e}^{\\kappa }\\mathbb {E}\\left(S_T {\\mathcal {N}}\\big (D_2(\\rho )\\big )\\right)\\\\=: &\\mathrm {e}^x \\mathrm {e}^{- \\frac{1}{2} \\sigma ^2 \\rho ^2 T}\\ F(\\rho )- \\mathrm {e}^{\\kappa } G(\\rho ).\\end{aligned}$ The function $F$ can be evaluated in closed form by means of the following Lemma 1 Let $p\\in \\mathbb {\\mathbb {R}}$ and $X\\sim N(\\mu ,\\nu ^2)$ , $(\\mu ,\\nu )\\in \\mathbb {R}\\times \\mathbb {R}^+$ , then $\\mathbb {E}(\\mathrm {e}^{pX}{\\mathcal {N}}(X))=\\mathrm {e}^{p\\mu +\\frac{(p\\nu )^2}{2}}{\\mathcal {N}}\\biggl (\\frac{\\mu +p\\nu ^2}{\\sqrt{1+\\nu ^2}}\\biggr ).$ Proof: See [21] for $p=0$ , the general case follows by a “completing the squares\" argument.$\\square $ Since $B_T^1 = \\big [D_1(\\rho ) - \\alpha _1(x,T,\\rho ) -\\gamma (T,\\rho )\\mathbb {E}(\\Lambda _T)\\big ] \\hat{\\beta }(T,\\rho )^{-1},$ we may rewrite $F$ as $\\begin{aligned}F(\\rho ) = & \\mathbb {E}\\left(\\mathrm {e}^{\\sigma \\rho (D_1(\\rho ) - \\alpha _1(x,T,\\rho ) - \\gamma (T,\\rho ) \\mathbb {E}(\\Lambda _T))\\hat{\\beta }(T,\\rho )^{-1} } {\\mathcal {N}}(D_1(\\rho ))\\right)\\\\= & \\mathrm {e}^{- \\sigma \\rho [\\alpha _1(x,T,\\rho ) +\\gamma (T,\\rho )\\mathbb {E}(\\Lambda _T)] \\hat{\\beta }(T,\\rho )^{-1} } \\mathbb {E}\\left(\\mathrm {e}^{\\sigma \\rho D_1(\\rho ) \\hat{\\beta }(T,\\rho )^{-1}}{\\mathcal {N}}(D_1(\\rho ) )\\right) \\\\= & \\mathrm {e}^{- \\sigma \\rho [\\alpha _1(x,T,\\rho ) +\\gamma (T,\\rho )\\mathbb {E}(\\Lambda _T)] \\hat{\\beta }(T,\\rho )^{-1} } \\mathrm {e}^{\\sigma \\rho \\mathbb {E}(D_1(\\rho ))\\hat{\\beta }(T,\\rho )^{-1} +\\frac{\\hat{\\beta }(T,\\rho )^{-2} \\sigma ^2 \\rho ^2\\mathrm {var}(D_1(\\rho ))}{2} } \\\\& \\times {\\mathcal {N}}\\Big (\\frac{\\mathbb {E}(D_1(\\rho ))+\\sigma \\rho \\mathrm {var}(D_1(\\rho ))\\hat{\\beta }(T,\\rho )^{-1}}{\\sqrt{1+\\mathrm {var}(D_1(\\rho ))}}\\Big ).\\end{aligned}$ From (REF ) and (REF ), we may conclude $ F(\\rho ) = \\mathrm {e}^{\\frac{\\sigma ^2 \\rho ^2 T}{2} } {\\mathcal {N}}\\left(\\frac{\\alpha _1(x,T,\\rho ) + \\sigma \\rho \\hat{\\beta }(T,\\rho ) T+ \\gamma (T,\\rho ) \\mathbb {E}(\\Lambda _T)}{\\sqrt{1+\\hat{\\beta }^2(T,\\rho ) T}} \\right).$ If $\\mathbb {E}(\\Lambda _T)$ and (REF ), () can be computed, then $F$ is totally explicit.",
"From now on, we denote $\\lambda (T):=\\mathbb {E}(\\Lambda _T)$ to point out this is a known constant.",
"On the contrary, the function $G$ cannot be evaluated in such a straightforward manner, as it involves a detailed knowledge of the joint distribution of $\\Lambda _T $ and $B^1_T$ and not only of their moments and covariance.",
"So, to represent $G$ , we suggest applying a a change-of-numeraire technique that allows us to exploit the bond pricing theory.",
"Let us define $ P(s,T):=\\mathbb {E}\\left((\\mathrm {e}^{-\\int _s^Tr_v dv}|{\\mathcal {F}}_s \\right),$ the Zero Coupon Bond price.",
"Again, since $r.$ is a Markov process, $P(s,T)$ is a deterministic function of the state variable, say $g(s,r_s)$ , which we assume to be ${\\mathcal {C}}^{1,2}([0,T]\\times \\mathbb {R}^+)$ .",
"For $0\\le s\\le T$ , we define the ${\\mathcal {F}}_s$ -martingale (we remark that is a true martingale thanks to the exponential integrability of $\\Lambda _T$ ) $L_s= \\frac{\\mathbb {E}(\\mathrm {e}^{-\\int _0^T r_v dv}|{\\mathcal {F}}_s)}{P(0,T)}= S_s\\frac{P(s,T)}{P(0,T)} = S_s\\frac{g(s,r_s)}{g(0,r)}, \\quad r_0=r.$ By applying Itô's formula, we have the dynamic of $L$ $\\begin{aligned}dL_s=&\\frac{S_s}{g(0,r)}\\Big [ \\frac{\\partial g}{\\partial t}(s,r_s)+ \\frac{1}{2} \\eta ^2(s,r_s) \\frac{\\partial ^2 g}{\\partial r^2} (s,r_s)+ \\mu (s,r_s) \\frac{\\partial g}{\\partial r} (s,r_s)-r_s g(s,r_s)\\Big ]ds\\\\+ &\\frac{S_s}{g(0,r)}\\eta (s,r_s) \\frac{\\partial g}{\\partial r} (s,r_s)dB^1_s=\\frac{S_s}{g(0,r)}\\eta (s,r_s) \\frac{\\partial g}{\\partial r}(s,r_s) dB^1_s, \\quad L_0=1\\end{aligned}$ and we may define the $T$ -forward measure on every $A\\in {\\mathcal {F}}$ by $Q^T(A):=\\mathbb {E}(L_T1_A)$ (see [4] for the method and [6] for a similar application).",
"Under $Q^T$ , we get $G(\\rho )=\\mathbb {E}\\left(S_T{\\mathcal {N}}(D_2(\\rho ))\\right)=P(0,T)\\mathbb {E}^{Q^T}\\left({\\mathcal {N}}(D_2(\\rho ))\\right),$ and by Girsanov theorem, by setting $\\xi _s:=\\int _0^s \\frac{\\eta (v,r_v) }{g(v,r_v)} \\frac{\\partial g }{\\partial r}(v,r_v) dv,$ we have that the process $\\tilde{B}^1_s:= B^1_s - \\xi _s$ is a $Q^T-$ Brownian motion.",
"When choosing an interest rate model that allows an explicit expression of the bond's price, $\\mathbb {E}^{Q^T}\\left({\\mathcal {N}}(D_2(\\rho ))\\right)$ will be the last quantity to compute.",
"Under $Q^T$ , $D_2(\\rho )$ has the expression $D_2(\\rho )=\\alpha _2(x,T,\\rho ) + \\xi _T \\hat{\\beta }(T,\\rho )+\\hat{\\beta }(T,\\rho ) \\tilde{B}_T^1 + \\gamma (T,\\rho ) \\lambda (T),$ whence its distribution is no longer known.",
"To compute the final expectation, we replace $D_2(\\rho )$ by the r.v.",
"$\\bar{D}_2(\\rho ):=\\alpha _2(x,T,\\rho ) + \\mathbb {E}(\\xi _T) \\hat{\\beta }(T,\\rho )+\\hat{\\beta }(T,\\rho ) \\tilde{B}_T^1 + \\gamma (T,\\rho ) \\lambda (T),$ where we are taking $\\epsilon (T):= \\mathbb {E}(\\xi _T) $ under the original probability $Q$ , so that $\\bar{D}_2(\\rho )$ is a Gaussian r.v.",
"and we may apply Lemma 1 once again to obtain $\\mathbb {E}^{Q^T}({\\mathcal {N}}(\\bar{D}_2(\\rho ))) = {\\mathcal {N}}\\left( \\frac{\\mathbb {E}^{Q^T}(\\bar{D}_2(\\rho ))}{\\sqrt{1+\\mathrm {var}^{Q^T}(\\bar{D}_2(\\rho ))}} \\right),$ with $\\mathbb {E}^{Q^T}(\\bar{D}_2(\\rho ))=\\alpha _2(x,T,\\rho ) +\\epsilon (T)\\hat{\\beta }(T,\\rho ) + \\gamma (T,\\rho ) \\lambda (T), \\ \\ \\mathrm {var}^{Q^T}(\\bar{D}_2(\\rho ))=\\hat{\\beta }^2(T,\\rho ) T.$ Hence we shall denote by $\\bar{G}(\\rho ):=P(0,T)\\mathbb {E}^{Q^T}({\\mathcal {N}}(\\bar{D}_2(\\rho )))$ the approximation of $G(\\rho )$ and we may define the final approximation of the call option price $u(x,r,T;\\rho )$ as $\\nonumber & &\\bar{u}(x,r,T;\\rho ) := \\mathrm {e}^{x- \\frac{1}{2} \\sigma ^2 \\rho ^2 T}F(\\rho ) - \\mathrm {e}^{\\kappa }\\bar{G}(\\rho ) \\nonumber \\\\&= &\\mathrm {e}^x {\\mathcal {N}}\\left(\\frac{\\alpha _1(x,T,\\rho )+ \\sigma \\rho \\hat{\\beta }(T,\\rho ) T+ \\gamma (T,\\rho ) \\lambda (T) }{\\sqrt{1+\\hat{\\beta }^2(T,\\rho ) T}}\\right)\\\\\\nonumber &- & \\mathrm {e}^{\\kappa } P(0,T) {\\mathcal {N}}\\left( \\frac{\\alpha _2(x,T,\\rho ) + \\epsilon (T) \\hat{\\beta }(T,\\rho ) +\\gamma (T,\\rho ) \\lambda (T)}{\\sqrt{1+\\hat{\\beta }^2(T,\\rho ) T}} \\right).$ As a conclusion, we summarize the key requirements to make the approximation (REF ) explicitly computable and hopefully efficient the distributions of $d_i(\\rho ), i=1,2$ should be close to a Gaussian distribution; the bond price $P(t,T)$ should be theoretically computable.",
"Moreover one can exploit the observed (today) bond price for $P(0,T)$ in (REF ) and for calibration purposes; the quantities $\\mathbb {E}(\\Lambda _T)$ , $\\mathrm {var}(\\Lambda _T)$ and $\\mathbb {E}(\\Lambda _TB^1_T)$ and/or their approximations, should be easily computable; the change of numeraire technique (Girsanov's theorem) should be applicable.",
"The performance of this approximation needs to be compared with Monte-Carlo simulated prices and then with other methods present in the literature.",
"This will be done in the next section."
],
[
"Numerics and comparison with other methodologies",
"In this section we employ an affine model for the interest rate.",
"This choice provides an explicit expression for the the ZCB's price (REF ).",
"So our market model is given by $\\begin{aligned}X_s =&X_t +\\!\\!",
"\\int _t^s(r_v- \\frac{\\sigma ^2}{2})dv+ \\sigma \\Big [\\rho (B_s^1\\!-B^1_t)+\\sqrt{1\\!-\\!\\rho ^2}(B^2_s-B_t^2)\\Big ], \\quad X_t =x\\\\r_s = &r_t +\\!\\!",
"\\int _t^s[a(v) r_v + b(v)]dv+ \\int _t^s[c(v) r_v + d(v)]^{1/2}dB^1_v,\\quad r_t=r,\\end{aligned}$ where $a,b,c,d:[0,T]\\longrightarrow \\mathbb {R}$ are bounded functions.",
"In this framework, we have for $r_t=r$ $P(t,T)= g(t,r)=A(t,T)\\mathrm {e}^{-r B(t,T)},$ for suitable deterministic functions $A(\\cdot ,T)$ and $B(\\cdot ,T)$ .",
"Two very classical models fall into this setting $\\begin{aligned}\\text{(Vasicek) }\\quad &a(v) = -\\gamma ,\\quad b(v) = \\gamma \\theta , \\quad c(v)=0, \\quad d(v) = \\eta ^2\\\\\\text{(CIR) }\\quad &a(v) = -\\gamma ,\\quad b(v) = \\gamma \\theta , \\quad c(v)= \\eta ^2, \\quad d(v) = 0\\end{aligned}\\quad \\gamma , \\theta >0,$ for which $A(t,T)$ and $B(t,T)$ are explicitly known ([5]), the same being true also for the Hull-White / Vasicek and Hull-White / CIR models, considering time dependent coefficients.",
"The functions $A$ and $B$ are usually characterized by the solution of a Riccati system of ODE's.",
"Unfortunately, when in presence of correlation, the same procedure cannot be applied to the pair $(X.,r.",
")$ , since its diffusion matrix $\\sigma (v,x,r)\\sigma (v,x,r)^T=\\begin{pmatrix}\\sigma ^2 & \\rho \\sigma [c(v) r + d(v)]^{1/2}\\\\\\rho \\sigma [c(v) r + d(v)]^{1/2}& c(v) r + d(v)\\end{pmatrix}$ may haves entries which are non-linear in the state variables, so that the joint diffusion is no longer affine, as it happens for the CIR model.",
"Hence, in this context it makes sense to apply the approximation procedure presented in the previous section.",
"As before we consider $t=0$ .",
"In this case (see e.g.",
"[5]), setting $ \\delta =\\sqrt{\\gamma ^2+2\\eta ^2}$ , we have $A(0,T)=\\mathrm {e}^{\\frac{2\\gamma \\theta }{\\eta ^2}}\\frac{2\\delta \\mathrm {e}^{\\gamma +\\delta T}}{\\delta -\\gamma +(\\delta +\\gamma )\\mathrm {e}^{\\delta T}},\\quad B(0,T)=\\frac{2(\\mathrm {e}^{\\delta T}-1)}{\\delta -\\gamma +(\\delta +\\gamma )\\mathrm {e}^{\\delta T}},$ and let us proceed to the computation of $\\mathbb {E}(\\Lambda _T)$ , $\\mathrm {var}(\\Lambda _T)$ and $\\mathbb {E}(\\Lambda _TB^1_T)$ .",
"Computation of $\\mathbb {E}(\\Lambda _T) $.",
"It is straightforward to see $\\mathbb {E}(\\Lambda _T) = \\int _0^T\\mathbb {E}(r_s) ds=\\int _0^T \\big [(r_0-\\theta )\\mathrm {e}^{-\\gamma s}+ \\theta \\big ] ds= \\theta T +(r_0-\\theta )\\frac{1- \\mathrm {e}^{-\\gamma T}}{\\gamma }.$ Computation of $\\mathrm {var}(\\Lambda _T) $ .",
"Taking into account the first point, we only have to compute the second moment $\\begin{aligned}\\mathbb {E}\\left(\\Big (\\!\\!",
"\\int _0^T \\!\\!\\!r_s ds\\Big )^2\\right)=& \\mathbb {E}\\left( \\int _0^T\\!\\!\\!\\int _0^T \\!\\!\\!r_s r_vds dv\\right)= \\int _0^T\\!\\!\\!\\int _0^T\\mathbb {E}( r_s r_v) ds dv\\\\=&\\int _0^T\\!\\!\\!\\int _0^t\\mathbb {E}( r_s r_v) ds dv+\\int _0^T\\!\\!\\!\\int _t^T\\mathbb {E}( r_s r_v) ds dv\\\\=&\\int _0^T\\!\\!\\!\\int _0^t\\!\\!\\!\\mathbb {E}( r_s r_v) ds dv+\\!\\!\\int _0^T\\!\\!\\!\\int _0^s\\!\\!\\!\\mathbb {E}( r_s r_v) dv ds= 2\\!\\!\\int _0^T\\!\\!\\!\\int _0^s\\!\\!\\!\\mathbb {E}( r_s r_v) dv ds.\\end{aligned}$ By the independence of the increments of the process $r$ , for $v<s$ we have $\\begin{aligned}&\\mathbb {E}( r_sr_v) =\\mathbb {E}\\left(( r_s- r_v) r_v+ r^2_v \\right)=\\mathbb {E}( r_s- r_v) \\mathbb {E}( r_v) + \\mathbb {E}( r_v^2)\\\\=&\\mathbb {E}\\Big (\\theta (s-v) +(r_v-\\theta )\\frac{1- \\mathrm {e}^{-\\gamma (s-v)}}{\\gamma }\\Big )\\mathbb {E}( r_v)+ \\mathbb {E}( r_v^2)\\\\=&\\theta \\Big [ (s-v)- \\frac{1- \\mathrm {e}^{-\\gamma (s-v)}}{\\gamma }\\Big ]\\mathbb {E}( r_v)+\\frac{1- \\mathrm {e}^{-\\gamma (s-v)}}{\\gamma }[\\mathbb {E}( r_v)]^2+ \\text{var} (r_v)+ [\\mathbb {E}( r_v)]^2\\end{aligned}$ Since $\\mathrm {var}(r_s) = r_0 \\frac{\\eta ^2}{\\gamma }(\\mathrm {e}^{-\\gamma s}-\\mathrm {e}^{-2 \\gamma s})+\\frac{\\theta \\eta ^2}{2 \\gamma }(1-\\mathrm {e}^{-\\gamma s})^2$ , all the integrals appearing in the second moment can be calculated analytically.",
"Computation of $\\mathbb {E}(B_T^1 \\Lambda _T)$.",
"By Itô's integration-by-parts formula, we get $\\mathbb {E}(B_T^1 \\Lambda _T)= \\int _0^T \\mathbb {E}(B_s^1 r_s ) ds$ and again by integration by parts we have $\\begin{aligned}B^1_s r_s =&\\int _0^s B^1_v dr_v+\\int _0^s r_v dB^1_v + \\langle B^1, r\\rangle _s\\\\=& \\int _0^s B^1_v \\gamma (\\theta -r_v) dt+ \\eta \\int _0^s B^1_t \\sqrt{r_v} dB^1_v + \\int _0^s r_v dB^1_v+ \\eta \\int _0^s \\sqrt{r_v} dv,\\end{aligned}$ so that $\\mathbb {E}(B^1_s r_s ) = -\\gamma \\int _0^s \\mathbb {E}(B^1_v r_v) dv + \\eta \\int _0^s \\mathbb {E}(\\sqrt{r_v}) dv.$ Solving this linear ODE for $h(s):=\\mathbb {E}(B^1_s r_s)$ , since $h(0)=0$ , we obtain $\\begin{aligned}\\mathbb {E}(B^1_s r_s) =& \\eta \\int _0^s \\mathrm {e}^{-\\gamma (s- v)} \\mathbb {E}(\\sqrt{r_v}) dv,\\\\\\mathbb {E}(B_T^1 \\Lambda _T)=& \\eta \\int _0^T \\int _0^s \\mathrm {e}^{-\\gamma (s- v)} \\mathbb {E}(\\sqrt{r_v}) dv ds.\\end{aligned}$ Thus the final crucial point is computing $ \\mathbb {E}(\\sqrt{r_v})$ , which is rather delicate (see [8]).",
"No explicit expression can be provided and we employ the approximation proposed in [9], that we are going to present in the next subsection $ \\mathbb {E}(\\sqrt{r_v}) \\approx a + b \\mathrm {e}^{-c v}$ where the parameters $a, b$ and $c$ are obtained by an ad hoc matching procedure, which proved to be numerically very efficient.",
"Finally, given the above three points, $\\hat{\\beta }(T,\\rho )$ is easily computed from (REF ).",
"As a last step, we have to approximate $\\mathbb {E}(\\xi _T)$ , which is readily done, given the last remark, since $\\begin{aligned}\\mathbb {E}(\\xi _T) = &\\mathbb {E}\\Big (- \\eta \\int _0^T\\!\\!\\!B(s,T) \\sqrt{r_s} ds\\Big )= \\mathbb {E}\\Big (- \\eta \\int _0^TB(s,T) \\mathbb {E}\\big (\\sqrt{r_s}\\big ) ds\\Big )\\\\\\approx & - \\eta \\int _0^TB(s,T) (a + b \\mathrm {e}^{-c s}) ds.\\end{aligned}$"
],
[
"The Grzelak-Oosterlee (GO) approximation",
"Here and in the next subsection, for completeness, we briefly describe the two approximation techniques, we are going to compare with.",
"The GO approximation consists simply in modifying the ${\\mathcal {A}}$ operator given in () by replacing the state variable in the coefficient with a constant, namely we define ${\\mathcal {A}}^{GO}u (s,x,r):=\\rho \\sigma \\mathbb {E}( \\eta (s, r_ts)\\frac{\\partial ^2u}{\\partial x\\partial r}.$ In the case of the CIR model, $\\eta $ is time-homogeneous and this operator becomes ${\\mathcal {A}}^{GO}u (s,x,r):=\\rho \\sigma \\eta \\mathbb {E}(\\sqrt{r_s}))\\frac{\\partial ^2u}{\\partial x\\partial r}\\approx \\rho \\sigma \\eta (a + b \\mathrm {e}^{-c s}) \\frac{\\partial ^2u}{\\partial x\\partial r}$ Once this replacement has been made then the Fourier transform methods apply, hence it is possible to compute approximated prices of the call option.",
"We shall denote this approximation by $u^{GO}(t,x,r,T;\\rho )$ .",
"To evaluate the accuracy of this approximation a comparison with the prices of the (non-affine) true model, obtained by MC simulations, must be performed.",
"Once again we specialize the formulas for $t=0$ for a direct comparison with our results, so $X_0=x$ and $r_0=r$ .",
"The discounted transform, for $\\zeta \\in , (see \\cite {DPS00}) for the affine approximation is$$\\phi (\\zeta , x, r, T) := \\mathbb {E}\\left(\\mathrm {e}^{-\\int _0^T r_s ds} \\mathrm {e}^{\\zeta X_T}\\right) = \\mathrm {e}^{A(\\zeta ,T)+B(\\zeta ,T)x + C(\\zeta ,T)r},$$where the functions $ A,B,C$ satisfy a system of solvable ODE^{\\prime }s, that give$$\\begin{aligned}B(\\zeta ,T) =& \\zeta ,\\\\C(\\zeta ,T) =& \\frac{1-\\mathrm {e}^{-d T}}{\\eta ^2(1-g \\mathrm {e}^{-dT})}, \\ \\ d=\\sqrt{\\gamma ^2+2 \\eta ^2(1-\\zeta )}, \\ \\ g=\\frac{\\gamma -d}{\\gamma +d},\\\\A(\\zeta ,T) = & -\\frac{\\sigma ^2}{2}T \\zeta (1+\\zeta ) + \\frac{\\gamma -d}{\\eta }\\int _0^T\\Big [\\frac{\\gamma \\theta }{\\eta } + \\rho \\sigma \\zeta \\overline{r^{sq}_s}\\Big ]\\frac{1-\\mathrm {e}^{-d s}}{(1-g \\mathrm {e}^{-d s})} ds,\\end{aligned}$$where $rsqs= E(rs)$ is approximated as in (\\ref {mean_sqrt}).$ Finally, by Lévy inversion formula as in [7] or Fourier inversion as in [13], one gets an integral representation for the price function: in our implementation we use the Fourier inversion $u^{GO}(x,r,T;\\rho ) = \\frac{\\mathrm {e}^{\\nu \\gamma }}{\\pi } \\int _0^{+\\infty } \\mathcal {R}\\left(\\frac{\\mathrm {e}^{-\\mathrm {i}\\zeta \\gamma }}{\\nu ^2-\\nu - \\zeta ^2+\\mathrm {i}\\zeta (1-2\\nu )} \\phi (\\zeta , x, r,T) \\right) d\\zeta ,$ where $\\nu <0$ is a dumping factor and $\\mathcal {R}(z)$ is the real part for $z \\in .$"
],
[
"The Kim-Kunimoto (KK) approximation",
"Kim and Kunimoto, in [11], consider a Taylor expansion of the process $r_s$ in powers of $\\eta $ around $\\eta =0$ .",
"Considering the first order polynomial and setting $\\varphi (s)=r \\exp (-\\gamma s)+\\theta (1- \\exp (-\\gamma s))$ , they obtain $r_s=\\varphi (s)+\\eta \\int _0^s \\mathrm {e}^{-\\gamma (s-v)}\\sqrt{\\varphi (v)}(\\rho dB^1_v+\\sqrt{1-\\rho ^2}dB^2_v)+o(\\eta ).$ Inserting the approximation (REF ) in the evaluation formula for the call option, after some manipulations one can approximate the option's price as $\\begin{aligned}u^{KK}(x,r,T;\\rho ) =& \\mathrm {e}^x {\\mathcal {N}}(d_1)- \\mathrm {e}^{\\kappa -\\int _0^T \\varphi (s) ds} {\\mathcal {N}}(d_2) \\\\+ & \\eta C_1 \\Big [d_2 \\mathrm {e}^x {\\mathcal {N}}^{\\prime }(d_1)-d_1 \\mathrm {e}^{\\kappa -\\int _0^T \\varphi (s) ds} {\\mathcal {N}}^{\\prime }(d_2) \\Big ]\\end{aligned}$ where $\\begin{aligned}C_1=& -\\frac{\\rho }{\\sigma T} \\frac{2 \\sqrt{\\theta } \\big [(1+2 \\mathrm {e}^{\\gamma T}) \\sqrt{r}-3\\gamma _K\\big ]+\\big [r-\\theta (1+2 e^{\\gamma T})\\big ] \\lambda _K}{2 \\mathrm {e}^{\\gamma T} \\gamma ^2 \\sqrt{\\theta }},\\\\d_1 =&\\frac{x-\\kappa +\\theta T+(r-\\theta ) (1-\\mathrm {e}^{-\\gamma T})/\\gamma +\\sigma ^2 T/ 2}{\\sqrt{\\sigma ^2 T}}, \\ \\ d_2=d_1-\\sigma \\sqrt{T},\\end{aligned}$ being $\\gamma _K=\\mathrm {e}^{\\gamma T/2} \\sqrt{r-\\theta (1-\\mathrm {e}^{\\gamma T})}$ and $\\lambda _K=\\log \\left(\\frac{ (\\sqrt{r}+\\sqrt{\\theta })^2}{r-\\theta (1-2 \\mathrm {e}^{\\gamma T})+2 \\gamma _K \\sqrt{\\theta })} \\right)$ ."
],
[
"Numerical results",
"We compare the results of the different approximations with the benchmark Monte Carlo method, applied to the price (REF ).",
"In particular this means that we only have to simulate the rate process to get samples from $d_1(\\rho )$ and $d_2(\\rho )$ .",
"The simulation was implemented by means of the Euler discretization with full truncation algorithm (see [14]).",
"In our numerical experiments we generated $M=10^6$ sample paths with a time step discretization equal to $10^{-3}$ for all the maturities.",
"All the algorithms were implemented in MatLab (R2019b) and ran on an Intel Core i7 2.40GHZ with 8GB RAM, by using the available building-in functions, in particular for the computation of all the integrals involved.",
"The average time to compute one price was (in secs) $32.1$ (MC), $0.055$ (GO), $0.005$ (KK) and $0.009$ (MM).",
"We chose different set of parameters $(\\kappa , \\theta , \\eta )$ and volatility scenarios: a low volatility $\\sigma _L=0.2$ and a high volatility $\\sigma _L=0.4$ ; hence we varied the correlation $\\rho $ , the rate volatility $\\eta $ and the maturity of the contract $T$ .",
"The initial price of the underlying was set to 100 as well as the strike price $K$ .",
"Numerical results are summarized in Tables (REF ) - (REF ).",
"At least in the CIR model, the numerical results show that the MM method produces the best approximations with respect to the benchmark Monte Carlo evaluation in most scenarios."
]
] | 2005.14063 |
[
[
"Quantum electromechanics with levitated nanoparticles"
],
[
"Abstract Preparing and observing quantum states of nanoscale particles is a challenging task with great relevance for quantum technologies and tests of fundamental physics.",
"In contrast to atomic systems with discrete transitions, nanoparticles exhibit a practically continuous absorption spectrum and thus their quantum dynamics cannot be easily manipulated.",
"Here, we demonstrate that charged nanoscale dielectrics can be artificially endowed with a discrete level structure by coherently interfacing their rotational and translational motion with a superconducting qubit.",
"We propose a pulsed scheme for the generation and read-out of motional quantum superpositions and entanglement between several levitated nanoparticles, providing an all-electric platform for networked hybrid quantum devices."
],
[
"Quantum electromechanics with levitated nanoparticles Lukas Martinetz Klaus Hornberger University of Duisburg-Essen, Faculty of Physics, Lotharstraße 1, 47048 Duisburg, Germany James Millen King's College London, Department of Physics, Strand, London WC2R2LS, United Kingdom M. S. Kim Imperial College London, Quantum Optics and Laser Science, Exhibition Road, London SW72AZ, United Kingdom Benjamin A. Stickler Imperial College London, Quantum Optics and Laser Science, Exhibition Road, London SW72AZ, United Kingdom University of Duisburg-Essen, Faculty of Physics, Lotharstraße 1, 47048 Duisburg, Germany Preparing and observing quantum states of nanoscale particles is a challenging task with great relevance for quantum technologies and tests of fundamental physics.",
"In contrast to atomic systems with discrete transitions, nanoparticles exhibit a practically continuous absorption spectrum and thus their quantum dynamics cannot be easily manipulated.",
"Here, we demonstrate that charged nanoscale dielectrics can be artificially endowed with a discrete level structure by coherently interfacing their rotational and translational motion with a superconducting qubit.",
"We propose a pulsed scheme for the generation and read-out of motional quantum superpositions and entanglement between several levitated nanoparticles, providing an all-electric platform for networked hybrid quantum devices.",
"Introduction Opto- and electromechanical systems are at the cutting edge of modern quantum devices [1], [2], [3], with great potential for technological application and fundamental tests [4], [5], [6].",
"Optically levitating nanoscale objects almost perfectly isolates them from their surroundings, enabling superior force sensitivity and coherence times [7].",
"Levitated nanoparticles have been successfully cooled into their motional quantum groundstate [8], opening the door to free-fall quantum experiments [9], [10], [11].",
"Quantum experiments with trapped nanoparticles require schemes to coherently control their rotational and translational quantum states.",
"Continuous-wave optical techniques are limited by the detrimental impact of photon scattering decoherence [12] and by internal heating due to photon absorption [13], [14].",
"In addition, the fact that nanoscale particles lack the discrete internal spectrum of atoms or other microscopic quantum systems makes it difficult to address them coherently with laser pulses.",
"Figure: The rotational and translational motion of a charged nanoparticle (blue) levitated in a Paul trap induces an electrical current between the superconducting endcap electrodes.",
"The latter can be coherently interfaced with a charge qubit formed by a superconducting island (red).",
"This Cooper-pair box can be used to generate and read-out spatial superpositions of the nanoparticle.",
"The system constitutes the basic element for networking levitated nanoparticles into hybrid quantum devices based on superconducting circuitry.Here, we demonstrate that such a beneficial discrete level structure can be artificially introduced by coherently interfacing a charged nanoparticle levitated in a Paul trap with a superconducting qubit, through which its quantum dynamics can be manipulated and read out (see Fig.",
"REF ).",
"In this all-electrical setup, the nanoparticle rotations decouple from the center-of-mass dynamics under experimentally realistic conditions, rendering it ideally suited for superposition experiments with a wide variety of particle geometries and charge distributions.",
"This paves the way for networking nanoscale objects with superconducting quantum technologies.",
"Quadrupole ion traps provide exceptionally stable confinement for charged nanoparticles [15].",
"Moreover, the particle motion induces an electric current in the endcap electrodes.",
"We propose to use this current for first cooling the nanoparticle to milliKelvin temperatures and then interfacing its motion with a superconducting circuit.",
"The resulting coupling between superconductor and particle scales as charge over root mass [16], [17] and can thus be as strong as for a single atomic ion for realistic charge distributions.",
"We show that the proposed all-electrical platform is ideally suited for nanoparticle cooling and interference experiments and for generating and reading-out entanglement between several particles and superconducting qubits, thus forming a building block of a larger quantum network.",
"The motional quantum state can be prepared and observed by qubit manipulations with an ultra-fast pulse scheme, operating on a timescale much shorter than the mechanical period and within the coherence time of the charge qubit.",
"This pulse scheme allows to speed-up the observation of nanoscale quantum interference in a variety of opto- and electromechanical setups [18], [19], [20], [21].",
"Ro-translational macromotion A charged nanoparticle is suspended in a hyperbolic Paul trap of endcap distance $2 z_0$ and radius $\\sqrt{2}z_0$ , where the ring electrode is put to the time-dependent potential $U_{\\rm PT}(t)=U_{\\rm dc} + U_{\\rm ac}\\cos (\\Omega _{\\rm ac}t)$ with respect to the floating endcaps, see Fig.",
"REF .",
"Due to the quadrupole symmetry of the electric field, the rotational and translational particle motion is fully determined by its total charge $q$ , orientation-dependent dipole vector ${\\text{$p$}}(\\Omega )$ , and quadrupole tensor ${\\sf Q}(\\Omega )$ .",
"Here, $\\Omega $ denotes the orientational degrees of freedom of the particle, e.g.",
"parametrized by Euler angles; its center-of-mass position is ${\\bf r}$ .",
"In general, the resulting time-dependent force and torque will lead to complicated and unstable dynamics of the nanoparticle.",
"However, if the trap is driven sufficiently fast, its micro-motion can be separated, and one obtains a time-independent effective trapping potential for the macro-motion (see Methods) $V_{\\rm eff}({\\bf r}, \\Omega ) & =&\\frac{U_{\\rm dc}}{2 z_0^2}\\left(\\frac{q}{2}\\mathbf {r}\\cdot \\mathsf {A}\\mathbf {r}+\\text{$p$}\\cdot \\mathsf {A}\\mathbf {r}-\\frac{1}{2}\\mathbf {e}_z\\cdot \\mathsf {Q}\\mathbf {e}_z\\right) \\\\& &+\\frac{U_{\\rm ac}^2}{16 z_0^4\\Omega _{\\rm ac}^2} \\sum _{i = 1}^3 \\frac{1}{I_i} \\left[ {\\bf n}_i \\cdot \\left(\\text{$p$}\\times \\mathsf {A}\\mathbf {r}+\\mathbf {e}_z\\times \\mathsf {Q}\\mathbf {e}_z\\right) \\right]^2 \\\\& &+\\frac{U_{\\rm ac}^2}{16 M z_0^4\\Omega _{\\rm ac}^2}\\left(q\\mathbf {r}+\\text{$p$}\\right)\\cdot \\mathsf {A}^2\\left(q\\mathbf {r}+\\text{$p$}\\right).$ Here, $M$ is the particle mass and the Paul trap symmetry axis is aligned with ${\\bf e}_z$ , such that $\\mathsf {A}=\\mathbb {1}-3\\mathbf {e}_z\\otimes \\mathbf {e}_z$ .",
"The $I_i$ denote the moments of inertia with ${\\bf n}_i$ the associated directions of the rotor principal axes.",
"We dropped the orientation dependence of ${\\text{$p$}}$ , ${\\sf Q}$ , and ${\\bf n}_i$ for compactness.",
"The effective potential (REF ) describes the coupled rotational and translational macromotion of an arbitrarily charged and shaped nanoparticle in a quadrupole ion trap, and is thus pertinent for ongoing nanoparticle experiments [15], [21].",
"It shows that stable trapping can be achieved for sufficiently small bias voltages $U_{\\rm dc}$ (with frequencies $\\omega _z = q U_{\\rm ac}/ \\sqrt{2} M \\Omega _{\\rm ac}z_0^2$ and $\\omega _{x,y} = \\omega _z/2$ ).",
"In addition, the rotational and translational motion decouple for particles with vanishing dipole moment.",
"Note that if the particle has a finite quadrupole moment its rotation dynamics can still be strongly affected by the trapping field.",
"Particle-circuit coupling The rotational and translational motion of the particle induces mirror charges in the endcap electrodes.",
"The latter can be quantified by extending the Shockley-Ramo theorem to arbitrary charge distributions (see Methods), yielding the capacitor charge $Q=-k\\mathbf {e}_z\\cdot (q\\mathbf {r}+\\text{$p$})/z_0+CV$ , given the endcap capacitance $C$ and voltage drop $V$ .",
"The geometry factor $k$ , with values of $0 < k \\lesssim 1/2$ for realistic electrode geometries, determines the approximately homogeneous field $-k V/z_0 \\,{\\mathbf {e}}_z$ close to the trap center in absence of the particle.",
"(A perfect plate capacitor corresponds to $k=1/2$ .)",
"The induced capacitor charge only depends on the total motional dipole moment $q \\mathbf {r} + \\text{$p$}$ along the Paul trap axis, and is thus independent of the particle quadrupole moment.",
"A circuit connecting the electrodes picks up the ro-translational motion of the particle via the current $I = dQ/dt$ .",
"At the same time, the particle feels a voltage-dependent electrostatic force and torque depending on the circuit state.",
"This can be used for resistive cooling and for coherently interfacing the particle with a superconducting qubit.",
"Nanoparticle resistive cooling can be achieved by joining the endcaps with a resistance $R$ .",
"The dissipation of the induced current in the resistor leads to thermalization of the particle motion at the circuit temperature.",
"The timescale of this resistive cooling can be tuned by adding an inductance in series to the circuit [17].",
"In the adiabatic limit, it reacts almost instantaneously to the particle motion.",
"Thus, the circuit degrees of freedom can be expanded to first order in the particle velocity and rotation speed, yielding the effective total cooling rate $\\gamma _{\\rm ad}=\\frac{Rk^2}{z_0^2}\\left(\\frac{q^2}{M}+ \\sum _{i = 1}^3 \\frac{1}{I_i} \\left[ {\\bf n}_i \\cdot (\\text{$p$}\\times \\mathbf {e}_z) \\right]^2\\right).",
"$ This quantifies how fast an initially occupied phase-space volume contracts, predicting the timescale of rotational and translational thermalization with the circuit.",
"The rate (REF ) is always positive and exhibits a $q^2/M$ -scaling, indicating that charged nanoparticles can be cooled as efficiently as atomic ions.",
"Interfacing nanoparticle and charge qubit The levitated nanoparticle can be coherently coupled to a superconducting Cooper-pair box by attaching the latter to the endcap electrodes (see Fig.",
"REF ).",
"The nanoparticle motion towards the endcaps modifies the voltage drop over the Cooper-pair box, whose charge state determines the force and torque acting on the particle.",
"Preparing the Cooper-pair box in a superposition of charge states thus entangles the nanoparticle motion with the circuit.",
"This can be used to generate and verify nanoscale motional superposition states.",
"The combined nanoparticle-Cooper-pair box Hamiltonian can be derived in a lengthy calculation from Kirchhoff's circuit laws (see Methods).",
"Operating the Cooper-pair box in the charge qubit regime of $N$ and $N+1$ Cooper pairs yields the nanorotor-qubit coupling $H_{\\rm int} = - \\frac{2 e k}{C_\\Sigma z_0} (N + \\sigma _+ \\sigma _- ) (q {\\bf r} + \\text{$p$} )\\cdot {\\bf e}_z,$ where $C_\\Sigma $ is the effective capacitance of the circuit and the qubit raising and lowering operators are denoted by $\\sigma _+$ , $\\sigma _-$ .",
"This interaction couples the charge eigenstates of the box to the motional dipole moment of the nanorotor, implying that charge states are conserved by the interaction and that rotations of the quadrupole or higher multipole moments of the nanoparticle are not coupled by the qubit.",
"In the experimentally realistic situation that the nanoparticle is almost homogeneously charged and inversion symmetric, its dipole moment is negligibly small.",
"The rotational and translational macromotion in the Paul trap (REF ) then decouple even for large quadrupole moments.",
"The center-of-mass motion along the Paul trap axis further decouples from the transverse degrees of freedom, since only the motion towards the electrode is affected by the Cooper-pair box.",
"The particle trapping potential in $z$ -direction is slightly shifted and stiffened due to the charge qubit (with $N\\ne 0$ ), yielding the effective Hamiltonian $H_{\\rm 1D}=E_{\\rm c}\\sigma _+\\sigma _-+\\hbar \\omega a^\\dagger a -\\hbar \\kappa \\sigma _+\\sigma _-\\left(a + a^\\dagger \\right),$ with charge energy $E_{\\rm c}$ and coupling strength $\\kappa =2ekq/C_\\Sigma z_0\\sqrt{2M\\hbar \\omega }$ , where the nanoparticle oscillation with frequency $\\omega ^2=\\omega _z^2 + q^2 k^2/C_\\Sigma M z_0^2$ is described by the ladder operators $a$ , $a^\\dagger $ (see Methods).",
"The Hamiltonian (REF ) demonstrates that the qubit can be used to generate nanoparticle quantum states.",
"The absence of discrete internal transitions can thus be compensated by the non-linearity provided by a superconducting circuit.",
"The nanoparticle-qubit coupling strength is proportional to charge over root mass, yielding appreciable coupling for highly charged nanoscale objects.",
"Note that a finite bias voltage $U_{\\rm dc}$ applied to the ring electrode will not affect the Cooper-pair box, but produce an additional, approximately linear potential at the shifted trap center $z_s$ .",
"It adds the term $-V_{\\rm ext} (a+a^\\dagger )$ to (REF ), with $V_{\\rm ext}=qU_{\\rm dc} z_s \\sqrt{\\hbar /2M\\omega z_0^4}$ .",
"This term will be used below to control the relative phase of the nanoparticle superposition state.",
"Figure: Position and momentum trajectories of the proposed interference scheme.",
"An initial π/2\\pi /2-pulse on the Cooper-pair box generates a charge superposition in the endcap electrodes, so that the charged nanoparticle feels a superposition of a spatially shifted and an unshifted harmonic potential.",
"The time evolution () gives rise to two wave packets traveling on separate trajectories.To verify this motional superposition state the wave packets must be reunited.This can be achieved by applying two π\\pi -pulses, each interchanging the potentials felt by the two branches, in such a way that the trajectories finally coincide in position and momentum.",
"(a)In the simplest case all pulses are separated by one sixth of the harmonic oscillation period and the particle is initially at rest.",
"The π/2\\pi /2-pulse then leaves one branch unaffected (red dashed line), while the other one is accelerated (blue line).",
"The first π\\pi -pulse accelerates the resting branch and decelerates the moving one to a standstill at the time of the second π\\pi -pulse.",
"After that, the blue trajectory remains at rest while the red one is decelerated until it reaches the blue one with zero velocity.",
"(b) Even for arbitrary pulse times τ\\tau theseparation Δ τ \\Delta _\\tau between the π\\pi -pulsescan be chosen such that the corresponding paths in phase space coalesce for all initial states at 2τ+Δ τ 2\\tau +\\Delta _\\tau .",
"(c) The scheme works for time durations much shorter than the oscillation period, which makes it particularly suitable for limited coherence times.",
"In this short time limit the accelerations are essentially constant and Δ τ =2τ\\Delta _\\tau =2\\tau .",
"Generating and observing superpositions Quantum interference of the nanoparticle motion on short timescales can now be performed by a rapid sequence of qubit rotations and measurements.",
"At the beginning of the interference scheme, the charge qubit is prepared in its groundstate $\\mathinner {|{N}\\rangle }$ , while the nanoparticle is cooled to temperature $T$ , $\\rho _0 = \\mathinner {|{N}\\rangle }\\mathinner {\\langle {N}|}\\otimes \\exp ( - \\hbar \\omega a^\\dagger a/k_{\\rm B} T)/Z $ .",
"After this initial state preparation $V_{\\rm ext}$ is switched to a constant value.",
"The free dynamics governed by (REF ) with the external potential is then intersected by $\\sigma _x$ -rotations of the qubit at four different times: (i) a $\\pi /2$ -pulse at $t=0$ , which prepares the qubit in a superposition of charge states, (ii) a $\\pi $ -pulse at $t=t_1$ , which flips the qubit state, (iii) another $\\pi $ -pulse at $t=t_2$ , and (iv) a $\\pi /2$ -pulse at $t=t_3$ with subsequent measurement of the qubit occupation ${\\sigma _+\\sigma _-}$ .",
"a $\\pi /2$ -pulse at $t=0$ , which prepares the qubit in a superposition of charge states, a $\\pi $ -pulse at $t=t_1$ , which flips the qubit state, another $\\pi $ -pulse at $t=t_2$ , and a $\\pi /2$ -pulse at $t=t_3$ with subsequent measurement of the qubit occupation ${\\sigma _+\\sigma _-}$ .",
"With a symmetric pulse scheme, i.e.",
"$t_1 = t_3-t_2=\\tau $ , it is always possible to find a $ \\Delta _\\tau \\equiv t_2-t_1 $ such that the nanoparticle state evolves into a superposition and then recombines with maximal overlap (see Methods).",
"The corresponding phase space trajectories are illustrated in Fig.",
"REF .",
"In this case, the particle motion is first entangled with the qubit, generating a motional quantum superposition.",
"This superposition is then reversed by steps (ii) and (iii), and finally recombined, undoing the entanglement.",
"Through this sequence, the phase imprinted on the nanoparticle motion through the external voltage $V_{\\rm ext}$ is transferred onto the qubit state and read-out via its population, $\\mathinner {\\langle {\\mathsf {\\sigma }^+\\mathsf {\\sigma }^-}\\rangle } &=&\\cos ^2\\left[\\left(\\frac{\\kappa ^2}{\\omega } + \\frac{2 \\kappa V_{\\rm ext}}{\\hbar \\omega }-\\frac{E_{\\rm c}}{\\hbar }\\right) \\left( \\tau - \\frac{\\Delta _\\tau }{2}\\right) \\right].$ Varying $V_{\\rm ext}$ and observing the corresponding modulation of the qubit population can thus be used to verify that the nanoparticle existed in a spatial superposition state.",
"The final population also oscillates as a function of the pulse time $\\tau $ .",
"This pulse scheme enables the generation and observation of nanoscale quantum superpositions in harmonic potentials with pulse separations $\\tau $ much shorter than the particle oscillation period.",
"It is therefore applicable to various opto- and electromechanical systems [18], [19], [21].",
"The required accuracy of the pulse times, determined by the qubit frequency and the particle temperature $T$ , must ensure that the phase $\\kappa V_{\\rm ext} (2\\tau -\\Delta _\\tau )/ \\hbar \\omega $ is measurable.",
"To illustrate that nano- to microsecond motional superpositions can be realistically prepared and observed on the coherence time scale of a charge qubit [22], we show in Fig.",
"REF the expected interference signal of a $10^6$ amu particle at $T=1$ mK.",
"The nanoparticle is assumed to be cylindrically shaped, with a homogeneous surface charge of $q=200$ e and a realistic dipole moment (see Methods).",
"It is stably levitated inside a sub-millimeter Paul trap, its motion well approximated by a harmonic oscillation with $\\omega =138\\,$ kHz.",
"We find that the resulting strong coupling to the Cooper-pair box of $\\kappa =16.8\\,$ MHz renders the nanoparticle particle sensitive to the presence or absence of a single Cooper pair.",
"A voltage of $U_{\\rm dc}=25\\,$ V then suffices to imprint a relative phase on the motional superposition that shifts the interference pattern by a full fringe.",
"Networking levitated nanoparticles The proposed interference protocol can be extended to transfer qubit entanglement to nanoparticles.",
"Levitated objects may thus be coherently integrated into superconducting quantum networks, e.g.",
"for sensing and metrology applications.",
"Here we illustrate how to entangle the nanoparticle with a second, separated charge qubit, or with another nanoparticle levitating in a distant Paul trap.",
"Entanglement of the nanoparticle with a second qubit is achieved by replacing the initial $\\pi /2$ -pulse with an operation which prepares a maximally entangled two-qubit state [23], [24].",
"To verify the involvement of the particle in the nonlocal dynamics one carries out the above interference protocol by performing all pulses on both qubits.",
"The occupation of the separated qubit, conditioned on having found the directly coupled one in the groundstate, is then given by $\\mathinner {\\langle {\\sigma ^+\\sigma ^-}\\rangle } &=&\\cos ^2\\left[\\chi \\left(\\tau - \\frac{\\Delta _\\tau }{2} \\right)\\right], $ with $\\chi =\\kappa ^2/\\omega + 2 \\kappa V_{\\rm ext} /\\hbar \\omega -(E_{{\\rm c1}}-E_{{\\rm c2}})/\\hbar $ .",
"This assumes that the qubits were initially prepared in the singlet state $\\mathinner {|{\\Psi ^-}\\rangle }$ .",
"The external potential $V_{\\rm ext}$ , acting only on the nanoparticle, thus serves to fully control the measurement outcome of the distant qubit.",
"Having established that the coherent dynamics extends from the particle to the distant qubit, entangled states of these two systems can be produced by carrying out step (iv) and the subsequent measurement of the directly coupled qubit at $t_3<2\\tau +\\Delta _\\tau $ , i.e.",
"before the particle wave packets overlap.",
"An all-electrical protocol to entangle two distant levitated nanoparticles works along the same lines: We consider two distant nanoparticle-qubit setups of identical frequency $\\omega $ , where the qubits are again initially in the state $\\mathinner {|{\\Psi ^-}\\rangle }$ .",
"To verify the involvement of both particles in the nonlocal dynamics, one carries out the protocol until the time $t_3=2\\tau +\\Delta _\\tau $ of wave packet overlap.",
"The occupation of the second qubit, conditioned on having found the first one in the ground state, is then given by (REF ), with $\\chi =\\chi _1-\\chi _2$ , where $\\chi _i = \\kappa _i^2/\\omega + 2 \\kappa _i V_i /\\hbar \\omega -E_{{\\rm c}i}/\\hbar $ .",
"The interference pattern thus depends on the difference of the local nanoparticle phases.",
"By measuring both qubits before the wave packets overlap, e.g.",
"at $\\tau +\\Delta _\\tau <t_3<2\\tau +\\Delta _\\tau $ the two oscillators can be projected onto an entangled motional state (see Methods).",
"Figure: The Cooper-pair box occupation ()shows a pronounced interference signal for experimentally realistic parameters if π\\pi -pulses are applied at t 1 =21.7t_1=21.7\\,ns and t 2 =65.1t_2=65.1\\,ns.",
"The interfering trajectories then coalesce at 86.986.9\\,ns.",
"(a) The envelope of the interference pattern is determined by the overlap of the two associated nanoparticle wave packets.",
"Its width decreases with increasing temperature.",
"Even at 1 mK, corresponding a mean phonon number of 945, more than a thousand fringes can be expected (assuming a qubit dephasing time 1/γ d =1001/\\gamma _{\\rm d}=100\\,ns, see methods).",
"(b) The frequency of the interference signal is mainly determined by the charge energy E c E_{\\rm c} of the qubit.",
"A bias voltage U dc =5U_{\\rm dc}=5\\,V applied on the ring electrode imprints a phase on the nanoparticle, which shifts the interference pattern by about 2π/52\\pi /5 (dashed line).",
"Conclusion The coherent control of charged nanoparticles by superconducting qubits offers a new avenue for quantum superposition experiments with massive objects.",
"The nanoparticle superposition state is generated and read-out by pulsed qubit rotations and measurements, enabling interference experiments on ultra-short time scales.",
"All-electric trapping, cooling, and manipulation avoids photon scattering and absorption, the dominant decoherence sources in laser fields.",
"In addition, the nanoparticle rotations decouple from the centre-of-mass motion for realistic particle shapes and charge distributions, rendering this setup widely applicable.",
"It holds the potential of bridging the mass gap in quantum superposition tests from current experiments with massive molecules [25] to future guided interferometers with superconducting microscale particles [26].",
"Moreover, these novel hybrid quantum devices can serve as building blocks for larger networks connected by superconducting circuitry, distributing entanglement between multiple nanoparticles.",
"The presented qubit-nanoparticle coupling scheme is feasible with available technology for realistic particles.",
"Beyond that, the degree of quantum control can be enhanced by fabricating particles with tailored dipole and quadrupole moments and by combining electric with optical techniques [27], [28].",
"This may give rise to the observation of coherent effects between the rotational and translational nanoparticle degrees of freedom, and provide a platform for studying charge-induced decoherence in an unprecedented mass and complexity regime.",
"Acknowledgements: LM, KH, and BAS acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 411042854.",
"BAS acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodovska-Curie grant agreement No 841040.",
"MSK thanks the Royal Society, the UK EPSRC (EP/R044082/1) and KIST Open Research Program.",
"JM is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.",
"803277), and by EPSRC New Investigator Award EP/S004777/1.",
"Author contributions: All authors contributed conceptually to the proposal.",
"LM, KH, and BAS performed the analytic calculations and wrote the manuscript with input from JM and MSK.",
"Data availability: No data sets were generated or analysed during the current study.",
"Competing interests: The authors declare no competing interests.",
"Methods Ro-translational macromotion in a Paul trap An arbitrarily charged nanoparticle moving and revolving at position ${\\mathbf {R}}$ and orientation $\\Omega $ in a hyperbolic Paul trap is subject to the time-dependent potential $V({\\mathbf {R}},\\Omega ,t) =\\frac{U_{\\rm PT}(t)}{2z_0^2}\\left(\\frac{q}{2}\\mathbf {R}\\cdot \\mathsf {A}\\mathbf {R}+\\text{$p$}\\cdot \\mathsf {A}\\mathbf {R}-\\frac{1}{2}\\mathbf {e}_z\\cdot \\mathsf {Q}\\mathbf {e}_z\\right),$ with $\\mathsf {A}=\\mathbb {1}-3\\mathbf {e}_z\\otimes \\mathbf {e}_z$ .",
"Here, the dipole moment $\\text{$p$}$ and the quadrupole tensor $\\mathsf {Q}$ depend on the principal axes ${\\bf N}_i$ of the nanoparticle with the associated moments of inertia $I_i$ .",
"The effective potential for the macromotion is obtained by setting ${\\mathbf {R}} = {\\bf r} + \\text{$\\epsilon $}$ , and ${\\bf N}_i = {\\bf n}_i + \\text{$\\delta $} \\times {\\bf n}_i$ , serving to separate the center-of-mass macromotion ${\\bf r}$ from the much faster micromotion $|{\\text{$\\epsilon $}}| \\ll |{\\bf r}|$ varying with zero mean.",
"Similarly, the rotational micromotion $|{\\text{$\\delta $}}| \\ll 1$ varies much faster than ${\\bf n}_i$ .",
"The center-of-mass and angular momentum obey $m\\ddot{\\mathbf {R}}=-\\frac{U_{\\rm PT}(t)}{2z_0^2}\\mathsf {A}\\left(q\\mathbf {R}+{\\text{$p$}}\\right) $ and $\\dot{\\bf J} =-\\frac{U_{\\rm PT}(t)}{2z_0^2}\\left(\\text{$p$}\\times \\mathsf {A}\\mathbf {R}+\\mathbf {e}_z\\times {\\sf Q} \\mathbf {e}_z\\right).$ Taking macromotion to be approximately constant on the time scale of the micromotion and neglecting all small terms yields $\\text{$\\epsilon $}\\approx \\frac{U_{\\rm ac}\\cos (\\Omega _{\\rm ac} t)}{2M z_0^2 \\Omega _{\\rm ac}^2}\\mathsf {A}\\left(q\\mathbf {r}+\\text{$p$}\\right)$ and $\\text{$\\delta $}\\approx \\frac{U_{\\rm ac}\\cos (\\Omega _{\\rm ac} t)}{2 z_0^2 \\Omega _{\\rm ac}^2}\\sum _{i = 1}^3 \\frac{1}{I_i} {\\bf n}_i [{\\bf n}_i \\cdot \\left(\\text{$p$}\\times \\mathsf {A}\\mathbf {r}+\\mathbf {e}_z\\times \\mathsf {Q}\\mathbf {e}_z\\right)],$ involving the familiar Mathieu parameter and its rotational analogues, respectively.",
"The dipole and quadrupole moments here only include the macromotion, i.e.",
"$\\mathbf {p}=\\sum p_i \\mathbf {n}_i$ and $\\mathsf {Q}=\\sum Q_{ij}\\mathbf {n}_i\\otimes \\mathbf {n}_j$ , in contrast to (REF ).",
"The effective force and torque of the macromotion can be obtained by inserting (REF ) into (REF ) and averaging over one micromotion cycle.",
"A lengthy but straightforward calculation demonstrates that they can be expressed through the time-independent effective potential (REF ).",
"We remark that the same potential (REF ) can also be derived quantum mechanically by adapting the method outlined in Ref.",
"[29] for the combined rotational and translational motion of the nanoparticle.",
"Generalized Shockley-Ramo theorem To calculate the current induced by an arbitrary, rigidly bound charge distribution moving and rotating between the endcap electrodes we use Green's reciprocity theorem, $\\int _{\\mathcal {V}}dV \\phi _{\\rm ref}\\rho +\\int _{\\partial \\mathcal {V}}dA \\phi _{\\rm ref}\\sigma =\\int _{\\mathcal {V}}dV \\phi \\rho _{\\rm ref}+\\int _{\\partial \\mathcal {V}}dA \\phi \\sigma _{\\rm ref}.$ It relates the particle charge density $\\rho $ , the electrode surface charge density $\\sigma $ and the electrostatic potential $\\phi $ to those of a reference system.",
"Choosing the reference system to have no particle in the trap volume $\\mathcal {V}$ , a vanishing potential on the ring electrode, and opposite potentials on the endcaps leads to an approximately linear potential $\\phi _{\\rm ref}$ near the trap center.",
"This results in $ \\frac{k}{z_0}\\mathbf {e}_{z}\\cdot \\int _\\mathcal {V} dV \\mathbf {x}\\rho (\\mathbf {x})+\\frac{Q_1-Q_2}{2}=CV,$ where $Q_1$ and $Q_2$ are the total charges on the top and bottom endcap and $\\mathbf {x}$ originates from the trap center.",
"The remaining integration yields the capacitance charge $Q = (Q_1 - Q_2)/2=-k\\mathbf {e}_z\\cdot (q\\mathbf {r}+\\text{$p$})/z_0+CV$ , and its time derivative the induced current.",
"Figure: Circuit diagram of the tunable Cooper-pair box coupled to the particle via the trap capacitor.The superconducting island to and from which Cooper pairs can tunnel is indicated by the dashed line.The arrowsindicate the direction of increasing electrostatic potential for positive UU's and of the electron flow for positive II's.",
"Cooper-pair box-nanoparticle Hamiltonian Figure REF shows how the Paul trap is connected with the circuit.",
"The latter consists of a superconducting loop with two Josephson junctions modeled as a capacitance $C_{\\rm J}$ and a tunneling junction in parallel.",
"The loop of vanishing inductance encloses an external magnetic flux $\\Phi $ .",
"The quantum state of the circuit is described by a macroscopic wave function whose phase jumps $\\varphi _1$ , $\\varphi _2$ at the Josephson junctions satisfy $\\varphi _2-\\varphi _1+{2\\pi \\Phi }/{\\hbar }=2\\pi m$ with $m$ an integer.",
"Josephson's equations $I_{{\\rm J}i}=I_{\\rm c}\\sin \\varphi _i$ and $U_{{\\rm J}i}=\\frac{\\hbar }{2e}\\dot{\\varphi _i}$ relate them to the tunneling current and the voltage drop in each junction $i = 1,2$ .",
"The loop is coupled capacitatively to the endcaps via $C_{\\rm c}$ , and can be controlled with the external voltage $U$ , applied via the gate capacitance $C_{\\rm g}$ .",
"The circuit equations of motion can be obtained starting from Kirchhoff's laws, $V+U_{{\\rm J1}}+U_{\\rm c1}+U_{\\rm c2} & = & 0,\\\\U_{\\rm J1}+U+U_{\\rm g} & = & 0, \\\\I_{\\rm c1}+I_{\\rm J1}+I_{\\rm c2}+I_{\\rm J2} & = & I+I_{\\rm g}.",
"$ Inserting the capacitance charge (REF ) into (REF ), differentiating with respect to time and using that $U_{{\\rm c}i}=Q_{{\\rm c}i}/C_{\\rm c}$ and that $I=\\dot{Q}=\\dot{Q}_{{\\rm c}i}$ yields $\\dot{Q}=-\\frac{k_0}{z_0}\\frac{C_{\\rm eff}}{C}\\left(q\\dot{{\\bf r}}+\\dot{\\text{$p$}}\\right)\\cdot {\\bf e}_z-\\frac{C_{\\rm eff}}{C_{\\rm J}}\\dot{Q}_{\\rm J1},$ with the effective capacitance $C_{\\rm eff}=CC_{\\rm c}/(C_{\\rm c}+2C)$ .",
"In addition, () and () yield the relations $\\dot{Q}_{\\rm g} & = & -\\frac{C_{\\rm g}}{C_{\\rm J}}\\dot{Q}_{\\rm J1}-C_{\\rm g}\\dot{U}, \\\\\\dot{Q}+\\dot{Q}_{\\rm g} & = & \\dot{Q}_{\\rm J1}+\\dot{Q}_{\\rm J2}+I_{\\rm c}\\sin {\\varphi _1}+I_{\\rm c}\\sin {\\varphi _2}.$ Inserting (REF ) and (REF ) into (), using flux quantization, and defining $\\varphi =\\varphi _1-e\\Phi /\\hbar $ finally yields the circuit equation of motion, $\\begin{split}\\ddot{\\varphi }=&-\\frac{4eI_{\\rm c}}{\\hbar C_\\Sigma }\\cos \\left(\\frac{e\\Phi }{\\hbar }\\right)\\sin \\varphi -\\frac{2 e C_{\\rm eff}k}{\\hbar CC_\\Sigma z_0}\\left(q\\dot{{\\bf r}}+\\dot{\\text{$p$}}\\right)\\cdot {\\bf e}_z\\\\&-\\frac{2e}{\\hbar C_\\Sigma }\\left[C_{\\rm g}\\dot{U}+\\left(C_{\\rm eff}+C_{\\rm g}\\right)\\frac{\\ddot{\\Phi }}{2}\\right],\\end{split}$ with $C_\\Sigma =C_{\\rm eff}+C_{\\rm g}+2C_{\\rm J}$ .",
"The ro-translational motion of the particle is driven by the endcap voltage $V$ via the force ${\\mathbf {F}}=-kVq\\mathbf {e}_z/z_0$ and the torque ${\\mathbf {N}}=kV\\mathbf {e}_z \\times \\text{$p$}/z_0$ , in addition to the Paul trap force and torque.",
"In the relevant limit of large $C_{\\rm c}$ the Hamiltonian generating the coupled dynamics of circuit and particle takes the form $ H &=&\\frac{2e^2}{C_\\Sigma }\\left[\\frac{\\Pi }{\\hbar }-\\frac{k}{2ez_0}(q {\\bf r}+{\\text{$p$}})\\cdot {\\bf e}_z-n_{\\rm g}\\right]^2-E_J\\cos \\varphi \\\\&&-\\frac{k \\dot{\\Phi }}{2z_0}\\left(q {\\bf r}+{\\text{$p$}}\\right) \\cdot {\\bf e}_z+H_{\\rm rb}+V_{\\rm eff}(\\mathbf {r},\\Omega )\\,,$ where the canonical momentum $\\Pi $ conjugate to $\\varphi $ quantifies the number of Cooper pairs on the island and $H_{\\rm rb}$ is the free rigid body Hamiltonian for the center-of-mass motion and rotation.",
"Eq.",
"(REF ) involves the voltage-induced number of Cooper pairs $n_{\\rm g}=C_{\\rm g}U/2e+\\left(C+C_{\\rm g}\\right)\\dot{\\Phi }/4e$ , and the Josephson energy $E_{\\rm J}=\\hbar I_{\\rm c}\\cos \\left(e\\Phi /\\hbar \\right)/e$ .",
"We choose the flux $\\Phi $ and applied voltage $U$ so that the Cooper-pair box can be treated as an effective two-level system [30] with $N$ or $N+1$ Cooper pairs on the island, $\\Pi = \\hbar (N + \\sigma _+\\sigma _-)$ .",
"Tuning $n_{\\rm g}$ , $E_{\\rm J}$ , and $\\dot{\\Phi }$ to zero yields the Hamiltonian $ H & = & \\frac{2 e^2}{C_\\Sigma } (2 N + 1) \\sigma _+ \\sigma _- - \\frac{2 e k}{C_\\Sigma z_0} (N + \\sigma _+ \\sigma _- ) (q {\\bf r} + \\text{$p$} )\\cdot {\\bf e}_z \\\\&& + \\frac{k^2}{2 C_\\Sigma z_0^2} [(q {\\bf r} + {\\text{$p$}})\\cdot {\\bf e}_z]^2 + H_{\\rm rb} + V_{\\rm eff}({\\bf r},\\Omega ),$ which leaves the charge eigenstates of the box unaffected.",
"The charge-dependent potential shift given by the second term will drive the particle into a ro-translational superposition if the charge states are superposed.",
"Neglecting the dipole moment and separating the nanoparticle transverse motion and rotations finally yields (REF ) with the potential minimum shifted to $z_{\\rm s}=2ek Nq/C_\\Sigma z_0 M\\omega ^2$ and the charge energy $E_c=2e^2(1+2N-kqz_{\\rm s}/ez_0)/C_\\Sigma $ .",
"Time evolution and measurement outcome The time evolution generated by (REF ) with external potential $V_{\\rm ext}$ can be written, up to a global phase, as a combination of a qubit-dependent phase, qubit-dependent particle displacements and the free time evolution of the harmonic oscillator, $ U(t)& = &\\exp \\left[-it\\left(\\frac{E_{\\rm c}}{\\hbar }-\\frac{\\kappa ^2}{\\omega } - \\frac{2 \\kappa V_{\\rm ext}}{\\hbar \\omega } \\right)\\sigma _+\\sigma _-\\right]\\\\&&\\times \\exp \\left[\\left(\\frac{\\kappa }{\\omega } \\sigma _+\\sigma _-+\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right)\\left(a^\\dagger - a\\right)\\right] \\\\&& \\times \\exp \\left(-i\\omega t a^\\dagger a \\right) \\\\&&\\times \\exp \\left[-\\left(\\frac{\\kappa }{\\omega } \\sigma _+\\sigma _-+\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right)\\left(a^\\dagger - a\\right)\\right].$ The qubit is initially prepared in its groundstate while the nanoparticle is in a thermal state of temperature $T$ .",
"A $\\pi /2$ -pulse rotates the qubit into the superposition $\\left(\\mathinner {|{N}\\rangle }+i\\mathinner {|{N+1}\\rangle }\\right)/\\sqrt{2}$ , so that the system after time $t$ is given by $\\rho _t = \\sum _{n=0}^{\\infty }\\exp \\left(-\\hbar \\omega n/k_{\\rm B}T \\right)\\mathinner {|{\\Psi _n}\\rangle }\\mathinner {\\langle {\\Psi _n}|} / Z$ with $\\mathinner {|{\\Psi _n}\\rangle }=\\left(\\mathinner {|{N}\\rangle }U_{\\rm g}(t)+i\\mathinner {|{N+1}\\rangle }U_{\\rm e}(t)\\right)\\mathinner {|{n}\\rangle }/\\sqrt{2}$ .",
"This involves particle time-evolution operators associated with the ground and the excited state of the qubit, $\\mathsf {U}_{\\rm g}(t)=\\mathsf {D}\\left(\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right)\\exp \\left(-i\\omega t a^\\dagger a\\right)\\mathsf {D}\\left(-\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right)$ and $\\mathsf {U}_{\\rm e}(t) & = & \\exp \\left[-it\\left(\\frac{E_{\\rm c}}{\\hbar }-\\frac{\\kappa ^2}{\\omega } - \\frac{2 \\kappa V_{\\rm ext}}{\\hbar \\omega } \\right)\\right]\\mathsf {D}\\left(\\frac{\\kappa }{\\omega } +\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right) \\\\&& \\times \\exp \\left(-i\\omega t a^\\dagger a\\right)\\mathsf {D}\\left(-\\frac{\\kappa }{\\omega } -\\frac{V_{\\rm ext}}{\\hbar \\omega }\\right),$ where $\\mathsf {D}(\\alpha )=\\exp \\left(\\alpha \\mathsf {a}^\\dagger -\\alpha ^*\\mathsf {a}\\right)$ is the displacement operator.",
"The scheme with $\\pi $ -pulses at times $t_1$ and $t_2$ then results in $\\mathinner {|{\\Psi _n}\\rangle }=\\left(\\mathinner {|{N}\\rangle }\\mathsf {U}_++i\\mathinner {|{N+1}\\rangle }\\mathsf {U}_-\\right)\\mathinner {|{n}\\rangle }/\\sqrt{2}$ at time $t_3$ , where $\\mathsf {U}_+ & = &\\mathsf {U}_{\\rm g}(t_3-t_2)\\mathsf {U}_{\\rm e}(t_2-t_1)\\mathsf {U}_{\\rm g}(t_1),\\\\\\mathsf {U_-} &= &\\mathsf {U}_{\\rm e}(t_3-t_2)\\mathsf {U}_{\\rm g}(t_2-t_1)\\mathsf {U}_{\\rm e}(t_1).$ The charge occupation of the box after a final $\\pi /2$ -pulse is thus given by $\\mathinner {\\langle {\\sigma _+\\sigma _-}\\rangle } = \\frac{1}{2}+\\frac{1}{2Z}\\sum _{n=0}^{\\infty } \\Re \\left[\\mathinner {\\langle {n}|}\\mathsf {U}_+^\\dagger \\mathsf {U}_-\\mathinner {|{n}\\rangle }\\right] \\exp \\left( -\\frac{\\hbar \\omega n}{k_{\\rm B}T}\\right).$ Noting that $\\mathsf {U}_+^\\dagger \\mathsf {U}_-$ displaces the particle state in phase space and using $\\begin{split}&\\frac{1}{Z} \\sum _{n=0}^{\\infty }\\mathinner {\\langle {n}|}\\mathsf {D}(\\alpha )\\mathinner {|{n}\\rangle }\\exp \\left(-\\frac{\\hbar \\omega n}{k_{\\rm B}T} \\right) \\\\&=\\exp \\left[-\\coth \\left( \\frac{\\hbar \\omega }{2 k_{\\rm B} T} \\right)\\frac{|\\alpha |^2}{2}\\right],\\end{split}$ one obtains $\\mathinner {\\langle {\\mathsf {\\sigma }^+\\mathsf {\\sigma }^-}\\rangle } &=&\\frac{1}{2}+\\frac{1}{2}\\exp \\left[-\\frac{\\kappa ^2}{2\\omega ^2}\\coth \\left( \\frac{\\hbar \\omega }{2 k_{\\rm B} T} \\right)\\vert d(t_1,t_2,t_3)\\vert ^2 \\right]\\\\&&\\times \\cos \\left[\\frac{\\kappa }{\\omega } \\left( \\frac{\\kappa }{\\omega }+ \\frac{2V_{\\rm ext}}{\\hbar \\omega } \\right) {\\rm Im}\\,d(t_1,t_2,t_3) \\right.",
"\\\\& &\\left.",
"-\\left(\\frac{\\kappa ^2}{\\omega } + \\frac{2\\kappa V_{\\rm ext}}{\\hbar \\omega }-\\frac{E_{\\rm c}}{\\hbar }\\right) ( 2 t_1 - 2 t_2 + t_3) \\right] ,$ where $d(t_1,t_2,t_3)=2 e^{i\\omega t_1}-2e^{i\\omega t_2}+e^{i\\omega t_3} - 1$ .",
"Qubit dephasing in Fig.",
"REF is modeled by taking $n_g$ in (REF ) to be a random number with Lorentzian distribution of width $\\gamma _{\\rm d}C_\\Sigma \\hbar /(2e)^2$ .",
"This adds the exponential factor $\\exp (-\\gamma _{\\rm d} t_3)$ to the second term in (REF ).",
"Equation (REF ) shows that the envelope of the qubit occupation assumes its maximum at $d(t_1,t_2,t_3) = 0$ and that the particle temperature determines the width of the peak.",
"Operating the interference protocol at the point of maximal envelope (corresponding to a maximal overlap in the particle state of both superposition branches) can be achieved with the symmetric choice $t_1 = \\tau $ , $t_2 = \\tau + \\Delta _\\tau $ and $t_3 = 2 \\tau + \\Delta _\\tau $ , where $\\Delta _\\tau = \\frac{1}{\\omega }\\arctan \\left[\\frac{2 \\sin (\\omega \\tau ) [2 - \\cos (\\omega \\tau )]}{[2 - \\cos (\\omega \\tau )]^2 - \\sin ^2(\\omega \\tau )}\\right]$ and $\\tau <\\pi /\\omega $ .",
"Evaluating (REF ) for these times finally yields (REF ).",
"Generation of nanoparticle entanglement The time evolution involving two nanoparticles can be described by means of the respective particle operators $\\mathsf {U}_+^{(i)}\\mathinner {|{n}\\rangle }&=&e^{i\\phi _+^{(i)}}\\mathsf {D}_i(\\alpha _{i})\\mathinner {|{n}\\rangle },\\\\\\mathsf {U}_-^{(i)}\\mathinner {|{n}\\rangle }&=&e^{i\\phi _-^{(i)}}\\mathsf {D}_i(\\beta _{i})\\mathinner {|{n}\\rangle }.$ where the ${\\sf D}_i$ are phase-space displacement operators acting on nanoparticle $i$ .",
"To entangle the particles one prepares the qubits in a Bell state and performs the trapped interference scheme on both subsystems.",
"Measuring both qubits before the wave packets overlap, e.g.",
"at $\\tau +\\Delta _\\tau <t_3<2\\tau +\\Delta _\\tau $ projects the two oscillators onto the outcome-conditioned state $ \\rho ^{\\prime }\\propto \\sum _{n,m=0}^\\infty \\exp \\left[-\\frac{\\hbar \\omega }{k_{\\rm B}T}(n+m) \\right]\\mathinner {|{\\Psi _{nm}}\\rangle }\\mathinner {\\langle {\\Psi _{nm}}|},$ where $\\mathinner {|{\\Psi _{nm}}\\rangle }=\\left[\\mathsf {D}_1(\\alpha _1)\\mathsf {D}_2(\\beta _2)\\pm e^{i\\phi }\\mathsf {D}_1(\\beta _1)\\mathsf {D}_2(\\alpha _2)\\right]\\mathinner {|{n}\\rangle }_1\\otimes \\mathinner {|{m}\\rangle }_2.$ The amplitudes $\\alpha _i$ , $\\beta _i$ and the phase $\\phi =\\phi _-^{(1)}+\\phi _+^{(2)}-\\phi _+^{(1)}-\\phi _-^{(2)}$ depend on the pulse times, whereas the sign in (REF ) is fixed by the outcome of the qubit measurements.",
"The values of $\\alpha _i$ and $\\beta _i$ determine the amount of entanglement of the state (REF ), as quantified by a suitable entanglement measure.",
"Experimental parameters For calculating the interference pattern in Fig.",
"REF we consider a cylindrically shaped silicon nanoparticle (diameter of $4.7\\,{\\rm nm}$ , length of $42\\,{\\rm nm}$ , homogeneously charged with $q = 200\\,$ e [31].",
"We assume a value of $\\vert \\text{$p$} \\vert = 200\\,{\\rm eÅ}$ for the dipole moment, based on previous studies reporting values of several 10 eÅ for neutral particles of the same size [32], [33], [34].",
"The Paul trap, with an endcap distance of $2z_0=0.5\\,{\\rm mm}$ and geometry factor $k=0.4$ [35], [17], is driven by an AC voltage of $U_{\\rm ac}=1\\,{\\rm kV}$ with frequency $\\Omega =2\\pi \\times 250\\,{\\rm MHz}$ .",
"The empty Cooper-pair box has a capacitance of $C_\\Sigma =4.4\\,{\\rm fF}$ , yielding a charge energy of $2e^2/C_\\Sigma \\approx 72\\,{\\rm \\mu eV}$ .",
"A relatively high occupation $N=10$ shifts the potential minimum by the distance $z_{\\rm s}=1.17\\,{\\rm \\mu m}$ from the trap center.",
"The fast box oscillations then require a measurement time resolution on the ${\\rm ps}$ -scale [30].",
"The total duration of the experiment of $87\\,{\\rm ns}$ is on the expected coherence time scale of a charge qubit [22].",
"An initial motional temperature of the particle of $T=1\\,{\\rm mK}$ is achievable via resistive cooling [17], [36], [37] (and potentially by electric feedback cooling [27], [28], [17] or optical techniques [8]).",
"Assuming a resistance of $R=100 \\,{\\rm M\\Omega }$ , the adiabatic cooling rate is $163\\,{\\rm Hz}$ , corresponding to $1.54\\times 10^5$ quanta per second in thermal equilibrium.",
"Our estimate of the surface noise [38] for the present system with superconducting endcap electrodes yields a heating rate of $170\\,\\hbar \\omega /$ s, which does not noticeably raise the temperature of the particle on the time scale of the experiment."
]
] | 2005.14006 |
[
[
"Active Fuzzing for Testing and Securing Cyber-Physical Systems"
],
[
"Abstract Cyber-physical systems (CPSs) in critical infrastructure face a pervasive threat from attackers, motivating research into a variety of countermeasures for securing them.",
"Assessing the effectiveness of these countermeasures is challenging, however, as realistic benchmarks of attacks are difficult to manually construct, blindly testing is ineffective due to the enormous search spaces and resource requirements, and intelligent fuzzing approaches require impractical amounts of data and network access.",
"In this work, we propose active fuzzing, an automatic approach for finding test suites of packet-level CPS network attacks, targeting scenarios in which attackers can observe sensors and manipulate packets, but have no existing knowledge about the payload encodings.",
"Our approach learns regression models for predicting sensor values that will result from sampled network packets, and uses these predictions to guide a search for payload manipulations (i.e.",
"bit flips) most likely to drive the CPS into an unsafe state.",
"Key to our solution is the use of online active learning, which iteratively updates the models by sampling payloads that are estimated to maximally improve them.",
"We evaluate the efficacy of active fuzzing by implementing it for a water purification plant testbed, finding it can automatically discover a test suite of flow, pressure, and over/underflow attacks, all with substantially less time, data, and network access than the most comparable approach.",
"Finally, we demonstrate that our prediction models can also be utilised as countermeasures themselves, implementing them as anomaly detectors and early warning systems."
],
[
"Introduction",
"Cyber-physical systems (CPSs), characterised by their tight and complex integration of computational and physical processes, are often used in the automation of critical public infrastructure [78].",
"Given the potential impact of cyber-attacks on these systems [46], [60], [51], ensuring their security and protection has become a more important goal than ever before.",
"The different temporal scales, modalities, and process interactions in CPSs, however, pose a significant challenge for solutions to overcome, and have led to a variety of research into different possible countermeasures, including ones based on anomaly detection [27], [45], [52], [69], [11], [15], [48], [58], [61], [66], [71], fingerprinting [12], [13], [44], [56], invariant-based monitoring [18], [8], [7], [24], [10], [25], [28], [39], [82], and trusted execution environments [74].",
"Assessing how effective these different countermeasures are at detecting and preventing attacks is another challenge in itself.",
"A typical solution is to use established benchmarks of attacks [2], [41], which have the advantage of facilitating direct comparisons between approaches, e.g.",
"as done so in [52], [58], [61].",
"Such benchmarks, unfortunately, are few and far between: constructing them manually requires a great deal of time and expertise in the targeted CPS (all while risking insider bias), and generalising them from one CPS to another is a non-starter given the distinct processes, behaviours, and complexities that different systems exhibit.",
"An alternative solution is to generate benchmarks using automated testing and fuzzing, with these techniques overcoming the complexity of CPSs by having access to machine learning (ML) models trained on their data (e.g.",
"logs of sensor readings, actuator states, or network traffic).",
"Existing solutions of this kind, however, tend to make unrealistic assumptions about an attacker's capabilities, or require a large body of training data that might not be available.",
"The fuzzer of [26], for example, can automatically identify actuator configurations that drive the physical states of CPSs to different extremes, but the technology assumes the attacker to have total control of the network and actuators, and is underpinned by a prediction model trained on complete sets of data logs from several days of operation.",
"Blindly fuzzing without such a model, however, is ineffective at finding attacks: first, because the search spaces of typical CPSs are enormous; and second, because of the wasted time and resources required to be able to observe the effects on a system's physical processes.",
"In this paper, we present active fuzzing, an automatic approach for finding test suites of packet-level CPS network attacks, targeting scenarios in which training data is limited, and in which attackers can observe sensors and manipulate network packets but have no existing knowledge about the encodings of their payloads.",
"Our approach constructs regression models for predicting future sensor readings from network packets, and uses these models to guide a search for payload manipulations that systematically drive the system into unsafe states.",
"To overcome the search space and resource costs, our solution utilises (online) active learning [63], a form of supervised ML that iteratively re-trains a model on examples that are estimated to maximally improve it.",
"We apply it to CPSs by flipping bits of existing payloads in a way that is guided by one of two frameworks: Expected Model Change Maximization [17], and a novel adaptation of it based on maximising behaviour change.",
"We query the effects of sampled payloads by spoofing them in the network, updating the model based on the observed effect.",
"We evaluate our approach by implementing it for the Secure Water Treatment (SWaT) testbed [4], a scaled-down version of a real-world water purification plant, able to produce up to five gallons of drinking water per minute.",
"SWaT is a complex multi-stage system involving chemical processes such as ultrafiltration, de-chlorination, and reverse osmosis.",
"Communication in the testbed is organised into a layered network hierarchy, in which we target the ring networks at the `lowest' level that exchange data using EtherNet/IP over UDP.",
"Our implementation manipulates the binary string payloads of 16 different types of packets, which when considered together have up to $2^{2752}$ different combinations.",
"Despite the enormous search space, we find that active fuzzing is effective at discovering packet-level flow, pressure, and over/underflow attacks, achieving comparable coverage to an established benchmark [41] and an LSTM-based fuzzer [26] but with substantially less training time, data, and network access.",
"Furthermore, by manipulating the bits of payloads directly, active fuzzing bypasses the logic checks enforced by the system's controllers.",
"These attacks are more sophisticated than those of the LSTM-based fuzzer [26], which can only generate high-level actuator commands and is unable to manipulate such packets.",
"Finally, we investigate the utility of the learnt models in a different role: defending a CPS directly.",
"We use them to implement anomaly detection and early warning systems for SWaT, finding that when models are suitably expressive, they are effective at detecting both random and known attacks.",
"Summary of Contributions.",
"We present active fuzzing, a black-box approach for automatically discovering packet-level network attacks on real-world CPSs.",
"By iteratively constructing a model with active learning, we demonstrate how to overcome enormous search spaces and resource costs by sampling new examples that maximally improve the model, and propose a new algorithm that guides this process by seeking maximally different behaviour.",
"We evaluate the efficacy of the approach by implementing it for a complex real-world critical infrastructure testbed, and show that it achieves comparable coverage to an established benchmark and LSTM-based fuzzer but with significantly less data, time, and network access.",
"Finally, we show that the learnt models are also effective as anomaly detectors and early warning systems.",
"Organisation.",
"In Section , we introduce the SWaT testbed, with a particular focus on its network and the structure of its packets.",
"In Section , we present the components of our active fuzzing approach, and explain how to implement it both in general and for SWaT.",
"In Section , we evaluate the efficacy of our approach at finding packet-level attacks, and investigate secondary applications of our models as anomaly detectors and early warning systems.",
"In Section , we discuss some related work, then conclude in Section ."
],
[
"Background",
"In the following, we present an overview of SWaT, a water treatment testbed that forms the critical infrastructure case study we evaluate active fuzzing on.",
"We describe in more detail its network hierarchy and the structure of its packets, before stating the assumptions we make about the capabilities of attackers.",
"SWaT Testbed.",
"The CPS forming the case study of this paper is Secure Water Treatment (SWaT) [4], a scaled-down version of a real-world water purification plant, able to produce up to five gallons of safe drinking water per minute.",
"SWaT (Figure REF ) is intended to be a testbed for advancing cyber-security research on critical infrastructure, with the potential for successful technologies to be transferred to the actual plants it is based on.",
"The testbed has been the subject of multiple hackathons [9] involving researchers from both academia and industry, and over the years has established a benchmark of attacks to evaluate defence mechanisms against [41].",
"Figure: The Secure Water Treatment (SWaT) testbedSWaT treats water across multiple distinct but co-operating stages, involving a variety of complex chemical processes, such as de-chlorination, reverse osmosis, and ultrafiltration.",
"Each stage in the CPS is controlled by a dedicated Allen-Bradley ControlLogix programmable logic controller (PLC), which communicates with the sensors and actuators relevant to that stage over a ring network, and with other PLCs over a star network.",
"Each PLC cycles through its program, computing the appropriate commands to send to actuators based on the latest sensor readings received as input.",
"The system consists of 42 sensors and actuators in total, with sensors monitoring physical properties such as tank levels, flow, pressure, and pH values, and actuators including motorised valves (for opening an inflow pipe) and pumps (for emptying a tank).",
"A historian regularly records the sensor readings and actuator commands during SWaT's operation.",
"SCADA software and tools developed by Rockwell Automation are available to support some analyses.",
"The sensors in SWaT are associated with manufacturer-defined ranges of safe values, which in normal operation, they are expected to remain within.",
"If a sensor reports a (true) reading outside of this range, we say the physical state of the CPS has become unsafe.",
"If a level indicator transmitter, for example, reports that the tank in stage one has become more than a certain percentage full (or empty), then the physical state has become unsafe due to the risk of an overflow (or underflow).",
"Unsafe pressure states indicate the risk of a pipe bursting, and unsafe levels of water flow indicate the risk of possible cascading effects in other parts of the system.",
"SWaT implements a number of standard safety and security measures for water treatment plants, such as alarms (reported to the operator) for when these thresholds are crossed, and logic checks for commands that are exchanged between the PLCs.",
"In addition, several attack defence mechanisms developed by researchers have been installed (see Section ).",
"Figure: A SWaT packet after dissection by ScapyThe network of the SWaT testbed is organised into a layered hierarchy compliant with the ISA99 standard [53], providing different levels of segmentation and traffic control.",
"The `upper' layers of the hierarchy, Levels 3 and 2, respectively handle operation management (e.g.",
"the historian) and supervisory control (e.g.",
"touch panel, engineering workstation).",
"Level 1 is a star network connecting the PLCs, and implements the Common Industrial Protocol (CIP) over EtherNet/IP.",
"Finally, the `lowest' layer of the hierarchy is Level 0, which consists of ring networks (EtherNet/IP over UDP) that connect individual PLCs to their relevant sensors and actuators.",
"Tools such as Wireshark [6] and Scapy [3] can be used to dissect the header information of a Level 0 SWaT packet, as illustrated in Figure REF .",
"Here, the source IP address ($\\mathtt {192.168.0.10}$ ) and target IP address ($\\mathtt {192.168.0.12}$ ) correspond respectively to PLC1 and its remote IO device.",
"Actuator commands (e.g.",
"“open valve MV101”) are encoded in the binary string payloads of these packets.",
"In Figure REF , the payload is 22 bytes long, but Level 0 packets can also have a payload length of 10 or 32 bytes.",
"Randomly manipulating the payloads has limited use given the size of the search space ($2^{2752}$ possibilities when considering the 16 types of packets we sample; see Section REF ).",
"Our solution uses active learning to overcome this enormous search space, establishing how different bits impact the physical state without requiring any knowledge of the encoding.",
"Attacker Model.",
"In this work, we assume that attackers have knowledge of the network protocol (e.g.",
"EtherNet/IP over UDP at Level 0 of SWaT), and thus are able to intercept (unencrypted) packets, dissect their header information, and manipulate their payloads.",
"We assume that the packet payloads are binary strings, but do not assume any existing knowledge about their meaning or encoding schemes.",
"We assume that attackers can always access the `true' sensor readings while the system is operating, in order to be able to observe the effects of a packet manipulation, or to judge whether or not an attack was successful.",
"These live sensor readings can be observed over several minutes at a time in order to perform some pre-training and active learning, but in contrast to other approaches (e.g.",
"[26]), we do not require access to extensive sets of data for offline learning, and we do not require the ability to arbitrarily issue high-level actuator commands across the system—we do so only by manipulating payloads."
],
[
"Active Fuzzing",
"Our approach for automatically finding packet-level network attacks in CPSs consists of the following steps.",
"First, data is collected: packets are sniffed from the network, their payloads are extracted, and (true) sensor readings are queried.",
"Second, we pre-train initial regression models, that take concatenations of packet payloads and predict the future effects on given sensors.",
"Third, we apply an online active learning framework, iteratively improving the current model by sampling payloads estimated to maximally improve it.",
"Finally, we search for candidate attacks by flipping important bits in packet payloads, and using our learnt models to identify which of them will drive the system to a targeted unsafe state.",
"[!t] High-Level Overview of Active Fuzzing Sensor $s$ , prediction time $t_s$ , pre-training time $t_p$ Prediction model $M_s$ Sniff packets and observe values of $s$ for $t_p$ minutes; Construct a sequence $P$ of feature (bit-)vectors from packet payloads; Construct a sequence $V$ such that each $V[i]$ contains the value of $s$ observed $t_s$ seconds after $P[i]$ was sniffed; (Pre-)train a regression model $M_s$ predicting $V$ from $P$ ; timeout Sample a new feature vector $p$ using an active learning framework (Section REF ); Wait for $t_s$ seconds then observe the value $v_s$ of $s$ ; $P := P^\\frown \\langle p\\rangle ^{t_s}$ ; [concatenation of $t_s$ copies] $V := V^\\frown \\langle v_s\\rangle ^{t_s}$ ; Re-train $M_s$ to predict $V$ from $P$ ; model $M_s$ ; Algorithm presents the high-level algorithm of these steps for active fuzzing.",
"Note that the notation in Line indicates concatenation of sequences.",
"In particular, $t_s$ copies of the vector $p$ are appended to sequence $P$ to add additional weight to the new example when the model is re-trained.",
"In the following, we describe the steps of the algorithm in more detail, and present the details of one particular implementation for the SWaT water purification testbed."
],
[
"Packet Sniffing and Pre-Training",
"Collecting Raw Data.",
"Both the pre-training and active learning phases of our approach require access to two types of data from the CPS under test.",
"First, they must be able to sniff the network packets and extract their binary string payloads.",
"Second, they must be able to access the true readings of any sensors under consideration, as the idea is to be able to observe the effects on sensor readings of different payload manipulations.",
"For SWaT, our approach extracts packets from Level 0 of the network hierarchy, i.e.",
"the packets exchanged between PLCs and remote IO devices.",
"By targeting this lowest level of the network, we ensure that our manipulations are altering the actuator states directly.",
"For our prototype, we physically connect some Raspberry Pis to the PLCs of SWaT and sniff the packets using a network connection bridge; in reality, an attacker might sniff packets by other means, e.g.",
"exploiting the wireless connection when enabled.",
"As Level 0 implements the EtherNet/IP protocol, we can use the tcpdump packet analyser to capture packets, and Scapy to further extract their contents.",
"For our prototype, we obtain the current sensor readings by querying SWaT's historian.",
"We assume that the historian's data is true, i.e.",
"that the system is not simultaneously under attack by another entity, and that it is operationally healthy.",
"In reality, an attacker might access this information through an exploit in the historian, e.g.",
"an EternalBlue exploit [1], or a backdoor connection (both of which were discovered in SWaT hackathons [9]).",
"Pre-Training Models.",
"A goal of our approach is to minimise the amount of data required to train an accurate prediction model for sensor readings.",
"We thus proceed in two phases: a pre-training phase, and an active learning phase.",
"The pre-training phase uses network data to construct an initial prediction model, the idea being that it provides a reasonable enough starting point such that active learning will later converge.",
"A key distinction between the two stages is how the attacker behaves: while pre-training, they sit silently to observe normal packets of the system; but while actively learning, they intervene by injecting (possibly) abnormal packets and then observe the effects.",
"It is thus important to minimise the amount of the time spent in the latter phase to avoid detection.",
"We require a series of regression models, one per sensor, that take as input the payloads of captured packets, and return as output a prediction of how the considered (true) sensor reading will evolve after a time period.",
"To achieve this goal requires a number of system-specific decisions to be made, for example, the types of packets to train the model on, and a fixed time period that is appropriate to the processes involved (some will change the physical state more quickly than others).",
"There are several types of regression models that are fit for the task.",
"In this work, we focus on two: linear models and gradient-boosting decision trees (GBDT) [35].",
"A linear model is the simplest possible choice and thus serves as our baseline, whereas the GBDT is a well-known and popular example of a non-linear model.",
"Both models can be integrated with existing active learning frameworks for regression, which was a key reason for their selection.",
"Several more expressive models, such as neural networks, do not have any good online active learning strategies (to the best of our knowledge).",
"In SWaT, packets are collected from Level 0 (see Section ) in the first four stages of the system.",
"By observing the network traffic, we identified four different types of packets in each stage based on payload lengths and headers: packets that have payloads of length (1) 10 bytes; (2) 32 bytes; (3) 22 bytes, with a source IP address of $\\mathtt {192.168.0.",
"}S\\mathtt {0}$ ; and (4) 22 bytes, with a source IP address of $\\mathtt {192.168.0.",
"}S\\mathtt {2}$ .",
"Here, $S$ is replaced with the given stage of the system (1, 2, 3, or 4).",
"Across these four stages, there are thus 16 different types of packets in total.",
"In constructing a feature vector for training, we make no assumptions about the meaning of these different packets, so select the first of each type of packet that is collected at a particular time point and concatenate their payloads together in a fixed order.",
"This leads to feature vectors containing a series of 2752 bits.",
"Figure: Input/output of a learnt model for sensor ssAlong with constructing a sufficient number of feature vectors (experimentally determined in Section ), we also query the historian for sensor values after fixed time periods have passed.",
"For flow and pressure sensors this time period is 5 seconds; for tank level sensors, it is 30 seconds, owing to the fact that they change state rather more slowly.",
"With this data collected, we train linear and GBDT models for each individual sensor in turn, such that a sensor reading can be predicted for a given bit vector of payloads from the 16 types.",
"An overview of the input/output of these models in given in Figure REF .",
"Note that for flow and pressure sensors, the corresponding models predict their future values, whereas for tank level sensors, the corresponding models predict by how much they will change.",
"This discrepancy is due to the fact that the effects of flow/pressure attacks stabilise at a final value very quickly."
],
[
"Active Learning and Attack Discovery",
"Active Learning.",
"After completing the pre-training phase, we should now have a model that is capable of making some reasonable predictions with respect to normal packets in the CPS network.",
"However, the attacks we need for testing the CPS are not necessarily composed of normal packets.",
"We need to train the model further on a broader set of examples, but cannot do it blindly owing to the expense of running the system and the enormity of the search space ($2^{2752}$ potential combinations of feature vectors in SWaT).",
"Our solution is to train the model further using (online) active learning [63], a supervised ML approach that iteratively improves the current model.",
"Theoretical studies have shown that active learning may exponentially reduce the amount of training data needed, e.g.",
"[33], [37], [38].",
"The idea is to reduce the amount of additional data by sampling examples that are estimated to maximally change the current model in some way.",
"In our case, we use one of two active learning frameworks to guide the construction of new feature vectors by flipping the bits of existing ones (this is more conservative than constructing payloads from scratch, but minimises the possibility of packet rejection).",
"Once new feature vectors have been sampled, we can decompose them into their constituent packets, spoof them in the network, observe their effects on true sensor readings, then re-train the model accordingly.",
"While active learning for classification problems is well-studied, there are limited active learning frameworks for regression, and some of the ones that exist make assumptions unsuitable for our application (e.g.",
"a Gaussian distribution [21]).",
"However, the Expected Model Change Maximization (EMCM) approach of Cai et al.",
"[17] avoids this assumption and is suitable for CPSs.",
"Their framework is based on the idea of sampling new examples that are estimated to maximally change the model itself, i.e.",
"the gradient in linear models, or a linear approximation of GBDTs based on `super features' extracted from the trees (see [17] for details).",
"Inspired by EMCM, and motivated by the fact we can query live behaviour of the system, we also propose a variant of the framework called Expected Behaviour Change Maximisation (EBCM).",
"Instead of sampling examples estimated to maximally change the model, EBCM attempts to identify examples that cause maximally different behaviour from what the system is currently exhibiting.",
"For example, if a considered sensor reading is increasing, then EBCM may identify examples that cause it to decrease as much as possible instead.",
"The intuition of the approach is that exploring different behaviour in a particular context is more informative.",
"It also seeks to check that unfamiliar packets predicted to cause that behaviour really do cause that behaviour, updating the model otherwise.",
"[!t] Expected Behaviour Change Maximisation Prediction model $M_s$ , prediction time $t_s$ , maximum number of bits to flip $n_m$ Feature (bit-)vector $p_f$ Sniff current packets and construct a feature vector $p_o$ based on their payloads; Wait for $t_s$ seconds then observe the value $v_s$ of $s$ ; Let $P := \\langle \\rangle ;$ [empty sequence] Let $D := \\langle \\rangle ;$ timeout Construct a new vector $p$ from $p_o$ by randomly selecting and flipping $n \\le n_m$ bits; $v_p := M_s(p)$ ; $P := P^\\frown \\langle p\\rangle $ ; $D := D^\\frown \\langle |v_s-v_p|\\rangle $ ; Select a feature vector $p_f$ from $P$ using Roulette Wheel Selection with corresponding fitness values in $D$ ; feature (bit-)vector $p_f$ ; Algorithm REF summarises the steps of EBCM, in which a new feature vector is constructed by sampling additional packets, randomly flipping the bits of several copies, and choosing a vector that would have led to a maximally different reading than the original.",
"Note that to ensure some variation, the feature vector is chosen from a set of several using Roulette Wheel Selection [43], which assigns to each candidate a probability of being selected based on its `fitness', here defined as the absolute difference between what the sensor reading actually became (with respect to the original packets) and what the current model predicted for the candidate.",
"If $f_i$ is the fitness of one of $n$ candidates, then its probability of being selected is $f_i / \\sum _{j=1}^n f_j$ .",
"A random number is generated between 0 and the sum of the candidates' fitness scores.",
"We then iterate through the candidates until the accumulated fitness is larger than that number, returning that final candidate as our chosen bit-vector.",
"For SWaT, we implemented both EMCM and EBCM, using the same construction of feature vectors (i.e.",
"a concatenation of the payloads of 16 types of packets).",
"Upon computing new feature vectors using these active learning frameworks, we then break the vectors down into their constituent packets, and spoof them in Level 0 of the network using Scapy.",
"After spoofing, we wait either 5 or 30 seconds (when targeting flow/pressure or tank level sensors respectively) before querying the latest sensor value, then re-train the model based on the new packets and readings observed.",
"This process is repeated until a suitable timeout condition (Section REF ).",
"Attack Discovery and Fuzzing.",
"In the final step of our approach, we use the learnt models (Figure REF ) to discover attacks, i.e.",
"packet manipulations that will drive targeted (true) sensor readings out of their safe operational ranges.",
"In particular, after choosing a sensor to target, the corresponding model is used to evaluate a number of candidate packet manipulations and reveal the one that is (predicted) to realise the attack most effectively.",
"The final part of our approach consists of generating those candidate packet manipulations for the model to evaluate.",
"[!t] Attack Discovery Prediction model $M_s$ , number of bits to flip $n$ , objective function $f$ A bit-vector $p_{max}$ Sniff current packets and construct a feature vector $p_o$ based on their payloads; Construct a sequence $\\Phi $ of (0-based) indices of $p_o$ , from the position with the highest feature importance (Section REF ) to the lowest; $k := n-1$ ; $Done := \\emptyset $ ; $f_{max} := 0$ ; $k==|\\Phi |$ or timeout $\\Phi _k := \\lbrace i \\mid i \\in \\Phi [0..k] \\rbrace $ ; $Combs := \\lbrace B \\mid B\\in 2^{\\Phi _k} \\wedge |B|=n \\wedge B \\notin Done \\rbrace $ ; $c \\in Combs$ Construct $p$ from $p_o$ by flipping $p_o[i]$ for every $i\\in c$ ; $f(M_s(p)) > f_{max}$ $f_{max} := f(M_s(p))$ ; $p_{max} := p$ $Done := Done \\cup c$ ; $k := k+1$ ; bit-vector $p_{max}$ ; Algorithm REF presents the steps of our packet manipulation procedure for attack discovery.",
"The idea of the algorithm is to identify the bits that are most important (i.e.",
"have the most influence in the prediction), generate candidates by flipping fixed numbers of those bits, before broadening the search to other, less important bits too.",
"As different candidates are generated, they are evaluated against a simple objective function that is maximised as the predicted sensor state becomes closer to an edge of its safe operational range.",
"Suppose that $v_s$ denotes a value of sensor $s$ , and that $L_s$ and $H_s$ respectively denote its lower and upper safety thresholds.",
"Let: $ d_s = \\left\\lbrace \\begin{array}{ll}\\mathrm {min}\\left( \\left| v_s - L_s \\right|, \\left| v_s - H_s \\right| \\right) & \\quad L_s \\le v_s \\le H_s \\\\0 & \\quad \\mathrm {otherwise}\\end{array}\\right.",
"$ A suitable objective function that is maximised by values approaching either of the thresholds would then be: $ f(v_s) = \\frac{1}{d_s / (H_s-L_s)} $ We calculate feature importance in one of two ways, depending on the model used.",
"For a linear model, the absolute value of the model's weight for that feature is taken as its importance.",
"For a GBDT model, since it is a boosting ensemble model with a bunch of decision trees, we average the feature importance scores of these trees to obtain the feature importance of the overall model.",
"For SWaT, we implemented attack discovery for multiple flow, pressure, and tank level sensors, and used instances of the objective function above for each of them.",
"The feature vectors returned by Algorithm REF are broken into their constituent packets, then spoofed in the network using Scapy.",
"If an attack successfully drives a targeted sensor out of its normal operational range (e.g.",
"over/underflow), we record this, adding the particular packet manipulation used to a test suite of attacks, and document it accordingly (see Section for an experimental evaluation).",
"Recall that in SWaT, the models for tank levels sensors do not predict future values directly, but rather the magnitude by which they will change by: as a consequence, Algorithm REF is adapted for these sensors by observing the current reading at the beginning, then using it to calculate the input for the objective function."
],
[
"Evaluation",
"We evaluate the effectiveness of active fuzzing for attack discovery and detection using the SWaT testbed (Section )."
],
[
"Research Questions",
"Our evaluation design is centred around the following key research questions (RQs): RQ1 (Training Time): How much time is required to learn a high-accuracy model?",
"RQ2 (Attack Discovery): Which model and active learning setup is most effective for attack discovery?",
"RQ3 (Comparisons): How does active fuzzing compare against other CPS fuzzing approaches?",
"RQ4 (Attack Detection): Can the learnt models be used for anomaly detection or early warnings?",
"RQ1 is motivated by our assumption that attackers do not have access to large offline datasets for training, and may need to evade anomaly detection systems.",
"How long would an attacker need to spend observing live sensor readings (pre-training) and spoofing packets (active learning) before obtaining a high-accuracy model?",
"RQ2 aims to explore the different combinations of our regression models with and without active learning, in order to establish which is most effective for discovering packet-level CPS attacks, and to quantify any added benefit of active learning in conquering the huge search space.",
"RQ3 is intended to check our work against a baseline, i.e.",
"its effectiveness in comparison to random search and another guided CPS network fuzzer.",
"Finally, RQ4 aims to explore whether our learnt models can have a secondary application as part of an anomaly detection or early warning system for attacks."
],
[
"Experiments and Discussion",
"We present the design of our experiments for each of the RQs in turn, as well as some tables of results and the conclusions we draw from them.",
"The programs we developed to implement these experiments on the SWaT testbed are all available online [5].",
"RQ1 (Training Time).",
"Our first RQ aims to assess the amount of time an attacker would require to learn a high-accuracy model from live packets.",
"To answer this question, we design experiments for the two phases of learning in turn.",
"First, we investigate how long the attacker must spend pre-training on normal live sensor readings (i.e.",
"without any manipulation).",
"Recall that our goal in this phase is not to obtain a highly accurate model, but rather to find a reasonable enough model as a starting point for active learning.",
"To do this, we compute the r2 scores of linear and GBDT regression models for individual sensors after training for different lengths of time.",
"An r2 score is the percentage of variation explained by the model, and reflects how well correlated the predictions of a sensor and their actual future values are.",
"Prior to training, we collect 230 minutes of packet and sensor data, splitting 180 minutes of it into a training set and the remaining 50 minutes into a test set.",
"For each sensor, we train linear and GBDT models using the full 180 minutes (our upper limit of the experiment), and compute their r2 scores using the test data.",
"We repeat this process for 10 minutes of data, then 20 minutes, ... up to 150 minutes at various intervals until it is clear that model is converging.",
"We judge that a model has converged when the importance scores of its features (see Section REF ) have stabilised up to a small tolerance (0.5% for flow/pressure sensors; 5% for level sensors) as the model is re-trained on new samples.",
"All steps are repeated ten times and medians are reported to reduce the effects of different starting states.",
"Second, given a model that has been pre-trained, we investigate how long the attacker must then spend actively learning before the model achieves a high accuracy.",
"To do this, we pre-train linear and GBDT prediction models for each sensor for the minimum amount of time previously determined (in the first experiment).",
"Then, for both variants of active learning (Section REF ), we sample new sensor data from the system and retrain the models every 5 minutes.",
"In this experiment, we record the amount of time it takes for a model to stabilise with a high r2 score, i.e.",
"above 0.9, using the same 50 minutes of test data to compute this.",
"We repeat these steps ten times and compute the medians.",
"Table: r2 scores (higher is better) of linear and GBDT sensor prediction models after different amounts of pre-trainingResults.",
"Table REF presents the results of our first experiment.",
"The columns correspond to the amount of training time (10 minutes through to 180), whereas the rows correspond to regression models for individual SWaT sensors, including Flow Indicator Transmitters (e.g.",
"FIT101), a Differential Pressure Indicator Transmitter (DPIT301), and Level Indicator Sensors (e.g. LIT101).",
"For the LITs, our models predict their values 30 seconds into the future (as tank levels rise very slowly), whereas for all other sensors our models make predictions for 5 seconds ahead.",
"The values reported in the table are r2 scores: here, a score of 1 indicates that the model and test data are perfectly correlated, whereas a score of -1 would indicate that there is no correlation at all.",
"When there is clear evidence of a model converging, we do not repeat the experiment for longer training periods (except 180 minutes, our upper limit).",
"All of our models eventually converge during pre-training, except the linear model for LIT401: the process involving this tank is too complicated to be represented as a linear model due to the multiple interactions and dependencies involving other stages of the testbed (the GBDT model does not suffer this problem).",
"Note that while pre-training leads to relatively high r2 scores for a number of the models (e.g.",
"the simpler processes involving flow), this does not necessarily imply that the models will be effective for attack discovery (as we investigate in RQ2).",
"For the goal of determining a minimum amount of pre-training time, we fix it at 40 minutes, as all models (except Linear-LIT401) exhibit some positive correlation by then ($\\ge \\!0.3$ ).",
"Some scores are still low, but this will allow us to assess whether active learning is still effective when applied to cases that lack a good pre-trained model.",
"Table: Median time (mins; lower is better) for active learning (AL) configurations to achieve an r2 score above 0.9Table REF presents the results of our second experiment.",
"Here, the columns contain the sensors that each regression model is targeting, whereas the rows contain the type of model and active learning variant considered.",
"The values reported are the number of minutes (accurate up to 5 minutes) of active learning that it takes before models achieve an r2 score above 0.9.",
"Note that with active learning, none of the linear models for tank level sensors were able to exceed our r2 threshold (although they did converge for LIT101 and LIT301 with lower scores).",
"All GBDT models were able to exceed the r2 threshold with active learning, indicating that the additional expressiveness is important for some processes of SWaT—likely because the actual processes are non-linear.",
"The amount of time required varied from 10 up to 45 minutes.",
"Taking the pre-training time into consideration: Once pre-trained on 40 minutes of data observations, attackers can accurately predict SWaT's sensor readings after 10–45 minutes of active learning.",
"This is a significantly reduced amount of time compared to SWaT's LSTM-based fuzzer [26], the model of which was trained for approximately two days on a rich dataset compiled from four days of constant operation.",
"RQ2 (Attack Discovery).",
"Our second RQ aims to assess which combinations of models and active learning setups (including no active learning at all) are most effective for finding attacks, i.e.",
"manipulations of packet payloads that would cause the true readings of a particular sensor to eventually cross one of its safety thresholds (e.g.",
"risk of overflow, or risk of bursting pipe).",
"To do this, we experimentally calculate the success rates at finding attacks for all variants of models covering the flow, pressure, and tank level sensors.",
"Furthermore, we do so while restricting the manipulation of the packets' payloads to different quantities of bit flips, from 1–5 and 10 such flips.",
"For each model variant, we calculate the success rate by running our active fuzzer 1000 times with the given model, and recording as a percentage the number of times in which the resulting modified packet would cause the physical state to crossWith the exception of the low threshold for flow sensors, which is 0. a safety threshold.",
"Note that it is important to flip existing payload bits, rather than craft packets directly, as the system's built-in validation procedures may reject them.",
"Table: Success rates (%s; higher is better) of different model configurations for finding packet manipulations (1-5 and 10 bit flips) that successfully drive SWaT's flow, pressure, and tank level readings to safety thresholdsResults.",
"Table REF presents the results of our experiment for RQ2.",
"Each sub-table reports on a restriction to a particular number of payload bit flips, ranging from 1–5 and then 10.",
"The columns contain the sensors we are attempting to drive into unsafe states, whereas the rows contain the type of model and active learning variant considered (if any).",
"The final row, Random (No Model), is discussed as part of RQ3.",
"The values recorded are success rates (%s), where 100% indicates that all 1000 model-guided bit flips would succeed, and where 0% indicates that none of them would do.",
"In the active learning models, pre-training was conducted for 40 minutes (as determined in RQ1).",
"We also include a model that was pre-trained only for 90 minutes—roughly the time to do both pre-training and active learning—to ensure a fair comparison.",
"We can draw a number of conclusions from these results.",
"First, linear models are not expressive enough in general for driving the bit flipping of payloads: their success rates for the LITs, for example, is mostly 0%, and even at 10 bit flips numbers for most sensors remain very low.",
"GBDT quite consistently outperforms the linear models, often approaching 100% success rates.",
"Like the linear models, GBDT struggled to attack the LITs for small numbers of bit flips (likely because multiple commands are needed to affect these sensors), but can attack them all once the restriction is lifted to ten bit flips.",
"The expressiveness of the underlying model is critically important: active learning alone is not enough to compensate for this.",
"Another conclusion that can be drawn from the tables relates to the significantly higher success rates for variants using active learning, both for linear models and GBDT models.",
"The combination of active learning with the expressiveness of GBDT, in particular, leads to attacks being found in all cases for the 10 bit flip restriction.",
"With active learning enabled, the difference is often significant (e.g.",
"0% vs. 100% for FIT401, 10 bit flips).",
"The results suggest that active learning is key for finding the `critical bits' in payloads, given its ability to sample and query new data.",
"Models that have only been pre-trained just recognise trends observed in normal data, and do not necessarily know which bits involved in the patterns are the critical ones for enacting an attack.",
"Active learning is effective at identifying critical bits in payloads, and can lead to significantly higher success rates in attack discovery.",
"RQ3 (Comparisons).",
"Our third RQ assesses how active fuzzing performs against two baselines.",
"First, for every sensor (as targeted in RQ2), we randomly generate 1000 $k$ -bit payload manipulations (where $k$ is 1–5 or 10) and assess for them the attack success rates (a percentage, as calculated in RQ2).",
"Second, we qualitatively compare our attacks against the ones identified by the LSTM-based fuzzer for SWaT [26] as well as an established benchmark of SWaT network attacks [41] that was manually crafted by experts.",
"Results.",
"The results of the random flipping baseline are given in the final rows of Table REF .",
"Clearly, this is not an effective strategy for finding attacks based on packet manipulation, as no success rate exceeds $0.5\\%$ .",
"This is unsurprising due to the huge search space involved.",
"Note also that for the more challenging over/underflow attacks, random bit flipping is unable to find any examples at all.",
"Regarding the LSTM-based fuzzer for SWaT [26], a side-by-side comparison is difficult to make as it does not manipulate packets but rather only issues high-level actuator commands (e.g.“OPEN MV101”).",
"Our approach is able to find attacks spanning the same range of sensed properties, but does so by manipulating the bits of packets directly (closer to the likely behaviour of a real attacker) and without the same level of network control (other than true sensor readings, which both approaches require).",
"In this sense our attacks are more elaborate than those of the LSTM fuzzer.",
"Our approach is also substantially faster: active fuzzing can train effective models in 50-85 minutes, whereas the underlying model used in [26] required approximately two whole days.",
"Our coverage of the SWaT benchmark [41] is comparable to that of [26], since both approaches find attacks spanning the same sensed properties.",
"However, all of the attacks in [41] and [26] are implemented at Level 1 of the network.",
"Active fuzzing instead generates packet-manipulating attacks at Level 0, which has the advantage of avoiding interactions with the PLC code, possibly making manipulations harder to detect (e.g.",
"bypassing command validation checks).",
"In this sense, the attacks that active fuzzing finds complement and enrich the benchmark.",
"Active fuzzing finds attacks covering the same sensors as comparable work, but with significantly less training time, and by manipulating packets directly.",
"RQ4 (Attack Detection).",
"Our final RQ considers whether our learnt models can be used not only for attack discovery but also attack prevention.",
"In particular, we investigate their use in two defence mechanisms: an anomaly detector and an early warning system.",
"We then assess how effective they are detecting attacks.",
"Table: Success rates (%s; higher is better) of different anomaly detector models at detecting injected sensor valuesTo perform anomaly detection, we continuously perform the following process: we read the current values of sensors, and then use our learnt models to predict their values 5 seconds into the future (30 seconds for tank levels).",
"After 5 (or 30) seconds have passed, the actual values $v_a$ are compared with those that were predicted $v_p$ , and an anomaly is reported if $|v_p-v_a|/v_m>0.05$ (or $|v_p-v_a|>5$ for tanks), where $v_m$ is the largest possible observable value for the sensor.",
"To evaluate the effectiveness of this detection scheme, we implement an experiment on actual historian data extracted from SWaT [41].",
"For each sensor in turn, we randomly generate 1000 spoofed sensor values by randomly adding or subtracting values (in the range 5-10 for LITs, or $0.05v_m$ through to $0.1v_m$ for the others) to sensor readings at different points of the data.",
"We then use our learnt models to determine what would have been predicted from the data 5 or 30 seconds earlier, comparing the actual and predicted values as described.",
"We record the success rates of our anomaly detectors at detecting these spoofed sensor readings.",
"Our early warning system is set up in a similar way, continuously predicting the future readings of sensors based on the current network traffic.",
"The key difference is that rather than comparing actual values with previously predicted values, we instead issue warnings at the time of prediction if the future value of a sensor is outside of its well-defined normal operational range.",
"To experimentally assess this, we manually subject the SWaT testbed to the Level 1 attacks identified in [26] (which itself covers more unsafe states that the SWaT benchmark [41]), targeting each sensor in turn.",
"When each attack is underway, we use our learnt models to predict the future sensor readings.",
"If a warning is issued at some point before a sensor is driven outside of its normal range, we record this as a success.",
"We repeat this ten times for each sensor.",
"Results.",
"Table REF contains the results of our anomaly detection experiment.",
"The columns indicate the sensors for which values in the data were manipulated, whereas the rows indicate the model and active learning variant used.",
"The values are the success rates, i.e.",
"the percentage of spoofed sensor values that were detected as anomalous.",
"Asterisks ($\\ast $ ) indicate where false positive rates were above $5\\%$ , meaning the anomaly detectors were not practically useful.",
"For the flow and pressure sensors, most variants of model and active learning were able to successfully detect anomalies, the main exception being FIT401 for which the linear model performed poorly.",
"The tank level sensors were more challenging to perform anomaly detection for, but the GBDT models have a clear edge over the linear ones.",
"Active learning made little difference across the experiments, except to improve the accuracy of the original 40 minute pre-trained models.",
"Table: Success rates (%s; higher is better) of different models at warning before sensors exit their safe rangesTable REF contains the results of our early warning detection experiment.",
"The columns indicate the sensed properties that were targeted by attacks (e.g.",
"drive LIT101 outside of its safe range), whereas the rows indicate the model and active learning variant used.",
"The values are the success rates, i.e.",
"the percentages of attacks that were warned about before succeeding.",
"Cells containing dashes (—) indicate that more than 5% of the warnings were false positive, and thus too unreliable.",
"The first thing to note is that the experiment only considered the tank level sensors: this is because the flow and pressure sensors can be forced into unsafe states very quickly, requiring more immediate measures than an early warning system.",
"The tanks however take time to fill up or empty, and thus are a more meaningful target for this solution.",
"Second, the model has a clear impact: GBDT models with either active learning or at least 90 minutes of pre-training are accurate enough to warn about 100% of the attacks, whereas the linear models are not expressive enough and suffer from false positives.",
"Again, active learning improves the accuracy of 40 minute pre-trained models sufficiently, but otherwise is not critical: its key role is not in prevention but in discovering attacks, through its ability to identify the critical bits to manipulate.",
"Our models can be repurposed as anomaly detectors or early warning systems, but active learning is not as critical here as in attack discovery."
],
[
"Threats to Validity",
"While our work has been extensively evaluated on a real critical infrastructure testbed, threats to the validity of our conclusions of course remain.",
"First, while SWaT is a fully operational water treatment testbed, it is not as large as the plants it is based on, meaning our results may not scale-up (this is difficult to assess, as access to such plants is subject to strict confidentially).",
"Second, it may not generalise to CPSs in domains that have different operational characteristics, motivating future work to properly assess this.",
"Finally, while our anomaly detector performed well, our sensor spoofing attacks were generated randomly, and may not be representative of a real attacker's behaviour (note however that the early warning system was assessed using previously documented attacks).",
"Similarly, our early warning detection systems performed well at detecting known over/underflow attacks, but these attacks are of the kind that active fuzzing itself can generate: how the models perform against different kinds of attacks requires further investigation."
],
[
"Related Work",
"In this section, we highlight a selection of the literature that is related to the main themes of this paper: learning from traffic (including active learning), defending CPSs, and testing/verifying CPSs.",
"Learning from Network Traffic.",
"The application of machine learning to network traffic is a vibrant area of research [68], but models are typically constructed to perform classification tasks.",
"To highlight a few examples: Zhang et al.",
"[84] combine supervised and unsupervised ML to learn models that can classify zero-day traffic; Nguyen and Armitage [67] learn from statistical features of sub-flows to classify between regular consumer traffic and traffic originating from online games; and Atkinson et al.",
"[16] use a classifier to infer personal information by analysing encrypted traffic patterns caused by mobile app usage.",
"All these examples are in contrast to active fuzzing, where regression models are learnt for predicting how a set of network packets will cause a (true) sensor reading of a CPS to change.",
"We are not aware of other work building regression models in a similar context.",
"Similar to active fuzzing, there are some works that apply active learning, but again for the purpose of classification, rather than regression.",
"Morgan [65], for example, uses active learning to reduce training time for streaming data classifiers, as do Zhao and Hoi [86] but for malicious URL classifiers.",
"Defending CPSs.",
"Several different research directions on detecting and preventing CPS attacks have emerged in the last few years.",
"Popular approaches include anomaly detection, where data logs (e.g.",
"from historians) are analysed for suspicious events or patterns [27], [45], [52], [69], [11], [15], [48], [58], [61], [66], [71]; digital fingerprinting, where sensors are checked for spoofing by monitoring time and frequency domain features from sensor and process noise [12], [13], [44], [56]; and invariant-based checks, where conditions over processes and components are constantly monitored [18], [8], [7], [24], [10], [25], [28], [39].",
"These techniques are meant to complement and go beyond the built-in validation procedures installed in CPSs, which typically focus on simpler and more localised properties of the system.",
"The strengths and weaknesses of different countermeasures has been the focus of various studies.",
"Urbina et al.",
"[77] evaluated several attack detection mechanisms in a comprehensive review, concluding that many of them are not limiting the impact of stealthy attacks (i.e.",
"from attackers who have knowledge about the system's defences), and suggest ways of mitigating this.",
"Cárdenas et al.",
"[19] propose a general framework for assessing attack detection mechanisms, but in contrast to the previous works, focus on the business cases between different solutions.",
"For example, they consider the cost-benefit trade-offs and attack threats associated with different methods, e.g.",
"centralised vs. distributed.",
"As a testbed dedicated for cyber-security research, many different countermeasures have been developed for SWaT itself.",
"These include anomaly detectors, typically trained on the publicly released dataset [41], [2] using unsupervised learning techniques, e.g.",
"[52], [58], [42].",
"A supervised learning approach is pursued by [24], [25], who inject faults into the PLC code of (a high-fidelity simulator) in order to obtain abnormal data for training.",
"Ahmed et al.",
"[12], [13] implemented fingerprinting systems based on sensor and process noise for detecting spoofing.",
"Adepu and Mathur [8], [7], [10] systematically and manually derived physics-based invariants and other conditions to be monitored during the operation of SWaT.",
"Feng et al.",
"[32] also generate invariants, but use an approach based on learning and data mining that can capture noise in sensor measurements more easily than manual approaches.",
"Testing and Verifying CPSs.",
"Several authors have sought to improve the defences of CPSs by constructing or synthesising attacks that demonstrate flaws to be fixed.",
"Liu et al.",
"[62] and Huang et al.",
"[50], for example, synthesise attacks for power grids that can bypass bad measurement detection systems and other conventional monitors.",
"Dash et al.",
"[30] target robotic vehicles, which are typically protected using control-based monitors, and demonstrate three types of stealthy attacks that evade detection.",
"Uluagac et al.",
"[76] presented attacks on sensory channels (e.g.",
"light, infrared, acoustic, and seismic), and used them to inform the design of an intrusion detection system for sensory channel threats.",
"Active fuzzing shares this goal of identifying attacks in order to improve CPS defences.",
"Fuzzing is a popular technique for automatically testing the defences of systems, by providing them with invalid, unexpected, or random input and monitoring how they respond.",
"Our active fuzzing approach does exactly this, guiding the construction of input (network packets) using prediction models, and then observing sensor readings to understand how the system responds.",
"The closest fuzzing work to ours is [26], which uses an LSTM-based model to generate actuator configurations, but requires vast amounts of data and system access to function effectively.",
"Fuzzing has also been applied for testing CPS models, e.g.",
"CyFuzz [29] and DeepFuzzSL [72], which target models developed in Simulink.",
"Outside of the CPS domain, several fuzzing tools are available for software: American fuzzy lop [83], for example, uses genetic algorithms to increase the code coverage of tests; Cha et al.",
"[22] use white-box symbolic analysis on execution traces to maximise the number of bugs they find; and grammar-based fuzzers (e.g.",
"[49], [40]) use formal grammars to generate complex structured input, such as HTML/JavaScript for testing web browsers.",
"Fuzzing can also be applied to network protocols in order to test their intrusion detection systems (e.g. [79]).",
"Our work, in contrast, assumes that an attacker has already compromised the network (as per Section ).",
"There are techniques beyond fuzzing available for analysing CPS models in Simulink.",
"A number of authors have proposed automated approaches for falsifying such models, i.e.",
"for finding counterexamples of formal properties.",
"To achieve this, Yamagata et al.",
"[14], [81] use deep reinforcement learning, and Silvetti et al.",
"[73] use active learning.",
"Chen et al.",
"[23] also use active learning, but for mining formal requirements from CPS models.",
"Note that unlike these approaches, active fuzzing is applied directly at the network packet level of a real and complex CPS, and therefore does not make any of the abstractions that modelling languages necessitate.",
"A number of approaches exist that allow for CPSs to be formally verified or analysed.",
"These typically require a formal specification or model, which, if available in the first place, may abstract away important complexities of full-fledged CPS processes.",
"Kang et al.",
"[55], for example, construct a discretised first-order model of SWaT's first three stages in Alloy, and analyse it with respect to some safety properties.",
"This work, however, uses high-level abstractions of the physical process, only partially models the system, and would not generalise to the packet-level analyses that active fuzzing performs.",
"Sugumar and Mathur [75] analyse CPSs using timed automata models, simulating their behaviour under single-stage single-point attacks.",
"Castellanos et al.",
"[20], McLaughlin et al.",
"[64], and Zhang et al.",
"[85] perform formal analyses based on models extracted from the PLC programs, whereas Etigowni et al.",
"[31] analyse information flow using symbolic execution.",
"If a CPS can be modelled as a hybrid system, then a number of formal techniques may be applied, including model checking [34], [80], SMT solving [36], reachability analysis [54], non-standard analysis [47], process calculi [59], concolic testing [57], and theorem proving [70].",
"Defining a formal model that accurately characterises enough of the CPS, however, is the hardest part, especially for techniques such as active fuzzing that operate directly at the level of packet payloads."
],
[
"Conclusion",
"We proposed active fuzzing, a black-box approach for automatically building test suites of packet-level CPS network attacks, overcoming the enormous search spaces and resource costs of such systems.",
"Our approach learnt regression models for predicting future sensor values from the binary string payloads of network packets, and used these models to identify payload manipulations that would achieve specific attack goals (i.e.",
"pushing true sensor values outside of their safe operational ranges).",
"Key to achieving this was our use of online active learning, which reduced the amount of training data needed by sampling examples that were estimated to maximally improve the model.",
"We adapted the EMCM [17] active learning framework to CPSs, and proposed a new version of it that guided the process by maximising behaviour change.",
"We presented algorithms for implementing active fuzzing, but also demonstrated its efficacy by implementing it for the SWaT testbed, a multi-stage water purification plant involving complex physical and chemical processes.",
"Our approach was able to achieve comparable coverage to an established benchmark and LSTM-based fuzzer, but with significantly less data, training time, and resource usage.",
"Furthermore, this coverage was achieved by more sophisticated attacks than those of the LSTM-based fuzzer, which can only generate high-level actuator commands and is unable to manipulate packets directly.",
"Finally, we showed that the models constructed in active learning were not only useful for attack discovery, but also for attack detection, by implementing them as anomaly detectors and early warning systems for SWaT.",
"We subjected the plant to a series of random sensor-modification attacks as well as existing actuator-manipulation attacks, finding that our most expressive learnt models were effective at detecting them.",
"We are grateful to the three anonymous ISSTA referees for their very constructive feedback.",
"This research / project is supported by the National Research Foundation, Singapore, under its National Satellite of Excellence Programme “Design Science and Technology for Secure Critical Infrastructure” (Award Number: NSoE_DeST-SCI2019-0008).",
"Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore.",
"It is also supported in part by a Major Scientific Research Project of Zhejiang Lab (2018FD0ZX01), Alibaba-Zhejiang University Joint Institute of Frontier Technologies, Zhejiang Key R&D Plan (2019C03133), the Fundamental Research Funds for the Central Universities (2020QNA5021), and by an SUTD-ZJU IDEA Grant for Visiting Professor (SUTD-ZJU VP 201901)."
]
] | 2005.14124 |
[
[
"Controlling vortical motion of particles in two-dimensional driven\n superlattices"
],
[
"Abstract We demonstrate the control of vortical motion of neutral classical particles in driven superlattices.",
"Our superlattice consists of a superposition of individual lattices whose potential depths are modulated periodically in time but with different phases.",
"This driving scheme breaks the spatial reflection symmetries and allows an ensemble of particles to rotate with an average angular velocity.",
"An analysis of the underlying dynamical attractors provides an efficient method to control the angular velocities of the particles by changing the driving amplitude.",
"As a result, spatially periodic patterns of particles showing different vortical motion can be created.",
"Possible experimental realizations include holographic optical lattice based setups for colloids or cold atoms."
],
[
"colorlinks=true, citecolor=blue, urlcolor=blue, linkcolor=blue red Controlling vortical motion of particles in two-dimensional driven superlattices Aritra K. Mukhopadhyay [email protected] Zentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Peter Schmelcher [email protected] Zentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany We demonstrate the control of vortical motion of neutral classical particles in driven superlattices.",
"Our superlattice consists of a superposition of individual lattices whose potential depths are modulated periodically in time but with different phases.",
"This driving scheme breaks the spatial reflection symmetries and allows an ensemble of particles to rotate with an average angular velocity.",
"An analysis of the underlying dynamical attractors provides an efficient method to control the angular velocities of the particles by changing the driving amplitude.",
"As a result, spatially periodic patterns of particles showing different vortical motion can be created.",
"Possible experimental realizations include holographic optical lattice based setups for colloids or cold atoms.",
"Due to their experimental controllability, driven lattice potentials have become an important test bed for the exploration of non-equilibrium physical phenomena [1], [2], [3].",
"The inherent non-linearity and tunable symmetries in these systems allow us to realize different non-equilibrium transport phenomena, the `ratchet effect' being one of them [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"A ratchet rectifies random particle motion into unidirectional particle transport in an unbiased non-equilibrium environment.",
"Certain spatio-temporal symmetries of the system need to be broken in order to realize it [16], [17], [18].",
"This leads to numerous applications across different disciplines, such as controlling the transport of atomic ensembles in ac-driven optical lattices [19], [20] both in the ultracold quantum [1] and classical regimes [2], [12], colloidal transport in driven holographic optical lattices [21], particle separation based on physical properties [22], [23], [24] and motion of vortices in type-II superconductors [25], [26], [27].",
"Due to the widespread applicability of such directed transport, there has been extensive research to control the strength and direction of the ratchet current.",
"Setups using one dimensional (1D) driven lattices have been shown to effectively accelerate, slow down or even completely reverse the direction of transport [18], [28], [29].",
"Two dimensional (2D) driven lattices on the other hand offer a higher variability in terms of transport direction and for particles to be transported parallel, orthogonal or at any arbitrary angle with respect to the direction of the driving force [21], [30], [31].",
"In contrast to 1D, the 2D ratchet setups also allow for the possibility to convert random particle motion into rotational or vortical motion leading to non-zero angular velocity of the particles.",
"This is particularly interesting since it provides a method to realize rotational motion of neutral particles analogous to the motion of charged particles in a magnetic field.",
"In fact, similar mechanisms have been used to generate artificial magnetic fields for exploring topological quantum states with cold neutral atoms in periodically modulated lattices [32], [33].",
"However, the extensive research on symmetry-breaking induced directed transport in the classical regime has mostly focused on translational currents and the control of rotational currents has remained largely unexplored.",
"The few existing setups either lead to a diffusive vortical motion over an extended space [34] or requires specially tailored potentials [35], [36] and temporally correlated colored noise [37], [38].",
"Furthermore, due to the lack of spatial tunability of the underlying lattice potential, these setups do not allow patterns of multiple vortices in space analogous to the different spatial configurations of artificial magnetic fluxes in the quantum regime [39].",
"In this work, we address these key limitations and present a setup to realize controllable rotational motion of classical particles along closed spatial paths in driven superlattices.",
"The individual lattices are modeled by a periodic arrangement of Gaussian potential wells whose depths can be individually modulated in a time-periodic manner.",
"We show that modulating different wells with the same driving amplitude but different driving phases allow us to break the relevant symmetries and generate non-zero average angular velocities for an ensemble of particles.",
"The angular velocities of individual trajectories can be controlled by varying the driving amplitude.",
"Additionally, we demonstrate periodic spatial arrangements of different types of rotational motion by modulating the different potential wells with different driving amplitudes and phases.",
"We consider $N$ non-interacting classical particles of mass $m$ in a 2D potential landscape $V\\left({\\bf r}\\equiv (x,y,0),t\\right)$ =$\\sum _{m,n=-\\infty }^{+\\infty } \\tilde{U}_{mn}(t) e^{-\\beta \\left( {\\bf r} - {\\bf r}_{mn}\\right)^2}$ formed by a lattice of 2D Gaussian wells centered at positions ${\\bf r}_{mn}=(mL,nL,0)$ , $m,n$ $\\in \\mathbb {Z}$ .",
"The depths of the wells are modulated periodically in time by the site-dependent driving law $\\tilde{U}_{mn}(t)=\\tilde{V}_{mn}\\left(\\cos (\\omega t +\\phi _{mn})-1\\right)$ with driving frequency $\\omega $ , driving amplitude $\\tilde{V}_{mn}$ and a temporal phase shift $\\phi _{mn}$ .",
"Introducing dimensionless variables ${\\bf r}^{\\prime }=\\frac{{\\bf r}}{L}$ and $t^{\\prime }=\\omega t$ and dropping the primes for simplicity, the equation of motion for a single particle at position ${\\bf r}=(x,y,0)$ with velocity ${\\bf \\dot{r}}=(\\dot{x},\\dot{y},0)$ reads $\\ddot{\\bf r}+\\gamma \\dot{\\bf r} =\\sum _{m,n=-\\infty }^{+\\infty } 2 \\alpha U_{mn}(t) \\left( {\\bf r} - {\\bf R}_{mn} \\right) e^{-\\alpha ({\\bf r} - {\\bf R}_{mn})^2} + \\xi (t)$ where $U_{mn}(t)=V_{mn}\\left(\\cos (t +\\phi _{mn})-1\\right)$ is the effective site dependent driving law with time period $T=2\\pi $ and driving amplitude $V_{mn}=\\frac{\\tilde{V}_{mn}}{m\\omega ^2 L^2}$ .",
"${\\bf R}_{mn}=(m,n,0)$ denotes the positions of the Gaussian wells, $\\gamma =\\frac{\\tilde{\\gamma }}{m\\omega }$ is the effective dissipation coefficient and the parameter $\\alpha =\\beta L^2$ is a measure of the widths of the wells.",
"$\\xi (t)=(\\xi _x,\\xi _y,0)$ denotes thermal fluctuations modeled by Gaussian white noise of zero mean with the property $\\langle \\xi _i (t) \\xi _j (t^{\\prime })\\rangle = 2 D\\delta _{ij}\\delta (t-t^{\\prime })$ where $i,j \\in {x,y}$ and $D=\\frac{\\tilde{\\gamma } k_B \\mathcal {T}}{m\\omega ^2 L^2}$ is the dimensionless noise strength with $\\mathcal {T}$ and $k_B$ denoting the temperature and Boltzmann constant respectively.",
"Unless mentioned otherwise, we choose $V_{mn}=V$ for all the wells, $\\alpha =3$ and $\\gamma =0.1$ .",
"The set of all wells arranged periodically in space with a specific value of the driving phase $\\phi _{mn}$ forms a sublattice of our system.",
"Our setup is hence a driven superlattice formed by the superposition of different sublattices, each driven with a distinct driving phase $\\phi _{mn}$ .",
"Possible experimental realizations of such a 2D potential include holographic optical lattices [40], [41], [42], [43], [21] or optical superlattices [44] with the lattice depth modulated via standard amplitude modulation techniques [45], [46].",
"The rotational dynamics of particles in such a setup could be observed with colloidal particles or with cold atoms in the classically describable regime of microkelvin temperatures [21], [12].",
"Figure: Schematic representation of the two superlattice setups A and B formed by the superposition of four square sublattices driven with an amplitude VV but at different phases φ i =(i-1)π 2\\phi _i=\\frac{(i-1)\\pi }{2}, i=1,2,3,4i=1,2,3,4.",
"Each colored (red) circle denotes the position of an individual Gaussian well.",
"The thick dashed lines in black denote the boundary of the lattice unitcells.",
"The spatial period of setup A is (2,2,0)(2,2,0) whereas that of setup B is (3,3,0)(3,3,0) due to the presence of empty sites without any wells.",
"The blue and green regions in Fig.",
"(a) denote plaquettes having clockwise and anti-clockwise chirality with respect to the spatial orientation of the wells with driving phases φ i \\phi _i.",
"Remaining parameters are: V=0.41V=0.41, α=3\\alpha =3, γ=0.1\\gamma =0.1.The asymptotic dynamics of particles in our setup can be either confined within a lattice unitcell such as in linear oscillatory motion or vortical motion along arbitrary closed spatial curves.",
"There can also be unconfined diffusive or ballistic motion throughout the lattice.",
"Different particles exhibiting vortical motion can, in general, possess different angular velocities.",
"Hence in order to distinguish vortical motion of a trajectory from ballistic, diffusive and vortical dynamics of other trajectories, we use the angular velocity ${\\bf \\Omega }(t) = \\left[{\\bf \\dot{r}}(t) \\times {\\bf \\ddot{r}}(t)\\right]/{\\bf \\dot{r}}^2(t)$ which is equivalent to the definition of curvature of planar curves measuring the speed of rotation of the velocity vector about the origin [34], [47].",
"Since the particle dynamics is confined to the $xy$ plane, the only possible non-zero component of ${\\bf \\Omega }(t)$ is along ${\\bf \\hat{z}}$ , the unit vector along the $z$ direction.",
"The mean angular velocity of a trajectory is defined as ${\\bf \\bar{\\Omega }}=\\frac{1}{t}\\lim \\limits _{t\\rightarrow \\infty } \\int _{0}^{t} {\\bf \\Omega }(t^{\\prime }) dt^{\\prime }$ .",
"For trajectories rotating along a closed spatial curve with period $\\eta T$ , the mean angular velocity can be expressed as ${\\bf \\bar{\\Omega }}=\\frac{2\\pi \\tau }{\\eta T} {\\bf \\hat{z}}=\\frac{\\tau }{\\eta } {\\bf \\hat{z}}$ (since $T=2\\pi $ ), where $2\\pi \\tau $ denotes the total curvature of the curve with the turning number $\\tau $ defined as the number of times the velocity vector winds about its origin [48].",
"The net rotational current, defined as the mean angular velocity of an ensemble of particles with different initial conditions, is given by ${\\bf J_{\\Omega }}=\\langle {\\bf \\bar{\\Omega }} \\rangle $ where $\\langle ... \\rangle $ denotes the average over all trajectories.",
"Since the only possible non-zero components of ${\\bf \\Omega }(t),{\\bf \\bar{\\Omega }}$ and ${\\bf J_{\\Omega }}$ is along ${\\bf \\hat{z}}$ , we drop the symbol ${\\bf \\hat{z}}$ henceforth.",
"The necessary condition for any setup to exhibit a net rotational current is to break the symmetries which keeps the system invariant but changes the sign of the angular velocity ${\\bf \\Omega }(t)$ [34].",
"There are only two symmetry transformations which can change the sign of ${\\bf \\Omega }(t)$ : (i) time reversal together with optional spatial inversion and space-time translations: $S_{t}$ : $t\\longrightarrow -t + t^{\\prime }$ , $\\mathbf {r}\\longrightarrow \\pm \\mathbf {r} + \\delta $ and (ii) parity or reflection $\\mathcal {P}$ about any plane perpendicular to the $xy$ plane with optional spatial rotation $\\mathcal {R}$ in the $xy$ plane and space-time translations: $S_{p}$ : $\\mathbf {r}\\longrightarrow \\mathcal {R}\\left(\\mathcal {P}\\mathbf {r}\\right) + \\delta $ , $t\\longrightarrow t + t^{\\prime }$ .",
"Since our setup is dissipative, $S_{t}$ is broken independent of our choice of the lattice potential $V\\left({\\bf r},t\\right)$ .",
"However, the superlattice potential allows us to preserve or break the symmetry $S_{p}$ by controlling the driving phases of the underlying sublattices.",
"In order to illustrate this, we consider two setups A and B (Figs.",
"REF (a,b)) each consisting of four square sublattices with the same driving amplitude $V=0.41$ but different phases $\\phi _{i}=\\frac{(i-1)\\pi }{2}$ , $i=1,2,3,4$ .",
"The sublattices in setup A have lattice vectors $(2,0,0)$ and $(0,2,0)$ , hence the setup has a spatial period ${\\bf L_A}=(2,2,0)$ .",
"In contrast, the setup B has a spatial period ${\\bf L_B}=(3,3,0)$ with the lattice vectors being $(3,0,0)$ and $(0,3,0)$ .",
"As shown in Fig.",
"REF (a), the arrangement of the sublattices allows us to consider the unitcell of the setup A as a collection of four distinct spatial domains or plaquettes.",
"The plaquettes are characterized by clockwise or counter-clockwise arrangement of Gaussian wells with driving phase $\\phi _{i}$ , i.e.",
"of opposite chirality.",
"Since the parity transformation $S_{p}$ reverses chirality, each of these plaquettes break the $S_{p}$ symmetry.",
"However since the unitcell has equal number of plaquettes with opposite chirality (two clockwise and two anti-clockwise), the unitcell and hence the entire setup A is symmetric with respect to $S_{p}$ .",
"This implies that although the setup A might allow trajectories with different mean angular velocities ${\\bf \\bar{\\Omega }}$ , the net rotational current ${\\bf J_{\\Omega }}$ must be zero.",
"In contrast, the entire unitcell of setup B has an anti-clockwise chirality which can be reversed by $S_{p}$ and hence the setup B breaks $S_{p}$ symmetry.",
"As a result one can expect ${\\bf J_{\\Omega }}$ to be non-zero.",
"Figure: Typical trajectories exhibiting rotational motion in (a) setup A and (c) setup B respectively over one time period of rotation (in colorbars).",
"The colored circles denote the positions of individual Gaussian wells with different driving phases φ i \\phi _i.",
"Figures (b) and (d) show the fraction of particles ρ(Ω ¯)\\rho ({\\bf \\bar{\\Omega }}) possessing mean angular momentum Ω ¯{\\bf \\bar{\\Omega }} for different noise strengths DD in setup A and B respectively.",
"The insets show the variation of the net rotational current 𝐉 Ω {\\bf J_{\\Omega }} with DD.",
"Remaining parameters are the same as in Fig.",
".In order to verify our symmetry analysis and explore the behavior of rotational current in our system, we initialize $N=10^4$ particles randomly within a square region $x,y \\in [-100,100]\\times [-100,100]$ in both setups A and B with small random velocities $v_x,v_y \\in [-0.1,0.1]$ .",
"Subsequently we time evolve our ensemble up to time $t_f= 10^4 T$ by numerical integration of Eq.",
"REF for different noise strength $D$ .",
"In the deterministic limit $D=0$ , all the particles in setup A exhibit only rotational motion along closed curves either with mean angular momentum ${\\bf \\bar{\\Omega }}=\\frac{1}{2} $ (vortex) or $-\\frac{1}{2} $ (antivortex).",
"Fig.",
"REF (a) shows a typical trajectory in this setup having ${\\bf \\bar{\\Omega }}=-\\frac{1}{2} $ .",
"The velocity vector winds around its origin in clockwise direction once during the period of rotation $2T$ , hence $\\tau =-1$ and $\\eta =2$ .",
"The vortical motion persists as the noise strength is increased to $D=0.001$ .",
"However most importantly, there exists an equal number of trajectories possessing ${\\bf \\bar{\\Omega }}=-\\frac{1}{2} $ and ${\\bf \\bar{\\Omega }}=\\frac{1}{2} $ signifying that the net rotational current ${\\bf J_{\\Omega }}=0$ (Fig.",
"REF (b)), as predicted by our symmetry analysis.",
"Even for higher noise strength up to $D=0.003$ , such a symmetry related cancellation of vortex-antivortex pairs with equal and opposite angular momentum persists, leading to a zero net rotational current.",
"Beyond $D>0.003$ , the vortical motion is destroyed resulting in a symmetric distribution of particles around ${\\bf \\bar{\\Omega }}=0$ and hence ${\\bf J_{\\Omega }}=0$ .",
"The particles in setup B also exhibit rotational motion, however unlike in setup A, all the particles in setup B possess a mean angular momentum ${\\bf \\bar{\\Omega }}=\\frac{3}{5} =0.6$ .",
"An example trajectory in setup B in the deterministic limit can be seen in Fig.",
"REF (c).",
"The velocity vector makes four anti-clockwise (at the four corners of the curve) and one clockwise (corresponding to one full rotation along the curve) winding around its origin during one period of rotation $5T$ , hence $\\tau =3$ and $\\eta =5$ .",
"For $D\\leqslant 0.002$ , the vortical motion is quite stable and almost all the particles in the setup rotate with ${\\bf \\bar{\\Omega }}=0.6$ resulting in ${\\bf J_{\\Omega }}=0.6 $ (Fig.",
"REF (d)) in accordance with our symmetry analysis.",
"For $D>0.002$ , the particles perform diffusive motion through the lattice and the vortical motion is gradually destroyed thus decreasing the value of ${\\bf J_{\\Omega }}$ .",
"Figure: (a) Bifurcation diagram of Ω ' (t){\\bf \\Omega ^{\\prime }}(t) as a function of the driving amplitude VV depicting the chaotic (broad blue bands) and regular (thin blue lines) attractors of the setup B (see Fig. (b)).",
"(b) The mean angular momentum Ω ¯{\\bf \\bar{\\Omega }} of the attractors in Fig.",
"(a) as a function of VV.",
"The values of Ω ¯{\\bf \\bar{\\Omega }} for the regular attractors denoting rotational motion and the turning number τ\\tau of the corresponding closed curves are labeled with arrows.",
"Remaining parameters are the same as in Fig.",
"(b).The question that naturally arises is that once we design a driven superlattice which breaks the $S_{p}$ symmetry, for e.g.",
"our setup B, can we predict the value of ${\\bf J_{\\Omega }}$ apriori?",
"Specifically, how does the mean angular momentum ${\\bf \\bar{\\Omega }}$ of the trajectories depend on the system parameters?",
"For a driven dissipative non-linear system like the present one, this can be answered by analyzing the asymptotic $t\\rightarrow \\infty $ particle dynamics in the deterministic limit $D=0$ .",
"The asymptotic dynamics of the particles is governed by the set of attractors underlying the phase space of the system, which can be of two types: (i) regular attractors denoting ballistic, linear oscillatory and rotational motions (ii) chaotic attractors denoting diffusive motion.",
"In order to distinguish between attractors corresponding to rotational motion as compared to the others, we introduce a slightly modified angular momentum vector ${\\bf \\Omega ^{\\prime }}(t) = \\left[{\\bf \\dot{r}}(t) \\times {\\bf \\ddot{r}}(t)\\right]/\\left[|{\\bf \\dot{r}}(t)||{\\bf \\ddot{r}}(t)|\\right]$ .",
"Note that ${\\bf \\Omega ^{\\prime }}(t)=\\sin \\vartheta (t) \\hspace{2.84544pt} {\\bf \\hat{z}}$ where $\\vartheta (t)$ denotes the instantaneous angle between the velocity and acceleration vectors of the particle.",
"${\\bf \\Omega ^{\\prime }}(t)$ transforms under $S_{p}$ and $S_{t}$ in exactly the same way as ${\\bf \\Omega }(t)$ .",
"However since the values of ${\\bf \\Omega ^{\\prime }}(t)$ are bounded in the interval $[-1,1]$ , as opposed to ${\\bf \\Omega }(t)$ which becomes large for small values of ${\\bf \\dot{r}}(t)$ , it is a good quantity to differentiate between chaotic and regular rotational dynamics of particles.",
"To illustrate this, we inspect the bifurcation diagram of ${\\bf \\Omega ^{\\prime }}(t)$ in Fig.",
"REF (a) as a function of the driving amplitude $V$ for our setup B by initializing particles with random position and velocities and stroboscopically monitoring ${\\bf \\Omega ^{\\prime }}(t)$ after an initial transient [49].",
"For certain ranges of values of $V$ , all the particles in the setup exhibit chaotic motion (broad blue bands in Fig.",
"REF (a)) such that ${\\bf \\Omega ^{\\prime }}(t)$ takes all possible values in the range $[-1,1]$ .",
"For all other values of $V$ , they perform regular periodic motion resulting in only specific values of ${\\bf \\Omega ^{\\prime }}(t)$ .",
"Most of these periodic motions correspond to particles performing rotational motion with different non-zero ${\\bf \\bar{\\Omega }}$ (except for $0.19\\lesssim V\\lesssim 0.25$ ) depending on the value of $V$ as shown in Fig.",
"REF (b).",
"This provides an efficient method to design and control the angular momentum of the trajectories in our setup by simply choosing the desired driving amplitude $V$ .",
"Our previous results (see Figs.",
"REF (c,d)) is such an example for the setup B with $V=0.41$ .",
"Figure: (a) Schematic representation of one unitcell of our setup consisting of four plaquettes 𝒟 1 \\mathcal {D}_1, 𝒟 2 \\mathcal {D}_2, 𝒟 3 \\mathcal {D}_3 and 𝒟 4 \\mathcal {D}_4 with the thick dashed lines denoting the plaquette boundaries.",
"The color filled circles denote the positions of individual Gaussian wells driven with amplitudes V 1 =0.51V_1=0.51 or V 2 =0.078V_2=0.078 and phases φ i \\phi _i.",
"𝒟 1 \\mathcal {D}_1 and 𝒟 4 \\mathcal {D}_4 (𝒟 2 \\mathcal {D}_2 and 𝒟 3 \\mathcal {D}_3) have anti-clockwise (clockwise) chirality with respect to the spatial orientation of the wells with driving phases φ i \\phi _i.",
"Trajectories of particles exhibiting vortical motion for D=0D=0 with positive (red) and negative (blue) Ω ¯{\\bf \\bar{\\Omega }} have been superimposed on the unitcell.",
"The trajectories in 𝒟 1 \\mathcal {D}_1, 𝒟 2 \\mathcal {D}_2, 𝒟 3 \\mathcal {D}_3 and 𝒟 4 \\mathcal {D}_4 have Ω ¯=-1{\\bf \\bar{\\Omega }}=-1, 1, -1 3-\\frac{1}{3} and 1 3\\frac{1}{3} respectively.",
"An extract of the spatial arrangements of the trajectories exhibiting vortical motion within different plaquettes for D=10 -4 D=10^{-4} and D=10 -3 D=10^{-3} is shown in (b) and (c) respectively.",
"Remaining parameters are the same as in Fig.",
".The ability to control the angular momentum of the particles with different driving amplitude $V$ allows us to design lattices with spatially periodic arrangements of multiple vortices.",
"In order to illustrate this, we consider a specific setup as shown in Fig.",
"REF (a).",
"It is designed such that the unitcell consists of a collection of four plaquettes $\\mathcal {D}_1$ , $\\mathcal {D}_2$ , $\\mathcal {D}_3$ and $\\mathcal {D}_4$ .",
"Each plaquette consists of four Gaussian wells driven at different phases $\\phi _{i}=\\frac{(i-1)\\pi }{2}$ , $i=1,2,3,4$ .",
"The plaquettes $\\mathcal {D}_1$ and $\\mathcal {D}_4$ possess an anti-clockwise chirality whereas $\\mathcal {D}_2$ and $\\mathcal {D}_3$ have clockwise chirality with respect to the spatial arrangement of the wells with driving phases $\\phi _{i}$ .",
"Additionally, the wells in $\\mathcal {D}_1$ and $\\mathcal {D}_2$ are driven with amplitude $V_1=0.51$ and those in $\\mathcal {D}_3$ and $\\mathcal {D}_4$ with $V_2=0.078$ .",
"Note that these specific values of driving amplitude are chosen by consulting the bifurcation diagram in Fig.",
"REF , so as to allow only vortex trajectories having specific angular momenta.",
"We initialize $N=10^4$ particles randomly in this setup within a square region $x,y \\in [-50,50]\\times [-50,50]$ with small random velocities $v_x,v_y \\in [-0.1,0.1]$ and propagate the ensemble up to time $t_f= 10^4 T$ .",
"For $D=0$ , the particles exhibit vortical motion at long timescales with their angular momentum being governed by the plaquette they are trapped within as shown in Fig.",
"REF (a).",
"The particles in $\\mathcal {D}_1$ and $\\mathcal {D}_4$ rotate with ${\\bf \\bar{\\Omega }}=-1$ and ${\\bf \\bar{\\Omega }}=\\frac{1}{3}$ respectively, as predicted by Fig.",
"REF (b).",
"Note that the plaquettes $\\mathcal {D}_2$ and $\\mathcal {D}_3$ can be obtained by a spatial parity transformation on $\\mathcal {D}_1$ and $\\mathcal {D}_4$ respectively.",
"Hence the mean angular momentum of the particles in $\\mathcal {D}_2$ and $\\mathcal {D}_3$ has an opposite sign as compared to the particles in $\\mathcal {D}_1$ and $\\mathcal {D}_4$ respectively.",
"Even for $D=10^{-4}$ , such rotational motion persists and we obtain a periodic arrangement of particles in space rotating with different angular momenta (Fig.",
"REF (b)).",
"For a higher strength $D=10^{-3}$ , the vortical motion of particles with ${\\bf \\bar{\\Omega }}=\\pm \\frac{1}{3}$ is destroyed and only the ones with ${\\bf \\bar{\\Omega }}=\\pm 1$ remain, yielding a different periodic arrangement (Fig.",
"REF (c)).",
"Noise strengths $D\\geqslant 4\\times 10^{-3}$ eventually destroy all the vortex trajectories.",
"We have demonstrated that superlatices of periodically driven localized wells provide highly controllable setups to realize different patterns of rotational motion of particles.",
"The spatial arrangement of the lattices is responsible for breaking the relevant symmetries, thus allowing for the non-zero average angular momentum of an ensemble of particles.",
"Our analysis of the underlying non-linear dynamical attractors provide an efficient method to control the angular momentum of the particles as well as create a variety of periodic arrangements of vortical motion with different angular momenta.",
"Future perspectives include investigation of rotational dynamics of particles operating in the purely Hamiltonian regime without dissipation, as well as in the quantum regime with the possibility to realize spatially varying artificial magnetic fluxes.",
"A.K.M acknowledges a doctoral research grant (Funding ID: 57129429) by the Deutscher Akademischer Austauschdienst (DAAD) and thanks J. Chen for insightful discussions."
]
] | 2005.14244 |
[
[
"3D Relativistic MHD Simulations of Pulsar Bow Shock Nebulae"
],
[
"Abstract Pulsars out of their parent SNR directly interact with the ISM producing so called Bow-Shock Pulsar Wind Nebulae, the relativistic equivalents of the heliosphere/heliotail system.",
"These have been directly observed from Radio to X-ray, and are found also associated to TeV halos, with a large variety of morphologies.",
"They offer a unique environment where the pulsar wind can be studied by modelling its interaction with the surrounding ambient medium, in a fashion that is different/complementary from the canonical Plerions.",
"These systems have also been suggested as the possible origin of the positron excess detected by AMS and PAMELA, in contrast to dark matter.",
"I will present results from 3D Relativistic MHD simulations of such nebulae.",
"On top of these simulations we computed the expected emission signatures, the properties of high energy particle escape, the role of current sheets in channeling cosmic rays, the level of turbulence and magnetic amplification, and how they depend on the wind structure and magnetisation."
],
[
"Introduction",
"Pulsar Wind Nebulae (PWNe) are synchrotron emitting sources powered by the wind of a pulsar (PSR).",
"Usually, they are observed inside the supernova remnant (SNR) of their parent progenitor, but for old pulsars they can also form as a consequence of the direct interaction of the pulsar wind with the interstellar medium (ISM) [1], [2], [3].",
"Pulsar winds are ultra-relativistic outflows, with typical Lorentz factors in the range $10^{4}-10^{7}$ , magnetised, and cold [4], [5], [6].",
"They are supposed to be mainly composed by electron-positron pairs [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"The interaction with the ambient medium, forces these supersonic winds to slow down in a a strong termination shock (TS).",
"It is there that particles are likely accelerated to a non-thermal distribution [16], [17], [18], [19].",
"The observed non-thermal radiation is produced via synchrotron and inverse Compton scattering, arising from the interaction of these particles with the magnetic field in the nebula and with the background photon field.",
"Given that between 10% and 50% of all the pulsars are born with kick-velocity of the order of $100-500$ km s$^{-1}$ [20], [21], [22], [23], while the supernova remnant expansion is decelerated [24], [25], [26], [27], they are fated to escape the supernova remnant shell on timescales of the order of a few tens of thousands of years.",
"At this point their associated nebulae acquire a cometary-like shape due to the ram pressure balance between the pulsar wind and the surrounding incoming ISM (in the reference frame of the PSR) [28], [29], [30].",
"The pulsar is now located in the head of these nebulae and an elongated tail forms that extends in the direction opposite to the pulsar motion.",
"These objects are known as bow shock PWNe (BSPWNe).",
"The formation of these systems was confirmed by numerical simulations in different regimes [30], [31], [32], [33].",
"In the last years BSPWNe have been observed at many different wavelengths, with a large, and sometimes unexpected, variety of structures at different scales: different shapes and elongation of the tails, different morphologies in the bow shock head, different polarisation properties, hard X-ray outflows misaligned with the pulsar velocity [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61].",
"Interestingly extended TeV halos have also been detected around BSPWNe [62].",
"Given that these nebulae could be one of the major contributors of leptonic anti-matter in the Galaxy, in competition with possible dark matter sources [63], [64], understanding the escape of particles from these systems, could have important consequences.",
"Contamination from neutral hydrogen can also modify the dynamics and morphology of the bow-shock tail, as predicted by theoretical model and verified numerically [65], [66], [67].",
"Beginning with analytical and semi-analytical works [28], [68], [29], dating back to more than one decade, the first numerical models have been presented in the non-relativistic hydrodynamical regime by [30], [69], and in relativistic hydrodynamics and magneto hydrodynamics with spin-kick alinement by [31].",
"However there is no reason to assume spin-kick alignment, and to cope with the relative inclination one needs to work in full 3D.",
"The third dimension is particularly important to correctly capture the structure of the magnetic field configuration [70], and in the study of the development of turbulence, which can be strongly affected by geometric constraints.",
"The first 3D models of BSPWNe were presented by [32] but limited to the classical HD regime.",
"Only recently 3D MHD simulations have been presented in the correct relativistic regime [33], [71], [72], [73].",
"More recently [74], [75], using a simplified, axisymmetric laminar semi-analytic model tuned on numerical simulations, have investigated for then first time how different magnetic field geometry act on the observed non-thermal synchrotron emission and polarisation, and on the propagation of high energy particles, showing the role played by current sheets and current layers.",
"Here we present the result of a detailed study of the dynamics, emission, and particle propagation properties of BSPWNe, done using 3D relativistic MHD simulations [74], [75], [72], [73], [76].",
"Various configurations were considered in terms of inclinations of the magnetic field with respect to the pulsar spin-axis, wind magnetisation, pulsar wind energy distribution, particle acceleration properties, in order to have a sample as complete as possible of the interaction conditions."
],
[
"Numerical Setup and Simulations",
"Our simulations were done with the numerical code PLUTO [77], [78].",
"PLUTO is a shock-capturing, finite-volume code for hyperbolic and parabolic partial differential equations.",
"Simulations were done with adaptive mesh refinement (AMR), corresponding to an effective resolution of $2048^3$ cells at the highest level, sufficient to capture simultaneously the pulsar wind injection region and the large scale of the cometary tail of the nebula.",
"A second order Runge-Kutta time integrator and an HLLD Riemann solver (the Harten-Lax-van Leer for discontinuities, [79], [80]) have been used, in order to better treat shear layers in the nebula.",
"The relativistic pulsar wind is cold $p/\\rho c^2\\approx 0.01$ , and has a Lorentz factor $\\gamma =10$ , high enough to ensure the correct relativistic regime.",
"The outer medium is modelled as a cold incoming flow both with and without magnetic field.",
"Solutions were computed for various relative inclinations, magnetisations $\\sigma $ , and pulsar wind energy distributions, making sure that the dynamics was relaxed to a quasi-steady regime.",
"For a more detailed description of the setup and of the models we refer the reader to [72], [73], [76], where they are discussed and presented in more details.",
"Please nota that the density and magnetic field can be scaled arbitrarily as far as the ratio, $B^2/\\rho c^2$ , is fixed.",
"The typical scale-length of these nebulae is the so called is the stand-off distance, where the wind ram pressure equilibrates the ISM ram pressure: $d_o=\\sqrt{\\frac{\\dot{E}}{4\\pi c\\rho _{\\rm ISM}v_{\\rm PSR}^2}}$ where $\\dot{E}$ is the pulsar spin-down luminosity, taken equal to the pulsar wind power, $\\rho _{\\rm ISM}$ is the ISM density, $v_{\\rm PSR}$ the speed of the pulsar with respect to the local medium.",
"To summarise the main model parameters are: The pulsar is rest at the origin in the Cartesian coordinates, while the ISM has the velocity, $(0, 0,-v_{\\rm PSR})= (0, 0, -0.1c)$ , along the $z$ -axis.",
"The pulsar spin axis lies in the $y-z$ plane and is offset by $\\phi _M$ (the spin-kick inclination) from the $z$ -axis.",
"The pulsar wind is injected steadily from $r = 0.2 d_o$ either with an isotropic energy distribution (cases $I$ ) or following the split-monopole prescription (cases $A$ ).",
"The magnetic field injected with the pulsar wind and has only toroidal component with respect to the spin axis, while the one in the ISM lies in the $y-z$ plane.",
"The magnetisation is defined as $\\sigma = B_o^2/(4\\pi \\rho _o \\gamma ^2c^2)$ where, $B_o$ is the strength of the magnetic field in the equatorial plane of the wind at a distance $r = 0.2d_o$ (for a split monopole solution), and $\\rho _o$ is the wind density at the same location [81].",
"In computing emission maps, the 3D structure of the velocity and magnetic field is provided by our 3D relativistic MHD models.",
"We assume, following [82], that the emitting pairs are distributed according to a power-law in energy, given the typical high flow speed found in numerical simulations, even X-ray emitting particles are only marginally affected by cooling [31], such that our results can reasonably apply even to higher energies.",
"For simplicity we assume that the power-law index is uniform in the nebula.",
"For the emitting particle density we adopt two different choices: either a uniform distribution, as was done in [75], or a density proportional to the local value of the thermal pressure, as it is customary in other PWNe models.",
"In a few cases we have also investigated a third possibility that the emission is concentrated in the current sheets that form in the BSPWN.",
"We build all of our maps using a spectral index $\\alpha =0$ .",
"This was shown by [75] to be a good average for the observed radio spectra, and changes in the typical observed range do not affect much the results.",
"The polarised fraction is always given in terms of the theoretical maximum, that is $0.7$ for $\\alpha =0$ .",
"Emission maps where computed for all simulations.",
"Emissivity toward the observer is computed taking into account all relativistic effects, including Doppler boosting and aberration.",
"For a complete description of the method used and the formalism adopted we refer to [82], [83], [84], [85].",
"We have also computed the polarisation properties, accounting for relativistic polarisation angle swing.",
"In computing particles escape, given that we are interested in the escape from the very head of these nebulae, as in [75], particles that move backward at a distance exceeding $10-20d_o$ , from the PSR are assumed to be lost in the tail.",
"It is still likely that those particles can escape from the tail, but given that the tail might be in general more turbulent than the head, we expect the escape there to be more isotropic.",
"Moreover, we assume that all high energy particles are injected from the PSR (or equivalently the pulsar wind) in the radial direction.",
"The electric field is given by the ideal MHD condition ${\\bf E}=-{\\bf V}\\times {\\bf B}/c$ where V and B are the flow speed and magnetic field given by our RMHD simulations.",
"All particles are injected with the same Lorentz factor, and we have investigated four cases with $\\gamma =[0.5,1.0,3.0,10.",
"]\\times 10^7$ , corresponding to values where we expect the transition from Larmor radii in the magnetic field smaller than the size of the bow-shock set by its stand-off distance $d_o$ , to larger than $d_o$ .",
"Particle trajectories in the electric and magnetic fields of this system are computed using an explicit Boris Pushing technique [86], [87], [88], which ensures energy and phase space number conservation in the non-radiative regime.",
"We verified, by computing radiative losses, that they are always negligible."
],
[
"Dynamics",
"We summarise here the results of our numerical study on the dynamics, morphology, and magnetic field structure in BSPWNe.",
"More details can be found in [72].",
"Comparing 3D with 2D HD runs we find a good agreement for isotropic winds, especially at intermediate values of the magnetisation ($\\sigma =0.1$ ).",
"This is unexpected, since in principle one would expect a larger similarity in the low magnetisation cases.",
"However for values of the magnetisation $\\sigma <0.1$ the dynamics is completely dominated by turbulence on small scales.",
"On the other hand for larger values, the conditions at injection become dominant.",
"In the tail it is quite clear the difference in the development of the local turbulence and in the level of mixing of the fluid, as shown in Fig.",
"REF .",
"In the direction aligned with the pulsar motion ($z-$ direction) the velocity structure shows a lower velocity channel around the $z$ axis ($\\sim 0.65c$ in 2D and $\\sim 0.7-0.8c$ in 3D) surrounded by an higher velocity flow layer (with $\\sim 0.85c$ in 2D and up to $0.9c$ in 3D).",
"In the $A$ cases the effect of turbulence is even more pronounced: injection matters only for $\\sigma >0.1$ , and the typical structures in the tail remain coherent on shorter scales than in the corresponding $I$ case.",
"There is also a higher level of mixing between the pulsar wind and ISM material.",
"In the $I$ cases the initial magnetic field configuration, and the current sheets separating field of opposite polarities, survive in the tail, while in the $A$ ones turbulent mixing tends to destroy them even at higher magnetisation.",
"We found that there is no efficient turbulent magnetic field amplification: in the presence of turbulence it tends just to reach equipartition with the turbulent kinetic energy, which is usually smaller than the thermal energy in the tail.",
"In 3D there is evidence of magnetic field enhancement near to the contact discontinuity (CD), possibly the effect of an efficient shear instability amplification acting at the CD.",
"The forward shock shape is almost the same, the only exception being the $A$ cases with the spin-kick inclination $\\phi _M=\\pi /4$ : differences appear as small extrusions and blobs, possibly resulting as periodic perturbation of the forward shock.",
"Fluctuations of the flow in the interior are not characterised by high values of the magnetic field, density or pressure.",
"Moreover they arise on very small spatial scales, possibly making it difficult to be revealed by actual instruments due to resolution limits.",
"Major differences among various cases are visible mainly in terms of collimation or broadness of the tails, arising as the effect of different magnetic inclination and especially of the wind anisotropy.",
"The average flow pattern in the tail between 2D and 3D runs is not significantly different (differences are largely in the head due to inclination and anisotropy).",
"In the low $\\sigma $ regime there is a strongly turbulent magnetic field structure, suggesting that perhaps a laminar model is not likely to fully capture the magnetic field.",
"For higher values of the magnetisation $\\sigma $ , the magnetic field is more coherent, qualitatively in agreement with the prediction of simplified laminar models [74].",
"Figure: Map of the magnetic field strength in the y=0y=0 plane (in code units), normalised to themaximal nominal value.",
"Upper panels: comparison for runs II(on the left) and AA (on the right), for in cases where the pulsarspin-axis is aligned to its kick velocity; maps are composed bytwo halves of different value of magnetisation, lower on theleft-side and higher on the right-side, respectively.",
"Bottom panels:maps of the magnetic field strength for magnetisation σ=1.0\\sigma =1.0 andfor the all the remaining inclinations φ M \\phi _M, both for II and AA cases.",
"From ."
],
[
"Emission",
"We summarise here the results of our numerical study on the emission and polarisation properties of BSPWNe.",
"More details can be found in [72].",
"Using our numerical simulations we analysed both cases of isotropic ($I$ ) and anisotropic ($A$ ) winds.",
"In the case of uniform injection and high magnetisation, we found a strong correlation between the conditions at injection, as the inclination $\\phi _M$ , and the surface brightness of our simulated maps.",
"The trend in general agrees with that of fully laminar semi-analytic models [74].",
"There is a variety of morphologies, from systems brighter in the head to ones dominated by the tail.",
"Some maps show bright wings.",
"In computing the maps one needs also to consider the viewing angle $\\chi $ , however the dependence on it of the observed properties is less marked and appreciable only at high resolution.",
"In the high $\\sigma $ regime the polarised fraction is quite high in the tail.",
"Once the magnetisation drops, turbulence begins to dominate the appearance of the maps, and it is far more difficult to find defined observational patterns.",
"In this cases it will be hard to distinguish between different inclinations $\\phi _M$ in a robust way.",
"Figure: Emission maps for the case of a uniform wind with pulsar spin-axis aligned withthe kick velocity and high magnetisation (upper row); thesame but for low magnetisation (middle row), andfinally for a pulsar spin-axis orthogonal to the pulsar the kickvelocity again at high magnetisation (bottom row; viewedfrom the pulsar axis direction).",
"All cases are computed assuminguniform local emissivity.",
"From left to right: square root of the totalsynchrotron intensity normalised to the maximum, square root of thepolarised intensity normalised to the maximum superimposed with thepolarised direction, and polarised fraction normalised to thetheoretical maximum for a power-law synchrotron with α=0\\alpha = 0.",
"Thecolour scale is linear between zero and one.",
"From .Interestingly in our maps, apart from the strongly turbulent cases, we do not observe major time variations along the tail direction.",
"This implies that time variability and temporal changes in the flow pattern are weak, and unlikely to give rise to major observational changes.",
"On the other hand, changes in the polarisation properties of a BSPWN could likely point to a strongly anisotropic energy injection, and spin axis misalignment.",
"Our models were computed according to different scalings for the emitting particles density.",
"If we consider a scaling proportional to the local pressure, typical of particles accelerated at the wind termination shock and then advected in the nebula, we find that the head is much brighter than the tail, even by a factor 10.",
"Only in systems dominated by turbulence this difference is less enhanced.",
"Indeed system like the Mouse nebula show a very bright head and a fainter tail, but in many others there is no evidence for such brightness difference [36], [46].",
"This could be the signature of an acceleration process diffused in the bulk of the nebula.",
"We also tried an emissivity scaled according to the strength of the currents, aiming at modelling the effect of reconnection and dissipation of currents.",
"However there seem to be no appreciable difference with respect to a uniform emissivity.",
"In almost all cases the direction of the polarisation (the inferred direction of the magnetic field) seems to be almost aligned with the tail.",
"This is an interesting aspect, most likely due to polarisation swing associated to the relativistic flow in the tail.",
"If one suppresses relativistic beaming and aberration, when computing maps, the structure of the polarisation pattern changes, and the polarisation looks less aligned.",
"Changes in the polarisation direction are observed in many BSPWNe, and they are usually associated with changes in brightness [44].",
"Our results suggest that these could originate when the flow decelerates maybe as a consequence of internal shocks, or because of mass loading from the ambient medium."
],
[
"Particle Escape",
"In few cases, non-thermal high energy emission is observed outside of the supposed location of the contact discontinuity separating the pulsar material from the ISM, where the canonical models predict there should be none.",
"This emission ranges from faint X-ray haloes, as in the case of IC443 [89] and the Mouse [35], or large non-thermal TeV haloes like in the case of Geminga [51], [90], to more structured features like the X-ray prongs observed ahead of the bow shock in G327.1-1.1 [91], or the one sided jets as seen in the Guitar Nebula [92] or in the Lighthouse nebula [93].",
"Here we illustrate the results obtained by computing the trajectories of high-energy particles in the electric and magnetic field of our simulations, to investigate their possible escape.",
"Results are shown in Fig.",
"REF .",
"The typical energy scale of pairs is set by the condition that their Larmor radius in the equipartition magnetic field in the head is equal to the typical size of the bow-shock [94], and this corresponds, for typical system, to a Lorentz factor $\\simeq 3\\times 10^7$ .",
"Our models show a transition in the properties of escaping particles, most of whom comes from the frontal polar region of the pulsar wind, while the others tend to remain confined in the tail.",
"At low energies the escape looks more likely due to reconnection of magnetic field lines between the pulsar wind and ISM: particles moving along those magnetic field lines can escape into the ISM.",
"The outflows are asymmetric (by a factor 4 to 5), likely a consequence of the reconnection at the magnetopause.",
"At higher energies the escape enters a different regime.",
"It becomes more charge separated.",
"Now current sheets and layers play a more important role, as suggested in [75].",
"The charge asymmetry is about a factor 1.5, while the spatial asymmetry is a factor 2.",
"At very high energies the regime becomes fully diffusive [95]: the charge asymmetry exceeds a factor 2, while the spatial asymmetry is reduced to less than 1.5.",
"This transition appears to take place within just an order of magnitude in the energy of the particles from $\\gamma =10^7$ to $\\gamma =10^8$ , suggesting that their outflow is quasi-monochromatic.",
"Both the asymmetry in the escaping flux and it being charge separated, can explain the presence of one-sided jet.",
"Unfortunately the relation between the number of escaping particles and the presence of bright features is non trivial: magnetic field amplification is required, together with turbulent particles confinement, in a non-linear regime [94].",
"This complex physics goes beyond what we can simulate at the moment.",
"For example, in the presence of a net current, the magnetic field can be amplified more efficiently in the non-resonant regime [96], [97].",
"This means that self confinement is more efficient.",
"Our results also show what could be a possible explanation for the X-ray morphology of G327.1-1.1 [91], and the two long tails seen in Geminga [51] (not due to limb brightening).",
"Figure: Projection on the plane transverse to the pulsar motion of the 3D-positions of particlesinjected in the wind with a Lorenz factor γ=3×10 7 \\gamma =3\\times 10^7.",
"Left and right panels refers to particles with differentsigns.",
"From top to bottom, particles injected within different rangesof the polar angle θ\\theta with respect to the pulsar spin-axis:[0,60] ∘ [0,60]^\\circ (green and red) indicates particles injected along thepolar current originating from the PSR spin-axis pointing toward thePSR motion, [60,120] ∘ [60,120]^\\circ (black) indicates particles injectedalong the equatorial current sheet, [120,180] ∘ [120,180]^\\circ (cyan andyellow) indicates particles injected along the polar current pointingin the opposite direction with respect to the PSR motion.",
"Thebackground image shows the log 10 \\log _{10} cut of the density (darker brownfor lower values, lighter brown for higher ones), in the tail at adistance from the pulsar z=-11.5d 0 z=-11.5 d_0within the position of the contact discontinuity with the ISM material, in order to mark the location of theshocked pulsar wind.",
"From ."
],
[
"Conclusion",
"We present here the summary of a detailed numerical study of bow-shock pulsar wind nebulae, carried in 3D using relativistic MHD.",
"Our study was focused on the development of a complete set of models in terms of injection properties, and ISM conditions, that allowed us to investigate not only how the dynamics changes in response to different injections (in terms of spin-axis inclinations, magnetisation, anisotropy) but also to compute on top of these fluid models realistic emission maps, using different prescriptions for the distribution of emitting particles, and to evaluate also the escape properties of high energy pairs, and the possible origin of extended non-thermal features.",
"Our results show that the large variety of observed morphologies, and structures in known pulsar bow-shocks, can be reasonably well accounted for in terms of differences in the injection conditions.",
"We plan in the future to continue our investigation of few selected configurations, that looks more representative of specific known objects."
],
[
"Acknowledgments",
"We acknowledge the “Accordo Quadro INAF-CINECA (2017-2019)” for high performance computing resources and support.",
"Simulations have been performed as part of the class-A project “Three-dimensional relativistic simulations of bow shock nebulae” (PI B. Olmi).",
"The authors acknowledge financial support from the “Accordo Attuativo ASI-INAF n. 2017-14-H.0 Progetto: on the escape of cosmic rays and their impact on the background plasma” and from the INFN Teongrav collaboration."
]
] | 2005.14079 |
[
[
"Systematically Measuring Ultra Diffuse Galaxies in HI: Results from the\n Pilot Survey"
],
[
"Abstract We present neutral hydrogen (HI) observations using the Robert C. Byrd Green Bank Telescope (GBT) of 70 optically-detected UDG candidates in the Coma region from the Systematically Measuring Ultra-Diffuse Galaxies survey (SMUDGes).",
"We detect HI in 18 targets, confirming 9 to be gas-rich UDGs and the remainder to be foreground dwarfs.",
"None of our HI-detected UDGs are Coma Cluster members and all but one are in low-density environments.",
"The HI-detected UDGs are bluer and have more irregular morphologies than the redder, smoother candidates not detected in HI, with the combination of optical color and morphology being a better predictor of gas richness than either parameter alone.",
"There is little visual difference between the gas-rich UDGs and the foreground dwarfs in the SMUDGes imaging, and distances are needed to distinguish between them.",
"We find that the gas richnesses of our HI-confirmed UDGs and those from other samples scale with their effective radii in two stellar mass bins, possibly providing clues to their formation.",
"We attempt to place our UDGs on the baryonic Tully-Fisher relation (BTFR) using optical ellipticities and turbulence-corrected HI linewidths to estimate rotation velocities, but the potential systematics associated with fitting smooth $\\mathrm{S\\acute{e}rsic}$ profiles to clumpy, low-inclination low surface brightness disks precludes a meaningful analysis of potential BTFR offsets.",
"These observations are a pilot for a large campaign now underway at the GBT to use the HI properties of gas-rich UDGs to quantitatively constrain how these galaxies form and evolve."
],
[
"Introduction",
"The study of the low surface brightness (LSB) galaxy population has been reinvigorated as a result of improvements in astronomical instrumentation [3], [5] and data reduction methods [36], [102], as well as the use of novel image searching algorithms [16], [74], [79], [117], [26].",
"Among this recent surge of LSB detections are populations of extended red LSB galaxies akin to those discovered in early LSB studies [91], [49], [20].",
"In their survey of the Coma Cluster, [108] presented the first significant sample of these extended LSBs, dubbing them ultra diffuse galaxies (UDGs) and proposing size $(R_{eff} > 1.5 \\,\\mathrm {kpc})$ and surface brightness $(\\mu _{0,g}\\gtrsim 24\\,\\mathrm {mag/arcsec^2})$ criteria that have since been widely adopted to define them.",
"To date, over 1000 UDG candidates have been discovered in subsequent searches of the Coma Cluster [58], [114], [117] and several other clusters [72], [12], [94], [111], [65], [60], as well as a growing number in lower density environments [71], [15], [86], [61], [103], [17], [85], [11].",
"Across these environments there exists a large diversity in the physical properties of UDGs, similar to that seen in the high surface brightness galaxy population.",
"Most UDGs seem to be embedded in dwarf galaxy-mass dark matter halos [13], [8], [29], [81], although there is evidence that at least some are in more massive halos ([106], [116], [64], [37], although see [89]).",
"While UDGs found in clusters tend to be red (i.e., quiescent) and smooth, those in lower density environments are bluer (i.e., star forming) and have more irregular morphologies [86], [80].",
"Some UDGs exhibit extreme properties that pose challenges to proposed galaxy formation mechanisms, such as high dark matter fractions [106], [12], dark matter deficiencies ([107], [105]; although see [104]), and offsets from established galaxy scaling relations such as the baryonic Tully-Fisher relation [66], [67].",
"Proposed UDG formation mechanisms generally fall into two categories: internally and externally-driven physics.",
"Isolated (i.e., field) UDGs may be formed through multiple internal mechanisms.",
"For example, [7] suggest that UDGs formed in dwarf dark matter halos with elevated angular momenta, naturally explaining their extended sizes.",
"Alternatively, using the NIHAO (Numerical Investigation of a Hundred Astrophysical Objects, [113]) suite of simulations, [33] show that UDG-like objects can form through bursty star-formation early in their evolution resulting in a more extended, diffuse matter distribution.",
"The red, smooth UDGs observed in groups and clusters may represent the population of field UDGs that formed through the aforementioned mechanisms and were subsequently quenched via ram-pressure and/or tidal effects [115], [63], [51], [25].",
"However, some may form initially as typical dwarf galaxies that are tidally disturbed after in-fall into a cluster or by a massive companion [17], [90].",
"In order to constrain which of these proposed formation mechanisms explains the origin of the detected UDGs, larger samples of UDGs with distance measurements are required, particularly in the field where inferring distances by projected separation from clusters or groups is not possible.",
"While some optical distances to UDGs have been obtained [109], [15], [52], [6], [35], [88], [29], [70], sample sizes are limited due the large spectroscopic integration times required at low surface brightnesses.",
"By contrast, the neutral hydrogen (H1) in gas-rich UDGs can not only provide a distance measure but also help distinguish among formation mechanisms.",
"The H1 redshift provides kinematic distances for candidates that can distinguish foreground dwarfs $(R_{eff} < 1.5\\,\\mathrm {kpc})$ from true UDGs $(R_{eff} > 1.5\\,\\mathrm {kpc})$ , and linewidths reflect their internal dynamics, and the H1 flux provides the gas mass.",
"H1 follow-up observations of optically-detected UDG candidates have been demonstrated to be feasible with single-dish radio telescopes [76], [98] and searches through extant blind H1 survey detections for diffuse stellar counterparts have also been fruitful [61].",
"The Systematically Measuring Ultra-Diffuse Galaxies [117] survey is uniquely positioned to produce samples of UDG candidates for H1 follow-up observations across a range of environments, as the combination of depth and coverage of the DECaLS data used to detect UDG candidates are unmatched.",
"The SMUDGes pilot survey searched publicly available DECaLS data (one of three DESI pre-imaging Legacy surveys, see [32] for details) for large $(r_{eff} > 5.3= 2.5 \\, \\mathrm {kpc\\,at\\,}D_{Coma}\\sim 100\\mathrm {Mpc})$ UDG candidates in a $290\\, \\mathrm {deg^2}$ region centered on the Coma Cluster.",
"The 275 UDG candidates resulting from that search (Z19) as well as subsequent SMUDGes detections provide ample targets to pilot a large follow-up campaign.",
"In this paper, we present pilot H1 observations along the lines-of-sight to 70 SMUDGes UDG candidates, which represent the first phase of a large H1 follow-up campaign using the Robert C. Byrd Green Bank Telescope (GBT).",
"We aim to obtain redshift measurements to UDG candidates and characterize the gas properties of confirmed UDGs to constrain their formation mechanisms.",
"These observations, which are part of a much larger GBT program that is currently underway, represent the largest H1 follow-up campaign of optically-selected UDG candidates ever reported.",
"The structure of this paper is as follows.",
"In Section , we describe our H1 target selection.",
"We outline our observations and data reduction procedure in Section .",
"In Section , we present the properties of our H1 detections and non-detections.",
"In Section , we discuss the environmental and morphological properties of H1 detections and non-detections, place initial constraints on UDG formation mechanisms, and discuss our UDGs in the context of the baryonic Tully-Fisher relation.",
"We conclude and outline future work in Section .",
"Throughout this work we use $D_{Coma} = 100\\,\\mathrm {Mpc}$ , $H_0 = 70 {\\mathrm {km\\,s^{-1}}}\\mathrm {Mpc^{-1}}$ , $\\Omega _\\Lambda = 0.7$ , and $ \\Omega _m = 0.3.$"
],
[
"Sample Selection",
"We select H1 follow-up targets from the SMUDGes pilot sample (Z19) and subsequent searches of the DECaLS data.",
"Focused on the $290\\, \\mathrm {deg^2}$ region centered on the Coma Cluster, the SMUDGes pilot survey employed a semi-automated UDG candidate identification procedure, described in detail in Z19.",
"Briefly, the DECaLS observations were preprocessed to remove any defects, and then foreground or background sources significantly brighter than UDGs were replaced with background noise.",
"Next, these processed images were spatially filtered to various scales using wavelet transforms, and diffuse objects are identified using SEP [10], [18].",
"In order to compare results with other studies [108], [114], their photometric properties were then modeled as exponential profiles using GALFIT [77] and only objects with $r_{eff} > 5.3(R_{eff}=2.5 \\, \\mathrm {kpc\\,at\\,}D_{Coma}\\footnote {We use r_{eff} for angular sizes and R_{eff} for physical sizes throughout this paper.",
"})$ and $\\mu _{0,g}> 24\\, \\mathrm {mag\\,arcsec^{-2}}$ were kept.",
"The remaining objects were examined by eye, and 275 classified as bona-fide UDG candidates.",
"We select 34 of the 275 SMUDGes UDG candidates with $m_g \\lesssim 19.5\\,\\mathrm {mag}$ to follow up in H1 (listed as Z19 in column 14 of Table ).",
"This magnitude limit combined with gas richness scaling relations for local dwarfs [21] implies that integration times of no more than a few hours are required to follow up each source (see Section ).",
"A subsequent optical search within the same region using an improved SMUDGes pipeline (Zaritsky et al., in prep) detected an additional 36 UDG candidates that satisfied the above magnitude limit and we include them in our H1 follow-up sample as well (listed as K20 in column 14 of Table ).",
"The DECaLS imaging for all targets was subsequently modeled as a $\\mathrm {S\\acute{e}rsic}$ profile with a variable $\\mathrm {S\\acute{e}rsic}$ index using GALFIT and the resulting parameters are listed in columns (4)-(11) of Table .",
"The parameter uncertainties are the GALFIT values which are derived using Poisson pixel noise; a more comprehensive error estimation method for SMUDGes photometery is being developed using simulated UDG recovery for use in the full survey [117].",
"The optical properties of the 36 previously unpublished UDG candidates are largely consistent with the sample selected from Z19, although there are a few candidates with smaller sizes $(r_{eff} > 4.7)$ and higher surface brightnesses $(\\mu _{0,g} > 23.7\\,\\mathrm {mag\\,arcsec^{-2}})$ .",
"Some of these candidates have $m_g > 19.5\\,\\mathrm {mag}$ (our H1 follow-up criterion) because initial estimates were used during the target selection.",
"There is some overlap of the UDG candidates in SMUDGes with other UDG samples (see Z19).",
"Of the 36 UDG candidates we present here, 6 have been either presented in other work and/or previously detected in H1.",
"We include references for these objects in column 14 of Table .",
"In total, our H1 follow-up sample consists of 69 UDG candidates in the Coma Cluster region and 1 outside of itThe exception is SMDG1103517+284118 which falls outside the Coma Cluster region and was nonetheless included as a target of interest.",
"Their projected spatial distribution relative to galaxies from the SDSS DR15 [4] with $5000 {\\mathrm {km\\,s^{-1}}}< cz < 9000 {\\mathrm {km\\,s^{-1}}}$ is shown in Figure REF .",
"We do not select on color in this work despite its accuracy for predicting gas richness in the high surface brightness galaxy population [28], [22].",
"Instead, we investigate the relationship between color and gas-richness in Sections REF and .",
"Figure: Projected sky distribution of our UDG candidate H1 follow-up sample in the Coma Cluster region (colored points), with galaxies from SDSS DR15 with 5000 km s -1 <cz<9000 km s -1 5000 {\\mathrm {km\\,s^{-1}}}< cz < 9000 {\\mathrm {km\\,s^{-1}}} plotted as small grey circles.",
"Our sample is subdivided into H1 detections of UDGs (blue stars), H1 detections of foreground dwarf galaxies (green squares), and H1 non-detections (red circles).",
"The orange open circle is centered on the Coma Cluster and has a radius of ∼3\\sim 3 Mpc that represents the virial radius of the Coma Cluster ."
],
[
"Observations and Data Reduction",
"We performed 88 hours of position-switched H1 observations between 2018 February and 2018 August using the GBT along the lines of sight (LOS) to the 70 UDG candidates in Table (program AGBT18B-239).",
"9 objects were observed with an offset between the optical centroid and the LOS in order to minimize contamination from known nearby objects (see Section REF ).",
"These objects are indicated with an asterisk next to their RA in Table .",
"Our observational setup and data reduction procedure are similar to those used in [55], which we briefly outline here.",
"We used the L-band receiver and the Versatile GBT Astronomical Spectrometer (VEGAS) with a spectral resolution of 3.1 kHz and a wide bandpass of 100 MHz which allows for the detection of H1 emission lines out to $V_{Helio} \\sim 14000 \\, {\\mathrm {km\\,s^{-1}}}$ .",
"We estimate the integration times for our targets using $m_g$ to reach a gas richness of $\\frac{M_{HI}}{L_{g}} \\sim $ 1 $\\frac{M_{\\odot }}{L_{\\odot }}$ with $S/N = 5$ in a single $50 \\, {\\mathrm {km\\,s^{-1}}}$ channel.",
"Gas richness is a distance-independent quantity since both $M_{HI}$ and $L_{g}$ scale with distance squared.",
"Therefore, a single spectrum allows us to search for an H1 reservoir in our targets anywhere within the wide bandpass.",
"The data were reduced using the standard GBTIDLhttp://gbtidl.nrao.edu/ procedure $getps$ .",
"We remove narrow-band and broadband radio frequency interference before smoothing our spectra to our desired resolutions, following the same procedure as [55].",
"Furthermore, we scale the fluxes in our final spectra up by 20% to account for the systematic offset in the GBT noise diode calibration values reported by [40].",
"The RMS noise, $\\sigma _{50}$ , for each spectrum at $\\Delta V = 50\\,{\\mathrm {km\\,s^{-1}}}$ resolution is given in column 13 of Table .",
"We examined the calibrated, RFI-excised spectra by-eye after smoothing to multiple resolutions from $5-50 \\, {\\mathrm {km\\,s^{-1}}}$ for statistically significant emission.",
"We detect H1 emission along the LOS to 18 UDG candidates (column 15 of Table ).",
"We show their spectra in Figure REF at $\\Delta V$ given in Tables and , which also lists other properties we have derived from these H1 detections.",
"We find no significant H1 emission associated with the 52 remaining targets and place stringent $5\\sigma $ the upper limits on H1 mass, ${M^{lim}_{HI}}$ , and gas richness, ${M^{lim}_{HI}}/{L_{g}}$ , which are listed in Table ."
],
[
"Properties of H1 Detections",
"We detect H1 along the LOS to 18 UDG candidates, and their spectra are shown in Figure REF .",
"At our observing frequency of $\\sim $ 1.4 GHz, the GBT beam (FWHM $\\sim \\mathrm {9.1^{\\prime }}$ ) response is well understood down to $\\approx -30 \\mathrm {dB}$ [99].",
"We therefore search through NEDThe NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"and the DESI Legacy Imaging Survey Sky Viewerhttp://legacysurvey.org/viewer for objects within $30^{\\prime }$ of the LOS that may present themselves as gas-rich interlopers in our spectra for all of our targets.",
"We find no such interlopers for any of our H1 detections, and conclude they are the H1 counterparts to the corresponding UDG candidates.",
"We derive distance-independent quantities from the spectra (systemic velocity, $V_{sys}$ , and velocity width, $W_{50}$ ) as described in [55] and briefly outline the method here.",
"Using a first-order polynomial fit to each edge of the H1 profile between 15% and 85% of the peak flux value, we find the velocities corresponding to the 50% flux value.",
"Their mean corresponds to $V_{sys}$ (column 4 of Tables and ) and the difference corresponds to $W_{50}$ .",
"The latter is corrected for instrumental and cosmological redshift broadening following [100] to produce $W_{50,c}$ (column 5 of Tables and ).",
"We assume an uncertainty of 50% for the instrumental broadening correction, which dominates the uncertainties on $V_{sys}$ and $W_{50,c}$ .",
"We note that we are conducting signal recovery simulations, similar to [100] but tailored to UDG-like H1 profile shapes, to more accurately understand how instrumental effects at the GBT affect our H1 detections in the full survey.",
"Distances are required to confirm candidates as true UDGs.",
"Using $V_{sys}$ and the Hubble-Lemaître Law, we estimate kinematic distances for all of our H1 detections and adopt a distance uncertainty of 5 Mpc [61], [98].",
"Interestingly, detections are almost equally split between foreground $(D_{HI} < 40 \\,\\mathrm {Mpc})$ and background objects $(D_{HI} > 80 \\,\\mathrm {Mpc})$ : this emphasizes how gas-rich diffuse objects at different distances can look similar on the sky (see also Figures REF and REF ), an issue that we discuss further in Section REF .",
"Based on these distances and the angular sizes listed in Table , we confirm 9 new UDGs with $R_{eff} > 1.5\\,\\mathrm {kpc}$ and $\\mu _{0,g}\\gtrsim 24 \\,\\mathrm {mag\\,arcsec^{-2}}$ , and give their H1 properties in Table .",
"The remaining detections are dwarfs in the foreground of Coma; their derived H1 properties are in Table .",
"To the best of our knowledge, the UDGs in Table represent the largest sample of optically-selected UDGs with follow-up H1 detections reported so far.",
"In the left panel of Figure REF , we compare the distribution of $W_{50,c}$ for our H1-confirmed UDGs (orange) to those of the H1-bearing ultra-diffuse sources (HUDs) samples, HUDs-B (green) and HUDs-R (purple), from [61].",
"The HUDs-B and -R samples are distinguished by their “broad\" and “restrictive\" optical selection criteria (see [61] for more details) with the latter sample using the same criteria used in this work.",
"We also include galaxies in the ALFALFA $\\alpha .40$ catalog [43].",
"Our H1-confirmed UDGs span a broad range in $W_{50,c}$ and are generally more consistent with the HUDs samples than galaxies in ALFALFA.",
"During our literature search for possible H1 interlopers we discovered that 5 of our 18 H1 detections were previously reported as gas-rich objects.",
"Of these previously detected objects, 1 is a UDG (SMDG1220188+280132) that has been detected by ALFALFA.",
"It was not included in the HUDs sample, likely due to their distance criteria [61].",
"Therefore, we present SMDG1220188+280132 as a UDG here for the first time.",
"Figure: H1 detections along the LOS to UDG candidates in our sample.",
"The first 9 panels show targets that satisfy the UDG size criterion of R eff >1.5 kpc R_{eff} > 1.5 \\mathrm {kpc} given their redshifts (confirming them as UDGs), while the last 9 panels show targets which do not (confirming them as foreground dwarfs).",
"Target names and classification (UDG or Dwarf) are in the top-right corner of each panel.",
"The black dotted line in each panel represents 0 mJy.",
"The spectral resolutions ΔV\\Delta V of the plotted spectra and the derived properties of the H1 detections are in Tables and for UDGs and foreground dwarfs, respectively.We calculate the H1 flux, $S_{HI}={\\int }S{\\delta V}$ , by integrating over the line profile, where uncertainties stem mainly from the noise statistics of the profile [100] and a 2% noise diode uncertainty [110].",
"We use these fluxes and our kinematic distances to determine H1 masses, $M_{HI}$ , using the standard equation for an optically thin gas [42]: $ M_{HI}=2.356\\times 10^{5}(D_{HI})^{2}S_{HI} \\, \\mathrm {M_{\\odot }},$ where the distance, $D_{HI}$ , is in Mpc and $S_{HI}$ is in Jy ${\\mathrm {km\\,s^{-1}}}$ .",
"H1 masses are listed in column 8 of Tables and .",
"Uncertainties are determined following the methods of [100] to which distance uncertainties are added in quadrature.",
"We calculate stellar masses, $M_{*}$ , for detections (column 9 of Tables and ) using $m_{g}$ and $g-r$ from Table in the relations of [118] and assuming $D_{HI}$ , propagating photometric and distance uncertainties along with those reported on the relations.",
"Finally, we estimate baryonic masses as $M_{bary}=1.33M_{HI}+M_{*}$ (column 10 of Tables and ).",
"In the right panel of Figure REF , we show the H1-confirmed UDGs in the $M_{HI}-M_{*}$ plane, along with the HUDs samples (-B: green circles and -R: purple squares) and galaxies from the $\\alpha .40$ catalog with SDSS and GALEX coverage from [46].",
"We find that our UDGs are broadly consistent with both the HUDs and $\\alpha .40$ samples, although Figure REF illustrates how the HUDs sample as a whole may be more gas-rich [61] compared to ours $(\\mathrm {mean\\,}M_{HI}/M_{*} \\sim 5)$ .",
"This may point to a difference in UDG samples drawn from H1 vs. optical searches, although their selection functions need to be understood before intrinsic population differences can be quantified.",
"In Figure REF , we show the relationship between gas-richness, $M_{HI}/M_{*}$ , and size, $R_{eff}$ , for our H1-confirmed UDGs and foreground dwarfs (filled stars).",
"We also include the 6 UDGs from [67] and 5 UDGs around Hickson Compact Groups from [98] (open symbols).",
"For consistency across samples we have calculated $R_{eff}$ from the exponential scale-lengths reported by [67].",
"Because of the large systematic differences between different color-mass-to-light-ratio prescriptions [83], we have also re-calculated stellar masses for the 5 UDGs followed up by [98] using the photometry of [86] and the [118] relations.",
"We also propagated the 5 Mpc distance uncertainties into the errorbars on $R_{eff}$ for all samples.",
"The colors of the symbols in Figure REF represent the stellar masses of the object.",
"There is some evidence that larger UDGs are more gas-rich within each stellar mass bin but little evidence for a similar trend among the foreground dwarfs; we discuss possible implications of this in Section REF .",
"Figure: Comparison of derived properties between our H1-confirmed UDGs and other similar samples.",
"Left:Left: Distribution of W 50,c W_{50,c} for UDGs in our sample (orange), the HUDs-B and -R samples (purple and green, respectively), and galaxies from the α.40\\alpha .40 catalog with SDSS and GALEX coverage (grey).",
"Right:Right: M HI -M * M_{HI}- M_{*} relation for the same samples as in the left panel.Figure: Gas-richness as a function of size for our 9 H1-confirmed UDGs and 9 foreground dwarfs as filled stars.",
"The gray dashed line shows the R eff =1.5 kpc R_{eff}=1.5\\, \\mathrm {kpc} size boundary between dwarfs and UDGs.",
"We also include the 5 UDGs around Hickson Compact groups from and 6 UDGs from .",
"The colors of the symbols represent the stellar mass bin of the objects.Figure: M HI /L g M_{HI}/L_{g} (blue stars and green squares for H1-confirmed UDGs and foreground dwarfs, respectively) and M HI lim /L g M^{lim}_{HI}/L_{g} (red downward arrows for non-detections) as a function of g-rg-r (left) and g-zg-z (right) color for our sample.",
"The dashed vertical lines in each panel show the median g-r=0.53g-r=0.53 and g-z=0.79g-z=0.79 colors for our sample.",
"For comparison, we show the median g-r=0.35g-r=0.35 color of the HUDs sample in the left panel with a vertical dash-dotted line."
],
[
"H1 Non-detections",
"We find no statistically significant H1 signals along the LOS to 52/70 targeted UDG candidates that can be attributed to these objects.",
"We smooth their spectra to $\\Delta V = 50\\, {\\mathrm {km\\,s^{-1}}}$ and list their representative RMS noise, $\\sigma _{50}$ , in column 13 of Table .",
"We modify equation REF to place stringent, $5\\sigma $ H1-mass upper limits $ M^{lim}_{HI}=5.89\\times 10^{7}(D_{lim})^{2}{\\sigma _{50}}\\, \\mathrm {M_{\\odot }},$ where $D_{lim}$ is the adopted distance in Mpc.",
"In most cases we assume the Coma Cluster distance of $D_{lim}=D_{Coma}=100\\,\\mathrm {Mpc}$ , aside from a few exceptions described below.",
"We also set (distance-independent) upper limits on the ratio of H1-mass to $g-$ band luminosity, $M^{lim}_{HI}/L_{g}$ , and list all of the calculated properties for non-detections in Table .",
"We briefly highlight a few of the H1 non-detections in our sample.",
"SMDG1221577+281436 was reported as the marginal H1 detection of a nearby, gas-rich dwarf galaxy with $V_{Helio} = 450\\pm 8\\,{\\mathrm {km\\,s^{-1}}}$ in [47].",
"However, when we smooth our spectra to match their velocity resolution we see no signal despite our deeper data.",
"SMDG1253151+274115 (first reported as DF30 in [108]) and SMDG1251013+274753 were confirmed as UDGs via optical spectroscopy in [52] and Kadowaki et al., in prep, with $V_{opt} = 7316 \\pm 81 \\, {\\mathrm {km\\,s^{-1}}}$ and $V_{opt} = 6118 \\pm 45 \\, {\\mathrm {km\\,s^{-1}}}$ , respectively.",
"The former was confirmed as a Coma Cluster member and we use $D_{lim}=100\\,\\mathrm {Mpc}$ to estimate H1 properties.",
"The latter was confirmed to lie outside the Coma Cluster, and therefore we estimate its distance using $V_{opt}$ and the Hubble-Lemaître Law to be $D_{lim}=87\\,\\mathrm {Mpc}$ .",
"In addition, Kadowaki et al.",
"(in prep) find velocities for SMDG1217378+283519 $(V_{opt} = 493\\pm 69\\,{\\mathrm {km\\,s^{-1}}})$ and SMDG1221086+292920 $(V_{opt} = 1024\\pm 66\\,{\\mathrm {km\\,s^{-1}}})$ that place them well in the foreground of Coma, and we use the corresponding $D_{lim}$ to compute $M^{lim}_{HI}$ .",
"SMDG1302417+215954 (IC 4107) has previously been reported as an H1 non-detection [92].",
"One SDSS spectrum of this object classifies it as a starhttp://cas.sdss.org/dr7/en/get/specById.asp?id=746142786461368320 with $V_{opt} = 267\\,{\\mathrm {km\\,s^{-1}}}$ [56], while another classifies it as a QSOhttp://skyserver.sdss.org/dr12/en/get/SpecById.ashx?id=2983699425022470144 with $V>100,000\\,{\\mathrm {km\\,s^{-1}}}$ .",
"We adopt $D_{lim}=3.8\\,\\mathrm {Mpc}$ using the lower SDSS velocity, consistent with both its morphology and the [54] association of this object with the NGC 4826 group.",
"In Figure REF , we show $M_{HI}/L_{g}$ for our H1 detections of UDGs (blue stars) and foreground dwarfs (green squares), and $M^{lim}_{HI}/L_{g}$ for our H1 non-detections (red downward arrows) as a function of $g-r$ (left panel) and $g-z$ (right panel).",
"The vertical dashed lines in each panel show the median colors of the follow-up sample as a whole: $g-r=0.53$ and $g-z=0.79$ .",
"We note that our upper limits are generally higher than the $M_{HI}/L_{g}= 1 M_{\\odot }/L_{\\odot }$ used to estimate the required integration times.",
"There are three potential reasons for this reduction in sensitivity: integrations/scans flagged due to RFI, 20% calibration adjustment, and/or noisier than expected data.",
"By and large, our H1 detections have colors that are bluer than the H1 non-detections but the scatter is large (see also Figures REF - REF ); in the left panel, we also show the median $g-r=0.35$ of the entire HUDs sample as the vertical dashed-dotted line.",
"Several of our H1 detections, including 6/9 UDGs, hover around this line and the vast majority of our non-detections lie on its redder side.",
"We discuss the differences between the optical properties of our H1-confirmed UDGs, foreground dwarfs, and H1 non-detections in Section REF ."
],
[
"Discussion",
"With our pilot sample of H1-confirmed UDGs, foreground dwarfs, and H1 non-detections in hand, we provide some initial insight on three main questions our survey aims to answer: 1.",
"Are there optical features that distinguish bona-fide gas-rich UDGs from foreground dwarfs or H1 non-detections among UDG candidates?",
"2.",
"What constraints, if any, do our H1-confirmed UDGs place on formation mechanisms?",
"3.",
"How unusual are UDGs in the context of local galaxy scaling relations?",
"We address these questions in Sections REF , REF , and REF , respectively."
],
[
"Comparing UDGs with H1 Detections and Non-detections",
"Our follow-up H1 observations of 70 SMUDGes UDG candidates have revealed 9 gas-rich UDGs and 9 gas-rich foreground dwarf galaxies, while the remaining 52 targets were not detected in H1.",
"In this section, we explore differences between the environment and optical/NUV properties of these subsamples both to improve our detection efficiency in the full survey as well as to constrain the properties of H1-rich and H1-poor objects in the LSB regime.",
"We first revisit the spatial distribution of the follow-up targets shown in Figure REF .",
"The projected distribution of our sample spans both high and low density regions around Coma, with no obvious difference in location relative to the large-scale filamentary structure (grey circles) between H1 detections (blue stars and green squares) and non-detections (red circles).",
"This qualitatively suggests that there is no strong correlation between H1 content and projected environment, implying that sky location is not a good predictor of gas richness among pilot sample galaxies.",
"Quantitatively, we find that none of the H1-confirmed UDGs are likely to be gravitationally bound to the Coma Cluster based on their redshifts and projected spatial separations.",
"Furthermore, only one of these objects (SMDG1248019+261236) has at least one massive companion $(M_g < -19\\,\\mathrm {mag})$ that projects within 300 kpc and within $\\pm 500\\,{\\mathrm {km\\,s^{-1}}}$ (Kadowaki et al., in prep).",
"While not obvious from Figure REF , our H1-confirmed UDGs reside in sparse environments.",
"These findings are generally consistent with previous work that has investigated the environmental dependence of gas content [23].",
"We next investigate whether or not discernible NUV emission in archival GALEX imaging predicts a detectable H1 reservoir among UDG candidates.",
"The vast majority of pilot survey targets that are in the GALEX footprint do not have detectable NUV emission, which is commensurate with the findings of [97] for the broader SMUDGes sample.",
"This is also the case for our H1 detections with GALEX All-sky Imaging Survey [73], [19] coverage, raising the possibility that AIS-depth NUV imaging is not sufficient to detect ongoing star formation in UDGs.",
"We therefore examine the subset of pilot survey targets with GALEX NUV exposures of at least 1000 seconds, i.e., Medium Imaging Survey (MIS) depth or $\\sim $ 5-10 times deeper than the AIS.",
"Of the 32 pilot survey targets in this category, 14 have discernible GALEX emission.",
"All of these objects for which our H1 spectra are sensitive to at least $M_{HI}/L_{g}=2\\,M_{\\odot }/L_{\\odot }$ across the observed band have been detected in H1, while many of the objects for which deep GALEX images reveal no NUV emission are H1 non-detections with $M^{lim}_{HI}/L_{g}< 1.5\\,M_{\\odot }/L_{\\odot }$ .",
"This suggests that MIS-depth NUV imaging is a good predictor of gas richness among SMUDGes UDG candidates.",
"Finally, we compare the DECaLS optical morphologies of the H1-confirmed UDGs, the UDG candidates that we did not detect in H1, and the H1-detected foreground dwarfs.",
"Figures REF -REF show color and grayscale $grz$ cutout pairs for these three subsets of our sample, where the adjusted contrast and brightness of the color image highlights the brighter emission in the object and the histogram equalization of the grayscale image highlights fainter emission.",
"In all panels, the dashed, white ellipses have the disk geometry and semi-major axis, $\\frac{a}{2}=r_{eff}$ , of the best-fitting GALFIT models reported in Table .",
"We note that Figures REF and REF show all of the H1-confirmed UDGs and foreground dwarfs in our sample, while Figure REF shows images of a subset of the 52 H1 non-detections with similar $r_{eff}$ and $m_g$ to the H1-confirmed UDGs.",
"Figures REF and REF demonstrate that on the whole, the H1-detected UDGs are bluer than the UDG candidates that we do not detect, although as illustrated in Figure REF the scatter in color is large (c.f.",
"SMDG1301005+210355 which we do detect in H1, and SMDG1223448+295949 which we do not).",
"This is consistent with the clear trends seen at higher surface brightnesses [46], [28], [22] as well as in other LSB studies [61], [41], [80], suggesting that star formation proceeds similarly in high and low surface brightness galaxies [14].",
"Figures REF and REF also illustrate that the H1-detected UDGs are more irregular in morphology both within and beyond $r_{eff}$ than the undetected UDG candidates, although there is some scatter (c.f.",
"SMDG1225185+270858 which we do detect in H1, and SMDG1253151+274115 which we do not).",
"On the other hand, the combination of DECaLS-depth color and morphology does appear to predict gas richness: blue and irregular objects in our pilot sample are almost invariably gas-rich, while red and smooth objects are invariably gas-poor.",
"The efficiency of future H1 follow-up UDG campaigns can therefore be increased relative to the statistics presented here by preferentially targeting candidates that are both blue and irregular.",
"Do our H1-confirmed UDGs differ in optical morphology from our gas-rich foreground dwarfs?",
"Comparing Figures REF and REF reveals that, among gas-rich objects, the foreground dwarfs tend to have larger angular sizes than the confirmed UDGs, consistent with Z19's hypothesis using a clustering analysis.",
"The bluest gas-rich objects that we detect are also foreground dwarfs and not confirmed UDGs.",
"While some stars in the very nearby dwarf SMDG1255412+191221 begin to appear resolved in the DECaLS imaging, we find no clear difference in optical morphology between bona-fide UDGs and foreground dwarfs in the pilot sample, making the two difficult to distinguish among follow-up targets.",
"Blue foreground dwarfs are therefore an important potential contaminant among gas-rich UDG candidates identified by their optical colors and morphologies alone.",
"Distance information is required to identify UDGs in the field.",
"Figure: 55×5555\\times 55 color and grayscale grzgrz image cutouts of H1-detected UDGs shown in pairs with the color image on the left and the grayscale image on the right.",
"The adjusted contrast and brightness of the color images highlights brighter emission in each object, while the histogram equalization of the grayscale images highlights the lower surface brightness emission.",
"In all panels, the dashed, white ellipses have the disk geometry and semi-major axis, a 2=r eff \\frac{a}{2}=r_{eff}, of the best-fitting GALFIT models reported in Table .",
"The object's color from Table is in the top-right corner of each image pair, and a scale bar that is 1 kpc across at the UDG distance is in the bottom-right corner.",
"For a subset of the objects, we also overlay red ellipses corresponding to the disk geometry of GALFIT models with lower inclinations as detailed in Section .",
"The inclinations of the corresponding disk, computed using Equation , are in the top-left corner of each image pair.Figure: Same as Figure , but for H1 non-detections.Figure: Same as Figure , but for H1-detected foreground dwarf galaxies and the scale bar represents 200 pc."
],
[
"Constraining Formation Mechanisms",
"The stellar masses and velocity widths of our H1-confirmed UDGs are commensurate with them being dwarf galaxies, in line with other estimates for UDGs in a variety of environments [96], [75], [117], [11].",
"How UDG-like field dwarfs could form is an active area of research (see Section ), and the small size of the H1-confirmed UDG sample from this pilot survey is too small for quantitative comparisons with theory.",
"Nonetheless, we briefly consider the gas richnesses and sizes of the H1-confirmed UDGs in the context of formation model predictions.",
"The star-formation feedback model presented by [33] predicts that UDGs in the field today have gas richnesses that scale with their sizes at fixed stellar mass.",
"As shown in Figure REF , we find evidence for a trend between $M_{HI}/M_*$ and $R_{eff}$ when the gas-rich UDGs are subdivided into two stellar mass bins.",
"This trend persists when the gas-rich UDG samples from [98] and [67] are also considered, but it is not evident in the foreground dwarf sample also plotted in Figure REF .",
"The correlation between gas richness and size for UDGs is qualitatively consistent with the predictions of [33], although a similar trend may also emerge from other UDG formation scenarios.",
"It is also possible that the correlations between gas richness and size exist in the broader galaxy population, and therefore that the trends in Figure REF do not constrain UDG formation mechanisms at all.",
"That the foreground dwarfs in our sample do not follow this trend argues against this possibility.",
"Examining gas richnesses and sizes for a larger sample of galaxies might further clarify this issue, as might be obtained by homogenizing measured properties across the SPARC [62], SHIELD [24], and LITTLE THINGS [48] samples along with samples of gas-rich UDGs.",
"More data are needed to quantify comparisons between gas richnesses and sizes predicted by UDG formation models and other mechanisms, which we anticipate undertaking with data from the full survey."
],
[
"Disk Geometry and the BTFR",
"We now discuss the H1-confirmed UDGs in the context of the baryonic Tully-Fisher (BTFR) in order to explore the possibility that our sample exhibits an offset from this relation similar to that found by [66], [67].",
"Because our H1 detections stem from spatially unresolved single-dish observations, we must resort to optical measures of the disk geometry to estimate H1 disk rotation velocities, $V_{rot}$ , from the measured velocity widths, $W_{50,c}$ , in Tables and .",
"We therefore proceed to derive $V_{rot}$ for our H1 detections, examine BTFR offsets in the context of the reliability with which we can estimate the disk geometry, and discuss the implications of these findings for UDG structure.",
"We first compute rotation velocities for our H1 detections using the relation for a flat axisymmetric disk: $V_{rot}^{GF} = \\frac{W_{50,c,t}}{2\\mathrm {sin}(i^{GF})},$ where $W_{50,c,t}$ is the profile velocity width that has been corrected for ISM turbulence (see below) in addition to the instrumental effects discussed in Section REF and $i^{GF}$ is the disk inclination implied by $b/a$ of the best-fitting GALFIT models of the optical UDG morphology given in Table and represented by the white ellipses in Figures REF .",
"We calculate $i^{GF}$ via the standard relation: $\\mathrm {cos^2}(i^{GF}) = \\frac{(b/a)^2 - q^2_{0}}{1 - q^2_{0}} \\, ,$ where $q_0$ is the intrinsic axial ratio.",
"We adopt $q_0 = 0.2$ in line with many previous studies [39], [38], [61], although for our intermediate and low-inclination systems values as large as $q_0 =0.5$ [87] only impact the derived $V_{rot}$ at the 10% level.",
"While the value of $q_{0}$ does not strongly impact the derived $V_{rot}$ , we emphasize that there are considerable uncertainties in $i^{GF}$ derived from $b/a$ in Table .",
"First, if the H1 disk is warped [53] or if the H1 and optical disks are misaligned [101], [67], $i^{GF}$ will not reflect the H1 disk geometry.",
"Second, Figure REF illustrates that the H1-confirmed UDGs have irregular morphologies, while the GALFIT models used to derive $b/a$ in Table assume a smooth distribution of light (Z19).",
"This raises the possibility that clumps in the disk systematically pull $b/a$ away from the value that reflects the underlying disk geometry, biasing $i^{GF}$ .",
"We therefore consider $i^{GF}$ to be only a rough approximation of the H1 disk inclination that are much more uncertain than $b/a$ from the smooth GALFIT models listed in Table , and list them as such in Table .",
"We note that, since $d(sinx)/dx = cosx$ is much larger for low $x$ than when $x$ approaches $90^{\\circ }$ , uncertainties in $i^{GF}$ in low- and intermediate-inclination systems have a larger impact on $V_{rot}^{GF}$ than uncertainties on $i^{GF}$ in high-inclination systems.",
"We follow the prescription of [112] to correct $W_{50,c}$ in Tables and for ISM turbulence to obtain $W_{50,c,t}$ , required in Equation REF , for the Gaussian profiles in Figure REF : $\\begin{split}W_{50,c,t} = W_{50,c}^2 + W_{T,50}^2[1-2\\mathrm {e}^{-(\\frac{W_{50,c}}{100})^2}] \\\\- 2W_{50,c}W_{T,50}[1-\\mathrm {e}^{-(\\frac{W_{50,c}}{100})^2}].\\ \\end{split}$ The factor of $100\\,{\\mathrm {km\\,s^{-1}}}$ in the exponential terms accounts for the profile shapes at 50% of their peak flux.",
"We set $W_{T,50}= 5 \\,{\\mathrm {km\\,s^{-1}}}$ in Eq.",
"REF , commensurate with estimates for systems with flat rotation curves by [112] and [57], since dwarf galaxies rarely have declining rotation curves [27], [62], and the UDG rotation curves from [67] are generally flat.",
"We have also not attempted to correct for asymmetric drift in our unresolved data, although this may be significant for $V_{rot}^{GF} \\lesssim 15 \\,{\\mathrm {km\\,s^{-1}}}$ [50], [82].",
"For these systems, $V_{rot}^{GF}$ is underestimated.",
"If our H1 detections have rising rotation curves at the edges of their H1 disks as is the case for many dwarfs and some UDGs, then our choice of $W_{T,50}$ results in an over-correction.",
"The resulting values of $V_{rot}^{GF}$ are given in Table , which we consider highly uncertain due to the uncertainties in $i^{GF}$ discussed above.",
"In Figure REF , we show the BTFR composed of two samples of galaxies with spatially-resolved H1 maps: SPARC [62] and LITTLE THINGS [48], [50].",
"In those samples, $V_{rot}$ has typically been measured using a standard tilted-ring approach [84], [95] that fits for the disk geometry and rotation simultaneously to break the degeneracy between $V_{rot}$ and sin$i$ in the line-of-sight velocities.",
"Figure REF also shows the 6 intermediate-inclination UDGs from the HUDs sample which deviate from the BTFRWe have calculated the $M_{bary}$ and its uncertainties using the values from Table 1 of [67].",
"We note that the error bars in our Figure REF for the UDGs from [66], [67] are smaller because they have propagated uncertainties in $M_{*}$ and $M_{HI}$ in logarithmic units instead of in linear units.",
"[66], and the 11 edge-on (i.e.",
"high-inclination) HUDs [45] which by and large do not [67].",
"We note that, because the H1 maps kinematically modeled by [66], [67] do not have sufficient spatial resolution to constrain $V_{rot}$ and $i$ simultaneously [34], [53], a novel method where $i$ is estimated separately from $V_{rot}$ is adopted.",
"On the other hand, any value of $i>75^{\\circ }$ for the high-inclination UDGs of [45] implies the same value of $V_{rot}$ since sin$i\\sim 1$ .",
"The orange and red stars in Figure REF show the locations of our H1-confirmed UDGs in the $M_{bary}-V_{rot}$ plane when $V_{rot}^{GF}$ is used to estimate rotation velocities, with the symbol colour denoting $i^{GF}$ as given by the colorbar.",
"The two UDGs with the largest $i^{GF}$ fall within the scatter of the relation defined by SPARC and LITTLE THINGS (dotted black line), while the rest do not.",
"Given the uncertainties in $i^{GF}$ particularly at low inclinations, we calculate the inclinations $i^{BTFR}$ required to bring the discrepant points onto the BTFR, connecting pairs of stars corresponding to the same galaxy in Figure REF with a horizontal line.",
"These values of $i^{BTFR}$ are also given in Table , the median $i^{BTFR}=14^{\\circ }$ .",
"As expected from Equation REF , the discrepant points move on to the BTFR if the H1 disks of the corresponding UDGs have inclinations below $i^{GF}$ .",
"To constrain the plausibility with which a disk with $i^{BTFR}$ can reproduce the optical morphologies of the UDGs, we compute $(b/a)^{BTFR}$ implied by $i^{BTFR}$ using Equations REF -REF and overplot ellipses corresponding to the best-fitting GALFIT models obtained with $b/a=(b/a)^{BTFR}$ held fixed in red in Figure REF .",
"In light of the above considerations, we conclude that interpreting the available observations to mean that the H1-confirmed SMUDGes UDGs deviate systematically from the BTFR is premature.",
"The uncertainties in $i^{GF}$ are large, particularly in low-inclination systems.",
"The white and red ellipses in Figure 6 demonstrate that in many cases, the GALFIT models that generated $i^{GF}$ and those produced holding $(b/a)^{BTFR}$ fixed produce nearly the same projected disk geometry.",
"Furthermore, the irregular optical morphologies of the H1-confirmed UDGs in Figure REF relative to our H1 non-detections evident in Figures REF and REF raise the possibility that clumpy emission systematically biases the GALFIT fits that generated $i^{GF}$ .",
"Since there are few clumps in each object and since those clumps are rarely symmetrically distributed about the object center, it seems plausible that the effect of fitting these irregular LSB objects with smooth GALFIT models is to systematically under-estimate $b/a$ such that $i^{GF}$ is biased high.",
"We emphasize that the SMUDGes UDG candidate selection criteria for low surface brightness and high ellipticity (Z19) favors low-inclination disks relative to high-inclination ones with the same $M_*$ and $R_eff$ , and therefore that low-inclination disks should be over-represented in the SMUDGes sample compared to samples with a random distribution of sky orientations with a mean $i \\sim 60^{\\circ }$ .",
"Furthermore, since our H1 follow-up sample is effectively selected on luminosity (see Section ) in a surface brightness-restricted sample, the objects in our sample are more likely still to be at low inclinations.",
"It is therefore possible that most of the H1-confirmed UDGs have low inclinations and that the $i^{GF}$ for those low-inclination systems is biased high.",
"A detailed investigation of potential biases in $i^{GF}$ for our H1-confirmed UDGs is beyond the scope of this pilot paper, but we are carrying out simulations to quantify biases in smooth GALFIT models of irregular LSB galaxies as a function of their asymmetry [1], [2], [30] for the full survey.",
"As a first check on our hypothesis, we estimate $V_{rot}^{GF}$ for the foreground dwarfs (which one would expect to lie within the scatter of the extrapolated BTFR, similar to other studies of the dwarf galaxy population; [50]; [24]) and overplot them on Figure REF .",
"We emphasize that $V_{rot}^{GF}$ for both the foreground dwarfs and the UDGs are only order-of-magnitude estimates that assume the optical and H1 disks are aligned, and we do not attempt to quantify these significant uncertainties either in Table or in Figure REF .",
"Nonetheless, at least some foreground dwarfs deviate from the BTFR similarly to the HI-confirmed UDGs, lending credence to our hypothesis that $i^{GF}$ is systematically overestimated.",
"The gas-rich, intermediate-inclination UDG outliers from the BTFR studied by [66], [67] imply that the underlying structure and baryonic composition of these systems differs fundamentally from that assumed in any of the UDG formation scenarios posited so far (see Section ).",
"As proposed by these authors, a high stellar specific angular momentum, low star formation feedback scenario is one possible explanation.",
"Examining Figure REF , however, it is curious that the consistency of gas-rich UDG samples with the BTFR defined by higher surface brightness systems seems to depend on how their inclinations were measured: the edge-on systems studied by [45] (where inclination uncertainties do not impact estimates of $V_{rot}$ ) are consistent with the BTFR, while the intermediate-inclination systems studied by [66], [67] (where $V_{rot}$ is measured independently from $i$ using a new technique) are outliers.",
"The sensitivity of the locations of our low- and intermediate-inclination H1-confirmed UDGs in the $M_{bary}-V_{rot}$ plane on the adopted inclination suggests that the effect of the viewing geometry should be carefully considered when inclination-dependent $V_{rot}$ are used to study the BTFR.",
"We emphasize that BTFR studies with SMUDGes UDGs that address the possible inclination dependence of offsets from this relation require H1 imaging with sufficient angular and spectral resolution to simultaneously model $V_{rot}$ and $i$ using standard tilted ring approaches.",
"This is feasible for a small subset of the H1 detections presented here, and work in this regard is underway.",
"Figure: Baryonic Tully-Fisher relation (M bary vs .V rot )(M_{bary}\\mathrm {vs.}\\, V_{rot}) formed by the SPARC and LITTLE THINGS samples, where V rot V_{rot} and ii are derived from standard tilted-ring kinematic modeling, with the best-fitting BTFR shown as a dotted black line.",
"The 6 UDGs from the HUDs sample where V rot V_{rot} is estimated separately using a new method to determine ii , lie off the BTFR, while the edge-on HUDs , by and large, lie within its scatter.",
"When we use the optically-derived inclinations, i GF i^{GF}, and turbulence-corrected H1 linewidths, W 50,c,t W_{50,c,t}, to estimate rotation velocities, V rot GF V_{rot}^{GF}, 7/9 H1-confirmed UDGs in our sample (orange and red stars) fall off this relation, as do some of our foreground dwarfs (purple crosses).",
"The UDG symbols are colored according to their axial ratios/inclinations as shown in the colorbar.",
"For the UDGs which fall off the BTFR, colored horizontal lines show how their axial ratios and inclinations change as they are brought onto the relation (from red to yellow).",
"Representations of best-fit GALFIT models with the axial ratios corresponding to each pair of stars are shown overlaid on stacked optical images in Figure .",
"It is plausible that the systematics of fitting smooth photometric models to clumpy, low inclination, LSB objects explains the offsets of the red stars from the BTFR.",
"See text for details."
],
[
"Conclusions",
"We have presented GBT H1 observations of 70 optically-detected SMUDGes UDG candidates (Z19) with $m_g \\lesssim 19.5\\,\\mathrm {mag}$ in the Coma region.",
"We detect H1 reservoirs in 18 of them (Figure REF ), measuring systemic velocities, $V_{sys}$ , velocity widths, $W_{50,c}$ , and flux integrals, $\\int S dv$ , directly from the spectra.",
"Using kinematic distances estimated from $V_{sys}$ , we compute H1 masses, $M_{HI}$ , from the spectra as well as stellar masses, $M_{*}$ , and half-light radii, $R_{eff}$ , from GALFIT models to the deep DECaLS imaging.",
"We use $R_{eff}$ to confirm that 9 of our H1 detections satisfy the size criterion defining UDGs, while the remainder are foreground dwarfs (Tables and ).",
"Although only a pilot for a much larger GBT program that is currently underway, these observations already represent the largest H1 follow-up campaign of optically-selected UDG candidates ever reported, and the 9 confirmed UDGs are the largest available sample of optically-selected UDGs with H1 detections.",
"Comparing the properties of our H1-detected UDGs, H1-detected foreground dwarfs and our H1 non-detections, we find similar sky distributions relative to the Coma large-scale structure (Figure REF ) but that 8/9 UDGs are in low-density environments with no massive ($M_g< -19\\,\\mathrm {mag}$ ) companions within $R_{proj}= 300\\,$ kpc or $\\Delta V_{sys}= \\pm 500\\,{\\mathrm {km\\,s^{-1}}}$ .",
"In addition, our H1 detections typically have counterparts in the NUV if the exposures are sufficiently deep ($\\gtrsim \\,1000\\,$ sec with GALEX).",
"In DECaLS-depth optical imaging, the gas-rich UDGs are bluer and smoother in morphology than the UDG candidates that we do not detect in H1 but the scatter is large in both properties (Figures REF , REF , and REF ).",
"On the other hand, targets that are both blue and irregular are gas-rich, while those that are both red and smooth are gas-poor: it is the combination of optical morphology and color that best predicts gas richness.",
"Although the angular sizes of the foreground dwarfs are typically larger than those of the H1-confirmed UDGs, there is little difference in optical morphology or color between these subsamples (Figures REF and REF ).",
"Without distance information, foreground dwarfs contaminate samples of optically blue, irregular UDG candidates.",
"Commensurate with tentative results for blue UDGs around galaxy groups [98], we find evidence for a correlation between the gas richness, $M_{HI}/M_{*}$ , and size, $R_{eff}$ , when our H1-confirmed UDGs as well as other gas-rich UDGs are divided into two stellar mass bins (Figure REF ).",
"The same trend is not obvious for the foreground dwarfs.",
"The correlation between UDG gas richness and size suggested by the data is broadly consistent with predictions from the star formation feedback model for UDG formation [33], although other mechanisms may also produce the trend.",
"We place our H1-confirmed UDGs on the BTFR using best-fitting inclinations, $i^{GF}$ , from smooth GALFIT models of DECaLS imaging and turbulence-corrected velocity widths to estimate rotation velocities $V_{rot}^{GF}$ .",
"We find that the 7/9 objects with the lowest $i^{GF}$ have lower $V_{rot}^{GF}$ than expected from the BTFR defined by high surface brightness, gas-rich galaxies with H1 rotation curves and disk geometries derived from kinematic models (Figure REF ), similar to that found by [66], [67] for a sample of marginally-resolved HUDs using a new technique for constraining $i$ separately from $V_{rot}$ via the H1 morphology.",
"For our sample, however, we find that plausible systematics resulting from the application of smooth GALFIT models to clumpy, low-inclination LSB objects are sufficient to reconcile these discrepancies (Figures REF and REF ) precluding a meaningful analysis of BTFR offsets.",
"We plan on investigating this trend and its implications in detail with our full follow-up sample.",
"The pilot survey results presented here provide some initial insight into the properties of gas-rich UDGs and the mechanisms by which they form.",
"Despite being the largest of its kind, our sample of confirmed gas-rich optically-detected UDGs remains small.",
"A much larger SMUDGes H1 follow-up campaign is underway at the GBT.",
"We ultimately plan on targeting over 200 objects, and expect to confirm at least 50 gas-rich UDGs.",
"This larger sample will enable quantitative investigations of the interplay between gas richness and UDG properties in order to understand how they form and evolve.",
"Furthermore, it will also provide predictive insight into the gas properties of UDG candidates in the eventual $\\sim 10,000\\,\\mathrm {deg^2}$ SMUDGes survey.",
"cccCCCCCCCCCCcc[!h] Target UDG Candidate Properties Name RA Dec $m_g$ $\\mu _{0,g}$ $g-r$ $g-z$ $r_{eff}$ $b/a$ ${\\theta }$ $n$ Int.",
"Time ${\\sigma _{50}}$ Ref HI H:M:S D:M:S (mag) ${(\\mathrm {\\frac{mag}{arcsec^{2}}})}$ (mag) (mag) (arcsec) (deg) (hours) (mJy) Det?",
"(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) SMDG1103517+284120 11:03:51.7 28:41:20 18.060.01 25.250.14 0.320.02 0.200.07 16.2 0.9 0.83 0.01 -29 3 0.71 0.04 0.2 1.39 K20 (M98) Y SMDG1217378+283519 12:17:38.0 28:35:20 18.980.01 24.950.05 0.540.01 0.870.02 10.2 0.2 0.70 0.01 47 1 0.73 0.02 0.5 0.58 Z19 SMDG1217443+332043 12:17:44.2 33:20:44 18.490.01 25.990.10 0.460.03 0.600.06 17.1 0.7 0.77 0.01 84 2 0.52 0.03 0.5 0.69 Z19 SMDG1217451+281724 12:17:45.0 28:17:25 20.040.04 25.710.12 0.660.05 1.110.08 7.3 0.3 0.82 0.03 0 7 0.56 0.04 2.4 0.38 Z19 SMDG1220188+280131 12:20:19.0 28:01:34 18.440.01 24.380.09 0.340.01 0.500.03 12.7 0.4 0.63 0.01 -67 1 0.96 0.03 0.2 0.79 K20 (H18) Y SMDG1220212+290831 12:20:21.0 29:08:34 19.480.01 24.300.06 0.620.02 0.880.04 6.1 0.1 0.90 0.01 -24 5 0.92 0.02 1.3 0.41 Z19 SMDG1221086+292920 12:21:17.0* 29:29:21 18.770.01 25.270.06 0.640.01 1.030.02 14.1 0.3 0.57 0.01 -4 1 0.70 0.02 0.4 0.73 Z19 SMDG1221235+303643 12:21:23.4 30:36:44 19.010.01 25.550.11 0.540.02 0.680.04 14.4 0.6 0.63 0.01 65 1 0.76 0.04 0.6 0.84 Z19 SMDG1221401+284346 12:21:40.0 28:43:47 19.260.01 25.000.06 0.640.02 0.910.03 9.5 0.2 0.60 0.01 77 1 0.67 0.02 0.9 0.53 Z19 SMDG1221497+283111 12:21:49.7 28:31:12 19.000.03 25.830.13 0.590.04 0.820.05 15.0 0.7 0.64 0.02 9 2 0.65 0.04 0.5 0.85 Z19 The first 10 rows of this table are shown here.",
"The full table is available online in machine readable format.",
"col.(1): Adopted SMUDGes UDG candidate name.",
"cols.",
"(2) and (3): J2000 position of optical centroid, which corresponds to our GBT LOS.",
"RA values with an asterisk (*) indicate an offset (in RA and/or Dec) in the GBT pointing position.",
"cols.",
"(4) and (5): $g-$ band apparent magnitude and central surface brightness.",
"cols.",
"(6) and (7): $g-r$ and $g-z$ colors.",
"col. (8)-(11): Best-fitting effective radius, axial ratio, position angle, and $\\mathrm {S\\acute{e}rsic}$ index in GALFIT model of UDG candidate.",
"col.(12): Total effective GBT integration time, including the ON+OFF positions and subtracting any time lost due to RFI.",
"col.(13): Representative RMS noise of the spectrum at a velocity resolution of $\\Delta V = 50 \\, {\\mathrm {km\\,s^{-1}}}$ .",
"col.(14): Reference from which UDG candidate is selected, alternative references in parentheses.",
"S97 = [93]; M98 = [69]; H18 = [44].",
"col.(15): H1 detection?",
"cCCCCCCCCCC[htb!]",
"Properties of UDG with H1 detections Name $\\Delta V$ $\\sigma _{\\Delta V}$ $V_{sys}$ $W_{50,c}$ $S_{HI}$ $D_{HI}$ log($M_{HI}$ ) log($M_{*}$ ) log$M_{bary}$ $R_{eff}$ (${\\mathrm {km\\,s^{-1}}}$ ) (mJy) (${\\mathrm {km\\,s^{-1}}}$ ) (${\\mathrm {km\\,s^{-1}}}$ ) (Jy$\\,{\\mathrm {km\\,s^{-1}}}$ ) (Mpc) (log[$M_{\\odot }$ ]) (log[$M_{\\odot }$ ]) (log[$M_{\\odot }$ ]) (kpc) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) SMDG1220188+280131 10 1.86 22832 322 0.610.10 32.6 8.190.15 7.510.24 8.370.15 2.000.30 SMDG1225185+270858 20 0.69 58885 637 0.170.07 84.1 8.440.19 8.010.21 8.670.16 2.630.18 SMDG1226040+241802 15 0.66 115853 344 0.260.05 165.5 9.220.08 8.570.20 9.410.08 4.480.17 SMDG1230359+273311 25 0.56 67943 1025 0.370.08 97.1 8.910.10 8.010.20 9.070.10 3.110.17 SMDG1241424+273353 10 1.52 77662 172 0.260.06 110.9 8.870.11 8.360.20 9.090.10 3.810.20 SMDG1248019+261236 25 0.51 60435 327 0.240.05 86.3 8.630.10 8.010.21 8.820.09 2.760.17 SMDG1301005+210355 25 0.53 70513 315 0.130.05 100.7 8.490.16 8.220.20 8.760.13 3.570.28 SMDG1312223+312320 20 1.05 74873 424 0.260.09 107.0 8.840.15 8.180.20 9.030.13 3.000.15 SMDG1315427+311846 25 0.99 74866 138 0.510.08 106.9 9.140.08 8.440.20 9.320.08 5.850.41 col.(2): Velocity resolution of spectrum used to compute H1 properties (see Figure REF ).",
"col.(3): RMS noise of spectrum at $\\Delta V$ in col.(2).",
"col.(4): Heliocentric systemic velocity.",
"col.(5): Velocity width of the H1 detection, corrected for cosmological redshift and instrumental broadening.",
"col.(6): Integrated H1 flux.",
"col.(7): Distance estimated using the Hubble-Lemaître Law, $V_{sys}$ and $\\mathrm {H}_0 = 70 \\, {\\mathrm {km\\,s^{-1}}}\\,\\mathrm {Mpc}^{-1}$ .",
"We adopt distance uncertainties of 5 Mpc.",
"col.(8): Logarithm of H1 mass calculated from Eq.REF using $S_{HI}$ in col.(6) and $D_{HI}$ in col.(7).",
"col.(9): Logarithm of stellar mass calculated using $m_g$ and $g-r$ from Table , $D_{HI}$ in col.(7), and the corresponding relation from [118].",
"col.(10): Logarithm of baryonic mass, $1.33M_{HI}+ M_{*}$ .",
"col.(11): Effective radius in physical units using $r_{eff}$ from Table and $D_{HI}$ in col.(7).",
"cCCCCCCCCCC[htb!]",
"H1 Properties of Dwarfs Name $\\Delta V$ $\\sigma _{\\Delta V}$ $V_{sys}$ $W_{50,c}$ $S_{HI}$ $D_{HI}$ log($M_{HI}$ ) log($M_{*}$ ) log$M_{bary}$ $R_{eff}$ (${\\mathrm {km\\,s^{-1}}}$ ) (mJy) (${\\mathrm {km\\,s^{-1}}}$ ) (${\\mathrm {km\\,s^{-1}}}$ ) (Jy$\\,{\\mathrm {km\\,s^{-1}}}$ ) (Mpc) (log[$M_{\\odot }$ ]) (log[$M_{\\odot }$ ]) (log[$M_{\\odot }$ ]) (kpc) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) SMDG1103517+284120 10 2.46 6682 263 0.970.13 9.5 7.320.45 6.580.50 7.490.46 0.750.40 SMDG1223451+283549 20 0.51 23774 595 0.340.05 34.0 7.970.15 7.190.23 8.140.14 0.780.02 SMDG1231329+232916 25 0.82 10606 88 0.440.07 15.1 7.380.29 6.680.35 7.560.30 0.770.02 SMDG1239050+323016 10 0.82 6131 322 0.190.04 8.8 6.540.50 5.790.53 6.720.50 0.420.24 SMDG1240017+261919 10 2.24 4522 133 0.720.10 6.5 6.850.68 5.640.70 6.990.67 0.300.23 SMDG1253571+291500 25 0.45 5024 675 0.230.05 7.2 6.450.61 6.020.64 6.680.61 0.220.15 SMDG1255412+191221 5 4.76 4201 192 4.370.21 6.0 7.570.72 6.380.75 7.710.72 0.750.62 SMDG1306148+275941 15 0.55 25593 424 0.150.04 36.6 7.670.17 6.940.23 7.840.16 1.280.17 SMDG1313188+312452 25 1.25 8025 138 0.500.10 11.5 7.190.39 7.140.42 7.530.39 1.240.54 All parameters have the same definitions as in Table .",
"cCCC H1 Properties of Non-detections Name $D_{lim}$ log$(M^{lim}_{HI})$ $({M^{lim}_{HI}}/{L_{g}})$ (Mpc) (log[$M_{\\odot }$ ]) $({M_{\\odot }}/{L_{\\odot }})$ (1) (2) (3) (4) SMDG1217378+283519 7a 6.22 1.27 SMDG1217443+332043 100 8.61 0.96 SMDG1217451+281724 100 8.35 2.22 SMDG1220212+290831 100 8.39 1.43 SMDG1221086+292920* 14.6a 6.96 1.32 SMDG1221235+303643 100 8.69 1.89 SMDG1221401+284346 100 8.49 1.50 SMDG1221497+283111 100 8.7 1.92 SMDG1221577+281436 100 8.8 1.06 SMDG1223448+295949 100 8.54 1.58 SMDG1224082+280544 100 8.41 0.82 SMDG1224166+291506 100 8.18 1.49 SMDG1225265+311646* 100 8.59 1.19 SMDG1231418+264433 100 8.3 2.49 SMDG1237277+333048 100 8.33 1.19 SMDG1239267+274736 100 8.41 1.81 SMDG1239503+244949 100 8.53 1.36 SMDG1240119+251447 100 8.4 2.76 SMDG1247233+180140 100 8.69 1.02 SMDG1249413+270645 100 8.27 1.49 SMDG1251013+274753* 87a 8.5 1.86 SMDG1253048+253121 100 8.44 1.97 SMDG1253151+274115* 100b 8.28 1.63 SMDG1253489+273934 100 8.79 2.86 SMDG1254252+194332 100 8.61 0.83 SMDG1254556+285846 100 8.45 1.89 SMDG1255336+213035 100 8.12 1.59 SMDG1307464+291230 100 8.44 2.51 SMDG1308296+271354 100 8.35 1.98 SMDG1322561+314804 100 8.61 2.06 SMDG1226306+220532 100 8.63 0.95 SMDG1231070+253508* 100 8.62 2.28 SMDG1232244+274043 100 8.22 1.46 SMDG1233516+234545 100 8.49 0.73 SMDG1234503+293313 100 8.49 1.34 SMDG1235065+263342 100 8.27 1.57 SMDG1240490+254406 100 8.45 3.50 SMDG1241097+221223 100 8.33 1.92 SMDG1245022+230956 100 8.28 1.44 SMDG1246029+255724 100 8.51 1.54 SMDG1248202+183824 100 8.7 0.37 SMDG1249353+253106 100 8.56 0.72 SMDG1251291+284433* 100 8.29 2.80 SMDG1251337+314240 100 8.44 1.12 SMDG1251371+244922 100 8.46 1.28 SMDG1252056+221556 100 8.2 1.71 SMDG1252402+262602* 100 8.48 0.85 SMDG1302417+215954 3.8c 5.94 0.25 SMDG1306158+273459 100 8.35 2.32 SMDG1312226+195525 100 8.27 1.68 SMDG1322538+220445* 100 8.4 1.72 SMDG1333509+275006 100 8.38 1.72 $^{\\mathrm {a}}$ Kadowaki et al., in prep,$^{\\mathrm {b}}$[52],$^{\\mathrm {c}}$[56] col. (2): Adopted distance in Eq.REF ; see text for details.",
"col. (3): $5\\sigma $ upper limit on $M_{HI}$ calculated from Eq.REF using $D_{lim}$ from col. (2) and $\\sigma _{50}$ from Table .",
"col.(4): Upper limit on the gas richness (which is distance independent).",
"cCCC Inclinations and Rotation Velocities Name $i^{GF}$ $V_{rot}^{GF}$ $i^{BTFR}$ (deg) (${{\\mathrm {km\\,s^{-1}}}}$ ) (deg) (1) (2) (3) (4) H1-confirmed UDGs SMDG1220188+280131 52 19 20 SMDG1225185+270858 53 38 - SMDG1226040+241802 37 26 11 SMDG1230359+273311 69 53 - SMDG1241424+273353 38 12 6 SMDG1248019+261236 36 25 15 SMDG1301005+210355 51 18 14 SMDG1312223+312320 42 29 17 SMDG1315427+311846 47 8 4 Foreground dwarfs SMDG1103517+284120 35 21 - SMDG1223451+283549 54 34 - SMDG1231329+232916 63 3 6 SMDG1239050+323016 33 27 - SMDG1240017+261919 43 8 17 SMDG1253571+291500 60 37 - SMDG1255412+191221 49 11 17 SMDG1306148+275941 65 22 - SMDG1313188+312452 39 9 12 col.(2): Inclination calculated using Eq.REF , $b/a$ from Table , and an intrinsic axial ratio of $q_0=0.2$ .",
"col.(3): Rotational velocity calculated using $W_{50,c}$ corrected for turbulence and $i$ in col.(2).",
"Given the systematics associated with measuring inclinations of clumpy low-inclination objects from smooth models, we consider $i^{GF}$ and $V_{rot}^{GF}$ to be rough estimates (see text).",
"cols.",
"(4): Inclinations required to lie on the BTFR for UDGs and dwarfs with $V_{rot}^{GF}$ lower than expected from the BTFR at their measured $M_{bary}$ .",
"We thank the anonymous referee for their detailed and useful feedback to help improve the original manuscript.",
"KS acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC).",
"The Green Bank Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; NOAO Proposal ID # 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Proposal ID # 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; NOAO Proposal ID # 2016A-0453; PI: Arjun Dey).",
"DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (NOAO); the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOAO.",
"The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation.",
"NOAO is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.",
"This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration.",
"Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.",
"The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC/CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University.",
"BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program \"The Emergence of Cosmological Structures\" Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance.",
"The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 11433005).",
"The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology.",
"NEOWISE is funded by the National Aeronautics and Space Administration.",
"The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No.",
"AST-0950945 to NOAO.",
"GBT (VEGAS) astropy[9], [78], GALFIT [77], GBTIDL [68], SEP [10], [18],"
]
] | 2005.14202 |
[
[
"Provably Good Solutions to the Knapsack Problem via Neural Networks of\n Bounded Size"
],
[
"Abstract The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence.",
"Against this background, we study the expressive power of neural networks through the example of the classical NP-hard Knapsack Problem.",
"Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a Knapsack instance and thereby compute optimal or provably good solution values.",
"We show that an RNN of depth four and width depending quadratically on the profit of an optimum Knapsack solution is sufficient to find optimum Knapsack solutions.",
"We also prove the following tradeoff between the size of an RNN and the quality of the computed Knapsack solution: for Knapsack instances consisting of $n$ items, an RNN of depth five and width $w$ computes a solution of value at least $1-\\mathcal{O}(n^2/\\sqrt{w})$ times the optimum solution value.",
"Our results build upon a classical dynamic programming formulation of the Knapsack Problem as well as a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the Knapsack Problem.",
"A carefully conducted computational study qualitatively supports our theoretical size bounds.",
"Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem."
],
[
"Introduction",
"Deep learning and neural networks (NNs) are at the heart of some of the greatest advances in modern computer science.",
"They enable huge breakthroughs in applications like computer vision, translation, speech recognition, and autonomous driving, to name just a few; see, e.g., [24].",
"While numerous computational studies present impressive empirical proof of neural networks' computational power, we are still far away from a more rigorous theoretical explanation of these observations.",
"Apart from the popular applications named above, it has been shown that NNs have high potential for practically solving combinatorial optimization (CO) problems or empirically improving classical solution methods [4].",
"For example, [48] and [47] utilize NNs in order to empirically enhance dynamic programming, a very classical CO method.",
"While the methods used in these papers indeed provide fast and empirically near-optimal solutions, their use of NNs makes it virtually impossible to give theoretical optimality or worst-case approximation guarantees.",
"Motivated by this imbalance, and focusing on the Knapsack Problem, which is a prime example of CO problems that can be solved via dynamic programming, we investigate the following question: Which neural network size is theoretically sufficient to find solutions of provable quality for the Knapsack Problem?",
"We give an answer to this question by presenting a class of carefully constructed NNs with provable quality guarantees and support our size bounds by a computational study.",
"Finally, we argue that our approach is not at all specific for the Knapsack Problem, but can be generalized to many other CO problems, e.g., various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem."
],
[
"The Knapsack Problem.",
"The Knapsack Problem constitutes one of the oldest and most studied problems in Combinatorial Optimization (CO).",
"Given a set of items with certain profit and size values, as well as a Knapsack capacity, the Knapsack Problem asks for a subset of items with maximum total profit such that the total size of the subset does not exceed the capacity.",
"The Knapsack Problem is one of Karp's 21 original NP-complete problems [19] and has numerous applications in a wide variety of fields, ranging from production and transportation, over finance and investment, to network security and cryptography.",
"It often appears as a subproblem at the core of more complex problems; see, e.g., [29], [20].",
"This fact substantiates the Knapsack Problem's prominent importance as one of the key problems in CO.",
"In particular, the Knapsack Problem is frequently being used as a testbed for measuring the progress of various exact and heuristic solution approaches and computational methods such as, e.g., integer programming, constraint programming, or evolutionary algorithms.",
"In integer programming, for example, the Knapsack Problem and so-called `Knapsack Inequalities' play a central role, both with respect to theory as well as in the development of modern computational methods; see, e.g., [5], [9].",
"The Knapsack Problem is therefore a natural and important object of study in order to advance our theoretical understanding of neural networks and get closer to a rigorous explanation of their stunning success in so many applications, including miscellaneous optimization problems."
],
[
"Related work.",
"The idea of using neural networks (NNs) to practically solve CO problems became popular with the work of [18].",
"Hopfield NNs are special versions of recurrent neural networks (RNNs) that find solutions to optimization problems by converging towards a minimum of an energy function.",
"[40] reviews this early stream of research.",
"While most authors mainly focus on the Traveling Salesperson Problem (TSP), [34] study a so-called mean field NN for (generalizations of) the Knapsack Problem and empirically assess the quality of its solutions.",
"While there has been less research at the intersection of CO and NNs in the 2000s, modern advances in the area of deep learning have boosted the interest in this direction again.",
"[4] review these developments from a practical perspective.",
"Common applications include speeding up solvers for mixed-integer linear programs, for instance, by automatically learning on which variables to branch in branch-and-bound algorithms; see [27] for a survey.",
"Machine learning has also been applied to modeling aspects of CO, as reviewed by [28], and to several specific CO problems, where the TSP is often one of them [44], [3], [21], [33], [8], [23].",
"The different methods used by these authors include feedforward and recurrent neural networks, reinforcement learning, attention mechanisms, pointer networks, graph embeddings, and graph neural networks.",
"For example, [3] utilize an RNN trained by reinforcement learning and present a computational study demonstrating their approach's empirical effectiveness for the TSP and the Knapsack Problem.",
"Particularly related to our work, [48] and [47] use NNs to speed up dynamic programming algorithms for CO problems.",
"The key difference to our work, however, is that NNs are used as heuristics in these papers, making it virtually impossible to give any meaningful worst-case performance guarantees.",
"The recent success of deep neural networks has also triggered a lot of research on their expressivity.",
"As we do in this paper, many authors focus on the simple but practically powerful model of feedforward NNs with activations in the form of rectified linear units (ReLU).",
"Since [10] corroborated their empirical success, such ReLU NNs have been established as a standard model in Machine Learning within the past decade.",
"ReLU NNs can compute any continuous piecewise linear function [11], [1].",
"This fact implies universal approximation properties.",
"A variety of results has been achieved on depth vs. width tradeoffs [41], [7], [42], [15], [26], [37], [49], [1], [32], [13].",
"Closely related are investigations concerning the number and structure of linear regions that NNs with certain size and depth may have [30], [35], [36], [14].",
"[38] use mixed-integer programming for precisely counting the number of such regions.",
"[31] prove size lower bounds to represent Boolean functions with NNs of limited depth."
],
[
"Our contribution.",
"We present a rigorous mathematical study on the expressivity of NNs through the example of the NP-hard Knapsack Problem.",
"To this end, we show that there is a class of feedforward ReLU NNs of bounded size that compute provably good solutions to the NP-hard Knapsack Problem.",
"In Section , we first present such an NN of depth $\\mathcal {O}(n)$ and width $\\mathcal {O}((p^*)^2)$ that always finds the exact value of an optimum Knapsack solution.",
"Here, $n$ is the number of items in the Knapsack instance, and $p^*$ is an a priori known upper bound on the value of an optimum solution.",
"More precisely, the optimum solution value is found by iteratively applying an RNN of depth four and width $\\mathcal {O}((p^*)^2)$ to the $n$ items of a Knapsack instance.",
"As $p^*$ can, e.g., be chosen as the total size of all items, the RNN's width is pseudo-polynomially bounded in the input size of the Knapsack instance.",
"Due to the Knapsack Problem's NP-hardness, however, there is no polynomial-size NN that always finds the optimum solution value, unless P$\\,=\\,$ NP.",
"In Section , we prove that the width of the NNs can be drastically decreased while still obtaining solution values of provable quality in the worst case.",
"We construct an RNN of depth five and fixed width $w$ which, when applied iteratively to the $n$ items of a Knapsack instance, always produces a solution value of at least $1-\\mathcal {O}(n^2/\\sqrt{w})$ times the optimum solution value.",
"In particular, an $\\varepsilon $ -approximate solution value can be guaranteed by choosing width $w\\in \\mathcal {O}(n^4/\\varepsilon ^2)$ .",
"The dependence of the width on $\\varepsilon $ is unavoidable, unless P$\\,=\\,$ NP.",
"To the best of our knowledge, our results establish the first rigorous tradeoff between the size of neural networks for CO problems and their worst-case solution quality.",
"Even though we cannot show theoretical lower bounds beyond what is directly implied by NP-hardness, we provide empirical evidence for the quadratic dependence on $p^*$ (and $1/\\varepsilon $ ) in Section .",
"The idea behind our construction of the NNs is to mimic the classical dynamic program for the Knapsack Problem.",
"More precisely, the output neurons of the RNN can be seen as elements of the dynamic programming state space while the hidden neurons and the network itself implement the recursive dynamic programming formula.",
"Here, the main technical difficulty is to always filter out the correct entries of the previous state space (input neurons) needed in the recursive formula.",
"In addition, our NNs of fixed width rely on a subtle variant of the rounding procedure that turns the pseudo-polynomial dynamic program into a fully polynomial-time approximation scheme for the Knapsack Problem.",
"In this paper, the Knapsack Problem mainly serves as a prominent showcase for a novel approach to the rigorous analysis of neural networks' expressivity.",
"This approach is by no means specific for the Knapsack Problem.",
"In Section , we discuss how it can be applied to NNs for other combinatorial optimization problems that can be solved via dynamic programming.",
"In Particular, we establish similar results for the Longest Common Subsequence Problem, the Single-Source and All-Pairs Shortest Path Problems, as well as the NP-hard Traveling Salesperson Problem and the Constrained Shortest Path Problem.",
"For the latter problem one can show similar results on the tradeoff between the size of NNs and their solution quality."
],
[
"Neural networks with rectified linear units.",
"We use definitions and notations similar to [39].",
"A feedforward neural network with rectified linear units, abbreviated by ReLU NN, or simply NN, is a finite directed acyclic graph $(V,E)$ , equipped with arc weights $w_{uv}\\in \\mathbb {R}$ , for each $(u,v)\\in E$ , and node biases $b_v\\in \\mathbb {R}$ , for each node $v\\in V\\setminus V_0$.",
"Here, $V_0$ is the set of nodes with in-degree zero.",
"The nodes in $V$ are called neurons.",
"The depth $k$ is the length of a longest path in the graph.",
"In the following we suppose that neurons are grouped into layers $V=V_0\\mathbin {\\cdot \\cup }V_1 \\mathbin {\\cdot \\cup }\\cdots \\mathbin {\\cdot \\cup }V_k$ such that the layer index strictly increases along each arc.Some authors only allow connections between successive layers.",
"One can create such a structure by adding additional neurons propagating the values of neurons from former layers through the network.",
"For our purposes, however, it is convenient to omit this restriction.",
"Further, we assume that $V_0$ and $V_k$ are precisely the sets of neurons with in-degree and out-degree zero, respectively.",
"Consequently, they are called input neurons and output neurons, respectively.",
"Neurons in $V\\setminus (V_0\\cup V_k)$ are called hidden neurons.",
"Let $n_\\ell =\\vert V_l\\vert $ be the number of neurons in the $\\ell $ -th layer.",
"The width and size of the NN are defined to be $\\max \\lbrace n_1,\\dots ,n_{k-1}\\rbrace $ and $\\sum _{\\ell =1}^{k-1} n_\\ell $ , respectively.",
"Every NN computes a function $\\mathbb {R}^{n_0}\\rightarrow \\mathbb {R}^{n_k}$ as follows.",
"Given an input vector $x\\in \\mathbb {R}^{n_0}$ , we associate an activation $a(v)$ with every neuron $v\\in V\\setminus V_0$ and an output $o(v)$ with every neuron $v\\in V\\setminus V_k$.",
"First, the output values $o(v)$ , $v\\in V_0$ , of the $n_0$ input neurons equal the $n_0$ components of input vector $x$ .",
"Second, the activation of a neuron $v\\in V\\setminus V_0$ is the weighted sum of outputs of all predecessors plus its bias, that is, $a(v)=b_v +\\sum _{u\\colon (u,v)\\in E} w_{uv} o(u)$ .",
"Third, for each hidden neuron $v\\in V\\setminus (V_0\\cup V_k)$ , the output is determined by $o(v)=\\sigma (a(v))$ , where $\\sigma $ is the so-called activation function.",
"In this paper, $\\sigma $ is always the rectifier function $\\sigma (z)=\\max \\lbrace 0,z\\rbrace $ .",
"Neurons having this activation function are called rectified linear units (ReLUs).",
"Finally, the output vector $y\\in \\mathbb {R}^{n_k}$ consists of the $n_k$ activation values $a(v)$ of the $n_k$ output neurons $v\\in V_k$ .",
"Figure REF gives an example, which will also be used as a subnetwork in later sections.",
"Figure: An NN with two input neurons, labeled x 1 x_1 and x 2 x_2, onehidden neuron, labeled with the shape of the rectifier function, and one output neuron, labeled yy.",
"The arcsare labeled with their weights and all biases are zero.",
"The network has depth 2, width 1, and size 1.",
"Itcomputes the function x↦y=x 2 -max{0,x 2 -x 1 }=min{x 1 ,x 2 }x \\mapsto y= x_2-\\max \\lbrace 0,x_2-x_1\\rbrace =\\min \\lbrace x_1,x_2\\rbrace .Since feedforward NNs have a fixed input size, a common way of handling sequential inputs of arbitrary length is to use recurrent neural networks (RNNs).",
"This type of NNs has become very popular, e.g., for tasks in language or speech processing.",
"Essentially, an RNN is a feedforward NN that is used repeatedly for every piece of the input sequence and maintains a hidden state by passing (part of) its output in each step as an additional input to the next step.",
"More precisely, in the $i$ -th step, the input of the RNN consists of the $i$ -th input vector $x_{i}$ , as well as, the previous hidden state vector $h_{i-1}$ .",
"In the same manner as a feedforward NN described above, it then computes the $i$ -th output vector $y_i$ , as well as, the new hidden state vector $h_i$ .",
"The basic structure of an RNN is shown in Figure REF .",
"Sometimes it holds that $y_i=h_i$ , that is, the $i$ -th output is actually equal to the $i$ -th hidden state.",
"Figure: Basic structure of an (unfolded) RNN."
],
[
"Notations and Algorithms for the Knapsack Problem.",
"An instance of the Knapsack Problem consists of $n$ items $1,2,\\dots ,n$ , where each item $i\\in [n]$ comes with a given profit $p_i\\in \\mathbb {N}$ and size $s_i\\in \\left]0,1\\right]$ , together with a Knapsack that can hold any subset $M\\subseteq [n]$ of items of total size $\\sum _{i\\in M}s_i$ at most 1.",
"The task is to find such a subset $M\\subseteq [n]$ that maximizes the total profit $\\sum _{i\\in M}p_i$ .",
"Here and in the following, we use $\\mathbb {N}\\lbrace 1,2,3,\\dots \\rbrace $ to denote the natural numbers (without zero), and for every $k\\in \\mathbb {N}$ , we let $[k]\\lbrace 1,2,\\dots ,k\\rbrace $ .",
"We outline a classical dynamic programming formulation for the Knapsack Problem.",
"Let $p^*$ be an upper bound on the optimum solution value, e.g., $p^*=\\sum _{i=1}^{n} p_i$ .",
"For $i\\in [n]$ and $p\\in [p^*]$ , let $f(p,i) \\min \\left\\lbrace \\sum \\nolimits _{j\\in M} s_j \\mathrel {}|\\mathrel {}M\\subseteq [i],~\\sum \\nolimits _{j\\in M} p_j \\ge p \\right\\rbrace $ be the minimum size of a subset of the first $i$ items with total profit at least $p$ .",
"With $f(p,i)0$ for $p\\le 0$ and $f(p,0)+\\infty $ for $p\\in [p^*]$ , the values of $f$ can be computed recursively by $f(p,i)=\\min \\bigl \\lbrace f(p,i-1), f(p-p_i,i-1)+s_i\\bigr \\rbrace $ for $i\\in [n]$ , $p\\in [p^*]$ , where the first option corresponds to not using the $i$ -th item, while the second option corresponds to using it.",
"The optimum solution value is $\\max \\lbrace p\\in [p^*]\\mid f(p,n)\\le 1\\rbrace $, and the optimum subset can easily be found by backtracking.",
"The runtime of the dynamic program is $\\mathcal {O}(np^*)$ , thus pseudo-polynomial in the input size.",
"Due to NP-hardness of the Knapsack Problem, one cannot expect to find an exact algorithm with polynomial running time.",
"However, by carefully downscaling and rounding the profit values in the dynamic program, for each $\\varepsilon >0$ , one can achieve a feasible solution with guaranteed profit of at least $1-\\varepsilon $ times the optimal profit, while the running time can be bound polynomially in the input size and $1/\\varepsilon $ .",
"Such a class of algorithms with arbitrary good approximation guarantees is called a fully polynomial-time approximation scheme (FPTAS).",
"For more details, we refer to the books by [17], [43], or [45].",
"Usually, the Knapsack Problem is defined with integer size values $s_i\\in \\mathbb {N}$ and some Knapsack capacity $S\\in \\mathbb {N}$ , bounding the total size of chosen items.",
"Dividing all item sizes by $S$ transforms such an instance into an instance of the type considered here.",
"For the case of integral item sizes, there is also a pseudo-polynomial dynamic programming formulation parameterized by the size instead of the profit values; see, e.g., [22].",
"Our construction in Section can analogously be applied to this formulation.",
"This variant, however, does not easily extend to an FPTAS.",
"We therefore stick to the variant parametrized by the profit values as introduced above."
],
[
"An Exact RNN for the Knapsack Problem",
"In this section we introduce the DP-NN, an NN that exactly executes the dynamic program described in Section .",
"In fact, the DP-NN is an RNN that receives the items one by one and computes the state space of the dynamic program for the items seen so far.",
"Like the dynamic program in Section , the DP-NN requires a fixed upper bound $p^*$ on the optimal objective value of the Knapsack Problem.",
"We relax this condition in Section , when we investigate how the FPTAS for the Knapsack Problem can be implemented as an NN.",
"In the $i$ -th step, the DP-NN receives $p^* + 2$ inputs, namely $f(p,i-1)$ for $p\\in [p^*]$ , as well as $p_{i}$ and $s_{i}$ .",
"It computes $p^*$ output values, namely $f(p,i)$ for $p\\in [p^*]$ .",
"Hence, overall it has $p^*+2$ input neurons and $p^*$ output neurons.",
"Figure REF illustrates the recurrent structure of the NN, which computes the state space of the dynamic program.",
"Figure: Recurrent structure of the DP-NN to solve the Knapsack Problem.In the following it is very important to distinguish fixed parameters of the NN from activation and output values of neurons that depend on the particular Knapsack instance.",
"We denote the latter by bold symbols in order to make the difference visible.",
"Moreover, in order to make the recurrent structure of our NN obvious, we do not use the index $i$ in the following description of the network.",
"Instead, we denote the $n_0=p^*+2$ input values by $\\mathbf {f}_\\mathrm {in}(p)$ for $p\\in [p^*]$ , as well as $\\mathbf {p}_\\mathrm {in}$ and $\\mathbf {s}_\\mathrm {in}$ .",
"The $p^*$ output values are denoted by $\\mathbf {f}_\\mathrm {out}(p)$ for $p\\in [p^*]$ .",
"The goal is to implement the recursion $\\mathbf {f}_\\mathrm {out}(p)=\\min \\bigl \\lbrace \\mathbf {f}_\\mathrm {in}(p),~\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})+\\mathbf {s}_\\mathrm {in}\\bigr \\rbrace \\quad \\text{for $p\\in [p^*]$}$ in an NN; cp.",
"(REF ).",
"It consists of an addition and taking a minimum, which are both simple operations for an NN.",
"Hence, ideally, we would like to have an architecture as depicted in Figure REF for computing $\\mathbf {f}_\\mathrm {out}(p)$ for every $p\\in [p^*]$ .",
"Figure: Desirable architecture for computing 𝐟 out (p)\\mathbf {f}_\\mathrm {out}(p), p∈[p * ]p\\in [p^*], from the inputs.",
"However, the existence of an edge (nonzero weight) depends critically on the input value 𝐩 in \\mathbf {p}_\\mathrm {in}, which is not allowed.The problem with this is, however, that the decision which component of $\\mathbf {f}_\\mathrm {in}$ is accessed in order to compute the sum with $\\mathbf {s}_\\mathrm {in}$ depends on the input value $\\mathbf {p}_\\mathrm {in}$ .",
"Since we aim for an architecture that is fixed and works for general input values $\\mathbf {p}_\\mathrm {in}$ , we have to extend our construction as depicted in Figure REF .",
"Figure: High-level idea how the DP-NN computes 𝐟 out (p)\\mathbf {f}_\\mathrm {out}(p) for p∈[p * ]p\\in [p^*] from the inputs.As we do not know the value of $\\mathbf {p}_\\mathrm {in}$ in advance, we connect every input neuron $\\mathbf {f}_\\mathrm {in}(p-p^{\\prime })$ , $p^{\\prime }\\in [p-1]$ , to the unit that computes the sum $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})+\\mathbf {s}_\\mathrm {in}$ .",
"Since we only want to take the value $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ into account, we need to add an additional unit that disables those connections if $p^{\\prime }\\ne \\mathbf {p}_\\mathrm {in}$ .",
"Due to the integrality of the profit values, this additional unit can be realized with two hidden layers and a constant number of neurons for every value of $p\\in [p^*]$ and $p^{\\prime }\\in [p-1]$ , as we show in Appendix .",
"Computing the minimum adds a third hidden layer.",
"Hence, the DP-NN has depth four while width and size are in $\\mathcal {O}((p^*)^2)$ .",
"Unfolding the RNN and viewing it as a single feedforward NN executing the whole dynamic program results in depth $\\mathcal {O}(n)$ and size $\\mathcal {O}(n(p^*)^2)$ .",
"In Appendix we provide a detailed construction of the DP-NN and prove the following theorem.",
"Theorem 1 For a Knapsack instance with capacity $S=1$ , $s_i\\in \\left]0,1\\right]$, and $p_i\\in \\mathbb {N}$ , for $i\\in [n]$ , with an upper bound $p^*$ on the optimal solution value, the corresponding dynamic programming values $f(p,i)$ , $i\\in [n]$ , $p\\in p^*$ , can be exactly computed by iteratively applying the DP-NN $n$ times.",
"Observe that due to the NP-hardness of the Knapsack Problem, the dependence of the network size on $p^*$ cannot be avoided if exact results are desired."
],
[
"Smaller RNNs with Provable Approximation Guarantees",
"In order to overcome the drawback due to the dependence of the network width on $p^*$ , we provide a construction, called FPTAS-NN, that uses less neurons, at the cost of losing optimality.",
"Instead, we prove an approximation ratio (i.e., a worst-case bound) for the solution value computed by the FPTAS-NN.",
"As in the standard Knapsack FPTAS [17], [43], [45], the idea of this construction is to round the profit values if $p^*$ becomes too large for an exact computation.",
"Our approximation result can be interpreted as a tradeoff between the width of the NN and the quality of the Knapsack solution obtained.",
"Let $P\\in \\mathbb {N}$ be a fixed number.",
"The FPTAS-NN computes values $g(p,i)$ for every $p\\in [P]$ and $i\\in [n]$ .",
"These values are similar to the values $f(p,i)$ of the previous section, there is, however, one major difference.",
"Let $p^*_i=\\sum _{j=1}^i p_j$ be the total profit of the first $i$ items.",
"As soon as $p^*_i$ exceeds $P$ , we can no longer store a required size value for every possible profit value but have to round profits instead.",
"The granularity we want to use for rounding is $d_i\\max \\lbrace 1,p^*_i/P\\rbrace $ .",
"We construct the FPTAS-NN to compute values $g(p,i)$ , $p\\in [P]$ , $i\\in [n]$ , such that we can guarantee the existence of a subset of $[i]$ that has size at most $g(p,i)$ and profit at least $p\\,d_i$ .",
"Moreover, this is done in such a way that the optimal solution cannot have a considerably higher profit value.",
"That is, we prove a worst-case approximation guarantee for the solution found by the FPTAS-NN.",
"In addition to the values of $g$ , the FPTAS-NN must also propagate the current total profit value $p^*_i$ through the network in order to determine the rounding granularity in each step.",
"Hence, in the $i$ -th step, it receives $P+3$ inputs, namely $g(p,i-1)$ for $p\\in [P]$ , $p^*_{i-1}$ , $p_i$ , and $s_i$ .",
"It computes $P+1$ outputs, namely $g(p,i)$ for $p\\in [P]$ and $p^*_{i}$ .",
"Figure REF illustrates the recurrent structure of this NN.",
"Figure: Recurrent structure of the FPTAS-NN for the Knapsack Problem.As in Section , we use bold symbols in order to distinguish input, activation, and output values that depend on the concrete Knapsack instance from fixed parameters of the network.",
"We again drop the index $i$ in order to make the recurrent structure obvious.",
"We denote the $n_0=P+3$ input parameters by $\\mathbf {g}_\\mathrm {in}(p)$ , for $p\\in [P]$ , as well as $\\mathbf {p}^*_\\mathrm {in}$ , $\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {s}_\\mathrm {in}$ .",
"The $P+1$ output values are denoted by $\\mathbf {g}_\\mathrm {out}(p)$ , for $p\\in [P]$ , and $\\mathbf {p}^*_\\mathrm {out}$ .",
"Similar to the DP-NN in Section , the basic idea is to implement a recursion of the type $\\mathbf {g}_\\mathrm {out}(p) = \\min \\bigl \\lbrace \\mathbf {g}_\\mathrm {in}(p^{(1)}), \\mathbf {g}_\\mathrm {in}(p^{(2)})+\\mathbf {s}_\\mathrm {in}\\bigr \\rbrace \\quad \\text{for $p\\in [P]$,}$ where the first argument of the minimum represents the option of not using item $i$ , while the second one corresponds to using it.",
"Notice, however, that $p^{(1)}$ and $p^{(2)}$ cannot simply be calculated as $p$ and $p-\\mathbf {p}_\\mathrm {in}$, respectively, since we may have to round with different granularities in two successive steps.",
"Therefore, the rough structure of the FPTAS-NN is as follows: first, $\\mathbf {p}^*_\\mathrm {in}$ and $\\mathbf {p}_\\mathrm {in}$ are used in order to calculate the old and new rounding granularities $\\mathbf {d}_\\mathrm {old}=\\max \\lbrace 1,\\mathbf {p}^*_\\mathrm {in}/P\\rbrace $ and $\\mathbf {d}_\\mathrm {new}=\\max \\lbrace 1,(\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in})/P\\rbrace $.",
"Since this computation consists of maxima and weighted sums only, it can easily be achieved by an NN with one hidden layer.",
"Second, the granularities are used in order to select $\\mathbf {g}_\\mathrm {in}(p^{(1)})$ and $\\mathbf {g}_\\mathrm {in}(p^{(2)})$ from the inputs.",
"Below we give some more details on how this is being done.",
"The value of $p^{(2)}$ also depends on $\\mathbf {p}_\\mathrm {in}$ .",
"Third, the final recursion is established as in the DP-NN.",
"In addition to $\\mathbf {g}_\\mathrm {out}(p)$ , for $p\\in [P]$ , we also output $\\mathbf {p}^*_\\mathrm {out}=\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}$ in order to keep track of the rounding granularities in subsequent steps.",
"An overview of the entire network structure is given in Figure REF .",
"Figure: High-level idea how the FPTAS-NN computes 𝐠 out (p)\\mathbf {g}_\\mathrm {out}(p), p∈[P]p\\in [P], and 𝐩 out * \\mathbf {p}^*_\\mathrm {out} from the inputs.Suppose we use the network for processing the $i$ -th item.",
"For each $p\\in [P]$ we want to determine a (preferably small) value $\\mathbf {g}_\\mathrm {out}(p)$ such that there is a subset of $[i]$ of total profit at least $p\\,\\mathbf {d}_\\mathrm {new}$ and total size at most $\\mathbf {g}_\\mathrm {out}(p)$ .",
"For each $p^{\\prime }\\in [P]$ , we know that there is a subset of $[i-1]$ of total profit at least $p^{\\prime }\\mathbf {d}_\\mathrm {old}$ and total size at most $\\mathbf {g}_\\mathrm {in}(p^{\\prime })$ .",
"We have two options: ignoring item $i$ or using it.",
"If we ignore it, then each $p^{(1)}$ with $p^{(1)}\\mathbf {d}_\\mathrm {old}\\ge p\\,\\mathbf {d}_\\mathrm {new}$ allows us to choose $\\mathbf {g}_\\mathrm {out}(p)=\\mathbf {g}_\\mathrm {in}(p^{(1)})$.",
"If we do use the $i$ -th item, however, then each $p^{(2)}$ with the property $p^{(2)}\\mathbf {d}_\\mathrm {old}+ \\mathbf {p}_\\mathrm {in}\\ge p\\,\\mathbf {d}_\\mathrm {new}$ allows us to choose $\\mathbf {g}_\\mathrm {out}(p)=\\mathbf {g}_\\mathrm {in}(p^{(2)}) + \\mathbf {s}_\\mathrm {in}$.",
"Hence, we want to choose $p^{(1)}$ and $p^{(2)}$ as small as possible such that these properties are fulfilled.",
"Therefore, the units labeled `Select $\\mathbf {g}_\\mathrm {in}(p^{(1)})$ ' and `Select $\\mathbf {g}_\\mathrm {in}(p^{(2)})$ ' in Figure REF are constructed by setting all other connections to zero except for those belonging to the smallest values of $p^{(1)}$ and $p^{(2)}$ satisfying the above properties.",
"Similar to how we computed $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ in the previous section, this requires two hidden layers and $\\mathcal {O}(P^2)$ neurons in total.",
"In total, the FPTAS-NN has depth 5.",
"The first hidden layer computes the rounding granularities, two hidden layers are required to select $\\mathbf {g}_\\mathrm {in}(p^{(1)})$ and $\\mathbf {g}_\\mathrm {in}(p^{(2)})$ and a final hidden layer computes the minimum in the actual recursion.",
"The width and size of the FPTAS-NN are in the order of $\\mathcal {O}(P^2)$ .",
"Unfolding the RNN and viewing it as a single feedforward NN executing the whole FPTAS results in depth $\\mathcal {O}(n)$ and size $\\mathcal {O}(nP^2)$ .",
"In Appendix we provide a formal description of the FPTAS-NN as well as proofs of the following two theorems.",
"The first one ensures that the FPTAS-NN produces only feasible Knapsack solutions, while the second one shows that the FPTAS-NN indeed provides a fully polynomial-time approximation scheme to solve the Knapsack Problem.",
"Theorem 2 Suppose the FPTAS-NN is applied to a Knapsack instance with capacity $S=1$ , $s_i\\in \\left]0,1\\right]$ , and $p_i\\in \\mathbb {N}$ , for $i\\in [n]$ .",
"For every $i\\in [n]$ and every $p\\in [P]$ , if $g(p,i)\\le 1$ , then there exists a subset of $[i]$ with profit at least $pd_i$ and size at most $g(p,i)$ .",
"Theorem 3 For a Knapsack instance with capacity $S=1$ , $s_i\\in \\left]0,1\\right]$ , $p_i\\in \\mathbb {N}$ , for $i\\in [n]$ , and for $\\varepsilon \\in \\left]0,1\\right]$ , set $P\\lceil n^2/\\varepsilon \\rceil $ .",
"Let $p^\\mathrm {OPT}$ be the profit of the optimal solution and $p^\\mathrm {NN}=\\max \\lbrace pd_n\\mid g(p,n)\\le 1\\rbrace $ be the best possible profit found by the FPTAS-NN.",
"Then $p^\\mathrm {NN}\\ge (1-\\varepsilon )p^\\mathrm {OPT}$ .",
"Theorem REF implies a tradeoff between the width of the NN and the precision of the Knapsack solution in the following sense.",
"For achieving an approximation ratio of $1-\\varepsilon $ , an NN of width $\\mathcal {O}(P^2)=\\mathcal {O}(n^4/\\varepsilon ^2)$ is required.",
"In other words, the FPTAS-NN with fixed width $w$ achieves a worst-case approximation ratio of $1-\\mathcal {O}(n^2/\\sqrt{w})$ .",
"Observe that, assuming $\\text{P}\\ne \\text{NP}$ , it is clear that the size of the NN must grow if $\\varepsilon $ tends to zero.",
"Hence, complexity theory implies that a width-quality trade-off cannot be avoided.",
"Still, it remains as an open question whether the growth rates implied by our construction are best possible."
],
[
"Empirical Evidence for Quadratic Width",
"While the running time of the classical Knapsack dynamic program depends only linearly on $p^*$ , the width of the DP-NN is $\\mathcal {O}((p^*)^2)$ .",
"In our construction, the quadratic factor arises from dynamically finding $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ in a hard-coded network, as explained in Section .",
"For similar reasons, the width of the FPTAS-NN grows with $1/\\varepsilon ^2$ instead of $1/\\varepsilon $ .",
"The natural question to ask is whether this quadratic dependence can be avoided by a different construction.",
"While this question remains open from a purely theoretical point of view, in this section, we provide empirical evidence that the quadratic factor might indeed be necessary due to inherent properties of ReLU feedforward NNs.",
"For details about the experimental setup, including used soft- and hardware, random data generation and systematic of seeds, training and testing setup, hyperparameters, as well as, the source code, please refer to Appendix .",
"Here we only include the necessary information to understand the key findings.",
"Similar to the DP-NN of Section , we train an NN with three hidden layers and variable width to execute one step of the Knapsack dynamic program, that is, to map $\\mathbf {f}_\\mathrm {in}$ , $\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {s}_\\mathrm {in}$ to $\\mathbf {f}_\\mathrm {out}$ , for random Knapsack instances.",
"For the 25 different values $\\lbrace 3,6,9,\\dots ,75\\rbrace $ of $p^*$ , we increase the width in steps of 25 until a mean squared error (MSE) loss of at most $0.005$ is reached.",
"The threshold $0.005$ is carefully chosen such that NNs with reasonable width are empirically able to achieve it.",
"In Appendix , we also show that other thresholds yield similar results.",
"Figure REF shows for each value of $p^*$ the required width to achieve an MSE of at most $0.005$ .",
"In order to statistically test whether a quadratic dependence is more likely than a linear relation, we use linear regression.",
"Assuming the required width is given by a function $\\text{width} = a_0 + a_1 p^* + a_2 (p^*)^2 + \\text{noise},$ the resulting least squares regression curve can be seen in Figure REF .",
"Testing the null hypothesis $a_2 = 0$ against the alternative $a_2\\ne 0$ , we obtain a p-value of $1.1\\,\\%$ , which we judge to be significant evidence that a quadratic relation is more likely than a linear one.",
"Of course, one should take this result with a grain of salt since the superlinear relation might have multiple reasons.",
"For instance, it is unclear, whether the difficulty to train larger networks has a stronger effect than the expressivity of ReLU NNs.",
"Still, we find that this computational study supports our theoretical size bounds.",
"Figure: Required network width to achieve a mean squared error of at most 0.0050.005 as a function of p * p^*."
],
[
"Neural Networks for Other CO Problems",
"In this section we demonstrate that our approach is by no means bound to the Knapsack Problem.",
"In fact, for many other CO problems it is possible to convert a dynamic programming solution method into a provably correct NN.",
"For certain NP-hard CO problems, a dynamic programming solution even implies the existence of a fully polynomial-time approximation scheme [46].",
"This, in turn, might shed light on the tradeoff between size of corresponding NNs and their solution quality, as for the Knapsack Problem in Section .",
"In the following we provide several examples in order to support these claims."
],
[
"Longest Common Subsequence.",
"First, consider the problem of finding the length of the longest common subsequence of two finite integer sequences $x_1, x_2, \\dots , x_m$ and $y_1, y_2, \\dots , y_n$ .",
"A standard dynamic programming procedure, see, e.g., [6], computes values $f(i,j)$ equal to the length of the longest common subsequence of the partial sequences $x_1, x_2, \\dots , x_i$ and $y_1, y_2, \\dots , y_j$ by applying the recursion $f(i,j)=\\left\\lbrace \\begin{array}{ll}f(i-1,j-1)+1&\\text{if }x_i=y_j,\\\\\\max \\bigl \\lbrace f(i-1,j),f(i,j-1)\\bigr \\rbrace &\\text{if }x_i\\ne y_j.\\end{array}\\right.$ Since the sequence consists of integers, the check whether $x_i$ equals $y_j$ can be performed similarly to checking whether $p^{\\prime }=\\mathbf {p}_\\mathrm {in}$ in Section .",
"The remainder of the recursion only consists of maxima and sums.",
"Hence, computing $f(i,j)$ from $f(i-1,j-1)$ , $f(i-1,j)$ , $f(i,j-1)$ , $x_i$ , and $y_j$ can be realized via an NN of constant size.",
"These basic units can be plugged together in a two-dimensional way for computing all values $f(i,j)$ , $i\\in [m]$ , $j\\in [n]$ .",
"The resulting NN can be seen as a two-dimensional RNN of constant size that is applied in an $m$ by $n$ grid structure, an architecture introduced by [12].",
"Unfolding the RNN results in a feedforward NN of depth $\\mathcal {O}(m+n)$ and size $\\mathcal {O}(mn)$ for computing the length of the longest common subsequence."
],
[
"Single-Source Shortest Path Problem.",
"As a second example, we consider the Bellman-Ford algorithm for the Single-Source Shortest Path Problem, see, e.g., [22].",
"If $(c_{uv})_{u,v\\in V}$ is the length matrix of a graph with vertex set $V$ and $s\\in V$ is the source vertex, this algorithm recursively computes values $f(i,v)$ determining the shortest possible length of a path from $s$ to $v$ using at most $i$ arcs by $f(i,v)=\\min _{u\\in V}\\lbrace f(i-1,u)+c_{uv}\\rbrace .$ Since this recursion consists only of sums and minima, it can be easily implemented in an NN.",
"The sequential time complexity of the Bellman-Ford algorithm on complete digraphs with $n=\\vert V\\vert $ is $\\mathcal {O}(n^3)$ , which can naturally be parallelized into $\\mathcal {O}(n)$ rounds.",
"Since the best known NNs for computing the minimum of $n$ numbers require $\\mathcal {O}(\\log n)$ depth [1], there exists an NN executing the Bellman-Ford algorithm with depth $\\mathcal {O}(n\\log n)$ and size $\\mathcal {O}(n^3\\log n)$ .",
"Observe that in each round $i\\in [n]$ , the computation mapping the values $f(i-1,v)$ , $v\\in V$ , to $f(i,v)$ , $v\\in V$ , is the same.",
"Therefore, this NN can also be seen as an RNN of depth $\\mathcal {O}(\\log n)$ and size $\\mathcal {O}(n^2\\log n)$ that is applied $n$ times."
],
[
"All-Pairs Shortest Path Problem.",
"Third, recall that the All-Pairs Shortest Path Problem can be solved by computing the $(n-1)$ -th min-plus matrix power of the length matrix $(c_{uv})_{u,v\\in V}$ , see, e.g., [25].",
"By repeated squaring, this can be achieved with only $\\mathcal {O}(\\log n)$ min-plus matrix multiplications.",
"For a single multiplication it is required to compute $\\mathcal {O}(n^2)$ times in parallel the minimum of $n$ numbers.",
"One of these minimum computations requires depth $\\mathcal {O}(\\log n)$ and size $\\mathcal {O}(n\\log n)$ .",
"Putting them in parallel to execute one min-plus matrix product results in depth $\\mathcal {O}(\\log n)$ and size $\\mathcal {O}(n^3\\log n)$ .",
"Note that the whole execution consists of $\\mathcal {O}(\\log n)$ repetitions of the same procedure, namely squaring a matrix in the min-plus sense.",
"Hence, this can again be viewed as an RNN with depth $\\mathcal {O}(\\log n)$ and size $\\mathcal {O}(n^3\\log n)$ that is applied $\\mathcal {O}(\\log n)$ times.",
"Unfolding results in a single feedforward NN with depth $\\mathcal {O}(\\log ^2 n)$ and size $\\mathcal {O}(n^3\\log ^2 n)$ for solving the All-Pairs Shortest Path Problem."
],
[
"Constrained Shortest Path Problem.",
"Next, consider a common generalization of the Shortest Path Problem and the Knapsack Problem, namely the NP-hard Constrained Shortest Path Problem.",
"Here, in addition to a (nonnegative) length matrix $(c_{uv})_{u,v\\in V}$ , the input graph is also equipped with a (nonnegative) resource matrix $(r_{uv})_{u,v\\in V}$ .",
"The task is to find a minimum length path $P$ from a source vertex $s$ to any other vertex, but this time subject to a resource constraint $\\sum _{(u,v)\\in P} r_{uv}\\le R$ for a given resource limit $R$ .",
"An extensive overview of solution approaches to this problem can be found, e.g., in the dissertation by [50].",
"Similar to the Knapsack Problem, there exist two natural pseudo-polynomial dynamic programs, one of them parametrized by length values and the other one by resource values.",
"Both can be implemented on an NN by combining the ideas from Section with the NN for the Bellmann-Ford algorithm above.",
"We showcase this for the variant parametrized by the length values.",
"This dynamic program recursively calculates values $f(c,v)$ representing the minimum amount of resource needed for a path from $s$ to $v$ with length at most $c$ by $\\text{\\small $f(c,v)=\\min \\bigl \\lbrace f(c-1,v),\\min \\nolimits _{u\\in V\\setminus \\lbrace v\\rbrace }\\lbrace f(c-c_{uv}, u) + r_{uv}\\rbrace \\bigr \\rbrace $.",
"}$ For fixed $c$ , $u$ , and $v$ , the term $f(c-c_{uv}, u) + r_{uv}$ can be calculated by a similar construction as we computed $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})+\\mathbf {s}_\\mathrm {in}$ in Section .",
"Assuming an upper bound $c^*$ on the optimal objective value, this can be achieved with constant depth and $\\mathcal {O}(c^*)$ width.",
"Hence, it remains to compute a minimum of at most $n$ numbers in order to compute $f(c,v)$ .",
"Thus, each single value $f(c,v)$ can be computed with depth $\\mathcal {O}(\\log n)$ and size $\\mathcal {O}(nc^*\\log n)$ .",
"We have to compute $\\mathcal {O}(nc^*)$ of these values, but for fixed $c$ , all these values can be computed in parallel.",
"Therefore, the whole dynamic program can be executed on an NN with depth $\\mathcal {O}(c^* \\log n)$ and a total size of $\\mathcal {O}(n^2(c^*)^2\\log n)$ .",
"This is pseudo-polynomial, which is the best we can hope for due to the NP-hardness of the problem.",
"Moreover, similar to the Knapsack Problem, this dynamic program can be turned into an FPTAS by downscaling and rounding the length values.",
"This observation can be used to obtain a width-quality tradeoff for the Constrained Shortest Path Problem similar to what we have shown in Section ."
],
[
"Traveling Salesperson Problem.",
"Finally, consider the Bellman-Held-Karp algorithm for solving the (asymmetric) Traveling Salesperson Problem (TSP), see [2], [16].",
"Given a (complete, directed) graph with vertex set $V$ and distances $c_{uv}$ from vertex $u\\in V$ to vertex $v\\in V$ , the TSP asks for the shortest round-trip visiting each vertex exactly once.",
"Choosing an arbitrary starting vertex $s\\in V$ , the Bellman-Held-Karp algorithm recursively computes values $f(T,v)$ for each $T\\subseteq V\\setminus \\lbrace s\\rbrace $ , $v\\in T$ , corresponding to the length of the shortest $s$ -$v$ -path visiting exactly the nodes in $T\\cup \\lbrace s\\rbrace $ by the formula $f(T,v)=\\min _{u\\in T\\setminus \\lbrace v\\rbrace }\\left\\lbrace f(T\\setminus \\lbrace v\\rbrace , u) + c_{uv}\\right\\rbrace .$ The length of the shortest TSP tour is then given by $\\min _{u\\in V\\setminus \\lbrace s\\rbrace }\\left\\lbrace f(V\\setminus \\lbrace s\\rbrace , u) + c_{us}\\right\\rbrace $ .",
"While the sequential time complexity of this algorithm on digraphs with $n=\\vert V\\vert $ is $\\mathcal {O}(n^22^n)$ , in an NN we can compute the values of $f$ for all $T$ with equal cardinality in parallel.",
"As before, computing the minimum introduces an additional factor of $\\log n$ in the depth and size of the network.",
"Hence, in total, the TSP can be solved with an NN of depth $\\mathcal {O}(n\\log n)$ and size $\\mathcal {O}(n^22^n\\log n)$ .",
"In particular, a polynomially deep NN suffices to solve the NP-hard (asymmetric) TSP."
],
[
"Conclusions and Future Work",
"An obvious open problem is to improve the obtained bounds on the required width of our neural network constructions.",
"In particular, an interesting question is whether meaningful lower bounds beyond those immediately implied by the NP-hardness of the Knapsack Problem can be obtained, as suggested by our experimental results.",
"Notice that our networks only output the solution value but not the corresponding solution, i.e., subset of items.",
"It is easy to see that, as for the dynamic program solving the Knapsack Problem, the subset of items can be obtained in a straightforward way via backtracking.",
"On the other hand, notice that it is impossible for a ReLU NN (without threshold gates) to output (the characteristic vector of) the optimum subset of items: while the function computed by a ReLU NN is piecewise linear and continuous [11], [1], already infinitesimal changes of the input (i.e., the profit values of items) might change the optimum subset of items.",
"Another interesting direction for future research is to generalize our results of Section by describing a general procedure to convert dynamic programs into ReLU NNs.",
"Ideally, one could exactly classify the type of dynamic programs that guarantee the existence of a corresponding ReLU NN.",
"Similar in spirit, [46] classifies dynamic programs that guarantee the existence of a fully polynomial-time approximation scheme."
],
[
"Detailed Construction of the DP-NN",
"In this section we formally describe the DP-NN and prove its correctness.",
"Note that for size values larger than the Knapsack capacity, which is equal to 1 by our definition, we do not really care how large they actually are.",
"Therefore, we define $\\tilde{f}(p,i)=\\min \\lbrace f(p,i), 2\\rbrace $ to be the values of the dynamic program truncated at 2.",
"In other words, we replace all values in the interval $[2,+\\infty ]$ by 2.",
"Note that the recursion $\\tilde{f}(p,i)=\\min \\bigl \\lbrace \\tilde{f}(p,i-1), \\tilde{f}(p-p_i,i-1)+s_i\\bigr \\rbrace $ is still valid with starting values $\\tilde{f}(p,i)=0$ for $p\\le 0$ and $\\tilde{f}(p,0)=2$ for $p\\in [p^*]$ .",
"Instead of computing the actual values of $f$ , the DP-NN computes the values of $\\tilde{f}$ .",
"The DP-NN has three hidden layers.",
"After the $n_0=p^*+2$ input neurons $\\mathbf {f}_\\mathrm {in}(p)$ for $p\\in [p^*]$ , $\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {s}_\\mathrm {in}$ , the first hidden layer consists of $n_1=2p^*$ neurons whose outputs are denoted by $\\mathbf {o}_1^{(+)}(k)$ and $\\mathbf {o}_1^{(-)}(k)$ for $k\\in [p^*]$ .",
"Its role is to detect whether $k=\\mathbf {p}_\\mathrm {in}$ .",
"If yes, then both $\\mathbf {o}_1^{(+)}(k)$ and $\\mathbf {o}_1^{(-)}(k)$ should be zero, otherwise at least one of them should be large (i.e., at least 2).",
"In the second hidden layer, we have $n_2=p^*(p^*-1)/2$ neurons, denoted by $\\mathbf {o}_2(p,k)$ for $p\\in [p^*]$ and $k\\in [p-1]$ .",
"A neuron in this layer should output $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ if $k=\\mathbf {p}_\\mathrm {in}$ and zero otherwise.",
"This way, the sum $\\sum _{k=1}^{p-1} \\mathbf {o}_2(p,k)$ equals $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ .",
"The third hidden layer has $n_3=p^*$ neurons, denoted by $\\mathbf {o}_3(p)$ for $p\\in [p^*]$ .",
"It is used for computing the minimum of $\\mathbf {f}_\\mathrm {in}(p)$ and $\\mathbf {s}_\\mathrm {in}+\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ as in Figure REF .",
"Finally, the $n_4=p^*$ output values are denoted by $\\mathbf {f}_\\mathrm {out}(p)$ for $p\\in [p^*]$ .",
"The following equations define the DP-NN.",
"$\\mathbf {o}_1^{(+)}(k) &= \\sigma (2(\\mathbf {p}_\\mathrm {in}-k)),&&k\\in [p^*],\\\\\\mathbf {o}_1^{(-)}(k) &= \\sigma (2(k-\\mathbf {p}_\\mathrm {in})),&&k\\in [p^*],\\\\\\mathbf {o}_2(p,k) &= \\sigma (\\mathbf {f}_\\mathrm {in}(p-k)-\\mathbf {o}_1^{(+)}(k)-\\mathbf {o}_1^{(-)}(k)),&&p\\in [p^*],~k\\in [p-1],\\\\\\mathbf {o}_3(p) &= \\sigma \\left(\\mathbf {f}_\\mathrm {in}(p)-\\left(\\mathbf {s}_\\mathrm {in}+\\sum \\nolimits _{k=1}^{p-1} \\mathbf {o}_2(p,k)\\right)\\right),&&p\\in [p^*],\\\\\\mathbf {f}_\\mathrm {out}(p) & = \\mathbf {f}_\\mathrm {in}(p)-\\mathbf {o}_3(p),&&p\\in [p^*].$ Our next goal is to prove Theorem REF , which states that the DP-NN indeed solves the Knapsack Problem exactly.",
"We do so by going through the NN layer by layer and show what the individual layers do.",
"As mentioned, the role of the first hidden layer is to detect the input value of $\\mathbf {p}_\\mathrm {in}$ and to provide a large value for every $k$ that is not equal to $\\mathbf {p}_\\mathrm {in}$ .",
"The following lemma follows immediately from the construction and the properties of the rectifier function $\\sigma $ .",
"Lemma 4 Let $\\mathbf {p}_\\mathrm {in}\\in \\mathbb {N}$ .",
"Then, for every $k\\in [p^*]$ , it holds that $\\mathbf {o}_1^{(+)}(k)+\\mathbf {o}_1^{(-)}(k)=0$ if and only if $k=\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {o}_1^{(+)}(k)+\\mathbf {o}_1^{(-)}(k)\\ge 2$ otherwise.",
"The role of the second layer is to compute $\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ , which is needed in the dynamic program.",
"The main difficulty of this step is that it depends on the input $\\mathbf {p}_\\mathrm {in}$ which neuron to access.",
"Therefore, this is computed for every possible value $k$ of $\\mathbf {p}_\\mathrm {in}$ and set to zero if $k\\ne \\mathbf {p}_\\mathrm {in}$ .",
"The following lemma explains how this is established.",
"Lemma 5 Let $\\mathbf {p}_\\mathrm {in}\\in \\mathbb {N}$ and $\\mathbf {f}_\\mathrm {in}(p)\\in \\left]0,2\\right]$ for every $p\\in [p^*]$ .",
"Then, for every $p\\in [p^*]$ and every $k\\in [p-1]$ , it holds that $\\mathbf {o}_2(p,k)=\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ if and only if $k=\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {o}_2(p,k)=0$ otherwise.",
"If $k=\\mathbf {p}_\\mathrm {in}$ , we obtain from Lemma REF that $\\mathbf {o}_1^{(+)}(k)+\\mathbf {o}_1^{(-)}(k)=0$ .",
"Thus, due to nonnegativity of $\\mathbf {f}_\\mathrm {in}(p-k)$ , we obtain $\\mathbf {o}_2(p,k) = \\sigma (\\mathbf {f}_\\mathrm {in}(p-k)) = \\mathbf {f}_\\mathrm {in}(p-k)= \\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ .",
"If $k\\ne \\mathbf {p}_\\mathrm {in}$ , we obtain from Lemma REF that $\\mathbf {o}_1^{(+)}(k)+\\mathbf {o}_1^{(-)}(k)\\ge 2$ .",
"Thus, due to $\\mathbf {f}_\\mathrm {in}(p-k)\\le 2$ , we obtain $\\mathbf {f}_\\mathrm {in}(p-k)-\\mathbf {o}_1^{(+)}(k)-\\mathbf {o}_1^{(-)}(k)\\le 0$ , which implies $\\mathbf {o}_2(p,k)=0$ .",
"The purpose of the third layer is to help calculating the final minimum.",
"More precisely, it computes how much $\\mathbf {f}_\\mathrm {out}(p)$ should be smaller than $\\mathbf {f}_\\mathrm {in}(p)$ in the following way.",
"Lemma 6 Let $\\mathbf {p}_\\mathrm {in}\\in \\mathbb {N}$ , $\\mathbf {s}_\\mathrm {in}\\in \\left]0,1\\right]$ , and $\\mathbf {f}_\\mathrm {in}(p)\\in \\left]0,2\\right]$ for every $p\\in [p^*]$ .",
"Then $\\mathbf {o}_3(p)=\\max \\lbrace 0,\\mathbf {f}_\\mathrm {in}(p)-\\mathbf {s}_\\mathrm {in}\\rbrace $ for every $p\\in [\\mathbf {p}_\\mathrm {in}]$ and $\\mathbf {o}_3(p)=\\max \\lbrace 0,\\mathbf {f}_\\mathrm {in}(p)-(\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})+\\mathbf {s}_\\mathrm {in})\\rbrace $ for every $p\\in \\lbrace \\mathbf {p}_\\mathrm {in}+1,\\mathbf {p}_\\mathrm {in}+2,\\dots ,p^*\\rbrace $ .",
"If $p\\le \\mathbf {p}_\\mathrm {in}$ , we obtain from Lemma REF that $\\sum _{k=1}^{p-1}\\mathbf {o}_2(p,k)=0$ .",
"If $p>\\mathbf {p}_\\mathrm {in}$ , Lemma REF implies $\\sum _{k=1}^{p-1}\\mathbf {o}_2(p,k)=\\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})$ .",
"Thus, the claim follows by construction of the third layer and definition of $\\sigma $ .",
"Now, after we have investigated the functionality of each of the hidden layers, we are able to prove Theorem REF .",
"Proof of Theorem REF .",
"Using Lemma REF , we obtain that $\\mathbf {f}_\\mathrm {out}(p)=\\min \\lbrace \\mathbf {f}_\\mathrm {in}(p), \\mathbf {s}_\\mathrm {in}\\rbrace $ if $p\\le \\mathbf {p}_\\mathrm {in}$ and $\\mathbf {f}_\\mathrm {out}(p)=\\min \\lbrace \\mathbf {f}_\\mathrm {in}(p), \\mathbf {f}_\\mathrm {in}(p-\\mathbf {p}_\\mathrm {in})+\\mathbf {s}_\\mathrm {in}\\rbrace $ if $p>\\mathbf {p}_\\mathrm {in}$ .",
"The claim follows due to the Recursion (REF ) with the respective starting values.",
"$\\Box $"
],
[
"Detailed Construction of the FPTAS-NN",
"In this section we formally describe the FPTAS-NN and prove that it provides provable approximation guarantees.",
"As for the DP-NN, we truncate the values of $g$ at 2, that is, instead of any value larger than 2 including $+\\infty $ , we just use the value 2.",
"The FPTAS-NN is applied to a Knapsack instance in the following way.",
"Using the initialization $g(p,0)=2$ for $p\\in [P]$ and $p^*_0=0$ , for $i=1,\\dots ,n$ , we feed the inputs $\\mathbf {g}_\\mathrm {in}(p)=g(p,i-1)$ for $p\\in [P]$ , $\\mathbf {p}^*_\\mathrm {in}=p^*_{i-1}$ , $\\mathbf {p}_\\mathrm {in}=p_i$ , and $\\mathbf {s}_\\mathrm {in}=s_i$ into the network and store the outputs as $g(p,i)=\\mathbf {g}_\\mathrm {out}(p)$ for $p\\in [P]$ and $p^*_i=\\mathbf {p}^*_\\mathrm {out}$ .",
"The FPTAS-NN has four hidden layers.",
"After the $n_0=P+3$ input neurons $\\mathbf {g}_\\mathrm {in}(p)$ for $p\\in [P]$ , $\\mathbf {p}^*_\\mathrm {in}$ , $\\mathbf {p}_\\mathrm {in}$ , and $\\mathbf {s}_\\mathrm {in}$ , the first hidden layer consists of $n_1=2$ neurons $\\mathbf {o}_1^\\mathrm {old}$ and $\\mathbf {o}_1^\\mathrm {new}$ which help to compute the rounding granularities $\\mathbf {d}_\\mathrm {old}$ and $\\mathbf {d}_\\mathrm {new}$ .",
"They are defined as follows: $\\mathbf {o}_1^\\mathrm {old} &= \\sigma \\left(\\frac{\\mathbf {p}^*_\\mathrm {in}}{P}-1\\right),\\\\\\mathbf {o}_1^\\mathrm {new} &= \\sigma \\left(\\frac{\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}}{P}-1\\right),\\\\\\mathbf {d}_\\mathrm {old}&= \\mathbf {o}_1^\\mathrm {old} +1,\\\\\\mathbf {d}_\\mathrm {new}&= \\mathbf {o}_1^\\mathrm {new} +1.$ The granularities $\\mathbf {d}_\\mathrm {old}$ and $\\mathbf {d}_\\mathrm {new}$ are just affine transformations of $\\mathbf {o}_1^\\mathrm {old}$ and $\\mathbf {o}_1^\\mathrm {new}$ .",
"Hence, they do not form an own hidden layer, because we do not apply the ReLU activation function there.",
"The correct functionality of the first layer is ensured by the following lemma.",
"Lemma 7 The first layer of the FPTAS-NN correctly computes the rounding granularities $\\mathbf {d}_\\mathrm {old}=\\max \\lbrace 1,\\frac{\\mathbf {p}^*_\\mathrm {in}}{P}\\rbrace $ and $\\mathbf {d}_\\mathrm {new}=\\max \\lbrace 1,\\frac{\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}}{P}\\rbrace $ .",
"This follows from the fact that $\\sigma (x-1)+1=\\max \\lbrace 0,x-1\\rbrace +1=\\max \\lbrace 1,x\\rbrace $ , where $x$ equals either $\\frac{\\mathbf {p}^*_\\mathrm {in}}{P}$ or $\\frac{\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}}{P}$ .",
"Hence, in the $i$ -th step, if we feed the inputs $\\mathbf {p}^*_\\mathrm {in}=p^*_{i-1}$ and $\\mathbf {p}_\\mathrm {in}=p_i$ into the network, $\\mathbf {d}_\\mathrm {old}$ and $\\mathbf {d}_\\mathrm {new}$ equal $d_{i-1}$ and $d_i$ , respectively.",
"In the second hidden layer, we have a total of $n_2=2 P^2 + 2 P$ hidden neurons, denoted by $\\mathbf {o}_2^{(1+)}(p,k)$ and $\\mathbf {o}_2^{(1-)}(p,k)$ for $p,k\\in [P]$ with $p\\le k$ , as well as, $\\mathbf {o}_2^{(2+)}(p,k)$ and $\\mathbf {o}_2^{(2-)}(p,k)$ for $p,k\\in [P]$ with $p\\ge k$ , defined as follows: $\\mathbf {o}_2^{(1+)}(p,k) &= \\sigma (2P(p\\mathbf {d}_\\mathrm {new}-k\\mathbf {d}_\\mathrm {old})),&&p,k\\in [P], p\\le k,\\\\\\mathbf {o}_2^{(1-)}(p,k) &= \\sigma (2P((k-1)\\mathbf {d}_\\mathrm {old}-p\\mathbf {d}_\\mathrm {new})+2),&&p,k\\in [P], p\\le k,\\\\\\mathbf {o}_2^{(2+)}(p,k) &= \\sigma (2P(p\\mathbf {d}_\\mathrm {new}-k\\mathbf {d}_\\mathrm {old}-\\mathbf {p}_\\mathrm {in})),&&p,k\\in [P], p\\ge k,\\\\\\mathbf {o}_2^{(2-)}(p,k) &= \\sigma (2P((k-1)\\mathbf {d}_\\mathrm {old}+\\mathbf {p}_\\mathrm {in}-p\\mathbf {d}_\\mathrm {new})+2),&&p,k\\in [P], p\\ge k.$ For a fixed $p\\in [P]$ , let $p^{(1)}$ and $p^{(2)}$ be the smallest possible integers with $p^{(1)}\\mathbf {d}_\\mathrm {old}\\ge p\\mathbf {d}_\\mathrm {new}$ and $p^{(2)}\\mathbf {d}_\\mathrm {old}+ \\mathbf {p}_\\mathrm {in}\\ge p\\mathbf {d}_\\mathrm {new}$ , respectively.",
"The task of the second layer is to detect the values $p^{(1)}$ and $p^{(2)}$ , as formalized by the following two lemmas.",
"Lemma 8 For each $p,k\\in [P]$ with $p\\le k$ , we have $\\mathbf {o}_2^{(1+)}(p,k)+\\mathbf {o}_2^{(1-)}(p,k)=0$ if and only if $k=p^{(1)}$ .",
"Otherwise, we have $\\mathbf {o}_2^{(1+)}(p,k)+\\mathbf {o}_2^{(1-)}(p,k)\\ge 2$ .",
"Obviously, it holds that $\\mathbf {o}_2^{(1+)}(p,k)=0$ if and only if $k\\mathbf {d}_\\mathrm {old}\\ge p\\mathbf {d}_\\mathrm {new}$ .",
"On the other hand, using that $\\mathbf {d}_\\mathrm {old}$ and $\\mathbf {d}_\\mathrm {new}$ are integer multiples of $\\frac{1}{P}$ , we obtain $& \\mathbf {o}_2^{(1-)}(p,k)=0\\\\\\Leftrightarrow \\quad & (k-1)\\mathbf {d}_\\mathrm {old}\\le p\\mathbf {d}_\\mathrm {new}- \\frac{1}{P}\\\\\\Leftrightarrow \\quad & (k-1)\\mathbf {d}_\\mathrm {old}< p\\mathbf {d}_\\mathrm {new}\\\\\\Leftrightarrow \\quad & \\text{no integer $k^{\\prime }<k$ satisfies $k^{\\prime }\\mathbf {d}_\\mathrm {old}\\ge p\\mathbf {d}_\\mathrm {new}$.",
"}$ This proves the first part of the claim.",
"The second part follows because, again, $\\mathbf {d}_\\mathrm {old}$ and $\\mathbf {d}_\\mathrm {new}$ are integer multiples of $\\frac{1}{P}$ and, hence, $\\mathbf {o}_2^{(1+)}(p,k)+\\mathbf {o}_2^{(1-)}(p,k)$ is an integer multiple of 2.",
"Lemma 9 For each $p,k\\in [P]$ with $p\\ge k$ , we have $\\mathbf {o}_2^{(2+)}(p,k)+\\mathbf {o}_2^{(2-)}(p,k)=0$ if and only if $k=p^{(2)}$ .",
"Otherwise, we have $\\mathbf {o}_2^{(2+)}(p,k)+\\mathbf {o}_2^{(2-)}(p,k)\\ge 2$ .",
"Analogous to Lemma REF .",
"The third hidden layer consists of $n_3=P^2 + P$ neurons $\\mathbf {o}_3^{(1)}(p,k)$ for $p,k\\in [P]$ with $p\\le k$ , as well as $\\mathbf {o}_3^{(2)}(p,k)$ for $p,k\\in [P]$ with $p\\ge k$ .",
"Moreover, we have again helping variables that do not form an own hidden layer because they are only affine transformations of the previous layers, namely $\\mathbf {h}^{(1)}(p)$ and $\\mathbf {h}^{(2)}(p)$ for $p\\in [P]$ .",
"$\\mathbf {o}_3^{(1)}(p,k) &= \\sigma (2-\\mathbf {g}_\\mathrm {in}(k)-\\mathbf {o}_2^{(1+)}(p,k)-\\mathbf {o}_2^{(1-)}(p,k)),&&p,k\\in [P], p\\le k,\\\\\\mathbf {o}_3^{(2)}(p,k) &= \\sigma (\\mathbf {g}_\\mathrm {in}(k)-\\mathbf {o}_2^{(2+)}(p,k)-\\mathbf {o}_2^{(2-)}(p,k)),&&p,k\\in [P], p\\ge k,\\\\\\mathbf {h}^{(1)}(p) &= 2-\\sum _{k=p}^{P} \\mathbf {o}_3^{(1)}(p,k), &&p\\in [P],\\\\\\mathbf {h}^{(2)}(p) &= \\sum _{k=1}^{p} \\mathbf {o}_3^{(2)}(p,k), &&p\\in [P].$ The idea of this layer is to compute $\\mathbf {g}_\\mathrm {in}(p^{(1)})$ and $\\mathbf {g}_\\mathrm {in}(p^{(2)})$ , as the following two lemmas show.",
"Lemma 10 For each $p\\in [P]$ , if $p^{(1)}\\le P$ , we have $\\mathbf {h}^{(1)}(p)=\\mathbf {g}_\\mathrm {in}(p^{(1)})$ .",
"If $p^{(1)}>P$ , we have $\\mathbf {h}^{(1)}(p)=2$ .",
"Note that $p^{(1)}$ is never smaller than $p$ .",
"If $p\\le p^{(1)}\\le P$ , then $\\mathbf {o}_3^{(1)}(p,p^{(1)})=2-\\mathbf {g}_\\mathrm {in}(p^{(1)})$ and $\\mathbf {o}_3^{(1)}(p,k)=0$ for each $k\\ne p^{(1)}$ by Lemma REF .",
"If $p^{(1)}>P$ , then $\\mathbf {o}_3^{(1)}(p,k)=0$ for each $k$ .",
"Thus, the claim follows by the definition of $\\mathbf {h}^{(1)}$ .",
"Lemma 11 For each $p\\in [P]$ , if $p^{(2)}\\ge 1$ , we have $\\mathbf {h}^{(2)}(p)=\\mathbf {g}_\\mathrm {in}(p^{(2)})$ .",
"If $p^{(2)}\\le 0$ , we have $\\mathbf {h}^{(2)}(p)=0$ .",
"We first show that $p^{(2)}$ is never larger than $p$ by proving that $p\\mathbf {d}_\\mathrm {old}+ \\mathbf {p}_\\mathrm {in}\\ge p\\mathbf {d}_\\mathrm {new}$ .",
"If $\\mathbf {d}_\\mathrm {new}=1$ , then also $\\mathbf {d}_\\mathrm {old}=1$ holds and this statement is true.",
"Otherwise, we have $\\mathbf {d}_\\mathrm {new}= \\frac{\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}}{P}$ and $\\mathbf {d}_\\mathrm {old}\\ge \\frac{\\mathbf {p}^*_\\mathrm {in}}{P}$ .",
"Hence, we obtain $p(\\mathbf {d}_\\mathrm {new}-\\mathbf {d}_\\mathrm {old}) \\le \\frac{p\\mathbf {p}_\\mathrm {in}}{P} \\le \\mathbf {p}_\\mathrm {in}$ .",
"Therefore, in any case, $p\\mathbf {d}_\\mathrm {old}+ \\mathbf {p}_\\mathrm {in}\\ge p\\mathbf {d}_\\mathrm {new}$ follows, and thus also $p^{(2)}\\le p$ .",
"If $1\\le p^{(2)}\\le p$ , then it follows that $\\mathbf {o}_3^{(2)}(p,p^{(2)})=\\mathbf {g}_\\mathrm {in}(p^{(2)})$ and $\\mathbf {o}_3^{(2)}(p,k)=0$ for each $k\\ne p^{(2)}$ by Lemma REF .",
"If $p^{(2)}\\le 0$ , then $\\mathbf {o}_3^{(2)}(p,k)=0$ holds for each $k$ .",
"Thus, the claim follows by the definition of $\\mathbf {h}^{(2)}$ .",
"The fourth hidden layer is used to compute the minimum in the recursion and consists of $n_4=P$ neurons $\\mathbf {o}_4(p)$ for $p\\in [P]$ .",
"Finally, we output the $P$ values $\\mathbf {g}_\\mathrm {out}(p)$ for $p\\in [P]$ , as well as $\\mathbf {p}^*_\\mathrm {out}=\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}$ .",
"$\\mathbf {o}_4(p) &= \\sigma (\\mathbf {h}^{(1)}(p)-(\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p))), &&p\\in [P],\\\\\\mathbf {g}_\\mathrm {out}(p) &= \\mathbf {h}^{(1)}(p)- \\mathbf {o}_4(p), &&p\\in [P],\\\\\\mathbf {p}^*_\\mathrm {out}&=\\mathbf {p}^*_\\mathrm {in}+\\mathbf {p}_\\mathrm {in}.$ The following lemma ensures that the output value $\\mathbf {g}_\\mathrm {out}(p)$ is indeed computed by the desired recursion, provided that $\\mathbf {h}^{(1)}$ and $\\mathbf {h}^{(2)}$ are computed properly.",
"Lemma 12 For each $p\\in [P]$ , we have $\\mathbf {g}_\\mathrm {out}(p)=\\min \\lbrace \\mathbf {h}^{(1)}(p),\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)\\rbrace $ .",
"This final layer of the FPTAS-NN is constructed exactly like the NN in Figure REF .",
"Equations (REF ) to (REF ) fully define the FPTAS-NN.",
"We have shown several lemmas concerning the functionality of the individual layers.",
"Now we turn towards the proofs of Theorems REF and REF .",
"Proof of Theorem REF .",
"We show that the claim even holds for all values of $p$ and $i$ with $g(p,i)<2$ and not only for those with $g(p,i)\\le 1$ .",
"We use induction on $i$ .",
"For the induction start ($i=0$ ), nothing is to show due to the initialization $g(p,0)=2$ for all $p\\in [P]$ .",
"For the induction step, suppose the claim is valid for all steps up to $i-1$ .",
"Fix some $p\\in [P]$ .",
"By Lemma REF , the output $g(p,i)=\\mathbf {g}_\\mathrm {out}(p)$ in the $i$ -th step equals $\\min \\lbrace \\mathbf {h}^{(1)}(p),\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)\\rbrace $ .",
"In the following, we distinguish two cases.",
"Recall that $p^{(1)}$ and $p^{(2)}$ are the smallest possible integers with $p^{(1)}\\mathbf {d}_\\mathrm {old}\\ge p\\mathbf {d}_\\mathrm {new}$ and $p^{(2)}\\mathbf {d}_\\mathrm {old}+ \\mathbf {p}_\\mathrm {in}\\ge p\\mathbf {d}_\\mathrm {new}$ , respectively.",
"Case 1: $\\mathbf {h}^{(1)}(p)\\le \\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)$ .",
"This implies $g(p,i)=\\mathbf {h}^{(1)}(p)$ .",
"If $\\mathbf {h}^{(1)}(p)=2$ , nothing is to show.",
"Otherwise, by Lemma REF , we have $p^{(1)}\\le P$ with $p^{(1)}\\mathbf {d}_\\mathrm {old}\\ge p\\mathbf {d}_\\mathrm {new}$ and $g(p,i)=\\mathbf {h}^{(1)}(p)=\\mathbf {g}_\\mathrm {in}(p^{(1)})=g(p^{(1)},i-1)$ .",
"By induction, we obtain that there exists a subset of $[i-1]$ with size at most $g(p,i)$ and profit at least $p^{(1)}\\mathbf {d}_\\mathrm {old}$ .",
"Hence, using the same items yields a subset of $[i]$ with size at most $g(p,i)$ and profit at least $p\\mathbf {d}_\\mathrm {new}$ .",
"Thus, the claim is proven in this case.",
"Case 2: $\\mathbf {h}^{(1)}(p)>\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)$ .",
"This implies $g(p,i)=\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)$ .",
"Note that this can only happen if $\\mathbf {h}^{(2)}(p)<2$ because $\\mathbf {h}^{(1)}(p)$ has at most value 2.",
"First, suppose $p^{(2)}\\le 0$ .",
"This implies $p_i=\\mathbf {p}_\\mathrm {in}\\ge p\\mathbf {d}_\\mathrm {new}$ .",
"Hence, by using just item $i$ , we obtain a subset of profit at least $p\\mathbf {d}_\\mathrm {new}$ and size at most $s_i=\\mathbf {s}_\\mathrm {in}\\le \\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p) = g(p,i)$ , and we are done.",
"Second, if $p^{(2)}\\ge 1$ , then Lemma REF implies that $g(p,i)=\\mathbf {s}_\\mathrm {in}+\\mathbf {h}^{(2)}(p)=\\mathbf {s}_\\mathrm {in}+\\mathbf {g}_\\mathrm {in}(p^{(2)})=s_i+g(p^{(2)},i-1)$ .",
"By induction, we obtain that there exists a subset of $[i-1]$ with size at most $g(p,i)-s_i$ and profit at least $p^{(2)}\\mathbf {d}_\\mathrm {old}$ .",
"Hence, adding item $i$ to this subset yields a subset of $[i]$ with size at most $g(p,i)$ and profit at least $p^{(2)}\\mathbf {d}_\\mathrm {old}+p_i\\ge p\\mathbf {d}_\\mathrm {new}$ .",
"Thus, the claim is also proven in this case.",
"$\\Box $ Proof of Theorem REF .",
"Let $M^\\mathrm {OPT}$ be an optimal solution to the Knapsack instance and $M^\\mathrm {OPT}_i=M^\\mathrm {OPT}\\cap [i]$ be the subset of $[i]$ chosen by the optimal solution.",
"Let $s^\\mathrm {OPT}_i=\\sum _{j\\in M^\\mathrm {OPT}_i} s_j$ be the size of $M^\\mathrm {OPT}_i$ and $p^\\mathrm {OPT}_i=\\sum _{j\\in M^\\mathrm {OPT}_i} p_j$ be the profit of $M^\\mathrm {OPT}_i$ .",
"The idea of the proof is that in each step, we lose at most a profit of $d_i$ compared to the optimal solution.",
"Formally, we prove the following claim by induction on $i$ : for every $i\\in [n]$ , and every $p\\le \\left\\lceil \\frac{p^\\mathrm {OPT}_i}{d_i}\\right\\rceil -i$ we have $g(p,i)\\le s^\\mathrm {OPT}_i$ .",
"The induction start is settled by extending the claim to $i=0$ , for which it is trivial.",
"For the induction step, suppose the claim is valid for all steps up to $i-1$ .",
"Fix a value $p\\le \\left\\lceil \\frac{p^\\mathrm {OPT}_i}{d_i}\\right\\rceil -i$ .",
"Let again $p^{(1)}$ and $p^{(2)}$ be the smallest possible integers with $p^{(1)}d_{i-1} \\ge pd_i$ and $p^{(2)}d_{i-1} + p_i \\ge pd_i$ , respectively.",
"We distinguish two cases.",
"Case 1: $i\\notin M^\\mathrm {OPT}$ , i.e., the optimal solution does not use item $i$ .",
"Observe that $pd_i&\\le \\left(\\left\\lceil \\frac{p^\\mathrm {OPT}_{i}}{d_{i}}\\right\\rceil -i\\right)d_{i}\\\\&\\le p^\\mathrm {OPT}_i-(i-1)d_i\\\\&= p^\\mathrm {OPT}_{i-1}-(i-1)d_i\\\\&\\le p^\\mathrm {OPT}_{i-1}-(i-1)d_{i-1}\\\\&\\le \\left(\\left\\lceil \\frac{p^\\mathrm {OPT}_{i-1}}{d_{i-1}}\\right\\rceil -(i-1)\\right)d_{i-1}.$ Hence, we obtain $p^{(1)}\\le \\left\\lceil \\frac{p^\\mathrm {OPT}_{i-1}}{d_{i-1}}\\right\\rceil -(i-1)$ by the definition of $p^{(1)}$ .",
"In particular, $p^{(1)}\\le \\frac{p^*_{i-1}}{d_{i-1}}\\le P$ by the definition of $d_{i-1}$ .",
"Therefore, Lemmas REF and REF imply $g(p,i)\\le g(p^{(1)},i-1)$ .",
"Due to Inequality (REF ), it further follows by induction that $g(p,i)\\le g(p^{(1)},i-1)\\le s^\\mathrm {OPT}_{i-1} = s^\\mathrm {OPT}_i$ , which settles the induction step in this case.",
"Case 2: $i\\in M^\\mathrm {OPT}$ , i.e., the optimal solution uses item $i$ .",
"Observe that $pd_i&\\le \\left(\\left\\lceil \\frac{p^\\mathrm {OPT}_{i}}{d_{i}}\\right\\rceil -i\\right)d_{i}\\\\&\\le p^\\mathrm {OPT}_i-(i-1)d_i\\\\&= p^\\mathrm {OPT}_{i-1}+p_i-(i-1)d_i\\\\&\\le p^\\mathrm {OPT}_{i-1}+p_i-(i-1)d_{i-1}\\\\&\\le \\left(\\left\\lceil \\frac{p^\\mathrm {OPT}_{i-1}}{d_{i-1}}\\right\\rceil -(i-1)\\right)d_{i-1}+p_i.$ Hence, we obtain $p^{(2)}\\le \\left\\lceil \\frac{p^\\mathrm {OPT}_{i-1}}{d_{i-1}}\\right\\rceil -(i-1)$ by the definition of $p^{(2)}$ .",
"If $p^{(2)}\\le 0$ , Lemmas REF and REF imply $g(p,i)\\le s_i \\le s^\\mathrm {OPT}_i$ .",
"If $p^{(2)}\\ge 0$ , Lemmas REF and REF imply $g(p,i)\\le g(p^{(2)},i-1)+s_i$ .",
"Due to Inequality (REF ), we can further conclude by induction that $g(p,i)\\le g(p^{(2)},i-1)+s_i\\le s^\\mathrm {OPT}_{i-1} +s_i= s^\\mathrm {OPT}_i$ , which finalizes the induction step.",
"Now, using the claim we have just proven by induction, we obtain that $g\\left(\\left\\lceil \\frac{p^\\mathrm {OPT}}{d_n}\\right\\rceil -n,n\\right)\\le s^\\mathrm {OPT}_n \\le 1$ .",
"Therefore, it holds that $p^\\mathrm {NN}\\ge \\left(\\left\\lceil \\frac{p^\\mathrm {OPT}}{d_n}\\right\\rceil -n\\right)d_n\\ge p^\\mathrm {OPT}-nd_n.$ If $d_n=1$ , that is, $p^*\\le P$ , then we have for all $i\\in [n]$ that $d_i=1$ .",
"Hence, in each step and for each $p\\in [P]$ , we have $p^{(1)}=p$ and $p^{(2)}=p-p_i$ .",
"Therefore, by Lemmas REF –REF , the FPTAS-NN behaves like the DP-NN from Section that executes the exact dynamic program and the theorem follows.",
"Otherwise, if $d_n>1$ , we have $d_n=\\frac{p^*}{P}$ .",
"Since there must exist one item with profit at least $\\frac{p^*}{n}$ , we obtain $p^\\mathrm {OPT}\\ge \\frac{p^*}{n}$ and, hence, $nd_n=\\frac{np^*}{P}\\le \\frac{n^2 p^\\mathrm {OPT}}{P}$ .",
"Together with (REF ), this implies $\\frac{p^\\mathrm {NN}}{p^\\mathrm {OPT}}\\ge 1-\\frac{n^2}{P}\\ge 1-\\varepsilon $ .",
"$\\Box $"
],
[
"Experiments with other MSE Thresholds",
"In order to show that our experimental results in Section do not depend on the particular choice of the MSE threshold, we conducted the experiments with other thresholds than $0.005$ as well.",
"In Figures REF and REF you can see the results for the thresholds $0.0025$ and $0.00375$ , respectively.",
"One can clearly observe that a quadratic dependence seems to be reasonable in these cases, too.",
"Testing the null hypothesis that the dependence is actually linear against the alternative of a quadratic relation yields p-values of $0.12\\,\\%$ and $0.0046\\,\\%$ , respectively, which is, again, a significant indication of a quadratic dependence.",
"Figure: Required network width to achieve a mean squared error of at most 0.00250.0025 as a function of p * p^*.Figure: Required network width to achieve a mean squared error of at most 0.003750.00375 as a function of p * p^*."
],
[
"Hard- and software.",
"All our experiments were conducted on a machine with an Intel Core i5-8500 6-Core 64-bit CPU and 15.5 GB RAM, using the openSUSE Leap 15.1 Linux distribution.",
"We use Python 3.8.5 with Numpy 1.19.1, Tensorflow 2.2.0 in CPU-only mode, and Statsmodels 0.11.1 for regression and statistical tests."
],
[
"Generation of random Knapsack instances.",
"For a given value of $p^*$ we sample a set of items of total profit $\\sum p_i = p^*$ in the following way: the profit of the $i$ -th item is always chosen uniformly at random among all integer values between 1 and $p^*-\\sum _{i^{\\prime }=1}^{i-1} p_{i^{\\prime }}$ .",
"This is repeated until all profits sum up to $p^*$ .",
"We chose this procedure in order to make it likely to have both, very profitable and less profitable items within one instance.",
"Finally, we shuffle the order of the items.",
"For each item, we then pick a size value uniformly in the interval $[0,1]$ and normalize these values such that their sum equals a uniformly at random chosen value $\\sum s_i\\in \\left]1,2\\right[$ .",
"We certainly want $\\sum s_i>1$ , because otherwise all items would fit into the Knapsack.",
"On the other hand, $\\sum s_i<2$ makes sense, because in our DP-NN (compare Section ), we use 2 as a placeholder for $+\\infty $ ."
],
[
"Preparation of the training set.",
"Since we can generate arbitrarily many random Knapsack instances, we use an infinite training set and never train on the same data point twice.",
"A Knapsack instance with $n$ items yields $n$ training points, namely one for each step of the dynamic program.",
"In order to avoid having the $n$ training points belonging to one instance in successive order, we generate training points belonging to several instances and shuffle them."
],
[
"Neural network architecture.",
"For a given value $p^*$ and width $w$ , the corresponding neural network used consists of an input layer with $p^*+2$ neurons (corresponding to the $p^*$ values of the previous dynamic programming state, as well as, the scalar profit and size values), three hidden layers with $w$ neurons each and ReLU activations, as well as an output layer with $p^*$ neurons (corresponding to the new state of the dynamic programming) without further activation function.",
"That way, we choose the same depth as in the DP-NN (Section ), but do not employ the specific knowledge about the widths of the three hidden layers.",
"As in the DP-NN, each layer is not only connected to the previous layer, but also receives direct connections from the input layer.",
"In total, by our results of Section and Appendix , this architecture is theoretically able to exactly represent the dynamic programming transition function if $w\\ge p^*(p^*-1)/2$ ."
],
[
"Training and testing a specific network.",
"For a given value $p^*$ and width $w$ , we train the neural network architecture described above as follows.",
"We train in epochs of 1000 batches with batch size 32 using mean squared error (MSE) loss and the Adam optimizer, which is a robust standard choice.",
"It makes sense to use MSE loss as it punishes errors in both directions equally hard and large errors harder than small errors.",
"All other (hyper-)parameters are left to the default settings of Tensorflow, which empirically works quite well for our problem type and size.",
"It takes between 8 and 30 seconds to train one epoch with our machine setup.",
"We train until there are two successive epochs without improvement in training loss, which empirically happens after 10 to 80 epochs.",
"Using a test set that is randomly generated in the same way as the training set, we evaluate the trained network on 1000 batches of size 32 each.",
"The resulting mean squared error is our final result for the given values of $p^*$ and $w$ ."
],
[
"Finding the required width.",
"For a given value $p^*$ and a given MSE threshold, we always train networks with increasing widths $25, 50, 75, \\dots $ in steps of 25 as described above until a network achieves a test MSE less or equal to the threshold."
],
[
"Seed generation.",
"In order to ensure reproducibility of our experiments, each time before we train and test an NN with given value $p^*$ and width $w$ , we reset the random seeds of both Numpy and Tensorflow to $257\\cdot p^* + w$ , where 257 is just an arbitrary prime number.",
"Note that these packages only guarantee the same result of random experiments if the same package versions are used."
],
[
"Regression and statistical tests.",
"For each of the three threshold values $0.005$ (Section ), as well as, $0.0025$ and $0.00375$ (Appendix ), we find the required width for achieving the respective threshold for 25 different values of $p^*$ .",
"With the help of the ordinary least squares (OLS) regression utility of the Statsmodels package, we find a quadratic regression line for the $p^*$ -width relation in each of the three cases.",
"The output of the OLS automatically contains the reported p-values for testing whether the coefficient of the quadratic term is zero."
],
[
"Source Code.",
"The source code is publicly available at https://github.com/ChristophHertrich/neural-knapsack.",
"There, the file README.md explains how the code can be used to reproduce the experiments of this paper."
]
] | 2005.14105 |
[
[
"Joint Modelling of Emotion and Abusive Language Detection"
],
[
"Abstract The rise of online communication platforms has been accompanied by some undesirable effects, such as the proliferation of aggressive and abusive behaviour online.",
"Aiming to tackle this problem, the natural language processing (NLP) community has experimented with a range of techniques for abuse detection.",
"While achieving substantial success, these methods have so far only focused on modelling the linguistic properties of the comments and the online communities of users, disregarding the emotional state of the users and how this might affect their language.",
"The latter is, however, inextricably linked to abusive behaviour.",
"In this paper, we present the first joint model of emotion and abusive language detection, experimenting in a multi-task learning framework that allows one task to inform the other.",
"Our results demonstrate that incorporating affective features leads to significant improvements in abuse detection performance across datasets."
],
[
"Introduction",
"Aggressive and abusive behaviour online can lead to severe psychological consequences for its victims [22].",
"This stresses the need for automated techniques for abusive language detection, a problem that has recently gained a great deal of interest in the natural language processing community.",
"The term abuse refers collectively to all forms of expression that vilify or offend an individual or a group, including racism, sexism, personal attacks, harassment, cyber-bullying, and many others.",
"Much of the recent research has focused on detecting explicit abuse, that comes in the form of expletives, derogatory words or threats, with substantial success [20].",
"However, abuse can also be expressed in more implicit and subtle ways, for instance, through the use of ambiguous terms and figurative language, which has proved more challenging to identify.",
"The NLP community has experimented with a range of techniques for abuse detection, such as recurrent and convolutional neural networks [27], [25], [34], character-based models [23] and graph-based learning methods [17], [1], [18], obtaining promising results.",
"However, all of the existing approaches have focused on modelling the linguistic properties of the comments or the meta-data about the users.",
"On the other hand, abusive language and behaviour are also inextricably linked to the emotional and psychological state of the speaker [26], which is reflected in the affective characteristics of their language [15].",
"In this paper, we propose to model these two phenomena jointly and present the first abusive language detection method that incorporates affective features via a multitask learning (MTL) paradigm.",
"MTL [3] allows two or more tasks to be learned jointly, thus sharing information and features between the tasks.",
"In this paper, our main focus is on abuse detection; hence we refer to it as the primary task, while the task that is used to provide additional knowledge — emotion detection — is referred to as the auxiliary task.",
"We propose an MTL framework where a single model can be trained to perform emotion detection and identify abuse at the same time.",
"We expect that affective features, which result from a joint learning setup through shared parameters, will encompass the emotional content of a comment that is likely to be predictive of potential abuse.",
"We propose and evaluate different MTL architectures.",
"We first experiment with hard parameter sharing, where the same encoder is shared between the tasks.",
"We then introduce two variants of the MTL model to relax the hard sharing constraint and further facilitate positive transfer.",
"Our results demonstrate that the MTL models significantly outperform single-task learning (STL) in two different abuse detection datasets.",
"This confirms our hypothesis of the importance of affective features for abuse detection.",
"Furthermore, we compare the performance of MTL to a transfer learning baseline and demonstrate that MTL provides significant improvements over transfer learning."
],
[
"Related Work",
"Techniques for abuse detection have gone through several stages of development, starting with extensive manual feature engineering and then turning to deep learning.",
"Early approaches experimented with lexicon-based features [10], bag-of-words (BOW) or n-gram features [32], [7], and user-specific features, such as age [5] and gender [35].",
"With the advent of deep learning, the trend shifted, with abundant work focusing on neural architectures for abuse detection.",
"In particular, the use of convolutional neural networks (CNNs) for detecting abuse has shown promising results [25], [34].",
"This can be attributed to the fact that CNNs are well suited to extract local and position-invariant features [36].",
"Character-level features have also been shown to be beneficial in tackling the issue of Out-of-Vocabulary (OOV) words [19], since abusive comments tend to contain obfuscated words.",
"Recently, approaches to abuse detection have moved towards more complex models that utilize auxiliary knowledge in addition to the abuse-annotated data.",
"For instance, [17], [18] used community-based author information as features in their classifiers with promising results.",
"[9] used transfer learning to fine-tune features from the author metadata network to improve abuse detection.",
"MTL, introduced by [3], has proven successful in many NLP problems, as illustrated in the MTL survey of [39].",
"It is interesting to note that many of these problems are domain-independent tasks, such as part-of-speech tagging, chunking, named entity recognition, etc.",
"[4].",
"These tasks are not restricted to a particular dataset or domain, i.e., any text data can be annotated for the phenomena involved.",
"On the contrary, tasks such as abuse detection are domain-specific and restricted to a handful of datasets (typically focusing on online communication), therefore presenting a different challenge to MTL.",
"Much research on emotion detection cast the problem in a categorical framework, identifying specific classes of emotions and using e.g., Ekman's model of six emotions [8], namely anger, disgust, fear, happiness, sadness, surprise.",
"Other approaches adopt the Valence-Arousal-Dominance (VAD) model of emotion [16], which represents polarity, degree of excitement, and degree of control, each taking a value from a range.",
"The community has experimented with a variety of computational techniques for emotion detection, including vector space modelling [6], machine learning classifiers [29] and deep learning methods [40].",
"In their work, [40] take an MTL approach to emotion detection.",
"However, all the tasks they consider are emotion-related (annotated for either classification or emotion distribution prediction), and the results show improvements over single-task baselines.",
"[2] use a multitask ensemble architecture to learn emotion, sentiment, and intensity prediction jointly and show that these tasks benefit each other, leading to improvements in performance.",
"To the best of our knowledge, there has not yet been an approach investigating emotion in the context of abuse detection."
],
[
"Datasets",
"The tasks in an MTL framework should be related in order to obtain positive transfer.",
"MTL models are sensitive to differences in the domain and distribution of data [24].",
"This affects the stability of training, which may deteriorate performance in comparison to an STL model [39].",
"We experiment with abuse and emotion detection datasetsWe do not own any rights to the datasets (or the containing tweets).",
"In the event of one who wishes to attain any of the datasets, to avoid redistribution infringement, we request them to contact the authors/owners of the source of the datasets.",
"that are from the same data domain — Twitter.",
"All of the datasets were subjected to the same pre-processing steps, namely lower-casing, mapping all mentions and URLs to a common token (i.e., _MTN_ and _URL_) and mapping hashtags to words."
],
[
"Abuse detection task",
"To ensure that the results are generalizable, we experiment with two different abuse detection datasets."
],
[
"OffensEval 2019 (",
"This dataset is from SemEval 2019 - Task 6: OffensEval 2019 - Identifying and Categorizing Offensive Language in Social Media [37], [38].",
"We focus on Subtask A, which involves offensive language identification.",
"It contains $13,240$ annotated tweets, and each tweet is classified as to whether it is offensive ($33\\%$ ) or not ($67\\%$ ).",
"Those classified as offensive contain offensive language or targeted offense, which includes insults, threats, profane language and swear words.",
"The dataset was annotated using crowdsourcing, with gold labels assigned based on the agreement of three annotators."
],
[
"Waseem and Hovy 2016 (",
"This dataset was compiled by [35] by searching for commonly used slurs and expletives related to religious, sexual, gender and ethnic minorities.",
"The tweets were then annotated with one of three classes: racism, sexism or neither.",
"The annotations were subsequently checked through an expert review, which yielded an inter-annotator agreement of $\\kappa =0.84$ .",
"The dataset contains $16,907$ TweetIDs and their corresponding annotation, out of which only $16,202$ TweetIDs were retrieved due to users being reported or tweets having been taken down since it was first published in 2016.",
"The distribution of classes is: $1,939$ ($12\\%$ ) racism; $3,148$ ($19.4\\%$ ) sexism; and $11,115$ ($68.6\\%$ ) neither, which is comparable to the original distribution: ($11.7\\%$ : $20.0\\%$ : $68.3\\%$ ).",
"It should be noted that racial or cultural biases may arise from annotating data using crowdsourcing, as pointed out by [31].",
"The performance of the model depends on the data used for training, which in turn depends on the quality of the annotations and the experience level of the annotators.",
"However, the aim of our work is to investigate the relationship between emotion and abuse detection, which is likely to be independent of the biases that may exist in the annotations."
],
[
"Emotion (",
"This dataset is from SemEval-2018 Task 1: Affect in Tweets [21], and specifically from Subtask 5 which is a multilabel classification of 11 emotion labels that best represent the mental state of the author of a tweet.",
"The dataset consists of around 11k tweets (training set: 6839; development set: 887; test set: 3260).",
"It contains the TweetID and 11 emotion labels (anger, anticipation, disgust, fear, joy, love, optimism, pessimism, sadness, surprise, trust) which take a binary value to indicate the presence or absence of the emotion.",
"The annotations were obtained for each tweet from at least 7 annotators and aggregated based on their agreement."
],
[
"Approach",
"In this section, we describe our baseline models and then proceed by describing our proposed models for jointly learning to detect emotion and abuse."
],
[
"Single-Task Learning",
"As our baselines, we use different Single-Task Learning (STL) models that utilize abuse detection as the sole optimization objective.",
"The STL experiments are conducted for each primary-task dataset separately.",
"Each STL model takes as input a sequence of words $\\lbrace w_{\\textrm {1}}, w_{\\textrm {2}}, ..., w_{\\textrm {n}}\\rbrace $ , which are initialized with $k$ -dimensional vectors $\\textrm {e}$ from a pre-trained embedding space.",
"We experiment with two different architecture variants:"
],
[
"Max Pooling and MLP classifier",
"We refer to this baseline as STL$_{maxpool+MLP}$ .",
"In this baseline, a two-layered bidirectional Long Short-Term Memory (LSTM) network [13] is applied to the embedding representations $\\textrm {e}$ of words in a post to get contextualized word representations $\\lbrace \\textrm {h}_{\\textrm {1}}, \\textrm {h}_{\\textrm {2}}, ..., \\textrm {h}_{\\textrm {n}}\\rbrace $ : $\\textrm {h}_{\\textrm {t}} = [\\overrightarrow{\\textrm {h}_{\\textrm {t}}};\\overleftarrow{\\textrm {h}_{\\textrm {t}}}]$ with $\\overrightarrow{\\textrm {h}_{\\textrm {t}}}, \\overleftarrow{\\textrm {h}_{\\textrm {t}}}\\in \\mathbb {R}^{l}$ and $\\textrm {h}_{\\textrm {t}} \\in \\mathbb {R}^{2\\cdot l}$ , where $l$ is the hidden dimensionality of the BiLSTM.",
"We then apply a max pooling operation over $\\lbrace \\textrm {h}_{\\textrm {1}}, \\textrm {h}_{\\textrm {2}}, ..., \\textrm {h}_{\\textrm {n}}\\rbrace $ : $r^{\\textrm {(p)}}_{\\textrm {i}} = {max_\\textrm {i}}(\\textrm {h}_{\\textrm {1}}, \\textrm {h}_{\\textrm {2}}, ..., \\textrm {h}_{\\textrm {n}})$ where $\\textrm {r}^\\textrm {(p)}\\in \\mathbb {R}^{2\\cdot l}$ and where the superscript $\\textrm {(p)}$ is used to indicate that the representations correspond to the primary task.",
"This is followed by dropout [33] for regularization and a 2-layered Multi-layer Perceptron (MLP) [12]: $\\textrm {m}^\\textrm {1(p)} &= BatchNorm(tanh(W^{l_1}{\\textrm {r}^{\\textrm {(p)}}})) \\\\ \\textrm {m}^\\textrm {2(p)} &= tanh(W^{l_2}{\\textrm {m}}^\\textrm {1(p)}) \\\\\\textrm {m}_{\\textrm {t}}^\\textrm {(p)} &= \\textrm {m}_{\\textrm {t}}^\\textrm {2(p)} $ where $W^{l_1}$ and $W^{l_2}$ are the weight matrices of the 2-layer MLP.",
"Dropout is applied to the output $\\textrm {m}^\\textrm {(p)}$ of the MLP, which is then followed by a linear output layer to get the unnormalized output $\\textrm {o}^\\textrm {(p)}$ .",
"For OffensEval, a sigmoid activation $\\sigma $ is then applied in order to make a binary prediction with respect to whether a post is offensive or not, while the network parameters are optimized to minimize the binary cross-entropy (BCE): $L_{BCE}=-\\frac{1}{N}\\sum ^N_{i=1}y_i\\cdot log(p(y_i)) + \\\\(1-y_i)\\cdot log(1-p(y_i))$ where $N$ is the number of training examples, and $y$ denotes the true and $p(y)$ the predicted label.",
"For Waseem&Hovy, a $log\\_softmax$ activation is applied for multiclass classification, while the network parameters are optimized to minimize the categorical cross-entropy, that is, the negative log-likelihood (NLL) of the true labels: $L_{NLL}=-\\frac{1}{N}\\sum ^N_{i=1}log(p(y_i))$ Figure: MTL Hard Sharing model.",
"The embedding representations {e 1 ,e 2 ,...,e n }\\lbrace \\textrm {e}_{\\textrm {1}}, \\textrm {e}_{\\textrm {2}}, ..., \\textrm {e}_{\\textrm {n}}\\rbrace are either a result of projection through the GloVe embedding layer or a concatenation of the projections through the GloVe and ELMo embedding layer.",
"The different arrows are used to indicate the different passes for the primary and auxiliary task.",
"The units on the left-hand side correspond to the primary task and the units on the right-hand side correspond to the auxiliary task with the Stacked BiLSTM Encoder and embedding layers shared by both tasks.",
"The model inside the dotted box corresponds to the STL BiLSTM+attn _{BiLSTM+attn} architecture."
],
[
"BiLSTM and Attention classifier",
"We refer to this model as STL$_{BiLSTM+attn}$ .",
"In this baseline (Figure REF ; enclosed in the dotted boxes), rather than applying max pooling, we apply dropout to $\\textrm {h}$ which is then followed by a third BiLSTM layer and an attention mechanism: $\\textrm {u}_{\\textrm {t}}^\\textrm {(p)} &= W^a\\textrm {r}_{\\textrm {t}}^\\textrm {(p)} \\\\a_{\\textrm {t}}^\\textrm {(p)} &= \\frac{exp(\\textrm {u}_{\\textrm {t}}^\\textrm {(p)})}{\\sum _{\\textrm {t}} exp(\\textrm {u}_{\\textrm {t}}^\\textrm {(p)})} \\\\\\textrm {m}^\\textrm {(p)} &= \\sum _{\\textrm {t}} a_{\\textrm {t}}^\\textrm {(p)} \\textrm {r}_{\\textrm {t}}^\\textrm {(p)} $ where $\\textrm {r}^\\textrm {(p)}$ is the output of the third BiLSTM.",
"We then apply dropout to the output of the attention layer $\\textrm {m}^\\textrm {(p)}$ .",
"The remaining components, output layer and activation, are the same as the STL$_{maxpool+MLP}$ model.",
"Across the two STL baselines, we further experiment with two different input representations: 1) GloVe (G), where the input is projected through the GloVe embedding layer [28]; 2) GloVe+ELMo (G+E), where the input is first projected through the GloVe embedding layer and the ELMo embedding layer [30] separately, and then the final word representation $\\textrm {e}$ is obtained by concatenating the output of these two layers.",
"Given these input representations, we have a total of 4 different baseline models for abuse detection.",
"We use grid search to tune the hyperparameters of the baselines on the development sets of the primary task (i.e., abuse detection).",
"Figure: MTL (Gated) Double Encoder architecture.",
"For the MTL Gated Double Encoder model we use two learnable parameters α\\alpha that control information flow.",
"For the MTL Double Encoder model, these are fixed and set to 1.",
"The dotted boxes represent the STL BiLSTM+attn _{BiLSTM+attn} architecture."
],
[
"Multi-task Learning",
"Our MTL approach uses two different optimization objectives: one for abuse detection and another for emotion detection.",
"The two objectives are weighted by a hyperparameter $\\beta $ [($1-\\beta $ ) for abuse detection and $\\beta $ for emotion detection] that controls the importance we place on each task.",
"We experiment with different STL architectures for the auxiliary task and propose MTL models that contain two network branches – one for the primary task and one for the auxiliary task – connected by a shared encoder which is updated by both tasks alternately."
],
[
"Hard Sharing Model",
"This model architecture, referred to as MTL$_{Hard}$ , is inspired by [3] and uses hard parameter sharing: it consists of a single encoder that is shared and updated by both tasks, followed by task-specific branches.",
"Figure REF presents MTL$_{Hard}$ where the dotted box represents the STL$_{BiLSTM+attn}$ architecture that is specific to the abuse detection task.",
"In the right-hand side branch – corresponding to the auxiliary objective of detecting emotion – we apply dropout to $\\textrm {h}$ before passing it to a third BiLSTM.",
"This is then followed by an attention mechanism to obtain $\\textrm {m}^\\textrm {(a)}$ and then dropout is applied to it.",
"The superscript $\\textrm {(a)}$ is used to indicate that these representations correspond to the auxiliary task.",
"Then, we obtain the unnormalized output $\\textrm {o}^{\\textrm {(a)}}$ after passing $\\textrm {m}^\\textrm {(a)}$ through a linear output layer with $\\textrm {o}^{\\textrm {(a)}}\\in \\mathbb {R}^{11}$ (11 different emotions in SemEval18), which is then subjected to a sigmoid activation to obtain a prediction $p(y)$ .",
"While the primary task on the left is optimized using either Equation REF or REF (depending on the dataset used), the auxiliary task is optimized to minimize binary cross-entropy."
],
[
"Double Encoder Model ",
"This model architecture, referred to as MTL$_{DEncoder}$ , is an extension of the previous model that now has two BiLSTM encoders: a task-specific two-layered BiLSTM encoder for the primary task, and a shared two-layered BiLSTM encoder.",
"During each training step of the primary task, the input representation $\\textrm {e}$ for the primary task is passed through both encoders, which results in two contextualized word representations $\\lbrace \\textrm {h}_{\\textrm {1}}^{\\textrm {(p)}}, \\textrm {h}_{\\textrm {2}}^{\\textrm {(p)}}, ..., \\textrm {h}_{\\textrm {n}}^{\\textrm {(p)}}\\rbrace $ and $\\lbrace \\textrm {h}_{\\textrm {1}}^{\\textrm {(s)}}, \\textrm {h}_{\\textrm {2}}^{\\textrm {(s)}}, ..., \\textrm {h}_{\\textrm {n}}^{\\textrm {(s)}}\\rbrace $ , where superscript (s) is used to denote the representations that result from the shared encoder.",
"These are then summed (Figure REF , where both $\\alpha ^{\\textrm {(p)}}$ and $\\alpha ^{\\textrm {(s)}}$ are fixed and set to 1) and the output representation is passed through a third BiLSTM followed by an attention mechanism to get the post representation $\\textrm {m}^{\\textrm {(p)}}$ .",
"The rest of the components of the primary task branch, as well as the auxiliary task branch are the same as those in MTL$_{Hard}$ ."
],
[
"Gated Double Encoder Model ",
"This model architecture, referred to as MTL$_{Gated DEncoder}$ , is an extension of MTL$_{DEncoder}$ , but is different in the way we obtain the post representations $\\textrm {m}^{\\textrm {(p)}}$ .",
"Representations $\\textrm {h}^{\\textrm {(p)}}$ and $\\textrm {h}^{\\textrm {(s)}}$ are now merged using two learnable parameters $\\alpha ^{\\textrm {(p)}}$ and $\\alpha ^{\\textrm {(s)}}$ (where $\\alpha ^{\\textrm {(p)}}+\\alpha ^{\\textrm {(s)}}=1.0$ ) to control the flow of information from the representations that result from the two encoders (Figure REF ): $\\alpha ^{\\textrm {(p)}}\\cdot \\textrm {h}^{\\textrm {(p)}} + \\alpha ^{\\textrm {(s)}}\\cdot \\textrm {h}^{\\textrm {(s)}}$ The remaining architecture components of the primary task and auxiliary task branch are the same as for MTL$_{DEncoder}$ ."
],
[
"Hyperparameters",
"We use pre-trained GloVe embeddingshttps://nlp.stanford.edu/projects/glove/ with dimensionality 300 and pre-trained ELMo embeddingshttps://allennlp.org/elmo with dimensionality 1024.",
"Grid search is performed to determine the optimal hyperparameters.",
"We find an optimal value of $\\beta =0.1$ that makes the updates for the auxiliary task 10 times less important.",
"The encoders consist of 2 stacked BiLSTMs with $hidden\\_size=512$ .",
"For all primary task datasets, the BiLSTM+Attention classifier and the 2-layered MLP classifier have $hidden\\_size=256$ .",
"For the auxiliary task datasets, the BiLSTM+Attention classifier and the 2-layered MLP classifier have $hidden\\_size=512$ .",
"Dropout is set to $0.2$ .",
"We use the Adam optimizer [14] for all experiments.",
"All model weights are initialized using Xavier Initialization [11].",
"For MTL$_{Gated DEncoder}$ , $\\alpha ^{\\textrm {(p)}}=0.9$ and $\\alpha ^{\\textrm {(s)}}=0.1$ ."
],
[
"Training",
"All models are trained until convergence for both the primary and the auxiliary task, and early stopping is applied based on the performance on the validation set.",
"For MTL, we ensure that both the primary and the auxiliary task have completed at least 5 epochs of training.",
"The MTL training process involves randomly (with $p=0.5$ ) alternating between the abuse detection and emotion detection training steps.",
"Each task has its own loss function, and in each of the corresponding task's training step, the model is optimized accordingly.",
"All experiments are run using stratified 10-fold cross-validation, and we use the paired t-test for significance testing.",
"We evaluate the models using Precision ($P$ ), Recall ($R$ ), and F1 ($F1$ ), and report the average $macro$ scores across the 10 folds."
],
[
"STL experiments",
"The STL experiments are conducted on the abuse detection datasets independently.",
"As mentioned in the STL section, we experiment with four different model configurations to select the best STL baseline.",
"Table REF presents the evaluation results of the STL models trained and tested on the OffensEval dataset, and Table REF on the Waseem and Hovy dataset.",
"The best results are highlighted in bold and are in line with the validation set results.",
"We select the best performing STL model configuration on each dataset and use it as part of the corresponding MTL architecture in the MTL experiments below.",
"Table: STL model comparisons.",
"In these tables, G denotes models that use GloVe embeddings and G+E denotes models in which word representations are concatenations of their corresponding GloVe and ELMo embeddings.",
"The best performing model is highlighted in bold."
],
[
"MTL experiments",
"In this section, we examine the effectiveness of the MTL models for the abuse detection task and explore the impact of using emotion detection as an auxiliary task.",
"We also compare the performance of our MTL models with that of a transfer learning approach."
],
[
"Emotion detection as an auxiliary task",
"In this experiment, we test whether incorporating emotion detection as an auxiliary task improves the performance of abuse detection.",
"Tables REF and REF show the results on OffensEval and Waseem and Hovy datasets ($\\dagger $ indicates statistically significant results over the corresponding STL model).",
"Learning emotion and abuse detection jointly proved beneficial, with MTL models achieving statistically significant improvement in F1 using the Gated Double Encoder Model MTL$_{Gated DEncoder}$ ($p<0.05$ , using a paired t-test).",
"This suggests that affective features from the shared encoder benefit the abuse detection task.",
"Table: STL vs. MTL with emotion detection as the auxiliary task.",
"†\\dagger indicates statistically significant improvement over STL.Table: MTL vs. transfer learning performance.",
"OE refers to the OffensEval dataset and W&H to the Waseem&Hovy dataset.",
"†\\dagger indicates statistically significant improvements.Table: STL vs. MTL: samples from Twitter - Waseem and Hovy and Twitter - OffensEval datasets, where superior performance of MTL is observed.",
"The `predicted emotion' column contains the emotion labels predicted on the abuse detection data.",
"The name of the politician in the fourth row is masked using the _POLITICIAN_ tag."
],
[
"MTL vs. transfer learning",
"Transfer learning is an alternative to MTL that also allows us to transfer knowledge from one task to another.",
"This experiment aims to compare the effectiveness of MTL against transfer learning.",
"We selected the MTL model with the best performance in abuse detection and compared it against an identical model, but trained in a transfer learning setting.",
"In this setup, we first train the model on the emotion detection task until convergence and then proceed by fine-tuning it for the abuse detection task.",
"Table REF presents the comparison between MTL and transfer learning, for which we use the same architecture and hyperparameter configuration as MTL.",
"We observe that MTL outperforms transfer learning and provides statistically significant $(p < 0.05)$ results on both OffensEval and Waseem and Hovy datasets."
],
[
"Auxiliary task",
"Our results show that emotion detection significantly improves abuse detection on both OffensEval and Waseem and Hovy datasets.",
"Table REF presents examples of improvements in both datasets achieved by the MTL$_{Gated DEncoder}$ model, over the STL model.",
"In the examples, the highlighted words are emotion evocative words, which are also found in the SemEval2018 Emotion dataset.",
"As the emotion detection task encourages the model to learn to predict the emotion labels for the examples that contain these words, the word representations and encoder weights that are learned by the model encompass some affective knowledge.",
"Ultimately, this allows the MTL model to determine the affective nature of the example, which may help it to classify abuse more accurately.",
"It is also interesting to observe that a controversial person or topic may strongly influence the classification of the sample containing it.",
"For example, sentences referring to certain politicians may be classified as Offensive, regardless of the context.",
"An example instance of this can be found in Table REF .We mask the name using the _POLITICIAN_ tag.",
"The MTL model, however, classifies it correctly, which may be attributed to the excessive use of “!” marks.",
"The latter is one of the most frequently used symbols in the SemEval2018 Emotion dataset, and it can encompass many emotions such as surprise, fear, etc., therefore, not being indicative of a particular type of emotion.",
"Such knowledge can be learned within the shared features of the MTL model."
],
[
"MTL vs. transfer learning",
"This experiment demonstrates that MTL achieves higher performance than transfer learning in a similar experimental setting.",
"The higher performance may be indicative of a more stable way of transferring knowledge, which leads to better generalization.",
"In the MTL framework, since the shared parameters are updated alternately, each task learns some knowledge that may be mutually beneficial to both related tasks, which leads to a shared representation that encompasses the knowledge of both tasks and hence is more generalized.",
"In contrast, in the case of transfer learning, the primary task fine-tunes the knowledge from the auxiliary task (i.e., in the form of pre-trained parameters) for its task objective and may be forgetting auxiliary task knowledge."
],
[
"Conclusion",
"In this paper, we proposed a new approach to abuse detection, which takes advantage of the affective features to gain auxiliary knowledge through an MTL framework.",
"Our experiments demonstrate that MTL with emotion detection is beneficial for the abuse detection task in the Twitter domain.",
"The mutually beneficial relationship that exists between these two tasks opens new research avenues for improvement of abuse detection systems in other domains as well, where emotion would equally play a role.",
"Overall, our results also suggest the superiority of MTL over STL for abuse detection.",
"With this new approach, one can build more complex models introducing new auxiliary tasks for abuse detection.",
"For instance, we expect that abuse detection may also benefit from joint learning with complex semantic tasks, such as figurative language processing and inference."
]
] | 2005.14028 |
[
[
"Machine Learning for Condensed Matter Physics"
],
[
"Abstract Condensed Matter Physics (CMP) seeks to understand the microscopic interactions of matter at the quantum and atomistic levels, and describes how these interactions result in both mesoscopic and macroscopic properties.",
"CMP overlaps with many other important branches of science, such as Chemistry, Materials Science, Statistical Physics, and High-Performance Computing.",
"With the advancements in modern Machine Learning (ML) technology, a keen interest in applying these algorithms to further CMP research has created a compelling new area of research at the intersection of both fields.",
"In this review, we aim to explore the main areas within CMP, which have successfully applied ML techniques to further research, such as the description and use of ML schemes for potential energy surfaces, the characterization of topological phases of matter in lattice systems, the prediction of phase transitions in off-lattice and atomistic simulations, the interpretation of ML theories with physics-inspired frameworks and the enhancement of simulation methods with ML algorithms.",
"We also discuss in detail the main challenges and drawbacks of using ML methods on CMP problems, as well as some perspectives for future developments."
],
[
"Introduction",
"src/introduction-revised"
],
[
"Overview of Machine Learning",
"src/machine-learning-revised"
],
[
"Overview of Condensed Matter Physics",
"src/condensed-matter-revised"
],
[
"Hard Matter and Machine Learning",
"src/hard-matter-revised"
],
[
"Physics-inspired Machine Learning Theory",
"src/theory-revised"
],
[
"Materials Modeling",
"src/materials-revised"
],
[
"Soft Matter",
"src/soft-matter-revised"
],
[
"Challenges and Perspectives",
"src/challenges-revised This work was financially supported by Conacyt grant 287067."
]
] | 2005.14228 |
[
[
"Two-dimensional extreme skin depth engineering for CMOS photonics"
],
[
"Abstract Extreme skin depth engineering (e-skid) can be applied to integrated photonics to manipulate the evanescent field of a waveguide.",
"Here we demonstrate that e-skid can be implemented in two directions in order to deterministically engineer the evanescent wave allowing for dense integration with enhanced functionalities.",
"In particular, by increasing the skin depth, we enable the creation of large gap, bendless directional couplers with large operational bandwidth.",
"Here we experimentally validate two-dimensional e-skid for integrated photonics in a CMOS photonics foundry and demonstrate strong coupling with a gap of 1.44 {\\mu}m."
],
[
"Evanescent Waves in Silicon Photonics",
"Evanescent waves in silicon photonic waveguides have the propensity to cause parasitic optical crosstalk.",
"In traditional photonic circuits, design strategies must consider minimum separation distances between any closely spaced waveguides to prevent unwanted coupling [1].",
"This problem is inhibiting for many photonic circuits due to cost and size constraints.",
"Many efforts have been made to overcome these issues, battling size constraints by employing inverse design [2], [3] or implementing metamaterials to increase performance [4], [5], [6].",
"Recent work introduced a new, metamaterial paradigm for waveguiding that fundamentally suppresses coupling between waveguides [7].",
"In this approach a subwavelength, multi-layer cladding is placed in plane and in parallel with the waveguide, decreasing the skin depth of the fundamental transverse-electric (TE) mode's evanescent field.",
"The concept is called extreme skin depth engineering, or e-skid.",
"E-skid has been employed as cross-talk suppresion [7], [8], and for high performance polarization splitting [9], [10].",
"The e-skid features are created in the same processing step as the waveguide itself, allowing this to be an innate no-cost addition to any design.",
"The addition of these features can reduce the crosstalk between waveguides by more than three orders of magnitude, which will dramatically reduce the photonic design footprint.",
"[7].",
"Here we expand on this work by using e-skid in two directions.",
"Using both a parallel [7] and perpendicular cladding we can engineer the coupling between waveguides throughout a photonic circuit.",
"Specifically, a perpendicular eskid cladding can increase coupling by up to four orders of magnitude.",
"Using this, we design a large gap, bendless directional coupler that operates over a large bandwidth ($\\ge 40$ nm).",
"Employing e-skid techniques to traditional photonic components allows an immediate decrease in overall system footprint, not just limited to waveguide routing.",
"Finally, we demonstrate these two direction e-skid large gap directional couplers in a complementary metal-oxide semiconductor (CMOS) photonic platform, thereby affirming the manufacturability of e-skid components and integration with foundry offerings.",
"Figure: (a) Here we demonstrate the parallel oriented subwavelength, multi-layer cladding.",
"The anisotropic permittivity tensor is displayed over the cladding, which follows the Rytov relations for each direction.",
"(b) The perpendicular cladding essentially swaps the xxxx and yyyy components of the permittivity tensor in (a).",
"The incident wave is reflected (shown by the two arrows in medium one) and the evanescent wave is strongly decaying in medium two in (a) and weakly decaying in (b).",
"(c) A plot of Eq.",
"for the two different cladding strategies.",
"We see that decay is increased over SiO 2 _2 for most of the fill factors of the parallel case, whereas we can see a variable decrease in decay by almost the whole scale between the two materials in the perpendicular cladding.Consider two media with index of refraction $n_1,n_2$ .",
"When an incoming wave from $n_1$ meets the boundary at $n_2$ and the angle is greater than the critical angle $\\theta _i < \\theta _c = \\operatorname{sin}^{-1}(n_2/n_1)$ an evanescent wave is formed in the second medium.",
"This wave does not carry power across the boundary; it exponentially decays into the second medium [11].",
"E-skid allows us to tune the decay constant of this evanescent wave by introducing subwavelength, periodic structures that transform wave's (specifically, a polarized wave's) momentum [12], [13].",
"These features change the second medium from an isotropic material to an anisotropic metamaterial.",
"The anisotropy here refers to the permittivity values of the dielectric tensor of the material (where we are assuming that the permittivity can be defined $\\epsilon _r = n^2$ ).",
"For deep subwavelength features, these component values are defined by the Rytov relations [14], [15]: $n_{\\parallel }^2 = n_{1}^2 \\rho + n_{2}^2 (1-\\rho ),$ $n_{\\bot }^{-2} = n_{1}^{-2} \\rho + n_{2}^{-2} (1-\\rho ),$ where $n_{1},n_{2}$ are the indices of the first and second medium, respectively, and $\\rho $ is the fill factor.",
"The parallel component, $n_{\\parallel }$ , is defined in the direction parallel to the periodic structure's orientation, and the perpendicular component, $n_{\\bot }$ , is oriented perpendicular to the periodic structure.",
"For features that are deep-subwavelength, these relations demonstrate how the second material transforms from isotropic to an anisotropic metamaterial, however any subwavelength structures will exhibit anisotropy albeit without these neat relations.",
"Figure: A comparison of the different cladding strategies discussed, with the note that they have a common cladding on the right hand side (normal isotropic SiO 2 _2).",
"(a,b,c) Here we have the top-down perspective of the waveguides, which shows the cladding for the none, parallel and perpendicular structures on the left hand side, respectively.",
"(d,e,f) The ZY plane cross section mode profiles corresponding to the cladding diagrams from (a,b,c), where the width of the waveguides is 400 nm and the height is 220 nm.",
"(g) A center slice through each of the mode profiles, which demonstrate, on a log-linear scale, the amount of control we can impose on the evanescent wave with these structures.",
"Simulations were done with an anisotropic material following Rytov relations (Eqs.",
",), where the core waveguide width was 400 nm, and fill factor was 0.6 for both orientations.The key result of the e-skid derivation leverages this anisotropy for the evanescent wave, which is characterized by the decay constant, $\\beta $ : $\\beta (\\theta _i;\\rho ) = \\frac{1}{\\delta (\\theta _i)} = k_0\\frac{n_{2x}(\\rho )}{n_{2y}(\\rho )}\\sqrt{n_1^2 sin^2(\\theta _i) - n_{2y}(\\rho )^2}.$ where $k_0$ is the wavevector and $\\theta _i$ is the angle of the incident wave to the boundary (we assume paraxial $\\theta _i \\approx \\pi /2$ ) [12].",
"The decay constant is now subject to a degree of variable tunability ($\\rho $ ), allowing for control of the evanescent wave [7], [12].",
"In Fig.",
"REF (a), we show the dielectric tensor for the e-skid structure, where the periodicity of the subwavelength features is parallel to the boundary ($y = 0$ ).",
"In this orientation, the diagonal components of the second material become $[n_{2x}^2,n_{2y}^2,n_{2z}^2] = [n_{\\parallel }^2,n_{\\bot }^2,n_{\\parallel }^2]$ in accordance with the Rytov relations (Eqs.",
"REF , REF ).",
"This structure will increase the decay constant of the evanescent wave, thereby decreasing the skin depth [7].",
"Without loss of generality, we recognize that we can rotate the optical axis by rotating the subwavelength features and realize e-skid in a second direction.",
"Due to the direction dependency outlined by the Rytov relations, when we rotate the periodicity of the features, we effectively swap the $xx$ and $yy$ components of the dielectric tensor of the parallel cladding such that we now see $[n_{2x}^2,n_{2y}^2,n_{2z}^2] = [n_{\\bot }^2,n_{\\parallel }^2,n_{\\parallel }^2]$ (Fig.",
"REF (b)).",
"The values of $n_{2x},n_{2y}$ in Eq.",
"REF control the decay constant, and by rotating the periodic structure we are able to dictate an increase or decrease.",
"We populated Eq.",
"REF with the new dielectric tensor values outlined in Fig.",
"REF (a,b) such that we show in Fig.",
"REF (c) the full range of decay constant tunability of e-skid in two directions.",
"Fig.",
"REF (c) shows clearly that both decreasing and increasing skin depth can be achieved by the parallel features, however applying this to CMOS photonics manufacturing, we generally omit the higher and lower fill factors due to resolution constraints.",
"[16].",
"We used a material platform consistent with CMOS photonics in (c), such that material one is silicon (Si) and material two is silicon dioxide (SiO$_2$ ), however, this is true for any optical material combination as long as $n_1>n_2$ ."
],
[
"Optical waveguiding is not fully described by the simple electromagnetic wave-at-a-boundary example above.",
"While it lends intuition, we must find the electromagnetic mode of the entire structure to get a clear picture of this effect.",
"We used a commercial finite difference eigenmode (FDE) solver to simulate three specific types of waveguides to demonstrate e-skid in two directions [17].",
"Fig.",
"REF (a) shows a top view of a single-mode strip waveguide, where the propagation is in the $\\hat{x}$ direction.",
"Next to the strip waveguide, Fig.",
"REF (d) shows a 2D profile of the fundamental TE propagating mode.",
"We introduce the wave supressing e-skid features on one side of the waveguide in Fig.",
"REF (b) and show the corresponding 2D mode in (e).",
"Finally, we introduce the wave enhancing e-skid features in Fig.",
"REF (c) and the corresponding 2D mode in (f).",
"We compiled the cross sections of all three modes in Fig.",
"REF (g) to demonstrate the effect of the features on the evanescent wave of the mode.",
"Fig.",
"REF (g) clearly demonstrates, with a log-scale in $y$ , that the decaying wave outside of the center of the waveguide is suppressed by the parallel features, and greatly enhanced by the perpendicular features.",
"Figure: (a) A conventional directional coupler with a coupling region characterized by the gap between waveguides and the length of the parallel section.",
"(b) The e-skid platform discussed in , where the period (Λ ∥ \\Lambda _{\\parallel }) and silicon fill (a ∥ a_{\\parallel }) characterize the subwavelength features.",
"(c) Our directional coupler which leverages the enhancing e-skid features in the coupling region, where the features outside the coupling region are the same as (b) and where the period (Λ ⊥ \\Lambda _{\\bot }) and silicon fill (a ⊥ a_{\\bot }) characterize the subwavelength features in the coupling region.",
"(d) An example of an integrated photonic circuit implementing two-dimensional e-skid.",
"Note the circuit maintains the size reduction of e-skid coupled with large gap, bendless couplers.",
"Colors throughout indicate the photonic waveguides (blue), parallel (orange) and perpendicular (green) e-skid features (where all are the same material - namely silicon), and the base is the buried oxide (black)."
],
[
"Large Gap, Bendless Directional Coupler Design",
"We propose and demonstrate a new coupler that leverages e-skid in two directions to create coupling in desired regions.",
"In a traditional integrated photonic platform, a directional coupler is created by bending two waveguides close to each other.",
"(Fig.",
"REF (a)).",
"The waveguides must otherwise be kept far apart in other parts of a circuit in order to avoid unwanted coupling, limiting the circuit density.",
"By using e-skid with parallel subwavelength features (Fig.",
"REF (b)), that suppress coupling, we overcome this limitation and keep two waveguides within close proximity, with negligible coupling.",
"Furthermore, when the design with e-skid needs coupling, we show that the introduction of perpendicular subwavelength features in the coupling region, as are seen in Fig.",
"REF (c), will significantly enhance coupling.",
"These features have tunable variables (i) period ($\\Lambda _{\\bot }$ ) and (ii) fill factor ($\\rho _{\\bot }$ ) which directly tune the amount of coupling experienced.",
"We introduce two-dimensional e-skid as a way to leverage the size reduction offered by the parallel features with the addition of the perpendicular features to create practical circuits as seen in Fig.",
"REF (d)."
],
[
"Coupled Modes for ",
"The evanescent wave of the mode, even though it carries no power across the boundary, causes coupling between parallel guides if the overlap between the evanescent waves of supported modes is large enough [18].",
"From coupled mode theory [18], we define the power in the bar and cross ports as $P_{bar}(L) = P_0 \\operatorname{cos}^2\\left(\\kappa L\\right) = P_0 \\operatorname{cos}^2\\left(\\frac{\\pi }{2}\\frac{L}{L_x}\\right),$ $P_{cross}(L) =P_0 \\operatorname{sin}^2\\left(\\kappa L\\right) = P_0 \\operatorname{sin}^2\\left(\\frac{\\pi }{2}\\frac{L}{L_x}\\right),$ for $L$ as the coupling length, $P_0$ as the injected power, $\\kappa $ as the coupling coefficient, $L_x$ as the crossover length such that $\\kappa = \\pi /(2 L_x)$ , and bar and cross refer to the light remaining in the injected waveguide or transitioning to the other waveguide, respectively.",
"The crossover length is defined so that when $L = L_x$ , there is complete power transfer from waveguide one to two.",
"This approach allows for an intuitive understanding of the device.",
"The crossover length is given as $L_x = \\frac{\\lambda }{2(n_{even,\\lambda } - n_{odd,\\lambda })},$ where $\\lambda $ is the free space wavelength and $n_{even,\\lambda }, n_{odd,\\lambda }$ are the effective indices of the even and odd modes, respectively [1].",
"The field of the odd mode is antisymmetric across the coupling region and it remains generally unaffected by symmetric features there [19].",
"However, by introducing features into the coupling region the even mode is affected, thereby enabling dispersion engineering of the directional coupler - specifically, controlling the directional couplers' optical bandwidth.",
"[19].",
"By crafting $L_x$ , we can dictate how the device performs according to Eqs.",
"REF ,REF .",
"Essentially, if we make the slope of $L_x$ as flat as possible over a span of $\\lambda $ , we ensure a useful operating bandwidth (e.g.",
"a 3 dB coupler) is preserved for that span.",
"Our design is fundamentally different than [19] due to the structural asymmetry, which encourages coupling, and the higher fill factor.",
"These parameters allow us to create a directional coupler with more than an order of magnitude shorter crossover length in comparison, at the penalty of reduced operating bandwidth.",
"In order to demonstrate the effect of dispersion engineering, we begin by simulating the photonic bandstructure of the directional coupler in a full wave 3D frequency time frequency domain (FDTD) solver with Bloch-periodic boundary conditions [17].",
"Fig.",
"REF (a,b,c) shows the photonic bandstructure of a traditional, e-skid and large gap directional coupler.",
"These directional couplers are fundamentally different from photonic crystals as they are not designed to work in the band gap, instead these subwavelength features allow for low loss propagation through the periodic structures [15].",
"The traditional and e-skid couplers exhibit similar bandstructures, but the large gap coupler's even mode is approaching the band edge just above 200 THz (1500 nm).",
"Because the even mode is near the band edge, dispersion is increased, which allows for flexibility in tuning the behavior.",
"Figure: (a,b,c) The photonic bandstructures of the conventional, e-skid and large gap directional couplers from Fig.",
"(a,b,c), respectively.",
"(d,e,f) Extracted effective indices of the bandstructures from (a,b,c), respectively.",
"(g,h,i) The crossover length, calculated from Eq.",
".",
"The insets of (g,h,i) show the device diagrams for each type of coupler.These devices were all simulated with the same gap to illustrate the effects on the same scale.In practice, the gap is limited by (i) the fabrication process, (ii) the circuit application, and (iii) the length of the waveguides.",
"Therefore, the gap is often larger than shown here.The e-skid design parameters were: gap =270=270 nm, Λ ∥ =50\\Lambda _{\\parallel } = 50 nm, ρ ∥ =50%\\rho _{\\parallel } = 50\\%, Λ ⊥ =270\\Lambda _{\\bot } = 270 nm, ρ ⊥ =50%\\rho _{\\bot } = 50\\%, and W=400W = 400 nm.From the bandstructures, we extract the dispersive effective index of both the fundamental even and odd supermodes (Fig.",
"REF (d,e,f)).",
"The effective indices exhibit a similar characteristic shape to their corresponding bandstructures.",
"The e-skid coupler brings the even mode effective index much closer to the odd mode in comparison with the traditional coupler, and the large gap coupler has increased the difference between the two effective indices.",
"Fig.",
"REF (g,h,i) show the crossover length given the corresponding effective indices.",
"The traditional coupler and e-skid behave as expected, with an increase in crossover length for the latter.",
"The large gap coupler exhibits a dramatically reduced crossover length, and in relation to dispersion engineering, a completely different shape.",
"It is important to note that the large gap directional coupler does support a higher-order mode (Fig.",
"REF (c)), however the coupling efficiency extracted from a modal overlap integral between the fundamental and the first higher-order mode is $<8\\%$ over the wavelength span for some designs, according to our 3D FDTD simulations [17].",
"While $8\\%$ is not insignificant, tweaking our parameters (specifically $\\rho _{\\bot }$ and $\\Lambda _{\\bot }$ ) can reduce this coupling efficiency into higher-order modes, increasing device performance [19].",
"For this experiment, the primary design choices were dictated from a perspective of manufacturability.",
"In the future, small decreases to $\\rho _{\\bot }$ and $\\Lambda _{\\bot }$ will result in higher performing couplers verified by our 3D FDTD simulations [17] and prior work [19].",
"Additionally, careful consideration of tapering, which is not investigated in this work, can also mitigate the excitation of higher-order modes [15], [20]."
],
[
"Device Design",
"We investigate the effect of different parameter variations of the large gap coupler.",
"Fig.",
"REF shows the results of varying the fill factor, period and the gap between waveguides.",
"These parameter variations indicate the substantial tunability offered by two-directional e-skid.",
"First, we selected the operating gap between the two waveguides to be $1.44$ $\\mu $ m in order to stay consistent with the parallel e-skid features ($\\rho _{\\parallel } = 60\\%, \\Lambda _{\\parallel }=225$ nm, 6 layers deep results in a $1.44$ $\\mu $ m gap).",
"We designed these devices for manufacturing with the American Institute of Manufacturing (AIM) Photonics CMOS foundary Multi-Project Wafer (MPW) offering.",
"For a photolithographic process like this one, we must take in to account the limitations of the processing, like feature size.",
"For example, many prior work designs with features smaller than 60 nm would not resolve with CMOS processing compared to electron beam lithography.",
"We chose to design our devices with $\\Lambda _{\\bot } = 275$ nm to remain beneath the Bragg limit but maintain high manufacturing quality.",
"The parallel features were designed with $\\Lambda _{\\parallel } = 225$ nm.",
"It should be noted that the parallel cladding structures are less challenging for a lithographic system because they are lines, not holes [16].",
"We targeted $\\rho _{\\bot } = 60 \\%$ for the majority of our devices because we assumed that the features would be over etched, a common practice in SOI fabrication, so that the fill factor would decrease [21]."
],
[
"Experiment",
"Our experimental setup is shown in Fig.",
"REF (a).",
"We placed the chip on a mount in between two 3-axis stages with bare fiber on either side for coupling in and out.",
"For the input, we connected the fiber to a tunable laser source (TLS), and a polarization controller (PC) to ensure TE polarization and we measured the output signal with an optical power meter (OPM).",
"The fibers were edge coupled to the chip, which routed the light through strip/wire waveguides to the devices.",
"To transition from the strip waveguide mode to the e-skid's, we slowly introduced the parallel and perpendicular claddings as seen in Fig.",
"REF (c) and depicted in the schematic in Fig.",
"REF (a).",
"We were careful to design simple tapers because when periodic, assymetric features are introduced there is a chance for radiative losses [15].",
"In the future we will optimize these tapers to improve insertion loss, which were measured under 2 dB per device.",
"We collected the transmission spectra (an example is shown in Fig.",
"REF (b)).",
"The spectrum shows a characteristic “chirp”-like behavior, which is due to the dispersive nature of the cross-over length.",
"This is seen in Fig.",
"REF where, based on the perpendicular e-skid parameters of the device, the cross-over length can exhibit a significant change with wavelength.",
"This will result in a quickly oscillating output (given by Eqs.",
"(4) and (5)).",
"With that said, using the experimental measurements in conjunction with the previously described models we were able to extract the parametric dependence of the coupler designs."
],
[
"Parameter Extraction Method",
"After extracting the transmission spectra from our fabricated devices, we used a dispersive model for extracting the cross-over length from the data.",
"Because inverse sine functions are multi-valued, we can not obtain $L_x$ directly from Eqs.",
"REF or REF .",
"We instead employed the behavioral model for characterizing directional couplers [22].",
"For the cross-over length, we are interested in the wavelength dependence, so we prepared the data by filtering the noise and normalizing the bar and cross measurements.",
"We used a polynomial expansion of the coupling coefficient coupled with a non-linear least squares (NLS) optimization algorithm to find the best fit for $L_x$ [17], [23], [24].",
"Because many of our devices exhibits a strong “chirp-like” behaviour (Fig.",
"REF (b)), we used a third-order polynomial expression for $L_x$ to determine the best fit, such that $L_x(\\lambda ) = L_{x,0} + L_{x,1}\\lambda + L_{x,2}\\lambda ^2 + L_{x,3}\\lambda ^3,$ where the curve is characterized by fitting parameters $L_{x,0},L_{x,1},L_{x,2},L_{x,3}$ pertaining to the wavelength, $\\lambda $ .",
"The NLS optimization aimed to minimize the difference between the measured and theoretical spectra by adjusting the fit parameters in Eq.",
"REF ."
],
[
"Experimental Results",
"In Fig.",
"REF (a), we show the devices dependence on fill factor variation.",
"Fill factors up to 60% were successfully fabricated in the CMOS process and will be reported below, while fabrication specific optimization needs to go into higher fill factor devices.",
"The fill factor variations resemble those from the simulations (Fig.",
"REF (a)), where higher fill factors increased $L_x$ .",
"In Fig.",
"REF (b) we investigated coupler-length variation for a fixed fill factor (60%) and gap ($1.44$ $\\mu $ m).",
"We fit nine different directional couplers with the exact same parameters changing only the coupling length.",
"According to Eqs.",
"REF and REF , the length is independent of the coupling coefficient, $\\kappa $ and therefore these couplers should exhibit identical $L_x$ measurements.",
"There are manufacturing variations, measurement errors, and fitting errors that reveal themselves in Fig.",
"REF (b).",
"The inset shows a 95 % confidence interval for these nine couplers' $L_x$ extraction and showcase expected similar behavior for all of these couplers.",
"This truly highlights the utility of these devices, as they are able to couple fully in $\\le 50$ $\\mu $ m, even with a large gap of $1.44$ $\\mu $ m. Finally, Fig.",
"REF (c) shows the $L_x$ extraction for varying gaps.",
"Even though there is a qualitative match between the simulations and experimental results, manufacturing variations account for the quantitative differences."
],
[
"Future Work",
"In this work we performed a comprehensive study of the parameter space of the large gap directional coupler using two-dimensional e-skid.",
"In the future, it will be desirable to realize devices with particular performance characteristics.",
"From our experimental data (Fig.",
"REF ), we extracted that we are able to realize a large gap, bendless coupler that achieves 100% coupling ($100/0$ splitting ratio) with a coupling length of $L=50$ $\\mu $ m as seen in Fig REF (a), using $\\rho _{\\bot } = 60\\%,\\ \\Lambda _{\\bot } = 275 $ nm and coupling gap of $1.44$ $\\mu $ m. We can take this to design a $50/50$ directional coupler.",
"Fig.",
"REF (b) shows the theoretical transmission spectrum of this device.",
"We set the coupling length $L = \\operatorname{avg}(L_x)/2 = 23.5$ $\\mu $ m, and we see broadband behaviour of nearly 40 nm.",
"Additionally, we can slightly vary parameters to tune the device to a more desirable center wavelength.",
"For example, by reducing the period of the large gap directional coupler to $\\Lambda _{\\bot } = 255$ nm, $\\rho _{\\bot } = 50\\%$ , and coupling gap of $1.44$ $\\mu $ m, the device's operating band can now be centered closer to 1.55 $\\mu $ m and achieves an even larger operating bandwidth of $>40$ nm (Fig.",
"REF (c)), which is sufficient for many applications."
],
[
"Conclusion",
"We introduced the deterministic, targeted control of the evanescent wave in strip waveguides by employing e-skid features in two directions.",
"We designed and demonstrated large gap, bendless directional couplers through a CMOS photonic chip fabricated by AIM Photonics.",
"All of the results were compared to simulations by extracting design parmaters using a NLS optimization technique coupled with a behavioral model of the directional coupler [22].",
"With the parameter extraction, we show experimentally that e-skid waveguides in general and the large gap, bendless directional coupler in particular are possible to realize in a CMOS platform.",
"Moving forward, we can design large gap (>1$\\mu $ m) directional couplers that operate with bandwidths over 50 nm."
],
[
"Appendix A: Verifying Mode Conversion Efficiency of 1D ",
"We verify the conversion efficiency between strip waveguides and e-skid waveguides by comparing the loss coefficients at the resonances of a racetrack resonator [25], [26].",
"We designed three strip waveguide racetrack resonators, Figure REF (a), with varying gaps, and then the exact same racetrack resonators in which we add two e-skid features into the ring, Figure REF (c).",
"The intent of this experiment is to quantify the additional loss created by these features, and thereby measure the mode conversion efficiency between strip and e-skid waveguides [7].",
"A simple ring resonator (shown in Fig.",
"REF (a)) can be paramaterized by two coefficients, $\\tau $ , the self-coupling coefficient that indicates how much light goes through the coupler, and $\\alpha $ , the loss coefficient which indicates how much light is lost into the ring.",
"We extract $\\alpha $ and $\\tau $ according to the method described by [25], such that F FSRFWHM, E TMAXTMIN, A = cos(/F)1+sin(/F), B = 1-(1-cos(/F)1+cos(/F))1E, (,) = [AB]1/2[AB-A]1/2.",
"The finesse, $\\mathcal {F}$ , is defined as the ratio between the free spectral range, $\\Delta \\lambda _{\\text{FSR}}$ , and the full width at half maximum, $\\Delta \\lambda _{\\text{FWHM}}$ , of each resonance.",
"The exctinction ratio, $\\mathcal {E}$ , is defined as the ratio between the transmission maximum, $T_{\\text{MAX}}$ , off resonance and the minimum, $T_{\\text{MIN}}$ , at each resonance.",
"We can decouple $\\alpha ,\\tau $ in Eq.",
"REF using the method further discussed in [25].",
"When we determine $\\alpha $ , we know that $\\alpha ^2$ indicates the percentage lost into the ring, which allows us to compare these two different resonators.",
"This resulted in average values of $\\alpha ^2 = 97.2\\%$ and $\\alpha ^2 = 95.7\\%$ for the two ring types (Fig.",
"REF (a,c)), respectively.",
"This leads to a mode conversion efficiency of $99.6$ %.",
"National Science Foundation (Award# 1810282), Air Force Research Laboratory (FA8650-15-2-5220 & FA8750-16-2-0140) This material is based upon work supported by the National Science Foundation under Grant No.",
"1810282 and AFRL awards (FA8650-15-2-5220 & FA8750-16-2-0140).",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.",
"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.",
"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government.",
"M.v.N.",
"and S.F.P would like to acknowledge Navin B. Lingaraju for sparking a collaboration between RIT and Purdue.",
"The authors declare that there are no conflicts of interest related to this article."
]
] | 2005.14265 |
[
[
"Using nonlocal surface transport to identify the axion insulator"
],
[
"Abstract The axion is a hypothetical but experimentally undetected particle.",
"Recently, the antiferromagnetic topological insulator MnBi$_2$Te$_4$ has been predicted to host the axion insulator, but the experimental evidence remains elusive.",
"Specifically, the axion insulator is believed to carry \"half-quantized\" chiral currents running antiparallel on its top and bottom surfaces.",
"However, it is challenging to measure precisely the half-quantization.",
"Here, we propose a nonlocal surface transport device, in which the axion insulator can be distinguished from normal insulators without a precise measurement of the half-quantization.",
"More importantly, we show that the nonlocal surface transport, as a qualitative measurement, is robust in realistic situations when the gapless side surfaces and disorder come to play.",
"Moreover, thick electrodes can be used in the device of MnBi$_2$Te$_4$ thick films, enhancing the feasibility of the surface measurements.",
"This proposal will be insightful for the search of the axion insulator and axion in topological matter."
],
[
"Using nonlocal surface transport to identify the axion insulator Rui Chen Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China School of Physics, Southeast University, Nanjing 211189, China Shuai Li Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China Hai-Peng Sun Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China Yue Zhao Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China Hai-Zhou Lu Corresponding author: [email protected] Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China X. C. Xie International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Beijing Academy of Quantum Information Sciences, Beijing 100193, China CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China The axion is a hypothetical but experimentally undetected particle.",
"Recently, the antiferromagnetic topological insulator MnBi$_2$ Te$_4$ has been predicted to host the axion insulator, but the experimental evidence remains elusive.",
"Specifically, the axion insulator is believed to carry “half-quantized\" chiral hinge currents running antiparallel on its top and bottom hinges.",
"However, it is challenging to measure precisely the half-quantization.",
"Here, we propose a nonlocal surface transport device, in which the axion insulator can be distinguished from normal insulators without a precise measurement of the half-quantization.",
"More importantly, we show that the nonlocal surface transport, as a qualitative measurement, is robust in realistic situations when the gapless side surfaces and disorder come to play.",
"Moreover, thick electrodes can be used in the device of MnBi$_2$ Te$_4$ thick films, enhancing the feasibility of the surface measurements.",
"This proposal will be insightful for the search of the axion insulator and axion in topological matter.",
"Introduction.– The axion is an elementary particle postulated to resolve the strong CP problem in quantum chromodynamics, but remains invisible in experiments [1].",
"In recent years, the axion insulator in condensed matter physics has attracted great attention because it shares the axionic electromagnetic response [2], [3], [4], [5], [6], [7], which modifies Maxwell's equations and may lead to a “half-quantized\" surface Hall conductance [8], [9], [10], [11] or the topological magnetoelectric effect [12], [13], [14], [15], [16], [17].",
"Unfortunately, these signatures require highly demanding precisions of measurement and device fabrication, so the experimental attempts to identify the axion insulator have yet to be successful [6], [5] (see also Sec.",
"IX of [18]).",
"Recently, in the experiments of the first antiferromagnetic topological insulator MnBi$_2$ Te$_4$ [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], , , , , , , , the signatures for the quantized anomalous Hall effect and axion insulator have been reported [28], [29].",
"Specifically, the axion (Chern) insulator has antiparallel (parallel) chiral hinge currents on the top and bottom surfaces.",
"Each hinge current carries a $e^2/2h$ Hall conductance.",
"The top and bottom hinge currents combine to yield a zero ($e^2/h$ ) Hall conductance as a signature for the axion (Chern) insulator [Figs.",
"REF (a)-(b)].",
"However, normal insulators also have a zero Hall conductance [Fig.",
"REF (c)].",
"Therefore, seeking signatures for the axion insulator is still an open problem.",
"Figure: For the antiferromagnetic topological insulator MnBi 2 _2Te 4 _4, (a) odd septuple layers host a Chern insulator with parallel “half-quantized\" chiral hinge currents on the top and bottom hinges, and (b) even septuple layers may support the axion insulator with opposite chiral hinge currents.",
"(c) The normal insulator has no chiral hinge currents.",
"(d-e) The proposed device for nonlocal surface measurements.",
"Red, blue, green, and grey correspond to the top and bottom electrodes, insulating shield, and antiferromagnetic topological insulator, respectively.",
"n t n_t is the thickness of the thick electrodes.In this Letter, we propose that the axion insulator can be distinguished from normal insulators by measuring nonlocal surface resistances in a device of the antiferromagnetic topological insulator MnBi$_2$ Te$_4$ [Fig.",
"REF (d)].",
"MnBi$_2$ Te$_4$ is formed by stacked septuple layers with out-of-plane magnetization.",
"The neighbouring septuple layers have opposite magnetizations.",
"For even (odd) layers, the chiral hinge currents on the top and bottom surfaces propagate along the opposite (same) directions [Figs.",
"REF (a)-(b)], forming an axion insulator (a thick Chern insulator).",
"Because of the chiral hinge currents, the axion insulator shows distinct nonlocal surface transport, compared to those in the Chern insulator and normal insulator.",
"We numerically calculate the nonlocal surface transport of MnBi$_2$ Te$_4$ with different septuple layers by using a lattice model and realistic parameters.",
"More importantly, we show that the nonlocal surface transport, as a qualitative measurement, is robust against disorder andside-surface transport.",
"Moreover, we show that thick films of MnBi$_2$ Te$_4$ allow thick electrodes to measure the surface transport [Fig.",
"REF (e)].",
"Therefore, the nonlocal surface transport provides a feasible approach to identify the axion insulator.",
"Our results will be helpful for the ongoing and future search for the axion insulator.",
"Figure: The numerically calculated nonlocal surface resistances defined as R=V/IR=V/I in the insets.This figure presents only ideal cases for demonstration.",
"The side surface, disorder, and thick electrodes are considered in Figs.",
", , and , respectively.",
"The red line and blue circle (red circle and blue line) correspond to the top and bottom surface of an even (odd) n z n_z, respectively.Nonlocal surface transport.– The axion insulator is characterized by the opposite “half-quantized\" chiral hinge currents [Fig.",
"REF (b)] due to its axion electromagnetic response.",
"Detecting the chiral hinge current can help to identify the axion insulator.",
"Compared to the ultrathin films of magnetically-doped Bi$_{2}$ Se$_3$ -like materials, bulk crystals of intrinsic antiferromagnetic insulator MnBi$_2$ Te$_4$ can be used directly for measurements, and allow electrodes to probe mostly the top or bottom surface, respectively, as shown in Fig.",
"REF (d).",
"Usually, a direct probe of the chiral hinge currents is prohibited by the side surfaces, which have metallic surface states that bury the signals of the chiral hinge currents.",
"To solve this problem, we introduce the nonlocal surface measurements.",
"The metallic states on the side surfaces are not chiral, that is, they go opposite directions with the same weight, thus do not play a significant role in the nonlocal transport compared to the chiral hinge currents.",
"Specifically, the top (bottom) surface of the device has six electrodes, 1-6 ($1^{\\prime }$ -$6^{\\prime }$ ) [Fig.",
"REF (d)].",
"As shown in Fig.",
"REF , a current $I$ is applied between electrodes 1 and 2 ($1^{\\prime }$ and $2^{\\prime }$ ) on the top (bottom) surface, and a nonlocal voltage $V$ is measured between other two electrodes on the same surface, to define a nonlocal resistance $R=V/I$ .",
"For a given surface, assuming that there are $n_L$ clockwise and $n_R$ anticlockwise chiral hinge currents ($n_L$ and $n_R$ may not be integers), the nonlocal resistances can be analytically found for the 4 configurations in Fig.",
"REF as $R^{12,26} &=&\\frac{h}{e^{2}}\\frac{n_{L}F_{2}}{F_{4}+G_{2}},R^{12,31}=\\frac{h}{e^{2}}\\frac{n_{R}F_{2}}{F_{4}+G_{2}}, \\\\R^{12,23} &=&\\frac{h}{e^{2}}\\frac{n_{L}^{4}}{F_{1}\\left( F_{4}+G_{2}\\right) },R^{12,36}=\\frac{h}{e^{2}}\\frac{n_{L}n_{R}}{F_{3}}$ with $F_{i}=n_{L}^{i}+n_{R}^{i}$ and $G_{i}=n_{L}^{i}n_{R}^{i}$ (details and more configurations can be found in Sec.",
"SII of [18]).",
"Before considering realistic situations with side surfaces, disorder, and thick electrodes, we first use ideal cases for illustration.",
"For odd layers, the half-quantized chiral hinge currents on the top and bottom surfaces are parallel, i.e., $n_L=0$ and $n_R=1/2$ for both top and bottom surfaces.",
"For even layers, the half-quantized chiral hinge currents are opposite, i.e., $n_L=0$ and $n_R=1/2$ on the top and $n_L=1/2$ and $n_R=0$ on the bottom surface.",
"Table REF shows the the ideal nonlocal resistances for the configurations in Fig.",
"REF .",
"For odd layers, the two surfaces always have the same nonlocal resistance.",
"For even layers, the nonlocal resistances on the two surfaces have different values for most of the cases, except for the case with $p,q=3,6$ .",
"Therefore, the axion insulator in even layers and Chern insulator in odd layers would show distinct nonlocal surface transport properties.",
"More importantly, the normal insulator has no such nonlocal resistance because it has no chiral hinge current (see Fig.",
"REF (c) and Sec.",
"SVI of [18]).",
"Therefore, the unique nonlocal resistance can be used to distinguish the axion insulator from normal insulators.",
"Above are the ideal cases.",
"In real materials with side surfaces, disorder, and thick electrodes, the chiral hinge currents $n_L$ and $n_R$ are not perfectly half-quantized.",
"Below we will verify the above proposal by using a realistic model and simulations.",
"Table: The ideal resistances R e/o 1 (') 2 (') ,p (') q (') R_{\\text{e/o}}^{1^{(\\prime )}2^{(\\prime )},p^{(\\prime )}q^{(\\prime )}} obtained from the analytical Eq.",
"() for the configurations in Fig.",
".Model.– For numerical calculations, we use an effective model for MnBi$_2$ Te$_4$ regularized on a stacked hexagonal lattice $H=\\begin{pmatrix}h +\\mathbf {M}_{A}\\cdot s\\otimes \\sigma _{0}& h_{AB} \\\\h_{AB}^{\\dag } & h+\\mathbf {M}_{B}\\cdot s\\otimes \\sigma _{0}\\end{pmatrix},$ with the intra-layer part $h $ = $[ \\tilde{C}$ - $(4/3)C_{2} $ $( \\cos k_{1}$ + $\\cos k_{2} $ + $\\cos k_{3} ) ] $ $ I_{4} $ +$(v/3) $ $ ( 2\\sin k_{1} $ + $\\sin k_{2}+\\sin k_{3}) $ $\\Gamma _{1} $ +$(v/\\sqrt{3}) $ $ ( \\sin k_{2}-\\sin k_{3}) $ $\\Gamma _{2}$ +$[ \\tilde{M}$ - $(4/3)M_{2} ( \\cos k_{1}+\\cos k_{2}+\\cos k_{3}) ]$$ \\Gamma _{4}$ , the inter-layer part $h_{AB}$ $=-2C_{1}\\cos k_{z}I_{4}$ + $v_{z}\\sin k_{z}\\Gamma _{3}-2M_{1}\\cos k_{z}$$ \\Gamma _{4}$ .",
"The lattice constants $\\mathbf {a}_{1}=\\left( 1,0,0\\right), \\mathbf {a}_{2}=\\left( 1,\\sqrt{3},0\\right)/2, \\mathbf {a}_{3}=\\left(0,0,1\\right)$ .",
"The wave vectors $k_{1}=k_{x}$ , $k_{2}=\\left( k_{x}+\\sqrt{3}k_{y}\\right)/2 \\,$ , $k_{3}=k_{1}-k_{2}=\\left(k_{x}-\\sqrt{3}k_{y}\\right)/2 $ .",
"$I$ is the identity matrix, $\\Gamma _{i}=s_{i}\\otimes \\sigma _{1}$ for $i=1,2,3$ , $\\Gamma _{4}=s_{0}\\otimes \\sigma _{3}$ , $\\tilde{M}=M_0+2M_1+4M_2$ , and $\\tilde{C}=C_0+2C_1+4C_2$ .",
"$s_{i}$ and $\\sigma _{i}$ are the Pauli matrices.",
"$\\mathbf {M}_{k}=m\\left( \\cos \\phi _{k}\\sin \\theta _{k},\\sin \\phi _{k}\\sin \\theta _{k},\\cos \\theta _{k}\\right) $ with angles $\\phi _{k}$ and $\\theta _{k}$ for $k=A,B$ .",
"The antiferromagnetic order is described by $\\left( \\phi _{A},\\theta _{A}\\right) =\\left(0,0\\right) $ and $\\left( \\phi _{B},\\theta _{B}\\right) =\\left( 0,\\pi \\right) $ .",
"For a realistic simulation, we use the parameters transformed from those in the $k\\cdot p$ model [24], [18], $C_{0}=C_{1}=C_{2}=0$ , $M_{0}=-0.1165$ eV, $M_{1}=11.9048$ eVÅ$^2$ , $M_{2}=9.4048$ eVÅ$^2$ , $v_{z}=2.7023$ eVÅ, $v=3.1964$ eVÅ, and $m=0.1$ eV, which describes the A-type antiferromagnetic topological insulator in MnBi$_2$ Te$_4$ .",
"The model describes a topological insulator with two gapped surface states on the top and bottom surfaces (Fig.",
"S1(b) of [18]), due to breaking of the combined antiferromagnetic time-reversal symmetry $\\Theta _M$ .",
"The gapped surfaces behave like “marginal\" 2D Chern insulators, characterized by “half-quantized\" chiral hinge currents (Fig.",
"REF ), in a sense that their Chern numbers are half-quantized as $\\pm 1/2$ , or in other words the Hall conductance $\\sigma _{xy}= \\pm e^2/2h$ [see, e.g., Fig.",
"REF (a) as $n_z\\rightarrow 200$ ].",
"The sign $\\pm $ depends on the magnetization, which reverses from one septuple layer to another in MnBi$_2$ Te$_4$ [9], [25].",
"For odd (even) layers, the top and bottom surfaces have the same (opposite) magnetization, so the half-quantized surface Hall conductances of the top and bottom surfaces add up (cancel) to give a Chern insulator characterized by $C=1$ (axion insulator with $C=0$ ) [24], [25], [26], [27], [28], [29], [30], , , , , (see also Fig.",
"S1(c) of [18]).",
"The chiral hinge currents, which are responsible for the half-quantized surface Hall conductance, can be numerically verified using first-principles calculations for MnBi$_2$ Te$_4$ .",
"Calculation of the nonlocal surface transport.–We calculate the nonlocal surface transport by using the Landauer-Büttiker-Fisher-Lee formula , , and the recursive Green's function method , .",
"At zero temperature, the current flowing into electrode $p$ is given by $I_p &=& \\frac{e^2}{h} \\sum _{q\\ne p} T_{pq}(E_F)(V_p-V_q),$ where $V_{p}$ is the voltage at electrode $p$ , $T_{pq}$ is the transmission coefficient from electrode $q$ to $p$ .",
"For the present case, there are 6 electrodes for each of top and bottom surfaces, so $T_{pq}$ is a 6$\\times $ 6 matrix (more details can be found in Sec.",
"SII and Sec.",
"SVI of [18]).",
"Figure REF shows the numerically calculated nonlocal resistances for 4 configurations as functions of film thickness (number of septuple layers $n_z$ ).",
"In odd layers, the resistances have the same values for the top and bottom surfaces.",
"In even layers, the resistances have distinct values for the two surfaces, except for those in Fig.",
"REF (d).",
"For thick films in Fig.",
"REF , there are deviations from the analytic values in Tab.",
"REF .",
"This is due to the gapless side surface states , and is one of the reasons why it is hard to measure the half-quantized Hall effect as the signature for the axion insulator.",
"Later, we will show the nonlocal surface transport, as a qualitative measurement, is robust against not only the side-surface transport but also disorder and thick electrodes.",
"Figure: (a-b) The surface Hall conductance on the top and bottom surface as functions of the number of septuple layers n z n_z with the Fermi energy E F =1E_F=1 meV and E F =10E_F=10 meV, respectively.",
"The inset in (a) shows the subbands of the side surface in the yy-zz plane.",
"The green lines indicate E F =1E_F=1 and 10 meV.",
"(c-f) The numerically calculated nonlocal surface resistancesfor E F =10E_F=10 meV.",
"The red lines and blue circles (red circles and blue lines) correspond to the top and bottom surface of an even (odd) n z n_z, respectively.",
"See the results for other configurations in Sec.",
"SIV of .Side-surface effects.– Figures REF (a-b) show the numerical results of the surface Hall conductance for different $E_F$ , which can be found from the left-chiral and right-chiral hinge currents, e.g., $e^2/h(T^{12}_{\\text{e}}-T^{21}_{\\text{e}})$ for the top surface in even layers.",
"For the ideal case, the top and bottom surface Hall conductances are half-quantized ($\\pm e^2/2h$ ) for thick samples ($n_z\\rightarrow 200$ ), and reduce to 0 and $e^2/h$ as $n_z\\rightarrow 0$ for even and odd layers, respectively, as a result of the top-bottom hybridization , , , , , , , , .",
"A zoom-in of Figure REF (a) shows that the surface Hall conductances are not precisely half-quantized because of the side surface states when the Fermi energy is away from the Dirac point.",
"This shows that a direct measurement of the half-quantized Hall effect is challenging, in accordance with the previous studies .",
"Figure REF (b) shows that the side surface states turn the surface Hall conductance into an oscillation with $n_z$ at higher Fermi energies ($E_F=10$ meV), as a result of the confinement-induced subbands of the side surface states [inset of Fig.",
"REF (a)].",
"Similar oscillations also present in the nonlocal surface transport in Figs.",
"REF (c-f).",
"Figure: The nonlocal surface resistances as functions of the disorder strength UU with E F =10E_F=10 meV.",
"The error bars show the standard deviation of the conductance for 100 samples.Red line and blue circle (red circle and blue line) correspond to the top and bottom surface of an even (odd) n z n_z, respectively.",
"See results for other configurations in Sec.",
"SIV of .Figure: Schematic illustration of the 3D device with thick electrodes.",
"Here the thickness of the electrodes is n t n_t.",
"(c-e) show the numerically calculated nonlocal surface resistances as functions of n t n_t with E F =10E_F=10 meV and U=100U=100 meV.",
"The error bars show the standard deviation of the conductance for 100 samples.Red line and blue circle (red circle and blue line) correspond to the top and bottom surface of an even (odd) n z n_z, respectively.",
"See more details in Sec.",
"SVIII of .Disorder effects.",
"– Now, we show that the nonlocal surface transport is robust against disorder.",
"We introduce the Anderson-type disorder to the central scattering region with $\\Delta H=\\sum _{i}{W_{i}\\sigma _{0}c_{i }^{\\dag }c_{i }}$ , where $W_{i}$ is uniformly distributed within $\\left[-U/2,U/2\\right]$ , with $U$ being the disorder strength.",
"In Fig.",
"REF , we show the nonlocal surface resistance as functions of the disorder strength $U$ for $n_z=200$ and with the side surfaces ($E_F=10$ meV).",
"Even for strong disorder ($U=0.2$ eV, about the size of the bulk band gap), the nonlocal resistances remain at the same order of magnitude.",
"Thick electrodes.",
"– Above, we assume that the electrodes are only attached to the topmost and bottommost septuple layers of the device.",
"Figure REF shows the numerical results for thick electrodes [Fig.",
"REF (e)].",
"We consider a realistic situation in which the side surface states ($E_F=10$ meV) and disorder ($U=100$ meV) are also taken into account.",
"The nonlocal resistances drop with increasing electrode thickness ($n_t$ ), but stay at the same order of magnitude up to 10 septuples (about 14 nm, thicker than the thinnest metal electrode that can be fabricated).",
"The chiral nature of the surface transport for even and odd layers are still stable with increasing the electrode thickness, showing the topological nature of the chiral hinge currents.",
"Experimental realization.– Finally, we discuss how to realize the nonlocal surface measurement in Fig.",
"REF (d) with the help of the existing nanotechnologies.",
"We propose the following fabrication process flow.",
"The bottom graphite electrodes can be prepatterned by electron beam lithography and plasma etching on Si/SiO$_2$ wafers.",
"To ensure the contact, the polymer residues on the graphite electrodes need to be removed by forming gas annealing in Ar/H$_2$ environment.",
"Thin flakes of MnBi$_2$ Te$_4$ are mechanically exfoliated.",
"Flakes with preferred thickness (e.g., 200 nm) and flatness can be selected by atomic force microscopy, then laminated to the prepatterned bottom electrodes via polymer-assisted dry transfer technique.",
"To electrically separate the bottom electrodes and avoid contact on the side edges of MnBi$_2$ Te$_4$ , lithography free contact in Ref.",
"can be used.",
"Such electrodes can be thin and flat so that it is non-invasive for probing just the surface septuple layer.",
"The process involving MnBi$_2$ Te$_4$ thin flakes has to be protected by an inert gas environment such as a glove-box, as surface reconstructions may occur .",
"Therefore, all of these nanotechnologies have been there, offering the possibility of detecting the axion insulator.",
"We thank helpful discussions with Qihang Liu, Jianpeng Liu, Chuizhen Chen, Hua Jiang, Donghui Xu and Bin Zhou.",
"This work was supported by the National Natural Science Foundation of China (11534001, 11974249, 11925402), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No.",
"XDB28000000), Guangdong province (2016ZT06D348), the National Key R & D Program (2016YFA0301700), the Natural Science Foundation of Shanghai (Grant No.",
"19ZR1437300), Shenzhen High-level Special Fund (No.",
"G02206304, G02206404), and the Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20190902092905285, ZDSYS20170303165926217, JCYJ20170412152620376, KYTDPT20181011104202253).",
"R.C.",
"acknowledges support from the project funded by the China Postdoctoral Science Foundation (Grant No.",
"2019M661678).",
"The numerical calculations were supported by Center for Computational Science and Engineering of Southern University of Science and Technology."
]
] | 2005.14074 |
[
[
"Unified framework for Early Dark Energy from $\\alpha$-attractors"
],
[
"Abstract One of the most appealing approaches to ease the Hubble tension is the inclusion of an early dark energy (EDE) component that adds energy to the Universe in a narrow redshift window around the time of recombination and dilutes faster than radiation afterwards.",
"In this paper, we analyze EDE in the framework of $\\alpha$-attractor models.",
"As well known, the success in alleviating the Hubble tension crucially depends on the shape of the energy injection.",
"We show how different types of energy injections can be easily obtained, thanks to the freedom in choosing the functional form of the potential inspired by $\\alpha$-attractor models.",
"To confirm our intuition we perform an MCMC analysis for three representative cases and find indeed that $H_0$ is significantly larger than in $\\Lambda$CDM like in other EDE models.",
"Unlike axion-driven EDE models with super Planckian decay constant, the curvature of EDE models required by the data is natural in the context of recent theoretical developments in $\\alpha$-attractors."
],
[
"Introduction",
"Recent low-redshifts distance-ladder measurements suggest a larger Hubble constant $H_0$ than the one determined from cosmic microwave background (CMB) data [1].",
"The value of $H_0$ inferred from the latest Planck 2018 data, $H_0 = (67.36 \\pm 0.54)$ km s$^{-1}$ Mpc$^{-1}$ [2], appears to be in a 4.4$\\sigma $ tension with the most recent distance-ladder measurement from the SH0ES team [3], $H_0 = (74.03 \\pm 1.42)$ km s$^{-1}$ Mpc$^{-1}$ which is determined by using type Ia supernovae (SNe Ia) as standard candles [4].",
"Other low-redshift methods to determine $H_0$ , such as from strong-lensing time delay [5] or from calibrating SNe Ia by the tip of the red giant branch [6], [7], [8], also point to a higher $H_0$ derived from the CMB.",
"In absence of unknown systematics, new physics seems necessary to solve this $H_0$ tension [9].",
"A common approach to model building consists of increasing the expansion rate at redshifts around matter-radiation equality in order to shrink the comoving sound horizon at baryon drag $r_s$ , which results in a higher $H_0$ inferred from CMBFor attempts to late time solutions, see e.g.",
"Refs.",
"[10], [11], [12], [13] [14], [15].",
"The addition of light relics is a typical example of physics that helps ease the tension by changing the early-time dynamics of the Universe [16], [17], [18], [19], [20].",
"Modified gravity models also lead to interesting solutions to the $H_0$ tension [21], [22], [23], [24], [25], [26], [27], [28], [29].",
"However, given the success of the $\\Lambda $ Cold Dark Matter ($\\Lambda $ CDM) concordance model in fitting CMB anisotropies, the early time deviation from it must be minimal.",
"To this end, the Early Dark Energy (EDE) scenario is perhaps the most minimal modification to the $\\Lambda $ CDM background dynamics that substantially alleviates the $H_0$ tension.",
"In this model, first proposed in [30], a very light scalar field $\\phi $ is frozen by Hubble friction during the radiation era, acting as DE with an equation of state $w_{\\rm EDE}\\equiv P_{\\rm EDE}/\\rho _{\\rm EDE}=-1$ and contributing negligibly to the energy budget of the Universe.",
"Eventually, when the Hubble rate becomes smaller than its effective mass $\\partial ^2 V(\\phi )/\\partial \\phi ^2$ , the scalar field quickly rolls down its potential and oscillates around its minimum, its energy diluting faster than radiation.",
"This results in a very sharp energy injection into the cosmic fluid, that for a suitable value of the mass of $\\phi $ occurs around the epoch of matter-radiation equality equivalence, successfully lowering $r_s$ .",
"Since the seminal work Ref.",
"[30], a substantial effort has been made in building new models of EDE [31], [32], [33], [34], [35], [36], [37], [38], [39], [40] and testing their predictions against larger datasets [41], [42].",
"In this work, we consider EDE in the framework of $\\alpha $ -attractors [43], [44], [45], in which the potential for the EDE scalar field is given by $V(\\phi )= f^2\\left[\\tanh (\\phi /\\sqrt{6 \\alpha } M_\\textup {pl})\\right].$ This potential arises naturally by turning a non-canonical kinetic pole-like term of the form $[\\alpha /(1 - \\varphi ^2/6 M_\\textup {pl}^2)^2] (\\partial \\varphi )^2/2$ into a canonical one, that is of the form $(\\partial \\varphi )^2/2$ .",
"Through the field redefinition $\\phi = \\sqrt{6 \\alpha } M_\\textup {pl} \\tanh ^{-1}(\\varphi /\\sqrt{6}M_\\textup {pl})$ , the kinetic term becomes canonical and the potential acquires the $\\tanh (\\phi /\\sqrt{\\alpha }M_\\textup {pl})$ dependence.",
"Due to this field redefinition, the potential flattens to a plateau at large values of $\\phi $ .",
"$\\alpha $ -attractor models were first introduced in the context of inflation, with predictions for the spectral index $n_s$ and tensor-to-scalar ratio $r$ , largely independent of the specific functional form of $V(\\phi )$ , hence the name “attractors\".",
"In the context of dark energy, $\\alpha $ -attractor models with an energy scale far below the one used in inflation were considered in [46], [47], [48].",
"An interesting connection between dark energy and inflation for $\\alpha $ -attractor models has also been investigated in [49], [50].",
"In our EDE proposal, however, the shape of the potential away from the plateau and around its minimum is crucial, as it regulates the shape of the energy injection.",
"One of the attractive features of $\\alpha $ -attractor models with the potential in Eq.",
"(REF ), is that they can easily accommodate various types of energy injection.",
"Indeed, we will show that, depending on the functional form of $V(\\phi )$ , a smooth or oscillating energy injection can be produced, reproducing results of representative earlier works in the field in a single framework [30], [31], [33].",
"This paper is organized as follows.",
"In Sec.",
", we describe the background evolution of the model and compare it to existing EDE models, focusing on the shape of the energy injection.",
"We confirm the capability of our model to alleviate the $H_0$ tension by performing an MCMC analysis in Sec.",
"and comment on our results in Sec. .",
"We end in the conclusions Sec.",
"."
],
[
"Background evolution and energy injection",
" Our model is described by the following Lagrangian $\\mathcal {L}=\\sqrt{-g}\\left[\\frac{M_\\textup {pl}^2}{2}R-\\frac{\\left(\\partial \\phi \\right)^2}{2}-V(\\phi )\\right] + \\mathcal {L}_m,$ where $\\mathcal {L}_m$ is the Lagrangian for matter (including baryons, CDM, photons and neutrinos) and the potential is: $V(\\phi )=\\Lambda +V_0\\frac{(1+\\beta )^{2n} \\tanh \\left(\\phi /\\sqrt{6\\alpha }M_\\textup {pl}\\right)^{2 p}}{\\left[1+ \\beta \\tanh \\left(\\phi /\\sqrt{6\\alpha }M_\\textup {pl}\\right)\\right]^{2n}} \\,,$ where $V_0,\\,p,\\,n$ , $\\alpha $ and $\\beta $ are constants.",
"The potential corresponds to the simple form $V(x) = \\Lambda +\\tilde{V}_0 x^{2 p}/(1+\\tilde{\\beta }x)^{2 n}$ for the field $x=\\phi /\\sqrt{6\\alpha }M_\\textup {pl}$ with a pole-like kinetic term and has an offset with respect to Refs.",
"[46], [47]Note that in our setting the field rolls towards the minimum of the potential in $\\phi =0$ and not towards infinity as in [46], [47]., which is admitted in the dark-energy context with $\\alpha $ -attractors [50].",
"We have inserted the normalization factor of $(1+\\beta )^{2n}$ to ensure the same normalization of the plateau at large $\\phi >0$ for every choice of $(p,\\,n)$ .",
"For definiteness we will consider $\\beta =1$ in the following and we will use a rescaled scalar field $\\Theta \\equiv \\phi /(\\sqrt{6 \\alpha } M_\\textup {pl})$ when useful.",
"Note that we have added an offset in the potential (REF ) as in Ref.",
"[50], so that the equation of state of the scalar field becomes $-1$ today.",
"Note however that the construction presented here does not provide an explanation of the magnitude of the offset, i.e.",
"does not address the cosmological constant problem.",
"We show the potential for three particular choices of $(p,\\,n)=\\lbrace (2,\\,0),\\,(2,\\,4),\\,(4,\\,2)\\rbrace $ , that we label $\\lbrace $A, B, C$\\rbrace $ respectively, in Fig REF .Since the purpose of this paper is to show that Early Dark Energy models can be incorporated in the $\\alpha $ -attractor framework, we have only restricted ourselves to three representative models $\\lbrace $A, B, C$\\rbrace $ which constitute a small sample of the full parameter space described by the potential in Eq.",
"(REF ).",
"We stress, however, that our choice is not the only possible one and other choices can lead to results similar to the ones shown in this paper.",
"The reason for this choice will become clear in the following.",
"Note that the potential is asymmetric around the origin in the last two cases for which $n\\ne 0$ .",
"As we will see, the potential in Eq.",
"(REF ) captures all the interesting phenomenological EDE models, but other functional forms can in principle be chosen according to Eq.",
"(REF ).",
"Figure: We plot the potential in Eq.",
"() for (p,n)={(2,0),(2,4),(4,2)}(p,\\,n)=\\lbrace (2,\\,0),\\,(2,\\,4),\\,(4,\\,2)\\rbrace .We now discuss the cosmological evolution of $\\alpha $ -attractor EDE.",
"The dynamics of the scalar field is similar to other models of EDE studied in the literature and is essentially that of an ultralight axion field [51].",
"The scalar field starts from its initial value $\\Theta _i$ deep in the radiation era and remains frozen because of the Hubble friction.",
"The energy density of the scalar field is subdominant in this regime and its equation of state $w_{\\rm EDE}\\equiv P_{\\rm EDE}/\\rho _{\\rm EDE}$ is equal to $-1$ , hence the name “Early Dark Energy\".",
"Eventually, the effective mass of the scalar field becomes comparable to the Hubble rate $H$ and $\\phi $ starts to thaw.",
"The redshift $z_c$ at which this occurs can be implicitly defined from the relation $\\frac{\\partial ^2 V(\\phi _i)}{\\partial \\phi ^2}\\simeq 9 H^2(z_c) $ [51].",
"After $z_c$ , the Hubble friction is too weak to keep the scalar field up its potential and it rolls down in a very short time.",
"When this happens, the potential energy of the scalar field is converted into a kinetic one and a certain amount of energy, parameterized by $f_{\\rm EDE}\\equiv \\rho _{\\rm EDE}(z_c)/3 M_{\\rm pl}^2 H^2(z_c)$ is injected into the cosmic fluid.",
"Depending on the slope of the potential and its structure around the minimum, the scalar field then starts to oscillate or simply freezes again once it has exhausted its inertia.",
"The critical redshift $z_c$ and the value of the energy injection $f_{\\rm EDE}$ are the key parameters describing all EDE models [52].",
"As we are going to discuss, the shape of the energy injection and $w_{\\rm EDE}$ crucially depend on the different possible dynamics of the scalar field after $z_c$ .",
"The scalar field energy density quickly redshifts away after $z_c$ and its contribution becomes subdominant with respect to the other components of the Universe.",
"We show in Fig.",
"REF the EDE dynamics for the three (A, B and C) cases mentioned above.",
"In particular, we plot the scalar field evolution, its equation of state and the energy injection in the left, central and right panels respectively (see the caption for the parameters used).",
"Figure: We plot the evolution of the normalized scalar field Θ\\Theta [Left], equation of state parameter w EDE w_{\\rm EDE} [Center] and the energy injection f EDE f_{\\rm EDE} [Right] for the three models with (p,n)={(2,0),(2,4),(4,2)}(p,\\,n)=\\lbrace (2,\\,0),\\,(2,\\,4),\\,(4,\\,2)\\rbrace .",
"For definiteness, we have chosen f EDE =0.1f_{\\rm EDE}=0.1, log 10 z c =3.5\\log _{10} z_c=3.5 and Θ i =0.4\\Theta _i=0.4.In the cases A and B , the scalar field oscillates at the bottom of its potential leading to a highly oscillatory equation of state.",
"In the A case, the potential is $\\tanh ^4 \\Theta \\sim \\Theta ^4$ around $\\Theta \\simeq 0$ and therefore the shape for the energy injection closely resembles the one obtained in the so-called rock'n'roll model of Ref.",
"[31] where $V(\\phi )\\propto \\phi ^4$ .",
"On the other hand, the B case looks more similar to the original EDE proposal of Ref.",
"[30] (see e.g.",
"Fig.",
"2 of Ref. [34]).",
"However, given the asymmetry of our potential for the B case, the oscillatory pattern in the energy injection shows an asymmetric amplitude of odd and even peaks in the oscillations.",
"Although this is barely visible in Fig.",
"REF , this effect is more pronounced for larger $\\Theta _i$ and might in principle lead to distinct results, as the CMB power spectrum is very sensitive to the shape of $f_{\\rm EDE}(z)$ [9].",
"Indeed, because of such a sensitivity, the oscillatory patterns of the scalar field in models A and B leave different imprints on the CMB angular spectra as shown in Refs.",
"[31] and [34].",
"Therefore, although at a first glance their background evolution might look similar, it is important to explore the phenomenology of both of them separately.",
"The case C is instead different.",
"Unlike the first two oscillatory models, for this choice of $p$ and $n$ , the bottom of the potential is very close to flat and the scalar field shows no oscillations.",
"As anticipated, this model looks indeed similar to the canonical Acoustic Dark Energy (cADE) model proposed in Ref. [33].",
"As in cADE (see also Ref.",
"[32]), the potential energy is suddenly converted to a kinetic one and the scalar field remains in a kination regime in which $w_{\\rm EDE}=1$ , and its energy is kinetically dominant until it redshifts away.",
"However, differently from cADE, where the potential was introduced by patching a quartic potential for positive values of $\\phi $ to $V(\\phi )=0$ for negative ones, our potential C is consistently embedded in the $\\alpha $ -attractor's construction.",
"Although we have focused on these three specific cases that well reproduce some cases in the literature, we stress that other possibilities can be obtained for other combinations of the potential parameters $(p,\\,n)$ ."
],
[
"Cosmological constraints and implications for the $H_0$ tension",
"In this Section, we perform a Markov-Chain Monte Carlo (MCMC) analysis with cosmological data and investigate the capability of $\\alpha $ -attractor EDE models to ease the $H_0$ tension.",
"We use the publicly available code MontePython-v3https://github.com/brinckmann/montepython_public [53], [54] interfaced with our modified version of CLASShttps://github.com/lesgourg/class_public [55], [56] We include several datasets in our analysis.",
"We consider CMB measurements from the Planck 2018 legacy release (P18) on temperature, polarization, and weak lensing CMB angular power spectra [57], [58].",
"The high-multipole likelihood $\\ell \\ge 30$ is based on the Plik likelihood.",
"We use the low-$\\ell $ likelihood combination at $2 \\le \\ell < 30$ : temperature-only Commander likelihood plus the SimAll EE-only likelihood.",
"For the Planck CMB lensing likelihood, we consider the conservative multipole range, i.e.",
"$8 \\le \\ell \\le 400$ .",
"To provide late-time information, complementary to the CMB anisotropies, we use the Baryon Spectroscopic Survey (BOSS) DR12 [59] “consensus\" results on baryon acoustic oscillations (BAO) in three redshift slices with effective redshifts $z_{\\rm eff} = 0.38,\\,0.51,\\,0.61$ [60], [61], [62].",
"Additionally, we use the Pantheon supernovae dataset [63], which includes measurements of the luminosity distances of 1048 SNe Ia in the redshift range $0.01< z <2.3$ .",
"Finally, we make use of a Gaussian prior based on the determination of the Hubble constant from from Hubble Space Telescope (HST) observations, i.e.",
"$H_0 = 74.03 \\pm 1.42$ [3].",
"We study the cosmological models denoted by A, B and C introduced in the previous section.",
"We sample the cosmological parameters $\\lbrace \\omega _b,\\,\\omega _{cdm}, \\,\\theta _s,\\,\\ln 10^{10}A_s,\\,n_s,\\,\\tau _\\textup {reio},\\,f_{\\rm EDE},\\,\\log _{10}\\,z_c,\\,\\Theta _i\\rbrace $ using a Metropolis-Hastings algorithm.",
"We consider the following flat priors on the EDE parameters: $f_{\\rm EDE}\\in [10^{-4},\\,0.4]$ , $\\log _{10}\\,z_c\\in [2.9,\\,4.2]$ as in other EDE studies, see e.g.",
"Ref.",
"[34], and $\\Theta _i\\in [0.05,\\,1.4]$ .",
"We have tested that larger priors on $\\Theta _i$ give the same results.",
"We consider the chains to be converged using the Gelman-Rubin criterion $R-1<0.01$ [64] and adopt the Planck convention for modeling free-streeming neutrinos as two massless species and one massive one with $M_\\nu =0.06$ eV.",
"Concerning the linearized perturbations, we impose adiabatic initial conditions and solve the exact evolution of the scalar field perturbations $\\delta \\phi (k,\\,\\tau )$ in the synchronous gauge [65].",
"Table: Constraints on main and derived parameters of the three examples in the main text consideringPlanck 2018 data (P18), BAO, Pantheon and SH0ES data.",
"We report mean values and the 68% CL, except for the subset of EDE parameters {Θ i ,log 10 V 0 / eV 4 ,log 10 α}\\lbrace \\,\\Theta _i,\\,\\log _{10}\\, V_0/{\\rm eV}^4,\\,\\log _{10}\\, \\alpha \\rbrace for which we report the 95% CL.Figure: Constraints on main parameters and H 0 H_0 of the α\\alpha -attractor models A, B and C from Planck 2018 data (P18), BAO, Pantheon and SH0ES data.",
"Parameters on the bottom axis are the standard cosmological parameters, and parameters on the left axis are the EDE parameters that we sample with flat priors, r s r_s in [Mpc] and H 0 H_0 in [km s -1 ^{-1}Mpc -1 ^{-1}].",
"Constraints for the Λ\\Lambda CDM model obtained with the same dataset are also shown.",
"Contours contain 68% and 95% of the probability.Our results are summarized in Table REF , where we report the reconstructed mean values and the 68% and 95% CL, and Fig.",
"REF , which has been obtained using GetDisthttps://getdist.readthedocs.io/en/latest [66], where we plot the reconstructed two-dimensional posterior distributions of the main and derived parameters.",
"We also plot in Fig.",
"REF the one-dimensional posterior distributions on the parameters of the potential $V_0$ and $\\alpha $ .",
"Figure: One dimensional derived posterior distribution of the potential parameters log 10 V 0 / eV 4 \\log _{10}\\, V_0/{\\rm eV}^4 and log 10 α\\log _{10}\\, \\alpha .",
"The convention for the colors used is the same of Fig.",
"."
],
[
"Results",
"We now comment the results of the previous Section.",
"As expected, all three cases lead to larger values for the Hubble parameter $H_0$ , as can be seen from Tab.",
"REF .",
"We find that the larger energy injection is allowed in the model B for which $f_{\\rm EDE}=0.082\\pm 0.029$ results in $H_0= (70.9\\pm 1.1)$ km s$^{-1}$ Mpc$^{-1}$ at 68% CL.",
"This is followed by model ${\\bf A}$ for which $f_{\\rm EDE}=0.065\\pm 0.026$ results in $H_0= (70.28\\pm 0.94)$ km s$^{-1}$ Mpc$^{-1}$ and model ${\\bf C}$ for which $f_{\\rm EDE}=0.048^{+0.029}_{-0.024}$ results in $H_0= (69.88\\pm 0.99)$ km s$^{-1}$ Mpc$^{-1}$ .",
"As in other EDE models, there is a clear degeneracy between the comoving sound horizon $r_s$ and $f_{\\rm EDE}$ which is responsible for the enhanced expansion rate around the epoch of matter-radiation equality and therefore the lower $r_s$ .",
"On the other hand, to successfully preserve the fit to the CMB data a shift to a larger CDM density $\\omega _c$ is needed, leading to the degeneracy between $r_s$ and $\\omega _c$ seen in Fig.",
"REF and to a worsening of the $\\sigma _8$ tension [41].",
"In all the three models considered, cosmological data require that the initial value for $\\Theta _i \\sim {\\cal O} (1)$ after nucleosynthesis, once the slow-roll regime is considered.",
"These values are typical also for other EDE models.",
"For these values, the scalar field is hung up in the descending slope of the potential shown in Fig.",
"REF after nucleosynthesis: this range of values does not exclude that the scalar field could have been in the plateau outside the slow-roll regime at the beginning of the relativistic era.",
"Our results are in agreement with the comparison between A, B and C and models in the literature made in the previous Section.",
"Indeed, the higher value of $H_0$ in the literature of EDE models can be found in the original EDE proposal of Ref. [30].",
"However, due to our use of more recent CMB data and perhaps the slightly asymmetric oscillations in the energy injection, our inferred value for $H_0$ is somewhat lower.",
"In fact Ref.",
"[41] also found an $H_0$ similar to ours when analyzing the model of Ref.",
"[30], adopting the same dataset used here.",
"Furthermore, contrary to Refs.",
"[31], [33], the potential in Eq.",
"(REF ) contains two free parameters, and despite the similar shape of the energy injection in the models A and C respectively, the enhanced number of degeneracies between parameters leads to a slightly different inferred $H_0$ also in this case.",
"We note an interesting difference between these EDE models based on $\\alpha $ -attractor-like potentials and those inspired by ultralight axionlike fields as Refs.",
"[52], [30], [34].",
"In models involving cosine potentials, the axion decay constant $f$ has to take values of order $\\mathcal {O}(M_\\textup {pl})$ to solve the $H_0$ tension [41], in contrast with the weak gravity conjecture [67].",
"Our results show instead that the allowed range for $\\alpha $ is natural in terms of model building and also includes the discrete values for $\\alpha $ motivated by maximal supersymmetry [68], [69].",
"It is also interesting to note that we have here an apparent non-zero detection of $\\alpha $ for EDE, whereas at present we have only upper bound on $\\alpha $ for inflationary models [70].",
"Note, however, that in order to correctly claim a non-zero detection of $\\alpha $ , we should also vary the potential parameters $n$ and $p$ in our MCMC and marginalize over them."
],
[
"Conclusions",
"In this paper we have studied a new model of Early Dark Energy (EDE) consisting of a minimally coupled scalar field in the framework of $\\alpha $ -attractors.",
"As is typical in EDE models, the scalar field remains frozen during the radiation era, until it becomes massive and quickly rolls down the potential, injecting energy into the cosmic fluid and temporarily enhancing the expansion rate of the Universe.",
"The shape of the energy injection in redshift is crucial to solve the $H_0$ tension and depends on the structure of the potential around its minimum.",
"The only constraint in $\\alpha $ -attractors is that the potential has to be of the form $V(\\phi )= f^2\\left[\\tanh (\\phi /\\sqrt{\\alpha })\\right]$ , giving in principle a large freedom to the model building.",
"Adopting the simple potential in Eq.",
"(REF ) as a working example, we have shown that it is indeed possible for the energy injection to take several different shapes in the single unified framework of $\\alpha $ -attractors and reproduce results from different studies in the literature.",
"To illustrate this, we have analyzed three example models.",
"In the first two (A and B) the scalar field oscillates at the bottom of the potential in a way that resembles the works [31] and [30] respectively.",
"Note however, that our second example slightly differs from [30] since the asymmetry of the potential around $\\phi =0$ leads to an asymmetric pattern of oscillations in the energy injection.",
"In our third model (C), instead, the scalar field never oscillates and quickly transfers its potential energy to kinetic one, undergoing a temporary phase of kination, as in Ref. [33].",
"We have used the latest Planck 2018 CMB temperature, lensing and polarization data together with a variety of high and low $z$ BAO measurements, SNe Ia data from Pantheon and the SH0ES estimate of the Hubble constant, and run an MCMC simulation to constrain the model parameters.",
"We have found that all the models can significantly alleviate the $H_0$ tension, the best being the model B, for which an energy injection of $f_{\\rm EDE}=0.082\\pm 0.029$ at the redshift $\\log _{10}\\,z_c=3.510^{+0.044}_{-0.053}$ leads to an inferred value of the Hubble rate today of $H_0= (70.9\\pm 1.1)$ km s$^{-1}$ Mpc$^{-1}$ at 68% CL.",
"As noticed in the literature, EDE models change the best-fit cosmological parameters from $\\Lambda $ CDM such as $n_s$ , $A_s$ and $\\omega _c$ .",
"This could lead to tension with large scale structure observations such as weak gravitational lensing [41], although the conclusion depends on the CMB data used in the analysis [42].",
"Another interesting consequence is the spectral index $n_s$ , which tends to be larger than the one obtained in $\\Lambda $ CDM, $n_s \\sim 0.98$ .",
"This has an interesting implication for inflationary models.",
"For example, some inflation models based on $\\alpha $ -attractors predict a larger $n_s$ if reheating occurs gravitationally [49], [50].",
"These $\\alpha $ -attractor inflation models can be combined with early dark energy models as a two field model or quintessential inflationary models as done in Refs.",
"[49], [50].",
"It will be interesting to explore this possibility and revisit inflation models in the light of new constraints on $n_s$ [71], [72], [70].",
"Note added: After this paper was completed, new studies involving LSS data appeared on the arXiv.",
"The results obtained in Ref.",
"[41] that first explored to which extent the shift towards larger values of $\\omega _c$ in EDE models can be tolerated by weak lensing and redshift-space distortions data, have been confirmed in more recent papers that used the full-shape of the power spectrum [73], [74], [75].",
"The conclusions of these papers, which do not include the SH0ES data in the analysis, is that LSS data break the degeneracies between the EDE and cosmological parameters mentioned in Sec.",
"and EDE models (which predict in general a higher sigma8) are not able to significantly ease the $H_0$ tension.",
"More optimistic results are found in Ref. [76].",
"MB acknowledges the Marco Polo program of the University of Bologna for supporting a visit to the Institute of Cosmology and Gravitation at the University of Portsmouth, where this work started.",
"FF acknowledges contribution from the contract ASI/INAF for the Euclid mission n.2018-23-HH.0 and by ASI Grant 2016-24-H.0.",
"AEG and KK received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.",
"646702 “CosTesGrav\").",
"KK is also supported by the UK STFC ST/S000550/1.",
"WTE is supported by an STFC consolidated grant, under grant no.",
"ST/P000703/1.",
"Numerical computations for this research were done on the Sciama High Performance Compute cluster, which is supported by the ICG, SEPNet, and the University of Portsmouth."
]
] | 2005.14053 |
[
[
"Semiclassical modeling of coupled quantum dot-cavity systems: From\n polariton-like dynamics to Rabi oscillations"
],
[
"Abstract Semiconductor quantum dots in photonic cavities are strongly coupled light-matter systems with prospective applications in optoelectronic devices and quantum information processing.",
"Here we present a theoretical study of the coupled exciton--light field dynamics of a planar quantum dot ensemble, treated as two-level systems, embedded in a photonic cavity modeled by Maxwell's equations.",
"When excited by coupling an external short laser pulse into the cavity, we find an exciton-polariton-like behavior for weak excitation and Rabi oscillations for strong excitation with a sharp transition between these regimes.",
"In the transition region we find highly non-linear dynamics involving high harmonics of the fundamental oscillation.",
"We perform a numerical study based on the Finite-Difference-Time-Domain method for the solution of Maxwell's equations coupled to Bloch equations for the quantum dots and also derive an analytical model to describe the coupled cavity-quantum dot system, which allows us to describe the light field dynamics in terms of a Newton-like dynamics in an effective anharmonic potential.",
"From the shape of this potential combined with the initial conditions the transition can be well understood.",
"The model is then extended to a broadened ensemble of quantum dots.",
"For weak excitation the polariton spectrum broadens and the lines slightly shift, however, the sharp transition to the Rabi oscillation regime is still present.",
"Furthermore, we find a second, lower threshold with additional lines in the spectra which can be traced back to Rabi oscillations driven by the polariton modes.",
"Our approach provides new insights in the dynamics of both quantum dot and light field in the photonic structure."
],
[
"Introduction",
"In state-of-the-art semiconductor nanostructures the electronic and optical properties can be tailored by spatial confinement of the electronic and/or photonic degrees of freedom.",
"The ultimate electronic confinement is reached in semiconductor quantum dots (QDs) which, due to the three-dimensional confinement of the electronic states, have a discrete energy spectrum.",
"These QDs can be embedded in photonic structures like micro cavities [1], [2], [3], [4], photonic crystal structures [5], [6], nano lenses [7], [8] or plasmonic structures [9].",
"By confining the light modes these structures change the local density of photon states leading to an increase of the light-matter coupling with the QDs [10].",
"This is a crucial aspect, when considering QDs to be used as single or entangled photon sources [11], [12], [13], but also for optoelectronic devices which are based on QD ensembles used, e.g., for lasing applications [14].",
"To model such structures it is required to account for both, the dynamics of the QD system and the photonic structure.",
"In the limit of strongly confined light, popular approaches are based on the Jaynes-Cummings model from cavity-quantum-electrodynamics [15], [16], [17], [18], which typically accounts for one quantized light mode and a single two-level system, but not explicitly for the spatio-temporal light-field dynamics in the photonic structure.",
"For QD-systems the Jaynes-Cummings model is often extended by considering additionally the electron-phonon interaction [19], [20], [21], [22], [23].",
"Instead of looking at a single QD in a microcavity, here we consider a planar ensemble of QDs in a one-dimensional photonic cavity formed by a pair of Bragg mirrors [1], [2].",
"In order to study the dynamics of the system we combine the equations of motion for the QD exciton with a Finite-Difference-Time-Domain (FDTD) method for the light field dynamics.",
"We take into account the spatial structure along the light propagation direction, perpendicular to a layered structure forming a photonic cavity.",
"In the cavity a QD ensemble is placed, as schematically sketched in fig1.",
"We are interested in the optical response of this QD ensemble in the cavity upon external driving of the cavity mode.",
"For this purpose, we assume that an external laser pulse excites the light field and then analyze the output of the system (cf.",
"Fig REF ).",
"We show that, depending on the excitation power, different regimes occur: For low intensities an exciton-polariton like spectrum is observed, while for high amplitudes a spectrum with Rabi splitting emerges.",
"Interestingly, we find that there is no smooth transition between the regimes but that the transition occurs abruptly at a certain strength of the driving.",
"Close to the transition nonlinearities strongly affect the dynamics leading to the appearance of a large number of higher harmonics in the spectrum.",
"We start by performing numerical calculations based on a FDTD method with the embedded few-level systems [24], [25], [26], [27], [28] and assume them to be identical.",
"To better interpret our findings, we then show that for the chosen conditions the problem can be reduced to a set of three ordinary non-linear differential equations for the light mode amplitude, the polarization and the occupation of the QD excitons.",
"These equations can be traced back to a Newton-type equation for an effective particle in a potential, where the shape of the potential depends on the initial excitation of the light field.",
"Different initial conditions thus lead to different behavior providing an intuitive picture for the transition from the exciton-polariton like dynamics to Rabi oscillations.",
"Finally we compare the results for identical few-level systems with QD ensembles with Gaussian shaped energetic distributions.",
"While in the polaritonic regime this leads, as expected, mainly to a broadening of the polariton lines associated with slight shifts [29], for stronger excitation, but still below the transition to the Rabi oscillation regime, we observe the appearance of additional lines which can be traced back to Rabi oscillations of sub-ensembles of the QDs.",
"Figure: Schematic drawing of the simulated situation.",
"A pulse from outside of the cavity excites the cavity mode, which interacts with the QD ensemble inside the layer leading to the emission of an output pulse."
],
[
"Theoretical background",
"Let us start by discussing the photonic structure.",
"We model a structure, which consists of two Bragg mirrors with $N$ AlAs/GaAs layer pairs on each side surrounding a GaAs cavity of length $l_\\text{cav}= \\lambda _0/n_\\text{GaAs}= 2\\pi c /(\\omega _0 n_\\text{GaAs})$ .",
"Here, $\\lambda _0$ and $\\omega _0$ are the vacuum wavelength and angular frequency of the cavity mode, respectively, $c=\\left( \\epsilon _0 \\mu _0 \\right)^{-1/2}$ is the vacuum speed of light, and $n_\\text{GaAs} = 3.535$ the refractive index of GaAs.",
"Each layer pair consists of a GaAs layer with a width of $l_\\text{GaAs} = \\lambda _0/(4n_\\text{GaAs})$ and an AlAs layer with a width of $l_\\text{AlAs} = \\lambda _0/(4n_\\text{AlAs})$ with the refractive index of AlAs $n_\\text{AlAs} = 2.956$ .",
"The cavity frequency is taken to be $\\hbar \\omega _0 = 1300\\, \\text{meV}$ , which is a typical transition energy for InGaAs QDs [1], [30].",
"The growth direction of the structure is taken to be the $z$ -direction.",
"In the center of the cavity, in the plane $z=z_0$ , a QD ensemble is placed consisting of $N_\\text{QD}$ QDs with transition energies $\\hbar \\omega _x^{(n)}$ , $n=1,\\dots ,N_\\text{QD}$ , located at the positions $r_n = (x_n, y_n, z_0)$ .",
"We assume that the spatial distribution of the QDs in the central plane is sufficiently homogeneous such that also the electromagnetic fields can be taken to be homogeneous in $x$ - and $y$ -direction.",
"The light field in the structure is described by Maxwell's equations ${t} D(z,t) &= \\nabla \\times H(z,t) - J_\\text{s}(z,t) \\\\\\mu _0 {t} H(z,t) &= -\\nabla \\times E(z,t) $ with the electric field $E$ , the magnetic field $H$ , the displacement field $D$ , and a source current density $J_\\text{s}$ , which will be used to model the external excitation of the system by a short laser pulse.",
"Note that we will consider only non-magnetic materials.",
"The displacement field is composed of the electric field $E$ and the macroscopic polarization $P$ of the system, $D(z,t) = \\epsilon _0 E(z,t) + P(z,t)$ , where $P(z,t)$ consists of the linear polarization of the material of the photonic structure, $P_\\text{mat}(z,t) = \\epsilon _0 \\left[n^2(z)-1\\right] E(z,t)$ with the space-dependent refractive index $n(z)$ , and the polarization of the QD ensemble $\\bar{P}_\\text{QD}(z,t)$ , averaged over the positions of the QDs.",
"In general, QDs interacting with a light field which has a frequency close to the lowest exciton transition can be described in terms of a four-level model consisting of the exciton ground state, two single-exciton states with either perpendicular linear or opposite circular polarization, and a biexciton state.",
"Here, however, we will concentrate on the excitation by circularly polarized light.",
"In this case the biexciton and the exciton with opposite circular polarization cannot be excited, such that each QD can be reduced to a two-level model consisting of the ground state $ {g^{(n)}}$ and the single exciton state ${x^{(n)}}$ .",
"The Hamiltonian of the QD ensemble including the light-matter coupling then reads $\\hat{H} = & \\sum _n \\hbar \\omega _x^{(n)} \\textstyle {{x^{(n)}}{x^{(n)}} }- E(z_0,t) \\nonumber \\\\& \\cdot \\displaystyle {\\sum _n} \\biggl ( M^{(n)} \\textstyle {{g^{(n)}}{x^{(n)}} } + M^{(n)*} \\textstyle {{x^{(n)}}{g^{(n)}} } \\biggr ).$ The dipole matrix element for the creation and annihilation of $\\sigma _{+}$ ($\\sigma _{-}$ ) polarized excitons is given by $M_{\\pm }^{(n)}= M^{(n)} e_{\\pm }$ with the unit polarization vector $e_{\\pm }=(e_x \\pm i e_y)/\\sqrt{2}$ , $e_x$ and $e_y$ being the unit vectors in $x$ - and $y$ -direction, $M^{(n)}$ refers to the dipole matrix element of the excitons with the polarization given by the polarization of the light field, and $M^{(n)}$ is taken to be real.",
"The microscopic state of the QD $n$ is specified by the microscopic polarization $p^{(n)} = \\langle {{g^{(n)}}{x^{(n)}}} \\rangle $ and the exciton occupation $f^{(n)} = \\langle {{x^{(n)}}{x^{(n)}}} \\rangle $ , which satisfy the Bloch-type equations of motion ${t} f^{(n)} &= -i \\frac{M^{(n)}E(z_0,t)}{\\hbar } \\cdot \\left[ e_{\\pm } p^{(n)} - e_{\\mp } p^{(n)*}\\right] \\\\{t} p^{(n)} &= -i\\omega _x^{(n)} p^{(n)} - i \\frac{M^{(n)} E(z_0,t) \\cdot e_{\\mp }}{\\hbar } \\left[ 2f^{(n)} - 1 \\right] \\, .$ The microscopic polarizations of the QDs then give rise to an average macroscopic polarization of the QD ensemble $\\bar{P}_\\text{QD} &= \\frac{1}{A} \\int _A {x} {y} \\sum _n \\delta (x - x_n) \\delta (y-y_n) \\delta (z-z_0) \\nonumber \\\\& \\hspace{14.22636pt}\\times \\int {\\omega _x} \\delta (\\omega _x-\\omega _x^{(n)}) \\left(M^{(n)} p^{(n)} + M^{(n)*} p^{(n)*}\\right) \\nonumber \\\\&= \\frac{N_\\text{QD}}{A} M \\delta (z-z_0) \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\nonumber \\\\& \\hspace{14.22636pt}\\times \\left[e_{\\pm } p(\\omega _x,t)+e_{\\mp } p^*(\\omega _x,t)\\right] \\nonumber \\\\&= \\tilde{P}(t) \\delta (z-z_0), $ where $\\tilde{P}(t) &= \\frac{MN_\\text{QD}}{A} \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\nonumber \\\\& \\quad \\times \\left[e_{\\pm } p(\\omega _x,t) + e_{\\mp } p^*(\\omega _x,t)\\right] $ with the normalization area $A$ and the number $N_\\text{QD}$ of QDs in the area $A$ .",
"Here, $\\rho _\\text{QD}(\\omega _x)= \\frac{1}{M N_\\text{QD}} \\sum _n \\delta (\\omega _x-\\omega _x^{(n)}) M^{(n)}$ is a normalized distribution function of QD transition energies $\\omega _x$ weighted by the dipole moments and $p(\\omega _x)$ is the microscopic polarization of the QDs with this transition energy.",
"Correspondingly, $f(\\omega _x)$ is the exciton occupation of these QDs.",
"The average dipole moment is defined by $M=\\frac{1}{ N_\\text{QD}} \\sum _n M^{(n)}.$ As mentioned above, in the following we will restrict ourselves to circularly polarized light.",
"However, since we use real electromagnetic fields some care has to be taken in identifying the respective contributions.",
"For a $\\sigma _{+}$ ($\\sigma _{-}$ ) circularly polarized light field with central frequency $\\omega _0$ traveling in $z$ -direction, the $y$ -component of the electric field follows (precedes) the $x$ -component by a quarter period, i.e., we have $E(z,t) = \\frac{1}{\\sqrt{2}} \\left[ E\\left(z,t\\right) e_x \\pm E\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e_y \\right].",
"$ The magnetic field then satisfies $H(z,t) = \\frac{1}{\\sqrt{2}} \\left[ H\\left(z,t\\right) e_y \\mp H\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e_x \\right].$ In order to excite such a field, also the source current density in eq:MaxwellDvect has to be of the same structure, i.e., $J_\\text{s}(z,t) = \\frac{1}{\\sqrt{2}} \\left[ J_\\text{s}\\left(z,t\\right) e_x \\pm J_\\text{s}\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e_y \\right].",
"$ As is shown in the appendix , as long as we restrict ourselves to light field amplitudes which vary on a time scale much longer than the oscillation period $2\\pi /\\omega _0$ , the electric field in eq:Ecirc can be rewritten as $E(z,t) = \\frac{1}{2} \\left[ e_{\\pm } \\tilde{E}\\left(z,t\\right) e^{-i\\omega _0 t} + e_{\\mp } \\tilde{E}^*\\left(z,t\\right) e^{i\\omega _0 t} \\right].",
"$ with the slowly varying complex amplitude $\\tilde{E}$ .",
"Inserting this field in eq:Bloch and using $e_{\\pm } \\cdot e_{\\mp }=1$ and $e_{\\pm } \\cdot e_{\\pm }=0$ , we see that indeed a $\\sigma _{+}$ ($\\sigma _{-}$ ) circularly polarized light field only excites the $\\sigma _{+}$ ($\\sigma _{-}$ ) circularly polarized exciton, except for negligible contributions resulting from counter-rotating terms $\\sim \\exp [\\pm 2 i\\omega _0 t]$ .",
"Following again the derivation in the appendix , we find that also the polarization in eq:polvect can be separated into $x$ - and $y$ -components according to $\\tilde{P}(t) &= \\frac{1}{\\sqrt{2}} \\left[ \\tilde{P}(t) e_x \\pm \\tilde{P}\\left( t-\\frac{\\pi }{2\\omega _x} \\right) e_y \\right],$ with $\\tilde{P}(t) = \\frac{MN_\\text{QD}}{A} \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\left[p(\\omega _x,t) + p^*(\\omega _x,t)\\right] .$ This shows that indeed for all vector fields entering Maxwell's equations () the $y$ -component agrees with the $x$ -component shifted by a quarter period.",
"Therefore, it is sufficient to solve the equations for the $x$ -component of the electric and the $y$ -component of the magnetic field, resulting in the final set of equations of motion ${t} E(z,t) &= - \\frac{1}{n^2(z) \\epsilon _0} \\left[ {z} H(z,t) \\right.",
"\\nonumber \\\\& \\hspace{14.22636pt}\\left.",
"{} + \\delta (z-z_0) {t} \\tilde{P}(t) - J_\\text{s}(z,t) \\right],\\\\{t} H(z,t) &= -\\frac{1}{\\mu _0} {z} E(z,t), \\\\{t} f(\\omega _x,t) &= 2\\frac{ME(z_0,t)}{\\hbar } \\Im \\left(p(\\omega _x,t)\\right), \\\\{t} p(\\omega _x,t) &= -i\\omega _x p(\\omega _x,t) - i \\frac{ME(z_0,t)}{\\hbar } \\left[ 2f(\\omega _x,t) - 1 \\right].",
"$ eq:eom,eq:macrPmean are solved numerically.",
"For eq:f,eq:p we use a standard fourth order Runge-Kutta method and eq:1DmaxwellE,eq:1DmaxwellH are implemented using a FDTD method in one spatial dimension [31].",
"The system is excited at a position $z_1$ outside of the cavity with a Gaussian current density $J_\\text{s}(z,t)=J \\delta \\left( z-z_1 \\right) \\exp \\left[-\\frac{4t^2 \\ln (2)}{\\tau ^2}\\right] \\cos (\\omega _0 t)$ with a full width at half maximum (FWHM) of $\\tau =200$ fs.",
"Reflections from the boundaries of the simulation region are avoided by using perfectly matched absorbing boundary layers [31].",
"The electric field in eq:1DmaxwellE has two driving terms, one involving the current density $J_\\text{s}$ and the other the QD polarization $\\tilde{P}(t)$ .",
"Due to the linearity of eq:1DmaxwellE,eq:1DmaxwellE the electric (and also the magnetic field) can be separated into two contributions, $E(z,t) = E_\\text{ext}(z,t) + E_\\text{ind}(z,t) $ with the external field $E_\\text{ext}(z,t)$ driven only by the current density and the induced field $E_\\text{ind}(z,t)$ driven only by the polarization.",
"We will come back to this separation below when discussing the results.",
"In the following sections we will first concentrate on the case of identical QDs in resonance with the light field, i.e., a sharp distribution $\\rho _\\text{QD}(\\omega _x) = \\delta (\\omega _x - \\omega _0)$ In Sec.",
"we will then analyze the influence of a non-vanishing width of the QD distribution on the results."
],
[
"Characterization of the system",
"Before considering the dynamics of the system, we briefly characterize the photonic cavity with the QDs by analyzing its transmission spectrum.",
"For this purpose, we excite the system with a small amplitude $J$ at the position $z_1$ on the left side and calculate the Fourier transformation of the electric field after passing the system, i.e., at a position $z_2$ on the right side.",
"Without a cavity, the pulse has a Gaussian spectral shape.",
"When the pulse interacts with the QDs, a dip in the spectrum is found as displayed in Fig.",
"REF (a).",
"The width of this absorption dip depends on the lifetime of the excited state.",
"The decay is an effect of the radiative interaction with the field induced by the QD polarization [32], [33] and here depends on $M \\frac{N_\\text{QD}}{A}$ (see eq:macrPmean.)",
"Now we put a cavity around the QD ensemble.",
"fig2 (b) shows the linear transmission spectrum of the QD ensemble in a cavity with Bragg mirrors of 30 layer pairs on each side.",
"Without QDs (dashed line) we see a sharp peak of the cavity mode at $\\omega = \\omega _0$ .",
"With QDs (solid line), this peak splits up in two separate peaks.",
"The splitting occurs due to the strong light confinement and enhanced light-matter interaction.",
"Accordingly, the double peak only exists if the confinement is strong, which depends on the number of Bragg layers.",
"This is demonstrated in fig2 (c), showing the transmission spectrum with QDs for different number of layer pairs on each side.",
"For small $N$ there is a broad spectrum with a dip as found in fig2 (a).",
"For values $N>10$ the formation of the coupled QD-light state can be observed.",
"We emphasize that the splitting is independent of the number of layers, once this state exists.",
"We have checked that the splitting does not depend on the amplitude as long as we stay in the limit of small amplitudes, but is proportional to $\\sqrt{N_\\text{QD}/A}$ and $M$ .",
"We will come back to this dependence below in Sec. .",
"We note that these peaks exhibit an anti-crossing behavior when tuning the transition frequency of the QDs through resonance of the cavity (not shown).",
"These properties are typical for exciton-polaritons [3].",
"Figure: Linear transmission spectra for excitations with low amplitude (a) without photonic structure and (b) for a cavity with N=30N=30 Bragg layer pairs on each side.The dashed lines correspond to the spectra without QDs, the solid lines show the spectra with QD ensemble.",
"(c) Transmission spectrum for increasing number of layer pairs of the Bragg mirror at each side of the cavity.",
"The spectra (b) and (c) are calculated using a small exponential damping for better visualization."
],
[
"Transition from low to high amplitudes",
"Now we study the spectrum, when increasing the amplitude $J$ of the external current.",
"The resulting spectra are shown in fig3, where we fixed the number of layer pairs to $N=30$ .",
"Note that here we have plotted the spectra of the field at the position of the QDs.",
"We checked that these agree qualitatively with the transmission spectra.",
"The lower part of this figure corresponds to the excitation with a small pulse amplitude and shows a double peak structure as discussed in the previous section.",
"For low amplitudes the splitting is essentially independent of the amplitude.",
"When increasing the amplitude further the splitting slightly reduces and additional side bands are formed.",
"These are nonlinear effects of the QD-cavity coupling, which result from the re-interaction with the induced light field.",
"At a certain amplitude $J_0$ of the driving current density a sudden transition takes place (see line (b) in fig3) and the behavior of the spectrum changes qualitatively.",
"For higher pulse amplitudes we now obtain three peaks in the spectrum with a main peak at $\\omega = \\omega _0$ and two satellite peaks.",
"The main peak stays at $\\omega = \\omega _0$ when increasing the amplitude, while the side peaks change their position with increasing $J$ .",
"For sufficiently strong excitation the splitting of the side peaks grows linearly with the $J$ .",
"Figure: Spectrum of the field inside the cavity for varying amplitude JJ of the exciting pulse, given in units of the threshold amplitude J 0 J_0.The marked horizontal lines correspond to the cases shown in fig4.To understand the difference in the two regimes, in fig4 we look at the dynamics of the occupation $f$ (red line), the amplitude $\\mathcal {E}=|\\tilde{E}(z_0,t)|$ of the slowly varying part of the electric field at the position of the QDs (green line) and the absolute value of the polarization $|p|$ (gray area) for three different cases: (a) low amplitude with a double peak, (b) at the transition and (c) in the high intensity regime with three peaks.",
"These cases are marked by white lines in fig3.",
"The field amplitude is normalized to $\\mathcal {E}_0$ being the maximal electric field amplitude at the transition (see line (b) in fig3).",
"All dynamics start with a switch on of the electric field around $t=0$ due to the pulsed current density $J_\\text{s}$ , followed by an oscillatory behavior.",
"At the switch on, first the electric field builds up, which then induces a polarization, and finally an occupation of the QD system is created.",
"This can also be seen in eq:f,eq:p where the electric field drives the polarization and the polarization drives the occupation.",
"For low amplitudes, shown in fig4 (a), we find oscillations for all three quantities.",
"The reason for this is that the electric field $E(z_0,t)$ builds up a polarization.",
"The polarization, in turn, creates an induced field $E_\\text{ind}(z_0,t)$ which is opposite to the external field $E_\\text{ext}(z_0,t)$ , such that the amplitude of the total field $\\mathcal {E}$ decreases (see eq:Eextind).",
"When the polarization reaches a maximum $\\mathcal {E}$ vanishes.",
"Then the polarization acts as a source for the electric field.",
"Due to the strong confinement, the emitted light is trapped in the cavity and builds up the cavity mode again.",
"This induces the oscillations between $|p|$ and $\\mathcal {E}$ .",
"The occupation follows the polarization.",
"In this regime of weak excitations none of the quantities reaches one.",
"The occupation is quadratic in the polarization and thus also in the electric field, therefore it remains small at all times.",
"Such a periodic oscillation of the energy between electric field and polarization is characteristic for polaritonic dynamics.",
"The two spectral lines reflect the lower and the upper polaritonic branch.",
"The dynamics for high amplitudes is shown in fig4 (c).",
"Also here we observe - after an initial transient - oscillations of all three quantities, field, polarization and occupation.",
"However, now the occupation oscillates between 0 and 1 and it is out of phase with the polarization.",
"This is a characteristic behavior for Rabi oscillations of a two-level system.",
"The amplitude of the electric field exhibits weak oscillations around a non-zero value, in contrast to the weak driving it never reaches zero.",
"This non-zero mean value is the reason for the central peak at $\\omega _0$ .",
"For high amplitudes the induced field is much smaller than the external one, because the driving term $\\bar{P}_\\text{QD}$ from eq:macrPmean is limited by the microscopic polarization which, in turn is limited by the number of QDs.",
"Therefore $E_\\text{ext}$ is the main contribution to the QD dynamics and approximately acts as a continuous wave excitation for the two-level system.",
"From the classical Rabi model one expects Rabi oscillations with Rabi frequency $\\Omega _\\text{R} = \\frac{\\left|\\tilde{E}_\\text{ext} M\\right|}{\\hbar }$ , where $\\tilde{E}_\\text{ext}$ denotes the amplitude of the external field.",
"In the dressed state picture this corresponds to two states, which are separated by the Rabi energy $\\hbar \\Omega _\\text{R}$ , when driven resonantly [34].",
"The oscillating polarization of the Rabi oscillations creates an induced field oscillating with the same frequency which contributes to the total field and leads to the frequency contributions at $\\omega = \\omega _0 \\pm \\Omega _\\text{R}$ .",
"This is the origin of the satellite peaks in the upper regime of fig3.",
"The three-peak structure of the spectrum with a central peak and two satellite peaks shifted by $\\pm \\Omega _\\text{R}$ reminds one of the Mollow triplet seen in the resonance fluorescence of a two-level system driven by a classical light field [35].",
"Indeed, it has the same physical origin.",
"However, while the Mollow triplet with its characteristic intensity ratio between the three peaks is a quantum optical effect seen in the power spectrum of a resonantly scattered additional light field, here, the peaks appear in the spectrum of the driving field itself, which is modified by the field induced self-consistently by the QD polarization.",
"The peaks have no fixed intensity ratio, instead the relative intensity of the central peak increases with increasing driving because, as discussed above, the induced field is limited by the number of QDs.",
"The oscillating part of the electric field couples back to the dynamics of occupation and polarization.",
"Due to the non-linearity of these equations this gives rise to the creation of higher harmonics of the Rabi frequency.",
"They are particularly pronounced slightly above the threshold field $\\mathcal {E}_0$ because here the induced field is of the same order as the external field, while its relative importance decreases with increasing external driving.",
"Finally we look at the transition region where $J \\approx J_0$ .",
"When increasing the pulse amplitude, the occupation increases likewise up to the point where the occupation almost reaches one.",
"The dynamics of field, occupation, and polarization in this regime are shown in fig4 (b).",
"We still observe an oscillatory behavior in all three quantities, but the oscillations strongly deviate from a sinusoidal shape.",
"The occupation now reaches its maximal value of unity, and around this maximum plateaus show up.",
"Furthermore, $\\mathcal {E}$ does not reach zero like in the low driving case, but gets close to zero when $f$ reaches its maximum.",
"If the total field is very small, the actual Rabi frequency is small too, and the occupation only varies slowly.",
"This results in the formation of the plateaus as seen in fig4 (b) and a longer oscillation periodicity resulting in the reduction of the splitting in the spectrum as well as the appearance of higher harmonics at line (b) in fig3.",
"The bottom panel in fig4 displays the maximum values of the occupation and the polarization as a function of the maximum value of the field.",
"At low fields we see the linear increase of the polarization and the quadratic increase of the occupation.",
"We also clearly see that the transition between the different regimes is reached when the maximum of the occupation reaches unity.",
"Figure: Occupation (red lines), polarization (gray area), and light field (green lines) dynamics inside a cavity with N=30N=30 Bragg layer pairs on each side for different driving pulse amplitudes JJ corresponding to the cases marked by (a), (b), and (c) in fig3.Bottom panel: maximum occupation and polarization depending on the maximum field amplitude ℰ max \\mathcal {E}_\\text{max} at the position of the QDs.ℰ 0 \\mathcal {E}_0 denotes the maximum field amplitude at the transition.Let us briefly compare our results for a QD ensemble in the plane $z=z_0$ with the case of a single QD in a microcavity.",
"In that case a quantized description of the light field is necessary, which is achieved by the Jaynes-Cummings model.",
"In our scenario, the cavity field is driven by an external laser pulse, which leads to the creation of a coherent state of the cavity with mean photon number depending on the driving strength.",
"In the low driving limit the mean photon number is much smaller than one, such that the field consists of a superposition of the zero-photon and the one-photon state with negligible contributions from higher photon states.",
"Since the QD is initially in its ground state, there are no dynamics in the zero-photon subspace, while in the one-photon subspace Rabi oscillations with the vacuum Rabi splitting set in.",
"This corresponds to the polariton regime with the vacuum Rabi splitting being the polariton splitting.",
"For very strong driving, on the other hand, a coherent state with a high mean photon number and thus small relative uncertainty in the photon number is generated.",
"This leads approximately to Rabi oscillations with the Rabi frequency corresponding to the mean photon number, giving rise to the three peak structure as seen in our case, too.",
"However, in the transition region the system is driven into the well-known collapse-and-revival regime because many different photon numbers and consequently different Rabi frequencies with in general irrational ratios are present.",
"This is in clear contrast to the findings in our case of a sharp transition between the two regimes and periodic oscillations at all driving amplitudes, however with strongly anharmonic shape in the region close to the transition.",
"Indeed, both the polaritonic and the Rabi oscillation have been seen in structures with a single QD.",
"The polariton-like behavior in the strong coupling limit [36] and the Mollow-triplet [37], [38], [39], [40], similar to the three-peak structure in the Rabi model, have been independently studied and have recently been connected in an experiment, where a QD in a micropillar cavity is optically excited from the side [41].",
"There the transition between the purely quantum model of the Jaynes-Cummings ladder with the semiclassical Autler-Townes ladder was investigated using a Jaynes-Cummings model showing a transition region between the two regimes, but no sharp transition.",
"In the next section we will show that our system can be mapped to the dynamics of a particle in an effective, anharmonic potential, which will allow us to obtain a deeper insight in the different regimes and the sharp transition between them."
],
[
"Analytical Model",
"The results in the previous section have been obtained for a cavity with a high quality factor, which means that cavity losses are negligible on the considered time scales.",
"In fact, from the FDTD simulation we can extract for the cavity with 30 layer pairs in each Bragg mirror a Q factor of 870,000 corresponding to a photon lifetime of 880 ps, which is much longer than the typical time scales of one or a few picoseconds (see fig4).",
"In this case, the light field in the cavity can be expanded in cavity modes.",
"We want to remark that this approach can be extended to leaky cavities by using an expansion into quasinormal modes of the cavity [42].",
"Combining Maxwells equations eq:1DmaxwellE,eq:1DmaxwellH we obtain the driven wave equation for the electric field $ \\frac{-c^2}{n^2(z)} [2]{z} E(z, t) + [2]{t} E(z, t) = \\frac{-1}{n^2(z)\\epsilon _0} [2]{t} \\bar{P}_\\text{QD} \\,.$ We have omitted the source current density because here we will replace the pulsed excitation by chosing an initial value of the electric field.",
"For the homogeneous part of the wave equation we can define the eigenvalue problem $\\hat{\\Phi } u_m = \\omega _m^2 u_m \\quad \\text{with} \\quad \\hat{\\Phi } =- \\frac{c^2}{n^2(z)} [2]{z} \\, ,$ where we have defined the differential operator $\\hat{\\Phi }$ with the eigenvalues $\\omega _m^2$ and eigenfunctions $u_m$ .",
"Note that $\\hat{\\Phi }$ is not self-adjoint.",
"However, by taking the derivative of the eigenvalue problem we obtain $\\hat{\\tilde{\\Phi }} \\tilde{u}_m = \\omega _m^2 \\tilde{u}_m$ with $\\hat{\\tilde{\\Phi }} =- {z} \\frac{c^2}{n^2(z)} {z}\\quad \\text{and} \\quad \\tilde{u}_m = {u_m}{z}.$ Now the operator $\\hat{\\tilde{\\Phi }}$ is self-adjoint, which ensures that the functions $\\tilde{u}_m(z)$ constitute a complete orthonormal set of eigenfunctions and the eigenvalues $\\omega _m^2$ are real.",
"Using a partial integration, the orthonormality condition can be rewritten as $\\delta _{km}&= \\int \\tilde{u}_k^* \\tilde{u}_m\\, {z} = \\int \\tilde{u}_k^* {u_m}{z} \\, {z} \\nonumber \\\\ &= - \\int {\\tilde{u}_k^*}{z} u_m(z)\\, {z}= - \\int [2]{u_k^*}{z} u_m(z)\\, {z},$ showing that the function $v_k^*(z) = - d^2u_k^*(z)/dz^2$ is the orthogonal eigenfunction to $u_k(z)$ .",
"Here we have assumed either periodic boundary conditions or vanishing mode functions far away from the cavity, which is well satisfied due to the high quality factor of the cavity.",
"Then we can expand the electric field into the eigenmodes $u_m(z)$ , $E(z, t) = \\sum _m E_m(t) u_m(z) \\, .$ Using eq:aeigenmodeexpansion and the orthogonality of $v_k$ and $u_m$ , we can rewrite the wave equation eq:awaveequation as $[2]{t} E_k(t) + \\omega _k^2 E_k(t) = -\\frac{v_k^*(z_0)}{n^2(z_0)\\epsilon _0} [2]{t} \\tilde{P}(t)\\, ,$ where we inserted the polarization $\\bar{P}_\\text{QD} = \\tilde{P}(t) \\delta (z-z_0)$ .",
"Assuming a sufficiently sharp QD ensemble with transition energies close to the frequency of one of the cavity modes and width much smaller than the mode separation to the other cavity modes, only a single cavity mode $E_c$ with frequency $\\omega _c= \\omega _0$ is effectively coupled to the QDs.",
"Defining $E_c(t) = \\frac{1}{2} \\left[\\tilde{E}_c(t) e^{-i\\omega _0 t}+\\tilde{E}^*_c(t) e^{i\\omega _0 t}\\right]$ and $p(\\omega _x, t) = \\tilde{p}(\\omega _x, t) e^{-i\\omega _0 t}$ we apply both the rotating-wave approximation (RWA) and the slowly-varying-amplitude approximation.",
"Within these approximations we obtain the equation of motion for the field amplitude ${t} \\tilde{E}_c(t) &= i \\tilde{M} \\lambda \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\tilde{p}(\\omega _x, t) \\,,$ which is complemented by the equations of motion for $f(\\omega _x, t)$ and $\\tilde{p}(\\omega _x, t)$ ${t} f(\\omega _x, t) &= -i \\tilde{M} \\left[\\tilde{E}_c^*(t) \\tilde{p}(\\omega _x, t) - \\tilde{E}_c(t) \\tilde{p}^*(\\omega _x, t) \\right] \\\\{t} \\tilde{p}(\\omega _x, t) &= -i(\\omega _x - \\omega _0)\\tilde{p}(\\omega _x, t) \\nonumber \\\\&-i\\tilde{M} \\tilde{E}_c(t) \\left[ 2f(\\omega _x, t) - 1 \\right] \\, .$ Here we have introduced the abbreviations $\\tilde{M} = M u_c(z_0)/(2\\hbar )$ and $\\lambda = \\frac{v_c^*(z_0)}{u_c(z_0)}\\frac{N_\\text{QD}}{A} \\frac{2\\hbar \\omega _0}{n^2(z_0)\\epsilon _0}$ .",
"The constants defined via material parameters are given in Sec.",
", while $u_c(z)$ and $v_c(z)$ can be either calculated using FDTD or by solving eq:adefu for the normal modes.",
"Equations () are a set of coupled nonlinear equations of motion, which define the dynamics of $\\tilde{E}_c(t)$ , $f(\\omega _x, t)$ and $\\tilde{p}(\\omega _x, t)$ by their initial values.",
"In this section, we consider $\\rho _\\text{QD}(\\omega _x) = \\delta (\\omega _x - \\omega _0)$ and the initial values at $t=0$ are $\\tilde{E}_c(0) = \\tilde{E}_0$ , $f(\\omega _x, 0) = 0$ and $\\tilde{p}(\\omega _x, 0) = 0$ in accordance with our numerical simulations.",
"Indeed, when we numerically solve eq:eomanalytical for different initial values $\\tilde{E}_0$ , we find basically the same dynamical and spectral behavior of this system as described in Sec.",
"when tuning the pulse amplitude.",
"Therefore we do not repeat the corresponding figures.",
"The set of equations can be further simplified by introducing nondimensionalized quantities $\\alpha (\\tau ) &= \\frac{\\tilde{M} \\tilde{E}_c(\\tau )}{\\sqrt{\\tilde{M}^2 \\lambda }},~~&\\beta (\\tau ) &= (2f(\\tau )-1), \\\\\\gamma (\\tau ) &= ip(\\tau ),~~&\\tau &= \\sqrt{\\tilde{M}^2 \\lambda }\\, t\\, .$ Thereby we assumed that $\\tilde{E}_c$ is real, such that the polarization is purely imaginary and $\\alpha $ , $\\beta $ , and $\\gamma $ are real functions.",
"Then the new equations of motion read ${\\tau } \\alpha (\\tau ) &= \\gamma (\\tau ) \\\\{\\tau } \\beta (\\tau ) &= -4\\alpha (\\tau ) \\gamma (\\tau ) \\\\{\\tau } \\gamma (\\tau ) &= \\alpha (\\tau ) \\beta (\\tau ) .$ It is interesting to note that we now have eliminated all system parameters in the equations of motion, which makes them a prototypical example of coupled, nonlinear differential equations.",
"The initial values at $\\tau =0$ are given by $\\alpha (\\tau =0) = \\alpha _0$ , $\\beta (\\tau =0) = \\beta _0=-1$ and $\\gamma (\\tau =0)=\\gamma _0=0$ .",
"Here $\\alpha $ is a measure for the electric field strength at the location of the QDs.",
"Let us start by discussing the dynamics in the limiting cases of a very low and a very high initial value of $\\alpha $ .",
"For low $\\alpha _0$ , we find that $2f = \\beta - \\beta _0$ is quadratic in the electric field $E \\sim \\alpha $ .",
"Restricting ourselves to the linear regime, we neglect the differential equation for $\\beta $ and set $\\beta = \\beta _0 = -1$ , resulting in two equations for $\\alpha $ and $\\gamma $ eq:aalpha,eq:agamma, which can be reduced to $[2]{\\tau } \\alpha (\\tau ) &= -\\alpha (\\tau ) \\nonumber \\,.$ This is the equation of a harmonic oscillator with a constant frequency of unity.",
"In the dimensionalized problem, the oscillation frequency is $\\Omega _p = \\pm \\tilde{M} \\sqrt{\\lambda }$ .",
"This is in agreement with the numerical findings in sec:resultsnum.",
"Indeed, this corresponds to the splitting in two exciton-polariton states as seen in the FDTD calculations (lower part of fig3).",
"In the case of a high $\\alpha _0$ the influence of the macroscopic polarization is small, such that we approximate ${\\tau } \\alpha \\rightarrow 0$ and $\\alpha \\rightarrow \\alpha _0$ .",
"Then eq:abeta,eq:agamma can be written as a harmonic oscillator $[2]{\\tau } \\beta (\\tau ) &= -4 \\alpha _0 \\beta (\\tau ) \\nonumber \\,.$ with frequency $\\Omega _\\text{R} = \\pm 2 {\\alpha _0}$ or transforming back with $ \\Omega _\\text{R}= \\pm 2 {\\tilde{M}\\tilde{E}}$ .",
"This oscillation weakly couples back to $\\alpha $ leading to the Rabi splitting, which has been found in the FDTD calculations in the upper part of fig3.",
"The analytical model allows us to furthermore understand the qualitative difference in the two regimes and also the transition region of fig3.",
"For this purpose, we first combine eq:aalpha and eq:abeta ${\\tau } \\beta (\\tau ) &= -4\\alpha (\\tau ) \\gamma (\\tau ) = -4\\alpha (\\tau ) {\\tau }\\alpha (\\tau ) \\nonumber $ and then solve by integration $\\Rightarrow \\beta (\\tau ) &= -1 - 4 \\int \\limits _{0}^{\\tau } {t^{\\prime }} \\alpha (t^{\\prime }) {t^{\\prime }}\\alpha (t^{\\prime }) \\nonumber \\\\&= -1 - 2 \\alpha ^2(\\tau ) + 2 \\alpha _0^2 \\, ,$ where we made use of the initial condition $\\beta _0 = -1$ .",
"Next we combine eq:aalpha and eq:agamma to $[2]{\\tau } \\alpha (\\tau ) = {\\tau } \\gamma (\\tau ) = \\alpha (\\tau ) \\beta (\\tau ) \\nonumber $ and use eq:betaint to obtain $[2]{\\tau } \\alpha (\\tau ) = -2\\alpha (\\tau )^3 + \\left(2 \\alpha _0^2 - 1 \\right) \\alpha (\\tau ) \\, .$ This is a Newton type equation of motion $[2]{\\tau } \\alpha = -{\\alpha } V(\\alpha )$ for a particle moving in a potential $V(\\alpha )$ defined as $V(\\alpha ) &= \\frac{1}{2} \\left( \\alpha ^4 - \\left( 2\\alpha _0^2 - 1 \\right) \\alpha ^2 \\right).",
"$ We stress that the potential $V$ itself already depends on the initial values of the system.",
"$V(x)=ax^4+bx^2$ is a typical potential leading to phase transitions when the single minimum splits into a double minimum.",
"However, in our case, we have no damping which would drive the system into one of the minima.",
"Instead, the initial condition determines both the potential shape and the starting point of the dynamics and thereby the dynamical behavior.",
"We have solved this equation numerically and extracted the spectrum of the electric field, i.e., $|\\alpha (\\omega _\\tau )|$ .",
"The resulting figure is indistinguishable from the one shown in fig3, confirming that the conditions for the reduction to the single cavity mode model and the various approximations discussed above are indeed almost perfectly satisfied.",
"Figure: Left: Potential VV for different initial values (red dots) of α 0 \\alpha _0.",
"The red dashed line marks the allowed values for the dynamics.Right: Resulting spectra of the corresponding dynamics at the values (a) α 0 =0.1\\alpha _0=0.1, (b) α 0 =0.8\\alpha _0=0.8, (c) α 0 ≈1\\alpha _0\\approx 1, (d) α 0 =1.1\\alpha _0=1.1.To get a deeper insight, in fig5 (left) we plot four distinct cases of the potential for different values of $\\alpha _0$ , the right column shows the corresponding spectrum $|\\alpha (\\omega )|$ .",
"Note that because $\\alpha $ describes the slowly varying envelope of the electric field the frequency axis is shifted such that the cavity mode now corresponds to the frequency zero.",
"The “particle” starts with zero velocity at the initial position $\\alpha _0$ marked by the red dot on the potential curve.",
"As in classical physics, every point of the graph below the red line can be reached during the dynamics.",
"In the case of a low initial field with $\\alpha _0 < {2}^{-1/2}$ (fig5 (a)), the potential has only one minimum and we have an oscillation between $-\\alpha _0$ and $\\alpha _0$ .",
"Accordingly the spectrum shows two separated peaks symmetrically around $\\omega =0$ .",
"For $\\alpha _0 > {2}^{-1/2}$ the potential splits and shows three extrema: a maximum at $\\alpha = 0$ with $V(0) = 0$ and two minima at $\\alpha = \\pm \\sqrt{\\frac{2\\alpha _0^2 - 1}{2}}$ .",
"For ${2}^{-1/2} < \\alpha _0 < 1$ (fig5 (b)) the initial condition is above the maximum, such that $\\alpha $ still oscillates symmetrically between $-\\alpha _0$ to $+\\alpha _0$ , however the velocity around $\\alpha =0$ is reduced, such the oscillation becomes strongly anharmonic.",
"The spectrum still consists of two main peaks symmetrically around zero and additional higher harmonics of these peaks which are, however, outside the plotted range.",
"The transition point is reached at $\\alpha _0 = \\alpha _\\text{t} = 1$ , where the unstable fixed point $\\alpha = 0$ (fig5 (c)) would be reached after infinite time.",
"The corresponding spectrum would be continuous, because no oscillation occurs.",
"Due to unavoidable numerical errors, however, this limiting case will in general not be exactly reached in numerical simulations.",
"Instead, after some rather long time either the barrier will be overcome, leading to a symmetric oscillation, or the motion turns around before reaching the unstable fixpoint, leading to an oscillation only in the range of positive alpha.",
"In both cases the spectrum consists of a series of equidistant peaks with small peak distance corresponding to the long oscillation period.",
"An example is shown on the right hand side of fig5 (c) where the simulation has been started nominally with $\\alpha _0 = 1$ .",
"The presence of a peak at $\\omega =0$ indicates that this spectrum actually corresponds to an asymmetric oscillation, i.e., the case slightly above the threshold.",
"If $\\alpha _0$ is further increased the dynamics take place only in the right valley and $\\alpha $ stays positive for all times (fig5 (d)).",
"The offset gives rise to a peak at $\\omega =0$ , in addition to the two side peaks from the oscillation.",
"The curvature at the minima $V^{\\prime \\prime }(\\alpha = \\sqrt{\\frac{2\\alpha _0^2 - 1}{2}}) = 2(2\\alpha _0^2 - 1)$ increases with $\\alpha _0$ and approaches the squared Rabi frequency $\\Omega _\\text{R}^2 = 4\\alpha _0^2$ for high $\\alpha _0$ .",
"This explains the transition to the Rabi-splitting in fig3, since the curvature equals the squared frequency of $\\alpha $ in harmonic approximation.",
"As discussed above, the analytical model gives us the possibility to identify the transition point $\\alpha _\\text{t}$ exactly by calculating $V(0) = V(\\alpha _\\text{t})$ , which is at $\\alpha _\\text{t}=1$ .",
"Going back to the electric field we find that the transition takes place when $\\alpha _\\text{t} = \\frac{\\tilde{M}\\tilde{E}_0}{\\sqrt{\\tilde{M}^2 \\lambda }} = \\frac{\\tilde{E}_0}{\\sqrt{\\lambda }} = 1 \\qquad \\Leftrightarrow \\tilde{E}_0 = \\sqrt{\\lambda } \\sim \\sqrt{N_\\text{QD}/A} \\,.",
"$ Interestingly, the transition point depends only on system parameters like the quantum dot density $N_\\text{QD}/A$ , the cavity frequency $\\omega _0$ or the refractive index at the position of the QDs $n(z_0)$ , but does not depend on the dipole matrix element."
],
[
"Influence of a QD distribution with non-vanishing width",
"The assumption of a QD ensemble where all QDs have the same transition energy is strongly idealized.",
"In a real sample there will always be a certain spread of transition frequencies.",
"Therefore, in this section we will investigate how the spectra change when the cavity mode interacts with a QD ensemble with a Gaussian distribution of transition frequencies with FWHM $\\Delta _{\\omega }$ according to $\\rho _\\text{QD}(\\omega ) = \\frac{2\\sqrt{\\ln (2)}}{\\sqrt{\\pi }\\Delta _\\omega } e^{-\\frac{4\\ln (2)(\\omega - \\omega _0)^2}{\\Delta _{\\omega }^2}}.$ The influence of such kind of inhomogeneous broadening on the polariton spectra has been investigated in Refs.",
"[43], [29] and an increasing broadening of the polariton lines with increasing width of the QD distribution has been found.",
"The question arises, whether also for increasing driving the spectra will just broaden and how the transition between the two regimes of low and high driving is affected by this broadened QD distribution.",
"Considering our analytical model (cf.",
"eq:analeom) and the nondimensionalization, the equations of motion change to ${\\tau } \\alpha (\\tau ) &= \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\gamma (\\omega _x,\\tau ) \\\\{\\tau } \\beta (\\omega _x, \\tau ) &= -2 \\left(\\alpha ^*(\\tau ) \\gamma (\\omega _x, \\tau ) + \\alpha (\\tau ) \\gamma ^*(\\omega _x, \\tau ) \\right) \\\\{\\tau } \\gamma (\\omega _x, \\tau ) &= - i\\frac{(\\omega _x-\\omega _0)}{\\sqrt{\\tilde{M}^2 \\lambda }} \\gamma (\\omega _x, \\tau ) + \\alpha (\\tau ) \\beta (\\omega _x, \\tau ) .$ Note that now $\\alpha $ and $\\gamma $ become complex quantities.",
"From these equations we calculate the spectra of the electric field $|E(\\omega )|$ depending on the initial value $\\alpha _0$ .",
"The results are shown in fig6 for two different values of the width of the QD distribution.",
"For a spectral width of $\\Delta _{\\omega } = 4\\,$ meV (fig6 (a)), similarly to fig3 we find a sharp transition at $\\alpha _0=\\alpha _\\text{t} \\approx 1$ between a polaritonic and a Rabi oscillation regime.",
"This shows that the transition found in the limiting case of a $\\delta $ -like ensemble is also present for finite widths.",
"However, in contrast to a simple broadening of the spectral lines we observe the appearance of additional narrow spectral lines below the threshold.",
"Also for a larger value of $\\Delta _{\\omega } = 7$ meV in fig6 (b), the threshold $\\alpha _\\text{t}$ is still present, but shifted to a lower value $\\alpha _\\text{t} < 1$ .",
"Below the threshold we find again additional lines; however, they appear only above another threshold $\\alpha _\\text{t2}$ .",
"In the following we will try to understand this quite surprising dependence of the spectra on the QD distribution.",
"Figure: Spectra |E(ω||E(\\omega | for initial values (a) α 0 =1.2\\alpha _0 = 1.2 and (b) α 0 =0.02\\alpha _0 = 0.02 and (c) spectrum at the resonance frequency |E(ω 0 )||E(\\omega _0)| as a function of the initial value α t \\alpha _\\text{t} for different ensemble widths as indicated above the plots.",
"(d) Transition point α t \\alpha _\\text{t} as a function of the ensemble width Δ ω \\Delta _{\\omega }.Let us start with the polaritonic regime, i.e., the case of a weak initial field $\\alpha _0 \\ll 1$ .",
"In this case, again the occupation can be neglected because it is quadratic in the field.",
"Thus, $\\beta \\approx \\beta _0 = -1$ .",
"The remaining equations (REF ) and () are then linear equations which can be solved by a Fourier-Laplace transform.",
"The formal result is rather lengthy and not very instructive, therefore we do not present the formula.",
"However, one can deduce that with increasing broadening the splitting between the polariton peaks slightly increases and the peaks broaden.",
"In addition, there is a broad maximum around $\\omega = \\omega _0$ which builds up with increasing broadening.",
"The spectra of the electric field at $\\alpha _0=0.02$ for different values of the broadening are shown in fig7 (b).",
"Here, two clear peaks appear for small ensemble widths $\\Delta _{\\omega }$ , but become broader and eventually vanish for larger $\\Delta _{\\omega }$ .",
"For $\\Delta _{\\omega }=15$ meV only a single broad peak is visible.",
"These spectra are in good agreement with spectra found for microcavity polaritons in disordered exciton lattices obtained on the basis of a model with a quantized light field [29], demonstrating that indeed in the present case quantum features of the light are of minor importance.",
"Above the threshold $\\alpha _\\text{t}$ again the Rabi oscillation regime is reached.",
"As shown in fig7 (a) for the case of $\\alpha _0=1.2$ , we obtain the three peak structure with a central peak at the cavity frequency and two side peaks split by the Rabi frequency.",
"Like in the polariton case, the splitting between the outer peaks slightly increases with increasing broadening.",
"This can be traced back to the fact that in a broadened ensemble there are QDs which are detuned from resonance and therefore exhibit a larger Rabi frequency than those at resonance.",
"The broadening of the polariton-like spectrum leads to the question, whether still a transition can be observed, even for ensembles with larger $\\Delta _{\\omega }$ .",
"To answer this question, we show in fig7 (c) the spectrum at the cavity frequency $|E(\\omega _0)|$ as function of the initial value $\\alpha _0$ .",
"For the $\\delta $ -like ensemble ($\\Delta _{\\omega }=0$ ) we find, as already discussed, a sharp increase of this amplitude at $\\alpha _0=1$ , which is also found for small ensemble widths.",
"Interestingly, also in the case of broad ensembles a sharp rise of the amplitude is found, indicating that still a transition between two regimes takes place.",
"We further observe that the transition $\\alpha _\\text{t}$ decreases to smaller values of $\\alpha _0$ with increasing $\\Delta _{\\omega }$ .",
"This is also quantified in fig7 (d), where we plot $\\alpha _\\text{t}$ as function of the ensemble width $\\Delta _{\\omega }$ .",
"As we have seen in the discussion of fig5 the transition is associated with the fact that $\\alpha $ (in other words, the envelope of the electric field) does not reach zero anymore but oscillates only in the region of positive values.",
"This leads to the appearance of the central peak.",
"To demonstrate that this occurs necessarily also in the ensemble, we can introduce the total inversion $B(\\tau )$ defined as $B(\\tau ) = \\int {\\omega _x} \\rho _\\text{QD}(\\omega _x) \\beta (\\omega _x, \\tau ) .$ With this definition we can combine eq:aalphae,eq:abetae to ${\\tau } \\left[2|\\alpha (\\tau )|^2 + B(\\tau ) \\right] = 0.$ Together with the initial conditions $\\alpha (0)=\\alpha _0$ and $B(0)=-1$ this leads to the energy conservation law $2|\\alpha (\\tau )|^2 + B(\\tau ) = 2\\alpha _0^2 -1 $ which is the direct generalization of eq:betaint to a broadened ensemble.",
"Since the inversion is limited to the range $-1 \\le B(\\tau ) \\le 1$ , we obtain $|\\alpha (\\tau )|^2 \\ge \\alpha _0^2 -1$ This is a proof that for $\\alpha _0 > 1$ the field envelope cannot vanish anymore and thus there is necessarily a peak at the cavity resonance $\\omega _0$ in $|E(\\omega )|$ .",
"In a broadened ensemble $B=1$ is not reached, because this would mean that all QDs are completely inverted, no matter how far they are detuned from resonance.",
"This explains our findings of fig7 (c) and (d).",
"Figure: (a) Maximum value of the inversion as a function of α 0 \\alpha _0 for ensembles of different widths Δ ω \\Delta _\\omega .",
"(b) Lower transition α t2 \\alpha _\\text{t2} (red squares) where the inversion at the polariton peaks reaches unity and upper transition α t \\alpha _\\text{t} (black circles) where the inversion at the cavity frequency reaches unity as functions of the ensemble width Δ ω \\Delta _\\omega .Finally we want to understand the origin of the additional lines in the spectra of fig6 starting from the polariton lines.",
"They are symmetric with respect to the polariton lines, which already indicates that they will originate from an additional modulation of the field amplitude.",
"Indeed it turns out that they are again related to Rabi oscillations, in contrast to the case above the threshold $\\alpha _\\text{t}$ , however, these are not Rabi oscillations of the QDs in resonance with the cavity mode but Rabi oscillations of the QDs in resonance with the polariton frequencies.",
"fig8 (a) shows the maximum of the inversion $\\beta _\\text{max}$ that is reached in the ensemble as a function of the initial field amplitude $\\alpha _0$ for different values of the ensemble width.",
"The maximum inversion increases with increasing $\\alpha _0$ until it reaches unity.",
"It turns out that this maximum is indeed reached at the polariton frequencies, since there the driving of the QDs is strongest.",
"As soon as the maximum inversion reaches unity, Rabi oscillations of the QDs in resonance with the polaritons set in leading to a modulation of the electric field and thus the observed side peaks of the polaritons.",
"The splitting between these side peaks increases with $\\alpha _0$ , as in the case of the Rabi oscillation above $\\alpha _\\text{t}$ .",
"The onset of these polariton Rabi oscillations, i.e., the point where the inversion reaches unity, occurs at a second, lower threshold $\\alpha _\\text{t2}$ which increases with increasing ensemble width, as is shown in fig8 (b) together with the upper threshold $\\alpha _\\text{t}$ already discussed in fig7 (d).",
"This increase reflects the fact that the number of QDs at the polariton frequency increases with increasing ensemble width, such that a larger initial field is required to invert all these QDs.",
"At a width of about 10 meV the two thresholds merge.",
"Here the broadening of the polariton lines becomes so strong (see fig7 (b)) that an inversion of unity is first reached at the cavity frequency, which defines the threshold $\\alpha _\\text{t}$ ."
],
[
"Conclusion",
"In this work we have calculated the dynamics of a combined QD-cavity system driven by an external laser pulse in a semiclassical model.",
"For the numerical calculations we employed an FDTD method with incorporated two-level systems.",
"First we studied a system with identical QDs having transition energies in resonance with the cavity mode.",
"Depending on the excitation power we found a fundamentally different behavior of the spectrum of the QD-cavity system when varying the driving strength: For low pulse amplitudes we found exciton-polariton-like states with two amplitude-independent peaks in the spectrum.",
"High pulse areas resulted in typical Rabi oscillations, where the cavity mode has the most dominant part in the spectra and the field induced by the QDs is small.",
"Accordingly, the spectrum shows three peaks, where the side peaks are given by the Rabi splitting.",
"Interestingly, in between these two regimes there is a sharp transition.",
"We identify the sharp transition in the numerical model by the point, where the envelope of the electric field does not reach zero anymore.",
"We furthermore have derived an analytical model resulting in a set of nonlinear coupled equations, which describe the dynamics found in almost perfect agreement with the FDTD simulation.",
"Our analytical model allowed us to interpret the different regimes.",
"In particular, we were able to explain the transition by showing that the field dynamics can be mapped to Newton-like dynamics in a fourth order potential.",
"The shape of the potential depends on the initial condition, changing from a single well potential for weak driving to a double well potential for strong driving.",
"The transition occurs when the driving reaches a value such that the oscillations remain in one of the two minima and thus, as found in the FDTD simulations, when the field amplitude does not reach zero anymore.",
"Close to the transition the dynamics are strongly anharmonic leading to higher harmonics in the spectrum.",
"We finally extended the analytical model to account for finite ensemble widths and showed that at low driving the polariton lines broaden and slightly shift.",
"However, the sharp transition to the Rabi oscillation regime exists independent of the ensemble width.",
"Interestingly, instead of seeing just a broadening, in the polariton regime we observe new lines above a second, lower threshold, which could be traced back to Rabi oscillations of the QDs in resonance with the polariton lines.",
"Our paper thus provides an intuitive model of the coupled cavity-QD system with and without broadening as a prototypical example of a nonlinear system."
],
[
"Reduction to the circularly polarized light field",
"According to eq:Ecirc a $\\sigma _{+}$ ($\\sigma _{-}$ ) circularly polarized light field with central frequency $\\omega _0$ is given by $E(z,t) = \\frac{1}{\\sqrt{2}} \\left[ E\\left(z,t\\right) e_x \\pm E\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e_y \\right].$ Separating the field in its positive and negative frequency components according to $E\\left(z,t\\right) = \\frac{1}{2} \\left[ \\tilde{E}(z,t) e^{-i \\omega _0 t} + \\tilde{E}^*(z,t) e^{i \\omega _0 t}\\right]$ the field reads $E(z,t) &= \\frac{1}{2\\sqrt{2}} \\biggl \\lbrace \\left[ \\tilde{E}\\left(z,t\\right) e^{-i \\omega _0 t}+ \\tilde{E}^*\\left(z,t\\right) e^{i \\omega _0 t} \\right] e_x \\nonumber \\\\& \\qquad \\pm \\left[ \\tilde{E}\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e^{-i \\omega _0 t +i\\frac{\\pi }{2}} \\right.",
"\\nonumber \\\\ & \\left.",
"\\qquad \\quad + \\tilde{E}^*\\left(z,t-\\frac{\\pi }{2\\omega _0}\\right) e^{i \\omega _0 t -i\\frac{\\pi }{2}} \\right] e_y \\biggr \\rbrace .$ Assuming that the envelope is slowly varying on the oscillation period of the field, i.e., $\\tilde{E}\\left(z,t\\right) \\approx \\tilde{E}\\left(z,t - \\frac{\\pi }{2\\omega _0}\\right) ,$ the real electric field can be written as $E(z,t) &= \\frac{1}{2\\sqrt{2}} \\left[ \\left( e_x \\pm ie_y \\right)\\tilde{E}\\left(z,t\\right) e^{-i \\omega _0 t} \\right.",
"\\nonumber \\\\& \\left.",
"\\qquad \\quad + \\left( e_x \\mp ie_y \\right)\\tilde{E}^*\\left(z,t\\right) e^{i \\omega _0 t} \\right] \\nonumber \\\\&= \\frac{1}{2} \\left[ e_\\pm \\tilde{E}\\left(z,t\\right) e^{-i \\omega _0 t}+ e_\\mp \\tilde{E}^*\\left(z,t\\right) e^{i \\omega _0 t} \\right] .$ This shows that indeed the positive frequency component is proportional to the polarization vector $e_\\pm $ .",
"The macroscopic polarization of the $\\sigma _{+}$ ($\\sigma _{-}$ ) circularly polarized exciton in a QD reads (for simplicity here we omit the superscript $(n)$ labeling the QD) $P(t) &= M \\left[ e_\\pm p + e_\\mp p^* \\right] \\nonumber \\\\&= \\frac{1}{\\sqrt{2}} M \\left[ e_x \\left( p + p^* \\right) \\pm i e_y \\left( p - p^* \\right) \\right]\\, .$ Using $p(t) = \\tilde{p}(t) e^{-i\\omega _x t}$ and assuming again a slowly varying envelope $\\tilde{p}(t) \\approx \\tilde{p}\\left(t-\\frac{\\pi }{2\\omega _x}\\right)$ we have $ip(t) &= \\tilde{p}(t) e^{-i\\omega _x t+i\\frac{\\pi }{2}} \\approx p\\left( t-\\frac{\\pi }{2\\omega _x} \\right) \\\\-ip^*(t) &= \\tilde{p}^*(t) e^{i\\omega _x t-i\\frac{\\pi }{2}} \\approx p^*\\left( t-\\frac{\\pi }{2\\omega _x} \\right) .$ This leads to $P(t) &= \\frac{1}{\\sqrt{2}} M \\biggl \\lbrace e_x \\biggl [ p(t) + p^*(t) \\biggr ] \\nonumber \\\\& \\pm e_y \\left[ p\\left( t-\\frac{\\pi }{2\\omega _x} \\right) + p^*\\left( t-\\frac{\\pi }{2\\omega _x} \\right) \\right] \\biggr \\rbrace $ showing that with $P(t) = M \\left[ p(t) + p^*(t) \\right]$ we obtain $P(t) &= \\frac{1}{\\sqrt{2}} \\left[ P(t) e_x \\pm P\\left( t-\\frac{\\pi }{2\\omega _x} \\right) e_y \\right],$ which has the same structure as eq:Ecirc for the electric field.",
"Therefore, $P(t)$ is the source for the field $E(t)$ ."
]
] | 2005.14129 |
[
[
"On bid and ask side-specific tick sizes"
],
[
"Abstract The tick size, which is the smallest increment between two consecutive prices for a given asset, is a key parameter of market microstructure.",
"In particular, the behavior of high frequency market makers is highly related to its value.",
"We take the point of view of an exchange and investigate the relevance of having different tick sizes on the bid and ask sides of the order book.",
"Using an approach based on the model with uncertainty zones, we show that when side-specific tick sizes are suitably chosen, it enables the exchange to improve the quality of liquidity provision."
],
[
"Introduction",
"The tick size is the smallest increment between two consecutive prices on a trading instrument.",
"It is fixed by the exchange or regulator and typically depends on both the price of the asset and the traded volume, see [16], [18].",
"It is a crucial parameter of market microstructure and its value is often subject of debates: a too small tick size leads to very frequent price changes whereas a too large tick size prevents the price from moving freely according to the investor's views.",
"In this article, we focus on so-called large tick assets, that is assets for which the spread is most of the time equal to one tick.",
"Such assets represent a large number of financial products, especially in Europe since MIFID II regulation, see [18].",
"The tick size has a major influence on the ecosystem of financial markets, in particular on the activity of high frequency traders.",
"Being usually considered as market makers, these agents are the main liquidity providers for most heavily traded financial assets.",
"This means that they propose prices at which they are ready to buy (bid price) and sell (ask price) units of financial products.",
"In [13], the authors investigate the behavior of high frequency traders with respect to the relative tick size, which is defined as the ratio between the tick size and the price level.",
"One of their findings is that everything else equal, stocks with a lower relative tick size attract a greater proportion of high frequency traders, see also [10], [19].",
"This is because they can rapidly marginally adjust their quotes to seize price priority.",
"In the case of a large tick asset, speed is still an important feature as market participants have to compete for queue priority in the order book, see [17], [20].",
"Market makers (typically high frequency traders) face a complex optimization problem: making money out of the bid-ask spread (the difference between the bid and ask prices) while mitigating the inventory risk associated to price changes.",
"This problem is usually addressed via stochastic control theory tools, see for example [2], [7], [8], [14], [15].",
"In classical market making models, the so-called efficient price, which represents the market consensus on the value of the asset at a given time, around which the market maker posts his quotes, is a continuous semi-martingale.",
"The quotes of the market maker are continuous in terms of price values and not necessarily multiple of the tick size.",
"However, in actual financial markets, transaction prices are obviously lying on the discrete tick grid.",
"This discreteness of prices is a key feature which cannot be neglected at the high frequency scale since it plays a fundamental role in the design of market making strategies in practice.",
"To get a more realistic market making model, one therefore needs to build a relevant continuous-time price dynamic with discrete state space to take into account this very important microstructural property of the asset.",
"To this end, we borrow the framework of the model with uncertainty zones introduced in [21], [22].",
"In this model, transaction prices are discrete and the current transaction price is modified only when the underlying continuous efficient price process crosses some predetermined zones.",
"In our approach, we also consider that there exists an efficient price that market participants have in mind when making their trading decisions.",
"Based on this efficient price, market participants build “fair” bid and “fair” ask prices.",
"These two prices are lying on the tick grid and represent the views of market participants on reasonable and tradable values for buying and selling, regardless of any inventory constraint.",
"In our setting, depending on his views and his inventory constraint, the market maker chooses whether or not to quote a constant volume at these fair bid and ask prices.",
"This is a stylized viewpoint as in practice the market maker will probably quote a larger spread rather than not quoting at all.",
"The market maker increases (resp.",
"decreases) his current “fair” bid price if the efficient price becomes “sufficiently” higher (resp.",
"lower) than his current fair bid price and similarly for the ask side.",
"The mechanism to determine whether the efficient price is sufficiently higher (resp.",
"lower) than the current price is that of the model with uncertainty zones, described in Section .",
"Usual market making models include a symmetric running penalty for the inventory process, often defined as $\\phi \\int _0^T Q_t^2 dt$ where $Q_t$ is the inventory of the market maker at time $t\\in [0,T]$ , $\\phi >0$ is a risk aversion parameter and $T$ is the end of the trading period.",
"It is well-known, see for example [1], that for regulatory and operational reasons, market participants and especially market makers are reluctant to have a short inventory at the end of the trading day.",
"This is mainly due to constraints imposed by the exchange/regulator and to the overnight repo rate that they have to pay.",
"This asymmetry between long and short terminal inventory of the market maker gives the intuition of the potential relevance of some kind of asymmetry in the market design between buy and sell orders.",
"If some kind of asymmetry is implemented at the microstructure level, it can have important consequences on the profit of exchanges, as it notably depends on the number of processed orders.",
"Typical ways to optimize the number of orders on platforms are the choice of relevant tick sizes and suitable fee schedules (which subsidize liquidity provision and tax liquidity consumption).",
"In [12], the authors highlight the importance of differentiating maker and taker fees in order to increase the trading rate.",
"In the more recent studies [3], [11], optimal make-take fees schedules are designed based on contract theory.",
"In this work, the asymmetry we consider is not between liquidity consumers and liquidity providers but between buyers and sellers.",
"The goal of this paper is to show the possible benefits for an exchange in terms of liquidity provision of side-specific tick sizes.",
"To this end, we build an agent-based model where a high frequency market maker acts on a large tick asset.",
"The exchange is mitigating the activity on its platform by choosing suitable tick sizes on the bid and ask sides.",
"This means we have a different tick grid for buy and sell orders.",
"For given the tick sizes chosen by the exchange, we formulate the stochastic control problem faced by the market maker who needs to maximize his Profit and Loss (PnL for short) while controlling his inventory risk, taking into account asymmetry between short and long inventory.",
"We show existence and uniqueness of a viscosity solution to the Hamilton-Jacobi-Bellman (HJB for short) equation associated to this problem.",
"Then, we derive a quasi-closed form for the optimal controls of the market maker (up to the value function).",
"In particular, the role of the tick size in the decision of whether or not to quote is explicit: essentially, a large tick size implies a large profit per trade for the market maker but less market orders coming from market takers, and conversely.",
"Next, we solve the optimization problem of the exchange which can select optimal tick sizes knowing the associated trading response of the market maker.",
"In our model, the exchange earns a fixed fee when a transaction occurs.",
"Therefore, its remuneration is related to the quality of the liquidity provided by the market maker on its platform.",
"Numerical results show that side-specific tick sizes are more suitable than symmetric ones both for the market maker and the exchange.",
"The former is able to trigger more alternations in the sign of market orders, which is beneficial both for spread pocketing and inventory management (in contrast with the case where sequences of buy orders are followed by sequences of sell orders).",
"The latter increases the number of transactions on its platform.",
"We also show that a tick size asymmetry can offset short inventory constraints, therefore increasing the gains of both the market maker and the exchange.",
"The paper is organized as follows.",
"In Section , we give a reminder on the model with uncertainty zones and explain how we revisit it for market making purposes.",
"The market maker and exchange's problems are described in Section .",
"We also state here our results about existence and uniqueness of a viscosity solution associated to the control problem of the market maker and derive its optimal controls.",
"Finally, Section is devoted to numerical results and their interpretations.",
"Proofs are relegated to an appendix."
],
[
"The model with uncertainty zones",
"In this section, we provide a reminder on the model with uncertainty zones introduced in [21], [22], and we adapt it to the framework of a market making problem with side-specific tick values.",
"It is commonly admitted that low frequency financial price data behave like a continuous Brownian semi-martingale.",
"However this is clearly not the case for high frequency data.",
"The model with uncertainty zones reproduces sparingly and accurately the behavior of ultra high frequency transaction data of a large tick asset.",
"It is based on a continuous-time semi-martingale efficient price and a one dimensional parameter $\\eta \\in [0,\\frac{1}{2}]$ .",
"The key idea of the model is that when a transaction occurs at some value on the tick grid, the efficient price is close enough to this value at the transaction time.",
"This proximity is measured through the parameter $\\eta $ .",
"We define the efficient price $(S_t)_{t\\in [0,T]}$ on a filtered probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ where $T$ is the trading horizon.",
"The logarithm of the efficient price $(Y_t)_{t\\in [0,T]}$ is an $\\mathcal {F}_t$ -adapted continuous Brownian semi-martingale of the form Yt=(St)=(S0)+0t asds + 0t s-dWs, where $W$ is an $\\mathcal {F}$ -Brownian motion, and $(\\sigma _t)_{t\\in [0,T]}$ is an $\\mathcal {F}$ -adapted process with càdlàg paths and $(a_t)_{t\\in [0,T]}$ is $\\mathcal {F}$ -progressively measurable.",
"Transaction prices lie on two fixed tick grids, defined by $\\lbrace k\\alpha ^a,k\\alpha ^b\\rbrace $ where $\\alpha ^a$ (resp.",
"$\\alpha ^b$ ) is the tick size on the ask (resp.",
"bid) side and $k\\in \\mathbb {N}$ .",
"For $0\\le \\eta ^i\\le \\frac{1}{2}$ and $i\\in \\lbrace a,b\\rbrace $ , we define the zone $U^i_k=[0,\\infty )\\times (d^i_k,u^i_k)$ with dik=(k+12-i)i, uik= (k+12+i)i.",
"Therefore $U_k^a$ is a band of size $2\\eta ^a\\alpha ^a$ around the ask mid-tick grid value $(k+\\frac{1}{2})\\alpha ^a$ and $U_k^b$ is a band of size $2\\eta ^b\\alpha ^b$ around the bid mid-tick grid value $(k+\\frac{1}{2})\\alpha ^b$ .",
"We call these bands the uncertainty zones.",
"The zones on the bid and ask sides are characterized by the parameters $\\eta ^b,\\eta ^a$ which control the width of the uncertainty zones.",
"We will see in the next section how the fair bid and ask prices are deduced from the efficient price dynamics across the uncertainty zones.",
"In particular, the larger $\\eta ^i$ , the farther from the last traded price (on the bid or ask side) the efficient price has to be so that a price change occurs.",
"The idea behind the model with uncertainty zones is that, in some sense, market participants feel more comfortable when the asset price is constant than when it is constantly moving.",
"However, there are times when the transaction price has to change because they consider that the last traded price value is not reasonable anymore.",
"For sake of simplicity, we assume that transaction prices cannot jump by more than one tick.",
"We also define the time series of bid and ask transaction times leading to a price change as $(\\tau ^b_j,\\tau _j^a)_{j\\ge 0}$ .",
"The last traded bid or ask price process is characterized by the couples of transaction times and transaction prices with price changes $(\\tau _j; P_{\\tau _j^i}^i)_{j\\ge 0}$ where $P_{\\tau ^i_j}^i=S_{\\tau ^i_j}^{(\\alpha ^i)}$ , the superscript $(\\alpha ^i)$ denoting the rounding to the nearest $\\alpha ^i$ .",
"The dynamics of the $(\\tau _j^i)$ will be described in Section .",
"One can actually show that the efficient price can be retrieved from transaction data using the equation Sji=Sji(i)-i (12-i)sgn(Sji(i)-Sj-1i(i)), i{a,b}, jN.",
"This formula is particularly useful in order to derive ultra high frequency estimators of volatility and covariation (see [22]).",
"The parameters $\\eta ^i$ can be estimated very easily.",
"Let $N_{\\alpha ^i,t}^{(a)}$ and $N_{\\alpha ^i,t}^{(c)}$ be respectively the number of alternations and continuationsAn alternation/continuation corresponds to two consecutive price changes in the opposite/same direction.",
"of one tick over the period $[0,t]$ .",
"Then, an estimator of $\\eta ^i$ over $[0,t]$ is given by i,t=Ni,t(c)2Ni,t(a).",
"We refer to [21], [22] for further details on these estimation procedures.",
"In this paper, we use the model with uncertainty zones for market making purposes rather than for statistical estimation."
],
[
"The market maker's problem",
"We consider a high frequency marker maker acting on an asset whose efficient price $S_t$ has the dynamics d St=dWt, where $\\sigma >0$ denotes the volatility of the asset.",
"He uses the model with uncertainty zones described earlier to materialize his views on the fair bid and ask prices.",
"He increases (resp.",
"decreases) his bid price if the efficient price is “sufficiently” higher (resp.",
"lower) than his current fair bid price.",
"The notion of “sufficiently” higher or lower is determined by the uncertainty zones parameters $\\eta ^a, \\eta ^b$ , and the tick sizes $\\alpha ^a,\\alpha ^b$ .",
"If $\\eta ^a$ is small (resp.",
"large), the market maker changes more (resp.",
"less) frequently his ask price, and similarly for the bid price with $\\eta ^b$ .",
"This leads to the following definition of fair bid and ask prices of the market maker $S^a,S^b$ :Note that we can have situations where the bid price is above the ask price.",
"However, recall that $S^a$ and $S^b$ are only views about the fair bid and ask prices under the constraint that they have to lie on the tick grids.",
"Sta=St-a+a 1{ St-Sat->(12+a)a}-a 1{ St-Sat-<-(12+a)a}, Stb=St-b+b 1{ St-Sbt->(12+b)b}-b 1{ St-Sbt-<-(12+b)b}.",
"Thus the fair bid (resp.",
"ask) is modified when the efficient price is close enough to a new tradable price on the tick grid with mesh $\\alpha ^b$ (resp.",
"$\\alpha ^a$ ).",
"Remark 3.1 Note that in the case $\\alpha ^a=\\alpha ^b,\\eta ^a=\\eta ^b$ , the fair best bid is equal to the fair best ask.",
"This means that at a given time, a buy or sell order would be at the same price.",
"In this situation, in our stylized view, the market maker would probably quote only on one side (bid or ask).",
"It is consistent with the standard form of the model with uncertainty zones, where, at a given time, transactions can only happen only on one side of the market, depending on the location of the efficient price.",
"Still, the market maker collects the spread from transactions occurring at different times as it is the case in practice.",
"We assume a constant volume of transaction equal to one.",
"The market maker can choose to be present or not for a transaction at the bid (with a price $S^b)$ or at the ask (with a price $S^a$ ).",
"The corresponding cash process at terminal time $T$ is given by XT=0T ( StadNta - StbdNtb), where the $N_t^i$ represent the number of transactions on the bid or ask side between 0 and $t$ .",
"In this framework, the inventory of the market maker is given by $Q_t = N_t^b - N_t^a \\in \\mathcal {Q}=[-\\tilde{q},\\tilde{q}]$ where $\\tilde{q}$ is the risk limit of the market maker.",
"For $i\\in \\lbrace a,b\\rbrace $ , the dynamics of $N_t^i$ is that of a point process with intensity (ti,Qt):=ti1+(i)21{(i)Qt > -q}, (i)= 1{i=a} - 1{i=b}.",
"The process $\\ell _t^i\\in \\lbrace 0,1\\rbrace $ is the market maker's control which lies in the set of $\\mathcal {F}-\\text{predictable}$ processes with values in $\\lbrace 0,1\\rbrace $ denoted by $\\mathcal {L}$ .",
"The parameter $\\kappa >0$ controls the sensitivity of the intensities to $\\alpha ^i$ , and $\\lambda >0$ is a scale parameter.",
"When the market maker does not want to be present on the bid (resp.",
"ask side) at the price $S^b$ (resp.",
"$S^a)$ he sets $\\ell ^b=0$ (resp.",
"$\\ell ^a=0$ ) and conversely.",
"In our large tick asset setting, the situation where the market maker is not present is a simplified way to model the case where the market maker's quote is higher than the best possible limit.",
"At a given time $t\\in [0,T]$ , when $\\ell _t^b=0$ (resp $\\ell _t^a=0)$ , the intensity of the point process $N_t^b$ (resp.",
"$N_t^a$ ) is equal to zero so that there are no incoming transactions.",
"In addition to this, market takers are more confident to send market orders when the tick size is small, as the market maker has more flexibility to adjust his bid and ask prices.When the tick size is smaller, the market takers are more willing to trade.",
"This does not necessarily lead to a higher number of orders as it depends on the market maker's presence.",
"This explains the decreasing shape of the intensities of market order arrivals from market takers with respect to the tick size.",
"The chosen parametric form for the intensities ensures no degenerate behavior when the tick size gets close to zero.",
"The marked-to-market value of the market maker's portfolio at time $t$ is defined as $Q_tS_t$ .",
"His optimization problem writes LE[XT +QT(ST-AQT) - 0T Qs2 ds - - 0T |Qs|21Qs<0ds], where $\\phi >0$ represents the risk-aversion parameter of the market maker, $\\phi _- >0$ is the additional risk aversion of the market maker toward short position on $[0,T]$ and $AQ_T^2$ , with $A>0$ , is a penalty term for the terminal inventory position regardless of its sign.",
"In this setting, the market maker wishes to hold a terminal inventory close to zero because of the quadratic penalty $AQ_T^2$ .",
"The term $\\phi \\int _0^T Q_s^2 ds$ penalizes long or short positions over the trading period.",
"Problem (REF ) can of course be rewritten as LE[ QT(ST-AQT) +0T {Ssa (as) - Ssb(bs) - Qs2 - - Qs21Qs<0 } ds ].",
"We define the corresponding value function $h$ defined on the open set D = { (Sa,Sb,S) a Z b Z R such that -( 12 + a)a < S-Sa < ( 12 + a)a).",
".",
"and -( 12 + b)b < S-Sb < ( 12 + b)b } by $h(t,S^a, S^b, S, q) = \\sup _{\\ell \\in \\mathcal {L}_t} \\mathbb {E}_{t,S^a, S^b, S,q}\\Big [ & Q_T(S_T-AQ_T)+\\!\\int _t^T\\!\\Big \\lbrace S_s^a \\lambda (\\ell ^a_s) - S_s^b \\lambda (\\ell ^b_s) \\\\& - \\phi Q_s^2 - \\phi _- Q_s^2\\mathbf {1}_{Q_s<0} \\Big \\rbrace \\!\\mathrm {d}s\\Big ],$ where $\\mathcal {L}_t$ denotes the restriction of admissible controls to $[t,T]$ .",
"We define the boundary $\\partial \\mathcal {D}$ of $\\mathcal {D}$ as D = { (Sa,Sb,S) a Z b Z R such that S-Sa = ( 12 + a)a and/or S-Sb = ( 12 + b)b .",
"}, and write $\\bar{\\mathcal {D}} = \\mathcal {D} \\cup \\partial \\mathcal {D} .$ For given $(S^a,S^b)$ , if $(S^a,S^b,S)\\in \\partial \\mathcal {D}$ , it means that $S$ corresponds to an efficient price value that triggers a modification of the fair bid or ask price.",
"The Hamilton-Jacobi-Bellman equation associated to this stochastic control problem is given by 0 = t h(t,Sa,Sb,S,q)-q2 - - q21q<0 + 122 SSh(t,Sa,Sb,S,q) + 1+(a)2a{0,1}{ a(Sa+h(t,Sa,Sb,S,q-a)-h(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b((-Sb)+h(t,Sa,Sb,S,q+b)-h(t,Sa,Sb,S,q))}, for $(t,S^a,S^b,S,q) \\in [0,T) \\times \\mathcal {D} \\times \\mathcal {Q},$ with terminal condition h(T,Sa,Sb,S,q)=q(S-Aq).",
"Let us consider the function $h$ defined in REF .",
"For $(t,S^a,S^b,S,q) \\in [0,T) \\times \\mathcal {\\partial }D \\times \\mathcal {Q}$ , and $(t_n,S^a_n,S^b_n,S_n,q_n)_{n\\in \\mathbb {N}}$ a sequence in $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ which converges to $(t,S^a,S^b,S,q)$ , we will show that $h(t_n,S^a_n,S^b_n,S_n,q_n)$ converges independently of the sequence and we denote by $h(t,S^a,S^b,S,q)$ its limit.",
"On $[0,T) \\times \\partial \\mathcal {D} \\times \\mathcal {Q}$ , we will show the following boundary conditions (which we will naturally impose for the solution of REF ): $0= &\\ \\mathbf {1}_{\\lbrace S-S^a=(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b<(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a+\\alpha ^a,S^b,S,q)-h(t,S^a,S^b,S,q) \\big )\\\\& + \\mathbf {1}_{\\lbrace S-S^a<(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b=(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a,S^b+\\alpha ^b,S,q)-h(t,S^a,S^b,S,q) \\big )\\\\& + \\mathbf {1}_{\\lbrace S-S^a=(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b=(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a+\\alpha ^a,S^b+\\alpha ^b,S,q)-h(t,S^a,S^b,S,q) \\big ) \\\\& + \\mathbf {1}_{\\lbrace S-S^a=-(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b>-(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a-\\alpha ^a,S^b,S,q)-h(t,S^a,S^b,S,q) \\big ) \\\\& + \\mathbf {1}_{\\lbrace S-S^a>-(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b=-(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a,S^b-\\alpha ^b,S,q)-h(t,S^a,S^b,S,q) \\big ) \\\\& + \\mathbf {1}_{\\lbrace S-S^a=-(\\frac{1}{2}+\\eta ^a)\\alpha ^a,\\ S-S^b=-(\\frac{1}{2}+\\eta ^b)\\alpha ^b\\rbrace }\\big (h(t,S^a-\\alpha ^a,S^b-\\alpha ^b,S,q)-h(t,S^a,S^b,S,q) \\big ).$ In other words, the value function varies continuously when the efficient price leaves an uncertainty zone and the prices $S^a$ and $S^b$ are modified.Note that, as the terminal condition does not depend on $S^a$ and $S^b$ , it also satisfies this boundary condition on $\\partial \\mathcal {D}$ .",
"In the following, we say that a function defined on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ satisfies the continuity conditions if it satisfies (REF ).",
"The following proposition is of particular importance for the existence and uniqueness of a viscosity solution associated to the control problem of the market maker.",
"Proposition 3.2 The function $h$ defined in Equation (REF ) is continuous on $\\mathcal {D}$ and satisfies the continuity conditions (REF ).",
"The proof is given in Appendix REF and relies on the specific structure of our model based on hitting times of a Brownian motion.",
"We now state the main theorem of this article, whose proof is relegated to Appendix REF .",
"Theorem 1 The value function $h$ is the unique continuous viscosity solution to Equation (REF ) on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ with terminal condition (REF ) and satisfying the continuity conditions.",
"The value function depends on five variables.",
"However, as $(S^a,S^b)$ takes value in $\\alpha ^a \\mathbb {N} \\times \\alpha ^b \\mathbb {N}$ , it can essentially be reduced to three variables as we now explain.",
"For any $(i,j)\\in \\mathbb {N} ^2 $ , we introduce the function $h^{i,j}$ defined on $[0,T] \\times \\underbrace{\\left(\\alpha ^a i - (\\frac{1}{2}+\\eta ^a)\\alpha ^a, \\alpha ^a i + (\\frac{1}{2}+\\eta ^a)\\alpha ^a \\right) \\cap \\left( \\alpha ^b j -(\\frac{1}{2}+\\eta ^b)\\alpha ^b, \\alpha ^b j + (\\frac{1}{2}+\\eta ^b)\\alpha ^b \\right)}_{=\\mathcal {D} _{i,j}} \\times \\mathcal {Q}$ by $h^{i,j}(t,S,q)= h(t,\\alpha ^a i, \\alpha ^b j,S,q)$ .",
"Then $h^{i,j}$ is the solution of the following HJB equation: 0 = t hi,j(t,S,q)-q2 - - (q)-21q<0 + 122 SShi,j(t,S,q) + 1+(a)2a{0,1}{ a(a i+hi,j(t,S,q-a)-hi,j(t,S,q))} + 1+(b)2b{0,1}{ b(-b j+hi,j(t,S,q+b)-hi,j(t,S,q))}, with terminal condition $h^{i,j}(T,S,q) = q(S-AQ)$ and natural Dirichlet boundary conditions for $S \\in \\partial \\mathcal {D} _{i,j}$ : hi,j(t,S,q) = hi+1,j(t,S,q) 1{S-a i=(12+a)a, S-b j<(12+b)b} + hi,j+1(t,S,q) 1{S-a i<(12+a)a, S-b j=(12+b)b} + hi+1,,j+1(t,S,q) 1{S-a i=(12+a)a, S-b j=(12+b)b} + hi-1,j(t,S,q) 1{S-a i=-(12+a)a, S-b j>-(12+b)b} + hi,j-1(t,S,q) 1{S-a i>-(12+a)a, S-b j=-(12+b)b} + hi-1,j-1(t,S,q) 1{S-a i=-(12+a)a, S-b j=-(12+b)b}.",
"From this, we derive the optimal controls of the market maker as a(t,i,j,S,q)=1{a i+hi,j(t,S,q-1)-hi,j(t,S,q)>0 }, b(t,i,j,S,q)=1{-b j+hi,j(t,S,q+1)-hi,j(t,S,q)>0 }.",
"The practical interest of Theorem REF is that it allows us to compute the value function and optimal controls based on a finite difference scheme.",
"Examples of computations of the value function are given in Section and Appendix REF .",
"Having described the problem of the market maker, we now turn to the optimization problem of the platform."
],
[
"The platform's problem",
"The market maker acts on a platform whose goal is to maximize the number of market orders on $[0,T]$ .",
"The intensities of arrival of market orders are functions of $\\ell ^a,\\ell ^b$ , which are themselves functions of $\\alpha ^a,\\alpha ^b$ .",
"We assume that the platform is risk-neutral and earns a fixed taker cost $c>0$ for each market order.More complex fee schedules can be handled in this framework.",
"We can for example add a component which is proportional to the amount of cash traded.",
"Therefore its optimization problem is defined as (a,b)R+2Ela,lb[XTp], given the optimal controls $(l^{\\star a},l^{\\star b})$ of the market maker and $X_t^p=c(N_t^a+N_t^b)$ .",
"It is easy to observe that this problem boils down to maximizing the function $v$ defined below over $\\mathbb {R}^2_+$ : v(a, b) := E [0T c{a(t,Sat, Sbt, St, qt)1+(a)2 + b(t,Sat, Sbt, St, qt)1+(b)2 }dt ].",
"Here we clearly see the tradeoff of the platform.",
"A small tick size $\\alpha ^a$ increase the term $(1+(\\kappa \\alpha ^a)^2)^{-1}$ .",
"This is because it attracts more buy market orders.",
"However, the optimal control $\\ell ^{\\star , a}$ is more often equal to zero: the gain of the market maker may be too small if he quotes at the price $S^a$ , therefore he regularly sets $\\ell ^{\\star ,a}=0$ .",
"The problem is similar on the bid side.",
"On the other hand, a large tick size increases the gain of the market maker if a transaction occurs, but decreases the number of market orders sent by market takers, hence decreasing the trading volume.",
"We study numerically this problem in the next section by computing the value of $v$ on a two dimensional grid and finding its maximum."
],
[
"Numerical results",
"In this section, we show from numerical experiments the benefits of side-specific tick values in terms of increase of their value function for both the market maker and the platform.",
"Also, we fix reference values $\\eta _0$ and $\\alpha _0$ .",
"From them, to choose the parameter $\\eta ^i$ associated to a given tick size $\\alpha ^i$ we use a result from [10] which gives the new value of the parameter $\\eta ^i$ in case of a change of tick size from $\\alpha _0$ to $\\alpha ^i$ .",
"This formula writes i = 00i.",
"In the following, we only consider values of $\\alpha ^a$ and $\\alpha ^b$ such that the underlying remains a large tick asset both on the bid and ask sides, that is $\\eta ^a\\le \\frac{1}{2},\\eta ^b\\le \\frac{1}{2}$ .",
"For the first experiments, we set $T=40 s$ , $\\overline{q}=5$ , $\\sigma = 0.01 s^{-1}$ , $A = 0.1,\\kappa =10,\\phi =0.005,\\lambda =4$ , $\\eta _0=0.3$ and $\\alpha _0=0.01$ which correspond to reasonable values to model a liquid asset.",
"To remain in the large tick regime, we investigate values of $\\alpha ^i$ satisfying $0.0045\\le \\alpha ^i\\le 0.05$ for $i=a,b$ ."
],
[
"Similar tick values on both sides",
"In this section we investigate the case where $\\alpha ^a=\\alpha ^b$ .",
"We plot in Figure REF the value functions of the market maker and the exchange, respectively $h$ and $v$ , for various values of $\\alpha = \\alpha ^a=\\alpha ^b$ .",
"We fix the efficient price $S=10.5$ , the inventory $q=0$ and we only consider values of $\\alpha $ so that $0.5/\\alpha \\in \\mathbb {N}$ .",
"Figure: Value function hh (on the left) and vv (on the right) for φ - =0\\phi _-=0 in blue, φ - =0.0005\\phi _-=0.0005 in orange, φ - =0.005\\phi _-=0.005 in green, as a function of α=α a =α b \\alpha = \\alpha ^a=\\alpha ^b.When $\\phi _- = 0$ , the value of the exchange reaches its maximum at $\\alpha \\simeq 0.012$ .",
"An increase of $\\phi _-$ leads to a reduction of the number of transactions.",
"However the optimal tick value for the exchange is not significantly modified.",
"The optimal tick value for the market maker is larger than that of the exchange.",
"This is because the exchange is only interested in attracting orders while the market maker's gain per trade (not taking into account the inventory risk) is linear with respect to the tick value.",
"The trade-off of the exchange is the following: on the one hand, he would like to implement a quite small tick value (to attract market orders) but on the other hand, he must ensure a reasonable presence of market maker.",
"When $\\phi _-$ increases, the value function of the market maker decreases, for all tick values.",
"This is no surprise since $\\phi _-$ corresponds to an inventory penalization, hence reducing the market maker's PnL.",
"In Figure REF , we substract the value function when $\\phi _-=0$ to the other value functions displayed in Figure REF .",
"We remark that for the market maker, the larger the tick the more significant the penalization of short inventory in terms of value function.",
"We observe the opposite phenomenon for the exchange: the difference is essentially slightly increasing with respect to $\\alpha $ .",
"In particular, we see a quite strong impact of the penalization on the value function of the exchange when the tick size is small.",
"Figure: Variation of the functions hh and vv (difference between φ - =0\\phi _-=0 in blue, φ - =0.0005\\phi _-=0.0005 in orange, φ - =0.005 \\phi _-=0.005 in green, and φ - =0\\phi _- = 0 as a function of α=α a =α b \\alpha = \\alpha ^a=\\alpha ^b.We now study the case of side-specific tick values."
],
[
"Side-specific tick values: additional opportunities for the market maker",
"We set $\\alpha ^b = 0.0124$ (optimal tick size in the non side-specific case) and let $\\alpha ^a$ vary.",
"We plot the value functions of the market maker and the exchange in Figure REF .",
"Figure: Value function hh (on the left) and vv (on the right) as functions of α a \\alpha ^a, for α b =0.0124\\alpha ^b = 0.0124, for φ - =0\\phi _-=0 in blue, φ - =0.0005\\phi _-=0.0005 in orange, φ - =0.005\\phi _-=0.005 in green.Again we observe that both value functions are decreasing with respect to $\\phi _-$ .",
"From the point of view of the market maker, having non side-specific tick values is sub-optimal, even in the case $\\phi _-=0$ .",
"This is because when the two tick values are different, it is possible for $S^a$ to be greater than $S^b$ and orders to arrive with the same intensities on both sides: the market maker can collect the spread.",
"It is not possible in the non side-specific case, where the market maker can only pocket the spread from buy and sell orders at two different times.",
"Side-specific tick values are also clearly beneficial for the exchange.",
"The transaction flow increases for $\\alpha ^a>\\alpha ^b$ because of the good liquidity provided by the market maker, and for $\\alpha ^a<\\alpha ^b$ because of the high number of incoming market orders.",
"Remark 4.1 Remark that with shifted grids (same tick values on both sides but with a grid shifted compared to the other), those additional opportunities for the market maker would remain.",
"In section REF , we will see however, that, from the point of view of the exchange, side-specific tick values are much more interesting."
],
[
"Side-specific tick values: effect of $\\phi _-$",
"We plot the two-dimensional value functions of the market maker and the exchange for side-specific tick values.",
"First we take $\\phi _-=0$ in Figure REF .",
"We note that the opportunity for the market maker mentioned above remains present for all tick values and that the value functions are symmetric around the axis $\\alpha ^b=\\alpha ^a$ (side-specific tick values are preferred).",
"Furthermore, we see that the exchange prefers smaller tick values than the market maker.",
"The optimal values for the exchange lie on an anti-diagonal which goes from $(\\alpha ^a=0.0045,\\alpha ^b=0.025)$ to $(\\alpha ^a=0.025,\\alpha ^b=0.0045)$ .",
"On this line the number of transactions varies little.",
"It seems however that the optimum is on the edges of the zone in which the asset remains large tick: the two couples $(\\alpha ^a,\\alpha ^b)$ mentioned above.",
"If the tick values are too large the intensities of the market orders become too small and the number of transactions diminishes.",
"If both ticks are too small, the market maker does not trade much because the gain per trade becomes too little compared to the inventory cost (recall that the intensity of market orders is upper bounded).",
"However, the case where one tick is quite small and the other is large is suitable for the market maker: for example, if $\\alpha ^a<\\alpha ^b$ his strategy is to be long and liquidate his long position fast if needed thanks to the small value of $\\alpha ^a$ which ensures a large number of incoming market orders.",
"This explains why the optimal tick values given by the exchange are side-specific and symmetric with respect to the axis $\\alpha ^a=\\alpha ^b$ .",
"More precisely, the choice of ticks $(\\alpha ^a=0.0045,\\alpha ^b=0.025)$ or $(\\alpha ^a=0.025,\\alpha ^b=0.0045)$ seems optimal.",
"Figure: Value function hh (on the left) and vv (on the right) as functions of α a \\alpha ^a and α b \\alpha ^b, for φ - =0 \\phi _-=0.We now plot in Figure REF the value function for $\\phi _- = 0.005$ .",
"This non-zero parameter implies a clear decrease of the value function of the market maker and the reduction of the number of transactions.",
"An important remark is that the value functions are no-longer symmetric around the axis $\\alpha ^b=\\alpha ^a$ .",
"Figure: Value function hh (on the left) and vv (on the right) as functions of α a \\alpha ^a and α b \\alpha ^b, for φ - =0.005\\phi _-=0.005.For clarity we plot in Figure REF the difference of the value functions when $\\phi _-=0.005$ and when $\\phi _-=0$ as a function of $\\alpha ^a,\\alpha ^b$ .",
"Figure: Difference between the value function hh (on the left) and vv (on the right) as functions of α a \\alpha ^a and α b \\alpha ^b, between the case φ - =0.005\\phi _-=0.005 and the case φ - =0\\phi _-=0.We see that the added component is not symmetric regarding to the axis $\\alpha ^b=\\alpha ^a$ and both the market maker and the exchange tend to prefer the case $\\alpha ^b>\\alpha ^a$ .",
"It is particularly clear for the market maker's problem where the difference between the values at $(\\alpha ^a=0.0045,\\alpha ^b=0.025)$ and $(\\alpha ^a=0.025,\\alpha ^b=0.0045)$ is approximately $0.03$ which is roughly $10\\%$ of the value function.",
"Indeed, as explained above, having $\\alpha ^b$ quite large and $\\alpha ^a$ rather small essentially ensures that the market maker can maintain a positive inventory all along the trading trajectory: attractive PnL for incoming buy orders and possibility to quickly reduce a positive inventory.",
"The exchange is also more satisfied by the choice $\\alpha ^b>\\alpha ^a$ .",
"To see that more clearly, we fix $\\alpha ^a=0.0045$ and plot in Figure REF the value functions $h$ and $v$ , as functions of $\\alpha ^b$ , for different values of $\\phi $ .",
"Figure: Value functions hh and vv for α a =0.0045\\alpha ^a=0.0045, as functions of α b \\alpha ^b, for different values of φ\\phi .The value function of the market maker is increasing in $\\alpha ^b$ .",
"This is the same phenomenon as already observed in Figure REF .",
"The value of the exchange has a maximum which is increasing in $\\phi _-$ : as the penalization gets more side-specific, the optimal tick values displayed by the exchange become more asymmetric.",
"Indeed, for $\\phi _-=0$ , the optimum is reached for $\\alpha ^b\\simeq 0.024$ , while for $\\phi _-=0.005$ , it is obtained for $\\alpha ^b\\simeq 0.034$ .",
"Note that a relevant tick value set by the exchange can compensate for his loss of value function due to an increase of $\\phi _-$ .",
"By choosing a new tick size optimally when going from $\\phi _-=0$ to $\\phi _-=0.005$ , the loss in value function is of $7\\%$ only.",
"Keeping $\\alpha ^b= 0.024$ would lead to a loss of $15\\%$ .",
"Note that the compensation can be total for the market maker (and even exceeds the loss) but is only partial for the exchange."
],
[
"Conclusion",
"A suitable choice of tick values by the exchange is a subtle equilibrium.",
"If the platform imposes the same tick value on the bid and ask sides, it has to be sufficiently large to ensure significant PnL per trade for the market maker and therefore good liquidity provision, and sufficiently small to attract market orders from market takers.",
"When allowing for side-specific tick values with no constraint on short inventory, the optimal tick values for the exchange are of the form $(\\alpha ^\\star _1,\\alpha ^\\star _2)$ or symmetrically $(\\alpha ^\\star _2,\\alpha ^\\star _1)$ with $\\alpha ^\\star _1<\\alpha ^\\star _2$ .",
"In this case, the market maker can take advantage from additional trading opportunities and increase his activity.",
"The exchange benefits from this situation because of the higher number of trades on his platform.",
"Moreover, when there is a penalty for short inventory positions of the market maker, there is only one optimal couple of tick values.",
"In this case, the market maker and subsequently the exchange prefer $\\alpha ^b>\\alpha ^a$ and the difference between $\\alpha ^a$ and $\\alpha ^b$ at the optimum becomes larger.",
"Finally, note that side-specific tick values could have subtle consequences in a multi-platform setting.",
"This issue is left for further study, as well as the situation where market takers are more strategic in their execution."
],
[
"Proof of Proposition ",
"First we prove the continuity of $h$ on ${\\cal D}\\times [0,T)$ .",
"Let $q\\in {\\cal Q}$ , $t_1\\in [0,T)$ , $(s^a, s^b, s_{1})\\in {\\cal D}$ .",
"Note that $\\lbrace s\\in \\mathbb {R}, (s^a,s^b,s)\\in {\\cal D}\\rbrace $ is an open interval containing $s_1$ , which we denote by $(s^{\\leftarrow },s^{\\rightarrow })$ .",
"If the process $S_t$ starts from a point $s\\in (s^{\\leftarrow },s^{\\rightarrow })$ with $S^a_t=s^a$ and $S^b_t=s^b$ , $S^a_t$ and $S^b_t$ will not jump as long as $S_t$ stays in $(s^{\\leftarrow },s^{\\rightarrow })$ .",
"We will prove that the function $(t,s)\\in [0,T)\\times (s^{\\leftarrow },s^{\\rightarrow })\\mapsto h(t,s^a,s^b,s,q)$ is continuous at $(t_1,s_1)$ .",
"We fix $\\eta >0$ .",
"There is a ball with positive diameter ${\\cal B}$ in $[0,T)\\times (s^{\\leftarrow },s^{\\rightarrow })$ centered on $(t_1,s_{1})$ and some $\\epsilon >0$ such that, if $(t_2,s_{2})\\in {\\cal B}$ , then E[1-t1|St1=s1]<, E[2-t2|St2=s2]<, P[1<T|St1=s1]>1-, P[2<T|St2=s2]>1-, andThese inequalities can be attained independently of the control $\\ell $ as $S$ is independent from $Q$ .",
"$&\\underset{\\ell \\in {\\cal L}}{\\inf } \\mathbb {P}[\\underset{t_1\\le s \\le \\tau ^1}{\\inf } Q_{s} = \\underset{t_1\\le s \\le \\tau ^1}{\\sup } Q_{s} = q|S_{t_1}=s_{1},Q_{t_1} = q]>1-\\eta , \\\\ &\\underset{\\ell \\in {\\cal L}}{\\inf }\\mathbb {P}[\\underset{t_2\\le s \\le \\tau ^2}{\\inf } Q_{s} = \\underset{t_2\\le s \\le \\tau ^2}{\\sup } Q_{s} = q|S_{t_2}=s_{2},Q_{t_2} = q]>1-\\eta ,$ where we write 1=T{tt1, Stt1,s1 = (s1s2)+ or Stt1,s1 = (s1s2)-}, 2=T{tt2, Stt2,s2 = (s1s2)+ or Stt2,s2 = (s1s2)-}.",
"The quantities $\\tau ^1$ and $\\tau ^2$ are stopping times such that $t_1\\le \\tau ^1\\le T$ a.s. and $t_2\\le \\tau ^2\\le T$ a.s. We impose s<(s1s2)-<s1<(s1s2)+< s, s<(s1s2)-<s2<(s1s2)+< s for any $(t_2,s_{2})\\in {\\cal B}$ by taking a smaller ball ${\\cal B}$ and a smaller $\\epsilon $ if necessary.",
"In particular, this tells us that if $(S_{t_1}, S^a_{t_1},S^b_{t_1}) = (s_{1},s^a,s^b)$ , $S^a_t$ does not jump between $t_1$ and $\\tau ^1$ .",
"Similarly, if $(S_{t_2}, S^a_{t_2},S^b_{t_2}) = (s_{2},s^a,s^b)$ , $S^b_t$ does not jump between between $t_2$ and $\\tau ^2$ .",
"Let some arbitrary $(t_2,s_{2})\\in {\\cal B}$ and $\\tau ^1$ and $\\tau ^2$ the associated stopping times.",
"Using the dynamic programming principle, we obtain h(t1,sa, sb, s1,q) = LE[ h(1,Sa1, Sb1, S1,Q1)+ t11 {- Qt2 - - Qt21Qt<0 } dt| St1 = s1, Sat1 = sa, Sbt1 = sb, Qt1=q].",
"This can be rewritten as h(t1,sa, sb, s1,q) = LE[qQ( h(1,sa, sb, S1,q)1{Q1=q}+ t11 {- Qt2 - - Qt21Qt<0 }1{Qt=q}) dt| St1 = s1, Qt1= q].",
"Recalling that $h$ is bounded, we deduce by REF that |h(t1,sa, sb, s1,q)- LE[ h(1,sa, sb, S1,q)+ t11 {- q2 - - (q)2- } dt| St1 = s1, Qt1=q]|C for a constant $C$ , independent from $(t_1,t_2,s_{1},s_{2})$ , and $(q)_-= q^2\\mathbf {1}_{q<0}$ .",
"The expectation above does not depend on the control $\\ell $ , hence we drop the supremum and fix an arbitrary control $\\ell =0$ .",
"We denote by $\\mathbb {E}^0$ the expectation under the probability measure given by this control.",
"The expectation neither depends on the process $Q_t$ , so we drop the conditioning with respect to $Q_{t_1}$ .",
"This leads to |h(t1,sa, sb, s1,q)- E0[ h(1,sa, sb, S1,q)+ t11 {- q2 - - (q)2- } dt| St1 = s1]|C.",
"Similarly, starting from $(t_2,s^a, s^b, s_{2},q)$ with $(t_2,s_2)\\in {\\cal B}$ , we get |h(t2,sa, sb, s2,q)- E0[ h(2,sa, sb, S2,q)+ t22 {- q2 - - (q)2- } dt| St2 = s2]|C, and we deduce that $&|h(t_1,s^a, s^b, s_{1},q)- h(t_2,s^a, s^b, s_{2},q)|\\le \\bigg |\\mathbb {E}^0\\Big [ h(\\tau ^1,s^a, s^b, S_{\\tau ^1},q)+ \\int _{t_1}^{\\tau ^1} \\left\\lbrace - \\phi q^2 - \\phi _- (q)^2_- \\right\\rbrace dt| S_{t_1} = s_{1}\\Big ]\\\\& -\\mathbb {E}^0\\Big [ h(\\tau ^2,s^a, s^b, S_{\\tau ^2},q)+ \\int _{t_2}^{\\tau ^2} \\left\\lbrace - \\phi q^2 - \\phi _- (q)^2_- \\right\\rbrace dt| S_{t_2} = s_{2}\\Big ]\\bigg | +2C\\eta \\\\&\\le \\bigg |\\mathbb {E}^0\\Big [ h(\\tau ^1,s^a, s^b, S_{\\tau ^1},q)| S_{t_1} = s_{1}\\Big ]-\\mathbb {E}^0\\Big [ h(\\tau ^2,s^a, s^b, S_{\\tau ^2},q)| S_{t_2} = s_{2}\\Big ]\\bigg |\\\\&+\\Big |\\phi q^2 - \\phi _- (q)^2_-\\Big |\\Big (\\mathbb {E}^0\\big [\\tau ^1-t_1| S_{t_1} = s_{1}\\big ]+\\mathbb {E}^0\\big [\\tau ^2-t_2| S_{t_2} = s_{2}\\big ]\\Big )+2C\\eta .$ Using (REF ), we get |E0[1-t1| St1 = s1]+E0[2-t2| St2 = s2]|<2.",
"Also, the conditional laws (1| St1 = s1, S1 = (s1s2)+, 1<T), (1| St1 = s1, S1 = (s1s2)-, 1<T), (2| St2 = s2, S2 = (s1s2)+, 2<T), (2| St2 = s2, S2 = (s1s2)-, 2<T), have bounded continuous densities, which we denote by $f^{1,+}$ , $f^{1,-}$ , $f^{2,+}$ and $f^{2,-}$ respectively (see for example [5], Formula $3.0.6$ ).",
"So, by decomposing the first term in (REF ) with respect to the values of $S_{\\tau ^1}$ and $S_{\\tau ^2}$ , we can write $&\\bigg |\\mathbb {E}^0\\Big [ h(\\tau ^1,s^a, s^b, S_{\\tau ^1},q)| S_{t_1} = s_{1}\\Big ]-\\mathbb {E}^0\\Big [ h(\\tau ^2,s^a, s^b, S_{\\tau ^2},q)| S_{t_2} = s_{2}\\Big ]\\bigg |\\\\\\le &\\Big |\\sum \\limits _{j\\in \\lbrace +,-\\rbrace }\\int _0^T h(t,s^a, s^b, s_j,q)(f^{1,j}(t)\\mathbb {P}^0[S_{\\tau ^1} = s_j, \\tau ^1<T|S_{t_1} = s_{1}]-f^{2,j}(t)\\mathbb {P}^0[S_{\\tau ^2} = s_j, \\tau ^2<T|S_{t_2} = s_{2}])dt\\Big | \\\\&+\\Big |\\mathbb {E}^0\\big [ h(\\tau ^1,s^a, s^b, S_{\\tau ^1},q)\\mathbf {1}_{\\lbrace S_{\\tau ^1} \\ne s_+, S_{\\tau ^1} \\ne s_-\\rbrace \\cup \\lbrace \\tau ^1=T\\rbrace } | S_{t_1} = s_{1}\\big ]\\Big |\\\\&+\\Big |\\mathbb {E}^0\\big [ h(\\tau ^2,s^a, s^b, S_{\\tau ^2},q)\\mathbf {1}_{\\lbrace S_{\\tau ^2} \\ne s_+, S_{\\tau ^2} \\ne s_-\\rbrace \\cup \\lbrace \\tau ^2=T\\rbrace } | S_{t_2} = s_{2}\\big ]\\Big |$ where $s_+ = s_{1}\\vee s_{2}+\\epsilon $ and $s_- = s_{1}\\wedge s_{2}-\\epsilon $ .",
"Remark that the event $S_{\\tau ^1} \\ne s_+, S_{\\tau ^1} \\ne s_- , S_{t_1} = s_{1}$ happens only if $\\tau ^1=T$ so that $\\mathbb {P}^0[\\lbrace S_{\\tau ^1} \\ne s_+, S_{\\tau ^1} \\ne s_-\\rbrace \\cup \\lbrace \\tau ^1=T\\rbrace | S_{t_1} = s_{1}]<\\eta $ by (REF ).",
"Similarly $\\mathbb {P}^0[\\lbrace S_{\\tau ^2} \\ne s_+, S_{\\tau ^2} \\ne s_-\\rbrace \\cup \\lbrace \\tau ^2=T\\rbrace | S_{t_2} = s_{2}]<\\eta $ by (REF ).",
"As a consequence, using again (REF ), (REF ) and the fact that $h$ is bounded, we get |h(t1,sa, sb, s1,q)- h(t2,sa, sb, s2,q)| |j{+,-}h(t,sa, sb, sj,q)(f1,j(t)P0[S1 = sj, 1<T|St1 = s1]-f2,j(t)P0[S2 = sj, 2<T|St2 = s2])dt| +(2|q2 - - (q)2-|+4C).",
"Recall that the $f^{1,+}$ , $f^{1,-}$ , $f^{2,+}$ and $f^{2,-}$ depend on $s_2$ and $t_2$ .",
"We have |P0[S2 = s+, 2<T|St2 = s2]f2,+-P0[S1 = s+, 1<T|St1 = s1]f1,+| (t2,s2)(t1,s1) 0, |P0[S2 = s-, 2<T|St2 = s2]f2,--P0[S1 = s-, 1<T|St1 = s1]f1,-| (t2,s2)(t1,s1) 0 point-wise on $[0,T]$ directly by [5] Formula 3.0.6 and Appendix 11.",
"Having fixed $\\epsilon $ and using again [5] Formula 3.0.6 and Appendix 11, we see that the above functions are uniformly bounded with respect to $(s_2, t_2)\\in {\\cal B}$ .",
"So, using that $h$ is bounded, we can apply the dominated convergence theorem to deduce that |j{+,-}0T h(t,sa, sb, sj,q)(f1,j(t)P0[S1 = sj, 1<T|St1 = s1]-f2,j(t)P0[S2 = sj, 2<T|St2 = s2])dt| (t2,s2)(t1,s1) 0.",
"Thus we have shown that $h$ is continuous at the point $(t_1,S^a,S^b,s_{1},q)$ .",
"The case $t_1=T$ is treated the same way.",
"The continuity conditions can be proved using the same lines: fixing $q\\in {\\cal Q}$ , $t_1\\in [0,T)$ and $(S^a,S^b,s_{1})\\in \\partial {\\cal D}$ , choosing $(t_2,s_2)$ close enough to $(t_1,s_1)$ and applying the dynamic programming principle between $t_1$ and $\\tau ^1$ , and $t_2$ and $\\tau ^2$ , for $\\tau ^1$ and $\\tau ^2$ two well-chosen stopping times (for example 1=T{t>t1, St=s1+ or St = ss12}, 2=T{t>t2, St=s1+ or St = ss12}.",
"with $\\epsilon >0$ small enough, for a boundary inducing an upward jump)."
],
[
"Proof of Theorem ",
"We first prove that the value function of the market maker's problem is indeed a viscosity solution of (REF ).",
"Proposition A.1 The value function $h$ is a continuous viscosity solution on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ of (REF ).",
"Furthermore, $h(T,S^a, S^b, S, q) = q(S - Aq)$ for all $(S^a, S^b, S, q) \\in \\mathcal {D} \\times \\mathcal {Q},$ and h(t,Sa, Sb, S, q) = 1{S-Sa=(12+a)a, S-Sb<(12+b)b} h(t,Sa+a,Sb,S,q) + 1{S-Sa<(12+a)a, S-Sb=(12+b)b}h(t,Sa,Sb+b,S,q) + 1{S-Sa=(12+a)a, S-Sb=(12+b)b}h(t,Sa+a,Sb+b,S,q) + 1{S-Sa=-(12+a)a, S-Sb>-(12+b)b}h(t,Sa-a,Sb,S,q) + 1{S-Sa>-(12+a)a, S-Sb=-(12+b)b}h(t,Sa,Sb-b,S,q) + 1{S-Sa=-(12+a)a, S-Sb=-(12+b)b}h(t,Sa-a,Sb-b,S,q), for all $(t,S^a, S^b, S, q) \\in [0,T) \\times \\partial \\mathcal {D} \\times \\mathcal {Q} .$ Let $(\\bar{S}^a, \\bar{S}^b, \\bar{q}) \\in \\alpha ^a \\mathbb {N} \\times \\alpha ^b \\mathbb {N} \\times \\mathcal {Q},$ and $(t_n, S_n)_{n\\in \\mathbb {N}} \\in [0,T] \\times \\mathbb {R}$ be a sequence such that $t_n \\underset{n\\rightarrow +\\infty }{\\rightarrow } \\hat{t} \\in [0,T),$ $S_n \\underset{n\\rightarrow +\\infty }{\\rightarrow } \\hat{S} \\in \\mathbb {R},$ $h(t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q}) \\underset{n\\rightarrow +\\infty }{\\rightarrow } h (\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}),$ with $(\\bar{S} ^a, \\bar{S} ^b, \\hat{S} ) \\in \\mathcal {D}.$ Without loss of generality we can assume that $(\\bar{S} ^a, \\bar{S} ^b, S_n ) \\in \\mathcal {D}$ for all $n\\in \\mathbb {N} .$ Let us first consider the case $\\hat{t} = T$ .",
"Let us take two arbitrary controls $\\ell ^a_s = \\ell ^b_s = 0$ , for all $s \\in [0,T),$ then for all $n \\in \\mathbb {N}$ we have h(tn, Sa, Sb, Sn,q) Etn,Sa, Sb, Sn,q[ QT(ST-AQT) - tnT Qs2 ds - - tnT Qs2 1Qs<0 ds], and by dominated convergence we can obtain $h(T,\\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}) \\ge \\bar{q}(\\hat{S} - A\\bar{q}).$ Now let us consider the case $\\hat{t}<T.$ Let $\\varphi : [0,T) \\times \\mathcal {D} \\times \\mathcal {Q} \\rightarrow \\mathbb {R}$ be a continuous function, $\\mathcal {C}^1$ in $t$ , $\\mathcal {C}^2$ in $S$ and such that $0 = \\underset{[0,T) \\times \\mathcal {D}}{\\min } (h - \\varphi ) = (h - \\varphi )(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}).$ We also assume that $h=\\varphi $ only at the point $(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ .",
"Let us assume that there exists $\\eta >0$ such that 2 t (t,Sa,Sb,S,q)-q2 - - q2 1q<0 + 122 SS(t,Sa,Sb,S,q) + 1+(a)2a{0,1}{ a(Sa+(t,Sa,Sb,S,q-a)-(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b(-Sb+(t,Sa,Sb,S,q+b)-(t,Sa,Sb,S,q))}.",
"Then we must have 0 t (t,Sa,Sb,S,q)-q2 - - q2 1q<0 + 122 SS(t,Sa,Sb,S,q) + 1+(a)2a{0,1}{ a(Sa+(t,Sa,Sb,S,q-a)-(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b(-Sb+(t,Sa,Sb,S,q+b)-(t,Sa,Sb,S,q))}, for all $(t,S) \\in B= \\left((\\hat{t}-r, \\hat{t}+r) \\cap [0,T) \\right) \\times \\left(\\hat{S}-r, \\hat{S}+r\\right)$ for some $r >0$ .",
"Without loss of generality, we can assume that $B$ contains the sequence $(t_n, S_n)_n$ and that for all $(t,S) \\in B,$ we have $(\\bar{S}^a, \\bar{S}^b, S) \\in \\mathcal {D}.$ We can choose the value of $\\eta $ such that $\\varphi (t,\\bar{S}^a,\\bar{S}^b,S,\\bar{q}) + \\eta \\le h(t,\\bar{S}^a,\\bar{S}^b,S,\\bar{q})$ on $\\partial _p B : = \\bigg (\\left((\\hat{t}-r, \\hat{t}+r) \\cap [0,T) \\right) \\times \\left(\\left\\lbrace \\hat{S}-r\\right\\rbrace \\cup \\left\\lbrace \\hat{S}+r\\right\\rbrace \\right)\\bigg ) \\cup \\bigg ( \\lbrace \\hat{t}+r \\rbrace \\times \\left[\\hat{S}-r, \\hat{S}+r\\right] \\bigg ).$ We can also assume that $\\varphi (t,S^a,S^b,S,q) + \\eta \\le h(t,S^a,S^b,S,q),$ for $(t, S^a,S^b,S,q) \\in \\tilde{B}$ with B = {(t,Sa,Sb,S,q) | (t,S) B, q{q-1, q+1 }Q }.",
"We introduce the set $B_{\\mathcal {D}} = \\left\\lbrace (t,\\bar{S}^a,\\bar{S}^b,S,\\bar{q}) \\big | (t,S) \\in B \\right\\rbrace $ and set $\\pi _n=\\inf \\lbrace t\\ge t_n | (t, S^a_t, S^b_t, S_t, q_t)\\notin B_{\\mathcal {D}}\\rbrace $ with $S^i_{t_n} = \\bar{S}^i,$ $q_{t_n} = \\bar{q},$ $S_{t_n} = S_n,$ where the processes are controlled by $\\ell ^a_t = \\mathbf {1}_{\\lbrace S^a_t + \\varphi (t, S^a_t, S^b_t, S_t, q_{t-}-1) - \\varphi (t, S^a_t, S^b_t, S_t, q_{t-}) >0\\rbrace },$ $\\ell ^b_t = \\mathbf {1}_{\\lbrace - S^b_t + \\varphi (t, S^a_t, S^b_t, S_t, q_{t-}+1) - \\varphi (t, S^a_t, S^b_t, S_t, q_{t-})>0 \\rbrace }.$ Using Itô's formula and noting that $S_t^a,S_t^b$ do not jump between $t_n$ and $\\pi _n$ , we derive (n, San, Sbn, Sn, qn)=(tn, Sa, Sb, Sn, q) + tnn{ t (t, Sat, Sbt, St, qt) + 12 2 S S (t, Sat, Sbt, St, qt) }dt + tnn (at){ (t, Sat, Sbt, St, qt- - at) - (t, Sat, Sbt, St, qt-) } dt + tnn (bt){ (t, Sat, Sbt, St, qt- + bt) - (t, Sat, Sbt, St, qt-) } dt + tnn S (t, Sat, Sbt, St, qt) dWt + tnn { (t, Sat, Sbt, St, qt- - at) - (t, Sat, Sbt, St, qt-) } dNat + tnn { (t, Sat, Sbt, St, qt- + bt) - (t, Sat, Sbt, St, qt-) } dNbt (tn, Sa, Sb, Sn, q) - tnn {Sta(at) - Stb (bt) - qt2 - - qt2 1qt<0} dt + tnn S (t, Sat, Sbt, St, qt) dWt + tnn { (t, Sat, Sbt, St, qt- - at) - (t, Sat, Sbt, St, qt-) } dNat + tnn { (t, Sat, Sbt, St, qt- + bt) - (t, Sat, Sbt, St, qt-) } dNbt.",
"Then by taking the expectation we get (tn, Sa, Sb, Sn, q) E [(n, San, Sbn, Sn, qn) + tnn {Sta (at) - Stb (bt) - qt2 - - qt21qt<0} dt ].",
"Thus (tn, Sa, Sb, Sn, q) -+ E [h(n, San, Sbn, Sn, qn) + tnn {Sta (at) - Stb (bt) - qt2 - - qt21qt<0 } dt ].",
"As (tn, Sa, Sb, Sn, q) n+(t,Sa, Sb, S, q) = h(t,Sa, Sb, S, q), h(tn, Sa, Sb, Sn, q) n+h(t,Sa, Sb, S, q), there exists $n_0\\in \\mathbb {N}$ such that for all $n\\ge n_0$ , $h(t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q}) - \\frac{\\eta }{2} \\le \\varphi (t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q})$ and we deduce h(tn, Sa, Sb, Sn, q) -2 + E [h(n, San, Sbn, Sn, qn) + tnn {Sta (at) - Stb (bt) - qt2 - - qt2 1qt<0} dt ], which contradicts the dynamic programming principle.",
"Therefore, 0 t (t,Sa,Sb,S,q)-q2 - - q21q<0+ 122 SS(t,Sa,Sb,S,q) + 1+(a)2a{0,1}{ a(Sa+(t,Sa,Sb,S,q-a)-(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b(-Sb+(t,Sa,Sb,S,q+b)-(t,Sa,Sb,S,q))}, and $h$ is a viscosity supersolution of the HJB equation on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}.$ The proof for the subsolution part is identical.",
"For a given $\\rho >0$ , we introduce the function $\\tilde{h}$ such that $\\tilde{h}(t,S^a, S^b, S, q) = e^{\\rho t}h(t,S^a, S^b, S, q) \\quad \\forall \\ (t,S^a, S^b, S, q) \\in [0,T] \\times \\mathcal {D} \\times \\mathcal {Q}.$ Then $\\tilde{h}$ is a viscosity solution of the following HJB equation: 0 = - h(t,Sa,Sb,S,q) + t h(t,Sa,Sb,S,q)-q2 - - q21q<0 + 122 SSh + 1+(a)2a{0,1}{ a(etSa+h(t,Sa,Sb,S,q-a)-h(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b(et(-Sb)+h(t,Sa,Sb,S,q+b)-h(t,Sa,Sb,S,q))}, and we see that proving a maximum principle for (REF ) is equivalent to proving one for (REF ).",
"Definition A.2 Let $U:[0,T) \\times \\mathcal {D} \\times \\mathcal {Q} \\rightarrow \\mathbb {R}$ be continuous with respect to $(t,S).$ For $(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}) \\in [0,T) \\times \\mathcal {D} \\times \\mathcal {Q},$ we say that $(y,p,A) \\in \\mathbb {R}^3$ is in the subjet $\\mathcal {P}^- U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ (resp.",
"the superjet $\\mathcal {P}^+ U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ ) if U(t,Sa, Sb, S, q) U(t, Sa, Sb, S, q) + y(t-t) + p(S-S) + 12 A(S-S)2 + o ( |t-t| + |S - S|2 ), (resp.",
"U(t,Sa, Sb, S, q) U(t, Sa, Sb, S, q) + y(t-t) + p(S-S) + 12 A(S-S)2 + o ( |t-t|+|S - S|2 )), for all $(t,S)$ such that $(t,\\bar{S}^a, \\bar{S}^b, S, \\bar{q}) \\in [0,T) \\times \\mathcal {D} \\times \\mathcal {Q}.$ We also define $\\bar{\\mathcal {P}}^- U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ as the set of points $(y,p,A) \\in \\mathbb {R}^3$ such that there exists a sequence $(t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q}, y_n, p_n, A_n) \\in [0,T) \\times \\mathcal {D} \\times \\mathcal {Q} \\times \\mathcal {P}^- U(t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q})$ satisfying $(t_n, \\bar{S}^a, \\bar{S}^b, S_n, \\bar{q}, y_n, p_n, A_n) \\underset{n \\rightarrow +\\infty }{\\rightarrow } (\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}, y, p, A).$ The set $\\bar{\\mathcal {P}}^+ U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ is defined similarly.",
"Let us recall one of the definitions of viscosity solutions which we are going to use for the proof of the uniqueness.",
"Lemma A.3 A continuous function $\\tilde{U}$ is a viscosity supersolution (resp.",
"subsolution) to (REF ) on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ if and only if for all $(\\hat{t},\\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}) \\in [0,T) \\times \\mathcal {D}\\times \\mathcal {Q}$ and all $(\\hat{y}, \\hat{p}, \\hat{A}) \\in \\bar{\\mathcal {P}}^- U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ (resp.",
"$\\bar{\\mathcal {P}}^+ U(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})$ ), we have -U(t, Sa, Sb, S, q) + y-q2 - - q21q<0 + 122 A + 1+(a)2a{0,1}{ a(etSa+U(t,Sa,Sb,S,q-a)-U(t,Sa,Sb,S,q))} + 1+(b)2b{0,1}{ b(et(-Sb)+U(t,Sa,Sb,S,q+b)-U(t,Sa,Sb,S,q))} 0 (resp.",
"$\\ge 0$ ).",
"We refer to [6] for a proof of this result.",
"We can now state a maximum principle from which the uniqueness can be easily deduced: Proposition A.4 Let $U$ (resp.",
"$V$ ) be a continuous viscosity supersolution (resp.",
"subsolution) of (REF ) with polynomial growth on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ and satisfying the continuity conditions (REF ).",
"If $U\\ge V$ on $\\lbrace T\\rbrace \\times \\mathcal {D} \\times \\mathcal {Q}$ , then $U\\ge V$ on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}.$ As before, we introduce the functions $\\tilde{U}$ and $\\tilde{V}$ such that $\\tilde{U}(t,S^a, S^b, S, q) = e^{\\rho t}U(t,S^a, S^b, S, q) \\qquad \\text{and} \\qquad \\tilde{V}(t,S^a, S^b, S, q) = e^{\\rho t}V(t,S^a, S^b, S, q).$ Then $\\tilde{U}$ and $\\tilde{V}$ are respectively viscosity supersolution and subsolution of Equation (REF ) on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}$ with $\\tilde{U} \\ge \\tilde{V}$ on $\\lbrace T\\rbrace \\times \\mathcal {D} \\times \\mathcal {Q}$ .",
"To prove the proposition, it is enough to prove that $\\tilde{U} \\ge \\tilde{V}$ on $[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}.$ We proceed by contradiction.",
"Let us assume that $\\underset{[0,T) \\times \\mathcal {D} \\times \\mathcal {Q}}{\\sup } \\tilde{V}-\\tilde{U} >0.$ Let $p\\in \\mathbb {N}^*$ such that $\\underset{\\Vert S\\Vert _2 \\rightarrow +\\infty }{\\lim }\\quad \\underset{{t \\in [0,T], q\\in \\mathcal {Q} \\\\ (S,S^a,S^b)\\in \\mathcal {D}}}{\\sup }\\quad \\frac{|\\tilde{U}(t, S^a, S^b, S, q)| + |\\tilde{V}(t,S^a, S^b, S, q)|}{1+\\Vert S\\Vert _2^{2p}} = 0, $ where $\\Vert \\cdot \\Vert _2$ is the Euclidian norm.",
"Then there exists $(\\hat{t}, \\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}) \\in [0,T] \\times {\\mathcal {D}}\\times \\mathcal {Q}$ such that 0< V(t, Sa, Sb, S, q) - U(t, Sa, Sb, S, q) - (t,S,S, q) = (t, Sa, Sb, S, q) V(t, Sa, Sb, S, q) - U(t, Sa, Sb, S, q) - (t,S,S,q), where $\\phi (t,S,R,q) := \\varepsilon e^{-\\mu t} (1 + \\Vert S\\Vert _2^{2p} + \\Vert R\\Vert _2^{2p} ),$ with $\\varepsilon >0,$ $\\mu >0$ .",
"The choice of the function $\\phi $ allows us to look for a supremum in a bounded set with respect to $(S,S^a,S^b)$ .",
"Then the supremum is either reached for a point in $[0,T] \\times {\\mathcal {D}}\\times \\mathcal {Q}$ or on $[0,T] \\times \\partial \\mathcal {D}\\times \\mathcal {Q}$ (recall that ${\\cal D}$ is open).",
"But the continuity conditions tell us that if the supremum is reached on $[0,T]\\times \\partial \\mathcal {D}\\times \\mathcal {Q}$ , it is also reached in $[0,T] \\times {\\mathcal {D}}\\times \\mathcal {Q}$ .",
"Since $\\tilde{U} \\ge \\tilde{V}$ on $\\lbrace T\\rbrace \\times \\mathcal {D} \\times \\mathcal {Q}$ , it is clear that $\\hat{t}<T$ .",
"Then, for all $n\\in \\mathbb {N}^*$ , we can find $(t_n,S_n,R_n) \\in [0,T] \\times \\mathbb {R}^2$ such that $(\\bar{S}^a,\\bar{S}^b,S_n),(\\bar{S}^a,\\bar{S}^b,R_n) \\in \\mathcal {D}$ and 0 <V(tn,Sa,Sn, Sn, q) - U(tn,Sa,Sb, Rn, q) - (tn,Sn,Rn,q) - n|Sn - Rn|2 - ( |tn-t|2 + |Sn - S|4 ) = (t,S,R) V(t,Sa,Sb, S, q) - U(t,Sa,Sb, R, q) - (t,S,R,q) - n|S - R|2 - ( |t-t|2 + |S - S|4 ).",
"Then, we have $(t_n,S_n,R_n) \\underset{n\\rightarrow +\\infty }{\\rightarrow } (\\hat{t},\\hat{S},\\hat{S}),$ and V(tn,Sa,Sb, Sn, q) - U(tn,Sa,Sb, Rn, q) - (tn,Sn,Rn) - n|Sn - Rn|2 - ( |tn-t|2 + |Sn - S|4 ) n+ V(t, Sa, Sb, S, q) - U(t, Sa, Sb, S, q) - (t,S,S).",
"For $n\\in \\mathbb {N}^*$ , let us write for $(t,S,R) \\in [0,T]\\times \\mathbb {R}^2$ $\\varphi _n(t,S,R):= \\phi (t,S,R) + n|S-R|^2 + |t-\\hat{t}|^2 + |S-\\hat{S}|^4.$ Then Ishii's Lemma (see [4], [9]) guarantees that for any $\\eta >0,$ we can find $(y^1_n,p^1_n,A^1_n) \\in \\bar{\\mathcal {P}}^+ \\tilde{V}(t_n,\\bar{S}^a, \\bar{S}^b, S_n,\\bar{q})$ and $(y^2_n,p^2_n,A^2_n) \\in \\bar{\\mathcal {P}}^- \\tilde{U}(t_n,\\bar{S}^a, \\bar{S}^b, R_n,\\bar{q})$ such that $y^1_n - y^2_n = \\partial _t \\varphi _n(t_n,S_n,R_n), \\quad (p^1_n,p^2_n) = \\left(\\partial _S \\varphi _n, -\\partial _R \\varphi _n \\right)(t_n,S_n,R_n)$ and $\\begin{pmatrix} A^1_n & 0\\\\ 0 & -A^2_n \\end{pmatrix} \\le H_{SR} \\varphi _n (t_n,S_n,R_n) + \\eta \\left( H_{SR} \\varphi _n (t_n,S_n,R_n) \\right)^2, $ where $H_{SR}\\varphi _n (t_n,.,.",
")$ denotes the Hessian of $\\varphi _n(t_n,.,.",
").$ Applying Lemma REF , we get ( V(tn,Sa, Sb, Sn, q) - U(tn,Sa, Sb, Rn, q) ) y1n - y2n + 12 2 (A1n - A2n) + 1+(a)2a{0,1}{ a(etnSa+V(tn,Sa,Sb,Sn,q-a)-V(tn,Sa,Sb,Sn,q))} + 1+(b)2b{0,1}{ b(etn(-Sb)+V(tn,Sa,Sb,Sn,q+b)-V(tn,Sa,Sb,Sn,q))} - 1+(a)2a{0,1}{ a(etnSa+U(tn,Sa,Sb,Rn,q-a)-U(tn,Sa,Sb,Rn,q))} - 1+(b)2b{0,1}{ b(etn(-Sb)+U(tn,Sa,Sb,Rn,q+b)-U(tn,Sa,Sb,Rn,q))}.",
"Moreover, we have $ H_{SR} \\varphi _n (t_n,S_n,R_n) = \\begin{pmatrix} \\partial ^2_{SS} \\phi (t_n,S_n,R_n) + 2n +12 (S_n-\\hat{S})^2 & \\partial ^2_{SR} \\phi (t_n,S_n,R_n)-2n\\\\ \\partial ^2_{SR} \\phi (t_n,S_n,R_n)-2n & \\partial ^2_{SR}\\phi (t_n,S_n,R_n) + 2n \\end{pmatrix}.$ It follows that ( V(tn,Sa, Sb, Sn, q) - U(tn,Sa, Sb, Rn, q) ) t (tn,Sn,Rn) + 2(tn-t) + 12 2 (2SS(tn,Sn,Rn) + 2RR(tn,Sn,Rn) + 22SR(tn,Sn,Rn) + 12(Sn-S)) + Cn + 1+(a)2a{0,1}{ a(etnSa+V(tn,Sa,Sb,Sn,q-a)-V(tn,Sa,Sb,Sn,q))} + 1+(b)2b{0,1}{ b(etn(-Sb)+V(tn,Sa,Sb,Sn,q+b)-V(tn,Sa,Sb,Sn,q))} - 1+(a)2a{0,1}{ a(etnSa+U(tn,Sa,Sb,Rn,q-a)-U(tn,Sa,Sb,Rn,q))} - 1+(b)2b{0,1}{ b(etn(-Sb)+U(tn,Sa,Sb,Rn,q+b)-U(tn,Sa,Sb,Rn,q))}, where $C_n$ does not depend on $\\eta .$ Therefore, as the maximums on the right-hand side are always positive, we deduce that for all $n \\in \\mathbb {N}^*$ , ( V(tn,Sa, Sb, Sn, q) - U(tn,Sa, Sb, Rn, q) ) t (tn,Sn,Rn) + 2(tn-t) + 12 2 (2SS(tn,Sn,Rn) + 2RR(tn,Sn,Rn) + 22SR(tn,Sn,Rn) + 12(Sn-S)) + 1+(a)2a{0,1}{ a(etnSa+V(tn,Sa,Sb,Sn,q-a)-V(tn,Sa,Sb,Sn,q))} + 1+(b)2b{0,1}{ b(etn(-Sb)+V(tn,Sa,Sb,Sn,q+b)-V(tn,Sa,Sb,Sn,q))}.",
"As $\\tilde{V}$ is continuous and $(t_n,S_n)_n$ converges to $(\\hat{t}, \\hat{S})$ , the last two terms are bounded from above by some constant $M.$ Then by sending $n$ to infinity, we get ( V(t,Sa, Sb, S, q) .",
".- U(t,Sa, Sb, S, q) ) t (t,S,S) + 12 2 (2SS(t,S,S) + 2RR(t,S,S) + 22SR(t,S,S) ) + M. For $\\mu >0$ large enough, the right-hand side is strictly negative, and as $\\rho >0$ we get $\\tilde{V}(\\hat{t},\\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q}) - \\tilde{U}(\\hat{t},\\bar{S}^a, \\bar{S}^b, \\hat{S}, \\bar{q})<0,$ hence the contradiction."
],
[
"Effects of the uncertainty zones on $h$",
"We keep the same parameters as in Section and take $\\alpha ^a=0.01$ and $\\alpha ^b=0.00625$ .",
"We plot the value function of the market maker's problem (the function $h$ ) on some small range of values of $S$ .",
"Note that $S=10.5$ is on both discrete grids.",
"Figure: Value function hh of the market maker for q=0q=0, as a function of SS.We distinguish 4 possible cases, depending on whether $S^a=\\alpha ^a\\left\\lfloor S/\\alpha ^a\\right\\rfloor $ and $S^b=\\alpha ^b\\left\\lfloor S/\\alpha ^b\\right\\rfloor $ (green dots), $S^a=\\alpha ^a\\left\\lfloor S/\\alpha ^a\\right\\rfloor $ and $S^b=\\alpha ^b\\left\\lceil S/\\alpha ^b\\right\\rceil $ (red dash-dots), $S^a=\\alpha ^a\\left\\lceil S/\\alpha ^a\\right\\rceil $ and $S^b=\\alpha ^b\\left\\lfloor S/\\alpha ^b\\right\\rfloor $ (orange dash), $S^a=\\alpha ^a\\left\\lceil S/\\alpha ^a\\right\\rceil $ and $S^b=\\alpha ^b\\left\\lceil S/\\alpha ^b\\right\\rceil $ (blue solid).",
"Note that depending on the value of $S$ , some of those cases can be excluded.",
"The solid vertical red and black lines represent respectively the values on the ask ($\\alpha ^a\\mathbb {N}$ ) and the bid grid ($\\alpha ^b\\mathbb {N}$ ).",
"The dotted vertical lines represent the limits of the uncertainty zones on each side.",
"In the uncertainty zones, the value function $h$ depends non-trivially on $S^a$ and $S^b$ .",
"Thanks to the continuity conditions at the boundaries of the uncertainty zones, we get a smooth behavior of $h$ when $S$ exits a zone.",
"Remark that when $S\\in [10 \\pm ((\\frac{1}{2}-\\eta ^a)\\alpha ^a)\\wedge ((\\frac{1}{2}-\\eta ^b)\\alpha ^b)]$ , necessarily $S^a=S^b=10$ .",
"In our example, $\\alpha ^a>\\alpha ^b$ and $(\\frac{1}{2}-\\eta _a)\\alpha ^a>(\\frac{1}{2}-\\eta _b)\\alpha ^b$ .",
"So, if $S$ is in $(10+(\\frac{1}{2}-\\eta _b)\\alpha ^b,(\\frac{1}{2}-\\eta _a)\\alpha ^a)$ , necessarily $S^a=10$ , but $S^b$ can take either the value 10 or $10+\\alpha ^b$ depending on whether $S$ comes from higher prices or lower prices.",
"This is why there are two curves in the interval $(10+(\\frac{1}{2}-\\eta _b)\\alpha ^b,(\\frac{1}{2}-\\eta _a)\\alpha ^a)$ .",
"At $(\\frac{1}{2}-\\eta _a)\\alpha ^a$ , two additional curves appear as $S^a$ can also be two different values."
]
] | 2005.14126 |
[
[
"Equation of state constraints from nuclear physics, neutron star masses,\n and future moment of inertia measurements"
],
[
"Abstract We explore constraints on the equation of state (EOS) of neutron-rich matter based on microscopic calculations up to nuclear densities and observations of neutron stars.",
"In a previous work we showed that predictions based on modern nuclear interactions derived within chiral effective field theory and the observation of two-solar-mass neutron stars result in a robust uncertainty range for neutron star radii and the EOS over a wide range of densities.",
"In this work we extend this study, employing both the piecewise polytrope extension from Hebeler et al.",
"as well as the speed of sound model of Greif et al., and show that moment of inertia measurements of neutron stars can significantly improve the constraints on the EOS and neutron star radii."
],
[
"Introduction",
"Recently, there has been significant progress in our understanding of the equation of state (EOS) of dense matter.",
"This was triggered by advances in nuclear theory, new constraints from precise measurements of heavy neutron stars, as well as astrophysical observations from the LIGO/Virgo [1], [2] and NICER [40], [43], [48] collaborations.",
"These offer complementary insights to the EOS.",
"While nuclear theory provides reliable predictions for neutron-rich matter up to densities around saturation density ($\\rho _0 = 2.8 \\times 10^{14} \\,\\text{g} \\, \\text{cm}^{-3}$), observations of neutron stars, and neutron star mergers probe the EOS over a higher range of densities but provide indirect constraints.",
"In nuclear physics the development of chiral effective field theory (EFT) has revolutionized our approach to nuclear forces.",
"The description of the interactions between neutrons and protons, both particles with a complex substructure, has been a challenge in nuclear theory for decades.",
"Pioneered by the seminal works of [55], [56], chiral EFT has now become the only known framework that allows a systematic expansion of nuclear forces at low energies [21], [39], [25] based on the symmetries of quantum chromodynamics (QCD), the fundamental theory of the strong interaction.",
"In addition, chiral EFT allows one to derive systematic estimates of uncertainties for observables.",
"Incorporating such chiral EFT interactions in microscopic many-body frameworks makes it possible to compute uncertainty bands for the pressure and energy density of matter [30], [13], [31], [52], [57], [19], [20], [18], [38].",
"As any effective low-energy theory, chiral EFT contains an intrinsic breakdown scale.",
"When approaching this breakdown scale with increasing energy or density the convergence of the effective expansion becomes slower until eventually it breaks down.",
"This breakdown scale translates into an upper density limit for such calculations.",
"The precise value for this upper density limit is still unknown, and also depends on details of the interactions.",
"In a previous work [29], we chose an upper density limit of $1.1 \\, \\rho _0$ for neutron-rich matter.",
"This limit represents a rather conservative choice and it might be possible to push this limit to somewhat higher densities [51], although a full understanding of the implied uncertainties is still an open problem.",
"Finally, for very high densities ($\\rho \\gtrsim 50 \\, \\rho _0$ ), there are model-independent constraints from perturbative QCD calculations of quark matter [33].",
"Neutron star observations provide powerful constraints on the EOS beyond the densities accessible by nuclear theory as well as laboratory experiments [53].",
"In particular, the precise mass measurements of the pulsars PSR J1614-2230 and PSR J0348+0432 with masses of $1.928 \\pm 0.017 \\,M_\\odot $ [22] and $2.01 \\pm 0.04 \\, M_\\odot $ [3] turned out to be a key discovery, as the existence of such heavy neutron stars puts tight constraints on the EOS and the composition of matter, ruling out a large number of EOSs with simple inclusion of exotic degrees of freedom like hyperons or deconfined quarks.",
"Recently, the mass of the pulsar PSR J0740+6620 was measured to be $2.14 \\begin{array}{c}+0.10 \\\\ -0.09\\end{array} \\, M_\\odot $ [15], which further tightens these constraints.",
"In this work, we study the EOS constraints that can be achieved from future moment of inertia measurements, in addition to the heavy mass constraint discussed above.",
"The moment of inertia has been suggested to provide complementary constraints for the EOS [46], [37], [35].",
"It can be obtained from measurements of the rate of advance of the periastron, $\\dot{\\omega }$ [16].",
"This advance is mainly caused by the relativistic spin-orbit coupling in a binary system [4], [58], [32], and the magnitude of the advance depends sensitively on the orbital period and the compactness of the binary system.",
"In 2003, the double neutron-star system PSR J0737–3039 was discovered [11], [37].",
"This system is particularly promising for such measurements, as it is extremely compact with an orbital period of only 2.4 hr [11], [12], [37].",
"In addition, due to the high orbital inclination [11], [12], the masses of the two neutron stars have been determined very precisely to be $1.3381(7) \\, M_\\odot $ and $1.2489(7) \\, M_\\odot $ [32].",
"Due to the compactness of the system, the moment-of-inertia correction to $\\dot{\\omega }$ is estimated to be an order of magnitude larger for PSR J0737-3039A (the heavier of the two pulsars) than for other systems like PSR B1913+16 [37].",
"Such a moment of inertia measurement has to be performed over a long period of time and an increase of timing precision would be beneficial [32].",
"Based on this, it was argued that a moment of inertia measurement with a relative uncertainty of about $10\\%$ may be achievable [16], [35], [32].",
"Previous works studied to what extent such measurements are able to provide constraints for different types of EOS [41], [7], [35].",
"In particular, [46] showed that the moment of inertia can be parameterized efficiently as a function of the compactness parameter, and [35] demonstrated that a universal relation between the moment of inertia and the compactness parameter exists, which can be used to provide constraints on neutron star radii.",
"More recently, [49], [23], and [36] studied the moment of inertia based on neutron star observations and EOS constraints, and [45] investigated the inference of neutron star radii from moment of inertia measurements.",
"In this work, we study how microscopic calculations based on chiral EFT interactions combined with neutron star masses and a future moment of inertia measurement can provide novel predictions for the EOS and neutron star radii.",
"In Section , we briefly review our approach employing both the piecewise polytrope extension from [29] as well as the speed of sound model of [24] and present uncertainty ranges for neutron star observables such as the mass, the radius, and the moment of inertia.",
"In Section , we present our results for neutron star radii, and how these can improve upon information from the neutron star merger GW170817 [2].",
"Moreover, we discuss the resulting EOS constraints and explore scaling relations for the dimensionless moment of inertia.",
"Finally, we conclude in Section ."
],
[
"Constraints from nuclear theory and neutron star masses",
"In [28], [29] we combined constraints from nuclear physics and neutron star masses to derive constraints for the EOS for all densities relevant for neutron stars.",
"We briefly review the strategy of this work and refer to [29] for details: a) The first constraint results from microscopic calculations of neutron-rich matter up to density $\\rho _1 = 1.1 \\, \\rho _0$ based on modern nuclear interactions derived from chiral EFT [30], [52].",
"These calculations resulted in uncertainty bands for the energy density and pressure.",
"For densities below $\\rho _{\\text{crust}} = 0.5 \\, \\rho _0$ the BPS crust EOS of [5] and [42] was used.",
"Remarkably, around the transition density $\\rho _{\\text{crust}}$ both EOSs overlap smoothly, so that our final results are insensitive to the particular choice for $\\rho _{\\text{crust}}$ .",
"Figure: Results for mass MM, radius RR, and moment of inertia II of neutronstars based on the EOS constraints (bands) derived with the piecewisepolytrope model based on chiral EFT calculations up to density ρ 1 =1.1ρ 0 \\rho _1 = 1.1 \\, \\rho _0, the new massconstraint M obs ⩾2.05M ⊙ M_{\\text{obs}} \\geqslant 2.05 \\, M_{\\odot }, and causality constraints.The individual panels (a), (b), and (c) show the mass-radius, moment ofinertia-radius, and moment of inertia-mass results, respectively.",
"Thegreen (dashed), yellow (solid), and red (dotted-dashed) lines correspond to the threerepresentative EOS (soft, intermediate, and stiff, respectively) from.",
"Note that the latter are for the oldmass constraint M obs ⩾1.97M ⊙ M_{\\text{obs}} \\geqslant 1.97 \\, M_{\\odot }, so that thesoft EOS leads to smaller radii.b) Based on the constraints from nuclear physics at low densities the EOS was extended in a general way to higher densities using piecewise polytropes, $P(\\rho )=K_i \\rho ^{\\Gamma _i}$, with the adiabatic indices $\\Gamma _i$ and constants $K_i$ (see also [47]).",
"The values for $\\Gamma _i$ are allowed to vary freely, whereas the values of $K_i$ are fixed by the constraint that the EOS should be continuous as a function of density.",
"For the extension beyond $\\rho _1$ , three polytropes characterized by exponents $\\Gamma _1$ , $\\Gamma _2$ (beyond $\\rho _{12}$ ), and $\\Gamma _3$ (beyond $\\rho _{23}$ ) allow one to control the softness or stiffness of the EOS in a given density region, and the transition densities $\\rho _{12}$ and $\\rho _{23}$ between polytropes are allowed to vary as well.",
"Sampling all possible EOSs using the step size $\\Delta \\Gamma _i = 0.5$ and $\\Delta \\rho _{12,23} = \\rho _0/2$ results in a very large number of possible EOSs (for details, see [29]), which include constructions that mimic first-order phase transitions.",
"The values of $\\Gamma _i$ , $\\rho _{12}$ , and $\\rho _{23}$ are then constrained by the condition that each EOS must be able to support a neutron star of at least $M_{\\text{obs}} = 2.05 \\, M_\\odot $ , which we take as the 68% lower limit of the mass of the heaviest precisely known pulsar [15].",
"This mass constraint provides an update compared to the $1\\sigma $ lower limit ($1.97 \\, M_\\odot $ ) of the mass of PSR J0348+0432 [3] used in [29].",
"c) As the final constraint we require that the speed of sound, $c_\\text{s}$ , remain smaller than the speed of light, $c$ , for all densities: $c_{\\text{s}}/c =\\sqrt{dP /d\\mathcal {E}} \\leqslant 1$ , where $P$ is the pressure and $\\mathcal {E}$ is the energy density.",
"Each EOS is followed in density until causality is violated or the maximum neutron star mass is reached when $dM/dR = 0$ .",
"The combination of these three conditions leads to mass-radius constraints on neutron stars shown in panel (a) of Fig.",
"REF .",
"In general, the boundaries of the band are spanned by a large number of different EOSs, but to distinguish soft and stiff EOSs, we show the three representative EOSs (soft, intermediate, and stiff) of [29], which span the radius range as shown in Fig.",
"REF , while the soft EOS leads to somewhat smaller radii due to the previous mass constraint $M_{\\text{obs}}\\geqslant 1.97 \\, M_{\\odot }$ .",
"For a typical $M=1.4 \\, M_{\\odot }$ star, the update gives a radius range of $R= 10.2$ –$13.6 \\, \\text{km}$ (taking the chiral EFT constraints from renormalization-group-evolved interactions, which have improved many-body convergence; [29]).",
"In order to explore the sensitivity to details of the high-density extension, we also employ the speed of sound model of [24] in addition to the piecewise polytrope extension.",
"The speed of sound model is based on the same crust EOS and chiral EFT band, but uses a parameterization of the speed of sound to high densities, which includes a maximum in the speed of sound $c_s^2/c^2 > 1/3$ and an asymptotic convergence to the conformal limit from below, for very high densities ($\\rho \\gtrsim 50 \\, \\rho _0$ ) suggested by the perturbative QCD calculations [33].",
"The two different extensions lead to small changes in the predicted ranges, e.g., for the radius of a neutron star.",
"These differences result from the choice of three polytropes and the particular functional form chosen for the speed of sound parameterization, and would be diminished for arbitrarily fine discretizations of the high-density part of the EOS.",
"In this work we build on our past mass-radius results [29], [24] and investigate how future moment of inertia measurements of neutron stars will be able to further constrain the EOS and neutron star radii.",
"To this end, we investigate rotating neutron stars and use the Hartle–Thorne slow-rotation approximation [26], [27].",
"Later studies have been more conservative, verifying the applicability of this treatment to frequencies up to $f \\approx 200 \\, \\text{Hz}$ [9], [14].",
"The heavier neutron star of the system PSR J0737–3039 has a period of about 23 ms [11], [37] and can hence reliably be treated within the slow-rotation approximation.",
"Figure: Mass MM as a function of radius RR.",
"The gray area depicts theentire region allowed by the general EOS construction using the piecewise polytropeextension.",
"The highlighted areas represent MM–RR pairs that reach values for thespeed of sound c s /c⩽1/3c_{\\text{s}}/c \\leqslant 1/\\sqrt{3} (purple), 0.65 (blue), 0.75 (orange),and 0.95 (dark gray).",
"The dashed lines mark the corresponding regions for thespeed of sound model.Figure: Moment of inertia II as a function of radius RR.",
"The gray bandgives the allowed II–RR range resulting from the general EOS bandfor the piecewise polytrope extension as shown in Fig. .",
"The dark gray, light blue, anddark blue areas show the allowed II–RR values for the particular neutronstar masses indicated, where the dark gray area includes all possible II–RRpairs for each mass, and the light blue (dark blue) area correspondsto an assumed measurement of the moment of inertia with central valueI c I_c given in Table with a relative uncertainty ofΔI=±10%\\Delta I = \\pm 10\\% (±20%)(\\pm 20\\%).",
"The three panels assume centralvalues I c I_c that approximately correspond to the soft (a), intermediate (int) (b),and stiff (c) EOS, see Table .",
"Note that for a 2.4M ⊙ 2.4 \\,M_\\odot neutron star, the soft EOS is ruled outand thus no compatible I c I_c exists in this case.Panels (b) and (c) of Fig.",
"REF show the results for the moment of inertia $I$ as a function of neutron star mass and radius based on our EOS bands from the piecewise polytrope extension.",
"The moment of inertia can reach values up to about $290 \\, M_{\\odot }\\, \\text{km}^2$ for very heavy neutron stars, where the maximal values are clearly correlated with the stiffness of the EOS.",
"In addition, it is manifest that the three EOSs which are representative with respect to the radius are also representative with respect to the moment of inertia and practically span the full moment-of-inertia range (with only minor modifications for the soft EOS due to the new mass constraint).",
"For the pulsar PSR J0737-3039A with $M = 1.338 \\, M_\\odot $ we find the moment of inertia to be in the range $I = 53.2$ –$85.7 \\, M_\\odot \\, \\text{km}^2$ .",
"Our predicted range is significantly smaller than that of [45], where $I = 21.1$ –$113.2 \\,M_\\odot \\, \\text{km}^2$ , and similar to the range obtained by [23] with $I= 60.3$ –$90.5 \\, M_\\odot \\, \\text{km}^2$ .",
"In addition, we show the speed of sound $c_\\text{s}$ reached in our general EOS bands.",
"In Fig.",
"REF the highlighted areas represent $M$ –$R$ pairs that reach particular values for $c_{\\text{s}}/c$ .",
"Note that $c_\\text{s}/c$ is small at low densities in the nonrelativistic chiral EFT calculations and reaches $1/\\sqrt{3} \\approx 0.577$ from below in the perturbative QCD regime [33].",
"Figure REF clearly demonstrates that $c_\\text{s}/c$ has to reach values of around 0.65 to be compatible with two-solar-mass neutron stars.",
"In particular, if one demands that $c_\\text{s}/c\\leqslant 1/\\sqrt{3}$ for all densities in neutron-star matter, no EOS exists in our general construction that is compatible with the observed heavy neutron stars.",
"This has also been pointed out by [6] and is consistent with the findings of [51] and [24].",
"Table: Radius Constraints Resulting from Mass andMoment of Inertia Measurements for the Same Star, Assuming the Mass UncertaintyIs Negligible and Using the Piecewise Polytrope ExtensionFigure: Same as Fig.",
"but using the speed of soundmodel from to extrapolate to high densities.Table: Same As Table but Corresponding toFig.",
"Using the Speed of SoundModel to Extrapolate to Higher Densities"
],
[
"Improved constraints from moment of inertia measurements",
"Based on the frameworks discussed in Section , we now investigate to what extent moment of inertia measurements can improve these constraints.",
"To this end, we assume that it is possible to measure simultaneously the neutron star mass (with negligible uncertainty) and the moment of inertia with central value $I_c$ and relative uncertainty of $\\Delta I = \\pm 10 \\%$ and $\\pm 20 \\%$ , respectively.",
"We consider three different masses, $M=1.338 \\, M_{\\odot }$ , $2.0 \\, M_{\\odot }$ , and $2.4 \\, M_{\\odot }$ , and for each mass, three possible central values $I_c$ , given by $I_{\\text{low}}$ , $I_{\\text{int}}$ , and $I_{\\text{high}}$ , which approximately correspond to the moment of inertia given by the three representative EOSs shown in panel (c) of Fig.",
"REF .",
"The values of $I_c$ for these assumed measurements are listed in Table REF , where we also give the improved radius ranges resulting from such a simultaneous measurement.",
"In addition, we show the allowed $I$ –$R$ areas in Fig.",
"REF , where the three panels correspond to the low, intermediate, and high $I_c$ cases.",
"For a $2.4 \\, M_\\odot $ neutron star, the soft EOS is ruled out (see Fig.",
"REF ), and no low $I_c$ scenario exists in this case.",
"We also note that the EOS can have a more intricate behavior in the general EOS band, e.g, going from soft to stiff and vice versa with higher slopes in the $M$ –$R$ diagram (see Fig.",
"REF ).",
"Moreover, we show in Table REF and Fig.",
"REF how these radius constraints change if one uses the speed of sound model instead of the piecewise polytrope extension.",
"The results show that the radius constraints are remarkably consistent, with the largest differences due to the underlying allowed bands (see the gray regions versus the area within the representative EOS in Fig.",
"REF ), occurring for heavy mass neutron stars and the high $I_c$ case.",
"Figure: Allowed values for the moment of inertia II as a function of radius RR (graybands) resulting from the general EOS construction using the piecewise polytropeextension (left panel) and the speed of sound model (right panel).",
"The darker grayregions indicate the I-RI-R pairs that are consistent with a 1.338M ⊙ 1.338 \\, M_{\\odot }neutron star, whereas the blue and green highlighted areas include in additionthe GW170817 constraints for the chirp mass, mass ratio, and binary tidal deformabilityfrom LIGO/Virgo , see the text for details.Figures REF and REF clearly show that a measurement of $I_c$ with a relative uncertainty of $\\Delta I = \\pm 10 \\%$ ($\\pm 20 \\%$ ) in (almost) all cases significantly improves the constraints on neutron star radii.",
"For a $\\pm 10 \\%$ measurement, if the measured value of $I_c$ is located close to the center of the EOS band, the radius range decreases by about 50%, whereas the radius becomes even more narrowly predicted when $I_c$ is close to low or high values.",
"In the latter cases, the radius spread in Table REF is only 0.9–1.2 km for the piecewise polytrope extension and 0.2–1.1 km for the speed of sound model.",
"Figure: Pressure PP as a function of mass density ρ/ρ 0 \\rho /\\rho _0 in unitsof the saturation density.",
"The gray region is the general EOS band based onthe piecewise polytrope extension.",
"The light and dark blueareas show the allowed EOS range for assumed simultaneous measurementsof the mass (different rows) and the moment of inertia (different columns), as inFig.",
"and Table ,with a relative uncertainty of ΔI=±10%\\Delta I = \\pm 10\\% (±20%)(\\pm 20\\%).Figure: Same as Fig.",
"but using the speed of soundmodel, corresponding to Fig.",
"andTable .Next, we focus on the neutron star PSR J$0737-3039$ A with mass $1.338 \\, M_{\\odot }$ , which is the target of a future moment of inertia measurement.",
"In Fig.",
"REF we show the allowed values for the moment of inertia as a function of radius resulting from the piecewise polytrope extension (left panel) and the speed of sound model (right panel), where the darker gray regions indicate the $I-R$ pairs that are consistent with a $1.338 \\, M_{\\odot }$ star.",
"The impact of an accurate $I$ measurement is clear from the representative cases in Tables REF and REF .",
"Figure REF shows again that the tightest radius constraints would result from $I_c$ values toward the extremes of our general EOS bands.",
"Figure: Dimensionless moment of inertia I/MR 2 I/MR^2 as a function ofcompactness M/RM/R.",
"The red-to-blue region in the upper panel and thelight blue region in the lower panel show our results for the general EOSband using the piecewise polytrope extension, with color coding accordingto the neutron star mass in the upper panel.",
"In addition, we also showthe results for the speed of sound model as the region enclosed by theblack dashed lines.",
"In the upper panel, this is compared to correlation bands from in orange as well as in gray.In the lower panel, we also show the three representative EOS (soft,intermediate, and stiff) of .",
"The red (blue) lines with down(up) triangle points are the individual EOS within the piecewise polytropeextension with minimal χ 2 \\chi ^2 of I/MR 2 I/MR^2 with respectto the lower (upper) boundary (from fits for M/R⩾0.1M/R \\geqslant 0.1).Figure: Pressure PP as a function of mass density ρ/ρ 0 \\rho /\\rho _0 in unitsof the saturation density.",
"The gray region is the general EOS band based onthe piecewise polytrope extension.",
"The lines correspondto the individual EOS shown in the lower panel of Fig.",
",where the red and blue lines extremize the I/MR 2 I/MR^2–compactnesscorrelation.In addition, we explore the constraints from the gravitational-wave signal of the neutron star merger GW170817 [1], [2].",
"In Fig.",
"REF , we have highlighted the $I-R$ regions in blue (green) for the general EOS construction based on the piecewise polytrope extension (speed of sound model) that are consistent with the LIGO/Virgo results [2] for the chirp mass $\\mathcal {M} = 1.186 \\pm 0.001 \\, M_{\\odot }$ , the mass ratio $q = 0.73 - 1.00$ , and the binary tidal deformability $\\widetilde{\\Lambda } = 300^{+ 420}_{- 230}$ (for the 90% highest posterior density interval).",
"These ranges are compatible with the analysis of [17], suggesting that they are robust with respect to assumptions about the underlying EOS and deformability priors.",
"The comparison to the general EOS regions without the GW10817 constraints (darker gray vs. blue and green regions) in Fig.",
"REF shows that the GW170817 observation is consistent with the general EOS band based on nuclear physics and the observation of $2 \\, M_\\odot $ neutron stars.",
"Figure: Same as Fig.",
", but including the individual EOSshown in the lower panel of Fig.",
", where the red andblue lines extremize the I/MR 2 I/MR^2–compactness correlation.In addition to the radius constraints based on a moment of inertia measurement, we can also study the corresponding constraints for the EOS.",
"The different $I_c$ and mass scenarios for the piecewise polytrope extension (corresponding to the radius constraints of Fig.",
"REF and Table REF ) are shown in Fig.",
"REF .",
"The gray region is again the general EOS band of [29] (updated for the maximum mass constraint), whereas the different panels show the constraints for the assumed simultaneous measurements of the mass (different rows) and the moment of inertia (different columns).",
"Naturally, we find that the constraints on the EOS are the strongest for those cases that also give the strongest radius constraints.",
"In addition, small values of $I$ tend to give stronger constraints on the EOS at higher densities, whereas large values for $I$ provide stronger constraints at lower densities.",
"Moreover, measurements of heavy neutron stars provide stronger constraints on the EOS than the scenarios for typical neutron stars.",
"Further, we give in Fig.",
"REF the EOS constraints for the speed of sound model (corresponding to the radius constraints of Fig.",
"REF and Table REF ).",
"This shows very similar constraints on the EOS, as for the piecewise polytrope extension.",
"Several studies based on different phenomenological EOS have shown that the dimensionless moment of inertia $I/M R^2$ correlates with the compactness $M/R$ to a good approximation [34], [8], [35], [10].",
"In Fig.",
"REF we present our results for the piecewise polytrope extension (color coded) and the speed of sound model (black dashed line) for the dimensionless moment of inertia, which yield a very similar correlation band, and compare these to the bands from [50] and [10].",
"Our results agree reasonably well with these for $M/R > 0.15 \\, M_\\odot /$ km, while we find a deviation for smaller compactness parameters and also a somewhat larger band for $M/R > 0.2 \\, M_\\odot /$ km.",
"This shows that, e.g., predictions for neutron stars with small mass and large radii based on the former correlation bands are not compatible with the general EOS band.",
"This is most likely due to low-density assumptions made that are incompatible with modern nuclear physics.",
"In addition, we show in the lower panel of Fig.",
"REF the three representative EOSs (soft, intermediate, and stiff) of [29].",
"These are representative with respect to radius and moment of inertia for all masses (see Fig.",
"REF ) but, as is clear from Fig.",
"REF , they do not capture the extremes of the dimensionless moment of inertia.",
"In order to investigate the band for the dimensionless moment of inertia in more detail, we determined the individual EOSs that represent the limits of the band in Fig.",
"REF for the piecewise polytrope extension, which provides the more conservative estimate.",
"To this end, we discretized $M/R$ for $M/R \\geqslant 0.1 \\, M_\\odot \\,$ km$^{-1}$ and determined the $\\chi ^2$ of each EOS for the deviation of $I/M R^2$ from the lower (upper) band.",
"The results for the individual EOSs with the minimal $\\chi ^2$ values are shown as red (blue) lines in the lower panel of Fig.",
"REF .",
"The corresponding EOSs for these extreme cases are shown in Fig.",
"REF .",
"We observe that the EOSs with a minimum $\\chi ^2$ with respect to the lower boundary of the dimensionless moment of inertia $I/M R^2$ (red lines) tend to be rather stiff at nuclear densities and soft at high densities, whereas the EOSs leading to large values of $I/M R^2$ tend to be soft at nuclear densities and stiff at high densities (blue lines).",
"These trends are also reflected in the results for the mass, radius, and moment of inertia in Fig.",
"REF , where these individual EOSs are clearly extreme but nevertheless very interesting cases.",
"The EOSs with the low values for the dimensionless moment of inertia predict large radii at small masses (and moment of inertia) and small radii at larger masses (red lines), while the ones corresponding to large values for $I/M R^2$ show the opposite trend."
],
[
"Summary and Outlook",
"We have explored new and improved constraints for the EOS of neutron-rich matter and neutron star radii.",
"Our work is based on four inputs: (a) microscopic calculations of the EOS up to $1.1 \\, \\rho _0$ based on state-of-the-art nuclear interactions derived from chiral EFT combined with the piecewise polytrope or speed of sound extension to high densities following [29] and [24], respectively, (b) the precise measurement of the mass of PSR J0740+6620 with $2.14 \\begin{array}{c}+0.10 \\\\ -0.09\\end{array} \\, M_\\odot $ [15], (c) causality constraints at all densities and an asymptotic behavior of the speed of sound consistent with perturbative QCD calculations at very high densities for the $c_\\text{s}^2$ model, and (d) constraints from future measurements of the mass and moment of inertia of the same star.",
"Note that this analysis does not rely on any assumptions regarding the composition and properties of matter beyond the density $1.1 \\, \\rho _0$ , and within the space of the piecewise polytrope and speed of sound extension includes EOS that mimic regions with a first-order phase transition.",
"For the moment of inertia measurements we considered different scenarios by assuming various values and uncertainties for the moment of inertia.",
"We find that measurements with an uncertainty of $10 \\%$ lead to a reduction of the radius range by about 50% compared to the general EOS band from [29] and [24] when the moment of inertia corresponds to an intermediate EOS.",
"If the moment of inertia corresponds to values predicted by a soft or stiff EOS the radius range is reduced by a factor of 3 or more.",
"For all $\\pm 10\\%$ measurements, the resulting radius range is smaller than $1.9\\,$ km for all considered masses $M=1.338, \\ 2.0$ , and $2.4 \\, M_{\\odot }$ .",
"Specifically, for a $1.338 \\, M_{\\odot }$ star, we find radius ranges of $R=10.2$ –$11.5\\,$ km for low values of the moment of intertia ($I_\\text{low}=55 \\, M_{\\odot }\\,$ km$^2$ with $\\Delta I = \\pm 10 \\%$ ; combining the ranges from the piecewise polytropic and speed of sound extensions), $R=11.3$ –$12.9\\,$ km for intermediate values ($I_\\text{int}=70 \\, M_{\\odot }\\,$ km$^2$ ), and $R=12.5$ –$13.6\\,$ km for high values ($I_\\text{high}=85 \\, M_{\\odot }\\,$ km$^2$ ).",
"These ranges need to be compared with $R=10.2$ –$13.6\\,$ km based on the combined general EOS bands for this mass, when no information about the moment of inertia is used.",
"We have also investigated the corresponding constraints for the EOS.",
"We found that large values for the moment of inertia provide stronger constraints at lower densities, whereas small values tend to constrain the EOS at higher densities.",
"Moreover, measurements of heavy neutron stars provide overall stronger constraints.",
"In addition, we have studied the dimensionless moment of inertia $I/M R^2$ and established the full uncertainty ranges based on our general piecewise polytrope and speed of sound extension.",
"We find very interesting extreme EOSs at the boundaries of the correlation with the compactness, which have not been considered before.",
"Finally, we showed that the gravitational-wave constraints from the neutron star merger GW170817 [1], [2] are consistent with the general EOS bands explored here (see also [44]).",
"We found that the latest analysis of GW170817 [2] only slightly reduces the radius range predicted by the general EOS bands from the piecewise polytrope and speed of sound extension, and only weakly narrows the range for the predicted moment of inertia for a $1.338 \\, M_{\\odot }$ star.",
"Therefore, additional future detections from LIGO/Virgo, as well as NICER and other X-ray timing observations [54], combined with measurements of neutron star masses and in particular the moment of inertia, are a powerful avenue to further constrain the EOS of dense matter in a model-independent way."
],
[
"Acknowledgments",
"We thank G. Raaijmakers, I. Tews, and A. Watts for useful discussions.",
"This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 279384907 – SFB 1245, and J.M.L.",
"acknowledges support from NASA through grant 80NSSC17K0554 and the U.S. DOE from grant DE-FG02-87ER40317."
]
] | 2005.14164 |
[
[
"Scheduling strategies for the ESPRESSO follow-up of TESS targets"
],
[
"Abstract Radial-velocity follow-up of stars harbouring transiting planets detected by TESS is expected to require very large amounts of expensive telescope time in the next few years.",
"Therefore, scheduling strategies should be implemented to maximize the amount of information gathered about the target planetary systems.",
"We consider myopic and non-myopic versions of a novel uniform-in-phase scheduler, as well as a random scheduler, and compare these scheduling strategies with respect to the bias, accuracy and precision achieved in recovering the mass and orbital parameters of transiting and non-transiting planets.",
"This comparison is carried out based on realistic simulations of radial-velocity follow-up with ESPRESSO of a sample of 50 TESS target stars, with simulated planetary systems containing at least one transiting planet with a radius below $4R_{\\oplus}$.",
"Radial-velocity datasets were generated under reasonable assumptions about their noise component, including that resulting from stellar activity, and analysed using a fully Bayesian methodology.",
"We find the random scheduler leads to a more biased, less accurate, and less precise, estimation of the mass of the transiting exoplanets.",
"No significant differences are found between the results of the myopic and non-myopic implementations of the uniform-in-phase scheduler.",
"With only about 22 radial velocity measurements per dataset, our novel uniform-in-phase scheduler enables an unbiased (at the level of 1\\%) measurement of the masses of the transiting planets, while keeping the average relative accuracy and precision around 16\\% and 23\\% respectively.",
"The number of non-transiting planets detected is similar for all the scheduling strategies considered, as well as the bias, accuracy and precision with which their masses and orbital parameters are recovered."
],
[
"Introduction",
"The radial-velocity (RV) follow-up of exoplanet candidates identified using the transit detection method is important to definitively establish their planetary nature, estimate their masses and further refine orbital parameters.",
"It also makes atmospheric studies more informative by constraining the scale height [3].",
"Modelling the internal structure of each exoplanet [14], [15], [53], and population-level studies, e.g.",
"the characterization of the mass-radius relation [57], [11], [43], [31], are other applications that benefit from the extra information brought by RV data.",
"In the next few years, RV follow-up of exoplanet transits will most likely be dominated by observations of TESS [Transiting Exoplanet Survey Satellite, e.g.",
"[52]] objects of interest (TOIs).",
"Over the two years of its primary mission, TESS is expected to discover more than 14,000 new transiting exoplanets around almost as many stars [2].",
"The RV measurements required to obtain precise mass measurements even for just a few tens of these planets will easily exceed the many hundreds.",
"Most will be part of concerted efforts by several groups, namely those taking part in the TESS Follow-Up Observing Program (TFOP), with access to large amounts of telescope time.",
"In particular, the ESPRESSO collaboration [46], [47], [48] plans to devote around 32% of its Guaranteed Time Observations (GTO) for TOI follow-up, amounting to almost 88 nights distributed across 4 years (N. C. Santos, private communication).",
"Often RV measurements for a sample of stars known to host transiting planets are performed in an almost random way, conditional on the target stars being visible at low airmass.",
"More commonly there is some prior planning of the observations, for example to ensure that the RV phase-curves are sampled as uniformly as possible, given the orbital periods inferred from the transit data [7].",
"The most usual stopping criterion for the RV measurements is reaching some relative precision with respect to the transiting exoplanets masses [42].",
"However, in any case, the observations are usually done in a myopic (or greedy) way, i.e.",
"which star is chosen to be observed at a certain time does not take into account all possible scheduling configurations for the future, given the time available and sample of stars to be observed.",
"In principle, this should lead to a less efficient use of available telescope time than non-myopic (also known as batch or block) scheduling.",
"Our main objective in this work is then to quantify, using mock but realistic RV simulations, the difference in efficiency, with respect to the information gathered about exoplanet masses and orbital parameters through RV measurements, between different scheduling strategies for the ESPRESSO follow-up of TESS targets.",
"We will consider algorithms whose objective function leads to a sampling of the RV phase-curves of the known transiting planets as uniform as possible, and compare their results with those obtained under an algorithm which just randomly samples the set of target stars that are visible at observation time.",
"The former, henceforth called uniform-in-phase, will be implemented in both a myopic and a non-myopic way.",
"We start by laying out the procedures used to construct a sample of simulated TOIs, and to generate mock distributions of the ESPRESSO GTO.",
"Next, we describe the scheduling algorithms that will be compared.",
"We then report the results obtained, discuss them, and present our conclusions.",
"The TESS observing strategy was modelled by [2], in order to identify the approximately 200,000 stars in the TESS Input Catalog Candidate Target List that should be observed at 2-minute cadence.",
"The remaining stars were assumed to be observed at 30-minute cadence in full-frame image data.",
"They then associated zero or more orbiting planets to each star, with specific physical and orbital characteristics, according to measured exoplanet occurrence rates [25], [16].",
"Finally, they used the TESS noise model to predict which exoplanets would be detected and their derived properties.",
"It was estimated that TESS would find around 1250 exoplanets in the 2-minute cadence mode, and about 13,100 planets in the full-frame image data.",
"A sample of stars for possible ESPRESSO follow-up observations was pre-selected among those stars considered in [2] by demanding: a declination in the interval $[-80^{\\rm o},\\,+30^{\\rm o}]$ , to ensure extended periods of visibility at low airmass from Paranal; an effective temperature, $T_{eff}$ , in the interval $[4000,\\,6000$ ] K, and high surface gravity, $\\log g > 4.0$ , i.e.",
"only G and K dwarf stars.",
"We then included in our final sample the 50 brightest stars among those pre-selected with at least one orbiting planet with a radius below $4R_{\\oplus }$ , 3 detected transits and a transit signal-to-noise greater than 10.",
"This final selection step effectively limits our sample to stars with a magnitude, $V$ , below 10.5, minimising the RV measurement uncertainty due to photon-noise.",
"It also aligns our sample with a TESS primary science requirement: the estimation of the mass of 50 exoplanets with radius smaller than $4R_{\\oplus }$ [52].",
"We ended up with 53 transiting planets orbiting 50 stars, with 3 systems having 2 transiting planets each.",
"We associated to each transiting planet the expected mass, given its radius, obtained using the Forecaster algorithm [11].",
"The radii were assumed to be known within an uncertainty of $10\\%$ (standard deviation), typical of what is expected by combining data from Gaia [5], [6] and TESS [7].",
"In the publicly available from [2] catalogue only planets that transit are identified.",
"But, in order to generate realistic simulations of a RV time series, we need to take into account all planets around each star in the sample.",
"Therefore, we added extra orbiting planets to each star, non-detectable by TESS.",
"In order to be coherent with the choice of occurrence rates made in [2], we used for such purpose the occurrence rates published in [25].",
"However, these do not extend to orbital periods long enough to include all planets capable of generating a RV semi-amplitude, $K$ , larger than 0.5 m/s, roughly the minimum value we expect our simulated follow-up survey to be sensitive to.",
"This expectation was fully supported a posteriori by the results of the analysis of the simulated RV datasets, we will later describe, as only planets with values for $K$ above 1 m/s were indeed detected.",
"Although the presence of planets with $K<0.5$ m/s could make the detection of planets with higher values for $K$ more difficult, the effect should be quite small given that we expect that almost always there will be a large difference with respect to $K$ between those extra low-$K$ planets and those that ended up being detected in the systems considered.",
"Therefore, we first extrapolated the occurrence rates in Table 2 of [25] up to orbital periods of 2 years, for radius in the intervals $[2,\\,4]$ , $[4,\\,6]$ and $[6,\\,22]$ $R_{\\oplus }$ , and to 418 days for radius in the interval $[1.25,\\,2]$ $R_{\\oplus }$ .",
"In order to achieve this, we assumed the occurrence rate density, as a function of orbital period, is described by a log-normal distribution [25], [56].",
"The joint posterior probability distribution for the parameters of such function was characterized within each of the four mentioned radius bins, given the occurrence rates provided in Table 2 of [25] for the available period bins.",
"The expected values for those log-normal parameters were then used to infer the integrated occurrence rates in the period bins: $[145,\\,245]$ and $[245,\\,418]$ days in the case of radius between $1.25$ and 2 $R_{\\oplus }$ ; $[245,\\,418]$ and $[418,\\,730]$ days in the case of radius between 2 and 4 $R_{\\oplus }$ ; $[418,\\,730]$ days in the case of radius in the intervals $[4,\\,6]$ and $[6,\\,22]$ $R_{\\oplus }$ .",
"These extrapolated occurrence rates can be found in Table REF , together with the values used from [25].",
"With this extrapolation, we are able to take into account all planets, with an orbital period smaller than 2 years, that are capable of inducing a RV signal with $K>0.5$ m/s, given their expected mass as estimated using the Forecaster algorithm [11].",
"Table: Average number of planets per star per radius and period bin (in percent) from .",
"Inside square brackets are extrapolated values by assuming that, inside each radius interval, the occurrence rate density is described by a log-normal function of the orbital period.The number of planets we associate with each star, within the radius-period bins identified in Table 2 of [25] plus those with extrapolated occurrence rates, was then randomly drawn from a Poisson distribution with mean 0.92 (expected number of planets across all such bins).",
"If the number obtained was greater than the number of transiting planets in the system, the radius-period bins where the extra planets are located were randomly drawn from the full radius-period bin distribution taking into account the respective occurrence rates.",
"Then, a specific radius and period was randomly drawn for each extra planet inside the associated bin, assuming a log-normal distribution (the same type that was considered in the radius-period bin occurrence rates extrapolation).",
"For each extra planet, the radius and orbital period drawing procedure is repeated until the transit signal-to-noise is lower than 10, or the period found is greater than twice the timespan of the scheduled TESS observations of the sector where the star is located.",
"This ensures that any extra planet associated with the stars in our sample would not have been detected in the simulations of TESS observations made by [2].",
"Since planets with radius above 4 $R_{\\oplus }$ can have $K>0.5$ m/s even with orbital periods greater than 2 years, we randomly added extra planets with orbital period between 2 and 10 years.",
"For this we used the occurrence rates in [29], respectively $0.24$ and $0.15$ in the radius ranges $[4.5,\\,9.5]$ and $[9.5,\\,15.7]$ $R_{\\oplus }$ .",
"We again associated to each extra planet the expected mass, given its radius, obtained using the Forecaster algorithm [11].",
"We ended up with 50 extra planets, distributed across 35 systems (only one of which has 3 extra planets).",
"Their orbital eccentricities, $e$ , were then randomly drawn from a Beta distribution with parameters $\\alpha = 1.03$ and $\\beta = 13.6 $ following [34].",
"We kept the assumption of [2] that all planets in any system are co-planar, and set the inclination of all systems to $90^{\\rm o}$ in order to enable a more direct comparison between true and estimated planetary masses.",
"At this point, we determined whether each possible pair of planets in any given system is Hill stable, by finding if the following inequality is true [26]: $\\left(\\mu _{1} + \\mu _{2} \\frac{a_1}{a_2} \\right) \\left(\\mu _{1} \\gamma _{1} + \\mu _{2}\\gamma _{2} \\sqrt{\\frac{a_{2}}{a_{1} }} \\right)^{2} >\\alpha ^{3} + 3^{4/3}\\mu _{1}\\mu _{2}\\alpha ^{5/3},$ where $\\mu _{i}$ , $a_{i}$ and $e_{i}$ are respectively the ratio between the planet mass and the mass of the star which it orbits, the orbital semi-major axis, and the orbital eccentricity, with $\\alpha =\\mu _{1}+\\mu _{2}$ and $\\gamma _{i}=\\sqrt{1-e_{i}^{2}}$ , for each planet $i=\\lbrace 1,\\,2\\rbrace $ in the pair being considered.",
"All the systems found to contain Hill unstable pairs of planets were re-simulated, keeping the number of extra planets but randomly re-drawing their radius and orbital parameters, until every simulated planetary system only contained Hill stable pairs.",
"In the process, we actually found that the pair of transiting planets in system with identification number 304142124 is not Hill stable.",
"In order to minimally change the catalogue published by [2], while ensuring this planet pair becomes Hill stable, we just decreased the eccentricity of the outer transiting planet from $0.15453$ to $0.142$ .",
"Finally, for both transiting and non-transiting planets, the mean anomaly $M_0$ at the time $t_0$ (when we start our scheduler), and the argument of periastron, $\\omega $ , were randomly drawn from a uniform distribution between 0 and $2\\pi $ .",
"With this it becomes possible to compute the overall planetary contribution to the RV time series for each star.",
"Figure REF shows the distributions for $P$ , $K$ and $e$ , for the simulated transiting and non-transiting planets.",
"A detailed description of the properties of every planet in our simulation is provided in a machine readable table, with a summary shown in Table REF .",
"Figure: From upper to lower panel, distributions of RV semi-amplitudes, KK, orbital eccentricities, ee, and periods, PP, for the transiting (blue) and non-transiting (orange) planets.Table: Summary of the properties of all stars and planets in the sample."
],
[
"ESPRESSO GTO simulations",
"The ESPRESSO GTO consists of 273 nights during 4 years, and began on the 1st of October 2018https://www.eso.org/sci/observing/policies/gto_policy.html.",
"Exoplanetary science occupies 80% of the time, 10% is allocated to fundamental constants time-variability studies and 10% is discretionary time at the disposal of the ESPRESSO consortium [46], [48].",
"The total amount of time available for exoplanetary science is in turn divided as follows: 30% for exoplanetary atmospheric characterization; 30% for TOI follow-up; 40% for a RV survey.",
"We simulated the scheduling of ESPRESSO GTO observations from the 1st of October 2019 until the 30th of September 2022, i.e.",
"only for 3 years.",
"Furthermore, we assumed that on the 1st of October 2019 all our TOIs would have been observed and characterized by TESS.",
"The 80% of the ESPRESSO GTO dedicated to exoplanetary science consists of close to 55 half-nights each semester.",
"We randomly spread them in such a way as to mimic the ESPRESSO GTO distribution in ESO periods 102 and 103, the only known at the time of writing, including aggregation of some half-nights into full nights.",
"Each full day is divided into 60 observation slots, all with a duration of 24 minutes (15 as integration time plus 9 for overheads), but due to seasonal variation, each astronomical night will have a different number of observation slots associated.",
"The integration time was defined to be 15 minutes in order to average out the RV variability induced by stellar oscillations in the G and K dwarf stars we are considering [18].",
"We will only consider observations slots with an associated airmass not greater than 2.0.",
"Thus, taking into account the magnitude and temperature ranges for the 50 stars we are considering, respectively, $[6.69,10.37]$ and $[4408,5978]$ K, the ESPRESSO ETC (Exposure Time Calculator) https://www.eso.org/observing/etc/bin/gen/form?INS.NAME=ESPRESSO+INS.MODE=spectro estimated RV variability due to photon-noise will range from $0.1$ to $0.5$ m/s, under normal atmospheric conditions.",
"The average value is close to $0.3$ m/s across all observational slots for which RV simulations were performed.",
"We further assumed that exoplanetary atmospheric characterization takes precedence, given that they are performed during transit and thus are time-critical.",
"For each semester we thus first randomly sampled, with repetition, the ESPRESSO consortium target list for this type of study, until 30% of the available time was reached.",
"Each scheduled transit observation is composed of enough sequential observational slots to cover the time interval from one hour before the transit starts until one hour after the transit ends.",
"Some of the half and full nights allocated to exoplanetary atmospheric characterization are not completely filled with these type of observations and thus the remaining slots are available for TOI follow-up and the RV survey.",
"We repeated this procedure 10 times, obtaining 10 different distributions for the 80% of the ESPRESSO GTO dedicated to exoplanetary science.",
"These simulations yielded between 2563 and 2628 (24-minute) slots that can be used for TOI follow-up and the RV survey.",
"Among these we decided to schedule a fixed number of 1102 slots for TOI follow-up, which we assume take precedence over the RV survey.",
"This number is very close to the fraction that can be used for TOI follow-up, i.e.",
"30% of the total number of slots associated with each GTO realization.",
"Although we could have let that number vary with each GTO simulation, we decided to fix it to the mean averaged over all simulations so that the results could be more easily compared."
],
[
"RV simulations",
"Stellar activity also induces variations in the radial velocity of a star [35], [17], [8], [9].",
"Their overall amplitude, $\\sigma _{\\rm act}$ , was determined, for all stars in our sample, by randomly drawing from a Normal distribution with mean given by Equation 4 in [10], and a standard deviation of $0.4$ .",
"The mean reproduces the observed correlation between RV variability and a measure of stellar flicker, $F_8$ , for stars with low levels of activity, while the value of $0.4$ is suggested by Figure 6 in [10].",
"The flicker parameter, $F_8$ , is determined using Equation 2 in [54], which depends on stellar mass, effective temperature and $\\log g $ , information we have for all the TESS target stars we consider.",
"The assumed value of $\\sigma _{\\rm act}$ for each star in our sample can be seen in Figure REF , in the Appendix.",
"The RV variations induced by stellar activity have been shown to be well modelled as a Gaussian Process, i.e.",
"their joint probability distribution is assumed to be a multivariate Normal with a number of dimensions equal to the number of RV measurements under analysis, and in particular as a Gaussian Process with a mean of zero and a covariance matrix, $\\Sigma $ , with entries calculated using a quasi-periodic covariance function or kernel [27], [19], [51], [1].",
"Nevertheless, as we will later see, none of the three scheduling strategies under study relies on the assumed model for the stellar activity induced RV variations to decide on the best schedule, thus any changes to such model should have little impact on the relative outcomes of those strategies.",
"In order to test this, we performed RV simulations where the impact of stellar activity was assumed to be the result of either a quasi-periodic Gaussian Process or Gaussian white noise, i.e.",
"randomly and independently generated (as a function of time) from a Normal distribution with a constant mean (zero) and standard deviation ($\\sigma _{\\rm wn}$ , equal to $\\sigma _{\\rm act}$ in our case).",
"Note that Gaussian white noise is equivalent to a Gaussian Process with a covariance matrix, $\\Sigma $ , whose entries are zero everywhere except in its diagonal, where they are equal to the square of the assumed standard deviation (i.e.",
"the variance).",
"The assumption of a quasi-periodic kernel gives rise to entries in the covariance matrix, $\\Sigma $ , of the form $\\Sigma _{ij} = \\eta _1^2\\exp \\left[ - \\frac{(t_i - t_j)^2}{2\\eta _2^2} - \\frac{2\\sin ^2\\left(\\frac{\\pi (t_i-t_j)}{\\eta _3}\\right)}{\\eta _4^2} \\right] + s^2 \\delta _{ij} \\:\\text{,}$ where $\\eta _1$ , $\\eta _2$ , $\\eta _3$ , and $\\eta _4$ are parameters that can be interpreted as the amplitude, timescale of decay, periodic timescale, and level of high-frequency variability (within the periodic timescale) of the RV variations.",
"Because these are caused by stellar active regions, whose appearance and disappearance along the line-of-sight is modulated by the stellar rotation period, $\\eta _3$ should be close to its value.",
"The parameter $s$ is usually known as jitter, and accounts for (apparent) non-correlated variability.",
"This can arise, for example, through sampling with a much lower frequency than that associated with correlated RV variability induced by one or more of the many physical process involved in stellar activity.",
"For each star, the value of $s$ was randomly drawn with equal probability from the interval $[0.1\\,{\\rm m/s},\\,\\sigma _{\\rm act}/2]$ , while the value for $\\eta _1$ was assumed to be the square-root of $\\sigma _{\\rm act}^2$ minus $s^2$ .",
"The assumed values of $s$ and $\\eta _1$ for each star in our sample can be seen in Figure REF .",
"We also set $\\eta _3$ equal to the rotation period of each star.",
"This was fixed using the following procedure.",
"First, we selected all stars with an effective temperature within $[4000,\\,6000$ ] K and $\\log g > 4.0$ , included in the catalogue of 34,000 Kepler main sequence stars assembled by [41].",
"Then, for each star, we identified its nearest 100 neighbours in the plane defined by effective temperature and $\\log g$ , determined the mean and standard deviation of the distribution of the rotation periods for those 100 stars, and randomly drew a value from a Gaussian with the derived mean and standard deviation.",
"Finally, we associated to the star $i$ under consideration the measured rotation period, $P_{{\\rm rot}\\,,i}$ (and its associated uncertainty, $\\sigma _{{\\rm rot}\\,,i}$ ) in the sample of 100 stars that is closest to the value previously drawn from the Gaussian.",
"The rotation period assumed for each star in our sample can be found in Table REF .",
"For all stars in our sample, the values of $\\eta _2$ were randomly drawn from a log-uniform distribution truncated at $\\eta _3$ and $5\\eta _3$ , while the values of $\\eta _4$ were randomly drawn from a Gaussian distribution with mean of $0.7$ and standard deviation of $0.05$ .",
"This ensures the values thus obtained are typical of those inferred from the analysis of RV data [19], [39], [12], [21].",
"We consider two other contributions to the variability of the RV time-series, which we assume can be characterized as Gaussian white noise: one due to photon-noise, $\\sigma _{\\rm ph}$ , which was calculated using the ESPRESSO ETC specifically for each star according to its magnitude, effective temperature and airmass at the time of observation; and another due to the RV variability induced by instrumental-noise, $\\sigma _{\\rm ins}$ , which we assumed to be constant and equal to $0.1$ m/s [47], [48].",
"Therefore, the full covariance matrix, $\\Sigma _v$ , associated with the multivariate Normal that describes the stochastic behaviour of each RV time-series becomes equal to the covariance matrix, $\\Sigma $ , associated with the specific Gaussian Process used to describe stellar activity induced RV variability (either quasi-periodic or white noise), to whose diagonal the squares of both $\\sigma _{\\rm ph}$ and $\\sigma _{\\rm ins}$ are added.",
"We also associated to every star a systemic RV relative to the centre of mass of the system, $v_{\\rm sys}$ , drawn from a random uniform distribution between $-100$ to 100 m/s, roughly the observed range for stars in the solar neighbourhood [37].",
"Thus, the RV time series associated with each star were generated based on the following model: $v_{\\rm r}(t) = v_{\\rm sys} + \\sum \\limits _{\\rm i=1}^{\\rm n_p} v_{\\rm r,i}(t) + \\epsilon (t)$ with $v_{\\rm r,i}(t)=K_{\\rm i}\\lbrace \\cos [\\phi _{\\rm i}(t)+\\omega _{\\rm i}]+e_{\\rm i}\\cos (\\omega _{\\rm i})\\rbrace $ $\\epsilon (t)\\sim N\\left(0,\\Sigma _v\\right)$ where ${\\rm n_p}$ is the number of planets orbiting the star, $K_{\\rm i}$ is the RV semi-amplitude, $\\omega _{\\rm i}$ is the argument of periastron, $e_{\\rm i}$ is the orbital eccentricity, and $\\phi _{\\rm i}(t)$ is the true anomaly as a function of time, $t$ , calculated from the other orbital parameters [49], all with respect to planet i.",
"Thus, we neglect any gravitational interactions between orbiting planets when calculating the instantaneous RV for every star."
],
[
"Scheduling strategies",
"We will consider three different scheduling strategies.",
"Two of them, labelled A, are myopic, i.e.",
"the best schedule is defined sequentially in time.",
"In strategy A1, the star chosen to be observed at any given time is randomly drawn from all stars in the sample which can be observed at that time, at an airmass equal or smaller than 2, and with a Moon separation greater than 30 degree, henceforth known as the observability constraint.",
"In strategy A2, this sub-sample of stars is further restricted to the stars that have a smaller number of observations than those associated with the sample star with the largest number of allocated observations at previous times, henceforth known as the equalizing condition.",
"Imposing the second condition leads to a more even distribution of the observational slots between the sample stars.",
"However, in the case of strategy A2, we also want the sampling of the RV phase-curves of the known transiting planets to be as uniform as possible, i.e.",
"to ensure as close as possible uniform-in-phase sampling.",
"This is achieved through the maximization of the following objective function, capable of measuring the overall dispersion of points in a given interval, $f(\\lbrace x_{\\rm i}\\rbrace )\\equiv \\left\\lbrace \\sum \\limits _{\\rm i=1}^{\\rm 1102} [d(x_{\\rm i})]^{-q}\\right\\rbrace ^{-1/q}$ where $d(x_{\\rm i})$ is the time distance between the observation $x_{\\rm i}$ and its nearest neighbour in the orbital phase-space of the transiting planet targeted by the observation (including across the phase-space boundary), as a fraction of the orbital period of such planet.",
"When more than one transiting planet exists around a star, $d(x_{\\rm i})$ equals the sum of the distances with respect to all transiting planets in the system, which favours the observation of stars for which multiple transiting planets are known.",
"We assume the orbital period and mid-transit time for each transiting planet to be perfectly known a priori.",
"In a real application, this would mean fixing them to e.g.",
"their expected values given the TESS data.",
"In the context of strategy A2, the best schedule is then also constructed sequentially in time.",
"First, the stars that fulfil both the observability constraint and the equalizing condition are identified.",
"The star chosen to be observed among these will be the star with the smallest number of allocated observations at previous times.",
"If several stars share this number of previous observations, then the star chosen to be observed is that which leads to the maximization of the objective function provided, $f(\\lbrace x_{\\rm i}\\rbrace )$ .",
"This same criterium is applied to all stars that satisfy the observability constraint in the situation where no stars fulfil the equality condition (i.e.",
"all have the same number of allocated observations at previous times).",
"However, these rules are applied only at those times for which all stars that satisfy the observability constraint and the equalizing condition have already been observed at least once, else the star chosen to be observed is randomly drawn among those that have not yet been so.",
"As a result of the this procedure, all 10 simulated schedules, according to strategy A2, associate between 17 and 24 observational slots to each star.",
"In contrast, strategy A1 always leads to some stars being observed only a few times, the minimum ranging from 2 to 8 across the 10 simulations, while some other stars end up being slotted for observation as many as 39 to 49 times.",
"The third strategy, labelled B, is non-myopic.",
"In this case, the aim is to compare all possible schedules, across the full time-span of 3 years, and then choose that which maximizes the objective function, $f(\\lbrace x_{\\rm i}\\rbrace )$ .",
"Given the form taken by such function, this procedure leads to what is known as $L^q$ relaxation of the points in the design space, in our case the orbital phase-space of each planet.",
"It yields a nearly optimal approximation to the maximin solution to the problem [50].",
"The larger the $q$ , the better should be this approximation.",
"But then the objective function becomes increasingly localised (in the space of all possible scheduling configurations) and it is more difficult to find the region where the function is maximised.",
"After extensive testing we decided to use $q=2$ (also in the case of strategy A2 to allow for easier comparison between myopic and non-myopic scheduling).",
"The maximin solution is a classical example of a space-filling strategy [50].",
"In our case, it corresponds to finding the schedule that maximizes the sum over all stars of the minimum (time) distance, normalized as a fraction of the orbital period(s) of the known transiting planet(s) around each star, between any observation and all others of the same star.",
"An alternative classical space-filling strategy is the minimax solution.",
"In this case, the objective would be to find the schedule that minimizes the sum over all stars of the maximum distance (as defined before) between any observation and all others of the same star.",
"However, the maximin solution is computationally faster to find, because it only requires the calculation of distances between neighbouring observations in the orbital phase-space of each planet.",
"Whereas finding the minimax solution would require the calculation of the distances between all observations with respect to each planet [50].",
"Nevertheless, we also implemented an algorithm to identify the minimax solution, and found it leads to schedules very similar to those obtained using strategy B.",
"Given the large number of time slots available for scheduling and the fact that the stars considered are observable during most of any given year, the number of possible scheduling configurations is very large.",
"Therefore, it is impossible to compare the values the objective function takes for all such configurations.",
"As a result, we used the acebayes R packagehttps://cran.r-project.org/web/packages/acebayes [45] to find the schedule that maximizes the objective function.",
"This is done via an approximate coordinate exchange (ACE) algorithm, where a sequence of conditional one-dimensional optimisation steps are used, as described in [44].",
"In our case, the objective function depends on the stellar label and the slot time, which will hence be our coordinates.",
"Each schedule, or design, can be viewed as a collection of points in this two-dimensional space.",
"The search for the maximum of the objective function then proceeds through the sequential change of the coordinates of each point in a given initial schedule.",
"In the case of the stellar label coordinate, a change occurs when it is found that, for the time slot associated with a particular design point, there is another star for which the objective function attains a higher value and each star in the sample continues to be observed within the pre-specified minimum number of times.",
"In the case of the time coordinate, a change occurs if there is another time slot for which the objective function reaches a higher value, among those which are not yet associated with a star and for which the observability constraint is obeyed.",
"The search for such optimal time slot is performed by first approximating the objective function with respect to observation time, for the star associated with the design point under consideration, through a Gaussian process (within acebayes), and then by identifying the time for which the objective function is maximized.",
"The initial schedule for strategy B is created randomly, with the only conditions being that the observability constraint is obeyed and each star in the sample is observed at least the pre-specified minimum number of times.",
"The closer the latter is to the average number of available observational slots per star, in our case $1102/50\\simeq 22$ , the harder it is for the ACE algorithm to optimize the schedule, and the smaller will be the value of the objective function at the end of the optimization process.",
"This means that there is a trade-off between ensuring an (almost) equal number of observations per star and an optimized sampling of the orbital phase-space of each transiting planet.",
"Somewhat arbitrarily, we set the required minimum number of observational slots per star to be 20.",
"If it was much smaller, there would be significant variations in the accuracy and precision with which the mass and orbital parameters of each transiting planet would be recovered.",
"In our case, each run of the ACE algorithm goes through a sequence of $2\\times 1102=2204$ conditional optimisation steps.",
"In order to consolidate the best schedule, we re-run 100 times the ACE algorithm within acebayes, using the output of each ACE run as input to the following one.",
"As the runs progress, we keep track of the objective function value, and choose the final best design (which is not necessarily the last) as that with the highest associated value for the objective function.",
"In Figure REF we show how many RV observations are scheduled, for all 10 simulations per strategy, as a function of where each observed planet is in the respective phase-curve.",
"This is equivalent to seeing the phase-curves in overlap, and as expected all scheduling strategies lead to almost uniform distributions.",
"However, this hides significant differences in the phase-space distribution of RV observations between transiting planets.",
"In particular, as expected, strategy A1 leads to the most irregular phase-curve coverage per planet and dataset, followed by strategy A2.",
"This can be clearly seen in Figure REF .",
"This shows the root mean square (rms) of the difference between the simulated orbital phase coverage of transiting planets and perfectly uniform phase sampling, averaged over all such planets in each system and the associated 10 simulated datasets per strategy.",
"This difference is obtained, for each planet and simulation, by summing the squares of the distances between the sorted orbital phases (between 0 and 1), with each distance subtracted by the inverse of the number of RV measurements (which is the distance between orbital phases under perfectly uniform phase sampling).",
"As expected, the orbital phase coverage is much consistently (low standard deviation) closer to uniform (low average) in the case of strategy B than in the case of the other two strategies, especially A1.",
"In practice, our assumption in the case of strategy B that the ESPRESSO GTO schedule can be known a priori for the full 3 years is unrealistic.",
"ESO will only inform the ESPRESSO consortium of its schedule for each semester close to its beginning.",
"Therefore, a more realistic implementation of strategy B would require re-scheduling every 6 months the remaining time for the completion of the 3 years.",
"This should not have a significant impact in the expected efficiency with which information is recovered about planets properties through the implementation of strategy B.",
"This is because what is expected to happen within each semester, as ESO relays the information about the available observational slots, is just an effectively random re-shuffling of their position within the semester.",
"Thus, the expected information gain guiding strategy B should remain essentially the same.",
"A more realistic implementation of this strategy should only suffer from some loss of coherence around the start/end of each semester, the more so the smaller the orbital periods of the systems scheduled to be observed at those times.",
"This near-randomization of the scheduler at such a small fraction of the available time should have a very small impact on the expected information gathered through strategy B.",
"Given the considerable amount of extra computing time required to simulate a re-scheduling every 6 months, we decided to implement strategy B in the more simplified manner previously presented."
],
[
"Bayesian analysis",
"We used the open-source software kimahttps://github.com/j-faria/kima [20] to perform Bayesian statistical analysis of all simulated RV datasets.",
"These were analysed assuming the meta-model described in sub-section 2.3, with ${\\rm n_p}$ now becoming effectively a label identifying mutually exclusive models.",
"We then have ${\\rm n_p}={\\rm n_t}+{\\rm n_{nt}}$ , where ${\\rm n_t}$ and ${\\rm n_{nt}}$ are, respectively, the number of transiting and non-transiting planets in each system.",
"While the former is fixed, to either 1 or 2, we let the latter vary between 0 and 5, with equal prior probability assigned to each possible value.",
"This means that we assume a priori all the planets detected in transit to have the status of confirmed planets from the point of view of the RV data analysis.",
"The orbital periods, $P$ , and times of mid-transit with respect to the transiting planets were assigned Gaussian priors, centred on the values provided by [2], and with standard deviations of $0.001$ days (which is the typical level of uncertainty expected from TESS data).",
"Knowledge about the time of mid-transit effectively constrains the mean anomaly at some particular time of choice, $M_0$ , given the other orbital parameters.",
"For the planets without transit information, the orbital periods were assigned log-uniform (often called Jeffreys) priors between 1 and 10 000 days.",
"For all planets, we assumed modified log-uniform distributions for the RV semi-amplitudes, $K$ , and the standard deviations associated with the Gaussian white noise contributions, both $s$ and $\\sigma _{\\rm wn}$ , with the knee located at 1 m/s, while limited above by 1000 m/s in the case of $K$ and the RV span, i.e.",
"the difference between the RV maximum and minimum, for each RV dataset $i$ , ${\\rm RV}_{{\\rm max},\\,i}-{\\rm RV}_{{\\rm min},\\,i}$ , in the case of both $s$ and $\\sigma _{\\rm wn}$ .",
"These modified distributions are defined until the lower limit of 0 m/s.",
"The prior for the orbital eccentricities was set to a half-Gaussian with $\\sigma =0.32$ for the transiting planet in systems with only one, and $\\sigma =0.083$ for both transiting planets in systems with two, as suggested by [55], and to a Kumaraswamy distribution [36], with shape parameters $\\alpha =0.867$ and $\\beta =3.03$ , for all the possible extra, non-transiting planets [33].",
"Finally, the priors we assumed for the Gaussian Process parameters $\\eta _1$ , $\\eta _2$ , $\\eta _3$ and $\\eta _4$ were, respectively, log-uniform within the interval $[0.1\\,{\\rm m/s},\\,{\\rm RV}_{{\\rm max},\\,i}-{\\rm RV}_{{\\rm min},\\,i}]$ , log-uniform between $P_{{\\rm rot},i}$ and $5P_{{\\rm rot},i}$ , Gaussian with mean equal to $P_{{\\rm rot},i}$ and standard deviation set to $5\\sigma _{{\\rm rot},i}$ , and log-uniform within the interval $[1/e,\\,e]$ .",
"Most other parameters are assigned uniform priors between sensible limits, as can be seen in Table REF .",
"From the computational point of view, the total amount of datasets is 3 (scheduling strategies ) $\\times $ 2 (stellar activity noise models) $\\times $ 10 (simulations) $\\times $ 50 (systems) = 3000, each containing between 4 and 49 measurements (most being around 22).",
"On a single processor, kima requires a few hours to yield converged posterior probability distributions with respect to all model parameters.",
"Thus, it would have been infeasible to perform the analysis sequentially on a single computer.",
"As a result, we adopted a full Cloud architecture by exploiting the services offered by the commercial platform Amazon Web Services (AWS).",
"In particular, since the analysis of each dataset by is independent from the others, we used the architecture described in [38] to run parallel applications by using clusters offered by AWS.",
"In this particular case, we used a cluster of 25 instances on AWS, each of them equipped with 64 vCPU and 256 GB of RAM.",
"This allowed the analysis of all 3000 datasets by kima in less than 5 hrs and consuming roughly 8000 CPU/hrs in the process.",
"This approach is particularly useful when the full analysis needs to be re-done, for some reason, since it can be performed quickly, while keeping the overall price low.",
"Our main objective with this work is to compare the different scheduling strategies with respect to: (1) the strength of the expected constraints on the values for the mass and orbital parameters of the planets that are known to transit; (2) the number of detected non-transiting planets, as well as the strength of the expected constraints on the values associated with the respective mass and orbital parameters.",
"These criteria are linked, given that a decision on how many extra planets have been detected is effectively equivalent to choosing the model, with some label ${\\rm n_p}$ , to be used for parameter estimation.",
"We choose to base such decision on the comparison between the Bayesian evidence or marginal likelihood, i.e.",
"the constant which normalizes the joint posterior distribution, for models with associated consecutive values for ${\\rm n_p}$ , starting with ${\\rm n_p}={\\rm n_t}$ , i.e.",
"${\\rm n_{nt}}=0$ .",
"Because we are assigning equal prior probabilities to all models with respect to the same star, comparing evidences is equivalent to determining the so-called Bayes Factor, $\\mathcal {B}$ , which is then just equal to their ratio.",
"Its value can be interpreted through the scale introduced by [30] (see also [32]), according to which a Bayes factor of at least 150 between models with associated consecutive values for ${\\rm n_p}$ is required in order to claim a planet detection [23], [22], [4].",
"Note that this procedure never leads to more detections of non-transiting planets than their true number, in any given system.",
"The correspondence between detected and existing non-transiting planets in a system is based on the proximity of values for $K$ and $P$ .",
"This criteria never lead to ambiguous cases in our simulations, as a result of the large difference in the values of one or both of these quantities in the few systems with more than one non-transiting planet.",
"Figure: In each plot, the upper panel shows the relation between the precision with which KK is estimated based on the simulated RV datasets, σ K( sim ) \\sigma _{K ({\\rm sim})}, and the theoretical precision expected under Eq.",
", σ K( the ) \\sigma _{K ({\\rm the})}, for the transiting planets in the 450 datasets pertaining to the 15 systems with only one (transiting) planet.",
"The result of the analysis for each of those datasets is represented by a point, whose colour is associated with the scheduling strategy used: magenta for strategy A1; orange for strategy A2; cyan for strategy B.",
"In each plot, the lower panel shows the residuals R≡[σ K( sim ) -σ K( the ) ]/σ K( the ) R\\equiv [\\sigma _{K ({\\rm sim})}-\\sigma _{K ({\\rm the})}]/\\sigma _{K ({\\rm the})}.",
"In the top figure, the RV datasets were simulated assuming the stellar activity induced RV variations are uncorrelated, while in the bottom figure those variations were assumed correlated.",
"Equality between ordinate and abscissa is represented by the dashed line.",
"Note that the only difference between both plots is in the values of the ordinate.Figure: Each point represents the relative difference between σ K( sim ) \\sigma _{K ({\\rm sim})} and σ K( the ) \\sigma _{K ({\\rm the})}, as represented in the lower panel of the top plot in Fig.",
", averaged over the 10 RV datasets simulated for each system per scheduling strategy (colour-coded as in Fig.",
"), as a function of the eccentricity of the transiting planet in the system.It should be noted that by using the full RV datasets in the analysis we are effectively assuming that there is neither partial or full loss of planned RV measurements due to adverse weather conditions or technical problems.",
"Here partial also means substantial degradation of the expected RV measurement uncertainty due to photon-noise, as a result of very bad seeing ($>1.3^{\\prime \\prime }$ ) or thick cirrus clouds.",
"Although such assumption is unrealistic, the loss should not amount to more than 10 percent of the expected data, according to the ESO annual reporting on the operational conditions at Paranalhttps://www.eso.org/public/products/annualreports/.",
"Therefore, on average this should affect only a couple of RV measurements per target in three years.",
"In any case, which RV measurements are affected or lost, as a result of such effects, will not be correlated with the actual scheduling strategy chosen to be implemented.",
"Therefore, the data loss will impact in a similar way the information about planetary masses and orbital parameters that can be recovered under each scheduling strategy.",
"Thus, we decided to ignore it in our analysis since our interest is on the comparison of the relative merit of different scheduling strategies.",
"However, in a practical context, this issue can be addressed and its impact minimised by rescheduling all future observations after some amount of the planned RV measurements are performed, and taking into account which were not or badly affected.",
"Figure: The top and bottom plots show the same information as in Fig.",
", but now each point refers to the results for the 38 transiting planets in the 35 systems with more than one planet.",
"Again, note that the only difference between both plots is in the values of the ordinate."
],
[
"Comparison with theoretical expectations",
"The RV variations induced by a transiting planet with known orbital period, $P$ , and time of mid-transit, $T_{\\rm transit}$ , depend only on the (unknown) value of $K$ if the orbit is assumed circular, $v_{\\rm r}(t_{\\rm i})=K\\sin \\phi (t_{\\rm i},P,T_{\\rm transit})\\,,$ with the true anomaly at time $t_{\\rm i}$ given by $\\phi (t_{\\rm i})=2\\pi (t_{\\rm i}-T_{\\rm transit})/P\\,.$ Further assuming that the measurements of such RV variations are affected by uncertainties that are independent and identically Gaussian distributed, i.e.",
"Gaussian white noise, it can then be shown [13] that the theoretically expected (a posteriori) absolute precision in the estimation of $K$ , which we will call $\\sigma _{K{\\rm (the)}}$ , is given by $\\sigma _{K{\\rm (the)}}=\\left\\lbrace \\sum \\limits _{\\rm i=1}^{N_{\\rm RV}}\\left\\lbrace \\frac{\\sin [\\phi (t_{\\rm i},P,M_0)]}{\\sigma _{\\rm eff}(t_{\\rm i})}\\right\\rbrace ^{2}\\right\\rbrace ^{-1/2}$ where $\\sigma _{\\rm eff}(t_{\\rm i})$ is the effective measurement uncertainty with respect to the planet-induced RV at time $t_{\\rm i}$ , and $N_{\\rm RV}$ is the number of RV measurements considered.",
"The former can result from several Gaussian white noise contributions, and is just the standard deviation associated with the Normal distribution that describes the full uncertainty with respect to the RVs induced just by the transiting planet.",
"Assuming the effective RV measurement uncertainty is approximately constant with time, the minimization of $\\sigma _K$ demands sampling the orbital phase space only when the radial velocity reaches its maximum absolute value.",
"This corresponds to what is usually known as quadrature sampling, whereby RV measurements are performed only when the transiting planet is at right-angles with respect to our line-of-sight to the star.",
"In this case, $\\sigma _{K{\\rm (the)}}=\\frac{\\sigma _{\\rm eff}}{\\sqrt{N_{\\rm RV}}}$ which is what one would expect under the central limit theorem if each RV measurement corresponds in fact to a direct estimation of $K$ .",
"Thus, quadrature sampling is the optimal procedure if (1) $\\sigma _{\\rm eff}$ is independent of the measurement time and (2) all the conditions under which Eq.",
"REF was derived are assured.",
"If the former is not true, but the later is, then some RV measurements off quadrature can actually yield more information about $K$ , if the associated RV measurement uncertainty is sufficiently smaller.",
"If condition (2) is not true, quadrature sampling can lead to biased results.",
"This can be easily seen in the case of non-circular orbits.",
"If these are only sampled in quadrature, the estimates for the eccentricity $e$ and $K$ will be significantly degenerate, reaching complete degeneracy when the argument of periastron is such that the orbital semi-major axis becomes aligned with our line-of-sight to the star.",
"These degeneracies allow very high values simultaneously for $e$ and $K$ , given the RV quadrature data.",
"Although such combinations could be disfavoured a priori, reducing their impact in a posteriori estimates of $e$ and $K$ , like their means, medians or modes, they will nevertheless tend to bias high any such estimates.",
"If the orbital phase-space is sampled uniformly, while still assuming the effective RV measurement uncertainty to be approximately constant with time, Eq.",
"REF then implies a decrease by a factor of $\\sqrt{2}$ in the absolute precision with which $K$ can be estimated, i.e.",
"$\\sigma _{K{\\rm (the)}}=\\sigma _{\\rm eff}\\sqrt{\\frac{2}{N_{\\rm RV}}}\\,,$ with respect to quadrature sampling.",
"This decrease can be understood by realizing that now a significant fraction of the RV measurements are made close to anti-quadrature, when star and planet are aligned along the line-of-sight, and thus the RV signal-to-noise ratio, i.e.",
"that between the expected RV amplitude and $\\sigma _{\\rm eff}$ , becomes much smaller with respect to RV measurements made in quadrature.",
"Therefore, less information about $K$ will be gathered, on average, per RV measurement.",
"However, in the case of non-circular orbits, uniform-in-phase sampling partially lifts the degeneracies between argument of periastron, $e$ and $K$ , so much so the denser the sampling.",
"Thus, this type of sampling does not lead to such strong biases as quadrature sampling.",
"We can then conclude that if there is some significant probability of $e$ being different from zero, one should opt for uniform-in-phase rather than quadrature sampling to ensure that estimates of $K$ in particular are as unbiased as possible.",
"Nevertheless, uniform-in-phase sampling is still not optimal if one wants to maximize the amount of information that can be gathered through RV measurements about the orbital parameters, when in the presence of non-circular orbits.",
"This is again, in part, the result of variations in the RV signal-to-noise ratio across the orbital phase-space.",
"The optimal sampling solution could be found using the tools of bayesian experimental design [24], [40], [28].",
"In the absence of any RV information it would depend critically on the assumed prior distributions for the orbital parameters.",
"But as each RV measurement is obtained, the optimal sampling solution can be continuously updated, converging to the same solution whatever the assumed prior distributions.",
"Unfortunately, bayesian experimental design carries a very high computational cost, which is why we did not consider using it in this work.",
"Our results present an unique opportunity to test the impact of different effects on the absolute precision with which $K$ can be estimated for transiting planets, $\\sigma _K$ .",
"We start by considering the results of the analysis of the $3\\times 15\\times 10=450$ RV datasets pertaining to systems which contain only one (transiting) planet and for which it was assumed just Gaussian white noise.",
"In the the top plot in Fig.",
"REF , the results of each of those 450 analysis is represented by a point.",
"The ordinates correspond to the values of $\\sigma _K$ that result from the analysis of the datasets, which we will denote as $\\sigma _{K{\\rm (sim)}}$ , while the abscissas are calculated using Eq.",
"REF given the characteristics of each dataset.",
"The later would correspond to the expected value of the former if the orbits were circular.",
"If this was the case, we should see only stochastic variations about zero for the deviations of the ordinates with respect to the abscissas.",
"Although the lower panel of the top plot in Fig.",
"REF seems to suggest otherwise, in particular for the datasets acquired under strategy A1, the variations seen are indeed statistically compatible with the expected value for $\\sigma _{K{\\rm (sim)}}$ being well approximated by Eq.",
"REF , even though the orbits of the planets considered are not circular: for strategies A1, A2 and B, the mean and standard deviation of the residuals shown in the lower panel of the top plot in Fig.",
"REF are $0.30\\pm 0.36$ , $0.20\\pm 0.25$ and $0.10\\pm 0.19$ .",
"Nevertheless, these values suggest that how well Eq.",
"REF predicts $\\sigma _K$ depends on how close the sampling of the phase-curve is to uniform.",
"It also seems to depend on how high is the information content of the RV measurements, given the decreasing scatter in the residuals as the expected value for $\\sigma _{K{\\rm (sim)}}$ decreases.",
"All these conclusions seem to be true irrespective of the eccentricity of the planets considered, given that this does not seem to be correlated with the magnitude of the residuals, as can be seen in Fig.",
"REF .",
"The results of the analysis of the RV datasets pertaining to the 35 systems which contain more than one planet, and for which it was assumed just Gaussian white noise, allow for the characterization of the impact of extra planets in the absolute precision with each $K$ can be recovered for the 38 transiting planets in those systems.",
"In the upper panel of the top plot in Fig.",
"REF , the results associated with each of those $3\\times 38\\times 10=1140$ analysis are represented by points, with the coordinates having the same meaning as in Fig.",
"REF .",
"It is perceptible an increase in scatter, as well as a higher systematic (positive) difference between the value for $\\sigma _K$ that results from each analysis and the value given by Eq.",
"REF .",
"Now, the mean and standard deviation of the residuals shown in the lower panel of the top plot in Fig.",
"REF are $0.95\\pm 0.82$ , $0.53\\pm 0.38$ and $0.42\\pm 0.36$ , for strategies A1, A2 and B, respectively.",
"Again, there seems to be no correlation between the magnitude of the residuals and the eccentricity of the planets considered, all below $0.3$ and with a mean value of $0.08$ , very similar to what is obtained ($0.07$ ) for the 15 lone transiting planets.",
"Finally, in the bottom plots of Figs.",
"REF and REF we show the results for the same two sets of planets, but when stellar activity induced RV variations are assumed correlated and jointly modelled as a GP.",
"The mean and standard deviation of the residuals shown in the bottom panels are, respectively, for strategies A1, A2 and B: $0.23\\pm 0.39$ , $0.18\\pm 0.29$ and $0.16\\pm 0.24$ , for the 15 lone transiting planets; $0.98\\pm 0.87$ , $0.58\\pm 0.53$ and $0.53\\pm 0.40$ , for the 38 transiting planets with companions.",
"Remarkably, these numbers are very similar to those obtained when the stellar activity induced RV variations are assumed non-correlated and modelled as Gaussian white noise.",
"This is just a reflection of the fact that the abscissas are the same, and the ordinates are very similar.",
"As we will later discuss, when $\\sigma _K$ is averaged over all transiting planets, it differs by $0.03$ at most (less than $10\\%$ ), for any of the three scheduling strategies, between what is obtained under the two contrasting assumptions about the characteristics of the stellar activity induced RV variations.",
"This indicates that, as long as these variations are correctly modelled (which is difficult to ascertain for any particular star besides the Sun), and there is enough information in the RV measurements (as seem to be the case in our simulated datasets), their impact on the absolute precision with which $K$ can be recovered for transiting planets is essentially independent of whether they are correlated or not (but dependent on their amplitude).",
"In summary, we find that Eq.",
"REF yields a good approximation to the expected absolute precision with which $K$ can be recovered for transiting planets, even in the presence of mild eccentricity (less than $0.3$ ) and realistic correlated RV variations due to stellar activity.",
"Nevertheless, large deviations (typically up to a $50\\%$ increase) from the expectation are possible, the more so the less uniform is the sampling of the orbital phase curves and the smaller the information content of the RV measurements.",
"As expected, the presence of extra planets in a system leads to an increase both in the magnitude and scatter of $\\sigma _K$ with respect to the value expected under Eq.",
"REF .",
"For the type of planetary systems considered, this increase is reflected in a typical underestimation of $\\sigma _K$ for the transiting planets, between $40\\%$ and $100\\%$ , and similar scatter, with higher values corresponding to orbital phase curves sampled less uniformly and less informative RVs.",
"In any case, we would like to stress that these conclusions are conditional on the assumption that the RV data generating mechanisms, in particular those associated with stellar activity and the instrumentation used, are well approximated by the assumed model."
],
[
"Results obtained assuming stellar activity correlated noise",
"We will now focus the discussion on the results obtained when the RV variations induced by stellar activity were assumed to be correlated, generated by a Gaussian Process with non-zero covariance terms, given that this constitutes the most realistic scenario.",
"In Appendix A, we present and compare with these, the results obtained when stellar activity induced RV variations were assumed to be uncorrelated, akin to Gaussian white noise.",
"In Figure REF we show the architecture of the 50 planetary systems we consider.",
"We identify which planets transit, and differentiate between the non-transiting planets that are never detected, sometimes detected or always detected, across all simulations and for all three scheduling strategies.",
"Averaging over the 10 simulations per strategy, a total of $8.2\\pm 0.6$ , $8.5\\pm 0.5$ and $8.8\\pm 0.5$ non-transiting planets are detected using strategies A1, A2 and B, respectively, out of the 50 that we simulated orbiting our sample of stars.",
"The numbers provided represent means and standard deviations, and are very similar.",
"The differences are not significant given the variation seen across the simulations.",
"In order to compare further the results, we define the following quantities, with respect to some planet characteristic $X$ , and to a given simulation: absolute bias, $\\mathbf {E}[X]-X_{true}$ relative bias, $(\\mathbf {E}[X]-X_{true})/X_{true}$ absolute accuracy, $\\mid \\mathbf {E}[X]-X_{true}\\mid $ relative accuracy, $\\mid \\mathbf {E}[X]-X_{true}\\mid /X_{true}$ absolute precision, $\\sigma _X$ relative precision, $\\sigma _X/\\mathbf {E}[X]$ where $X_{true}$ , $\\mathbf {E}[X]$ and $\\sigma _X$ represent, respectively, the true, expected value and standard deviation of $X$ .",
"The latter two are estimated given all values for $X$ present in the MCMC output from the kima analysis of the dataset associated with the simulation being considered.",
"In Figures REF , REF and REF we show the means and standard deviations for the distributions of absolute bias associated with the orbital parameters $K$ and $e$ , as well as mass, $M$ , for the planets that are known to transit and the three scheduling strategies.",
"The averaging is performed over the 10 simulations per system and strategy.",
"Because the absolute bias differs from the expected value by a constant, their distributions have the same standard deviations.",
"Expected values are more scattered (with respect to the true values) and uncertain for some planets in the case of strategy A1 mostly as a result of the respective host stars being systematically under-observed (and others over-observed) with respect to average, due to having shorter (longer) visibility windows.",
"The marginal posteriors used for this exercise, and those that follow, are those associated with the model chosen using the detection procedure for the non-transiting planets previously described, given the result of the Bayesian analysis of each simulated dataset.",
"Figure: Absolute bias, i.e.",
"the difference between the marginal posterior mean and the true value, as a function of the later, for the RV semi-amplitude, KK, and with respect to the transiting planets.",
"Results averaged over 10 simulations are shown, with the associated standard deviation, for the three scheduling strategies, A1 (upper panel), A2 (middle panel ) and B (lower panel).",
"Colour indicates the number of RV measurements per host star, averaged over the 10 simulations: red, less than 15; blue, between 15 and 25; purple, more than 25.Figure: Absolute bias, i.e.",
"the difference between the marginal posterior mean and the true value, as a function of the later, for the orbital eccentricity, ee, and with respect to the transiting planets.",
"Results averaged over 10 simulations are shown, with the associated standard deviation, for the three scheduling strategies, A1 (upper panel), A2 (middle panel ) and B (lower panel).",
"The colour code is the same as in Fig.",
".Figure: Absolute bias, i.e.",
"the difference between the marginal posterior mean and the true value, as a function of the later, for the mass, MM, and with respect to the transiting planets.",
"Results averaged over 10 simulations are shown, with the associated standard deviation, for the three scheduling strategies, A1 (upper panel), A2 (middle panel ) and B (lower panel).",
"The colour code is the same as in Fig.",
".In the upper panel of Table REF , the absolute and relative bias, accuracy and precision with which $K$ , $e$ and $M$ are recovered, averaged over all transiting planets and simulations, is shown for the three strategies.",
"The uncertainties provided are standard deviations, and characterise the dispersion of such values taking into account all transiting planets.",
"They should not be confused with the uncertainties associated with the estimates of $K$ , $e$ and $M$ for individual planets, and thus should not be used to draw any conclusions regarding confidence or credible intervals for those quantities.",
"This is particularly true in the case of quantities whose marginal posterior distributions are heavily skewed, like the eccentricity.",
"The same quantities shown in the upper panel of Table REF are provided in the lower panel, including with respect to the orbital period, $P$ , but now averaged over the detected non-transiting planets.",
"In Table REF , the absolute and relative bias, accuracy and precision with which $\\eta _1$ , $\\eta _2$ , $\\eta _3$ , $\\eta _4$ and $s$ are recovered, averaged over all simulations, is shown for the three strategies.",
"Table: In the upper panel it is shown the absolute and relative bias, accuracy and precision with which KK, ee and mass, MM, are recovered, averaged over all transiting planets and simulations, for the three strategies.",
"The uncertainties provided are standard deviations, and characterise the dispersion of such values taking into account all transiting planets.",
"The same quantities, as well as the orbital period, PP, are provided in the lower panel with respect to all detected non-transiting planets.",
"The absolute quantities with respect to KK, MM and PP are in units of m/s, M ⊙ M_{\\odot } and days, respectively.Table: Absolute and relative bias, accuracy and precision with which η 1 \\eta _1, η 2 \\eta _2, η 3 \\eta _3, η 4 \\eta _4 and ss, are recovered, averaged over all simulations, for the three strategies.",
"The absolute quantities with respect to η 1 \\eta _1 and s are in units of m/s, in the case of η 2 \\eta _2 and η 3 \\eta _3 are in units of days, while η 4 \\eta _4 is dimensionless.The estimation of $M$ is most dependent of $K$ , but it is also contingent on the values for $e$ , $P$ and the stellar mass.",
"Thus, it is not straightforward to extrapolate results for $K$ to what would be expected with respect to $M$ .",
"In order to estimate $M$ one also needs to assume an inclination for the orbital plane.",
"We will assume this to be known, and set it to $90^{\\rm o}$ , the same value assumed for all systems when the RV measurements were simulated.",
"Although this situation is not realistic, it allows for a direct comparison between true and estimated planetary masses.",
"Overall, the two uniform-in-phase scheduling strategies, myopic, A2, and non-myopic, B, lead to very similar results.",
"In the case of the transiting planets, the values estimated for $K$ and $M$ are significantly less biased, as well as more accurate and precise than those obtained through the random strategy, A1.",
"However, there are no significant differences between the three strategies with respect to how well the true values of $e$ are recovered.",
"Given that most of these are about $0.1$ or smaller, as can be seen in Figure REF , it is not surprising to find that all scheduling strategies lead to values around $0.1$ or smaller for the absolute bias, accuracy and precision, and thus much higher values for the relative counterparts to these quantities.",
"With respect to the detected non-transiting planets, all scheduling strategies lead to the acquisition of similar amounts of information about the true values of $K$ , $e$ , $P$ and $M$ .",
"This is not surprising, given that none of the strategies was designed with the aim of detecting such planets.",
"The same happens with respect to the parameters associated with the Gaussian process model that is used to describe the stellar activity induced RV variations.",
"Interestingly, the expected values for $K$ , as well as for the mass, $M$ , derived for the detected non-transiting planets given the simulated datasets are typically smaller by a factor of about 10% with respect to the true values.",
"However, the expected values for the orbital period, $P$ , are essentially unbiased with respect to the true values.",
"As expected, in particular given the discussion in Subsection 3.2, the most important factors affecting the detection probability of a non-transiting planet in our simulations are the number of RV measurements available and the value of $K/\\sigma _{\\rm act}$ .",
"In strategies A2 and B, the former is almost always 22, which is enough for the detection of 8 non-transiting planets in all simulations.",
"But the non-transiting planet in system 40, which has a significantly lower value for $K/\\sigma _{\\rm act}$ , just 5.6, is only detected 50% and 70% of the times in strategies A2 and B, respectively.",
"This suggests that for $N_{\\rm RV}$ around 22 only planets with $K$ in excess of roughly $5.6\\times \\sigma _{\\rm act}$ can be detected with a probability greater than 50%.",
"On the other hand, although the planets in systems 11 and 45, which have very similar values for $K/\\sigma _{\\rm act}$ (close to 68), are always detected in the simulations of strategies A2 and B, they are only detected 70% and 60% of the times, respectively, under strategy A1.",
"This is due to $N_{\\rm RV}$ falling below 12 in the simulations were detection did not occur.",
"In the case of the transiting planets, all the distributions associated with the bias, accuracy and precision are positively skewed, except those for the bias and precision with respect to $e$ for which the skew is negative.",
"The non-zero mean and positive skew in the distribution of the bias for $K$ and $M$ , seems to be the result of the existence of undetected (non-transiting) planets.",
"The mean bias gets closer to zero and the skew greatly diminishes, if only datasets whose analysis lead to the detection of all non-transiting planets in the associated systems are considered in the calculation of these statistics (and the opposite occurs for the other systems).",
"Interestingly, in the case of the uniform-in-phase strategies, the sampling of the phase-curves of the transiting planets seems to be so close to optimal in terms of information gathering, that even in the presence of undetected (non-transiting) planets the bias is very close to zero and the skew small.",
"The differences between the results obtained for each scheduling strategy, regarding both the transiting and non-transiting planets, should increase as the average number of possible RV measurements per star, $N_{\\rm RV}$ , decreases, and vice-versa.",
"For example, if this number was about half of what was assumed, i.e.",
"around 10, we would still expect strategy B, as well as A2 to a lesser extent, to yield fairly strong constraints on the masses and orbital parameters of the transiting planets, but it would be hard to detect any non-transiting planet.",
"On the contrary, in this situation, strategy A1 would probably fail to deliver reliable constraints for the transiting planets around the least observed stars, but some non-transiting planets would end up being detected around the most observed stars.",
"Although the mean $N_{\\rm RV}$ is, by construction, exactly the same for all the scheduling strategies, the associated standard deviation is $8.39$ for strategy A1, while only $1.44$ for A2 and $1.05$ for B.",
"In the Appendix, we present the results obtained by assuming the RV variations induced by stellar activity are uncorrelated, and can be described as Gaussian white noise.",
"As expected, given that none of the three scheduling strategies considered relies on the assumed model for such variations to decide on the best schedule, the conclusions that can be drawn are very similar to the ones just described."
],
[
"Conclusions",
"We implemented three different scheduling strategies for the ESPRESSO GTO allocated to radial velocity follow-up of TOIs.",
"Our main objective was to compare a novel uniform-in-phase scheduling algorithm with a random scheduler, and determine whether a non-myopic implementation of the former offered any advantage with respect to the more common myopic way.",
"The scheduling strategies were compared with respect to the amount of information gathered about the masses and orbital parameters of all planets in the TOIs host systems.",
"In particular, we considered a sample of 50 TESS target stars, with simulated planetary systems containing at least one transiting planet with a radius below $4R_{\\oplus }$ [2].",
"We found that both uniform-in-phase scheduling strategies lead to an unbiased (at the level of 1%) measurement of the masses of the transiting planets, while keeping the average accuracy and precision around 16% and 23%, respectively.",
"This is significantly better than what can be achieved with random scheduling, which does not only lead to more biased (about 2%) estimates of the mass of the simulated TOIs, but also to less accurate and precise estimates, respectively about 19% and 28% on average.",
"The number of non-transiting planets detected is similar for all the scheduling strategies considered, as well as the bias, accuracy and precision with which their masses and orbital parameters are recovered.",
"Although we have not found any significant difference between the results obtained with the two uniform-in-phase scheduling strategies, myopic and non-myopic, this may be due to an assumed timespan for the observations (3 years) that is much larger than the orbital periods of the target transiting planets (below 50 days).",
"As this difference decreases, a myopic scheduling strategy should lead to increasingly larger deviations with respect to uniform sampling of the phase curves, given that less than optimal choices early on become more difficult to compensate later in the observation schedule."
],
[
"Acknowledgements",
"We thank Nuno Santos for insightful discussions.",
"We acknowledge the excellent open-source acebayes R package made available to the community by Antony Overstall.",
"This work was supported by Fundação para a Ciência e a Tecnologia (FCT) through national funds (PIDDAC) and the research grants UID/FIS/04434/2019, UIDB/04434/2020 and UIDP/04434/2020.",
"This work was also supported by FCT through national funds (PTDC/FIS-AST/28953/2017, PTDC/FIS-AST/32113/2017) and by FEDER - Fundo Europeu de Desenvolvimento Regional through COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI-01-0145-FEDER-028953, POCI-01-0145-FEDER-032113).",
"All the data underlying this article will be shared on request to the corresponding author."
],
[
"Results obtained assuming only Gaussian white noise",
"Here we present the results of the analysis of the RV datasets generated assuming the stellar activity induced RV variations are akin to Gaussian white noise, and compare them with the results previously discussed.",
"In Figure REF we highlight the non-transiting planets that are never detected, sometimes detected or always detected, across all simulations and for all three scheduling strategies.",
"Averaging over the 10 simulations per strategy, a total of $9.8\\pm 0.6$ , $9.9\\pm 0.8$ and $10.5\\pm 1.2$ non-transiting planets are detected using strategies A1, A2 and B, respectively, out of the 50 that we simulated orbiting our sample of stars.",
"As before, these numbers are very similar, and the differences not significant given the variation seen across the simulations.",
"They are also 15 to 20% higher than those obtained for the datasets with correlated stellar activity noise.",
"This was expected, given that it is more difficult to disentangle correlated noise than uncorrelated noise from a signal.",
"In Table REF , the absolute and relative bias, accuracy and precision with which $K$ , $e$ and $M$ are recovered, averaged over all simulations and either all transiting planets (upper panel) or all detected non-transiting planets (lower panel), are shown for the three strategies.",
"In the lower panel the same quantities are shown with respect to the orbital period, $P$ , of the detected non-transiting planets.",
"Table: In the upper panel it is shown the absolute and relative bias, accuracy and precision with which KK, ee and mass, MM, are recovered, averaged over all transiting planets and simulations, for the three strategies.",
"The uncertainties provided are standard deviations, and characterise the dispersion of such values taking into account all transiting planets.",
"The same quantities, as well as the orbital period, PP, are provided in the lower panel with respect to all detected non-transiting planets.",
"The absolute quantities with respect to KK, MM and PP are in units of m/s, M ⊙ M_{\\odot } and days, respectively.Overall, the results are very similar to the ones previously obtained under the assumption of correlated stellar activity induced RV variations.",
"However, now the non-myopic uniform-in-phase strategy, B, seems consistently, though only slightly, better on average than the myopic strategy, A2, in terms of the information recovered about the true values of $K$ and $M$ for the transiting planets.",
"Again, both these strategies lead on average to significantly less biased, as well as more accurate and precise values for $K$ and $M$ than strategy A1, with little difference between the three strategies with respect to how well the true values of $e$ are recovered.",
"Somewhat counter-intuitively, the estimates for the mass and orbital parameters of the non-transiting planets seem now to be on average significantly more biased, less accurate, and less precise, than the estimates for the same quantities previously obtained under the assumption of correlated stellar activity induced RV variations.",
"This is due to the fact that most of the 15 to 20% extra non-transiting planets that are now being detected have substantially lower values for $K$ .",
"And given that the average number of RV measurements per system is independent of its characteristics, significantly less information is recovered about their mass and orbital parameters.",
"In turn, this brings down the information recovered about such quantities when averaged over all detected non-transiting planets, making the averaged bias, accuracy and precision obtained under the assumption of non-correlated stellar activity noise seem worse than when such noise is assumed correlated."
]
] | 2005.14008 |
[
[
"Higher Order Temporal Analysis of Global Terrorism Data"
],
[
"Abstract Temporal networks are a fundamental and flexible way of describing the activities, relationships, and evolution of any complex system.",
"Global terrorism is one of the biggest concerns of recent times.",
"It is also an example of a temporal network that evolves over time.",
"Graph analytics can be used to explore salient properties of the terrorism network to understand its modus operandi, which can be used by the global alliance of security and government entities to form a co-ordinated response to this threat.",
"We present graph based analysis to understand temporal evolution of global terrorism using the Global Terrorism Database (GTD)."
],
[
"Introduction",
"Networks are a fundamental and flexible way of representing entities, relationships, and behaviors in many real-world domains such as power grids [1], social networks [2], modeling adversarial activities [3], and terrorist networks [4].",
"Temporal evolution of such networks is of great interest to understand how a specific network changes over the course of time and whether can we predict the changes expected in the future.",
"Graph theoretic metrics that are sufficient to model a static network fail to capture non-linear dynamic behavior of a temporal network.",
"Global terrorism involves a network of terrorist organizations and their sympathizers, and has a long history of perpetrating attacks on the social, political, and economic stability of different regions of the world.",
"We present a graph based approach to investigate the relationships and behavior of such organizations which cannot be captured using naive count-based tabular analysis.",
"We model these organizations based on their involvement in a set of terrorist events.",
"We use the Global Terrorism Database (GTD) [5] [6], which is an open-source database including information on terrorist events around the world from 1970 through 2017.",
"It contains information about 181,000 terrorist events, with a total of 135 attributes for each event, including date, location, group, weapon, casualty and so on.",
"Some of the more useful variables in the GTD include: ID, Date, Location, Summary, Attack type, Target type (for example, Police Checkpoint), Target subtypes (for example, Iraqi Police Service), and specific targets, Group name if known, and Weapon types.",
"When the attack is part of a coordinated multi part incident, then the related incident IDs are also listed.",
"It is maintained by the National Consortium for the Study of Terrorism and Responses to Terrorism (START)."
],
[
"Related Work",
"Steve Ressler [7] presents a survey of social network analysis approaches to combat terrorism.",
"It distinguishes terrorist organizations from hierarchical, state-sponsored appointments in characteristics such as leadership and organizational structure.",
"It uses networks to analyze recruitment, evolution, and the diffusion of radical ideas.",
"Fellman et al.",
"[8] uses non-linear dynamical systems modeling to explore centrality and hierarchy of 9-11 hijackers network.",
"Carley et al.",
"[9] presents dynamic network analysis to understand evolution of a network to destabilize a terrorist network.",
"Our work presents a real-world use case and uses temporal graph patterns to model evolution of a terrorist network.",
"We present tools to analyze large scale graphs."
],
[
"Static Graph Analysis",
"The tabular nature of these datasets makes it hard to bring out the relationships between the entities involved in the system.",
"We propose to construct a different view of the tabular datasets in terms of a graph to analyze them.",
"We focus on the affinity of a terrorist organization to other organizations, attack types, targets, and weapons.",
"We start with creating some small networks from 1,000 incidences in GTD to demonstrate some of the possible node-type/edge-type combinations that may be useful in detecting potential Chemical, Biological, Radiological, Nuclear, and high yield Explosives (CBRNE) Weapon of Mass Destruction (WMD) activities.",
"We filter events to include only US incidents past 1990.",
"Figure REF shows a graph with nodes representing perpetrator groups, weapon types, attack types, targets, and events.",
"Figure: GTD Event GraphWe are also interested in finding the weapon profile of a terrorist organization and finding groups of organizations using similar weapons.",
"This is high value information which can be used to identify common sourcing of such weapons and to disrupt supply-chain of such organizations.",
"Figure REF shows a graph where nodes are the perpetrators, joined by the weapon types used in at least one incident.",
"A bipartite graph view of the same network in figure REF allows us to use graph connectivity of terror networks to compare their similarity in terms of modus operandi.",
"Figure: Weapon Profile of Terrorist organizationsFigure: Bipartite View: Weapon Profile of Terrorist organizations"
],
[
"Temporal Graph Analysis",
"Temporal analysis of a complex system reveals many interesting and non-intuitive phenomena.",
"Counting based measures such as Figure REF present limited information about the evolution of the network.",
"Graphs are a powerful modeling tool to understand much more complex and latent properties of the system.",
"We use small temporal dyads [10] to find terrorist organizations that are identified as multi-perpetrators of an event.",
"In order to ensure consistency in the usage of group names for the database, the GTD database uses a standardized list of group names that has been established by project staff to serve as a reference for all subsequent entries [5].",
"Multiple perpetrator group attributions do not necessarily indicate that perpetrator groups collaborated to execute an attack.",
"This could represent competing attributions, competing claims of responsibility, competing accusations, or a combination of these.",
"We construct an association graph between main perpetrators of every event.",
"For the sample GDT data beyond 1990, we get a clear indication of different communities of the terrorist organizations in figure REF .",
"The terrorist groups in a community may have co-ordinated some attacks together or claimed responsibilities for it.",
"Figures REF , REF , and REF show a temporal shift in the active terrorist organization communities in terms of associated attack events.",
"These types of higher order graph analyses of the terrorism network give much more insight into the operations of these groups, which is not available using lower order elements of the graph such as vertices and edges.",
"We can clearly see which organization is central to facilitating such collaborations among smaller groups to conduct terrorist attacks.",
"As shown in figure REF , Al-Qaida, Lashkar-e-Taiba (LeT), and PIJ were the hub of terrorist association between 1990-2000, but were overshadowed by ISIL and TTP after 2010.",
"Similarly, edge thickness shows the strength of association between two groups.",
"We also observe silos of operation for each terrorist group which gets bigger and denser over the course of time.",
"LeT is such an example which can be seen growing its strength, association, and longevity from 1990 to 2017.",
"The average degree of the association graph between 1990-2000 is 1.2, which increases to 1.97 during 2001-2010, and is maximum 2.5 during 2011-2017.",
"Another interesting trend is observed before and after 2010, as modularity and average path length of the network decrease, indicating an increase in the number of isolated, scattered terror modules around the world.",
"Similarly, a temporal network analysis allows us to examine the change in a group's behavior over time and events leading up to a significant event such as a chemical attack.",
"Figure REF shows a change in capabilities, target types, and activeness of ISIS over an 18 month period of time.",
"Graphs also allow us to measure the impact of a terrorist organization in terms of number of casualties inflicted by an attack.",
"We construct a higher order weighted graph between events and perpetrators with casualty count as an edge weight.",
"We compute the lethality of a terror network from 1970 to 2017 using PageRank centrality measure.",
"Figure REF shows top-10 lethal terror organizations between 1970-2017.",
"Figure: Yearly Frequency of GTD EventsFigure: Communities of accomplice terrorist organizationFigure: Communities of accomplice terrorist organization: 1990-2000Figure: Communities of accomplice terrorist organization: 2001-2010Figure: Communities of accomplice terrorist organization: 2011-2017Figure: Top-k lethal terror organizationsFigure: Temporal Network of ISIS"
],
[
"Conclusion",
"International terrorism is a complex, ever-shifting threat and one of the biggest concerns of recent times.",
"The global terrorism environment can be also described as a temporal network that evolves over time.",
"We present a case study of analyzing the Global Terrorism Database (GTD) using graph based temporal analysis to reveal insights about different terror groups and their relationships with each other."
]
] | 2005.14002 |
[
[
"A class of higher inductive types in Zermelo-Fraenkel set theory"
],
[
"Abstract We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals.",
"This class includes the example of unordered trees of any arity."
],
[
"Introduction",
"Higher inductive types are one of the key ideas in homotopy type theory [18].",
"We think of an (ordinary) inductive type as the smallest type closed under certain algebraic operations or point constructors.",
"For instance, we define the type of countably branching trees $T$ to be the smallest type closed under the following operations.",
"$\\mathtt {leaf} &: T \\\\\\mathtt {node} &: (\\omega \\rightarrow T) \\rightarrow T$ Within type theory we formalise the idea that $T$ is the smallest type with the above point constructors using recursion or induction terms.",
"However, semantically, it is often more convenient to think in terms of initial algebras.",
"We say an algebra for the above constructors is a type $X$ together with a map $1 + X^\\omega \\rightarrow X$ .",
"$T$ is then the initial object in the category of alegbras.",
"This is a classic example of a $W$ -type, as defined by Moerdijk and Palmgren [15].",
"For higher inductive types, one not only has point constructors, but also path constructors, which add proofs of identities of terms.",
"Higher inductive types are usually considered within HoTT and have well understood semantics within models of HoTT [13], [7], [6].",
"However, since are stated within the language of type theory, one might also consider whether they hold in interpretations of extensional type theory in locally cartesian closed categories, and in particular the category of sets, $\\mathsf {Set}$ .",
"One of the simplest examples of higher inductive type is pushouts.",
"In $\\mathsf {Set}$ these can be implemented as pushouts in the usual categorical sense.",
"It follows that $\\mathsf {Set}$ contains all of the $n$ -dimensional spheres, although there is not much you can say about them without the univalence axiom, and indeed they turn out to be trivial in $\\mathsf {Set}$ .",
"Quotients and image factorisations are examples of simple colimits that play a useful role even within models of extensional type theory [14], [1].",
"There are also more complicated examples of higher inductive types that are non trivial in extensional type theory, and even $\\mathsf {Set}$ , within the framework of quotient inductive types [3].",
"In fact our examples of interest fall within a smaller class with a simpler definition and clearer semantics.",
"This class was studied by Blass [5] under the name free algebras subject to identities and by Fiore, Pitts and Steenkamp in [8] under the name $QW$ -types (we will refer to them by the latter name).",
"A well known example of such a type is that of “unordered countably branching trees.” We modify the definition of $T$ above to get the higher inductive type $T_{\\operatorname{Sym}}$ by adding an equation as follows, where we write $\\operatorname{Sym}(\\omega )$ for the type of permutations $\\omega \\rightarrow \\omega $ .",
"$\\mathtt {leaf} &: T_{\\operatorname{Sym}} \\\\\\mathtt {node} &: (\\omega \\rightarrow T_{\\operatorname{Sym}}) \\rightarrow T_{\\operatorname{Sym}} \\\\\\mathtt {perm} &: \\prod _{f \\colon \\omega \\rightarrow T_{\\operatorname{Sym}}}\\prod _{\\pi : \\operatorname{Sym}(\\omega )} \\mathtt {node}(f) =\\mathtt {node}(f \\circ \\pi )$ Altenkirch, Capriotti, Dijkstra, Kraus and Forsberg include this in [2], as a non trivial example of a quotient inductive(-inductive) type.",
"As they remark, the obvious construction of $T_{\\operatorname{Sym}}$ as a quotient of $T$ requires the axiom of choice.For the special case above, countable choice would be enough.",
"Fiore, Pitts and Steenkamp showed that in fact it is an example of a $QW$ -type [8].",
"Blass showed that all $QW$ -types can be constructed in $\\mathsf {Set}$ under the assumption that regular cardinals are unbounded in the class of all ordinals.",
"More generally free algebras can be constructed in cocomplete categories from the existence of regular cardinals of sufficiently high cardinality via the general techniques of Kelly [11].",
"For example this plays an important role in the construction of higher inductive types by Lumsdaine and Shulman [13].",
"The existence of an unbounded class of regular ordinals is usually a reasonable one.",
"It follows from very weak versions of choice, such as $\\mathbf {WISC}$ [19] and a variant is often assumed in constructive set theory [4].",
"It is also the case that every inaccessible cardinal is in particular regular.Note, however that even in the presence of inaccessible cardinals it can be useful to have proofs that are valid in $\\mathbf {ZF}$ without further assumption: if $\\kappa $ is inaccessible, then $V_\\kappa $ is a transitive model of $\\mathbf {ZF}$ , and so any proof valid in $\\mathbf {ZF}$ can be carried out inside it (e.g.",
"to construct HITs that belong to $V_\\kappa $ ), but $V_\\kappa $ itself does not contain inaccessible cardinals without further assumptions on $\\kappa $ .",
"However, Gitik [9] has constructed a model of $\\mathbf {ZF}$ in which $\\omega $ is the only regular cardinal.under certain large cardinal assumptions Moreover, Blass showed that the assumption is strictly necessary, by constructing a $QW$ -type which is isomorphic to the collection of ordinals of countable cofinality, if it exists.",
"He deduced by Gitik's result that this gives an example of a $QW$ -type that does not provably exist in $\\mathsf {Set}$ under the assumptions of $\\mathbf {ZF}$ .",
"Fiore, Pitts and Steenkamp in loc.",
"cit.",
"gave an electronically verified proof that $QW$ -types can be constructed using Agda sized types and universes closed under inductive-inductive types.",
"We can see from Blass' counterexample that some combination of these assumptions for $\\mathsf {Set}$ must lead to the existence of uncountable regular cardinals.",
"On the other hand, some higher inductive types can be constructed in $\\mathsf {Set}$ without choice or unbounded regular cardinals.",
"In addition to colimits, as mentioned above, the author showed in [17] that $W$ -types with reductions exist in any boolean topos, including $\\mathsf {Set}$ .",
"A similar argument shows that Sojakova's notion of $W$ -suspensions [16] also exist in all boolean toposes.A proof is left as an exercise for the reader.",
"In this paper we will see a new class of $QW$ -types that can be constructed in $\\mathsf {Set}$ under $\\mathbf {ZF}$ , without any assumptions of choice or existence of regular cardinals, that we call image preserving $QW$ -types.",
"This will included the example of unordered countably branching trees above, and more generally unordered trees of any arity.",
"The proof is based on a construction of hereditarily countable sets due to Jech [10]."
],
[
"Image preserving $QW$ -types",
"We now define our class of higher inductive types that will be construct in $\\mathsf {Set}$ .",
"It will be clear by the definition that this is a special case of $QW$ -types [8].",
"Definition 2.1 A $\\emph {polynomial}$ is a type $A$ together with a family of types $(B_a)_{a \\in A}$ .",
"We will refer to elements of $A$ as constructors and say $B_a$ is the arity of the constructor $a : A$ .",
"Definition 2.2 Given a polynomial $(B_a)_{a \\in A}$ , an image preserving equation over $(B_a)_{a \\in A}$ consists of a type $V$ , and $a, b \\in A$ together with $l \\colon B_a \\rightarrow V$ and $r \\colon B_{b} \\rightarrow V$ such that the image of $l$ is equal to the image of $r$ .",
"A family of image preserving equations consists of a type $E$ together with a family of image preserving equations $(V_e, a_e,b_e, l_e, r_e)_{e \\in E}$ .",
"Remark 2.3 One might also consider pairs of functions $l \\colon B_a \\rightarrow T V_e$ and $r \\colon B_{b} \\rightarrow T V_e$ that have the same image in the free algebra on $V_e$ , $T V_e$ .",
"However, this seems to complicate the proof without adding any interesting examples.",
"Definition 2.4 Given a polynomial $(B_a)_{a \\in A}$ and a family of image preserving equations, $(V_e, a_e, b_e, l_e, r_e)_{e \\in E}$ , an algebra is a type $X$ , together with a function $s \\colon \\sum _{a \\in A} X^{B_a} \\rightarrow X$ such that for every $e \\in E$ and every function $h \\colon V_e \\rightarrow X$ we have $s(a_e, h \\circ l_e) = s(b_e, h \\circ r_e)$ .",
"Example 2.5 Suppose we are given a set $B$ .",
"We consider the polynomial with two constructors of arity 0 and $B$ .",
"We consider the set of image preserving equations with set of variables $B$ , and $l, r \\colon B \\rightarrow B$ defined by $l = 1_B$ and $r = \\pi $ for each permutation $\\pi \\in \\operatorname{Sym}(B)$ .",
"The initial algebra is then the set of unordered trees of arity $B$.",
"In particular, we can take $B = \\omega $ to get unordered countably branching trees.",
"Example 2.6 We consider the polynomial with two constructors with arities 0 and $\\omega $ .",
"We consider all image preserving equations with set of variables $\\omega $ .",
"Note that two maps $f, g \\colon \\omega \\rightarrow X$ have the same image if and only if they factor through some map $h \\colon \\omega \\rightarrow X$ , say $f = h \\circ l$ and $g = h \\circ r$ where $l$ and $r$ have the same image in $\\omega $ .",
"One can check that the set of all hereditarily countable sets is an initial algebra in $\\mathsf {Set}$ , if it exists.",
"Conversely, one can also show that if the initial algebra exists in $\\mathsf {Set}$ , then it is isomorphic to the class of hereditarily countable sets (e.g.",
"one can define the necessary function to hereditarily countable sets by defining a partial function to the set of hereditarily countable sets of rank $\\alpha $ for each $\\alpha $ and taking the limit).",
"Example 2.7 We will be able to deduce from the main theorem that Blass' example of a $QW$ -type that cannot be constructed in $\\mathbf {ZF}$ cannot be viewed as an image preserving $QW$ -type.",
"However, for illumination we will give a more intuitive direct reason why it does not satisfy the definition.",
"In Blass' example, the initial algebra is expected to behave like the collection of all ordinals of countable cofinality.",
"In particular there is an operation $\\sup $ which takes a sequence $(\\alpha _n)_{n < \\omega }$ and is expected to behave like the supremum of the countable sequence of ordinals $(\\alpha _n)_{n < \\omega }$ .",
"In particular we should identify $\\sup ((\\alpha _n)_{n < \\omega })$ and $\\sup ((\\beta _n)_{n < \\omega })$ whenever $\\alpha _n$ is cofinal in $\\beta _n$ and vice versa.",
"However, this is much weaker than $\\alpha _n$ and $\\beta _n$ containing exactly the same elements (possibly in a different order)."
],
[
"Some useful propositions",
"We recall some basic classical set theory that will be useful for the proof.",
"We fill in some of the details, with the remainder left as an exercise for the reader.",
"Proposition 3.1 For any ordinals $0 < \\alpha < \\beta $ , there is a canonical surjection $\\beta \\twoheadrightarrow \\alpha $ .",
"If there is a surjection $X \\twoheadrightarrow \\beta $ for some set $X$ , then there is also a surjection $X \\twoheadrightarrow \\alpha $ .",
"Proposition 3.2 For any well ordered set $(X, <)$ (and in particular for sets of ordinals ordered by $\\in $ ), there is a unique ordinal $\\beta $ with a unique order isomorphism $(X, <) \\cong (\\beta , \\in )$ .",
"We refer to $\\beta $ as the order type of $(X, <)$ .",
"Proposition 3.3 For any family of sets $(X_i)_{i \\in I}$ , there is an ordinal $\\aleph ((X_i)_{i \\in I})$ which is the smallest for which there is no surjection $X_i \\twoheadrightarrow \\aleph ((X_i)_{i \\in I})$ for any $i \\in I$ .",
"It is precisely the set of all ordinals $\\alpha $ for which there is a surjection $X_i \\twoheadrightarrow \\alpha $ for some $i \\in I$ .",
"Note that whenever $X_i \\twoheadrightarrow \\alpha $ , there is an equivalence relation $\\sim $ on $X$ , and a well ordering $<$ on $X/{\\sim }$ such that the order type of $(X/{\\sim }, <)$ is $\\alpha $ .",
"However there is clearly a set of such well orders by power set, and so there is a set of all such ordinals $\\alpha $ .",
"Since this is a downwards closed set of ordinals, it is an ordinal itself.",
"Since the set cannot contain itself, it is the least ordinal for which there is no surjection from $X_i$ for any $i$ .",
"Proposition 3.4 If $\\kappa $ is a cardinal number (i.e.",
"an ordinal that is not in bijection with any lower ordinal), then one can define surjections $\\kappa \\twoheadrightarrow \\kappa \\times \\kappa $ $\\kappa \\twoheadrightarrow \\kappa ^n$ for any $n < \\omega $ $\\kappa \\twoheadrightarrow \\omega \\times \\kappa $ $\\kappa \\twoheadrightarrow \\sum _{n < \\omega } \\kappa ^n$ For REF , see e.g.",
"[12].",
"For REF , suppose we are given a bijective pairing function $(-,-) \\colon \\omega \\times \\omega \\rightarrow \\omega $ .",
"Any ordinal $\\alpha $ can be written uniquely as $\\alpha = \\lambda + (m, n)$ where $\\lambda $ is a limit ordinal and $m, n \\in \\omega $ .",
"We then decode this as the pair $(m, \\lambda + n)$ , which clearly gives a bijection.",
"Deriving the other parts from these two is straightforward."
],
[
"The proof",
"We now construct image preserving $QW$ -types in $\\mathsf {Set}$ .",
"This is based on a construction of the set of hereditarily countable sets due to Jech [10].",
"Definition 4.1 We will define a functor $Q \\colon \\mathsf {On}\\rightarrow \\mathsf {Set}$ sending all maps to monomorphisms by recursion on ordinals.",
"We define $Q(0)$ to be $\\emptyset $ and for limit ordinals $\\lambda $ , we define $Q(\\lambda )$ to be $\\operatorname{colim}_{\\alpha < \\lambda }Q(\\alpha )$ .",
"We define $Q(\\alpha + 1)$ as follows.",
"Let $X$ be the set of pairs $(a, f)$ where $a \\in A$ and $f$ is a function from $B_a$ to $Q(\\alpha )$ that does not factor through the monomorphism $Q(\\beta ) \\rightarrowtail Q(\\alpha )$ for any $\\beta < \\alpha $ .",
"We then take $\\sim $ to be the equivalence relation on $X$ generated by identifying $(a_e, t \\circ l_e)$ and $(b_e, t \\circ r_e)$ whenever $t \\colon V_e \\rightarrow Q(\\alpha )$ for $e \\in E$ .",
"Finally we define $Q(\\alpha + 1)$ to be $X/{\\sim } + Q(\\alpha )$ .",
"We now give a series of definitions and lemmas that apply at any stage $\\alpha \\in \\mathsf {On}$ .",
"Definition 4.2 Note that we only identify $(a, f)$ and $(b, g)$ when $f$ and $g$ have the same image in $Q(\\alpha )$ .",
"Hence we have a well defined image function $\\operatorname{im}\\colon Q(\\alpha ) \\rightarrow \\mathcal {P}(Q(\\alpha ))$ , such that whenever $x = [(a, f)]$ , $\\operatorname{im}(x)$ is the image of $f$ in $Q(\\alpha )$ .",
"Definition 4.3 Given an element $x$ of $Q(\\alpha )$ of the form $[(a, f)]$ , we defined the rank of $x$ , $\\operatorname{rank}(x)$ to be the smallest ordinal $\\beta $ such that $f$ factors through the monomorphism $Q(\\beta ) \\rightarrowtail Q(\\alpha )$ .",
"To check this is a well defined, note that it depends only on the image of $f$ .",
"Note that $\\operatorname{rank}(x) + 1$ is the smallest ordinal $\\beta $ such that $x \\in Q(\\beta )$ .",
"Definition 4.4 Given a set $X \\subseteq Q(\\alpha )$ , we define the union $\\cup X$ by $\\cup X := \\bigcup _{x \\in X} \\operatorname{im}(x)$ We define the transitive closure of $x \\in Q(\\alpha )$ , $\\operatorname{TC}(x)$ by $\\operatorname{TC}(x) := \\bigcup _{1 \\le n < \\omega } {\\cup }^n \\lbrace x\\rbrace $ Lemma 4.5 For all $x \\in Q(\\alpha )$ , we have the following equation.",
"$\\operatorname{rank}(x) = \\lbrace \\operatorname{rank}(y) \\;|\\; y \\in \\operatorname{TC}(x) \\rbrace $ It is clear that whenever $y \\in \\operatorname{TC}(x)$ we must have $\\operatorname{rank}(y) < \\operatorname{rank}(x)$ since this is the case for any $n < \\omega $ and any $y \\in \\cup ^n \\lbrace x\\rbrace $ by induction on $n$ .",
"It remains to show that for any $\\beta < \\operatorname{rank}(x)$ , we have $\\beta = \\operatorname{rank}(y)$ for some $y \\in \\operatorname{TC}(x)$ .",
"By the definition of rank, $\\operatorname{im}(x)$ cannot be contained in $Q(\\beta )$ .",
"Hence we must have $x = [(a, f)]$ and $b \\in B_a$ such that $f b \\notin Q(\\beta )$ .",
"For this $b$ we have $\\beta \\le \\operatorname{rank}(f b)$ .",
"If $\\beta = \\operatorname{rank}(f b)$ , then $f b \\in \\cup \\lbrace x\\rbrace \\subseteq \\operatorname{TC}(x)$ and so $\\beta $ is as required.",
"Otherwise, $\\beta < \\operatorname{rank}(f b)$ and so by induction on rank we may assume $\\beta = \\operatorname{rank}(y)$ for some $y \\in \\operatorname{TC}(f b)$ .",
"However, $\\operatorname{TC}(f b) \\subseteq \\operatorname{TC}(x)$ , so $y \\in \\operatorname{TC}(x)$ and $\\beta $ is again as required.",
"Definition 4.6 For $x \\in Q(\\alpha )$ , we write $R_n(x)$ for the set $\\lbrace \\operatorname{rank}(z) \\;|\\; z\\in \\cup ^n\\lbrace x\\rbrace \\rbrace $ .",
"Lemma 4.7 For all $x \\in Q(\\alpha )$ , $\\operatorname{rank}(x) = \\bigcup _{1 \\le n < \\omega } R_n(x)$ By lemma REF .",
"Definition 4.8 We define $\\kappa $ to be $\\aleph ((B_a)_{a \\in A})$ .",
"We define $\\kappa ^+$ to be the smallest limit ordinal for which there is no surjection $\\kappa \\twoheadrightarrow \\kappa ^+$ .",
"Lemma 4.9 We define for each $x \\in Q(\\alpha )$ and each $1 \\le n < \\omega $ , a surjection $F_{x, n} \\colon \\kappa ^n \\twoheadrightarrow R_n(x) \\cup \\lbrace \\emptyset \\rbrace $ .",
"We first consider the case $n = 1$ .",
"Suppose that $x = [(a,f)]$ .",
"Note that $R_1(x) := \\lbrace \\operatorname{rank}(f b) \\;|\\; b \\in B_a \\rbrace $ is a set of ordinals, and so it has an order type $\\beta \\in \\mathsf {On}$ , and in particular we have a unique order isomorphism with $\\beta $ , say $\\theta \\colon \\beta \\stackrel{\\cong }{\\rightarrow } R_1(x)$ .",
"Furthermore, by definition, there is clearly a surjection from $B_a$ to $R_1(x)$ .",
"It follows that $\\beta < \\kappa $ .",
"Hence we can define a canonical surjection $F_{x, 1} \\colon \\kappa \\twoheadrightarrow R_1(x) \\cup \\lbrace \\emptyset \\rbrace $ as follows.",
"$F_{x, 1}(\\alpha ) :={\\left\\lbrace \\begin{array}{ll}\\theta (\\alpha ) & \\alpha < \\beta \\\\\\emptyset & \\text{otherwise}\\end{array}\\right.",
"}$ Now suppose $n = m + 1$ .",
"We fix $m$ ordinals less than $\\kappa $ , say $\\beta _1, \\ldots , \\beta _m$ and consider the set $Y$ below.",
"$Y := \\lbrace F_{f b, m}(\\beta _1, \\ldots , \\beta _m) \\;|\\; b\\in B_a \\rbrace $ This is again a set of ordinals with a surjection from $B_a$ for some $a \\in A$ , and so as before, we have a canonical surjection $G \\colon \\kappa \\twoheadrightarrow Y \\cup \\lbrace \\emptyset \\rbrace $ .",
"We take $F_{x, n}( \\beta _1, \\ldots , \\beta _m, \\beta _{m + 1} )$ to be $G(\\beta _{m + 1})$ .",
"We now simultaneously check that $F_{x, n}$ has the correct codomain and is surjective.",
"$\\operatorname{im}(F_{x, n})&= \\bigcup _{\\beta _1, \\ldots , \\beta _m < \\kappa } (\\lbrace F_{fb,m}( \\beta _1, \\ldots , \\beta _m) \\;|\\; b \\in B_a \\rbrace \\cup \\lbrace \\emptyset \\rbrace ) \\\\&= \\bigcup _{b \\in B_a} \\lbrace F_{fb,m}( \\beta _1, \\ldots , \\beta _m ) \\;|\\;\\beta _1, \\ldots , \\beta _m < \\kappa \\rbrace \\; \\cup \\; \\lbrace \\emptyset \\rbrace \\\\&= \\bigcup _{b \\in B_a} \\lbrace \\operatorname{rank}(z) \\;|\\; z \\in \\cup ^m\\lbrace f b \\rbrace \\rbrace \\; \\cup \\; \\lbrace \\emptyset \\rbrace \\\\&= R_n(x) \\; \\cup \\; \\lbrace \\emptyset \\rbrace $ Lemma 4.10 For any $x \\in Q(\\alpha )$ we have $\\operatorname{rank}(x) < \\kappa ^+$ .",
"First note that this is clear when $\\operatorname{rank}(x) = 0$ .",
"Hence we may assume for the rest of the proof $\\operatorname{rank}(x) > 0$ .",
"By the definition of $\\kappa ^+$ , it suffices to define a surjection $\\kappa \\twoheadrightarrow \\operatorname{rank}(x)$ .",
"By proposition REF it suffices to define a surjection $\\sum _{1 \\le n < \\omega } \\kappa ^n \\twoheadrightarrow \\operatorname{rank}(x)$ .",
"However, by lemma REF we can express $\\operatorname{rank}(x)$ as $\\bigcup _{1 \\le n < \\omega } R_n(x)$ .",
"Since $\\operatorname{rank}(x) > 0$ , this is the same as $\\bigcup _{1 \\le n < \\omega } (R_n(x) \\cup \\lbrace \\emptyset \\rbrace )$ , and so we can just combine the surjections defined in lemma REF .",
"Theorem 4.11 All image preserving $QW$ -types exist in $\\mathsf {Set}$ .",
"We show that $Q(\\kappa ^+)$ is an initial algebra.",
"We first need to show how to define an algebra structure.",
"Suppose we are given $a \\in A$ and a map $f \\colon B_a \\rightarrow Q(\\kappa ^+)$ .",
"Then $[(a, f)]$ is an element of $Q(\\kappa ^+ + 1)$ .",
"By lemma REF we have $\\operatorname{rank}([(a, f)]) < \\kappa ^+$ , and so $f$ factors through $Q(\\beta )$ for some $\\beta < \\kappa ^+$ .",
"We can then take $\\sup (a, f)$ to be $[(a, f)] \\in Q(\\beta + 1)$ .",
"We check that this structure respects the equations.",
"Suppose that we are given $g \\colon V_e \\rightarrow Q(\\kappa ^+)$ .",
"Note that $g \\circ l_e$ and $g \\circ r_e$ have the same image, and so $\\operatorname{rank}([(a_e, g \\circ l_e)]) = \\operatorname{rank}([(b_e, g \\circ r_e)])$ .",
"Hence $[(a_e, g \\circ l_e)]$ and $[(b_e, g \\circ r_e)]$ must have been first added at the same stage, $\\alpha + 1$ .",
"We can now see that they were identified in the definition of $Q(\\alpha + 1)$ .",
"Finally, it is clear that for any other algebra structure, we can define a unique structure preserving map out of $Q(\\kappa ^+)$ by recursion on ordinals."
],
[
"Conclusion",
"We have constructed a class of HITs in $\\mathbf {ZF}$ .",
"Although the proof is somewhat elaborate, the results of Jech [10] suggest that the complication is necessary.",
"He showed that the transfinite construction of hereditarily countable sets does not, provably in $\\mathbf {ZF}$ , converge at stage $\\omega _1$ , and that in fact if $\\omega _1$ is singular then there are hereditarily countable sets of rank $\\alpha $ for any $\\alpha < \\omega _2$ .",
"Our proof makes essential use of the fact that any set of ordinals is order isomorphic to an ordinal, which in turn uses classical logic.",
"We therefore leave it as an open problem to construct image preserving $QW$ -types in $\\mathsf {Set}$ under the assumptions of $\\mathbf {IZF}$ , or to find an independence proof.",
"The same question is still open for the case of $W$ -types with reductions.",
"We also leave the open problem of finding more interesting examples of HITs that can be constructed in $\\mathsf {Set}$ under $\\mathbf {ZF}$ ."
]
] | 2005.14240 |
[
[
"Bayesian Restoration of Audio Degraded by Low-Frequency Pulses Modeled\n via Gaussian Process"
],
[
"Abstract A common defect found when reproducing old vinyl and gramophone recordings with mechanical devices are the long pulses with significant low-frequency content caused by the interaction of the arm-needle system with deep scratches or even breakages on the media surface.",
"Previous approaches to their suppression on digital counterparts of the recordings depend on a prior estimation of the pulse location, usually performed via heuristic methods.",
"This paper proposes a novel Bayesian approach capable of jointly estimating the pulse location; interpolating the almost annihilated signal underlying the strong discontinuity that initiates the pulse; and also estimating the long pulse tail by a simple Gaussian Process, allowing its suppression from the corrupted signal.",
"The posterior distribution for the model parameters as well for the pulse is explored via Markov-Chain Monte Carlo (MCMC) algorithms.",
"Controlled experiments indicate that the proposed method, while requiring significantly less user intervention, achieves perceptual results similar to those of previous approaches and performs well when dealing with naturally degraded signals."
],
[
"Introduction",
"Afairly common degradation found in old vinyl and gramophone recordings are long pulses with significant low-frequency content produced by the nonlinear response of the arm-needle system of the playback device when it passes through a deep scratch or even a breakage on the surface of the respective media.",
"More precisely, this degradation can be split into two complementary parts.",
"An initial discontinuity that arises when the needle passes exactly over the physical damage is immediately followed by the tail, a low-frequency oscillation whose amplitude and frequency decay slowly.",
"The discontinuity (lasting typically less than 10 ms) behaves as a high-variance noise added to the original signal that almost hides the underlying information, whereas the smooth tail (which in the most severe cases can be about 1-second long) is clearly superimposed to the original signal.",
"Since the physical restoration of the media is almost impossible, one must resort to numerical algorithms that process a digitized version of the degraded recording.",
"A pioneering method tailored to tackle this type of degradation was proposed in [1], [2]: it is based on the hypothesis of similarity among the pulses present in a signal, i.e., every physical damage found during reproduction is supposed to evoke a similar response of the system, differing only in location and amplitude.",
"These two quantities are estimated by correlating the degraded signal with prototypical pulses of a reference database.",
"The method achieves good results when this hypothesis is valid, but its scope is limited to pulses similar to the templates present in the database.",
"Moreover, if two or more pulses overlap, it fails.",
"A statistical approach capable of dealing with more general cases can be found in [3], [4].",
"This method assumes that both the underlying signal and the pulse can be modeled by superimposed auto-regressive (AR) processes.",
"The original signal is then estimated by separating the two processes.",
"Some limitations are requiring the location of the pulse to be known and the unrealistic assumption about the AR model for the pulse.",
"In [5] a much simpler method is proposed, based on a nonlinear filtering technique called Two-Pass Split Window (TPSW).",
"This filtering is employed to obtain a rough estimate of the pulse shape, which is then smoothed by a piece-wise polynomial fitting.",
"Although this method requires less computational power, the location of the pulse must be known in advance.",
"In [6] the authors introduce a restoration method based on the Empirical Mode Decomposition, which decomposes a signal waveform into a set of simpler Intrinsic Mode Functions, but also requires the location of the pulse.",
"In general, the aforementioned methods only deal with the pulse tail, demanding also some de-clicking technique to interpolate the signal underlying the initial discontinuity.",
"The exception is the AR separation based method [3], [4], which assumes that the initial discontinuity is modeled by the same AR process as the tail but with a higher excitation variance—a somewhat unrealistic hypothesis.",
"In [7], an innovative approach is introduced to jointly estimate the location of the pulse and restore the audio signal, including the excerpt underlying the initial discontinuity, that adopts a model for the pulse shape whose parameters are then estimated via Bayesian inference, by sampling from their posterior distribution.",
"However, this method has three drawbacks, the first two being reconsidered in the present work: 1) the algorithm requires that several hyperparameters be manually tuned in order to correctly estimate the pulse tail parameters; 2) despite being capable of estimating precisely the location of the pulse, the method requires a good initialization; 3) it is not capable of dealing with superimposed pulses, because in this scenario the posterior distributions become very complicated to handle, even in the context of Markov-Chain Monte Carlo methods.",
"This work introduces an improvement of the method proposed in [7] that circumvents the first two issues above: 1) the pulse tail is modeled via a Gaussian Process, requiring much less hyperparameters to be tuned; 2) an efficient initialization procedure based on [8] is adopted, which provides good initial estimates of both location and duration of the initial discontinuity.",
"But even if the posterior distribution does not increase in complexity in the case of overlapping pulses thanks to the Gaussian Process modeling of the pulse tail, the problem of estimating the underlying signal when a new initial discontinuity is superimposed to an unfinished tail still requires further investigation.",
"The paper is organized as follows: after this introduction, Section recalls the shape-based model for the pulse and introduces the proposed Gaussian Process model for the pulse tail, followed by a brief presentation of the AR-based model assumed for the underlying signal in Section ; in Section we present the complete hierarchical model on which the inference will be based, and specify prior distributions for the parameters under consideration; Section briefly describes the inference algorithm employed, followed by the computation of the marginal likelihood in Section , and by the computation of the conditional distributions and description of the sampling procedures in Section ; results are presented in Section , and conclusions are drawn in Section ."
],
[
"A model for the degradation",
"As already mentioned, a single pulse of the type considered in this work can be described by two contiguous parts: the initial discontinuity (when the needle passes through the physical degradation) and the tail (dumped oscillations of decaying frequency due to the non-linear response of the playback device to the impulsive excitation).",
"These two parts of the degradation are denoted, respectively, by vectors $v̭_{\\mathrm {d}}$ and $v̭_{\\mathrm {t}}$ .",
"When necessary, a superscript “s\" or “G\" will be added to vector $v̭_{\\mathrm {t}}$ in order to make explicit whether the shape-based or the Gaussian Process model is being used, respectively.",
"There is no superscript when this distinction is not necessary.",
"Assuming the audio signal is processed in time frames of length $N$ , denote the corresponding original and corrupted signal blocks by $x̭$ and $y̭$ , respectively.",
"In order to describe the relationship between these vectors and the degradation in vectors $v̭_{\\mathrm {d}}$ and $v̭_{\\mathrm {t}}$ , three sets of indexes are defined: $i̭_0$ , $i̭_1$ and $i̭_2$ , indicating the time samples in $y̭$ that belong to the region preceding the degradation, to the initial discontinuity, and to the tail, respectively.",
"Sub-vectors $x̭_0, y̭_0, x̭_1, y̭_1, x̭_2$ , and $y̭_2$ contain the time samples corresponding to their respective sets of indexes, such that $\\begin{split}y̭_0 &= x̭_0, \\\\y̭_1 &= x̭_1 + v̭_{\\mathrm {d}}, \\\\y̭_2 &= x̭_2 + v̭_{\\mathrm {t}}.\\end{split}$ Historically, despite their simplicity, additive degradation models have been used successfully in audio restoration problems [1], [3].",
"By defining the set of auxiliary matrices $K̭$ , $Ṷ_1$ , and $Ṷ_2$ , containing the columns of an $N \\times N$ identity matrix indexed by $i̭_0$ , $i̭_1$ , and $i̭_2$ , respectively, one can write $x̭ = K̭x̭_0 + Ṷ_1x̭_1 + Ṷ_2x̭_2.$ In the following, the models for the initial discontinuity and the tail of the pulse are described."
],
[
"Initial discontinuity",
"The initial discontinuity, stored in vector $v̭_{\\mathrm {d}}$ , can be modeled by Gaussian white noise superimposed to the underlying original signal in vector $x̭_1$ .",
"This part of the pulse begins at sample $n_0$ and lasts for $M$ samples, with fixed variance $\\sigma _{\\mathrm {d}}^2$ : $v̭_{\\mathrm {d}}(n) = r(n)[u(n - n_0) - u(n - n_0 - M)],$ where $u(n)$ is the unit step function, $r(n) \\sim \\mathcal {N}(0, \\sigma _{\\mathrm {d}}^2)$ , and $n_0$ , $M$ , and $\\sigma _{\\mathrm {d}}^2$ are unknown.",
"These three parameters are more concisely denoted by vector $\\mathbf {\\theta }_{\\mathrm {d}} = [n_0 ~~ M ~~ \\sigma _{\\mathrm {d}}^2]^T$ ."
],
[
"Shape-based model for the tail",
"Here the model adopted in [7] for the tail is briefly recalled.",
"Based on [5], it is mathematically described by $\\begin{split}v̭_{\\mathrm {t}}^{\\mathrm {s}}(n) = V_{\\mathrm {t}}{\\mathrm {e}}^{-n/(f^{\\prime }\\tau _m)}\\sin \\left( 2 \\pi n \\frac{f_n}{f^{\\prime }} + \\phi \\right) \\times \\\\ [u(n - n_0 - M - 1)],\\end{split}$ where $f_n = (f_{\\text{max}}- f_{\\text{min}}){\\mathrm {e}}^{-n/(f^{\\prime }\\tau _f)} + f_{\\text{min}}.$ This model is motivated by visual inspection of pulses present in silent parts of degraded audio signals [3], which exhibit a similar behavior: a decay in amplitude described by the exponential function, a decay in frequency modeled by variable $f_n$ inside the sinusoidal function.",
"Variables $n_0$ and $M$ are defined as before, and the remaining ones are precisely defined below: $V_{\\mathrm {t}}$ is related to the tail amplitude; $f^{\\prime }$ is the signal sampling rate (usually 44.1 kHz); $\\tau _m$ is the time constant (in seconds) associated with the pulse envelope decay; $\\tau _f$ is the time constant (in seconds) associated with the pulse frequency decay; $f_{\\text{max}}$ and $f_{\\text{min}}$ are, respectively, the maximum and minimum tail oscillation frequencies (in Hz); $\\phi $ is the initial phase of the pulse.",
"All these quantities (except for $f^{\\prime }$ ) are also unknown beforehand.",
"Let vector $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}} = [V_{\\mathrm {t}} ~~ \\tau _m ~~ \\tau _f ~~ f_{\\text{max}}~~ f_{\\text{min}}~~ \\phi ]^T$ store such tail parameters.",
"They can be visualized in Figure REF , where a prototypical pulse that follows the model above is depicted.",
"Figure: Model for the pulse shape following Equation ."
],
[
"Gaussian Process model for the tail",
"Gaussian Processes are a widely employed technique in Statistics, Machine Learning and Linear Regression [9], [10], [11].",
"Before presenting its application to the current problem, some important facts should be recalled.",
"A stochastic process $G̭ = \\lbrace G_t\\rbrace _{t \\in \\mathcal {T}}$ , indexed by some set $\\mathcal {T}$ , is said to be a Gaussian Process if for any finite subset $\\lbrace t_1, \\dots , t_n\\rbrace \\in \\mathcal {T}$ , the joint distribution of the random vector $(G_{t_1}, \\dots , G_{t_n})$ is Gaussian.",
"Note that Gaussian distributions are completely determined by their first and second order statistics, so by assuming (without loss of generality) zero mean, several features of the process are encoded in its covariance kernel, denoted by $K(t,t^{\\prime })$ , for $t, t^{\\prime } \\in \\mathcal {T}$ .",
"This function describes the dependence between any two random variables (i.e.",
"time instants) of the process, and there are several commonly used covariance kernels tailored to impose some desired structure, like stationarity, smoothness, and periodicity, among others [12].",
"For a detailed discussion about Gaussian processes, see [10].",
"Returning to the modeling of the pulse tail, stored in vector $v̭_{\\mathrm {t}}$ , it can be seen as a generic function superimposed to the underlying signal in vector $x̭_2$ .",
"This function is assumed to be much smoother than the underlying signal, and this assumption was implicitly taken into account in the design of the shape-based model.",
"In order to give more flexibility to the pulse tail description, the deterministic function is replaced by a sample from a Gaussian Process with a squared-exponential covariance kernel given by $K_{\\mathrm {SE}}(t,t^{\\prime }) = \\sigma _f^2 \\exp \\left(-\\frac{|t -t^{\\prime }|^2}{2\\sigma _{\\ell }^2}\\right).$ Parameters $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ control the amplitude of the pulse and the effective extent of the covariance kernel, respectively.",
"The choice of this covariance kernel to model the tail of a long pulse is arguable, since it models a stationary process, which is clearly not the case of the typical pulse tail.",
"A more precise model would require the definition of a covariance kernel specific to this application, with some additional parameters to encode the desired behavior.",
"The drawback of such a choice would be to excessively increase the model complexity.",
"The good results obtained with the squared-exponential covariance kernel consolidate our option for simplicity.",
"Now, in this framework, in addition to the parameters $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ of the covariance kernel, the whole vector $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ containing the tail of the pulse must be estimatedNote that $v̭_{\\mathrm {d}}$ is not included in $\\mathbf {\\theta }_{\\mathrm {d}}$ , since the corresponding underlying signal $x̭_1$ is directly estimated instead of $v̭_{\\mathrm {d}}$ ; and $v̭_{\\mathrm {t}}^{\\mathrm {s}}$ is implicitly defined by the estimated $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ .",
"However, $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ must be directly estimated in the Gaussian Process model, due to its non-parametric nature..",
"Such quantities are assembled in vector $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {G}} = [(v̭_{\\mathrm {t}}^{\\mathrm {G}})^T ~~ \\sigma _f^2 ~~ \\sigma _{\\ell }^2]^T.$ The estimation of parameters $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ can be performed by exploring their respective posterior distributions, in the proposed Bayesian framework.",
"However, preliminary tests indicated that, with a proper initialization, keeping these parameters constant during the rest of the procedure is enough to reach a good pulse estimate, with the advantage of decreasing the computational cost of the algorithm.",
"The intuition behind this observation is that with a good initialization of variables $n_0$ and $M$ , guaranteed by the procedure described in Section , the overall shape of the pulse can be easily estimated, requiring only minor adjustments during the sampling procedure.",
"Since parameters $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ are responsible for the overall shape of the pulse, they are not expected to change significantly during the entire procedure.",
"In the context of audio restoration it is common to model the underlying original signal, here denoted by $x̭$ , as an auto-regressive (AR) process of fixed order $P$ [3].",
"More precisely, the samples of signal $x̭$ satisfy the following difference equation: $x(n) = \\sum _{i = 1}^{P} a_i x(n-i) + e(n),$ where $e(n)$ is the innovation error, modeled as a white Gaussian noise with variance $\\sigma _e^2$ .",
"This model is an easy way to encode the predictability of audio signals in a short scale together with some unpredictable relation between successive samples.",
"From the signal processing viewpoint, Equation (REF ) describes signal $x̭$ as the output of an all-pole linear filter whose input is signal $ḙ = [e(1) ~~ \\dots ~~ e(N)]^T$ , with transfer function given by $A(z) = \\frac{1}{1 - \\sum \\limits _{i = 1}^{P} a_i z^{-i}}.$ Therefore, the coefficients in vector $a̭$ are related to the most prominent frequencies in signal $x̭$ , through the poles of function $A(z)$ .",
"It is important to discuss how long the block under analysis can be in order to be accurately described by an AR model.",
"A reasonable choice is the time interval during which $x̭$ can be considered stationary, which for audio signals is usually accepted as around 20 ms (approximately 1,000 samples at a frequency sampling rate of 44,100 Hz, for instance).",
"However, as mentioned before, the overall pulse duration can be much longer.",
"The most natural way to deal with this issue would be to split signal $x̭$ into contiguous blocks of approximately 20 ms each and describe each of them with the corresponding AR model.",
"However, this choice would greatly increase the complexity of the model.",
"The solution proposed here is to describe the original signal immediately preceding the degradation and during the initial discontinuity (i.e.",
"vector $[x̭_0^T ~~ x̭_1^T]^T$ ) by a single AR model of order about 40; and during the pulse tail (i.e.",
"vector $x̭_2$ ) as white Gaussian noise—a degenerated AR model of order zero.",
"This apparent oversimplification can be justified: since by hypothesis the pulse tail is much smoother than the underlying signal and consists essentially of low frequency content, in the large time-scale of the tail the signal is almost indistinguishable from white noise.",
"Notice that this model switching makes $x̭$ also dependent on $n_0$ and $M$ .",
"In contrast, a high-order AR model must be kept for the first part, since this information will be crucial to restore the virtually missed $x̭_1$ .",
"Here, an additional simplification can be envisaged: since estimating the coefficients of the AR model from their respective posterior distribution together with the other parameters would make the computational time very large, they are estimated beforehand from $y̭_0$ , the region preceding the degradation, and kept constant through the whole procedure.",
"This approximation is justifiable, since these parameters will only be used to estimate $x̭_1$ , the signal underlying the initial discontinuity.",
"Being $y̭_0 = x̭_0$ and contiguous to $x̭_1$ , it is reasonable to assume they follow the same AR model.",
"We denote the coefficients of the AR model by $a̭ = [a_1 ~~ \\dots ~~ a_P]^T$ , which, together with $\\sigma _e^2$ , form vector $\\mathbf {\\theta }_x = [a̭^T ~~ \\sigma _e^2]^T$ containing the modeling parameters of the underlying signal.",
"As discussed in [3], [7], by constructing matrix $\\hat{A̭} =\\begin{bmatrix}-a_P & \\hdots & -a_1 & 1 & 0 & \\hdots & 0 \\\\0 & -a_P & \\hdots & -a_1 & 1 & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\hdots & 0 & -a_P & \\hdots & -a_1 & 1\\end{bmatrix}$ of size $N \\times (N + P)$ and considering the distribution of the innovation error $ḙ = \\hat{A̭}x̭$ as Gaussian, by a simple change of variables one is able to write the conditional distribution for $x̭$ , also incorporating the model switching, as $p(x̭ | n_0, M, \\mathbf {\\theta }_x) \\propto \\exp \\left(-\\frac{1}{2\\sigma _e^2} {\\bf x}^T{\\bf A}^T{\\bf A}{\\bf x}\\right),$ being matrix $A̭$ the vertical concatenation of $A̭_0$ , $A̭_1$ , and $A̭_2$ —matrices containing the columns of $\\hat{A̭}$ indexed by $i̭_0$ , $i̭_1$ , and $i̭_2$ , respectively, with all $a_j$ terms in $A̭_2$ set to zero, according to the simplified model adopted for $x̭_2$ ."
],
[
"Description of the model",
"We are now ready to present the whole hierarchical generative model for the degraded signal, together with the prior distributions of the parameters under consideration.",
"First, we can summarize the discussion up to this point about the models for the degradation and the underlying signal in the following distributions: $\\begin{split}y̭_0 | n_0, x̭_0 &\\sim \\delta (y̭_0 - x̭_0) \\\\y̭_1 | \\mathbf {\\theta }_\\mathrm {d}, x̭_1 &\\sim \\mathcal {N}(y̭_1 | x̭_1, \\sigma _d^2 I̭_M) \\\\y̭_2 | n_0, M, \\mathbf {\\theta }_\\mathrm {t}, x̭_2 &\\sim \\delta ( y̭_2 - (x̭_2 + v̭_{\\mathrm {t}})) \\\\x̭ | n_0, M, \\mathbf {\\theta }_x &\\sim \\mathcal {N}(x̭ | , \\sigma _e^2(A̭^TA̭)^{-1}).\\end{split}$ The two multi-dimensional Dirac's delta distributions for $y̭_0$ and $y̭_2$ are due to the fact that they depend deterministically on $x̭_0$ and $x̭_2 + v̭_{\\mathrm {t}}$ , respectively, as stated in Equation (REF ), whereas the distribution of $y̭_1$ is Gaussian as the noise superimposed on $x̭_1$ during the initial discontinuity.",
"We also assume that $y̭_0$ , $y̭_1$ , and $y̭_2$ are conditionally independent given $\\mathbf {\\theta }_{\\mathrm {d}}$ , $\\mathbf {\\theta }_{\\mathrm {t}}$ , and $x̭$ .",
"Finally, the Gaussian distribution on $x̭$ follows from Equation (REF ).",
"To complete the specification of the hierarchical model, we impose prior distributions to $\\mathbf {\\theta }_{\\mathrm {d}}$ and $\\mathbf {\\theta }_{\\mathrm {t}}$ , which are assumed to be prior-independent.",
"The components of vector $\\mathbf {\\theta }_{\\mathrm {d}}$ are assumed to be prior-independent as well.",
"Uniform discrete distributions over the whole set of samples, denoted by $\\mathcal {U}\\lbrace \\cdot | 1, \\dots , N\\rbrace $ , were imposed on $n_0$ and $M$ .",
"For $\\sigma _{\\mathrm {d}}^2$ , an Inverse Gamma prior parameterized by shape $\\alpha _{\\mathrm {d}}$ and scale $\\beta _{\\mathrm {d}}$ , denoted by $\\mathcal {I}\\mathcal {G}(\\sigma _{\\mathrm {d}}^2 | \\alpha _{\\mathrm {d}}, \\beta _{\\mathrm {d}})$ , was adopted; to make this prior vague, $\\alpha _{\\mathrm {d}} = \\beta _{\\mathrm {d}} = 10^{-4}$ were chosen.",
"The prior distribution for $\\mathbf {\\theta }_{\\mathrm {t}}$ , generically written as $p(\\mathbf {\\theta }_\\mathrm {t})$ for now, depends on the specific model under consideration, and will be detailed later.",
"The overall prior structure can be summarized as: $\\begin{split}n_0 &\\sim \\mathcal {U}\\lbrace n_0 | 1, \\dots , N\\rbrace \\\\M &\\sim \\mathcal {U}\\lbrace M | 1, \\dots , N\\rbrace \\\\\\sigma _{\\mathrm {d}}^2 | \\alpha _{\\mathrm {d}}, \\beta _{\\mathrm {d}} &\\sim \\mathcal {I}\\mathcal {G}(\\sigma _{\\mathrm {d}}^2 | \\alpha _{\\mathrm {d}}, \\beta _{\\mathrm {d}}) \\\\\\mathbf {\\theta }_\\mathrm {t} &\\sim p(\\mathbf {\\theta }_\\mathrm {t}).\\end{split}$ The dependence relations between signals and parameters are shown as a graphical model in Figure REF , where arrows indicate direct dependencies; solid lines denote the splitting of a vector into its components; and the dotted lines after vector $\\mathbf {\\theta }_\\mathrm {t}$ refer to the two possible models for the pulse tail.",
"Figure: Graphical dependence structure in the proposed model.",
"Arrows, solid lines and dotted lines indicate, respectively: direct dependence, display of vector components, and both modeling possibilities for θ t \\mathbf {\\theta }_\\mathrm {t}."
],
[
"Prior distribution for $\\mathbf {\\theta }_\\mathrm {t}^{\\mathrm {s}}$",
"The dependence between $y̭_2$ and $\\mathbf {\\theta }_\\mathrm {t}^{\\mathrm {s}}$ is quite complex when considering the shape-based model for the pulse tail.",
"For this reason, we chose to impose a simple non-informative prior to the components of $\\mathbf {\\theta }_\\mathrm {t}^{\\mathrm {s}}$ .",
"After assuming their mutual prior-independence, improper uniform priors in their respective domains were adopted: $p(\\mathbf {\\theta }_\\mathrm {t}^{\\mathrm {s}}) \\propto \\mathbb {1}(\\tau _m > 0) \\mathbb {1}(\\tau _f > 0) \\mathbb {1}(f_{\\text{max}}> 0) \\mathbb {1}(f_{\\text{min}}> 0),$ where $\\mathbb {1}$ denotes an indicator function.",
"Note that variables $V_\\mathrm {t}$ and $\\phi $ can assume any real value.",
"Even if this choice of prior distribution prioritize simplicity over accuracy, it was verified in preliminary tests that improper priors does not negatively impact the estimation procedure."
],
[
"Prior distribution for $\\mathbf {\\theta }_\\mathrm {t}^{\\mathrm {G}}$",
"Since in this scenario we are assuming that the pulse tail, stored in vector $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ , is well described by a Gaussian Process with a squared-exponential covariance kernel, its prior distribution is given by $v̭_{\\mathrm {t}}^{\\mathrm {G}} | \\sigma _f^2, \\sigma _{\\ell }^2 \\sim \\mathcal {N}(v̭_{\\mathrm {t}}^{\\mathrm {G}} | , C̭),$ where matrix $C̭$ is the Gram matrix of the covariance kernel function, computed from Equation (REF ).",
"No prior distributions are assigned to $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ , since (as stated in Section ) they will be estimated beforehand and kept constant during the rest of the estimation procedure."
],
[
"Description of the algorithm",
"Having defined the model in Equations (REF ) and (REF ), we now describe the inference strategy adopted.",
"First, the initialization procedure for $n_0$ and $M$ is presented, followed by a description of the estimation of $\\mathbf {\\theta }_x$ , $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ .",
"This section closes with an overview of the sampling algorithm."
],
[
"Initialization procedure for $n_0$ and {{formula:6298bbbe-ec32-450a-a4c9-f44b747bd89c}}",
"Previous experiments indicated that an accurate initialization of variables $n_0$ and $M$ was important for the sampling procedure.",
"Performing this initialization by hand (i.e., choosing the respective initial values by a visual inspection of the signal) may not be practical, and some method for automatically doing this task was required.",
"An adaptation of the method for detection of long pulses in audio signals proposed in [8] was able to provide a quite accurate first estimate of the desired variables.",
"The initialization procedure looks for sudden high-amplitude impulses in time (typical of the initial discontinuity of the pulse), whose energy splits over the whole spectrum of a time frame around it.",
"More precisely, the corrupted signal is split in contiguous blocks of length $L$ with an overlap of 50% between adjacent blocks, and the discrete Fourier Transform (DFT) [13] of each block is computed.",
"Denote the DFT of block $b$ by $\\hat{y̭}_b$ , for $b = 1, \\dots B$ .",
"Since typically the power of the audio spectrum is concentrated below some medium-high frequency, it is more convenient to look for unusual information in vectors $\\hat{y̭}_b$ above some frequency defined by the user.",
"Denote this cut-off frequency by $f_\\mathrm {co}$ and its respective frequency bin by $\\alpha _\\mathrm {co}$ .",
"Define the function $\\mu (b) = \\frac{1}{\\beta - \\alpha _\\mathrm {co} + 1}\\sum _{k = \\alpha _\\mathrm {co}}^{\\beta } |\\hat{y̭}_b(k)|,$ where $\\beta $ indexes the last bin in the DFT.",
"This function is an arithmetic mean of the high-frequency content in block $b$ , starting at frequency $f_\\mathrm {co}$ .",
"It is expected to reach a high value when the initial discontinuity of a pulse occurs in a given block of the degraded signal.",
"However, as reported in [8], if the considered signal exhibits a broad dynamic range with substantial high-frequency content (for example, brass or percussive instruments), the value of $\\mu $ can be high without necessarily implying the presence of long pulses.",
"In order to circumvent this issue, here a median filter is applied to function $\\mu $ .",
"As it is known in the literature, the median filter is capable of removing local occurrences of unusual values within a sequence, and is widely used in Image Processing as a tool to remove impulsive noise while preserving edges [14].",
"The overall procedure is described below: Pad function $\\mu $ with $\\left\\lfloor {c/2}\\right\\rfloor $ zeros before its first and after its last samples, respectively.",
"Define a new function $\\mu _m$ resulting of median filtering $\\mu $ with an odd-sized window of length $c$ , that is, by replacing each value of $\\mu $ by the empirical median of the $c$ values around it.",
"Define function $\\Delta \\mu (b)$ as the difference between $\\mu $ and $\\mu _m$ normalized by its highest value, that is, $\\Delta \\mu (b) = \\frac{\\mu (b) - \\mu _m(b)}{\\max \\limits _{b^{\\prime }} [\\mu (b^{\\prime }) - \\mu _m(b^{\\prime })]},$ for $b = 1, \\dots , B$ .",
"This ensures that the maximum absolute value of $\\Delta \\mu $ is one, thus allowing an easier definition of the threshold specified below.",
"Define a threshold $\\xi $ such that block $b^*$ is considered corrupted by the initial discontinuity of a long pulse if $|\\Delta \\mu (b^*)| \\ge \\xi $ .",
"This procedure defines a set of causally ordered candidate blocks $b_1^*, \\dots , b_M^*$ .",
"Each contiguous subset of blocks $b_i^*,\\dots ,b_j^*$ is attributed to the initial discontinuity of a long pulse, for which $n_0$ is chosen as the first time sample of block $b_i^*$ , and $M$ as the gap size between the last time sample of block $b_j^*$ and $n_0$ .",
"In summary, in the initialization step the values of $L$ , $f_\\mathrm {co}$ , and $\\xi $ are left to the user's choice.",
"However, in order to reach a useful initialization, the value of $L$ , representing the length of each block prior to the computation of its DFT, must be carefully chosen.",
"For simplicity, all the block lengths here refer to a signal sampled at the frequency rate of 44,1 kHz.",
"In [8] the authors adopted $L = 2048$ ($\\approx $ 46 ms).",
"Since the portion of the signal being analyzed that contains the initial discontinuity and the pulse tail typically spans around 10,000 samples ($\\approx $ 226 ms), this choice of $L$ would imply a crude time resolution, thus providing loose initial estimates for $n_0$ and $M$ .",
"In order to increase time resolution, a value of $L$ between 16 and 64 is suggested here.",
"Experiments indicate that when the degraded signal is also contaminated with broadband additive noise, at low SNR, values of $L$ near the upper-range allow for better estimates.",
"Besides $L\\in \\lbrace 16,32,64\\rbrace $ , $\\xi $ around $0.3$ is recommended, and $f_{\\mathrm {co}}=\\frac{f^{\\prime }}{2}$ is a good starting point.",
"It is important to remark that this initialization procedure is specifically tailored to be applied on signals effectively containing at least one pulse, mainly due to the normalization in Equation (REF ).",
"Indeed, tests indicate that with an undistorted signal, any higher-frequency content is likely to be detected as a pulse with this procedure.",
"This issue can be circumvented by dropping the normalization in Equation (REF ); however, it makes the definition of the threshold $\\xi $ more subtle, requiring a deeper examination of the signal under consideration.",
"Since the existence of a pulse is easily confirmed by informal listening of the signal, we have opted for keeping the normalization."
],
[
"Estimation of $\\mathbf {\\theta }_x$ , {{formula:a732e245-446d-4ab7-a89b-8304df3b66a3}} and {{formula:d3643751-fc49-4f46-b33b-8a1d4bde17a3}}",
"After initialization of variables $n_0$ and $M$ , we can consider the excerpts $y̭_0^{(0)}$ and $y̭_2^{(0)}$ of the degraded signal, before and after the initial estimate of the initial discontinuity respectively, from which $\\mathbf {\\theta }_x$ , $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ will be estimated and kept constant during the execution of the sampling algorithm.",
"Since $y̭_0^{(0)}$ is assumed to be uncorrupted, the parameters of the AR model, stored in vector $\\mathbf {\\theta }_x$ , are estimated from this portion of the signal by the covariance method [15].",
"As for the Gaussian Process model for the pulse tail, $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ , along with parameters $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ , are initialized by fitting a Gaussian Process with the chosen squared-exponential kernel to $y̭_2^{(0)}$ , via a standard maximum likelihood procedure [10].",
"Note that in the shape-based model for the pulse tail the initialization of $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ is done by the user.",
"In Section a suggested set of parameters produces a shape that resembles an actual pulse."
],
[
"Sampling algorithm",
"At this point, the goal is to recover the underlying signal $x̭$ from the observed degraded signal $y̭$ .",
"This can be achieved by estimating the auxiliary quantities stored in vectors $\\mathbf {\\theta }_{\\mathrm {d}}$ and $\\mathbf {\\theta }_{\\mathrm {t}}$ —the latter related to either the shape-based or the Gaussian Process model for the pulse tail, as well as the almost missing signal $x̭_1$ underlying the initial discontinuity.",
"After some manipulation of Equation (REF ), it is possible to obtain the likelihood $p(y̭ | x̭, \\mathbf {\\theta }_x, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_{\\mathrm {t}})$ .",
"Bayes' Theorem leads to the posterior distribution of the desired quantities $p(x̭, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_{\\mathrm {t}} | \\mathbf {\\theta }_x, y̭).$ Since this expression is analytically intractable, this distribution is sampled from via the Gibbs sampler [16], [17], [18] (eventually with some Metropolis steps if the corresponding conditional distribution is not from a known family of distributions), implemented as follows: Initialize values $n_0^{(0)}$ , $M^{(0)}$ , $\\sigma _{\\mathrm {d}}^{2^{(0)}}$ , $\\mathbf {\\theta }_{\\mathrm {t}}^{(0)}$ and $x̭^{(0)}$ .",
"For $k$ from 1 to $N_{\\text{iter}}$ : Sample $n_0^{(k)}$ and $M^{(k)}$ from distribution $p(n_0, M | \\sigma _{\\mathrm {d}}^{2^{(k-1)}}, \\mathbf {\\theta }_{\\mathrm {t}}^{(k-1)}, x̭^{(k-1)}, \\mathbf {\\theta }_x, y̭).$ Sample $\\mathbf {\\theta }_{\\mathrm {t}}^{(k)}$ and $x̭^{(k)}$ from distribution $p(\\mathbf {\\theta }_{\\mathrm {t}}, x̭ | n_0^{(k)}, M^{(k)}, \\sigma _{\\mathrm {d}}^{2^{(k-1)}}, \\mathbf {\\theta }_x, y̭).$ Sample $\\sigma _{\\mathrm {d}}^{2^{(k)}}$ from distribution $p(\\sigma _{\\mathrm {d}}^2 | n_0^{(k)}, M^{(k)}, \\mathbf {\\theta }_{\\mathrm {t}}^{(k)}, x̭^{(k)}, \\mathbf {\\theta }_x, y̭).$ Variables $n_0$ and $M$ are jointly sampled, since block Gibbs sampling is empirically known to improve the convergence speed of the algorithm [19].",
"The joint sampling of $\\mathbf {\\theta }_{\\mathrm {t}}$ and $x̭$ is adopted for the same reason, in addition to the fact that their joint distribution can be rewritten, via Bayes' Theorem and recalling the prior independence between $\\mathbf {\\theta }_{\\mathrm {d}}$ and $\\mathbf {\\theta }_{\\mathrm {t}}$ , in the following form, which will simplify some computations in Section : $\\begin{split}\\!\\!p(\\mathbf {\\theta }_{\\mathrm {t}},& x̭ | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) = p(\\mathbf {\\theta }_{\\mathrm {t}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭)p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) \\propto \\\\& [p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)p(\\mathbf {\\theta }_{\\mathrm {t}})] p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭).\\end{split}$ The order of the sampling is quite arbitrary, since it does not affect the convergence properties of the algorithm [19].",
"This particular order was implemented since it seems more natural to first sample $n_0$ and $M$ , the pulse location variables, and then sample the other variables.",
"Finally, the mean of the posterior distribution, estimated by averaging the samples obtained after the burn-in time, is used to perform the restoration procedure.",
"Notice that the sampling of $\\mathbf {\\theta }_{\\mathrm {t}}$ depends on the considered model.",
"For the shape-based model, the posterior distribution $p(\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}} | x̭, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭)$ is very complicated and does not belong to any known family of distributions; therefore, in this case, a Metropolis-Hastings step is performed within the Gibbs sampler—for more details, see [7].",
"Hereafter, the focus will be on the computation of the Gaussian Process model only, detailing the derivation of each conditional distribution of interest.",
"For the sake of completeness, the corresponding results for the shape-based model are also reported."
],
[
"Computation of the marginal likelihood",
"This section develops the marginal likelihood $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)$ , which will be needed later.",
"Firstly, two important results about Gaussian distributions must be recalled [3].",
"Consider an integral of the form $I = \\int _{\\mathbb {R}^D} \\exp \\left\\lbrace -\\frac{1}{2}(a + b̭^Tz̭ + z̭^TC̭z̭)\\right\\rbrace \\mathrm {d}z̭.$ Since the term inside the exponential function is a quadratic form on $z̭$ , one could complete the squares and compare the obtained expression with the probability density function of a multivariate Gaussian.",
"One can then prove that [3] $I = \\frac{(2\\pi )^{D/2}}{\\det (C̭)^{1/2}}\\exp \\left\\lbrace -\\frac{1}{2}\\left(a - \\frac{b̭^TC̭^{-1}b̭}{4}\\right)\\right\\rbrace .$ Now, consider the product of two multivariate Gaussian probability density functions, that is, $f(z̭) = f_1(z̭)f_2(z̭),$ where $f_1(z̭) = \\mathcal {N}(z̭ | \\mathbf {\\mu }_1, \\mathbf {\\Sigma }_1)$ and $f_2(z̭) = \\mathcal {N}(z̭ | \\mathbf {\\mu }_2, \\mathbf {\\Sigma }_2)$ .",
"Also after completing the squares inside the exponentials, one is able to prove that $f(z̭)$ is also the probability density function of a multivariate Gaussian distribution, but with covariance matrix given by $\\mathbf {\\Sigma } = (\\mathbf {\\Sigma }_1^{-1} + \\mathbf {\\Sigma }_2^{-1})^{-1}$ and mean $\\mathbf {\\mu } = \\mathbf {\\Sigma }^{-1}(\\mathbf {\\Sigma }_1^{-1}\\mathbf {\\mu }_1 + \\mathbf {\\Sigma }_2^{-1}\\mathbf {\\mu }_2)$ .",
"Proceeding to the derivation of the marginal likelihood, it can be rewritten as $\\begin{split}p(y̭ |& \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) = \\int _{\\mathbb {R}^N} p(x̭, y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)~\\mathrm {d}x̭ =\\\\& \\int _{\\mathbb {R}^N} p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) p(y̭ | x̭, \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)~\\mathrm {d}x̭.\\end{split}$ Note that the dependence of $x̭$ on $\\mathbf {\\theta }_{\\mathrm {t}}$ and $\\sigma _{\\mathrm {d}}^2$ can be dropped, and from Equation (REF ), $p(x̭ | n_0, M, \\mathbf {\\theta }_x)$ is Gaussian with mean $$ and covariance matrix $\\sigma _e^2(A̭^TA̭)^{-1}$ .",
"From Equation (REF ) and recalling the conditional independence of $y̭_0$ , $y̭_1$ and $y̭_2$ given $\\mathbf {\\theta }_{\\mathrm {d}}$ , $\\mathbf {\\theta }_{\\mathrm {t}}$ and $x̭$ , we can obtain $\\begin{split}p(y̭ |& x̭, \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) =\\\\&\\delta (y̭_0 - x̭_0) \\mathcal {N}(y̭_1 | x̭_1, \\sigma _{\\mathrm {d}}^2I̭_M) \\delta (y̭_2 - (x̭_2 + v̭_{\\mathrm {t}})),\\end{split}$ and therefore $\\begin{split}p(x̭,&y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) =\\mathcal {N}(x̭ | , \\sigma _e^2(A̭^TA̭)^{-1}) \\times \\\\ &[\\delta (y̭_0 - x̭_0)\\mathcal {N}(y̭_1 | x̭_1, \\sigma _{\\mathrm {d}}^2I̭_M)\\delta (y̭_2 - (x̭_2 + v̭_{\\mathrm {t}}))].\\end{split}$ Notice that the integral in Equation (REF ) is calculated with respect to $x̭$ , and the second Gaussian in Equation (REF ) depends on $x̭_1$ only via its mean.",
"In order to make explicit the dependence on $x̭_1$ , one can use the symmetry of the Gaussian distribution and the fact that $\\mathcal {N}(y̭_1 | x̭_1, \\sigma _{\\mathrm {d}}^2I̭_M) = \\mathcal {N}(x̭_1 | y̭_1, \\sigma _{\\mathrm {d}}^2I̭_M)$ (i.e., both PDFs have the same formula).",
"Using the decomposition of $x̭$ given in Equation (REF ): $\\begin{split}p(x̭,& y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) = \\\\ &\\mathcal {N}(K̭x̭_0 + Ṷ_1x̭_1 + Ṷ_2x̭_2 | , \\sigma _e^2(A̭^TA̭)^{-1}) \\times \\\\& [\\delta (y̭_0 - x̭_0) N(x̭_1 | y̭_1, \\sigma _{\\mathrm {d}}^2I̭_M) \\delta (y̭_2 - (x̭_2 + v̭_{\\mathrm {t}}))],\\end{split}$ and finally, $ \\begin{split}&p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) = \\\\&\\!\\int _{\\mathbb {R}^N} \\!\\!\\!",
"\\mathcal {N}(K̭x̭_0 + Ṷ_1x̭_1 + Ṷ_2x̭_2 | , \\sigma _e^2(A̭^TA̭)^{-1}) \\times \\\\ &~~[\\delta (y̭_0 -x̭_0) \\mathcal {N}(x̭_1 | y̭_1, \\sigma _{\\mathrm {d}}^2I̭_M) \\delta (y̭_2 - (x̭_2 \\!",
"+ \\!",
"v̭_{\\mathrm {t}}))]\\mathrm {d}x̭ = \\\\ &\\!\\int _{\\mathbb {R}^M} \\!\\!\\!",
"\\mathcal {N}(K̭y̭_0 \\!",
"+ \\!",
"Ṷ_1x̭_1 \\!",
"+ \\!",
"Ṷ_2(y̭_2 - v̭_{\\mathrm {t}}) | , \\sigma _e^2(A̭^TA̭)^{-1}) \\times \\\\ &~~\\mathcal {N}(x̭_1 | y̭_1, \\sigma _{\\mathrm {d}}^2I̭_M)~\\mathrm {d}x̭_1.\\end{split}$ The integral can be computed by using the Gaussian PDFs along with the results in the beginning of this Section with $z̭ = x̭_1$ : $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) = \\frac{\\lambda ^M}{(2\\pi \\sigma _e^2)^{\\frac{N - P}{2}}\\det (\\mathbf {\\Phi })}\\exp \\left\\lbrace -\\frac{E_{\\text{min}}}{2\\sigma _e^2}\\right\\rbrace ,$ where $\\begin{split}E_{\\text{min}} &= \\lambda ^2y̭_1^Ty̭_1 + z̭^T\\begin{bmatrix}A̭_0^T \\\\A̭_2^T\\end{bmatrix}[A̭_0 ~ A̭_2]z̭ + (x̭_1^{\\text{MAP}})^T\\mathbf {\\Theta }, \\\\z̭ &= \\begin{bmatrix}y̭_0 \\\\ y̭_2 - v̭_{\\mathrm {t}} \\end{bmatrix}, \\\\x̭_1^{\\text{MAP}} &= \\mathbf {\\Phi }^{-1}\\mathbf {\\Theta }, \\\\\\mathbf {\\Phi } &= \\lambda I̭_M + A̭_1^TA̭_1, \\\\\\mathbf {\\Theta } &= \\lambda y̭_1 - A̭_1^T[A̭_0 ~ A̭_2]z̭, \\\\\\lambda &= \\sigma _e^2/\\sigma _{\\mathrm {d}}^2.\\end{split}$ This expression can be simplified by noting that $\\lambda $ is likely to be very small, since $\\sigma _{\\mathrm {d}}^2$ is usually several orders of magnitude greater than $\\sigma _e^2$ .",
"In the argument of the exponential in Equation (REF ) this quantity multiplies $y̭_1$ , whose entries do not typically exceed $3\\sigma _{\\mathrm {d}}$ beyond the underlying signal, due to their Gaussian distribution.",
"One can then ignore all terms involving $\\lambda $ inside the exponential in Equation (REF ), which becomes proportional to $\\exp (-\\frac{1}{2}z̭^TR̭z̭)$ , where $R̭ = \\frac{1}{\\sigma _e^2}\\begin{bmatrix}A̭_0^T \\\\A̭_2^T\\end{bmatrix}S̭[A̭_0 ~ A̭_2],$ with $S̭ = I̭_{N-P} - A̭_1(A̭^{-1}A̭)^{-1}A̭_1^T.$ In several steps of the algorithm the distribution $p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭)$ , which was indirectly obtained above, will be necessary.",
"Notice that, by Bayes' Theorem, $\\begin{split}p(x̭ |& \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) \\propto \\\\& p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x) =\\\\& p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)p(x̭ | n_0, M, \\mathbf {\\theta }_x) = \\\\& [\\delta (y̭_0 - x̭_0) \\mathcal {N}(x̭_1 | y̭_1, \\sigma _{\\mathrm {d}}^2I̭_M) \\delta (y̭_2 - (x̭_2 + v̭_{\\mathrm {t}}))] \\times \\\\&~~~~\\mathcal {N}(K̭x̭_0 + Ṷ_1x̭_1 + Ṷ_2x̭_2 | , \\sigma _e^2(A̭^TA̭)^{-1}),\\end{split}$ as obtained in Equation (REF ).",
"This expression can be further simplified by noting that the second Gaussian depends essentially only on $x̭_1$ , since its dependence on $x̭_0$ and $x̭_2$ is defined by the two Dirac's deltas.",
"After using again the fact that a product of Gaussians is also Gaussian and calculating its mean and covariance matrix, one obtains $\\begin{split}\\!\\!p(x̭ &| \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) =\\\\&\\delta (y̭_0 - x̭_0) \\mathcal {N}(x̭_1 | x̭_1^{\\text{MAP}}, \\sigma _e^2\\mathbf {\\Phi }^{-1}) \\delta (y̭_2 - (x̭_2 + v̭_{\\mathrm {t}})).\\end{split}$"
],
[
"Computation of the conditional distributions and sampling procedure",
"In this section, the conditional distributions which must be sampled from are explicitly computed, and the sampling procedure itself is detailed for each case."
],
[
"Sampling $n_0$ and {{formula:c82a25ce-76f4-4617-aa3a-50dc15e9e595}}",
"By using Bayes' Theorem and recalling the prior independence between $n_0$ and $M$ , we have $\\!\\!\\!\\!p(n_0, M | \\sigma _{\\mathrm {d}}^2, \\mathbf {\\theta }_{\\mathrm {t}}, x̭, \\mathbf {\\theta }_x, y̭) \\!\\propto \\!",
"p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)p(n_0)p(M),\\!\\!$ where $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)$ is given by Equation (REF ).",
"Note that this equation depends implicitly on $n_0$ and $M$ , which influence the size of vectors $y̭_0$ , $y̭_1$ and $y̭_2$ .",
"Therefore, this formula defines a complicated distribution that is not easy to sample from, requiring a Metropolis-Hastings step within the Gibbs sampler.",
"The proposal distribution employed here is uniform over an interval whose length can be controlled by the user, centered at its respective last accepted values.",
"Experiments indicated that small lengths are preferable, and a value of 10 is suggested."
],
[
"Sampling $\\mathbf {\\theta }_{\\mathrm {t}}$ and {{formula:c5fded9b-1ed6-467b-8ae5-025ac0f71c1f}}",
"In order to sample from the joint posterior distribution of $\\mathbf {\\theta }_{\\mathrm {t}}$ and $x̭$ we recall the decomposition of $p(\\mathbf {\\theta }_{\\mathrm {t}}, x̭ | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭)$ in Equation (REF ).",
"By using Bayes' Theorem, it implies that this joint sampling is performed by first sampling $\\mathbf {\\theta }_{\\mathrm {t}}$ from $p(\\mathbf {\\theta }_{\\mathrm {t}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) \\propto p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)p(\\mathbf {\\theta }_{\\mathrm {t}})$ and then sampling $x̭$ from $p(x̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭)$ .",
"The second step is quite straightforward, since this distribution was already computed and is given by Equation (REF ).",
"Therefore, we only set $x̭_0 = y̭_0$ and $x̭_2 = y̭_2 - v̭_{\\mathrm {t}}$ , and sample $x̭_1$ from a Gaussian distribution with mean $x̭_1^{\\text{MAP}}$ and covariance matrix $\\sigma _{\\mathrm {e}}^2\\mathbf {\\Phi }^{-1}$ .",
"Note that $x̭_1$ is initialized simply with zeros, meaning that no previous knowledge about the signal underlying the initial discontinuity is available.",
"The first step, sampling from $p(\\mathbf {\\theta }_{\\mathrm {t}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) \\propto p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)p(\\mathbf {\\theta }_{\\mathrm {t}})$ is more complicated, and depends on whether the shape-based or the Gaussian Process model is being considered.",
"The former is briefly recalled below for the sake of completeness, followed by an exposition of the latter."
],
[
"Shape-based model",
"Note that the distribution $p(\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) \\propto p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)p(\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}})$ , seen as a function of $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ , is very complicated, since $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)$ is given by Equation (REF ) and parameters in $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ influence it through $y̭_2$ and the model in Equation (REF ).",
"Except for $V_{\\mathrm {t}}$ , whose posterior distribution can be verified to be Gaussian [7], [20], another Metropolis-Hastings step within the Gibbs sampler must be employed to provide samples of the components of $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ .",
"The chosen proposal distribution is multivariate Gaussian centered on the previous accepted value of the components of $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ except $V_{\\mathrm {t}}$ .",
"The diagonal covariance matrix of this proposal distribution is tuned by the user to keep the acceptance rate of this particular step around 50%, as suggested by some authors to guarantee that the sample space is well explored in a reasonable computational time [19]; particular values are recommended in Section .",
"This tuning can be made by increasing or decreasing the corresponding variances based on short runs of the simulated chain; despite being a classical procedure in Computational Statistics, this procedure was verified in experiments to be difficult to be performed, highly signal-dependent, and time-consuming, also possibly impacting in the convergence of the Gibbs sampler."
],
[
"Gaussian Process model",
"Recall that in this scenario the only component of $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {G}}$ to be sampled within the Gibbs sampler is $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ , since $\\sigma _f^2$ and $\\sigma _{\\ell }^2$ are kept constant.",
"However, even if its conditional distribution is readily available, due to its high-dimensionality (in the order of thousands of time samples), sampling from it is computationally very expensive.",
"Therefore, in this step we approximate a sample of this distribution by its mean, denoted by $v̭_{\\mathrm {t}, \\mathrm {mean}}^{\\mathrm {G}}$ , derived as follows.",
"By using Bayes' Theorem, we have that $\\begin{split}p(\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {G}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, y̭) &= p(v̭_{\\mathrm {t}}^{\\mathrm {G}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, \\sigma _f^2, \\sigma _{\\ell }^2, y̭) \\\\&\\propto p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {G}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)p(v̭_{\\mathrm {t}}^{\\mathrm {G}} | \\sigma _f^2, \\sigma _{\\ell }^2),\\end{split}$ where $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {G}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x)$ has already been determined in Equations (REF ) and (REF ), further simplified as Equations (REF ) and (REF ).",
"By applying the Gaussian Process prior of Equation (REF ) in Equation (REF ), the conditional posterior distribution for $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ is given by $\\begin{split}p(v̭_{\\mathrm {t}}^{\\mathrm {G}} &| \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, \\sigma _f^2, \\sigma _{\\ell }^2, y̭) \\propto \\\\ & \\exp \\left(-\\frac{1}{2}(v̭_{\\mathrm {t}}^{\\mathrm {G}})^TC̭v̭_{\\mathrm {t}}^{\\mathrm {G}}\\right)\\exp \\left(-\\frac{1}{2}z̭^TR̭z̭\\right)= \\\\&\\exp \\left\\lbrace -\\frac{1}{2}\\left((v̭_{\\mathrm {t}}^{\\mathrm {G}})^TC̭v̭_{\\mathrm {t}}^{\\mathrm {G}} + z̭^TR̭z̭\\right)\\right\\rbrace .\\end{split}$ We must then compute the term $z̭^TR̭z̭$ in order to make explicit its dependence on $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ .",
"To this end, note that it can be rewritten as: $\\begin{split}z̭^T&R̭z̭ = [y̭_0^T ~ (y̭_2 - v̭_{\\mathrm {t}}^{\\mathrm {G}})^T] \\!\\!\\left[\\begin{array}{c|c}R̭_{11} & R̭_{12} \\\\\\hline R̭_{21} & R̭_{22}\\end{array}\\right] \\!\\!\\!",
"\\begin{bmatrix}y̭_0 \\\\ y̭_2 - v̭_{\\mathrm {t}}^{\\mathrm {G}} \\end{bmatrix} \\!",
"= \\\\& -y̭_0^TR̭_{12}v̭_{\\mathrm {t}}^{\\mathrm {G}} - (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{21}y̭_0 - y̭_2^TR̭_{22}v̭_{\\mathrm {t}}^{\\mathrm {G}} \\\\ &- (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{22}y̭_2 + (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{22}v̭_{\\mathrm {t}}^{\\mathrm {G}} \\\\ &+ \\text{terms not depending on } v̭_{\\mathrm {t}}^{\\mathrm {G}}.\\end{split}$ Therefore, we have that $\\begin{split}(v̭_{\\mathrm {t}}^{\\mathrm {G}}&)^TC̭v̭_{\\mathrm {t}}^{\\mathrm {G}} +z̭^TR̭z̭ = \\\\ &-y̭_0^TR̭_{12}v̭_{\\mathrm {t}}^{\\mathrm {G}} - (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{21}y̭_0 - y̭_2^TR̭_{22}v̭_{\\mathrm {t}}^{\\mathrm {G}} \\\\&- (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{22}y̭_2 + (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TR̭_{22}v̭_{\\mathrm {t}}^{\\mathrm {G}} + (v̭_{\\mathrm {t}}^{\\mathrm {G}})^TC̭v̭_{\\mathrm {t}}^{\\mathrm {G}} \\\\&+\\text{terms not depending on } v̭_{\\mathrm {t}}^{\\mathrm {G}}.\\end{split}$ Since this expression is quadratic in $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ , the conditional posterior distribution for $v̭_{\\mathrm {t}}^{\\mathrm {G}}$ is a Gaussian whose mean vector and covariance matrix can be easily computed by completing the squares on the expression above, as indicated in the beginning of Section .",
"We then have that $p(v̭_{\\mathrm {t}}^{\\mathrm {G}} | \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, \\sigma _f^2, \\sigma _{\\ell }^2, y̭) = \\mathcal {N}(v̭_{\\mathrm {t}, \\mathrm {mean}}^{\\mathrm {G}}, \\mathbf {\\Sigma })$ , where $\\begin{split}v̭_{\\mathrm {t}, \\mathrm {mean}}^{\\mathrm {G}} = [R̭_{22} + R̭_{22}^T + C̭^{-1} + C̭^{-T}]^{-1} \\times \\\\ [(R̭_{12}^T + R̭_{21})y̭_0 + (R̭_{22}^T + R̭_{22})y̭_2]\\end{split}$ $\\mathbf {\\Sigma } &= \\left[\\frac{1}{2}(R̭_{22} + R̭_{22}^T + C̭^{-1} + C̭^{-T})\\right]^{-1}.$ Finally, note that if we have two overlapping pulses, their respective tails are being modeled by Gaussian Processes, and when the respective tails overlap, the posterior distribution of the superimposed pulse will still be Gaussian.",
"Therefore, this modeling may allow a simpler treatment of this heretofore complicated scenario.",
"However, preliminary tests indicated that the estimation of the AR model parameters in order to interpolate the signal underlying an initial discontinuity superimposed to an unfinished pulse tail is problematic; this issue is left to be addressed in a future work."
],
[
"Sampling $\\sigma _{\\mathrm {d}}^2$",
"This is the last step of the Gibbs sampler.",
"To compute the required posterior distribution, we use Bayes' Theorem together with the prior independence once again to obtain $p(\\sigma _{\\mathrm {d}}^2 | n_0, M, \\mathbf {\\theta }_{\\mathrm {t}}, x̭, \\mathbf {\\theta }_x, y̭) \\propto p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)p(\\sigma _{\\mathrm {d}}^2).$ Now, the dependence of $p(y̭ | \\mathbf {\\theta }_{\\mathrm {t}}, \\mathbf {\\theta }_{\\mathrm {d}}, \\mathbf {\\theta }_x, x̭)$ on $\\sigma _{\\mathrm {d}}^2$ is very simple: as can be seen in Equation (REF ), it is just a scale parameter for the distribution.",
"Therefore, the Inverse Gamma prior for $\\sigma _{\\mathrm {d}}^2$ implies that $p(\\sigma _{\\mathrm {d}}^2 | n_0, M, \\mathbf {\\theta }_{\\mathrm {t}}, x̭, \\mathbf {\\theta }_x, y̭)$ is also an Inverse Gamma, with parameters given by $\\alpha = \\alpha _{\\mathrm {d}} + \\frac{M}{2}$ and $\\beta = \\beta _{\\mathrm {d}} + \\frac{1}{2}\\sum \\limits _{i = 0}^{M-1}v̭_{\\mathrm {d}}(n_0 + i)^2.$"
],
[
"Results",
"The performance of the proposed algorithm was evaluated through tests performed in three distinct scenarios, namely: (A) A real signal artificially distorted by a pulse following Equation REF , in order to assess the method's accuracy and convergence, by means of a complete statistical analysis of the simulated chains; (B) Three real signals artificially degraded by pulses following Equation REF , whose restoration results are compared with other methods from the literature; (C) Two real degraded signals, informing us about the method's capability of dealing with real distortions.",
"The tests were run in a PC with a quadcore processor operating at 1.60 GHz clock and 8 GB of RAM.",
"All signals are monophonic, sampled at 44.1 kHz with 16-bit precision, and implementations are in MATLAB™The MathWorks, Inc., http://www.mathworks.com/.."
],
[
"Statistical analysis of the simulated chains",
"This test was performed over the first 3 s of a 13-s excerpt from a musical track, which consists of orchestral music containing a slowly varying string passage with some percussion in the last 5 seconds.",
"The signal was degraded by a single pulse following Equation REF .",
"The main goal of this test was to simulate long runs of the proposed algorithm, considering both the shape-based and Gaussian Process modeling for the pulse tail, and evaluate its convergence properties.",
"Table REF summarizes the results obtained, which will be discussed along this Section.",
"Its first nine lines refers to the shape-based model parameters, and the last three ones to the Gaussian Process counterparts.",
"The parameters common to both models are distinguished by the superscripts “s\" and “G\", respectively.",
"Table: Summary of statistical properties of the chains simulated in experiment (A).The 3-s degraded signal was given as input to the initialization procedure with parameters $L = 16$ , $\\xi = 0.3$ , $c = 5$ , and $f_{\\mathrm {co}} = 3$ kHz.",
"Once the single pulse had been localized through the initialization procedure, a segment of $N = 8,\\!000$ samples containing the pulse was processed.",
"Within this block, the initial estimate of the pulse start was indexed as $n_0=500$ (its true value is 3 samples earlier), and the parameters of the order-40 AR model used to infer the underlying signal in the initial discontinuity were estimated from the first 450 samples.",
"The initial estimates for $M$ and $\\sigma _{\\mathrm {d}}^2$ are displayed in the third column of Table REF , which also contains the initial values for $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ : despite seeming numerically close to their respective true values (displayed in the second column of Table REF ), they model quite distinct pulses, as illustrated in Figure REF .",
"A single chain with $10,\\!000$ iterations was simulated for the proposed algorithm with the shape-based model, being the first $5,\\!000$ discarded as burn-in time.",
"Each iteration lasted for approximately $1.6$ s. Since the autocorrelation plots of the sampled values after the burn-in time showed a highly correlated chain, of every 50 samples only one was considered and the rest were discarded, which resulted in 100 approximately uncorrelated samples for the posterior distribution of $x̭$ , $\\mathbf {\\theta }_{\\mathrm {d}}$ , and $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ .",
"Recall that estimates are obtained by averaging these sampled values, as mentioned in Section .",
"The estimates for $\\mathbf {\\theta }_{\\mathrm {d}}$ and $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ are displayed in the column named “Estimate I\" of Table REF , whose left column contains the respective estimated credible intervals at 95%.",
"Note that, except for $\\phi $ , all credible intervals contain the respective true values, indicating that the algorithm was capable of correctly identifying most of the degrading parameters.",
"It is also important to remark that after the burn-in time all proposed values for $n_0$ and $M$ distinct from the actual ones were rejected, meaning that their respective posterior distributions are highly concentrated around these values.",
"Such behavior is expected in this scenario, since the artificial degradation was introduced in a signal without other defects, such as broadband additive noise or clicks, which could increase the variances of the distributions.",
"The penultimate column of Table REF displays the variance of the proposal distribution for the components of $\\mathbf {\\theta }_{\\mathrm {d}}$ and $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ , except for $\\sigma _{\\mathrm {d}}^2$ and $V_{\\mathrm {t}}$ , which have closed form and well-known full conditional distributions.",
"For $n_0$ and $M$ , the value corresponds to the aforementioned length-10 discrete uniform proposal.",
"The values for the other variables were manually tuned by simulating shorter chains, and yield the acceptance rates displayed in the last column of Table REF .",
"Note that these rates are in a broad range around the desired 50%, a behavior also observed when using the same variances with other signals in experiments (B) and (C).",
"In fact, the choice of such values allowed satisfactory restored signals to be obtained in all scenarios tested without the need to fine-tune the algorithm separately for each situation.",
"Finally, in an attempt to circumvent the computational burden of the algorithm, another estimate was performed by considering only $1,\\!000$ iterations of the chain: the first 500 samples were discarded as burn-in phase, and the last 500 were averaged without discarding any sample, thus not masking the chain correlation.",
"These estimates are displayed in the column “Estimate II\" of Table REF ; it can be noticed that they are also close to their respective true values and within the respective credible intervals at 95%, except for $\\phi $ .",
"This last procedure will be preferred over the previous in the next tests with the shape-based model, since it was found that it produces equally satisfactory results with less computational effort.",
"As for the Gaussian Process model, a chain of $1,\\!000$ iterations was simulated, each iteration lasting for approximately 9 s, with the same initialization procedure described before.",
"Note that in addition to the initialization procedure parameters, the only parameters to be tuned in this algorithm are the supports of the proposal distributions for $n_0$ and $M$ , both set to 10.",
"The behavior of the simulated chain for these variables, as well as for $\\sigma _{\\mathrm {d}}^2$ , is similar to that observed in the shape-based case, and the first 200 iterations are displayed in Figure REF .",
"It can be seen that the sampled values of $n_0$ and $M$ rapidly converge to their respective true values, and contrarily to the shape-based case, it was verified that the simulated chain for $\\sigma _{\\mathrm {d}}^2$ does not exhibit the autocorrelation problem, dispensing with the thinning procedure.",
"A summary of the simulation is displayed in the last three lines of Table REF : the estimates obtained by averaging the samples after the burn-in time of 500 iterations is displayed in the column named “Estimate I\", and the estimated credible intervals at 95% are in its left column.",
"In order to reduce the computational impact of this model, it would be enough to simulate a chain with 200 iterations and discard the first 150 as burn-in time, leading to the estimates in the “Estimate II\" column of Table REF .",
"This procedure will be adopted in the next tests with the Gaussian Process model.",
"Figure: First 200 iterations of the simulated chain for n 0 n_0, MM, and σ d 2 \\sigma _{\\mathrm {d}}^2 in experiment (A) for the Gaussian Process model.",
"The rest of the iterations show similar behavior: n 0 n_0 and MM are constant, and analogous oscillations occur for σ d 2 \\sigma _{\\mathrm {d}}^2.The true pulse and the pulses estimated by both the shape-based and Gaussian Process models are displayed in Figure REF .",
"In the upper panel it can be seen that, despite the inaccuracy in the estimate of $\\phi $ in the shape-based case, both true and estimated pulses are visually similar, indicating the capability of the algorithm to suitably identify the degradation; the pulse generated by the initial value for $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ is also displayed.",
"As shown in the lower panel, the Gaussian Process algorithm is also capable of satisfactorily estimating the pulse, despite the stationary squared-exponential covariance kernel employed.",
"Despite not being the case illustrated in Figure REF , due to the stationary nature of the squared-exponential covariance kernel employed in the Gaussian Process model, the estimated pulse can be distant from zero at the end of the processed block.",
"In this situation, the restored excerpt, when replaced on the original signal, will exhibit a discontinuity that may generate audible artifacts.",
"This issue can be solved by a simple heuristics: fading-out the last samples of the estimated pulse linearly to zero before being subtracted from the degraded signal.",
"For a block of the size considered in experiments (A) and (B), fading-out the last $1,\\!000$ samples showed good results; this procedure was not necessary in experiment (C).",
"Figure: Comparison between true and estimated pulses in experiment (A): light continuous and dark dashed lines represent the underlying distorted signal and the superimposed pulse, respectively, in both graphs; in the upper panel, the solid dark continuous line is the pulse estimate by the shape-based algorithm, and the point-dashed line is the pulse generated by the initial values of θ t s \\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}; in the lower panel, the solid dark line is the pulse estimated by the Gaussian Process algorithm.In short, these experiments illustrate the following: 1) the initialization procedure is capable of providing accurate estimates of the pulse location, which are promptly corrected by the algorithm under both shape-based and Gaussian Process modeling; 2) when the variances of the proposal distributions for the Metropolis-Hastings steps in the shape-based case are properly tuned and sufficiently long chains are simulated, the algorithm is capable of correctly identifying the pulse parameters; 3) despite the inaccuracy of the squared-exponential covariance kernel employed on the Gaussian Process model, it is also capable of identifying the pulse."
],
[
"Comparison with previously proposed methods",
"Three signals were evaluated in this test, with both the shape-based and Gaussian Process models: the same 13-second long excerpt of orchestral music mentioned in Experiment (A), an 8-second long excerpt of jazz quartet music with drums, bass, guitar and clarinet, and an 11-second long excerpt of Brazilian music with singing male voice and percussion.",
"These signals are referred to as “classical\", “jazz\", and “Brazilian\", and were artificially corrupted with 17, 11, and 14 non-overlapping and uniformly spaced pulses, respectively, following Equation REF .",
"All signals were given as input to the initialization procedure with the same setup as before.",
"For illustration, its output $\\Delta \\mu $ is shown for the “jazz\" signal in Figure REF .",
"Note that all pulses are identified; the same was verified in the other two test signals.",
"An excerpt of size $N = 8,\\!000$ containing each pulse was separately processed following the shorter procedures previously described, namely: for the shape-based model, $1,\\!000$ iterations with burn-in of 500 for each pulse, $\\mathbf {\\theta }_{\\mathrm {t}}^{\\mathrm {s}}$ initialized as in the third column of Table REF , and variances of proposal distributions (except for $V_{\\mathrm {t}}$ ) as in the penultimate column of Table REF ; for the Gaussian Process model, 200 iterations with burn-in of 150.",
"The estimates are obtained by averaging the samples obtained after the burn-in time, and the initial estimates for $n_0$ , $M$ , and $\\sigma _{\\mathrm {d}}^2$ are the outputs of the initialization procedure.",
"Figure: Function Δμ\\Delta \\mu – output of the initialization procedure (with associated threshold) for the “jazz\" signal in experiment (B).In order to objectively assess the quality of the restored signal, we evaluated it through the PEAQ (Perceptual Evaluation of Audio Quality) [21], [22] algorithm, tailored to compare the perceived quality of a wideband audio signal against a reference signal after mapping them to a perceptual domain that models the operation of the human auditory system.",
"The output score is a number from 0 (imperceptible difference from the reference signal) to -4 (very annoying difference).",
"Table REF compares the PEAQ scores received by the proposed method with both modeling possibilities with those of [5] (based on the TPSW filtering), [6] (based on the EMD), and [4] (based on the AR separation technique).",
"The original signals and their restored counterparts using these methods are available from [23].",
"As required by the PEAQ algorithm, signals were previously upsampled to 48 kHz.",
"Note that two instances of the “classical\" signal were considered: its entirety and its first 8 s, which excludes the mainly percussive part.",
"Table: PEAQ grades for the artificially degraded signals in experiment (B) and its respective restored versions.",
"The closer the grade is to zero, the cleaner the signal.Note that the proposed algorithm with both modeling possibilities returns signals with substantially improved PEAQ grades than its corrupted counterparts, being the performance of the Gaussian Process model always better than the shape-based one.",
"Both models show their better results in the initial excerpt of the “classical\" signal, a behavior explained by two aspects: 1) in this case the underlying signal easily fulfills the hypothesis of being similar to white noise in the time-scale of the tail, as can be seen in Figure REF ; and 2) the tuning parameters of the shape-based algorithm were the same as in experiment (A), being then tailored to a similar underlying signal.",
"For the sake of completeness, the PEAQ scores of the degraded and restored (by both the shape-based and Gaussian Process algorithms) signals in experiment (A) were -2.6513, -0.1873, and -0.1471, respectively, indicating that the shape-based model is capable of producing results similar to the Gaussian Process model, when properly tuned.",
"Despite not providing much information from the perceptual viewpoint, the signal-to-noise ratio (SNR) measured in dB for the same set of signals is presented in Table REF .",
"It can be seen that the proposed method always return restored signals with improved SNR, but this improvement is substantially lower for signals with substantial percussive content.",
"Let us expand this discussion by considering the “Brazilian\" signal as an example.",
"Informal listening tests indicates that its restoration when using the Gaussian Process model sounds identical to the non-degraded signal, whereas when employing the shape-based model several remnants of the initial discontinuities are noticeable.",
"The respective PEAQ grades reflects this impression, but their SNR indicates the opposite.",
"Table: SNR (in dB \\mathrm {dB}) for the artificially degraded signals in experiment (B) and its respective restored versions.",
"The higher the value, the closer to the cleaner signal.To illustrate this case, in Figure REF the input excerpt containing the fifth pulse of the degraded version of the “Brazilian\" signal is displayed, as well as the estimated pulse from both models.",
"It can be seen that the Gaussian Process estimate also captures fluctuations intrinsic to the underlying signal that are not generated by the pulse, implying the observed low SNR.",
"This can be seen as a disadvantage of this model when compared to the shape-based one.",
"However, the latter has not showed itself capable of correctly interpolating the signal underlying the initial discontinuity, whereas the former leaves no audible remnant of the degradation, being the only exception noticed on the percussive part of the “classical\" signal.",
"Figure: Excerpt processed containing the fifth pulse of the “Brazilian\" signal (clear solid line), and pulse estimates using the shape-based (solid dark line) and Gaussian Process (dashed dark line).In summary, the proposed algorithm with the Gaussian Process model yields a signal with marginally better or similar PEAQ score when compared to its competitors.",
"In practice, these numbers mean that the restored signals sound almost identical, a fact that was verified by informal listening tests.",
"However, our proposed algorithm automatically locates the initial discontinuity and its duration, improves these initial estimates and interpolates the underlying signal; in addition, it produces a very accurate estimate of the pulse tail.",
"Even so, it requires significantly less user intervention than the shape-based model."
],
[
"Real degraded signals",
"The two signals considered in this scenario are: an excerpt of 3 seconds from cylinder recording number 154 from [24], approximately from 1907, containing a severe pulse at the beginning when the music that will be played is introduced; and an excerpt of 11 seconds from a piece by Chopin for solo piano, containing six pulses.",
"Both are sampled at 44.1 kHz with 16-bit resolution.",
"Parameters for the initialization procedure for each signal are shown in Table REF .",
"They were manually tuned in order to correctly identify the pulses only, and not other defects also present within the signal, such as clicks and crackles.",
"Essentially, signals with more background noise require a higher value of $f_{\\mathrm {co}}$ and $c$ , but since the initialization procedure is very fast, lasting less than 1 s, some trial and error is not tedious or time-consuming.",
"Table: Parameters of the initialization procedure for the “cylinder\" and “Chopin\" signals in experiment (C).In both cases, since the tail of the pulse is not too long, an excerpt of only 2,000 samples containing the degradation was processed.",
"When considering the Gaussian Process model, for each pulse the Gibbs sampler ran for 500 iterations (lasting $\\approx 0.5$ s each), of which the first 400 were discarded; regarding the shape-based model, for each pulse $5,\\!000$ iterations (lasting $\\approx 0.2$ s each) were run, of which the first $2,\\!500$ were discarded.",
"The initial value of $\\theta _{\\mathrm {t}}^{\\mathrm {s}}$ was the same as in the third column of Table REF .",
"Since in this scenario we do not have a reference signal, it is not possible to compute the PEAQ grade, so we will present a brief discussion about the restoration on both cases: In the cylinder excerpt, the restoration seems almost perfect with the Gaussian Process model.",
"A very light click remains where the pulse originally started, although almost masked by the high background noise present throughout the signal.",
"On the other hand, there is no vestige of the low-frequency content of the pulse tail.",
"The shape-based algorithm was also capable of removing the pulse tail and attenuate the initial discontinuity, but its remnant is much more perceptible.",
"The processed excerpt and the restored signals can be seen in Figure REF .",
"In the Chopin excerpt, the restoration in both modeling scenarios sounded almost identical: the remnant of the initial discontinuities, this time in the form of a short-duration and low-level hiss, is more apparent because of the lower level of background noise in the signal.",
"However, once more there is also no perceptible vestige of the low-frequency content of the pulse in the restored signal.",
"Figure: Processed excerpt of “Cylinder\" signal (solid line) and its restored versions (dashed lines): shape-based in the upper panel and Gaussian Process in the lower one.This small residue left after the interpolation of the signal underlying the initial discontinuity is much probably due to the impairment of the AR modeling by the presence of background noise, as discussed in [3].",
"This issue could be worked around by including the estimation of its parameters in the Gibbs sampler instead of keeping them constant, or by employing a more sophisticated interpolation scheme.",
"However, the high impact on the overall computational load found in preliminary tests suggested that the simplified approach should be adopted and that this point should be further investigated with a view to a new version of the algorithm.",
"However, it should be stressed that, as in the controlled experiments, the proposed algorithm was shown to be capable of identifying the location and extent of the initial discontinuity as well as the tail of the pulse when dealing with real degraded signals, a much more challenging scenario where defects other than the long pulses may also be present."
],
[
"Conclusion",
"In this paper we presented a novel algorithm for the restoration of audio signals degraded by long pulses with significant low-frequency content that circumvents two main drawbacks of previous works: a large quantity of hyper-parameters difficult to interpret that must be adjusted manually, and the need to know the location of the pulse beforehand.",
"These issues were addressed through modeling of the pulse tail via Gaussian Process and a Bayesian framework that includes location as a random quantity to be estimated.",
"The signal underlying the initial discontinuity is also interpolated along the procedure.",
"In order to accelerate the convergence of the Gibbs sampler, we employed an efficient initialization algorithm based on the time-frequency content of the degraded signal that roughly locate the beginning of the degradation.",
"In controlled experiments, when compared to previous approaches our algorithm shows a slightly better or similar performance, according to the PEAQ grades.",
"These results are confirmed by the experiments with real degraded signals.",
"Some improvements to the proposed algorithm will be addressed in future works: 1) make it able to handle overlapping pulses; and 2) refine the restored signal interpolation in the region of the initial discontinuity, which is likely to be impaired by other degradations such as background noise.",
"Both points are linked to the difficulty of estimating the AR model parameters and further interpolate the restored signal in the region of the discontinuities in noisy scenarios; this is the next research target.",
"Another possible improvement is to sub-sample the pulse tail before the Gibbs sampler.",
"This will significantly decrease the computational cost of the algorithm, since at each iteration a matrix of size approximately $N \\times N$ must be inverted.",
"The sub-sampling can be justified by the fact that it implies no relevant loss of information on the tail, since it usually contains much lower frequencies than the typical underlying signal.",
"Finally, from the statistical viewpoint the adoption of Variational Inference instead of MCMC in the estimation procedure should be addressed in a future work aiming at an acceleration of the entire procedure."
],
[
"Acknowledgment",
"The authors would like to thank Paulo Antonio Andrade Esquef for providing the signals used in the experiment with artificially degraded signals, and the “Chopin\" signal employed on the real degraded scenario."
]
] | 2005.14181 |
[
[
"Planar Graphs that Need Four Pages"
],
[
"Abstract We show that there are planar graphs that require four pages in any book embedding."
],
[
"Introduction",
"A book embedding of a graph consists of a linear order of its nodes and a partitioning of its edges, so that the nodes can be placed in order on a straight line (the “spine\" of the book) and the edges in each part can be drawn on a separate half-plane bordered by the line (a “page\" of the book) so that the edges on the same page do not intersect.",
"The objective is to find a book embedding that uses the minimum number of pages.",
"This minimum number is called the pagenumber (or book thickness) of the graph.",
"Book embeddings were introduced in [12] and [3].",
"They were studied in connection with an approach to fault-tolerant VLSI design [13], [5], and have applications also in sorting using stacks, in graph drawing, complexity theory, and other areas.",
"Computationally, the problem of computing the pagenumber of a graph is hard: it is NP-complete to decide whether a planar graph has pagenumber 2 [14], [5].",
"Note that in the case of two pages, the crux of the problem is the node-embedding part (the linear ordering of the nodes): once the node ordering is fixed, it is easy to test whether two pages suffice.",
"In general, the subproblem of minimizing the number of pages for a given fixed node ordering is itself also NP-hard [8].",
"Graphs with pagenumber one are exactly the outerplanar graphs.",
"Graphs with pagenumber two are exactly the planar subhamiltonian graphs, i.e.",
"the subgraphs of planar Hamiltonian graphs [3].",
"In [15], [16] we showed that all planar graphs can be embedded in four pages, and gave a linear time algorithm for this purpose (references [4] and [9] gave earlier algorithms that embed planar graphs in nine and seven pages respectively).",
"In the conference paper [15] we stated also that there are planar graphs that require four pages, and outlined briefly the approach and the structure of the construction.",
"The present paper gives the full details of the construction and the proof that the constructed planar graph has pagenumber at least four.",
"Besides planar graphs, there has been extensive work on book embeddings for various other classes of graphs, for example graphs of bounded genus [10], [11], bounded treewidth [7], 1-planar graphs [1].",
"Before getting into the technical details of the construction and the proof, we make a few remarks regarding the issues involved and our approach to address them.",
"To show a lower bound of four on the pagenumber of planar graphs means finding a planar graph $G$ and showing that no matter how we order its nodes and how we partition its edges into three pages, there will be a violation (two conflicting edges on the same page).",
"A major obstacle in this regard stems from the computational complexity of the problem: The problem is NP-complete, which means that if $NP \\ne coNP$ , as is widely believed to be the case, there is in general no short way to prove the nonexistence of a suitable node ordering and edge partitioning for a given graph; that is, any proof has to be in general at least superpolynomially long in the size of the graph, and will likely amount to examining essentially all (or at least a large number of) possibilities to ensure that they do not work.",
"An important difference here is that we do not have to deal with an arbitrary general graph, but with a specific graph $G$ of our own design that is suitable for our purpose.",
"If there was a very small suitable graph $G$ , then the complexity would not be an issue.",
"This is the case for example in the graph coloring problem, where there is an extremely small planar graph (namely, $K_4$ ) that needs 4 colors, so the lower bound is trivial.",
"Unfortunately, this does not seem to be the case in the book embedding problem.",
"For example, [2] studied experimentally the book embedding problem by formulating it in terms of the SAT (Boolean formula satisfiability) problem and using a SAT solver; the software handles graphs with up to 500-700 nodes.",
"In their experiments searching for planar graphs that require four pages (they tried both random graphs and crafted graphs in certain families) none was encountered, which led the authors to hypothesize that perhaps three pages are enough for all planar graphs.",
"The way we deal with the complexity of the problem in our construction is by building up the graph (and the proof) in stages, to control the exponential explosion in case analysis.",
"As in NP-completeness reductions, we design and use gadgets (`small' graphs that have useful properties) as building blocks.",
"We start with a small graph $Q_1$ (10 nodes) which we analyze in some detail to characterize its book embeddings under some strong restrictions.",
"Then we use this gadget to analyze a somewhat larger graph $Q_2$ , and this is used in turn for a larger gadget $Q$ under weaker restrictions.",
"The gadgets are used in the construction of the final graph $G$ that cannot be embedded in three pages.",
"The properties we showed for the gadgets restrict significantly the possible book embeddings of $G$ and make the analysis tractable.",
"The rest of the paper is organized as follows.",
"Section 2 provides basic definitions and some simple observations.",
"In Section 3 we design the gadgets and prove their properties.",
"Section 4 gives the definition of the graph $G$ and proves that $G$ requires four pages."
],
[
"Preliminaries",
"Let $G=(N,E)$ be a (undirected) graph, and $\\pi $ a linear ordering (a permutation) of its nodes.",
"We say that two edges $(a,b), (c,d)$ conflict in the ordering $\\pi $ if $\\pi (a) < \\pi (c) < \\pi (b) < \\pi (d)$ or $\\pi (c) < \\pi (a) < \\pi (d) < \\pi (b)$ ; that is, if we place the nodes on a line ordered according to $\\pi $ and draw the edges (as curves) on a half-plane bordered by the line, two edges conflict iff they intersect.",
"A book embedding of $G$ in $k$ pages consists of (1) a linear ordering $\\pi $ of its nodes, and (2) a coloring of its edges with $k$ colors (the “pages\") so that conflicting edges in $\\pi $ receive different colors.",
"The pagenumber of $G$ is the minimum number $k$ of pages such that $G$ has a book embedding in $k$ pages.",
"We will usually show in figures a (partial) book embedding in 3 pages by showing the positions of the nodes on a line and showing edges of color 1 as red dashed curves, color 2 as solid blue, and color 3 as dotted green.",
"To simplify notation, we will identify the positions of the nodes on the line with the nodes, and refer for example to node $a$ on the line (instead of point $\\pi (a)$ ), to interval $(a,b)$ of the line (instead of interval $(\\pi (a),\\pi (b))$ ), and so forth.",
"Alternatively, a book embedding of a graph $G$ can be defined by a mapping of its nodes to (distinct) points on a circle, drawing the edges as straight line segments (chords) inside the circle and coloring them with $k$ colors so that intersecting edges receive different colors.",
"From such a circle embedding one can obtain a linear embedding by cutting the circle at any point and then ordering the nodes on the line in either of the two directions, clockwise or counterclockwise (the edges retain their colors).",
"Conversely, given a linear embedding, one can obtain a circle embedding by connecting the two ends of the line to form a circle that encloses all the edges.",
"Note that a circle embedding corresponds to $2|N|$ linear embeddings that are essentially equivalent.",
"Given a book embedding of a graph $G$ , we will say that an edge exits an interval $(a,b)$ if one node of the edge is in the open interval $(a,b)$ and the other node is outside the closed interval $[a,b]$ .",
"Similarly, for a circle embedding, an edge exits an arc $(a,b)$ if the two nodes of the edge are on the two different open arcs of the circle between $a$ and $b$ .",
"The following proposition gives some basic simple observations that are used throughout the paper, usually without making explicit reference to the proposition.",
"Proposition 1 (1) If in a book embedding of a graph $G$ there is a path $p_i$ between two nodes $a, b$ all of whose edges have the same color $i$ , then there is no edge of color $i$ that exits the interval $(a,b)$ and connects two nodes that are not on the path $p_i$ .",
"(2) If in a 3-page embedding of $G$ there are paths $p_1, p_2, p_3$ of all 3 colors $1, 2, 3$ respectively between two nodes $a,b$ , then there is no edge that exits the interval $(a,b)$ and connects two nodes that are not in any of these paths.",
"(3) If in a 3-page embedding of $G$ there are paths $p_1, p_2, p_3$ of all 3 colors $1, 2, 3$ between two nodes $a,b$ , then for every connected component $C$ of the graph $G \\setminus (p_1 \\cup p_2 \\cup p_3)$ obtained by deleting the nodes of $p_1, p_2, p_3$ from $G$ , either all the nodes of $C$ are in the interval $(a,b)$ or they are all outside $(a,b)$ .",
"(1) Let $p_i$ be the path $a=v_0, v_1, \\ldots , v_m=b$ , and let $(x,y)$ be a color-$i$ edge whose nodes are not on $p_i$ .",
"If a node $v_j$ of the path is in the interval $(x,y)$ , then the next node $v_{j+1}$ must be also in $(x,y)$ because the edges $(v_j,v_{j+1}), (x,y)$ have the same color, hence they do not conflict.",
"Thus, by induction, if $a=v_0$ is in the interval $(x,y)$ then all the nodes of the path are in $(x,y)$ , hence also $b$ .",
"Similarly, if $a=v_0$ is not in the interval $(x,y)$ then $b$ is not in the interval $(x,y)$ either.",
"In either case, $(x,y)$ does not exit the interval $(a,b)$ .",
"(2) Follows from (1).",
"(3) Any two nodes $x, y$ of $C$ are connected by a path $x=u_0, u_1, \\ldots , u_l=y$ that does not contain any node of the paths $p_i$ .",
"By (2), $u_j$ is in the interval $(a,b)$ iff $u_{j+1}$ is also in $(a,b)$ .",
"Hence by induction, $x=u_0$ is in $(a,b)$ iff every node of the path, and in particular $y$ , is in $(a,b)$ .",
"We will say that a node $u$ reaches an interval $(a,b)$ (or any subset of the line) if there is an edge from $u$ to a node in the interval $(a,b)$ (resp., in the subset).",
"Thus, for example by the above Proposition (part 2), if in a 3-page embedding there are paths $p_1, p_2, p_3$ of all three colors between nodes $a, b$ , then the nodes embedded in the interval $(a,b)$ cannot reach any nodes outside $(a,b)$ except possibly the nodes of the paths $p_i$ ."
],
[
"The Gadgets",
"Figure REF shows our first “gadget\" $Q_1$ .",
"We will refer to nodes 1, 2 as the outer terminals, to nodes $a,b$ as the inner terminals, and to nodes $c_1, c_2$ as the centers.",
"We will use $Q_1+12$ to denote the graph $Q_1$ with the additional edge $(1,2)$ connecting the outer terminals, and we will use $Q_1 +ab$ to denote the graph with the additional edge $(a,b)$ connecting the inner terminals.",
"Figure: The graph Q 1 Q_1.Lemma 1 There is no 3-page embedding of $Q_1+12$ or of $Q_1+ab$ such that (1) the inner terminals $a, b$ lie in the interval $(1,2)$ between the outer terminals, (2) the centers $c_1, c_2$ of $Q_1$ lie in the interval $(a,b)$ , and (3) all edges from node 1 to the closed interval $[a,b]$ use the same color, say color 1, and all edges from node 2 to $[a,b]$ use color 2.",
"The proof is by contradiction.",
"Consider any 3-page embedding of $Q_1$ that satisfies conditions 1-3 of the lemma.",
"Suppose without loss of generality that the nodes $a,b, c_1, c_2$ are laid out in the order shown in Figure REF ; the other possible embeddings with the order of $a,b$ and/or $c_1, c_2$ reversed, are symmetric.",
"The outer terminals 1,2 are not shown explicitly in the figure: one of them, node $s \\in \\lbrace 1,2\\rbrace $ is at the left end of the line and the other outer terminal $t$ is to the right of $b$ , but we do not specify which is which so that we do not have to distinguish cases; it is irrelevant for the following arguments which one of 1,2 is $s$ at the left end and which is $t$ on the right.",
"The edges to 1, 2 are shown in the figure as `dangling' segments with only one node.",
"Thus, the red dashed dangling edges have color 1 and go to node 1, and the solid blue dangling edges have color 2 and go to terminal 2.",
"We shall prove that edge $(a,c_2)$ has color 2 and $(b,c_1)$ has color 1, as shown in the figure.",
"Figure: Centers in (a,b)(a,b).Edge $(a,c_2)$ conflicts with the edge $(1,c_1)$ which is colored 1 by the hypothesis, hence $(a,c_2)$ is colored 2 or 3.",
"Similarly edge $(b,c_1)$ conflicts with $(2,c_2)$ , hence $(b,c_1)$ is colored 1 or 3.",
"The two edges $(a,c_2)$ , $(b,c_1)$ conflict with each other, so they cannot both be colored 3.",
"Suppose without loss of generality that $(a,c_2)$ is not colored 3, hence it is colored 2.",
"Suppose that edge $(b,c_1)$ has color 3.",
"We shall argue that it is impossible then to embed legally node $d_{2a}$ .",
"Observe that node $c_2$ cannot reach any node, other than 1,2, outside the interval $[a,b]$ because of the color-3 edge $(b,c_1)$ , the color-1 path $(a,1,b)$ and the color-2 path $(a,2,b)$ .",
"Hence $d_{2a}$ must lie in the interval $(a,b)$ .",
"Since the edges from node 2 to this interval are colored 2 and $(a,c_2)$ is also colored 2, node $d_{2a}$ must lie in the interval $(c_2,b)$ .",
"Then there is no legal color available for the edge $(a,d_{2a})$ because it conflicts with the color-1 edge $(c_1,1)$ , the color-2 edge $(c_2,2)$ and the color-3 edge $(b,c_1)$ .",
"We conclude that the edge $(b,c_1)$ does not have color 3, hence it has color 1.",
"Consider now the possible positions of nodes $d_{1b}$ and $d_{2a}$ .",
"We show first that they must be outside the interval $(a,b)$ .",
"Suppose that $d_{1b}$ is inside the interval $(a,b)$ , to derive a contradiction.",
"Since $d_{1b}$ is adjacent to node 1 and edge $(c_1,b)$ is colored 1, node $d_{1b}$ cannot be in the interval $(c_1,b)$ , hence it must be in $(a,c_1)$ .",
"The edge $(b,d_{1b})$ must be colored 3 (since it conflicts with the color-1 edge $(1,c_1)$ and the color-2 edge $(2,c_2)$ ).",
"Hence, node $c_2$ cannot reach any node, other than 1, 2, outside the interval $(a,b)$ because of the color-1 path $(a,1,b)$ , the color-2 path $(a,2,b)$ , and the color-3 edge $(b,d_{1b})$ .",
"Therefore, node $d_{2a}$ must then also be inside the interval $(a,b)$ .",
"However, node 2 cannot reach the subinterval $(a,c_2)$ because of the color-2 edge $(a,c_2)$ , and node $a$ cannot reach the subinterval $(c_2,b)$ because of the color-1 edge $(c_1,b)$ , the color-2 path $(c_2,2, b)$ and the color-3 edge $(b,d_{1b})$ Thus, there is no legal position for node $d_{2a}$ , contradiction.",
"We conclude that $d_{1b}$ must be outside the interval $(a,b)$ .",
"By a symmetric argument, $d_{2a}$ also lies outside the interval $(a,b)$ .",
"The edges $(c_1,d_{1b})$ and $(c_2,d_{2a})$ must be colored 3, because they exit the interval $(a,b)$ and there are color-1 and -2 paths $(a,1,b)$ and $(a,2,b)$ connecting $a$ and $b$ .",
"We can prove the lemma now for $Q_1+ab$ : The edge $(a,b)$ connecting the inner terminals cannot be colored legally because it intersects the color-1 edge $(1,c_1)$ , the color-2 edge $(2,c_2)$ and the color-3 edge $(c_1,d_{1b})$ .",
"This proves the claim for the graph $Q_1+ab$ .",
"It remains to prove the claim for $Q_1+12$ .",
"Since the edges $(c_1,d_{1b})$ and $(c_2,d_{2a})$ have the same color, either $d_{1b}$ is to the left of $c_1$ and hence of $a$ , or $d_{2a}$ is to the right of $c_2$ and hence of $b$ (or both).",
"Suppose without loss of generality that $d_{2a}$ is to the right of $b$ .",
"If it is left of $t$ , i.e., within the same arc $(1,2)$ in the corresponding embedding on a circle, then the edge $(a,d_{2a})$ must be also colored 3 (because it conflicts with the edges $(1,b), (2,b)$ ), but then either $(a,d_{2a})$ or $(c_2,d_{2a})$ conflicts with the edge $(c_1,d_{1b})$ , a contradiction.",
"We conclude that $d_{2a}$ is right of $t$ , i.e., lies in the opposite arc $(1,2)$ .",
"By a symmetric argument, the same holds for the node $d_{1b}$ .",
"Since $(c_1,d_{1b})$ and $(c_2,d_{2a})$ have the same color (3), hence do not intersect, nodes $s, a, c_1, c_2, b, t, d_{2a}, d_{1b}$ appear in this order.",
"Consider the three edges $(d_{1b},b), (d_{2a},a), (s,t)$ .",
"They all intersect a color-3 edge, namely $(c_2,d_{2a}), (c_1,d_{1b}), (c_1,d_{1b})$ respectively, hence they can only use the colors 1, 2.",
"However, the three edges $(d_{1b},b), (d_{2a},a), (s,t)$ conflict with each other, hence they cannot all be colored with two colors.",
"This proves the lemma for $Q_1+12$ .",
"Lemma 2 It is not possible to embed $Q_1$ in three pages, such that (1) the inner terminals $a, b$ lie in a subinterval $(u,v)$ between the outer terminals 1,2, (2) both centers $c_1, c_2$ lie outside the interval $[a,b]$ , (3) all edges from node 1 to the interval $(u,v)$ use the same color, say color 1, and all edges from node 2 to $(u,v)$ use color 2, and (4) all other edges exiting the interval $(u,v)$ (not connecting to nodes 1, 2) are colored 3.",
"The proof is by contradiction.",
"Assume an embedding as in the lemma.",
"If $c_1$ is outside $(u,v)$ then both edges $(c_1,a)$ , $(c_1,b)$ must have color 3 (by condition (4)), and similarly for $c_2$ .",
"We cannot have both $c_1, c_2$ outside the interval $(u,v)$ because then (at least) one of the edges $(c_1,a)$ , $(c_1,b)$ would intersect one of the edges $(c_2,a)$ , $(c_2,b)$ .",
"Therefore, at least one of $c_1, c_2$ must be in the interval $(u,v)$ .",
"Assume without loss of generality that $c_1$ is in the interval $(u,v)$ , and, since it is not in the interval $[a,b]$ , assume wlog that it is left of $a$ , i.e., that it is in the subinterval $(u,a)$ .",
"The edge $(c_1,b)$ must be colored 3 because of the edges $(1,a), (2,a)$ .",
"Node $a$ cannot have any edge exiting the interval $(u,b)$ to a node other than 1,2 because of condition (4), the color-3 edge $(c_1,b)$ and the color-1 and -2 edges $(1,b), (2,b)$ .",
"Therefore, $c_2$ lies in the interval $(u,b)$ , and since it is not in $[a,b]$ , it is in $(u,a)$ .",
"Assume without of loss of generality that $c_1$ is closer to $u$ than $c_2$ ; see Figure REF .",
"Figure: Centers outside (a,b)(a,b).The edge $(c_2,b)$ is colored 3, because of the edges $(1,a), (2,a)$ .",
"This implies that the edge $(c_1,a)$ must be colored 1 because of the color-2 edge $(2,c_2)$ .",
"Consider the possible position of node $d_{1a}$ .",
"Node $a$ cannot exit the interval $(c_1,b)$ because of the color-1 path $(c_1,1,b)$ , the color-2 path $(c_2,2,b)$ , and the color-3 edge $(c_1,b)$ .",
"On the other hand, node 1 cannot reach the interval $(c_1,a)$ because of the color-1 edge $(c_1,a)$ .",
"Furthermore, $c_1$ cannot reach the interval $(a,b)$ because of the color-1 path $(a,1,b)$ , the color-2 path $(a,2,b)$ and the color-3 edge $(c_2,b)$ (see Figure REF ).",
"Therefore, there is no legal position for node $d_{1a}$ , a contradiction.",
"Let $Q_2$ be the graph shown in Figure REF where each internal face (triangle) is stellated twice more.",
"That is, inside each triangle (e.g.",
"$(1,a,d_{1a}), (1,d_{1a},c_1)$ etc.)",
"we insert a center node with edges to the nodes of the triangle, and repeat this once more for each resulting triangle; these additional nodes are not shown in the Figure so that it will not become too cluttered.",
"Note that $Q_2$ contains many copies of $Q_1$ .",
"For example there is one copy with outer terminals 1,2 and inner terminals $a, b$ .",
"Another copy of $Q_1$ has outer terminals $1, c_2$ and inner terminals $a, c_1$ ; another has outer terminals $d_{1a}, c_2$ and inner terminals $a, c_1$ .",
"And so forth.",
"We will use again $Q_2 +12$ to denote the graph $Q_2$ with the additional edge $(1,2)$ .",
"Figure: The graph Q 2 Q_2 (the triangles are stellated twice more)Lemma 3 Suppose that in a 3-page embedding of $Q_2+12$ , (1) the inner terminals $a, b$ lie in a subinterval $(u,v)$ between the outer terminals 1,2, (2) all edges from node 1 to the interval $(u,v)$ use the same color, say color 1, and all edges from node 2 to $(u,v)$ use color 2, and (3) all other edges exiting the interval $(u,v)$ (not connecting to nodes 1, 2) are colored 3.",
"Then one of the center nodes $c_1, c_2$ is inside the interval $(a,b)$ and the other one is outside $(a,b)$ but inside $(u,v)$ .",
"Furthermore, the embedding is as shown in Figure REF (up to reversing the order, switching $a,b$ and/or switching the indices $1,2$ ).",
"That is, if the center node $c_1$ is outside $(a,b)$ and it is in the interval $(u,a)$ , then $d_{1b}$ is in the interval $(b,v)$ and the edges $(c_1,c_2), (c_1,b), (c_1,d_{1b}), (a,c_2)$ all have color 3.",
"Figure: Centers in and out.By Lemmas REF , REF , one of the centers $c_1, c_2$ must lie outside the interval $(a,b)$ and one inside.",
"Assume without loss of generality that $c_1$ is outside $(a,b)$ and $c_2$ is inside.",
"The edge $(c_1,c_2)$ must have color 3 because of the color-1 path $(a,1,b)$ and the color-2 path $(a,2,b)$ .",
"We will show that the embedding conforms to the more detailed Figure REF (or a symmetric one obtained by reversing the order, switching $a,b$ and/or switching $1,2$ ).",
"We will do this in several steps.",
"First we will show that $c_1$ is in the interval $(u,v)$ .",
"Second we will show that $e_b$ and $d_{2b}$ have the indicated positions.",
"Third, we will identify the positions of $e_a$ and $d_{1a}$ .",
"Finally, we will identify the position of $d_{1b}$ to complete the proof.",
"Figure: Centers in and out: Detailed embedding.Claim 1 Node $c_1$ is in the interval $(u,v)$ .",
"Suppose that $c_1$ is outside $(u,v)$ .",
"Then both edges $(c_1,a), (c_1,b)$ have color 3 by condition (3).",
"Since $a$ and $b$ are connected by paths $(a,1,b)$ , $(a,2,b)$ , $(a,c_1,b)$ of all three colors, no edge can exit the interval $(a,b)$ to any node other than 1, 2, $c_1$ .",
"Since $c_2$ is inside the interval $(a,b)$ , it follows that all the nodes of $Q_2$ in the interior of the graph bounded by the cycle $(a,c_1, b, 2)$ (which is connected) must be inside the interval $(a,b)$ .",
"Note that the subgraph bounded by the cycle $(a, c_1, c_2, 2)$ contains $Q_1$ with $c_1, 2$ as the outer terminals and $a, c_2$ as the inner terminals, which are adjacent.",
"The inner terminals $a, c_2$ lie in the same arc $(c_1, 2)$ (in a cyclic ordering), all edges from $c_1$ to the interval $[a,c_2]$ must be colored 3 and all edges from 2 to $[a,c_2]$ must be colored 2.",
"By Lemma REF , at least one of the two centers $e_a, d_{2a}$ must be outside the interval $[a,c_2]$ .",
"Hence it must be in the interval $(c_2,b)$ , and the edge connecting it to node $a$ must be colored 1 because it conflicts with the edges $(c_1,c_2)$ and $(2,c_2)$ .",
"Similarly, the subgraph of $Q_2$ bounded by the cycle $(b,c_1, c_2, 2)$ contains $Q_1$ with $c_1, 2$ as the outer terminals and $b, c_2$ as the inner terminals, which are adjacent.",
"By a symmetric argument, at least one of the two centers $e_b, d_{2b}$ must be in the interval $(a,c_2)$ , and the edge that connects it to $b$ is also colored 1.",
"Thus, there are two color-1 edges that intersect, a contradiction.",
"We conclude that $c_1$ is in the interval $(u,v)$ .",
"Since $c_1$ is outside the interval $(a,b)$ (see the beginning of the proof) and is inside the interval $(u,v)$ (by Claim 1), it follows that $c_1$ is either in the interval $(u,a)$ or in $(b,v)$ .",
"Assume without loss of generality that $c_1$ is in the interval $(u,a)$ as in Fig.",
"REF .",
"Claim 2 Node $e_b$ is in the interval $(c_2,b)$ and node $d_{2b}$ is in the interval $(a,c_2)$ .",
"Edge $(b,d_{2b})$ has color 1 and edge $(a,c_2)$ has color 3.",
"Nodes $e_b$ , $d_{2b}$ are both adjacent to nodes $c_2$ and $b$ .",
"Node $c_2$ cannot reach outside the interval $(c_1,b)$ because of the color-1 path $(a,1,b)$ , color-2 path $(a,2,b)$ and the color-3 edge $(c_1,b)$ .",
"Node $b$ cannot reach the interval $(c_1,a)$ because of the edges $(1,a), (2,b)$ and $(c_1,c_2)$ .",
"Hence $e_b$ and $d_{2b}$ must be in the interval $(a,b)$ .",
"Suppose that $e_b$ is in $(a,c_2)$ , to derive a contradiction.",
"Then the edge $(e_b,b)$ must have color 1 and $(c_1,e_b)$ color 3, hence the edge $(a,c_2)$ must have color 2.",
"Then there is no legal position for node $d_{2a}$ : $c_2$ cannot reach outside the interval $(a,b)$ (because of the color-1 path $(a,1,b)$ , the color-2 path $(a,2,b)$ and the color-3 path $(e_b,c_1,b)$ ), node 2 cannot reach the interval $(a,c_2)$ with color 2, and node $a$ cannot reach the interval $(c_2,b)$ (because of the color-1 edge $(e_b,b)$ , the color-2 edge $(c_2,2)$ and the color-3 edge $(c_1,e_b)$ ).",
"We conclude that $e_b$ is in interval $(c_2,b)$ .",
"The subgraph of $Q_2$ bounded by the cycle $(c_1, c_2, 2, b)$ contains a copy of $Q_1$ with $c_1, 2$ as the outer terminals and $c_2, b$ as the inner terminals.",
"The inner terminals $c_2, b$ are adjacent, they lie in the same ($c_1,2$ ) arc (in the cyclic order), node $c_1$ can reach the interval $[c_2,b]$ only with color 3 and node 2 can reach $[c_2,b]$ only with color 2.",
"The conditions of Lemma REF are satisfied, therefore the interval $(c_2,b)$ cannot contain both centers $e_b, d_{2b}$ .",
"Since $e_b$ is in the interval $(c_2,b)$ , the other center $d_{2b}$ is not, thus it is in the interval $(a,c_2)$ .",
"The edge $(b,d_{2b})$ must have color 1 (because of the conflicting edges $(2,c_2)$ and $(c_1,c_2)$ ), and the edge $(2,d_{2b})$ has color 2, therefore the edge $(a,c_2)$ must have color 3.",
"Claim 3 Node $e_a$ is in the interval $(c_1,a)$ .",
"Node $d_{1a}$ is in interval $(u,c_1)$ .",
"Edge $(a,d_{1a})$ has color 2.",
"Node $e_a$ is adjacent to nodes $a, c_1, c_2$ .",
"Node $c_2$ can only reach the interval $(c_1,b)$ .",
"Node $a$ cannot reach the subinterval $(c_2,b)$ (because of the color-1 edge $(d_{2b},b)$ , the color-2 edge $(c_2,2)$ and the color-3 edge $(c_1,c_2)$ ) and node $c_1$ cannot reach the subinterval $(a,c_2)$ (because of the edges $(a,1), (a,2), (a,c_2)$ ).",
"Therefore, $e_a$ is in the interval $(c_1,a)$ .",
"The subgraph of $Q_2$ bounded by the cycle $(1, a, c_2, c_1)$ contains a copy of $Q_1$ with outer terminals $1, c_2$ , and inner terminals $a, c_1$ .",
"The inner terminals are adjacent and are in the same $(1,c_2)$ arc.",
"Furthermore, all edges from 1 to the interval $[c_1,a]$ are colored 1, and all edges from $c_2$ to $[c_1,a]$ must be colored 3.",
"The conditions of Lemma REF are satisfied, therefore we cannot have both centers $e_a, d_{1a}$ in the interval $[c_1,a]$ .",
"Since $e_a$ is in the interval, it follows that $d_{1a}$ must be outside the interval $[c_1,a]$ .",
"Node $a$ cannot reach outside the interval $(u,c_2)$ (because of the color-1 path $(c_1, 1, b, d_{2b})$ , the color-2 edge $(c_2,2)$ and the the color-3 edge $(c_1,c_2)$ ), and node $c_1$ cannot reach the interval $(a,c_2)$ .",
"Therefore, $d_{1a}$ must be in the interval $(u,c_1)$ .",
"It follows that the edge $(a,d_{1a})$ must have color 2, since it conflicts with edges $(1,c_1)$ and $(c_1,c_2)$ .",
"Claim 4 Node $d_{1b}$ is in interval $(b,v)$ .",
"Edge $(c_1,d_{1b})$ is colored 3.",
"Node $d_{1b}$ is adjacent to $b, 1, c_1$ .",
"Node $b$ cannot reach the interval $(c_1,a)$ , node $c_1$ cannot reach the interval $(a,c_2)$ and 1 cannot reach $(c_2,b)$ (because of the edge $(d_{2b},b)$ ).",
"Therefore $d_{1b}$ lies outside the interval $(c_1,b)$ .",
"First, we claim that it must be in the interval $(d_{1a},v)$ .",
"Suppose to the contrary that it lies outside the interval $(d_{1a},v)$ (possibly even outside the interval $(u,v)$ ).",
"Then it is easy to see that both edges $(d_{1b},c_1)$ and $(d_{1b},b)$ must have color 3: If $d_{1b}$ is outside the interval $(u,v)$ then this holds because of condition (3), and if $d_{1b}$ is in $(u,d_{1a})$ this holds because of the color-1 edge $(1,d_{1a})$ and the color-2 edges $(d_{1a},a)$ and $(2,a)$ .",
"Then there is no legal position to place the center $f$ of the triangle $(b,c_1,d_{1b})$ : If $f$ is outside $(u,v)$ then $(f,b), (f,c_1)$ must both have color 3 and one of them intersects one of $(d_{1b},c_1)$ , $(d_{1b},b)$ .",
"Similarly, if $f$ is in $(u,c_1)$ , the edge $(f,b)$ must have color 3 and intersects $(d_{1b},c_1)$ .",
"If $f$ is in the interval $(c_1,b)$ then $(d_{1b},f)$ must be colored 3 (by condition (3) or because of the edges $(1,d_{1a}), (d_{1a},a), (2,a)$ ) and it intersects the edge $(c_1,b)$ .",
"If $f$ is in the interval $(b,v)$ then $(c_1,f)$ must have color 3 (because of the edges $(1,a), (2,a)$ ) and intersects the edge $(b,d_{1b})$ .",
"We conclude that $d_{1b}$ is in the interval $(d_{1a},v)$ .",
"Hence it is either in $(d_{1a},c_1)$ or in $(b,v)$ .",
"Suppose that $d_{1b}$ is in the interval $(d_{1a},c_1)$ , to derive a contradiction.",
"The edge $(d_{1b},1)$ is colored 1 and the edge $(d_{1b},b)$ must be colored 3, hence the edge $(d_{1a},c_1)$ must be colored 2.",
"Consider the subgraph of $Q_2$ bounded by the cycle $(a, c_2, c_1, d_{1a})$ and recall that all the triangles in Figure REF are stellated twice.",
"The subgraph contains a copy of $Q_1$ with $c_2, d_{1a}$ as the outer terminals, $a, c_1$ as the inner terminals, which are adjacent, and are embedded inside the interval $(d_{1a},c_2)$ .",
"Node $d_{1a}$ can reach the interval $[a,c_1]$ only with color 2 (because of the color-1 edge $(d_{1b},1)$ and the color-3 edge $(d_{1b},b)$ ), and $c_2$ can reach $[a,c_1]$ only with color 3.",
"The center $e_a$ of the triangle $(c_2, c_1, a)$ is inside the interval $(a,c_1)$ .",
"Hence, by Lemma REF , the center $g$ of the other triangle $(d_{1a}, c_1, a)$ cannot be in the interval $(c_1,a)$ .",
"This leaves no possible position for the center $g$ of $(d_{1a}, c_1, a)$ : Node $c_1$ cannot reach outside $(u,v)$ or left of $d_{1a}$ (because of the edges $(d_{1b},1), (d_{1a},a), (d_{1b},b)$ and condition 3), node $a$ cannot reach inside the interval $(d_{1a},c_1)$ or $(b,v)$ , and $d_{1a}$ cannot reach the interval $(a,b)$ .",
"We conclude that $d_{1b}$ is not in the interval $(d_{1a},c_1)$ , hence it must be in the interval $(b,v)$ .",
"The edge $(c_1,d_{1b})$ must have color 3 because it conflicts with the edges $(a,1), (a,2)$ .",
"This completes the proof of the lemma.",
"Let $Q$ be the graph formed by taking 15 copies of $Q_2$ , identifying their outer terminals 1,2, and identifying terminal $b$ of the $i$ -th copy with terminal $a$ of the $(i+1)$ -th copy; i.e., $Q$ is formed by glueing together back-to-back 15 copies of $Q_2$ with the same outer terminals, see Figure REF .",
"We call $Q$ a quad, nodes 1,2 the outer terminals of quad $Q$ and call the inner terminals $a_1, a_2, \\ldots , a_{16}$ of the copies of $Q_2$ the inner terminals of $Q$ .",
"We use $Q_2^i$ to denote the $i$ th copy of $Q_2$ .",
"Let $Q+12$ denote the graph consisting of $Q$ and the edge $(1,2)$ .",
"Figure: The quad QQ.Lemma 4 There is no embedding of $Q+12$ in three pages such that all the inner terminals are embedded in a subinterval of the interval $(1,2)$ , and all edges from nodes 1 and 2 to this subinterval use only two (the same two) colors (i.e., one of the three colors is not used by any edge connecting nodes 1 and 2 to this subinterval).",
"Consider a book embedding of the nodes of the quad $Q$ with all the inner terminals embedded in a subinterval of the interval $(1,2)$ and edges from 1, 2 to this subinterval using two colors.",
"Let $u^{\\prime }, u$ be the inner terminals closest to 1 and $v^{\\prime }, v$ the inner terminals closest to 2; thus, if we assume wlog that the embedding on the line starts with terminal 1, then $u^{\\prime },u$ are the first two inner terminals and $v, v^{\\prime }$ the last two.",
"The order of these nodes is $1, u^{\\prime }, u, v, v^{\\prime }, 2$ .",
"Let 1 be the color of the edge $(1,v^{\\prime })$ and let 2 be the color of the edge $(2,u^{\\prime })$ .",
"By the hypothesis, all edges from nodes 1 and 2 to the interval $(u^{\\prime },v^{\\prime })$ are colored 1 or 2.",
"Since the edge $(2,u^{\\prime })$ is colored 2, all edges from node 1 to the interval $[u,v]$ (including nodes $u, v$ ) must be colored 1.",
"Similarly, since the edge $(1,v^{\\prime })$ is colored 1, all edges from node 2 to the interval $[u,v]$ (including $u$ and $v$ ) must be colored 2.",
"Because of the color-1 path $(u,1,v)$ and the color-2 path $(u,2,v)$ , all other edges that exit the interval $(u,v)$ (and are not going to 1,2) must be colored 3.",
"Thus, the conditions of Lemma REF are satisfied for every copy of $Q_2$ whose inner terminals are in the interval $(u,v)$ .",
"Since there are 16 inner terminals, there are at least three consecutive terminals $a_i, a_{i+1}, a_{i+2}$ that are not in the set $\\lbrace u,u^{\\prime },v,v^{\\prime } \\rbrace $ , i.e.",
"that are embedded in the interval $(u,v)$ .",
"Thus, there are two consecutive copies of $Q_2$ , for which Lemma REF holds.",
"Consider any copy of $Q_2$ with inner terminals in the interval $(u,v)$ .",
"Observe from the conclusion of Lemma REF (see Fig.",
"REF ) that the inner terminals $a,b$ are connected by paths of all 3 colors, $(a,1,b), (a,2,b), (a, c_2, c_1, b)$ , and thus no edge can exit the interval $(a,b)$ unless it connects to $1, 2$ , $c_1$ or $c_2$ .",
"Second, observe that there are nodes of $Q_2$ outside the interval $(a,b)$ on both sides, e.g.",
"the nodes $c_1, d_{1b}$ .",
"Applying these observations to the $i$ -th and $(i+1)$ -th copy of $Q_2$ , tells us that $a_i$ cannot lie inside the interval $(a_{i+1}, a_{i+2})$ , and similarly $a_{i+2}$ cannot lie inside the interval $(a_i, a_{i+1})$ .",
"For, suppose to the contrary that $a_i$ is in $(a_{i+1}, a_{i+2})$ .",
"Since no edge of the $i$ -th copy $Q_2^i$ can exit the interval $(a_{i+1}, a_{i+2})$ , all nodes of $Q_2^i$ must lie inside the interval $(a_{i+1}, a_{i+2})$ , contradicting the fact that some nodes must lie on both sides outside $(a_i,a_{i+1})$ .",
"Therefore, $a_i$ is outside the interval $(a_{i+1}, a_{i+2})$ .",
"Similarly $a_{i+2}$ is outside the interval $(a_i, a_{i+1})$ .",
"Thus, we may assume without loss of generality that the nodes $a_i, a_{i+1}, a_{i+2}$ appear in this order.",
"Observe from Lemma REF for each of copy of $Q_2$ that there is a color-3 edge $e$ (edge $(c_1,d_{1b})$ in Fig.",
"REF ) that connects two nodes of $Q_2$ that lie outside, and on both sides, of the interval $(a,b)$ and that furthermore, one of these two nodes of the edge $e$ (node $c_1$ in Fig.",
"REF ) has a color-3 edge to an inner terminal ($b$ in Fig.",
"REF ).",
"Let $e^i$ be the edge $e$ for $Q_2^i$ , and $ e^{i+1}$ for $Q_2^{i+1}$ .",
"The left endpoint of $e^i$ is left of $a_i$ and the right endpoint is right of $a_{i+1}$ , and since it cannot be in the interval $(a_{i+1}, a_{i+2})$ (no edge that does not belong to $Q_2^{i+1}$ can exit this interval unless it goes to 1 or 2), it must be right of $a_{i+2}$ .",
"Similarly, the left endpoint of $e^{i+1}$ must be left of $a_i$ and the right endpoint right of $a_{i+2}$ .",
"Since the edges $e^i, e^{i+1}$ both have color 3, they are nested.",
"Suppose without loss of generality that $e^i$ is nested inside $e^{i+1}$ .",
"Then neither endpoint of $e^{i+1}$ can have a color-3 edge to $a_{i+1}$ or $a_{i+2}$ , because it would conflict with $e^i$ .",
"This contradicts Lemma REF .",
"We remark that the properties of Lemmas REF and REF depend crucially on the fact that in $Q_2$ we included the edge $(c_1,c_2)$ between the two centers rather than the edge $(a,b)$ between the two inner terminals.",
"It can be shown that with the edge $(a,b)$ instead, even after stellating the faces an arbitrary number of times, and gluing back-to-back an arbitrary number of copies of the resulting graph, yields a graph that does not have the property of Lemma REF : the graph can be embedded between the two outer terminals so that all edges to each outer terminal have the same color.",
"Attaching quads to the edges of a graph restricts the possible embeddings into 3 pages.",
"The following lemma illustrates how $Q$ can be used.",
"The lemma will be used often in the sequel.",
"Lemma 5 Consider a graph $H$ formed by taking a triangle $(A,B,C)$ on the plane, attaching (adding) a quad to each edge of the triangle, by identifying the outer terminals of the quad with the nodes of the edge.",
"There is no embedding of $H$ in three pages such that all inner terminals of the quads are embedded in the arc $(B,C)$ that does not contain $A$ and all edges from $A$ to the inner terminals have the same color.",
"Suppose that there is such a 3-page embedding, consider its linearization with $A$ lying outside the interval $(B,C)$ .",
"Assume first that among all the inner terminals of the quads, the one embedded closest to $B$ does not belong to the $AB$ quad, i.e., it belongs to the $AC$ or the $BC$ quad.",
"Let $u$ be this terminal and assume wlog that the edge $(u,C)$ has color 1.",
"All the inner terminals of the $AB$ quad are in the interval $(u,C)$ ; this interval can be reached from $A$ and $B$ only with colors 2 and 3.",
"By Lemma REF this is impossible.",
"Therefore, the inner terminal $u$ closest to $B$ belongs to the $AB$ quad.",
"By a symmetric argument, the inner terminal $v$ closest to $C$ belongs to the $AC$ quad.",
"The edges $(A,u)$ and $(A,v)$ have the same color, say color 1.",
"Then all inner terminals of the $BC$ quad are in the interval $(u,v)$ and $B$ and $C$ can reach this interval only with colors 2 and 3.",
"This is impossible by Lemma REF ."
],
[
"The Graph",
"Our `hard' graph $G$ is constructed as follows.",
"Take a long path $p=(x_1, x_2, \\ldots , x_n)$ , of $n$ nodes, where $n$ is sufficiently large, say $n=1000$ .",
"Take two other nodes 1, 2 and connect them to all the nodes $x_i$ of the path, as in Fig.",
"REF .",
"This forms 2$(n-1)$ triangles, which we call the big triangles.",
"Subdivide each big triangle to three small triangles by inserting a center node and connecting it to the 3 nodes of the triangle.",
"Inside each small triangle, attach a copy of the quad $Q$ to each edge of the triangle, and add a central node connecting it to the three nodes of the triangle and the innermost inner terminals of the three quads.",
"The construction is shown in Figure REF (except for the quads attached to the edges, which we omitted for clarity).",
"Note that all small triangles as well as all big triangles satisfy the conditions of Lemma REF .",
"Note also the copies of $Q_1$ with outer terminals 1,2 and inner terminals $x_i, x_{i+1}$ .",
"We call the nodes 1, 2 the terminals of $G$ , we call the $n$ nodes $x_1, \\ldots , x_n$ the vertical nodes of $G$ , and call the edges $(x_i,x_{i+1})$ the vertical edges.",
"Figure: The Graph GG.",
"(There are quads attached to the edges.",
")We will show that $G$ cannot be embedded in three pages.",
"For this purpose, fix any 3-page embedding of $G$ .",
"We will show that the embedding has to satisfy a sequence of properties, and derive eventually a contradiction.",
"Given a 3-page circle embedding of $G$ , we designate one of the two arcs between the two terminals 1,2 as the major (1,2) arc, and the other as the minor (1,2) arc as follows: the major arc is an arc (1,2) that contains at least half of the vertical nodes, and the other (1,2) arc is the minor arc (if both (1,2) arcs contain exactly half of the vertical nodes, we arbitrarily designate one as the major and the other as the minor arc).",
"Let $z_1$ be the node in the major arc that is closest to node 1 and adjacent to 2, and let $z_2$ be the node in the major arc that is closest to node 2 and adjacent to 1 - see Fig.",
"REF .",
"We show first that there are not many vertical edges $(x_i,x_{i+1})$ with nodes on both (1,2) arcs.",
"Figure: Crossing vertical edges.Lemma 6 There are at most 4 vertical nodes $x_i$ in the arc $(z_1,z_2)$ , whose successor $x_{i+1}$ on the path $p$ is in the minor arc (1,2).",
"By contradiction.",
"Suppose that there are 5 such vertical nodes $x_i$ in $(z_1,z_2)$ whose successor $x_{i+1}$ is in the minor arc (1,2).",
"Denote these 5 nodes as $a_1, \\ldots , a_5$ in the order that they appear in $(z_1,z_2)$ , and let their successors be $b_1, \\ldots ,b_5$ respectively.",
"Assume wlog that edge $(1,z_2)$ has color 1 and edge $(2,z_1)$ has color 2 (they intersect so they must have different colors).",
"The 5 vertical edges $(a_1,b_1), \\ldots , (a_5,b_5)$ intersect both edges $(1,z_2), (2,z_1)$ , so they must all have color 3, hence they do not intersect each other; see Fig.",
"REF .",
"The edges $(1,a_2), \\ldots (1,a_5)$ must have color 1 because they intersect $(a_1,b_1)$ and $(2,z_1)$ .",
"Similarly the edges $(2,a_1), \\ldots (2,a_4)$ must have color 2 because they intersect $(a_5,b_5)$ and $(1,z_2)$ .",
"Note that a node inside the arc $(a_2 , a_3)$ can reach nodes other than nodes 1,2 only in the same arc or the arc $[b_2,b_3]$ , because of the color-1 path $(a_2,1,a_3)$ , the color-2 path $(a_2,2,a_3)$ and the color-3 edges $(a_2,b_2), (a_3,b_3)$ .",
"Similarly, a node inside the arc $(a_3,a_4)$ can reach nodes other than nodes 1,2 only in the same arc or the arc $[b_3,b_4]$ .",
"Node $a_3$ can only reach nodes in the arcs $[a_2,a_3,a_4]$ and $[b_2,b_3,b_4]$ .",
"Similar observations hold for the edges connecting nodes 1, 2 to the vertical nodes $b_1, \\ldots , b_5$ on the minor arc (1,2).",
"Edges $(1,b_5), (2,b_1)$ , must have color 1 or 2 because they intersect $(a_2,b_2)$ .",
"If $(1,b_5)$ has color 1 then $(2,b_1)$ must have color 2, hence all edges $(1,b_2), \\ldots (1,b_5)$ must have color 1 and all the edges $(2,b_1), \\ldots (2,b_4)$ must have color 2.",
"If $(1,b_5)$ has color 2 then $(2,b_1)$ must have color 1, all edges $(1,b_2), \\ldots (1,b_5)$ have color 2 and the edges $(2,b_1), \\ldots (2,b_4)$ have color 1.",
"(Figure REF depicts the latter case.)",
"In either case, note again that a node in the arc $(b_2,b_3)$ can reach nodes other than 1,2 only in the same arc or the arc $[a_2,a_3]$ .",
"Similarly, a node inside the arc $(b_3,b_4)$ can reach nodes other than 1, 2 only in the same arc or the arc $[a_3,a_4]$ .",
"Node $b_3$ can only reach nodes in the arcs $[b_2,b_3,b_4]$ and $[a_2,a_3,a_4]$ .",
"Consider the big triangle $(1,a_3,b_3)$ of $G$ .",
"From the observations in the previous two paragraphs it follows that either all the internal nodes of this triangle are in the strip $(a_2,a_3) \\cup (b_2,b_3)$ or they are all in the strip $(a_3,a_4) \\cup (b_3,b_4)$ .",
"Assume without loss of generality that they are in the strip $(a_2,a_3) \\cup (b_2,b_3)$ (the argument is the same in the other case).",
"Any edge from $a_3$ to the arc $(b_2,b_3)$ must be colored 3 and likewise any edge from $b_3$ to the arc $(a_2,a_3)$ must be colored 3, therefore there cannot exist both kinds of edges.",
"Hence, either all the inner terminals of the quads attached to edges $(1,a_3), (a_3,b_3)$ are in arc $(a_2,a_3)$ or all inner terminals of the quads attached to the edges $(1,b_3), (a_3,b_3)$ are in arc $(b_2,b_3)$ .",
"Assume wlog that the former holds, i.e.",
"all the inner terminals of the quads attached to edges $(1,a_3), (a_3,b_3)$ are in arc $(a_2,a_3)$ .",
"If the inner terminal closest to $a_3$ belongs to the $(1,a_3)$ quad, then the edge connecting it to 1 must be colored 1, hence all edges from the inner terminals of the $(a_3,b_3)$ quad to $a_3$ must be colored 2 or 3 and all edges from these inner terminals to $b_3$ must be colored 3, contradicting Lemma REF .",
"On the other hand, if the inner terminal closest to $a_3$ belongs to the $(a_3,b_3)$ quad, then the edge to $b_3$ is colored 3, hence all edges from the inner terminals of the $(1,a_3)$ quad to $a_3$ must be colored 1 or 2 and all edges from these inner terminals to 1 must be colored 1, contradicting again Lemma REF .",
"The lemma follows.",
"We view the 3-page embedding as an embedding on the line, starting with one of the terminals, followed by the major (1,2) arc (which is now an interval on the line), then the other terminal, followed by the minor arc (1,2).",
"We will focus on the interval $(1,2)$ corresponding to the major arc.",
"Let $z_1$ (resp.",
"$z_2$ ) be again the node in the interval $(1,2)$ that is closest to node 1 (resp.",
"2) and is adjacent to 2 (resp.",
"1).",
"Define the stretch between two points $u, v$ , of the linear embedding, denoted $str(u,v)$ , to be the number of vertical nodes in the interval $(u,v)$ .",
"Lemma 7 (1) If $u, v$ are two nodes in the interval $(1,2)$ connected by an edge then $str(u,v) \\le 15$ .",
"(2) If $u, v$ are two nodes in the interval $(z_1,z_2)$ that have incident edges that exit the interval $[1,2]$ then $str(u,v) \\le 15$ .",
"The proof is essentially the same for both parts.",
"Suppose that $u,v$ are two nodes as in part (1) or (2) of the lemma such that $str(u,v) > 15$ .",
"We will derive a contradiction.",
"If there is an edge $(u,v)$ , say of color 3, then all edges from the terminals 1, 2 to all the nodes in the interval $(u,v)$ must use the other two colors 1, 2.",
"On the other hand, if $u , v$ are in the interval $(z_1,z_2)$ and there are two edges incident to them that exit the interval $[1,2]$ , then both edges intersect the edges $(1,z_2)$ and $(2,z_1)$ , which themselves intersect each other.",
"Therefore, the two exiting edges incident to $u$ and $v$ must have the same color, say color 3, and thus again all edges from the terminals 1, 2 to all the nodes in the interval $(u,v)$ must use the other two colors 1, 2.",
"Let $y_1, y^{\\prime }_1$ (respectively, $y_2, y^{\\prime }_2$ ) be the vertical nodes in the interval $(u,v)$ that are closest to 1 (resp.",
"to 2).",
"So, if we assume wlog that $u$ is closer to 1 (than $v$ is) and $v$ is closer to 2, then the order of these nodes is $1, u, y_1, y^{\\prime }_1, y^{\\prime }_2, y_2, v, 2$ (or the reverse).",
"The other vertical nodes in the interval $(u,v)$ lie between $y^{\\prime }_1$ and $y^{\\prime }_2$ .",
"Assume without loss of generality that $(1,y_2)$ has color 1 and $(2,y_1)$ has color 2.",
"Then all edges from 1 to the vertical nodes in the interval $(u,v)$ , except possibly $y_1$ have color 1, and all edges from 2 to the vertical nodes in $(u,v)$ , except possibly $y_2$ , have color 2.",
"Let $x_i$ be any vertical node in the interval $(u,v)$ such that $x_i,\\notin \\lbrace y_1, y^{\\prime }_1, y_2, y^{\\prime }_2, x_1, x_{n}\\rbrace $ , $x_{i+1} \\notin \\lbrace y_1, y^{\\prime }_1, y_2, y^{\\prime }_2, x_{n}\\rbrace $ and $x_{i+1}$ is in the interval $(1,2)$ (i.e., $x_{i+1}$ is not in the minor arc (1,2)).",
"By Lemma REF there are at most 4 vertical nodes $x_i$ in the interval $(z_1,z_2)$ whose successor $x_{i+1}$ is in the minor arc (1,2).",
"Since there are at least 16 vertical nodes in $(u,v)$ , we can choose $x_i$ to be a vertical node in $(u,v)$ that satisfies the above conditions.",
"Claim 5 Node $x_{i+1}$ is in $(u,v)$ and there is no other vertical node embedded between $x_i$ and $x_{i+1}$ .",
"Suppose that $x_{i+1}$ is not in $(u,v)$ .",
"Then the edge $(x_i,x_{i+1})$ must have color 3 because it conflicts with the color-1 path $(y^{\\prime }_1,1,y^{\\prime }_2)$ and the color-2 path $(y^{\\prime }_1,2,y^{\\prime }_2)$ .",
"In case (1) of the statement of the lemma, where there is an edge $(u,v)$ of color 3, we get a contradiction.",
"In case (2) where there are color-3 edges incident to $u$ and $v$ that exit the interval $(1,2)$ , there is again a contradiction because $x_{i+1}$ lies in the interval $(1,2)$ , and therefore $(x_i,x_{i+1})$ intersects one of the exiting edges incident to $u, v$ .",
"In either case we conclude that $x_{i+1}$ is in $(u,v)$ .",
"Figure: NO_CAPTIONSince neither $x_i$ nor $x_{i+1}$ is $y_1$ or $y_2$ , the edges $(1,x_i), (1,x_{i+1})$ are colored 1 and the edges $(2,x_i), (2,x_{i+1})$ are colored 2.",
"Suppose that there is another vertical node $x_j$ , in the interval between $x_i$ and $x_{i+1}$ (see Fig.",
"REF ).",
"Then the edge $(1,x_j)$ has color 1, and the edge $(2,x_j)$ has color 2.",
"Therefore, the edge $(x_i,x_{i+1})$ must have color 3.",
"No edge can exit the interval $(x_i,x_{i+1})$ , other than edges to 1, 2, because of the color-1 path $(x_i, 1, x_{i+1})$ , the color-2 path $(x_i, 2, x_{i+1})$ and the color-3 edge $(x_i,x_{i+1})$ .",
"Since $x_j$ is in this interval, then all vertical nodes of the path $p$ between $x_j$ and $x_i$ or $x_{i+1}$ must be in this interval.",
"That is, if $j<i$ , then $x_{j+1}, \\ldots , x_{i-1}$ must be in the interval $(x_i,x_{i+1})$ ; if $j>i+1$ , then $x_{i+2}, \\ldots , x_j$ are in the interval $(x_i,x_{i+1})$ .",
"Suppose without loss of generality that $j<i$ , hence $x_{i-1}$ is in $(x_i,x_{i+1})$ .",
"Consider the triangles $(1,x_{i-1},x_{i})$ , $(2,x_{i-1},x_{i})$ .",
"Their interior nodes must all be in the interval $(x_i,x_{i+1})$ because $x_{i-1}$ is in this interval (refer to Fig.",
"REF with $x_{i-1}$ in place of $x_j$ ).",
"If all the inner terminals of the quad attached to the edge $(1,x_i)$ are in the interval $(x_{i-1}, x_{i+1})$ , then all edges connecting them to 1 must have color 1 and all edges to $x_i$ must have color 3, contradicting Lemma REF .",
"Similarly, if all the inner terminals of the quad attached to the edge $(2,x_i)$ are in the interval $(x_{i-1}, x_{i+1})$ , then all edges connecting them to 2 must have color 2 and all edges to $x_i$ must have color 3, contradicting again Lemma REF .",
"It follows that at least one inner terminal from each of the two quads attached to edges $(1,x_i), (2,x_i)$ must be in the interval $(x_i, x_{i-1})$ ; the edges connecting these inner terminals to 1 and 2 respectively are colored 1, 2.",
"Hence edge $(x_i, x_{i-1})$ is colored 3.",
"Because of the color-1 and -2 paths $(x_i,1,x_{i-1})$ and $(x_i,2,x_{i-1})$ , no edge to a node other than 1, 2, can exit the interval $(x_i, x_{i-1})$ , hence all the interior of the triangle $(1,x_{i-1},x_{i})$ (as well as $(2,x_{i-1},x_{i})$ ) must lie in this interval.",
"All the edges from node 1 to the inner terminals must use color 1, so this contradicts Lemma REF .",
"Assume without loss of generality that $x_{i+1}$ is embedded right of $x_i$ .",
"Let $y$ be the vertical node left of $x_i$ and $z$ the vertical node right of $x_{i+1}$ .",
"Since $x_i, x_{i+1}$ are in $(u,v)$ and are not among $\\lbrace y_1, y^{\\prime }_1, y_2, y^{\\prime }_2\\rbrace $ , the nodes $y, z$ exist, and the edges from $y, x_i, x_{i+1}, z$ to 1 are colored 1 and the edges to 2 are colored 2.",
"Claim 6 If a node adjacent to both $x_i, x_{i+1}$ (other than 1, 2) is outside the interval $(x_i, x_{i+1})$ then it must be either in the interval $(y,x_i)$ or in the interval $(x_{i+1},z)$ .",
"In the former case $y=x_{i-1}$ and in the latter case $z=x_{i+2}$ .",
"Let $w$ be a node adjacent to both $x_i, x_{i+1}$ (other than 1, 2) that is outside the interval $(x_i, x_{i+1})$ .",
"Note that $w \\ne z$ , since $w$ is adjacent to both $x_i, x_{i+1}$ , hence it is not a vertical node, and $z$ is a vertical node.",
"Assume without loss of generality that $w$ is to the right of the interval $(x_i, x_{i+1})$ .",
"(The argument is symmetric if $w$ is to the left of the interval.)",
"If $w$ is right of $z$ , then both edges $(z,x_i), (z,x_{i+1})$ have color 3 (because they intersect the color-1 and -2 paths $(y,1,z), (y,2,z)$ ), thus there is a color-3 path between $x_i$ and $x_{i+1}$ .",
"There are also color-1 and -2 paths $(x_i,1,x_{i+1})$ , $(x_i,2,x_{i+1})$ , hence no edge can exit the interval $(x_i, x_{i+1})$ to a node other than 1, 2, $w$ , and any other node adjacent to both $x_i, x_{i+1}$ must be inside $(x_i, x_{i+1})$ .",
"This implies in particular that the center of either the triangle $(1,x_i,x_{i+1})$ or the triangle $(2,x_i,x_{i+1})$ is in the interval $(x_i, x_{i+1})$ , and hence all the nodes of the triangle are in this interval.",
"This contradicts Lemma REF because the interval is reachable from node 1 only with color 1 (and from node 2 only with color 2).",
"We conclude that $w$ is in the interval $(x_{i+1},z)$ - see Fig.",
"REF .",
"The edge $(x_i,w)$ has color 3 because it intersects $(1,x_{i+1}), (2,x_{i+1})$ .",
"Node $x_{i+1}$ cannot reach any node other than 1, 2 left of $x_i$ nor right of $z$ because of the color-1 and -2 paths $(x_i,1,z), (x_i,2,z)$ and the color-3 edge $(x_i,w)$ .",
"Since $x_{i+1} \\ne x_n$ from our choice of $x_i$ , node $x_{i+1}$ has a successor $x_{i+2}$ on the path $p$ , and $x_{i+2}$ must lie in the interval $(x_i,z)$ .",
"By Claim REF , $x_{i+2}$ cannot be in the interval $(x_i,x_{i+1})$ , hence it must be $z$ .",
"Figure: NO_CAPTIONWe can finish now the proof of the lemma.",
"Applying Lemma REF to the copy of $Q_1$ with outer terminals $1, 2$ and inner terminals $x_i, x_{i+1}$ , we deduce that at least one of the centers of the triangles $(1,x_i,x_{i+1})$ , $(2,x_i,x_{i+1})$ must be outside the interval $(x_i, x_{i+1})$ .",
"Let $w$ be this center.",
"By Claim REF , $w$ is either in $(y,x_i)$ or in $(x_{i+1},z)$ .",
"Assume wlog that $w$ is in in $(x_{i+1},z)$ - see Fig.",
"11.",
"Then $z=x_{i+2}$ by Claim REF .",
"The edge $(x_i,w)$ has color 3 (because it intersects $(1,x_{i+1}), (2,x_{i+1})$ ), hence $x_{i+1}$ cannot reach outside the interval $(x_i,x_{i+2})$ and $x_{i+2}$ cannot reach inside $(x_i,x_{i+1})$ .",
"Therefore, all nodes (other than 1, 2) adjacent to both $x_{i+1},x_{i+2}$ , e.g.",
"the centers of the triangles $(1,x_{i+1},x_{i+2})$ and $(2,x_{i+1},x_{i+2})$ must be inside the interval $(x_{i+1},x_{i+2})$ .",
"This contradicts Lemma REF for the copy of $Q_1$ with outer terminals $1, 2$ and inner terminals $x_{i+1}, x_{i+2}$ .",
"We will concentrate on a region of the interval between the terminals that has the properties indicated in the following lemma.",
"Lemma 8 There is a subinterval $I$ of the interval $(1,2)$ such that (1) no edge from a node in $I$ exits the interval $(1,2)$ , (2) all edges from one terminal to $I$ use only one color, and all edges from the other terminal use the other two colors, and (3) $I$ contains at least 240 vertical nodes.",
"Let again $z_1$ (resp.",
"$z_2$ ) be the node in the interval $(1,2)$ that is closest to 1 (resp.",
"2) and is adjacent to 2 (resp.",
"1).",
"Assume wlog that the edge $(1,z_2)$ has color 1 and the edge $(2,z_1)$ has color 2.",
"All edges from 1 to nodes in the interval $(z_1,2)$ have color 1 or 3, and similarly all edges from 2 to nodes in $(1,z_2)$ have color 2 or 3.",
"Let $g_2$ be the node in the interval $(z_1,z_2)$ closest to 2 that has a color-3 edge to 1 if there is such a node (see Fig.",
"REF ); otherwise let $g_2=z_1$ .",
"Similarly, let $g_1$ be the node in the interval $(z_1,z_2)$ closest to 1 that has a color-3 edge to 2, if there is such a node, otherwise let $g_1=z_2$ .",
"From the definitions, all edges from 1 to the (open) interval $(g_2,2)$ have color 1, and all edges from 2 to the interval $(1,g_1)$ have color 2.",
"Clearly, $g_2$ must be left of (or equal to) $g_1$ because two color-3 edges cannot intersect.",
"If there is an edge incident to a node $u $ in the interval $(z_1,z_2)$ that exits the interval $(1,2)$ , it must have color 3 and thus $u$ must be between $g_2$ and $g_1$ .",
"If there are such exiting edges, then let $f_1$ be the leftmost node in $(z_1,z_2)$ that has such an edge and $f_2$ be the rightmost such node.",
"By Lemma REF , $str(f_1,f_2) \\le 15$ .",
"If there are no such edges that connect a node in $(z_1,z_2)$ to a node outside $(1,2)$ , then let $f_1=g_1$ , $f_2=g_2$ .",
"Figure: NO_CAPTIONThe intervals $(z_1,f_1)$ , $(f_2,z_2)$ have the following properties: (1) No edge from a node in these intervals exits $(1,2)$ .",
"(2) All edges from node 1 to $(f_2,z_2)$ have color 1, and all edges from node 2 to $(z_1,f_1)$ have color 2.",
"(3) The total number of vertical nodes in the two intervals is at least $\\frac{n}{2}-20 = 480$ .",
"The interval among $(z_1,f_1)$ , $(f_2,z_2)$ that has the maximum number of vertical nodes satisfies the conditions of the lemma.",
"We call the subinterval $I$ of Lemma REF the prime region.",
"Assume wlog for the remainder that all edges from the prime region $I$ to terminal 1 have color 1, and edges to terminal 2 have color 2 or 3; there are no edges from $I$ that exit the interval $(1,2)$ .",
"By Lemma REF , if a node in the prime region $I$ has stretch distance more than 15 from the endpoints of $I$ , then all its adjacent nodes (except 1,2) are also in the prime region.",
"Let $x_i, i \\ne n$ be any vertical node in the prime region that has stretch distance more than 50 from its endpoints.",
"Then all the nodes that are within distance 3 from $x_i$ in the graph $G \\setminus \\lbrace 1,2\\rbrace $ are also in the prime region.",
"This implies in particular that $x_{i+1}$ is also in the prime region, and so are all nodes adjacent to $x_i$ or $x_{i+1}$ in $G$ (except 1,2) and their adjacent nodes.",
"The edges from 1 to $x_i$ , $x_{i+1}$ are both colored 1.",
"The edges from 2 to $x_i$ , $x_{i+1}$ are colored 2 or 3; we show next that they have the same color.",
"Lemma 9 Let $x_i, i \\ne n$ be any vertical node in the prime region that has stretch distance more than 50 from its endpoints.",
"The edges $(2,x_i), (2,x_{i+1})$ have the same color.",
"Suppose that the edges $(2,x_i), (2,x_{i+1})$ have different colors, say 2, 3 respectively.",
"We will derive a contradiction.",
"Assume wlog that $x_i$ is left of $x_{i+1}$ .",
"We distinguish cases depending on the color of the edge $(x_i,x_{i+1})$ .",
"Case 1: $(x_i,x_{i+1})$ has color 1.",
"Consider the inner terminals of the quads attached to the edges $(1,x_i)$ and $(1,x_{i+1})$ .",
"They must be in the prime region (since they are adjacent to $x_i$ or $x_{i+1}$ ), but they cannot be in the interval $(x_i,x_{i+1})$ (since it has color 1).",
"Observe from the definition of $Q_2$ that the two inner terminals $a, b$ are connected by two paths of length 2, where the intermediate node of each path is adjacent to one of the terminals.",
"Therefore, any two inner terminals of a quad $Q$ are connected by a path consisting of nodes that are adjacent to either outer terminal, where every other node of the path is an inner terminal.",
"In particular, any two inner terminals of the quads attached to the edges $(1,x_i)$ and $(1,x_{i+1})$ are connected by a path of nodes that are adjacent to node 1.",
"All of the nodes of these paths must be in the prime region because they have distance at most 2 in the graph $G \\setminus \\lbrace 1,2\\rbrace $ from $x_i$ or $x_{i+1}$ .",
"None of these nodes can be in the interval $(x_i,x_{i+1})$ because the edge $(x_i,x_{i+1})$ is colored 1.",
"Furthermore, there cannot be an edge connecting a node in the prime region left of $x_i$ to a node right of $x_{i+1}$ because of the edges $(1,x_i), (2,x_i), (2,x_{i+1})$ that have colors 1, 2, 3 respectively.",
"Therefore, either all the inner terminals of both quads are left of $x_i$ or they are all right of $x_{i+1}$ .",
"In the former case, the quad of $(1,x_{i+1})$ contradicts Lemma REF because all edges from the inner terminals to 1 have color 1 and the edges to $x_{i+1}$ must have color 3 (because they intersect $(1,x_i), (2,x_i)$ ).",
"In the latter case, the quad of $(1,x_i)$ contradicts Lemma REF because all edges from the inner terminals to 1 have color 1 and the edges to $x_i$ must have color 2 (because they intersect $(1,x_{i+1}), (2,x_{i+1})$ ).",
"Case 2: $(x_i,x_{i+1})$ has color 2 or 3.",
"Assume wlog that $(x_i,x_{i+1})$ has color 2 (the case of color 3 is symmetric).",
"Consider the triangles $(1,x_i,x_{i+1})$ , $(2,x_i,x_{i+1})$ and the quads attached to their edges.",
"If one of the inner terminals is right of $x_{i+1}$ then all of them must be there (within the prime region), because the edge $(1,x_{i+1})$ , the path $(2,x_i,x_{i+1})$ and the edge $(2,x_{i+1})$ use all 3 colors, hence there cannot be an edge from a node right of $x_{i+1}$ to a node (other than $x_i$ ) left of $x_{i+1}$ .",
"Node $x_i$ can reach nodes right of $x_{i+1}$ only with color 2.",
"By Lemma REF , the inner terminals of the quads of the triangles $(1,x_i,x_{i+1})$ , $(2,x_i,x_{i+1})$ cannot be right of $x_{i+1}$ , hence they are all left of $x_{i+1}$ .",
"If there is an edge from $x_{i+1}$ to a node left of $x_i$ , then the edge must be colored 3 (because of the edges $(1,x_i), (2,x_i)$ ), in which case node 2 cannot reach the interval $(x_i,x_{i+1})$ (because the edge $(x_i,x_{i+1})$ was assumed to have color 2).",
"Then all inner terminals of the quad for the edge $(2,x_{i+1})$ must be left of $x_i$ and their edges to 2 and $x_{i+1}$ are colored 2 or 3, contradicting Lemma REF .",
"Therefore, there is no edge from $x_{i+1}$ to a node left of $x_i$ , hence all inner terminals of the quads for the edges $(1,x_{i+1})$ and $(2,x_{i+1})$ are in the interval $(x_i,x_{i+1})$ .",
"All edges from this interval to 1 have color 1 and to 2 have color 3 (because of the edge $(x_i,x_{i+1})$ ).",
"Suppose that the closest inner terminal to $x_{i+1}$ belongs to the $(1,x_{i+1})$ quad, then its edge to 1 has color 1.",
"Consider the quad of the edge $(2,x_{i+1})$ : the edges from the inner terminals to 2 and $x_{i+1}$ have color 2 or 3, contradicting Lemma REF .",
"Similarly, if the closest inner terminal to $x_{i+1}$ belongs to the $(2,x_{i+1})$ quad, then its edge to 2 has color 3, hence the edges from the inner terminals of the quad of the edge $(1,x_{i+1})$ to 1 and $x_{i+1}$ have colors 1 and 2, contradicting again Lemma REF .",
"Let $x_i, i \\notin \\lbrace 1, n-1, n \\rbrace $ be a vertical node in the prime region that is at stretch distance more than 80 from the endpoints of the prime region.",
"Then both its successor $x_{i+1}$ on the path $p$ and its predecessor $x_{i-1}$ are also in the prime region, at stretch distance more than 60 from its endpoints.",
"Therefore, the edges from node 1 to $x_{i-1}, x_i, x_{i+1}$ have all color 1, and the edges from node 2 to $x_{i-1}, x_i, x_{i+1}$ have all the same color (2 or 3) by Lemma REF .",
"Assume without loss of generality that $x_i$ is left of $x_{i+1}$ , and that the edges from 2 to $x_{i-1}, x_i, x_{i+1}$ have color 2.",
"Using the same argument as in Claim REF (in the proof of Lemma REF ), we can deduce that $x_{i-1}$ is not embedded between $x_i$ and $x_{i+1}$ .",
"(If $x_{i-1}$ is in $(x_i, x_{i+1})$ then the edge $(x_i,x_{i+1})$ must have color 3 (because $(1,x_{i-1})$ has color 1 and $(2,x_{i-1})$ has color 2), hence nodes 1 and 2 can reach the interval $(x_i, x_{i+1})$ only with colors 1 and 2 respectively.",
"The argument in the last paragraph of the proof of Claim REF applies then verbatim.)",
"Similarly, $x_{i+1}$ is not between $x_i$ and $x_{i-1}$ .",
"Therefore, $x_{i-1}$ is left of $x_i$ .",
"Similarly, since $x_{i+1}$ is at stretch distance more than 60 from the endpoints of the prime region, its successor $x_{i+2}$ is also in the prime region, and it is right of $x_{i+1}$ .",
"All edges from $x_{i-1}, x_i, x_{i+1}, x_{i+2}$ to terminal 1 are colored 1, and all their edges to terminal 2 have the same color by Lemma REF , say color 2.",
"Lemma 10 Let $x_i, i \\notin \\lbrace 1, n-1, n \\rbrace $ be any vertical node in the prime region that is at stretch distance more than 80 from the endpoints of the prime region.",
"There is an index $j \\in \\lbrace i-1, i, i+1 \\rbrace $ , such that the edges $(x_j,x_{j+1}), (2,x_j), (2,x_{j+1})$ all have the same color, say 2, and the centers of both triangles $(1,x_j, x_{j+1})$ , $(2,x_j, x_{j+1})$ are in the interval $(x_j, x_{j+1})$ .",
"As we observed before the lemma, the nodes $x_{i-1}, x_i, x_{i+1}, x_{i+2}$ are in the prime region in this order (or the reverse), all edges from $x_{i-1}, x_i, x_{i+1}, x_{i+2}$ to terminal 1 are colored 1, and all their edges to terminal 2 have the same color by Lemma REF , say color 2.",
"Suppose that both centers $c_{1i}, c_{2i}$ of the triangles $(1,x_i, x_{i+1})$ , $(2,x_i, x_{i+1})$ are in the interval $(x_i, x_{i+1})$ .",
"Consider the copy of $Q_1$ with outer terminals 1, 2 and inner terminals $x_i, x_{i+1}$ .",
"Node 1 can reach the interval $(x_i, x_{i+1})$ only with color 1.",
"If $(x_i, x_{i+1})$ is colored 3, then node 2 can reach the interval only with color 2, contradicting Lemma REF .",
"Therefore, $(x_i, x_{i+1})$ must be colored 2.",
"Thus, the claim holds for $j=i$ .",
"On the other hand, suppose that a center $w$ of one of the triangles $(1,x_i, x_{i+1})$ , $(2,x_i, x_{i+1})$ is outside the interval $(x_i, x_{i+1})$ .",
"Then by (the proof of) Claim REF , $w$ must be either in the interval $(x_{i-1},x_i)$ or in the interval $(x_{i+1}, x_{i+2})$ .",
"Suppose wlog that $w$ is in $(x_{i+1}, x_{i+2})$ - see Fig.",
"REF ; $z=x_{i+2}$ in the figure.",
"The edge $(x_i,w)$ has color 3 because it intersects the edges $(1,x_{i+1})$ and $(2,x_{i+1})$ .",
"Node $x_{i+1}$ cannot reach any node, other than 1,2, outside the interval $(x_i, x_{i+2})$ because of the color-1 path $(x_i,1,x_{i+2})$ , the color-2 path $(x_i,2,x_{i+2})$ , and the color-3 edge $(x_i,w)$ .",
"Similarly, node $x_{i+2}$ cannot reach any node in the interval $(x_i,x_{i+1})$ because of the color-1 path $(x_i,1,x_{i+1})$ , the color-2 path $(x_i,2,x_{i+1})$ , and the color-3 edge $(x_i,w)$ .",
"Therefore, the centers $c_{1,i+1}, c_{2,i+1}$ of both triangles $(1,x_{i+1}, x_{i+2})$ , $(2,x_{i+1}, x_{i+2})$ must be in the interval $(x_{i+1}, x_{i+2})$ .",
"The edge $(1,c_{1,i+1})$ is colored 1 and the edge $(x_i,w)$ is colored 3, therefore the edge $(x_{i+1}, x_{i+2})$ must be colored 2.",
"Thus, the claim holds for $j=i+1$ .",
"Since the edge $(x_j,x_{j+1})$ has color 2, all edges from node 2 to the open interval $(x_j,x_{j+1})$ must have color 3.",
"We finish the proof by showing that it is impossible to embed and color the edges of the triangles $(1,x_j, x_{j+1})$ , $(2,x_j, x_{j+1})$ so that both centers are in $(x_j,x_{j+1})$ as required by Lemma REF .",
"This statement is similar to Lemma REF for the copy of $Q_1$ with outer terminals 1, 2, and inner terminals $x_i, x_{i+1}$ , but the important difference to Lemma REF , is that here the color available from node 2 to the open interval $(x_j, x_{j+1})$ is different than the color of the edges from 2 to the nodes $x_j, x_{j+1}$ .",
"On the other hand, the graph here is more involved and has quads attached to the edges.",
"Lemma 11 There is no 3-page embedding in which nodes $x_j, x_{j+1}$ are in the prime region, with color-1 edges to terminal 1, color-2 edges to terminal 2, the edge $(x_j, x_{j+1})$ has color 2, all edges from the open interval $(x_j, x_{j+1})$ to 1 have color 1 and to node 2 have color 3, and the centers of both triangles $(1,x_j, x_{j+1})$ , $(2,x_j, x_{j+1})$ are in the interval $(x_j, x_{j+1})$ .",
"Assume without loss of generality that the center $c_{1j}$ of the triangle $(1,x_j, x_{j+1})$ is left of the center $c_{2j}$ of the triangle $(2,x_j, x_{j+1})$ - see Figure REF .",
"Figure: NO_CAPTIONIf $x_j$ has a color-2 edge to a node in the interval $(c_{1j},x_{j+1})$ (for example, to $c_{2j}$ ) then the edge $(c_{1j},x_{j+1})$ must have color 1 (because of the color-3 edge $(2,c_{2j})$ ).",
"Then there is no position for the center of the triangle $(1,x_{j+1},c_{1j})$ : The center must be in the prime region because it is adjacent to $x_{j+1}$ .",
"Node $x_{j+1}$ cannot reach left of $c_{1j}$ (because of the color-1 edge $(1,c_{1j})$ , the color-2 edge from $x_j$ to the interval $(c_{1j},x_{j+1})$ and the edge $(2,x_j)$ , and the color-3 edge $(2,c_{2j})$ ); node $c_{1j}$ cannot reach right of $x_{j+1}$ (because of the color-1 edge $(1,x_{j+1})$ , the color-2 edge $(2,x_{j+1})$ and the color-3 edge $(2,c_{2j})$ ); and 1 cannot reach the interval $(c_{1j},x_{j+1})$ (because of the color-1 edge $(c_{1j},x_{j+1})$ ).",
"Therefore, there is no color-2 edge from $x_j$ to the interval $(c_{1j},x_{j+1})$ .",
"In particular, the edge $(x_j,c_{2j})$ must be colored 3 (it cannot be colored 1 because of the edge $(1,c_{1j})$ ).",
"Now consider the triangle $(2,x_j,c_{2j})$ and the quads attached to its edges.",
"Since there is no color-2 edge from $x_j$ to the interval $(c_{1j},x_{j+1})$ , there is no edge (of any color) from $x_j$ to the right of $c_{2j}$ because of the color-1 edge $(1,c_{1j})$ , the color-2 edge $(2,x_{j+1})$ and the color-3 edge $(2,c_{2j})$ .",
"Node 2 cannot reach the interval $(x_j,c_{2j})$ because of the color-3 edge $(x_j,c_{2j})$ .",
"Therefore, the center of the triangle $(2,x_j,c_{2j})$ (and the inner terminals of the quad for the edge $(2,x_j)$ ) must be left of $x_j$ .",
"Note that no edge can connect a node left of $x_j$ to a node right of $x_j$ (within the prime region), other than $c_{2j}$ , because of the color-1 edge $(1,x_j)$ , the color-2 edge $(2,x_j)$ , and the color-3 path $(x_j,c_{2j},2)$ .",
"Therefore all the inner terminals of the quads attached to all the edges of the triangle $(2,x_j,c_{2j})$ are left of $x_j$ (and within the prime region).",
"Their edges to $c_{2j}$ must be colored 3 because of the conflicting edges $(1,x_j), (2,x_j)$ .",
"This contradicts Lemma REF .",
"It follows that there is no 3-page embedding satisfying the conditions of the lemma.",
"Lemmas REF and REF provide the desired contradiction to the assumption that $G$ can be embedded in three pages.",
"We conclude: Theorem 1 There is no 3-page embedding of the graph $G$ .",
"Acknowledgment.",
"Work supported by NSF Grants CCF-1703925, CCF-1763970.",
"We thank the anonymous referees for their helpful comments."
]
] | 2005.14111 |
[
[
"The ASAS-SN Catalog of Variable Stars VIII: \"Dipper\" Stars in the Lupus\n Star-Forming Region"
],
[
"Abstract Some young stellar objects such as T Tauri-like \"dipper\" stars vary due to transient partial occultation by circumstellar dust, and observations of this phenomenon inform us of conditions in the planet-forming zones close to these stars.",
"Although many dipper stars have been identified with space missions such as $Kepler$/$K2$, ground-based telescopes offer longer term and multi-wavelength perspectives.",
"We identified 11 dipper stars in the Lupus star forming region in data from the All-Sky Automated Survey for SuperNovae (ASAS-SN), and further characterized these using observations by the Las Cumbres Global Observatory Telescope (LCOGT) and the Transiting Exoplanet Survey Satellite $TESS$, as well as archival data from other missions.",
"Dipper stars were identified from a catalog of nearby young stars and selected based on the statistical significance, asymmetry, and quasi-periodicity or aperiodicity of variability in their ASAS-SN light curves.",
"All 11 stars lie above or red-ward of the zero-age main sequence and have infrared excesses indicating the presence of full circumstellar disks.",
"We obtain reddening-extinction relations for the variability of 7 stars using our combined ASAS-SN-$TESS$ and LCOGT photometry.",
"In all cases the slopes are below the ISM value, suggesting larger grains, and we find a tentative relation between the slope (grain size) and the $\\text{K}_\\text{s}-[22\\:\\mu\\text{m}]$ infrared color regarded as a proxy for disk evolutionary state."
],
[
"Introduction",
"Exoplanet science has seen much progress in recent years from both space- and ground-based observations alike (see the reviews by [75], , and [74]).",
"Discoveries by these surveys challenge us to reconcile the layout of the Solar System with what we observe in other planetary systems [64], [76].",
"One of the biggest obstacles to progress is our limited knowledge of protoplanetary disk structure and the interactions between the disk and the central Young Stellar Object (YSO) [63].",
"While Kepler found many planets within $\\lesssim 1$ AU of stars, protoplanetary disks are generally not resolved at this scale by current telescopes, including the Atacama Large Millimeter Array ($\\sim $ 5-10 au).",
"Observations of YSO variability offer us a complementary view of phenomena around these stars at separations corresponding to the equivalent Keplerian orbital periods [19], [36], [8].",
"YSO variability is divided into three classes [38].",
"Each class provides different insight into conditions on or around young stars: Type I, sinusoidal patterns are caused by the rotation of star spots; Type II, episodic “bursting\" events are caused by increases in accretion onto the star; and Type III, dimming events are caused by occulting circumstellar matter.",
"In this paper we focus on Type III variables.",
"UX Orionis stars (UXORs), Herbig Ae/Be stars with occulting circumstellar dust, are a well-known example of this phenomenon [22].",
"Lower mass ($M\\lesssim 1$ M$_\\odot $ ) T Tauri stars also exhibit dimming due to occulting dust.",
"These were first found in ground-based observations, [14] but surveys by space telescopes such as Kepler/K2 have led to the discovery of many more such variables, which have become colloquially known as “dipper\" stars [18], [62], [20], , [4], [6], [19], [36].",
"Unlike UXORs, dipper stars are T Tauri stars which exhibit dimming events ranging from quasi-periodic to completely episodic.",
"The observable difference between UXORs and dippers are the dip timescale, occurrence, and depth, which for dippers are shorter, more frequent, and shallower, respectively, possibly because the inner edge of the dust disk is closer to lower-luminosity dippers [53].",
"Dippers typically fade from tens of percent to multiple mag for about one day.",
"Dipper variability is believed to be caused by transiting dust, probably associated with the inner regions of a nearly edge-on disk , [13].",
"Alternative explanations such as exocomets [15] or dusty disk winds , [10] have also been proposed.",
"Additional studies of dipper stars are needed to distinguish between the differing models.",
"While space telescopes such as Kepler provided high precision and cadence, their temporal baselines are limited to weeks (Spitzer and TESS) or months (K2), and survey only a small fraction of the sky at a time.",
"CoRoT, Spitzer, and TESS conducted individual pointed observations, and the K2 mission was restricted to the ecliptic plane, and the Taurus and Upper Scorpius star-forming regions.",
"Ground-based observatories such as the Kilodegree Extremely Little Telescope (KELT; [73]) or the Palomar Transient Factory (PTF; [55]) have been used to survey YSOs, e.g.",
"[80], [81], [82], [67], [6] and , [26], respectively.",
"KELT had the advantage of all-sky coverage, but was limited to bright stars (8<V<12 mag).",
"PTF had a deeper limiting magnitude ($\\sim 20$ mag), but surveyed only the North American Nebula complex and the 25 Ori region.",
"The successor to PTF, the Zwicky Transient Factory [33], [11], has expanded coverage to the entire night sky visible by Palomar.",
"Here we perform a search for variable YSOs in the Lupus region using the All-Sky Automated Survey for SuperNovae (ASAS-SN; [85], [54]) as part of a larger survey of transient objects.",
"While ASAS-SN has discovered variable YSOs in the past (e.g., [41], [86]), the standard pipeline intentionally does not trigger on known objects or low amplitude variability.",
"ASAS-SN offers photometric data of the entire sky from late 2012 to mid 2018 in the V band, and from late 2017 to present in the g band.",
"The combination of a long baseline and all-sky coverage allows for large surveys of many different types of variability.",
"While the ASAS-SN cadence is slower than K2 or KELT (typically once per day), ASAS-SN data provides a far wider sky coverage than K2, and has a deeper limiting magnitude than KELT.",
"Additionally, ASAS-SN's large longitude coverage allows for observations of the same source by multiple telescopes.",
"As an example, ASAS-SN data has already been successfully used to identify variable stars [46], [50], [49], [68], [51], derive period–luminosity relationships for $\\delta $ Scuti stars [47], identify M-dwarf flares [79], study contact binaries [48], and to examine the long-term variability of Boyajian's Star and compare it with the dipper phenomenon [84].",
"Finally, ASAS-SN has recently identified a potential new “Boyajian's Star\" analog exhibiting rapid dimming events, nicknamed Zachy's Star .",
"The constellation Lupus comprises multiple 1–3 Myr-old regions of low-mass star formation near the Scorpius Centaurus OB association (see the review by [21]).",
"At a distance of $189\\pm 13$ pc, it is one of the closest star-forming regions .",
"The Lupus III cloud in particular is home to many well-studied T Tauri stars such as RU Lup and EX Lup.",
"Lupus was not surveyed by K2, KELT, or PTF.",
"While there have been several recent studies on disk structure (e.g., [5], [7]) and accretion (e.g., [65]) in Lupus YSOs, no dippers have been identified.",
"Here we use ASAS-SN observations to identify and classify Lupus variable stars in sec:IdentifyingDippers.",
"We discuss follow-up observations of dipper sources in sec:Followup, and investigate their infrared excesses and dust properties using multi-band photometry in sec:analysis."
],
[
"Identifying Dipper Candidates",
"As the starting point of this investigation, we compiled a catalog of all known candidate members of young associations, open clusters and moving groups within 150 pc based on the member lists of , [58], [78], and [29], complemented by , [27], [66], [24], [60], , [28], and [30].",
"Of these sources, 307 are members of the Lupus Star-Forming Region."
],
[
"ASAS-SN Observations",
"The ASAS-SN network consists of 20 telescopes mounted on 5 fully-robotic mounts located at the Haleakalā Observatory, the Cerro Tololo International Observatory, McDonald Observatory, and the South African Astrophysical Observatory.",
"Observations span from late 2012 to mid-2018 in the V band, and from late 2017 in the g band, providing $\\sim 800$ epochs per source on average.",
"Each science image consists of three dithered 90 s exposures taken using 14-cm aperture Nikon telephoto lenses and thinned back-illuminated CCDs with $8.\\!\\!^{\\prime \\prime }0$ pixels.",
"Images from ASAS-SN are processed by the fully-automated ASAS-SN pipeline using the ISIS image subtraction package [2], [1].",
"The IRAF apphot package is used with a 2-pixel ($\\approx 16.\\!\\!^{\\prime \\prime }0$ ) aperture to perform aperture photometry on the subtracted images to generate a differential light curve.",
"The same aperture is used to perform photometry in the reference images, and the resulting fluxes are added back to the differential light curve.",
"The photometry is then calibrated using stars in the AAVSO Photometric All-Sky Survey [37].",
"The greatest proper motion measured by Gaia for the Lupus YSOs is only $53.68\\pm 0.05$ mas/yr, so there was no need to take the proper motions into account given our temporal baseline.",
"There are known systematic offsets between ASAS-SN telescopes [46].",
"To combine data from different telescopes for a given filter, we use the SciPy interp1d package [52] to interpolate the most sampled curve.",
"We then calculate the median difference between this interpolated light curve and the dataset with the highest number of contemporaneous points.",
"We add this difference to the second dataset, combine the light curves, and repeat the process until the data from all telescopes are aligned.",
"This results in each source having a g-band and V-band light curve.",
"To decrease the likelihood of false positives in our variability search, the light curve of each star is processed as follows.",
"First, we remove data points corresponding to images where the FHWM of the point spread function is greater than the 90th percentile value of each telescope.",
"Next, non-detections are omitted from statistical calculations.",
"Lastly, we remove the top and bottom 1% of detections to clip potential outliers.",
"Stars brighter than $10.5$ mag in $V$ and 11 mag in $g$ are at risk of saturation in ASAS-SN.",
"Approximately 40% of the stars in our target list are saturated.",
"We removed these brighter sources from our sample.",
"Inconsistent reference fluxes between telescopes yield discrepant magnitude ranges during variable events.",
"This most often took the form of one telescope reporting a large range of magnitudes while simultaneous observations from another appear to “flatline\" by comparison.",
"Any sources that noticeably demonstrated this pattern were omitted.",
"Applying these cuts reduces our sample size from 307 to 177.",
"The V-band curves of all remaining sources are processed and analyzed as outlined in sec:searchvar.",
"Our selection process is summarized in Table REF .",
"Figure: Peak-to-peak variability vv as a function of ASAS-SN V for our sample of YSOs.",
"Stars flagged as dippers in Upper Scorpius and Taurus that were present in our catalog are also shown.",
"The dashed line corresponds to the 90th percentile of vv at a given magnitude based on the ASAS-SN V-band light curves of about 115,000 randomly selected field stars.",
"Notice how saturation increases the average vv at magnitudes brighter than ∼11.5\\sim 11.5.",
"YSOs above the dashed line are considered variable.",
"Out of the 177 Lupus YSOs analyzed, 91 pass this criterion for variability.",
"The eleven Lupus dippers described in sec:classifications are marked by orange stars.Figure: Flux Asymmetry vs Quasi-Periodicity for our sample of variable YSOs.",
"The Flux Asymmetry MM is a measure of variability that is predominantly brightening (M<0M<0), dimming (M>0M>0), or symmetric (M=0M=0).",
"The Quasi-Periodicity QQ measures whether the variability is more periodic (Q=0Q=0) or stochastic (Q>0Q>0).",
"The Q-MQ-M space is divided into nine distinct regions to classify variability .",
"Stars in Lupus that are in the Quasi-Periodic Dipper region are shown with orange stars.",
"The area of each point is proportional to the peak-to-peak variability metric vv.",
"Stars flagged as dippers in Table 2 of and Table 3 of that were present in our sample are also shown."
],
[
"Searching for Variability",
"In order to quantify variability, we use the peak-to-peak variability metric $v=\\frac{(m_i-\\sigma _i)_\\text{max}-(m_i+\\sigma _i)_\\text{min}}{(m_i-\\sigma _i)_\\text{max}+(m_i+\\sigma _i)_\\text{min}},$ from , where $m_i$ is a measured magnitude and $\\sigma _i$ is its associated uncertainty.",
"For a comparison sample, we measured $v$ for the ASAS-SN V-band light curves of approximately $115,000$ randomly selected field stars.",
"We calculate the 90th percentile of $v$ as a function of magnitude and selected the 91 out of 177 Lupus YSOs above this threshold as variable stars and retained them for further scrutiny.",
"Figure REF illustrates our selection of these sources based on $v$ .",
"Table: Summary of cuts applied to Lupus YSO sample."
],
[
"Classifications",
"Now that we have a selection of variable YSOs, we next classify their variability using the Quasi-Periodicity and Flux Asymmetry statistics defined in [20].",
"The Quasi-Periodicity $Q=\\frac{(\\sigma ^2_\\text{resid}-\\sigma _\\text{phot}^2)}{(\\sigma ^2_m-\\sigma _\\text{phot}^2)},$ quantifies the degree to which variability is periodic or stochastic.",
"Here $\\sigma _\\text{phot}$ is the estimated photometric uncertainty, $\\sigma ^2_m$ is the variance of the original light curve, and $\\sigma ^2_\\text{resid}$ is the variance of the residual light curve after subtracting the dominant periodic signal.",
"For a perfectly periodic signal, the variance of the residual is equivalent to photometric noise, yielding $Q\\approx 0$ .",
"To produce the residual curve, we create a window function (Dirac Comb) out of the original light curve, i.e.",
"values of 1 when an observation takes place and values of 0 elsewhere.",
"We utilize Astropy LombScargle , to find the periodograms of both the original curve and the Dirac Comb window function.",
"We find all frequencies in the window function with a power greater than two standard deviations above the mean and ignore any peaks in the periodogram of the light curve within 0.03 Hz of the peaks in the window function.",
"The curve is then phased using the remaining period with the highest power.",
"We boxcar smooth this folded light curve with a window size of $25\\%$ of the period and subtract the smoothed model from the folded curve.",
"We define $\\sigma _\\text{resid}$ as the standard deviation of this residual curve.",
"The second metric to classify variability is Flux Asymmetry $M=\\frac{\\left<m_{10\\%}\\right>-m_\\text{med}}{\\sigma _m},$ where $\\left<m_{10\\%}\\right>$ is the mean of the top and bottom $10\\%$ of magnitude measurements, $m_\\text{med}$ is the median of all measurements, and $\\sigma _m$ is the standard deviation of the light curve.",
"$M$ determines whether the variability is predominantly brightening ($M$ < 0) or predominantly dimming ($M$ > 0).",
"Figure: ASAS-SN light curves for the first six dippers in Lupus, in order of ascending RA.",
"Measurements in the VV-band are in green, while measurements in the gg-band are in blue.",
"The purple and gray shaded regions show the TESS (T) and LCOGT (L) observing periods, respectively, if applicable to the system.",
"The vertical offset between the filters was found by interpolating the V-band data in the overlapping region.",
"The correction is purely for presentation purposes.Figure: NO_CAPTIONWe calculate $Q$ and $M$ for the V-band data to separate the YSOs by variability type.",
"The results are shown in Figure REF .",
"The $Q-M$ space is divided into nine regions following [19].",
"All eleven stars in the Quasi-periodic Dipping region ($0.15<Q<0.85$ , $-0.25<M<0.25$ ) of Figure REF were examined in detail, and their light curves are shown in Figure REF .",
"The designations of all 91 variable Lupus YSOs are included in Table REF .",
"We also performed the same analysis for the ASAS-SN light curves of the dippers identified in [19] and [82] that were present in our catalog.",
"These stars are also included in Figures REF , REF , and REF .",
"The values of $Q$ and $M$ appear to be inconsistent between ASAS-SN and K2, in that there are Upper Scorpius stars identified in [19] as having dipper-like $Q$ and $M$ values based on K2 data that have non-dipper values based on ASAS-SN data.",
"We suspect that the discrepant $Q$ and $M$ values are the result of differences in instrumental error and cadence.",
"Ground telescopes such as ASAS-SN sample at timescales comparable to that of individual dipping episodes ($\\sim $ 1 day), whereas K2 provides a much higher cadence (30 min) and greater precision.",
"A more detailed comparison of $Q$ and $M$ in ASAS-SN as compared to K2 is left for future work.",
"The ASAS-SN light curves of the flagged dippers (Figure REF ) contain multiple episodic dimming events of up to 2-3 magnitudes, consistent with this variable type.",
"Table: Average ASAS-SN VV magnitudes, peak-to-peak variability, quasi-periodicity, flux asymmetry, and variability classification for all Lupus YSO members present in our catalog, as well as additional information for quasi-periodic dippers.",
"Names of sources other than quasi-periodic dippers are given as ASAS-SN IDs, generated by their coordinates from the source catalogs (see sec:IdentifyingDippers).",
"The classifications are P, QP, and AP for periodic, quasi-periodic, and aperiodic respectively, and D, S, and B for dipper, symmetric, and burster, respectively.",
"A complete list is available electronically.Obscuration by dust produces reddening as well as extinction, and the relation between these depends on the grain size distribution and composition.",
"Simultaneous multi-band observations of dipping events allow this to be investigated and compared between dipper stars and with the ISM.",
"For example, monitoring in both Sloan $g$ and $i$ (or equivalently TESS $T$ ) bands allow the relation between the reddening $\\Delta (g-i)$ with dimming $\\Delta g$ to be measured, with $\\frac{\\Delta (g-i)}{\\Delta g}=1-\\frac{A_i}{A_g}$ where $A_g$ and $A_i$ (or $A_T$ ) are the inferred extinction coefficients for the dust in the Sloan $g$ and $i$ (or TESS) bands (see sec:Dust).",
"To perform a higher-precision examination of individual dimming events and compare with contemporaneous ASAS-SN measurements, we utilize the observations made by TESS during Sector 12 and the Las Cumbres Observatory Global Telescope [16] 0.4 m telescopes in the $g$ and $i$ bands."
],
[
"We extracted TESS light curves for five of the dipper candidates: HW Lup, RY Lup, MY Lup, Sz 131, and Sz 133.",
"These observations were obtained during Sector 12, near the end of TESS's initial survey of the Southern Hemisphere.",
"The other dipper candidates were not observed by TESS.",
"We produce light curves from the TESS observations using an image subtraction pipeline optimized for use with the full-frame images (FFIs).",
"This pipeline is very similar to that used to process ASAS-SN images and is also based on the ISIS package [2], [1].",
"This method has become a standard technique for using TESS to study extragalactic transients like supernovae , [25] and the exceptional tidal disruption event ASASSN-19bt [42].",
"It has also been previously applied to the study of the EXor variable ESO-H$\\alpha $ 99 by [40].",
"For each source, we construct a reference image using 750-pixel-wide postage stamps cut from the first 100 Sector 12 FFIs of good quality.",
"When building the reference images, we exclude FFIs with PSF widths or sky background levels above average for the sector as well as any FFIs associated with non-zero data quality flags.",
"Using these reference images, we produce raw differential light curves for the five sources.",
"However, Both TESS and ASAS-SN lightcurves these light curves only provide the change in flux relative to the value in the reference image.",
"Equation REF evaluated at two epochs $m$ and $n$ can be written as: $\\frac{A_T}{A_g} = - \\log _{10}\\left(\\frac{\\Delta f_{T,m}+f_{T,0}}{\\Delta f_{T,n}+f_{T,0}}\\right)\\left[ \\log _{10}\\left(\\frac{\\Delta f_{g,m}+f_{g,0}}{\\Delta f_{g,n}+f_{g,0}}\\right)\\right]^{-1}$ This shows that the reddening-extinction slope depends not only on the TESS and ASAS-SN differential fluxes $df_{T,m}$ and $f_{g,m}$ but also on the corresponding reference fluxes $f_{T,0}$ and $f_{g,0}$ .",
"However, TESS's PSF is about 16 times larger in area than ASAS-SN's, so crowding affects TESS reference fluxes more often.",
"Crowding is not a problem for RY Lup and MY Lup, which are sufficiently bright ($T\\sim 10$ mag) and well-isolated that we were able to obtain good measurements of the reference fluxes using aperture photometry.",
"However, HW Lup, Sz 131, and Sz 133 are fainter and have nearby stars of comparable brightness.",
"For these sources we estimated the reference flux using the ticgen software package [45], .",
"This method yields TESS magnitude estimates for HW Lup, Sz 131, and Sz 133 of 12.26, 11.56, and 13.86 mag, respectively.",
"These magnitude estimates were converted into fluxes using a standard instrumental zero point of 20.44 electrons per second .",
"We then added flux to the raw differential light curves such that the median of the HW Lup, Sz 131, and Sz 133 observations matched this estimated reference value.",
"This method has been used previously by [47] when studying $\\delta $ Scuti stars with TESS.",
"However, we caution that for non-periodic, inherently variable sources such as dippers, estimating the reference flux from ticgen should be made with higher-resolution contemporaneous observations with the TESS reference image.",
"Because no such observations were made, the values of $1-A_T/A_g$ for HW Lup, Sz 131, and Sz 133 are influenced by the difference between the ticgen reported flux and the actual flux of the reference image.",
"Future investigations can resolve this problem with a single epoch of higher resolution, multi-band, ground-based observations during a TESS campaign to calibrate the differential flux light curve.",
"The resultant TESS photometry is shown in Figure REF , along with the simultaneous ASAS-SN g-band observations.",
"We note that the two short-duration features visible in the TESS light curve of HW Lup around MJD 58627 and MJD 58641 (marked with grey vertical bands) are almost certainly artifacts produced by scattered Earthshine.",
"Excess background light due to scattered Earth and Moon light is a well-known nuisance characteristic of TESS observations, and the two spikes in the HW Lup light curve coincide with epochs of documented high background flux (see the Sector 12 TESS Data Release Noteshttps://archive.stsci.edu/missions/tess/doc/tess_drn/tess_sector_12_drn17_v02.pdf for more details).",
"Aside from these artifacts, the ASAS-SN and TESS data sets show generally excellent agreement.",
"Figure: Left Panels: Simultaneous TESS (red) and ASAS-SN gg (blue) observations of five of the dipper candidates.",
"The data from ASAS-SN has been vertically scaled to demonstrate the agreement with TESS on variability structure, with the right axis reporting the actual measured ASAS-SN magnitudes.",
"The vertical gray regions indicate epochs when scattered Earthshine artifacts are dominant.",
"TESS data from these epochs is excluded from the subsequent analysis.",
"Right Panels: Reddening vs Dimming between ASAS-SN and TESS.",
"The slope of the fitted line is equivalent to 1-A T /A g 1-A_T/A_g.",
"The value for interstellar dust is 1-A T /A g =0.381-A_T/A_g=0.38.",
"Sources marked with ticgen were located in crowded fields and could not have their reference flux measured directly.",
"Consequently the slopes of these sources should be met with skepticism."
],
[
"LCOGT",
"We also obtained $g$ , $r$ , and $i$ band observations of five Lupus dippers with the LCOGT 0.4-m telescope network, specifically the nodes at Suderland South Africa, CTIO, and Siding Spring, from 21 July, 2019 to 28 August, 2019.",
"Images were acquired with the SBIG detector.",
"Bias removal, image trimming, flat fielding, and photometry were performed with the BANZAI pipeline [59].",
"Further analysis of the photometry was performed with custom differential scripts, choosing a set of reference stars with similar magnitudes in each case.",
"RY Lup and Sz 117 were observed for 77 and 80 epochs, respectively, and Sz 90, Sz 96, and HK Lup were observed for 85 epochs (Figure REF )."
],
[
"Dippers in the Color-Magnitude Diagram",
"At the age of the Lupus molecular cloud, low-mass T Tauri stars such as these dippers have not reached the zero-age main sequence and should appear above the Zero-Age Main Sequence (ZAMS) in a color-magnitude diagram.",
"We computed absolute magnitudes in the 2MASS $K_s$ band using parallaxes from Gaia DR2 [32] and show these with Gaia $B_P-R_p$ colors as orange stars in Figure REF .",
"Note that Sz 133 lacks a DR2 parallax and does not appear.",
"We compare these with predictions from Isochrones and Stellar Tracks [23], [17], [69], [70], [71], [72].",
"Extinction by interstellar dust is comparatively low in the $K_s$ band relative to Gaia $B_P$ or $R_p$ and interstellar reddening will cause a star to move nearly horizontally to the right in this diagram, and so appear further above the ZAMS, giving the impression that they are both younger and less massive.",
"Without spectroscopic follow-up, we cannot robustly estimate the $T_{\\rm eff}$ and hence intrinsic $B_p-R_p$ color of these stars to reliably place them in this diagram to constrain their ages and masses.",
"Moreover, the time-averaged visible-wavelength $B_P - R_P$ color reported by DR2 will be affected both by the mean level of circumstellar dust as well as spots on these young, active stars, which are not accounted for by the MIST models , .",
"However, we can obtain an approximate estimate of the stellar masses if we assume a minimum co-eval age.",
"By adopting 2.2 Myr as the minimum co-eval age of the dipper stars, as well as a reddening relation for the ISM ($A_{\\text{K}_\\text{s}}=0.194\\,\\,\\text{E}(B_p-R_p)$ ; ) we can project (dashed black lines and red dots in Figure REF ) these stars onto the isochrone (dashed blue line) and estimate the minimum reddening and minimum masses of the stars.",
"Most of the dippers have $M\\lesssim 1\\,\\text{M}_\\odot $ with the notable exception of RY Lup, where we obtain $M\\sim 2.3\\,\\text{M}_\\odot $ using this method.",
"This is consistent with the picture of dipper stars as low mass T Tauri stars, to be distinguished from their higher mass UXOR Ae/Be-type cousins.",
"Figure: Gaia B P -R P B_P-R_P color vs. absolute M K s \\text{M}_{\\text{K}_\\text{s}} color-mag diagram for ten of the dippers.",
"Absolute magnitudes are obtained using distances from .",
"The isochrone models are from MIST.",
"By assuming an age of 2.2 Myr for the dippers and iteratively applying a reddening correction, we obtain rough estimates for stellar masses."
],
[
"Infrared Excess and Disks",
"For young stars, excess emission in the infrared is indicative of the presence of a disk.",
"By comparing a star's absolute flux in the 2MASS $\\text{K}_\\text{s}$ band, which will be dominated by the photosphere, with their 12 ($W3$ ) and 22 ($W4$ ) $\\mu $ m fluxes from the Wide-field Infrared Survey Explorer , [57], we can ascertain the presence of a disk and investigate its evolutionary state [56].",
"Figure REF shows the $\\text{K}_\\text{s}-[12\\:\\mu \\text{m}]$ vs. $\\text{K}_\\text{s}-[22\\:\\mu \\text{m}]$ colors of the Lupus YSOs along with other reported dippers in Upper Scorpius and Taurus.",
"The boundaries for disk types defined by [56] are included.",
"Young stars with “full\" or classical T Tauri disks occupy the upper right hand portion of the figure.",
"As the disk evolves, the star will lose its infrared excess.",
"The relative rates at which $\\text{K}_\\text{s}-[22\\:\\mu \\text{m}]$ and $\\text{K}_\\text{s}-[12\\:\\mu \\text{m}]$ decrease are indicative of how and where the disk is evolving.",
"All eleven dipper candidates show infrared excesses consistent with that of full disk-bearing YSOs, along with all but one of the previously reported dippers in our sample.",
"This further supports a connection between the presence of a disk and the dipper phenomenon.",
"Figure: 2MASS–WISE color–color diagram.",
"Zones characteristic of the different evolutionary stages of disks as defined by are demarcated by the dashed lines.",
"The Lupus dippers are labelled alongside other known dippers.",
"Other Lupus stars in our YSO catalog are shown in blue.",
"The infrared excess of the dippers is consistent with dipping being related to the presence of a disk.Figure: Top Panel: The dependence of the A g /A i A_g/A_i extinction ratio on the source's IR excess.",
"IR Excess is an indicator of excess emission due to a hot inner disk.",
"We see a potential relation between the two, though additional data is needed.",
"Slopes for HW Lup, Sz 131, and Sz 133 are shown as open points to reflect the possible unreliability of the values (see text).",
"Bottom Panel: A zoomed view of the dippers with directly measured references fluxes."
],
[
"Multi-Band Variability and Dust Properties",
"Simultaneous observations of young stellar variability in two or more band-passes provides a means of differentiating between different phenomena (e.g., flares, accretion, circumstellar dust).",
"In the case of obscuration and reddening by circumstellar dust, it serves as an indicator of the grain size distribution [38], [14], [43].",
"The relative change (slope) in color (reddening) and magnitude (dimming) (Equation REF ) depends on the characteristic grain size $d$ and composition.",
"Dust dominated by small grains will produce steeper slopes (more reddening for a given amount of dimming) compared to dust dominated by larger grains, which in the limit of $d \\gg \\lambda $ will produce no reddening at all.",
"The grain size distribution is affected by grain growth in the disks, and the slope of the trend in a color–magnitude plot is indicative of that history [35].",
"UX Orionis stars exhibit reddening up to a point where the direct line of sight to the star is almost entirely obscured, causing scattered light that is comparatively blue to dominate, switching the sign of the slope [12], [34].",
"In contrast, the lack of reddening (and in fact slight bluing) by the repetitive dipper AA Tau has been explained by grains $>1\\mu $ m [14].",
"Note that these plots are insensitive to zero-point offsets in magnitudes, since all values are differential.",
"Five Lupus dippers (HW Lup, RY Lup, MY Lup, Sz 131, Sz 133) were monitored simultaneously by the TESS mission and ASAS-SN.",
"The TESS band-pass $T$ spans 600–1000 nm and is centered on the Cousins $I$ -band [77].",
"Figure REF shows the $g$ -band ASAS-SN and TESS data, and the change in the $g-T$ color with $g$ .",
"Linear least-$\\chi ^2$ fits are shown as red lines.",
"Based on the reddening $R$ values in and , the expected slope of reddening due to interstellar dust is 0.38.",
"Light curves and $g-i$ versus $i$ reddening diagrams for the five stars (RY Lup, Sz 90, Sz 96, HK Lup, and Sz 117) observed with LCOGT are shown in Figure REF .",
"In this case, the expected slope from interstellar dust is 0.48.",
"In all cases the slope is less than, and sometimes significantly less than, the expected interstellar dust value.",
"Only one star, RY Lup, was observed by all three telescope networks, but not all simultaneously.",
"The TESS and LCOGT observations are not contemporaneous, with the former ending about 33 days before the beginning of the latter.",
"The LCOGT slope (0.29) and TESS–ASAS-SN slope (0.25) are roughy consistent, with the smaller TESS–ASAS-SN slope (larger $A_T$ compared to $A_i$ ) consistent with the extension of the TESS bandpass to shorter wavelengths than the SDSS $i$ band filter.",
"Values of $A_g/A_i$ and $A_g/A_T$ , derived from these slops, are included in Table REF .",
"We can find no obvious correlation between the slope of the color–magnitude variation from our TESS–ASAS-SN or LCOGT observations and other properties of our disks such as the relative luminosity, the minimum wavelength of the infrared excess [61], the disk inclination or physical extent of dust and gas (if resolved) or the estimated dust/gas masses [5], [7].",
"However, we do see a possible correlation between the reddening slope and the infrared excess of the disk (WISE 22 $\\mu $ m emission) relative to the photosphere (2MASS $\\text{K}_\\text{s}$ emission).",
"HK Lup, MY Lup, and RY Lup have highest values of $A_g/A_i$ and ostensibly the smallest grains and also the largest infrared excess (Figure REF ).",
"The system with the lowest 22 $\\mu $ m excess, Sz 117, also has little or no reddening, with $A_i/A_g$ near unity.",
"We have included our TESS measurements of $A_g/A_i$ for HW Lup, Sz 131, and Sz 133, but we use open symbols and arrows for them in Figure REF since the reference flux estimated using ticgen may not be reliable.",
"Excluding these objects, the Spearman correlation coefficient of the data indicates a 97% probability that the two parameters, $A_g/A_i$ and $\\text{K}_\\text{s}-[22\\:\\mu \\text{m}]$ , are correlated.",
"A correlation between extinction coefficients and infrared excess could be explained if either (a) coalescence of a fixed mass of dust into larger grains produced both a flatter slope and less emitting area (and hence less emission), or (b) the growth of grains also proceeded contemporaneously with clearing of the disk.",
"The two scenarios are mutually exclusive.",
"With our small sample size (11 systems) and limited spectrophotometric data set we are unable to choose between these two possible explanations; more observations of a larger sample are clearly needed."
],
[
"Summary",
"We have identified 11 dipper stars in the Lupus molecular cloud using ASAS-SN observations and a series of quantitative selection criteria that identified YSO members of Lupus, variable Lupus YSOs, and Lupus YSOs that varied in a manner consistent with dippers identified in other star-forming regions [20].",
"All eleven stars are pre-main sequence, lying above the theoretical zero-age main sequence in a color-magnitude diagram.",
"Their positions in the diagram are consistent with expectations for Lupus-age ($\\sim $ 2 Myr) low-mass ($\\lesssim $ 1 $M_{\\odot }$ ) stars, but we are unable to estimate individual ages or masses due both to the lack of spectra and uncertainties in the effect of spots on theoretical models.",
"All 11 dippers have infrared excesses consistent with a “full\" T Tauri disk, as has been observed for other dipper stars [4].",
"Our findings expand the scope of the dipping phenomenon to yet another star-forming environment and epoch.",
"We used multi-wavelength time-series photometry of these dipper stars, combining ASAS-SN, TESS, and LCOGT observations, to measure the reddening by the occulting dust and thus infer something about the grain size.",
"In all cases the amount of reddening was less, and thus the grain sizes larger, than that expected from ISM-like dust.",
"Moreover, we find a tentative trend of decreasing reddening, and hence increasing grain size, with decreasing infrared excess, or equivalently more advanced evolutionary stages of the disk.",
"This is consistent with a picture of grain growth with time in protoplanetary disks .",
"These results need to be confirmed with larger and more in-depth studies, but highlight the synergy of ground- and space-based observations in the effort to better understand the structure and evolution of circumstellar disks and the planetary systems they spawn."
],
[
"Acknowledgements",
"We thank the Las Cumbres Observatory and its staff for its continuing support of the ASAS-SN project.",
"LCOGT observations were performed as part of DDT award 2019B-003 to EG.",
"ASAS-SN is supported by the Gordon and Betty Moore Foundation through grant GBMF5490 to the Ohio State University and NSF grant AST-1515927.",
"Development of ASAS-SN has been supported by NSF grant AST-0908816, the Mt.",
"Cuba Astronomical Foundation, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Chinese Academy of Sciences South America Center for Astronomy (CASSACA), the Villum Foundation, and George Skestos.",
"JWB acknowledges support from Research Experience for Undergraduate program at the Institute for Astronomy, University of Hawaii–Manoa funded through NSF grant 6104374.",
"He would also like to thank the Institute for Astronomy for their kind hospitality during the course of this project.",
"This research was also supported through Hawaii Space Grant Consortium fellowships.",
"JWB would like to thank Jason Hinkle and Emily Heckle for constructive criticism of the manuscript.",
"This work was supported by NASA grant 80NSSC19K1717.",
"BJS, CSK, and KZS are supported by NSF grant AST-1907570/AST-1908952.",
"BJS is also supported by NSF grants AST-1920392 and AST-1911074.",
"EG acknowledges support from NSF award AST-187215.",
"EG conducted part of this research during the Exostar19 program at the Kavli Institute for Theoretical Physics at UC Santa Barbara, which was supported in part by the National Science Foundation under Grant No.",
"NSF PHY-1748958.",
"EG was also supported as a visiting professor to the University of Göttingen by the German Science Foundation through DFG Research 644 Unit FOR2544 “Blue Planets around Red Stars\".",
"CSK and KZS are also supported by NSF grants AST-1515927 and AST-181440.",
"Support for JLP is provided in part by FONDECYT through the grant 1191038 and by the Ministry of Economy, Development, and Tourism's Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS.",
"SD acknowledges Project 11573003 supported by NSFC.",
"TAT is supported in part by NASA grant 80NSSC20K0531.",
"PJV is supported by the National Science Foundation Graduate Research Fellowship Program Under Grant No.",
"DGE-1343012.",
"We thank Ethan Kruse for uploading videos of the TESS FFIs to YouTube, as these are an invaluable aide when examining the images.",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology.",
"WISE and NEOWISE are funded by the National Aeronautics and Space Administration.",
"This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.",
"This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST).",
"Funding for the TESS mission is provided by NASA's Science Mission directorate.",
"This work is based on observations made by ASAS-SN.",
"We wish to extend our special thanks to those of Hawaiian ancestry on whose sacred mountain of Haleakalā, we are privileged to be guests.",
"Without their generous hospitality, the observations presented herein would not have been possible."
]
] | 2005.14201 |
[
[
"On Sustainable Equilibria"
],
[
"Abstract Following the ideas laid out in Myerson (1996), Hofbauer (2000) defined a Nash equilibrium of a finite game as sustainable if it can be made the unique Nash equilibrium of a game obtained by deleting/adding a subset of the strategies that are inferior replies to it.",
"This paper proves two results about sustainable equilibria.",
"The first concerns the Hofbauer-Myerson conjecture about the relationship between the sustainability of an equilibrium and its index: for a generic class of games, an equilibrium is sustainable iff its index is $+1$.",
"Von Schemde and von Stengel (2008) proved this conjecture for bimatrix games; we show that the conjecture is true for all finite games.",
"More precisely, we prove that an isolated equilibrium has index +1 if and only if it can be made unique in a larger game obtained by adding finitely many strategies that are inferior replies to that equilibrium.",
"Our second result gives an axiomatic extension of sustainability to all games and shows that only the Nash components with positive index can be sustainable."
],
[
"1"
]
] | 2005.14094 |
[
[
"One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory"
],
[
"Abstract We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues.",
"Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions.",
"We then use non-degenerate quantum-mechanical perturbation theory to compute the leading $1/N^2$ corrections to operator dimensions and as an example compute the large $R$-charge limit for two-excitation states through subleading order in the $R$-charge.",
"Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite $N$ to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite $N$."
],
[
"Introduction",
"The eigenvalue problem for the dilatation operator, $\\mathfrak {D}$ , acting on the set of gauge-invariant local operators, $\\mathcal {O}_i$ , in $\\mathcal {N}=4$ super Yang-Mills (SYM) theory, $\\mathfrak {D}\\cdot \\mathcal {O}_i =\\Delta _i \\mathcal {O}_i$ has been of continued interest due to its role as a proving ground for novel calculational techniques and because of its importance in the AdS/CFT correspondence.",
"The operator dimensions, $\\Delta _i =\\Delta _i(g_{\\rm YM}, N)$ , are non-trivial functions of the coupling $g_{\\rm YM}$ and $N$ , the rank of the gauge group.",
"In perturbation theory we can expand the dilatation operator in powers of the 't Hooft coupling $\\lambda =g_{\\rm YM}^2N$ $\\mathfrak {D}=\\sum _k g^{2k}\\mathfrak {D}_{2k}~~~\\text{with}~~~~g^2=\\frac{\\lambda }{16\\pi ^2}$ and at each order in $g^2$ we can further consider the large-$N$ expansion of the operator dimensions.",
"A key development [1] was the insight that for the $\\mathfrak {so}(6)$ sector of operators the one-loop, $\\mathcal {O}(g^2)$ , leading large-$N$ anomalous dimensions can be computed by means of an integrable spin chain.",
"Single-trace operators composed of $L$ scalar fields were identified with closed spin chains of length $L$ and the planar dilatation operator with a spin-chain Hamiltonian that can be diagonalised by use of the Bethe ansatz.",
"This was subsequently extended to the full one-loop theory [2] and to higher orders in perturbation theory [3] as well as being observed at strong coupling [4].",
"This prompted a great deal of work and ultimately lead to non-perturbative results for the spectrum of planar anomalous dimensions first through the thermodynamic Bethe ansatz and subsequently by means of the Quantum Spectral Curve (QSC), see [5], [6], [7], [8] for reviews.",
"While less is known about non-planar anomalous dimensions there are a number of impressive perturbative results for specific operators.",
"For example, twist-two operators at four loops were studied using standard Feynman diagramatics [9], [10] as well as twistor methods [11], and the four-loop non-planar correction to the cusp anomalous dimension was computed in [12].",
"Additionally, the Hexagon formalism for correlation functions, [13], [14], [15], [16], provides an integrability-based method to studying non-planar $\\mathcal {N}=4$ SYM.",
"The framework can be used to evaluate higher-point functions at higher genus and one can then in principle extract non-planar dimensions from OPE limits.",
"A second approach, which was applied at tree-level in [17], is to directly compute the non-planar two-point function by considering the four-point function with two operators taken to be the identity.",
"Moreover, there are results for the complete dilatation operator including its non-planar parts which was found at one-loop for the $\\mathfrak {so}(6)$ sector in [18], [19], [20] and for the full theory in [21].",
"Also at one-loop, a spin-bit model that captures some features of the string-bit formalism for interacting strings [22] was used in [23] to compute non-planar corrections for operators in the $\\mathfrak {su}(2)$ and $\\mathfrak {sl}(2)$ sectors.",
"The full two-loop dilatation operator in the $\\mathfrak {su}(2)$ sector was found in [3] and extended to the non-compact $\\mathfrak {su}(1,1|2)$ sector in [24] by using the superconformal symmetry of the theory.",
"The problem of diagonalising the non-planar dilatation operator has itself been studied using a number of techniques.",
"One approach makes use of group-theoretic insights, initially developed in the context of 1/2-BPS operators [25], to make an appropriate choice of basis operators diagonalising two-point functions.",
"Alternatively, one may attempt to perturbatively compute $1/N$ corrections about the planar limit using quantum-mechanical perturbation theory.",
"One splits the dilatation operator into a leading planar part and subleading off-diagonal terms, which mix single- and multi-trace operators, and then computes matrix elements of the subleading terms in the basis of planar eigenstates.",
"Such an approach was used to compute the large $R$ -charge limit of non-planar dimensions of two-impurity BMN operators in the $\\mathfrak {su}(2)$ sector at one-loop and at two-loop level [18], [19], [20].",
"In this work we will generalise this approach for the one-loop dilatation operator in the $\\mathfrak {su}(2)$ sector by making use of Bethe states describing arbitrary numbers of excitations or magnons.",
"Figure: The action of the non-planar dilatation operator on single-trace states (a) can be viewed as a simultaneous splitting of the spin-chain and an application of the planar Hamiltonian density, H j (0) H_j^{(0)}, on pairs of non-adjacent spins.",
"The action on double-trace operators (b) is given by applying, H j (0) H_j^{(0)}, to pairs of spins, one drawn from each spin-chain, while joining the two chains together.In Sec.",
"we compute the action of the off-diagonal terms on Bethe states, schematically shown in Fig.",
"REF $(a)$ , and write the overlap of the resulting state with the tensor product of two Bethe states, corresponding to a double-trace operator, in terms of off-shell scalar products.",
"The latter can be computed efficiently using the algebraic Bethe ansatz [26], [27] or, equivalently at this order, a Hexagon-like formalism [13].",
"We similarly find the action on tensor products of Bethe states, see Fig.",
"REF $(b)$ , and compute the overlap with single-trace Bethe states in terms of off-shell scalar products.",
"These overlaps can in principle be used to find corrected anomalous dimensions, or equivalently, spin-chain energies $\\Delta (g, N)=L+g^2 E(N)+\\mathcal {O}(g^4)~,~~~\\text{with}~~~E(N)=\\sum _{k=0}^\\infty \\frac{1}{N^k} E^{(k)}$ by performing the sum over all such overlaps.",
"Naively, as the overlaps are of order $1/N$ , we expect the leading correction to be of order $1/N^2$ .",
"However, due to degeneracies in the planar spectrum, non-degenerate perturbation can fail and can result in order $1/N$ corrections to dimensions.",
"The issue of planar degeneracies between operators with different numbers of traces was noted in [28], [20], [3], see also [29], [30], and requires solving a non-trivial mixing problem.",
"We will instead consider deformations of $\\mathcal {N}=4$ SYM theory for which these degeneracies are lifted.",
"In particular, we consider $\\beta $ -deformed $\\mathcal {N}=4$ SYM which is a marginal deformation of the maximally supersymmetric theory [31] preserving $\\mathcal {N}=1$ supersymmetry.",
"We concentrate on the case $\\beta \\in \\mathbb {R}$ for which the theory is exactly conformal to all loop orders in the planar limit [32].",
"Most importantly, the planar spectral problem for the $\\beta $ -deformed theory is described by an integrable twist of the undeformed spin chain.",
"Twisted asymptotic Bethe equations at one and higher loops were found in [33], while the twisted QSC was found in [34] and used in [35] to study the anomalous dimensions of operators corresponding to the Konishi multiplet.",
"In Sec.",
"we fix the form of the deformed non-planar dilatation operator, which includes additional double-trace terms, and then expand the action on operators in powers of $1/N$ .",
"In addition to the leading planar piece and the subleading $1/N$ terms, which are similar to those occuring in the undeformed theory, there are $1/N^2$ and $1/N^3$ terms which contribute to non-planar dimensions.",
"We compute the matrix elements of the subleading dilatation operator in the basis of Bethe states and study the BMN limit of large $R$ -charge, $J=L-2\\rightarrow \\infty $ , where perturbation theory can be rewritten in terms of the effective loop- and genus-counting parameters [36] $g^{\\prime }=\\frac{g^2_{\\text{YM}} N}{16\\pi ^2 J^2}~~~\\text{and}~~~ g_2=\\frac{J^2}{N}~.$ The anomalous dimensions of two-impurity BMN operators, which additionally depend on an integer parameter $n$ and rescaled deformation parameter $b=\\beta L/\\pi $ , can be written in terms of rescaled energies $\\tilde{E}^{(k)}=J^{2-2k} E^{(k)}$ $\\Delta _n(g^{\\prime },g_2, b, J)=L+ g^{\\prime } \\big [ \\tilde{E}^{(0)}_n(b,J)+ g_2^2 \\tilde{E}^{(2)}_n(b, J)+\\mathcal {O}(g_2^4)\\big ]+\\mathcal {O}((g^{\\prime })^2)~$ and we compute $\\tilde{E}^{(2)}_n(b, J)$ through $\\mathcal {O}(J^{-1})$ .",
"Besides the lifting of degeneracies, there are a number of further advantages to considering the deformed theory.",
"At a technical level it allows us to consider singular solutions to the undeformed Bethe equations.",
"Such solutions correspond to finite energies but have singular wavefunctions and so matrix elements in the undeformed theory are not well defined.",
"Instead they can be computed in the deformed theory, where the deformation parameter regularises singularities in the wavefunctions [37], and the limit of vanishing deformation parameter can be smoothly taken.",
"More conceptually, the dependence of the spectrum on a continuous parameter allows for the phenomenon of level crossing whereby eigenvalues become degenerate at special values of the parameter.",
"As was noted in the early days of quantum mechanics by von Neumann and Wigner [38], given a generic theory depending on a number of parameters it is necessary to tune at least two parameters to cause energy levels to cross and produce a degeneracy.",
"Subsequently it was shown by Teller [39] that the surfaces $E=E(\\beta _1, \\beta _2)$ representing energy levels depending on two such parameters $\\beta _1, \\beta _2$ are connected at points like the two sheets of a degenerate cone.",
"In [40] the dimensions of local operators in $\\mathcal {N}=4$ SYM were analysed as functions of the 't Hooft coupling and it was shown, by re-summing the large-$N$ expansion, that when $N$ is held fixed anomalous dimensions do not cross as $\\lambda $ is varied.",
"Studying the one-loop spectrum as a function of the deformation parameter $\\beta $ , we similarly find that at finite $N$ the anomalous dimensions repel and it is only at large $N$ that they cross.",
"Level repulsion is characteristic of a chaotic system where energy levels are correlated and so avoid each other, while in an integrable system they are uncorrelated and move independently, crossing on occasion.",
"The phenomenon of (non-)repulsing energy levels can be studied by looking at the distribution of spacings between neighbouring energy levels.",
"If one computes the probability, $P(s)ds$ , that the normalised spacing between adjacent levels lies in the interval between $s$ and $s+ds$ , one finds that for a generic, chaotic, quantum system $P(s)\\rightarrow 0$ as $s\\rightarrow 0$ .",
"For integrable systems it is generally the case that $P(s)$ goes to a constant as $s\\rightarrow 0$ which reflects the presence of hidden symmetries in these models.",
"In Sec.",
"we numerically study the spectrum of both the deformed and undeformed theories and show that in the planar limit the spectral distribution is Poisson, consistent with integrability, while at finite $N$ the distribution is Wigner-Dyson and corresponds to that of the Gaussian Orthogonal Ensemble (GOE) random matrix theory.",
"We are thus able to numerically study the transition from quantum-integrable to quantum-chaotic systems in the context of interacting four-dimensional gauge theory as we vary $N$ .",
"An analysis of the transition from Poisson to Wigner-Dyson statistics in a context similar to planar $\\mathcal {N}=4$ SYM was performed in [41] which considered and integrability breaking deformation of the XXZ spin-chain.",
"The appearance of quantum chaos in the spectrum is in fact quite natural if we view the dilatation operator as the Hamiltonian of the theory defined on $\\mathbb {R}\\times S^3$ and multi-trace operators as defining states somewhat analogous to large nuclei in QCD.",
"Indeed it was the work of Wigner [42] and Porter & Rosenzweig [43] on the statistical properties of the energy levels of highly-excited nuclei that lead to much of the initial interest of physicists in random matrix theory [44]."
],
[
"Non-Planar Dilatation Operator",
"In order to fix our conventions and notations we briefly review the one-loop dilatation operator of $\\mathcal {N}=4$ SYM.",
"We follow closely the work [3] where more details regarding the calculations and generalisations to higher loops can be found.",
"$\\mathcal {N}=4$ SYM theory contains six scalar fields, $(\\phi _I)^a{}_b$ , $I=1,\\dots 6$ , $a,b=1,\\dots N$ which transform as a vector of the $SO(6)\\simeq SU(4)$ R-symmetry and in the adjoint representation of the $SU(N)$ gauge group.",
"We will restrict ourselves to the $\\mathfrak {su}(2)$ sector comprising operators made of products of traces of two complex scalar fields, $Z=\\tfrac{1}{\\sqrt{2}}(\\phi _{5}+i \\phi _6)$ and $X=\\tfrac{1}{\\sqrt{2}}(\\phi _{1}+i \\phi _2)$ , and so we consider operators such as $\\text{Tr}(Z^{\\ell _1})~,~~~\\text{Tr}(XZ^{\\ell _1} XZ^{\\ell _2})~,~~~\\text{Tr}(XZ^{\\ell _1}XZ^{\\ell _2} Z^{\\ell _3})\\text{Tr}(X Z^{\\ell _4})\\text{Tr}(XX)~.$ These operators can be organised into $SO(6)$ representations with Dynkin labels $[M,L-2M,M]$ .",
"This sector is known to be closed under the action of the dilatation operator and does not mix with operators containing other scalars, field strengths or fermions.",
"To describe the action of the dilatation operator it is useful to make use of the notation for functional derivatives of fields, for example $(\\check{Z})^a{}_b\\equiv \\frac{\\delta }{\\delta (Z)^b{}_a}$ such that $(\\check{Z})^a{}_b(Z)^c{}_d=\\delta _b^c\\delta ^a_d-N^{-1}\\delta ^a_b\\delta ^c_d~, ~~~\\text{and}~~~(\\check{Z})^a{}_b (X)^c{}_d=0~.$ This can be used to derive the fusion and splitting formulae $& &\\text{Tr}(A \\check{Z})~ \\text{Tr}(B Z)=\\text{Tr}(AB)-N^{-1} \\text{Tr}A~\\text{Tr}B~,\\nonumber \\\\& & \\text{Tr}(A\\check{Z}B Z)=\\text{Tr}A~ \\text{Tr}B -N^{-1}\\text{Tr}(AB)~,$ where it is assumed that $A$ and $B$ do not contain any $Z$ 's.",
"The $N^{-1}$ terms are due to the fact that we are considering the $SU(N)$ gauge theory.",
"This is not particularly important for $\\mathcal {N}=4$ SYM and we could equally well consider a $U(N)$ gauge group, however it will become relevant when we subsequently consider the $\\beta $ -deformed theory.",
"Using this notation, the tree-level dilatation operator in the $\\mathfrak {su}(2)$ sector can be written as $\\mathfrak {D}_0=\\text{Tr}(Z\\check{Z})+\\text{Tr}(X\\check{X})$ and simply counts the number of fields present in a given operator.",
"The one-loop correction to the dilatation operator is then given by [18] $\\mathfrak {D}_2=-\\frac{2}{N} :\\text{Tr}([X,Z ][\\check{X},\\check{Z}]):$ where the normal-ordering markers $:\\,\\,:$ indicate that the functional derivatives do not act on the fields in $\\mathfrak {D}_2$ itself.",
"We can find the action on multi-trace operators by repeated use of identities (REF ).",
"For example on the length-six single-trace operator, $\\text{Tr}(X^2 Z^4)$ , we have $\\mathfrak {D}_2\\text{Tr}(X^2 Z^4)=4\\Big (\\text{Tr}(X^2Z^4)-\\text{Tr}(XZXZ^3)\\Big )+\\frac{4}{N}\\Big (\\text{Tr}(X^2 Z^2)\\text{Tr}(Z^2)-\\text{Tr}(X ZXZ)\\text{Tr}(Z^2)\\Big )~,$ where we see that the leading term in a large-$N$ expansion corresponds to a superposition of single-trace operators and the subleading term is a double-trace contribution.",
"In general, we can decompose the action of the one-loop dilatation operator on multi-trace operators into planar and non-planar pieces $\\mathfrak {D}_2=H^{(0)}+\\frac{1}{N}H^-+\\frac{1}{N}H^+~.$ The planar piece, $H^{(0)}$ , leaves the number of traces in an operator unchanged, while the non-planar corrections, $H^{\\pm }$ , which are suppressed by a factor of $1/N$ , increase/reduce the number of traces in a given operator.",
"In order to find the eigenvalues of $\\mathfrak {D}_2$ , one can first solve the planar problem using integrability and then attempt to use perturbation theory to find the $1/N^k$ corrections."
],
[
"Planar Theory and Integrability",
"Mapping the problem of computing anomalous dimensions to that of computing integrable spin-chain energies proved to be an important step in solving the planar spectral problem.",
"We will make use of the spin-chain notation and the results from integrability to organise the computation of non-planar corrections.",
"We thus review the coordinate Bethe-ansatz approach to integrable spin chains here.",
"Single-trace operators with $M$ insertions of $X$ fields in a background of $(L-M)$ $Z$ 's will be denoted as $\\text{Tr}(\\overbrace{Z\\dots Z}^{n_1-1}X\\overbrace{Z\\dots Z}^{n_2-n_1-1}X\\dots )&\\equiv &\\mathopen {|}\\overbrace{\\uparrow \\dots \\uparrow }^{n_1-1}\\downarrow \\overbrace{\\uparrow \\dots \\uparrow }^{n_2-n_1-1}\\downarrow \\dots \\mathclose {\\rangle }_L\\nonumber \\\\&\\equiv &\\mathopen {|}n_1, n_2,\\dots ,n_M\\mathclose {\\rangle }_{L}~.$ Multi-trace operators with $K$ traces and $M$ insertions of $X$ fields, where $M=\\sum _{k=1}^K M_{k}$ , can be denoted by products of such states $\\prod _{k=1}^K\\mathopen {|}n_1^{(k)},\\dots , n_{M_{k}}^{(k)}\\mathclose {\\rangle }_{L_k}$ which is an element of the symmetrized tensor product.",
"It will often be convenient to use the compressed notation $\\mathopen {|}\\lbrace n\\rbrace \\mathclose {\\rangle }$ and $\\prod _k \\mathopen {|}\\lbrace n^{(k)}\\rbrace \\mathclose {\\rangle }$ .",
"We can write general single-trace states as linear combinations of this basis in terms of a wavefunction $\\psi _{\\lbrace n\\rbrace }$ which can depend on the quantum numbers describing the particular state.",
"For example we will consider states with $M$ impurities characterised by the momenta $\\lbrace p\\rbrace =\\lbrace p_1,p_2, \\dots , p_M\\rbrace $ of $M$ excitations, or magnons $\\mathopen {|}\\lbrace p\\rbrace \\mathclose {\\rangle }=\\sum _{\\lbrace n\\rbrace }\\psi _{\\lbrace n\\rbrace }^{\\lbrace p\\rbrace }\\mathopen {|}\\lbrace n\\rbrace \\mathclose {\\rangle }~.$ The sum is over the positions of excitations ranging over the nested values $1\\le n_1< n_2 <\\dots <n_M\\le L$ .",
"We will compute overlaps of such spin-chain states and so we define the natural dual basisThis choice does not take into account the cyclicity of the traces which we thus need to impose as a separate condition.",
"$\\mathopen {\\langle }m_1, m_2, \\dots , m_M|n_1,n_2, \\dots , n_M\\mathclose {\\rangle }=\\prod _{j=1}^M\\delta _{m_j, n_j}$ so that the scalar product for states $\\mathopen {|}\\tilde{\\psi }\\mathclose {\\rangle }$ and $\\mathopen {|}\\psi \\mathclose {\\rangle }$ is given by $\\mathopen {\\langle }\\tilde{\\psi }|\\psi \\mathclose {\\rangle }=\\sum _{\\lbrace n\\rbrace } \\tilde{\\psi }^\\ast _{\\lbrace n\\rbrace } {\\psi }_{\\lbrace n\\rbrace }~.$ The action of the planar dilatation operator can be defined in this basis and is given by the well-known formula $H^{(0)}\\mathopen {|}n_1,n_2,\\dots \\mathclose {\\rangle }_L=2 \\sum _{j=1}^M \\Big (2\\mathopen {|} \\dots , n_j, \\dots \\mathclose {\\rangle }-\\mathopen {|}\\dots , n_j-1,\\dots \\mathclose {\\rangle }-\\mathopen {|}\\dots ,n_j+1, \\dots \\mathclose {\\rangle }\\Big )~.$ This spin-chain Hamiltonian can be diagonalised by means of the Bethe ansatz.",
"The ferromagnetic vacuum state with no impurities $\\mathopen {|}\\emptyset \\mathclose {\\rangle }=\\mathopen {|}\\uparrow \\uparrow \\dots \\uparrow \\mathclose {\\rangle }$ is an eigenstate with zero energy.",
"Eigenstates with $M$ impurities have wavefunctions given as sums over $\\sigma \\in \\mathcal {S}_M$ , permutations of $M$ objects, $\\psi _{\\lbrace n\\rbrace }^{\\lbrace p\\rbrace }\\equiv \\psi _{n_1,\\dots , n_M} ^{p_1, \\dots , p_M}=\\frac{1}{\\prod _{j<k}\\sqrt{S(p_j,p_k)}}\\sum _{\\sigma \\in \\mathcal {S}_M}e^{i \\sum _{j=1}^M p_{\\sigma (j)}n_j}\\prod _{\\begin{array}{c}j>k \\\\ \\sigma (j)<\\sigma (k)\\end{array}} S(p_{\\sigma (j)},p_{\\sigma (k)})$ where $S(p_j,p_k)$ is the two-magnon S-matrix $S(p_j,p_k)=-\\frac{e^{ip_j+ip_k}+1-2e^{i p_k}}{e^{ip_j+ip_k}+1-2e^{i p_j}}~.$ In this definition we have made a particular choice for the overall, non-physical, phase of the wavefunction which is convenient for our subsequent purposes.",
"For these states to satisfy periodic boundary conditions the momenta must satisfy the Bethe equations, i.e.",
"for each $j=1,\\dots , M$ $e^{i \\phi _j}=1~,~~~~\\text{where }~~~e^{i \\phi _j}\\equiv e^{ip_j L} \\prod _{k\\ne j}^M S(p_j, p_k)~,$ which implies that the wavefunctions satisfy the condition $\\psi ^{\\lbrace p\\rbrace }_{n_1, n_2, \\dots , n_M}=\\psi ^{\\lbrace p\\rbrace }_{n_2,\\dots , n_M, n_1+L}~.$ Each eigenstate corresponds to a solution of the algebraic equations (REF ) and the energy eigenvalue is given as a sum over individual magnon energies $E^{(0)}(\\lbrace p\\rbrace )= \\sum _{j=1}^M \\varepsilon (p_j)~,~~~~ \\varepsilon (p_j)=8 \\sin ^2 \\frac{p_j}{2}~.$ The cyclicity of the trace for gauge-theory operators becomes the condition that the spin chain is invariant under the shift $n_j \\rightarrow n_j+1$ and so we consider only states which satisfy the condition $\\prod _{j=1}^Me^{i p_j}=1~.$ It is useful to introduce rapidity variables $u_j=\\frac{1}{2}\\cot \\tfrac{p_j}{2}$ , or $e^{ip_j}=\\frac{u_j+i/2}{u_j-i/2}~,$ for each excitation, which we can use to rewrite the S-matrix and the individual magnon energies as $S(u_j, u_k)=\\frac{u_j-u_k-i}{u_j-u_k+i}~~~\\text{and}~~~\\varepsilon (u_j)=\\frac{2}{u_j^2+\\tfrac{1}{4}}~.$ It will be useful to define the quantity $h(u_j,u_k)= \\frac{u_j-u_k}{u_j-u_k+i}$ so that the S-matrix is given as $S(u_j,u_k)= \\frac{h(u_j,u_k)}{h(u_k,u_j)}\\,.$ Additionally, it is convenient to define the normalisation factors for states involving momenta $p_1, p_2, \\dots $ corresponding to rapidities $u_1, u_2, \\dots $ as $\\mathcal {N}(p(u))=\\frac{\\prod _{i<j} h(u_i, u_j)}{\\prod _{j<k}\\sqrt{S(u_j,u_k)}}$ and generalisations such as $\\mathcal {N}(p, q)=\\mathcal {N}(p)\\mathcal {N}(q)$ .",
"Finally, similar to [45], we will use the following short-hand notation for products $f^{\\lbrace a\\rbrace } = \\prod _{i} f(a_i)\\,, \\qquad f^{\\lbrace a\\rbrace }_< = \\prod _{i<j} f(a_i,a_j)\\,, \\qquad h^{\\lbrace a\\rbrace \\lbrace b\\rbrace } = \\prod _{i,j} h (a_i,b_j)\\,$ and $\\lbrace z\\rbrace _{\\hat{a}} = \\lbrace z_1, \\ldots , \\hat{z}_a, \\ldots , z_n\\rbrace =\\lbrace z_1, \\ldots , z_{a-1}, z_{a+1}, \\ldots , z_n\\rbrace $ for lists with a missing element.",
"Using this notation a Bethe state can be written as $\\mathopen {|}\\lbrace p\\rbrace \\mathclose {\\rangle }=\\mathcal {N}(p)\\sum _{\\lbrace n\\rbrace }\\sum _{\\sigma \\in S_M} \\frac{1}{h^{\\lbrace p_\\sigma \\rbrace }_< } e^{i p_\\sigma \\cdot n} \\mathopen {|}\\lbrace n\\rbrace \\mathclose {\\rangle }\\,$ and its conjugate as $\\mathopen {\\langle }\\lbrace p\\rbrace \\mathclose {|}=\\mathcal {N}(p^\\ast )\\sum _{\\lbrace n\\rbrace }\\sum _{\\sigma \\in S_M} \\frac{1}{h^{\\lbrace p^*_\\sigma \\rbrace }_> } e^{-i p^*_\\sigma \\cdot n} \\mathopen {\\langle }\\lbrace n\\rbrace \\mathclose {|}\\,$ where the functions $h^{\\lbrace p\\rbrace }_<$ etc.",
"should be understood as being defined in terms of the set of rapidities $\\lbrace u\\rbrace $ corresponding to the momenta $\\lbrace p(u)\\rbrace $ ."
],
[
"Perturbative Non-Planar Anomalous Dimensions",
"In this section we study the action of the non-planar dilatation operator on the planar eigenstates, the Bethe vectors, and subsequent overlaps with other Bethe states.",
"We consider first the action of $H^-$ on a double-trace operator corresponding to the product of a length $L_q$ Bethe state $|\\lbrace q\\rbrace \\rangle $ with $Q$ excitations and a length $L_r$ state $|\\lbrace r\\rbrace \\rangle $ with $R$ excitations.",
"There are $L_q L_r$ terms corresponding to the action of the dilatation operator on each pair of sites of the two spin chains.",
"However, the terms where it acts on a $Z$ field at the $i$ -th site of the spin chain can be rewritten using the Bethe equations, so that they become equivalent to the action on a $Z$ at the first site.",
"This can be seen by gathering all the terms in the state (REF ) with a $Z$ field at the $i$ -th site and using the cyclicity and on-shell condition to write them with the $Z$ field at the first position: $\\sum _{l=0}^M \\sum _{\\begin{array}{c}1\\le n_1<\\ldots < n_l \\le i-1\\\\i+1\\le n_{l+1}<\\ldots < n_M \\le L\\end{array}}\\psi ^{\\lbrace p\\rbrace }_{\\lbrace n\\rbrace } |n_1,\\ldots ,n_l\\rangle _{i-1} \\otimes \\mathopen {|}Z\\mathclose {\\rangle }\\otimes |n_{l+1},\\dots , n_M\\rangle _{L-i}&\\nonumber \\\\&\\hspace{-100.0pt}= \\sum _{2\\le n_1<\\ldots < n_M\\le L } \\psi _{\\lbrace n\\rbrace }^{\\lbrace p\\rbrace } \\;\\mathopen {|}Z\\mathclose {\\rangle }\\otimes | \\lbrace n\\rbrace \\rangle _{L-1}\\,.$ Analogously, the action on an $X$ field at the $i$ -th site can be rewritten as the action on the same chain with the $X$ field placed at the first site $\\sum _{l=1}^{M} \\sum _{\\begin{array}{c}1\\le n_1<\\ldots < n_l=i\\\\i=n_{l}<\\ldots < n_{M} \\le L\\end{array}}\\psi ^{\\lbrace p\\rbrace }_{n_1,\\ldots ,n_l=i, \\ldots , n_{M}} |n_1,\\ldots ,n_l=i,\\ldots , n_M\\rangle =\\sum _{1= n_1<\\ldots < n_{M}\\le L} \\psi _{\\lbrace n\\rbrace }^{\\lbrace p\\rbrace } \\;| \\lbrace n\\rbrace \\rangle \\,.$ With these and similar simplifications the action of $H^-$ on the double-trace state can be written as $H^-|\\lbrace q\\rbrace \\rangle |\\lbrace r\\rbrace \\rangle = 2 L_q L_r&\\Bigg [~~\\sum _{\\begin{array}{c}1\\le m_1 < \\ldots < m_{Q}= L_q\\\\2\\le n_1 < \\ldots < n_{R}\\le L_r\\end{array}} \\psi ^{\\lbrace q\\rbrace }_{\\lbrace m\\rbrace } \\psi ^{\\lbrace r\\rbrace }_{\\lbrace n\\rbrace } |\\lbrace m\\rbrace _{\\hat{Q}}\\rangle \\otimes |[X,Z]\\rangle \\otimes |\\lbrace n+L_q\\rbrace \\rangle \\nonumber \\\\&+\\sum _{\\begin{array}{c}1\\le m_1 < \\ldots < m_{Q}\\le L_q-1\\\\1= n_1 < \\ldots < n_{R}\\le L_r\\end{array}} \\psi ^{\\lbrace q\\rbrace }_{\\lbrace m\\rbrace } \\psi ^{\\lbrace r\\rbrace }_{\\lbrace n\\rbrace } |\\lbrace m \\rbrace \\rangle \\otimes |[Z,X]\\rangle \\otimes |\\lbrace n+L_q\\rbrace _{\\hat{1}}\\rangle \\nonumber \\\\&+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace ~~ \\Bigg ]\\,$ where the terms on the right hand side all correspond to single-trace operators.",
"The overlap with a dual state $\\mathopen {\\langle }\\lbrace p\\rbrace \\mathclose {|}$ , of length $L_p=L_q+L_r$ and with $P=Q+R$ excitations, can then be computed $\\langle \\lbrace p\\rbrace | H^- | \\lbrace q\\rbrace \\rangle | \\lbrace r\\rbrace \\rangle &= 2L_q L_r \\, \\mathcal {N}(p^\\ast , q, r)\\sum _{\\rho ,\\sigma ,\\tau } \\frac{1}{h_>^{\\lbrace p^*_\\rho \\rbrace } h_<^{\\lbrace q_\\sigma \\rbrace } h_<^{\\lbrace r_\\tau \\rbrace }}\\nonumber \\\\&\\hspace{-35.0pt}\\times \\Bigg ( \\delta _{Q\\ne 0} (e^{i p^*_{\\rho (Q)}}-1) e^{i L_q q_{\\sigma (Q)}} e^{-i(L_q+1)(p^*_\\rho )_{Q}^{Q+R}} P_{L_q}\\left(\\lbrace q_{\\sigma }-p^*_\\rho \\rbrace _1^{Q-1} \\right) P_{L_r}\\left(r_\\tau -\\lbrace p^*_\\rho \\rbrace _{Q+1}^{Q+R} \\right)\\nonumber \\\\&\\ + \\delta _{Q\\ne 0} (1- e^{i p^*_{\\rho (R+1)} } )e^{-i(L_r+1) (p_\\rho ^*)_{R+1}^{Q+R}} P_{L_q}\\left(\\lbrace q_{\\sigma }\\rbrace _{2}^Q-\\lbrace p^*_\\rho \\rbrace _{R+2}^{R+Q}\\right) P_{L_r}\\left(r_\\tau -\\lbrace p^*_\\rho \\rbrace _1^R\\right) \\nonumber \\\\&\\ +\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace \\Bigg ) \\,, $ where we define the sets $\\lbrace t_\\lambda \\rbrace _a^b = \\lbrace t_{\\lambda (a)},\\ldots ,t_{\\lambda (b)}\\rbrace $ and denote products of exponentials over such sets using the notation $e^{i L (t_\\lambda )_a^b}=\\prod _{i=a}^b e^{i L t_{\\lambda (i)}}$ .",
"The factors of $P_L(z)$ in (REF ) correspond to the geometric sums of exponentials in the wavefunctions which can be rewritten as sums over ordered partitions $P_L(z)= \\sum _{1\\le n_1 < \\ldots < n_{|z|} \\le L-1} e^{i z\\cdot n} = \\sum _{l=0}^{|z|} \\prod _{k=1}^l \\frac{1}{e^{-i \\sum _{j=k}^l z_l} -1} \\prod _{k=l+1}^{|z|} \\frac{e^{i z_k L}}{e^{i \\sum _{j=l+1}^k z_l}-1}\\,.$ Using these notations we can write a similar expression for overlaps of $H^+$ as sums over ordered partitions: $\\langle \\lbrace r\\rbrace |\\langle \\lbrace q\\rbrace | H^+ | \\lbrace p\\rbrace \\rangle &=2 L_p\\, \\mathcal {N}_+(p,q^\\ast , r^\\ast ) \\sum _{\\rho ,\\sigma ,\\tau } \\frac{1}{h_<^{\\lbrace p_\\rho \\rbrace } h_>^{\\lbrace q^\\ast _\\sigma \\rbrace } h_>^{\\lbrace r^*_\\tau \\rbrace }} \\nonumber \\\\&\\hspace{-60.0pt}\\times \\Bigg ( \\delta _{Q\\ne 0} (e^{i q^*_{\\sigma (Q)}}-1) e^{-i L_q q^*_{\\sigma (Q)}} e^{i(L_q-1) \\lbrace p_\\rho \\rbrace _{Q}^{Q+R}} P_{L_q-1}\\left(\\lbrace p_\\rho - q^*_{\\sigma }\\rbrace _1^{Q-1} \\right) P_{L_r+1}\\left(\\lbrace p_\\rho \\rbrace _{Q+1}^{Q+R} -r^*_\\tau \\right)\\nonumber \\\\&+ \\delta _{Q\\ne 0} (1- e^{i q^*_{\\sigma (1)}} ) e^{i(L_r+1) (p_\\rho )_{R+1}^{R+Q}} P_{L_q-1}\\left(\\lbrace p_\\rho \\rbrace _{R+2}^{R+Q}-\\lbrace q_\\sigma ^*\\rbrace _2^Q\\right) P_{L_r+1}\\left(\\lbrace p_\\rho \\rbrace _1^R-r^*_\\tau \\right) \\nonumber \\\\&+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace \\Bigg ) \\,.$ The normalisation in this case is defined slightly differently with $\\mathcal {N}_+=\\mathcal {N}/S$ , where $S$ is a symmetry factor that equals 2 when the states in the double trace are equal and 1 otherwise.",
"Carrying out the geometric sums via (REF ) makes these formulae useful for analysing states of arbitrary lengths.",
"However, while these expressions are reasonably compact, they involve sums over permutations for each of the sets of external momenta and so they rapidly become impractical as the number of excitations grows.",
"The same growth is known from the computation of spin-chain scalar products in the coordinate Bethe ansatz and by making use of known results in this case we can find further simplifications."
],
[
"Matrix Elements from Spin-Chain Scalar Products",
"The scalar product of two Bethe states $\\langle \\lbrace l\\rbrace | \\lbrace k\\rbrace \\rangle _L&=& \\sum _{1\\le n_1 < \\dots n_{|k|} \\le L} \\psi ^\\ast {}^{\\lbrace l\\rbrace }_{\\lbrace n\\rbrace } \\psi ^{\\lbrace k\\rbrace }_{\\lbrace n\\rbrace }= \\mathcal {N}(k,l^\\ast ) \\sum _{\\rho ,\\sigma } \\frac{P_{L+1}(k_\\rho - l^*_\\sigma )}{h_>^{\\lbrace l^*_\\sigma \\rbrace }h_<^{\\lbrace k_\\rho \\rbrace }}$ involves double-sums over permutations and so is generally complicated to evaluate.",
"Fortunately, there are well-known formulae for such scalar products which were developed in the Algebraic Bethe Ansatz (ABA) approach to integrable spin chains (see Sec.",
"for a brief review).",
"In the case where both sets of momenta $ \\lbrace k\\rbrace $ and $\\lbrace l\\rbrace $ do not satisfy the Bethe equations, i.e.",
"they are off-shell, the scalar product can be written as a sum over partitions of the sets of momenta into subsets of equal cardinality [46], see (REF ).",
"Similar simplifications can be used to rewrite the expressions of the overlaps (REF ) and (REF ).",
"Each term in the formulae for the overlaps non-trivially involves one momentum of an excitation from the single-trace operator, which we label $p_j$ , and one excitation momentum from the double-trace operator, i.e.",
"from either $\\lbrace q\\rbrace $ or $\\lbrace r\\rbrace $ , which we label as $q_i$ or $r_i$ .",
"The remaining momenta are simply contracted using a rescaled spin-chain scalar product $(\\lbrace l\\rbrace |\\lbrace k\\rbrace )_L\\equiv \\frac{\\mathopen {\\langle }\\lbrace l\\rbrace |\\lbrace k\\rbrace \\mathclose {\\rangle }_L}{\\mathcal {N}(k,l^\\ast )}~.$ We can thus write the overlaps (REF ) and (REF ) in terms of the off-shell scalar products by splitting the single-trace excitation momenta into three subsets, $\\lbrace p \\rbrace = s \\cup t \\cup \\lbrace p_j \\rbrace $ , with the cardinality of $s$ equal to that of $\\lbrace q \\rbrace _{\\hat{i}}$ (or $\\lbrace r \\rbrace _{\\hat{i}}$ ) and the cardinality of $t$ equal to that of $\\lbrace r\\rbrace $ (resp.",
"$\\lbrace q\\rbrace $ ).",
"In terms of off-shell scalar products, the overlap of $H^-$ can then be written as a sum over all such splittings $\\langle \\lbrace p\\rbrace | H^- | \\lbrace q\\rbrace \\rangle | \\lbrace r\\rbrace \\rangle &=2 L_q L_r \\, \\mathcal {N}(p^*,q,r)\\sum _{\\begin{array}{c}i,j \\\\ s \\cup t = \\lbrace p\\rbrace _{\\hat{j}}\\end{array}}\\frac{e^{i p^*_j}-1}{h^{q_i q_{\\hat{i}}}}\\big [s^{L_q+1\\, \\ast }_\\circlearrowleft -t^{L_r+1\\, \\ast }_\\circlearrowright \\big ](s| \\lbrace q\\rbrace _{\\hat{i}})_{L_q-1}(t|\\lbrace r\\rbrace )_{L_r-1}\\nonumber \\\\&\\hspace{150.0pt}+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace $ and that of $H^+$ as $\\langle \\lbrace r\\rbrace |\\langle \\lbrace q\\rbrace | H^+ | \\lbrace p\\rbrace \\rangle &=2 L_p\\, \\mathcal {N}_+(p,q^\\ast ,r^*)\\sum _{\\begin{array}{c}i,j \\\\ s \\cup t = \\lbrace p\\rbrace _{\\hat{j}}\\end{array}}\\frac{e^{i q^*_i}-1}{h^{q^\\ast _{\\hat{i}}q_i^\\ast }}\\big [s^{L_q-1}_{j\\,\\circlearrowleft }-t^{L_r+1}_{j\\, \\circlearrowright }\\big ]( \\lbrace q \\rbrace _{\\hat{i}}|s)_{L_q-2}( \\lbrace r\\rbrace |t)_{L_r}\\nonumber \\\\&\\hspace{180.0pt}+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace ~.$ In addition to the scalar products of Bethe states these expressions involve $(e^{ip}-1)$ factors, which are essentially the same as arise in the planar dilatation operator, and ordering factors for which we introduced the notations $s^{L}_{j\\, \\circlearrowleft }= \\frac{e^{-i L s} }{h^{ p_j t} h^{s p_j} h^{s t} }~,~~~ t^{L}_{j\\, \\circlearrowright } =\\frac{e^{-i L t} }{h^{ p_j s } h^{t p_j} h^{ t s } }~.$ These terms account for the phase acquired by the $p_j$ magnon as it is shifted around the chain before being contracted with a magnon on the double-trace operator.",
"For each configuration there are two different ways to carry out this reordering and the overlap is a linear combination of both.",
"Spin-chain scalar products have previously appeared in the context of $\\mathcal {N}=4$ SYM in the computation of structure constants.",
"In the all-order hexagon approach, [13], structure constants are written as sums over partitions of the magnon excitations and it was noted that this formulation is related to the scalar-product formula of Korepin [46].",
"It is therefore convenient to use a tree-level version of the hexagon formulation of scalar products $(\\lbrace l(v)\\rbrace | \\lbrace k(u)\\rbrace )&=(-1)^M\\prod _{j=1}^M (u_j+i/2)(v^*_j-i/2) \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }=\\lbrace k\\rbrace \\\\ \\beta \\cup \\bar{\\beta }= \\lbrace l^*\\rbrace \\end{array}} \\frac{e^{i L (\\bar{\\alpha }-\\bar{\\beta })} G(\\alpha ,\\beta ) G(\\bar{\\beta },\\bar{\\alpha })}{h^{\\alpha \\bar{\\alpha }} h^{\\bar{\\beta }\\beta } } \\,,$ where $G(\\alpha (u), \\beta (v^\\ast )) = \\frac{\\det \\left[ \\frac{i}{(u_j-v^*_k)(u_j-v^*_k+i)}\\right] \\prod _{j,k}(u_j-v^*_k+i) }{\\prod _{j<k} (u_k-u_j)(v^*_j-v^*_k)}\\,,$ in order to rewrite the overlaps (REF ,REF )."
],
[
"A Hexagon-like Formulation",
"In the previous section we obtained the non-planar dilatation operator overlaps as sums over partitions of the rapidities, in a way that is reminiscent of the Hexagon formulation of three-point correlation functions [13].",
"In that context, the partitions of the rapidities arise naturally in the large-volume regime where the correlation function is broken down to its simplest building blocks, the Hexagon form factors.",
"Crucially, these form factors satisfy a set of axioms which, together with the diagonal symmetries and some educated guesswork, can be used to obtain an all-loop description of structure constants.",
"A particular feature of the Hexagon is its conical defect which is associated with the existence of three asymptotic regions and corresponds to a monodromy composed by three crossing operations.",
"In the context of non-planar overlaps between a single-trace and a double-trace operator, a similar role seems to be played by the three distinct traces.",
"In this section we investigate the properties of the objects arising from the action of $H^+$ and $H^-$ and find that they satisfy the same form factor axioms as appear in the context of correlation functions.",
"The sum over determinants occuring in our rewriting of off-shell scalar products (REF ) can be found in a straighforward way from the hexagonalization of three-point functions [13].",
"To be precise we consider the three-point function of two unprotected operators in the $SU(2)$ sector, one with $X$ excitations and the other with $\\bar{X}$ , and one rotated half-BPS operator.",
"The $X$ and $\\bar{X}$ fields must be Wick contracted at tree level in order to produce a non-vanishing contribution.",
"If there are $l$ Wick contraction between the excited operators, then the structure constant is $C^{\\bar{X}| X}_{\\lbrace p\\rbrace |\\lbrace q\\rbrace } \\propto \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }= \\lbrace p\\rbrace \\\\\\ \\beta \\cup \\bar{\\beta }= \\lbrace q\\rbrace \\end{array}} \\omega _l(\\alpha ,\\bar{\\alpha }) \\omega _{L_q-l}(\\beta ,\\bar{\\beta }) \\mathcal {H} (\\alpha | \\beta ) \\mathcal {H} (\\bar{\\beta }| \\bar{\\alpha })\\,,$ with the splitting factor defined as $\\omega _l(\\alpha ,\\bar{\\alpha }) = e^{i \\bar{\\alpha }l } \\prod _{\\begin{array}{c}u_i \\in \\bar{\\alpha }, u_j \\in \\alpha \\\\i<j\\end{array}} S(u_i,u_j)\\,.$ The Hexagon function $\\mathcal {H}$ in this particular configuration is simply related to our determinant expression (REF ) by $\\mathcal {H}(\\alpha |\\beta ) = h_<^{\\alpha } h_<^{\\beta } G(\\alpha , \\beta ) \\,.$ The Hexagon description of three-point functions allows the evaluation of general configurations where all three operators are excited.",
"If we now let two of the operators have $\\bar{X}$ excitations, while the other is composed of $X$ fields, then the structure constant becomes $C^{\\bar{X}| X| \\bar{X}}_{\\lbrace p\\rbrace |\\lbrace q\\rbrace |\\lbrace r\\rbrace } \\propto \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }= \\lbrace p\\rbrace \\\\ \\beta \\cup \\bar{\\beta }= \\lbrace q\\rbrace \\\\ \\gamma \\cup \\bar{\\gamma }= \\lbrace r\\rbrace \\end{array}} \\omega _{l_{pq}}(\\alpha ,\\bar{\\alpha }) \\omega _{l_{qr}}(\\beta ,\\bar{\\beta })\\omega _{l_{pr}}(\\gamma ,\\bar{\\gamma })\\mathcal {H}(\\alpha | \\beta | \\gamma ) \\mathcal {H}(\\bar{\\gamma }| \\bar{\\beta }| \\bar{\\alpha })\\,,$ where $l_{ij}$ denote the number of Wick contraction between operators $i$ and $j$ at tree level and the sum over partitions is further restricted by the fact that $\\mathcal {H}(\\alpha |\\beta |\\gamma )$ is non-vanishing only when the cardinality of $\\beta $ matches that of $\\alpha \\cup \\gamma $ .",
"It is interesting to note that while (REF ) is given as a sum over partitions of three sets of rapidities, a naive tree-level evaluation would give rise to geometric sums naturally yielding a sum over five partitions $C^{\\bar{X}| X| \\bar{X}}_{\\lbrace p\\rbrace |\\lbrace q\\rbrace |\\lbrace r\\rbrace } \\propto \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }= \\lbrace p\\rbrace \\\\ \\gamma \\cup \\bar{\\gamma }= \\lbrace r\\rbrace \\end{array}} \\sum _{\\begin{array}{c}s \\cup t = \\lbrace q\\rbrace \\\\\\beta \\cup \\bar{\\beta }= s \\\\ \\delta \\cup \\bar{\\delta }= t\\end{array}} \\frac{e^{i(\\bar{\\alpha }-\\bar{\\beta })l_{12}} e^{i(\\bar{\\gamma }-\\bar{\\delta })l_{23}} e^{i s l_{12}}}{h^{\\alpha \\bar{\\alpha }}h^{\\gamma \\bar{\\gamma }} h^{\\bar{\\delta }\\delta } h^{\\bar{\\beta }\\beta } h^{ts}} G(\\alpha ,\\beta ) G(\\bar{\\beta },\\bar{\\alpha }) G(\\gamma ,\\delta ) G(\\bar{\\delta },\\bar{\\gamma }) \\,.$ The equivalence of these descriptions follows from the following tree-level relation $\\mathcal {H}(\\alpha |\\beta |\\gamma )= h_<^\\alpha h_<^\\beta h_<^\\gamma \\sum _{\\mu \\cup \\nu = \\beta } \\frac{G(\\alpha ,\\mu )G(\\nu ,\\gamma )}{h^{\\mu \\nu }}\\,.$ We should stress that, computationally speaking, (REF ) is not necessarily a more efficient version of (REF ) as the objects $\\mathcal {H}(\\alpha |\\beta |\\gamma )$ do not have a known compact determinant description.",
"Nevertheless, while the computational gain might not be considerable, there is a conceptual advantage due to the fact that the Hexagon functions can be bootstrapped.",
"First, they obey the Watson equation $\\mathcal {H}( \\ldots | \\ldots , \\beta _i, \\beta _{i+1},\\ldots | \\ldots ) = S(\\beta _i,\\beta _{i+1}) \\mathcal {H}(\\ldots | \\ldots , \\beta _{i+1},\\beta _i \\ldots |\\ldots ) \\,,$ which holds also for the exchange of excitations in the other edges, and they also satisfy the decoupling conditions $-i \\underset{\\alpha _{|\\alpha |} = \\beta _1}{\\mathrm {Res}} \\left[\\mathcal {H}(\\ldots , \\alpha _{|\\alpha |} | \\beta _1, \\ldots | \\ldots ) \\right] &= \\mathcal {H}(\\ldots ,\\alpha _{|\\alpha |-1}| \\beta _2,\\ldots | \\ldots )\\,, \\nonumber \\\\-i \\underset{\\beta _{|\\beta |} = \\gamma _1}{\\mathrm {Res}} \\left[\\mathcal {H}(\\ldots | \\ldots , \\beta _{|\\beta |} | \\gamma _1, \\ldots ) \\right] &= \\mathcal {H}(\\ldots ,| \\ldots ,\\beta _{|\\beta |-1}| \\gamma _2,\\ldots )\\,.$ Together with the diagonal symmetries of three-point functions, these form-factor axioms allowed the determination of the hexagon functions at any value of the coupling [13].",
"With this in mind, we can attempt a similar rewriting of the dilatation operator overlaps.",
"It is useful to work with normalised spin-chain states where we divide by the norms of on-shell Bethe states $\\Vert \\lbrace p\\rbrace \\Vert =\\sqrt{\\mathopen {\\langle }\\lbrace p\\rbrace |\\lbrace p\\rbrace \\mathclose {\\rangle }}$ .",
"These can be conveniently calculated using the Gaudin formula (REF ) which for the coordinate Bethe-ansatz normalisation is $\\Vert \\lbrace p(u)\\rbrace \\Vert ^2=(-1)^M \\prod _j (u_j+i/2)(u_j-i/2)\\, \\text{det}\\, \\partial _u \\phi (u)$ with $\\phi $ defined in (REF ).",
"This can be combined with the normalisation factors of the overlaps to define a new normalisation factor $\\widetilde{\\mathcal {N}}(p(u),q(v),r(w))= \\frac{\\mathcal {N}(p,q,r)}{h_<^{\\lbrace p\\rbrace } h_<^{\\lbrace q\\rbrace } h_<^{\\lbrace r\\rbrace } \\sqrt{\\text{det}\\, \\partial _u \\phi (u)\\, \\text{det}\\, \\partial _v \\phi (v)\\, \\text{det}\\, \\partial _w \\phi (w)} }~,$ where we used the fact that solutions of the Bethe equations are invariant under complex conjugation, e.g.",
"$\\lbrace u^\\ast \\rbrace =\\lbrace u\\rbrace $ , [47], and the cyclicity condition to simplify the expressionsThere is a potential ambiguity in our simplifications arising from square roots of S-matrices in $\\mathcal {N}$ .",
"There are in principle combinations of rapidities such that products of S-matrices cross the square-root branch cut resulting in additional minus signs in the normalisation.",
"The same ambiguity seems to appear in the Gaudin norm and so will cancel.",
"Moreover, these signs will appear symmetrically in the overlaps of $H^-$ and $H^+$ and thus certainly cancel in the calculation of energies..",
"The overlap with normalised external states can then be written as $V^-(q,r;p) =~& 2 L_q L_r \\,\\widetilde{\\mathcal {N}}(p,q,r) \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }= \\lbrace q\\rbrace \\\\ \\beta \\cup \\bar{\\beta }= \\lbrace p\\rbrace \\\\ \\gamma \\cup \\bar{\\gamma }= \\lbrace r\\rbrace \\end{array}} \\omega _{L_q}(\\alpha , \\bar{\\alpha }) \\omega _{L_r}(\\beta , \\bar{\\beta }) \\omega _0(\\gamma , \\bar{\\gamma }) \\times \\nonumber \\\\&\\hspace{-30.0pt}\\times \\Bigg [\\mathcal {H}(\\alpha |\\beta |\\gamma ) \\Big (\\mathcal {H}^-_1(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha }) +\\mathcal {H}^-_2(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha })\\Big )+ \\Big ( \\mathcal {H}^-_1(\\alpha |\\beta |\\gamma ) +\\mathcal {H}^-_2(\\alpha |\\beta |\\gamma )\\Big )\\mathcal {H}(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha })\\Bigg ]\\,,$ where $\\mathcal {H}$ is the same as in (REF ), and we define the new functions $\\mathcal {H}^-_i$ as $\\mathcal {H}^-_1(\\alpha |\\beta |\\gamma ) &= h_<^{\\alpha } h_<^{\\beta } h_<^{\\gamma } \\sum _{\\begin{array}{c}i,j \\\\\\mu \\cup \\nu =\\beta _{\\hat{j}}\\end{array}} \\frac{(e^{i\\beta _j}-1)(e^{-i\\beta _j}-1)(e^{i \\alpha _i}-1)G(\\alpha _{\\hat{i}},\\mu ) G(\\nu ,\\gamma )}{e^{i (\\mu +\\gamma -\\alpha _{\\hat{i}}-\\nu )}h^{\\alpha _{i} \\alpha _{\\hat{i}}} h^{\\mu \\beta _j} h^{\\beta _j \\nu } h^{\\mu \\nu }}\\,,\\nonumber \\\\\\mathcal {H}^-_2(\\alpha |\\beta |\\gamma ) &= h_<^{\\alpha } h_<^{\\beta } h_<^{\\gamma } \\sum _{\\begin{array}{c}i,j\\\\ \\mu \\cup \\nu =\\beta _{\\hat{j}}\\end{array}} \\frac{(e^{i\\beta _j}-1)(e^{-i\\beta _j}-1)(e^{-i \\gamma _{i}}-1)G(\\alpha ,\\mu ) G(\\nu ,\\gamma _{\\hat{i}})}{e^{i (\\mu +\\gamma _{\\hat{i}}-\\alpha -\\nu )}h^{\\gamma _{\\hat{i}} \\gamma _i} h^{\\mu \\beta _j} h^{\\beta _j \\nu } h^{\\mu \\nu }}\\,.$ We remind the reader that $\\mu _{\\hat{k}}$ denotes the set of rapidities $\\mu $ without $\\mu _k$ , following the notation introduced earlier in (REF ), while the short-hand notations for products are defined in (REF ).",
"This decomposition of the overlap in (REF ) seems to fit the splitting of Figure REF (b) particularly well.",
"By cutting the pair of pants depicted in that figure, one would naively expect the side facing away to be represented by the original hexagon $\\mathcal {H}$ of (REF ), while the side facing forward should lead to something new as it contains the action of the commutator from the dilatation operator.",
"Similarly, the overlap $V^+$ can be rewritten as $V^+(p;q,r) =~& 2 L_p \\,\\widetilde{\\mathcal {N}}_+(p,q,r) \\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }= \\lbrace q\\rbrace \\\\ \\beta \\cup \\bar{\\beta }= \\lbrace p\\rbrace \\\\ \\gamma \\cup \\bar{\\gamma }= \\lbrace r\\rbrace \\end{array}} \\omega _{L_q}(\\alpha , \\bar{\\alpha }) \\omega _{L_r}(\\beta , \\bar{\\beta }) \\omega _0(\\gamma , \\bar{\\gamma }) \\times \\nonumber \\\\&\\hspace{-50.0pt}\\times \\Bigg [\\mathcal { H}^+_0(\\alpha |\\beta |\\gamma ) \\Big (\\mathcal { H}^+_1(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha }) +\\mathcal { H}^+_2(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha })\\Big )+ \\Big ( \\mathcal { H}^+_1(\\alpha |\\beta |\\gamma ) +\\mathcal { H}^+_2(\\alpha |\\beta |\\gamma )\\Big )\\mathcal { H}^+_0(\\bar{\\gamma }|\\bar{\\beta }|\\bar{\\alpha })\\Bigg ]\\,,$ where the normalization is now $\\widetilde{\\mathcal {N}}_+ = \\widetilde{\\mathcal {N}}/S$ , with $S$ a symmetry factor that equals 2 when the states in the double-trace are the same, and 1 otherwise, and we have further defined the functions $\\mathcal {H}^+_i$ $\\mathcal { H}^+_0(\\alpha |\\beta |\\gamma ) &= h_<^{\\alpha } h_<^{\\beta } h_<^{\\gamma } \\sum _{\\mu \\cup \\nu =\\beta } e^{i (\\alpha -\\mu )}\\frac{G(\\alpha ,\\mu ) G(\\nu ,\\gamma )}{h^{\\mu \\nu }}\\,,\\nonumber \\\\\\mathcal { H}^+_1(\\alpha |\\beta |\\gamma ) &= h_<^{\\alpha } h_<^{\\beta } h_<^{\\gamma } \\, e^{i(\\alpha +\\gamma -\\beta )}\\sum _{\\begin{array}{c}i,j\\\\\\mu \\cup \\nu =\\beta _{\\hat{j}}\\end{array}} \\frac{(e^{i \\alpha _i}-1)(e^{-i \\alpha _i}-1)(e^{-i \\beta _j}-1)G(\\alpha _{\\hat{i}},\\mu ) G(\\nu ,\\gamma )}{e^{i (\\mu -\\alpha _{\\hat{i}})}h^{\\alpha _i \\alpha _{\\hat{i}}} h^{\\mu \\beta _j} h^{\\beta _j \\nu } h^{\\mu \\nu }}\\,,\\nonumber \\\\\\mathcal { H}^+_2(\\alpha |\\beta |\\gamma ) &= h_<^{\\alpha } h_<^{\\beta } h_<^{\\gamma } \\sum _{\\begin{array}{c}i,j\\\\\\mu \\cup \\nu =\\beta _{\\hat{j}}\\end{array}} \\frac{(e^{i \\gamma _i}-1)(e^{-i \\gamma _i}-1)(e^{i \\beta _j}-1)G(\\alpha ,\\mu ) G(\\nu ,\\gamma _{\\hat{i}})}{e^{i (\\gamma _{\\hat{i}}-\\nu )}h^{ \\gamma _{\\hat{i}} \\gamma _i} h^{\\mu \\beta _j} h^{\\beta _j \\nu } h^{\\mu \\nu }}\\,.$ Unfortunately, in this case we are not able to write any of the new objects in terms of the original hexagon function $\\mathcal {H}$ , since the partitions of the rapidities $\\beta $ into $\\mu $ and $\\nu $ appear with a distinct structure.",
"The decomposition is however very similar to that of $V^-$ , and seems to match once again the intuition derived from Figure REF (a), with the cutting producing a product between a simpler structure with a more complex ones.",
"We wish to emphasize that while these formulae appear quite involved, they can be straightforwardly evaluated once the rapidities are known, by using, for example, Mathematica.",
"While these expressions are a post hoc massaging of the expressions in (REF ,REF ), when written in this form they clearly resemble the formulas for structure constants.",
"Importantly, the new objects $\\mathcal {H}_i^+$ and $\\mathcal {H}^-_j$ also obey the Watson equations (REF ) and decoupling conditions (REF ).",
"Note that these properties of $\\mathcal {H}^+_i$ and $\\mathcal {H}^-_i$ follow from those of the object $\\mathcal {H}(\\alpha |\\beta )$ defined in (REF ).",
"This is non-trivial nevertheless, as it occurs only for certain functions of the rapidities in the summands of (REF ,REF ).",
"This hints at the possibility that the non-planar dilatation operator overlaps can be written in terms of Hexagon-like objects and potentially determined even at higher orders in perturbation theory.",
"The corrections to the energies of single-trace operators are obtained through the action of $H^+$ and $H^-$ and a sum over intermediate double-trace operators, which has the natural representation of cutting the torus into two pairs of pants.",
"The fact that the overlaps themselves have a decomposition into hexagon-like objects therefore seems to indicate a possible tesselation of the torus as depicted in Figure REF .",
"Figure: Both V + V^+ and V - V^- can be seen as a pair of pants where the asymptotic regions correspond to the three distinct traces involved in the overlap.",
"We have found that each of them can be decomposed into hexagon-like objects satisfying the Watson and decoupling conditions.",
"By glueing them together one can reconstruct the torus, thus finding the non-planar corrections to two-point functions.There is an implicit notion of crossing that comes with the decoupling condition.",
"It is natural to imagine that, once such an operation is defined, the excitations can be moved around, so that we relate the hexagon-like objects to a single function where all rapidities are on the same edge.",
"It is upon crossing of the excitations in (REF ) to the same edge that a particle-antiparticle pair $\\bar{X}(u^{2\\gamma }) X(u)$ can form in a manifest way and decouple from the corresponding form factor.",
"Such a formulation of $\\mathcal {H}^+_i$ and $\\mathcal {H}^-_i$ with all excitations on the same edge would also be the ideal setup for implementing a bootstrap of those objects.",
"Unfortunately, crossing operations do not commute with the perturbative expansion, and since our one-loop analysis gives access only to the more complicated form of these objects, we were not able to explore further the possibility of such a bootstrap programme."
],
[
"Anomalous Dimensions from Overlaps",
"Our main goal in calculating the above overlaps is to perturbatively compute the leading non-planar correction to operator anomalous dimensions.",
"The general idea is to apply first-order quantum-mechanical perturbation theory.",
"We denote the planar energies as $E^{(0)}$ and their non-planar corrections at order $N^{-k}$ as $E^{(k)}$ .",
"Given a single-trace operator characterised by momenta $\\lbrace p(u)\\rbrace $ solving the Bethe equations and with planar energy $E^{(0)}(\\lbrace p\\rbrace )$ , the non-planar correction is $E^{(2)}(\\lbrace p\\rbrace )=\\sum _{\\lbrace I\\rbrace }\\frac{ V^-(p; I)V^+(I; p)}{E^{(0)}(\\lbrace p\\rbrace )-E^{(0)}(\\lbrace I\\rbrace )}\\,.$ The sum over $I$ is taken over all intermediate double-trace states $\\mathopen {|}I\\mathclose {\\rangle }=\\mathopen {|}\\lbrace q\\rbrace \\mathclose {\\rangle }_{L_q}\\mathopen {|}\\lbrace r\\rbrace \\mathclose {\\rangle }_{L_r}$ where we must sum over all lengths $1<L_q<L_p-1$ and for each length sum also over all solutions $\\lbrace q\\rbrace $ , $\\lbrace r\\rbrace $ of the Bethe equations corresponding to operators of lengths $L_q$ and $L_r$ with planar energy $E^{(0)}(\\lbrace I\\rbrace )=E^{(0)}(\\lbrace q\\rbrace )+E^{(0)}(\\lbrace r\\rbrace )$ .",
"As a simple example let us consider the unprotected operators of length six in the $[2,2,2]$ $SO(6)$ representation.",
"There are two single-trace operators with planar energies and rapidities given by $E^{(0)}_{(6,2a)}&=&2(5+ \\sqrt{5})~, ~~~ u_{(6,2a),1}=-{u_{(6,2a),2}}=\\frac{1}{2}\\sqrt{1-\\frac{2}{\\sqrt{5}}}~,\\nonumber \\\\E^{(0)}_{(6,2b)}&=&2(5- \\sqrt{5})~, ~~~ u_{(6,2b),1}=-{u_{(6,2b),2}}=\\frac{1}{2}\\sqrt{1+\\frac{2}{\\sqrt{5}}}~,$ both of which mix with the double-trace operator with rapidities $u_{(4,2),1}=-u_{(4,2),2}=1/2\\sqrt{3}$ and planar energy $E_{(4,2)}^{(0)}=12$ .",
"The overlaps can be simply found from the general formulae (REF ) and (REF ) $& & V^-({ u_{(6,2a)}}; u_{(4,2)}, \\emptyset )=\\frac{4}{3}(5+3\\sqrt{5})~,~~~V^-({u_{(6,2b)}}; u_{(4,2)}, \\emptyset )=\\frac{4}{3}(5-3\\sqrt{5})$ and $V^+({u_{(4,2)}}, \\emptyset ;u_{(6,2a)})=V^+({u_{(4,2)}}, \\emptyset ;u_{(6,2b)})=6\\sqrt{2}$ .",
"The resulting non-planar corrections are $E_{(6,2a)}^{(2)}=8(5+ 2\\sqrt{5})~~~\\text{and}~~~E_{(6,2b)}^{(2)}=8(5- 2\\sqrt{5})~.$ These results are in agreement with those found by direct calculation [48], and follow from diagonalising the dilatation operator [3].",
"In the case where there are only two magnons we can in fact solve the Bethe equation for any length, $L$ , if we consider only cyclic solutions with $\\prod _j e^{i p_j}=1$ .",
"In terms of the momenta such solutions are given by $p_1=-p_2=\\frac{2 \\pi n}{L-1}~,~~~$ with $n\\in \\mathbb {Z}$ and $0<n<\\frac{L-1}{2}$ .",
"Given such a complete set of solutions it is possible to numerically carry out the sum over intermediate states so that we can compute quite efficiently the corrections to energies even for states with quite long lengths, e.g.",
"$n=1$ for $L=100, 250, 400$ , which to six digits gives $E_{L=\\lbrace 100,250,400\\rbrace }^{(2)}=L^2 \\lbrace 0.758732, 0.770021, 0.772582\\rbrace ~.$ From this and similar numerical examples it can be seen that the corrections to the energies of long operators scale as $L^2/N^2$ .",
"This is essentially the well-known BMN limit [36] where one considers operators with large R-charge, $J$ .",
"The non-planar corrections to two-magnon states in the BMN limit were computed in [18], [28], see also [49], [20] and shown to be $\\Delta _n=L+g^{\\prime }\\Big [16\\pi n^2+g_2^2\\left(\\frac{1}{3}+\\frac{35}{8\\pi ^2 n^2}\\right)\\Big ]~.$ It is straightforward to check that our general expressions reproduce this result by substituting the two-magnon rapidities, $u_{n,1}=-u_{n, 2}=\\tfrac{L-1}{2\\pi n}$ , into (REF ) and (REF ) and taking the large $L$ limit.",
"We must consider the overlaps with all double-trace operators consisting of a vacuum state of length $(1-r)L$ and two-magnon states with rapidities $u_{m,1}=-u_{m, 2}=\\tfrac{r L-1}{2\\pi m}$ .",
"Following [20], we then expand in $L$ , sum over $m=0, \\dots , \\infty $ and approximate the sum over intermediate lengths by an integral over $r$ from 0 to 1.",
"At leading order in $J=L-2$ this reproduces (REF ), while at subleading orders we find the same result but with $J$ replaced with $L-1=J+1$ which is the natural parameter from the perspective of the Bethe equations.",
"It is naturally interesting to consider higher numbers of excitations.",
"For example at $L=7$ with three excitations, i.e.",
"for states in the $[3,1,3]$ representation, we have two single-trace operators with planar dimensions $E^{(0)}_{(7,3a/b)}=10$ .",
"Due to the degeneracy of the states, a naive application of relation (REF ) will fail as it is not clear which linear combination of the Bethe states to use as planar eigenstates.",
"We may use the fact that the two degenerate states are distinguished by their transformation under the parity operation [3].",
"This operator, $\\mathcal {P}$ , reverses the order of fields within each trace, for example $\\mathcal {P}:\\text{Tr}(XZXXZZ)\\mapsto \\text{Tr}(ZZXXZX)~,$ and commutes with the complete non-planar dilatation operator.",
"Thus the non-planar eigenoperators must have definite parity, and consequently also their planar limits.",
"The rapidities for the two $L=7$ and $M=3$ Bethe solutions $u_{(7,3a)}$ and $u_{(7,3b)}$ can be easily found using the method (and Mathematica programme) of [50].",
"They can be seen to transform into each other under parity which acts on finite rapidities by $u_i\\rightarrow -u_i$ while rapidities at infinity are left invariant.",
"The two parity eigenstates can then be formed from the corresponding Bethe eigenstates as $\\mathopen {|}\\pm \\mathclose {\\rangle }=\\tfrac{1}{\\sqrt{2}}(\\mathopen {|}u_{(7,3a)}\\mathclose {\\rangle }\\pm \\mathopen {|}{u_{(7,3b)}}\\mathclose {\\rangle } )$ .",
"Having identified the proper planar linear combinations, we can proceed by computing the mixing with double-trace operators.",
"We choose as our basis of double-trace operators $\\mathopen {|} u_{(5,3)} \\mathclose {\\rangle }_5\\mathopen {|} \\emptyset \\mathclose {\\rangle }_2~,~~~ \\mathopen {|} u_{(5,2)} \\mathclose {\\rangle }_5\\mathopen {|} \\infty \\mathclose {\\rangle }_2~,~~~\\mathopen {|} u_{(4,2)} \\mathclose {\\rangle }_4\\mathopen {|} \\infty \\mathclose {\\rangle }_3$ where we have labelled the Bethe states by the magnon rapidities rather than the momenta and $u_{(5,3)}=\\lbrace \\tfrac{1}{2}, -\\tfrac{1}{2}, \\infty \\rbrace $ , and $u_{(5,2)} =\\lbrace \\tfrac{1}{2}, -\\tfrac{1}{2} \\rbrace $ .",
"Both of these operators have positive parity and the linear combination $\\sqrt{\\tfrac{2}{3}}\\mathopen {|} u_{(5,3)} \\mathclose {\\rangle }_5\\mathopen {|} \\emptyset \\mathclose {\\rangle }_2-\\sqrt{\\tfrac{1}{3}}\\mathopen {|} u_{(5,2)} \\mathclose {\\rangle }_5\\mathopen {|} \\infty \\mathclose {\\rangle }_2$ is the remaining non-protected operator in the $[3,1,3]$ representation.",
"The other linear combination is a descendant of a two-excitation double-trace operator from $[2,3,2]$ .",
"The non-vanishing overlaps following from (REF ) and (REF ) are $& & V^-(u_{(7,3a)}; u_{(5,3)},\\emptyset )=V^-(u_{(7,3b)}; u_{(5,3)},\\emptyset )=2\\sqrt{\\tfrac{14}{3}}\\nonumber \\\\& & V^-(u_{(7,3a)}; u_{(5,2)},\\infty )=V^-(u_{(7,3b)}; u_{(5,2)},\\infty )=-2\\sqrt{\\tfrac{7}{3}}\\nonumber \\\\& & V^+(u_{(5,3)},\\emptyset ;u_{(7,3a)})=V^+(u_{(5,3)},\\emptyset ;{u_{(7,3b)}})=40\\sqrt{\\tfrac{2}{21}}\\nonumber \\\\& & V^+(u_{(5,2)},\\infty ;u_{(7,3a)})=V^+(u_{(5,2)},\\infty ;{u_{(7,3b)}})=-40\\sqrt{\\tfrac{1}{21}}~,$ while the overlaps involving $\\mathopen {|} u_{(4,2)} \\mathclose {\\rangle }_4\\mathopen {|} \\infty \\mathclose {\\rangle }_3$ are all zero, which is expected since primary operators cannot mix with descendants.",
"Now by applying (REF ) for the parity eigenstates, we find that the non-planar corrections arise from the mixing of the positive-parity eigenstate $\\mathopen {|}+\\mathclose {\\rangle }$ with the double-trace state (which has planar energy $E^{(0)}=8$ ) and are given by ${ E_{(7,3+)}^{(2)}}=80~,~~~{E_{(7,3-)}^{(2)}}=0$ which agrees with [51], [3].",
"The occurrence of degenerate parity pairs in the planar limit is quite general and so to use non-degenerate perturbation theory we must work within sectors of definite parity.",
"Unfortunately, as has been noted by several authors, for example in [28], [20], [3], the energies following from the Bethe equations demonstrate an additional degeneracy which is relevant to the mixing problem between multi-trace operators.",
"For example, if we consider the two-excitation states with Bethe solution (REF ), states with different lengths, $L_a$ , $L_b$ , and mode numbers, $n_a$ , $n_b$ , but equal ratios $\\tfrac{n_a}{L_a-1}=\\tfrac{n_b}{L_b-1}$ will have equal energies.",
"Correspondingly, in the planar limit, the single-trace state corresponding to the spin-chain state $\\mathopen {|}\\lbrace \\tfrac{2\\pi m}{L-1},-\\tfrac{2\\pi m}{L-1}\\rbrace \\mathclose {\\rangle }_L$ is degenerate with the double-trace state $\\mathopen {|} \\lbrace \\tfrac{2\\pi \\tilde{m}}{L-L_1-1},\\tfrac{2\\pi \\tilde{m}}{L-L_1-1} \\rbrace \\mathclose {\\rangle }_{ L-L_1} \\mathopen {|}\\emptyset \\mathclose {\\rangle }_{L_1}$ with $\\tfrac{ \\tilde{m}}{L-L_1-1}=\\tfrac{ m}{L-1}$ .",
"While it is less straightforward to show, analogous degeneracies generally also occur for higher excitation numbers.",
"As just one example, if we consider the $L=8$ operators with $M=3$ , we see that there are three solutions to the Bethe equations.",
"Two of which, whose rapidities we denote $u_{(8,3a)}$ and $u_{(8,3b)}$ , are degenerate parity pairs with energy ${E_{(8,3a/b)}^{(0)}}=8$ , while the third is a singular solution with energy ${E_{(8,3s)}^{(0)}}=12$ .",
"There is in this case a positive parity double-trace state which is degenerate $\\sqrt{\\tfrac{3}{4}}\\mathopen {|} u_{(5,3)} \\mathclose {\\rangle }_5\\mathopen {|} \\emptyset \\mathclose {\\rangle }_3-\\sqrt{\\tfrac{1}{4} }\\mathopen {|} u_{(5,2)} \\mathclose {\\rangle }_5\\mathopen {|} \\infty \\mathclose {\\rangle }_3~$ and which mixes with the positive-parity linear combination of single traces.",
"The mixing matrix can be computed from the overlaps and is $\\begin{pmatrix}0 & -4\\sqrt{15} \\\\-\\tfrac{32}{\\sqrt{15}} & 0\\end{pmatrix}$ from which we can compute the leading corrections to the energies ${ E^{(1)}}=\\pm 8\\sqrt{2}$ .",
"We can now proceed to use the corresponding eigenstates to find the subleading $1/N^2$ corrections.",
"As we proceed to longer lengths and more impurities, the need to diagonalise the mixing matrix will rapidly become difficult.",
"One way to avoid this problem is to deform the theory to remove such degeneracies.",
"In principle, if we can completely solve the deformed problem, one can then hope to remove the deformation parameter however as this requires resumming the $1/N$ corrections before removing the deformation we will only be able to make preliminary steps in this direction.",
"There is another reason for considering the deformed theory which has to do with the singular solutions of the Bethe equations.",
"Already at $L=6$ and $M=3$ there is a solution $u_1=i/2$ , $u_2=-i/2$ and $u_3=0$ for which the Bethe wavefunction is singular and naive application of the above formulae will lead to unphysical infinities.",
"It is possible to regularize the Bethe equations by the introduction of a twist, see [37] for a useful discussion and further references, which is equivalent to the deformation parameter we introduce below.",
"We can use the solutions of the twisted Bethe equations and the overlaps of the deformed theory to compute non-planar energies which reproduce the undeformed results in the limit of vanishing deformation."
],
[
"$\\beta $ -deformed SYM Theory",
"We now turn to the $\\beta $ -deformed $\\mathcal {N}=4$ SYM theory preserving $\\mathcal {N}=1$ supersymmetry.",
"The theory's Lagrangian may be obtained from the undeformed Lagrangian by replacing all products of fields by a Moyal-like $\\star $ -product where the non-commutativity occurs in the internal $SU(4)$ R-symmetry directions [52].",
"Using $\\mathcal {N}=1$ superspace formalism this corresponds to adding a single-trace deformation to the superpotential, however when written in terms of the component fields this results in both single-trace and double-trace deformations of the Lagrangian [53], [54].",
"For this theory the $U(N)$ gauge group is no longer conformal at the quantum level due to the couplings of $U(1)$ scalars.",
"These degrees of freedom decouple at the infrared fixed point corresponding to the $SU(N)$ theory and we will thus consider only the $SU(N)$ gauge group.",
"In the remainder of this section we use blue colour to denote how the $\\beta $ -deformation changes terms existing in the undeformed theory, and use purple to emphasize terms that are new."
],
[
"$\\beta $ -deformed Dilatation Operator",
"The planar dilatation operator for the deformed theory has been previously studied using both integrable methods [33] and direct field theory computations [55].",
"The non-planar dilatation operator can in principle be directly computed from the deformed Lagrangian using standard Feynman diagrammatics or perhaps more efficiently using on-shell methods [56].",
"We instead fix its form by using symmetries and known one-loop results.",
"The form of the single-trace part of the dilatation operator is simply inherited from the undeformed theory and is fixed by the planar theory.",
"It is found by replacing the commutators in (REF ) by the $\\beta $ -deformed commutator $[.,.",
"]_\\beta $ defined via the R-charges of the fields.",
"In the $\\mathfrak {su}(2)$ sector spanned by $X=\\phi _{14}$ and $Z=\\phi _{12}$ the only relevant commutator is $[X,Z]_{\\color {blue} {}\\beta {}}={\\color {blue} {}e^{i\\beta }{}}XZ-{\\color {blue} {}e^{-i\\beta }{}}ZX.$ This is supplemented by a double-trace term which is necessary to make the theory exactly conformal [54].",
"The form of this term follows from the deformed action [53], [54] and in the $\\mathfrak {su}(2)$ sector it becomes $:\\text{Tr}[X,Z]_{\\color {blue} {}\\beta {}} \\text{Tr} [\\check{X},\\check{Z}]_{\\color {blue} {}\\beta {}}:~.$ We fix the coefficient of this term by imposing that the operator $\\text{Tr}(XZ)$ is a protected operator $\\mathfrak {D}_2 \\text{Tr}(XZ)=0~.$ This has been shown perturbatively at one- and two-loop level by direct calculation [57], [58].",
"Using these conditions we find that the deformation leaves the tree-level dilatation operator (REF ) unchanged, while the one-loop correction (REF ) gets deformed to $\\mathfrak {D}_2=-\\frac{2}{N}\\left(:\\text{Tr}([X,Z]_{\\color {blue} {}\\beta {}} [\\check{X},\\check{Z}]_{\\color {blue} {}\\beta {}}):{\\color {purple} {}-\\frac{(e^{i \\beta }-e^{-i \\beta })^2}{N}:\\text{Tr}(XZ) \\text{Tr}(\\check{X}\\check{Z}):{}}\\right)~.$ It is important to note that although the double-trace term is suppressed by $1/N$ , it can be relevant at leading order when acting on short operators and results in the vanishing anomalous dimension of $\\text{Tr}(XZ)$ .",
"For longer operators in the planar limit this term is not relevant, however it is essential in understanding the non-planar corrections.",
"The fusion and splitting formulas (REF ) imply for the action of the one-loop dilatation operator on single-trace states $\\mathfrak {D}_2 \\text{Tr}(XAZB)&= \\frac{2}{N}\\Big ( {\\color {blue} {}e^{-i \\beta }{}}~\\text{Tr}(A)~\\text{Tr}([X,Z]_{\\color {blue} {}\\beta {}} B)-{\\color {blue} {}e^{i\\beta }{}}~\\text{Tr}([X,Z]_{\\color {blue} {}\\beta {}} A)~\\text{Tr}(B)\\Big )\\nonumber \\\\&\\quad {\\color {purple} {}+\\frac{2(e^{i\\beta }-e^{-i \\beta })}{N^2}\\Big (\\text{Tr}([X,Z]_{\\beta } \\lbrace A,B\\rbrace )+\\text{Tr}([X,Z]_{\\beta }) ~\\text{Tr}(A)~\\text{Tr}(B)\\Big ){}}\\nonumber \\\\&\\quad {\\color {purple} {}-\\frac{4(e^{i\\beta }-e^{-i\\beta })}{N^3}~\\text{Tr}([X,Z]_{\\beta })~\\text{Tr}(AB){}}~,$ where the double-trace part of the dilatation operator contributes the triple-trace term at order $1/N^2$ .",
"For the action of the dilatation operator on double-trace states we find $\\mathfrak {D}_2\\text{Tr}(XA)\\text{Tr}(ZB)&=\\frac{2}{N}\\Big ({\\color {blue} {}e^{-i\\beta }{}}\\text{Tr}([X,Z]_{\\color {blue} {}\\beta {}} BA)-{\\color {blue} {}e^{i\\beta }{}}\\text{Tr}([X,Z]_{\\color {blue} {}\\beta {}} AB)\\Big )\\nonumber \\\\&\\quad {\\color {purple} {}+\\frac{2(e^{i\\beta }-e^{-i\\beta })}{N^2}\\Big (\\text{Tr}(A)~\\text{Tr}([X,Z]_{\\beta } B)+\\text{Tr}([X,Z]_{\\beta } A)~\\text{Tr}(B){}}\\nonumber \\\\&\\quad {\\color {purple} {}+\\text{Tr}([X,Z]_{\\beta })~\\text{Tr}(AB)\\Big )-\\frac{4(e^{i\\beta }-e^{-i\\beta })}{N^3}\\text{Tr}([X,Z]_{\\beta })~\\text{Tr}(A)~\\text{Tr}(B){}}~.$ Relations (REF ) and (REF ) suggest that the deformed one-loop dilatation operator can be decomposed into planar and non-planar pieces similar to the undeformed case (REF ), however we now find subleading contributions and so we decompose (REF ) as $\\mathfrak {D}_2=H^{(0)}_{\\color {blue} {}\\beta {}}+\\frac{1}{N}H^-_{\\color {blue} {}\\beta {}}+\\frac{1}{N}H^+_{\\color {blue} {}\\beta {}} {\\color {purple} {}+\\frac{1}{N^2}H_\\beta ^{(2)}+\\frac{1}{N^3}H_\\beta ^{(3)} {}}.$ As for the undeformed case $H_\\beta ^{(0)}$ leaves the number of traces in an operator unchanged while $H^{\\pm }_\\beta $ increases/reduces the number of traces.",
"$H_\\beta ^{(2)}$ and $H_\\beta ^{(3)}$ are subleading terms which only arise in the deformed theory.",
"In particular, $H_\\beta ^{(2)}$ has a contribution which leaves the number of traces unchanged and so we have diagonal overlaps which we consider in Sec.",
"REF ."
],
[
"Deformed Planar Theory",
"The action of the planar dilatation operator on single-trace operators of length $L>2$ is quite similar to the undeformed action and is given by $H_{\\beta }^{(0)}\\mathopen {|}n_1,n_2,\\dots \\mathclose {\\rangle }_L=2\\sum _{j=1}^M \\Big (2\\mathopen {|}\\dots , n_j, \\dots \\mathclose {\\rangle }-{\\color {blue} {}e^{2i\\beta }{}}\\mathopen {|}\\dots , n_j-1,\\dots \\mathclose {\\rangle }-{\\color {blue} {}e^{-2i\\beta }{}}\\mathopen {|}\\dots ,n_j+1, \\dots \\mathclose {\\rangle }\\Big )~.$ It can be related to the integrable deformation of the Heisenberg XXX-Hamiltonian [59], [60] $H_D=\\sum _{i=1}^L \\big [ \\mathbb {1}_{i,i+1}-\\sigma _i^z \\sigma _{i+1}^z-2{\\color {blue} {} e^{2i \\beta }{}}\\sigma _i^- \\sigma _{i+1}^+ -2 {\\color {blue} {}e^{-2i \\beta }{}}\\sigma _i^+ \\sigma _{i+1}^-\\big ]~,$ so that the planar spectrum can still be solved using integrability.",
"As the Hamiltonian $H_D$ commutes with $\\sum _i \\sigma _i^z$ we can still consider sectors with fixed excitation number $M=L-\\sum _i \\sigma _i^z$ .",
"The vacuum corresponding to $M=0$ is the same as in the undeformed theory, i.e.",
"(REF ), and has energy $E^{(0)}(\\emptyset )=0$ .",
"Similarly, the one-excitation eigenstate is given by the usual Bethe state, but its energy becomes $E^{(0)}(p)=4(1-\\cos (p+{\\color {blue} {}2\\beta {}}))~$ which we see is no longer degenerate with $E_0$ when $p=0$ .",
"For more excitations the spectrum and wavefunctions are still given by the Bethe ansatz (REF ) but now with the deformed S-matrix $S_{\\beta }(p_j,p_k)=-\\frac{e^{i (p_j+p_k)} {\\color {blue} {}e^{2i\\beta }{}}+{\\color {blue} {}e^{-2i\\beta }{}}-2 e^{i p_k}}{e^{i (p_j+p_k)} {\\color {blue} {}e^{2i\\beta }{}}+{\\color {blue} {}e^{-2i\\beta }{}}-2 e^{i p_j}}~.$ The Bethe equations determining the momenta are as in the undeformed theory (REF ) but with the S-matrix replaced with $S_\\beta $ and the trace cyclicity condition still requires that the momenta satisfy ${\\rm exp}(i \\sum _{j}p_j)=1$ .",
"The dependence of the S-matrix on the deformation parameter can be removed by defining the shifted momenta $\\tilde{p}_j=p_j+2\\beta $ so that $S_{\\beta }(p_j,p_k)=-\\frac{e^{i (\\tilde{p}_j+\\tilde{p}_k)}+1-2 e^{i \\tilde{p}_k}}{e^{i (\\tilde{p}_j+\\tilde{p}_k)}+1-2 e^{i \\tilde{p}_k}}.$ However, this new parametrisation makes the parameter $\\beta $ manifest in the Bethe equations and cyclicity condition.",
"Introducing the rapidity variable $u=\\tfrac{1}{2}\\cot \\tfrac{\\tilde{p}}{2}$ they are given by $\\left(\\frac{u_j+\\tfrac{i}{2}}{u_j-\\tfrac{i}{2} }\\right)^L \\prod _{k\\ne j}^M \\frac{u_j-u_k-i}{u_j-u_k+i}={\\color {blue} {}e^{2 i L \\beta }{}}~~~~\\text{and}~~~~\\prod _{j=1}^M\\frac{u_j+\\tfrac{i}{2}}{u_j-\\tfrac{i}{2}}={\\color {blue} {}e^{2 i M\\beta }{}}~,$ thus in terms of the rapidity variable both $S_\\beta $ and the corresponding function $h_\\beta $ can be defined as in the undeformed case, i.e.",
"via () and (REF ).",
"One consequence of the deformation is that the degeneracy occurring in the undeformed theory between single-trace and double-trace operators (REF ) is lifted.",
"This can be seen directly in the case of single-trace operators corresponding to two-magnon states, $\\mathopen {|}\\lbrace k,-k\\rbrace \\mathclose {\\rangle }_L$ , by solving the Bethe equations to the first non-vanishing order in the deformation parameter $k(m,L)=\\frac{2\\pi m}{L-1} {\\color {purple} {}-\\frac{2\\beta ^2}{L-1}\\cot \\left(\\frac{m \\pi }{L-1}\\right)+\\mathcal {O}(\\beta ^4){}}~, ~~~m\\in \\mathbb {Z}~.$ For generic real values of $\\beta $ there will be no integers $\\tilde{m}$ and $L_1$ such that $k(\\tilde{m}, L-L_1)=k(m, L)$ and hence no double-trace operator will be degenerate with the single-trace operator.",
"While we do not have a similar proof for states with more excitations, direct diagonalisation of the dilatation matrix for operators with short lengths shows that the degeneracy of excited states is lifted in all cases of operators which were unprotected in the undeformed theory.",
"This reduced degeneracy increases the number of operators for which we can compute the non-planar corrections to the energies by using non-degenerate perturbation theory."
],
[
"Matrix Elements and Dimensions",
"The action of the non-planar dilatation operator on Bethe states and the corresponding overlaps can be computed by essentially the same method as for the undeformed theory.",
"For the deformed theory, if we wish to compute the corrections to the energies to order $\\mathcal {O}(1/N^2)$ we must consider not only the off-diagonal contributions from $H^\\pm _\\beta $ but also the diagonal contributions from $H^{(2)}_\\beta $ ."
],
[
"Off-diagonal Overlaps.",
"We can write the overlaps of $H^\\pm _\\beta $ using the notation of Sec.",
"REF .",
"As for the undeformed theory, the solutions of the deformed Bethe equations are invariant under complex conjugation which can be used to simplify the expressions.",
"For $H^-_\\beta $ the overlaps are almost identical to (REF ) $\\hspace{-10.0pt}\\langle \\lbrace p\\rbrace | H_\\beta ^- | \\lbrace q\\rbrace \\rangle | \\lbrace r\\rbrace \\rangle &=2 L_q L_r \\, \\mathcal {N}(p^\\ast ,q,r)\\times \\\\&\\hspace{-80.0pt}\\Bigg [\\sum _{\\begin{array}{c}i,j \\\\ s \\cup t = \\lbrace p\\rbrace _{\\hat{j}}\\end{array}}\\frac{e^{i p^\\ast _j} {\\color {blue} {}e^{ 2i\\beta }{}}-1}{h^{q_i q_{\\hat{i}}}}\\big [ {\\color {blue} {}e^{-2i \\beta } {}}s^{L_{q+1}\\, \\ast }_{j\\, \\circlearrowleft } - t^{L_r+1\\, \\ast }_{j\\, \\circlearrowright }\\big ] (s|\\lbrace q\\rbrace _{\\hat{i}})_{L_q-1} (t|\\lbrace r\\rbrace )_{L_r-1}+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace \\Bigg ]\\nonumber ~.$ The function $h$ in this formula has exactly the same form as in the undeformed theory (REF ) when written in terms of the on-shell rapidities, and similarly for the S-matrices implicit in the scalar products.",
"We should remember however that the rapidities themselves depend on the deformation parameter through the Bethe equations.",
"The overlaps of $H_\\beta ^+$ involve additional contributions in the case where one of the traces has length two and are given by $\\langle \\lbrace r\\rbrace |\\langle \\lbrace q\\rbrace | H_\\beta ^+ | \\lbrace p\\rbrace \\rangle &= 2 L_p\\, \\mathcal {N}_+ (p,q^\\ast ,r^\\ast ) \\, \\Bigg [\\sum _{\\begin{array}{c}i,j \\\\ s \\cup t = \\lbrace p\\rbrace _{\\hat{j}}\\end{array}}\\frac{1}{h^{q^\\ast _{\\hat{i}}q_i^\\ast }}\\Big [(e^{iq^*_i} {\\color {blue} {}e^{2i\\beta }{}}-1)( {\\color {blue} {}e^{-2i \\beta } {}} s^{L_q-1}_{j\\,\\circlearrowleft }-t^{L_r+1}_{j\\, \\circlearrowright })\\\\&\\hspace{-30.0pt}{\\color {purple} {}-4\\delta _{Q,1} \\delta _{L_q,2} \\sin ^2 \\beta (e^{i q^*_i} s^{L_q-1}_{j\\,\\circlearrowleft }+ t^{L_r+1}_{j\\, \\circlearrowright }) {}} \\Big ]( \\lbrace q \\rbrace _{\\hat{i}}|s)_{L_q-2}( \\lbrace r\\rbrace |t)_{L_r}+\\lbrace \\text{terms~with~} q \\rightleftarrows r\\rbrace \\Bigg ]~.\\nonumber $ As in the undeformed theory, dividing by the norms of the external states we can define the normalised overlaps $V_\\beta ^\\pm $ ."
],
[
"Diagonal Overlaps.",
"The contribution $H_\\beta ^{(2)}$ , which does not occur in the undeformed theory, to the dilatation operator (REF ) contains both length-preserving and -changing parts.",
"Here we are interested in the former, since the computation of non-planar corrections at order $1/N^2$ to the anomalous dimensions requires solely the diagonal overlap $\\mathopen {\\langle }\\lbrace p\\rbrace |H^{(2)}_\\beta |\\lbrace p\\rbrace \\mathclose {\\rangle }$ .",
"Using (REF ) one finds for the action of $H_\\beta ^{(2)}$ on a Bethe state (REF ) $&{\\color {purple} {}H^{(2)}_\\beta \\mathopen {|}\\lbrace p\\rbrace \\mathclose {\\rangle }{}}=2L_p(e^{i\\beta }-e^{-i\\beta })\\sum _{x=2}^{L_p}\\sum _{l=1}^P\\sum _{\\begin{array}{c}2\\le n_1<...<n_{l-1}<n_l= x \\\\ x<n_{l+1}<...<n_P\\le L_p\\end{array}}\\psi ^{\\lbrace p\\rbrace }_{\\lbrace n\\rbrace }~\\nonumber \\\\&\\hspace{28.45274pt}\\times \\Big (\\mathopen {|}[X,Z]_\\beta \\mathclose {\\rangle }\\otimes \\mathopen {|}n_1+1,...,n_{l-1}+1\\mathclose {\\rangle }_{x-2}\\otimes \\mathopen {|}n_{l+1},...,n_P\\mathclose {\\rangle }_{L_p-x}\\nonumber \\\\&\\hspace{56.9055pt}+\\mathopen {|}n_1-1,...,n_{l-1}-1\\mathclose {\\rangle }_{x-2}\\otimes \\mathopen {|}[X,Z]_\\beta \\mathclose {\\rangle }\\otimes \\mathopen {|}n_{l+1},...,n_P\\mathclose {\\rangle }_{L_p-x}-2\\delta _{L_p,2}\\mathopen {|}[X,Z]_\\beta \\mathclose {\\rangle }\\Big )\\nonumber \\\\&\\hspace{28.45274pt}+\\lbrace \\text{double-trace~terms}\\rbrace ,$ where the $\\delta _{L_p,2}$ -term arises from the enhanced contribution of the last double-trace term in (REF ).",
"The diagonal overlap can then be written in terms of ordered partitions (REF ) as $&{\\color {purple} {}\\mathopen {\\langle }\\lbrace p\\rbrace |H^{(2)}_\\beta |\\lbrace p\\rbrace \\mathclose {\\rangle }{}}=2L_p\\delta _{L_p\\ne 2}~\\mathcal {N}(p^*,p)(e^{i\\beta }-e^{-i\\beta })\\sum _{\\rho ,\\sigma }\\frac{e^{i\\beta } e^{i p_{\\rho (1)}^*}-e^{-i\\beta }}{h_{>}^{\\lbrace p_\\rho ^*\\rbrace }h_{<}^{\\lbrace p_\\sigma \\rbrace }}\\nonumber \\\\&\\hspace{93.89418pt}\\times \\sum _{x=2}^{L_p}\\sum _{l=1}^P \\left(e^{i(x-1) p_{\\sigma (1)}}\\prod _{k=2}^l S_\\beta \\left(p_{\\sigma (1)},p_{\\sigma (k)}\\right)+1\\right)e^{i(x-1)(p_\\sigma )_{l+1}^P}e^{-i(x-2)(p^*_\\rho )_{l+1}^P}\\nonumber \\\\&\\hspace{93.89418pt}\\times P_{x-1}\\left(\\lbrace p_{\\sigma }-p^*_{\\rho }\\rbrace _2^l\\right)P_{L_p-x+1}\\left(\\lbrace p_{\\sigma }-p^*_{\\rho }\\rbrace _{l+1}^P\\right)~.$ Note that $P_L(z)$ vanishes for $|z|\\ge L$ , cf.",
"(REF ).",
"In terms of scalar products of normalised Bethe states (REF ) the overlap can be written as $&{\\color {purple} {}\\mathopen {\\langle }\\lbrace p\\rbrace |H^{(2)}_\\beta |\\lbrace p\\rbrace \\mathclose {\\rangle }{}}=2L_p\\delta _{L_p\\ne 2}~\\mathcal {N}(p^*, p)(e^{i\\beta }-e^{-i\\beta })\\nonumber \\\\&\\hspace{34.14322pt}\\times \\sum _{\\begin{array}{c}k,l=1\\\\\\kappa \\cup \\bar{\\kappa }=\\lbrace p\\rbrace _{\\hat{k}}\\\\ \\lambda \\cup \\bar{\\lambda }=\\lbrace p^*\\rbrace _{\\hat{l}}\\end{array}}^P\\sum _{x=2}^{L_p}\\frac{e^{i\\beta } e^{ip_l^*}-e^{-i\\beta }}{h^{p_k\\bar{\\kappa }}h^{\\kappa \\bar{\\kappa }}}\\left(\\frac{e^{-i(x-1) \\kappa }}{h^{\\kappa p_k}}+\\frac{e^{i(x-1) \\bar{\\kappa }}}{h^{p_k\\kappa }}\\right)\\frac{ e^{-i(x-2)\\bar{\\lambda }}}{h^{p^*_{\\hat{l}}p^*_l}h^{\\bar{\\lambda }\\lambda }}(\\lambda |\\kappa )_{x-2}(\\bar{\\lambda }|\\bar{\\kappa })_{L_p-x}~,$ where $\\kappa $ and $\\lambda $ are of the same cardinality, which must be smaller than $x-1$ .",
"Finally we can divide by the square of the norm of the external state, $\\Vert p\\Vert ^2$ , to defined normalised overlaps ${\\color {purple} {}V^{(2)}_\\beta (\\lbrace p\\rbrace ){}}$ which can be then used to compute the energy corrections of single trace states."
],
[
"Anomalous Dimensions.",
"As the deformation lifts many of the degeneracies present in $\\mathcal {N}=4$ SYM we can use the overlaps in the deformed theory to compute the corrections to energies for a wide range of states by using the deformed analogue of (REF ) $E_{{\\color {black} {}\\beta {}}}^{(2)}(\\lbrace p\\rbrace )=\\sum _{\\lbrace I\\rbrace }\\frac{ V_{{\\color {black} {}\\beta {}}}^-(p; I)V_{{\\color {black} {}\\beta {}}}^+(I; p)}{E_{{\\color {black} {}\\beta {}}}^{(0)}(\\lbrace p\\rbrace )-E_{{\\color {black} {}\\beta {}}}^{(0)}(\\lbrace I\\rbrace )}+{\\color {purple} {}V^{(2)}_\\beta (\\lbrace p\\rbrace ){}}\\,.$ The additional input to such a calculation are the solutions to the deformed Bethe equations.",
"Solving the deformed Bethe equations is generally a non-trivial task, however for short lengths it can be done either for specific numerical values of $\\beta $ or by starting with the undeformed result and perturbatively solving for $\\beta \\ll 1$ .",
"The latter is particularly useful when we wish to use the deformation as a regulator of singular solutions of the undeformed Bethe equations.",
"One must be careful with the order of limits as the one-loop anomalous dimensions are functions of both $\\beta $ and $N$ and we may choose to first expand in large $N$ and then small $\\beta $ , $E(\\beta \\gg N^{-1})$ , or alternatively first in small $\\beta $ and then large $N$ , $E(\\beta \\ll N^{-1})$ .",
"In general these expansions will not commute.",
"For example, let us consider the $L=6$ , $M=3$ single-trace operator described in the planar undeformed theory by the roots $\\lbrace u_1=0, u_2=-i/2, u_3=i/2\\rbrace $ with planar energy $E^{(0)}=12$ .",
"This solution is singular as it has rapidities separated by $i$ .",
"It has a vanishing $1/N^2$ correction to the energy due to the $\\mathfrak {su}(2)$ symmetry which ensures there is no other operator with which it can mix.",
"We can study the same operator in the deformed theory where the mixing problem is non-trivial and we find from direct diagonalisation that through $\\mathcal {O}(N^{-4})$ , and keeping only the leading terms in the $\\beta $ -expansion, we have $E(\\beta \\gg N^{-1})=(12-72 \\beta ^2 +\\mathcal {O}(\\beta ^4))+\\frac{1}{N^2}\\big [-\\frac{2304}{23} +\\mathcal {O}(\\beta ^2)\\big ]+\\frac{1}{N^4}\\big [\\frac{400896}{12167\\beta ^2}+\\mathcal {O}(\\beta ^0)\\big ]~.$ In this expression we can see that the leading non-planar term does not reduce to the vanishing $1/N^2$ undeformed answer in the $\\beta \\rightarrow 0$ limit and in fact the $1/N^4$ term is singular.",
"There will be additional singular terms at subsequent powers in the $1/N$ expansion that would need to be resummed to recover the smooth limit.",
"As the wavefunction in the deformed theory is perfectly regular we can use (REF ,REF ) and (REF ) to compute the overlaps, then take the $\\beta \\rightarrow 0$ limit and use these expressions to perturbatively compute the undeformed non-planar correction.",
"To be explicit, we need to solve for the deformed rapidities to $\\mathcal {O}(\\beta ^6)$ to find a non-singular wavefunction since $u^\\beta _3-u^\\beta _2=i+24576 i \\beta ^6+\\mathcal {O}(\\beta ^{8})~$ and in general for length $L$ singular solutions we need $\\mathcal {O}(\\beta ^L)$ to resolve the singularity.",
"This solution mixes with the double-trace operator $\\mathopen {|} u_{(4,2)} \\mathclose {\\rangle }_4\\mathopen {|}u_{(2,1)}\\mathclose {\\rangle }_2$ of planar energy $E^{(0)}_{(4, 2)}=12-32/3\\beta ^2+\\mathcal {O}(\\beta ^4)$ with normalised overlaps $& & V^+_\\beta =-48\\sqrt{2}\\beta ~~~\\text{and}~~~ V^-_\\beta =-64 \\sqrt{2}\\beta ~$ where we have kept only the first leading term in the $\\beta $ -expansion.",
"The deformation is particularly important when calculating the norm of the singular state, which diverges in the $\\beta \\rightarrow 0$ limit.",
"However the overlaps themselves are smoothly vanishing in this limit and so, consistent with the symmetries, there is no mixing in the undeformed theory.",
"If we instead use these overlaps in the perturbative formula (REF ) we find a cancellation between the powers of $\\beta $ in the overlaps and the energy differences.",
"As the diagonal contribution is of order $\\beta ^2$ , it gives no leading contribution and we find $E^{(2)}(\\beta \\gg N^{-1})= -\\frac{2304}{23}+\\mathcal {O}(\\beta ^2)$ in agreement with the result from direct diagonalisation.",
"Let us now turn to the $L=8, M=3$ singular Bethe state $\\mathopen {|}u_{(8,3s)}\\mathclose {\\rangle }$ with planar energy $E^{(0)}_{(8,3s)}=12$ .",
"This operator is not protected by symmetry in the undeformed theory.",
"Instead it mixes with the double-trace operator $\\mathopen {|}u_{(6,3s)}\\mathclose {\\rangle }_6\\mathopen {|}\\emptyset \\mathclose {\\rangle }_2$ made up of the length-6 singular state and length-2 vacuum.",
"From directly diagonalising the corrected energies are $E_\\pm =12\\pm \\frac{12}{N}~,$ where we see that due to the degeneracy the correction is $\\mathcal {O}(N^{-1})$ .",
"Due to the singular nature of the Bethe solution we cannot directly use the overlap formulae of $\\mathcal {N}=4$ SYM but again we can compute the mixing matrix using the regularised singular states in the deformed theory.",
"We find that in this case they are non-vanishing in the $\\beta \\rightarrow 0$ limit $V^+_\\beta =4\\sqrt{6}+\\mathcal {O}(\\beta ^2)~~~\\text{and}~~~ V^-_\\beta =6\\sqrt{6}+\\mathcal {O}(\\beta ^2)~$ and give the correct $1/N$ corrections.",
"This corresponds to the case where $\\beta \\ll N^{-1}$ .",
"Alternatively, in the deformed theory as the degeneracy between the two states is lifted, with the planar energy of the single-trace state becoming $E^{(0)}_{(8,3s)}=12-36\\beta ^2+\\mathcal {O}(\\beta ^4)$ , we can use the same overlaps in non-degenerate perturbation theory in the small $\\beta $ -limit with $\\beta \\gg N^{-1}$ to find the $1/N^2$ corrections in the deformed theory.",
"The contribution of the overlaps between the two regularised singular states to $E_{(8,3s)}^{(2)}$ is $+4/\\beta ^2$ , i.e.",
"it is singular in the limit $\\beta \\rightarrow 0$ .",
"There are additional overlaps with other double trace states however they are subleading as is the diagonal contribution which is $\\mathcal {O}(\\beta ^4)$ .",
"Thus for $\\beta \\gg N^{-1}$ the non-planar corrections start at order $N^{-2}$ but are singular in the $\\beta \\rightarrow 0$ limit.",
"This demonstrates that, in general, the two limits $\\beta \\rightarrow 0$ and $N^{-1}\\rightarrow 0$ do not commute."
],
[
"BMN Limit",
"We will now look at the two-excitation single-trace solutions and their non-planar corrections in the BMN limit [36] of the deformed theory.",
"First of all, we need to analyse carefully the rapidities that solve the Bethe equations.",
"The solutions are periodic in the deformation parameter with period $\\pi $ and they are symmetric around $\\beta =\\pi /2$ .",
"In general the solutions are parametrized by an integer $n$ which is given as $2 \\pi n =L p_1 - i \\log (S_{\\color {blue} {}\\beta {}}(p_1,p_2)) \\,.$ As can be seen in Figure REF , all but one of the energy levels become degenerate when the deformation parameter equals $\\pi /4$ .",
"Figure: Planar energy levels for L=25L=25 and two excitations in the deformed theory.",
"While in the BMN regime we take the limit of small β\\beta , this plot already hints at the different nature of the zero-mode solution corresponding to the lowest energy.Before that point the mode number $n$ is in the range $[0,\\lfloor L/2 \\rfloor -1]$ .",
"The solutions with positive mode numbers correspond to deformations of the primary operators in $\\mathcal {N}=4$ SYM, while the zero mode becomes a descendant in the undeformed theory.",
"After the crossing point the mode number $n$ takes values in the range $[1,\\lfloor L/2\\rfloor ]$ , with the lowest-energy state now corresponding to $n=\\lfloor L/2\\rfloor $ .",
"Here we focus on the BMN limit where the deformation parameter scales as $\\beta =\\pi b /L\\,, \\qquad \\mbox{with $b$ fixed.",
"}$ Effectively we thus concentrate on a regime of small deformations, with the mode number $n$ in the range $[0,\\lfloor L/2 \\rfloor -1]$ as described above.",
"We can solve the Bethe equations perturbatively, and find that the rapidity for a strictly positive mode number $n$ is given by $u_n = \\frac{L}{2({\\color {blue} {}b{}}+n)\\pi } \\left(1+\\frac{{\\color {blue} {}b{}}-n}{n L} + \\frac{({\\color {blue} {}b{}}+n)\\left({\\color {blue} {}3b(b-n){}}-({\\color {blue} {}b{}}+n)n^3\\pi ^2\\right)}{3 n^3 L^2} +\\ldots \\right)\\,,$ with momentum conservation requiring the other excitation in the solution to be $u_{-n}$ .",
"Meanwhile, the zero-mode solutions have a distinct expression where the expansion parameter becomes the square root of the length ${\\color {purple} {}u_0^\\pm {}} = \\frac{L}{2 {b} \\pi } \\left(1 \\pm \\frac{i }{L^{1/2}} - \\frac{1}{L} \\pm \\frac{i({b^2} \\pi ^2-3)}{6 L^{3/2}}-\\frac{2 {b^2} \\pi ^2}{3 L^2}\\ldots \\right) \\,.$ The planar energies in the BMN limit can then be computed through (REF ), yielding $E_n^{(0)} &= \\frac{16 \\pi ^2({\\color {blue} {}b^2{}}+n^2)}{L^2}\\left(1 +\\frac{2(n^2{\\color {blue} {}-b^2{}})}{({\\color {blue} {}b^2{}}+n^2)L}+ \\frac{3(3n^4{\\color {blue} {}-2b^2n^2-b^4{}})-(n^4{\\color {blue} {}+6b^2n^2+b^4{}})n^2\\pi ^2}{3({\\color {blue} {}b^2{}}+n^2)n^2 L^2}+\\ldots \\right)\\,,\\nonumber \\\\{\\color {purple} {}E_0^{(0)}{}} & {\\color {black} {}= \\frac{16 {b^2}\\pi ^2 }{L^2} \\left( 1-\\frac{1}{L}-\\frac{(3+2{b^2}\\pi ^2)}{3L^2}+\\ldots \\right){}}\\,.$ Note that despite the unusual expansion of the zero-mode rapidities $u_0^\\pm $ , the expansion of the corresponding energy is free of any square roots.",
"Finally, while at the leading order the rapidities $u_0^\\pm $ seem to be only a particular case of $u_{\\pm n}$ , it is important to note that the expression for the Gaudin norm, denoted here by $N_\\psi =||\\psi ||^2$ , differs already at the leading order by a factor of two $N_n &= L^2 \\left(1-\\frac{{\\color {blue} {}b^2{}} +n^2}{n^2 L} {\\color {purple} {}+\\frac{2 b^2(n^2-b^2)}{n^4 L^2}+\\ldots {}}\\right)\\,, \\nonumber \\\\{\\color {purple} {} N_0\\, {}} & {\\color {black} {} = 2 L^2 \\left(1 + \\frac{ b^2 \\pi ^2-3}{3 L}+ \\frac{2b^2\\pi ^2(4b^2 \\pi ^2-15)}{45 L^2}+\\ldots \\right){}}\\,.$ Now that we understand the behaviour of the Bethe solutions, we can study the non-planar corrections to the energies in the BMN limit.",
"The strategy is to expand the dilatation operator overlaps obtained in section REF and plug them into (REF ) written explicitly as $E_{\\psi }^{(2)}= \\sum _{\\psi ^{\\prime }}\\frac{\\langle \\psi |H^-_{\\color {blue} {}\\beta {}} | \\psi ^{\\prime }\\rangle \\langle \\psi ^{\\prime } |H^+_{\\color {blue} {}\\beta {}} | \\psi \\rangle }{N_\\psi N_{\\psi ^{\\prime }} \\left(E_\\psi ^{(0)}- E_{\\psi ^{\\prime }}^{(0)} \\right)}+{\\color {purple} {}\\frac{\\langle \\psi | H_\\beta ^{(2)} | \\psi \\rangle }{N_\\psi }{}} \\,.$ In the deformed theory there are three contributions that we need to consider: Off-diagonal overlaps $H_\\beta ^\\pm $ with double-trace operators where: Both excitations end up on the same trace, The excitations split over the two traces.",
"Diagonal overlap $H_\\beta ^{(2)}$ .",
"In general the two-excitation overlaps corresponding to $H^-_\\beta $ and $H^+_\\beta $ scale at most as $L$ and $L^2$ , respectively.",
"Meanwhile, the sum over intermediate states $\\psi ^{\\prime }$ includes a sum over the splitting of the lengths $(L^{\\prime },L-L^{\\prime })$ in the double-trace operator, which can be approximated by the Euler-MacLaurin formula $\\sum _{L^{\\prime }=a}^b f(L^{\\prime }) \\approx L \\int _{a/L}^{b/L} \\mathrm {d} r f(r) + \\frac{f(a)+f(b)}{2} + \\ldots \\,,$ thus leading to a further factor of $L$ .",
"Therefore, the one-loop non-planar energies $E_\\psi ^{(2)}$ scale at most as $L^2$ which, combined with the colour factor $1/N^2$ , produces at leading order in the BMN expansion a factor of $g^{\\prime } g_2^2$ , with the relevant expansion parameters introduced in (REF ).",
"While in principle we can find the BMN corrections to the overlaps at any subleading order, the expansion of the energies eventually breaks down due to the approximation of the summation over intermediate states by an integration.",
"More precisely, we find that for mode number $n>1$ the sub$^k$ -leading BMN correction to the integrand with $k>1$ has simple poles at lengths $L^{\\prime }/L = n^{\\prime } /n$ with $n^{\\prime }=1,\\ldots ,n-1$ .",
"As explained in Figure REF this failure of the BMN expansion is in fact expected and agrees with the numerical experiments performed.",
"Figure: We compute the non-planar correction to the energies E n (2) E_n^{(2)} from the dilatation operator overlaps.",
"We focus here on mode number n=1,2n=1,2 for single-trace operators up to length 100.",
"We observe that for n=2n=2 the large LL limit is approached differently for even- and odd-length operators.",
"However, fitting the two curves with polynomials in 1/L1/L we find that the mismatch in the coefficients starts only at the subsubleading order.Let us then start with the configuration of uneven splitting REF .",
"We consider first an external state with positive mode number $n$ while the intermediate double-trace operator has two excitations on the trace of size $L^{\\prime } = r L$ and is described by another positive mode number $n^{\\prime }$ .",
"While $L^{\\prime }$ is smaller than $L$ , the deformation parameter for the double-trace solution is still expanded in terms of the length $L$ of the single-trace operator as in (REF ), so the rapidities, energies and norms of the double-trace states are written as $u_{n^{\\prime }}^{\\prime }(L,b) = u_{n^{\\prime }}( r L, r b) \\,, \\qquad {E^{\\prime }}_{n^{\\prime }}^{(0)}(L,b)= E_{n^{\\prime }}^{(0)}(r L, r b)\\,, \\qquad N_{n^{\\prime }}^{\\prime }(L,b) = N_{n^{\\prime }}(r L, r b)\\,,$ and analogously for the zero-mode expressions.",
"The overlaps in the BMN limit then become $H_{nn^{\\prime }}^- &= \\frac{32 (r-1) r^3 n^2\\sin ^2(\\pi r n)L^2}{(n^{\\prime 2}-r^2 n^2)} \\left(1+ 2\\frac{(r-1) n^2 n^{\\prime 2} {\\color {blue} {}-r(n^{\\prime 2}-r n^2)b^2{}}}{rn^2(n^{\\prime 2} - r^2 n^2 )L} \\right.",
"\\nonumber \\\\& \\hspace{153.6447pt}\\left.",
"{}+ \\pi \\cot (\\pi r n)\\frac{(2r-1)n^2{\\color {blue} {}+(3-2r) b^2{}}}{nL} + \\ldots \\right)\\,,\\nonumber \\\\H_{n^{\\prime }n}^+ &= \\frac{32 n^{\\prime 2}\\sin ^2(\\pi r n)L}{(n^{\\prime 2}-r^2 n^2)} \\left(1 + 2r\\frac{(r-1) n^2 n^{\\prime 2} {\\color {blue} {}-r(n^{\\prime 2}-r n^2)b^2{}}}{n^{\\prime 2}(n^{\\prime 2} - r^2 n^2 )L} \\right.",
"\\nonumber \\\\& \\hspace{108.12054pt} \\left.",
"{}+ \\pi \\cot (\\pi r n)\\frac{(2r-3)n^2{\\color {blue} {}+(1-2r)b^2{}}}{n L} + \\ldots \\right)\\,.$ On the other hand, if the intermediate double-trace operator consists of a zero-mode solution, then the $H_\\beta ^+$ overlap becomes suppressed and we have instead $H_{n0}^- &= -32 (r-1) r\\sin ^2(\\pi r n) L^2 + \\ldots \\,,\\nonumber \\\\{\\color {purple} {}H_{0n}^+{}} &{\\color {black} {}=\\frac{32 b^2 \\sin ^2(\\pi r n)}{r n^2 }+\\ldots {}}\\,.$ The contribution to the non-planar energies is the combination of those two cases, leading to $E_{n,A}^{(2)} &= L\\int _{2/L}^{(L-2)/L} \\mathrm {d} r \\left( \\frac{H_{n0}^- H_{0n}^+}{N_n N^{\\prime }_0 \\left(E_n^{(0)}-{E^{\\prime }}_0^{(0)}\\right)}+ \\sum _{n^{\\prime }=1}^\\infty \\frac{H_{nn^{\\prime }}^- H_{n^{\\prime }n}^+}{N_n N^{\\prime }_{n^{\\prime }} \\left(E_n^{(0)}-{E^{\\prime }}_{n^{\\prime }}^{(0)}\\right)} \\right) \\nonumber \\\\&=\\left(\\frac{1}{3} + \\frac{35}{8 \\pi ^2 n^2}\\right) L^2\\left(1 + \\frac{{\\color {blue} {}4 b^2{}}-2 n^2}{n^2 L}+\\ldots \\right) \\,.$ Note that in this particular case the subleading correction of the Euler-McLaurin formula is vanishing.",
"Furthermore, the contribution of the intermediate zero-mode is crucial for the simplicity of this formula, which would otherwise be plagued by more complex functions such as $\\int _0^{z} \\mathrm {d}t \\sin (t)/t$ .",
"Finally, taking $b=0$ yields the non-planar correction to the two-excitation energies of $\\mathcal {N}=4$ SYM.",
"In that theory the rapidities are more naturally written as an expansion in even powers of $1/(L-1)$ , thus matching the result obtained here.",
"As explained above, the integral approximation at the subsubleading order is not well defined for $n>1$ , and that is manifested here by the presence of poles in the integrand.",
"However, the expression for $n=1$ appears to be well defined at any order, thus allowing us to obtain $E_{1,A}^{(2)} =&\\frac{105(1{\\color {blue} {}-12 b^2+15 b^4{}})-(1{\\color {blue} {}+144 b^2-9 b^4{}})\\pi ^2-8(3{\\color {blue} {}+4 b^2+b^4{}}) \\pi ^4-288 (1+{\\color {blue} {}b^2{}})^2 \\pi ^2\\zeta _3}{24 \\pi ^2 } \\nonumber \\\\&+ \\frac{(105+8 \\pi ^2)({\\color {blue} {}2b^2{}}-1)}{12\\pi ^2}L +\\left(\\frac{1}{3} + \\frac{35}{8 \\pi ^2}\\right) L^2\\,.$ This expression matches the coefficients in the fit to the data of Figure REF to 8 digits of precision.",
"Still considering the configuration of uneven splitting REF , we focus now on the case when the external state is a zero-mode.",
"If the intermediate operator is a zero-mode as well, then each of the overlaps is suppressed by a power of $L$ and given by ${\\color {purple} {}H_{00}^-{}} & {\\color {black} {}= 32 \\pi ^2(r-1)(r-2)r^2 b^2 L\\left(1+\\frac{6r^2-6(r-2)+\\pi ^2 r (r-2)(3+r(2r-3))}{6r(r-2)L}+ \\ldots \\right){}}\\,,\\nonumber \\\\{\\color {purple} {}H_{00}^+ {}}& {\\color {black} {}=32 \\pi ^2 r b^2\\left(1+\\frac{-12+ \\pi ^2 r (3+r(2r-3))}{6 rL}+\\ldots \\right){}}\\,.$ Notice that in this case the difference of planar energies $E_0^{(0)}-{E^{\\prime }}_0^{(0)}$ vanishes at the leading order, therefore enhancing the contribution of the intermediate zero-mode to $E_0^{(2)}$ by a factor of $L$ .",
"In principle we would need to consider also the off-diagonal overlaps with a positive mode state, $H_{0n^{\\prime }}^-$ and $H_{n^{\\prime }0}^+$ , but each of them starts contributing at $L^0$ .",
"For that configuration the difference $E_0^{(0)}-{E^{\\prime }}_{n^{\\prime }}^{(0)}$ is once again $\\mathcal {O}(L^2)$ and so we can safely ignore the contribution of these modes at the order we wish to consider.",
"The non-planar correction to the energy of the zero-mode solution is then given by ${\\color {purple} {}E_{0,A}^{(2)}{}} &= L\\int _{2/L}^{(L-2)/L} \\mathrm {d} r \\left( \\frac{H_{00}^- H_{00}^+}{N_0 N^{\\prime }_0 \\left(E_0^{(0)}-{E^{\\prime }}_0^{(0)}\\right)}\\right) +8 \\pi ^2 b^2\\nonumber \\\\&{\\color {black} {}=\\frac{20 \\pi ^2 b^2 L}{3} \\left(1+\\frac{2 \\pi ^2 b^2-225}{25 L}+\\ldots \\right) {}}\\,,$ where the second term of the first line corresponds to the subleading correction in the Euler-MacLaurin formula (REF ).",
"As expected $E_{0,A}^{(2)}$ vanishes in the undeformed theory, where the operator becomes a descendant of the chiral primary.",
"In order to study the second splitting configuration REF , where $H_\\beta ^+$ leads to double-trace operators with an excitation in each of the traces, we need to consider the single-excitation solution in more detail.",
"The rapidity in that case is given by ${\\color {purple} {}u{}} = \\frac{L}{2 b \\pi } \\left(1- \\frac{b^2 \\pi ^2}{3 L^2} +\\ldots \\right)\\,.$ Note that $L$ here is again the length of the external single-trace operator as this solution turns out to be independent of the length of the trace it describes and is defined solely in terms of the scaled deformation parameter from (REF ).",
"The energy of this state is ${\\color {purple} {}E^{(0)}{}}=\\frac{8 b^2\\pi ^2 }{L^2} \\left(1-\\frac{b^2 \\pi ^2}{3 L^2} +\\ldots \\right)\\,,$ and the norm is simply the length of the operator.",
"It is important to remember that the single-excitation solution of length 2, $\\text{Tr}(ZX)$ , is an exception to this formula.",
"The operator is in fact protected due to the contribution of the double-trace term to the planar dilatation operator.",
"When the single-trace operator corresponds to a non-zero mode, the overlaps are ${\\color {purple} {}H_n^+ {}}& {\\color {black} {}= \\frac{16 \\pi ^2 b^2}{L} \\left( \\cos (2 \\pi r n)-\\frac{b^2}{n^2}\\right) + \\ldots {}}\\,,\\nonumber \\\\H_n^- & = 32 r (r-1) \\sin ^2(\\pi r n)L^2 + \\ldots \\,.$ We can already see that this will contribute at the subsubleading order, so it suffices to consider the leading integral approximation which gives ${\\color {purple} {}E_{n,B}^{(2)}{}}&= \\frac{L}{2} \\int _{0}^{1} \\mathrm {d} r \\frac{H_n^- H_n^+}{N_n r(1-r) L^2 (E_n^{(0)}-2 E^{(0)})} \\nonumber \\\\&{\\color {black} {}= \\frac{4 b^2}{n^4 } \\left(n^2+2b^2 \\right) + \\ldots {}} \\,.$ Meanwhile, for an external zero mode, the overlaps are ${\\color {purple} {}H_0^-{}} &{\\color {black} {}= 16 \\pi ^2 b^2(1-r) r L\\left(1+2r(r-1)+\\frac{12 r (r-1) + (1+6r(r-1)+4 r^2(r-1)^2)b^2 \\pi ^2}{6L} +\\ldots \\right){}}\\,,\\nonumber \\\\{\\color {purple} {}H_0^+{}} & {\\color {black} {}= 16 \\pi ^2 b^2 \\left(1+\\frac{\\pi ^2 b^2}{6 L}+ \\ldots \\right){}}\\,.$ We now wish to perform the sum over operators as in equation (REF ).",
"Looking at the expressions for the planar energies (REF ,REF ), we see that $E_0^{(0)}-2 E^{(0)}$ vanishes at leading order, which effectively enhances the leading order non-planar correction of the energy to $\\mathcal {O}(L^1)$ (note that both $H_0^+$ and $H_0^-$ are subleading).",
"However, this reasoning does not apply when one of the traces has length 2 due to the double-trace term of the dilatation operator, and so that particular double-trace operator contributes at a further subleading order.",
"Therefore the non-planar correction to the energy from this splitting configuration becomes ${\\color {purple} {}E_{0,B}^{(2)}{}}&=\\frac{L}{2}\\int _{3/L}^{(L-3)/L} \\mathrm {d}r \\frac{H_0^- H_0^+}{N_0 r(L-r) L^2\\left(E_0^{(0)}-2 E^{(0)}\\right)} -4 \\pi ^2 b^2 \\nonumber \\\\&{\\color {black} {}=-\\frac{8 \\pi ^2 b^2 L}{3} \\left(1-\\frac{120+7 \\pi ^2 b^2}{15L}+\\ldots \\right) {}}\\,,$ where, as before, the second term of the first line corresponds to a non-vanishing subleading contribution in the Euler-McLaurin approximation.",
"Taking $b$ to zero we see that both $E_{n,B}^{(2)}$ and $E_{0,B}^{(2)}$ vanish as expected.",
"In that limit the single-excitation solutions correspond to descendants of the undeformed theory, and so these splitting configurations are not expected to contribute to the non-planar corrections of energies in $\\mathcal {N}=4$ SYM.",
"Finally, in the deformed theory we should also consider the diagonal contribution of the dilatation operator REF .",
"However, the overlap grows linearly with $L$ , and so its normalized contribution to the non-planar energy starts only at $\\mathcal {O}(1/L)$ , which goes beyond the order we wish to consider here.",
"Therefore, the non-planar correction to the energy of two-excitation single-trace operators is, at $\\mathcal {O}(L^0)$ in the BMN limit, given by the sum of the off-diagonal uneven splittings from (REF ,REF ) with the symmetric ones in (REF ,REF ) $E_{n}^{(2)} &= E_{n,A}^{(2)} + E_{n,B}^{(2)}\\,, \\nonumber \\\\E_{0}^{(2)} &= E_{0,A}^{(2)} + E_{0,B}^{(2)}\\,.$ We can then write the scaling dimensions of two-excitation states in the BMN limit of large $R$ -charge $J=L-2$ , which for the non-zero modes yields $\\Delta _n = L + g^{\\prime } &\\Bigg [16 \\pi ^2 (n^2{\\color {blue} {}+b^2{}} )\\left(1 - \\frac{2 (n^2{\\color {blue} {}+3b^2{}})}{(n^2 {\\color {blue} {}+b^2{}})J}+\\mathcal {O}(J^{-2})\\right) \\nonumber \\\\&{}+ g_2^2 \\left(\\frac{1}{3}+\\frac{35}{8\\pi ^2 n^2}\\right)\\left(1+\\frac{2(n^2 {\\color {blue} {}+2b^2{}})}{n^2 J}+\\mathcal {O}(J^{-2})\\right)+ \\mathcal {O}(g_2^4)\\Bigg ] + \\mathcal {O}((g^{\\prime })^2)\\,,$ reproducing the result of [18] at leading order, while for the zero mode we have $\\Delta _0 = L +{}&{\\color {purple} {} g^{\\prime } \\Bigg [16 \\pi ^2 b^2 \\left(1 - \\frac{5 }{J}+\\frac{51-2 \\pi ^2 b^2}{3J^2}+\\mathcal {O}(J^{-3})\\right) {}}\\nonumber \\\\ &\\qquad {\\color {purple} {}+ 4 \\pi ^2 b^2 g_2^2 \\left(\\frac{1}{J}+\\frac{4 \\pi ^2 b^2 -69}{9 J^2}+\\mathcal {O}(J^{-3})\\right)+ \\mathcal {O}(g_2^4)\\Bigg ] + \\mathcal {O}((g^{\\prime })^2){}}\\,.$"
],
[
"Level-Crossing and Spectral Statistics",
"The lifting of the degeneracies in the spectrum of one-loop dimensions by the $\\beta $ -deformation is related to the phenomenon of level repulsion.",
"In general, energy surfaces depending on a number of parameters are connected only at special points where multiple parameters are tuned and the energy surfaces possess a diabolo-like geometric structure (such points were thus called “diabolic” by Berry and Wilkinson [61]).",
"The situation is quite different for systems with additional symmetries, such as integrable systems, where degeneracies occur even as only a single parameter is varied.",
"For the spectrum of $\\mathcal {N}=4$ SYM this implies that operator dimensions depending on parameters $\\lambda $ and $N$ avoid crossing for generic fixed values of $N$ as $\\lambda $ is varied, as was borne out in [40].",
"In our case, being at one-loop, the $\\lambda $ -dependence is trivial however we can study the spectrum as a function of both the deformation parameter, $\\beta $ , and the rank, $N$ , of the gauge group.",
"By numerically solving for the eigenvalues of specific families of operators we can see the behaviour of the scaling dimensions as we vary $\\beta $ and as an example we consider the length-six, three impurity states in Fig.",
"REF .",
"For finite values of $N$ the energy levels mostly repel and even at points where they appear to come close they do not in fact cross, maintaining a separation of $\\sim 1/N^2$ .",
"There is one obvious exception which clearly does cross other levels at finite $N$ .",
"This is a double trace state that does not mix with other operators, receives no $1/N$ corrections and so is effectively uncorrelated with the other states.",
"This is due to the fact that at half-filling the charge conjugation transformation $Z \\leftrightarrow X$ combined with the parity transformation (REF ) is a symmetry which commutes with both the impurity number and the one-loop non-planar dilation operator.",
"The double-trace operator is the only $L=6$ , $M=3$ state with negative charge with respect to this transformation.",
"This points to the fact that in order to avoid trivial crossings we must consider operators which have the same quantum numbers.",
"At large values of $N$ we can see the appearance of crossings which occur at special values where $\\beta /\\pi \\in \\mathbb {Q}$ ; for example in Fig.",
"REF there are crossings in the planar limit at $\\beta =\\pi /4$ and $\\beta =\\pi /6$ .",
"These points correspond to values where the $\\beta $ -deformed theory becomes equivalent to an orbifold of $\\mathcal {N}=4$ SYM, e.g.",
"[62], which are known to have enhanced structures such as additional regions on the Coulomb branch.",
"Figure: Eight eigenvalues corresponding to states with L=6L=6 and M=3M=3 as functions of β∈[0,π]\\beta \\in [0, \\pi ] with (a) N=7N=7 (b) N=10 6 N=10^6.",
"The top, purple, line and fourth from the top, brown, correspond in the undeformed, planar limit to descendants of two single trace two-impurity states.",
"The second and third lines, yellow and light blue, correspond to the single trace three-impurity singular solution and a degenerate double trace operator.",
"The remaining operators are protected in the undeformed theory but acquire non-vanishing anomalous dimensions for non-vanishing β\\beta .The level repulsion at finite-$N$ suggests the transistion from a quantum integrable model to a chaotic system.",
"To further explore this it is interesting to compute the distribution of level spacings.",
"Given a spectrum of energy levels one can easily show that if we assume they are uncorrelated the spacing of successive levels satisfies Poisson statistics, $P_P(s)=e^{-s}$ .",
"That this distribution is a good description of integrable sytems has been numerically shown in a range of models including many-body systems such as the Heisenberg spin chain, the t-J model at its integrable supersymmetric point and the Hubbard model [63], [64].",
"There are significantly fewer analytical results, however one important result by Berry and Tabour [65] showed that for a “generic” integrable, semi-classical system the distribution of energy levels is indeed Poisson.",
"There are known examples of integrable models for which this is not the case, such as [66] where by considering finely tuned multi-parameter Richardson-Gaudin models, integrable models with non-integrable statistics were found.",
"However in that case even small changes in the parameters resulted in a restoration of the integrable distribution.",
"It is well known that in Random Matrix Theory (RMT) the joint probability distribution for the eigenvalues, $x_1, x_2, \\dots , x_S$ , of $S\\times S$ Hermitian random matrices is given by $P_\\alpha (x_1, x_2, \\dots , x_S)=C_{S_\\alpha } \\prod _{j<k} |x_j-x_k|^\\alpha e^{-\\tfrac{\\alpha }{2} \\sum _{j=1}^Sx_j^2}$ with $\\alpha =1, 2, 4$ corresponding to orthogonal, unitary and symplectic ensembles, respectively.",
"Furthermore, the distribution of spacings between eigenvalues, normalised so that the mean spacing is unity, can be well approximated by the Wigner-Dyson distribution $P_{WD}(s)&=&A(\\alpha ) s^\\alpha e^{-B(\\alpha )s^2}~,~~~~ A(\\alpha )=2 \\frac{\\Gamma (1+\\tfrac{\\alpha }{2})^{1+\\alpha }}{\\Gamma (\\tfrac{1+\\alpha }{2})^{2+\\alpha }}~,~~~~B(\\alpha )=\\frac{\\Gamma (1+\\tfrac{\\alpha }{2})^{2}}{\\Gamma (\\tfrac{1+\\alpha }{2})^{2}}~.$ A particularly important feature is that when approximating a physical system by random matrices the appropriate ensemble can be determined from the symmetries of the Hamiltonian, such as rotational or time-reversal symmetry, and does not depend on the specifics of the interactions.",
"In particular, for a Hamiltonian with time-reversal and rotational symmetry it is appropriate to choose the Gaussian Orthogonal Ensemble (GOE) corresponding to $\\alpha =1$ .",
"Figure: The level-spacing statistics for L=16L=16, β=0.9\\beta =0.9 states at N=17N=17.",
"The grey dots are the numerically calculated values and the solid lines correspond to the Wigner-Dyson distribution, P WD (s)P_{WD}(s).In order to similarly analyse the spectrum of one-loop anomalous dimensions we must first focus on a specific sector comprising states which have the same quantum numbers for any operators which commute with the dilatation operator.",
"For the $\\beta $ -deformed theory we thus consider states with fixed bare dimension, $L$ , and excitation number, $M$ .",
"Additionally we remove all zero energies and we do not consider the sector with $M=L/2$ .",
"The former correspond to protected states whose dimensions are fixed by supersymmetry and, as previously mentioned, the latter contains states which are related by symmetry and so are degenerate.",
"We then numerically compute the spectrum and order the dimensions in this sector $E_1, E_2, \\dots , E_S$ so that $E_1\\le E_2 \\le \\dots \\le E_S$ .",
"One important technicality is that the spectra of RMT are normalised so that the mean level density is unity.",
"So in comparing with physical systems the spectrum must be rescaled to remove the overall dependence on the energy.",
"This procedure is called unfolding and as we do not know the mean level distribution we find a way to approximate it.",
"We describe our procedure in App.",
", and find that the final result is relatively insensitive to the specific details of the unfolding.",
"After carrying out this step we label the unfolded spectrum of anomalous dimensions from smallest to largest: $x_1\\le x_2\\le \\dots \\le x_S$ .",
"From the unfolded spectrum we estimate the distribution of level spacings between consecutive levels by computing $s_i=x_{i+1}-x_i$ , then binning the data and calculating the fraction that occur in each bin.",
"The results naturally depend on the bin size and a choice is made such that small changes do not significantly affect the overall results.",
"The estimate of the probability distribution naturally improves with larger numbers of states and so one must compute the dimensions for relatively long operators.",
"Table: L=16L=16 states with MM excitations for N=17N=17 and β=0.9\\beta =0.9.",
"The GOE Wigner-Dyson distribution corresponds to ω=1\\omega =1 and α=1\\alpha =1.In Fig.",
"REF and Tab.",
"REF we present the results for $L=16$ states with $N=17$ and $\\beta =0.9$ .",
"By visual inspection it is apparent that the GOE Wigner-Dyson distribution ($\\alpha =1$ ) closely matches the data for most values of $M$ .",
"To be more quantitative, one can fit the data to the Brody distribution $P_B(s) =\\Gamma \\left( \\tfrac{\\omega +2}{\\omega +1}\\right)^{1+\\omega }(1+\\omega ) s^\\omega \\text{exp}\\left(-\\Gamma \\left( \\tfrac{\\omega +2}{\\omega +1}\\right)^{1+\\omega } s^{1+\\omega }\\right)$ which is a one-parameter generalisation smoothly interpolating between the GOE Wigner-Dyson distribution ($\\omega =1$ ) and the Poisson distribution ($\\omega =0$ ).",
"In Tab.",
"REF we show the best fit values of $\\omega $ for different values of $M$ which are generally close to one suggesting that this is the appropriate value for the distribution at relatively small values of $N$ .",
"This fit captures the Gaussian behaviour of the exponential decay of the tail and the fact that the distribution goes to zero as $s\\rightarrow 0$ .",
"If we assume the distribution is of the Wigner-Dyson form we can perform a fit to the general form (REF ) and find the best fit value of $\\alpha $ which from Tab.",
"REF can again be seen to be approximately one.",
"It is clear that the fit is better for higher excitation number as the values for $M=2$ are furthest from those of the GOE.",
"The values for $M=0$ , which are protected operators, and $M=1$ , which are protected in the undeformed theory, clearly do not fit the Wigner-Dyson distribution but we do not have a clear explanation for why the $M=2$ fit is so poor.",
"In general however we find that the Gaussian Orthogonal Ensemble describes the non-planar distribution of energy levels in the $\\mathfrak {su}(2)$ sector of deformed $\\mathcal {N}=4$ SYM.",
"We can repeat the computation for the strict planar spectrum, however in this case there are additional symmetries that we must account for.",
"In particular the number of traces in a given operator is conserved under the action of the dilatation operator and so we must work at fixed number of traces.",
"In the single-trace sector this reduces the problem to essentially that of the integrable twisted XXX spin chain which is known to satisfy Poisson statistics while the multi-trace sectors are uncorrelated tensor products and so also have the same distribution, see Fig.",
"REF .",
"We can again compare the planar spectrum to the Brody distribution (REF ) and we find that for most impurity numbers the fit is best for a value of $\\omega \\simeq 0$ , though there are a small number of cases where the value is larger.",
"If we combine the distributions of single-trace states with $M\\ge 3$ together we find $-0.4 \\le \\omega \\le 0.2$ depending on how we bin the data while for the double-trace operators we find $-0.4 \\le \\omega \\le 0.4$ with the results generally close to zero.",
"Thus the spectrum appears to be well described by the Poisson distribution.",
"Figure: The level-spacing statistics for L=16L=16, β=0.9\\beta =0.9 single-trace (on the left) and double-trace (on the right) states in the planar limit.",
"The blue markers are the numerically calculated values of unfolded spacings computed for states with M=3M=3 and similarly for the other impurity numbers.",
"The solid line corresponds to the Poisson distribution, P P (s)P_{P}(s).One can see how the distributions change as the spectrum transitions from chaotic to integrable by considering large a sequences of values of $N$ .",
"In Fig.",
"REF we plot such a sequence of distributions of spacings for $L=15$ states.",
"For $N=15$ we find the expected Wigner-Dyson distribution while for $N=50$ and $N=100$ we find distributions between Wigner-Dyson and Poisson with $\\omega \\simeq 0.71$ and $\\omega \\simeq 0.39$ respectively while for $N=200$ we find $\\omega \\simeq 0.05$ and the distribution appears to be approaching Poisson.",
"However this is not quite the case with the value of $\\omega $ further decreasing as we increase $N$ giving $\\omega \\simeq -0.61$ for $N=10^6$ .",
"This is due to an excess of points occurring toward $s=0$ due to the decoupling of sectors with different numbers of traces which, as explained above, should be considered separately.",
"Figure: The level spacing statistics for L=15L=15 states with β=0.9\\beta =0.9 and different values of NN.",
"The spacings in each impurity sector are separately computed and then the combined distribution is plotted.",
"The yellow solid line corresponds to the Poisson distribution and the blue line to the Wigner-Dyson distribution.We can of course ask about the statistics of the spectrum for the undeformed theory.",
"In this case there are additional symmetries even in the non-planar theory and, in order to find the Wigner-Dyson distribution, we must carefully desymmetrize the spectrum.",
"As mentioned previously, the parity operation, (REF ), commutes with the full dilation operator in the undeformed theory and so non-planar eigenstates with different parity are uncorrelated.",
"Additionally the full global $\\mathfrak {su}(2)$ symmetry is present in the undeformed theory and so it is necessary to work with only highest-weight states.",
"This means that at fixed length and excitation number there are fewer available states and consequently the statistics are of poorer quality.",
"For example, if we consider positive parity $L=16$ states with $M=3$ we find only 315 distinct states.",
"Nonetheless, the general features of the Wigner-Dyson distribution can be seen in a plot of the level spacings, Fig.",
"REF , and the best fit to the Brody distribution occurs for $\\omega \\simeq 0.9$ .",
"This suggests that the statistics of the undeformed spectrum are similarly described by GOE random matrix theory.",
"Figure: The level spacing distribution for L=16L=16, M=3M=3 and positive parity highest weight states with N=17N=17 in the undeformed theory."
],
[
"Conclusions",
"We have considered the problem of computing non-planar anomalous dimensions in $\\mathcal {N}=4$ SYM and its deformations.",
"The approach we have followed involves two steps: first one must obtain the mixing matrix, and then find its eigenvalues with the method of quantum-mechanical perturbation theory.",
"In this work we have mostly focused on the first half of the problem by finding the matrix elements of the one-loop dilatation operator in terms of the Bethe rapidities.",
"While direct application of the dilatation operator can in many cases yield the mixing matrix in a similarly efficient fashion, our formulas are given in terms of partitions of the Bethe rapidities, and therefore they are especially advantageous when the number of excitations is small.",
"In those cases we are able to easily evaluate the overlaps, even for long operators where direct diagonalisation would be infeasible, and the bottleneck in computing anomalous dimensions is the determination of the Bethe rapidities.",
"While there are tools for efficiently computing such rapidities, most notably the Baxter Q-function method of [50], carrying out the sums over solutions is still non-trivial.",
"At a more conceptual level, we found that the matrix elements can be written in terms of Hexagon-like objects satisfying both the Watson and decoupling conditions.",
"While our methods are not obviously related to the hexagonalization of the torus, this decomposition hints at the possibility of an approach similar to [67], where four-point functions are built through the OPE, but the OPE data itself is computed within an integrable framework.",
"Similarly, the matrix elements of the dilatation operator might have a more general description which determines their form at higher orders in the perturbative expansion.",
"In order to study this further it would be useful to determine the overlaps at higher loops and to investigate if hexagon-like objects can be found for other sectors of the theory.",
"One issue with our approach to the diagonalization of the mixing matrix is that it assumes a non-degenerate spectrum of excited states.",
"There are however many degeneracies in the planar spectrum of $\\mathcal {N}=4$ SYM, and so we also considered the $\\beta $ -deformed theory where these degeneracies are lifted.",
"A second advantage of the $\\beta $ -deformation is that it provides a useful regularization for the singular solutions occuring in the $\\mathfrak {su}(2)$ sector of the $\\mathcal {N}=4$ spin chain.",
"The action of the dilatation operator in the deformed theory yields several new structures and for the purpose of evaluating $1/N^2$ corrections to the spectrum it is necessary to include an additional diagonal overlap and the contribution of the double-trace term.",
"As an application of our method we computed the anomalous dimension of two-excitation states in the BMN limit through subleading order.",
"We extracted the corresponding coefficients from fits to numerical data at lower lengths and found the results agreed with at least 8 digits of precision.",
"As the problem of degeneracies occurs in other sectors of the theory additional twists will be needed.",
"For example to study the $\\mathfrak {sl}(2)$ sector it may be useful to consider the integrable dipole deformation [68].",
"The problem of summing over intermediate states increases with the excitation number of the operators under consideration and will rapidly become unfeasible.",
"To compute such sums in the deformed theory it would be of great advantage to generalise the Baxter Q-function method for determining rapidities to the twisted case.",
"It may also be possible that the sum over solutions is simpler than the individual terms and that the computational methods based on algebraic geometry discussed in [69] can be fruitfully applied.",
"It will likely be of interest to study the semi-classical limit of the non-planar corrections where both the number of excitations and the spin-chain lengths are taken to be large.",
"For the planar theory this limit proved to be of great use in making contact with the strong-coupling classical string description.",
"One tool to carry out the sum over intermediate states in this thermodynamic limit is the Quench Action [70], [71], where the sum over Bethe solutions is replaced by a functional integral over root densities which can then be evaluated by saddle-point approximation.",
"There are of course alternative methods for studying non-planar anomalous dimensions such as Hexagonalisation.",
"There are in principle two approaches that can be taken within this formalism.",
"On the one hand, hexagonalisation can be used to compute four-point functions on the torus [16] and OPE limits then used to extract anomalous dimensions.",
"While in this method one can restrict to correlation functions of protected operators, it gives access only to sum rules of OPE data.",
"On the other hand, it is possible to consider the two-point function on the torus [17] by taking the four-point function with two identity operators, and while this approach does not solve the problem of diagonalizing the mixing matrix, pursuing it beyond tree-level might provide an alternative way of finding the matrix elements of the dilatation operator.",
"In addition to computing specific operator dimensions it is also of interest to understand the general properties of the spectrum.",
"To this end we analysed the distribution of level spacings and found that at infinite $N$ the one-loop spectra of both $\\mathcal {N}=4$ SYM and its deformation were well described by the Poisson distribution which is characterisic of integrable systems.",
"This could be seen to transition at finite-$N$ to the Wigner-Dyson distribution of chaotic quantum many-body systems which suggests that the statistical properties of the finite-$N$ spectrum can be well described by a GOE random matrix model.",
"Quantum chaos has in recent years been studied extensively in the context of the holographic duality between the SYK-model of $N$ $(0+1)$ -dimensional Majorana fermions and Jackiw-Teitelboim gravity on AdS$_2$ [72], [73], [74], [75].",
"The distribution of the level spacings for the SYK model was numerically computed in [76], see also [77], [78], [79], and it was shown that it is Wigner-Dyson with all three ensembles, GOE, GUE and GSE, occuring depending on the value of $N$ .",
"It would be naturally interesting to study this chaotic behaviour at higher loop-orders in $\\mathcal {N}=4$ SYM and whether, by the holographic correspondence, we can describe the properties of interacting quantum strings on anti-de Sitter space by RMT.",
"Acknowledgements We thank Marius De Leeuw, Sergey Frolov, and Pak Hang Chris Lau for useful conversations.",
"We also thank João Caetano, Thiago Fleury and Pedro Vieira for comments on an earlier draft of this work.",
"T. McL.",
"would particularly like to thank Samuel Bateman, Joe Davis and Denis Murphy for their contributions to summer projects out of which this work originated.",
"This work was supported by the Science Foundation Ireland through grant 15/CDA/3472 and has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.",
"764850 \"SAGEX\"."
],
[
"Overlaps from the Algebraic Bethe Ansatz",
"The algebraic Bethe ansatz (ABA), see [80], [81], [82] for introductions, provides a powerful framework for studying integrable systems such as the spin chains arising in the one-loop planar dilatation operator.",
"Of particular interest in this work are the computationally efficient formulae for scalar products of Bethe states [26], [27].",
"These scalar products have previously appeared in the context of $\\mathcal {N}=4$ SYM structure constants and we will mostly follow the conventions of [45].",
"Central to the ABA approach is the monodromy matrix, $\\hat{T}_a(u)$ , which is an operator depending on the spectral parameter, $u\\in \\mathbb {C}$ , and acting on the tensor product of the $L$ spin-chain sites, $(\\mathbb {C}^2)^{\\otimes L}$ , and an extra auxiliary space, $V\\simeq \\mathbb {C}^2$ , labelled by the index $a$ .",
"Considering $\\hat{T}_a(u)$ as a $2\\times 2$ matrix, whose entries are operators acting on the spin chain, we can write $\\hat{T}_a(u)=\\begin{pmatrix}\\mathcal {A}(u) & \\mathcal {B}(u) \\\\\\mathcal {C}(u) & \\mathcal {D}(u)\\end{pmatrix}~.$ The commutation relations of these entries can be found from the relations $R_{a_1 a_2}(u-v) \\hat{T}_{a_1}(u)\\hat{T}_{a_2}(v)=\\hat{T}_{a_2}(v)\\hat{T}_{a_1}(u)R_{a_1 a_2}(u-v)$ where the R-matrix, $R_{a_1a_2}(u-v)$ , is an operator acting on the two auxiliary spaces labelled by $a_1$ and $a_2$ and, for the theories we consider, depending on the difference of the spectral parameters $u$ and $v$ .",
"Viewed as a $4\\times 4$ matrix, mapping $(\\mathbb {C}^2)^{\\otimes 2}\\rightarrow (\\mathbb {C}^2)^{\\otimes 2}$ , we can write $R_{a_1 a_2}(u-v)=\\begin{pmatrix}f(u,v) & 0 & 0 & 0 \\\\0 & 1 & g(u,v) & 0 \\\\0 & g(u,v) & 1 & 0 \\\\0 & 0 & 0 & f(u,v)\\end{pmatrix}$ where we have introduced the functions $f(u,v)\\equiv f(u-v) =1+\\frac{i}{u-v} ~~~\\text{and}~~~ g(u,v)\\equiv g(u-v)=\\frac{i}{u-v }~.$ The trace of the monodromy matrix over the auxiliary space defines the transfer matrix, $\\hat{T}(u)=\\text{Tr}\\, \\hat{T}_a(u)$ , and it follows from (REF ) that transfer matrices with different spectral parameters commute.",
"The Hamiltonian of the spin chain is given by the log derivative of the transfer matrix evaluated at $u=i/2$ while the higher conserved charges can be found by further expanding the logarithm of the transfer matrix near $u=i/2$ .",
"The eigenstates of the transfer matrix thus simultaneously diagonalise the Hamiltonian and all higher charges.",
"One can define Bethe states as $\\mathopen {|}\\lbrace u\\rbrace \\mathclose {\\rangle }^{\\text{al}}=\\prod _{i=1}^M\\mathcal {B}(u_i)\\mathopen {|}0\\mathclose {\\rangle }$ where the pseudovacuum is defined by $\\mathcal {C}(u)\\mathopen {|}0\\mathclose {\\rangle }=0$ and satisfies $\\mathcal {A}(u)\\mathopen {|}0\\mathclose {\\rangle }=a(u)\\mathopen {|}0\\mathclose {\\rangle }~~~\\text{and}~~~ \\mathcal {D}(u)\\mathopen {|}0\\mathclose {\\rangle }=d(u)\\mathopen {|}0\\mathclose {\\rangle }$ with $a(u)=(u+i/2)^L$ and $d(u)=(u-i/2)^L$ .",
"When the rapidities $\\lbrace u_i\\rbrace $ in (REF ) satisfy the Bethe equations (REF ), using the parametrisations (REF ) and (REF ), the Bethe states are eigenstates of the transfer matrix with eigenvalues $\\hat{T}(v)\\mathopen {|}\\lbrace u\\rbrace \\mathclose {\\rangle } =T(v,\\lbrace u\\rbrace )\\mathopen {|}\\lbrace u\\rbrace \\mathclose {\\rangle }~~~\\text{with}~~~T(v,\\lbrace u\\rbrace )=a(v)\\prod _{i=1}^M f(v,u_i)+d(v)\\prod _{i=1}^M f(u_i, v)~.$ The operators $\\mathcal {B}(u_i)$ can thus be viewed as creating excited states whose relative normalisation is given by, see [45], $\\mathopen {|}\\lbrace p\\rbrace \\mathclose {\\rangle }=\\frac{1}{\\sqrt{ S}^{\\lbrace u\\rbrace }_<f_<^{\\lbrace u\\rbrace } d^{\\lbrace u\\rbrace } g^{\\lbrace u+i/2\\rbrace }}~\\mathopen {|}\\lbrace u\\rbrace \\mathclose {\\rangle }^{\\text{al}}~,$ where we use the product notation (REF ).",
"The dual states in the ABA are defined by ${}^{\\text{al}}\\mathopen {\\langle }\\lbrace u\\rbrace \\mathclose {|}=(-1)^M \\mathopen {\\langle }0\\mathclose {|} \\prod _{i=1}^M \\mathcal {C}(u^\\ast _i)$ where the dual vacuum satisfies $\\mathopen {\\langle }0\\mathclose {|}\\mathcal {B}(u)=0$ and $\\mathopen {\\langle }0\\mathclose {|}\\mathcal {A}(u)=\\mathopen {\\langle }0\\mathclose {|}a(u)~~~\\text{and}~~~ \\mathopen {\\langle }0\\mathclose {|}\\mathcal {D}(u)=\\mathopen {\\langle }0\\mathclose {|}d(u)~.$ These dual states are related to Bethe states by Hermitian conjugation using the definition $\\mathopen {|}0\\mathclose {\\rangle }=\\mathopen {\\langle }0\\mathclose {|}^\\dagger ~,~~~ \\text{and}~~~\\mathcal {C}(u^\\ast )=-\\mathcal {B}^\\dagger (u)$ and are dual eigenstates of $\\hat{T}(u)$ when the rapidities satisfy the Bethe equations.",
"We will be interested in the quantity $I_M(\\lbrace v\\rbrace ,\\lbrace u\\rbrace )$ which is related to the scalar products of Bethe states by the definition $I_M(\\lbrace v\\rbrace ,\\lbrace u\\rbrace )&\\equiv &\\mathopen {\\langle }0\\mathclose {|}\\prod _{j=1}^M \\mathcal {C}(v_j)\\prod _{j=1}^M\\mathcal {B}(u_j)\\mathopen {|}0\\mathclose {\\rangle }\\\\&=&(-1)^M~ {}^{\\text{al}}\\mathopen {\\langle } \\lbrace v^\\ast \\rbrace | \\lbrace u\\rbrace \\mathclose {\\rangle }^{\\text{al}}~$ and, following [46], can be written as a sum over partitions of the excitations.",
"The partitions are defined by splitting each set of excitations, $\\lbrace u\\rbrace $ and $\\lbrace v\\rbrace $ , into subsets, $\\alpha \\cup \\bar{\\alpha }=\\lbrace u\\rbrace $ and $\\beta \\cup \\bar{\\beta }=\\lbrace v\\rbrace $ , with the cardinality of $\\alpha $ is equal to that of $\\beta $ .",
"The scalar product is then given as $I_M(\\lbrace v\\rbrace ,\\lbrace u\\rbrace )&=&{g_<}^{\\lbrace u\\rbrace }g_>^{\\lbrace v\\rbrace }\\sum _{\\begin{array}{c}\\alpha \\cup \\bar{\\alpha }=\\lbrace u\\rbrace \\\\\\beta \\cup \\bar{\\beta }=\\lbrace v\\rbrace \\end{array}}\\text{sgn}(\\alpha )\\text{sgn}(\\beta )d^\\alpha a^{\\bar{\\alpha }}a^\\beta d^{\\bar{\\beta }} k^{\\alpha \\beta }k^{\\bar{\\beta }\\bar{\\alpha }}k^{\\alpha \\bar{\\alpha }}k^{\\bar{\\beta }\\beta } \\text{det}\\, t^{\\alpha \\beta }\\text{det}\\, t^{\\bar{\\beta }\\bar{\\alpha }}$ where $k(u,v)=\\frac{f(u,v)}{g(u,v)}=1-i(u-v)~,~~~\\text{and}~~~t(u,v)=\\frac{g^2(u,v)}{f(u,v)}=\\frac{-1}{(u-v)(u-v+i)}$ and $\\text{sgn}(\\alpha )$ is the signature of the permutation required to put $\\alpha \\cup \\bar{\\alpha }$ into the canonical order $\\lbrace u\\rbrace $ .",
"This formula is valid for arbitrary Bethe states, even those whose rapidities do not satisfy the Bethe equations and which are thus said to be \"off-shell\".",
"In the case where one set of rapidities does satisfy the Bethe equations, they are said to be \"on-shell\", the formula can be dramatically simplified to the calculation of a single determinant [27].",
"There is a further simplification when both sets of rapidities are on-shell and equal.",
"In this case, as the set of rapidities is invariant under complex conjugation, the quantity $I_M$ is related to the norm of the Bethe state and is given by Gaudin's formula: $I_M(\\lbrace u\\rbrace ,\\lbrace u\\rbrace )&=&d^{\\lbrace u\\rbrace }a^{\\lbrace u\\rbrace }f_>^{\\lbrace u\\rbrace } f_<^{\\lbrace u\\rbrace }~\\text{det}_{j,k}~\\partial _{u_j}\\phi _k$ where $\\phi _k$ is defined in (REF )."
],
[
"Unfolding Procedure",
"For an ordered spectrum $E_1\\le E_2 \\le \\dots \\le E_N$ we define the level density function $n(E)=\\sum _{i=1}^N \\delta (E-E_i)~.$ and the cumulative spectral function, or staircase function, $I(E)=\\int _{0}^{E} n(E^{\\prime })dE^{\\prime }= \\sum _{i=1}^N \\Theta (E-E_i)~.$ We now separate these spectral functions into smooth and fluctuating parts $n(E)=n_{\\text{a}v}(E)+n_{\\text{f}l}(E)~, ~~~\\text{and}~~~I(E)=I_{\\text{a}v}(E)+I_{\\text{f}l}(E)$ and then define new unfolded variables $x_i=I_{\\text{av}}(E_i)~, ~~~\\text{for}~~~i=1, 2, \\dots , N$ so that for small separations $x_{i+1}-x_i\\simeq \\frac{(E_{i+1}-E_i)}{D}~,$ where $D=1/n_{\\text{av}}(E_i)$ is the local mean spacing.",
"These new variables thus capture the nature of the spectral fluctuations about the smoothed behaviour.",
"Figure: The unfolded spectrum of anomalous dimension for L=16L=16, M=4M=4, N=17N=17, β=0.9\\beta =0.9 using linear interpolations based on choosing n=2,10,200,400n=2, 10,200, 400.",
"The different unfoldings are essentially identical and are all similar to the original spectrum.",
"Here we include the nn extrapolated values at the each end of the spectrum where the difference in the unfolding procedure is large and which we neglect in our computations.However, without a priori knowledge of the smooth or mean level density for a physical system we must use approximate methods to compute the unfolded spectrum.",
"There does not appear to be an optimal procedure and so we use a relatively straightforward method.",
"We select each $n$ -th energy from the spectrum $\\lbrace E_i\\rbrace $ and then perform a piece-wise linear interpolation to define $I_{\\text{av}}$ .",
"Fortunately the final result does not appear to be particularly sensitive to the choice of method.",
"For example, we took $n=10$ but alternative choices such as $n= 2, 200, 400$ all give similar results, and so the values for the unfolded spectrum are likely reasonably robust, see Fig.",
"REF .",
"The procedure does cut-off the first and last $n$ -elements and so has edge effects, however as we are interested in differences of energies the overall shift has no effect and the differences in the tails of the unfolded distribution do not modify the final results significantly.",
"In fact for the anomalous dimensions the unfolding process has only a very minor effect and could have been neglected."
]
] | 2005.14254 |
[
[
"A First Principle Study on Magneto-Optical Effects and Magnetism in\n Ferromagnetic Semiconductors Y$_3$Fe$_5$O$_{12}$ and Bi$_3$Fe$_5$O$_{12}$"
],
[
"Abstract The magneto-optical (MO) effects not only are a powerful probe of magnetism and electronic structure of magnetic solids but also have valuable applications in high-density data-storage technology.",
"Yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$) (YIG) and bismuth iron garnet (Bi$_3$Fe$_5$O$_{12}$) (BIG) are two widely used magnetic semiconductors with strong magneto-optical effects and have also attracted the attention for fundamental physics studies.",
"In particular, YIG has been routinely used as a spin current injector.",
"In this paper, we present a thorough theoretical investigation on magnetism, electronic, optical and MO properties of YIG and BIG, based on the density functional theory with the generalized gradient approximation plus onsite Coulomb repulsion.",
"We find that both semiconductors exhibit large MO effects with their Kerr and Faraday rotation angles being comparable to that of best-known MO materials such as MnBi.",
"Especially, the MO Kerr rotation angle for bulk BIG reaches -1.2$ ^{\\circ}$ at photon energy $\\sim2.4$ eV, and the MO Faraday rotation angle for BIG film reaches -74.6 $ ^{\\circ}/\\mu m$ at photon energy $\\sim2.7$ eV.",
"Furthermore, we also find that both valence and conduction bands across the MO band gap in BIG are purely spin-down states, i.e., BIG is a single spin semiconductor.",
"These interesting findings suggest that the iron garnets will find valuable applications in semiconductor MO and spintronic nanodevices.",
"The calculated optical conductivity spectra, MO Kerr and Faraday rotation angles agree well with the available experimental data.",
"The main features in the optical and MO spectra of both systems are analyzed in terms of the calculated band structures especially by determining the band state symmetries and the main optical transitions at the $\\Gamma$ point in the Brillouin zone."
],
[
"Introduction",
"Yttrium iron garnet (Y$_3$ Fe$_5$ O$_{12}$ , YIG) is a ferrimagnetic semiconductor with excellent magnetic properties such as high curie temperature $T_c$ [1], low Gilbert damping $\\alpha \\sim 6.7\\times 10^{-5}$ [4], [2], [3] and long spin wave propragating length [5].",
"Various applications such as spin pumping require a non-metallic magnet.",
"YIG is thus routinely used for spin pumping purposes [4].",
"It is also widely used as a magnetic insulating substrate for purposes such as introducing magnetic proximity effect while avoiding electrical short-cut.",
"[6] YIG has high Curie temperature, which is good for applications across a wide temperature range.",
"The low Gilbert damping of YIG also makes it a good microwave material.",
"YIG thus becomes a famous material in the field of spintronics, where coupling between magnetism, microwave and spin current becomes possible.",
"Magneto-optical (MO) effects are important examples of light-matter interactions in magnetic phases.",
"[7], [8] When a linearly polarized light beam is shined onto a magnetic material, the reflected and transmitted light becomes elliptically polarized.",
"The principal axis is rotated with respect to the polarization direction of incident light beam.",
"The former and latter effects are termed MO Kerr (MOKE) and MO Faraday (MOFE) effects, respectively.",
"MOKE allowes us to detect the magnetization locally with a high spatial and temporal resolution in a non-invasive fashion.",
"Furthermore, magnetic materials with large MOKE would find valuable MO storage and sensor applications [10], [9].",
"Thus it has been widely used to probe the electronic and magnetic properties of solids, surface, thin films and 2D magnets [8].",
"On the other hand, MOFE can be used as a time-reversal symmetry-breaking element in optics [11], and its applications such as optical isolators are consequenses of time-reversal symmetry-breaking [12].",
"Magnetic materials with large Kerr or Faraday rotation angles have technological applications.",
"YIG is also known to be MO active [13].",
"Various experiments have been carried out to study the MOKE and MOFE of iron garnets in the visible and near-UV regime [14], [15].",
"Substituting yttrium with bismuth results in bismuth iron garnet (Bi$_3$ Fe$_5$ O$_{12}$ ) (BIG).",
"BIG has approximately 7 times larger Faraday rotation angles than that of YIG.",
"The effect of doping bismuth into YIG on the MOFE spectrum was studied [16], [17].",
"The large radius of bismuth atoms seems to make bulk BIG unstable.",
"Thus high quality BIG film is difficult to synthesize [18].",
"Though numerous experimental studies have been done on these systems, first-principle calculations are scarce.",
"This is probably due to the complexity of the structures of BIG and YIG.",
"As shown in Fig.",
"1(a), they have a total of 80 atoms in the primitive cell.",
"Although the electronic structures of YIG and BIG have been theoretically studied [19], [20], no first principle calculation on the MOKE or MOFE spectra of YIG and BIG have been reported.",
"Therefore, here we carry out a systematic first-principle density functional study on the optical and MO properties of YIG and BIG.",
"The rest of this paper is organized as follows.",
"A brief description of the crystal structures of YIG and BIG as well as the theoretical methods used is given in Sec.",
"II.",
"In Sec.",
"III, the calculated magnetic moments, electronic structure, optical conductivities, MO Kerr and Faraday effects are presented.",
"Finally, the conclusions drawn from this work are given in section IV."
],
[
"CRYSTAL STRUCTURE AND COMPUTATIONAL METHODS ",
"YIG and BIG crystalize in the cubic structure with space group $Ia3d$ [21], [22], as illustrated in Fig.",
"1(a).",
"In each unit cell, there are 48 oxygen atoms at the Wyckoff 96h positions, 8 octahedrally coordinated iron atoms (Fe$^O$ ) at the 16a positions, and 12 tetrahedrally coordinated iron atoms (Fe$^T$ ) at the 24d positions in the primitive cell.",
"In other words, there are two Fe$^O$ ions and three Fe$^T$ ions per formula unit (f.u.).",
"The experimental lattice constant $a=12.376$ Å, and the experimental Wyckoff parameters for oxygen atoms are $(x, y, z) = (0.9726, 0.0572, 0.1492)$ .",
"[21] The experimental lattice constant for BIG $a=12.6469$ Å.",
"[22] Accurate oxygen position measurement for BIG is still on demand and under debate [18].",
"Therefore we use the experimental lattice constant for BIG with the atomic positions determined theoretically (see Table I), as described next.",
"We use the experimental lattice constant and atomic positions for all YIG calculations, Table: Structural parameters of Y 3 _3Fe 5 _5O 12 _{12} and Bi 3 _3Fe 5 _5O 12 _{12}.For YIG, experimental lattice constant a=12.376a=12.376 Å and oxygen positions are used.For BIG, experimental lattice constant a=12.6469a=12.6469 Å is used while the oxygen positionsare determined theoretically.Figure: (a) 1/8 of BIG conventional unit cell.",
"Oxygen atoms are shown as red balls;bismuth atoms are shown as purple balls; Fe T ^T atoms are shown as yellow balls; Fe O ^O atoms are shown as blue balls.",
"(b) Brillouin zone of both YIG and BIG.",
"The red lines denote the high symmetry lines where the calculatedenergy bands will be plotted.Our first principle calculations are based on the density functional theory with the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof formula [23] to the electron exchange-correlation potential.",
"Furthermore, we use the GGA + $U$ method to have a better description for on-site interaction for Fe $d$ electrons.",
"[24] Here we set $U = 4.0$ eV, which was found to be rather appropriate for iron oxides [25].",
"Indeed, as we will show below, the optical and MO spectra calculated using this $U$ value agree rather well with the available experimental spectra.",
"All the calculations are carried out by using the accurate projector-augmented wave [26] method, as implemented in Vienna ab initio Simulation Package (VASP).",
"[27], [28] A large energy cutoff of 450 eV for the plane-wave basis is used.",
"A $6 \\times 6 \\times 6$ $k$ -point mesh is used for both systems in the self-consistent charge density calculations.",
"The density of states (DOS) calculation is performed with a denser $k$ -point mesh of $10 \\times 10 \\times 10$ .",
"We first calculate the optical conductivity tensor which determine the MOKE and MOFE.",
"We let the magnetization of our systems be along (001) ($z$ ) direction.",
"In this case, our systems have the four-fold rotational symmetry along the $z$ axis and thus the optical conductivity tensor can be written in the following form [29]: $\\sigma =\\begin{pmatrix}\\sigma _{xx} & \\sigma _{xy} & 0 \\\\-\\sigma _{xy}& \\sigma _{xx} & 0 \\\\0 & 0 & \\sigma _{zz}\\end{pmatrix}.$ The optical conductivity tensor can be formulated within the linear response theory.",
"Here the real part of the diagonal elements and imaginary part of the off-diagonal elements are given by [30], [31], [29]: $\\sigma _{aa}^{1} (\\omega ) = \\frac{\\pi e^2}{\\hbar \\omega m^2}\\sum _{i,j}\\int _{BZ}\\frac{d{\\bf k}}{(2\\pi )^3}|p_{ij}^{a}|^{2}\\delta (\\epsilon _{{\\bf k}j}-\\epsilon _{{\\bf k}i}-\\hbar \\omega ),$ $\\sigma _{xy}^{2} (\\omega ) = \\frac{\\pi e^2}{\\hbar \\omega m^2}\\sum _{i,j}\\int _{BZ}\\frac{d{\\bf k}}{(2\\pi )^3}\\text{Im}[p_{ij}^{x}p_{ji}^{y}]\\delta (\\epsilon _{{\\bf k}j}-\\epsilon _{{\\bf k}i}-\\hbar \\omega ),$ where $\\hbar \\omega $ is the photon energy, and $\\epsilon _{\\textbf {k}i(j)}$ are the energy eigenvalues of occupied (unoccupied) states.",
"The transition matrix elements $p_{ij}^{a} = \\langle \\textbf {k}\\emph {j}|\\hat{p}_{a}|\\textbf {k}i\\rangle $ where $|\\textbf {k}i(j)\\rangle $ are the $i$ ($j$ )th occupied(unoccupied) states at $k$ -point $\\textbf {k}$ , and $\\hat{p}_a$ is the Cartesian component $a$ of the momentum operator.",
"The imaginary part of the diagonal elements and the real part of the off-diagonal elements are then obtained from $\\sigma _{aa}^{1} (\\omega )$ and $\\sigma _{xy}^{2} (\\omega )$ , respectively, via the Kramers-Kronig transformations as follows: $\\sigma _{aa}^{2} (\\omega ) = -\\frac{2\\omega }{\\pi }P \\int _{0}^{\\infty }\\frac{\\sigma _{aa}^{1}(\\omega ^{\\prime })}{\\omega ^{^{\\prime }2}-\\omega ^{2}}d\\omega ^{^{\\prime }},$ $\\sigma _{xy}^{1} (\\omega ) = \\frac{2}{\\pi }P \\int _{0}^{\\infty }\\frac{\\omega ^{^{\\prime }}\\sigma _{xy}^{2}(\\omega ^{\\prime })}{\\omega ^{^{\\prime }2}-\\omega ^{2}}d\\omega ^{^{\\prime }},$ where $P$ denotes the principal value of the integration.",
"We can see that Eq.",
"(2) and Eq.",
"(3) neglect transitions across different $k$ -points since the momentum of the optical photon is negligibly small compared with the electron crystal momentum and thus only the direct interband transitions need to be considered.",
"In our calculations $p^a_{ij}$ are obtained in the PAW formalism [32].",
"We use a $10 \\times 10 \\times 10$ $k$ -point mesh and the Brillouin zone integration is carried out with the linear tetrahedron method (see [33] and references therein), which leads to well converged results.",
"To ensure that the $\\sigma _{aa}^{2} (\\omega )$ and $\\sigma _{xy}^{1} (\\omega )$ in the optical frequency range (e.g., $\\hbar \\omega < 8$ eV) obtained via Eqs.",
"(4) and (5) are converged, we include the unoccupied states at least 21 eV above the Fermi energy, i.e., a total of 1200 (1300) bands are used in the YIG (BIG) calculations.",
"For a bulk magnetic material, the complex polar Kerr rotation angle is given by [34], [35], $\\theta _{K}+i\\epsilon _{K}=\\frac{-\\sigma _{xy}}{\\sigma _{xx}\\sqrt{1+i(4\\pi /\\omega )\\sigma _{xx}}}.$ Similarly, the complex Faraday rotation angle for a thin film can be written as [39] $\\theta _{F}+i\\epsilon _{F}=\\frac{\\omega d}{2c}(n_{+}-n_{-}),$ where $n_+$ and $n_-$ represent the refractive indices for left- and right-handed polarized lights, respectively, and are related to the corresponding dielectric function (or optical conductivity via expressions $n_{\\pm }^{2}=\\varepsilon _{\\pm }=1+{\\frac{4\\pi i}{\\omega }}\\sigma _{\\pm }=1+{\\frac{4\\pi i}{\\omega }}(\\sigma _{xx}\\pm i \\sigma _{xy})$ .",
"Here the real parts of the optical conductivity $\\sigma _{\\pm }$ can be written as $\\sigma _{\\pm }^{1} (\\omega ) = \\frac{\\pi e^2}{\\hbar \\omega m^2}\\sum _{i,j}\\int _{BZ}\\frac{d{\\bf k}}{(2\\pi )^3}|\\Pi _{ij}^{\\pm }|^{2}\\delta (\\epsilon _{{\\bf k}j}-\\epsilon _{{\\bf k}i}-\\hbar \\omega ),$ where $\\Pi _{ij}^{\\pm } = \\langle \\textbf {k}\\emph {j}|\\frac{1}{\\sqrt{2}}(\\hat{p}_{x}\\pm i\\hat{p}_{y})|\\textbf {k}i\\rangle $ .",
"Clearly, $\\sigma _{xy} = \\frac{1}{2i}(\\sigma _{+}-\\sigma _{-})$ , and this shows that $\\sigma _{xy}$ would be nonzero only if $\\sigma _{+}$ and $\\sigma _{-}$ are different.",
"In other words, magnetic circular dichroism is the fundamental cause of the nonzero $\\sigma _{xy}$ and hence the MO effects."
],
[
"Magnetic moments",
"Here we first present calculated total and atom-decomposed magnetic moments in Table I.",
"As expected, Y$_3$ Fe$_5$ O$_{12}$ is a ferrimagnet in which Fe ions of the same type couple ferromagnetically while Fe ions of different types couple antiferromagnetically.",
"Since there are two Fe$^O$ ions and three Fe$^T$ ions in a unit cell, Y$_3$ Fe$_5$ O$_{12}$ is ferrimagnetic with a total magnetic moment per f.u.",
"being $\\sim $ 5.0 $ \\mu _B$ (see Table I).",
"The calculated spin magnetic moments of Fe ions of both types are $\\sim $ 4.0 $ \\mu _B$ , being consistent with the high spin state of Fe$^{+2}$ ($d^{5\\uparrow }t_{2g}^{1\\downarrow }$ ) ions in either octahedral or tetrahedral crystal field.",
"We note that the orbital magnetic moments of Fe are parallel to their spin magnetic moments.",
"Nonetheless, the calculated orbital magnetic moments of Fe are small, because of strong crystal field quenching.",
"Interestingly, there is a significant spin magnetic moment on each O ion, and this together with the spin magnetic moment of one net Fe ion per f. u. leads to the total spin magnetic moment per f.u.",
"of $\\sim $ 5.0 $ \\mu _B$ .",
"The calculated Fe magnetic moments for both symmetry sites agree rather well with the measured ones of $\\sim 4.0$ $ \\mu _B$ .",
"[36] The calculated total magnetization of $\\sim 5.0$ $\\mu _B$ /f.u.",
"is also in excellent agreement with the experiment.",
"[36] Bi$_3$ Fe$_5$ O$_{12}$ is also predicted to be ferrimagnetic, although the calculated magnetic moments of both Fe$^O$ and Fe$^T$ ions are slightly smaller than the corresponding ones in Y$_3$ Fe$_5$ O$_{12}$ (see Table I).",
"The total magnetization and local magnetic moments of the other ions in BIG are almost identical to that in YIG.",
"However, the experimental $m_{tot}$ for BIG is only $4.4$ $\\mu _B$ , [37] being significantly smaller than the calculated value.",
"As mentioned before, stable high quality BIG crystals are hard to grow.",
"Consequently, this notable discrepancy in total magnetization between the calculation and the previous experiment [37] could be due to the poor quality of the samples used in the experiment."
],
[
"Electronic structure",
"Here we present the calculated scalar-relativistic band structures of YIG and BIG in Fig.",
"2(a) and Fig.",
"3(a), respectively.",
"The calculated band structures show that YIG and BIG are both direct band-gap semiconductors, where the conduction band minimum (CBM) and valence band maximum (VBM) are both located at the $\\Gamma $ point.",
"For BIG, both CBM and VBM are purely spin-up bands.",
"This means that BIG is a single-spin semiconductor, which may find applications for spintronic and spin photovoltaic devices.",
"The origin of the MO effects is the magnetic circular dichorism [see Eq.",
"(8)], as mentioned above, which cannot occur without the presence of the spin-orbit coupling (SOC).",
"Therefore, it is useful to examine how the SOC influence the band structures.",
"The fully relativistic band structures for YIG and BIG are presented in Fig.",
"2(b) and Fig.",
"3(b), respectively.",
"First, we notice that with the inclusion of the SOC, YIG and BIG are still direct band-gap semiconductors, where the CBM and VBM are both located at the $\\Gamma $ point.",
"Second, Fig.",
"3(b) indicates that when the SOC is considered, the BIG band structure changes significantly, while the YIG band structure hardly changes [see Fig.",
"2(b)].",
"For example, the band gap for BIG decreases from 2.0 to 1.8 eV after the SOC is included.",
"Also, the gap, which was at 3.4 to 3.7 eV above the Fermi energy [see Fig.",
"3(a)], now becomes from 3.9 to 4.5 eV above the Fermi energy [see Fig.",
"3(b)].",
"Interestingly, the substitution of yittrium by bismuth not only enhances the SOC but also changes the electronic band structure significantly, as can be seen by comparing Figs.",
"2 and 3.",
"We also calculate total as well as site-, orbital-, and spin-projected densities of states (DOS) for YIG and BIG, as displayed in Fig.",
"(4) and (5), respectively.",
"First, Figs.",
"(4) and (5) show that in both YIG and BIG, the upper valence bands ranging from -4.0 to 0.0 eV, are dominated by O $p$ -orbitals with minor contributions from Fe $d$ -orbitals as well as Y $d$ -orbitals in YIG and Bi $sp$ -orbitals in BIG.",
"Second, the lower conduction band manifold, ranging from 1.8 to $\\sim $ 3.9 eV in YIG (Fig.",
"4) and from 2.0 to 3.4 eV in BIG (Fig.",
"5), stems predominately from Fe $d$ -orbitals with small contributions from O $p$ -orbitals.",
"Therefore, the semiconducting band gaps in YIG and BIG are mainly of the charge transfer type.",
"Furthermore, on the Fe$^T$ sites, the $d$ -DOS in this conduction band is almost fully spin-down [see Figs.",
"4(e) and 5(e)].",
"On the Fe$^O$ sites, on the other hand, the $d$ -DOS in this conduction band is almost purely spin-up [see Figs.",
"4(d) and 5(d)].",
"Here, the DOS peak marked $a$ mostly consists of $t_{2g}$ orbital while that marked $b$ above peak $a$ , is made up of mainly $e_g$ orbital.",
"The gap between peaks $a$ and $b$ is thus caused by the crystal field splitting.",
"Figure 4 indicates that in YIG, the upper conduction bands from 4.4 to 6.0 eV are mainly of Y $d$ orbital character with some contribution from O $p$ orbitals.",
"In BIG, on the other hand, the upper conduction bands from 3.6 to 6.0 eV are mainly the Bi and O $p$ orbital hybridized bands (see Fig.",
"5).",
"Notably, there is sizable Bi $sp$ DOS in the lower conduction band region from 2.0 to 3.4 eV (see Fig.",
"5(c)], indicating that the lower conduction bands in BIG are significantly mixed with Bi $sp$ orbitals, as noticed already by Oikawa et al.",
"[20], Since the SOC of the Bi $p$ orbitals are very strong, this explains why the band width of the lower conduction bands in BIG increases from $\\sim $ 1.4 to 2.1 eV when the SOC is included (see Fig.",
"3).",
"In contrast, the band width of the lower conduction bands in YIG remains unaffected by the SOC (see Fig.",
"2).",
"This also explains why the MO effects in BIG are much stronger than in YIG, as reported in Sec.",
"III.D.",
"below.",
"Figure: Calculated optical conductivity of Y 3 _3Fe 5 _5O 12 _{12}.",
"(a) Real part and (b) imaginary partof the diagonal element; (c) imaginary part and (d) real part of the off-diagonal element.All the spectra have been convoluted with a Lorentzian of 0.3 eV to simulate the finite quasiparticle lifetime effects.Red lines are the optical conductivity derived from the experimental dielectric constant.",
"Figure: Calculated optical conductivity of Bi 3 _3Fe 5 _5O 12 _{12}.",
"(a) Real part and (b) imaginary partof the diagonal element; (c) imaginary part and (d) real part of the off-diagonal element.All the spectra have been convoluted with a Lorentzian of 0.3 eV to simulate the finite quasiparticle lifetime effects.Red lines are the optical conductivity derived from the experimental dielectric constant."
],
[
"Optical Conductivity",
"Here we present the optical and magneto-optical conductivities for YIG and BIG which are ingredients for calculating the Kerr and Faraday rotation angles [see Eq.",
"(6) and Eq.",
"(7)].",
"In particular, the MO conductivity (i.e., the off-diagonal element of the conductivity tensor $\\sigma _{xy}$ ) is crucial, as shown by Eq.",
"(8).",
"Calculated optical conductivity spectra of YIG and BIG are plotted as a function of photon energy in Fig.",
"6 and Fig.",
"7, respectively.",
"For YIG, the real part of the diagonal element of the conductivity tensor ($\\sigma ^1_{xx}$ ) starts to increase rapidly from the absorption edge ($\\sim $ 2.3 eV) to $\\sim $ 4.0 eV, and then further increases with a smaller slope up to $\\sim $ 5.6 eV [see Fig.",
"6(a)].",
"It then decreases slightly until 6.6 eV and finally increases again with a much steeper slope up to $\\sim $ 8.0 eV.",
"Similarly, in BIG, $\\sigma ^1_{xx}$ increases steeply from the absorption edge ($\\sim $ 2.0 eV) to $\\sim $ 4.0 eV, and then further increases with a smaller slope up to $\\sim $ 6.0 eV [see Fig.",
"7(a)].",
"It then decrease steadily from $\\sim $ 6.0 eV to $\\sim $ 8.0 eV.",
"The behaviors of the imaginary part of the diagonal element ($\\sigma ^2_{xx}$ ) of YIG and BIG are rather similar in the energy range up to 5.0 eV [see Figs.",
"6(b) and 7(b)].",
"The $\\sigma ^2_{xx}$ spectrum has a broad valley at $\\sim $ 3.5 eV ($\\sim $ 3.0 eV) in the case of YIG (BIG).",
"However, the $\\sigma ^2_{xx}$ spectra of YIG and BIG differ from each other for energy $>$ 5.0 eV.",
"There is a sign change in $\\sigma ^2_{xx}$ occuring at $\\sim $ 5.8 eV for BIG, while there is no such a sign change in $\\sigma ^2_{xx}$ of YIG up to 8.0 eV.",
"The striking difference in the off-diagonal element of the conductivity ($\\sigma _{xy}$ ) (i.e., magneto-optical conductivity or magnetic circular dichroism) between YIG and BIG is that $\\sigma _{xy}$ of BIG is almost ten times larger than that of YIG (see Figs.",
"6 and 7).",
"Nonetheless, the line shapes of the off-diagonal element of YIG and BIG are rather similar except that their signs seem to be opposite and their peaks appear at quite different energy positions.",
"In particular, in the low energy range up to $\\sim $ 4.4 eV, the line shape of the imaginary part of the off-diagonal element ($\\sigma ^2_{xy}$ ) of BIG looks like a \"W\" [see Fig.",
"7(c)], while that of YIG in the energy region up to $\\sim $ 7.0 eV seems to have the inverted \"W\" shape [see Fig.",
"6(c)], The main difference is that the $\\sigma ^2_{xy}$ of BIG decreases oscillatorily from 4.4 to 8.0 eV.",
"On the other hand, the line shape of the real part of the off-diagonal element ($\\sigma ^1_{xy}$ ) of BIG looks like a \"sine wave\" between 2.0 and 4.7 eV [see Fig.",
"7(d)], while that of YIG appears to be an inverted \"sine wave\" between 2.6 and 6.4 eV [see Fig.",
"6(d)].",
"The largest magnitude of $\\sigma ^2_{xy}$ of YIG is $\\sim 1.6 \\times 10^{13}s^{-1}$ at $\\sim $ 4.3 eV, while that of BIG is $\\sim 1.9 \\times 10^{14}s^{-1}$ at $\\sim $ 3.1 eV.",
"The largest magnitude of $\\sigma ^1_{xy}$ of YIG is $\\sim 1.2 \\times 10^{13}s^{-1}$ at $\\sim $ 4.8 eV, while that of BIG is $\\sim 1.9 \\times 10^{14}s^{-1}$ at $\\sim $ 2.6 eV.",
"In order to compare with the available experimental data, we also plot the experimental optical conductivity spectra [14], [17] in Figs.",
"6 and 7.",
"The theoretical spectra of the diagonal element of the optical conductivity tensor for both YIG and BIG match well with that of the experimental ones in the measured energy range [see Figs.",
"6(a) and 6(b) as well as Figs.",
"7(a) and 7(b)].",
"Interestingly, we note that the relativistic GGA+U calculations give rise to the band gaps of YIG and BIG that are smaller than the experimental ones (see Table II), and yet the calculated and measured optical spectra agree rather well with each other.",
"This apparently contradiction can be resolved as follows.",
"In YIG, for example, the lowest conduction bands at $E = 1.8 \\sim 2.4$ eV above the VBM are highly dispersive (see Fig.",
"2) and thus have very low DOS (see Fig.",
"4).",
"This results in very low optical transition.",
"Therefore, the main absoption edge that appears in the optical spectrum ($\\sigma _{xx}^1$ ) is $\\sim 2.2$ eV, which is close to the experimental absorption edge of 2.5 eV, instead of 1.8 eV as determined by the calculated band structure (see Table II).",
"In contrast, no such highly dispersive bands appear at the CBM in BIG, Thus the calculated band gap agrees better with the measured band gap [17] (Table II).",
"Figures 7(c) and 7(d) show that the calculated $\\sigma _{xy}^1$ and $\\sigma _{xy}^2$ of BIG agree almost perfectly with the experimental data [17].",
"The peak positions, peak heights and overall trend of the theoretical spectra are nearly identical to that of the experimental ones [17].",
"On the other hand, the calculated $\\sigma _{xy}^1$ and $\\sigma _{xy}^2$ for YIG do not agree so well with the experimental data [14] [Figs.",
"6(c) and 6(d].",
"For example, there is a sharp peak at $\\sim $ 4.8 eV in the experimental $\\sigma _{xy}^1$ spectrum, which seems to be shifted to a higher energy at 5.6 eV with much reduced magnitude in the theoretical $\\sigma _{xy}^1$ spectrum [see Fig.",
"6(c)].",
"Also, for $\\sigma _{xy}^2$ spectrum, there is a sharp peak at $\\sim $ 4.5 eV in the experimental $\\sigma _{xy}^2$ spectrum, which appears at $\\sim 4.8$ eV with considerably reduced height [see Fig.",
"6(d)].",
"Nonetheless, the overall trend of the theoretical $\\sigma _{xy}$ spectra of YIG is in rather good agreement with that of the measured ones [14].",
"Equations (2), (3), and (8) indicate that the absorptive parts of the optical conductivity elements ($\\sigma _{xx}^1, \\sigma _{zz}^1, \\sigma _{xy}^2$ and $\\sigma _{\\pm }^1$ ) are directly related to the dipole allowed interband transitions.",
"Thus, we analyze the origin of the main features in the magneto-optical conductivity ($\\sigma _{xy}^2$ ) spectrum by determining the symmetries of the involved band states and the dipole selection rules (see the Appendix for details).",
"The absorptive optical spectra are usually dominated by the interband transitions at the high symmetry points where the energy bands are generally flat (see, e.g., Figs.",
"2 and 3), thus resulting in large joint density of states.",
"As an example, here we consider the interband optical transitions at the $\\Gamma $ point where the band extrema often occur.",
"Based on the determined band state symmetries and dipole selection rules (see Table III in the Appendix) as well as calculated transition matrix elements [Im$(p_{ij}^xp_{ji}^y)$ ], we assign the main features in $\\sigma _{xy}^2$ [labelled in Figs.",
"6(c) and 7(c)] to the main interband transitions at the $\\Gamma $ point as shown in Figs.",
"8 and 9.",
"The details of these assignments, related interband transitions and transition matrix elements for YIG and BIG are presented in Tables IV and V in the Appendix, respectively.",
"Since there are too many possible transitions to list, we present only those transitions whose transition matrix elements $|$ Im$(p_{ij}^xp_{ji}^y) | > 0.010 $ a.u.",
"in YIG (Table IV) and $|$ Im$(p_{ij}^xp_{ji}^y) | > 0.012$ a.u.",
"in BIG (Table V).",
"Figure 8 shows that nearly all the main optical transitions in YIG are from the upper valence bands to the upper conduction bands, and only one main transition (P$_3$ ) to the lower conduction bands.",
"Consequently, these transitions contribute to the main features in $\\sigma _{xy}^2$ at photon energy $> 4.0$ eV [see Fig.",
"6(c)].",
"In contrast, in BIG, a large number of the main transitions (e.g., P1-5, P7, N1-4, N5-8) are from the upper valence bands to lower conduction bands (see Fig.",
"9).",
"This gives rise to the main features in $\\sigma _{xy}^2$ for photon energy $< 4.0$ eV [see Fig.",
"7(c)], whose magnitudes are generally one order of magnitude larger than that of $\\sigma _{xy}^2$ in YIG, as mentioned above.",
"The largely enhanced MO activity in BIG stems from the significant hybridization of Bi $p$ -orbitals with Fe $d$ -orbitals in the lower conduction bands, as mentioned above.",
"Since heavy Bi has a strong spin-orbit coupling, this hybridization greatly increases the dichroic interband transitions from the upper valence bands to the lower conduction bands in BIG.",
"As mentioned above, Y $sd$ orbitals contribute significantly only to the upper conduction bands in YIG, and this results in the pronounced magneto-optical transitions only from the upper valence bands to the upper conduction bands (Fig.",
"8).",
"Furthermore, Y is lighter than Bi and thus has a weaker SOC than Bi.",
"The discussion in the proceeding paragraph clearly indicates that the significant hybridization of heavy Bi $p$ orbitals with Fe $d$ orbitals in the lower conduction bands just above the band gap is the main reason for the large MO effect in BIG.",
"The magnetism in BIG is mainly caused by the iron $d$ orbitals which have a rather weak SOC.",
"However, through the hybridization between Bi $p$ orbitals and Fe $d$ orbitals, the strong SOC effect is also transfered to the lower conduction bands.",
"Large exchange splitting and strong spin-orbit coupling in the valence and conduction bands below and above the band gap are crucial for strong magnetic circular dichroism and hence large MO effects.",
"Therefore, in search of materials with strong MO effects, one should look for magnetic systems that contain heavy elements such as Bi and Pt [38].",
"Figure: Relativistic band structures of Y 3 _3Fe 5 _5O 12 _{12}.",
"Horizontal dashed lines denote the top of valance band.The principal interband transitions at the Γ\\Gamma point and the corresponding peaks in the σ xy \\sigma _{xy} in Fig.",
"6 (c)are indicated by red and blue arrows.Figure: Relativistic band structures of Bi 3 _3Fe 5 _5O 12 _{12}.",
"Horizontal dashed lines denote the top of valance band.The principal interband transitions at the Γ\\Gamma point and the corresponding peaks in the σ xy \\sigma _{xy} in Fig.",
"7 (c)are indicated by red and blue arrows."
],
[
"Magneto-optical Kerr and Faraday effect ",
"Finally, let us study the polar Kerr and Faraday effects in YIG and BIG.",
"The complex Kerr and Faraday rotation angles for YIG and BIG are plotted as a function of photon energy in Figs.",
"8 and 9, respectively.",
"First of all, we notice that the Kerr rotation angles of BIG [Fig.",
"10(c)] are many times larger than that of YIG [Fig.",
"10(a)].",
"For example, the positive Kerr rotation maximum of 0.10 ${}^{\\circ }$$ $ in YIG occurs at $\\sim 3.6$ eV, while that (0.80 ${}^{\\circ }$ ) for BIG appears at $\\sim 3.5$ eV.",
"The negative Kerr rotation maximum (-0.12 ${}^{\\circ }$ ) of YIG occurs at $\\sim 4.8$ eV, while that (-1.21 ${}^{\\circ }$ ) for BIG appears at $\\sim 2.4$ eV.",
"This may be expected because Kerr rotation angle is proportional to the MO conductivity ($\\sigma _{xy}^{1}$ ) [Eq.",
"(6)], which in BIG is nearly ten times larger than in YIG, as mentioned in the proceeding subsection.",
"Similarly, the Kerr ellipticity maximum (0.16 ${}^{\\circ }$ ) of YIG occurs at $\\sim 4.1$ eV [Fig.",
"10(b)], whereas that (0.54 ${}^{\\circ }$ ) of BIG [Fig.",
"10(d)] appears at $\\sim 1.9$ eV.",
"The negative Kerr ellipticity maximum (-0.07 ${}^{\\circ }$ ) of YIG occurs at $\\sim 5.7$ eV while that (-1.16${}^{\\circ }$ ) of BIG is located at $\\sim 2.9$ eV.",
"Let us now compare our calculated Kerr rotation angles with some known MO materials such as $3d$ transition metal alloys and compound semiconductors.",
"[8] For magnetic metals, ferromagnetic $3d$ transition metals and their alloys are an important family.",
"Among them, manganese-based pnictides are known to have strong MO effects.",
"In particular, MnBi thin films were reported to have a large Kerr rotation angle of 2.3 ${}^{\\circ }$ .",
"[40], [39] Platinum alloys such as FePt, Co$_2$ Pt [38] and PtMnSb [41] also possess large Kerr rotation angles.",
"It was shown that the strong SOC on heavy Pt in these systems is the main cause of the strong MOKE.",
"[38] Among semiconductor MO materials, diluted magnetic semiconductors Ga$_{1-x}$ Mn$_x$ As were reported to show Kerr rotations angle as large as 0.4 ${}^{\\circ }$ at 1.80 eV.",
"[42] Therefore, the strong MOKE effect in YIG and BIG could have promising applications in high density MO data-storage devices or MO nanosensors with high spatial resolution.",
"Figure 9 shows that as for the Kerr rotation angles, the Faraday rotation angles of BIG are generally up to ten times larger than that of YIG.",
"The Faraday rotation maximum (7.2 ${}^{\\circ }$ /$\\mu $ m) of YIG occurs at $\\sim 3.9$ eV, while that (51.2${}^{\\circ }$ /$\\mu $ m) of BIG is located at $\\sim 3.7$ eV.",
"The Faraday ellipticity maximum (7.9 ${}^{\\circ }$ /$\\mu $ m) for YIG appears at $\\sim 4.4$ eV, whereas that (54.1${}^{\\circ }$ /$\\mu $ m) of BIG occurs at $\\sim 2.3$ eV.",
"On the other hand, the negative Faraday rotation maximum (-5.7${}^{\\circ }$ /$\\mu $ m) occurs at $\\sim 5.4$ eV, while that (-74.6${}^{\\circ }$ /$\\mu $ m) for BIG appears at $\\sim 2.7$ eV.",
"The negative Faraday ellipticity maximum (-3.6${}^{\\circ }$ /$\\mu $ m) of YIG occurs at $\\sim 6.6$ eV, while that (-70.2${}^{\\circ }$ /$\\mu m$ ) for BIG is located at $\\sim 3.2$ eV.",
"For comparision, we notice that MnBi films are known to possess large Faraday rotation angles of $\\sim 80$${}^{\\circ }$ /$\\mu $ m at 1.8 eV.",
"[40], [39] Finally, we compare our predicted MOKE and MOFE spectra with the available experiments in Figs.",
"10 and 11.",
"All the predicted MOKE and MOFE spectra are in rather good agreement with the experimental ones in the experimental photon energy range [15], [14], [17], [43].",
"Nonetheless, our theoretical predictions would have a better agreement with the experiments if all the calculated spectra are blue-shifted slightly by $\\sim $ 0.3 eV, thus suggesting that the theoretical band gaps are slightly too small.",
"Figure: Calculated complex Kerr rotation angles (blue curves).",
"(a) Kerr rotation (θ K \\theta _K)and (b) Kerr ellipticity (ε K \\varepsilon _K) spectraof Y 3 _3Fe 5 _5O 12 _{12}; (c) Kerr rotation (θ K \\theta _K) and (d) Kerr ellipticity (ε K \\varepsilon _K) spectraof Bi 3 _3Fe 5 _5O 12 _{12}.",
"Red circles in (a) and (b) denote the experimental values from Ref.",
".Figure: Calculated complex Faraday rotation angles (blue curves).",
"(a) Faraday rotation (θ F \\theta _F) and(b) Faraday ellipticity (ε F \\varepsilon _F) spectra of Y 3 _3Fe 5 _5O 12 _{12};(c) Kerr rotation (θ F \\theta _F) and(d) Kerr ellipticity (ε F \\varepsilon _F) spectraof Bi 3 _3Fe 5 _5O 12 _{12}.Red dashed line in (a) denotes the measured values from Ref.",
".Black circles in (a) and (b) are the experimental values from Ref.",
".Red (green) circles in (c) and (d) are the experimental valuesfrom Ref.",
"()To summarize, we have systematically studied the electronic structure, magnetic, optical and MO properties of cubic iron garnets YIG and BIG by performing GGA+U calculations.",
"We find that YIG exhibits significant MO Kerr and Faraday effects in UV frequency range that are comparable to cubic ferromagnetic iron.",
"Strikingly, we find that BIG shows gigantic MO effects in the visible frequency region that are several times larger than YIG.",
"In particular, the Kerr rotation angle of BIG becomes as large as -1.2${}^{\\circ }$ at photon energy 2.4 eV, and the Faraday rotation angle for the BIG film reaches -75 ${}^{\\circ }$ /$\\mu m$ at 2.7 eV.",
"Calculated MO conductivity ($\\sigma _{xy}^2$ ) spectra reveal that these distinctly different MO properties of YIG and BIG result from the fact that the magnitude of $\\sigma _{xy}^2$ of BIG is nearly ten times larger than that of YIG.",
"Our calculated Kerr and Faraday rotation angles of YIG agree well with the available experimental values.",
"Our calculated Faraday rotation angles of BIG are in nearly perfect agreement with the measured ones.",
"Thus, we hope that our predicted giant MO Kerr effect in BIG will stimulate further MOKE experiments on high quality BIG crystals.` Principal features in the optical and MO spectra are analyzed in terms of the calculated band structures especially the symmetry of the band states and optical transition matrix elements at the $\\Gamma $ point of the BZ.",
"We find that in YIG, Y $sd$ orbitals mix mainly with the upper conduction bands that are $\\sim 4.5$ eV above the VBM, and thus leave the Fe $d$ orbital dominated lower conduction bands from 1.8 to 3.8 eV above the VBM almost unaffected by the SOC on the Y atom.",
"In contrast, Bi $p$ orbitals in BIG hybridize significantly with Fe $d$ orbitals in the lower conduction bands and this leads to large SOC-induced band splitting and much increased band width of the lower conduction bands.",
"Consequently, the MO transitions between the upper valence bands and lower conduction bands are greatly enhanced when Y is replaced by heavier Bi.",
"This finding thus provides a guideline in search for materials with desired MO effects, i.e., one should look for magnetic materials with heavy elements such as Bi whose orbitals hybridize significantly with the MO active conduction or valence bands.",
"Finally, our findings of strong MO effects in these iron garnets and also single-spin semiconductivity in BIG suggest that cubic iron garnets are an useful playground of exploring the interplay of microwave, spin current, magnetism, and optics degrees of freedom, and also have promising applications in high density semiconductor MO data-storage and low-power consumption spintronic nanodevices."
],
[
"Acknowledgments",
"The authors thank Ming-Chun Jiang for many valuable discussions throughout this work.",
"The authors acknowledge the support from the Ministry of Science and Technology and the National Center for Theoretical Sciences (NCTS) of The R.O.C.",
"The authors are also grateful to the National Center for High-performance Computing (NCHC) for the computing time.",
"G.-Y.",
"Guo also thanks the support from the Far Eastern Y.",
"Z. Hsu Science and Technology Memorial Foundation in Taiwan.",
"In this Appendix, to help identify the origins of the main features in the magneto-optical conductivity $\\sigma _{xy}(\\omega )$ spectra of YIG and BIG, we provide the dipole selection rules and the symmetries of the band states at the $\\Gamma $ as well as the main optical transitions between them.",
"Both YIG and BIG have the Ia$\\bar{3}$ d space group and thus they have the $C_{4h}$ ($4/mm^{\\prime }m^{\\prime }$ ) point group at the $\\Gamma $ point in the Brillouin zone.",
"Based on the character table of the $C_{4h}$ point group [44], we determine the dipole selection rules for the optical transitions between the band states at the $\\Gamma $ point, as listed in Table III.",
"We calculate the eigenvalues for all symmetry elements of each eigenstate of the $\\Gamma $ point using the Irvsp program [45] and then determine the irreducible representation and hence the symmetry of the state.",
"Based on the obtained symmetries of the band states and also calculated optical matrix elements [Im$(p_{ij}^xp_{ji}^y)$ ] [see Eq.",
"(3)], we assign the peaks in the $\\sigma _{xy}(\\omega )$ spectra of YIG [see Fig.",
"6(c)] and BIG [see Fig.",
"7(c)] to the main optical transitions at the $\\Gamma $ point (see Fig.",
"8 and 9, respectively), as listed in Tables IV and V, respectively.",
"Table: Dipole selection rules for the C 4h C_{4h} point group at the Γ\\Gamma point in the Brillouin zone of YIG and BIG.Table: Main optical transitions between the states at the Γ\\Gamma point of the Brillouin zone of YIG.Symbols in the first column denote the assigned peaks in the magneto-optical conductivity (σ xy 2 \\sigma _{xy}^{2}) spectrum(Figs.",
"6 and 8).",
"ii and jj denote the initial and final states, respectively.Im(p ij x p ji y )(p_{ij}^xp_{ji}^y) denote the calculated transition matrix element (in atomic units) [see Eq.",
"(3)].E i E_i and E j E_j represent the initial and final state energies (in eV),respectively.",
"ΔE ij =E j -E i \\Delta E_{ij} = E_j-E_i is the transition energy.Table: Main optical transitions between the states at the Γ\\Gamma point of the Brillouin zone of BIG.Symbols in the first column denote the assigned peaks in the magneto-optical conductivity (σ xy 2 \\sigma _{xy}^{2}) spectrum(Figs.",
"7 and 9).",
"ii and jj denote the initial and final states, respectively.Im(p ij x p ji y )(p_{ij}^xp_{ji}^y) denote the calculated transition matrix element (in atomic units) [see Eq.",
"(3)].E i E_i and E j E_j represent the initial and final state energies (in eV),respectively.",
"ΔE ij =E j -E i \\Delta E_{ij} = E_j-E_i is the transition energy."
]
] | 2005.14133 |
[
[
"Compact Halo around the Sun Accreted after Dark Matter Dissipative Self\n Interaction"
],
[
"Abstract If dark matter particle can be decelerated due to its dissipative self scattering, except for sinking at the galaxy scale to speed up structure formation, it can also be accreted onto local celestial bodies such as the Sun, forming a compact halo.",
"With some simplified assumptions we develop the Boltzmann equation set based on the partition function of the elliptical orbits, and numerically solve it for the accretion process.",
"We find that the orbited dark matter particles will form a halo around the Sun, with the density profile well fitted to be proportional to $r^{-1.6}$ in a wide range of radius.",
"While around the earth such local halo contribution is always several orders below the galactic component, in a very small region centered around the Sun the sunk dark matter particles can lead to a halo density several orders larger than the background galactic component, in particular in the parameter region of small deceleration speed and large cross section, which is still consistent with current constraints.",
"Such potential dark matter local halo with significantly enhanced density will be a very interesting source for dark matter indirect detection if the corresponding channel exists, we discuss the possibility of the gamma-ray spectrum in the solar direction in some detail as an example."
],
[
"Introduction",
"The model that dark matter (DM) has self interaction with a cross section of $\\mathcal {O}(1)~\\text{cm}^2/\\text{g}$ [1] is originally suggested as a solution to the core vs. cusp problem [2], [3], and later possibly to other small scale problems such as the “too-big-to-fail” problem [4], [5] and the missing satellite problem [6], [7].",
"Nowadays the improved simulation with the baryonic feedback effect [8] and the accumulated observation gradually became consistent with each other, reducing the room for any new physics beyond the vanilla cold DM model such as this self-interacting DM (SIDM) model.",
"But the SIDM model has found other motivations such as interpreting the diversities of the rotation curves [9] and the dwarf galaxies [10].",
"If elastic DM self scattering is acceptable, then the postulation of the DM self scattering being inelastic seems a natural alternative, given the freedom of model building in a non-minimal dark sector.",
"The case that DM up-scatters onto an excited state [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40] and promptly decays back to the original ground state (emitting a dark photon which is not our object) is simple but particularly interesting, in which the DM particle decelerates and consequently varializes to a deeper position in the gravitational well to accelerate the structure formation.",
"On galaxies scales, such process will lead to a dark disk [18], [19], or greatly accelerate the core collapse process of the DM halo.",
"For example, the latter effect is suggested as a solution to the mysterious origin of super massive black hole at a very high redshift [30], [31], [35], [36], [37], [38], [40] that seems to violate the Eddington limit of accretion.",
"Except for the galaxy center or the galactic disk, in a much smaller region [29] the known local celestial bodies such as the Sun or the earth also provides a similar local “deep” position of the gravitational field to collect the decelerated DM particle.",
"To the knowledge of the author, such possibility has not been explored in detail in literature.",
"A similar DM deceleration followed by gravitational capture process has already been spotted in the context of the traditional DM candidate of the Weakly Interacting Massive Particle (WIMP) [41].",
"However, such process is based on the WIMP nucleon scattering cross section for the initial deceleration, and now get severe constraint [42].",
"On the other hand, the DM self scattering cross section can be many orders larger, which actually can overwhelm the shortage of the target DM particle number.",
"A quick estimation with a solar mass of $2\\times 10^{30}~\\text{kg}$ (or $1.2\\times 10^{57}$ nucleon) gives a scattering/capture rate of $1.4\\times 10^{16}(\\frac{\\sigma _{\\chi n}}{10^{-46}~\\text{cm}^2})(\\frac{\\rho _\\chi }{0.39~\\text{GeV}/\\text{cm}^3})(\\frac{\\langle \\Delta v\\rangle }{240~\\text{km}/\\text{s}})~\\text{s}^{-1}$ for an $80~\\text{GeV}$ WIMP, and if we switch to our interested dissipative DM (DDM) model In literature similar models have been named as eXciting DM in [12] or Double-Disk DM in [18]., similarly the scattering/capture rate for the DDM in the volume of the Sun is already $1.2\\times 10^{14}(\\frac{\\sigma /m_\\chi }{1~\\text{cm}^2/\\text{g}})(\\frac{\\rho _\\chi }{0.39~\\text{GeV}/\\text{cm}^3})^2(\\frac{\\langle \\Delta v\\rangle }{240~\\text{km}/\\text{s}})~\\text{s}^{-1}$ .",
"The initial collisional deceleration can happen far away from the Sun and the DDM is still accreted onto the Sun, so the accretion volume is many orders larger than the volume of the Sun, making the DDM accretion larger than the WIMP accretion.",
"This is a simple demonstration that the DDM accretion can be very significant.",
"In this paper we develop the first calculation based on the Boltzmann equation for such accretion process.",
"The key to our calculation is the usage of number of DDM particles on each elliptical orbit characterized by its energy and angular momentum as the partition function, which enables robust calculation for the outer part of the resultant halo.",
"Due to the limitation of such approximation, the DDM particles very close to or even inside the Sun cannot be studied directly, but can still be inferred indirectly with potentially large uncertainty.",
"However, the possibility of the existence of a solar DDM halo with number density several orders larger than the galactic component is established, which will be very interesting to the DM indirect detection field.",
"We will define the model and the approximation schemes in section , and further do some definition and review in section .",
"Then we write down the Boltzmann equation terms for each process in section , and numerically solve it with the results presented and discussed in section .",
"Section provides a brief discussion of the found overdense in the context of DM indirect detection experiments, in particular the possibility towards the interpretation of the unexpected gamma-ray spectrum dip.",
"Finally we conclude in section ."
],
[
"The Model and Approximation Scheme",
"The simplified model is the same as the one used in fluid model for cooling [36] and N-body simulation [40] (see a variant in [37]).",
"We assume a velocity independent DDM self scattering cross section (actually the ratio of the cross section over the DDM mass) $\\sigma /m_\\chi $ if and only if in the center-of-mass frame the incoming speed of each DDM particle satisfies $\\Delta v/2>v_\\text{th}$ (the threshold velocity), then the final state DDM speed in the same frame is $\\Delta v^{\\prime }/2=((\\Delta v)^2/4-v_\\text{th}^2)^{\\frac{1}{2}}$ .",
"Namely in the up scattering and followed prompt decay, in the center-of-mass frame and for each DDM particle a constant kinetic energy of $m_\\chi v_\\text{th}^2/2$ is transferred to the dark radiation and dissipated at a constant cross section, once there is such kinetic energy to dissipate.",
"The recoil due to the dark radiation emission is $\\mathcal {O}(v_\\text{th}^2/c)$ and negligible.",
"We also assume all DM is our interested DDM.",
"For simplicity, in the calculation we will further ignore Any DDM elastic self-scattering, as a key simplification.",
"Any DDM nucleon scattering, which is much smaller as aforementioned.",
"Any DDM annihilation temporarily.",
"The annihilation rate will always be too small to modify the local halo accretion noticeably.",
"However, it may be manifested in a DM indirect detection experiment.",
"The size of the Sun to approximate it by a point mass $M$ , in order always to use a closed elliptical orbit.",
"But we still use the true density profile of the Sun (BS2005-AGS, OP model in Ref.",
"[43]) to calculate the inner gravitational field and the speed of DDM if it is submerged, for the purpose of the scattering kinematics.",
"We can see that the total accreted DM mass is still many orders (18 orders as the extreme case we have ever calculated) below the mass of the Sun finally, so its gravity can be ignored.",
"Moreover, unlike the WIMP case whose orbit will always be partially inside the Sun since the scattering deceleration happens there, in our DDM case the whole orbit can be completely outside the Sun.",
"In such cases the bounded DDM particle initially being on a perfect elliptical orbit is not an approximation but exact.",
"However, later the DDM further scatters to decelerate onto lower orbit, and eventually sink to an orbit mostly or completely submerged into the Sun.",
"Such elliptical orbit approximation enable us the trace the DDM to a region sufficiently close to the Sun.",
"the planets and all other celestial bodies in the solar system while studying the accretion of the Sun.",
"For a microscopic dissipative self-scattering $\\vec{u}_1+\\vec{u}_2\\rightarrow \\vec{u}^{\\prime }_1+\\vec{u}^{\\prime }_2$ event viewed in the solar frame (in this paper we will generally use $\\vec{v}$ for a velocity in the Milky Way (MW) rest frame, and $\\vec{u}$ for a velocity in the solar frame), we assume The scattering is $s$ wave, so the outgoing particles are always isotropic in angular distribution statistically.",
"Consequently if the final state particles become gravitationally bounded, against the angular momentum $L=m_\\chi ru\\sin \\theta =L_\\text{max}\\sin \\theta $ with a certain speed $u$ ($u^{\\prime }_1$ or $u^{\\prime }_2$ above) at certain radius $r$ , the expected number $dN$ is $\\propto \\sin \\theta d\\theta $ which is the solid angle, so the differential spectrum can be determined $\\frac{dN}{dL}&\\propto \\frac{\\sin \\theta d\\theta }{d\\sin \\theta }=\\frac{\\sin \\theta }{\\sqrt{1-\\sin ^2\\theta }}=\\frac{L}{L_\\text{max}\\sqrt{L_\\text{max}^2-L^2}},\\nonumber \\\\L_\\text{max}&=m_\\chi ru=m_\\chi r{\\textstyle \\sqrt{2(\\frac{E}{m_\\chi }+\\frac{GM}{r})}}.$ By integration $dL$ over 0 to $L_\\text{max}$ we can check that it is already normalized.",
"In fact such isotropic assumption holds only in the center-of-mass frame.",
"In the free-free process or the bound-free process which will be discussed soon, the free incoming DDM particle will have a nonzero expectation of velocity in the solar frame, therefore the scattered DM particle has an overall velocity expectation superposed on its isotropic distribution.",
"Later we will calculate the free-free process directly in more detail, taking this subtlety into account.",
"As for the bound-free process, such an overall velocity expectation due to the solar motion itself is not harmful, since the initial angular momentum is also randomly distributed due to the isotropic distribution of collision positions surrounding the Sun.",
"The outgoing DM particles have speed $u^{\\prime }_1=u^{\\prime }_2=\\sqrt{\\frac{1}{2}(u_1^2+u_2^2)-v_\\text{th}^2}$ .",
"Exact kinematics gives the final state velocity of $\\frac{1}{2}(\\vec{u}_1+\\vec{u}_2)\\pm \\hat{n}\\sqrt{\\frac{1}{4}|\\vec{u}_1-\\vec{u}_2|^2-v_\\text{th}^2}$ , where $\\hat{n}$ is the random unit direction vector of one outgoing DM particle.",
"According to the above isotropic assumption, statistically there will be no overall interference for the magnitude of the velocity, namely $(u^{\\prime }_1)^2=\\frac{1}{4}|\\vec{u}_1+\\vec{u}_2|^2+\\frac{1}{4}|\\vec{u}_1-\\vec{u}_2|^2-v_\\text{th}^2=\\frac{1}{2}(u_1^2+u_2^2)-v_\\text{th}^2=(u^{\\prime }_2)^2$ .",
"By a bit misuse of terminology we will refer to the assumption $u^{\\prime }_1=u^{\\prime }_2$ as equipartition.",
"This is of course not held in every microscopic dissipative scattering, but an unbiased estimation with correct energy conservation relation.",
"One can introduce some broadening of the final state energy instead of using the (Dirac) delta energy approximation, and in our following specific calculation for the free-free process we will indeed do so.",
"But for simplicity we will ignore such broadening for the other two processes.",
"In the same spirit, the relative speed for the two colliding particle is statistically $\\Delta v=\\sqrt{u_1^2+u_2^2}$ in the scattering rate calculation.",
"Again, later in the free-free process we will go beyond such statistical expectation, and calculate the relative speed for dissipative scattering rate in the full phase space."
],
[
"5 Processes and 3 Components",
"For a specific elliptical orbit with energy $E$ and angular momentum $L$ , there are only three possibilities for a microscopic dissipative DM self scattering event, or 5 processes while considering particle filling or removal on a specific energy state: Free-free process: two DDM particles are added onto orbits of the same energy (free-free-in) according to our previous equipartition assumption (therefore the two final state DDM particles are either both bounded at orbits with the same energy or both free, which also applies in the following processes); while initially they are both unbounded galactic ones.",
"Bound-free process: either a DDM particle is kicked out from the initial orbit (bound-free-out), or two DDM particles are added onto orbits of the same energy (bound-free-in); while in the initial state one of the two DDM particles belongs to a bound orbit and the other is an unbounded galactic one.",
"Bound-bound process: either one of the two DDM particles is kicked out from the initial orbit (bound-bound-out), or two DDM particles are added onto orbits of the same energy (bound-bound-in); while initially they are both already bounded at certain orbits and such scattering process only redistributes them onto different orbits.",
"We will sort the gravitationally bounded DDM particles into the orbited part and the sunk part, that the orbited part can be well described by an aforementioned closed elliptical orbit, with a partition function only depending on $E$ and $L$ , while the sunk part cannot due to the failure of such assumption or approximation.",
"Then in our later calculation the above reference of “bound” actually means the orbited component.",
"The reason for such categorizing is that we do not have a way to directly study the inner sunk component, due to the failure of the basic tool of elliptical orbit partition function.",
"Our strategy is record the sunk component by the transfer at the lower cutoff of the orbited component.",
"The third component is the free DDM particle which is not gravitationally bounded.",
"Then the free component, orbited component and sunk component are in the descending order of energy."
],
[
"The Free Galactic Component",
"For the free galactic DM particle far away from the Sun, we use the standard halo model, and the partition function is the escape velocity $v_\\text{esc}$ truncated Maxwell-Boltzmann distribution $f_0(\\vec{v})d^3v&=&\\frac{1}{N_\\text{esc}(\\sqrt{\\pi }v_0)^3}e^{-\\frac{v^2}{v_0^2}}d^3v~H(v_\\text{esc}-v).$ Here $H$ is the heaviside step function.",
"The Maxwell-Boltzmann distribution uses 1D velocity dispersion, which has been switched to the circle velocity $v_0$ at the solar radius and confined on a two-dimensional plane as its estimator.",
"$N_\\text{esc}=\\frac{4}{\\sqrt{\\pi }v_0^3}\\int ^{v_\\text{esc}}_0\\exp (-\\frac{v^2}{v_0^2})v^2dv$ gives the normalization when the escape velocity truncation presents.",
"Numerically for the standard halo model, we use the recent value $v_0=240~\\text{km}/\\text{s}$ [44] and $v_\\text{esc}=580~\\text{km}/\\text{s}$ [45], as well as the recent local DM energy density $\\rho _\\chi =0.39~\\text{GeV}/\\text{cm}^3$ [46] For historical reason, $v_0=220~\\text{km}/\\text{s}$ and $\\rho _\\chi =0.3~\\text{GeV}/\\text{cm}^3$ are widely used such as in the DM direct detection literature.. As seen by an observer at rest to the Sun and therefore moving with velocity $\\vec{v}_\\odot $ in the MW rest frame, the velocity has relation $\\vec{u}+\\vec{v}_\\odot =\\vec{v}$ and the partition function now becomes $&f_\\odot (\\vec{u})d^3u=\\frac{2e^{-\\frac{u^2+v_\\odot ^2}{v_0^2}}}{N_\\text{esc}\\sqrt{\\pi }v_0^3}u^2du\\int (e^{\\frac{2uv_\\odot }{v_0^2}\\cos \\theta }d\\cos \\theta )\\\\=&\\left\\lbrace \\begin{array}{ll}{\\displaystyle \\frac{udu}{\\sqrt{\\pi }N_\\text{esc}v_0v_\\odot }e^{-\\frac{u^2+v_\\odot ^2}{v_0^2}}2\\sinh \\frac{2uv_\\odot }{v_0^2}}, & u<v_\\text{esc}-v_\\odot ,\\smallskip \\\\{\\displaystyle \\frac{udu}{\\sqrt{\\pi }N_\\text{esc}v_0v_\\odot }\\Big (e^{-\\frac{(u-v_\\odot )^2}{v_0^2}}-e^{-\\frac{v_\\text{esc}^2}{v_0^2}}\\Big )}, & \\begin{array}{c}v_\\text{esc}-v_\\odot <u\\\\u<v_\\text{esc}+v_\\odot ,\\end{array}\\medskip \\\\0, & u>v_\\text{esc}+v_\\odot .\\end{array}\\right.\\nonumber $ In cases that the escape velocity is not considered or effectively taken to be infinity (and $N_\\text{esc}\\rightarrow 1$ ), the first expression reduces to the well-known form found by Gould [41].",
"As the process studied here spans almost $5~\\text{Gyr}$ of the whole age of the Sun, after time average the observer velocity $v_\\odot $ should go back to $v_0$ , while at the moment $v_\\odot $ may differ from $v_0$ by a few components.",
"At last, the gravitational field will also accelerate the free DM particle to distort the phase space distribution.",
"In analogy to Ref.",
"[41], at the position $r$ inside a gravitational field, we can see the free DM particle partition function is $&f_\\odot (u,r)d^3u=\\frac{uf_\\odot (\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)})d^3\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}}{\\sqrt{u^2-u_\\text{esc}^2(r)}}\\\\=&\\left\\lbrace \\begin{array}{l}{\\displaystyle \\frac{u^2du~e^{-\\frac{u^2-u_\\text{esc}^2(r)+v_\\odot ^2}{v_0^2}}}{\\sqrt{\\pi }N_\\text{esc}v_0v_\\odot \\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}}2\\sinh \\frac{2\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}v_\\odot }{v_0^2}},\\smallskip \\\\\\qquad \\qquad \\sqrt{u^2-u_\\text{esc}^2(r)}<v_\\text{esc}-v_\\odot , \\smallskip \\\\{\\displaystyle \\frac{u^2du}{\\sqrt{\\pi }N_\\text{esc}v_0v_\\odot \\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}}\\Big (e^{-\\frac{(\\sqrt{u^2-u_\\text{esc}^2(r)}-v_\\odot )^2}{v_0^2}}-e^{-\\frac{v_\\text{esc}^2}{v_0^2}}\\Big )}, \\smallskip \\\\ \\qquad \\qquad v_\\text{esc}-v_\\odot <\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}<v_\\text{esc}+v_\\odot \\smallskip \\\\0,~~\\qquad \\sqrt{u^2-u_\\text{esc}^2(r)}>v_\\text{esc}+v_\\odot .\\end{array}\\right.\\nonumber $ where we have used $\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}d\\sqrt{u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)}=udu$ .",
"$u_\\text{esc}(r)$ is the escape velocity from the local gravitational field, as mentioned earlier it is determined by the true density profile of the Sun but outside it is simply given by $u_\\text{esc}^2(r)=2GM/r$ .",
"Such argument shift can be viewed as implementing the Liouville's theorem of phase space, and the factor $u/\\sqrt{u^2-u_\\text{esc}^2(r)}$ is from the rate calculation within a shell of target collision rate."
],
[
"The Orbited Component",
"As for the orbited DM particle, given spherical symmetry, the distribution with the orbital parameters $E$ and $L$ of $\\frac{d^2N}{dEdL}$ plays the role of partition function.",
"The energy $E$ and angular momentum $L$ can be related to the elliptical Kepler orbit semi-major axis $a$ and semi-minor axis $b$ by $a=\\frac{GMm_\\chi }{-2E},\\qquad b=\\frac{L}{\\sqrt{-2Em_\\chi }}.$ We also define $c=\\sqrt{a^2-b^2}$ for short, then the DM particle has a radial range from $a-c$ to $a+c$ (Throughout this paper we will always suppress the $E$ and $L$ dependence of $a$ , $b$ and $c$ , but it should be understood that the corresponding ones should share the same subscript, e.g., $a_1(E_1)$ ).",
"From the two variables $E=\\frac{1}{2}m_\\chi ((\\frac{dr}{dt})^2+r^2(\\frac{d\\theta }{dt})^2)-\\frac{GMm_\\chi }{r}$ and $L=m_\\chi r^2\\frac{d\\theta }{dt}$ conserved in the orbital motion, we can solve $\\frac{dr}{dt}=\\sqrt{\\frac{2GM}{r}-\\frac{-2E}{m_\\chi }-\\frac{L^2}{r^2m_\\chi ^2}}$ .",
"Now we are interested in the probability of the DM particle being in a radius interval $r\\rightarrow r+dr$ .",
"It is proportional to the time the particle spends on it, or $d~\\text{probability}\\propto dt(r)=dr/(\\frac{dr}{dt})$ .",
"Eventually with normalization we find $d~\\text{probability}=\\frac{dr}{\\pi a\\sqrt{\\frac{2a}{r}-1-\\frac{b^2}{r^2}}}.$ It can be also rewritten as $rdr/(\\pi a\\sqrt{c^2-(r-a)^2})$ , which implies the region of $a-c<r<a+c$ ."
],
[
"The Boltzmann Equation",
"With the last probability differential we can calculate the bound-free or bound-bound process collision rate.",
"It is useful to at first summarize our constraints in the phase space (Statistical) energy conservation $E_1+E_2-m_\\chi v_\\text{th}^2=2E$ , with the help of the equipartition assumption.",
"Elliptical orbits radial position $a_i-c_i<r<a_i+c_i,~~\\forall ~i$ .",
"Threshold for dissipation $\\Delta v=\\sqrt{u_1^2+u_2^2}>2v_\\text{th}$ .",
"Free “really free” $u>u_\\text{esc}(r)$ for free particle.",
"Cannot escape the MW $u^2+u_\\text{esc}^2(r)<(v_\\text{esc}+v_\\odot )^2$ .",
"The rate for a general bound-free-out process to remove particles on an $(E,L)$ elliptical orbit reads $\\text{BFO}(E,L)=&-\\frac{d^2N}{dEdL}\\int _{a-c}^{a+c}\\frac{dr}{\\pi a\\sqrt{\\frac{2a}{r}-1-\\frac{b^2}{r^2}}}\\nonumber \\\\&\\times \\int _{\\text{max}(\\sqrt{4v_\\text{th}^2-\\frac{2E}{m_\\chi }-u_\\text{esc}^2(r)},u_\\text{esc}(r))}^{\\sqrt{(v_\\text{esc}+v_\\odot )^2+u_\\text{esc}^2(r)}}f_\\odot (u,r)d^3u\\nonumber \\\\&\\times \\frac{\\rho _\\chi }{m_\\chi }\\sigma \\Delta v_\\text{bf}(u,E,r).$ Here the two lines inside the two integrations are both dimensionless probabilities, to scan over all contributing phase space of the incoming free DDM particles.",
"The $f_\\odot (u,r)d^3u$ is given in Eq.",
"REF .",
"The two lower bound for velocity integration are “threshold for dissipation” and “free really free” conditions respectively, and the upper bound is the “cannot escape the MW” condition.",
"And in the last line of we have used the notation for the relative speed of the two initial state DDM particles $\\Delta v=\\left\\lbrace \\begin{array}{ll}\\sqrt{u^2+\\frac{2E}{m_\\chi }+u_\\text{esc}^2(r)} & \\text{bound-free}, \\\\\\sqrt{\\frac{2E_1}{m_\\chi }+\\frac{2E_2}{m_\\chi }+2u_\\text{esc}^2(r)} & \\text{bound-bound},\\end{array}\\right.$ as our last overall approximation point.",
"Then $\\frac{\\rho _\\chi }{m_\\chi }\\sigma \\Delta v$ gives the correct dimension of a rate, for kicking particles off the target orbit.",
"The rate for a general bound-bound-out process reads $\\text{BBO}(E,L)=&-\\frac{d^2N}{dEdL}\\int _{E_\\text{min}}^{E_\\text{max}}dE_1\\int _0^{L_\\text{max}(E_1)}dL_1\\frac{d^2N}{dE_1dL_1}\\nonumber \\\\&\\hspace{-40.0pt}\\times \\int _{\\text{max}(a-c,a_1-c_1)}^{\\text{min}(a+c,a_1+c_1)}\\frac{dr~H(\\Delta v_\\text{bb}(E,E_1,r)-2v_\\text{th})}{\\pi ^2aa_1\\sqrt{\\frac{2a}{r}-1-\\frac{b^2}{r^2}}\\sqrt{\\frac{2a_1}{r}-1-\\frac{b_1^2}{r^2}}}\\nonumber \\\\&\\times \\frac{1}{4\\pi r^2}\\sigma \\Delta v_\\text{bb}(E,E_1,r).$ Now we have to integrate over all possible $(E_1,L_1)$ orbits for the other orbited incoming DDM, which in practical numerical calculation are to be sampled by a finite number of bins.",
"The minimal energy $E_\\text{min}$ corresponds to a minimal semi-major axis for which the point mass approximation still effectively holds, and the maximal energy $E_\\text{max}$ corresponds to a maximal semi-major axis which should probably be related to whether at such scale the target celestial body can still be viewed as isolated.",
"Here for convenience, in practice we somewhat arbitrarily choose a range with $\\frac{E_\\text{min}}{m_\\chi }=-2^{16}~(\\text{km}/\\text{s})^2,\\qquad \\frac{E_\\text{max}}{m_\\chi }=-2^{3}~(\\text{km}/\\text{s})^2,$ and sample the (negative value of) energy states by every power of 2 in this range.",
"Such minimal semi-major axis is $1.455~R_\\odot $ and the information extracted based on this orbit will suffer some error, while the maximal semi-major axis is $55.4~\\text{AU}$ which should have sufficient coverage of the interested halo outskirt.",
"As for angular momentum, for a certain energy, the largest available angular momentum is achieved at a perfectly circular orbit, which is $L_\\text{max}(E)=m_\\chi a(E)\\sqrt{-\\frac{2E}{m_\\chi }}$ where the radius $a(E)$ is given by Eq.",
"REF and $\\sqrt{-2E/m_\\chi }$ is the circular velocity.",
"In practice we use 13 bins on the $L/L_\\text{max}(E)$ dimension.",
"Moreover, the cut from the “threshold for dissipation” condition is expressed in heaviside step function $H$ for convenience, and there are also double “elliptical orbits radial position” conditions.",
"The change compared with the first line of Eq.",
"REF can be understood as the number density from an $(E_1,L_1)$ elliptical orbit being $\\frac{Ndr}{\\pi a_1\\sqrt{2a_1/r-1-b_1^2/r^2}}/(4\\pi r^2dr)$ , namely the number of DDM particles divided by the volume, with the $dr$ s canceled.",
"The bound-free-in process rate to add DDM particles on an $(E,L)$ elliptical orbit reads $\\text{BFI}(E,L)=2&\\int _{E_\\text{min}}^{E_\\text{max}}dE_1\\int _0^{L_\\text{max}(E_1)}dL_1\\frac{d^2N}{dE_1dL_1}\\nonumber \\\\\\times &\\int _{\\text{max}(a-c,a_1-c_1)}^{\\text{min}(a+c,a_1+c_1)}\\frac{dr}{\\pi a_1\\sqrt{\\frac{2a_1}{r}-1-\\frac{b_1^2}{r^2}}}\\nonumber \\\\\\times &\\int _{\\text{max}(\\sqrt{4v_\\text{th}^2-\\frac{2E}{m}-u_\\text{esc}^2(r)},u_\\text{esc}(r))}^{\\sqrt{(v_\\text{esc}+v_\\odot )^2+u_\\text{esc}^2(r)}} f_\\odot (u,r)d^3u\\nonumber \\\\\\times &~\\frac{\\rho _\\chi }{m_\\chi }\\sigma \\Delta v_\\text{bf}(u,E_1,r)\\nonumber \\\\&\\hspace{-60.0pt}\\times {\\textstyle \\delta (\\frac{m_\\chi }{2}(u^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)\\hspace{-1.99997pt}-\\hspace{-1.99997pt}2v_\\text{th}^2)\\hspace{-1.99997pt}+\\hspace{-1.99997pt}E_1\\hspace{-1.99997pt}-\\hspace{-1.99997pt}2E)}~\\frac{dN}{dL}(E,L,r).$ The factor 2 in front corresponds to that two DDM particles are added (the equipartition assumption).",
"The first four lines are quite similar in structure to the previous ones.",
"The new fifth lines contains the differential spectrum to bring the contribution onto the desired $(E,L)$ orbit.",
"For energy the Dirac delta function is our “statistical energy conservation” condition, and for angular momentum the $dN/dL$ is given by Eq.",
"REF .",
"In terms of $a$ and $b$ we can rewrite the latter as $\\frac{dN}{dL}=\\frac{1}{\\big (m_\\chi \\frac{r^2}{b}\\sqrt{\\frac{GM}{a}}\\big )\\sqrt{\\frac{2a}{r}-1}\\sqrt{\\frac{2a}{r}-1-\\frac{b^2}{r^2}}},$ and the last factor on the denominator implies a similar radial position cutoff.",
"As for the former delta function, we can trivially integrate it out with the $du$ from the third line.",
"Quite similarly, the bound-bound-in process rate reads $\\text{BBI}(E,L)=&\\int _{E_\\text{min}}^{E_\\text{max}}dE_1\\int _0^{L_\\text{max}(E_1)}dL_1\\frac{d^2N}{dE_1dL_1}\\nonumber \\\\\\times &\\int _{E_\\text{min}}^{E_\\text{max}}dE_2\\int _0^{L_\\text{max}(E_2)}dL_2\\frac{d^2N}{dE_2dL_2}\\nonumber \\\\&\\hspace{-60.0pt}\\times \\int _{\\text{max}(a-c,a_1-c_1,a_2-c_2)}^{\\text{min}(a+c,a_1+c_1,a_2+c_2)}\\frac{dr~H(\\Delta v_\\text{bb}(E_1,E_2,r)-2v_\\text{th})}{\\sqrt{\\frac{2a_1}{r}-1-\\frac{b_1^2}{r^2}}\\sqrt{\\frac{2a_2}{r}-1-\\frac{b_2^2}{r^2}}}\\nonumber \\\\\\times &\\frac{1}{\\pi ^2a_1a_2~4\\pi r^2}\\sigma \\Delta v_\\text{bb}(E_1,E_2,r)\\nonumber \\\\\\times &{\\textstyle \\delta (E_1\\hspace{-1.99997pt}+\\hspace{-1.99997pt}E_2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}m_\\chi v_\\text{th}^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}2E)}~\\frac{dN}{dL}(E,L,r).$ Two DDM particles are added but the $(E_i,L_i)$ space has been doubly counted, so the overall factor is 1.",
"In real calculation the delta function of “statistical energy conservation” is also trivially integrated out by $dE_1$ or $dE_2$ .",
"At last we will work out the free-free-in term with more care.",
"Since this process has no bounded/orbited particle in the initial state, the velocities can be treated exactly with full respection to their directional information, and later we can see that this will be the most significant channel.",
"Unlike the above one we do not always accomplish the integration over the angle $\\theta _i$ of each relative velocity $i=1,2$ with the observer $\\vec{v}_\\odot $ , then Eq.",
"REF becomes $f_\\odot (\\vec{u}_i,r)d^3u_i=&\\frac{u_i^2du_i\\sin \\theta _id\\theta _id\\phi _i}{N_\\text{esc}(\\sqrt{\\pi }v_0)^3}\\nonumber \\\\\\times &e^{-\\frac{u_i^2-u_\\text{esc}^2(r)+v_\\odot ^2-2\\sqrt{u_i^2-u_\\text{esc}^2(r)}v_\\odot \\cos \\theta _i}{v_0^2}}.$ Here the $\\theta _i$ angle should be asymptotically defined at infinity for the local gravitational field, but we implicitly approximate it to be the local one.",
"Without loss of generality we choose the DDM particle 1 to be at azimuth $\\phi =0$ , then the general partition function multiplication have 5 variables $u_1$ , $u_2$ , $\\theta _1$ , $\\theta _2$ and $\\phi $ .",
"In order to determine the final state energy, we write down the exact center-of-mass and the relative speeds $u_c=\\frac{1}{2}&\\sqrt{u_1^2\\hspace{-1.99997pt}+\\hspace{-1.99997pt}u_2^2\\hspace{-1.99997pt}+\\hspace{-1.99997pt}2u_1u_2(\\cos \\theta _1\\hspace{-1.99997pt}\\cos \\theta _2\\hspace{-1.99997pt}+\\hspace{-1.99997pt}\\sin \\theta _1\\hspace{-1.99997pt}\\sin \\theta _2\\hspace{-1.99997pt}\\cos \\phi )},\\nonumber \\\\\\Delta v_\\text{ff}=&\\sqrt{u_1^2\\hspace{-1.99997pt}+\\hspace{-1.99997pt}u_2^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}2u_1u_2(\\cos \\theta _1\\hspace{-1.99997pt}\\cos \\theta _2\\hspace{-1.99997pt}+\\hspace{-1.99997pt}\\sin \\theta _1\\hspace{-1.99997pt}\\sin \\theta _2\\hspace{-1.99997pt}\\cos \\phi )},\\nonumber \\\\\\Delta v^{\\prime }_\\text{ff}=&\\sqrt{(\\Delta v_\\text{ff})^2-4v_\\text{th}^2}.$ After the dissipative scattering denoting the angle between $\\vec{u}_c$ and $\\Delta \\vec{v}^{\\prime }$ as $\\alpha $ , then the probability at such angle is proportional to $\\sin \\alpha d\\alpha $ , and the kinetic energy is $\\frac{m}{2}(u_c^2+\\frac{1}{4}\\Delta v_\\text{ff}^{\\prime 2}+u_c\\Delta v_\\text{ff}^{\\prime }\\cos \\alpha )$ so the energy differential is $dE^{\\prime }=\\frac{1}{2}mv_c\\Delta v^{\\prime }_\\text{ff}d\\cos \\alpha $ .",
"Eventually $d~\\text{probability}/dE^{\\prime }\\propto d\\cos \\alpha /d\\cos \\alpha =$ constant, namely the final state energy is evenly distributed in an interval of $mu_c\\Delta v_\\text{ff}^{\\prime }$ .",
"With such broadening of the final state energy distribution, the “statistical energy conservation” condition can be better replaced by the Real spectrum energy conservation $u_c^2+\\frac{1}{4}\\Delta v_\\text{ff}^{\\prime 2}-u_c\\Delta v_\\text{ff}^{\\prime }<\\frac{2E}{m_\\chi }+u_\\text{esc}^2(r)<u_c^2+\\frac{1}{4}\\Delta v_\\text{ff}^{\\prime 2}+u_c\\Delta v_\\text{ff}^{\\prime }$ , that at some $\\alpha $ value such final state energy is achieved.",
"Eventually the free-free-in rate reads $\\text{FFI}(E,L)=&\\int _{a-c}^{a+c} dr~4\\pi r^2\\nonumber \\\\&\\hspace{-60.0pt}\\times \\iint \\iiint _{u_\\text{esc}(r)}^{\\sqrt{(v_\\text{esc}+v_\\odot )^2+u_\\text{esc}^2(r)}}f_\\odot (\\vec{u}_1,r)d^3u_1f_\\odot (\\vec{u}_2,r)d^3u_2\\nonumber \\\\\\times &\\frac{\\rho _\\chi ^2}{m_\\chi ^2}\\sigma \\Delta v_\\text{ff}~H(\\Delta v_\\text{ff}-2v_\\text{th})\\nonumber \\\\\\times &{\\textstyle H(2E-m_\\chi (u_c^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)\\hspace{-1.99997pt}+\\hspace{-1.99997pt}\\frac{1}{4}\\Delta v^{\\prime 2}\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_c\\Delta v^{\\prime }))}\\nonumber \\\\\\times &{\\textstyle H(m_\\chi (u_c^2\\hspace{-1.99997pt}-\\hspace{-1.99997pt}u_\\text{esc}^2(r)\\hspace{-1.99997pt}+\\hspace{-1.99997pt}\\frac{1}{4}\\Delta v^{\\prime 2}\\hspace{-1.99997pt}+\\hspace{-1.99997pt}u_c\\Delta v^{\\prime })-2E)}\\nonumber \\\\\\times &\\frac{1}{m_\\chi u_c\\Delta v_\\text{ff}^{\\prime }}~\\frac{dN}{dL}(E,L,r),$ with the functions $f_\\odot (\\vec{u}_i,r)d^3u_i$ , $u_c$ , $\\Delta v_\\text{ff}$ and $\\Delta v_\\text{ff}^{\\prime }$ given just above in Eq.",
"REF to REF .",
"Again two DDM particles are added but the free particle phase space has been doubly counted, so the overall factor is 1.",
"In the second line the integration over the direction of $\\phi _1$ which is chosen should give $2\\pi $ .",
"We use the Monte Carlo integrator vegas to explore the whole phase space spanned by the 5 velocity variables as well as the radius $r$ in the local gravitational field, with respect to all the cuts.",
"Eventually with all terms at hand, we can write down the Boltzmann equation for the partition function $\\frac{d^2N}{dEdL}$ $\\frac{d}{dt}\\Big (\\frac{d^2N}{dEdL}\\Big )=\\big (\\text{FFI}+\\text{BFO}+\\text{BFI}+\\text{BBO}+\\text{BBI}\\big )(E,L).$"
],
[
"Results",
"Before we show the sample results, we can see that the energy density has no dependence on the DDM mass $m_\\chi $ , given that we actually use the value of the ratio $\\sigma /m_\\chi $ for the DDM self scattering cross section.",
"In fact when multiplying the Boltzmann Eq.",
"REF by $m_\\chi $ on both sides, in each term all the $m_\\chi $ factors originally in $\\rho _\\chi /m_\\chi $ can be absorbed either into the partition function $\\frac{d^2(m_\\chi N)}{dEdL}\\equiv \\frac{d^2m}{dEdL}$ or into the cross section $\\sigma /m_\\chi $ ."
],
[
"The Rates",
"In Fig.",
"REF we first show the sample rates for all the five processes.",
"Except for the constant free-free-in rate, the other four rates depend on at least one partition function of an elliptical orbit so are time dependent, here we choose to show them at the present time of the history of the Sun, which is roughly $t=1.5\\times 10^{17}~\\text{s}$ .",
"The energy range is given in Eq.",
"REF ; and for the angular momentum range which we have integrated over, we in fact sample them by a set of angular momentum values $L$ which satisfy $0<L/L_\\text{max}(E)<1$ .",
"We can see that very generally in a large $E$ range (or large radius range), the free-free-in rate (red filled diamond) is the largest, in particular for an energy state not very deep in the gravitational potential well, or at the outer part of the halo.",
"But this large rate tends to get balanced by the bound-free-out (blue filled circle) rate, and for a large cross section such cancellation is really good.",
"Later we will see that an equilibrium configuration should have been achieved, so the net partition function after cancellation are many orders smaller.",
"The remaining bound-bound-out rate, bound-free-in rate and bound-bound-in rate are much smaller and somewhat comparable themselves, and their dependence on the depth in the gravitational potential well or radial position is much milder.",
"The bound-free-in rate is much smaller than the bound-free-out process because the latter also contains the events that the final state DDM particles are kicked out of the halo by the energetic incoming DDM particles, and this process is actually dominant.",
"For small $v_\\text{th}$ the scattering is almost elastic and the partition function change due to scattering should be small, then the bound-bound-out process should be balanced by the bound-bound-in process, and we can see that it is indeed the case within numerical precisions.",
"But the bound-bound-in rate at a shallow position of the gravitational field is cutoff, since the target energy state must be at least deeper than $-v_\\text{th}^2/2+E_\\text{max}/m_\\chi $ .",
"While for a $5~\\text{km}/\\text{s}$ threshold speed such cutoff only affect the rightmost 3 bins in energy, for a $240~\\text{km}/\\text{s}$ threshold speed the effect can be very deep.",
"On the other hand, due to the low cutoff energy $E_\\text{min}/m_\\chi $ we expect the leftmost rate to suffer some error, so we use dotted curves to indicate the possible errors."
],
[
"The Orbit Component and the Outer Halo",
"With the numerical solution to the coupled Boltzmann equation set, we can calculate the halo profile from the contribution of the orbited component $&\\rho (r)=\\\\&\\int _{E_\\text{min}}^{E_\\text{max}}dE\\int _0^{L_\\text{max}(E)}dL\\frac{d^2m}{dEdL}\\frac{H(r-a+c)~H(a+c-r)}{\\pi a\\sqrt{\\frac{2a}{r}-1-\\frac{b^2}{r^2}}~4\\pi r^2}.\\nonumber $ Figure: Sample spatial and temporal slices of the halo around the Sun, using the orbited component only.",
"In the left panel we also fix the cross section, and in the right panel we also fix the threshold velocity.",
"Note that on the radial slice the sunk component can greatly further enhance the DDM density, but this is not included here, so the profile only from the orbited component in the corresponding region is plotted by dotted curves.",
"The evolution of the density show clear equilibrium configuration and the tendency to approach such equilibrium at different speed.We show the resultant halo by two slices in Fig.",
"REF .",
"In the left panel of the radial slice, we have fixed the time as well as the cross section, and choose four representative threshold speeds spanning a wide range of interest.",
"We can see that in a very large radial range the halo can be well fitted by a power law $\\rho \\propto r^n$ with $n\\sim -1.6$ for $v_\\text{th}\\lesssim 300~\\text{km}/\\text{s}$ .",
"In this region the dependence of the halo on the threshold speed is very weak, for example, the $v_\\text{th}=5~\\text{km}/\\text{s}$ curve overlaps with the $v_\\text{th}=80~\\text{km}/\\text{s}$ curve very well.",
"As $v_\\text{th}$ further increases the halo becomes more cuspy.",
"The halo density at the position of the earth is about $2.5$ orders below the galactic component, so no effect should be seen such as in the DM direct detection experiment.",
"But at a radius of $r\\sim 0.01-0.03~\\text{AU}$ the halo density increases to equal to the galactic component, and inside the local halo one is even higher.",
"However, later we will see that the sunk component will give an even much more enhanced contribution.",
"In the right panel of the temporal slice, we have fixed the radius as well as the $v_\\text{th}$ , and choose three representative cross sections to see the comparison of the accretion speed.",
"We can see that there is an equilibrium configuration, and all the densities corresponding to different cross sections tend to approach such equilibrium configuration upon accumulations.",
"If the cross section is large the current density will be very close to the equilibrium one, and if the cross section is small the current density can still be quite far from the equilibrium one.",
"At last, we can see that the equilibrium configuration will not be far away from the one calculated with $\\sigma /m_\\chi =1~\\text{cm}^2/\\text{g}$ , namely the solar halo contribution to the terrestrial DM density will always be at least about $2.5$ orders below the galactic contribution and unimportant."
],
[
"The Sunk Component",
"As mentioned before, while we can directly sample the orbited component by a set of discretized bins for the differential equation in the $(E,L)$ space, we cannot do so directly for the sunk component due to the intrinsic failure of the elliptical orbit approximation.",
"Such component has to be counted by the transition at the lower boundary $E_\\text{min}$ of our energy range.",
"Here we will naively estimate the sunk component as the particles sunk in the bound-bound processes, in particular $&N_\\text{sunk}=\\\\&-\\int dt\\int _{E_\\text{min}}^{E_\\text{max}}dE\\int _0^{L_\\text{max}(E)}dL(\\text{BBO}(E,L)+\\text{BBI}(E,L)).\\nonumber $ This estimation goes as follows.",
"In our previous definition of 5 processes, we do not specify the final state being free or bound (orbit) or sunk for an “out” process, which suffices the removal rate calculation for a specific orbit.",
"If we always specify the final state, the previous 3 possibilities in the initial state categorizing should by enumeration be expanded to the free-free to bound-bound processes, the bound-bound to bound-bound and bound-bound to sunk-sunk processes, as well as the bound-free to free-free and bound-free to bound-bound processes.",
"We have actually ignored the free-free to sunk-sunk process since it need to go beyond the equipartition assumption and use the broadened spectrum (the real spectrum energy conservation) of the final state kinetic energy, although from Fig.",
"REF we can see that by doing this a small rate can be really extrapolated.",
"Then we can see that the previous BBO process corresponds to the sum of the bound-bound to bound-bound and bound-bound to sunk-sunk processes, subtracting the bound-bound to bound-bound (BBI) process is indeed the only channel contributing to the sunk component.",
"Figure: The contour of the DDM average density enhancement factors from the sunk component only, compared to the DM galactic component.",
"The sunk component is counted by Eq.",
"at a minimal semi-major axis of a min =1.455R ⊙ a_\\text{min}=1.455~R_\\odot , corresponding to our minimally tracked energy state E min /m χ =-2 16 (km/s) 2 E_\\text{min}/m_\\chi =-2^{16}~(\\text{km}/\\text{s})^2 in Eq.",
", as the boundary between the bound (orbited) component and the sunk component.",
"Then the enhancement factor is calculated as m χ N sunk /(ρ χ 4 3πa min 3 )m_\\chi N_\\text{sunk}/(\\rho _\\chi \\frac{4}{3}\\pi a_\\text{min}^3).",
"Note that further channels that cannot captured in our calculation may further significantly affect the enhancement, see the text.",
"We also plot the constraint from for reference, which differs from our scenario by always assuming an elastic cross section of 3cm 2 /g3~\\text{cm}^2/\\text{g}.In such definition our results in Fig.",
"REF shows the possibility of a great enhancement to the local DM density, due to the sunk DDM component which is inside or extremely close to the Sun.",
"In general we can see that the sunk component will increase in the direction of small $v_\\text{th}$ as well as large $\\sigma /m_\\chi $ , and the average density can be several orders larger than the local DM density of the galactic component.",
"We warn the reader that this is far from a satisfying accurate estimation, but actually only the numerical guess we can reach in our approach.",
"If we can manage to include the sunk component in the initial state in our calculation, by the same enumeration we should have the additional sunk-sunk to sunk-sunk process (which just redistribute the sunk component and is not directly interesting in counting the total sunk DDM number), the sunk-free initial state to free-free, bound-bound or sunk-sunk processes, and the sunk-bound initial state to bound-bound or sunk-sunk processes.",
"They are all beyond our current calculation.",
"Among them, only the ones with the sunk-free initial state can change the total particle number of the bounded halo, while the other processes do not.",
"The sunk-free to free-free process is in the direction to reduce the sunk DDM number, but it should be limited from the point of view of the available initial free DDM phase space, which should be only an energetic corner.",
"On the other hand, the sunk-free to bound-bound process and the sunk-free to sunk-sunk process will further accrete the free DDM particles onto the bounded halo and increase the sunk component eventually.",
"Although not directly contributing DDM particles from connection to the free component, the sunk-bound ones and the sunk-sunk ones are helping the deceleration of the bounded halo particles and the accretion, which is also in the direction of further increasing the sunk component.",
"Since in the last two processes the second bounded DDM particle for the first sunk DDM to scatter has already a density much larger than the free galactic component density in the dominant central region (from Fig.",
"REF and REF ), we can expect the effect of further increasing the sunk component will dominate over the effect of reducing the sunk component by the sunk-free to free-free process.",
"Namely we expect the true DDM halo density inside or in the close neighborhood of the Sun is even more enhanced than what we guess from the Eq.",
"REF and show in Fig.",
"REF .",
"Such processes have the sunk component in the initial state, therefore the further increase behavior can be exponential, and the final DDM density can be further boosted by several orders.",
"Moreover, till now we have completely neglected the dark radiation which is assumed as promptly going away after the up-scattering.",
"For low DDM density it should be fine, however, if the DDM dissipative scattering happens at a high enough rate, the dark radiation will exert a pressure on the DDM particles, expelling them from the center of the Sun.",
"And this effect will presumably depend on other parameters beyond the cross section and threshold velocity, which further complicates the determination of the very inner region of the halo.",
"In Fig.",
"REF we have also plotted the constraint from Ref. [36].",
"Note the model assumption therein differs from ours by always assuming an elastic cross section of $3~\\text{cm}^2/\\text{g}$ .",
"We can find an area avoiding the constraint, that still has large enhancement and will be particularly interesting for DM indirect detection.",
"One may generally think that the DDM accretion at the galaxy scale to speed up structure formation such as the core collapse and the accretion at the solar scale to form the compact halo are quite similar, so they should be optimized at exactly the same $\\sigma /m_\\chi $ and $v_\\text{th}$ combination, and the most optimistic parameter for the solar halo enhancement should have been ruled out at the galaxy scale if the structure observed there is not that extreme.",
"But in fact the solar case has the Sun as the external gravitational source which is many orders larger in providing gravity, and the accretion is not purely self-driven as the galaxy case.",
"At the galaxy scale the optimistic $v_\\text{th}$ for core collapse should be close to the characteristic circular velocity, and even on the low mass tail of the dwarf galaxy distribution such characteristic circular velocity is in practical bounded from below.",
"On the other hand, the $v_\\text{th}\\rightarrow 0$ limit is actually the elastic SIDM limit, on the solar scale side eventually the free-free-in process as the source should vanish if the DM self-scattering turns into purely elastic, and consequently all the accrete rate.",
"But numerically we have not yet achieved that far in Fig.",
"REF .",
"With the Sun providing the external driving of the accretion, arguably the region left to the exclusion in Fig.",
"REF or its expected update in the future is a viable parameter region to achieve significant density enhancement."
],
[
"Comment on Other Cases",
"We have also performed similar calculations with the earth as the central gravitational source.",
"Except for adopting different values for the earth, the other key difference is that there is also the effect of the solar gravitational field, which is equivalent to an extra gap of $-GMm_\\chi /\\text{AU}$ between the local infinity to the free galactic component in energy.",
"We found that with a reasonable cross section as currently taken, the resultant halo will always be several orders below the galactic component, so it is less interesting.",
"Given the values of the standard halo model, similar calculation can be performed to other nearby celestial bodies, e.g., a black hole.",
"In such cases the validity of the elliptical orbit approximation can extend far deeper than that in the solar case, even if the black hole has a similar mass and consequently a similar DDM halo outskirt.",
"On the other hand, on the model side, the constant cross section is easily substituted by a velocity dependent cross section such as $\\sigma /m_\\chi \\propto v^{-2}$ (Sommerfeld enhancement) or $v^{-4}$ (Coulomb scattering), up to model preferences."
],
[
"Gamma-Ray Emission from the Sun",
"The Sun is known to be a strong $\\gamma $ -ray source due to the hadronic interactions between cosmic ray nuclei and the solar atmosphere, as well as the inverse Compton scattering (ICS) between cosmic ray electrons and the sunlight [47], [48].",
"The Fermi-LAT observations do reveal a disk component as expected from the solar atmospheric interactions and a more extended component from the ICS interactions [49].",
"The measured $\\gamma $ -ray fluxes and energy spectra are different from the naive expectation based on the cosmic ray spectra, implying possible significant roles of the magnetic fields around the Sun [49], [50], [51].",
"More detailed analyses of the Fermi-LAT data revealed even more complicated temporal and spectral features of the solar $\\gamma $ -rays [52], [53].",
"Surprisingly, a statistically significant “dip” feature has been detected at around $30\\sim 50~\\text{GeV}$ with a significance higher than $5\\sigma $ , which is lack of a reasonable explanation yet [53].",
"Motivated by the possibility of compact DDM halo with significantly enhanced density, we consider the DM origin of the dip structure.",
"We assume that the DDM annihilation products are $e^+e^-$ leptons.",
"Then there are three relevant major contributing mechanisms to the observed $\\gamma $ -ray spectrum.",
"The internal bremsstrahlung emission associated with the charged lepton final state (the final state radiation or FSR) and the ICS emission from $e^+e^-$ scattering off the sunlight contribute to the $\\gamma $ -ray spectrum mainly below the “dip”.",
"Since the magnetic field around the Sun is strong enough to confine charged particles below $\\sim \\text{TeV}$ energies [51], in estimation we adopt the in situ cooling approximation.",
"On the other hand, in order to give the upper half of the “dip”, we further consider the contribution from the virtual internal bremsstrahlung (VIB) process via the exchange of a virtual charged particle [54], [55].",
"The spectrum of the VIB emission is very hard, which can mimic the monochromatic $\\gamma $ -ray emission given finite energy resolution of the detector [55].",
"Its amplitude depends on the mass splitting between the mediator $\\eta $ and the DM $\\chi $ , which is characterized by $\\mu \\equiv (m_\\eta /m_\\chi )^2$ with the best fit of $1.5$ .",
"Matching the “dip” position, the DDM mass should be $90~\\text{GeV}$ .",
"And the velocity weighted average annihilation cross section is adopted $\\langle \\sigma v\\rangle =3\\times 10^{26}~\\text{cm}^3\\text{s}^{-1}$ , namely the value consistent with the thermal freeze-out mechanism.",
"For the background contribution, we adopt the simulation results given in Ref.",
"[56] with the potential field source surface [57] magnetic field model, together with an enhancement of the BIFROST model near the Sun [58].",
"In all, the observational spectrum can be reasonably well fitted, only given an extremely large DDM density enhancement around the Sun.",
"As the key, the DDM halo profile relevant to the phenomena is the region $r\\gtrsim R_\\odot $ .",
"For example, if the DDM density profile outside the Sun is adopted to be a power-law form $\\propto r^{-1.6}$ as given in Fig.",
"REF , then the best fit DDM density at the Sun's surface is $\\rho _\\chi (R_\\odot )=3.8\\times 10^7\\rho _\\chi $ with $\\rho _\\chi =0.39~\\text{GeV}/\\text{cm}^3$ as mentioned earlier; and (since we also do not know the profile of the DDM halo) if we switch the profile to a power law continuously connecting $\\rho _\\chi (R_\\odot )$ and $\\rho _\\chi (2R_\\odot )\\approx \\rho _\\chi $ (assumed), in order to get the same spectrum we need roughly $\\rho _\\chi (R_\\odot )=2.2\\times 10^8\\rho _\\chi $ .",
"This surface DDM density is still several orders larger than what we can get from our incomplete calculation such as in Fig.",
"REF , so we consider this fitting to be premature and not a rigorous interpretation for the $\\gamma $ -ray spectrum dip.",
"However, as we commented earlier, we expect the true DDM halo density inside or in the close neighborhood of the Sun is even more enhanced (potentially by several orders) than what we guessed there, so we still consider this effort as a reasonable and interesting step towards a completely satisfying interpretation."
],
[
"Other Constraints",
"For canonical DM density profiles (e.g., the Navarro-Frenk-White distribution [59] or NFW) and $m_\\chi \\sim 100$ GeV, the Fermi-LAT $\\gamma $ -ray observations constrain the annihilation cross section to the $e^+e^-$ channel (or the $\\mu ^+\\mu ^-$ channel which gives quite similar constraints to that of the $e^+e^-$ channel) to be a few times of $\\sim 10^{-26}$ cm$^3$ s$^{-1}$ [60] and $\\sim 10^{-25}$ cm$^{-3}$ s$^{-1}$ [61], from the Galactic center region and the combination of a population of dwarf spheroidal galaxies, respectivelyThe PLANCK observations of the cosmic microwave background anisotropies give similar constraints [64].",
"They are all consistent with the adopted value $\\langle \\sigma v\\rangle =3\\times 10^{26}~\\text{cm}^3\\text{s}^{-1}$ .",
"However, in our DDM model the astrophysical profile of the DM halo should be different from the NFW one, and the accretion of the DDM in the MW center or the dwarf galaxies may also enhance the density distributions at these places, and change the constraints.",
"Note that in reaching such a significant solar DM density enhancement, the favorite parameter region has a small $v_\\text{th}$ from the implication of Fig.",
"REF , which is close to the $v_\\text{th}\\rightarrow 0$ conventional elastic SIDM case as mentioned earlier.",
"Pure SIDM simulation gives halo structure characterized by a constant density core in its central region, rather than the NFW cuspy proportional to $r^{-1}$ [62], [63].",
"In such cases in the center of the MW or the dwarf galaxies, DM can be in fact less concentrated as their canonical DM density profile counterparts, and the above DM annihilation cross section bound should even be relaxed instead.",
"On the other hand, the possibility that the DM concentration is more cuspy than its NFW counterpart can be indeed achieved.",
"In particular in the specific MW satellite galaxy Draco, as shown in Ref.",
"[10], an $\\mathcal {O}(10)~\\text{cm}^2/\\text{g}$ purely elastic SIDM model with the MW tidal effect will lead to a core collapse configuration, which is indeed consistent with the observation.",
"However, as mentioned earlier, the DM annihilation cross section constraint from Draco alone is not so strong.",
"Simply put, we believe that the DM annihilation cross (as well as other parameters) adopted earlier in our calculation is consistent with the other observational constraints."
],
[
"Conclusion",
"In this paper working in the dissipative self-interacting dark matter (DM) model, we have pointed out the possible existence of a compact DM halo around the Sun and other celestial bodies.",
"We have defined a scheme of approximations to enable such calculation, and develop the Boltzmann equation set based on the partition function of the elliptical orbits characterized by its energy and angular momentum.",
"Then we numerically solve the Boltzmann equation set to demonstrate the existence of the DM halo.",
"Our results show that the DM density enhancement can be several orders in a compact region centered around the Sun.",
"As an application, we use such possibility to study the DM origin of the unexpected “dip” structure in the observed solar $\\gamma $ -ray spectrum, which is reproduced by the inverse Compton scattering emission on the low energy side and the virtual internal bremsstrahlung mechanism on the high energy side, respectively.",
"Our results are incomplete and limited by the validity of the elliptical orbit approximation.",
"Therefore we cannot directly sample all the bounded DM particles, but have to truncate at the lower boundary of the elliptical orbits, which renders the “sunk” component calculation, the inner region halo determination as well as the attempt of the DM origin of the solar $\\gamma $ -ray “dip” structure not rigorous.",
"One may consider to completely numerically study the “sunk” orbits, but we reserve this for a future work.",
"Acknowledgments: The author is grateful to useful discussion with Kenny Ng, Hai-bo Yu, Qiang Yuan and Yi-Ming Zhong.",
"The credit towards interpreting the solar gamma-ray spectrum dip should totally go to Qiang Yuan.",
"The author acknowledge the computational facilities of the chepfarm cluster at Tsinghua University."
]
] | 2005.14158 |
[
[
"A non-perturbative approach to the scalar Casimir effect with Lorentz\n symmetry violation"
],
[
"Abstract We determine the effect of Lorentz invariance violation in the vacuum energy and stress between two parallel plates separated by a distance $L$, in the presence of a massive real scalar field.",
"We parametrize the Lorentz-violation in terms of a symmetric tensor $h^{\\,\\mu\\nu}$ that represents a constant background.",
"Through the Green's function method, we obtain the global Casimir energy, the Casimir force between the plates and the energy density in a closed analytical form without resorting to perturbative methods.",
"With regards to the pressure, we find that $\\mathcal{F}_c(L)=\\mathcal{F}_0(\\tilde{L})/\\sqrt{-{\\rm det}\\, h^{\\,\\mu\\nu}}$, where $\\mathcal{F}_0$ is the Lorentz-invariant expression, and $\\tilde{L}$ is the plate separation rescaled by the component of $h^{\\,\\mu\\nu}$ normal to the plates, $\\tilde{L}=L/\\sqrt{-h^{nn}}$.",
"We also analyze the Casimir stress including finite-temperature corrections.",
"The local behavior of the Casimir energy density is also discussed."
],
[
"Introduction",
"The existence of a zero-point vacuum energy is one of the main tenets of the quantum formulation of the laws that we believe govern our Universe.",
"In a Quantum Field Theory (QFT), the presence of fluctuating zero-point fields implies the existence of a non-vanishing macroscopic force between the boundaries that delimit a spatial region [1], due to the difference in the spectrum of quantized field modes inside and outside this region.",
"When the boundaries of this delimited spatial domain take the form of two parallel plates, this manifestation of the vacuum fluctuation is known as the Casimir effect [2].",
"The computation of the Casimir force in QFT is a standard textbook exercise [3], [4], [5], and its existence, in the case of Quantum Electrodynamics, has been verified to a high precision [6], [7], [8], [9].",
"The Casimir effect is now behind many experimental and theoretical pursuits.",
"It is used as a tool to place constraints on Yukawa-type interactions [10], [11], and it has been suggested as a potential probe for the detection of feebly-interacting axion-like dark matter [12].",
"Casimir forces cannot be neglected at the nanoscale, and must be accounted for in the design of microelectromechanical systems [13].",
"Among theoretical extensions one can list its generalization to spacetimes with non-trivial topologies [14], [15], dynamical boundary conditions [16], and non-Euclidean space-times [17], [18], [19].",
"In the latter case it offers an independent derivation of the Hawking temperature from particle production from black holes [20], [21].",
"Modifications of the Casimir effect in the presence of weak gravitational fields have been extensively studied [22], [23], [24], [25], [26], [27], [28].",
"In this context, the Casimir effect can potentially provide clues on the connection between zero-point fluctuations and the cosmological constant [29], [30], [31], [32].",
"The Casimir effect stands as a potential handle to distinguish between Lorentz-invariant and Lorentz-violating formulations of QFT.",
"Lorentz invariance (LI) is one of the cornerstones behind QFT and general relativity, and to date there are no experimental signs of a departure from it [33].",
"Nevertheless, the quantum nature of the spacetime at distances of the order of the Planck length ($\\ell _P$ ) has been shown to provide mechanisms that can lead to violation of LI in certain formulations of quantum gravity [34], [35], [36].",
"As an example, spontaneous LI breaking can occur within some string theories [35].",
"Therefore, a better understanding on the consequences of the breakdown of LI at scales larger than $\\ell _P$ would provide valuable information about the microscopic structure of spacetime.",
"In this Letter we explore the manifestation of the spontaneous breakdown of LI, induced by a constant background tensor, on the Casimir effect for a real massive scalar field between two parallel conductive plates in flat spacetime.",
"We also explore how thermal corrections are affected by the Lorentz symmetry breaking.",
"Our work provides a generalization of previous studies of LI violation in the Casimir context [37], [38]."
],
[
"The model",
"Arguably, the most straightforward way to implement Lorentz violation is by means of the introduction of a tensor field with a non-zero vacuum expectation value (VEV).",
"When coupled to the Standard Model fields the spontaneous symmetry breaking, induced by the non-zero VEV, is manifested as preferential directions on the spacetime, leading to a breakdown of LI.",
"In the case of a real scalar field in flat spacetime, with the Minkowski metric with signature $(+,-,-,-)$ , we parametrize this coupling in the following form, $\\mathcal {L} = \\frac{1}{2} h^{\\,\\mu \\nu }\\partial _{\\mu } \\phi \\,\\partial _{\\nu } \\phi - \\frac{1}{2} m ^{2} \\phi ^{2}\\,.$ Here $h^{\\,\\mu \\nu }$ is a symmetric tensor that represents a constant background, independent of the spacetime position, and which does not transform as a second order tensor under active Lorentz transformationsSingle derivative terms, such as $i\\phi u^\\mu \\partial _\\mu \\phi $ with $u^\\mu $ a constant 4-vector, can be reduced to surface terms, which in absence of topological effects do not have physical contributions [39]..",
"Naturally, causality, the positive energy condition and stability impose restrictions on the components of $h^{\\, \\mu \\nu }$ .",
"Consider now the following set-up: a pair of parallel, conductive plates, orthogonal to the $\\hat{z}$ -direction, located at $z=0$ and $z=L$ , on which Dirichlet boundary conditions apply for the field $\\phi $ .",
"That is, $\\phi (z=0)=\\phi (z=L)=0$ .",
"We now solve for the scalar field between the plates, applying the Green's function technique [40].",
"Namely, we are interested in computing the time-ordered, vacuum two-point correlation function $G(x,x^{\\prime })=-i\\langle 0| \\mathcal {T} \\phi (x) \\phi (x^{\\prime })|0\\rangle $ , which as is well known (see e.g.",
"[41]) satisfies the Green's function (GF) equation $\\mathcal {O}_{\\vec{x}}\\,G (x,x^{\\prime }) = \\delta ^{(4)} (x - x ^{\\prime }).$ Here, in the configuration space, the modified Klein-Gordon operator has the following explicit form, $\\mathcal {O}_{\\vec{x}}=h^{00}\\partial _0^2 + 2h^{0\\bar{i}}\\partial _{0}\\partial _{\\,\\bar{i}}+h^{\\bar{i}\\bar{j}}\\partial _{\\,\\bar{i}}\\partial _{\\,\\bar{j}}+2h^{03}\\partial _{0}\\partial _{3}+2h^{\\bar{i}3}\\partial _{\\,\\bar{i}}\\partial _{3} +h^{33}\\partial ^2 _{z}+m^2$ with $\\bar{i},\\bar{j}=1,2$ .",
"In the chosen coordinate system, the GF is invariant under translations in the $(\\hat{x},\\hat{y})$ -plane.",
"Taking advantage of this symmetry, we can express the GF in terms of the Fourier transform in the direction parallel to the plates, $G(x,x^{\\prime })=\\int \\frac{d^2 \\vec{k}_\\bot }{(2\\pi )^2} e^{i\\vec{k}_\\bot \\cdot (\\vec{x}_\\bot -\\vec{x}_\\bot ^{\\prime })}\\int \\frac{d\\omega }{2\\pi }e^{-i\\omega (t-t^{\\prime })}g\\left(z,z^{\\prime };\\omega ,\\vec{k}_\\bot \\right),$ where $\\vec{k}_\\bot =(k_x,k_y)$ and $\\vec{x}_\\bot =(x,y)$ .",
"Henceforth we will drop the explicit dependence on $\\omega $ , $\\vec{k}_\\bot $ of $g$ for simplicity.",
"After substitution of (REF ) into (REF ), and straightforward integration of the resulting 1D boundary problem,$\\endcsname $An analogous step-by-step procedure can be found in [40].",
"an exact solution for the reduced GF between the plates can be found, $g _{\\parallel } (z,z ^{\\prime }) \\;&=\\; e ^{- i \\xi _{0} (z ^{\\prime } - z)} \\frac{\\sin ( \\xi _{1} z _{<}) \\sin [ \\xi _{1} (z _{>} -L) ]}{ h^{33} \\xi _{1} \\sin (\\xi _{1} L) }\\,.",
"$ Here $z _{>}$ ($z _{<}$ ) is the greater (lesser) between $z$ and $z ^{\\prime }$ .",
"The coefficients $\\xi _{0,1}$ denote the following combinations of energy-momenta and the Lorentz-violating tensor, $\\xi _0 \\;&=\\; \\frac{1}{h^{33}}(h^{03}\\omega -h^{\\bar{i}3}\\vec{k}_{\\bot \\bar{i}})\\,,\\\\ \\xi _1 \\;&=\\;\\frac{1}{|h^{33}|} \\left[ (h^{03}\\omega -h^{\\bar{i}3}\\vec{k}_{\\bot \\bar{i}})^2 - h^{33} \\gamma ^2 \\right]^{1/2}\\,,$ where $\\gamma ^{2} \\;=\\; h^{00}\\omega ^{2} - 2h^{0\\bar{i}}\\omega k _{\\perp _{\\bar{i}}} + h^{\\bar{i}\\bar{j}}\\vec{k}_{\\perp _{\\bar{i}}}\\vec{k}_{\\perp _{\\bar{j}}}-m^2\\,.$ The LI limit is recovered by taking $h ^{\\, \\mu \\nu } \\rightarrow \\eta ^{\\, \\mu \\nu }$ , which implies $\\xi _0\\rightarrow 0$ and $\\xi _1^2\\rightarrow \\omega ^{2} - k _{\\perp } ^{2} - m ^{2}$ .",
"It is worth noting that the case with Neumann conditions can be trivially recovered by replacing $\\sin \\rightarrow \\cos $ in the numerator of (REF ).",
"The determination of the Casimir energy and stress requires not only the GF for the two plate setup, but also the GF in the presence of no plates and a single plate.",
"For the former, we find $g _{v} (z,z ^{\\prime }) \\;&=\\; - \\frac{i}{2\\xi _1} \\frac{e^{i\\xi _0(z-z^{\\prime })}}{h^{33}} e^{i\\xi _1 (z_> - z<)} \\,, $ while for the latter, $g _{|} (z,z ^{\\prime }) \\;&=\\; -\\frac{e^{i\\xi _0(z-z^{\\prime })}}{\\xi _1h^{33}} \\sin [\\xi _1(z_<-L)] e^{i\\xi _1 (z_> - L)} \\,.",
"$ In order to quantify the Casimir effect, we need an expression for the vacuum expectation value for the stress-energy tensor of the scalar field, $T ^{\\mu \\nu } = h ^{\\,\\mu \\alpha }\\partial _\\alpha \\phi \\,\\partial ^\\nu \\phi -\\eta ^{\\,\\mu \\nu }\\mathcal {L}$ .",
"In terms of the GF, it can be generically computed as [40] $\\langle T^{\\mu \\nu }\\rangle = -i\\lim _{x\\rightarrow x^\\prime }\\left[h^{\\mu \\alpha }\\partial _\\alpha \\,\\partial ^{^{\\prime }\\nu }\\right]G(x,x^\\prime ) - \\eta ^{\\,\\mu \\nu } \\langle \\mathcal {L}\\rangle \\; ,$ while the VEV of the Lagrangian density can be written as $\\langle \\mathcal {L}\\rangle = -i \\lim _{x\\rightarrow x^\\prime }\\frac{1}{2}\\left( h^{\\,\\mu \\nu }\\partial _\\mu \\,\\partial _{^{\\prime }\\nu }-m^2\\right) G(x,x^\\prime )$ .",
"Substitution of (REF ) leads to the following expressions for the energy density and the pressure in the $\\hat{z}$ -direction, $\\langle T^{00}\\rangle \\;=\\; &- i \\lim _{z ^{\\prime } \\rightarrow z} \\int \\frac{d \\omega }{2 \\pi } \\int \\frac{d ^{2} \\vec{k} _{\\perp }}{(2 \\pi ) ^{2}}\\left[h^{00} \\omega ^{2} - h^{0\\bar{i}}\\omega \\vec{k}_{\\bot _{\\bar{i}}}\\right.\\nonumber \\\\&\\left.+ih^{03}\\omega \\partial _z\\right] g (z,z ^{\\prime }) - \\langle \\mathcal {L} \\rangle , \\\\ \\langle T^{33} \\rangle \\;=\\; &- \\frac{i}{2} \\lim _{z ^{\\prime } \\rightarrow z} \\int \\frac{d \\omega }{2 \\pi } \\int \\frac{d ^{2} \\vec{k} _{\\perp }}{(2 \\pi ) ^{2}} \\left[ \\gamma ^{2} - h^{33} \\partial _{z} \\partial _{z ^{\\prime }} \\right] g (z , z ^{\\prime } ) .",
"$ where $ \\langle \\mathcal {L}\\rangle \\;=\\; &- \\frac{i}{2} \\lim _{z ^{\\prime } \\rightarrow z} \\int \\frac{d \\omega }{2 \\pi } \\int \\frac{d ^{2} \\vec{k} _{\\perp }}{(2 \\pi ) ^{2}} \\bigg [ \\gamma ^{2} + h^{33} \\partial _{z} \\partial _{z ^{\\prime }} \\\\&-ih^{33}\\xi _0(\\partial _{z^\\prime }-\\partial _z) \\bigg ] \\; g (z , z ^{\\prime } ) \\; .",
"$"
],
[
"Casimir Effect with Lorentz symmetry violation",
"With the VEV of the stress-energy tensor at hand, we now proceed to compute the global Casimir energy and the Casimir stress upon the plates in the presence of Lorentz-invariance violation."
],
[
"Global Casimir energy",
"The renormalized vacuum energy stored between the parallel plates can be computed formally as the difference between the zero-point energy in the presence of the boundary, $\\langle T^{00} \\rangle _{\\parallel }$ , and that of the free vacuum, $\\langle T^{00} \\rangle _{v}$ .",
"Namely, $\\mathcal {E} _{C} (L) = \\int _{0} ^{L} \\left( \\langle T^{00} \\rangle _{\\parallel } - \\langle T^{00} \\rangle _{v} \\right) \\, dz\\,.$ We begin by evaluating $\\langle T^{00} \\rangle _{\\parallel }$ .",
"As a first step, it can be noted after a cursory computation that the contribution from the VEV of $\\mathcal {L}$ in (REF ) is $L$ -independent and will therefore not contribute to the Casimir pressure.$\\endcsname $More precisely, $\\int _{0} ^{L} \\langle \\mathcal {L} \\rangle _{\\parallel } \\, dz = (1/2i) \\int \\frac{d \\omega }{2 \\pi } \\int \\frac{d ^{2} \\vec{k} _{\\perp }}{(2 \\pi ) ^{2}}$ .",
"After simplification, the remaining terms in (REF ) can be rearranged to lead to the following expression, $\\langle T^{00} \\rangle _{\\parallel } = - i \\int \\frac{d \\omega }{2 \\pi } \\int \\frac{d ^{2} \\vec{k} _{\\perp }}{(2 \\pi ) ^{2}} \\Big [h^{00} \\omega ^{2} &-h^{0\\bar{i}}\\omega (\\vec{k}_\\bot )_{\\,\\bar{i}} \\\\& -h^{03}\\omega \\,\\xi _0 \\Big ] g _{\\parallel } (z,z)\\,.",
"$ The term inside the brackets in the previous equation is a quadratic form in $(\\omega ,k _{x} , k _{y})$ , with coefficients given by the components of $h^{\\,\\mu \\nu }$ .",
"This quadratic form is different from that appearing in the argument of the GF, $|\\,h^{33}|\\,\\xi _1^2$ , and this makes the evaluation of (REF ) a non-trivial task.",
"However, a closed-form solution may be obtained by diagonalization of the latter quadratic form, mapping it into a mimic of the LI case, $|\\,h^{33}|\\,\\xi _1^2=\\omega ^{ \\prime \\, 2} - k _{x} ^{\\prime \\, 2} - k _{y} ^{\\prime \\, 2} - m ^{2}$ , where primed quantities correspond to the rotated frequency and momenta.",
"Further performing a Wick rotation $\\omega ^{\\prime } \\rightarrow i\\zeta $ , it can be shown that (REF ) is equivalent to the following expression, $\\langle T^{00} \\rangle _{\\parallel } &= \\frac{1}{\\sqrt{-h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k^{\\prime }} _{\\perp }}{(2 \\pi ) ^{2}} \\, \\zeta ^{2} \\, \\frac{\\sinh ( \\gamma \\tilde{z} ) \\sinh [ \\gamma (\\tilde{z}-\\tilde{L}) ]}{\\gamma \\sinh (\\gamma \\tilde{L}) } , $ where now $\\gamma ^{2} = \\zeta ^{2} + k _{\\perp } ^{\\prime \\, 2} + m ^{2}$ , $h\\equiv \\textrm {det}\\,h^{\\,\\mu \\nu }$ , and $\\tilde{z} =\\frac{ z}{ \\sqrt{-h^{33}}}\\,, \\quad \\tilde{L} = \\frac{L}{ \\sqrt{-h^{33}} }\\,.$ An entirely analogous procedure can be followed to evaluate the vacuum energy density $\\langle T^{00} \\rangle _{v}$ , making use in this case of the corresponding GF (REF ).",
"For it we obtain $\\langle T^{00} \\rangle _{v} &= - \\frac{1}{\\sqrt{-h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k}^{\\prime } _{\\perp }}{(2 \\pi ) ^{2}} \\, \\frac{\\zeta ^{2}}{2 \\gamma } .",
"$ Finally, substituting into (REF ), integrating with respect to $z$ and dropping an $L$ -independent constant term leads to the following expression for the vacuum energy between the plates, $\\mathcal {E} _{C} (L) = - \\sqrt{\\frac{h^{33}}{h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k}^{\\prime }_{\\perp }}{(2 \\pi ) ^{2}} \\frac{\\zeta ^{2}}{2 \\gamma } \\tilde{L} \\, [ \\coth {(\\gamma \\tilde{L})} - 1 ] .$ This resulting integral can be recognized as the LI result, $\\mathcal {E} _{0}$ , rescaled by the factor $\\sqrt{h^{33}/h}$ , with a rescaled separation between the plates (REF ) [40].",
"Integration gives $\\mathcal {E} _{C} (L)= \\sqrt{\\frac{h^{33}}{h}} \\mathcal {E} _{0} (\\tilde{L})= - \\dfrac{m ^{2}}{8 \\pi ^{2} \\tilde{L}} \\sqrt{\\frac{h^{33}}{h}} {\\displaystyle \\sum } _{n = 1} ^{\\infty } \\dfrac{1}{n ^{2}} K _{2} (2 m n \\tilde{L}), $ where $K _{2} (x)$ is the second-order Bessel function of the second kind.",
"Note that the Lorentz-violating result reduces trivially to the LI one as $h^{33}, h \\rightarrow -1$ , which would be the case for $h^{\\mu \\nu }\\rightarrow \\eta ^{\\mu \\nu }$ .",
"Although the sum which appears in (REF ) does not have an analytical closed form, it can be reduced to simple expressions in the large and small mass limits, $\\mathcal {E} _{C} (L) \\;\\simeq \\; - \\sqrt{\\frac{h^{33}}{h}} \\times {\\left\\lbrace \\begin{array}{ll} \\dfrac{\\pi ^{2}}{1440 \\tilde{L} ^{3}} - \\dfrac{m ^{2}}{96 \\tilde{L}} \\,, & m\\tilde{L}\\ll 1\\,,\\\\[8pt] \\dfrac{m ^{2}}{16 \\pi ^{2} \\tilde{L}} \\sqrt{\\dfrac{\\pi }{m \\tilde{L}}} e ^{- 2m \\tilde{L}} \\,, & m\\tilde{L}\\gg 1\\,.",
"\\end{array}\\right.",
"}$ The massless case is trivially recovered taking the $m\\rightarrow 0$ limit in the previous equation."
],
[
"Stress on the plates",
"We now proceed to determine the Casimir stress upon the plate at $z = L$ by direct evaluation of the normal-normal component of the stress-energy tensor ().",
"Denoting by $\\langle T ^{33} \\rangle _{\\parallel }$ the vacuum stress due to the confined scalar field, and by $\\langle T ^{33} \\rangle _{\\vert }$ the stress due to the field above the plate, we can write $\\mathcal {F} _{C} (L) = \\langle T ^{33} \\rangle _{\\parallel } - \\langle T ^{33} \\rangle _{\\vert } .",
"$ In a similar fashion to the previous computation of the Casimir energy, all it takes to calculate these stresses is to substitute the corresponding reduced GFs into (), and to repeat the quadratic form diagonalization procedure.",
"A key difference in this analysis is the fact that the Lagrangian density does contribute to the stress.$\\langle \\mathcal {L}\\rangle $ also plays a fundamental role regarding the behavior of the field near the boundaries, see Section REF .",
"Nevertheless, despite this relative complication, a straightforward calculation using (REF ) and (REF ) yields $\\begin{aligned}\\langle T^{33} \\rangle _{\\parallel } \\;&=\\; - \\frac{1}{\\sqrt{-h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k}^{\\prime } _{\\perp }}{(2 \\pi ) ^{2}} \\frac{\\gamma }{2} \\coth (\\gamma \\tilde{L})\\,,\\\\\\langle T^{33} \\rangle _{\\vert } \\;&=\\; - \\frac{1}{\\sqrt{-h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k}^{\\prime }_{\\perp }}{(2 \\pi ) ^{2}} \\frac{\\gamma }{2}\\, .\\end{aligned}$ Each stress contains an $L$ -independent divergent term that is canceled by the regularization provided by (REF ).",
"Substitution of these expressions into (REF ), and following the same steps that lead to Eq.",
"(REF ), produces $\\mathcal {F} _{C} (L)&= \\frac{1}{\\sqrt{-h}} \\mathcal {F} _{0} (\\tilde{L})= \\frac{1}{\\sqrt{-h}} \\dfrac{1}{4 \\pi ^{2}} {\\displaystyle \\int _{0} ^{\\infty }} \\dfrac{\\tau ^{2} \\sqrt{\\tau ^{2} + m ^{2}}}{e ^{2\\tilde{L} \\sqrt{\\tau ^{2} + m ^{2}}} - 1} d \\tau .",
"$ In this expression $\\mathcal {F} _{0}$ denotes the LI result.",
"Expectedly, the stress in the LI violating result is proportional to the stress in the absence of Lorentz violation, but evaluated at the rescaled length $\\tilde{L}$ .",
"For a vanishing scalar field mass, Eq.",
"(REF ) reduces to $ \\mathcal {F} _{C} (L)|_{m=0}=- \\pi ^2/(480 \\tilde{L}^4\\sqrt{-h})$ .",
"As a consistency check, one can verify that the Casimir energy (REF ) and the stress (REF ) are connected by the elementary relation $\\mathcal {F} _{C} (L) = - \\frac{\\partial \\mathcal {E}_{C} (L)}{\\partial L} .$"
],
[
"Local effects",
"In Section REF we derived an expression for the global Casimir energy by computing the integral of $\\langle T^{00} \\rangle _{\\parallel } - \\langle T^{00} \\rangle _{v}$ in the region between the plates by means of the GF method.",
"Although alternative methods exist to evaluate $\\mathcal {E}_C$ [42], the power of the GF procedure arises clearly when studying the local energy density, which in turn reveals the divergence structure of the theory.",
"The computation of $\\langle T^{\\mu \\nu }\\rangle $ is the goal of this section.",
"We begin with the energy density per unit volume between the plates.",
"Without dropping in this case the contribution of $\\langle \\mathcal {L}\\rangle $ (which was discarded in the global analysis due to its $L$ -independence after integration), the same analysis that led to (REF ) in this case gives $ \\langle T^{00} \\rangle \\;=\\; &- \\frac{1}{\\sqrt{-h}} \\int \\frac{d \\zeta }{2 \\pi } \\int \\frac{d ^{2} \\vec{k}^{\\prime }_{\\perp }}{(2 \\pi ) ^{2}} \\\\&\\times \\left\\lbrace \\frac{\\zeta ^{2}}{2 \\gamma } \\coth (\\gamma \\tilde{L}) + \\frac{k_{\\perp } ^{\\prime \\,2} + m ^{2}}{2 \\gamma } \\frac{\\cosh [\\gamma (2 \\tilde{z} - \\tilde{L})]}{\\sinh (\\gamma \\tilde{L})} \\right\\rbrace .$ The introduction of the polar coordinates $k _{\\perp } = \\rho \\cos \\theta $ , $\\zeta = \\rho \\sin \\theta $ , where $\\rho \\in [0, \\infty )$ and $\\theta \\in [- \\pi / 2 , \\pi / 2]$ , leads to the following result $\\langle T^{00} \\rangle &=& - \\frac{1}{12 \\pi ^{2}} \\frac{1}{\\sqrt{-h}} \\int _{0} ^{\\infty } \\left\\lbrace \\frac{\\rho ^{4}}{\\gamma ^{\\ast }} \\frac{2}{e ^{2 \\gamma ^{\\ast } \\tilde{L}} - 1} \\right.",
"\\nonumber \\\\&&\\left.",
"+ \\frac{\\rho ^{2}}{\\gamma ^{\\ast }} (2 \\gamma ^{\\ast \\, 2} + m ^{2}) \\frac{e ^{2 \\gamma ^{\\ast } \\tilde{z}} + e ^{2 \\gamma ^{\\ast } ( \\tilde{L} - \\tilde{z} )}}{e ^{2 \\gamma ^{\\ast } \\tilde{L}} - 1} \\right\\rbrace d \\rho \\,.",
"$ Here $\\gamma ^{\\ast } = \\sqrt{\\rho ^{2} + m ^{2}}$ , and we have discarded an $L$ -independent term.",
"Denoting by $\\mathfrak {U}$ the $z$ -independent term in the previous expression, one can easily show that $\\mathfrak {U}=\\mathcal {E}_C/L$ .",
"Similarly, a straightforward change of variables allows us to write the $z$ -dependent term of (REF ), which we denote by $f(z)$ , as follows, $f (z) &=& - \\frac{1}{192 \\pi ^{2} \\tilde{L} ^{4}} \\frac{1}{\\sqrt{-h}} \\int _{2 m \\tilde{L}} ^{\\infty } \\sqrt{y ^{2} - (2 m \\tilde{L}) ^{2}} \\nonumber \\\\&&\\times \\left[ 2y ^{2} + (2 m \\tilde{L}) ^{2} \\right] \\frac{e ^{y z/L} + e ^{y(1-z/L)}}{e ^{y} -1} dy .",
"$ In the massless limit, this function can be expressed in terms of the Hurwirtz zeta function, $\\zeta (s,a) = \\sum _{n = 0} ^{\\infty } (n+a) ^{-s}$ , $f (z) \\;&=\\; - \\frac{1}{16 \\pi ^{2} L ^{4}} \\frac{(h^{33}) ^{2}}{\\sqrt{-h}} \\left[ \\zeta (4, z/L) + \\zeta (4, 1 - z/L) \\right]\\,.",
"$ Therefore we have found that $ \\langle T^{00} \\rangle = \\mathfrak {U} + f (z)$ .",
"$\\mathfrak {U}$ encodes the part of the vacuum energy resulting in an observable force, whereas $f (z)$ corresponds to a local, divergent effect that does not contribute to the pressure, as the $L$ -independence of the following integral confirms $\\int _{0} ^{L} f(z) dz = - \\frac{1}{48 \\pi ^{2}} \\sqrt{\\frac{h^{33}}{h}} \\int _{2 m} ^{\\infty } \\sqrt{x ^{2} - 4 m ^{2}} ( x ^{2} + 2 m ^{2} ) \\frac{dx}{x}.$ In the massless case this divergence is quartic as $z$ approaches the plates, as can be appreciated from Eq.",
"(REF ).",
"For a generic mass the complex form of (REF ) prevents us from analytically determining the degree of divergence.",
"We turn now to the evaluation of the VEV for the remaining components of $T^{\\mu \\nu }$ .",
"Owing to the symmetry of the setup, these components can be easily determined.",
"For example, rotational invariance around the $z$ -axis immediately implies that $\\langle T^{11} \\rangle =\\langle T^{22} \\rangle $ .",
"Moreover, after an explicit calculation we find that $\\langle T^{11} \\rangle = - \\langle T^{00} \\rangle $ .",
"The off-diagonal components of $T^{\\mu \\nu }$ vanish in the LI limit, but in the presence of a non-trivial $h^{\\,\\mu \\nu }$ they are in general non-zero, although they can also be related to the 00 and 33 components by symmetry arguments.",
"A cursory computation provides the following general expression for the VEV of the stress-energy tensor, $\\langle T^{\\mu \\nu } \\rangle &=& - \\frac{2 h^{\\alpha 3}}{h^{33}} (\\eta ^{\\,\\mu \\alpha } + n ^{\\mu } n ^{\\alpha }) n ^{\\nu } \\left[\\langle T^{00} \\rangle - \\langle T^{33} \\rangle - f(z) \\right] \\nonumber \\\\&& + (\\eta ^{\\,\\mu \\nu } + n ^{\\mu } n ^{\\nu }) \\, \\langle T^{00}\\rangle + n ^{\\mu } n ^{\\nu } \\, \\langle T^{33} \\rangle \\,.",
"$ Here $n ^{\\mu } = (0,0,0,1)$ is the unit vector perpendicular to the plates, and $\\langle T^{00} \\rangle $ and $\\langle T^{33} \\rangle $ are given by Eqs.",
"(REF ) and (REF ), respectively.",
"Clearly, in the LI limit the first term vanishes and we recover the usual structure of the vacuum stress [43]."
],
[
"Finite temperature effects",
"The Casimir effect, as described in the previous sections, is a manifestation of the fluctuations of the $\\phi $ field in the vacuum.",
"However, any realistic parallel plate setup will necessarily be immersed in a bath with a temperature above absolute zero.",
"It is therefore crucial to determine the effect that thermal fluctuations would have in the Casimir stress.",
"Luckily, in our relatively simple scenario, the stress at $T>0$ case can be determined in a straightforward manner.",
"In the Matsubara formalism of finite temperature QFT, the Casimir stress at nonzero temperature can be obtained from Eq.",
"(REF ) upon the replacement $\\int d\\zeta /2\\pi \\rightarrow \\beta ^{-1} \\sum _{n=-\\infty }^{\\infty }$ , together with mapping the imaginary frequency $\\zeta $ to the discrete Matsubara frequency $\\zeta _n \\equiv 2\\pi n/\\beta $ [44].",
"Here $\\beta = 1 / k _{B} T$ , with $k _{B}$ the Boltzmann constant.",
"These substitutions yield $\\mathcal {F} _{C} (L;T) = - \\frac{1}{\\beta \\sqrt{h}}\\sum _{n = - \\infty } ^{+ \\infty } \\int \\frac{d ^{2}\\vec{k} _{\\perp }}{(2 \\pi ) ^{2}} \\frac{\\gamma _{n}}{e ^{2 \\gamma _{n}\\tilde{L}} -1}\\, , $ where $\\gamma _{n} = \\sqrt{\\zeta _{n} ^{2} + k _{\\perp } ^{2} + m ^{2}}$ .",
"Although this expression lacks a closed form in terms of elementary functions, we can gain some insight of its behavior in the massless case for small temperature and large temperature (classical) limits.",
"For low temperature, the above expression for the pressure takes the form $\\mathcal {F} _{C} (L;T \\ll 1) \\approx - \\frac{\\pi ^{2}}{480 \\tilde{L} ^{4} \\sqrt{h}} \\left( 1 + \\frac{1}{48 \\pi ^{4}} s ^{4} - \\frac{60}{\\pi ^{2}} s e ^{- 4 \\pi ^{2} / s} \\right) , $ where $s = 4 \\pi k _{B} T \\tilde{L} \\ll 1$ .",
"Clearly this result is consistent with the Nernst heat theorem, since the associated entropy vanishes as $s$ goes to zero.",
"In the opposite regime, at high temperatures, all terms in the sum of Eq.",
"(REF ) except the $n=0$ term are exponentially suppressed, resulting in $\\mathcal {F} _{C} (L;T\\gg 1) \\approx - \\frac{\\zeta (3) k _{B} T}{8 \\pi \\tilde{L} ^{3} \\sqrt{h}} - \\frac{k _{B} T}{4 \\pi \\tilde{L} ^{3} \\sqrt{h}} \\left( 1 + s + \\frac{s ^{2}}{2} \\right) e ^{-s} , $ where here $s \\gg 1$ .",
"The leading term can also be obtained from the Helmholtz free energy for Lorentz-violating massless bosons.",
"The results of equations (REF ) and (REF ) exhibit an interesting behaviour as a function of the Lorentz violating parameter $h ^{33}$ through the rescaled length $\\tilde{L} = L / \\sqrt{- h ^{33}}$ .",
"When Lorentz invariance is mildly broken, $h ^{33} \\approx - 1$ , and hence the conditions $s \\ll 1$ and $s\\gg 1$ correspond to low and high temperatures, respectively.",
"However, when Lorentz symmetry breaking is not negligible, such conditions are relaxed and possibly flipped.",
"For example, when $h ^{33} \\approx 0 ^{-}$ , the condition $s \\gg 1$ can be fulfilled even for low temperatures.",
"In the present work we have obtained explicit expressions for the Casimir energy and force between two parallel conductive plates, arising from the vacuum fluctuations of a massive real scalar field, in the presence of a generic background defined by the tensor $h^{\\,\\mu \\nu }$ in Eq.",
"(REF ).",
"This background is motivated by theories in which the breakdown of Lorentz invariance manifests itself as the non-vanishing vacuum expectation value of a fundamental field.",
"Since no deviation from Lorentz invariance has been experimentally observed yet, the perturbative expansion $h^{\\,\\mu \\nu }=\\eta ^{\\,\\mu \\nu }+k^{\\mu \\nu }$ is justified.",
"Here $\\eta ^{\\,\\mu \\nu }$ is the Minkowski metric and $k^{\\mu \\nu }$ is a constant tensor whose components are much smaller than one $\\vert k^{\\, \\mu \\nu }\\vert \\ll 1$ .",
"Working to first order in $k^{\\, \\mu \\nu }$ , it is possible to prove that the Lorentz-violating theory described by Eq.",
"(REF ) can be transformed into the standard Lorentz-invariant theory by an appropriate change of spacetime coordinates $x^{\\prime \\, \\mu }=x ^{\\, \\mu } - \\frac{1}{2}k\\, ^\\mu \\,_\\nu x^{\\nu }$ [45].",
"In this new coordinate system it is relatively straightforward to evaluate the Casimir energy.",
"It is given by the Lorentz-invariant result, albeit with a redefinition of the separation between the plates and a global multiplicative factor arising from the Jacobian of the transformation.",
"Let us discuss our result in Eq.",
"(REF ) in this approximation.",
"One can verify that the global multiplicative factor, $1/\\sqrt{-h}$ in Eq.",
"(REF ), corresponds to the square root of the Jacobian, whereas $\\tilde{L}\\approx L(1+\\frac{1}{2}k^{33})$ is precisely the transformed distance between plates.",
"This confirms that our result, valid to all orders in $k^{\\mu \\nu }$ , correctly reduces to the expected result in the limit $\\vert k^{\\mu \\nu }\\vert \\ll 1$ .",
"Focusing on the massless case for simplicity, the (measurable) Casimir force explicitly reduces to first order in $k^{\\mu \\nu }$ to $\\mathcal {F}_C(L)=(1-2k^{33} - \\frac{1}{2} \\eta ^{\\, \\mu \\nu } k _{\\mu \\nu }) \\, \\mathcal {F}_0(L)$ .",
"For the sake of comparison, if we consider the present experimental measurements of the Casimir force between parallel plates for the electromagnetic case (15% precision in the 0.5-3 $\\mu $ m range), the bound that can be obtained from this result is $|\\, 2k^{33} + \\frac{1}{2} \\eta ^{\\, \\mu \\nu } k _{\\mu \\nu }|<10^{-2}$ .",
"Note that the leading-order modification to the Lorentz invariant result only involves the component of $k^{\\mu \\nu }$ perpendicular to the plates and the trace of $k^{\\mu \\nu }$ .",
"We also note that in this Letter we have assumed that Dirichlet boundary conditions apply at the plates location.",
"Nevertheless, other types of boundary conditions, such as Neumann conditions, can be treated in a completely analogous manner since they only directly modify the Green's function form.",
"We have found that for this parallel plate setup, the form of the Casimir energy and force are independent of the choice of Dirichlet or Neumann conditions, as happens in the LI case.",
"It is worth mentioning that in Refs.",
"[37], [38] the Casimir effect and its corresponding thermal corrections for the scalar field were studied for a particular case where $h^{\\,\\mu \\nu }=\\eta ^{\\,\\mu \\nu }+\\lambda u\\,^\\mu u\\,^\\nu $ , being $\\lambda $ a LV parameter and $u\\,^\\mu $ a four-vector that specifies the direction in which the Lorentz symmetry is broken.",
"There, the authors considered separately different choices of the four-vector $u\\,^\\mu $ and analyzed, by means of the mode-summation method, the Casimir effect.",
"One can verify that our results in Eqs.",
"(REF ) and (REF ) for the global Casimir energy and thermal corrections to the Casimir stress respectively reduce to the ones reported in Refs.",
"[37], [38] by setting $h^{\\,\\mu \\nu }=\\eta ^{\\,\\mu \\nu }+\\lambda u\\,^\\mu u\\,^\\nu $ .",
"However, the local approach adopted here provides additional information regarding the local behavior of the theory, besides the generalization and flexibility that the second-rank tensor $h ^{\\, \\mu \\nu }$ gives to the model.",
"We finish by emphasizing that our method allowed us to determine the effect of $h^{\\,\\mu \\nu }$ on the Casimir energy and stress in a non-perturbative way and did not require a smallness condition on the magnitude of the components of $h^{\\,\\mu \\nu }$ .",
"Although this appears to be an overkill in the context of Lorentz invariance violation, our computation can be relevant for condensed matter physics and materials science because therein the internal structure of media, which generically leads to anisotropies, will play an analogous role to that of a background in empty space."
],
[
"Acknowledgements",
"A. M.-R. acknowledges support from DGAPA-UNAM Project No.",
"IA101320.",
"C. A. E. is supported by a UNAM- DGAPA postdoctoral fellowship and Project PAPIIT No.",
"IN111518.",
"M. A. G. G. is supported by the Spanish Agencia Estatal de Investigación through the grants FPA2015-65929-P (MINECO/FEDER, UE), PGC2018095161-B-I00, IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and Red Consolider MultiDark FPA2017-90566-REDC.",
"O. J. F. acknowledges support from DGAPA-UNAM Project No.",
"IN103319."
]
] | 2005.14217 |
[
[
"Understanding and monitoring the evolution of the Covid-19 epidemic from\n medical emergency calls: the example of the Paris area"
],
[
"Abstract We portray the evolution of the Covid-19 epidemic during the crisis of March-April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94).",
"Our study reveals strong dissimilarities between these departments.",
"We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time.",
"This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS.",
"This also leads to an algorithm, allowing one to detect epidemic resurgences.",
"We rely on a transport PDE epidemiological model, and we use methods from Perron-Frobenius theory and tropical geometry."
],
[
"Introduction",
"The outbreak of Covid-19 in France has put the national Emergency Medical System (EMS), the SAMU, in the front line.",
"In the Île-de-France region, one most affected by the epidemic, the SAMU centers of Paris and its inner suburbs experienced a major increase in the number of calls received and of the number of ambulance dispatches for Covid-19 patients.",
"We show that indicators based on EMS calls and vehicle dispatches allow to analyze the evolution of the epidemic.",
"In particular, we show that EMS calls are early signals, allowing one to anticipate vehicle dispatch.",
"We provide a method of short term prediction of the evolution of the epidemic, based on mathematical modeling.",
"This leads to early detection and early alarm mechanisms allowing one either to confirm that certain sanitary measures are strong enough to contain the epidemic, or to detect its resurgence.",
"These mechanisms rely on simple data generally available in EMS: numbers of patient records tagged as Covid-19, and among these, numbers of records resulting in medical advice, ambulance dispatch, or Mobile Intensive Care Unit dispatch.",
"We also provide a comparative description of the evolution of the epidemic in the four central departments of the Paris area, showing spatial dissimilarities, including a strong variation of the doubling time, depending on the department.",
"Our approach relies on several mathematical tools in an essential way.",
"Indeed, the Covid-19 epidemic has unprecedented characteristics, and, given the lack of experience of similar epidemics, one needs to rely on mathematical models.",
"We use transport PDE to represent the dynamics of Covid-19 epidemic.",
"Transport PDE capture epidemics with a significant time interval between contamination and the start of the infectious phase (in contrast, ODE models without time delays allow instantaneous transitions from contamination to the infectious phase).",
"In the early stage of the epidemic, in which the majority of the population is susceptible, this dynamics becomes approximately linear and order preserving.",
"Then, it can be analyzed by methods of Perron–Frobenius theory.",
"Our main theoretical result shows that the logarithm of epidemic observables can be approximated by a piecewise linear map, with as many pieces as there are phases of the epidemic (i.e., periods with different contamination conditions), see th-1.",
"This methods allows us to identify, the phases of the epidemic evolution, and also to evaluate the time interval between sanitary measures and their impact on epidemic observables, like vehicle dispatch.",
"The idea of piecewise linear approximation and of “log glasses”, a key ingredient of the present approach, arises from tropical geometry.",
"The present work started on March 13th, and led to the algorithm presented here.",
"A preliminary version of this algorithm was used, on March 20th, to forecast the epidemic wave, anticipating that the peak load of SAMU (which occurred around March 27th) would be different depending on the department of the Paris area.",
"We subsequently applied our method to provide Assistance Publique – Hôpitaux de Paris (AP-HP), on April 5th, with an early report, quantifying the efficiency of the lockdown measures from the estimation of the contraction rate of the epidemic in the different departments.",
"This algorithm is now deployed operationally in the four SAMU of AP-HP.",
"This work may be quickly reproduced in any EMS.",
"Although it was developed for Covid-19 and for EMS calls, the present monitoring method is generic.",
"It may also apply to other medical indicators, see ssec-alarm, and to other epidemics, for instance, influenza.",
"This paper is a crisis report, giving a unified picture of a work done jointly by a team of physicians of the SAMU of AP-HP and applied mathematicians from INRIA and École polytechnique.",
"Medical, epidemiological, and mathematical aspects are intricated in this work.",
"We received help from several physicians, researchers and engineers, not listed as authors, and also help from several organizations.",
"They are thanked in the acknowledgments section.",
"This paper should be understood as an announce.",
"The results will be subsequently developed in several papers, with different subsets of coauthors.",
"It is intended to be read both by a medical and a mathematical audience.",
"The first part of the paper, up to sec-discuss included, and the conclusion, are intended to a broad audience.",
"Mathematical tools are presented in sec-pde, sec-tropical, sec-proba and in the appendix.",
"The present work shows the epidemiological significance of the calls received by the EMS, it focuses on the mathematical modeling aspects, on the description of the evolution of the epidemy in the Paris area, and on prediction algorithms.",
"The current workCOVID19 APHP-Universities-INRIA-INSERM, Emergency calls are early indicators of ICU bed requirement during the COVID-19 epidemic, medRxiv:2020.06.02.20117499, June 2020. with an intersecting set of authors, is coordinated with the present one.",
"It focuses on medical aspects.",
"It makes a case study of the Covid-19 crisis of March-April 2020, in Paris, considering the EMS and the hospital services in a unified perspective.",
"It shows that the calls received by SAMU are early predictors of the future load on ICU."
],
[
"Context",
"The mission of the SAMU centers is to provide an appropriate response to calls to the number 15, the French toll-free phone number dedicated to medical emergencies.",
"This service is based on the medical regulation of emergency calls, in the sense that for each patient, a physician decides which response is most appropriate.",
"Thus, depending on the evaluation over the phone of the severity of the case and the circumstances, the response may be a medical advice, a home visit by a general practitioner, the dispatch of a team of EMTs (Emergency Medical Technicians) of either a first aid association or the Fire brigade, or an ambulance of a private company.",
"A Mobile Intensive Care Unit (MICU), staffed by a physician, a nurse and an EMT, is sent to the scene as a second or a first tier, when a life threatening problem is suspected.",
"The role of the SAMU in the management of disasters or mass casualties has been described elsewhere [18], [4].",
"The city of Paris and its inner suburbs are covered by 4 departmental SAMU Center-15 : Paris (75), Hauts-de-Seine (92), Seine Saint-Denis (93), and Val-de-Marne (94), see the map on fig-all.",
"They serve a population of 6.77 million inhabitants.",
"These four Center-15 are part of the public hospital administration, AP-HP (Assistance Publique – Hôpitaux de Paris).",
"They operate identically and use the same computerized call management system.",
"Since the outbreak of the Covid-19 epidemic, the French government instructed the public that anyone with signs of respiratory infection or fever should not go directly to the hospital emergency room to limit overcrowding, but should call number 15 for orientation.",
"To comply with the recommendations of the health care authorities, the four Center-15 applied the same procedures: after medical call regulation, only patients with signs of severity or significant risk factors were transported by EMTs and ambulances to hospitals, either to Emergency Room (ER) or newly created Covid-19 Units.",
"The cases presenting a life-threatening emergency, mostly respiratory distress, were managed by a MICU team and then admitted directly in Intensive Care Unit (ICU).",
"All other cases were advised to stay at home and isolate themselves.",
"When necessary, these patients were also eligible for a home visit by a general practitioner or a consultation appointment the following days.",
"Figure: Flowchart: from calls to Center 15 to admission in hospital units.",
"The numbers are summed over the departments 75, 92, 93 and 94 of the Paris area.In order to maintain a rapid response when a major increase in the number of calls was observed, the four Center-15 implemented specific procedures.",
"Switchboard operators and medical staff was reinforced, and for calls related to Covid-19 an interactive voice server —triaging the calls to dedicated computer stations— was developed.",
"Patient evaluation and management were improved by introducing video consultation, sending of instruction using SMS, giving the patient the option to be called back.",
"Prehospital EMT teams were also significantly reinforced by first aid volunteers, and additional MICU were created.",
"Since January 20th 2020 all calls and patient records related to Covid-19 were flagged in the information system of Center-15 and a daily automated activity report was produced."
],
[
"Methods",
"In this section, we describe the methods used in this work, in a way adapted to a general audience.",
"Mathematical developments appear in sec-pde,sec-tropical,sec-proba and in the appendix."
],
[
"Classification of calls",
"In order to develop a mathematical analysis of the evolution of the epidemic, we classified the calls tagged as Covid-19 in three categories, according to the decision taken: [align=left] Class 1: calls resulting in the dispatch of a Mobile Intensive Care Unit; Class 2: calls resulting in the dispatch of an ambulance staffed with EMT; Class 3: calls resulting in no dispatch decision.",
"Such calls correspond to different forms of medical advice (recommendation to consult a GP, specific instructions to the patient, etc.).",
"We shall denote by $Y_{\\mathrm {MICU}}(t)$ (resp.",
"$Y_{\\mathrm {EMT}}(t)$ and $Y_{\\mathrm {adv}}(t)$ ) the number of MICU transports (resp.",
"the number of ambulances transport and the number of medical advices) on day $t$ , for patients tagged with suspicion of Covid-19.",
"We shall call these functions of time the observables, in contrast with $C(t)$ , the actual number of new contaminations on day $t$ , which cannot be measured.",
"We developed a piece of software that computes these observables by analyzing the medical decisions associated with the patient records, made accessible daily by AP-HP."
],
[
"Mathematical properties of the observables",
"To analyze these observables, we shall rely on a mathematical model.",
"A standard approach represents the evolution of an epidemic by an ordinary differential equation (SEIR ODE), representing the evolution of the population in four compartments: “susceptible” (S), “exposed” but not yet infectious (E), “infectious” (I), and finally, “removed” from the contamination chain (R), either by recovery or death.",
"A refinement of the SEIR model splits the S and E compartments in sub-compartments corresponding to different age classes.",
"It includes a contact matrix, providing differentiated age-dependent infectiosity rates [29].",
"Another refinement includes additional compartments, representing, for instance, patients at hospital [13], or individuals with mild symptoms [28].",
"In contrast with such ODE models, we use a partial differential equation (PDE), i.e., an infinite dimensional dynamical system, described in sec-pde.",
"Our approach is inspired by the PDE model of Kermack and McKendrick [22].",
"We use PDE, rather than ODE, to take into account the presence of delays in the contamination process: the median incubation period of Covid-19 is estimated of 5.1 days, with a 95% confidence interval of 4.5-5.8, and 95% of patients in the range [2.2,11.5], see [25], in line with other human coronaviruses having also long incubation times, like SARS [37] and MERS [36].",
"(This may be compared with a median incubation time of 1.4 days [95% CI, 1.3–1.6] for the toxigenic Cholera [2], or with an interval of 36 hours between infection by pneumonic plague and first symptoms in Brown Norway rats, with rapid letality, 2-4 days after infection [1].)",
"ODE models assume exponentially distributed transitions times from one compartment to another.",
"This entails that the interval elapsed between contamination and the time an individual becomes infectious can be arbitrarily small, so ODE models are more adapted to epidemics with a short incubation time.",
"Using transport PDE, as is done in sec-pde, takes delays into account, allowing also one to recover ODE models as special cases.",
"We shall limit here our analysis to the early stage of the epidemic, assuming that the population that has been infected is much smaller than the susceptible population.",
"This approximation is reasonable at least in the initial part of the epidemic, according to the study [34] which gives an estimate of 5.7% for the proportion of the population in France that has been infected prior to May 11th, 2020.",
"Then, the dynamics becomes linear and order-preserving.",
"The latter property entails that the observables are an increasing function of the size of the initial population that is either exposed or infected.",
"Results of Perron–Frobenius and of Krein-Rutman theory, which we recall in sec-epidemio, entail that, if the sanitary measures stay unchanged, there is a rate $\\lambda $ , such that the number of newly contaminated individuals at day $t$ grows as $C(t)\\simeq K_C \\exp (\\lambda t)$ , as $t\\rightarrow \\infty $ , where $K_C$ is a positive constant.",
"The number $\\delta := (\\log 2)/\\lambda $ , when it is positive, represents the doubling time: every $\\delta $ days, the number of new contaminations per day doubles.",
"When $\\delta $ is negative, the epidemic is in a phase of exponential decay.",
"Then, the opposite of $\\delta $ yields the time after which the number of new contaminations per day is cut by half.",
"For the analysis which follows, it is essential to consider, instead of $C(t)$ , its logarithm, $\\log C(t)\\simeq \\log K_C + \\lambda t$ .",
"The exponential growth or decay of $C(t)$ corresponds to a linear growth or decay of the logarithm.",
"We shall also see in sec-pde that all the epidemiological observables evolve with the same rate.",
"E.g., assuming that all the patients transported by MICU were contaminated $\\tau _{\\mathrm {MICU}}$ days before the transport, and that a proportion $\\pi _{\\mathrm {MICU}}$ of the contaminated individuals will require MICU transport, we arrive at $Y_{\\mathrm {MICU}}(t) = \\pi _{\\mathrm {MICU}}C(t-\\tau _{\\mathrm {MICU}})$ , and so $\\log Y_{\\mathrm {MICU}}(t) \\simeq \\log \\pi _{\\mathrm {MICU}}+ \\log K_C + \\lambda (t-\\tau _{\\mathrm {MICU}})$ .",
"Similar formulæ apply to $Y_{\\mathrm {EMT}}$ and $Y_{\\mathrm {adv}}$ , and to other observables based for instance on ICU admissions or deceases.",
"A finer model of observables, taking into account a distribution of times $\\tau _{\\mathrm {MICU}}$ , instead of a single value, is presented in subsec-observables.",
"For the analysis which follows, we shall keep in mind that the logarithm of all the observables is asymptotically linear as $t\\rightarrow \\infty $, and that the rate, $\\lambda $ , is independent of the observable."
],
[
"Piecewise linear approximation of the logarithm of the observables",
"When the sanitary measures change, for instance, when lockdown is established, the rate $\\lambda $ changes.",
"So, the logarithm of the observables cannot be approximated any more by a linear function.",
"However, a general result, stated as th-1 below, shows that this logarithm can be approximated by a piecewise linear function with as many linear pieces as there are phases of sanitary policy.",
"This result stems from the order preserving and linear character of the epidemiological dynamics, and so, it holds for a broad class of epidemiological models; several examples of such models are discussed in sec-pde.",
"In the Paris area, there are three relevant sanitary phases to consider from February to May, 2020: initial growth (no restrictions); “stade 2” (stage 2) starting on Feb. 29th (prevention measures), and then lockdown from March 17th to May 11th.",
"Sanitary phases are further described in sec-delay.",
"Since the number $\\nu $ of sanitary phases is known (here $\\nu =3$ ), we can infer the different values of $\\lambda $ attached to each of these phases, by computing the best piecewise linear approximation, $\\mathcal {L}(t)$ with at most $\\nu $ pieces of the logarithm of an observable $Y(t)$ .",
"To compute a robust approximation, we minimize the $\\ell _1$ norm, $\\sum _{t} |\\mathcal {L}(t)- \\log Y(t)|$ , where the sum is taken over the days $t$ in which the data are available.",
"Finding the best approximation $\\mathcal {L}$ is a difficult optimization problem, for the objective function is both non-smooth and non-convex.",
"Methods to solve this problem are discussed in appendix-2."
],
[
"Epidemic alarms based on doubling times",
"To construct epidemic alarms, we shall compute a linear fit, $\\mathcal {L}(t)=\\alpha +\\beta t$ , to the variables $\\log Y(t)$ , where $Y$ is an epidemic observable.",
"The principle is to trigger an alarm when the doubling time becomes positive, or equivalently, when the slope $\\beta $ becomes positive.",
"Assuming that values of $Y(t)$ are known over a temporal window, there are simple ready-to-use methods for computing estimates $\\hat{\\beta }$ for the slope $\\beta $ .",
"We can also determine the probability $p^+$ that the slope is positive.",
"These methods are detailed in sec-appendixC.",
"On their basis, we propose the following alarm raising mechanism, allowing one to deploy a gradual response.",
"This mechanism relies on the two following observables, $Y_{\\mathrm {adv}}$ , the number of calls resulting in medical advice, and $Y_{\\mathrm {disp}}:=Y_{\\mathrm {EMT}}+Y_{\\mathrm {MICU}}$ , the number of dispatched vehicles.",
"The consolidation of the observables $Y_{\\mathrm {EMT}}$ and $Y_{\\mathrm {MICU}}$ is justified, because the two time series both correspond to the stage of aggravation, albeit with different degrees, and so they evolve more or less at the same time.",
"First define a temporal window of days $t$ over which the linear fit $\\mathcal {L}_{\\mathrm {adv}}(t)=\\alpha _{\\mathrm {adv}}+\\beta _{\\mathrm {adv}} t$ to $\\log Y_{\\mathrm {adv}}(t)$ is made.",
"By default we consider the last ten days prior to the current day.",
"Similarly, we compute a linear fit $\\mathcal {L}_{\\mathrm {disp}}(t)=\\alpha _{\\mathrm {disp}}+\\beta _{\\mathrm {disp}} t$ to $\\log Y_{\\mathrm {disp}}(t)$ over the same time window.",
"Our algorithm will generate both a warning and alarms.",
"A warning is a mere incentive to be careful.",
"An unjustified warning is bothersome but generally harmless, so we accept a high probability of false positive for warnings.",
"An alarm may imply some actions, so we wish to avoid false alarms.",
"For this reason, we shall consider two different probability thresholds, $\\vartheta _\\mathrm {alarm}$ and $\\vartheta _\\mathrm {warn}$ , say $\\vartheta _\\mathrm {alarm}= 75\\%$ and $\\vartheta _\\mathrm {warn}=25\\%$ .",
"With this setting, we will be warned as soon as the probability of the undesirable event is $\\ge 25\\%$ , and we will be alarmed when the same probability becomes $\\ge 75\\%$ .",
"Of course, these thresholds can be changed, depending on the risk level deemed to be acceptable.",
"We shall denote by $p^+_\\mathrm {adv}$ the probability that the slope $\\beta _{\\mathrm {adv}}$ is positive, and by $p^+_{\\mathrm {disp}}$ the probability that $\\beta _{\\mathrm {disp}}$ is positive.",
"These probabilities are evaluated on the basis of statistical assumptions detailed in sec-appendixC.",
"A warning is provided when $p^+_\\mathrm {adv}\\ge \\vartheta _\\mathrm {warn}$ , meaning that the probability that the slope $\\beta _\\mathrm {adv}$ of the curve of the logarithm of the calls for medical advice over the corresponding time window be positive is at least $\\vartheta _\\mathrm {warn}$ .",
"This should be interpreted as a mere warning of epidemic risk: choosing $\\vartheta _\\mathrm {warn}$ as above, the odds are at least 25% that the epidemic is growing.",
"This warning is subsequently transformed into an alarm when $p^+_{\\mathrm {adv}}\\ge \\vartheta _\\mathrm {alarm}$ .",
"Choosing $\\vartheta _\\mathrm {alarm}$ as above, the odds that the epidemic is growing are now at least $75\\%$ .",
"Such an alarm is then subsequently transformed into a confirmed alarm if we still have $p^+_{\\mathrm {adv}}\\ge \\vartheta _\\mathrm {alarm}$ , and if, in addition, $p^+_{\\mathrm {disp}}\\ge \\vartheta _\\mathrm {alarm}$ , meaning that the probability that the slope of the logarithm of the curve of ambulances and MICU dispatches be positive is now above $\\vartheta _\\mathrm {alarm}$ .",
"Again, this estimate is defined in terms of a time window over which $\\beta _{\\mathrm {disp}}$ is estimated.",
"We use the same default values of ten days and $\\vartheta _\\mathrm {alarm}$ as above.",
"As shown in sec-data, the indicators based on vehicle dispatch are by far less noisy than the indicators based on calls for medical advices, but their evolution is delayed.",
"This is the rationale for using medical advice for an early warning and early alarm, and then vehicle dispatch for confirmation.",
"Instead of considering the probability $p^+$ , we could consider the upper and lower bounds of a confidence interval $[\\beta ^-_\\epsilon ,\\beta ^+_\\epsilon ]$ for the estimated slope $\\beta $ , with a probability threshold $\\epsilon $ .",
"Then we may, trigger a warning when $\\beta ^+_\\epsilon \\ge 0$ , and an alarm when $\\beta ^-_\\epsilon \\ge 0$ .",
"This leads to an essentially equivalent mechanism.",
"We prefer the algorithm above as it allows to interpret the thresholds in terms of false positives and false negatives.",
"Given the severity of the risk implied by Covid-19, it may be desirable to complete the previous alarm, based only on tail probabilities of the slope, by a different type of alarm, based on a threshold of doubling time, $D$ .",
"The alarm will be triggered if the odds that the doubling time be positive and smaller than $D$ are at least one half.",
"An indicative value of $D$ might be 14 days: a doubling of the number of arrivals of Covid-19 patients in hospital services every 14 days may be quite challenging, justifying an alarm, and the slope corresponding to this doubling time seems significant enough to avoid false alarms.",
"Again, the value of $D$ can be changed arbitrarily depending on the acceptable level of risk.",
"Moreover, this other type of alarm can still be implemented in two stages: early alarm, with the medical advice signal, and then confirmed alarm, with the vehicle dispatch signal.",
"In addition, sec-appendixC provides more sophisticated ready-to-use methods for obtaining sharper confidence intervals or probabilities for the slope $\\beta $ , resulting in more precise alarm mechanisms, when different time series are available.",
"We require, however, that these series correspond to events occurring approximately at the same stage in the pathology unfolding.",
"Here, we used the trivial aggregator, $Y_{\\mathrm {disp}}=Y_{\\mathrm {EMT}}+Y_{\\mathrm {MICU}}$ .",
"There is an optimal way to mix different series to minimize the variance of the composite estimator, explained in sec-appendixC.",
"This methodology is generic.",
"It could thus also apply to obtain a sharper confidence interval for the early indicator by combining its estimate $\\hat{\\beta }_{\\mathrm {adv}}$ with that of other time series associated with signals that correspond to the same stage in pathology unfolding.",
"Specifically, the count $Y_{\\mathrm {GP}}(t)$ of patients consulting general practitioners for recently developed Covid-19 symptoms, if available, provides such a signal.",
"A linear fit to $\\log Y_{\\mathrm {GP}}(t)$ would then yield an estimate $\\hat{\\beta }_{\\mathrm {GP}}$ which can be combined with $\\hat{\\beta }_{\\mathrm {adv}}$ to refine the corresponding confidence interval.",
"In this way, we can mix several early but noisy indicators to get an early but less noisy consolidated indicator."
],
[
"Key figures and graphs",
"From February 15th to May 15th, we counted a total of [group-separator=,]170166 patient files tagged with a suspicion of Covid-19, distributed as follows in the different departments: [group-separator=,]53646 in Dep.",
"75; [group-separator=,]36721 in Dep.",
"92; [group-separator=,]49703 in Dep.",
"93; and [group-separator=,]30096 in Dep. 94.",
"The flow of calls to the SAMU of the Paris area, and its impact on ER and ICU, is shown on Figure REF .",
"The data concerning the ER and the ICU are taken from the governmental website SPF (Santé Publique France) [15], it is available only from March 19th.",
"On Figure REF , we represent, in logarithmic ordinates, the numbers of events of different types, summed over the four departments of the Paris area (75, 92, 93 and 94): (i) the number of patients calling the SAMU (including patients not calling for Covid-19 suspicion); (ii) the number of calls tagged as Covid-19 not resulting in a vehicle dispatch (i.e., as discussed in §REF , all kinds of medical advices); (iii) the number of calls tagged as Covid-19 resulting in an ambulance or MICU dispatch, We obtained the data (i) by analyzing the phone operator log files.",
"Since a patient may call the Center 15 several times, we eliminated multiple calls to count unique patients.",
"To compute data (ii) and (iii), we developed a software to analyze the “medical decision” field of the regulation records.",
"Using logarithmic ordinates is essential on fig-differenttypes, as it allows to visualize on the same graph signals of different orders of magnitude (e.g, there is a ratio of 20 between the peak number of patients calling and the peak number of vehicles dispatched).",
"Figure: Number of patients calling Center-15, of MICU and ambulances dispatch for Covid-19 suspicionin the Paris area (departments 75, 92, 93 and 94)The evolution of the number of vehicles dispatched (MICU and ambulances) is shown on Figure REF , for each department of the Paris area (still with logarithmic ordinates).",
"Figure: Comparison of the evolution of the epidemic in the different departments of the Paris area: numbers of vehicles dispatch by department.",
"The figure inset displays the same curves in usual linear ordinates to keep in mind the different magnitudes at stake.",
"A map of the Paris area, showing the departments 75, 92, 93, 94, is at the bottom right of the figure.opacity = .6] at (0,0) ;opacity = .6] at (0,0) ;opacity = .6] at (0,0) ;opacity = .6] at (0,0) ;white] at (-0.12,-0.02) 75;black] at (-0.57,-0.45) 92;black] at ( 0.1, 0.6) 93;black] at ( 0.8, -0.4) 94;We provide in Table REF the doubling times of the number of vehicles dispatched (ambulances and MICU), for the different departments, measured in days (abbreviation “d”).",
"Table: Doubling time of the number of MICU and ambulances dispatched, for different periods, for each department, obtained by a least squares approximation of the logarithm of this number.",
"The opposite of a negative doubling time yields the halving time.We now draw several conclusions from the previous analysis."
],
[
"The increase in the number of calls for medical advice provides an early, but noisy, indicator\nof the epidemic growth",
"As shown in fig-differenttypes, the peak of the number of calls for medical advice was on March 13th.",
"However, this date, four days before the lockdown (March 17th), is not consistent with epidemiological modeling.",
"This peak seems rather to be caused by announcements to the population, see the discussion in sec-announce."
],
[
"The epidemic kinetics vary strongly across neighboring departments",
"In the initial phase of the epidemic (Feb. 28th–March 15th), the doubling time was significantly shorter in the 93 department (4.2 d) than in central Paris (5.9 d).",
"The 93 department, with 1.6M inhabitants, is less populated than central Paris (2.1M inhabitants).",
"Another difference between the departments concerns mobility.",
"Movement from the population from central Paris to smaller towns and cities or to countryside were observed, after March 12th, the date of the first presidential address concerning the Covid-19 crisis.",
"In order to quantify this mobility, we requested information from Enedis, the company in charge of the electricity distribution network in France, and also from Orange and SFR, two operators of mobile phone networks.",
"Enedis provided us with an estimation of the departure rates of households, based on a variation of the volume of electricity consumed, aggregated at the level of departments and districts (i.e., arrondissements).",
"SFR provided us with estimates of daily flows from the Paris area to other regions, again aggregated at the scale of the departments or districts, based on mobile phone activity, confirming this decrease of population.",
"Orange Flux Vision provided us with daily population estimates, at the scale of department, based on mobile phone activity.",
"By March 30th, the population, during the night, was estimated to be 1.6M inhabitants in central Paris, versus 1.35M in the 93.",
"However, the epidemic peak was higher in the 93 than in the 75 (338 dispatches versus 296).",
"The contraction rate in the period after the peak (March 29th–Apr.",
"24th) was also smaller in the 93, with a halving time of 10.2 days, to be compared with 9.4 days in the 75.",
"Possible explanations for these strong spatial discrepancies are discussed in sec-discrepancies."
],
[
"Delay between implementation of sanitary policies and its effect on hospital admissions",
"We explained in sec-piecewise, based on th-1 below, that the logarithm of an epidemic observable can be approached by a piecewise linear map with as many pieces as there are stages of sanitary measures.",
"So, we look for the best approximation, in the $\\ell _1$ norm, of the logarithm of the number of vehicles dispatched (ambulances and MICU), by a piecewise linear map with at most three pieces.",
"This best approximation is shown on p-phases.",
"It is computed by the method of appendix-2.",
"In order to evaluate the influence of a sanitary measure on the growth of the epidemic, an approach is to compare the date of the measure with the date of the change of slope of the logarithmic curve, consecutive to the measure.",
"This method is expected to be more robust than, for instance, a comparison of peak values, because the best piecewise-linear approximation is obtained by an optimization procedure taking the whole sequence into account.",
"Indeed, a local corruption of data will not change significantly the date of change of slope, if the problem is well conditioned.",
"This is the case in particular if the difference between consecutive slopes is sufficiently important.",
"In other words, we can identify in a more robust manner the time of effect of a strong measure than of a mild one.",
"Let us recall the main changes of sanitary measures in the Paris area, between February and May 2020.",
"We may distinguish the following phases: - Initial development of the epidemic, no general sanitary measures in the Paris area, until Feb 29th, first day of so-called “stade 2” by the authorities (following “stade 1” in which measures intended to prevent the introduction of the virus in France – like quarantine in specific cases– were taken).",
"- “Stade 2” (stage 2) measures: general instructions of social distancing given to the population (e.g., not shaking hands), ban on large gatherings.",
"Moreover, some large companies created crisis committees, and decided to take more restrictive measures than the ones required by the authorities, including for instance banning meetings with more of 10 people, and banning business travels.",
"Restrictive measures in companies were deployed gradually during the work week from March 2nd to March 6th.",
"- School closure on March 16th.",
"- Lockdown on March 17th.",
"The lockdown ended on May 11th, throughout the country.",
"Hence, we may interpret the variations in the slope in the piecewise linear approximation of the logarithm of the number of ambulances and MICU dispatched, shown on p-phases, as the effect of sanitary measures.",
"The dates where the slope changes are represented in the figure by dotted lines.",
"Thus, the latest breakpoint of the piecewise linear approximation of the 75 curve (in blue) arises on March 26th, to be compared with March 30th in the 93 (red curve).",
"The dates of breakpoints in the 92 and 94 are intermediate.",
"Given the first strong measure (closing of schools) was taken on March 16th, we may evaluate the delay between a sanitary measure and its effect on the ambulances and MICU dispatch to be between 10 and 14 days.",
"This corresponds to a delay between contamination and occurrence of severe symptoms.",
"Figure: Logarithm of the number of ambulances dispatched: the effect of the successive sanitary measures"
],
[
"Construction of statistical indicators of epidemic resurgence based on emergency calls",
"We implemented the alarm mechanism based on the inference of doubling times described in ssec-alarm and further explained in sec-appendixC.",
"The method is illustrated on fig-predictor.",
"Given a time period where the data is known, we perform a linear regression on the number of medical advices and vehicles dispatched.",
"The light shaded, tubular areas around the curves are based on confidence intervals for the fluctuations of the observed log-counts.",
"The dark-shaded, trapezoidal areas prolongate the tubular areas with straight lines, the slopes of which correspond to confidence intervals for the slope of the linear regression.",
"We display such confidence domains for the last known data in May, performing a 6-day forecast based on the last ten days.",
"In order to validate the method, we also display these domains for older data in March and April, performing for each a 6-day forecast based on a number of past days.",
"For these time-periods, the short-term confidence domains are seen to satisfactorily contain the data of the following days.",
"Observe how the shape of the confidence trapezoids depends on the number of points and the variance of the data used to compute them.",
"The needle-shaped (or clock hand) indicators depicted in each trapezoidal confidence domain illustrate the alarm mechanism of ssec-alarm.",
"For each slope inference, there is a $\\vartheta _\\mathrm {warn}=25\\%$ probability that the real-value of the slope we estimated using recent data is greater than the slope of the thin needle on the picture.",
"Likewise there is a $\\vartheta _\\mathrm {alarm}=75\\%$ probability that the real slope is greater than the slope of the fat hand.",
"As a result, a warning (resp.",
"an alarm) on the dynamics of the medical advice curve should be triggered as soon as the thin needle (resp.",
"the fat needle) has a positive angle with respect to the horizontal.",
"We have depicted the horizontal with dashed lines to enhance readability.",
"On May 25nd, no warning nor alarm is triggered, since all the needle-indicators are below the positivity threshold, indicating with at least a 75% confidence level that based on the last ten days, the two observed signals are on a decreasing trend.",
"Note that due to the relative stagnation of the medical advice curve in the two first weeks of May, computing our indicators a few days earlier (such as May 22nd) would have raised an warning to arouse vigilance due to the uncertainty on the future trend, but no alarm.",
"The numbers $\\vartheta _\\mathrm {warn}$ and $\\vartheta _\\mathrm {alarm}$ need to be carefully calibrated, and for this, additional data for the forecoming weeks may be helpful."
],
[
"Calls resulting in medical advice are highly influenced by the instructions given to the population",
"The blue curve on fig-differenttypes,fig-predictor counts the number of times medical advices was given, of all kinds (calls resulting in recommendations given to the patient but no vehicle dispatched).",
"It is generally associated with early events in the unfolding of the pathology, and in particular occurrence of the first symptoms.",
"An estimate of 5.1 days between the date of contamination and the date of the first symptoms is given in [25], so we may assume that calls for medical advice are made by patients 5-8 days after contamination.",
"We observed in sec-noisy that the peak in the number of calls resulting in medical advice was on March 13th.",
"Hence, assuming the peak of contaminations was just before lockdown, the peak of the curve of new symptom occurrences should occur only several days after the lockdown time (March 17th).",
"This indicates that the curve of calls resulting in a medical advice did not give a reliable picture of the epidemic growth around March 13th.",
"Indeed, this curve is very sensitive to changes in the instructions given to the population and to political announcements, notable examples of which include the following: recommendation to patients to call emergency number 15, instead of going directly to emergency departments (to avoid contamination and overcrowding); – the presidential announcement on March 12th of more restrictive measures to be deployed from March 16th, making the population more aware of the growth of the epidemic."
],
[
"The indicators of medical advice given and ambulances and MICU dispatch can be used to monitor the epidemic",
"Setting aside perturbations due to political announcements or changes in the policy for calling SAMU, the curve of calls for medical advice should be a reliable and early estimate of the curve of ambulances dispatched, which is triggered at a later stage in the unfolding of the pathology when symptom severity increases.",
"Thus, it gives an early signal allowing both SAMU and hospitals to anticipate by several days an increase in load.",
"We can give a rough estimate of this delay by considering the peak dates in fig-differenttypes.",
"Epidemiological modeling indicates that the number of new contaminations grows exponentially until a sanitary measure that is strong enough to contain the epidemic is taken.",
"In the present case, the candidates for such strong measures are the school closing (March 16th) or the lockdown (March 17th).",
"Considering the last day previous to the measures, we may assume that the peak date for new contaminations was on March 15th or March 16th.",
"As mentioned above, according to [25], the time between contamination and first symptoms is estimated to be 5.1 days.",
"Hence, if calls for medical advice were representative of first symptoms, the peak value for these calls would have been between around March 20th or March 21st.",
"The peak of the number of dispatches of ambulances and MICU was on March 27th.",
"This leads to an estimate of 6-7 days for the delay between the curve of the true need for medical advice and the curve of vehicles dispatch.",
"The alarm mechanism was developed during the crisis, after March 20th.",
"Had it been available before, in the 75, the early warning would have occurred by February 24th (for both Covid-19 indicators on medical advice and vehicle dispatch), alarm would have been been triggered on February 25th based on medical advice, and a confirmation alarm would have occurred on February 27th based on ambulance and MICU dispatch.",
"Tracking the same signal at a finer spatial resolution (a neighborhood rather than a department) may enable epidemic surveillance during the period following the lifting of measures such as travel bans.",
"Indeed in view of the spatial differentiation in doubling times at the department level, it appears plausible that disparities may also be present at much finer spatial granularities, with resurgences localized to towns or neighborhoods.",
"Deployment of alarm mechanisms constructed from event counts at finer spatial granularities could be used to identify clusters of resurgence early on, and guide subsequent action.",
"Seasonal influenza, whose early symptoms can be mistaken for those of Covid-19, can also trigger a linear growth of the logarithms of the number of calls, or vehicle dispatched.",
"However the slope of this logarithmic curve is expected to be shallower, owing to the lower contagiosity of influenza.",
"One could thus distinguish, on the basis of the observed slope, whether one is confronted with seasonal flu or with an outbreak of Covid-19."
],
[
"Jumps of the curves of the number of calls may be caused by large clusters or influenced by neighboring countries",
"The curves of the number of calls for medical advice, and of vehicles dispatched (fig-differenttypes), both jump between February 23rd and 25th.",
"Epidemiogical models are unlikely to produce shocks of this type in the absence of exceptional factors.",
"Several significant epidemic events occurred at nearby dates, including: The development of the Covid-19 epidemic in the north of Italy, a region closely tied to France (the first lockdowns occurred around February 21st in the province of Lodi).",
"The school vacation ended on February 23thin the Paris area.",
"A significant number of parisians went back from Italy during the week-end of February 22nd-23th(end of school vacation).",
"A large Evangelist meeting (Semaine de Carême de l’Eglise La Porte Ouverte Chrétienne de Bourtzwiller, Haut-Rhin), from February 17th to February 21st, identified by Agence Régionale de Santé Grand Est as the source of a cluster phenomenon [12].",
"The potential influence of the north Italy epidemic was pointed out by Paul-Georges Reuter (private communication).",
"At this stage, the essential factors are not yet known.",
"The influence of mobility on the development of the epidemic in the Paris area will be studied in a further work."
],
[
"Patients from different areas tend to call the SAMU at different stages of the pathology",
"Considering the piecewise linear curves in p-phases, we note that in the 93, the date of the break point is shifted of 3 days, by comparison with the 75, suggesting that by this time, the patients of 93 were calling Center 15 at a later stage of the evolution of the disease.",
"This hypothesis is confirmed by an examination of the ratio of the number of MICU dispatched over the number of ambulances dispatched.",
"For instance, on March 30th, there were 9 MICU dispatches and 276 other dispatches in the 75, to be compared with 25 MICU dispatches and 262 other dispatches in the 93, i.e., ratios of 3.3% in the 75 and 9.5% in the 93.",
"In the same way, the first breakpoints of the curves give an indication of the times at which “stade 2” measures influence the epidemic growth.",
"These dates range from March 15th (for 92) to March 22nd (for 75).",
"It may be the case that these dates are dispersed because the change of slope is relatively small, meaning that the effect of stade 2 is mild.",
"Indeed, the milder the slope change, the more the estimation of the corresponding date is sensitive to noise.",
"Another effect which may have perturbed the curves is the important mobility of the population in the 4 departments, between March 12th and March 16th."
],
[
"The strong spatial heterogeneity of the evolution of the epidemic may be explained by local conditions",
"We observed in sec-spatial-varies that in the initial phase of the epidemic (Feb. 28th–March 15th), the doubling time was significantly shorter in the 93 department, whereas in the contraction phase, the halving time was significantly higher.",
"One may speculate that the contraction rate in the lockdown phase is influenced by intra-familial contaminations.",
"In this respect, according to a survey of INSEE, the national institute of statistics, the average size of a household is of 2.6 in the 93, versus 1.9 in the 75 (values in 2016 [19]).",
"We also remark that just after the peak on March 27, and up to April 6, the curve of the department 93 on fig-all has the shape of a high, oscillating plateau, decaying more slowly than the curve of the department 75.",
"This may be caused by changes in the nature of the dominant mode of contaminations, intra-familial contamination becoming an essential part of the kinetics during lockdown.",
"One may also speculate that the blowup rate in the initial phase is higher when the population is more dependent on public transport, or working in jobs with more contamination risk.",
"These aspects will be further studied elsewhere.",
"After Stade 2 was announced, during the period from March 2nd to March 6th, a number of large companies took specific measures (e.g., forbidding avoidable small group meetings, enforcing travel restrictions, restricting office access), in addition to the general measures (not shaking hands, forbidding large meetings) enforced by the authorities.",
"This may have led to a decrease of the number of contamination on the workplace, and one explanation for the increase of the doubling time."
],
[
"Taking delays into account: a transport PDE SEIR model",
"We now introduce a multi-compartment transport PDE model, representing the dynamics of Covid-19.",
"As explained in sec-model, in contrast to ODE models, that assume that the transition time from a compartment to the next one has an exponential distribution, PDE models capture transition delays bounded away from zero, an essential feature of Covid-19.",
"An interest of this PDE model also lies in its unifying character: it includes as special cases, or as variations, SEIR ODE models that have been considered [29], [13].",
"We shall keep the traditional decomposition of individuals in compartments, “susceptible” (S), “exposed” (E), “infectious” (I), and “removed” from the contamination chain (R), as explained in sec-model, but the state variables attached to the $E$ and $I$ compartments will take the time elapsed in the compartment into account, and thus, will be infinite dimensional.",
"For all $t\\ge 0$ , we denote by $n_E(x,t)$ the density of the number of individuals that were contaminated $x$ time units before time $t$ , and that are not yet infectious at time $t$ , i.e., the number of exposed individuals that began to be exposed at time $t-x$ .",
"Then, the size of the exposed population at time $t$ is given by $E(t) &= \\int _0^\\infty n_E(x,t)\\,\\mathrm {d}x \\hspace{5.0pt}.$ Similarly, we denote by $n_I(x,t)$ the density of the number of individuals that became infectious $x$ time units before time $t$ , and that are not yet removed from the contamination chain at that time.",
"Then, the size of the infectious population at time $t$ is given by $I(t) &= \\int _0^\\infty n_I(x,t)\\,\\mathrm {d}x \\hspace{5.0pt}.$ Finally, we denote by $S(t)$ the number of susceptible individuals at time $t$ , and by $R(t)$ the number of individuals that have been removed from the contamination chain before time $t$ .",
"The total population at time $t$ is given by $ N(t):= S(t) + E(t) + I(t) + R(t) \\hspace{5.0pt}.$ We consider the following system of PDE and ODE, with integral terms in the boundary conditions: $& \\frac{\\mathrm {d}S}{\\mathrm {d}t} = - \\frac{S(t)}{N(t)}\\int _0^\\infty K_{I\\rightarrow E}(x,t) {n_I(x,t)} \\,\\mathrm {d}x \\hspace{5.0pt}, \\\\&n_E(0,t) = \\frac{S(t)}{N(t)}\\int _0^\\infty K_{I\\rightarrow E}(x,t) {n_I(x,t)}\\,\\mathrm {d}x \\hspace{5.0pt},\\qquad \\frac{\\partial n_E}{\\partial t} (x,t)+ \\frac{\\partial n_E}{\\partial x}(x,t) + K_{E\\rightarrow I}(x,t) n_E(x,t) = 0 \\hspace{5.0pt}, \\\\& n_I(0,t) = \\int _0^\\infty K_{E\\rightarrow I}(x,t) n_E(x,t)\\,\\mathrm {d}x \\hspace{5.0pt},\\qquad \\frac{\\partial n_I}{\\partial t}(x,t) + \\frac{\\partial n_I}{\\partial x}(x,t) + K_{I\\rightarrow R}(x,t) n_I(x,t) = 0 \\hspace{5.0pt},\\\\& \\frac{\\mathrm {d}R}{\\mathrm {d}t} = \\int _0^\\infty K_{I\\rightarrow R}(x,t) n_I(x,t)\\,\\mathrm {d}x \\hspace{5.0pt}.$ We assume that an initial condition at time 0, $S(0)$ , $n_E(\\cdot ,0)$ , $n_I(\\cdot ,0)$ and $R(0)$ is given.",
"This is inspired by the so called “age structured models” considered in population dynamics.",
"Kermack and McKendrick developed the first model of this kind to analyze the Plague epidemy of Dec. 1905 – July 1906 in Mumbai [22].",
"Von Forster [39] studied a similar model.",
"Nowadays, these models are used as a general tool in population dynamics, with applications to biology and ecology [40], [33], [30], In these models, “age” refers to the age elapsed in a compartment – each transition to a new compartment resets to zero the “age” of an individual.",
"In contrast, in the classical SEIR literature based on ODE, the standard notion of age (time elapsed since birth) is taken into account, via a contact matrix tabulating age-dependent infectiosity rates [29].",
"These two notions of age should not be confused.",
"In the sequel, we shall use quotes, as in “age”, to denote the age in a compartment, and will omit quotes to denote the ordinary age (since birth).",
"We suppose that $K_{I\\rightarrow E}$ , $K_{E\\rightarrow I}$ and $K_{I\\rightarrow R}$ are given nonnegative functions.",
"The value $K_{E\\rightarrow I}(x,t)$ gives the departure rate from the compartment $E$ to the compartment $I$ , for individuals of “age” $x$ in the compartment $E$ , at time $t$ .",
"Similarly, $K_{I\\rightarrow R}(x,t)$ gives the departure rate from the compartment $I$ to the compartment $R$ .",
"As in the classical SEIR model, the departure term from the susceptible compartment, i.e., the right-hand-side of (REF ) is bilinear in the number $S(t)$ of susceptible individuals and in the population of infectious individuals $n_I(\\cdot ,t)$ , and we normalize by the size of the population $N(t)$ .",
"The term $K_{I\\rightarrow E}(x,t)$ can be interpreted as an infection rate.",
"Differentiating $N(t)$ with respect to time, using the system above, and assuming that for all $t\\ge 0$ , $n_E(x,t)$ and $n_I(x,t)$ vanish when $x$ tends to infinity, we verify that the total population $N(t)$ is independent of time.",
"When the functions $K_{I\\rightarrow E}, K_{E\\rightarrow I}$ and $K_{I\\rightarrow R}$ are constant, taking into account (REF ) and (REF ), we recover the classical SEIR model from the dynamics (REF ): $& \\dot{S} = - \\frac{S}{N} K_{I\\rightarrow E} I\\hspace{5.0pt}, \\\\&\\dot{E} = \\frac{S}{N} K_{I\\rightarrow E} I - K_{E\\rightarrow I} E \\hspace{5.0pt}, \\\\&\\dot{I}= K_{E\\rightarrow I} E - K_{I\\rightarrow R} I \\hspace{5.0pt}, \\\\& \\dot{R} = K_{I\\rightarrow R} I \\hspace{5.0pt}.$ In the sequel, we shall consider (REF ) instead of (REF ), and we shall assume that the rates $K_{E\\rightarrow I}(x,t)=K_{E\\rightarrow I}(x)$ and $K_{I\\rightarrow R}(x,t)=K_{I\\rightarrow R}(x)$ are functions of $x$ , independent of time.",
"The rate $K_{I\\rightarrow E}$ will have the product form $K_{I\\rightarrow E}(x,t) =\\mu (t) \\psi (x) \\hspace{5.0pt}.$ The function $\\psi (\\cdot )$ is fixed, it is nonnegative and not a.e. zero.",
"In this way, the infectiosity of an individual depends on his “age” in the infectious phase, whereas the term $\\mu (t)$ represents the control of the epidemic by sanitary measures (social distancing, wearing masks, closing schools, lockdown, etc.).",
"We shall assume that the infectiosity rate $K_{I\\rightarrow E}(x,t)$ is the only parameter which can be controlled, hence, $\\mu (\\cdot )$ is a decision variable.",
"A variant of the ODE model (REF ), in which $K_{I\\rightarrow E}$ depends on time, but not on $x$ , is considered in [9].",
"Other versions, including a modification of the SEIR model leading to a time delay differential equation, are discussed in [27].",
"For epidemics in their early stages, i.e., when the number of individuals in the exposed, infectious, or removed compartments is negligible with respect to the number of susceptible individuals, the classical SEIR model is well-approximated by a linear system (see e.g.",
"[22], [3]) tracking only the populations in the (E) and (I) compartments.",
"As noted in sec-model, the fraction of the French population exposed prior to May 11 is estimated of 5.7% (see [34]), which justifies reliance on this linear approximation in our context.",
"The same approximation applies to the present PDE model.",
"This is translated to the assumption $S(t)/N(t)\\simeq 1$ , and we are reduced to the following system: $&n_E(0,t) = \\int _0^\\infty \\!\\!\\mu (t) \\psi (x) {n_I(x,t)}\\,\\mathrm {d}x \\hspace{5.0pt},\\qquad \\frac{\\partial n_E}{\\partial t} (x,t)+ \\frac{\\partial n_E}{\\partial x}(x,t) + K_{E\\rightarrow I}(x) n_E(x,t) = 0 \\,, \\quad \\text{for }x>0 \\,,\\\\& n_I(0,t) = \\int _0^\\infty \\!\\!K_{E\\rightarrow I}(x) n_E(x,t)\\,\\mathrm {d}x \\,,\\qquad \\frac{\\partial n_I}{\\partial t}(x,t) + \\frac{\\partial n_I}{\\partial x}(x,t) + K_{I\\rightarrow R}(x) n_I(x,t) = 0 \\hspace{5.0pt}, \\quad \\text{for } x>0 \\hspace{5.0pt}.$ This is a two-compartment generalization of the renewal equation, studied in Chapter 3 of [33].",
"In the sequel, we shall assume that there is a maximal “age” $x^*_E$ of an individual in the exposed state.",
"Similarly, we shall assume that there is a maximal “age” $x^*_I$ of an individual in the infectious state.",
"These assumptions, which are consistent with epidemiological observations [25], will be incorporated in our model by forcing all remaining exposed individuals of “age” $x_E^*$ to become infectious, with “age” 0.",
"Similarly, all the remaining infectious individuals are removed when reaching “age” $x_I^*$ .",
"So, the function $n_E$ is now only defined on the interval $[0,x_E^*]$ , and similarly, $n_I$ is only defined on $[0,x_I^*]$ .",
"This leads to the following system: $&\\!\\!\\!n_E(0,t) = \\int _0^{x_I^*}\\!\\!\\!",
"\\mu (t) \\psi (x) {n_I(x,t)}\\,\\mathrm {d}x \\,,\\quad \\frac{\\partial n_E}{\\partial t} (x,t)+ \\frac{\\partial n_E}{\\partial x}(x,t) + K_{E\\rightarrow I}(x) n_E(x,t) = 0 \\,,\\quad \\text{for }0<x<x_E^* \\,,\\\\&\\!\\!\\!",
"n_I(0,t) = \\int _0^{x_E^*} K_{E\\rightarrow I}(x) n_E(x,t)\\,\\mathrm {d}x + n_E(x^*_E,t) \\,,\\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\frac{\\partial n_I}{\\partial t}(x,t) + \\frac{\\partial n_I}{\\partial x}(x,t) + K_{I\\rightarrow R}(x) n_I(x,t) = 0 \\,, \\quad \\text{for }0<x<x_I^*\\,,\\\\&\\!\\!\\!\\frac{\\mathrm {d}R}{\\mathrm {d}t}(t) = \\int _0^{x_I^*} K_{I\\rightarrow R}(x) n_I(x,t)\\mathrm {d} x + n_I(x_I^*) \\hspace{5.0pt}.$ This system may be obtained as a specialization of (REF ), in which $K_{E\\rightarrow I}(x) $ is replaced by $K_{E\\rightarrow I}(x) \\mathbbold {1}_{[0,x_E^*]}(x)+ \\delta _{x_E^*}(x)$ , where $\\mathbbold {1}$ denotes the indicator function of a set, and $\\delta $ denotes Dirac's delta function.",
"Note that the above model is still relevant when $x^*_E=0$ .",
"Then, the partial differential equation in (REF ) disappears, and we are left with a PDE model modelling an infinite dimensional compartement of infectious individuals, without an explicit “exposed but not yet infectious” compartment.",
"This is similar to the original model of [22].",
"In contrast, the maximal time elapsed by an individual in the infectious state, $x^*_I$ , must be positive.",
"Otherwise, the integral term in (REF ), representing contaminations, vanishes.",
"We shall assume, in the sequel, that the following assumption holds.",
"Asssumption 1 The functions $K_{E\\rightarrow I}(\\cdot )$ , defined on $[0,x_E^*]$ , and $\\psi (\\cdot )$ and $K_{I\\rightarrow R}(\\cdot )$ , defined on $[0,x_I^*]$ , are nonnegative, measurable and bounded.",
"Moreover, the function $\\psi $ does not vanish a.e.",
"and the point $x_I^*>0$ is the maximum of the essential support of the function $\\psi $ .",
"Indeed, considering the boundary condition in (REF ), we see that a population of “age” $x> \\max \\operatorname{ess\\, supp}\\psi $ in the infected ($I$ ) compartment will not participate any more to the contamination chain.",
"Hence, the last part of Assumption REF is needed to interpret $R$ has the number of all the removed individuals.",
"Systems of PDE of this nature have been studied in particular by Michel, Mischler and Perthame, see [30], [33], and also, with an abstract semigroup perspective, in the work by Mischler and Scher [31].",
"Then, using the boundedness of the coefficients (Assumption REF ), and arguing as in the proof of Theorem 3.1 of [33] – which concerns the case of a single compartment – one can show that the system (REF ) admits a unique solution in the distribution sense $n:=(n_E,n_I)$ with $n_E \\in \\mathcal {C}(\\mathbb {R}_{\\ge 0}, L^1([0,x_E^*]))$ and $n_I\\in \\mathcal {C}(\\mathbb {R}_{\\ge 0}, L^1([0,x_I^*]))$ .",
"Hence, we can associate to the PDE (REF ) a well defined family of time evolution linear operators $(T_{s,t})_{t\\ge s\\ge 0}$ , acting on the space $L^1([0,x^*_E])\\times L^1([0,x^*_I])$ .",
"The operator $T_{s,t}$ maps an initial condition at time $s\\ge 0$ , that is a couple of functions $n(\\cdot ,s):=(n_E(\\cdot ,s),n_I(\\cdot ,s))$ , to the couple of functions $n(\\cdot ,t):=(n_E(\\cdot ,t),n_I(\\cdot ,t))$ at $t\\ge s$ .",
"These operators are order preserving, meaning that, if $n^1(\\cdot ,s)$ and $n^2(\\cdot ,s)$ are two initial conditions such that $n^1_E(x,s)\\le n^2_E(x,s)$ and $n^1_I(x,s)\\le n^2_I(x,s)$ for all $x\\ge 0$ , then the inequalities $n^1_E(x,t)\\le n^2_E(x,t)$ and $n^1_I(x,t)\\le n^2_I(x,t)$ hold for all $x\\ge 0$ and for all $t\\ge s$ .",
"An alternative modeling, more in the spirit of [22], would be to consider a single compartment, describing the evolution of the density $n(x,t)$ of individuals that were contaminated at time $t-x$ by the system: $&n(0,t) = \\int _0^{\\infty }\\!\\!",
"\\mu (t) \\psi (x) {n(x,t)}\\,\\mathrm {d}x \\,,\\quad \\frac{\\partial n}{\\partial t} (x,t)+ \\frac{\\partial n}{\\partial x}(x,t) + K(x) n(x,t) = 0 \\,,\\quad \\text{for }0<x<x^* \\,,$ where $x^*>x^*_E$ is fixed, and $\\psi (x)=0$ for $x<x^*_E$ .",
"Then, $E(t) = \\int _0^{x_E^*} n(x,t) \\mathrm {d}x $ yields the size of the exposed compartment.",
"However, we prefer the model (REF ) as it allows us to represent variable incubation times.",
"The system () can be extended to represent infectiosity rates that depends on the ages (time elapsed since birth) of individuals, with infectiosity rates given by a contact matrix, as in [29].",
"It suffices to split each compartment in sub-compartments, corresponding to different age groups.",
"This will be detailed in a further work."
],
[
"A Perron-Frobenius Eigenproblem for Transport PDE",
"When the control $\\mu (t)$ is constant and positive, the family of time evolution operators $(T_{s,t})_{t\\ge s\\ge 0}$ is determined by the semigroup $(S_{t}=T_{0,t})_{t\\ge 0}$ , and the long term evolution of the dynamical system (REF ) can be studied by means of the Perron–Frobenius eigenproblem $&\\bar{n}_E(0) = \\int _0^{x_I^*} \\mu \\psi (x) {\\bar{n}_I(x)}\\,\\mathrm {d}x \\hspace{5.0pt},\\quad \\frac{\\mathrm {d} \\bar{n}_E}{\\mathrm {d} x}(x) + (\\lambda + K_{E\\rightarrow I}(x)) \\bar{n}_E(x) = 0 \\, \\quad \\text{for }0<x<x_E^*\\, ,\\\\& \\bar{n}_I(0) = \\int _0^{x_E^*} K_{E\\rightarrow I}(x) \\bar{n}_E(x)\\,\\mathrm {d}x + n_E(x_E^*)\\hspace{5.0pt},\\quad \\frac{\\mathrm {d} \\bar{n}_I}{\\mathrm {d} x}(x) +( \\lambda + K_{I\\rightarrow R}(x) )\\bar{n}_I(x) = 0 \\,\\quad \\text{for }0<x<x_I^* \\,,$ where $\\bar{n}:=( \\bar{n}_I(\\cdot ),\\bar{n}_E(\\cdot ))$ is a nonnegative eigenvector, and $\\lambda $ is the eigenvalue.",
"We make a general observation from Perron-Frobenius theory.",
"Lemma 1 Let $w=(w_E,w_I)$ , with $w_E\\in L^1([0,x_E^*])$ and $w_I\\in L^1([0,x_I^*])$ , be such that $\\alpha \\bar{n} \\le w \\le \\beta \\bar{n}$ for some $\\alpha ,\\beta >0$ .",
"Then, $\\alpha \\exp (\\lambda t) \\bar{n} \\le S_t w \\le \\beta \\exp (\\lambda t) \\bar{n} ,\\qquad \\text{for all } t\\ge 0 \\hspace{5.0pt}.$ This follows from the order preserving and linear character of the semigroup $S_t$ , together with $S_t\\bar{n} =\\exp (\\lambda t)\\bar{n}$ .",
"This shows that the eigenvalue $\\lambda $ determines the growth rate of $n(t,x)$ as $t\\rightarrow \\infty $ , under the assumption that the initial population $w:=n(0,\\cdot )$ is comparable with the eigenvector $\\bar{n}$ , meaning that inequality (REF ) below holds for some $\\alpha ,\\beta >0$ .",
"Since the functions $K_{E\\rightarrow I}$ and $K_{I\\rightarrow R}$ are independent of time, the existence of a positive eigenvector is an elementary result: Proposition 1 Suppose $\\mu >0$ and that Assumption REF holds.",
"Then, the eigenproblem (REF ) has a solution $(\\bar{n},\\lambda )$ , where $\\bar{n}=(\\bar{n}_E,\\bar{n}_I)$ , the functions $\\bar{n}_E$ and $\\bar{n}_I$ are continuous and positive, and $\\lambda \\in \\mathbb {R}$ .",
"Moreover, the eigenvalue $\\lambda $ is unique, and the eigenvector $\\bar{n}$ satisfying the latter conditions is unique up to a multiplicative constant.",
"The proof of this proposition exploits a classical argument in renewal theory, see Lemma 3.1 p. 57 of [33].",
"We give the proof, leading to a semi-explicit representation of the eigenvector, which we shall need in sec-tropical.",
"We next provide a semi-explicit formula for the eigenvector.",
"We set $F^\\lambda _{E\\rightarrow I}(x):= \\int _0^x (\\lambda + K_{E\\rightarrow I}(z))\\mathrm {d}z\\,,\\text{ for }0\\le x\\le x_E^*\\, \\quad F^\\lambda _{I\\rightarrow R}(x):= \\int _0^x (\\lambda + K_{I\\rightarrow R}(z))\\mathrm {d}z\\,,\\text{ for }0\\le x\\le x_I^*\\, .$ Let $\\bar{n}=(\\bar{n}_E,\\bar{n}_I)$ be an eigenvector associated to the eigenvalue $\\lambda $ , so that it satisfies (REF ), and assume that $\\bar{n}_E(\\cdot )$ is continuous on $(0,x_E^*)$ , and $\\bar{n}_I(\\cdot )$ is continuous on $[0,x_I^*]$ .",
"Integrating the differential equations in (REF ), and using the continuity of $\\bar{n}$ and of the following expressions, we see that $\\bar{n}$ necessarily satisfies $\\bar{n}_E(x) &= \\exp \\big ( - F^\\lambda _{E\\rightarrow I}(x)\\big ) \\bar{n}_E(0)\\,,\\text{ for } 0\\le x \\le x_E^*\\,\\quad \\bar{n}_I(y)= \\exp \\big ( - F^\\lambda _{I\\rightarrow R}(y)\\big ) \\bar{n}_I(0)\\,,\\text{ for } 0 \\le y\\le x_I^*\\,.$ Conversely, if $\\bar{n}$ satisfies the above expressions, then it satisfies the differential equations in (REF ) and is continuous.",
"Using (REF ), together with the boundary conditions in (REF ), and the assumption that $ \\psi $ does not vanish a.e., and that $\\mu >0$ , we deduce that if $\\bar{n}_E(x)> 0$ for some $x\\in [0,x_E^*]$ or $\\bar{n}_I(y)> 0$ for some $y\\in [0,x_I^*]$ , then $\\bar{n}_E(x)> 0$ for all $x\\in [0,x_E^*]$ and $\\bar{n}_I(y)> 0$ for all $y\\in [0,x_I^*]$ .",
"Then, if a continuous nonnegative eigenvector $\\bar{n}$ exists, it is everywhere positive.",
"Moreover, (REF ) and the boundary condition in () entail that the eigenvector $\\bar{n}$ is unique, up to a scalar multiple.",
"Using also the boundary condition in (REF ), and specializing (REF ) to $x=x_E^*$ and $y=x_I^*$ , we deduce that $\\mu G^\\lambda \\bar{n}_E(0)= \\bar{n}_E(0)$ , where $ G^\\lambda = \\Big (\\int _0^{x_I^*}\\psi (x)\\exp \\big ( - F_{I\\rightarrow R}^\\lambda (x) \\big )\\mathrm {d}x\\Big )\\Big (\\int _0^{x_E^*} K_{E\\rightarrow I}(y)\\exp \\big ( - F^\\lambda _{E\\rightarrow I} (y)\\big )\\mathrm {d}y+\\exp \\big ( - F^\\lambda _{E\\rightarrow I}(x_E^*)\\big )\\Big )\\,.$ Therefore, for an eigenvector $\\bar{n}$ to exist, we must solve the equation $\\mu G^\\lambda =1$ (the so-called “characteristic equation” in renewal theory).",
"Since the functions $K_{E\\rightarrow I}$ , $K_{I\\rightarrow R}$ and $\\psi $ are nonnegative and integrable, and $\\psi $ is nonzero on a set of positive measure, we deduce that $\\lim _{\\lambda \\rightarrow -\\infty } G^{\\lambda }=+\\infty $ .",
"We also have $\\lim _{\\lambda \\rightarrow +\\infty }G^{\\lambda }=0$ .",
"Moreover, the map $\\lambda \\mapsto G^\\lambda $ is continuous.",
"Since $\\mu >0$ , by the intermediate value theorem, we can find $\\lambda $ such that $\\mu G^\\lambda =1$ , and this $\\lambda $ is the eigenvalue.",
"We showed that any nonnegative continuous eigenvector is positive.",
"Hence, any two nonnegative eigenvectors $\\bar{n}^1$ and $\\bar{n}^2$ with eigenvalues $\\lambda _1$ and $\\lambda _2$ satisfy $\\alpha \\bar{n}^1\\le \\bar{n}^2\\le \\beta \\bar{n}^1$ for some $\\alpha ,\\beta >0$ , and it follows from prop-unique that $\\alpha \\exp (\\lambda _1 t)\\bar{n}^1\\le \\exp (\\lambda _2 t) \\bar{n}^2\\le \\beta \\exp (\\lambda _2 t) \\bar{n}^1$ , which entails that $\\lambda _1=\\lambda _2$ , showing that the eigenvalue associated with a nonegative eigenvector is unique.",
"Alternatively, the uniqueness of this eigenvalue follows from the strictly decreasing character of the map $\\lambda \\mapsto G^\\lambda $ .",
"The asymptotic bound (REF ) can be reinforced, by showing that, for all positive initial conditions $w$ , $S_tw = C_1(w) \\bar{n} \\exp (\\lambda t)+ O(\\exp (\\lambda _2 t)) \\hspace{5.0pt}, \\qquad \\text{as } t\\rightarrow \\infty \\hspace{5.0pt},$ for some positive constant $C_1(w)$ , and $\\lambda _2<\\lambda $ .",
"This result, with an explicit control of $\\lambda _2$ , can be obtained as follows.",
"We make a diagonal scaling, using the positive eigenvector, and we normalize the semigroup to make the Perron eigenvalue $\\lambda $ equal to zero.",
"This leads to the semigroup $(\\tilde{S}_t w)(x) := \\exp (-\\lambda t)\\bar{n}^{-1}(x) [S_t ( w\\bar{n})](x)\\, .$ In potential theory, a version of this scaling is known as Doob's h-transform (see e.g. [14]).",
"The semigroup $\\tilde{S}_t$ obtained in this way is associated with a Markov process, and, so, the spectral gap of this semigroup can be bounded in terms of Doeblin's ergodicity coefficient [16], [5], leading to (REF ).",
"These aspects will be detailed elsewhere.",
"Alternatively, the relative entropy inequality technique of [30] allows one to establish the convergence of $n(\\cdot ,t)$ to the eigenvector, modulo multiplicative constants, as $t$ tends to infinity."
],
[
"Universality of the log-rate of epidemic observables",
"Epidemic observables are obtained by applying a continuous linear form to the state variable.",
"Supposing that $n_I(\\cdot ,t)$ is a continuous function, an epidemic observable will be of the form $Y_\\kappa (t) =\\varphi (n(\\cdot ,t)):=\\int _0^{x_I^*} n_I(x,t)\\,\\mathrm {d}\\kappa (x)\\hspace{5.0pt},$ where $\\mathrm {d}\\kappa (x)$ is a nonnegative nonzero Borel measure.",
"Epidemic events anterior to the infectious phase, like contamination, are by nature hard to detect, so the observable depends only on $n_I$ .",
"Proposition 2 Suppose that Assumption REF holds, let $(\\lambda ,\\bar{n})$ denote the solution of the Perron-Frobenius eigenproblem (REF ), and suppose that for some $T>0$ , there exist positive constants $\\alpha ,\\beta $ such that $\\alpha \\bar{n}\\le n(\\cdot ,T)\\le \\beta \\bar{n}$ .",
"Then, for all epidemic observables of the form (REF ), the map $t\\mapsto \\log Y_\\kappa (t)- \\lambda t$ is bounded.",
"A fortiori, $\\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\log Y_\\kappa (t) = \\lambda \\hspace{5.0pt}.$ Taking $w=n(\\cdot ,T)$ in prop-unique, we deduce that $\\alpha \\exp (\\lambda t) \\bar{n}\\le S_{T+t} n(\\cdot ,0)=n(\\cdot ,T+t)\\le \\beta \\exp (\\lambda t) \\bar{n}\\,, \\text{ for }t>0 \\hspace{5.0pt}.$ It follows that $\\log \\alpha + \\lambda t + \\log \\varphi (\\bar{n})\\le \\log Y_\\kappa (T+t)\\le \\log \\beta + \\lambda t + \\log \\varphi (\\bar{n})$ .",
"A simple example of observable, discussed in sec-model, consists of pure delays.",
"For instance, we assumed that the number of dispatches of MICU is given by $Y_{\\mathrm {MICU}}(t) = \\pi _{\\mathrm {MICU}} C(t-\\tau _{\\mathrm {MICU}})$ where $C(t)$ is the number of contaminations at time $t$ , $\\pi _{\\mathrm {MICU}}$ the proportion of contaminated patients who will need a MICU transport, and $\\tau _{\\mathrm {MICU}}$ a fixed delay.",
"This can be obtained as a special case of (REF ), taking $K_{E\\rightarrow I}\\equiv 0$ , so that the transition tom $E$ to $I$ occurs always at time $x_E^*$ , and $\\mathrm {d}\\kappa :=\\pi _{\\mathrm {MICU}}\\delta _{\\tau _{\\mathrm {MICU}}-x_E^*}$ , where $\\delta $ is the Dirac $\\delta $ function.",
"Other events can be considered: medical advice, EMT dispatch, admission to ICU, or decease.",
"These events corresponds to different values of the proportion $\\pi $ and of the delay $\\tau $ .",
"By prop-universal, the rate $\\lim _{t\\rightarrow \\infty } t^{-1}\\log Y(t)$ will be the same for all the corresponding observables, although the convergence of the function $t^{-1}\\log Y(t)$ to its limit will be observed in a delayed manner, for observables corresponding to the latest stages of the pathology."
],
[
"Discrete versions of the epidemiological model",
"The reader interested in ODE model of epidemics might wish to note that the previous analysis applies to such finite dimensional models.",
"Instead of the transport PDE (REF ), we may consider an ODE of the form $\\dot{v} = Mv $ where $v(t)\\in \\mathbb {R}^n$ and $M$ is a $n\\times n$ matrix with non-negative off-diagonal terms, a so-called Metzler matrix.",
"In the original SEIR model [3], the matrix $M$ , obtained by considering the $(E,I)$ -block equations (), (), with $S/N\\simeq 1$ , is of dimension 2.",
"In the generalizations of the SEIR model considered in [29], [13], the dimension $n$ is increased to account for other compartments.",
"One can also discretize the PDE system (REF ) using a monotone (upwind) finite difference scheme, and this leads to a system of the form (REF ).",
"In all these finite dimensional models, the matrix $M$ is Metzler and irreducible.",
"Then, the Perron–Frobenius theorem for linear, order-preserving semigroups (see [7]) implies that $M$ admits a unique eigenvalue $\\lambda $ of maximal real part.",
"Furthermore $\\lambda $ is algebraically simple and real, and its associated eigenvector $u$ has strictly positive coordinates.",
"Then, it follows from the spectral theorem that $ v(t)= \\exp (\\lambda t)u + o(\\exp (\\lambda _2 t))$ as $t\\rightarrow \\infty $ , where $\\lambda _2$ is the maximal real part of an eigenvalue of $M$ distinct from $\\lambda $ .",
"Again, in this discrete model, an epidemic observable $Y(t)$ is obtained by applying a nonnegative linear form to the vector $v(t)$ , i.e, $Y(t)=\\varphi ^\\top v(t)$ , for some nonnegative column vector $\\varphi $ ."
],
[
"Hilbert's geometry applied to piecewise linear approximation",
"We introduce an abstract setting, which captures epidemiological models in which most individuals are susceptible.",
"This setting applies, in particular, to the transport PDE model of (REF ), when the transition functions are supported by compact intervals, and to the general finite dimensional Metzler model (REF ).",
"We consider $(V,\\le )$ , a partially ordered Banach space, with topological dual $V^{\\prime }$ .",
"We denote by $V_{\\ge 0}:= \\lbrace v\\in V\\mid v\\ge 0\\rbrace $ the set of nonnegative elements of $V$ , which is a convex cone.",
"This cone must be pointed (i.e., $V_{\\ge 0} \\cap (-V_{\\ge 0})=\\lbrace 0\\rbrace $ ), since the relation $\\le $ is a partial order.",
"We require this cone to be closed.",
"We consider a sequence of $m$ semigroups $S^i=(S^i_t)_{t\\ge 0}$ , for $i\\in [m]$ , where $[m]:=\\lbrace 1,\\dots ,m\\rbrace $ .",
"We assume that for all $i\\in [m]$ , and for all $t\\ge 0$ , $S^i_t$ is a bounded linear operator from $V$ to itself, and that the semigroup property holds, i.e., $S^i_{t+s}= S^i_t\\circ S^i_s$ .",
"We shall say that the semigroup $S^i$ is order preserving if, for all $v\\in V_{\\ge 0}$ , and for all $t\\ge 0$ , $S^i_t v\\in V_{\\ge 0}$ .",
"We shall consider commutation instants, $t_0:=0<t_1<\\dots < t_{m-1}$ , These instants will correspond to significant epidemiological dates, for instance, dates at which sanitary measures are taken.",
"We set $t_m:=+\\infty $ .",
"We select an initial condition $v_0 \\in V_{\\ge 0}$ , and consider the abstract dynamical system obtained by switching between the evolutions determined by the semigroups $S^1,\\dots ,S^m$ , at the successive times $t_1,\\dots , t_{m-1}$ .",
"The state of this dynamical system, at time $t\\in [t_j, t_{j+1})$ , is given by $v_t := S^{j+1}_{t-t_j} \\circ S^{j}_{t_j -t_{j-1}}\\circ \\dots \\circ S^1_{t_1-t_0}(v_0)\\hspace{5.0pt}.$ Recall that a part of the closed convex cone $V_{\\ge 0}$ is an equivalence class for the relation $\\sim $ such that, for $v,w\\in V_{\\ge 0}$ , we have $v\\sim w$ if and only if there exists two positive constants $\\alpha $ and $\\beta $ such that $\\alpha v \\le w \\le \\beta v$ .",
"A part is trivial if it is reduced to the equivalence class of the zero vector.",
"Hilbert's projective metric $d_H$ is defined on every non-trivial part of $V_{\\ge 0}$ by the following formula $d_H(v,w) = \\log \\inf \\bigg \\lbrace \\frac{\\beta }{\\alpha } : \\alpha ,\\beta >0, \\; \\alpha v \\le w \\le \\beta v\\bigg \\rbrace \\hspace{5.0pt}.$ The infimum is achieved, since $V_{\\ge 0}$ is closed.",
"The map $d_H$ is nonnegative, it satisfies the triangular inequality, and $d_H(v,w)$ vanishes if, and only if, $v$ and $w$ are proportional – this justifies the term “projective metric”.",
"This metric plays a fundamental role in Perron–Frobenius theory and in metric geometry, and also in tropical geometry, see [26], [32], [10] for background.",
"When $V_{\\ge 0}=(\\mathbb {R}_{\\ge 0})^n$ is the standard orthant, and when all the entries of the vectors $v$ and $w$ are positive, we have $d_H(v,w) = \\max _{k\\in [n]} (\\log v_k-\\log w_k )-\\min _{k\\in [n]}(\\log v_k-\\log w_k )\\hspace{5.0pt}.$ Denoting by $e$ the unit vector of $\\mathbb {R}^n$ , we observe that $d_H(v,w)= \\Vert \\log v -\\log w\\Vert _H$ where the notation $\\log v$ is understood entrywise, and $ \\Vert z\\Vert _H= 2 \\min _{c\\in \\mathbb {R}} \\Vert x-c e\\Vert _\\infty \\hspace{5.0pt}.$ In other words, up to a logarithmic change of variables, $d_H$ arises by modding out the normed space $(\\mathbb {R}^n,\\Vert \\cdot \\Vert _\\infty )$ by the one-dimensional space $\\mathbb {R}e$ .",
"We shall suppose that every semigroup $S^i$ has an eigenvector $u^i\\ge 0$ , with eigenvalue $\\lambda ^i$ , meaning that $S^i_t u^i = \\exp (\\lambda ^i t) u^i,\\quad \\forall t\\ge 0 \\hspace{5.0pt}.$ Since $S^i_t $ preserves $V_{\\ge 0}$ , this entails that $\\lambda ^i$ is real.",
"We choose a linear form $\\varphi \\in V^{\\prime }$ which we require to take nonnegative values on $V_{\\ge 0}$ .",
"We shall think of $V$ as the state space and $\\varphi $ as an observable.",
"We consider the following scalar observation of the dynamics $ Y_t: = \\varphi (v_t) \\hspace{5.0pt}.$ We shall assume, in addition, that $\\varphi $ does not vanish on $v_t$ , for all $t\\ge 0$ .",
"Then, we can define the image of the observation by the logarithmic map $y_t:= \\log Y_t,\\qquad \\forall t\\ge 0 \\hspace{5.0pt}.$ The following result shows that the logarithm of the observation stays at finite distance from a piecewise linear map.",
"Theorem 1 Suppose that the semigroups $S^1,\\dots ,S^m$ are order preserving.",
"Suppose in addition that the initial condition $v_0$ and the eigenvectors $u^1,\\dots , u^m$ all lie in the same non-trivial part of $V_{\\ge 0}$ , and that the linear form $\\varphi $ takes positive values on this part.",
"Then, there exists a constant $\\gamma $ such that the piecewise linear map $t\\mapsto y^\\mathrm {trop}_t$ defined, for $t\\in [t_j, t_{j+1})$ , by $y^\\mathrm {trop}_t := \\lambda _{j+1}(t-t_j)+ \\lambda _{j} (t_j-t_{j-1})+ \\dots + \\lambda _1 (t_1-t_0) + \\gamma \\hspace{5.0pt},$ satisfies $|y_t - y^{\\mathrm {trop}}_t |\\le \\frac{\\Delta }{2}, \\qquad \\forall t\\ge 0 \\hspace{5.0pt},$ where $\\Delta = d_H(v_0,u^1)+ d_H(u^1,u^2)+ \\dots +d_H(u^{m-1},u^m)\\hspace{5.0pt}.$ By definition of Hilbert's projective metric, we can find positive constants $\\alpha _0,\\beta _0$ , such that $\\alpha _0 u^1\\le v_0 \\le \\beta _0 u^1$ and $d_H(v_0,u^1)=\\log (\\beta _0/\\alpha _0)$ .",
"Similarly, for all $i\\in [m-1]$ , we can find positive constants $\\alpha _i,\\beta _i$ , such that $\\alpha _{i} u^{i+1}\\le u^{i} \\le \\beta _{i} u^{i+1}$ , and $d_H(u^{i},u^{i+1})=\\log (\\beta _i/\\alpha _i)$ .",
"For all $j\\ge 0$ , with $j\\le m-1$ , and for all $t\\in [t_j,t_{j+1})$ , we set $z_t:= \\lambda _{j+1}(t-t_j) + \\lambda _{j}(t_j-t_{j-1})+\\dots + \\lambda _1(t_1-t_0) \\hspace{5.0pt}.$ Since the semigroups $S^i$ are linear and order preserving, we prove by induction $\\exp (z_t )\\alpha _j\\dots \\alpha _0 u^{j+1}\\le v_t \\le \\exp (z_t)\\beta _j\\dots \\beta _0 u^{j+1} \\hspace{5.0pt}.$ We observe that $\\alpha _{m-1}\\dots \\alpha _{j+1} u^m\\le u^{j+1} \\le \\beta _{m-1}\\dots \\beta _{j+1} u^m$ and so $\\exp (z_t )\\alpha _{m-1}\\dots \\alpha _0 u^{m}\\le v_t \\le \\exp (z_t)\\beta _{m-1}\\dots \\beta _0 u^{m} \\hspace{5.0pt}.$ Applying the linear form $\\varphi $ to latter inequalities, taking the image by the log map, and setting $\\gamma := \\log \\varphi (u^m) + \\frac{1}{2}\\sum _{j=0}^{m-1}\\log (\\beta _j \\alpha _j) \\hspace{5.0pt},$ we arrive at the bound of the theorem.",
"A general principle from tropical geometry states that using “logarithmic glasses” reveals a piecewise linear structure [38], [20].",
"th-1 is inspired by this principle.",
"This motivates the notation $y^\\mathrm {trop}$ , for the “tropicalization” of the logarithm of the observable $Y$ .",
"Remark 1 If the space $V$ is of dimension 1, then the bound $\\Delta $ appearing in th-1 is zero, implying that the approximation of the logarithm of observables by a piecewise linear curve is exact.",
"This occurs if one considers a SIR ODE model: then, in the early stage of the epidemics, the dynamics can be written only in terms of the population of the one-dimensional I compartment.",
"Remark 2 th-1 carries over to discrete time systems in a straightforward manner."
],
[
"Application to the transport PDE model",
"th-1 applies in particular to the transport model (REF ).",
"Then, as noted above, the evolution operator of the system (REF ) preserves the space $V=L^1([0,x^*_E])\\times L^1([0,x^*_I])$ .",
"Moreover, when the epidemiological control term $\\mu (t)$ is constant, prop-eig shows that the eigenproblem (REF ) has a positive and continuous solution $\\bar{n}$ , with a real eigenvalue $\\lambda $ .",
"Different stages of sanitary policies correspond to successive values $\\mu ^1,\\ldots ,\\mu ^m$ of $\\mu (t)$ , leading to different semigroups $S^i$ , $i\\in [m]$ .",
"Then, the solution $v_t:= n(\\cdot ,t)$ of (REF ) is determined as in (REF ).",
"Each semigroup $S^i$ yields a continuous and positive eigenvector $u^i:= \\bar{n}^i$ satisfying (REF ) associated with a real eigenvalue $\\lambda ^i$ of $S^i$ .",
"Two continuous and positive functions defined on a compact interval are always in the same part of the cone of nonnegative functions of $V$ , so th-1 applies to this model.",
"We next give an explicit estimate for the Hilbert projective distances between eigenvectors, arising in th-1.",
"Proposition 3 Suppose that Assumption REF holds, and that for $i=1,2$ , $(\\lambda ^i,\\bar{n}^i)$ is the solution $(\\lambda ,\\bar{n})$ of the Perron-Frobenius eigenproblem (REF ) when $\\mu =\\mu ^i$ .",
"Then, we have $ d_H(\\bar{n}^1, \\bar{n}^{2})\\le |\\lambda _1-\\lambda _{2}| (x^*_E+x^*_I)\\hspace{5.0pt}.$ The term $x^*_E+x^*_I$ is the maximal time elapsed between contamination and the end of infectiosity.",
"[Proof of prop-bound] Suppose, without loss of generality, that $\\bar{n}^i_E(0)=1$ for $i=1,2$ .",
"Let $F^\\lambda _{E\\rightarrow I}$ and $F^\\lambda _{I\\rightarrow R}$ be defined as in (REF ).",
"We have $F^\\lambda _{E\\rightarrow I}(x)=\\lambda x+ F^0_{E\\rightarrow I}(x)$ and $F^\\lambda _{I\\rightarrow R}(x)= \\lambda x+ F^0_{I\\rightarrow R}(x)$ .",
"Then, (REF ) and the boundary condition in () yield $\\bar{n}_E^i(x)&= \\exp (-\\lambda ^i x)\\exp \\big ( - F^0_{E\\rightarrow I}(x)\\big ) \\,,\\text{ for } 0\\le x \\le x_E^*\\,,\\\\\\bar{n}^i_I(0)& = \\int _0^{x_E^*} K_{E\\rightarrow I}(x)n^i_E(x)\\mathrm {d}x + n_E^i(x_E^*)\\,,\\\\\\bar{n}_I^i(y)&= \\exp (-\\lambda ^i y) \\exp \\big ( - F^0_{I\\rightarrow R}(y)\\big ) \\bar{n}_I^i(0)\\,,\\text{ for } 0\\le y\\le x_I^*\\,.$ Let $j\\in \\lbrace 1,2\\rbrace $ be distinct from $i$ , and set $t^+:=\\max (t,0)$ .",
"Bounding $\\exp (-\\lambda ^i x)$ by $\\exp (-\\lambda ^j x) \\exp \\big ( (\\lambda ^j-\\lambda ^i)^+ x_E^*\\big )$ in (REF ), we obtain: $ \\bar{n}_E^i(x)\\le \\exp \\big ( (\\lambda ^j-\\lambda ^i)^+ x_E^*\\big ) \\bar{n}_E^j(x)\\,, \\text{ for } 0\\le x \\le x_E^*\\hspace{5.0pt}.$ Applying this inequality in (), we deduce $ \\bar{n}_I^i(0)\\le \\exp \\big ( (\\lambda ^j-\\lambda ^i)^+ x_E^*\\big ) \\bar{n}_I^j (0)\\hspace{5.0pt}.$ Now applying this inequality and bounding $\\exp (-\\lambda ^i y)$ by $\\exp (-\\lambda ^j y) \\exp \\big ( (\\lambda ^j-\\lambda ^i)^+ x_I^*\\big )$ in (), we obtain: $\\bar{n}_I^i(y)\\le \\exp \\big ( (\\lambda ^j-\\lambda ^i)^+ (x_E^*+x_I^*)\\big ) \\bar{n}_I^j (y)\\,, \\text{ for } 0\\le y\\le x_I^*\\hspace{5.0pt}.$ So, $d_H( (\\bar{n}_E^i,\\bar{n}_I^i), (\\bar{n}_E^j,\\bar{n}_I^j))\\le (\\lambda ^j-\\lambda ^i)^+ (x_I^*+x_E^*)+(\\lambda ^i-\\lambda ^j)^+ (x_I^*+x_E^*)= |\\lambda ^i-\\lambda ^j| (x_I^*+x_E^*) \\, .$ Note that applying (REF ) to the boundary condition in (REF ), we also deduce from the previous proof that $\\bar{n}_E^i (0) \\le \\frac{\\mu ^i}{\\mu ^j}\\exp \\big ( (\\lambda ^j-\\lambda ^i)^+(x_I^*+x_E^*)\\big ) \\bar{n}_E^{j}(0)\\,.$ Since $\\bar{n}_E^i (0)=1$ for $i=1,2$ , we get the following bound $\\mu ^j/\\mu ^i \\le \\exp \\big ((\\lambda ^j-\\lambda ^i)^+\\big )(x_I^*+x_E^*)\\,,$ from which one recover that $\\mu $ is a nondecreasing function of $\\lambda $ , and which gives an estimation of the rate $\\mu $ in terms of the eigenvalue $\\lambda $ .",
"The bound of th-1 may be refined.",
"This is left for further work."
],
[
"Short term predictions",
"We now describe the basic methodology we propose to build confidence intervals for future occurrences of medical events related to epidemic progression, and raise alarms about its potential resurgence.",
"We first consider a single time series of numbers of event occurrences.",
"We then describe how to consolidate several time series corresponding to distinct medical events in order to construct improved alarm criteria.",
"The simpler case of least squares fitting is considered first, the more robust $\\ell _1$ alternative is described next."
],
[
"Time series for a single type of events:",
"Let $X(1),\\ldots ,X(n)$ be indices of days, and we aim to do a forecast based on observations made on these days.",
"Typically, on day $d_0$ , we may select $n=7$ and let $X(1)=d_0-n,\\ldots ,X(n)=d_0-1$ to perform a forecast on the basis of the last seven days.",
"Let $Y(t)$ denote the count of medical events (for instance, dispatches of ambulances) on day $X(t)$ , and let $Z(t)=\\log Y(t)$ .",
"Based on the previous discussion (epidemiological modeling) we assume that for all $t=1,\\ldots ,n$ , $Z(t)=\\alpha + \\beta X(t) +\\epsilon _t$ for constants $\\alpha $ , $\\beta $ , where $\\epsilon _t$ denotes some random noise.",
"For simplicity we assume here i.i.d.",
"noise sequence $\\epsilon _1,\\ldots ,\\epsilon _n$ , and that each $\\epsilon _t$ admits a Gaussian distribution $\\mathcal {N}(0,\\sigma ^2)$ with zero mean and variance $\\sigma ^2$ .",
"Least-square estimates for the parameters $\\alpha $ , $\\beta $ are then provided by $\\hat{\\beta }=\\frac{\\sum _{t=1}^n (X(t)-\\bar{X})(Z(t)-\\bar{Z})}{\\sum _{t=1}^n(X(t)-\\bar{X})^2},\\quad \\hat{\\alpha }=\\bar{Z}-\\hat{\\beta }\\bar{X},$ where $\\bar{X}=\\frac{1}{n}\\sum _{t=1}^nX(t),\\quad \\bar{Z}=\\frac{1}{n}\\sum _{t=1}^nZ(t).$ The variance $\\sigma ^2$ can be estimated as $\\hat{\\sigma }^2=\\frac{1}{n-2}\\sum _{t=1}^n(Z(t)-\\hat{Z}(t))^2,$ where $\\hat{Z}(t):=\\hat{\\alpha }+\\hat{\\beta }X(t).$ Under the assumptions of i.i.d.",
"Gaussian errors $\\epsilon _t$ , we have that, for each $t$ corresponding to a future day $X(t)$ (in particular, $t\\notin \\lbrace 1,\\ldots ,n\\rbrace $ ), the three following variables: $\\displaystyle \\frac{\\hat{\\alpha }-\\alpha }{\\hat{\\sigma }\\sqrt{\\frac{1}{n}+\\frac{\\bar{X}^2}{\\sum _{t^{\\prime }=1}^n (X(t^{\\prime })-\\bar{X})^2}}},\\;\\frac{\\hat{\\beta }-\\beta }{\\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}},\\;\\;,\\frac{Z(t)-\\hat{Z}(t)}{\\hat{\\sigma }\\sqrt{1+\\frac{1}{n}+\\frac{(X(t)-\\bar{X})^2}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}},$ all admit a bilateral Student distribution with $n-2$ degrees of freedom (see [21] or [11]).",
"Denote by $t^{n-2}_{\\gamma }$ the $\\gamma $ -th quantile of this distribution.",
"For $\\epsilon \\in [0,1]$ , this provides us with the following confidence intervals with confidence $1-\\epsilon $ : $\\begin{array}{lll}\\alpha &\\in & \\left[\\hat{\\alpha }-\\hat{\\sigma }\\sqrt{\\frac{1}{n}+\\frac{\\bar{X}^2}{\\sum _{t^{\\prime }=1}^n (X(t^{\\prime })-\\bar{X})^2}} t^{n-2}_{1-\\epsilon /2},\\hat{\\alpha }+\\hat{\\sigma }\\sqrt{\\frac{1}{n}+\\frac{\\bar{X}^2}{\\sum _{t^{\\prime }=1}^n (X(t^{\\prime })-\\bar{X})^2}} t^{n-2}_{1-\\epsilon /2}\\right],\\\\\\beta &\\in & \\left[\\hat{\\beta }-\\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}t^{n-2}_{1-\\epsilon /2},\\hat{\\beta }+\\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}t^{n-2}_{1-\\epsilon /2} \\right]\\\\Z(t)&\\in & \\left[\\hat{Z}(t)-\\hat{\\sigma }\\sqrt{1+\\frac{1}{n}+\\frac{(X(t)-\\bar{X})^2}{\\sum _{t^{\\prime }=1}^n(X_(t^{\\prime })-\\bar{X})^2}}t^{n-2}_{1-\\epsilon /2}, \\hat{Z}(t)+\\hat{\\sigma }\\sqrt{1+\\frac{1}{n}+\\frac{(X(t)-\\bar{X})^2}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}t^{n-2}_{1-\\epsilon /2} \\right]\\end{array}$ As an illustration, for $n=7$ and $\\epsilon =5\\%$ , we can plug in $t^{5}_{0.975}=2.571$ in the last interval, and thus obtain a $95\\%$ -confidence interval centered around $\\hat{Z}(t)$ for $Z(t)=\\log Y(t)$ , the logarithm of the count $Y(t)$ on a future day $X(t)$ , that is: $Z(t)\\in \\left[\\hat{Y}(t)-2.571\\times \\hat{\\sigma }\\sqrt{1+\\frac{1}{n}+\\frac{(X(t)-\\bar{X})^2}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}, \\hat{Z}(t)+2.571\\times \\hat{\\sigma }\\sqrt{1+\\frac{1}{n}+\\frac{(X(t)-\\bar{X})^2}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}\\right]$ Although we could extend this definition of the confidence interval for the short terms predictions of the value of $Z(t)$ , we propose a more conservative confidence domain, in the shape of a trapezoid.",
"It is obtained by extending the upper-bound $Z(t)^+$ (resp.",
"the lower bound $Z(t)^-$ of the $95\\%$ confidence interval on $Z(t)$ by a line with slope equal to the upper-bound $\\beta ^+$ (resp.",
"lower-bound $\\beta ^-$ ) of $95\\%$ confidence interval on $\\beta $ .",
"For a given day $t$ , the upper and lower envelopes of the trapezoid have ordinates $ \\bigg (\\hat{\\beta }(t-t_n)+\\hat{Z}_n\\bigg )\\pm \\displaystyle \\left(\\sqrt{\\mathrm {Var}(\\hat{\\beta })}(t-t_n)+\\sqrt{\\widehat{\\sigma }^2+\\mathrm {Var}(\\hat{Z}_n)}\\right) t^{n-2}_{1-\\epsilon /2}\\hspace{5.0pt}.$ If instead of the count $Y(t)$ on a particular day $X(t)$ , we are interested in the trend of the epidemic, whether exploding or contracting, we should then consider the confidence interval for parameter $\\beta $ .",
"Again for $n=7$ and $\\epsilon =5\\%$ this gives $\\beta \\in \\left[\\hat{\\beta }-2.571\\times \\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}},\\hat{\\beta }+2.571\\times \\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}} \\right]\\\\$ One-sided confidence intervals may also be provided, and are in fact more natural for the definition of alarm indicators.",
"For concreteness, assume we want to raise an alarm when the doubling time, $\\delta =(\\log 2)/\\beta $ , is $\\delta ^*$ days or less, where $\\delta ^*$ could be 10 for instance.",
"This is equivalent to $\\beta $ exceeding $(\\log 2)/\\delta ^*$ .",
"Thus $\\delta $ is less than $\\delta ^*$ days with confidence $1-\\epsilon $ when $\\frac{\\log 2}{\\delta ^*} < \\hat{\\beta }- t^{n-2}_{1-\\epsilon } \\sqrt{V},$ where $V=\\hat{\\sigma }\\sqrt{\\frac{1}{\\sum _{t^{\\prime }=1}^n(X(t^{\\prime })-\\bar{X})^2}}.$ Raising an alarm under this condition then amounts to calibrating the false positive probability at $\\epsilon $ .",
"For instance, for $\\epsilon = 5\\%$ , and $n=7$ , we would plug in $t^7_{0.95}=2.015$ in the above expression.",
"Alternatively, raising an alarm under the condition $\\frac{\\log 2}{\\delta ^*} < \\hat{\\beta }+t^{n-2}_{1-\\epsilon } \\sqrt{V},$ corresponds to calibrating the false negative probability (probability of not raising an alarm while $\\delta \\le 10$ ) at $\\epsilon $ .",
"Our alarm indicators correspond to the first choice, i.e.",
"calibration of a false positive rate, with $\\delta ^*$ set to $+\\infty $ ."
],
[
"Alarm indicators based on multiple types of events:",
"Assume that several types $j$ of events are available, and let $J$ denote the corresponding set of events.",
"For instance, we could distinguish between dispatches of ambulances bringing patients to Intensive Care Units as opposed to Non-intensive Care Units, thereby producing two distinct time series.",
"Let $X_j(t)$ , $t=1,\\ldots ,n_j$ denote the days on which counts $Y_j(t)$ of type $j$ event occurrences are to be used.",
"Let $Z_j(t)=\\log Y_j(t)$ .",
"We assume as before the linear regression model $Z_j(t)= \\alpha _j +\\beta _j X_j(t)+\\epsilon _j(t),\\; t=1,\\ldots ,n^j \\hspace{5.0pt}.$ Now for each of these times series, we can produce, based on the previous discussion, the estimator $\\hat{\\beta }_j:=\\frac{\\sum _{t=1}^{n_j} (X_j(t)-\\bar{X}_j)(Z_j(t)-\\bar{Z}_j)}{\\sum _{t=1}^{n_j}(X_j(t)-\\bar{X}_j)^2},$ where $\\bar{X}_j=\\frac{1}{n_j}\\sum _{t=1}^{n_j}X_j(t),\\quad \\bar{Z}_j=\\frac{1}{n_j}\\sum _{t=1}^{n_j}Z_j(t).$ Suppose in addition that the noise terms $\\epsilon _j(t)$ are mutually independent, Gaussian, with zero mean and variance $\\sigma _j^2$ for errors $\\epsilon _j(t)$ .",
"Suppose finally that the exponents $\\beta _j$ all coincide with $\\beta $ , the exponent that is characteristic of the epidemic's progression.",
"Denote by $V_j:=\\hat{\\sigma }_j^2\\sqrt{\\frac{1}{\\sum _{t=1}^{n_j}(X_j(t)-\\bar{X}_j)^2}},$ where, reproducing the computations for a single time series, we let $\\hat{\\sigma }_j^2:=\\frac{1}{n_j-2}\\sum _{t=1}^{n_j}(Z_j(t)-\\hat{Z}_j(t))^2,$ and $\\hat{Z}_j(t):=\\hat{\\alpha }_j+\\hat{\\beta }_jX_j(t).$ As previously, $V_j$ is our estimate of the variance of estimate $\\hat{\\beta }_j$ .",
"We finally propose to combine the individual estimators $\\hat{\\beta }_j$ into $\\hat{\\beta }:=\\frac{\\sum _{j\\in J} \\frac{1}{V_j}\\hat{\\beta }_j}{\\sum _{j\\in J}\\frac{1}{V_j}}\\cdot $ For the sake of simplicity, let us approximate the bilateral Student distribution with $n-2$ degrees of freedom by the standard distribution $\\mathcal {N}(0,1)$ .",
"We then have the approximate distributions $\\hat{\\beta }_j\\approx \\mathcal {N}(\\beta , V_j)$ , and hence the approximate distribution $\\hat{\\beta }-\\beta \\approx \\mathcal {N}(0,V)$ , $V:=\\frac{1}{\\sum _{j\\in J} \\frac{1}{V_j}}\\cdot $ Weighing the individual estimators $\\hat{\\beta }_j$ by the reciprocal of their variances as just done minimizes the variance of the resulting estimator.",
"The same approach as previously considered then leads to the following conditions for alarm raising: To raise an alarm when the doubling time $\\delta =(\\log 2)/\\beta $ exceeds $\\delta ^*$ days (e.g., $\\delta ^*=10$ ), if we target a false alarm probability of $\\epsilon $ , we are led to raise an alarm when Condition $\\frac{\\log 2}{\\delta ^*}< \\hat{\\beta } -g_{1-\\epsilon }\\sqrt{V},$ where $g_{1-\\epsilon }$ is the $1-\\epsilon $ -quantile of the standard Gaussian distribution.",
"If instead we target a false negative probability (probability of not raising an alarm) at $\\epsilon $ , we would then raise an alarm when $\\frac{\\log 2}{\\delta ^*}< \\hat{\\beta }+g_{1-\\epsilon } \\sqrt{V},$"
],
[
"More robust $\\ell _1$ -based approach:",
"The previous estimators and derived alarm conditions have the appeal of simplicity, but can be advantageously replaced by more robust versions, that are less sensitive to the presence of outliers.",
"A popular alternative is the following $\\ell _1$ criterion.",
"We again consider $Z_j(t):=\\log Y_j(t)$ , where $Y_j(t)$ is the number of type $j$ events on day $X_j(t)$ .",
"We then let $\\hat{\\alpha }_j$ , $\\hat{\\beta }_j$ achieve the minimum of the criterion $\\sum _{t=1}^{n^j} |\\alpha +\\beta X^j_t -Y^j_t|$ .",
"They are obtained by solving a linear program.",
"Here we assume that observations $Z_j(t)$ are mutually independent and distributed according to density $f_{j,t}(z)=\\frac{1}{2\\lambda ^j}\\exp (-|z-\\alpha _j-\\beta _j X_j(t)|/\\lambda _j)$ .",
"In other words this corresponds to adding a Laplace observation noise with density $\\frac{1}{2\\lambda ^j}\\exp (-|z|/\\lambda _j)$ to the signal of interest $\\alpha _j+X_j(t)\\beta _j$ .",
"The above $\\ell _1$ minimization criterion corresponds to maximum likelihood estimation of $\\alpha _j$ , $\\beta _j$ in this observational noise model, as its log-likelihood is given by $-|T|\\log (2 \\lambda _j) -\\sum _{t=1}^{n_j}\\frac{|Z_j(t)-\\alpha _j-\\beta _j X_j(t)|}{\\lambda _j}.$ A rich theory for the performance of the resulting estimators is available, see for instance [23].",
"The latter work treats general i.i.d.",
"errors, and do not restrict itself to e.g.",
"Laplacian distribution of errors; recent work like [35] experiments techniques to obtain confidence intervals when distribution of errors is unknown.",
"Here we make the choice of Laplace-distributed errors for sake of simplicity.",
"In particular, the asymptotic theory in [23] suggests the approximation $\\hat{\\beta }_j\\sim \\mathcal {N}( \\beta _j,V_j)$ where $\\hat{\\lambda }_j:=\\frac{1}{n_j}\\sum _{t=1}^{n_j}|Z_j(t)-\\hat{\\alpha }_j-\\hat{\\beta }_j X_j(t)|,$ and $V_j:=(\\hat{\\lambda }_j)^2\\frac{1}{\\sum _{t=1}^{n_j}X_j(t)^2 -\\frac{1}{n_j}(\\sum _{t=1}^{n_j} X_j(t))^2}\\cdot $ We again consider that multiple types $j\\in J$ of time series are conjointly available, and that each $\\beta ^j$ coincides with $\\beta $ , the parameter to be estimated.",
"Assuming the $\\hat{\\beta }_j$ to be independent with $\\hat{\\beta }\\sim \\mathcal {N}( \\beta _j,V_j)$ , leads us to define the estimator $\\hat{\\beta }:=\\frac{\\sum _{j\\in J}\\frac{\\hat{\\beta }_j}{(\\hat{\\lambda }_j)^2}}{\\sum _{j\\in J}\\frac{1}{(\\hat{\\lambda }_j)^2}},$ whose distribution is then given by $\\hat{\\beta }\\sim \\mathcal {N}(\\beta , V)$ where $V=\\frac{1}{\\sum _{j\\in J}\\frac{\\sum _{t=1}^{n_j} X_j(t)^2 -(\\sum _{t=1}^{n_j}X_j(t))^2/n_j}{(\\hat{\\lambda }_j)^2} }\\cdot $ A symmetric $(1-\\epsilon )$ -confidence interval for $\\beta $ is then provided by $\\beta \\in I:=\\left[\\hat{\\beta }-g_{1-\\epsilon /2} \\sqrt{V},\\hat{\\beta }+g_{1-\\epsilon /2} \\sqrt{V}\\right].$ Similarly, $(1-\\epsilon )$ -confidence one-sided intervals for $\\beta $ are obtained by letting $\\beta \\in I^{\\prime }:=[\\hat{\\beta }-g_{1-\\epsilon } \\sqrt{V},+\\infty ), \\; \\beta \\in I^{\\prime \\prime }:=(-\\infty ,\\hat{\\beta }+g_{1-\\epsilon }\\sqrt{V}].$ The doubling time $\\delta $ is given by $(\\log 2)/\\beta $ if $\\beta >0$ , and $+\\infty $ otherwise.",
"This gives the $1-\\epsilon $ -confidence conditions for $\\delta $ : $\\hbox{if } \\hat{\\beta }-g_{1-\\epsilon } \\sqrt{V}>0,\\; \\delta \\in I_1=\\left[0,\\frac{\\log 2}{\\hat{\\beta }-g_{1-\\epsilon } \\sqrt{V}}\\right],$ and $\\delta \\in I_2=\\left[\\frac{\\log 2}{\\max (0,\\hat{\\beta }+g_{1-\\epsilon } \\sqrt{V})},+\\infty \\right).$ For concreteness assume we want to raise an alarm when $\\delta $ is $\\delta ^*$ days or less, where $\\delta ^*$ could be 10.",
"From the above consideration, $\\delta $ is below $\\delta ^*$ days with confidence $1-\\epsilon $ when $\\frac{\\log 2}{\\delta ^*} < \\hat{\\beta }-g_{1-\\epsilon } \\sqrt{V}.$ Raising an alarm under this condition then amounts to calibrating the false positive probability at $\\epsilon $ .",
"Alternatively, we may consider to raise an alarm under the condition $\\frac{\\log 2}{\\delta ^*} < \\hat{\\beta }+g_{1-\\epsilon } \\sqrt{V}.$ This would correspond to calibrating the false negative probability (probability of not raising an alarm while $\\delta \\le \\delta ^*$ ) at $\\epsilon $ ."
],
[
"Conclusion",
"We have shown that monitoring of emergency calls to EMS allows to anticipate the evolution of an epidemic by providing several early signals, each with specific characteristics in terms of time lag and reliability.",
"Our study illustrates the spatially differentiated nature of the epidemic kinetics, with significant doubling time differences between neighboring departments.",
"Such spatial differentiation, if present at a granularity finer than that of departments considered here, could be exploited using the methods described in the present work in order to detect potential epidemic resurgences at the corresponding spatial granularity.",
"This shows great promise in enabling detection of so-called epidemic clusters.",
"There is thus huge potential in the extension of this work and its application to finer spatial resolution.",
"Notwithstanding such extensions, monitoring epidemic kinetics through EMS calls at regional levels can already be exploited to define region-specific sanitary measures, such as lifting of travel bans, proportionate to the regional situation, and to allow early detection of epidemic resurgence.",
"Importantly, we expect this finding to be applicable in full generality to EMS organizations worldwide.",
"Thus the methods introduced here may be of wide applicability to combat Covid-19.",
"Beyond Covid-19, EMS organizations have a unique role to play in early detection of sanitary crises."
],
[
"Acknowledgments",
"We thank the operational team of DSI of AP-HP, who helped to extract information records, especially Stéphane Crézé, Laurent Fontaine, Pierre Cabot, François Planeix, Fabrice Tordjman, Grégory Terrell and Martine Spiegelmann.",
"We thank Pr.",
"Renaud Piarroux for very helpful remarks.",
"We thank Pr.",
"Bruno Riou for his suggestion to include quantitative statistical estimates in the present article.",
"We thank Pr.",
"Frédéric Batteux for having provided epidemiological information.",
"We thank Dr. François Braun (SAMU 57) and Dr. Vincent Bounes (SAMU 31) for providing comparison elements between their departments.",
"We thank Dr. Nicolas Poirot for introducing us to SAMU 31.",
"We thank Dr. Paul-Georges Reuter (SAMU 92) for useful comments on the interpretation of SAMU data relative to the Covid crisis.",
"We thank Ayoub Foussoul, for having developed a robust dynamic programming algorithm, allowing one to consolidate the results of this manuscript concerning the best piecewise linear approximation of the log of observables.",
"We thank Jérôme Bolte, for providing insights on non-convex and non-smooth best-approximation problems.",
"We thank Tania Lasisz for her help in the administration of the project, and Guillermo Andrade Barroso, Thomas Calmant and Matthieu Simonin for their contribution to software development.",
"We thank NXO France Integrator of communication solutions team and SIS Centaure15 solution from GFI World team for the help they provided and their availability for the project.",
"We thank Orange Flux Vision (especially Jean-Michel Contet) for having provided daily population estimates, at the scale of the department, helping to calibrate our models.",
"We thank Enedis (especially Pierre Gotelaere and his team) for having provided an estimation of the departure rate of households, aggregated at the scale of departments and districts, helping us to refine our model.",
"We thank SFR Geostatistic Team (especially Loic Lelièvre) for having provided estimates of flows between Paris and province, aggregated at the scale of departments and districts, allowing us to incorporate mobility in our model.",
"Stéphane Gaubert thanks Nicolas Bacaër for a decisive help, concerning epidemiological and mathematical analysis, provided during the week of March 16th-20th.",
"He thanks Cormac Walsh for improvements of the text.",
"He also thanks Thomas Lepoutre for very helpful mathematical comments and suggestions concerning sec-pde.",
"The INRIA–École polytechnique team thanks the Direction de Programme de la Plate Forme d'Appels d'Urgences – PFAU at Préfecture de Police, DOSTL (Régis Reboul), and Brigade de Sapeurs Pompiers de Paris (especially Gen. Jean-Marie Gontier and Capt.",
"Denis Daviaud) for having provided precious elements of comparison concerning the calls received at the emergency numbers 17-18-112."
],
[
"Appendix: algorithms to compute a best approximation of the logarithm of the number of events by a piecewise linear map",
"Given an epidemiologic observable $Y(t)$ , we need to approximate $\\log Y(t)$ by a function $ \\mathcal {L}(t):=\\min _{1\\le j\\le \\nu }(\\lambda _j t + c_j)\\,,$ where $\\nu $ is the number of phases with constant sanitary policy during the considered time period.",
"The parameters $\\lambda _j$ , $c_j$ are assumed without loss of generality to satisfy $\\lambda _1\\le \\lambda _2\\cdots \\le \\lambda _{\\nu }$ .",
"The concavity constraint imposed on the approximating function $\\mathcal {L}(t)$ makes the problem different from standard function approximation problems, and contributes to the robustness of the fitting procedure by reducing the amount of overfitting.",
"The two most natural criteria for fitting function $\\mathcal {L}(t)$ to observations $\\log Y(t)$ are to minimize either a least squares, or $\\ell _2$ loss function $\\sum _{t\\in \\mathcal {T}} |\\mathcal {L}(t)-\\log Y(t)|^2$ , or an $\\ell _1$ loss function $\\sum _{t\\in \\mathcal {T}} |\\mathcal {L}(t)-\\log Y(t)|$ , where $\\mathcal {T}$ is a finite set of time instants at which observations have been made.",
"As discussed in sec-proba, the $\\ell _1$ formulation is more robust in being less sensitive to outliers, and is the one used on p-phases.",
"The corresponding optimization problem over parameters $\\lambda _i$ , $c_i$ is non-convex as soon as $\\nu \\ge 2$ .",
"A straightforward option is to use a derivative free procedure, like the Nelder-Mead [24] algorithm.",
"Depending on the initial point, this algorithm may converge to a local minimum, which may not be epidemiologically significant.",
"So, a possibility is to guide the algorithm by providing it a initial guess of the optimal solution.",
"To do, we start by an a priori selection of the time periods over which function $\\mathcal {L}(t)$ is linear (which could be obtained by prior knowledge of delay parameters $\\tau $ and times of policy changes, or found by brute force search).",
"We then determine a minimum cost linear fit of target function $\\log Y(t)$ over each such period, and use the concave envelope of the resulting function as our initial condition for local search.",
"This is how we initially obtained the best $\\ell _1$ approximation shown on p-phases.",
"We also used CMA-ES for comparison [17].",
"Both Nelder-Mead and CMA-ES algorithms appear to be sensitive to the initial conditions.",
"Notice in this respect that the objective function is linear on the cells of a polyhedral complex and that it can be constant on certain unbounded cells of this complex, so a local search algorithm may be trapped in a cell in which the function is constant.",
"Another perspective is to observe that this best approximation problem is equivalent to a learning problem, looking for the parameters of a neural networks with a single hidden layer and min-type activation functions, see [8].",
"This allows one to apply (nonsmooth) optimization algorithms used in learning, still leading in general to a local optimum.",
"An approach leading to the global optimum is dynamic programming, originating from Bellman [6].",
"Ayoub Foussoul (École polytechnique) provided us with a dynamic programming solver, implementing several refinements, and allowing us to certify the global optimality of the approximation shown in p-phases, up to a fixed precision."
]
] | 2005.14186 |
[
[
"Quantum and semi-classical aspects of confined systems with variable\n mass"
],
[
"Abstract We explore the quantization of classical models with position-dependent mass (PDM) terms constrained to a bounded interval in the canonical position.",
"This is achieved through the Weyl-Heisenberg covariant integral quantization by properly choosing a regularizing function $\\Pi(q,p)$ on the phase space that smooths the discontinuities present in the classical model.",
"We thus obtain well-defined operators without requiring the construction of self-adjoint extensions.",
"Simultaneously, the quantization mechanism leads naturally to a semi-classical system, that is, a classical-like model with a well-defined Hamiltonian structure in which the effects of the Planck's constant are not negligible.",
"Interestingly, for a non-separable function $\\Pi(q,p)$, a purely quantum minimal-coupling term arises in the form of a vector potential for both the quantum and semi-classical models."
],
[
"Introduction",
"The aim of this work is to study the extent to which the mass of a system depends on the quantum regime on geometric confinement in the configuration space.",
"The procedure is illustrated with one of the most elementary examples in mechanics, namely the motion of a particle in a subset $E$ of the line $\\mathbb {R}$ .",
"The quantum versions of this classical model are established by following the so-called covariant Weyl-Heisenberg (WH)integral quantization [1], [2], [3], a procedure which is built from a function $\\Pi (q,p)$ whose the symplectic Fourier transform $\\mathfrak {F_s}[\\Pi ](q,p):= \\int _{\\mathbb {R}^2}e^{-\\frac{\\mathsf {i}}{\\hbar }(qp^{\\prime }-pq^{\\prime })}\\, \\Pi (q^{\\prime },p^{\\prime })\\,\\frac{\\mathrm {d}q^{\\prime }\\,\\mathrm {d}p^{\\prime }}{2\\pi \\hbar }$ is assumed to be a distribution probability on the plane viewed as the phase space for the motion of a particle on the line.",
"In a nutshell, the canonical quantization of classical observables $f(q,p)$ consists in the replacements $\\mathbb {R}^2\\ni (q,p) &\\mapsto \\ \\mbox{self-adjoint}\\ (Q,P)\\,, \\\\f(q,p) &\\mapsto f(Q,P) \\mapsto (\\mathrm {Sym}f)(Q,P)\\, ,$ where $\\mathrm {Sym}$ stands for a certain choice of symmetrisation of the operator-valued function.",
"One should notice that the canonical commutation relation (CCR) $[Q,P] \\equiv QP-PQ= \\mathsf {i}\\hbar \\mathbb {1}\\, ,$ holds true with (essentially) self-adjoint $Q$ , $P$ , only if both have continuous spectrum $(-\\infty ,+\\infty )$ .",
"Clearly, this scheme does not encompass the cases of other phase space geometries, like barriers or other impassable boundaries.",
"Moreover, the quantization of a Hamiltonian with variable mass, $H(q,p) = p^2/2m(q) + \\cdots $ raises the well known ordering problem, see for instance [4], [5], [6] and references therein.",
"This concept of variable mass is a natural outcome of the so-called shadow Galilean invariance [7].",
"For the motion of an interacting massive particle on the line, it is reasonable to impose that no discrimination is possible instantaneously between a free and an interacting system.",
"After introducing the evolution parameter $t$ , one shows that shadow Galilean dynamics is ruled by Hamiltonians of the general form, $\\begin{split}\\mathsf {H}^{\\mathrm {gen}}(q,p;t) &= \\frac{1}{2m(q)}(p - A(q;t))^2 + U(q;t)\\\\&= \\frac{p^2}{2m(q)} -\\frac{p}{m(q)}\\,A(q;t) + \\frac{A^2(q;t)}{2m(q)} + U(q;t) \\\\ &\\equiv L_2(q;t)\\,p^2 + L_1(q;t)\\,p + L_0(q;t)\\, ,\\end{split}$ to which the WH integral quantization applies easily, and plays in general a regularizing role, depending on the choice of the function $\\Pi (q,p)$ .",
"Note that this choice will dispel the ordering ambiguity due to the presence of the variable mass $m(q)$ .",
"Of course, changing $\\Pi (q,p)$ yields in general another ordering.",
"Now, imposing to a particle with a constant mass $m_0$ to move inside a subset $E\\subset \\mathbb {R}$ entails that its Hamiltonian has to be multiplied by the characteristic function $\\chi _{E} (q)=\\left\\lbrace \\begin{array}{cl}1 & q\\in E \\\\0 & {\\rm otherwise}\\end{array} \\, .\\right.$ Hence, the Hamiltonian becomes the truncated observable $H(q,p) = \\frac{p^2}{2m_0} + \\cdots \\quad \\mapsto \\quad \\chi _{E} (q)H(q,p)= \\frac{p^2}{2m(q)} + \\cdots \\, ,\\,.$ where $\\quad m(q)= \\frac{m_0}{\\chi _{E} (q)}\\,.$ Clearly, this variable mass $m(q)$ is a singular function, since it becomes infinite at the boundary of $E$ .",
"Generalising the procedure introduced in the previous work [6], we show in this paper that our quantization method based on the choice of a, at least, continuous function $\\Pi (q,p)$ may regularise this singular mass and yield a soft semi-classical Hamiltonian mechanics, say à la Klauder [8], [9], allowing the particle to escape its confining geometry $E$ , at the order $\\hbar $ .",
"Covariant Weyl-Heisenberg quantization is presented in Section .",
"The quantum counterparts issued from the procedure are made explicit in Section for the most common observables.",
"In Section we examine the outcomes yielded by the choice of some functions $\\Pi (q,p)$ .",
"The semi-classical side of the quantization and its probabilistic aspects are described in Section .",
"In Section we adapt and generalize the Klauder's formalism of enhanced quantization through the use of the function $\\Pi (q,p)$ .",
"The general formalism of the quantization of truncated observables and subsequent semi-classical portraits are developed in Section .",
"The elementary example of the motion in an interval is comprehensively treated in Section .",
"Applications to problems involving specific potentials are considered in Section .",
"In Section some ideas of future works are sketched.",
"Appendix is devoted to some useful properties of the symplectic Fourier transform."
],
[
"Covariant integral quantization of the motion on the line",
"In this section we remind the content of Reference [2] in which was described the covariant Weyl-Heisenberg integral quantization of the motion on the line.",
"Precisely, we transform a function $f(q,p)$ into an operator $A_f$ in some separable Hilbert space $\\mathcal {H}$ through a linear map which sends the function $f=1$ to the identity operator in $\\mathcal {H}$ and which respects the basic translational symmetry of the phase space.",
"A probabilistic content is one of the most appealing outcomes of the procedure."
],
[
"The quantization map",
"We define the integral quantization of the motion on the line as the linear map $f(q,p) \\mapsto A_f = \\int _{\\mathbb {R}^2} f(q,p)\\, \\mathfrak {Q}(q,p)\\, \\frac{\\mathrm {d}q\\, \\mathrm {d}p}{2\\pi \\hbar }\\, .$ where $\\mathfrak {Q}(q,p)$ is a family of operators which solves the identity in $\\mathcal {H}$ with respect to the measure $\\mathrm {d}q\\,\\mathrm {d}p/(2\\pi \\hbar )$ , $\\int _{\\mathbb {R}^2} \\mathfrak {Q}(q,p)\\, \\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar } = \\mathbb {1}\\,.$ Hence the identity $\\mathbb {1}$ is the quantized version of the function $f=1$ .",
"In addition to (REF ), we impose the family $ \\mathfrak {Q}(q,p)$ to obey a symmetry condition issued from the homogeneity of the phase space.",
"Indeed, the choice of the origin in $\\mathbb {R}^2$ is arbitrary.",
"Hence we must have translational covariance in the sense that the quantization of the translated of $f$ is unitarily equivalent to the quantization of $f$ $U(q_0,p_0)\\,A_f \\,U(q_0,p_0)^{\\dag }= A_{\\mathcal {T}(q_0,p_0)f}\\, ,$ where $\\left(\\mathcal {T}(q_0,p_0)f\\right)(q,p):= f\\left(q-q_0,p-p_0\\right)\\,.$ So $(q,p)\\mapsto U(q,p)$ has to be a unitary, possibly projective, representation of the abelian group $\\mathbb {R}^2$ .",
"Then, from (REF ) and the translational invariance of $\\mathrm {d}q\\,\\mathrm {d}p$ , the operator valued function $\\mathfrak {Q}(q,p)$ has to obey $U(q_0,p_0)\\,\\mathfrak {Q}(q,p) \\,U^{\\dag }(q_0,p_0)= \\mathfrak {Q}\\left(q+ q_0,p+p_0\\right) \\,.$ A solution to (REF ) is found by picking an operator $\\mathfrak {Q}_0\\equiv \\mathfrak {Q}(0,0)$ and write $\\mathfrak {Q}\\left(q,p\\right) := U(q,p)\\,\\mathfrak {Q}_0 \\,U^{\\dag }(q,p) \\,.$ Then the resolution of the identity holds from Schur's Lemma [10] if $U$ is irreducible, and if the operator-valued integral (REF ) makes sense, i.e., if the choice of the fixed operator $\\mathfrak {Q}_0$ is valid."
],
[
"From $\\mathbb {R}^2$ to the Weyl-Heisenberg group and its UIR",
"In order to find the non-trivial unitary operator $U(q,p)$ involved in (REF ), we deal with the Weyl-Heisenberg (WH) group, $\\begin{split}\\mathrm {WH} &= \\lbrace (s,q,p)\\, , \\, s\\in \\mathbb {R}\\, , \\, (q,p)\\in \\mathbb {R}^2\\rbrace \\, , \\\\(s,q,p)(s^{\\prime },q,p) &= \\left(s+s^{\\prime } - \\frac{1}{2\\hbar }(qp^{\\prime }-pq^{\\prime }), q+q^{\\prime },p+p^{\\prime }\\right)\\, ,\\end{split}$ instead of just the group $\\mathbb {R}^2$ .",
"From von Neumann [11], [12], [13], WH has a unique non trivial unitary irreducible representation (UIR), up to equivalence corresponding precisely to the arbitrariness of a constant $k\\ne 0$ , that we put equal to $\\hbar $ : $(s,q,p) \\mapsto \\mathcal {U}(s,q,p)= e^{\\mathsf {i}s}\\,U(q,p) \\, ,$ where $U(q,p)= e^{\\frac{\\mathsf {i}}{\\hbar } (pQ-qP)}\\,.$ is the displacement operator, and where $Q$ and $P$ are the two above-mentionned self-adjoint operators in $\\mathcal {H}$ such that $[Q,P]= \\mathsf {i}\\, \\hbar \\mathbb {1}$ ."
],
[
"WH covariant integral quantization(s)",
"From Schur's Lemma applied to the WH UIR $\\mathcal {U}$ , or equivalently to $U$ since $e^{\\mathsf {i}s}$ is just a phase factor, we confirm the resolution of the identity $\\int _{\\mathbb {R}^2} \\mathfrak {Q}(q,p)\\, \\frac{\\mathrm {d}q\\, \\mathrm {d}p}{2\\pi \\hbar } = \\mathbb {1}\\,, \\quad \\mathfrak {Q}(q,p)= U(q,p)\\mathfrak {Q}_0 U^{\\dag }(q,p)\\, ,$ where $\\mathfrak {Q}_0$ is the fixed operator introduced in (REF ), whose choice is left to us.",
"Let us prove that this is possible if $\\mathfrak {Q}_0$ is trace class with unit trace i.e., $\\mathrm {Tr}(\\mathfrak {Q}_0)=1$ .",
"Indeed, let us introduce the function $\\Pi (q,p) = \\mathrm {Tr}\\left(U(-q,-p)\\mathfrak {Q}_0 \\right)\\, .$ This is interpreted as the Weyl-Heisenberg transform of operator $\\mathfrak {Q}_0$ .",
"The inverse WH-transform exists due to two remarkable properties [1], [3] of the displacement operator $U(q,p)$ , $\\int _{\\mathbb {R}^2} U(q,p) \\,\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar } = 2{\\sf P}\\ \\mbox{and}\\ \\mathrm {Tr}\\left(U(q,p)\\right)= 2\\pi \\hbar \\delta (q,p)\\,,$ with $\\delta (q,p)\\equiv \\delta (q)\\delta (p)$ the two-dimensional Dirac delta distribution [14], [15], and ${\\sf P}= {\\sf P}^{-1}$ the parity operator defined as ${\\sf P}U(q,p){\\sf P}= U(-q,-p)$ , and the trace of ${\\sf P}$ is put equal to $1/2$ .",
"One derives from (REF ) the inverse WH-transform of $\\Pi (q,p)$ : $\\mathfrak {Q}_0 = \\int _{\\mathbb {R}^2} U(q,p) \\, \\Pi (q,p)\\,\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar }\\, ,$ The function $\\Pi (q,p)$ is like a weight, not necessarily normalisable, or even positive.",
"It can be viewed as an apodization [16] on the plane, or a kind of coarse graining of the phase space.",
"Equipped with one choice of a traceclass $\\mathfrak {Q}_0$ , we can now proceed with the corresponding WH covariant integral quantization map $f(q,p) \\mapsto A_f \\left(\\equiv A^{\\mathfrak {Q}_0}_f\\right) = \\int _{\\mathbb {R}^2} f(q,p) \\mathfrak {Q}(q,p)\\, \\frac{\\mathrm {d}q \\mathrm {d}p}{2\\pi \\hbar }\\, .$ In this context, the operator $\\mathfrak {Q}_0$ is the quantum version (up to a constant) of the origin of the phase space, identified with the $2\\pi \\times $ Dirac distribution at the origin.",
"$\\begin{split}& 2\\pi \\hbar \\delta (q,p) \\mapsto A_{\\delta }= \\mathfrak {Q}_0\\, , \\\\& 2\\pi \\hbar \\delta (q-q_0,p-p_0) \\mapsto A_{\\delta _{(q_0,p_0)}}= \\mathfrak {Q}(q_0,p_0) \\, .\\end{split}$ The probabilistic content of our quantization procedure is better captured if one uses an alternative quantization formula through the symplectic Fourier transform, $\\mathfrak {F_s}[f](q,p)= \\int _{\\mathbb {R}^2}e^{-\\frac{\\mathsf {i}}{\\hbar } (qp^{\\prime }-pq^{\\prime })}\\, f(q^{\\prime },p^{\\prime })\\,\\frac{\\mathrm {d}q^{\\prime }\\,\\mathrm {d}p^{\\prime }}{2\\pi \\hbar } \\, .$ It is involutive, $\\mathfrak {F_s}\\left[\\mathfrak {F_s}[f]\\right]= f$ like its dual defined as $\\overline{\\mathfrak {F_s}}[f](q,p)= \\mathfrak {F_s}[f](-q,-p)$ .",
"See Appendix for details.",
"The equivalent form of the WH integral quantization (REF ) reads as $f\\mapsto A_f= \\int _{\\mathbb {R}^2} U(q,p)\\, \\overline{\\mathfrak {F_s}}[f](q,p)\\, \\Pi (q,p) \\,\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar }\\, .$ From (REF ) or (REF ) is derived the action of $A_f$ as the integral operator $L^2(\\mathbb {R},\\mathrm {d}x) \\ni \\phi (x) \\mapsto (A_f \\phi )(x) =\\int _{-\\infty }^{+\\infty }\\mathrm {d} x^{\\prime }\\, \\mathcal {A}_f(x,x^{\\prime })\\, \\phi (x^{\\prime })\\, ,$ with kernel given by $\\mathcal {A}_f(x,x^{\\prime })= \\frac{1}{2\\pi \\hbar }\\int _{-\\infty }^{+\\infty }\\mathrm {d} q\\, \\widehat{f}_{p}(q,x^{\\prime }-x)\\, \\widehat{\\Pi }_{p}\\left(x-x^{\\prime },q- \\frac{x+x^{\\prime }}{2}\\right)\\,.$ Here the symbol $\\widehat{f}_{p}$ stands for partial Fourier transform of $f$ with respect its second variable $p$ : $\\widehat{f}_{p}(q,y)= \\frac{1}{\\sqrt{2\\pi \\hbar }}\\int _{-\\infty }^{+\\infty }\\mathrm {d}p \\, f(q,p)\\, e^{-\\mathsf {i}\\frac{yp}{\\hbar }}\\,.$"
],
[
"Permanent outcomes of WH covariant integral quantizations",
"By permanent outcomes we mean that some basic rules managing the quantum model have a kind of universality, almost whatever the choice of admissible $\\mathfrak {Q}_0$ , or its corresponding apodization $\\Pi (q,p)$ ."
],
[
"Symmetric operators and self-adjointness",
"First, we have the general important outcome: if $\\mathfrak {Q}_0$ is symmetric, i.e.",
"$[\\Pi (-q,-p)]^{*}=\\Pi (q,p)$ with $z^{*}$ the complex-conjugate of $z$ , a real function $f(q,p)$ is mapped to a symmetric operator $A_f$ .",
"Moreover, if $\\mathfrak {Q}_0$ is a positive operator, then a real semi-bounded function $f(q,p)$ is mapped to a self-adjoint operator $A_f$ through the Friedrich extension [17] of its associated semi-bounded quadratic form.",
"Canonical commutation rule is preserved: $A_q = Q + c_0\\, , \\quad A_p= P+d_0\\,, \\ \\Rightarrow \\ \\left[A_q,A_p\\right]= \\mathsf {i}\\hbar \\mathbb {1}\\, .$ This result is actually the direct consequence of the underlying Weyl-Heisenberg covariance when one expresses (REF ) on the level of infinitesimal generators.",
"In the above formulas, the constants $c_0,d_0,e_0,e_1,$ can be easily removed by imposing mild constraints on $\\Pi (q,p)$ .",
"Moreover, constant $f_0$ can be fixed to $-\\mathsf {i}/2$ in order to get the symmetric dilation operator $(QP + PQ)/2$ .",
"A potential energy $V(q)$ becomes a multiplication operator in position representation.",
"$A_{V} = \\mathfrak {V}(Q)\\, , \\quad \\mathfrak {V}(Q)= \\frac{1}{\\sqrt{2\\pi \\hbar }}\\,V\\ast \\overline{\\mathcal {F}}[\\Pi (0,\\cdot )](Q)\\, ,$ where $ \\overline{\\mathcal {F}}$ is the inverse 1-D Fourier transform $\\mathcal {F}[f](q)=\\frac{1}{\\sqrt{2\\pi \\hbar }}\\int _{-\\infty }^{+\\infty }f(p)\\,e^{-\\frac{\\mathsf {i}}{\\hbar } qp}\\,\\mathrm {d}p\\,, \\quad \\overline{\\mathcal {F}}[f](q)=\\mathcal {F}[f](-q)\\,,$ and “$\\ast $ ” stands for convolution, both with respect to the second variable.",
"If $F(q,p)\\equiv v(p)$ is a function of $p$ only, then $A_v$ depends on $P$ only through the convolution: $A_v= \\frac{1}{\\sqrt{2\\pi \\hbar }}\\,v\\ast \\mathcal {F}[\\Pi (\\cdot ,0)](P)\\, ,$ where the 1-D Fourier transform $ \\mathcal {F}$ and the convolution hold with respect to the first variable."
],
[
"Examples of $\\Pi $ functions",
"The simplest choice is $\\Pi (q,p) = 1$ , of course.",
"Then $\\mathfrak {Q}_0 = 2 \\mathsf {P}$ .",
"This no filtering choice yields the popular Weyl-Wigner integral quantization (see [13] and references therein), equivalent to the standard ($\\sim $ canonical) quantization.",
"No regularisation of space or momentum singularity present in the classical model is possible since $V(q) \\mapsto A_V = V(Q)\\, , \\quad v(p) \\mapsto A_v = v(P)\\, .$ This quantization yields the so-called Weyl ordering [3].",
"Another choice is the Born-Jordan function, $\\Pi (q,p) = \\dfrac{\\hbar \\sin (qp/\\hbar )}{qp}$ , which presents appealing aspects [18], [19].",
"With this choice Eqs.",
"(REF ) also hold true.",
"If $\\mathfrak {Q}_0= |\\psi \\rangle \\langle \\psi |$ , with $\\Vert \\psi \\Vert =1$ , then $\\Pi (q,p)= e^{-\\mathsf {i}\\frac{qp}{2\\hbar }}\\left(\\mathcal {F}[\\psi ]\\ast \\mathcal {F}[\\mathrm {t}_{-q}\\psi ]\\right)(p)\\, ,$ where $\\mathcal {F}[\\mathrm {t}_{-q}\\psi ](p)= e^{\\mathsf {i}\\frac{qp}{\\hbar }}\\mathcal {F}[\\psi ](p)$ for $\\psi (x)\\in L^2(\\mathbb {R},\\mathrm {d}x)$ .",
"The corresponding integral kernel is given by $\\mathcal {A}_f(x,x^{\\prime })= \\frac{1}{\\sqrt{2\\pi \\hbar }}\\int _{-\\infty }^{+\\infty }\\mathrm {d} q\\, \\widehat{f}_{p}(q,x^{\\prime }-x)\\, \\psi (x-q)\\,[\\psi (x^{\\prime }-q)]^{*}$ For instance, let us consider the squeezed vacuum state $\\vert \\xi \\rangle =S(\\xi )\\vert \\psi _{\\ell } \\rangle \\,, \\quad S(\\xi )=e^{-\\frac{\\xi }{2}a^{\\dagger 2}+\\frac{\\xi ^{*}}{2}a^{2}} \\,, \\xi \\in \\mathbb {C} \\,,$ where $S(\\xi )$ stands for the squeezing operator [20], $\\vert \\psi _{\\ell }\\rangle $ the harmonic oscillator ground state $\\langle x|\\psi _\\ell \\rangle = \\psi _{\\ell }(x)= \\left( \\frac{1}{\\pi \\ell ^2} \\right)^{1/4} e^{-\\frac{x^2}{2\\ell ^2}}\\,,$ and $\\lbrace a,a^{\\dagger }\\rbrace $ denote the set of boson operators.",
"The squeezed vacuum state is given, in coordinate representation, by the normalized complex-valued Gaussian function $\\psi _{\\ell ,\\eta }(x)\\equiv \\langle x \\vert \\xi \\rangle = \\left( \\frac{1-\\vert \\eta \\vert ^{2}}{\\pi \\ell ^2(1-\\eta )^{2}} \\right)^{1/4} e^{-\\frac{x^2}{2\\ell ^2}\\left(\\frac{1+\\eta }{1-\\eta }\\right)} \\, , \\quad \\eta =\\frac{\\xi }{\\vert \\xi \\vert }\\tanh \\vert \\xi \\vert \\, ,$ where $\\vert \\eta \\vert <1$ , and $\\ell $ is a constant that has dimension of position.",
"Its Fourier transform is $\\mathcal {F}[\\psi _{\\ell ,\\eta }](p)=\\left( \\frac{1-\\vert \\eta \\vert ^{2}}{\\pi \\wp ^2(1+\\eta )^{2}}\\right)^{1/4}e^{-\\frac{p^2}{2\\wp ^2}\\left(\\frac{1-\\eta }{1+\\eta }\\right)}\\,,$ where $\\wp =\\hbar /\\ell $ has dimension of momentum.",
"Its associate function $\\Pi (q,p)$ is given through (REF ) by $\\begin{split}&\\Pi _{\\ell ,\\eta }(q,p)= e^{-\\frac{q^{2}}{4\\ell ^{2}}\\left(\\kappa _{R}+\\frac{\\kappa _{I}^{2}}{\\kappa _{R}}\\right)}e^{-\\frac{p^{2}}{4\\wp ^{2}}\\frac{1}{\\kappa _{R}}}e^{-\\frac{\\kappa _{I}}{\\kappa _{R}}\\frac{qp}{2\\hbar }}\\\\&\\kappa =\\frac{1+\\eta }{1-\\eta } \\, , \\quad \\kappa _{R}=\\frac{\\kappa +\\kappa ^{*}}{2} \\, , \\quad \\kappa _{I}=\\frac{\\kappa -\\kappa ^{*}}{2i} \\,.\\end{split}$ It is worth noting that $\\xi =\\eta =0$ leads to the Gaussian ground state $\\vert \\psi _{\\ell } \\rangle $ , and consequently to a function $\\Pi (q,p)$ of the form $\\Pi _\\ell (q,p)\\equiv \\Pi _{\\ell ,\\eta =0}=e^{-\\frac{q^2}{4\\ell ^2}}e^{-\\frac{p^2}{4\\wp ^2}}\\,.$ Its symplectic Fourier transform reads $\\mathfrak {F_s}\\left[\\Pi _{\\ell }\\right](q,p)= 2e^{-\\frac{q^2}{\\ell ^2}-\\frac{p^2}{\\wp ^2}} \\,, \\quad \\ell \\wp =\\hbar \\,.$ $\\Pi _\\ell (q,p)$ is precisely the function yielding the coherent state or Berezin or anti-Wick quantization, where two parameters are present, $\\ell $ and $\\hbar $ , or equivalently $\\wp $ and $\\hbar $ .",
"The corresponding operator $\\mathfrak {Q}_0$ is the orthogonal projector $|\\psi _\\ell \\rangle \\langle \\psi _\\ell |$ on the harmonic oscillator ground state.",
"An easily manageable generalisation of (REF ) concerns separable functions $\\Pi (q,p)= \\lambda (q)\\, \\mu (p)\\,,$ where $\\lambda $ and $\\mu $ are preferably regular, e.g., rapidly decreasing smooth functions.",
"Note its symplectic Fourier transform: $\\mathfrak {F_s}[\\Pi ](q,p)= \\mathcal {F}[\\mu ](q)\\,\\overline{\\mathcal {F}}[\\lambda ](p)\\,.$ Such an option is suitable for physical Hamiltonians which are sums of terms like $L(q)\\,p^n$ , where it allows regularisations through convolutions if functions $\\lambda $ and $\\mu $ are regular enough.",
"$\\begin{split}A_{L(q)\\,p^n}&= \\sum _{\\begin{array}{c}r,s,t\\\\r+s+t=n\\end{array}} 2^{-s}\\, \\binom{n}{r\\,s\\,t}\\,(\\mathsf {i}\\hbar )^r\\,\\lambda ^{(r)}(0)\\times \\\\&\\times (-\\mathsf {i}\\hbar )^s \\frac{1}{\\sqrt{2\\pi \\hbar }}\\,\\left(\\overline{\\mathcal {F}}[\\mu ]\\ast L \\right)^{(s)}(Q)\\, P^t\\, .\\end{split}$ Note that $\\left(\\overline{\\mathcal {F}}[\\mu ]\\ast L \\right)^{(s)}= \\left(\\overline{\\mathcal {F}}[\\mu ]\\right)^{(s)}\\ast L = \\overline{\\mathcal {F}}[\\mu ]\\ast L^{(s)}$ , relations whose validity depends on the derivability of the factors.",
"For the cases $n=0$ , $n=1$ and $n=2$ , i.e.",
"the most relevant to Galilean physics, we have, with $T(x):= \\frac{1}{\\sqrt{2\\pi \\hbar }}\\,\\left(\\overline{\\mathcal {F}}[\\mu ]\\ast L \\right)(x)$ , $A_{L(q)} = \\lambda (0)\\, T(Q)\\, ,$ $\\begin{split}A_{L(q)\\,p} &= \\lambda (0)\\, T(Q)\\, P + \\mathsf {i}\\hbar \\,\\lambda ^{\\prime }(0)\\, T(Q) -\\frac{\\mathsf {i}\\hbar }{2}\\,\\lambda (0)\\, T^{\\prime }(Q)\\\\&= \\lambda (0)\\, \\frac{T(Q)\\, P + P\\,T(Q)}{2} + \\mathsf {i}\\hbar \\, \\lambda ^{\\prime }(0)\\, T(Q)\\, ,\\end{split}$ $ A_{L(q)\\,p^2} &= \\lambda (0)\\, T(Q)\\, P^2 + \\mathsf {i}\\hbar \\,(2\\lambda ^{\\prime }(0)\\, T(Q)- \\lambda (0)\\, T^{\\prime }(Q))\\,P \\\\\\nonumber &+\\hbar ^2\\left(-\\lambda ^{\\prime \\prime }(0)\\, T(Q)+ \\lambda ^{\\prime }(0)\\, T^{\\prime }(Q)-\\frac{\\lambda (0)}{2}\\, T^{\\prime \\prime }(Q)\\right)\\\\\\nonumber &= \\lambda (0)\\, \\frac{T(Q)\\, P^2 + P^2\\,T(Q)}{2} + 2\\mathsf {i}\\hbar \\, \\lambda ^{\\prime }(0)\\, T(Q)\\,P\\\\ \\nonumber &+\\hbar ^2\\left(-\\lambda ^{\\prime \\prime }(0)\\, T(Q)+ \\lambda ^{\\prime }(0)\\, T^{\\prime }(Q) + \\frac{\\lambda (0)}{4}\\, T^{\\prime \\prime }(Q)\\right)\\, .$ We observe that the operators (REF ) and (REF ) are symmetric under the condition $\\lambda ^{\\prime }(0) = 0\\, .$ Note the appearance, in the expression of the operator (REF ), of a potential built from derivatives of the regularisation of $L(q)$ .",
"This feature is typical of quantum Hamiltonians with variable mass (see the discussion in [5]).",
"The function $\\Pi (q,p)$ in (REF ) can be generalized as the separable Gaussian $\\Pi _{\\sigma _{\\ell } \\sigma _{\\wp }}(q,p) = e^{-\\frac{q^2}{2\\sigma _{\\ell }^2}-\\frac{p^2}{2\\sigma _{\\wp }^2}}\\,,\\ \\mathfrak {F_s}\\left[\\Pi _{\\sigma _{\\ell } \\sigma _{\\wp }}\\right](q,p)= \\frac{\\sigma _{\\ell }\\sigma _{\\wp }}{\\hbar }e^{-\\frac{\\sigma _{\\wp }^2q^2}{2\\hbar ^2}-\\frac{\\sigma _{\\ell }^2p^2}{2\\hbar ^2}}\\,,$ with independent widths $\\sigma _{\\ell }$ and $\\sigma _{\\wp }$ , which yields to a simple formulae with familiar probabilistic content of the symplectic Fourier transform.",
"Moreover they satisfy condition (REF ).",
"As was proved above, standard coherent state (or Berezin or anti-Wick) quantization corresponds to the particular values $\\sigma _{\\ell }=\\sqrt{2}\\ell , \\sigma _{\\wp }= \\sqrt{2}\\wp $ , with the constraint $\\wp = \\hbar /\\ell $ .",
"On the other hand, the limit Weyl-Wigner case holds as the widths $\\sigma _{\\ell }$ and $\\sigma _{\\wp }$ are infinite.",
"Weyl-Wigner is singular in this respect and the symplectic Fourier transform is the Dirac probability distribution centred at the origin of the phase space.",
"Recent applications of the present formalism to quantum cosmology is found in [21], [22]."
],
[
"Quantum phase portrait and its probabilistic content",
"The quantization formula (REF ) allows to prove the trace formula (when applicable to $f$ ): $\\mathrm {Tr}\\left(U(q,p)\\right)= 2\\pi \\hbar \\delta (q,p) \\, ,$ and so $\\mathrm {Tr}\\left(A_f\\right)= \\overline{\\mathfrak {F_s}}[f](0,0)= \\int _{\\mathbb {R}^2} f(q,p)\\, \\,\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar }\\, .$ Thus, the operator $A_f$ is traceclass if its classical counterpart $f(q,p)$ ) has finite average on the phase space.",
"By using (REF ) we derive the quantum phase space, i.e.",
"semi-classical, portrait of the operator as an autocorrelation averaging of the original $f$ .",
"More precisely, starting from a function (or distribution) $f(q,p)$ , one defines through its quantum version $A_f$ the new function ${f}(q,p)$ as $\\begin{split}{f}(q,p) &= \\mathrm {Tr}\\left(\\mathfrak {Q}(q,p)A_f\\right)\\\\ &=\\int _{\\mathbb {R}^2} \\, \\mathrm {Tr}\\left(\\mathfrak {Q}(q,p)\\,\\mathfrak {Q}(q^{\\prime },p^{\\prime })\\right)\\, f(q^{\\prime },p^{\\prime })\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar }\\, .\\end{split}$ The map $(q^{\\prime },p^{\\prime })\\mapsto \\mathrm {Tr}\\left(\\mathfrak {Q}(q,p)\\,\\mathfrak {Q}(q^{\\prime },p^{\\prime })\\right)$ might be a probability distribution if this expression is non negative.",
"Now, this map is better understood from the equivalent formulas, ${f}(q,p) &= \\int _{\\mathbb {R}^2} \\mathfrak {F_s}\\left[\\Pi \\,\\widetilde{\\Pi }\\right](q^{\\prime }-q,p^{\\prime }-p)\\, f(q^{\\prime },p^{\\prime }) \\,\\frac{\\mathrm {d}q^{\\prime }\\,\\mathrm {d}p^{\\prime }}{2\\pi \\hbar } \\\\&= \\frac{1}{2\\pi \\hbar } \\left(\\overline{\\mathfrak {F_s}}\\left[\\Pi \\,\\widetilde{\\Pi }\\right]\\ast f\\right)(q,p) \\\\ &=\\int _{\\mathbb {R}^2} \\left(\\mathfrak {F_s}\\left[\\Pi \\right]\\ast \\mathfrak {F_s}\\left[\\widetilde{\\Pi }\\right]\\right)(q^{\\prime }-q,p^{\\prime }-p)\\, f(q^{\\prime },p^{\\prime }) \\,\\frac{\\mathrm {d}q\\,\\mathrm {d}p}{4\\pi ^2\\hbar ^2}\\, ,$ where $\\widetilde{\\Pi }(q,p):=\\Pi (-q,-p)$ .",
"In particular, and with mild conditions on $\\Pi (q,p)$ , we have for the coordinate functions, ${q}= q\\, , \\quad {p}= p\\,.$ Eq.",
"(REF ) represents the convolution ($\\sim $ local averaging) of the original $f$ with the autocorrelation of the symplectic Fourier transform of the (normalised) function $\\Pi (q,p)$ .",
"For instance, with a separable function $\\Pi (q,p)= \\lambda (q)\\,\\mu (p)$ , the quantum phase space portrait of a separable function $f(q,p) = u(q)\\,v(p)$ is given by the product of two 1d-convolutions: ${f}(q,p)= \\left[\\overline{\\mathcal {F}}[\\lambda \\widetilde{\\lambda }]\\ast u\\right](q)\\,\\left[\\overline{\\mathcal {F}}[\\mu \\widetilde{\\mu }]\\ast v\\right](p)\\,.$ Note that taking the dual symplectic Fourier of () and applying (REF ) yields the equality $\\overline{\\mathfrak {F_s}}\\left[{f}\\right](q,p) = \\Pi (q,p)\\Pi (-q,-p)\\, \\overline{\\mathfrak {F_s}}\\left[f\\right](q,p)\\,.$ This alternative expression might provide the inversion formula ${f} \\mapsto f$ , only if all employed formal manipulations have been mathematically justified, essentially in the framework of distributions, e.g., convolution algebra: $f(q,p)=\\overline{\\mathfrak {F_s}}\\left[ \\frac{\\overline{\\mathfrak {F_s}}\\left[{f}\\right]}{\\Pi \\,\\widetilde{\\Pi }}\\right](q,p)\\,.$ For instance, suppose that we start from a function ${f}$ which is Lebesgue integrable on $\\mathbb {R}^2$ , i.e.",
"${f}\\in L^1(\\mathbb {R}^2)$ .",
"Then $\\overline{\\mathfrak {F_s}}\\left[{f}\\right]$ is continuous, bounded, and goes to 0 at the infinity.",
"Suppose that $1/(\\Pi \\,\\widetilde{\\Pi }) $ is locally integrable and slowly increasing on $\\mathbb {R}^2$ .",
"Then (REF ) defines a tempered distribution on the plane.",
"The classical limit ${f} \\rightarrow f$ of the above semi-classical formalism is also a fundamental point to be considered.",
"In view of the formula (REF ), it is sufficient to assert that the classical limit is reached if $\\mathfrak {F_s}\\left[\\Pi \\,\\widetilde{\\Pi }\\right](q,p) \\rightarrow 2\\pi \\hbar \\delta (q,p)=2\\pi \\hbar \\delta (q) \\delta (p)\\,,$ as a certain set of introduced dimensional parameters, e.g.",
"$\\hbar $ , $\\ell $ , $\\wp \\cdots $ , go to zero.",
"For instance, with the example of separable Gaussians (REF ), this limit is reached at $\\sigma _{\\ell } \\rightarrow 0$ and $\\sigma _{\\wp } \\rightarrow 0$ .",
"In view of the convolution formulas (REF )-()-(), we are inclined to choose windows $\\Pi (q,p)$ , or equivalently $\\mathfrak {Q}_0$ , such that $\\mathfrak {F_s}\\left[\\Pi \\right]$ is a probability distribution on the phase space $\\mathbb {R}^2$ equipped with the measure $\\dfrac{\\mathrm {d}q\\,\\mathrm {d}p}{2\\pi \\hbar }$ .",
"Note that the uniform Weyl-Wigner choice $\\Pi (q,p)= 1$ yields $\\mathfrak {F_s}\\left[1\\right](q,p)= 2\\pi \\hbar \\,\\delta (q,p)$ and $\\mathfrak {Q}_0 = 2\\mathrm {P}$ , and $f= f$ in this case.",
"Also note that the celebrated Wigner function $\\mathcal {W}_{\\rho }(q,p)$ for a density operator or mixed quantum state $\\rho $ , defined by $\\mathcal {W}_{\\rho }(q,p) = \\mathrm {tr}\\left(U(q,p)2{\\sf P}U^{\\dag }(q,p)\\rho \\right)= \\mathrm {tr}\\left(U(2q,2p)2{\\sf P}\\rho \\right)\\, ,$ is a normalised quasi-distribution which can assume negative values.",
"With a true probabilistic content, the meaning of the convolution $\\mathfrak {F_s}\\left[\\Pi \\right]\\ast \\mathfrak {F_s}\\left[\\widetilde{\\Pi }\\right]$ is clear: it is the probability distribution for the difference of two vectors in the phase plane, viewed as independent random variables, and thus is perfectly adapted to the abelian and homogeneous structure of the classical phase space.",
"We can conclude that a quantum phase space portrait in this probabilistic context is like a measurement of the intensity of a diffraction pattern resulting from the function $\\Pi (q,p)$ or $\\mathfrak {F_s}\\left[\\Pi \\right](q,p)$ , the coarse graining of the idealistic phase space $\\mathbb {R}^2$ ."
],
[
"Extending Klauder's formalism with function $\\Pi $",
"Klauder's approach [8], [9] is based upon the comparison between the two action functionals, the classical and the quantum ones, $\\texttt {A}_C &= \\int _0^T [p(t)\\dot{q}(t) - H_c(q(t),p(t))] \\,\\mathrm {d}t\\,,\\\\ \\texttt {A}_Q & = \\int _0^T [\\langle \\psi (t)| \\mathsf {i}\\hbar \\partial /\\partial t - \\mathcal {H}(Q,P)]|\\psi (t)\\rangle \\,\\mathrm {d}t\\, ,$ where the quantum Hamiltonian $\\mathcal {H}(Q,P)$ is deduced from its classical counterpart $H_c$ through canonical quantization, and unit norm state $|\\psi \\rangle $ is in the domain of $\\mathcal {H}(Q,P)$ .",
"Stationary variations of $\\texttt {A}_C$ and $\\texttt {A}_Q$ yield their respective dynamical equations, the Hamilton ones and the Shrödinger equation, $&\\dot{q} = \\lbrace q,H_c\\rbrace = \\frac{\\partial H_c}{\\partial p}\\, , \\quad \\dot{p} = \\lbrace p,H_c\\rbrace = -\\frac{\\partial H_c}{\\partial q}\\, , \\\\ &\\mathsf {i}\\hbar \\partial \\psi /\\partial t = \\mathcal {H}(Q,P) \\psi \\,,$ with solutions determined by initial conditions at $t=0$ , $(q_0,p_0)$ and $|\\psi _0\\rangle $ respectively.",
"Hence, () results from stationary variation of () over arbitrary histories of pure states $\\lbrace |\\psi (t)\\rangle \\rbrace _0^T$ modulo fixed end points.",
"The main point raised by Klauder is that “macroscopic observers studying a microscopic system cannot vary histories over such a wide range”.",
"Instead they are confined to moving the system to a new position $(q)$ or changing its velocity $(p)$ through $\\dot{q} = \\partial H_c(q,p)/\\partial p$ .",
"Therefore, they are restricted to vary only the coherent states $|q(t),p(t)\\rangle = U(q(t),p(t))|0\\rangle $ , and this leads to a restricted (R) version of $\\texttt {A}_Q$ given by $\\begin{split}\\texttt {A}_{Q(R)} & = \\int _0^T [\\langle q(t),p(t) | \\mathsf {i}\\hbar \\partial /\\partial t - \\mathcal {H}(Q,P)]|q(t),p(t)\\rangle \\,\\mathrm {d}t\\\\& = \\int _0^T [ p(t)\\dot{q}(t) - H(q(t),p(t))] \\,\\mathrm {d}t\\,, \\ H(q,p)= \\langle q,p | \\mathcal {H}(Q,P)|q,p\\rangle \\,.\\end{split}$ Here we have used the relations $\\hbar \\dfrac{\\partial }{\\partial q} U(q,p) &= \\left(-\\mathsf {i}\\,P + \\dfrac{\\mathsf {i}}{2} \\,p \\right) U(q,p) = -U(q,p) \\left( \\mathsf {i}P + \\dfrac{\\mathsf {i}}{2}\\,p \\right)\\, , \\\\\\hbar \\dfrac{\\partial }{\\partial p} U(q,p) &= \\left(\\mathsf {i}\\, Q - \\dfrac{\\mathsf {i}}{2} \\,q \\right) U(q,p) = U(q,p) \\left( \\mathsf {i}\\,Q + \\dfrac{\\mathsf {i}}{2}\\,q \\right)\\, ,$ to prove that $\\mathsf {i}\\hbar \\langle q,p | \\mathrm {d}|q,p\\rangle = \\langle 0| [(P +p )\\,\\mathrm {d}q - Q\\,\\mathrm {d}p|0\\rangle = p \\,\\mathrm {d}q\\,.$ We now extend this formalism in the spirit of the present work by replacing $\\langle q(t),p(t) |\\cdot |q(t),p(t)\\rangle $ in the expression (REF ) with $\\mathrm {Tr}\\left(\\mathfrak {Q}(q(t),p(t))\\cdot \\right)$ in accordance with (REF ).",
"We also replace $\\mathcal {H}(Q,P)$ with $A_{H_c}$ , the WH quantized version of the classical $H_c$ .",
"By using (REF ) or (), and (REF ) we now deal with the action $\\begin{split}\\texttt {A}^{\\Pi }_{Q(R)} & = \\int _0^T \\mathrm {Tr}\\left[\\mathfrak {Q}(q(t),p(t))\\, \\left( \\mathsf {i}\\hbar \\partial /\\partial t - A_{H_c}\\right)\\right] \\,\\mathrm {d}t\\\\& = \\int _0^T [ p(t)\\dot{q}(t) - {H_c}(q(t),p(t))] \\,\\mathrm {d}t \\,.\\end{split}$ Stationary variations of this $\\texttt {A}^\\Pi _Q$ now yield the semi-classical Hamilton equations $\\dot{q} = \\lbrace q,{H_c}\\rbrace = \\frac{\\partial {H_c}}{\\partial p}\\, , \\quad \\dot{p} = \\lbrace p,{H_c}\\rbrace = -\\frac{\\partial {H_c}}{\\partial q}\\, .$"
],
[
"Quantization and semi-classical portraits of truncated observables",
"We consider classical motions which are geometrically restricted to hold in some subset $E$ of the configuration space, i.e.",
"the real line $\\mathbb {R}$ , and we truncate all classical observables to $E$ as $f(q,p)\\mapsto \\chi _{E}(q)f(q,p)\\equiv f_{\\chi }(q,p)\\, ,$ where $\\chi _{E}$ is the characteristic (or indicator) function of set $E$ .",
"Although these truncated observables are generally discontinuous, they can be quantized through the Weyl-Heisenberg quantization (REF ) or (REF ).",
"We obtain the $E$ -modified operators, $f_{\\chi }(q,p)\\mapsto A_{f_\\chi } &= \\frac{1}{2\\pi \\hbar }\\int _{E} \\mathrm {d}q\\int _{\\mathbb {R}} \\mathrm {d}p \\,f_\\chi (q,p) \\mathfrak {Q}(q,p)\\\\ &= \\frac{1}{2\\pi \\hbar }\\int _{\\mathbb {R}^2} \\mathrm {d}q\\,\\mathrm {d}p\\, \\Pi (q,p)\\, \\overline{\\mathfrak {F_s}}[f_\\chi ](q,p)\\,U(q,p)\\,.$ All quantization formulae given in Section (for general $\\Pi $ ) and in Section (for separable $\\Pi $ ) apply here with the change $f\\mapsto \\chi f$ .",
"In particular, the quantization of the singular $f_\\chi (q,p)= \\chi _{E}(q)$ yields what we call the “window” operator, $A_{\\chi _{E}} = \\frac{1}{\\sqrt{2\\pi \\hbar }}\\int _{\\mathbb {R}} \\mathrm {d}p \\, \\Pi (0,p)\\, \\widehat{\\chi }(p)\\,e^{\\mathsf {i}pQ}= \\overline{\\mathcal {F}}[\\Pi (0,\\cdot )\\, \\widehat{\\chi }](Q)\\equiv \\mathcal {E}(Q)\\, ,$ where $\\widehat{\\chi }(p)=\\mathcal {F}[\\chi ](p)$ .",
"Note that the Hilbert space in which act these “$E$ -modified” operators is left unchanged.",
"Thus, in position representation, one continues to deal with $\\mathcal {H}= L^2(\\mathbb {R}, \\mathrm {d}x)$ .",
"Nevertheless, if the function $\\Pi (q,p)$ is regular enough (resp.",
"smooth), our approach gives rise to a regularisation (rep. smoothing) of the constraint boundary, i.e., a “fuzzy” boundary, and also a regularisation (resp.",
"smoothing) of all discontinuous restricted observable $f_\\chi (q,p)$ introduced in the quantization map (REF )-().",
"Indeed, there is no mechanics outside the set $E\\times \\mathbb {R}$ defined by the position constraint on the classical level.",
"It is not the same on the quantum level since our quantization method allows to go beyond the boundary of this set in a way which can be smoothly rapidly decreasing, depending on the function $\\Pi (0,p)$ .",
"Consistently, the semi-classical phase space portrait of the operator (REF ) is given by (REF )-()-().",
"Let us retain (), which appears to be the most condensed ${f}_\\chi (q,p) = \\frac{1}{2\\pi \\hbar } \\left(\\overline{\\mathfrak {F_s}}\\left[\\Pi \\,\\widetilde{\\Pi }\\right]\\ast f_\\chi \\right)(q,p) \\, .$ This function, which is expected to be concentrated on the classical $E\\times \\mathbb {R}$ , should be viewed as a new classical observable defined on the full phase space $\\mathbb {R}^2$ where $q$ and $p$ keep their status of canonical variables.",
"Thus, we have here the meaningful sequence $\\begin{split}\\mathrm {virtual} \\ f(q,p) \\rightarrow \\ \\mathrm {truncated} \\ &f_\\chi (q,p)\\\\&{[{{}={}}][c]{{\\downarrow }}} \\\\\\mathrm {regularised} \\ {f}_\\chi (q,p) \\leftarrow \\ \\mathrm {quantum}\\ &A_{f_{\\chi }}\\,,\\end{split}$ allowing to establish a semi-classical dynamics à la Klauder [8], mainly concentrated on $E\\times \\mathbb {R}$ ."
],
[
"Operators in an interval and their semi-classical portraits",
"As an elementary illustration of the formalism and as it was done with coherent states in [6], we restrict our study to the bounded open interval $E= (a,b)$ , i.e., $\\chi _E(q)= \\chi _{(a,b)}(q) =\\Theta (q-a) - \\Theta (q-b)\\, ,$ where $\\Theta (q)= \\chi _{(0,+\\infty )}(q)$ is the Heaviside function.",
"At this point, we should be aware that the motion in our bounded geometry is determined by a confinement potential, such as an oscillator-like interaction, and the presence of position-dependent mass terms.",
"Before considering the details of the classical model under consideration, it is worth to discuss first the general properties of the functions $\\Pi (q,p)$ used throughout the rest of the text.",
"To this end, let us consider a function $\\Pi _{G}(q,p)$ , together with the respective symplectic Fourier transform $\\mathfrak {F_s}[\\Pi _{G}\\,\\widetilde{\\Pi }_{G}](q,p)$ , of the form $\\begin{split}\\Pi _{G}(q,p)&=e^{-\\frac{q^{2}}{2\\sigma _{\\ell }^{2}}}e^{-\\frac{p^{2}}{2\\sigma _{\\wp }^{2}}}e^{\\frac{\\gamma }{2}qp} \\, , \\\\ \\mathfrak {F_s}[\\Pi _{G}\\,\\widetilde{\\Pi }_{G}](q,p)&=\\frac{\\sigma _{\\ell }\\sigma _{p}}{2\\Lambda \\hbar }e^{-\\frac{\\Lambda ^{2}_{\\ell }q^{2}}{4\\hbar ^2}}e^{-\\frac{\\Lambda _{\\wp }^{2}p^{2}}{4\\hbar ^2}}e^{\\frac{\\Lambda _{0}}{\\hbar ^{2}}qp} \\, ,\\end{split}$ where the parameters in $\\mathfrak {F_s}[\\Pi _{G}\\,\\widetilde{\\Pi }_{G}](q,p)$ are given by $&\\Lambda _{\\ell }^{2}=\\frac{\\sigma _{\\wp }^{2}}{\\Lambda ^{2}}, \\quad \\Lambda _{\\wp }^{2}=\\frac{\\sigma _{\\ell }^{2}}{\\Lambda ^{2}} , \\quad \\Lambda _{0}=\\frac{\\sigma ^{2}_{\\ell }\\sigma _{\\wp }^{2}\\gamma }{4\\Lambda ^{2}}, \\\\&\\Lambda ^{2}=\\frac{4-\\sigma _{\\ell }^{2}\\sigma _{\\wp }^{2}\\gamma ^{2}}{4}>0,$ with $\\sigma _{\\ell }$ and $\\sigma _{\\wp }$ positive constants with dimension of length and momentum, respectively, and the real coupling constant $\\gamma $ with inverse action dimension constrained to $\\vert \\gamma \\vert <2/(\\sigma _{\\ell }\\sigma _{\\wp })$ .",
"Notice that (REF ) is a generalization of the separable function (REF ), we thus call $\\Pi _{G}(q,p)$ as the non-separable Gaussian function.",
"In general, the quantum operator $\\hat{\\chi }^{(G)}_{(a,b)}$ related to the characteristic function and the semi-classical portrait ${\\chi }_{(a,b)}^{(G)}(q)$ are respectively defined as $\\hat{\\chi }^{(G)}_{(a,b)}(Q)\\equiv B_{\\frac{\\sigma _{\\wp }}{\\sqrt{2}}}(a,b;Q), \\quad {\\chi }^{(G)}_{(a,b)}(q)\\equiv B_{\\frac{\\sigma _{\\wp }}{2}}(a,b;q),$ with $B_{\\sigma _\\wp }(a,b;x):= \\frac{1}{2}\\left( \\operatorname{Erfc}\\left[ \\frac{\\sigma _{\\wp }}{\\hbar }(x-b)\\right] - \\operatorname{Erfc}\\left[ \\frac{\\sigma _{\\wp }}{\\hbar }(x-a)\\right] \\right) \\, ,$ where Erfc$(z)$ stands for the complementary error function [14], [15].",
"Notice that both expressions in (REF ) are independent of the coupling factor $\\gamma $ , that is, the non-separability of $\\Pi _{G}(q,p)$ plays no role in the quantization of the characteristic function.",
"Nevertheless, as we discuss in the sequel, the effects of $\\gamma $ are clear once we consider more general classical observables.",
"Interestingly, the regularization introduced by $\\Pi (q,p)$ leads to a smooth function ${\\chi }^{(G)}_{(a,b)}(q)$ that approximates the original characteristic function (REF ).",
"The latter allows defining the appropriate quantum operator $\\hat{\\chi }^{(G)}_{(a,b)}(Q)$ , and consequently any truncated observable.",
"The behavior of the characteristic function is depicted in Fig.",
"REF for several values of $\\sigma _{\\wp }/\\hbar $ , where it is clear that ${\\chi }^{(G)}_{(a,b)}(q)$ converges to (REF ) at $\\sigma _{\\wp }/\\hbar \\rightarrow \\infty $ .",
"Figure: (In units of ℏ=1\\hbar =1) Semi-classical portraits χ (a,b) (G) (q){\\chi }^{(G)}_{(a,b)}(q) of the characteristic function fixed at a=1a=1 and b=5b=5, together with σ ℘ =3,5,10\\sigma _{\\wp }=3,5,10 (blue-dotted, red-dashed and black-solid, respectively)."
],
[
"Some applications: Position-dependent mass models in a bounded interval",
"A well-known problem in the quantization procedure arises in determining the quantum operators related to classical observables of the form $f(q,p)=q^{n}p^{m}$ .",
"Given that the canonical position and momentum operators do not commute (REF ), the operator ordering of the resulting quantum model is not unique [23], [24], [25], [26], [27], [28].",
"A prime example is given by the quantization of classical models associated with position-dependent mass (PDM) terms [4], [5], where different quantization rules lead to different quantum Hamiltonians with different spectrum, see [29] for details.",
"Despite the ordering problem, PDM models find interesting application in the study of heterostructures [7], [30], [31], and recently in the study of supersymmetric models in quantum mechanics [32], [33], [34], [35].",
"Throughout this section, to illustrate the usefulness of the WH covariant integral quantization, we consider a classical PDM model corresponding to the family of nonlinear oscillators of the form $H(q,p)=\\frac{p^{2}}{2m(q)}+\\frac{V_{0}}{2}(q-q_{0})^{2}, \\quad m(q)=\\frac{m_{0}L^{2}}{(q-a)(b-q)}, \\quad m_{0}>0.$ The latter arises as a modification of the nonlinear oscillator discussed in [36].",
"The constant $L$ is a characteristic length, e.g.",
"$L=b-a$ .",
"In this form, the related quantum operators are determined with the appropriate use of the $\\Pi (q,p)$ function.",
"Although, from (REF ), we notice that $m(q)$ is a physical mass-term only in the interval $q\\in (a,b)$ , that is, $m(q)$ is a positive function.",
"To preserve the physical structure of the model (REF ), we introduce the truncated classical Hamiltonian of the form $H_{\\chi }(q,p)\\equiv \\chi _{(a,b)}(q) H(q,p)=\\frac{p^{2}}{2m_{\\chi }(q)}+V_{\\chi }(q),$ where the truncated position-dependent mass and potential terms are respectively given by $\\begin{split}&\\frac{1}{m_{\\chi }(q)}\\equiv \\mathfrak {m}_{\\chi }(q)=\\frac{(q-a)(b-q)}{m_{0}L^{2}}\\chi _{(a,b)}(q), \\\\&V_{\\chi }(q)=\\frac{V_{0}}{2}(q-q_{0})^{2}\\chi _{(a,b)}(q).\\end{split}$ From the constraint in the bounded interval, we realize that the shape of the truncated oscillator potential $V_{\\chi }(q)$ is steered by the potential minimum $q_{0}$ .",
"For details, see Fig.",
"REF .",
"Figure: (a) Classical truncated-potential V χ (q)V_{\\chi }(q) given in () for V 0 =3V_{0}=3, together with q 0 =3q_{0}=3 (solid-black), q 0 =3.5q_{0}=3.5 (dashed-blue), and q 0 =5q_{0}=5 (dotted-red).",
"(b) (In units of ℏ=1\\hbar =1) Regularized semi-classical mass term m χ (G) (q)m^{(G)}_{\\chi }(q), given in (), for σ ℘ =3,5,10\\sigma _{\\wp }=3,5,10 (dotted-blue, dashed-red and solid-black, respectively).Before proceeding, it is useful to compute the quantum operator and semi-classical portrait related to the mass-term given in (REF ).",
"Following () and (REF ), we notice that the function $\\begin{split}\\mathfrak {M}_{\\sigma _{\\wp }}(x)&=\\frac{1}{m_{0}L^2}\\left[(x-a)(b-x)-\\frac{\\hbar ^2}{\\sigma _{\\wp }^{2}} \\right]B_{\\frac{\\sigma _{\\wp }}{\\sqrt{2}}}(a,b;x) \\\\&+\\frac{1}{m_{0}L^2}\\frac{\\hbar }{\\sqrt{2\\pi \\sigma _{\\wp }^{2}}}\\left[ (x-a)e^{-\\frac{\\sigma _{\\wp }^{2}}{2\\hbar ^2}(x-b)^{2}}-(x-b)e^{-\\frac{\\sigma _{\\wp }^{2}}{2\\hbar ^2}(x-a)^{2}} \\right],\\end{split}$ leads to the respective quantum and semi-classical mass terms conveniently written in the form $\\frac{1}{M^{(G)}_{\\chi }(Q)}\\equiv \\widehat{\\mathfrak {m}}_{\\chi }(Q)=\\mathfrak {M}_{\\sigma _{\\wp }}(Q), \\quad \\frac{1}{m^{(G)}_{\\chi }(q)}\\equiv {\\mathfrak {m}}_{\\chi }(q)=\\mathfrak {M}_{\\frac{\\sigma _{\\wp }}{\\sqrt{2}}}(q)\\,.$ The behavior of $m^{(G)}_{\\chi }(q)$ is depicted in Fig.",
"REF for several values of the Gaussian width $\\frac{\\sigma _{\\wp }}{\\hbar }$ .",
"Notice that, due the regularization effects of $\\Pi _{G}(q,p)$ , the singularities of the initial mass-term $m(q)$ in (REF ) have been smoothed in the semi-classical portrait, leading to a function that extends beyond the bounded interval $q\\in (a,b)$ .",
"In turn, the classical limit is recovered for $\\sigma _{\\wp }/\\hbar \\rightarrow \\infty $ , where the function $B_{\\sigma }(a,b;q)$ converges, in the sense of a distribution, to the characteristic function (REF ).",
"For the non-separable Gaussian function $\\Pi _{G}(q,p)$ , it can be shown that a truncated classical function $f_{\\chi }(q,p)=p^{2}h(q)\\chi _{(a,b)}(q)$ gives rise to the semi-classical portrait ${f}_{p^{2}h_{\\chi }}$ of the form ${f}_{p^{2}h_{\\chi }(q)}={f}_{h_{\\chi }}\\left(p+\\hbar ^{2}\\gamma \\frac{{f}_{h_{\\chi }}^{\\prime }}{{f}_{h_{\\chi }}} \\right)^{2}+{f}_{h_{\\chi }}\\left[\\hbar ^{4}\\gamma ^{2}\\left(\\frac{{f}_{h_{\\chi }}^{\\prime }}{{f}_{h_{\\chi }}}\\right)^{\\prime }+ \\frac{2\\hbar ^{2}}{\\sigma _{\\ell }^{2}} \\right]$ with ${f}_{h_{\\chi }}\\equiv {f}_{h_{\\chi }}(q)$ the semi-classical portrait of $f(q,p)=h(q)\\chi _{(a,b)}(q)$ , and the notation ${f}^{\\prime }=\\frac{\\partial {f}}{\\partial q}$ denotes the partial derivative of any semi-classical function ${f}$ with respect to $q$ .",
"From (REF ), together with $1/m^{(G)}_{\\chi }$ in (REF ), we can easily compute the semi-classical portrait of the truncated Hamiltonian (REF ).",
"After some calculations we get ${H}^{(G)}_{\\chi }(q,p):=\\frac{1}{2m^{(G)}_{\\chi }}\\left(p-\\hbar ^{2}\\gamma \\frac{[m_{\\chi }^{(G)}]^{\\prime }}{m_{\\chi }^{(G)}} \\right)^{2}+{V}_{eff}^{(G)}(q),$ with $[m^{(G)}_{\\chi }]^{\\prime }\\equiv \\partial m^{(G)}_{\\chi }/\\partial q$ , and ${V}_{eff}^{(G)}(q)$ an effective potential term of the form ${V}_{eff}^{(G)}(q):=\\frac{V_{0}}{2}\\left[(q-q_{0})^{2}+\\frac{2\\hbar ^{2}}{\\sigma _{\\wp }^{2}}\\right]{\\chi }^{(G)}_{(a,b)}(q)\\\\+\\frac{1}{2m^{(G)}_{\\chi }}\\left[-\\hbar ^{4}\\gamma ^{2}\\left(\\frac{[m^{(G)}_{\\chi }]^{\\prime }}{m^{(G)}_{\\chi }}\\right)^{\\prime }+\\frac{2\\hbar ^{2}}{\\sigma _{\\ell }^{2}}\\right]\\\\-\\frac{V_{0}\\hbar }{\\sqrt{4\\pi \\sigma _{\\wp }^{2}}}\\left[(q+b-2q_{0})e^{-\\frac{\\sigma _{\\wp }^{2}}{4\\hbar ^{2}}(q-b)^{2}}-(q+a-2q_{0})e^{-\\frac{\\sigma _{\\wp }^{2}}{4\\hbar ^{2}}(q-a)^{2}}\\right].$ The effective potential is composed of several parts.",
"Besides the effects of the truncated oscillator-like interaction, the effective potential contains the effects of the geometrical constraint imposed on the classical model, which is reflected in the regularized characteristic function ${\\chi }_{(a,b)}(q)$ .",
"Also, the initial mass $m(q)$ contributes with an additional regularized term in the potential through $m^{(G)}_{\\chi }(q)$ , and its derivatives.",
"Finally, the effects of the coupling factor $\\gamma $ are present in the potential as well, which also induces a minimal coupling term [38] in the kinetic energy term of the semi-classical Hamiltonian (REF ).",
"This is an interesting phenomenon that can be interpreted as the existence of an effective vector potential of the form $\\vec{A}(q)=\\hbar ^{2}\\gamma [m^{(G)}_{\\chi }]^{\\prime }/m^{(G)}_{\\chi } \\hat{n}$ , where $\\hat{n}$ is a unit vector in the same direction as the canonical coordinate $q\\hat{n}$ .",
"Since we are dealing with a one-dimensional system, the direction of the vector potential is not relevant, and henceforth will be omitted.",
"We want to stress out that a striking feature of the semi-classical portrait relies upon the Hamiltonian structure, that is, we can determine the dynamics of the semi-classical model through the Hamilton equations of motion (REF ).",
"Although the equations of motion would lead to a nonlinear differential equation associated with the position $q(t)$ , as it happens with the classical PDM model of [36], we may still get information about the dynamics of the system from its respective phase-space trajectories.",
"The latter is achieved by fixing the total energy of the system as a constant ${H}_{\\chi }^{(G)}(q,p)=E$ .",
"Such a procedure is a legitimate operation, since the semi-classical Hamiltonian is time-independent, and thus the energy is a conserved quantity [37], [38].",
"It is worth notice that, from the phase-space trajectories themselves, we can not obtain information about the direction of motion of the particle.",
"We thus compute the vector quantity $\\vec{\\mathcal {F}}$ defined in terms of the canonical variables as [38] $\\vec{\\mathcal {F}}=(\\dot{q}(q,p),\\dot{p}(q,p)),$ where $\\dot{q}(q,p)$ and $\\dot{p}(q,p)$ are determined from the Hamilton equations of motion (REF ).",
"We have $\\dot{q}(q,p)=\\frac{1}{m^{(G)}_{\\chi }}\\left(p-\\hbar ^{2}\\gamma \\frac{[m^{(G)}_{\\chi }]^{\\prime }}{m^{(G)}_{\\chi }}\\right),$ $\\begin{split}\\dot{p}=\\frac{[m^{(G)}_{\\chi }]^{\\prime }}{2[m^{(G)}_{\\chi }]^2}&\\left[ p-\\hbar ^{2}\\gamma \\left(3\\frac{[m^{(G)}_{\\chi }]^{\\prime }}{{m}_{\\chi }}-2\\frac{[m^{(G)}_{\\chi }]^{\\prime \\prime }}{[m^{(G)}_{\\chi }]^{\\prime }}\\right)\\right] \\\\&\\times \\left(p-\\hbar ^{2}\\gamma \\frac{[m^{(G)}_{\\chi }]^{\\prime }}{m^{(G)}_{\\chi }}\\right)-\\frac{\\partial {V}_{eff}^{(G)}}{\\partial q}.\\end{split}$ In this form, by combining the phase-space trajectories for a fixed total energy, together with the vector field $\\vec{\\mathcal {F}}$ , we obtain a complete description of the system dynamics, we discuss such details in the sequel.",
"Alternatively, from (REF ), the canonical momentum $p$ is determined in terms of the position-dependent mass term and the particle velocity.",
"This allows us to obtain an equivalent representation of the system dynamics through the reparametrized Hamiltonian ${\\mathfrak {h}}_{\\chi }^{(G)}(q,\\dot{q})\\equiv {H}^{(G)}_{\\chi }(q,p(q,\\dot{q}))=\\frac{m^{(G)}_{\\chi }\\dot{q}^{2}}{2}+{V}_{eff}^{(G)}(q) \\, .$ From the general results obtained so far, it is worth separating the discussion in some particular cases."
],
[
"Separable case $\\gamma =0$",
"As we have previously pointed out, for $\\gamma =0$ , the function $\\Pi _{G}(q,p)$ reduces to the separable one introduced in (REF ).",
"Additionally, let us also consider a null oscillator-like interaction $V_{0}=0$ .",
"The latter leads, in general, to a non-null effective potential that arises from the bounded interval of the initial model, as in the case discussed in [6], but in our model we also have the effects of the regularized PDM term $1/m^{(G)}_{\\chi }$ .",
"The behavior of the effective potential is depicted in Fig.",
"REF , from where it is evident that ${V}_{\\chi }^{(G)}$ is regularized beyond the initial bounded interval $q\\in (a,b)$ .",
"Given the lack of returning points of the potential, the particle is not allowed to perform a bounded motion, this fact is confirmed by the information obtained from the phase-space trajectories (in units of $\\hbar =1$ ), with a fixed total energy ${H}_{\\chi }^{(G)}(q,p)=E=1/4$ , depicted in Fig.",
"REF .",
"For $\\sigma _{\\wp }=2$ , the particle can not pass from one region of the interval to the other, since the potential energy prohibits such a transition.",
"Moreover, the particle tends to be attracted to either of the regularized walls generated by the mass-term.",
"At the same energy $E=1/4$ , but for higher values of $\\sigma _{\\wp }$ , the particle has enough energy to transit from one wall to the other, nevertheless, the particle never returns to the initial wall, that is, the system does not allow bouncing behavior.",
"Notice that in the classical limit, $\\sigma _{\\wp }/\\hbar \\rightarrow \\infty $ and $\\sigma _{\\ell }/\\hbar \\rightarrow \\infty $ , the effective potential vanishes inside the bounded interval.",
"Figure: (In units of ℏ=1\\hbar =1) Regularized semi-classical effective potential V eff (G) (q)V_{eff}^{(G)}(q) with the regularization parameters σ ℓ =σ ℘ =4\\sigma _{\\ell }=\\sigma _{\\wp }=4 (a) and σ ℓ =σ ℘ =10\\sigma _{\\ell }=\\sigma _{\\wp }=10 (b).",
"The rest of parameters are V 0 =3V_{0}=3, a=1a=1, b=5b=5, m 0 =1m_{0}=1.",
"The position of the potential minimum takes the values q 0 =3q_{0}=3 (solid-black), q 0 =3.5q_{0}=3.5 (dashed-blue), and q 0 =5q_{0}=5 (dotted-red).In turn, for $V_{0}>0$ , we have different scenarios.",
"For instance, if $a<q_{0}<b$ , the oscillator-like interaction gets truncated in such a way that classical confinement is possible for finite range of energies $E_{m}<E<E_{th}$ , with $E_{m}$ and $E_{th}$ the minimum and threshold energies, respectively.",
"The behavior of the effective potential is depicted in Fig.",
"REF for several values of the potential minimum position $q_{0}$ and the regularization parameters $\\sigma _{\\ell }$ and $\\sigma _{wp}$ .",
"In particular, in Fig.",
"REF , for $q_{0}=3$ (solid-black) we can see the effects of the regularized truncated potential, where a symmetric potential is obtained, and energies of classical confinement are present.",
"For $q_{0}=3.5$ (dashed-blue), the effective potential becomes asymmetric, and the energy threshold $E_{th}$ is lower compared with the symmetric case.",
"For increasing ratios $\\sigma _{\\wp }/\\hbar $ and $\\sigma _{\\ell }/\\hbar $ the effective potential displays a behavior more similar to the classical one, as shown in Fig.",
"REF .",
"Clearly, in the classical limit, we recover the initial truncated-potential previously shown in Fig.",
"REF .",
"The classical confinement in the potential is interpreted in the phase-space description as closed trajectories.",
"The latter is depicted in Figs.",
"REF -REF for several different energies.",
"Contrary to an oscillator interaction, in our case the bounded motion depends on the initial condition as well.",
"Additionally, comparing the dashed-blue trajectories for $q_{0}=3$ (Fig.",
"REF ) and that for $q_{0}=3.5$ (Fig.",
"REF ), with both being fixed at the same energy $E=2$ , it is clear that the presence of classical confinement strongly depends on the position of the potential minimum $q_{0}$ .",
"Moreover, in the critical cases $q_{0}=a$ and $q_{0}=b$ , the effective potential prohibits the existence of confinement, as depicted in Fig.",
"REF .",
"Alternatively, we can analyze the dynamics of the particle when it approaches the regularized walls.",
"To this end, we depict in Fig.",
"REF the trajectories of ${\\mathfrak {h}}_{\\chi }^{(G)}$ in the $q$ -$\\dot{q}$ plane, computed from (REF ).",
"In the latter, for the sake of clarity, we have used the same parameters as in Fig.",
"REF .",
"As mentioned above, the particle never returns from the walls, such a conclusion is confirmed from Figs.",
"REF -REF , where the velocity $\\dot{q}$ drops close to zero as the particle approaches either of the walls.",
"Also, the particle is allowed to travel beyond the initial bounded interval $[a,b]$ , because of the regularization effects of $\\Pi _{G}(q,p)$ , where it spends an infinite amount of time, that is, the particle never returns.",
"Additional details are provided in the sequel.",
"Figure: (In units of ℏ=1\\hbar =1) Phase-space trajectories of H χ (G) (q,p)≡E{H}_{\\chi }^{(G)}(q,p)\\equiv E, given in (), for the separable case γ=0\\gamma =0 and the parameters a=1a=1, b=5b=5, σ ℓ =σ ℘ =4\\sigma _{\\ell }=\\sigma _{\\wp }=4, V 0 =3V_{0}=3, m 0 L 2 =1m_{0}L^{2}=1.",
"The curves correspond to energies E=0.5E=0.5 (solid-black), E=2E=2 (dashed-blue), and E=3.5E=3.5 (dotted-red).",
"The arrows depict ℱ →\\vec{\\mathcal {F}} given in ().",
"The position minimum is being fixed as q 0 =3q_{0}=3 (a), q 0 =3.5q_{0}=3.5 (b) and q 0 =5q_{0}=5.Figure: (In units of ℏ=1\\hbar =1) Trajectories 𝔥 χ (G) (q,q ˙)=E{\\mathfrak {h}}_{\\chi }^{(G)}(q,\\dot{q})=E, given in (), for the separable case γ=0\\gamma =0 and the parameters a=1a=1, b=5b=5, σ ℓ =σ ℘ =4\\sigma _{\\ell }=\\sigma _{\\wp }=4, V 0 =3V_{0}=3, m 0 L 2 =1m_{0}L^{2}=1.",
"The curves correspond to energies E=0.5E=0.5 (solid-black), E=2E=2 (dashed-blue), and E=3.5E=3.5 (dotted-red).",
"The position minimum is being fixed as q 0 =3q_{0}=3 (a), q 0 =3.5q_{0}=3.5 (b) and q 0 =5q_{0}=5."
],
[
"Non-separable case $\\gamma \\ne 0$",
"In this case, the global behavior of the effective potential is similar to that of the previous case depicted in Fig.",
"REF .",
"Nevertheless, the most relevant change comes from the kinetic energy term of the Hamiltonian (REF ) in the form of a minimal coupling.",
"To get more insight in the dynamics of the system under the influence of such a minimal coupling, we depict the phase-space trajectories in Fig.",
"REF , with the same parameters as in Fig.",
"REF .",
"It is worth to recall that the coupling factor is bounded by the inequality ().",
"Given that the Gaussian widths have been fixed (in units of $\\hbar =1$ ) as $\\sigma _{\\wp }=4$ and $\\sigma _{\\ell }=4$ , we thus choose the coupling factor as $\\gamma =0.1<1/8$ .",
"With the latter, we can see the differences that arises from the non-separability of the function $\\Pi _{G}(q,p)$ .",
"From Figs.",
"REF -REF , it is clear that the trajectories, both open and closed, are deviated with respect to the $\\gamma =0$ case.",
"As we have pointed out previously, the minimal coupling is related to a vector potential of the form $\\vec{A}(q)=\\gamma [m^{(G)}_{\\chi }]^{\\prime }/m^{(G)}_{\\chi } \\hat{n}$ , for which there is an associated magnetic field $\\vec{B}=\\nabla \\times \\vec{A}$ .",
"It is worth to recall that the minimal coupling term is a purely quantum effect, for it is proportional to $\\hbar $ that vanishes in the classical limit.",
"We complement this section by discussing the trajectories in the $q$ -$\\dot{q}$ plane, depicted in Fig.",
"REF .",
"From ${\\mathfrak {h}}^{(G)}_{\\chi }(q,\\dot{q})$ given in (REF ) we see that the minimal coupling term $A(q)$ does not contribute in the dynamics, but the effects of the non-separability due the coupling factor $\\gamma $ are still present in the effective potential ${V}_{eff}(q)$ .",
"To further understand the dynamics, we can compute from (REF ) the equation of motion for $q(t)$ with ease, but the resulting equation is nonlinear, and determining an analytic solution for $q(t)$ is not possible.",
"Nevertheless, we can overcome such an issue by performing some numerical calculations.",
"Such information is presented in Fig.",
"REF -REF for the initial conditions $A$ and $B$ depicted in Figs.",
"REF -REF , respectively.",
"From those figures we see that, in Fig.",
"REF , the initial conditions allow a closed trajectory, whereas in Fig.",
"REF the particle spends an infinite amount of time traveling through the wall.",
"Figure: (In units of ℏ=1\\hbar =1) Phase-space trajectories of H χ (G) (q,p)=E{H}_{\\chi }^{(G)}(q,p)=E, given in (), for the non-separable case γ=0.3\\gamma =0.3 and the parameters a=1a=1, b=5b=5, σ ℓ =σ ℘ =4\\sigma _{\\ell }=\\sigma _{\\wp }=4, V 0 =3V_{0}=3, m 0 L 2 =1m_{0}L^{2}=1.",
"The curves correspond to energies E=0.5E=0.5 (solid-black), E=2E=2 (dashed-blue), and E=3.5E=3.5 (dotted-red).",
"The arrows depict ℱ →\\vec{\\mathcal {F}} given in ().Figure: (In units of ℏ=1\\hbar =1) (a) Trajectories of 𝔥 χ (G) (q,q ˙)=E{\\mathfrak {h}}_{\\chi }^{(G)}(q,\\dot{q})=E , determined from (), for the non-separable function Π(q,p)\\Pi (q,p) with the parameters a=1a=1, b=5b=5, V 0 =3V_{0}=3, m 0 =L 2 =1m_{0}=L^{2}=1, together with σ ℓ =σ ℘ =4\\sigma _{\\ell }=\\sigma _{\\wp }=4 and γ=0.1\\gamma =0.1.",
"The curves correspond to energies E=0.5E=0.5 (solid-black), E=2E=2 (dashed-blue), and E=3.5E=3.5 (dotted-red).",
"(b-c) Numerical solutions for the position q(t)q(t) (solid-blue) and velocity q ˙(t)\\dot{q}(t) (dashed-red) for the initial conditions AA and BB depicted in (a)"
],
[
"Quantum regularized PDM model",
"For completeness, we determine the quantum model related to the truncated Hamiltonian (REF ).",
"We proceed similarly as in the semi-classical approach, and after some calculations we get $\\widehat{H}_{\\chi }^{(G)}:=\\frac{1}{2} \\left\\lbrace \\frac{1}{2M^{(G)}_{\\chi }(Q)},\\left( P-\\frac{\\hbar ^{2}\\gamma }{2}\\frac{[M^{(G)}_{\\chi }(Q)]^{\\prime }}{M^{(G)}_{\\chi }(Q)}\\right)^{2} \\right\\rbrace + \\hat{V}_{eff}^{(G)}(Q),$ where $\\lbrace \\hat{A},\\hat{B}\\rbrace =\\hat{A}\\hat{B}+\\hat{B}\\hat{A}$ stands for the anti-commutator, and we have used the notation $[M^{(G)}_{\\chi }]^{\\prime }\\equiv \\left.",
"\\frac{\\partial }{\\partial x} \\frac{1}{\\mathfrak {M}_{\\sigma _{\\wp }}(x)}\\right|_{x\\rightarrow Q} \\, ,$ with $\\mathfrak {m}_{\\sigma _{\\wp }}(x)$ given in (REF ).",
"The term $\\hat{V}_{eff}^{(G)}(Q)$ stands for a quantum effective potential term of the form $\\hat{V}^{(G)}_{eff}(Q):=\\frac{V_{0}}{2}\\left[(Q-q_{0})^{2}+\\frac{\\hbar ^{2}}{\\sigma _{\\wp }^{2}} \\right]\\hat{\\chi }_{(a,b)}^{(G)}(Q)\\\\+\\frac{1}{M^{(G)}_{\\chi }(Q)}\\left[-\\frac{\\hbar ^{4}\\gamma ^{2}}{4}\\left( \\frac{[M^{(G)}_{\\chi }(Q)]^{\\prime }}{M^{(G)}_{\\chi }(Q)} \\right)^{\\prime } +\\frac{\\hbar ^{2}}{\\sigma _{\\ell }^{2}}\\right]\\\\-\\frac{V_{0}\\hbar }{\\sqrt{2\\pi \\sigma _{\\wp }^{2}}}\\left[(Q+b-2q_{0})e^{-\\frac{\\sigma _{\\wp }^{2}}{2\\hbar ^{2}}(Q-b)^{2}}-(Q+a-2q_{0})e^{-\\frac{\\sigma _{\\wp }^{2}}{2\\hbar ^{2}}(Q-a)^{2}}\\right].$ Notice that, the kinetic energy term in (REF ) corresponds to a symmetrical PDM ordering introduced by Gora-Williams [30] in the study of band-structures in slowly graded semiconductors, and later generalized by von-Roos [31].",
"Nevertheless, our model possesses an additional minimal coupling term that depends on the regularized quantized mass-term $M^{(G)}_{\\chi }$ .",
"As discussed in Section , the apodization $\\Pi (q,p)$ is equivalent to a trace-class density operator $\\mathfrak {Q}_{0}$ , where both quantities are related by (REF ).",
"From our model, the non-separable Gaussian function (REF ) allows determining a unique density operator $\\mathfrak {Q}^{(G)}_{0}$ .",
"In the Fock basis $\\lbrace \\vert n \\rangle \\rbrace _{n=0}^{\\infty }$ , the upper-diagonal matrix elements $\\mathfrak {Q}^{(G)}_{0;(n+2M,n)}\\equiv \\langle n+2M\\vert \\mathfrak {Q}^{(G)}_{0}\\vert n\\rangle $ , with $n,M=0,1,\\cdots $ , are given in units of $\\hbar =1$ by $\\begin{aligned}\\mathfrak {Q}^{(G)}_{0;(n+2M,n)}&=\\frac{\\sqrt{n!(n+2M)!}(2M)!",
"}{2^{2M}\\Delta _{\\ell }\\Delta _{\\wp }\\Delta _{\\wp }^{2M}} \\left( \\mathfrak {G}_{1}-\\frac{2i\\gamma }{\\Delta _{\\ell }^{2}}\\mathfrak {G}_{2} \\right) \\, ,\\end{aligned}$ where $\\nonumber \\begin{aligned}\\mathfrak {G}_{1}:=\\sum _{k=0}^{M}&\\sum _{r=0}^{n}\\sum _{s=0}^{r}\\frac{(-1)^{r+k}\\left(\\tfrac{1}{2}+k\\right)_{s}\\left(\\tfrac{1}{2}+N-k\\right)_{r+s}}{s!k!(r-s)!(M-k)!(n-r)!(2M+r)!",
"}\\left(\\frac{\\Delta _{\\wp }}{\\Delta _{\\ell }}\\right)^{2k+2s}\\\\&\\times \\frac{1}{\\Delta _{\\wp }^{2r}}\\,{}_{2}F_{1}\\left(\\left.\\begin{split} \\tfrac{1}{2}+s+k,\\tfrac{1}{2}+M+r-s-k \\\\ \\tfrac{1}{2} \\end{split}\\right|\\frac{\\gamma ^{2}}{4\\Delta _{\\ell }^{2}\\Delta _{\\wp }^{2}} \\right)\\end{aligned}$ $\\nonumber \\begin{aligned}\\mathfrak {G}_{2}:=\\sum _{k=0}^{M}&\\sum _{r=0}^{n}\\sum _{s=0}^{r}\\frac{(-1)^{r+k}\\left(\\tfrac{3}{2}+k\\right)_{s}\\left(\\tfrac{1}{2}+N-k\\right)_{r+s}}{s!k!(r-s)!(M-k-1)!(n-r)!(2M+r)!",
"}\\left(\\frac{\\Delta _{\\wp }}{\\Delta _{\\ell }}\\right)^{2k+2s}\\\\&\\times \\frac{1}{\\Delta _{\\wp }^{2r}}\\,{}_{2}F_{1}\\left(\\left.\\begin{split} \\tfrac{3}{2}+s+k,\\tfrac{1}{2}+M+r-s-k \\\\ \\tfrac{3}{2} \\end{split}\\right|\\frac{\\gamma ^{2}}{4\\Delta _{\\ell }^{2}\\Delta _{\\wp }^{2}} \\right)\\end{aligned}$ with $\\Delta ^{2}_{\\ell }=\\tfrac{1}{2}+\\tfrac{1}{\\sigma _{\\ell }^{2}}$ , $\\Delta ^{2}_{\\wp }=\\tfrac{1}{2}+\\tfrac{1}{\\sigma _{\\wp }^{2}}$ .",
"The self-adjointness of $\\mathfrak {Q}_{0}$ implies that the lower-diagonal elements are $\\mathfrak {Q}^{(G)}_{0;(n,n+2M)}:=\\left[\\mathfrak {Q}^{(G)}_{0;(n+2M,n)}\\right]^{*}$ .",
"In particular, after fixing the parameters to $\\sigma ^{2}_{\\ell }=2\\ell ^{2}\\frac{\\kappa _{R}}{\\vert \\kappa \\vert ^{2}} \\, ,\\quad \\sigma ^{2}_{\\wp }=2\\wp ^{2}\\kappa _{R} \\, , \\quad \\gamma =-\\frac{\\kappa _{I}}{\\kappa _{R}} \\, ,$ with $\\kappa $ given in (REF ), we recover the density operator related to the squeezed vacuum state, that is, $\\mathfrak {Q}_{0}^{(\\xi )}=S(\\xi )\\vert \\psi _{\\ell } \\rangle \\langle \\psi _{\\ell } \\vert S^{\\dagger }(\\xi )$ , with $S(\\xi )$ defined in (REF ).",
"We still have freedom in the remaining parameters in (REF ), from which an interesting case is achieved by considering $\\kappa =1$ ($\\xi =\\eta =0$ ) and $\\sigma _{\\ell }=\\sigma _{\\wp }=\\sigma >0$ , this leads to the diagonal density operator $\\mathfrak {Q}_{0}^{th}=\\frac{1}{\\Delta ^{2}}\\sum _{n=0}^{\\infty }\\left(1-\\frac{1}{\\Delta ^{2}}\\right)^{n}\\vert n \\rangle \\langle n\\vert \\, , \\quad \\Delta ^{2}=\\frac{1}{2}+\\frac{1}{\\sigma ^{2}} \\, ,$ which is nothing but a thermal state, as discussed in [1].",
"Finally, the most simple case is determined after fixing $\\sigma =\\sqrt{2}$ ($\\Delta =1$ ), such that $\\mathfrak {Q}^{0}_{0}=\\vert \\psi _{\\ell } \\rangle \\langle \\psi _{\\ell } \\vert $ .",
"In this form, we recover the coherent state quantization previously discussed in the literature [1]."
],
[
"Conclusion",
"In this work, we have explored a quantization mechanism for classical systems defined in a finite interval based on the Weyl-Heisenberg covariant integral quantization.",
"The essential feature of this approach lies in the use of a smooth function $\\Pi (q,p)$ , defined in the whole classical phase-space $\\mathbb {R}^{2}$ , that regularizes the discontinuities present in the classical model.",
"As a result, we obtain quantum models defined in terms of genuinely self-adjoint operators in the respective Hilbert space, and therefore there is no need to include self-adjoint extensions in terms of unbounded operators.",
"Instead, we have at our disposal a set of regularization parameters such that the quantum model can be adjusted to the physical system under consideration while preserving the regularity of the operators.",
"The latter is an essential feature of our approach.",
"On the other hand, the $\\Pi (q,p)$ function defines a unique density operator $\\mathfrak {Q}$ such that, after averaging the operators, leads to a semi-classical portrait of the quantum model, that is, a classical-like model in which $\\hbar $ is still present.",
"In this form, the formalism of Klauder [8], [9] is extended such that the action, defined in terms of the semi-classical portrait of the quantum Hamiltonian, is minimized.",
"From the latter, a set of Hamilton equations is determined, which allows us to treat the semi-classical system within the well-known Hamiltonian formalism.",
"An interesting application of the quantization procedure is provided by a classical model with an oscillator interaction and a position-dependent mass term whose physicality (positive-definitive mass) is only valid in a finite interval.",
"In this form, the truncation of the domain to a constrained geometry allows defining the resulting classical model properly, from which we obtain effective infinite-walls at the end of the interval domain.",
"In such a case, a non-separable Gaussian function $\\Pi (q,p)$ regularizes the discontinuities in the truncated potential and the mass-term.",
"For instance, in the semi-classical level, an oscillatory motion is obtained for total mechanical energies lower than the effective potential threshold.",
"The latter depends strongly on the regularization parameters and the domain of the finite-interval.",
"It is explicitly shown that, for some intervals, oscillatory motion is completely forbidden, since the energy threshold is zero.",
"The particle travels toward either of the walls, after which it never returns.",
"The maximal trapping is achieved for finite-intervals such that the effective potential becomes symmetric with respect to its minimum, that is, the energy threshold reaches its maximum value.",
"In those cases, for mechanical energies below the threshold, the particle performs harmonic motion.",
"In contrast, for mechanical energies higher than the threshold, the particle escapes the confinement potential and travels toward one of the walls.",
"A striking feature of the non-separable function $\\Pi (q,p)$ is given by the appearance of a minimal coupling in the form of an effective magnetic potential term for both the regularized quantum and semi-classical Hamiltonians.",
"This is a purely quantum effect that arises from the non-separability of $\\Pi (q,p)$ , which is controlled by a coupling parameter.",
"In this form, the resulting quantum model is governed by a generalization of the PDM Hamiltonian with the Gora-Williams ordering [30].",
"Therefore, the minimal coupling vanishes in either the classical limit or by tuning the coupling the parameter in such a way that $\\Pi (q,p)$ becomes a separable function.",
"In this regard, it is worth exploring in detail which other properties would arise from the non-separability of the function $\\Pi (q,p)$ .",
"Results in this direction will be reported elsewhere."
],
[
"Harmonic analysis on $\\mathbb {C}$ or {{formula:95ebb3ab-90d4-4488-a041-8442cacf0b99}} and symbol calculus",
"We give here all formulas with $\\hbar =1$ ."
],
[
"In terms of $z=\\frac{q+\\mathsf {i}p}{\\sqrt{2}}$ and {{formula:c33b44e3-1530-4e41-993d-06323cb626b6}}",
" Symplectic Fourier transform on $\\mathbb {C}$ (for a sake of simplicity, we write $f(z,\\bar{z}) \\equiv f(z)$ ) $ \\mathfrak {f_s}[f](z)&=\\int _{\\mathbb {C}} e^{ z \\bar{\\xi }-\\bar{z} \\xi } f(\\xi )\\, \\frac{\\mathrm {d}^2 \\xi }{\\pi }= \\int _{\\mathbb {C}} e^{2\\mathsf {i}\\,\\mathrm {Im}(z \\bar{\\xi })} f(\\xi )\\, \\frac{\\mathrm {d}^2 \\xi }{\\pi }\\\\ &=\\int _{\\mathbb {C}} e^{z\\circ \\xi } f(\\xi )\\, \\frac{\\mathrm {d}^2 \\xi }{\\pi }\\quad \\mbox{(notation often used here)}\\, .$ Dirac-Fourier formula $\\mathfrak {f_s}[1](z)= \\int _{\\mathbb {C}}e^{z \\circ \\xi }\\, \\dfrac{\\mathrm {d}^2 \\xi }{\\pi } = \\int _{\\mathbb {R}^2}e^{-\\mathsf {i}(qy-px)}\\, \\dfrac{\\mathrm {d}x\\,\\mathrm {d}y}{2\\pi } = 2\\pi \\delta (q)\\,\\delta (p)=\\pi \\delta ^{2}(z)\\, .$ The symplectic Fourier transform is its inverse: it is an involution $\\mathfrak {f_s}[\\mathfrak {f_s}[f]](z) = f(z) \\ \\Leftrightarrow \\ \\mathfrak {f_s}\\,\\mathfrak {f_s} = \\mathfrak {f_s}^2 = I\\, .$ The symplectic Fourier transform commutes with the parity operator $\\mathfrak {f_s}= \\mathsf {P}\\,\\mathfrak {f_s}\\,\\mathsf {P}\\, , \\quad (\\mathsf {P}\\,f)(z)=f(-z)= \\tilde{f}(z)\\,, \\ \\tilde{f}(z):= f(-z)\\, .$ Reflected symplectic Fourier transform $ \\overline{\\mathfrak {f_s}}[f](z)=\\int _{\\mathbb {C}} e^{-z\\circ \\xi } f(\\xi )\\, \\frac{\\mathrm {d}^2 \\xi }{\\pi }= \\mathfrak {f_s}[f](-z)= \\mathfrak {f_s}\\left[\\tilde{f}\\right](z)= \\overline{\\mathfrak {f_s}\\left[\\bar{f}\\right](z)}\\, .$ The reflected symplectic Fourier transform is its inverse $\\overline{\\mathfrak {f_s}}\\,\\overline{\\mathfrak {f_s}}= I \\,.$ Factorization of the parity operator $\\overline{\\mathfrak {f_s}}\\mathfrak {f_s}= \\mathfrak {f_s}\\overline{\\mathfrak {f_s}}= \\mathsf {P}\\,.$ Symplectic Fourier transform and translation with $(\\mathsf {t}_z\\,f)(z^{\\prime }):= f(z^{\\prime }-z)\\, ,$ $\\mathfrak {f_s}\\left[\\mathsf {t}_z\\,f\\right](z^{\\prime })= e^{z^{\\prime }\\circ z}\\, \\mathfrak {f_s}[f](z^{\\prime })\\,,\\quad \\overline{\\mathfrak {f_s}}\\left[\\mathsf {t}_{-z}\\,f\\right](z^{\\prime })= e^{z^{\\prime }\\circ z}\\, \\overline{\\mathfrak {f_s}}[f](z^{\\prime }) \\,.$ Symplectic Fourier transform and derivation $\\frac{\\partial ^k}{\\partial z^k} \\,\\mathfrak {f_s}[f](z)&= \\mathfrak {f_s}\\left[\\bar{\\xi }^k\\, f\\right](z)\\, , \\quad &\\frac{\\partial ^k}{\\partial z^k} \\,\\overline{\\mathfrak {f_s}}[f](z)= \\overline{\\mathfrak {f_s}}\\left[\\left(-\\bar{\\xi }\\right)^k\\, f\\right](z) \\, , \\\\ \\frac{\\partial ^k}{\\partial \\bar{z}^k} \\,\\mathfrak {f_s}[f](z)&= \\mathfrak {f_s}\\left[(-\\xi )^k\\,f\\right](z)\\, , \\quad &\\frac{\\partial ^k}{\\partial \\bar{z}^k} \\,\\overline{\\mathfrak {f_s}}[f](z)= \\overline{\\mathfrak {f_s}}\\left[\\xi ^k\\,f\\right](z)\\, , \\\\ \\mathfrak {f_s}\\left[\\frac{\\partial ^k}{\\partial \\xi ^k} \\,f\\right](z)&= \\bar{z}^k\\,\\mathfrak {f_s}\\left[f\\right](z)\\, , \\quad &\\overline{\\mathfrak {f_s}}\\left[\\frac{\\partial ^k}{\\partial \\xi ^k} \\,f\\right](z)=(- \\bar{z})^k\\,\\overline{\\mathfrak {f_s}}\\left[f\\right](z)\\, , \\\\ \\mathfrak {f_s}\\left[\\frac{\\partial ^k}{\\partial \\bar{\\xi }^k} \\,f\\right](z)&= (-z)^k\\,\\mathfrak {f_s}\\left[f\\right](z)\\, , \\quad &\\overline{\\mathfrak {f_s}}\\left[\\frac{\\partial ^k}{\\partial \\bar{\\xi }^k} \\,f\\right](z)=z^k\\,\\overline{\\mathfrak {f_s}}\\left[f\\right](z)\\, .$ Convolution product with complex variables $(f\\ast g)(z):= \\int _{\\mathbb {C}}\\mathrm {d}^2z^{\\prime }\\, f(z-z^{\\prime }) \\, g(z^{\\prime })= (g\\ast f)(z)\\, .$ Symplectic Fourier transform of convolution products $\\mathfrak {f_s}[f\\ast g] (z)&= \\pi \\, \\mathfrak {f_s}[f] (z)\\, \\mathfrak {f_s}[ g] (z)\\, , \\\\ \\mathfrak {f_s}[f\\, g] (z)&= \\frac{1}{\\pi } \\, (\\mathfrak {f_s}[f] \\ast \\mathfrak {f_s}[ g]) (z)\\, .$ Symplectic Fourier transform of Gaussian $\\mathfrak {f_s}\\left[e^{\\nu \\, \\vert \\xi \\vert ^2}\\right](z) = \\frac{1}{(-\\nu )}\\, e^{ \\frac{\\vert z\\vert ^2}{\\nu }}= \\overline{\\mathfrak {f_s}}\\left[e^{\\nu \\, \\vert \\xi \\vert ^2}\\right](z)\\, , \\quad \\mathrm {Re}(\\nu ) < 0\\, .$ Symplectic Fourier transform of $D$ $\\int _{\\mathbb {C}} e^{z \\circ z^{\\prime }} \\, D(z^{\\prime })\\,\\frac{\\mathrm {d}^2 z^{\\prime }}{\\pi } = 2\\, D(2z)\\,{\\sf P} = 2 {\\sf P}\\, D(-2z) \\,.$"
],
[
"In terms of $q$ and {{formula:1d94b2f2-ab6e-4ddb-980f-5b756cbc7cd8}}",
" In terms of coordinates $z= (q+\\mathsf {i}p)/\\sqrt{2}$ , $\\xi = (x+\\mathsf {i}y)/\\sqrt{2}$ , $\\mathfrak {f_s}[f](z)\\equiv \\mathfrak {F_s}[F](q,p)= \\int _{\\mathbb {R}^2}e^{-\\mathsf {i}(qy - px)}\\, F(x,y)\\,\\frac{\\mathrm {d}x\\,\\mathrm {d}y}{2\\pi } = \\mathfrak {F}[F](-p,q)\\, ,$ where $ \\mathfrak {F}$ denotes the standard two-dimensional Fourier transform, $\\mathfrak {F}[F](k_x,k_y)= \\int _{\\mathbb {R}^2}e^{-\\mathsf {i}(k_x x + k_y y)}\\, F(x,y)\\,\\frac{\\mathrm {d}x\\,\\mathrm {d}y}{2\\pi }\\, ,$ with inverse $\\overline{\\mathfrak {F}}[F](k_x,k_y)= \\int _{\\mathbb {R}^2}e^{\\mathsf {i}(k_x x + k_y y)}\\, F(x,y)\\,\\frac{\\mathrm {d}x\\,\\mathrm {d}y}{2\\pi }= \\mathfrak {F}[F](-k_x,-k_y)\\, .$ $\\mathfrak {F_s}$ is involutive, $\\mathfrak {F_s}\\left[\\mathfrak {F_s}[F]\\right]= \\mathfrak {F_s}^2[F]= F$ like its “dual” defined as $\\overline{\\mathfrak {F_s}}[F](q,p)= \\mathfrak {F_s}[F](-q,-p)=\\int _{\\mathbb {R}^2}e^{\\mathsf {i}(qy - px)}\\, F(x,y)\\,\\frac{\\mathrm {d}x\\,\\mathrm {d}y}{2\\pi } = \\mathfrak {F}[F](p,-q)\\, ,$ Symplectic Fourier transform and derivation $\\frac{\\partial ^k}{\\partial q^k} \\,\\mathfrak {F_s}[F](q,p)&= (-\\mathsf {i})^k \\mathfrak {F_s}\\left[y^k\\, F\\right](q,p)\\, , &\\frac{\\partial ^k}{\\partial q^k} \\,\\overline{\\mathfrak {F_s}}[F](q,p)= \\mathsf {i}^k\\mathfrak {F_s}\\left[y^k\\, F\\right](q,p) \\, , \\\\ \\frac{\\partial ^k}{\\partial p^k} \\,\\mathfrak {F_s}[F](q,p)&= \\mathsf {i}^k\\mathfrak {F_s}\\left[x^k\\, F\\right](q,p)\\, , &\\frac{\\partial ^k}{\\partial p^k} \\,\\overline{\\mathfrak {F_s}}[F](q,p)= (-\\mathsf {i})^k\\overline{\\mathfrak {F_s}}\\left[x^k\\,F\\right](q,p)\\, , \\\\ \\mathsf {i}^k\\mathfrak {F_s}\\left[\\frac{\\partial ^k}{\\partial x^k} \\,F\\right](q,p)&= p^k\\,\\mathfrak {f_s}\\left[F\\right](q,p)\\, , &(-\\mathsf {i})^k\\overline{\\mathfrak {F_s}}\\left[\\frac{\\partial ^k}{\\partial x^k} \\,f\\right](q,p)=p^k\\,\\overline{\\mathfrak {F_s}}\\left[F\\right](q,p)\\, , \\\\ (-\\mathsf {i})^k \\mathfrak {F_s}\\left[\\frac{\\partial ^k}{\\partial y^k} \\,F\\right](q,p)&= q^k\\,\\mathfrak {F_s}\\left[F\\right](q,p)\\, , &\\mathsf {i}^k\\overline{\\mathfrak {F_s}}\\left[\\frac{\\partial ^k}{\\partial y^k} \\,F\\right](q,p)=q^k\\,\\overline{\\mathfrak {F_s}}\\left[F\\right](q,p)\\, .$ Convolution product $(F\\ast G)(q,p):= \\int _{\\mathbb {C}}\\mathrm {d}q^{\\prime }\\,\\mathrm {d}p^{\\prime }\\, F(q-q^{\\prime }, p-p^{\\prime }) \\, G(q-q^{\\prime },p-p^{\\prime })= (G\\ast F)(z)\\, .$ Symplectic Fourier transform of convolution products $\\mathfrak {F_s}[F\\ast G] (q,p)&= 2\\pi \\, \\mathfrak {F_s}[F] (q,p)\\, \\mathfrak {F_s}[ G] (q,p)\\, , \\\\ \\mathfrak {F_s}[F\\, G] (q,p)&= \\frac{1}{2\\pi } \\, (\\mathfrak {F_s}[F] \\ast \\mathfrak {F_s}[ G]) (q,p)\\, .$ Same formulae for $\\overline{\\mathfrak {F_s}}$"
],
[
"Acknowledgments",
"J.-P. Gazeau acknowledges partial support of CNRS-CRM-UMI 3457.",
"V. Hussin acknowledges the support of research grants from NSERC of Canada.",
"K. Zelaya acknowledges the support from the Mathematical Physics Laboratory of the Centre de Recherches Matématiques, through a postdoctoral fellowship.",
"He also acknowledges the support of Consejo Nacional de Ciencia y Tecnología (Mexico), grant number A1-S-24569."
]
] | 2005.14231 |
[
[
"Evolving to learn: discovering interpretable plasticity rules for\n spiking networks"
],
[
"Abstract Continuous adaptation allows survival in an ever-changing world.",
"Adjustments in the synaptic coupling strength between neurons are essential for this capability, setting us apart from simpler, hard-wired organisms.",
"How these changes can be mathematically described at the phenomenological level, as so called \"plasticity rules\", is essential both for understanding biological information processing and for developing cognitively performant artificial systems.",
"We suggest an automated approach for discovering biophysically plausible plasticity rules based on the definition of task families, associated performance measures and biophysical constraints.",
"By evolving compact symbolic expressions we ensure the discovered plasticity rules are amenable to intuitive understanding, fundamental for successful communication and human-guided generalization.",
"We successfully apply our approach to typical learning scenarios and discover previously unknown mechanisms for learning efficiently from rewards, recover efficient gradient-descent methods for learning from target signals, and uncover various functionally equivalent STDP-like rules with tuned homeostatic mechanisms."
],
[
"Introduction",
"How do we learn?",
"Whether we are memorizing the way to the lecture hall at a conference or mastering a new sport, somehow our central nervous system is able to retain the relevant information over extended periods of time, sometimes with ease, other times only after intense practice.",
"This acquisition of new memories and skills manifests at various levels of the system, with changes of the interaction strength between neurons being a key ingredient.",
"Uncovering the mechanisms behind this synaptic plasticity is a key challenge in understanding brain function.",
"Most studies approach this monumental task by searching for phenomenological models described by symbolic expressions that map local biophysical quantities to changes of the connection strength between cells (fig:intro-microcircuitA, B).",
"Figure: Artificial evolution of synaptic plasticity rules in spiking neuronal networks.",
"(A) Sketch of cortical microcircuits consisting of pyramidal cells (orange) and inhibitory interneurons (blue).Stimulation elicits action potentials in pre- and postsynaptic cells, which, in turn, influence synaptic plasticity.",
"(B) Synaptic plasticity leads to a weight change (Δw\\Delta w) between the two cells, here measured by the change in the amplitude of post-synaptic potentials.The change in synaptic weight can be expressed by a function ff that in addition to spike timings (t pre ,t post t_\\text{pre}, t_\\text{post}) can take into account additional local quantities, such as the concentration of neuromodulators (ρ\\rho , green dots in A) or postsynaptic membrane potentials.",
"(C) For a specific experimental setup, an evolutionary algorithm searches for individuals representing functions ff that maximize the corresponding fitness function ℱ\\mathcal {F}.An offspring is generated by modifying the genome of a parent individual.Several runs of the evolutionary algorithm can discover phenomenologically different solutions (f 0 ,f 1 ,f 2 f_0,f_1,f_2) with comparable fitness.",
"(D) An offspring is generated from a single parent via mutation.",
"Mutations of the genome can, for example, exchange mathematical operators, resulting in a different function ff.Approaches to deciphering synaptic plasticity can be broadly categorized into bottom-up and top-down.",
"Bottom-up approaches typically rely on experimental data [2], [22], [8], [57] to derive dynamic equations for synaptic parameters that lead to functional emergent macroscopic behavior if appropriately embedded in networks [31], [33], [13].",
"Top-down approaches proceed in the opposite direction: from a high-level description of network function, e.g., in terms of an objective function [74], [20], [37], [42], [68], [29], dynamic equations for synaptic changes are derived and biophysically plausible implementations suggested.",
"Evidently, this demarcation is not strict, as most approaches seek some balance between experimental evidence, functional considerations and model complexity.",
"However, the relative weighting of each of these aspects is usually not made explicit in the communication of scientific results, making it difficult to track by other researchers.",
"Furthermore, the selection of specific tasks to illustrate the effect of a suggested learning rule is usually made only after the rule was derived based on other considerations.",
"Hence, this typically does not consider competing alternative solutions, as an exhaustive comparison would require significant additional investment of human resources.",
"A related problem is that researchers, in a reasonable effort to use resources efficiently, tend to focus on promising parts of the search space around known solutions, leaving large parts of the search space unexplored [64].",
"Automated procedures, in contrast, can perform a significantly less biased search.",
"We suggest an automated approach to discover learning rules in spiking neuronal networks that explicitly addresses these issues.",
"Automated procedures interpret the search for biological plasticity mechanisms as an optimization problem [4], an idea typically referred to as meta-learning or learning-to-learn.",
"These approaches make the emphasis of particular aspects that guide this search explicit and place the researcher at the very end of the process, supporting much larger search spaces and the generation of a diverse set of hypotheses.",
"Furthermore, they have the potential to discover domain-specific solutions that are more efficient than general-purpose algorithms.",
"Early experiments focusing on learning in artificial neural networks (ANNs) made use of gradient descent or genetic algorithms to optimize parameterized learning rules [6], [4], [5] or genetic programming to evolve less constrained learning rules [3], [64], rediscovering mechanisms resembling the backpropagation of errors [43], [32], [67].",
"Recent experiments demonstrate how optimization methods can design optimization algorithms for recurrent ANNs [1], evolve machine learning algorithms from scratch [65], and optimize parametrized learning rules in neuronal networks to achieve a desired function [14].",
"We extend these meta-learning ideas to discover free-form, yet interpretable plasticity rules for spiking neuronal networks.",
"The discrete nature of spike-based neuronal interactions endows these networks with rich dynamical and functional properties [21], [36], [39].",
"In addition, with the advent of non-von Neumann computing systems based on spiking neuronal networks with online learning capabilities [54], [16], [9], efficient learning algorithms for spiking systems become increasingly relevant for non-conventional computing.",
"Here, we employ genetic programming [41] as a search algorithm for two main reasons.",
"First, genetic programming can operate on analytically tractable mathematical expressions describing synaptic weight changes that are interpretable.",
"Second, an evolutionary search does not need to compute gradients in the search space, thereby circumventing the need to estimate a gradient in non-differentiable systems.",
"We successfully apply our approach, which we refer to as “evolving-to-learn” (E2L), to three different learning paradigms for spiking neuronal networks: reward-driven, error-driven and correlation-driven learning.",
"For the reward-driven task our approach discovers new plasticity rules with efficient reward baselines perform that perform competively and even outperform previously suggested methods.",
"The analytic form of the resulting expressions suggests experimental approaches that would allow us to distinguish between them.",
"In the error-driven learning scenario, the evolutionary search discovers a variety of solutions which, with appropriate analysis of the corresponding expressions, can be shown to effectively implement stochastic gradient descent.",
"Finally, in the correlation-driven task, our method generates a variety of STDP kernels and associated homeostatic mechanisms that lead to similar network-level behavior.",
"This sheds new light onto the observed variability of synaptic plasticity and thus suggests a reevaluation of the reported variety in experimentally-measured STDP curves with respect to their possible functional equivalence.",
"Our results demonstrate the significant potential of automated procedures in the search for plasticity rules in spiking neuronal networks, analogous to the transition from hand-designed to learned features that lies at the heart of modern machine learning."
],
[
"Setting up an evolutionary search for plasticity rules",
"We introduce the following recipe to search for biophysically plausible plasticity rules in spiking neuronal networks.",
"First we determine a task family of interest and an associated experimental setup which includes specification of the network architecture, e.g., neuron types and connectivity, as well as stimulation protocols or training data sets.",
"Crucially, this step involves defining a fitness function to guide the evolutionary search towards promising regions of the search space.",
"It assigns high fitness to those individuals, i.e., learning rules, that solve the task well and low fitness to others.",
"The fitness function may additionally contain constraints implied by experimental data or arising from computational considerations.",
"We determine each individual's fitness on various examples from the given task family, e.g., different input spike train realizations, to discover plasticity rules that generalize well [12], [71].",
"Finally, we specify the neuronal variables available to the plasticity rule, such as low-pass-filtered traces of pre- and postsynaptic spiking activity or neuromodulator concentrations.",
"This choice is guided by biophysical considerations, e.g., which quantities are locally available at a synapse, as well as by the task family, e.g., whether reward or error signals are provided by the environment.",
"We write the plasticity rule in the general form $\\Delta w = \\eta \\, f(\\dots )$ , where $\\eta $ is a fixed learning rate, and employ an evolutionary search to discover functions $f$ that lead to high fitness.",
"We propose to use genetic programming (GP) as an evolutionary algorithm to discover plasticity rules in spiking neuronal networks.",
"GP applies mutations and selection pressure to an initially random population of computer programs to artificially evolve algorithms with desired behaviors [40], [41].",
"Here we consider the evolution of mathematical expressions.",
"We employ a specific form of GP, Cartesian genetic programming [51], [53], that uses an indexed graph representation of programs.",
"The genotype of an individual is a two-dimensional Cartesian graph (fig:cgp-sketchA, top).",
"Figure: Representation and mutation of mathematical expressions in Cartesian genetic programming.",
"(A) The genotype of an individual is a two-dimensional Cartesian graph (top).In this example, the graph contains three input nodes (0-20-2), six internal nodes (3-83-8) and a single output node (9).In each node the genes of a specific genotype are shown, encoding the operator used to compute the node's output and its inputs.Each operator gene maps to a specific mathematical function (bottom).Special values (-1,-2-1, -2) represent input and output nodes.For example, node 4 uses the operator 1, the multiplication operation “**”, and receives input from nodes 0 and 2.This node's output is hence given by x 0 *x 2 x_0 * x_2.The number of input genes per node is determined by the operator with the maximal arity (here two).Fixed genes that cannot be mutated are highlighted in red.∅\\emptyset denotes non-coding genes.",
"(B) The computational graph (phenotype) generated by the genotype in A.Input nodes (x 0 ,x 1 ,x 2 {x_0, x_1, x_2}) represent the arguments of the function ff.Each output node selects one of the other nodes as a return value of the computational graph, thus defining a function from input xx to output y=f(x)y = f(x).Here, the output node selects node 4 as a return value.Some nodes defined in the genotype are not used by a particular realization of the computational graph (in light gray, e.g., node 6).",
"Mutations that affect such nodes have no effect on the phenotype and are therefore considered “silent”.",
"(C) Mutations in the genome either lead to a change in graph connectivity (top, green arrow) or alter the operators used by an internal node (bottom, green node).Here, both mutations affect the phenotype and are hence not silent.Over the course of an evolutionary run, this graph has a fixed number of input, output, and internal nodes.",
"The operation of each internal node is fully described by a single function gene and a fixed number of input genes.",
"A function table maps function genes to mathematical operations (fig:cgp-sketchA, bottom), while input genes determine from where this node receives data.",
"A given genotype is decoded into a corresponding computational graph (the phenotype, fig:cgp-sketchB) which defines a function $f$ .",
"During the evolutionary run, mutations of the genotype alter connectivity and node operations, which can modify the encoded function (fig:cgp-sketchC).",
"The indirect encoding of the computational graph via the genotype supports variable-length phenotypes, since some internal nodes may not be used to compute the output (fig:cgp-sketchB).",
"The size of the genotype, in contrast, is fixed, thereby restricting the maximal size of the computational graph and ensuring compact, interpretable mathematical expressions.",
"Furthermore, the separation into genotype and phenotype allows the buildup of “silent mutations”, i.e., mutations in the genotype that do not alter the phenotype.",
"These silent mutations can lead to more efficient search as they can accumulate and in combination lead to an increase in fitness once affecting the phenotype [51].",
"A $\\mu + \\lambda $ evolution strategy [7] drives evolution by creating the next generation of individuals from the current one via tournament selection, mutation and selection of the fittest individuals (sec:methods-evolutionary-algorithm).",
"Prior to starting the search, we choose the mathematical operations that can appear in the plasticity rule and other (hyper)parameters of the Cartesian graph and evolutionary algorithm.",
"For simplicity, we consider a restricted set of mathematical operations and additionally make use of nodes with constant output.",
"After the search has completed, e.g., by reaching a target fitness value or a maximal number of generations, we analyze the discovered set of solutions.",
"In the following, we describe the results of three experiments following the recipe outlined above."
],
[
"Evolving an efficient reward-driven plasticity rule",
"We consider a simple reinforcement learning task for spiking neurons.",
"The experiment can be succinctly described as follows: $N$ inputs project to a single readout modeled by a leaky integrator neuron with exponential postsynaptic currents and stochastic spike generation (for details see sec:methods-reinforcement-learning-task).",
"We generate a finite number $M$ of frozen-Poisson-noise patterns of duration $T$ and assign each of these randomly to one of two classes.",
"The output neuron should learn to classify each of these spatio-temporal input patterns into the corresponding class using a spike/no-spike code (fig:results-reinforcement-learning-fitnessA, B).",
"Figure: Cartesian genetic programming evolves various efficient reward-driven learning rules.",
"(A) Network sketch.Multiple input neurons with Poisson activity project to a single output unit.Pre- and postsynaptic activity generate an eligibility trace in each synapse.Comparison between the output activity and the target activity generates a reward signal.R ¯\\bar{R}, and R ¯ + \\bar{R}^+, R ¯ - \\bar{R}^- represent the expected reward, the expected positive and the expected negative reward, respectively.Depending on the hyperparameter settings either the former or the latter two are provided to the plasticity rule.",
"(B) Raster plot of the activity of input neurons (small black dots) and output neuron (large golden dots).Gray (white) background indicate patterns for which the output should be active (inactive).Top indicates correct classifications (++) and incorrect classifications (--).We show 10 trials at the beginning (left) and the end of training (right) using the evolved plasticity rule: Δw j =η(R-1)E j r \\Delta w_j = \\eta \\, (R-1)E_j^\\text{r}.",
"(C) Fitness of best individual per generation as a function of the generation index for multiple example runs of the evolutionary algorithm with different initial conditions but identical hyperparameters.Labels show the expression ff at the end of the respective run for three runs resulting in well-performing plasticity rules.Gray lines represent runs with functionally identical solutions or low final fitness.",
"(D) Fitness of a selected subset of evolved learning rules on the 10 experiments used during the evolutionary search (blue) and additional 30 fitness evaluations, each on 10 new experiments consisting of sets of frozen noise patterns and associated class labels not used during the evolutionary search (orange).Horizontal boxes represent mean, error bars indicate one standard deviation over fitness values.Gray line indicates mean fitness of LR0 for visual reference.See main text for the full expressions for all learning rules.The fitness $\\mathcal {F}(f)$ of an individual encoding the function $f$ is measured by the mean reward per trial averaged over a certain number of experiments $n_\\text{exp}$ , each consisting of $n$ classification trials $\\mathcal {F}(f) := \\frac{1}{n_\\text{exp}} \\sum _{k=1}^{n_\\text{exp}} R_k(f) \\; ,$ where $R_k(f):=\\frac{1}{n} \\sum _{i=1}^{n} R_{k,i}(f)$ is the mean reward per trial obtained in experiment $k$ .",
"The reward $R_{k,i}$ is the reward obtained in the $i$ th trial of experiment $k$ .",
"It is 1 if the classification is correct and $-1$ otherwise.",
"In the following we drop the subscripts from $R_{k,i}$ where its meaning is clear from context.",
"Each experiment contains different realizations of frozen-noise input spike trains with associated class labels.",
"Previous work on reward-driven learning [83] has demonstrated that in policy-gradient-based approaches [72], subtracting a so called “reward baseline” from the received reward can improve the convergence properties by adjusting the magnitude of weight updates.",
"However, choosing a good reward baseline is notoriously difficult [82], [18], [79].",
"For example, in a model for reinforcement learning in spiking neurons, [77] suggest the expected positive reward as a suitable baseline.",
"Here, we consider plasticity rules which, besides immediate rewards, have access to expected rewards.",
"These expectations are obtained as moving averages over a number of consecutive trials during one experiment and can either be estimated jointly ($\\bar{R}\\in [-1, 1]$ ) or separately for positive ($\\bar{R}^+\\in [0, 1]$ ) and negative ($\\bar{R}^-\\in [-1, 0]$ ) rewards, with $\\bar{R}= \\bar{R}^++ \\bar{R}^-$ (for details, see sec:methods-reinforcement-learning-task).",
"In the former case, the plasticity rule takes the general form $\\Delta w_j = \\eta \\, f \\left( R, E_j^\\text{r}(T), \\bar{R}\\right) \\; ,$ while for seperately estimated positive and negative rewards it takes the form $\\Delta w_j = \\eta \\, f \\left( R, E_j^\\text{r}(T), \\bar{R}^+, \\bar{R}^-\\right) \\; .$ In both cases, $\\eta $ is a fixed learning rate and $E_j^\\text{r}(t)$ is an eligibility trace that contains contributions from the spiking activity of pre- and post-synaptic neurons which is updated over the course of a single trial (for details see sec:methods-reinforcement-learning-task).",
"The eligibility trace arises as a natural consequence of policy-gradient methods aiming to maximize the expected reward [83] and is a common ingredient of reward-modulated plasticity rules for spiking neurons [77], [24].",
"It is a low-pass filter of the product of two terms: the first is positive if the neuron was more active than expected by synaptic input; this can happen because the neuronal output is stochastic (sec:methods-reinforcement-learning-task), to promote exploration.",
"The second is a low-pass filter of presynaptic activity.",
"A simple plasticity rule derived from maximizing the expected rewards would, for example, change weights according to the product of the received reward and the eligibility trace: $\\Delta w_j = R E_j^\\text{r}$ .",
"If by chance a neuron is more active than expected, and the agent receives a reward, all weights of active afferents are increased, making it more likely for the neuron to fire in the future given identical input.",
"Reward and eligibility trace values at the end of each trial ($t=T$ ) are used to determine synaptic weight changes.",
"In the following we drop the time argument of $E_j^\\text{r}$ for simplicity.",
"Using CGP with three ($R$ , $E_j^\\text{r}, \\bar{R}$ ), or four inputs ($R$ , $E_j^\\text{r}, \\bar{R}^+, \\bar{R}^-$ ), respectively, we search for plasticity rules that maximize the fitness $\\mathcal {F}(f)$ (eq:reward-learning-fitness).",
"None of the evolutionary runs with access to the expected reward ($\\bar{R}$ ) make use of it as a reward baseline (see Appendix sec:app-reward-learning for full data).",
"Some of them discover high-performing rules identical to that suggested by [75]: $\\Delta w_j = \\eta \\, (R - 1)E_j^\\text{r}$ (LR0, $\\mathcal {F}=216.2$ , fig:results-reinforcement-learning-fitnessC,D).",
"This rule uses a fixed base line, the maximal reward ($R_\\text{max}=1$ ), rather than the expected reward.",
"Some runs discover a more sophisticated variant of this rule with a term that decreases the effective learning rate for negative rewards as the agent improves, i.e., when the expected reward increases: $\\Delta w_j = \\eta \\, (1 + R\\bar{R})(R - 1)E_j^\\text{r}$ (LR1, $\\mathcal {F}=234.2$ , fig:results-reinforcement-learning-fitnessC,D).",
"Using this effective learning-rate, this rule achieve higher fitness than the vanilla formulation at the expense of requiring the agent to keep track of the expected reward.",
"Using the expected reward as a baseline, e.g., $\\Delta w_j = \\eta \\, (R - \\bar{R})E_j^\\text{r}$ , is unlikely to yield high-performing solutions: an agent may get stuck in weight configurations in which it continuously receives negative rewards, yet, as it is expecting negative rewards, does not significantly change its weights.",
"This intuition is supported by our observation that none of the high-performing plasticity rules discovered by our evolutionary search make use of such a baseline, in contrast to previous studies [24].",
"Subtracting the maximal reward, in contrast, can be interpreted as an optimistic baseline sutton2018reinforcement, which biases learning to move away from weight configurations that result in negative rewards, while maintaining weight configurations that lead to positive rewards.",
"However, a detrimental effect of such an optimistic baseline is that learning is sparse, as it only occurs upon receiving negative rewards, an assumption at odds with behavioral evidence.",
"In contrast, evolutionary runs with access to separate estimates of the negative and positive rewards discover plasticity rules with efficient baselines, resulting in increased fitness (see Appendix sec:app-reward-learning for the full data).",
"In the following we discuss four such high-performing plasticity rules with at least $10\\%$ higher fitness than LR0 (fig:results-reinforcement-learning-fitnessD).",
"We first consider the rule (LR2, $\\mathcal {F}=242.0$ , fig:results-reinforcement-learning-fitnessD) $\\Delta w_j = \\eta \\, [R - (\\bar{R}^+- \\bar{R}^-)]E_j^\\text{r}= \\eta (R - \\bar{R}_\\text{abs})E_j^\\text{r}\\; ,$ where we introduced the expected absolute reward $\\bar{R}_\\text{abs}:= \\bar{R}^+- \\bar{R}^-= |\\bar{R}^+| + |\\bar{R}^-|$ , with $\\bar{R}_\\text{abs}\\in [0, 1]$ .",
"Note the difference to the expected reward $\\bar{R}= \\bar{R}^++ \\bar{R}^-$ .",
"Since the absolute magnitude of positive and negative rewards is identical in the considered task, $\\bar{R}_\\text{abs}$ increases in each trial, starting at zero and slowly converging to one.",
"Instead of keeping track of the expected reward, the agent can thus simply optimistically increase its baseline with each trial.",
"Behind this lies the, equally optimistic, expectation that the agent improves its performance over trials.",
"Starting out as $RE_j^\\text{r}$ and converging to $(R-1)E_j^\\text{r}$ this rule combines efficient learning from both positive and negative rewards initially, with improved convergence towards successful weight configuration during late learning.",
"Note that such a strategy may lead to issues with un- or re-learning.",
"This may be overcome by the agent resetting the expected absolute reward $\\bar{R}_\\text{abs}$ upon encountering a new task, similar to a “novelty” signal.",
"Furthermore, our algorithm discovers a variation of this rule (LR3, $\\mathcal {F}=256.0$ , fig:results-reinforcement-learning-fitnessD), which replaces $\\eta $ with $\\eta /(1 + \\bar{R}^+)$ to decrease the magnitude of weight changes in regions of the weight space associated with high performance.",
"This can improve convergence properties.",
"We next consider the rule (LR4, $\\mathcal {F}=247.2$ , fig:results-reinforcement-learning-fitnessD): $\\Delta w_j = \\eta \\left[ \\, (R - 1)E_j^\\text{r}+ (R - 1)(R + 2 \\bar{R}^+) \\right] \\; .$ This rule has the familiar form of LR0 and LR1, with an additional homeostatic term.",
"Due to the prefactors $R-1$ , this rule only changes weights on trials with negative reward.",
"Initially, the expected reward $\\bar{R}^+$ is close to zero and the homeostatic term results in potentiation of all synapses, causing more and more neurons to spike.",
"In particular if initial weights are chosen poorly, this can make learning more robust.",
"As the agent improves and the expected positive rewards increases, the homeostatic term becomes negative.",
"In this regime it can be interpreted as pruning all weights until only those are left that do not lead to negative rewards.",
"This term can hence be interpreted as an adapting action baseline [72].",
"Finally, we consider the rule (LR5, $\\mathcal {F}=254.8$ , fig:results-reinforcement-learning-fitnessD): $\\Delta w_j = \\eta \\left\\lbrace \\, 2[R - (\\bar{R}^+- R \\bar{R}^-)] E_j^\\text{r}- [R - (\\bar{R}^+- R \\bar{R}^-)] R \\bar{R}^-\\right\\rbrace \\; .$ To analyze this seemingly complex rule, we consider the expression for trials with positive and trials with negative reward separately: $R=1: \\;\\, \\Delta w_j^+ =& \\, \\eta \\left\\lbrace \\, 2 (1 - \\bar{R}_\\text{abs}) E_j^\\text{r}- (1 - \\bar{R}_\\text{abs}) \\bar{R}^-\\right\\rbrace \\; , \\\\R=-1: \\;\\, \\Delta w_j^- =& \\, \\eta \\left\\lbrace \\, 2(-1 - \\bar{R}) E_j^\\text{r}- (1 + \\bar{R}) \\bar{R}^-\\right\\rbrace \\; .$ Both expressions contain a “causal” term depending on pre- and postsynaptic activity via the eligibility trace, as well as, and a “homeostatic” term.",
"Aside from the constant scaling factor, the causal term of $\\Delta w_j^+$ is identical to LR2 (eq:results-reward-learning-lr2), i.e., it only causes weight changes early during learning, and converges to zero for later times.",
"Similarly, the causal term of $\\Delta w_j^-$ is initially identical to that of LR2 (eq:results-reward-learning-lr2), decreasing weights for connections contributing to wrong decisions.",
"However it increases in magnitude as the agent improves and the expected reward increases.",
"The homeostatic term of $\\Delta w_j^+$ is potentiating, similarly to LR4 (eq:results-reward-learning-lr4): it encourages spiking by increasing all synaptic weights during early learning and decreases over time.",
"The homeostatic term for negative rewards is also potentiating, but does not vanish for long times unless the agent is performing perfectly ($\\bar{R}^-\\rightarrow 0$ ).",
"Over time this plasticity rule hence reacts less and less to positive rewards, while increasing weight changes for negative rewards.",
"The reward-modulated potentiating homeostatic mechanisms can prevent synaptic weights from vanishing and thus encourage exploration for challenging scenarios in which the agent mainly receives negative rewards.",
"In conclusion, by making use of the separately estimated expected negative and positive rewards in precise combinations with the eligibility trace and the instantaneous reward, our evolving-to-learn approach discovered a variety of reward-based plasticity rules, many of then outperforming previously known solutions [75].",
"The evolution of closed-form expressions allowed us to analyze the computational principles that allow these newly discovered rules to achieve high fitness.",
"This analysis suggests new mechanisms for efficient learning, for example from “novelty” and via reward-modulated homeostatic mechanisms.",
"Each of these new hypotheses for reward-driven plasticity rules makes specific predictions about behavioral and neuronal signatures that potentially allow us to distinguish between them.",
"For example LR2, LR3 and LR5 suggest that agents initially learn both from positive and negative rewards, while later they mainly learn from negative rewards.",
"In particular the initial learning from positive rewards distinguishes these hypotheses from LR0, LR1, and LR4, and previous work [75].",
"As LR2 does not make use of the, separately estimated, expected rewards, it is potentially employed in settings in which such estimates are difficult to obtain.",
"Furthermore, LR4 and LR5 suggest that precisely regulated homeostatic mechanisms play a crucial role besides weight changes due to pre- and post-synaptic activity traces.",
"During early learning, both rules implement potentiating homeostatic mechanisms triggered by negative rewards, likely to explore many possible weight configurations which may support successful behavior.",
"During late learning, LR4 suggests that homeostatic changes become depressing, thus pruning unnecessary or even harmful connections.",
"In contrast, they remain positive for LR5, potentially avoiding catastrophic dissociation between inputs and outputs for challenging tasks.",
"Besides experimental data from the behavioral and neuronal level, different artificial reward-learning scenarios could further further select for strengths or against weaknesses of the discovered rules.",
"Furthermore, additional considerations, for example achieving small variance in weight updates [81], [18], may lead to particular rules being favored over others.",
"We thus believe that our new insights into reinforcement learning are merely a forerunner of future experimental and theoretical work enabled by our approach."
],
[
"Evolving an efficient error-driven plasticity rule",
"We next consider a supervised learning task in which a neuron receives information about how far its output is from the desired behavior, instead of just a scalar reward signal as in the previous task.",
"The widespread success of this approach in machine learning highlights the efficacy of learning from errors compared to correlation- or reward-driven learning [30].",
"It has therefore often been hypothesized that evolution has installed similar capabilities in biological nervous systems [44], [80].",
"[76] introduced an efficient, flexible and biophysically plausible implementation of error-driven learning via multi-compartment neurons.",
"For simplicity, we consider an equivalent formulation of this learning principle in terms of two point neurons modeled as leaky integrator neurons with exponential postsynaptic currents and stochastic spike generation.",
"One neuron mimics the somatic compartment and provides a teaching signal to the other neuron acting as the dendritic compartment.",
"Here, the difference between the teacher and student membrane potentials drives learning: $\\Delta w_j(t) = \\eta \\, \\left[ v(t) - u(t) \\right] \\bar{s}_j(t) \\; ,$ where $v$ is the teacher potential, $u$ the student membrane potential, and $\\eta $ a fixed learning rate.",
"$\\bar{s}_j(t)=(\\kappa * s_j)(t)$ represents the the presynaptic spike train $s_j$ filtered by the synaptic kernel $\\kappa $ .",
"eq:results-error-driven-us can be interpreted as stochastic gradient descent on an appropriate cost function [76] and can be extended to support credit assignment in hierarchical neuronal networks [68].",
"For simplicity we assume the student has direct access to the teacher's membrane potential, but in principle one could also employ proxies such as firing rates as suggested in [61], [76].",
"We consider a one-dimensional regression task in which we feed random Poisson spike trains into the two neurons (fig:results-error-driven-learningA).",
"Figure: Cartesian genetic programming evolves efficient error-driven learning rules.",
"(A) Network sketch.Multiple input neurons with Poisson activity project to two neurons.One of the neurons (the teacher) generates a target for the other (the student).The membrane potentials of teacher and student as well as the filtered pre-synaptic spike trains are provided to the plasticity rule that determines the weight update.",
"(B) Root mean squared error between the teacher and student membrane potential over the course of learning using the evolved plasticity rule: Δw j (t)=ηv(t)-u(t)s ¯ j (t)\\Delta w_j(t) = \\eta \\, \\left[ v(t) - u(t) \\right] \\bar{s}_j(t).",
"(C) Synaptic weights over the course of learning corresponding to panel B.Horizontal dashed lines represent target weights, i.e., the fixed synaptic weights onto the teacher.",
"(D) Fitness of the best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions.Labels represent the rule at the end of the respective run.Colored markers represent fitness of each plasticity rule averaged over 15 validation tasks not used during the evolutionary search; error bars indicate one standard deviation.The teacher maintains fixed input weights while the student's weights should be adapted over the course of learning such that its membrane potential follows the teacher's (fig:results-error-driven-learningB, C).",
"The fitness $\\mathcal {F}(f)$ of an individual encoding the function $f$ is measured by the root mean-squared error between the teacher and student membrane potential over the simulation duration $T$ , excluding the initial $10\\%$ , averaged over $n_\\text{exp}$ experiments: $\\mathcal {F}(f) := \\frac{1}{n_\\text{exp}} \\sum _{k=1}^{n_\\text{exp}} \\sqrt{\\int _{0.1 T}^T \\text{d}t \\left[ v_k(t) - u_k(t) \\right]^2} \\; .$ Each experiment consists of different input spike trains and different teacher weights.",
"The general form of the plasticity rule for this error-driven learning task is given by: $\\Delta w_j = \\eta \\, f(v, u, \\bar{s}_j) \\; .$ Using CGP with three inputs ($v, u, \\bar{s}_j$ ), we search for plasticity rules that maximize the fitness $\\mathcal {F}(f)$ .",
"Starting from low fitness, about half of the evolutionary runs evolve efficient plasticity rules (fig:results-error-driven-learningD) closely matching the performance of the stochastic gradient descent rule of [76].",
"While two runs evolve exactly eq:results-error-driven-us, other solutions with comparable fitness are discovered, such as $\\Delta w_j =& \\eta (v - u)\\bar{s}_j \\frac{2u - 1}{v} \\; \\text{, and}\\\\\\Delta w_j =& \\eta \\bar{s}_j (v + u)\\frac{v (v - u) - \\bar{s}_j}{v^2} \\; .$ At first sight, these rules may appear more complex, but for the considered parameter regime under the assumptions $v \\approx u; v, u \\gg 1$ , we can write them as (see Appendix ): $\\Delta w_j = \\eta \\, c_1 (v - u) \\bar{s}_j + c_2 \\; ,$ with a multiplicative constant $c_1 = \\mathcal {O}(1)$ and a negligible additive constant $c_2$ .",
"Elementary manipulations of the expressions found by CGP thus demonstrate the similarity of these superficially different rules to eq:results-error-driven-us.",
"Consequently, they can be interpreted as approximations of gradient descent.",
"The constants generally fall into two categories: fine-tuning of the learning rate for the set of task samples encountered during evolution ($c_1$ ), which could be responsible for the slight increase in performance, and factors that have negligible influence and would most likely be pruned over longer evolutionary timescales ($c_2$ ).",
"This pruning could be accelerated, for example, by imposing a penalty on the model complexity in the fitness function, thus preferentially selecting simpler solutions.",
"In conclusion, the evolutionary search rediscovers variations of a human-designed efficient gradient-descent-based learning rule for the considered error-driven learning task.",
"Due to the compact, interpretable representation of the plasticity rules we are able to analyze the set of high-performing solutions and thereby identify phenomenologically identical rules despite their superficial differences."
],
[
"Evolving an efficient correlation-driven plasticity rule",
"We now consider a task in which neurons do not receive any feedback from the environment about their performance but instead only have access to correlations between pre- and postsynaptic activity.",
"Specifically, we consider a scenario in which an output neuron should discover a repeating frozen-noise pattern interrupted by random background spikes using a combination of spike-timing-dependent plasticity and homeostatic mechanisms.",
"Our experimental setup is briefly described as follows: $N$ inputs project to a single output neuron (fig:results-corr-learning-taskA).",
"Figure: Cartesian genetic programming evolves diverse correlation-driven learning rules.",
"(A) Network sketch.Multiple inputs project to a single output neuron.The current synaptic weight w j w_j and the eligibility trace E j c E_j^\\text{c} are provided to the plasticity rule that determines the weight update.",
"(B) Membrane potential uu of the output neuron over the course of learning using eq:stdphomeostatis.",
"Gray boxes indicate presentation of the frozen-noise pattern.",
"(C) Fitness (eq:corr-learning-fitness) of the best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions.Blue and red curves correspond to the two representative plasticity rules selected for detailed analysis.Blue and red markers represent fitness of the two representative rules and the orange marker the fitness of the homeostatic STDP rule , respectively, on 20 validation tasks not used during the evolutionary search.Error bars indicate one standard deviation over tasks.",
"(D, E): Learning rules evolved by two runs of CGP (D: LR1, eq:correlation-lr1; E: LR2, eq:correlation-lr2).",
"(F): Homeostatic STDP rule eq:stdphomeostatis suggested by .Top panels: STDP kernels Δw j \\Delta w_j as a function of spike timing differences Δt j \\Delta t_j for three different weights w j w_j.Bottom panels: homeostatic mechanisms for those weights.The colors are specific to the respective learning rules (blue for LR1, red for LR2), with different shades representing the different weights w j w_j.The learning rate is η=0.01\\eta =0.01.The activity of all inputs is determined by a Poisson process with a fixed rate.",
"A frozen-noise activity pattern of duration $T_\\text{pattern}$ is generated once and replayed every $T_\\text{inter}\\,\\mathrm {ms}$ (fig:results-corr-learning-taskB) while inputs are randomly spiking in between.",
"We define the fitness $\\mathcal {F}(f)$ of an individual encoding the function $f$ by the minimal average signal-to-noise ratio ($\\mathrm {SNR}$ ) across $n_\\text{exp}$ experiments: $\\mathcal {F}(f) &:=& \\min _k \\left\\lbrace \\mathrm {SNR}_k \\,, k\\in [1, n_\\text{exp}] \\right\\rbrace \\; .$ The signal-to-noise ratio $\\mathrm {SNR}_k$ , following [45], is defined as the difference between the maximal free membrane potential during pattern presentation averaged over multiple presentations ($\\langle u_{k,i,\\text{max}}\\rangle $ ) and the mean of the free membrane potential in between pattern presentations ($\\langle u_{k,\\text{inter}} \\rangle $ ) divided by its variance ($\\mathrm {Var}(v_{k,\\mathrm {inter}})$ ): $\\mathrm {SNR}_k &:=& \\frac{\\langle u_{k,i,\\max } \\rangle - \\langle u_{k,\\mathrm {inter}} \\rangle }{\\mathrm {Var}(u_{k,\\mathrm {inter}})} \\; .$ The free membrane potential is obtained in a separate simulation with frozen weights by disabling the spiking mechanism for the output neuron.",
"This removes measurement noise in the signal-to-noise ratio arising from spiking and subsequent membrane-potential reset.",
"Each experiment consists of different realizations of a frozen-noise pattern and background spiking.",
"We evolve learning rules of the following general form, which includes a dependence on the current synaptic weight in line with previously suggested STDP rules [31]: $\\Delta w_j^\\text{STDP} = \\eta {\\left\\lbrace \\begin{array}{ll}f_{\\mathrm {dep}}(w_j, E_j^\\text{c}) &\\Delta t_j <0 \\\\f_{\\mathrm {fac}} (w_j, E_j^\\text{c}) &\\Delta t_j \\ge 0 \\; .\\end{array}\\right.",
"}$ Here, $E_j^\\text{c}:=\\mathrm {e}^{- | \\Delta t_j | / \\tau }$ represents an eligibility trace that depends on the relative timing of post- and presynaptic spiking ($\\Delta t_j= t_{\\mathrm {post}} - t_{\\mathrm {pre}, j}$ ) and is represented locally in each synapse [56].",
"$\\eta $ represents a fixed learning rate.",
"The synaptic weight is bound such that $w_j \\in [0, 1]$ .",
"We additionally consider weight-dependent homeostatic mechanisms triggered by pre- and postsynaptic spikes, respectively.",
"These are implemented by additional functions of the general form: $\\Delta w^{\\mathrm {hom}}_j = \\eta {\\left\\lbrace \\begin{array}{ll}f_\\mathrm {pre}^\\mathrm {hom}(w_j) & \\text{upon presynaptic spike} \\\\f_\\mathrm {post}^\\mathrm {hom}(w_j) & \\text{upon postsynaptic spike} \\\\\\end{array}\\right.",
"}$ Weight changes are determined jointly by eq:correlationbasedgenerallr and eq:correlationbasedgeneralhom as $\\Delta w_j = \\Delta w_j^\\text{STDP} + \\Delta w^{\\mathrm {hom}}$ .",
"Using CGP, we search for functions $f_\\text{dep}$ , $f_\\text{fac}$ , $f_\\text{pre}^\\text{hom}$ , and $f_\\text{post}^\\text{hom}$ that maximize the fitness $\\mathcal {F}(f_\\text{dep}, f_\\text{fac})$ (eq:corr-learning-fitness).",
"As a baseline we consider a rule described by [45] (fig:results-corr-learning-taskC).",
"It is a simple additive spike-timing-dependent plasticity (STDP) rule that replaces the depression branch of traditional STDP variants with a postsynaptically-triggered constant homeostatic term $w^{\\mathrm {hom}}<0$ [38].",
"The synaptic weight of the projection from input $j$ changes according to (fig:results-corr-learning-taskG): $\\Delta w_j^\\text{STDP} = \\eta {\\left\\lbrace \\begin{array}{ll}0 &\\Delta t_j <0 \\text{ (anticausal interaction)}\\\\E_j^\\text{c} &\\Delta t_j \\ge 0 \\text{ (causal interaction)} \\; ,\\end{array}\\right.",
"}$ with homeostatic mechanisms: $\\Delta w^{\\mathrm {hom}}_j = \\eta {\\left\\lbrace \\begin{array}{ll}0 & \\text{upon presynaptic spike} \\\\w^{\\mathrm {hom}}& \\text{upon postsynaptic spike} \\; .\\end{array}\\right.",
"}$ To illustrate the result of synaptic plasticity following eq:stdphomeostatis and eq:homhomeostatis, we consider the evolution of the membrane potential of an output neuron over the course of learning (fig:results-corr-learning-taskC).",
"While the target neuron spikes randomly at the beginning of learning, its membrane potential finally stays subthreshold in between pattern presentations and crosses the threshold reliably upon pattern presentation.",
"After 2000 generations, half of the runs of the evolutionary algorithm discover high-fitness solutions (fig:results-corr-learning-taskD).",
"These plasticity rules lead to synaptic weight configurations which cause the neuron to reliably detect the frozen-noise pattern.",
"From these well-performing learning rules, we pick two representative examples (fig:results-corr-learning-taskD, E) to analyze in detail.",
"Learning rule 1 (LR1, fig:results-corr-learning-taskD) is defined by the following equations: $\\Delta w_j^\\text{STDP} = \\eta {\\left\\lbrace \\begin{array}{ll} -(w_j - 1) E_j^\\text{c} &\\quad \\Delta t_j < 0\\\\E_j^\\text{c} &\\quad \\Delta t_j \\ge 0\\end{array}\\right.",
"}\\;,\\quad \\quad \\Delta w^{\\mathrm {hom}}_j = \\eta {\\left\\lbrace \\begin{array}{ll}w_j & \\text{upon presyn.~spike} \\\\-w_j & \\text{upon postsyn.~spike} \\; .\\end{array}\\right.",
"}$ Learning rule 2 (LR2, fig:results-corr-learning-taskE) is defined by the following equations: $\\Delta w_j^\\text{STDP} = \\eta {\\left\\lbrace \\begin{array}{ll} -E_j^\\text{c}/w_j &\\quad \\Delta t_j < 0\\\\(w_j E_j^\\text{c})^{w_j} &\\quad \\Delta t_j \\ge 0\\end{array}\\right.",
"}\\;,\\quad \\quad \\Delta w^{\\mathrm {hom}}_j = \\eta {\\left\\lbrace \\begin{array}{ll}w_j & \\text{upon presyn.~spike} \\\\-1 & \\text{upon postsyn.~spike} \\; .\\end{array}\\right.",
"}$ The form of these discovered learning rules and associated homeostatic mechanisms suggests that they use distinct strategies to detect the repeated spatio-temporal pattern.",
"LR1 causes potentiation for small time differences, regardless of whether they are causal or anticausal (note that $-(w_j - 1) \\ge 0$ since $w_j \\in [0, 1]$ ).",
"In the Hebbian spirit, this learning rule favors correlation between presynaptic and postsynaptic firing.",
"Additionally, it potentiates synaptic weights upon presynaptic spikes, and depresses them for each postsynaptic spike.",
"In contrast, LR2 implements a similar strategy as the learning rule of [45]: it potentiates synapses only for small, positive (causal) time differences.",
"Additionally, however, it pronouncedly punishes anticausal interactions.",
"Similarly to LR1, its homeostatic component potentiates synaptic weights upon presynaptic spikes, and depresses them for each postsynaptic spike.",
"Note how both rules reproduce important components of experimentally established STDP traces [11].",
"Despite their differences both in the form of the STDP kernel as well as the associated homeostatic mechanisms, both rules lead to high fitness, i.e., comparable system-level behavior.",
"Unlike the classical perception of homeostatic mechanisms as merely maintaining an ideal working point of neurons [17], in both discovered plasticity rules these components support the computational goal of detecting the repeated pattern.",
"By potentiating large weights more strongly than small weights, the pre-synaptically triggered homeostatic mechanisms support the divergence of synaptic weights into strong weights, related to the repeated pattern, and weak ones, providing background input.",
"This observation suggests that homeostatic mechanisms and STDP work hand in hand to achieve desired functional outcomes.",
"Experimental approaches hence need to take both factors into account and variations in observed STDP curves should be reconsidered from a point of functional equivalence when paired with data on homeostatic changes.",
"In conclusion, for the correlation-driven task, the evolutionary search discovered a wide variety of plasticity rules with associated homeostatic mechanisms supporting successful task learning, thus enabling new perspectives for learning in biological substrates."
],
[
"Discussion",
"Uncovering the mechanisms of learning via synaptic plasticity is a critical step towards understanding brain (dys)function and building truly intelligent, adaptive machines.",
"We introduce a novel approach to discover biophysically plausible plasticity rules in spiking neuronal networks.",
"Our meta-learning framework uses genetic programming to search for plasticity rules by optimizing a fitness function specific to the respective task family.",
"Our evolving-to-learn approach discovers high-performing solutions for various learning paradigms, reward-driven, error-driven, and correlation-driven learning, yielding new insights into biological learning principles.",
"Moreover, our results from the reward-driven and correlation-driven task families demonstrate that homeostatic terms and their precise interation with plasticity play an important role in shaping network function, highlighting the importance of considering both mechanisms jointly.",
"This is, to the best of our knowledge, the first demonstration of the power of genetic programming methods in the search for plasticity mechanisms in spiking neuronal networks.",
"The experiments considered here were mainly chosen due to their simplicity and prior knowledge about corresponding plasticity rules that provided us with a high-performance reference for comparison.",
"Additionally, in each experiment, we restricted ourselves to a constrained set of possible inputs to the plasticity rule.",
"Here, we chose quantities which have been previously shown to be linked to synaptic plasticity in various learning paradigms, such as reward, low-pass filtered spike trains, and correlations between pre- and postsynaptic activities.",
"This prior knowledge avoids requiring the evolutionary algorithm to rediscover these quantities but limits the search space, thus potentially excluding other efficient solutions.",
"A key point of E2L is the compact representation of the plasticity rules.",
"We restrict the complexity of the expressions by three considerations.",
"First, we assume that effective descriptions of weight changes can be found that are not unique to each individual synapse.",
"This is is a common assumption in computational neuroscience and based on the observation that nature must have found a parsimonious encoding of brain structure, as not every connection in the brain can be specified in the DNA of the organism [84]; rather, genes encode general principles by which the neuronal networks and subnetworks are organized and reorganized [66].",
"Our approach aims at discovering such general principles for synaptic plasticity.",
"Second, physical considerations restrict the information available to the plasticity rule to local quantities, such as pre- and post-synaptic activity traces or specific signals delivered via neuromodulators [15], [49].",
"Third, we limit the maximal size of the expressions to keep the resulting learning rules interpretable and avoid overfitting.",
"We explicitly want to avoid constructing an opaque system that has high task performance but does not allow us to understand how the network structure is shaped over the course of learning.",
"Since we obtain analytically tractable expressions for the plasticity rule, we can analyze them with conventional methods, in contrast to approaches representing plasticity rules with ANNs [66], [59], [10], for which it is challenging to fully understand their macroscopic computation.",
"This analysis generates intuitive understanding, facilitating communication and human-guided generalization from a set of solutions to different network architectures or task domains.",
"In the search for plasticity rules suitable for physical implementations in biological systems, these insights are crucial as the identified plasticity mechanisms can serve as building blocks for learning rules that generalize to the actual challenges faced by biological agents.",
"Rather than merely applying the discovered rules to different learning problems, researchers may use the analytic expressions and prior knowledge to distill general learning principles – such as the computational role of homeostasis emerging from the present work – and combine them in new ways to extrapolate beyond the task families considered in the evolutionary search.",
"Therefore, our evolving-to-learn approach is a new addition to the toolset of the computational neuroscientist in which human intuition is paired with efficient search algorithms.",
"Moreover, simple expressions highlight the key interactions between the local variables giving rise to plasticity, thus providing hints about the underlying biophysical processes and potentially suggesting new experimental approaches.",
"From a different perspective, while the learning rules found in the experiments described above were all evolved from random expressions, one can also view the presented framework as a hypothesis-testing machine.",
"Starting from a known plasticity rule, our framework would allow researchers to address questions like: assuming the learning rule would additionally have access to variable $x$ , could this be incorporated into the weight updates such that learning would improve?",
"The automated procedure makes answering such questions much more efficient than a human-guided manual search.",
"Additionally, the framework is suitable to find robust biophysically plausible approximations for complex learning rules containing quantities that might be non-local, difficult to compute, and/or hard to implement in physical substrates.",
"In particular, multi-objective optimization is suitable to evolve a known, complex rule into simpler versions while maintaining high task performance.",
"Similarly, one could search for modifications of general rules that are purposefully tuned to quickly learn within a specific task family, outperforming more general solutions.",
"In each of these cases, prior knowledge about effective learning algorithms provides a starting point from which the evolutionary search can discover powerful extensions.",
"The automated search can discover plasticity rules for a given problem that exploit implicit assumptions in the task.",
"It therefore highlights underconstrained searches, be this due to scarcity of biological data, the simplicity of chosen tasks or the omission of critical features in the task design.",
"For instance, without asserting equal average spike rates of background and pattern neurons in the correlation-driven task, one could discover plasticity rules that exploit the rate difference rather than the spatio-temporal structure of the input.",
"Evolved Plastic Artificial Neural Networks [71] and in particular adaptive HyperNEAT [66], represent an alternative approach to designing plastic neural networks.",
"In contrast to our method, however, these approaches include the network architecture itself into the evolutionary search, alongside synaptic plasticity rules.",
"While this can lead to high-performance solutions due to a synergy between network architecture and plasticity, this interplay has a an important drawback, as in general it is difficult to tease apart the contribution of plasticity from that of network structure to high task performance [26].",
"In addition, the distributed, implicit representation of plasticity rules in HyperNEAT can be difficult to interpret, which hinders a deeper understanding of the learning mechanisms.",
"In machine-learning-oriented applications, this lack of credit assignment is less of an issue.",
"For research into plasticity rules employed by biological systems, however, it presents a significant obstacle.",
"Future work needs to address a general issue of any optimization method: how can we systematically counter overfitting to reveal general solutions?",
"A simple approach would increase the number of sample tasks during a single fitness evaluation.",
"However, computational costs increase linearly in the number of samples.",
"Another technique penalizes the complexity of the resulting expressions, e.g., proportional to the size of the computational graph.",
"Besides avoiding overfitting, such a penalty would automatically remove “null terms” in the plasticity rules, i.e., trivial subexpressions which have no influence on the expressions' output.",
"Since it is a priori unclear how this complexity penalty should be weighted against the original fitness measures, one should consider multi-objective optimization algorithms [19].",
"Another issue to be addressed in future work is the choice of the learning rate.",
"Currently, this value is not part of the optimization process and all tasks assume a fixed learning rate.",
"The analysis of the reward- and error-driven learning rules revealed that the evolutionary algorithm tried to optimize the learning rate using the variables it had access to, partly generating complex terms that that amount to a variable scaling of the learning rate.",
"The algorithm may benefit from the inclusion of additional constants which it could, for example, use for an unmitigated, permanent scaling of the learning rate.",
"However, the dimensionality of the search space scales exponentially in the number of operators and constants, and the feasibility of such an approach needs to be carefully evaluated.",
"One possibility to mitigate this combinatorial explosion is to combine the evolutionary search with gradient-based optimization methods that can fine-tune constants in the expressions [73], [34].",
"Additionally, future work may involve less preprocessed data as inputs while considering more diverse mathematical operators.",
"In the correlation-driven task, one could for example provide the raw times of pre- and postsynaptic spiking to the graph instead of the exponential of their difference, leaving more freedom for the evolutionary search to discover creative solutions.",
"We expect particularly interesting applications of our framework to involve more complex tasks that are challenging for contemporary algorithms, such as life-long learning, which needs to tackle the issue of catastrophic forgetting [25] or learning in recurrent spiking neuronal networks.",
"In order to yield insights into information processing in the nervous system, the design of the network architecture should be guided by known anatomical features, while the considered task families should fall within the realm of ecologically relevant problems.",
"The evolutionary search for plasticity rules requires a large number of simulations, as each candidate solution needs to be evaluated on a sufficiently large number of samples from the task family to encourage generalization [12], [4].",
"Due to silent mutations in CGP, i.e., modifications of the genotype that do not alter the phenotype, we use caching methods to significantly reduce computational cost as only new solutions need to be evaluated.",
"However, even employing such methods, the number of required simulations remains large, in the order of $10^3-10^4$ per evolutionary run.",
"For the experiments considered here, the computational costs are rather low, requiring $24-48$ node hours for a few parallel runs of the evolutionary algorithms, easily within reach of a modern workstation.",
"The total time increases linearly with the duration of a single simulation.",
"When considering more complex tasks which would require larger networks and hence longer simulations, one possibility to limit computational costs would be to evolve scalable plasticity rules in simplified versions of the tasks and architectures.",
"Such rules, quickly evolved, may then be applied to individual instances of the original complex tasks, mimicking the idea of “evolutionary hurdles” that avoid wasting computational power on low-quality solutions [70], [65].",
"A proof of concept for such an approach is the delta rule: originally in used in small-scale tasks, it has demonstrated incredible scaling potential in the context of error backpropagation.",
"Similar observations indeed hold for evolved optimizers [47].",
"Neuromorphic systems – dedicated hardware specifically designed to emulate neuronal networks – provide an attractive way to speed up the evolutionary search.",
"To serve as suitable substrates for the approach presented here, these systems should be able to emulate spiking neuronal networks in an accelerated fashion with respect to real time and provide on-chip plasticity with a flexible specification of plasticity mechanisms [16], [9], [46].",
"We view the presented methods as a machinery for generating, testing, and extending hypotheses on learning in spiking neuronal networks driven by problem instances and prior knowledge and constrained by experimental evidence.",
"We believe this approach holds significant promise to accelerate progress towards deep insights into information processing in physical systems, both biological and biologically inspired, with immanent potential for the development of powerful artificial learning machines."
],
[
"Evolutionary algorithm",
"We use a $\\mu + \\lambda $ evolution strategy [7] to evolve a population of individuals towards high fitness.",
"In each generation, $\\lambda $ new offsprings are created from $\\mu $ parents via tournament selection [50] and subsequent mutation.",
"From these $\\mu + \\lambda $ , individuals the best $\\mu $ individuals are selected as parents for the next generation (alg:evo-alg).",
"Initial random parent Population $P_0 = \\lbrace p_i \\rbrace $ of size $\\mu $ $t \\leftarrow 0$ $t < n_{\\mathrm {generations}}$ Create new offspring population $Q_{t} = \\text{CreateOffspringPopulation}(P_{t})$ Combine parent + offspring populations $R_t = P_t \\cup Q_t$ Evaluate fitness of each individual in $R_t$ Pick $P_{t+1} \\subset R_t$ best individuals as new parents $t \\leftarrow t + 1$ Function $\\text{CreateOffspringPopulation}(P)$ Offspring population $Q = \\lbrace \\rbrace $ $|Q| < \\lambda $ Choose random subset of $P$ of size $N_{\\mathrm {tournament}}$ Choose best individual in the subset and append to $Q$ $q_i \\in Q$ Mutate each gene of $q_i$ with mutation probability $p_\\text{mutation}$ Return $Q$ Variant of $\\mu + \\lambda $ evolution strategies used in this study.",
"Note the absence of a crossover step.",
"In this work we use a tournament size of one and a fixed mutation probability $p_\\text{mutate}$ for each gene in an offspring individual.",
"Since in CGP crossover of individuals can lead to significant disruption of the search process due to major changes in the computational graphs [52], we avoid it here.",
"In other words, new offspring are only modified by mutations.",
"We use neutral search [51], in which an offspring is preferred over a parent with equal fitness, to allow the accumulation of silent mutations that can jointly lead to an increase in fitness.",
"As it is computationally infeasible to exhaustively evaluate an individual on all possible tasks from a task family, we evaluate individuals only on a limited number of sample tasks and aggregate the results into a scalar fitness, either by choosing the minimal result or averaging.",
"We manually select the number of sample tasks to balance computational costs and sampling noise for each task.",
"In each generation, we use the same initial conditions to allow a meaningful comparison of results across generations.",
"If an expression is encountered that can not be meaningfully evaluated, such as division by zero, the corresponding individual is assigned a fitness of $-\\infty $ ."
],
[
"HAL-CGP",
"HAL-CGP [69] (https://github.com/Happy-Algorithms-League/hal-cgp) is an extensible pure Python library implementing Cartesian genetic programming to represent, mutate and evaluate populations of individuals encoding symbolic expressions targeting applications with computationally expensive fitness evaluations.",
"It supports the translation from a CGP genotype, a two-dimensional Cartesian graph, into the corresponding phenotype, a computational graph implementing a particular mathematical expression.",
"These computational graphs can be exported as pure Python functions, NumPy-compatible functions [78], SymPy expressions [48] or PyTorch modules [60].",
"Users define the structure of the two-dimensional graph from which the computational graph is generated.",
"This includes the number of inputs, columns, rows, and outputs, as well as the computational primitives, i.e., mathematical operators and constants, that compose the mathematical expressions.",
"Due to the modular design of the library, users can easily implement new operators to be used as primitives.",
"It supports advanced algorithmic features, such as shuffling the genotype of an individual without modifying its phenotype to introduce additional drift over plateus in the search space and hence lead to better exploration [28].",
"The library implements a $\\mu + \\lambda $ evolution strategy to evolve individuals (see sec:methods-evolutionary-algorithm).",
"Users need to specify hyperparameters for the evolutionary algorithm, such as the size of parent and offspring populations and the maximal number of generations.",
"To avoid reevaluating phenotypes that have been previously evaluated, the library provides a mechanism for caching results on disk.",
"Exploiting the wide availability of multi-core architectures, the library can parallelize the evaluation of all individuals in a single generation via separate processes."
],
[
"NEST simulator",
"Spiking neuronal network simulations are based on the 2.16.0 release of the NEST simulator [27] (https://github.com/nest/nest-simulator; commit 3c6f0f3).",
"NEST is an open-source simulator for spiking neuronal networks with a focus on large networks with simple neuron models.",
"The computationally intensive propagation of network dynamics is implemented in C++ while the network model can be specified using a Python API [23], [85].",
"NEST profits from modern multi-core and multi-node systems by combining local parallelization with OpenMP threads and inter-node communication via the Message Passing Interface (MPI) [35].",
"The standard distribution offers a variety of established neuron and plastic synapse models, including variants of spike-timing-dependent plasticity, reward-modulated plasticity and structural plasticity.",
"New models can be implemented via a domain-specific language [63] or custom C++ code.",
"For the purpose of this study, we implemented a reward-driven [75] and an error-driven learning rule [76], as well as a homeostatic STDP rule [45] via custom C++ code.",
"Due to the specific implementation of spike delivery in NEST, we introduce a constant in the STDP rule that is added at each potentiation call instead of using a separate depression term.",
"To support arbitrary mathematical expressions in the error-driven (eq:results-error-driven-general) and correlation-driven synapse models (eq:correlationbasedgenerallr), we additionally implemented variants in which the weight update can be specified via SymPy compatible strings [48] that are parsed by SymEngine (https://github.com/symengine/symengine), a C++ library for symbolic computation.",
"All custom synapse models and necessary kernel patches are available as NEST modules in the repository accompanying this study (https://github.com/Happy-Algorithms-League/e2l-cgp-snn)."
],
[
"Computing systems",
"Experiments were performed on JUWELS (Jülich Wizard for European Leadership Science), an HPC system at the Jülich Research Centre, Jülich, Germany, with 12 Petaflop peak performance.",
"The system contains 2271 general-purpose compute nodes, each equipped with two Intel Xeon Platinum 8168 processors ($2\\text{x}24$ cores) and $12\\text{x}8$ GB main memory.",
"Compute nodes are connected via an EDR-Infiniband fat-tree network and run CentOS 7.",
"Additional experiments were performed on the multicore partition of Piz Daint, an HPC system at the Swiss National Supercomputing Centre, Lugano, Switzerland with $1.731$ Petaflops peak performance.",
"The system contains 1813 general-purpose compute nodes, each equipped with two Intel Xeon E5-2695 v4 processors ($2\\text{x}18$ cores) and 64GB main memory.",
"Compute nodes are connected via Cray Aries routing and communications ASIC with Dragonfly network topology and run Cray Linux Environment (CLE).",
"Each experiment employed a single compute node."
],
[
"Reward-driven learning task",
"We consider a reinforcement learning task for spiking neurons inspired by [75].",
"Spiking activity of the output neuron is generated by an inhomogeneous Poisson process with instantaneous rate $\\phi $ determined by its membrane potential $u$ [62], [75]: $\\phi (u) := \\rho \\, e^{\\frac{u - u_\\text{th}}{\\Delta u}} \\; .$ Here, $\\rho $ is the firing rate at threshold, $u_\\text{th}$ the threshold potential, and $\\Delta u$ a parameter governing the noise amplitude.",
"In contrast to [75], we consider an instantaneous reset of the membrane potential after a spike instead of an hyperpolarization kernel.",
"The output neuron receives spike trains from sources randomly drawn from an input population of size $N$ with randomly initialized weights ($w_\\text{initial} \\sim \\mathcal {N}(0, \\sigma _w)$ ).",
"Before each pattern presentation, the output neurons membrane potential and synaptic currents are reset.",
"The eligibility trace in every synapse is updated in continuous time according to the following differential equation [75], [24]: $\\tau _\\text{M}\\dot{E_j^\\text{r}}= -E_j^\\text{r}+ \\frac{1}{\\Delta u} \\left[ \\sum _{s\\in y}\\delta (t - s) - \\phi (u(t)) \\right] \\bar{s}_j(t) \\; ,$ where $\\tau _\\text{M}$ governs the time scale of the eligibility trace and has a similar role as the decay parameter $\\gamma $ in policy-gradient methods [72], $\\Delta u$ is a parameter of the postsynaptic cell governing its noise amplitude, $y$ represents the postsynaptic spike train, and $\\bar{s}_j(t) = (\\kappa * s_j)(t)$ the presynaptic spike train $s_j$ filtered by the synaptic kernel $\\kappa $ .",
"The learning rate $\\eta $ was manually tuned to obtain high performance with the suggested by [75].",
"Expected positive and negative rewards in trial $i$ are separately calculated as moving averages over previous trials [77]: $\\bar{R}_{i}^{+/-} = (1 - \\frac{1}{m_\\text{r}})\\bar{R}_{i-1}^{+/-} + \\frac{1}{m_\\text{r}} [R_{i-1}]_\\text{+/-} \\; ,$ where $m_\\text{r}$ determines the number of relevant previous trials and $[x]_+ := \\text{max}(0, x), [x]_- := \\text{min}(0, x)$ .",
"Note that $\\bar{R}^+\\in [0, 1]$ and $\\bar{R}^-\\in [-1, 0]$ , since $R \\in \\lbrace -1, 1\\rbrace $ .",
"We obtain the average reward as a sum of these separate estimates $\\bar{R}= \\bar{R}^++ \\bar{R}^-; \\bar{R}\\in [-1, 1]$ , while the expected absolute reward is determined by their difference $\\bar{R}_\\text{abs}= \\bar{R}^+- \\bar{R}^-; \\bar{R}_\\text{abs}\\in [0, 1]$ ."
],
[
"Error-driven learning task",
"We consider an error-driven learning task for spiking neurons inspired by [76].",
"$N$ Poisson inputs with constant rates ($r_i \\sim \\mathcal {U}[r_\\text{min}, r_\\text{max}], i \\in [1, N]$ ) project to a teacher neuron and, with the same connectivity pattern, to a student neuron.",
"As in sec:methods-reinforcement-learning-task, spiking activity of the output neuron is generated by an inhomogeneous Poisson process.",
"In contrast to sec:methods-reinforcement-learning-task, the membrane potential is not reset after spike emission.",
"Fixed synaptic weights from the inputs to the teacher are uniformly sampled from the interval $[w_\\text{min}, w_\\text{max}]$ , while weights to the student are all initialized to a fixed value $w_0$ .",
"In each trial we randomly shift all teacher weights by a global value $w_\\text{shift}$ to avoid a bias in the error signal that may arise if the teacher membrane potential is initially always larger or always smaller than the student membrane potential.",
"Target potentials are read out from the teacher every $\\delta t$ and provided instantaneously to the student.",
"The learning rate $\\eta $ was chosen via grid search on a single example task for high performance with eq:results-error-driven-us.",
"Similar to [76], we low-pass filter weight updates with an exponential kernel with time constant $\\tau _\\text{I}$ before applying them."
],
[
"Correlation-driven learning task",
"We consider a correlation-driven learning task for spiking neurons similar to [45]: a spiking neuron, modeled as a leaky integrate-and-fire neuron with delta-shaped post-synaptic currents, receives stochastic spike trains from $N$ inputs via plastic synapses.",
"To construct the input spike trains, we first create a frozen-noise pattern by drawing random spikes $\\mathcal {S}_i^{\\mathrm {pattern}} \\in [0, T_{\\mathrm {pattern}}], i \\in [0, N-1]$ from a Poisson process with rate $\\nu $ .",
"Neurons that fire at least once in this pattern are in the following called “pattern neurons”, the remaining are called “background neurons”.",
"We alternate this frozen-noise pattern with random spike trains of length $T_{\\mathrm {inter}}$ generated by a Poisson process with rate $\\nu $ (fig:results-corr-learning-taskB).",
"To balance the average rates of pattern neurons and background neurons, we reduce the spike rate of pattern neurons in between patterns by a factor $\\alpha $ .",
"Background neurons have an average rate of $\\nu _{\\mathrm {inter}} = \\nu \\frac{T_{\\mathrm {inter}}}{T_{\\mathrm {inter}}+ T_{\\mathrm {pattern}}}$ .",
"We assume that pattern neurons spike only once during the pattern.",
"Thus, they have an average rate of rate of $\\nu = \\alpha \\nu _{\\mathrm {inter}} + \\frac{1}{T_{\\mathrm {inter}}+ T_{\\mathrm {pattern}}} = \\alpha \\nu _{\\mathrm {inter}} + \\nu _{\\mathrm {pattern}}$ .",
"Plugging in the previous expression for $\\nu _\\text{inter}$ and solving for $\\alpha $ yields $\\alpha = 1 - \\frac{\\nu _{\\mathrm {pattern}}}{\\nu _{\\mathrm {inter}}}$ .",
"We choose the same learning rate as [45].",
"Due to the particular implementation of STDP-like rules in NEST [55], we do not need to evolve multiple functions describing correlation-induced and homeostatic changes separately, but can evolve only one function for each branch of the STDP window.",
"Terms in these functions which do not vanish for $E_j^\\text{c} \\rightarrow 0$ are effectively implementing pre-synaptically triggered (in the acausal branch) and post-synaptically triggered (in the causal branch) homeostatic mechanisms."
],
[
"Acknowledgments",
"We gratefully acknowledge funding from the European Union, under grant agreements 604102, 720270, 785907, 945539 (HBP) and the Manfred Stärk Foundation.",
"We further express our gratitude towards the Gauss Centre for Supercomputing e.V.",
"(www.gauss-centre.eu) for co-funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC).",
"We acknowledge the use of Fenix Infrastructure resources, which are partially funded from the European Union's Horizon 2020 research and innovation programme through the ICEI project under the grant agreement No.",
"800858.",
"We would like to thank all participants from the HBP SP9 Fürberg meetings for stimulating interactions and Tomoki Fukai for initial discussions and support.",
"We also thank Henrik Mettler and Akos Kungl for helpful comments on the manuscript.",
"All network simulations carried out with NEST (www.nest-simulator.org)."
],
[
"Competing interests",
"The authors declare no competing interests."
],
[
"Full evolution data for different CGP hyperparameter choices",
"|p|X| 2|l| CGP hyperparameter set 0 Population $\\mu =1, p_\\text{mutation}=0.035$ Genome $n_\\text{inputs}=3, n_\\text{outputs}=1, n_\\text{rows}=1, n_\\text{columns}=24, l_\\text{max}=24$ Primitives Add, Sub, Mul, Div, Const(1.0), Const(0.5) EA $\\lambda =4, n_\\text{breeding}=4, n_\\text{tournament}=1, \\text{reorder} = \\text{true}$ Other $\\text{max generations}=1000, \\text{minimal fitness}=500.0$ |p|p|X| 3|l| Discovered plasticity rules for hyperparameter set 0 Label Fitness $\\mathcal {F}$ Expression $f$ LR0 $216.2$ $-E_j^\\text{r}+ E_j^\\text{r}/R$ LR1 $73.0$ $(R + {E_j^\\text{r}}^2)/\\bar{R}$ LR2 $216.2$ $E_j^\\text{r}(R - 1.0)$ LR3 $221.6$ $E_j^\\text{r}/(2R + (R + 1.0)(R + \\bar{R}) + 1.0)$ LR4 $234.2$ $-E_j^\\text{r}(R - 1)(R + \\bar{R})$ LR5 $216.2$ $E_j^\\text{r}(R - 1)$ LR6 $69.2$ $4.0{E_j^\\text{r}}^2/\\bar{R}+ 2.0E_j^\\text{r}$ LR7 $234.2$ $E_j^\\text{r}(R - 1)(R + \\bar{R})/R$ Figure: Fitness of best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions for hyperparameter set 0.|p|X| 2|l| CGP hyperparameter set 1 Population $\\mu =1, p_\\text{mutation}=0.035$ Genome $n_\\text{inputs}=\\leavevmode {\\color {red}\\bf 4}^*, n_\\text{outputs}=1, n_\\text{rows}=1, n_\\text{columns}=\\leavevmode {\\color {red}\\bf 12}, l_\\text{max}=\\leavevmode {\\color {red}\\bf 12}$ Primitives Add, Sub, Mul, Div, Const(1.0), Const(0.5) EA $\\lambda =4, n_\\text{breeding}=4, n_\\text{tournament}=1, \\text{reorder} = \\text{true}$ Other $\\text{max generations}=1000, \\text{minimal fitness}=500.0$ $^*$ Red highlights values changed with respect to hyperparameter set 0.",
"|p|p|X| 3|l| Discovered plasticity rules for hyperparameter set 1 Label Fitness $\\mathcal {F}$ Expression $f$ LR0 $238.6$ $(-E_j^\\text{r}(R + \\bar{R}^-(R + \\bar{R}^-)) + E_j^\\text{r}+ \\bar{R}^-)/(R + \\bar{R}^-(R + \\bar{R}^-))$ LR1 $233.4$ $E_j^\\text{r}(R - 1)/(R(R - \\bar{R}^+))$ LR2 $217.2$ $-E_j^\\text{r}(-R + \\bar{R}^-+ 1.0)$ LR3 $227.6$ $R\\bar{R}^-- E_j^\\text{r}+ E_j^\\text{r}/R$ LR4 $247.2$ $(R - 1.0)(R + E_j^\\text{r}+ 2\\bar{R}^+)$ LR5 $198.2$ $(E_j^\\text{r}- \\bar{R}^+- \\bar{R}^-)/(R + \\bar{R}^+)$ LR6 $216.2$ $E_j^\\text{r}(R - 1)$ LR7 $225.8$ $-E_j^\\text{r}- \\bar{R}^-+ (R - \\bar{R}^-)(E_j^\\text{r}+ \\bar{R}^-)$ Figure: Fitness of best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions for hyperparameter set 1.|p|X| 2|l| CGP hyperparameter set 2 Population $\\mu =1, p_\\text{mutation}=0.035$ Genome $n_\\text{inputs}=4, n_\\text{outputs}=1, n_\\text{rows}=1, n_\\text{columns}=\\leavevmode {\\color {red}\\bf 24}^*, l_\\text{max}=\\leavevmode {\\color {red}\\bf 24}$ Primitives Add, Sub, Mul, Div, Const(1.0), Const(0.5) EA $\\lambda =4, n_\\text{breeding}=4, n_\\text{tournament}=1, \\text{reorder} = \\text{\\leavevmode {\\color {red}\\bf false}}$ Other $\\text{max generations}=1000, \\text{minimal fitness}=500.0$ $^*$ Red highlights values changed with respect to hyperparameter set 1.",
"|p|p|X| 3|l| Discovered plasticity rules for hyperparameter set 2 Label Fitness $\\mathcal {F}$ Expression $f$ LR0 $127.2$ $E_j^\\text{r}/(R + \\bar{R}^+- \\bar{R}^-)$ LR1 $192.0$ $E_j^\\text{r}/(R + \\bar{R}^+)$ LR2 $216.2$ $E_j^\\text{r}(R - 1)$ LR3 $170.6$ $(2E_j^\\text{r}\\bar{R}^-(R - \\bar{R}^-) + E_j^\\text{r}- 1)/(R - \\bar{R}^-)$ LR4 $237.6$ $(-RE_j^\\text{r}(\\bar{R}^-+ 1) + E_j^\\text{r}+ \\bar{R}^-)/(R(\\bar{R}^-+ 1))$ LR5 $233.4$ $E_j^\\text{r}(1 - R)/(R - \\bar{R}^+)$ LR6 $120.8$ $(R + \\bar{R}^-)(E_j^\\text{r}- \\bar{R}^+)$ LR7 $254.8$ $(-R\\bar{R}^-+ 2E_j^\\text{r})(R\\bar{R}^-+ R - \\bar{R}^+)$ Figure: Fitness of best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions for hyperparameter set 2.|p|X| 2|l| CGP hyperparameter set 3 Population $\\mu =1, p_\\text{mutation}=0.035$ Genome $n_\\text{inputs}=4, n_\\text{outputs}=1, n_\\text{rows}=1, n_\\text{columns}=24, l_\\text{max}=24$ Primitives Add, Sub, Mul, Div, Const(1.0), Const(0.5) EA $\\lambda =4, n_\\text{breeding}=4, n_\\text{tournament}=1, \\text{reorder} = \\text{\\leavevmode {\\color {red}\\bf true}}^*$ Other $\\text{max generations}=1000, \\text{minimal fitness}=500.0$ $^*$ Red highlights values changed with respect to hyperparameter set 2.",
"|p|p|X| 3|l| Discovered plasticity rules for hyperparameter set 3 Label Fitness $\\mathcal {F}$ Expression $f$ LR0 $236.0$ $E_j^\\text{r}(-R^3(\\bar{R}^-+ 1) + 1)/R$ LR1 $242.0$ $E_j^\\text{r}(R - \\bar{R}^++ \\bar{R}^-)$ LR2 $242.0$ $E_j^\\text{r}(R - \\bar{R}^++ \\bar{R}^-)$ LR3 $227.6$ $R(E_j^\\text{r}+ \\bar{R}^-) - E_j^\\text{r}$ LR4 $256.0$ $E_j^\\text{r}(R - \\bar{R}^++ \\bar{R}^-)/(\\bar{R}^++ 1.0)$ LR5 $71.0$ $(\\bar{R}^+(-R + E_j^\\text{r}+ \\bar{R}^-(R + \\bar{R}^-) + \\bar{R}^-) - \\bar{R}^-)/\\bar{R}^+$ LR6 $216.2$ $E_j^\\text{r}(R - 1.0)$ LR7 $227.8$ $(E_j^\\text{r}- \\bar{R}^-\\,^2)(R + \\bar{R}^-\\,^2 - 1.0)$ Figure: Fitness of best individual per generation as a function of the generation index for multiple runs of the evolutionary algorithm with different initial conditions for hyperparameter set 3."
],
[
"Error-driven learning – simplification of the discovered rules",
"As in the main manuscript $v$ is the teacher potential, $u$ the student membrane potential, and $\\eta $ a fixed learning rate.",
"$\\bar{s}_j(t)=(\\kappa * s_j)(t)$ represents the the presynaptic spike train $s_j$ filtered by the synaptic kernel $\\kappa $ .",
"We first consider eq:results-error-driven-learning-lr0: $\\Delta w_j =& \\eta (v - u)\\bar{s}_j \\frac{2u - 1}{v} \\\\=& \\eta (v - u)\\bar{s}_j \\frac{2(v - \\delta ) - 1}{v} \\\\=& \\eta (v - u)\\bar{s}_j \\left( 2 - 2\\underbrace{\\frac{\\delta }{v}}_{\\ll 1} - \\underbrace{\\frac{1}{v}}_{\\approx 0} \\right) \\\\\\approx & 2 (v - u)\\bar{s}_j \\; ,$ where we introduced $\\delta := v - u$ .",
"From the third to the fourth line we assumed that the mismatch between student and teacher potential is much smaller than their absolute magnitude and that their absolute magnitude is much larger than one.",
"For our parameter choices and initial conditions this is a reasonable assumption.",
"We next consider eq:results-error-driven-learning-lr1: $\\Delta w_j =& \\eta \\bar{s}_j (v + u)\\frac{v (v - u) - \\bar{s}_j}{v^2} \\\\=& \\eta \\bar{s}_j (2v - \\delta ) \\left( \\frac{v - u}{v} - \\frac{\\bar{s}_j}{v^2} \\right) \\\\=& \\eta \\bar{s}_j (2 - \\frac{\\delta }{v}) \\left( (v - u) - \\frac{\\bar{s}_j}{v} \\right) \\\\=& \\eta \\bar{s}_j \\left( \\left( 2 - \\underbrace{\\frac{\\delta }{v}}_{\\ll 1} \\right)(v - u) - 2\\underbrace{\\frac{\\bar{s}_j}{v}}_{\\ll 1} + \\underbrace{\\frac{\\delta }{v}}_{\\ll 1} \\underbrace{\\frac{\\bar{s}_j}{v}}_{\\ll 1} \\right) \\\\\\approx & 2 (v - u)\\bar{s}_j$ As previously, from the third to fourth line we assumed that the mismatch between student and teacher potential is much smaller than their absolute magnitude and that their absolute magnitude is much larger than one.",
"This implies $\\frac{\\bar{s}_j}{v} \\ll 1$ as $\\bar{s}_j \\approx \\mathcal {O}(1)$ for small input rates.",
"The additional terms in both eq:results-error-driven-learning-lr0 and eq:results-error-driven-learning-lr1 hence reduce to a simple scaling of the learning rate and thus perform similarly to the purple rule in fig:results-error-driven-learning."
],
[
"Simulation details",
"tab:supp-nordlie-reward-learning, tab:supp-nordlie-error-learning, and tab:supp-nordlie-corr-learning summarize the network models used in the experiments [58].",
"Table: Description of the network model used in the reward-driven learning task (sec:methods-reinforcement-learning-task).Table: Description of the network model used in the error-driven learning task (sec:methods-error-driven-learning-task).Table: Description of the network model used in the correlation-driven learning task (sec:methods-correlation-driven-learning-task)."
]
] | 2005.14149 |
[
[
"Uncertainty Evaluation Metric for Brain Tumour Segmentation"
],
[
"Abstract In this paper, we develop a metric designed to assess and rank uncertainty measures for the task of brain tumour sub-tissue segmentation in the BraTS 2019 sub-challenge on uncertainty quantification.",
"The metric is designed to: (1) reward uncertainty measures where high confidence is assigned to correct assertions, and where incorrect assertions are assigned low confidence and (2) penalize measures that have higher percentages of under-confident correct assertions.",
"Here, the workings of the components of the metric are explored based on a number of popular uncertainty measures evaluated on the BraTS 2019 dataset."
],
[
"Introduction",
"Deep Neural Networks (DNN) have shown to outperform traditional machine learning methods on a variety of automatic medical image segmentation tasks [5], [6], [11], including tumour segmentation, as depicted by the highest ranking results on recent BraTS challenges [1].",
"However, errors in brain tumour segmentation deter the adoption of DNN frameworks in clinical contexts, particularly those that rely on high voxel-level accuracy, such as in image-guide neurosurgery.",
"Although popular DNNs [2], [7] for brain tumour segmentation provide the \"sigmoid\"/\"softmax\" predictions for tumour labels, it is the overall model uncertainties which would more informative in assisting clinicians in making more informed decisions.",
"Although, several recent methods [3], [8], [9] have been proposed to estimate uncertainties in deep neural networks, there is no established strategy in which their usefulness can be assessed and compared for particular clinical contexts.",
"In this paper, we develop metrics to measure the quality of different uncertainty measures for the task of brain tumour segmentation, with the objectives: (1) when the network is correct it is confident in the predicted labels, and (2) when they are incorrect, it is not confident.",
"These metrics were combined and used as the basis of ranking the uncertainties produced by participating teams in the BraTS 2019 sub-challenge on uncertainty quantification."
],
[
"Metric for Assessing and Comparing Uncertainty Measures",
"In the context of the BraTS 2019 challenge, each team provided their multi-class brain tumour segmentation output labels and the voxel-wise uncertainties for each of the associated tasks: whole tumour (WT), tumour core (TC) and enhanced tumour (ET) segmentations.",
"For each task, the uncertain voxels were filtered out at several predetermined (N) number of uncertainty thresholds, $\\tau $ , and the model performance was assessed based on the contextual metric of interest (here, Dice score) on the remaining voxels at each of the thresholds.",
"For example, at $\\tau = 0.75$ , all voxels with uncertainty values $\\ge 0.75$ are marked as uncertain.",
"The associated predictions are filtered out, and Dice values are calculated for the remaining predictions based on the unfiltered voxels.",
"This evaluation rewards models where the confidence in the incorrect assertions (False Positives - FPs, and False Negatives - FNs) is low and high for correct assertions (True Positives - TPs and True Negatives - TNs).",
"For these models, it is expected that as more uncertain voxels are filtered out, the Dice score should increase on the remaining predictions.",
"Figure: Effect of uncertainty thresholding on two examples for whole tumour segmentations (Top and bottom rows).",
"(a) FLAIR MRI, (b) \"Ground truth\" labels, (c) Sample prediction, (d) Prediction with no filtering, and (e)-(g) Filtering with uncertainty thresholds (τ\\tau ) of 0.75, 0.5 and 0.25.Table: Changes in Dice, Filtered True Positives (FTP), and Filtered True Negatives (FTN) with different uncertainty thresholds (τ\\tau ) for two different examples.Figure: Effect of changing uncertainty threshold (τ\\tau ) on whole tumour for entropy measure.The proposed strategy does not keep track of the number of correctly labeled voxels that are filtered at each threshold level along with the uncertain incorrect labels.",
"In order to penalize filtering out many correctly predicted voxels (TPs, TNs) when attaining high Dice values, an additional assessment component is added to keep track of the filtered TP and TNs voxels.",
"Given that tumour segmentation is expected to have a high-class imbalance between tumour and healthy tissues, the system keeps track of the filtered TPs and TNs separately.",
"The ratio of filtered TPs (FTP) at different thresholds ($\\tau $ ) is measured relative to the unfiltered values ($\\tau = 1.00$ ) such that FTP = ($\\text{TP}_{\\text{1.00}}$ - $\\text{TP}_\\tau $ ) / $\\text{TP}_{\\text{1.00}}$ .",
"The ratio of filtered TNs is calculated in a similar manner.",
"This evaluation essentially penalizes approaches that filter out a large percentage of TP or TN relative to $\\tau =1.00$ voxels in order to attain the reported Dice value.",
"Figure REF and Table REF shows the workings of the assessment metric for example cases based on images from BraTS 2019.",
"Decreasing the threshold ($\\tau $ ) leads to filtering out voxels with incorrect assertions, leading to an increase in the Dice value for the remaining voxels.",
"Case 2 shows a marginally higher Dice value than Case 1 at uncertainty thresholds $\\tau $ = 0.50 and 0.25.",
"However, the Ratio of Filtered TPs and TNs indicates that this is at the expense of marking more TPs and TNs as uncertain.",
"Finally, different uncertainty measures are ranked according to a unified score which combines the area under three curves: 1) Dice vs $\\tau $ , 2) FTP vs $\\tau $ , and 3) FTN vs $\\tau $ , for different values of $\\tau $ .",
"The unified score is calculated as follows: $score = \\frac{{AUC}_1 + (1-{AUC}_2) + (1-{AUC}_3)}{3}.$"
],
[
"Experiments and Results",
"A modified 3D U-Net architecture [2], [10] generates the segmentation outputs and corresponding uncertainties.",
"We train (228), validate (57), and test (50) this network based on the publicly available Brain Tumour Segmentation (BraTS) challenge 2019 training dataset (335) [1].",
"The performances of whole tumour segmentation with the Entropy uncertainty measure [4], which captures the average amount of information contained in the predictive distribution, using MC-Dropout [3], Deep Ensemble [8], Dropout Ensemble [12], Bootstrap, Dropout Bootstrap, and Deterministic, are shown in Figure REF .",
"Please refer to supplementary material for more results.",
"Dropout bootstrap shows the best Dice performance (highest AUC), but also has the worst performance for Filtered True Positive and Filtered True Negative curves (highest AUC).",
"This result shows that, here, the higher performance in Dice is at the expense of a higher number of filtered correct voxels.",
"Overall, the metric is working in line with the objectives.",
"However, there is no clear winner amongst these uncertainty methods in terms of rankings."
],
[
"Conclusion",
"This paper provides a rationale behind the design of a metric presented at the MICCAI BraTS 2019 sub-challenge to evaluate and rank uncertainties produced by different methods for brain tumour segmentation.",
"Using two different examples it was demonstrated that the designed metric is able to reward methods which convey higher uncertainty for incorrect assertions and penalize methods which have higher uncertainties for correct assertions."
]
] | 2005.14262 |
[
[
"Spectral halo for Hilbert modular forms"
],
[
"Abstract Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$.",
"We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\\mathfrak{p}$ of $F$ over $p$.",
"Applying Hansen's $p$-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties.",
"In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its $U_p$ slope goes to zero.",
"In the case of eigencurves, this completes the proof of Coleman-Mazur's `halo' conjecture."
],
[
"Introduction",
"The theory of $p$ -adic analytic families of modular forms grew out of the study of congruences of modular forms.",
"In [24], Serre gave the first example of a $p$ -adic family of modular forms-an Eisenstein family.",
"After more than one decade, Hida made the first major step towards the construction of $p$ -adic families of cuspforms.",
"In [13] and [14], he constructed $p$ -adic families of ordinary modular forms.",
"His theory led Mazur to develop his general theory of deformations of Galois representations in [21], which turns out to be a crucial ingredient of Wiles' proof of Fermat's last theorem.",
"The theory of $p$ -adic families of modular forms reached a culmination in Coleman and Mazur's celebrated work [9] in which they constructed eigencurves.",
"An eigencurve is a rigid analytic curve whose points correspond to finite slopes normalized overconvergent eigenforms.",
"When pursuing a description of the geometry of the eigencurves, Coleman and Mazur raised the following question based on detailed computations for small primes: does the slope always tend to 0 as one approaches the `boundary' of the weight space?",
"This problem has been made into a much more delicate conjecture based on the computation of Buzzard and Kilford in [7].",
"It is expected that over the boundary of the weight space, the eigencurve is a disjoint union of infinitely many connected components that are finite flat over the weight space.",
"We refer to [20] for the precise statement of the conjecture.",
"Figure: Spectral halo of eigencurvesFor eigencurves associated to a definite quaternion algebra $D$ over $\\mathbb {Q}$ , this conjecture was studied by R. Liu, D. Wan, and L. Xiao in [20].",
"To be more precise, we fix an odd prime $p$ and denote by $\\mathrm {Spc}_D$ the spectral curve associated to the overconvergent automorphic forms on $D/\\mathbb {Q}$ (with some tame level).",
"It admits a weight map $\\mathrm {wt}:\\mathrm {Spc}_D\\rightarrow \\mathcal {W}$ to the weight space $\\mathcal {W}$ , where $\\mathcal {W}$ is the rigid analytification of the Iwasawa algebra $\\mathbb {Z}_p\\llbracket \\mathbb {Z}_p^\\times \\rrbracket $ and a slope map $a_p:\\mathrm {Spc}_D\\rightarrow \\mathbb {G}_m$ .",
"For $r\\in (0,1)$ , we use $\\mathcal {W}^{>r}$ to denote the subspace of $\\mathcal {W}$ where the fixed parameter $T:= [\\exp (p)]-1$ satisfies $|T|_p\\in (r,1)$ and let $\\mathrm {Spc}_D^{>r}=\\mathrm {wt}^{-1}(\\mathcal {W}^{>r})$ .",
"Under the above notations, Liu-Wan-Xiao proved the following theorem.",
"Theorem 1.0.1 The space $\\mathrm {Spc}_D^{>1/p}$ can be decomposed into a disjoint union $X_0\\coprod X_{(0,1)}\\coprod X_1\\coprod X_{(1,2)}\\coprod X_2\\coprod \\cdots $ of rigid analytic spaces which are finite and flat over $\\mathcal {W}^{>1/p}$ via $\\mathrm {wt}$ such that for each point $x \\in X_I$ with $I$ denoting the interval $n=[n,n]$ or $(n, n+1)$ , we have $v_p(a_p(x)) \\in (p-1)v_p(T_{\\mathrm {wt}(x)})\\cdot I.$ In particular, as $x$ varies on each irreducible component of $\\mathrm {Spc}_D$ with $\\mathrm {wt}(x)$ approaching the boundary of weight space, i.e.",
"$|T_{\\mathrm {wt}(x)}|_p\\rightarrow 1^-$ , the slope $v_p(a_p(x))\\rightarrow 0$ .",
"By $p$ -adic family version of the Jacquet-Langlands correspondence (see [8]), they also deduce the similar results for most of the components of the original Coleman-Mazur eigencurves.",
"The goal of this paper is to generalize the result in [20] to the eigenvarieties associated to ($p$ -adic) overconvergent automorphic forms for a definite quaternion algebra over a totally real field $F$ in which $p$ splits completely.",
"Combining with the $p$ -adic family versions of base change and Jacquet-Langlands correspondence, this result allows us to determine the boundary behavior for the entire Coleman-Mazur eigencurves, and hence answers the first part of original question raised by Coleman and Mazur (see [20]) in complete generality.",
"We set up a few notations before stating our main result.",
"Let $F$ be a totally real field of degree $g$ and $p$ be an odd prime number which splits completely in $F$ .",
"Let $\\mathcal {O}_F$ be the ring of integers of $F$ and set $\\mathcal {O}_p:= \\mathcal {O}_F\\otimes _{\\mathbb {Z}}\\mathbb {Z}_p$ .",
"Let $I:=\\operatorname{Hom}(F,\\bar{\\mathbb {Q}}_p)$ and for each $i\\in I$ , we choose a uniformizer $\\pi _i$ of the completion $F_{\\mathfrak {p}_i}$ of $F$ at the prime $\\mathfrak {p}_i$ induced by $i$ .",
"The weight space $\\mathcal {W}$ is the rigid analytification of $\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ .",
"It has a full set of parameters $\\big \\lbrace (T_i:= [\\exp (\\pi _i),1]-1)_{i\\in I},T:= [1,\\exp (p)]-1 \\big \\rbrace $ .",
"For $r\\in (0,1)$ , we use $\\mathcal {W}^{>r}$ to denote the subspace of $\\mathcal {W}$ where $|T_i|_p\\in (r,1)$ for all $i\\in I$ .",
"Let $D/F$ be a totally definite quaternion division algebra over $F$ , which is split at all places over $p$ .",
"Fix a tame level structure.",
"Let $\\mathcal {X}_D$ (resp.",
"$\\mathcal {Z}_D$ ) be the eigenvariety (resp.",
"spectral variety) associated to the overconvergent automorphic forms for $D^\\times $ constructed in [5].",
"We put $w:\\mathcal {X}_D\\rightarrow \\mathcal {W}$ be the weight map and denote by $\\mathcal {X}_D^{>r}$ the preimage $w^{-1}(\\mathcal {W}^{>r})$ .",
"For each $x\\in \\mathcal {X}_D(\\mathbb {C}_p)$ , it corresponds to a $\\mathbb {C}_p$ -valued system of eigenvalues (we refer to [5] for the precise definition).",
"For each $i\\in I$ , there is a Hecke operator $U_{\\pi _i}$ acting on the space of overconvergent automorphic forms and we use $a_i(x)$ to denote the eigenvalue of the $U_{\\pi _i}$ -operator for $x$ .",
"The constructions of the eigenvarieties and Hecke operators will be carefully recalled in §REF and §REF .",
"$@=1.5cm{\\mathcal {X}_D [r]^{(a_i)_{i\\in I}} [d]^{w} & (\\mathbb {G}_m)^g \\\\\\mathcal {W}&}$ Under the above notations, we have the following theorem.",
"Theorem 1.0.2 We denote by $\\Sigma $ the subset $\\lbrace 0 \\rbrace \\bigcup \\lbrace 1+2k|k\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $\\mathbb {Z}$ .",
"The eigenvariety $\\mathcal {X}_D^{>1/p}$ is a disjoint union $\\mathcal {X}_D^{>1/p}=\\bigsqcup _{l\\in \\Sigma ^I,\\ \\sigma \\in \\lbrace \\pm \\rbrace ^I}\\mathcal {X}_{l,\\sigma }$ of (possibly empty) rigid analytic spaces which are finite over $\\mathcal {W}^{>1/p}$ via $w$ , such that for each closed point $x\\in \\mathcal {X}_{l,\\sigma }(\\mathbb {C}_p)$ with $l=(l_i)_{i\\in I}\\in \\Sigma ^I$ and $\\sigma =(\\sigma _i)_{i\\in I}\\in \\lbrace \\pm \\rbrace ^I$ , we have ${\\left\\lbrace \\begin{array}{ll}v_p(a_i(x)) = (p-1)v_p(T_{i,w(x)})\\cdot l_i,&\\textrm {for}\\ \\sigma _i=-,\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (l_i,l_i+2),&\\textrm {for}\\ \\sigma _i=+ \\textrm {~and~} l_i\\ne 0,\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (0,1),&\\textrm {for}\\ \\sigma _i=+ \\textrm {~and~} l_i=0,\\end{array}\\right.",
"}$ for all $i\\in I$ .",
"In particular, as $x$ varies on each irreducible component of $\\mathcal {X}_D$ with $w(x)$ approaching the boundary of the weight space, i.e.",
"$v_p(T_{i,w(x)})\\rightarrow 0$ , the slopes $v_p(a_i(x))\\rightarrow 0$ for all $i\\in I$ .",
"Remark 1.0.3 In our main theorem, the decomposition of the eigenvariety $\\mathcal {X}_D^{>1/p}$ is characterized by all the $U_{\\pi _i}$ -operators ($i\\in I$ ).",
"We cannot work solely with the spectral varieties since only the eigenvalues of $U_{\\pi }:=\\prod _{i\\in I}U_{\\pi _i}$ can be read from the spectral varieties.",
"This is a significant difference to the eigencurve case.",
"There have been several generalizations of Liu-Wan-Xiao's results to more general eigenvarieties.",
"In [27], Ye generalized Liu-Wan-Xiao's estimation on slopes in eigencurves to eigenvarieties for definite unitary groups of arbitrary rank.",
"Ye gave a lower bound and upper bound of the Newton polygon of characteristic power series for the $U_p$ -operator, but they do not match at any point on the Newton polygon.",
"So she cannot prove a similar result as in Theorem REF for eigenvarieties associated to definite unitary groups.",
"There are also generalizations to Hilbert modular eigenvarieties.",
"Let $F$ and $p$ be as above.",
"In [17], Johansson and Newton defined a one-dimensional `partial' eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place above $p$ .",
"They proved that over the boundary annulus of the weight space, the partial eigenvarieties decompose as a disjoint union of components that are finite over weight space, and the components have a similar property as described in the main result of Liu-Wan-Xiao.",
"Our result agrees with theirs when restricting to their `partial' eigenvarieties.",
"The assumption that $p$ splits completely in $F$ is essential in our argument.",
"In fact, the philosophical analogy between the Artin–Schreier–Witt tower and the Igusa tower of Hilbert modular Shimura varieties explained in [23] and the main theorem there suggests that further modifications are needed in order to formulate a `reasonable' conjecture that generalizes Theorem REF to general $F$ .",
"We refer to §REF for more discussion towards the (conjectural) generalization of our theorem."
],
[
"Idea of the proof of Theorems ",
"We now explain how we deduce our main result.",
"Going back to Liu-Wan-Xiao's work in [20], the way they deduce their main results over $\\mathbb {Q}$ relies crucially on two key ingredients: a sharp estimate of the action of the $U_p$ -operator on the space of overconvergent automorphic forms, in the form of providing a lower bound of the Hodge polygon, and the observation that, at classical weights, the subspaces of classical automorphic forms provide known points of the corresponding Newton polygon which happens to lie on the Hodge polygon.",
"While we choose to work with automorphic forms associated to a definite quaternion algebra over $F$ as opposed to the usual overconvergent modular forms, we circumvent the complication of the geometry of the Hilbert modular Shimura varieties.",
"We will define the spaces of integral $p$ -adic automorphic forms.",
"It contains the spaces of overconvergent automorphic forms and have the same characteristic power series of the $U_\\pi $ -operator.",
"An important observation is that the $U_\\pi $ -operator on the spaces of integral $p$ -adic automorphic forms can be written reasonably explicitly, as explained in §REF .",
"This was inspired by the thesis [16] of D. Jacobs (a former student of Buzzard), and the generalization in [20].",
"The major difficulty we encounter here is that the action of each individual $U_{\\pi _i}$ -operator on the space of overconvergent automorphic forms is not compact, whereas the action of their product is.",
"So if we generalize the method in [20] naively, one would have to work with the operator $U_\\pi =\\prod \\limits _{i\\in I}U_{\\pi _i}$ .",
"It is possible to give a lower bound of the corresponding Hodge polygon similar to $\\textbf {I}(a)$ , which is associated to inequalities regarding the sum of all the $U_{\\pi _i}$ -slopes.",
"But for $\\textbf {I}(b)$ , the space of classical forms are characterized by the properties of the type: the $U_{\\pi _i}$ -slope is less than or equal to $k_i-1$ , for every individual $i\\in I$ .",
"This does not provide a known point on the Newton polygon for the action of $U_\\pi $ , because the ordering mechanism of bases are incompatible.",
"More precisely, there exists no orthonormal basis, in the sense of [10], of the space of continuous function $\\mathcal {C}(\\mathcal {O}_p,\\mathbb {C}_p)$ , such that the first $d:= \\prod \\limits _{i\\in I}(k_i-1)$ elements of $\\Omega $ , gives a basis of $\\mathcal {C}(\\mathcal {O}_p,\\mathbb {C}_p)^{\\deg \\le k-2}$ for all $k=(k_i)_{i\\in I}\\in \\mathbb {Z}_{\\ge 2}^I$ , where $\\mathcal {C}(\\mathcal {O}_p,\\mathbb {C}_p)^{\\deg \\le k-2}$ is the $\\mathbb {C}_p$ -subspace of $\\mathcal {C}(\\mathcal {O}_p,\\mathbb {C}_p)$ spanned by the polynomial functions $\\prod \\limits _{i\\in I}z_i^{l_i}$ , for all $0\\le l_i\\le k_i-2$ , $i\\in I$ , and $z_i:\\mathcal {O}_p=\\prod \\limits _{i\\in I}\\mathcal {O}_{\\mathfrak {p}_i}\\rightarrow \\mathbb {C}_p$ is the projection to the $i$ th component.",
"Therefore, the lower bound of the Hodge polygon of the $U_\\pi $ -operator does not touch its actual Newton polygon in general when the degree $g=[F:\\mathbb {Q}]\\ge 2$ .",
"To circumvent this difficulty, we work with the space of generalized integral $p$ -adic automorphic forms.",
"In the introduction, we explain our idea in the simplest case when $[F:\\mathbb {Q}]=2$ .",
"Let $\\mathfrak {p}_1,\\mathfrak {p}_2$ be the two places of $F$ over $p$ .",
"Let $A$ be a topological ring in which $p$ is topologically nilpotent.",
"For a continuous homomorphism $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ , we define the space of integral $p$ -adic automorphic forms $S_{\\kappa ,I}^D(K^p,A)$ and the space of generalized $p$ -adic automorphic forms $S_{\\kappa ,1}^D(K^p,A)$ .",
"The latter space consists of generalized automorphic forms that are like automorphic forms at the place $\\mathfrak {p}_1$ , but are like the continuous dual of the completed homology at the place $\\mathfrak {p}_2$ .",
"We have explicit isomorphisms of these spaces: $S_{\\kappa ,I}^D(K^p,A)\\cong \\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}(\\mathcal {O}_p,A)\\textrm {~and~}S_{\\kappa ,1}^D(K^p,A)\\cong \\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}_1(\\kappa ,A),$ where $\\mathcal {C}_1(\\kappa ,A)$ is a suitably defined subspace of $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_1}\\times \\mathrm {Iw}_{\\pi _2},A)$ that contains $\\mathcal {C}(\\mathcal {O}_p,A)$ .",
"We use $S_{\\kappa ,I}^D(K^p,A)^\\vee $ (resp.",
"$S_{\\kappa ,1}^D(K^p,A)^\\vee $ ) to denote the continuous $A$ -dual space of $S_{\\kappa ,I}^D(K^p,A)$ (resp.",
"$S_{\\kappa ,1}^D(K^p,A)$ ).",
"Our argument can be exhibited in the following diagram: $@=1.2cm{S_{\\kappa ,1}^D(K^p,A)^\\vee @{-}[r]^{dual} [d] &S_{\\kappa ,1}^D(K^p,A) & \\curvearrowleft U_{\\pi _1} \\\\S_{\\kappa ,I}^D(K^p,A)^\\vee @{-}[r]^{dual}& S_{\\kappa ,I}^D(K^p,A) [u] & \\curvearrowleft U_{\\pi _1}, U_{\\pi _2}.",
"}$ Here the right vertical arrow is an embedding $S_{\\kappa ,I}^D(K^p,A)\\hookrightarrow S_{\\kappa ,1}^D(K^p,A)$ , which identifies $S_{\\kappa ,I}^D(K^p,A)$ as the invariant subspace of $S_{\\kappa ,1}^D(K^p,A)$ under an action of the Borel subgroup $B(\\mathcal {O}_{\\mathfrak {p}_2})$ of $\\operatorname{GL}_2(\\mathcal {O}_{\\mathfrak {p}_2})$ ; and the left vertical arrow is the dual of this embedding, i.e.",
"taking the $B(\\mathcal {O}_{\\mathfrak {p}_2})$ -coinvariants of $S_{\\kappa ,1}^D(K^p,A)^\\vee $ .",
"We first embed $S_{\\kappa ,I}^D(K^p,A)$ into the larger space $S_{\\kappa ,1}^D(K^p,A)$ .",
"The latter space carries an extra structure of right $A\\llbracket P^{\\prime }_2\\rrbracket $ -modules, where $P^{\\prime }_2\\subset \\mathrm {SL}_2(\\mathcal {O}_{\\mathfrak {p}_2})$ is some explicit open compact pro-$p$ -subgroup.",
"The sacrifice of doing this is that there is only $U_{\\pi _1}$ -operator, but no $U_{\\pi _2}$ -operator defined on $S_{\\kappa ,1}^D(K^p,A)$ .",
"Thus the space $S_{\\kappa ,1}^D(K^p,A)^\\vee $ is an infinite free left $A\\llbracket P^{\\prime }_2\\rrbracket $ -module and the induced $U_{\\pi _1}$ -operator on it is $A\\llbracket P^{\\prime }_2\\rrbracket $ -linear.",
"Under a suitable chosen basis of $S_{\\kappa ,1}^D(K^p,A)^\\vee $ , we verify that the infinite matrix $M$ corresponding to the $U_{\\pi _1}$ -operator admits a similar estimation as obtained in [20].",
"On the other hand, we have a characterization of the image of the spaces of classical automorphic forms in $S_{\\kappa ,1}^D(K^p,A)$ (when $A=\\mathbb {C}_p$ and $\\kappa $ is locally algebraic).",
"An analogous argument of $\\textbf {I}(b)$ provides us known points on the Newton polygon of $M$ modulo the augmentation ideal of $A\\llbracket P^{\\prime }_2\\rrbracket $ .",
"In §, we prove a Newton-Hodge decomposition theorem for infinite matrices over certain noncommutative rings, which is a generalization of the Newton-Hodge decomposition theorem over valuation rings.",
"We apply this theorem to the matrix $M$ , and obtain a filtration $\\lbrace \\tilde{F}_{\\alpha } \\rbrace $ of the space $S_{\\kappa ,1}^D(K^p,A)^\\vee $ .",
"Under the surjective map $S_{\\kappa ,1}^D(K^p,A)^\\vee \\rightarrow S_{\\kappa ,I}^D(K^p,A)^\\vee $ , we obtain a filtration $\\lbrace F_\\alpha \\rbrace $ of $S_{\\kappa ,I}^D(K^p,A)^\\vee $ .",
"We will show that this filtration is stable under the $U_{\\pi _1}$ and $U_{\\pi _2}$ -operators on $S_{\\kappa ,I}^D(K^p,A)^\\vee $ .",
"When $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ is associated to a point $x\\in \\mathcal {W}^{>1/p}(\\mathbb {C}_p)$ , the graded pieces of this filtration are characterized by the condition in Theorem REF for the single place $i=\\mathfrak {p}_1$ .",
"We run the above argument to the graded pieces of the filtration $\\lbrace F_\\alpha \\rbrace $ and the $U_{\\pi _2}$ -operator on it.",
"We get a filtration for every graded piece, and the graded pieces of all these filtrations are characterized by the desired property in Theorem REF .",
"Our main theorem follows form the existence of such filtrations."
],
[
"Applications",
"Let $F$ and $p$ be as above and $\\mathfrak {n}$ be an ideal of $F$ prime to $p$ .",
"We use $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})$ to denote the Hilbert modular eigenvariety of tame level $\\mathfrak {n}$ , which admits a weight map $w:\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})\\rightarrow \\mathcal {W}$ .",
"Similar as before, we can define $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})^{>1/p}$ .",
"An important consequence of Theorem REF is the following description of the full Hilbert modular eigenvarieties over the boundary of the weight space.",
"Theorem 1.2.1 The eigenvariety $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})^{>1/p}$ is a disjoint union $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})^{>1/p}=\\bigsqcup _{l\\in \\Sigma ^I, \\sigma \\in \\lbrace \\pm \\rbrace ^I}\\mathcal {X}_{l,\\sigma }$ of (possibly empty) rigid analytic spaces which are finite over $\\mathcal {W}^{>1/p}$ via $w$ , such that for each closed point $x\\in \\mathcal {X}_{l,\\sigma }(\\mathbb {C}_p)$ with $l=(l_i)_{i\\in I}\\in \\Sigma ^I$ and $\\sigma =(\\sigma _i)_{i\\in I}\\in \\lbrace \\pm \\rbrace ^I$ , we have ${\\left\\lbrace \\begin{array}{ll}v_p(a_i(x)) = (p-1)v_p(T_{i,w(x)})\\cdot l_i,&\\textrm {for}\\ \\sigma _i=-,\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (l_i,l_i+2),&\\textrm {for}\\ \\sigma _i=+ \\textrm {~and~} l_i\\ne 0,\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (0,1),&\\textrm {for}\\ \\sigma _i=+ \\textrm {~and~} l_i=0,\\end{array}\\right.",
"}$ for all $i\\in I$ .",
"This theorem will be proved in §.",
"The main tool of our proof is Hansen's $p$ -adic interpolation theorem ([12]).",
"When the degree $[F:\\mathbb {Q}]$ is even, there is an isomorphism between $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})$ and the eigenvariety $\\mathcal {X}_D(\\mathfrak {n})$ for the totally definite quaternion algebra $D$ over $F$ with discriminant 1.",
"The theorem above follows directly from Theorem REF .",
"When $[F:\\mathbb {Q}]$ is odd, we take a quadratic extension $F^{\\prime }/F$ such that $F^{\\prime }$ is totally real and $p$ splits completely in $F^{\\prime }$ .",
"Set $\\mathfrak {n}^{\\prime }=\\mathfrak {n}\\mathcal {O}_F$ .",
"We show that there exists a morphism $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})\\rightarrow \\mathcal {X}_{\\operatorname{GL}_{2/F^{\\prime }}}(\\mathfrak {n}^{\\prime })$ that interpolates the quadratic base change from $F$ to $F^{\\prime }$ on non-critical classical points.",
"Then the theorem for $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})$ follows from that for $\\mathcal {X}_{\\operatorname{GL}_{2/F^{\\prime }}}(\\mathfrak {n}^{\\prime })$ .",
"A notable consequence of the above theorem is that every irreducible component of the (full) Hilbert modular eigenvarieties contains a classical point of parallel weight 2.",
"This result has been proven for most irreducible components by Johansson-Newton in [17].",
"They use this result to prove the parity conjecture for Hilbert modular forms unconditionally when the degree $[F:\\mathbb {Q}]$ is even, and when the degree $[F:\\mathbb {Q}]$ is odd, they need to impose the assumption that the automorphic representation corresponding to the Hilbert modular form is not principal series at some place prime to $p$ .",
"The last assumption is to guarantee that the Hilbert modular form corresponds to an automorphic form for a definite quaternion algebra over $F$ under the Jacquet-Langlands correspondence.",
"This assumption can be removed now."
],
[
"Further questions",
"Theorem REF is not a complete description of the boundary behavior of the eigenvariety $\\mathcal {X}_D$ .",
"Inspired by Coleman-Mazur-Buzzard-Kilford conjecture ([20]), we make the following conjecture.",
"Conjecture 1.3.1 When $r\\in (0,1)$ is sufficiently close to $1^-$ , there exists a sequence of rational numbers $\\Sigma _i=\\lbrace \\alpha _{i,0}\\le \\alpha _{i,1}\\le \\dots \\rbrace $ for every $i\\in I$ , such that if we denote $\\Sigma :=\\prod \\limits _{i\\in I}\\Sigma _i$ , then the eigenvariety $\\mathcal {X}_D^{>r}$ is a disjoint union $\\mathcal {X}_D^{>r}=\\bigsqcup _{\\alpha =(\\alpha _i)_{i\\in I}\\in \\Sigma } \\mathcal {X}_\\alpha $ of (possibly empty) rigid analytic spaces which are finite over $\\mathcal {W}^{>r}$ via $w$ .",
"For every $\\alpha \\in \\Sigma $ and each closed point $x\\in \\mathcal {X}_\\alpha (\\mathbb {C}_p)$ , we have $v_p(a_i(x))=(p-1)v_p(T_{i,w(x)})\\cdot \\alpha _i$ for all $i\\in I$ .",
"Moreover, each sequence $\\Sigma _i$ is a disjoint union of finitely many arithmetic progressions, counted with multiplicities.",
"When $F=\\mathbb {Q}$ , Conjecture REF has been proved for the eigenvariety $\\mathcal {X}_D^{>r}$ for some (explicit) rational number $r\\in (\\frac{1}{p},1)$ in [20].",
"Their proof is based on a careful analysis of the characteristic power series of the $U_p$ -operator and the fact that the coefficients of the characteristic power series belong to the Iwasawa algebra $\\Lambda =\\mathbb {Z}_p\\llbracket \\mathbb {Z}_p^\\times \\rrbracket $ .",
"It seems to us that more ideas are needed in order to generalize their result to the case of Hilbert modular eigenvarieties.",
"In [3], Birkbeck made a similar conjecture on the $p$ -adic slopes of overconvergent Hilbert modular forms without any assumption on the totally real field $F$ .",
"Our Conjecture REF makes the assumption that $p$ splits in $F$ , and hence gives a more precise characterization of the decomposition of the eigenvarieties.",
"We also refer to Birkbeck's thesis [4] for numerical evidence towards Conjecture REF .",
"Finally, we make a comment about the case for $p=2$ .",
"Most of our arguments work for $p=2$ with mild modifications.",
"However, there is one subtle place in Notation REF that the oddness of $p$ is crucial.",
"We explain the case for $F=\\mathbb {Q}$ as an example and refer to Notation REF for details.",
"Let $\\mathrm {Iw}_p=\\big ( {\\begin{matrix}\\mathbb {Z}_p^\\times &\\mathbb {Z}_p\\\\p\\mathbb {Z}_p&\\mathbb {Z}_p^\\times \\end{matrix}}\\big )$ be the Iwahori subgroup of $\\operatorname{GL}_2(\\mathbb {Z}_p)$ and $D(\\mathbb {Z}_p^\\times )=\\left\\lbrace \\big ( {\\begin{matrix}\\alpha &0\\\\0&\\alpha \\end{matrix}}\\big )|\\alpha \\in \\mathbb {Z}_p^\\times \\right\\rbrace $ be the subgroup of $\\mathrm {Iw}_p$ consisting of scalar matrices.",
"When $p>2$ , we show that the inclusion map $D(\\mathbb {Z}_p)\\hookrightarrow \\mathrm {Iw}_p$ has a section, i.e.",
"there exists a subgroup $P$ of $\\mathrm {Iw}_p$ such that the multiplication map $D(\\mathbb {Z}_p)\\times P\\rightarrow \\mathrm {Iw}_p$ is an isomorphism.",
"We encourage the audience to explore how to modify the argument for $p=2$ ."
],
[
"Structure of the paper",
"The § is devoted to prove a Newton-Hodge decomposition theorem for infinite matrices over certain noncommutative rings.",
"This section is technical and the readers can assume the main result Theorem REF and skip its lengthy proof at first.",
"In §, we first recall the notions of overconvergent automorphic forms, Hecke operators and eigenvarieties.",
"Then we construct a space of generalized integral $p$ -adic automorphic forms and explain its relation with the spaces of classical automorphic forms.",
"In §, we give an explicit expression of the Hecke operator $U_{\\pi _i}$ on the space of generalized integral $p$ -adic automorphic forms, and use the Newton-Hodge decomposition Theorem REF to obtain a filtration of this space, whose graded piece has a nice description in terms of the $U_{\\pi _i}$ -action, for a fixed $i\\in I$ .",
"We complete the proof of Theorem REF in §.",
"The idea is to inductively apply the argument in the previous section to all places of $F$ over $p$ , and then get a filtration of the space of integral $p$ -adic automorphic forms, whose graded pieces can be described in terms of all the $U_{\\pi _i}$ -operators.",
"The decomposition of the eigenvarieties follows from the existence of such a filtration.",
"In §, we use Hansen's $p$ -adic interpolation theorem to translate our results to Hilbert modular eigenvarieties."
],
[
"Acknowledgment",
"We thank Liang Xiao for sharing his ideas and answering many questions on this topic.",
"We would also like to thank Yiwen Ding, Yongquan Hu and Daqing Wan for helpful comments and conversations.",
"Finally we would like to thank the anonymous referees for their impressively helpful report that significantly improved the exposition of this paper."
],
[
"Notations",
"For every prime $p$ , we fix an embedding $\\iota _p:\\bar{\\mathbb {Q}}\\rightarrow \\mathbb {C}_p$ .",
"We use $v_p(\\cdot )$ (resp.",
"$|\\cdot |_p$ ) to denote the $p$ -adic valuation (resp.",
"$p$ -adic norm) on $\\mathbb {C}_p$ , normalized by $v_p(p)=1$ (resp.",
"$|p|_p=p^{-1}$ ).",
"For two topological spaces $X$ and $Y$ , we use $\\mathcal {C}(X,Y)$ to denote the set of continuous maps from $X$ to $Y$ ."
],
[
"Notations",
" Let $\\mathbb {N}=\\lbrace 1,2,\\dots \\rbrace $ .",
"For an integer $n>0$ , we denote by $[n]:= \\lbrace 1,2,\\dots , n \\rbrace \\subset \\mathbb {N}$ and $[\\infty ]:=\\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ .",
"For an increasing sequence $\\underline{\\lambda }=(\\lambda _1,\\lambda _2,\\dots )$ of real numbers, we denote by $\\underline{\\lambda }^{[n]}:=(\\lambda _1,\\dots ,\\lambda _n )$ the $n$ -th truncated subsequence of $\\underline{\\lambda }$ .",
"More generally, for two integers $0<m<n$ , we put $\\underline{\\lambda }^{(m,n]}:=(\\lambda _{m+1},\\dots , \\lambda _n)$ .",
"For $m,n\\in [\\infty ]$ and a ring $R$ , we denote by $\\mathrm {M}_{m\\times n}(R)$ the set of $m\\times n$ matrices with entries in $R$ and abbreviate it by $\\mathrm {M}_n(R)$ when $m=n$ .",
"For any $n\\in [\\infty ]$ , $M\\in \\mathrm {M}_n(R)$ and $I,J$ two subsets of $[n]$ with cardinalities $|I|=k$ , $|J|=l$ , we denote by $M_{I,J}$ the $k\\times l$ matrix consisting of entries of $M$ with row indices in $I$ and column indices in $J$ , and abbreviate it by $M_I$ when $I=J$ .",
"We denote by $v_T(-)$ the $T$ -adic valuation of the formal power series ring $\\mathbb {F}_p\\llbracket T \\rrbracket $ normalized by $v_T(T)=1$ .",
"We denote by $\\mathbf {C}$ an algebraic closure of the fractional field $\\mathbb {F}_p(\\!(T)\\!",
")$ of $\\mathbb {F}_p\\llbracket T\\rrbracket $ .",
"The valuation $v_T(-)$ extends uniquely to $\\mathbf {C}$ , which is still denoted by $v_T(-)$ ."
],
[
"Newton-Hodge decomposition for matrices in $ M_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$",
"Definition 2.2.1 The Newton and Hodge functions of a matrix $M\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ are defined by $N(M, k):= v_T\\left(\\sum \\limits _{N : \\textrm {~principal~} k\\times k \\textrm {~minor of~} M} \\det (N)\\right)$ and $H(M, k):=\\min \\lbrace v_T(\\det (N))\\;|\\; N\\ \\textrm {is a}\\ k\\times k\\ \\textrm {minor of}\\ M\\rbrace $ for every $k\\in [n]$ .",
"Moreover, we set $H(M,0):=0$ and $N(M,0):=0$ .",
"The Hodge polygon (resp.",
"Newton polygon) of $M$ is the lower convex hull of the points $\\lbrace (k,H(M,k))\\rbrace _{k=0}^n$ (resp.",
"$\\lbrace (k, N(M,k))\\rbrace _{k=0}^n$ ).",
"Remark 2.2.2 Let $\\alpha _1,\\dots ,\\alpha _n$ be the eigenvalues of $M$ in $\\mathbf {C}$ (counted with multiplicities) such that $v_T(\\alpha _1)\\le \\dots \\le v_T(\\alpha _n)$ .",
"Then the slopes of the Newton polygon are exactly $v_T(\\alpha _1),\\dots , v_T(\\alpha _n)$ .",
"Let $\\underline{\\lambda }=(\\lambda _1, \\lambda _2, \\dots )$ be a (not necessarily strictly) increasing sequence of nonnegative real numbers such that $\\lim \\limits _{n\\rightarrow +\\infty }\\lambda _n=+\\infty .$ For every integer $n>0$ , we denote by $D(\\underline{\\lambda }^{[n]})$ (resp.",
"$D(\\underline{\\lambda })$ ) the diagonal matrix $\\mathrm {Diag}(T^{\\lambda _1},\\dots ,T^{\\lambda _n})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ (resp.",
"$\\mathrm {Diag}(T^{\\lambda _1},\\dots )\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ ).",
"For every two integers $0<m<n$ , we denote by $D(\\underline{\\lambda }^{(m,n]})$ the diagonal matrix $\\mathrm {Diag}(T^{\\lambda _{m+1}},\\dots ,T^{\\lambda _n})\\in \\mathrm {M}_{n-m}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"Definition 2.2.3 For $\\ell \\in \\mathbb {N}$ and $n\\in \\mathbb {N}$ , a matrix $M=(m_{i,j})_{1\\le i\\le n, 1\\le j\\le \\ell }\\in \\mathrm {M}_{n\\times \\ell }(\\mathbb {F}_p\\llbracket T \\rrbracket )$ is called $\\underline{\\lambda }^{[n]}$ -Hodge bounded if $M=D(\\underline{\\lambda }^{[n]})M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_{n\\times \\ell }(\\mathbb {F}_p\\llbracket T \\rrbracket )$ , or equivalently, $v_T(m_{ij})\\ge \\lambda _i$ for all $1\\le i\\le n$ and $ 1\\le j\\le \\ell $ .",
"When $\\ell =n$ and $M^{\\prime }\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ , we call $M$ as strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"A matrix $M\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ is called $\\underline{\\lambda }$ -Hodge bounded if $M=D(\\underline{\\lambda })M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_{\\infty }(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"A matrix $A\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ is called $\\underline{\\lambda }^{[n]}$ -stable if for every $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix $B\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ , $AB$ is also $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"A matrix $A\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ is called $\\underline{\\lambda }$ -stable if for every $\\underline{\\lambda }$ -Hodge bounded matrix $B\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ , $AB$ is also $\\underline{\\lambda }$ -Hodge bounded.",
"Remark 2.2.4 For $A,B\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $B$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, then $BA$ is also $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"Let $M\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ be a $\\underline{\\lambda }$ -Hodge bounded matrix.",
"Since $\\lambda _i\\xrightarrow{} \\infty $ , the functions (REF ) and (REF ) are well-defined on $M$ , which are called the Newton and Hodge functions for $M$ , respectively.",
"Moreover, we define the Newton and Hodge polygons for $M$ in a similar way to the ones for matrices of finite dimensions.",
"Lemma 2.2.5 For $n\\in [\\infty ]$ and a matrix $A=(a_{ij})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ , the following statements are equivalent: $A$ is $\\underline{\\lambda }^{[n]}$ -stable; $D(\\underline{\\lambda }^{[n]})^{-1}AD(\\underline{\\lambda }^{[n]})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ ; $v_T(a_{ij})\\ge \\lambda _i-\\lambda _j$ for every $1\\le i\\le n$ and $1\\le j\\le n$ .",
"$(1)\\Rightarrow (2)$ Since $A$ is $\\underline{\\lambda }^{[n]}$ -stable and $D(\\underline{\\lambda }^{[n]})$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, $AD(\\underline{\\lambda }^{[n]})$ is also $\\underline{\\lambda }^{[n]}$ -Hodge bounded, and hence $AD(\\underline{\\lambda }^{[n]})=D(\\underline{\\lambda }^{[n]})B$ for some $B\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"This proves $D(\\underline{\\lambda }^{[n]})^{-1}AD(\\underline{\\lambda }^{[n]})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"$(2)\\Rightarrow (3)$ Suppose that we have $AD(\\underline{\\lambda }^{[n]})=D(\\underline{\\lambda }^{[n]})B$ for some $B=(b_{ij})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"Comparing the $(i,j)$ -entries of the matrices on the two sides of this equality, we have $T^{\\lambda _j}a_{ij}=T^{\\lambda _i}b_{ij}$ , and hence $v_T(a_{ij})=v_T(b_{ij})+\\lambda _i-\\lambda _j\\ge \\lambda _i-\\lambda _j.$ $(3)\\Rightarrow (1)$ For every $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix $M=(m_{ij})\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ , we have $v_T(m_{ij})\\ge \\lambda _i$ for all $1\\le i\\le n$ and $1\\le j\\le n$ .",
"Then the $(i,j)$ -entry of $AM$ is $\\sum \\limits _{k=1}^n a_{ik}m_{kj}.$ Since $v_T(a_{ik})\\ge \\lambda _i-\\lambda _k$ , we have $v_T(a_{ik}m_{kj})\\ge \\lambda _i$ , and hence $v_T(\\sum \\limits _{k=1}^n a_{ik}m_{kj})\\ge \\lambda _i.$ Therefore, we conclude that $AM$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded and $A$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Lemma 2.2.6 Let $M\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ be strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"For a matrix $A\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $AM$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, then $A$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Since $M$ is strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded and $AM$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, there are matrices $M^{\\prime }\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ and $B\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $M=D(\\underline{\\lambda }^{[n]})M^{\\prime }$ and $AM=D(\\underline{\\lambda }^{[n]})B$ .",
"Therefore, we have $A=D(\\underline{\\lambda }^{[n]})BM^{\\prime -1}D(\\underline{\\lambda }^{[n]})^{-1}$ .",
"Combined with Lemma REF , this equality implies that $A$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Lemma 2.2.7 For $n\\in \\mathbb {N}$ and $M\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded and satisfies $N(M,n)=\\sum \\limits _{i=1}^n\\lambda _i$ , then $H(M,k)=\\sum \\limits _{i=1}^k\\lambda _i \\text{~for every~} 1\\le k\\le n,$ i.e.",
"the slopes of the Hodge polygon of $M$ are $\\lambda _1,\\lambda _2,\\dots ,\\lambda _n$ .",
"In particular, $M$ is strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"Since $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, for every $1\\le k\\le n$ we have $H(M,k)\\ge \\sum \\limits _{i=1}^k \\lambda _i.$ Suppose $H(M,k)>\\sum \\limits _{i=1}^k \\lambda _i \\text{~for some~} k.$ Let $I:=[k]$ .",
"By Laplace expansion, we have $\\det (M)=\\sum _{J\\subset [n], |J|=k} \\mathrm {sgn}(I,J)\\det (M_{I,J})\\det (M_{I,J}^{comp}),$ where $\\mathrm {sgn}(I,J)=(-1)^{\\sum _{i\\in I}i+\\sum _{j\\in J}j}$ is the signature of the permutation determined by $I$ and $J$ , and $M_{I,J}^{comp}$ is the complement $(n-k)\\times (n-k)$ minor of $M_{I,J}$ .",
"Since $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, we have $v_T(\\det (M_{I,J}))\\ge H(M,k)>\\sum \\limits _{i=1}^k \\lambda _i \\textrm {\\ and\\ } v_T(\\det (M_{I,J}^{comp}))\\ge \\sum _{i=k+1}^n\\lambda _i,$ which together imply $ N(M,n)=v_T(\\det (M))>\\sum _{i=1}^n\\lambda _i,$ a contradiction.",
"Therefore we have $H(M,k)=\\sum \\limits _{i=1}^k\\lambda _i, \\text{~for all~} 1\\le k\\le n.$ Since $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, $M=D(\\underline{\\lambda }^{[n]})M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"Taking determinants and then $T$ -adic valuations on both sides of this equality, we have $v_T(\\det (M^{\\prime }))=0$ , and hence $M^{\\prime }\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"So $M$ is strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"Corollary 2.2.8 Let $M\\in M_{\\infty }(\\mathbb {F}_p\\llbracket T \\rrbracket )$ be $\\underline{\\lambda }$ -Hodge bounded.",
"If there exists an strictly increasing infinite sequence $\\underline{s}=(s_1, s_2,\\dots )$ of positive integers such that $N(M,s_n)=\\sum \\limits _{i=1}^{s_n}\\lambda _i \\text{~for every~} n\\ge 1,$ then for every $k\\ge 1$ we have $H(M,k)=\\sum \\limits _{i=1}^k\\lambda _i.$ Since $M$ is $\\underline{\\lambda }$ -Hodge bounded, for every $k\\ge 1$ we have $H(M,k)\\ge \\sum \\limits _{i=1}^k\\lambda _i.$ For a fixed integer $k\\ge 1$ , we can choose $n\\ge 1$ such that $s_n>k$ .",
"From our hypotheses that $N(M,s_n)=\\sum \\limits _{i=1}^{s_n}\\lambda _i$ and that $M$ is $\\underline{\\lambda }$ -Hodge bounded, there exists an $s_n\\times s_n$ principal minor $M_1$ of $M$ such that $v_T(\\det (M_1))=\\sum \\limits _{i=1}^{s_n}\\lambda _i.$ Since $\\underline{\\lambda }$ is increasing, $M_1$ is $\\underline{\\lambda }^{[s_n]}$ -Hodge bounded.",
"Combined with (REF ) and Lemma REF for $M_1$ , this implies $H(M_1,k)=\\sum \\limits _{i=1}^k\\lambda _i$ .",
"Note that every $k\\times k$ minor of $M_1$ is also a $k\\times k$ minor of $M$ .",
"We have $H(M,k)\\le H(M_1,k)=\\sum \\limits _{i=1}^k\\lambda _i$ , and hence $H(M,k)=\\sum \\limits _{i=1}^k\\lambda _i.$ Since we choose $k$ arbitrarily, the last equality completes the proof.",
"Definition 2.2.9 For every $n\\in [\\infty ]$ and $M\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ we call an integer $k$ a touching vertex of $M$ if it satisfies $(k,N(M,k))$ is a vertex of the Newton polygon of $M$ , and $N(M,k)=H(M,k)$ .",
"Theorem 2.2.10 ([19] Theorem 4.3.11) For every $n\\in \\mathbb {N}$ and $M\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $1\\le k<n$ is a touching vertex of $M$ , then there exists an invertible matrix $P\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $PMP^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12} \\\\0 & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle k};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle k};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ where $M_{11}$ and $M_{22}$ are $k\\times k$ and $(n-k)\\times (n-k)$ matrices such that the matrix $M_{11}$ accounts for the first $k$ slopes of the Hodge and Newton polygon of $M$ , while $M_{22}$ accounts for the others.",
"See [19].",
"Lemma 2.2.11 For every $n\\in \\mathbb {N}$ and $M\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $\\mu _1\\le \\mu _2\\le \\dots \\le \\mu _n$ are the slopes of the Hodge polygon of some matrix, then there exists an invertible matrix $P\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $PMP^{-1}$ is $\\underline{\\mu }$ -Hodge bounded, where $\\underline{\\mu }=(\\mu _1, \\mu _2, \\dots , \\mu _n)$ .",
"By [19], there exist matrices $U,V\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $UMV=D(\\underline{\\mu }).$ Right-multiplying $V^{-1}U^{-1}$ to both sides of this equality, we prove that $UMU^{-1}=D(\\underline{\\mu })V^{-1}U^{-1}$ is $\\underline{\\mu }$ -Hodge bounded.",
"Theorem 2.2.12 Let $M\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ be a $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix.",
"Assume that $1<s<n$ is a touching vertex of $M$ and $H(M,n)=\\sum \\limits _{i=1}^n\\lambda _i.$ Then there exists a matrix $W\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $WMW^{-1}$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded and a block upper triangular matrix of the form $WMW^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12} \\\\0 & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Moreover, $W$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"By Lemma REF , for every $1\\le k\\le n$ we have $H(M,k)=\\sum \\limits _{i=1}^{k}\\lambda _i.$ By Theorem REF , there exists a matrix $P_1\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $P_1MP_1^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M^{\\prime }_{11} & M^{\\prime }_{12} \\\\0 & M^{\\prime }_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ and $M^{\\prime }_{11}$ accounts for the first $s$ -th slopes of the Newton and Hodge polygons of $M$ , while $M^{\\prime }_{22}$ accounts for the others.",
"By Lemma REF , there exist matrices $Q_1\\in \\operatorname{GL}_s(\\mathbb {F}_p\\llbracket T \\rrbracket )$ and $Q_2\\in \\operatorname{GL}_{n-s}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $M_{11}=Q_1M^{\\prime }_{11}Q_1^{-1}$ and $M_{22}=Q_2M^{\\prime }_{22}Q_2^{-1}$ are $\\underline{\\lambda }^{[s]}$ and $\\underline{\\lambda }^{(s,n]}$ -Hodge bounded, respectively.",
"Set $P_2:= \\begin{bmatrix}Q_1&0\\\\0&Q_2\\end{bmatrix} \\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )\\textrm {\\ and \\ } N:= P_2(P_1MP_1^{-1})P_2^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12} \\\\0 & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ where $M_{12}=Q_1M^{\\prime }_{12}Q_2^{-1}$ .",
"Let $v_1,\\dots ,v_s$ be the column vectors of $M_{11}$ and $w_1,\\dots ,w_{n-s}$ be the column vectors of $M_{12}$ .",
"From $\\det (M_{11})\\ne 0$ we know that $\\lbrace v_1,\\dots ,v_s\\rbrace $ forms a basis of the vector space $(\\mathbb {F}_p(\\!(T)\\!",
"))^s$ .",
"Hence, for each integer $1\\le k\\le n-s$ there exist a set of scalars $\\lbrace a_1,\\dots , a_s\\rbrace $ in $ \\mathbb {F}_p(\\!(T)\\!",
")$ such that $w_k=\\sum \\limits _{j=1}^s a_jv_j.$ By Cramer's rule, we have $a_j=\\frac{\\det (M_{11,k})}{\\det (M_{11})}$ , where $M_{11,k}$ is the $s\\times s$ matrix obtained by replacing the $k$ -th column vector of $M_{11}$ by $w_k$ .",
"Since $M_{11,k}$ is an $s\\times s$ minor of $N$ , by definition of Hodge functions, we have $v_T(\\det (M_{11,k}))\\ge v_T(\\det (M_{11})),$ and hence $v_T(a_{j})\\ge 0$ for every $j=1,\\dots , s,$ which implies that $M_{12}$ is $\\underline{\\lambda }^{[s]}$ -Hodge bounded.",
"Let $W=P_2P_1\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"From the above computation, we obtain that $WMW^{-1}$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, and hence so is $WM$ .",
"Combined with Lemma REF and that $M$ is strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded, this implies that $W$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Lemma 2.2.13 Fix an integer $n\\ge 1$ .",
"Let $A\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ and $\\underline{\\alpha }:= \\lbrace \\alpha _1,\\dots ,\\alpha _n\\rbrace $ be the set of eigenvalues of $A$ in $\\mathbf {C}$ .",
"Then there exists $b\\in \\mathbb {R}$ such that $H(A^k,1)\\ge ks+b \\text{~and ~} H(A^{-k},1)\\ge -kS+b$ for every $k\\ge 1$ , where $S:=\\max _{1\\le i\\le n}\\lbrace v_T(\\alpha _i) \\rbrace \\textrm {\\ and \\ } s:=\\min _{1\\le i\\le n}\\lbrace v_T(\\alpha _i) \\rbrace .$ There exists a matrix $P\\in \\operatorname{GL}_n(\\mathbf {C})$ such that $J=PAP^{-1}$ is in Jordan canonical form with diagonal entries $\\alpha _1,\\dots ,\\alpha _n$ .",
"Let $D\\in \\mathrm {M}_n(\\mathbf {C})$ be the diagonal matrix with $\\alpha _1,\\dots ,\\alpha _n$ on the diagonal and set $N:=J-D$ .",
"Note that $N$ is nilpotent and commutes with $D$ .",
"Let $t:= \\min \\lbrace m|N^m=0 \\rbrace $ .",
"From the decomposition $(D+N)^k=\\sum _{i=0}^{k}\\binom{k}{i} D^iN^{k-i},$ we have the following estimations $H(J^k,1)\\ge {\\left\\lbrace \\begin{array}{ll}sk &\\text{~if~} H(N,1)\\ge s,\\\\s(k-t+1)+(t-1)H(N,1)& \\text{~if~} H(N,1)< s.\\end{array}\\right.",
"}$ Hence there exists $b_1\\in \\mathbb {R}$ such that $H(J^k,1)\\ge ks+b_1$ for every $k\\ge 1$ .",
"Similarly, there exists $b_2\\in \\mathbb {R}$ such that $H(J^{-k},1)\\ge -kS+b_2$ for every $k\\ge 1$ .",
"From $A^k=P^{-1}J^kP$ and $A^{-k}=P^{-1}J^{-k}P$ , we have $H(A^k,1)\\ge H(P^{-1},1)+H(P,1)+H(J^k,1)\\ge H(P^{-1},1)+H(P,1)+b_1+ks,$ and $H(A^{-k},1)\\ge H(P^{-1},1)+H(P,1)+H(J^{-k},1)\\ge H(P^{-1},1)+H(P,1)+b_2-kS.$ Setting $b:=\\min (b_1,b_2)+ H(P^{-1},1)+H(P,1)$ , we complete the proof.",
"Lemma 2.2.14 Let $1\\le s<n$ be two integers and $M=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A & B \\\\C & D \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ be a $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix.",
"Assume that $s$ is a touching vertex of $M$ ; $H(C,1)-\\lambda _s>0$ ; $A$ is strictly $\\underline{\\lambda }^{[s]}$ -Hodge bounded.",
"Then we have $\\max \\lbrace v_T(\\alpha )\\;|\\; \\alpha \\text{~is an eigenvalue of ~} A \\rbrace <\\min \\lbrace v_T(\\beta )\\;|\\; \\beta \\text{~is an eigenvalue of ~} D \\rbrace .$ Write $a:=\\max \\lbrace v_T(\\alpha )\\;|\\; \\alpha \\text{~is an eigenvalue of ~} A \\rbrace \\textrm {\\ and\\ } b:=\\min \\lbrace v_T(\\beta )\\;| \\;\\beta \\text{~is an eigenvalue of ~} D \\rbrace .$ From Remark REF , $a$ (resp.$b$ ) is the largest slope (resp.",
"smallest slope) of the Newton polygon of the matrix $A$ (resp.",
"$D$ ).",
"Since Newton polygon always lies on or above Hodge polygon, we have $a\\le \\lambda _s$ and $b\\ge \\lambda _{s+1}\\ge \\lambda _s$ .",
"Set $M_0:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A & B \\\\0 & D \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T\\rrbracket ).$ Suppose that (REF ) fails.",
"Then we have $a=b=\\lambda _s$ .",
"Denote by $k_1$ and $k_2$ with $k_1<k_2$ the $x$ -coordinates of two endpoints of the line segment with the slope $\\lambda _{s}$ on the Newton polygon of $M_0$ .",
"Then we have $k_1<s<k_2$ , $\\lambda _i=\\lambda _s$ for $k_1+1\\le i\\le k_2$ and $N(M_0, k_t)=H(M, k_t)=\\sum _{i=1}^{k_t}\\lambda _i \\textrm {\\ for}\\ t=1,2,$ where the second equality follows from the hypothesis (2).",
"Now we prove $N(M,k_t)=N(M_0,k_t) \\textrm {\\ for\\ } t=1,2.$ For an integer $m\\in [n]$ , we denote by $F(M,m)$ the sum of all the principal $m\\times m$ minors of $M$ and define $F(M_0,m)$ in the same way.",
"We write $M=(m_{i,j})_{1\\le i,j\\le n}$ and have $F(M,k_t)= \\sum _{\\begin{array}{c}I\\subset [n]\\\\|I|=k_t\\end{array}}\\sum _{\\sigma \\in S_{k_t}}\\mathrm {Sgn}(\\sigma )\\prod _{i\\in I}m_{i,\\sigma (i)}.$ Note that we have a similar description for $F(M_0,k_t)$ .",
"From our construction of $M_0$ , the difference $F(M,k_t)-F(M_0,k_t)$ is the sum of terms $\\mathrm {Sgn}(\\sigma )\\prod \\limits _{i\\in I}m_{i,\\sigma (i)}$ such that at least one of the entries $\\lbrace m_{i,\\sigma (i)}|i\\in I \\rbrace $ belonging to the block $C$ .",
"Combined with the hypothesis (2) and that $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, this implies $v_T(F(M,k_t)-F(M_0,k_t))>\\sum \\limits _{i=1}^{k_t}\\lambda _i=H(M,k_t).",
"$ Together with (REF ), this inequality proves (REF ).",
"Therefore, the point $(s,N(M,s))$ lies on the line segment with endpoints $(k_t,N(M_0,k_t))$ for $t=1,2$ , and hence cannot be a vertex of the Newton polygon of $M$ , which is a contradiction to our hypothesis (1).",
"Lemma 2.2.15 For every integer $n\\ge 1$ and $A\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ if $A$ is strictly $\\underline{\\lambda }^{[n]}$ -Hodge bounded, then for every integer $m\\ge 1$ we have $H(A^{-m},1)\\ge -m\\lambda _n$ .",
"We denote by $A^{ad}=(a^{\\prime }_{k,l})\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ the adjoint matrix of $A$ , i.e.",
"$a^{\\prime }_{k,l}=(-1)^{k+l}\\det (A_{k,l})$ , where $A_{k,l}$ is the $(n-1)\\times (n-1)$ minor of $A$ by removing the $k$ -th row and $l$ -th column.",
"So we have $H(A^{ad},1)\\ge H(A, n-1)$ and hence for every $m\\ge 1$ , $H((A^{ad})^m,1)\\ge mH(A^{ad},1)\\ge mH(A,n-1)=m\\sum \\limits _{i=1}^{n-1}\\lambda _i.$ Combined with $A^{-1}=\\det (A)^{-1}A^{ad}$ , this inequality implies $H(A^{-m},1)=H((A^{ad})^m,1)-mv_T(\\det (A))\\ge m\\sum \\limits _{i=1}^{n-1}\\lambda _i-m\\sum \\limits _{i=1}^{n}\\lambda _i=-m\\lambda _n.$ Lemma 2.2.16 For every $n\\in \\mathbb {N}$ if $A\\in \\operatorname{GL}_n(\\mathbb {F}_p[\\![T]\\!",
"])$ and $B\\in M_n(\\mathbb {F}_p[\\![T]\\!",
"])$ such that $H(B,1)>0$ , then $A+B\\in \\operatorname{GL}_n(\\mathbb {F}_p[\\![T]\\!",
"])$ .",
"Let $C:=\\left(\\sum _{i=0}^\\infty (-A^{-1}B)^{i}\\right)A^{-1}.",
"$ From the hypothesis $H(B,1)>0$ , we know that $C$ is well-defined and $(A+B)C=C(A+B)=I_n.$ Lemma 2.2.17 Let $1\\le s<n$ be two integers and $M=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A & B \\\\C & D \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in M_n(\\mathbb {F}_p\\llbracket T \\rrbracket ).$ be a $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix.",
"Assume that $M$ satisfies all the three hypotheses in Lemma REF .",
"Then there exists a matrix $X\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ with $H(X,1)\\ge H(C,1)-\\lambda _{s}$ such that $\\begin{bmatrix}I_{s}&0\\\\X& I_{n-s}\\end{bmatrix} \\begin{bmatrix}A&B\\\\C& D\\end{bmatrix} \\begin{bmatrix}I_{s}&0\\\\-X& I_{n-s}\\end{bmatrix}=\\ \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A^{\\prime } & B^{\\prime } \\\\0\\ & D^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ and $\\begin{bmatrix}I_{s}&0\\\\X& I_{n-s}\\end{bmatrix}$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"We first prove the following claim.",
"Claim Let $\\epsilon := H(C,1)-\\lambda _{s}$ .",
"For any $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix $M_k:=\\ \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_k & B_k \\\\C_k & D_k\\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]$ that satisfies all three hypotheses in Lemma REF and $H(C_k,1)-\\lambda _{s}\\ge k\\epsilon $ for some integer $k>0$ , there exists a matrix $X_k\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that if we set $M_{k+1}:=\\ \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{k+1} & B_{k+1} \\\\C_{k+1} & D_{k+1} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]=\\begin{bmatrix}I_{s}&0\\\\X_k& I_{n-s}\\end{bmatrix} M_k \\begin{bmatrix}I_{s}&0\\\\-X_k& I_{n-s}\\end{bmatrix},$ then $H(X_k,1)\\ge k\\epsilon $ ; $H(C_{k+1},1)-\\lambda _{s}\\ge (k+1)\\epsilon $ ; the matrix $ \\begin{bmatrix}I_{s}&0\\\\X_k& I_{n-s}\\end{bmatrix}$ is $\\underline{\\lambda }^{[n]}$ -stable; the matrix $M_{k+1}$ also satisfies all the three hypotheses in Lemma REF .",
"We now prove that $X_k:=-\\sum \\limits _{i=0}^\\infty D_k^iC_kA_k^{-i-1}$ satisfies all desired properties.",
"We first check that $X_k$ is well-defined.",
"Let $S_{A_k}:=\\max \\lbrace v_T(\\alpha )\\;|\\; \\alpha \\text{~is an eigenvalue of ~} A_k\\rbrace \\text{~and~} s_{D_k}:=\\min \\lbrace v_T(\\beta )\\;|\\; \\beta \\text{~is an eigenvalue of ~} D_k\\rbrace .$ By Lemmas REF and REF , we have $S_{A_k}<s_{D_k}$ and that there exists $b_k\\in \\mathbb {R}$ such that $H(A_k^{-i},1)\\ge -iS_{A_k}+b_k \\textrm {\\ and\\ } H(D_k^i,1)\\ge is_{D_k}+b_k \\textrm {\\ for every\\ } i\\ge 0.$ Combining them, we have $H(D_k^iC_kA_k^{-i-1},1)\\ge H(D_k^i,1)+H(C_k,1)+H(A_k^{-i-1},1)\\\\\\ge is_{D_k}+b_k+\\lambda _{s}+k\\epsilon -(i+1)S_{A_k}+b_k\\xrightarrow{} \\infty .$ Therefore, the series $X_k=-\\sum \\limits _{i=0}^\\infty D_k^iC_kA_k^{-i-1}$ converges and hence $X_k$ is well-defined.",
"By Lemma REF , we have $H(A_k^{-i-1},1)\\ge -(i+1)\\lambda _{s}$ .",
"Since $H(C_k,1)\\ge \\lambda _{s}+k\\epsilon $ and $H(D_k,1)\\ge \\lambda _{s+1}\\ge \\lambda _{s}$ , we have $H(D_k^iC_kA_k^{-i-1},1)\\ge i\\lambda _{s}+\\lambda _{s}+k\\epsilon -(i+1)\\lambda _{s}=k\\epsilon ,$ and hence $H(X_k,1)\\ge k\\epsilon $ , which proves $(i)$ .",
"From $X_kA_k+C_k-D_kX_k=0$ , we have $M_{k+1}=\\begin{bmatrix}A_k-B_kX_k&B_k\\\\-X_kB_kX_k& D_k+X_kB_k\\end{bmatrix}.$ Since $A_k$ is strictly $\\underline{\\lambda }^{[s]}$ -Hodge bounded and $B_k$ is $\\underline{\\lambda }^{[s]}$ -Hodge bounded, we have $A_k^{-1}B_k\\in \\mathrm {M}_{s\\times (n-s)}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"Combined with $X_kB_k=-\\sum \\limits _{i=0}^{\\infty }D_k^iC_kA_k^{-i}(A_k^{-1}B_k)$ and $H(D_k^iC_kA_k^{-i}(A_k^{-1}B_k),1)\\ge i\\lambda _{s}+\\lambda _{s}+k\\epsilon -i\\lambda _{s}+0=\\lambda _{s}+k\\epsilon \\textrm {\\ for every\\ } i\\ge 0,$ this implies $H(X_kB_k,1)\\ge \\lambda _{s}+k\\epsilon ,$ and hence $H(C_{k+1},1)=H(-X_kB_kX_k,1)\\ge \\lambda _{s}+k\\epsilon +\\epsilon =\\lambda _{s}+(k+1)\\epsilon .$ This completes the proof of $(ii)$ .",
"By Lemma REF , to show that the matrix $\\begin{bmatrix}I_{s}&0\\\\X_k& I_{n-s}\\end{bmatrix}$ is $\\underline{\\lambda }^{[n]}$ -stable, it is enough to show that the matrix $\\begin{bmatrix}D(\\underline{\\lambda }^{[s]})&0\\\\0& D(\\underline{\\lambda }^{(s,n]})\\end{bmatrix}^{-1}\\begin{bmatrix}I_{s}&0\\\\X_k& I_{n-s}\\end{bmatrix}\\begin{bmatrix}D(\\underline{\\lambda }^{[s]})&0\\\\0& D(\\underline{\\lambda }^{(s,n]})\\end{bmatrix}=\\begin{bmatrix}I_{s}&0\\\\D(\\underline{\\lambda }^{(s,n]})^{-1}X_kD(\\underline{\\lambda }^{[s]})& I_{n-s}\\end{bmatrix}$ belongs to $\\mathrm {M}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ , or equivalently, $D\\left(\\underline{\\lambda }^{(s,n]}\\right)^{-1}X_kD\\left(\\underline{\\lambda }^{[s]}\\right)\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T \\rrbracket ).$ Note that $D(\\underline{\\lambda }^{(s,n]})^{-1}X_kD(\\underline{\\lambda }^{[s]})=-\\sum \\limits _{i=0}^\\infty D(\\underline{\\lambda }^{(s,n]})^{-1}D_k^iC_kA_k^{-i}A_k^{-1}D(\\underline{\\lambda }^{[s]}).$ Since $A_k$ is strictly $\\underline{\\lambda }^{[s]}$ -Hodge bounded, we have $A_k^{-1}D(\\underline{\\lambda }^{[s]})\\in \\operatorname{GL}_{s}(\\mathbb {F}_p\\llbracket T\\rrbracket ),\\quad (\\textrm {i.e.",
"}H(A_k^{-1}D(\\underline{\\lambda }^{[s]}),1)\\ge 0).",
"$ Combined with $H(D_k^{i-1}C_kA_k^{-i},1)\\ge (i-1)\\lambda _{s}+\\lambda _{s}+k\\epsilon -i\\lambda _{s}=k\\epsilon >0 \\textrm {\\ for every\\ } i\\ge 1$ and $H(D(\\underline{\\lambda }^{(s,n]})^{-1}D_k,1)\\ge 0,$ we have $D(\\underline{\\lambda }^{(s,n]})^{-1}D_k^iC_kA_k^{-i}A_k^{-1}D(\\underline{\\lambda }^{[s]})\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T\\rrbracket ) \\textrm {\\ for every\\ } i\\ge 1.$ Note that by $H(D(\\underline{\\lambda }^{(s,n]})^{-1}C_k,1)\\ge 0 $ , the belonging relation (REF ) holds for $i=0$ as well.",
"Therefore, further combining it with (REF ), we complete the proof of $(iii)$ .",
"Since $M_k$ satisfies the hypothesis (1) and conjugating by matrices in $\\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T \\rrbracket )$ does not change either its Newton or Hodge polygons, we prove the hypothesis (1) for $M_{k+1}$ .",
"The hypothesis (2) for $M_{k+1}$ follows directly from $(ii)$ .",
"Now we are left to prove hypothesis (3) for $M_{k+1}$ .",
"Note that $A_{k+1}=A_k+B_kX_k=A_k(I_{s}+A_k^{-1}B_kX_k).",
"$ From our previous discussion, we know that $A_k^{-1}B_k\\in \\mathrm {M}_{s\\times (n-s)}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Combined with hypothesis (2) on $M_k$ and Lemma REF , this implies $H(A_k^{-1}B_kX_k,1)\\ge \\epsilon >0$ and that $I_{s}+A_k^{-1}B_kX_k$ is invertible.",
"Therefore, $A_{k+1}$ is strictly $\\underline{\\lambda }^{[s]}$ -Hodge bounded, which proves $(iv)$ .",
"Note that the matrix $M_1:=M$ satisfies all the conditions in our claim for $k=1$ .",
"Therefore, inductively using this claim, we obtain two infinite sequences of matrices $\\underline{X}\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ and $\\underline{M}\\in \\mathrm {M}_{(n-s)\\times n}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $H(X_k,1)\\ge k\\epsilon $ ; $M_k:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_k & B_k\\\\C_k & D_k \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]=Y_kM_{k-1}Y_k^{-1}$ , where $Y_k:= \\begin{bmatrix}I_{s}&0\\\\X_k& I_{n-s}\\end{bmatrix}$ ; $Y_k$ is $\\underline{\\lambda }^{[n]}$ -stable; $H(C_k,1)\\ge \\lambda _{s}+k\\epsilon $ for all $k\\ge 1$ .",
"Notice that $Y_kY_{k-1}\\dots Y_1= \\begin{bmatrix}I_{s}&0\\\\\\sum \\limits _{i=1}^k X_i& I_{n-s}\\end{bmatrix}.$ Since $H(X_k,1)\\ge k\\epsilon $ , the series $\\sum \\limits _{k=1}^\\infty X_k$ converges to a matrix $X\\in \\mathrm {M}_{(n-s)\\times s}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"From the above construction, we see that $X$ satisfies all the required properties.",
"Now we state the Newton-Hodge decomposition theorem for matrices in $\\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Theorem 2.2.18 Let $M\\in M_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ be a $\\underline{\\lambda }$ -Hodge bounded matrix and $\\Omega =\\lbrace 0=s_0<s_1<\\cdots \\rbrace $ be an infinite subset of touching vertices of $M$ .",
"If $H(M,s)=\\sum _{i=1}^{s}\\lambda _i \\text{~for all~} s\\in \\Omega ,$ then there exists $W\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $W$ is $\\underline{\\lambda }$ -stable, and in particular $WMW^{-1}$ is $\\underline{\\lambda }$ -Hodge bounded.",
"$WMW^{-1}$ is a block upper triangular matrix of the form $WMW^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){N_{11} & N_{12}\\\\\\ 0\\ \\ & N_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Remark 2.2.19 Although the conclusion of Theorem REF only refers to the touching vertex $s_1$ , the assumption that there exist infinitely many touching vertices of $M$ is necessary, at least according to our proof.",
"The reason is because in the Newton-Hodge Decomposition Theorem REF for a finite matrix $M\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ , we have an automatic condition $H(M,n)=N(M,n)$ , i.e.",
"$(n,N(M,n))$ is always a touching vertex of $M$ , while we have no such condition for infinite matrix $M\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Therefore we have to impose extra condition on the infinite $M$ to obtain the desired decomposition.",
"Here we assume that $M$ has infinitely many touching vertices, which holds in our applications.",
"It is an interesting question whether this assumption can be weakened.",
"Since $\\lim \\limits _{n\\rightarrow \\infty }\\lambda _n=\\infty $ , replacing $\\Omega $ by a subset if necessary, we can assume that $\\lambda _{s_1}<\\lambda _{s_2}$ .",
"We will make this assumption throughout the proof of Theorem REF .",
"Remark 2.2.20 One must be careful when compute the infinite product of infinite matrices.",
"It is not true in general that the product of an countable infinitely many invertible matrices is invertible, even if the product is well-defined.",
"For example, we denote by $Q_n\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ the infinite matrix obtained by switching the 1-st and $n$ -th rows of the identity matrix $\\mathrm {I}_\\infty $ .",
"In particular, set $Q_1=\\mathrm {I}_\\infty $ .",
"Then the sequence $(Q_n\\cdots Q_1\\;|\\;n\\ge 1)$ converges to the infinite matrix $ \\begin{bmatrix}0&0\\\\\\mathrm {I}_\\infty &0\\end{bmatrix}$ , which is obviously not invertible.",
"What we will use is the following result: Let $\\lbrace Q_n \\rbrace _{n\\ge 1}$ be a family of matrices in $\\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Assume that there exists a sequence of integers $(\\alpha _n)_{n\\ge 1}$ such that $\\lim \\limits _{n\\rightarrow \\infty }\\alpha _n=\\infty $ and $Q_n-\\mathrm {I}_\\infty \\in T^{\\alpha _n}\\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Then the sequence $(Q_n\\cdots Q_1\\;|\\;n\\ge 1)$ converges to a matrix $Q\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"In fact, the condition on $Q_n$ 's implies that $Q_n^{-1}-\\mathrm {I}_\\infty \\in T^{\\alpha _n}\\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ and hence $(Q_1^{-1}\\cdots Q_n^{-1}|n\\ge 1 )$ converges to a matrix which gives the two-sided inverse of $Q$ .",
"Moreover, if $Q_n$ is $\\underline{\\lambda }$ -stable for all $n\\ge 1$ , the matrix $Q$ is also $\\underline{\\lambda }$ -stable.",
"Lemma 2.2.21 Let $M$ be the matrix in Theorem REF .",
"Then for every $k\\ge 1$ there exists a $\\underline{\\lambda }$ -stable matrix $P_k\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that if we write $P_kMP_k^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12}\\\\M_{21} & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_k};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_k};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $v_T(\\det (M_{11}))=\\sum \\limits _{i=1}^{s_k}\\lambda _i.$ There exists an integer $\\ell _M$ such that for every $n\\ge \\ell _M$ and every $\\underline{\\lambda }$ -stable matrix $P\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p[\\![T]\\!])",
"$ the Newton polygon of $(PMP^{-1})_{[n]}$ coincides with the Newton polygon of $M$ when one restricts the range of the $x$ -coordinates to $[0,s_2]$ .",
"In particular, $s_1$ is a touching vertex of $M_{[n]}$ .",
"(1) Since $s_k$ is a touching vertex of $M$ , we have $N(M,s_k)=H(M,s_k)=\\sum \\limits _{i=1}^{s_k}\\lambda _i.$ Let $m_1:=\\min \\lbrace j\\;|\\;\\lambda _j=\\lambda _{s_k} \\rbrace $ and $m_2:=\\max \\lbrace j\\;|\\;\\lambda _j=\\lambda _{s_k} \\rbrace .$ Since $M$ is $\\underline{\\lambda }$ -Hodge bounded, there exists a principal $s_k\\times s_k$ minor $N$ of $M$ such that $v_T(\\det (N))=\\sum \\limits _{i=1}^{s_k}\\lambda _i.$ Moreover, $N$ contains $M_{[m_1-1]}$ as a principal minor and is contained in the principal minor $M_{[m_2]}$ .",
"So there exists a permutation matrix $P^{\\prime }\\in \\operatorname{GL}_{m_2-m_1}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ (that is, a matrix obtained by permuting the row vectors of the identity matrix) such that if we set $N^{\\prime }:=\\begin{bmatrix}I_{m_1}&0\\\\0&P^{\\prime }\\end{bmatrix} M_{[m_2]} \\begin{bmatrix}I_{m_1}&0\\\\0&P^{\\prime }\\end{bmatrix}^{-1},$ then $N^{\\prime }_{[s_k]}=N$ .",
"Set $P:=\\begin{bmatrix}I_{m_1}&0& 0\\\\0&P^{\\prime }&0\\\\0&0& I_\\infty \\end{bmatrix}.$ By Lemma REF , $P$ is $\\underline{\\lambda }$ -stable.",
"Clearly, the matrix $P$ has the desired property.",
"(2) From our hypothesis $\\lim \\limits _{n\\rightarrow \\infty }\\lambda _n=\\infty $ , there exists an integer $\\ell _M> s_2$ such that $\\lambda _{\\ell _M}> \\sum \\limits _{i=1}^{s_2}\\lambda _i$ , and $(\\ell _M, \\sum \\limits _{i=1}^{\\ell _M} \\lambda _i)$ lies above the straight line determined by the line segment in the Hodge polygon of $M$ with slope $\\lambda _{s_2}$ .",
"Combined with that for every $\\underline{\\lambda }$ -stable matrix $P\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p[\\![T]\\!])",
"$ , $PMP^{-1}$ has the same Newton polygon to $M$ and is $\\underline{\\lambda }$ -Hodge bounded, these hypotheses on $\\ell _M$ implies that for every $n\\ge \\ell _M$ and for every $0\\le j<s_2$ we have $N((PMP^{-1})_{[n]}, j){\\left\\lbrace \\begin{array}{ll}=N(PMP^{-1}, j)=N(M, j), & \\textrm {if\\ } N(M, j)\\le \\sum _{i=1}^{s_2}\\lambda _i, \\\\>\\sum _{i=1}^{s_2}\\lambda _i, &\\textrm {else,}\\end{array}\\right.",
"}$ and $N((PMP^{-1})_{[n]}, s_2)=N(PMP^{-1}, s_2)=N(M, s_2)=\\sum _{i=1}^{s_2}\\lambda _i,$ and hence complete the proof.",
"Notation 2.2.22 From now on, we denote by $s$ a fixed vertex of the Newton polygon of $M$ such that $s\\ge \\ell _M$ as in Lemma REF .",
"We also set $\\Delta :=\\lambda _s-\\lambda _{s_1}$ .",
"Lemma 2.2.23 Under the notations and conditions in Theorem REF , there exists a $\\underline{\\lambda }$ -stable matrix $Q\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ such that if we set $QMQ^{-1}:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{11} & A_{12}\\\\A_{21} & A_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then we have $H(A_{21},1)\\ge \\lambda _{s}$ and $A_{11}$ is strictly $\\lambda ^{[s_1]}$ -Hodge bounded.",
"By Lemma REF (1), we may assume $v_T(\\det (M_{[s]}))=\\sum _{i=1}^s \\lambda _i.$ Therefore, $M_{[s]}$ satisfies all the conditions in Theorem REF .",
"By Theorem REF , there exists a $\\underline{\\lambda }^{[s]}$ -stable matrix $P$ such that $PM_{[s]}P^{-1}$ is of the form $PM_{[s]}P^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){B_{11} & B_{12} \\\\0 & B_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ and $B_{11}$ is strictly $\\lambda ^{[s_1]}$ -Hodge bounded.",
"Let $Q:=\\begin{bmatrix}P&0\\\\0&I_\\infty \\end{bmatrix}$ .",
"Clearly, $Q$ is $\\underline{\\lambda }$ -stable as $P$ is $\\underline{\\lambda }^{[s]}$ -stable.",
"Since $M$ is $\\underline{\\lambda }$ -Hodge bounded, the constructed matrix $Q$ satisfies the required properties.",
"Lemma 2.2.24 Let $M:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12}\\\\M_{21} & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in M_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ be the matrix in Theorem REF .",
"If $M_{11}$ is strictly $\\lambda ^{[s_1]}$ -Hodge bounded and $H(M_{21},1)\\ge \\lambda _{s}+i\\Delta $ for some integer $i\\ge 0$ , then there exists a $\\underline{\\lambda }$ -stable matrix $Q:=\\begin{bmatrix}I_{s_1}&0\\\\Q^{\\prime }&I_\\infty \\end{bmatrix}\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T \\rrbracket )$ such that $H(Q^{\\prime },1)\\ge (i+1)\\Delta $ , and if we set $QMQ^{-1}:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{11} & A_{12}\\\\A_{21} & A_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(A_{21},1)\\ge \\lambda _s+(i+1)\\Delta $ .",
"We set $s^{\\prime }:=s-s_1$ and rewrite $M$ as $M:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12} &M_{13}\\\\M_{21} & M_{22} &M_{23} \\\\M_{31} & M_{32} &M_{33}\\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-3-1.south east);[dotted]([xshift=0.5ex]m-1-2.north east) -- ([xshift=0.5ex]m-3-2.south east);[dotted](m-1-1.south west) -- (m-1-3.south east);[dotted](m-2-1.south west) -- (m-2-3.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-2.north) {\\scriptstyle s};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-2-1.west) {\\scriptstyle s};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Let $P_1:=\\begin{bmatrix}I_{s_1} & 0 &0\\\\0& I_{s^{\\prime }} & 0 \\\\-M_{31}M_{11}^{-1} & 0 & I_\\infty \\end{bmatrix}.$ Then we have $M^{\\prime }:= \\begin{bmatrix}M^{\\prime }_{11} & M^{\\prime }_{12} &M^{\\prime }_{13}\\\\M^{\\prime }_{21} & M^{\\prime }_{22} &M^{\\prime }_{23} \\\\M^{\\prime }_{31} & M^{\\prime }_{32} &M^{\\prime }_{33}\\end{bmatrix}= P_1MP_1^{-1}\\\\=\\begin{bmatrix}M_{11}+M_{13}M_{31}M_{11}^{-1} & M_{12} &M_{13}\\\\M_{21}+M_{23}M_{31}M_{11}^{-1} & M_{11} &M_{23} \\\\(M_{33}-M_{31}M_{11}^{-1}M_{13})M_{31}M_{11}^{-1} & M_{32}-M_{31}M_{11}^{-1}M_{12} & M_{33}-M_{31}M_{11}^{-1}M_{13}\\end{bmatrix}.$ From our hypotheses that $M_{11}$ is strictly $\\underline{\\lambda }^{[s_1]}$ -Hodge bounded and that $M_{12}$ , $M_{13}$ are $\\underline{\\lambda }^{[s_1]}$ -Hodge bounded, we have $M_{11}^{-1}M_{12}\\in \\mathrm {M}_{s_1\\times s}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ and $M_{11}^{-1}M_{13}\\in \\mathrm {M}_{s_1\\times \\infty }(\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Combining $H(M_{31},1)\\ge \\lambda _{s}+i\\Delta $ with Lemma REF , we have $H(M_{31}M_{11}^{-1},1)\\ge \\lambda _{s}+i\\Delta -\\lambda _{s_1}=(i+1)\\Delta .$ Therefore, $M^{\\prime }$ is $\\underline{\\lambda }$ -Hodge bounded; and we have the estimations $H( M_{21}+M_{23}M_{31}M_{11}^{-1},1)\\ge \\lambda _{s}+i\\Delta $ and $H(M^{\\prime }_{31},1)=H((-M_{31}M_{11}^{-1}M_{13}+M_{33})M_{31}M_{11}^{-1},1)\\\\\\ge H(-M_{31}M_{11}^{-1}M_{13}+M_{33},1)+H(M_{31}M_{11}^{-1},1)\\ge \\lambda _{s}+(i+1)\\Delta .$ By Lemmas REF and REF (2), the matrix $M^{\\prime }_{[s]}= \\begin{bmatrix}M^{\\prime }_{11}&M^{\\prime }_{12}\\\\M^{\\prime }_{21}&M^{\\prime }_{22}\\end{bmatrix}$ satisfies all the conditions in Lemma REF with $n:=s$ and $s:=s_1$ .",
"Therefore, there exists a matrix $X\\in \\mathrm {M}_{s\\times s_1}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ such that $H(X,1)\\ge H(M^{\\prime }_{21},1)-\\lambda _{s_1}\\ge \\lambda _{s}+i\\Delta -\\lambda _{s_1}=(i+1)\\Delta $ ; The matrix $ \\begin{bmatrix}I_{s_1}&0\\\\X&I_{s}\\end{bmatrix}$ is $\\underline{\\lambda }^{[s]}$ -stable; $\\begin{bmatrix}I_{s_1}&0\\\\X&I_{s}\\end{bmatrix}\\begin{bmatrix}M^{\\prime }_{11}&M^{\\prime }_{12}\\\\M^{\\prime }_{21}&M^{\\prime }_{22}\\end{bmatrix}\\begin{bmatrix}I_{s_1}&0\\\\-X&I_{s}\\end{bmatrix}= \\begin{bmatrix}M^{\\prime \\prime }_{11}&M^{\\prime \\prime }_{12}\\\\0&M^{\\prime \\prime }_{22}\\end{bmatrix}$ .",
"Set $P_2:=\\begin{bmatrix}I_{s_1} & 0 &0\\\\X& I_{s} &0 \\\\0 & 0 & I_{\\infty }\\end{bmatrix}.$ Then $P_2\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ is $\\underline{\\lambda }$ -stable and $P_2MP^{-1}_2:=\\begin{bmatrix}M^{\\prime \\prime }_{11} & M^{\\prime \\prime }_{12} &M^{\\prime }_{13}\\\\0& M^{\\prime \\prime }_{22} & M^{\\prime }_{23}+XM^{\\prime }_{33} \\\\M^{\\prime }_{31}-XM^{\\prime }_{32} & M^{\\prime }_{32} & M^{\\prime }_{33}.\\end{bmatrix}.$ Combining the inequalities $H(XM^{\\prime }_{32},1)\\ge H(X,1)+H(M^{\\prime }_{32},1)\\ge (i+1)\\Delta +\\lambda _{s}\\textrm {\\ and\\ }H(M^{\\prime }_{31},1)\\ge \\lambda _{s}+(i+1)\\Delta ,$ we have $H(M^{\\prime }_{31}-XM^{\\prime }_{32},1)\\ge \\lambda _{s}+(i+1)\\Delta $ .",
"Setting $Q:= P_2P_1=\\begin{bmatrix}I_{s_1} & 0 &0\\\\X& I_{s} &0 \\\\-M_{31}M_{11}^{-1} & 0 & I_{\\infty }\\end{bmatrix},$ from the above discussion we know that $Q$ satisfies all the required properties.",
"Now we can prove Theorem REF , which is an easy consequence of the above lemmas.",
"By Lemma REF , there exists a $\\underline{\\lambda }$ -stable matrix $Q_0\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ such that if we write $M_0:= Q_0MQ_0^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{0,11} & M_{0,12}\\\\M_{0,21} & M_{0,22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(M_{0,21},1)\\ge \\lambda _{s}=\\lambda _{s}+0\\cdot \\Delta $ and $M_{0,11}$ is strictly $\\lambda ^{[s_1]}$ -Hodge bounded.",
"Then we can apply Lemma REF inductively on $k$ , and get a sequence of $\\underline{\\lambda }$ -stable matrices $\\left(Q_k= \\begin{bmatrix}I_{s_1}&0\\\\Q_k^{\\prime }&I_{\\infty }\\end{bmatrix}\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )\\; \\Big | \\; k\\ge 1\\right)$ with $H(Q_k^{\\prime },1)\\ge (k+1)\\Delta $ such that we write $M_k:=(Q_k\\dots Q_1Q_0)M(Q_k\\dots Q_1Q_0)^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{k,11} & M_{k,12}\\\\M_{k,21} & M_{k,22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(M_{k,21},1)\\ge \\lambda _{s}+k\\Delta $ .",
"By Remarks REF and REF , the product $Q_k\\dots Q_1Q_0$ converges as $k\\rightarrow \\infty $ , and we denote by $W$ this limit.",
"Then $W\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ is $\\underline{\\lambda }$ -stable and $WMW^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){N_{11} & N_{12} \\\\0 & N_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$"
],
[
"Newton-Hodge decomposition for matrices over certain non-commutative rings.",
"In this subsection, we will denote by $R$ a ring with unit but not necessarily commutative.",
"Let $\\mathfrak {m}\\subset R$ be a two-sided ideal, and $\\tilde{T}\\in R$ a central element which is not a zero divisor of $R$ .",
"We assume that $R$ is complete under the $(\\mathfrak {m},\\tilde{T})$ -adic topology, where $(\\mathfrak {m},\\tilde{T})$ is the (two-sided) ideal of $R$ generated by $\\mathfrak {m}$ and $\\tilde{T}$ , i.e.",
"we have $R\\cong \\varprojlim \\limits _{n\\rightarrow \\infty } R/(\\mathfrak {m}, \\tilde{T})^n.$ There exists a surjective ring homomorphism $\\pi :R\\rightarrow \\mathbb {F}_p\\llbracket T \\rrbracket $ with $\\ker (\\pi )=\\mathfrak {m}$ and $\\pi (\\tilde{T})=T$ .",
"For $x\\in R\\setminus \\lbrace 0\\rbrace $ , we define $v_{\\tilde{T}}(x):=\\sup \\lbrace n|x\\in \\tilde{T}^nR \\rbrace $ and $v_{\\tilde{T}}(0):=\\infty $ .",
"Then for each $\\bar{x}\\in \\mathbb {F}_p\\llbracket T \\rrbracket $ , there exists $x\\in R$ such that $\\pi (x)=\\bar{x}$ and $v_{\\tilde{T}}(x)=v_T(\\bar{x})$ .",
"The element $x$ with such properties is called a special lift of $\\bar{x}$ .",
"Remark 2.3.1 The number $v_{\\tilde{T}}(x)=\\sup \\lbrace n|x\\in \\tilde{T}^nR \\rbrace $ is finite for $x\\ne 0$ , and we have the inequality $v_{\\tilde{T}}(x)\\le v_T(\\bar{x})$ with $\\bar{x}=\\pi (x)$ .",
"Since $\\tilde{T}$ is central and not a zero divisor of $R$ , we can consider the localization $R[\\frac{1}{\\tilde{T}}]$ of $R$ with respect to the multiplicative set $\\lbrace \\tilde{T}^n|n\\ge 0 \\rbrace $ .",
"The natural homomorphism $R\\rightarrow R[\\frac{1}{\\tilde{T}}]$ is injective.",
"Since $\\tilde{T}$ is not a zero divisor of $R$ , if $x\\in \\tilde{T}^k R$ , then there exists a unique $x^{\\prime }\\in R$ with $x=\\tilde{T}^kx^{\\prime }$ .",
"We write $x^{\\prime }=\\tilde{T}^{-k}x$ .",
"Lemma 2.3.2 Let $x\\in \\tilde{T}^kR$ for some $k\\ge 1$ .",
"If $x\\in \\mathfrak {m}$ , then $\\tilde{T}^{-k}x\\in \\mathfrak {m}$ .",
"We write $x=\\tilde{T}^ky$ for some $y\\in R$ .",
"Applying the homomorphism $\\pi $ on both sides of this equality, we get $\\pi (x)=T^k\\pi (y)=0$ , and hence $\\pi (y)=0$ .",
"This implies $y\\in \\mathfrak {m}$ .",
"Definition 2.3.3 For a matrix $\\bar{A}=(\\bar{a}_{ij})\\in \\mathrm {M}_{m\\times n}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ , we call $A=(a_{ij})\\in \\mathrm {M}_{m\\times n}(R)$ a special lift of $\\bar{A}$ if $a_{ij}$ is a special lift of $\\bar{a}_{ij}$ for every $1\\le i\\le m$ and $1\\le j\\le n$ .",
"We denote by $D_R(\\underline{\\lambda }^{[n]})$ (resp.",
"$D_R(\\underline{\\lambda })$ ) the diagonal matrix $\\mathrm {Diag}(\\tilde{T}^{\\lambda _1},\\dots ,\\tilde{T}^{\\lambda _n})\\in \\mathrm {M}_n(R)$ (resp.",
"$\\mathrm {Diag}(\\tilde{T}^{\\lambda _1},\\dots )\\in \\mathrm {M}_\\infty (R)$ ).",
"For two integers $m<n$ we denote by $D_R(\\underline{\\lambda }^{(m,n]})$ the diagonal matrix $\\mathrm {Diag}(\\tilde{T}^{\\lambda _{m+1}},\\dots ,\\tilde{T}^{\\lambda _n})\\in \\mathrm {M}_{n-m}(R)$ .",
"Similar to Definition REF , we define $\\underline{\\lambda }$ -Hodge bounded matrices and $\\underline{\\lambda }$ -stable matrices over $R$ as follows.",
"Definition 2.3.4 Let $m,n$ be two positive integers.",
"A matrix $M\\in \\mathrm {M}_{n\\times m}(R)$ is called $\\underline{\\lambda }^{[n]}$ -Hodge bounded with respect to $\\tilde{T}\\in R$ if $M=D_R(\\underline{\\lambda }^{[n]})M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_{n\\times m}(R)$ .",
"More generally, for two integers $0\\le n_1<n_2$ a matrix $M\\in \\mathrm {M}_{(n_2-n_1)\\times m}(R)$ is called $\\underline{\\lambda }^{(n_1,n_2]}$ -Hodge bounded with respect to $\\tilde{T}\\in R$ if $M=D_R(\\underline{\\lambda }^{(n_1,n_2]})M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_{(n_2-n_1)\\times m}(R)$ .",
"A matrix $M\\in \\mathrm {M}_\\infty (R)$ is called $\\underline{\\lambda }$ -Hodge bounded with respect to $\\tilde{T}\\in R$ if $M=D_R(\\underline{\\lambda })M^{\\prime }$ for some $M^{\\prime }\\in \\mathrm {M}_{\\infty }(R)$ .",
"A matrix $A\\in \\mathrm {M}_n(R)$ is called $\\underline{\\lambda }^{[n]}$ -stable with respect to $\\tilde{T}\\in R$ if for every $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix $B\\in \\mathrm {M}_n(R)$ , $AB$ is also $\\underline{\\lambda }^{[n]}$ -Hodge bounded.",
"A matrix $A\\in \\mathrm {M}_\\infty (R)$ is called $\\underline{\\lambda }$ -stable with respect to $\\tilde{T}\\in R$ if for every $\\underline{\\lambda }$ -Hodge bounded matrix $B\\in \\mathrm {M}_\\infty (R)$ , $AB$ is also $\\underline{\\lambda }$ -Hodge bounded.",
"Convention 2.3.5 In the rest of this section, we will fix an element $\\tilde{T}\\in R$ .",
"When we say that a matrix is $\\underline{\\lambda }^{[n]}$ -Hodge bounded or $\\underline{\\lambda }^{[n]}$ -stable, we mean that it is so with respect to $\\tilde{T}$ .",
"Since determinants do not behave well for matrices over noncommutative rings, we do not define general Newton functions and Hodge functions for matrices over $R$ .",
"But we make the following.",
"Definition 2.3.6 For a nonzero matrix $A=(a_{i,j})_{1\\le i\\le m,1\\le j\\le n}\\in \\mathrm {M}_{m\\times n}(R)$ , we set $H(A,1):=\\min \\left\\lbrace v_{\\tilde{T}}(a_{i,j})\\;|\\;1\\le i\\le m,1\\le j\\le n \\right\\rbrace ,$ which is a well-defined integer.",
"If $H(A,1)\\ge k$ for some $k\\in \\mathbb {Z}_{\\ge 0}$ , we denote by $\\tilde{T}^{-k}A$ the (unique) matrix $N$ in $\\mathrm {M}_{m\\times n}(R)$ satisfying $\\tilde{T}^k N=A$ .",
"We also have a similar criterion of $\\underline{\\lambda }$ -stability as Lemma REF : Lemma 2.3.7 For $n\\in [\\infty ]$ and a matrix $A=(a_{ij})\\in \\mathrm {M}_n(R)$ , the following statements are equivalent: $A$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"There exists a matrix $B\\in \\mathrm {M}_n(R)$ such that $AD_R(\\underline{\\lambda }^{[n]})=D_R(\\underline{\\lambda }^{[n]})B$ .",
"For all $i>j$ , we have $v_{\\tilde{T}}(a_{ij})\\ge \\lambda _i-\\lambda _j$ , i.e.",
"$a_{ij}\\in \\tilde{T}^{\\lambda _i-\\lambda _j}R$ .",
"The proof is almost identical to that of Lemma REF so we omit the proof here, and it has the following direct consequence.",
"Corollary 2.3.8 Let $\\bar{A}\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ be a $\\underline{\\lambda }^{[n]}$ -stable matrix.",
"If $A\\in \\mathrm {M}_n(R)$ is a special lift of $\\bar{A}$ , then $A$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Lemma 2.3.9 Let $n\\in [\\infty ]$ and $\\bar{P}\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ .",
"Then any lift matrix $P\\in \\mathrm {M}_n(R)$ of $\\bar{P}$ is invertible.",
"From $\\bar{P}\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ , there exists $\\bar{Q}\\in \\operatorname{GL}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ such that $\\bar{P}\\bar{Q}=\\bar{Q}\\bar{P}=\\mathrm {I}_n$ .",
"Let $Q$ be an arbitrary lift of $\\bar{Q}$ .",
"Then $B:=\\mathrm {I}_n-PQ\\in \\mathrm {M}_n(\\mathfrak {m})$ and the series $B^{\\prime }:=\\sum \\limits _{i=0}^\\infty B^i$ converges.",
"By a direct computation, we have $PQB^{\\prime }=\\mathrm {I}_n$ .",
"Similarly, let $C:=\\mathrm {I}_n-QP\\in \\mathrm {M}_n(\\mathfrak {m})$ and the series $C^{\\prime }:=\\sum \\limits _{i=0}^\\infty C^i$ converges.",
"So we have $C^{\\prime }QP=\\mathrm {I}_n$ and hence $P$ is invertible in $\\mathrm {M}_n(R)$ .",
"Proposition 2.3.10 Let $n_1<n$ be two positive integers, $M:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A & B \\\\C & D \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle n_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle n_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in \\mathrm {M}_n(R)$ be a $\\underline{\\lambda }^{[n]}$ -Hodge bounded matrix, and $\\bar{M}\\in \\mathrm {M}_n(\\mathbb {F}_p\\llbracket T\\rrbracket )$ be its reduction by $\\mathfrak {m}$ .",
"Fix an integer $\\alpha \\ge \\lambda _{n_1}$ .",
"Assume that $H(C,1)\\ge \\alpha $ and $C\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}}$ ; $n_1$ is a touching vertex of $\\bar{M}$ , and $\\bar{A}$ is strictly $\\underline{\\lambda }^{[n_1]}$ -Hodge bounded.",
"Then there exists a matrix $X\\in \\mathrm {M}_{(n-n_1)\\times n_1}(\\mathfrak {m})$ with $H(X,1)\\ge \\alpha -\\lambda _{n_1}$ such that if we denote $Y:= \\begin{bmatrix}I_{n_1}&0\\\\X&I_{n-n_1}\\end{bmatrix} $ , then $Y$ is $\\underline{\\lambda }^{[n]}$ -stable and $YMY^{-1}$ is of the form $YMY^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A^{\\prime } & B^{\\prime } \\\\0\\ & D^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle n_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle n_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ First we need the following lemma.",
"Lemma 2.3.11 Let $M\\in \\mathrm {M}_n(R)$ be the matrix in Proposition REF .",
"Assume further that $\\tilde{T}^{-\\alpha }C\\equiv 0 \\mathfrak {}\\pmod {\\mathfrak {m}^k}$ for some integer $k\\ge 1$ .",
"Then there exists a matrix $X_k\\in \\mathrm {M}_{(n-n_1)\\times n_1}(R)$ such that $H(X_k,1)\\ge \\alpha -\\lambda _{n_1}$ and $\\tilde{T}^{\\lambda _{n_1}-\\alpha }X_k\\equiv 0 \\mathfrak {}\\pmod {\\mathfrak {m}^k}$ .",
"The matrix $Y_k:= \\begin{bmatrix}I_{n_1}&0\\\\X_k&I_{n-n_1}\\end{bmatrix}$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"If we write $Y_kMY_k^{-1}:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A^{\\prime } & B^{\\prime } \\\\C^{\\prime } & D^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle n_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle n_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(C^{\\prime },1)\\ge \\alpha $ and $\\tilde{T}^{-\\alpha }C^{\\prime }\\equiv 0 \\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}$ .",
"Since $M$ is $\\underline{\\lambda }^{[n]}$ -Hodge bounded, so is its reduction $\\bar{M}$ .",
"By Lemma REF , we have $S_{\\bar{A}}<s_{\\bar{D}}$ , where $S_{\\bar{A}}:=\\max \\lbrace v_T(\\beta )\\;|\\;\\beta \\text{~is an eigenvalue of ~} \\bar{A}\\rbrace \\textrm {\\ and\\ }s_{\\bar{D}}:=\\max \\lbrace v_T(\\beta )\\;|\\;\\beta \\text{~is an eigenvalue of ~} \\bar{D}\\rbrace .$ By Lemma REF , there exists $b\\in \\mathbb {R}$ such that for every $i\\ge 1$ we have $H(\\bar{A}^{-i},1)\\ge -iS_{\\bar{A}}+b\\textrm {\\ and\\ }H(\\bar{D}^i,1)\\ge is_{\\bar{D}}+b.",
"$ For every integer $i\\ge 1$ , we set $t_i:=i\\lambda _{n_1}$ and construct matrices as follows.",
"Since $T^{t_i}\\bar{A}^{-i}$ belongs to $\\mathrm {M}_{n_1}(\\mathbb {F}_p\\llbracket T\\rrbracket )$ , we can find a special lift $A^{\\prime }(-i)\\in \\mathrm {M}_{n_1}(R)$ of $T^{t_i}\\bar{A}^{-i}$ and set $A(-i):=\\tilde{T}^{-t_i}A^{\\prime }(-i)\\in \\mathrm {M}_{n_1}(R[\\frac{1}{\\tilde{T}}])$ .",
"We fix a special lift $D(i)\\in \\mathrm {M}_{n-n_1}(R)$ of $\\bar{D}^i\\in \\mathrm {M}_{n-n_1}(\\mathbb {F}_p\\llbracket T \\rrbracket )$ .",
"To simplify our notations, we set $A(0):=I_{n_1}\\in \\mathrm {M}_{n_1}(R)$ and $D(0):=I_{n-n_1}\\in \\mathrm {M}_{n-n_1}(R)$ .",
"Since $A$ is $\\underline{\\lambda }^{[n_1]}$ -Hodge bounded, we can write $A=D_R(\\underline{\\lambda }^{[n_1]})\\cdot A^{\\prime }$ for some $A^{\\prime }\\in \\mathrm {M}_{n_1}(R)$ .",
"As the reduction $\\bar{A}$ of $A$ by $\\mathfrak {m}$ is strictly $\\underline{\\lambda }^{[n_1]}$ -Hodge bounded, the reduction $\\bar{A}^{\\prime }$ of $A^{\\prime }$ by $\\mathfrak {m}$ is invertible.",
"It follows from Lemma REF that $A^{\\prime }\\in \\operatorname{GL}_{n_1}(R)$ .",
"We define $\\tilde{A}^{-1}:= A^{\\prime -1}\\cdot D_R(\\underline{\\lambda }^{[n_1]})^{-1}\\in \\mathrm {M}_{n_1}(R[\\frac{1}{\\tilde{T}}])$ .",
"Although $\\tilde{A}^{-1}$ satisfies $A\\cdot \\tilde{A}^{-1}=\\tilde{A}^{-1}\\cdot A=I_{n_1}$ , $A$ is not invertible in $\\mathrm {M}_{n_1}(R)$ and $\\tilde{A}^{-1}$ is the inverse of $A$ in $\\mathrm {M}_{n_1}(R[\\frac{1}{\\tilde{T}}])$ .",
"To emphasize this fact, we use the notation $\\tilde{A}^{-1}$ instead of $A^{-1}$ .",
"Finally we remark that $\\tilde{T}^{\\lambda _{n_1}}\\tilde{A}^{-1}\\in \\mathrm {M}_{n_1}(R)$ .",
"The construction above give us $H(\\tilde{T}^{t_i}A(-i),1)=H(A^{\\prime }(-i),1)=H(T^{t_i}\\bar{A}^{-i},1)=t_i+H(\\bar{A}^{-i},1)\\textrm {\\ and\\ }H(D(i),1)=H(\\bar{D}^i,1).$ Set $X_k:=-\\left(C\\tilde{A}^{-1}+\\sum _{i=1}^\\infty DD(i-1)CA(-i)\\tilde{A}^{-1}\\right)\\\\=-\\left(C+\\sum _{i=1}^\\infty DD(i-1)CA(-i)\\right)\\tilde{A}^{-1}.$ We first verify that each term $S$ in the sum of (REF ) satisfies $H(S,1)\\ge \\alpha -\\lambda _{n_1}$ and in particular belongs to $\\mathrm {M}_{(n-n_1)\\times n_1}(R)$ .",
"In fact, for every integer $i\\ge 1$ we have $H(D(i-1)CA(-i),1)\\ge H(D(i-1),1)+H(C,1)+H(A(-i),1)\\\\=H(\\bar{D}^{i-1},1)+H(C,1)+H(\\bar{A}^{-i},1)\\ge (i-1)\\lambda _{n_1}+\\alpha -i\\lambda _{n_1}=\\alpha -\\lambda _{n_1},$ where the second inequality is from Lemma REF .",
"Combined with the fact that $\\tilde{T}^{-\\lambda _{n_1}}D\\in \\mathrm {M}_{n-n_1}(R)$ and $\\tilde{T}^{\\lambda _{n_1}}\\tilde{A}^{-1}\\in \\mathrm {M}_{n_1}(R)$ , this chain of inequality implies $H(DD(i-1)CA(-i)\\tilde{A}^{-1},1)\\ge \\alpha -\\lambda _{n_1}$ .",
"On the other hand, from $H(C,1)\\ge \\alpha \\ge \\lambda _{n_1}$ and $\\tilde{T}^{\\lambda _{n_1}}\\tilde{A}^{-1}\\in \\mathrm {M}_{n_1}(R)$ , we have $C\\tilde{A}^{-1}\\in \\mathrm {M}_{(n-n_1)\\times n_1}(R)\\textrm {\\ and\\ }H(C\\tilde{A}^{-1},1)\\ge \\alpha -\\lambda _{n_1}.",
"$ Now we verify the convergence of the infinite series in (REF ).",
"From $s_{\\bar{D}}>S_{\\bar{A}}$ , we have $H(DD(i-1)CA(-i)\\tilde{A}^{-1},1)\\ge \\lambda _{n_1}+(i-1)s_{\\bar{D}}+b+\\alpha -iS_{\\bar{A}}+b-\\lambda _{n_1}\\\\=i(s_{\\bar{D}}-S_{\\bar{A}})-s_{\\bar{D}}+2b+\\alpha \\xrightarrow{} \\infty .$ Therefore, the series in (REF ) converges to a matrix $X_k\\in \\mathrm {M}_{(n-n_1)\\times n_1}(R)$ , and $H(X_k, 1)\\ge \\alpha -\\lambda _{n_1}.$ Moreover, from $\\tilde{T}^{-\\alpha }C\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}$ , we have $\\tilde{T}^{\\lambda _{n_1}-\\alpha }X_k\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}\\unknown.",
"\\textrm {\\ and\\ }\\tilde{T}^{-\\alpha }X_kD_R(\\underline{\\lambda }^{(n_1,n]})\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}.$ Now we set $Y_k:= \\begin{bmatrix}I_{n_1}&0\\\\X_k&I_{n-n_1}\\end{bmatrix}$ , and have $Y_kD_R(\\underline{\\lambda }^{[n]})=\\begin{bmatrix}D_R(\\underline{\\lambda }^{[n_1]})&0\\\\X_kD_R(\\underline{\\lambda }^{[n_1]})&D_R(\\underline{\\lambda }^{(n_1,n]})\\end{bmatrix}.$ From $\\tilde{A}^{-1}=A^{\\prime -1}D_R(\\underline{\\lambda }^{[n_1]})^{-1}$ , we have $X_kD_R(\\underline{\\lambda }^{[n_1]})=-\\left(C+\\sum \\limits _{i=1}^\\infty DD(i-1)CA(-i)\\right)A^{\\prime -1}.$ Combined with (REF ) and that both $C$ and $D$ are $\\underline{\\lambda }^{(n_1,n]}$ -Hodge bounded, this equality implies that $X_kD_R(\\underline{\\lambda }^{[n_1]})$ is $\\underline{\\lambda }^{(n_1,n]}$ -Hodge bounded and $H(X_kD_R(\\underline{\\lambda }^{[n_1]}),1)\\ge \\alpha .$ Hence there exists $Y^{\\prime }_k\\in \\mathrm {M}_n(R)$ such that $Y_kD_R(\\underline{\\lambda }^{[n]})=D_R(\\underline{\\lambda }^{[n]})Y^{\\prime }_k$ .",
"By Lemma REF , $Y_k$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"Note that $Y_kMY_k^{-1}= \\begin{bmatrix}A-BX_k&B\\\\X_kA+C-DX_k-X_kBX_k&X_kB+D\\end{bmatrix}.$ We set $C^{\\prime }:=X_kA+C-DX_k-X_kBX_k$ and need to prove that $H(C^{\\prime },1)\\ge \\alpha \\textrm {\\ and \\ } \\tilde{T}^{-\\alpha }C^{\\prime }\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}.$ The first inequality follows directly from $H(C,1)\\ge \\alpha $ , (REF ) and (REF ).",
"It remains to prove the second congruence relation.",
"From our construction, for every $i\\ge 1$ we have $\\begin{split}&\\tilde{A}^{-1}A\\equiv I_{n_1}\\mathfrak {}\\pmod {\\mathfrak {m}}\\\\&\\tilde{T}^{(i+1)\\lambda _{n_1}}A(-i)\\tilde{A}^{-1}\\equiv \\tilde{T}^{(i+1)\\lambda _{n_1}}A(-i-1)\\tilde{A}^{-1}A \\mathfrak {}\\pmod {\\mathfrak {m}},\\\\&\\tilde{T}^{-(i+1)\\lambda _{n_1}}D^2D(i-1)\\equiv \\tilde{T}^{-(i+1)\\lambda _{n_1}}DD(i)\\mathfrak {}\\pmod {\\mathfrak {m}}.\\end{split}$ Combining these congruence relations above with $\\tilde{T}^{-\\alpha }C\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}$ , we have $\\tilde{T}^{-\\alpha }D^2D(i-1)CA(-i)\\tilde{A}^{-1}\\equiv \\tilde{T}^{-\\alpha }DD(i)CA(-i-1)\\tilde{A}^{-1}A \\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}$ and $\\tilde{T}^{-\\alpha }C\\equiv \\tilde{T}^{-\\alpha }C\\tilde{A}^{-1}A \\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}.$ By $\\tilde{T}^{\\lambda _{n_1}}A(-1)\\tilde{A}^{-1}A\\equiv \\tilde{T}^{\\lambda _{n_1}}\\tilde{A}^{-1}\\mathfrak {}\\pmod {\\mathfrak {m}},$ we have $\\tilde{T}^{-\\alpha }DC\\tilde{A}^{-1}\\equiv \\tilde{T}^{-\\alpha }DCA(-1)\\tilde{A}^{-1}A\\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}.$ By (REF ), we have $X_k\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}$ and $\\tilde{T}^{-\\alpha }X_kB\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}$ .",
"Combined with our hypothesis $k\\ge 1$ , the above two congruence relations imply $\\tilde{T}^{-\\alpha }X_kBX_k\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}.",
"$ Combined with (REF ), (REF ), (REF ) and (REF ), this congruence implies $\\tilde{T}^{-\\alpha }C^{\\prime }\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}.$ Remark 2.3.12 We emphasize that in the above argument, all the congruence relations modulo powers of $\\mathfrak {m}$ are for matrices with entries in $R$ instead of $R[\\frac{1}{\\tilde{T}}]$ .",
"The reason is because $\\mathfrak {m}^k\\cdot R[\\frac{1}{\\tilde{T}}]\\cap R$ and $\\mathfrak {m}^k$ are not necessary to be the same.",
"Equivalently, we do not know whether $\\tilde{T}$ is a non-zero divisor in the ring $R/\\mathfrak {m}^k$ in general.",
"However, this is true for $k=1$ by Lemma REF .",
"Now we prove Proposition REF .",
"Combining our hypotheses $C\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}}$ and $H(C,1)\\ge \\alpha $ with Lemma REF , we have $\\tilde{T}^{-\\alpha }C\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}}$ .",
"We can apply Lemma REF inductively to get a sequence of matrices $(X_k)_{k\\ge 1}$ in $\\mathrm {M}_{(n-n_1)\\times n_1}(R)$ with the following properties: $H(X_k,1)\\ge \\alpha -\\lambda _{n_1}$ , and $\\tilde{T}^{\\lambda _{n_1}-\\alpha }X_k\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^k}$ .",
"The matrix $Y_k=\\begin{bmatrix}I_{n_1}&0\\\\X_k&I_{n-n_1}\\end{bmatrix}$ is $\\underline{\\lambda }^{[n]}$ -stable.",
"If we set $(Y_k\\dots Y_1)M(Y_k\\dots Y_1)^{-1}:=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_k^{\\prime } & B_k^{\\prime } \\\\C_k^{\\prime } & D_k^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle n_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle n_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(C_k^{\\prime },1)\\ge \\alpha $ and $\\tilde{T}^{-\\alpha }C_k^{\\prime }\\equiv 0\\mathfrak {}\\pmod {\\mathfrak {m}^{k+1}}$ .",
"The series $\\sum \\limits _{i=1}^\\infty X_i$ converges to a matrix $X\\in \\mathrm {M}_{(n-n_1)\\times n_1}(R)$ with $H(X,1)\\ge \\alpha -\\lambda _{n_1}$ .",
"Therefore the products $\\lbrace Y_k\\dots Y_1|k\\ge 1\\rbrace $ converge to the matrix $Y=\\begin{bmatrix}I_{n_1}&0\\\\X&I_{n-n_1}\\end{bmatrix}$ .",
"By our construction as above, it is straightforward to verify that $Y$ is $\\underline{\\lambda }^{[n]}$ -stable and $YMY^{-1}$ is of the form $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A^{\\prime } & B^{\\prime } \\\\0\\ & D^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle n_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle n_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Theorem 2.3.13 Let $M\\in \\mathrm {M}_\\infty (R)$ be a $\\underline{\\lambda }$ -Hodge bounded matrix and $\\bar{M}\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ be its reduction by $\\mathfrak {m}$ .",
"Suppose that $\\Omega =\\lbrace s_1<s_2<\\dots \\rbrace $ is a set of touching vertices of $\\bar{M}$ such that $N(\\bar{M},s)=\\sum \\limits _{i=1}^{s}\\lambda _i$ , for all $s\\in \\Omega $ .",
"Then there exists a $\\underline{\\lambda }$ -stable matrix $Q\\in \\operatorname{GL}_\\infty (R)$ with the following properties: $Q$ is $\\underline{\\lambda }$ -stable, and in particular $QMQ^{-1}$ is $\\underline{\\lambda }$ -Hodge bounded; and $QMQ^{-1}$ is of the following block upper triangular shape: $QMQ^{-1}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){M_{11} & M_{12}\\\\\\ 0\\ \\ & M_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ By Theorem REF , there exists a $\\underline{\\lambda }$ -stable matrix $\\bar{Q}^{\\prime }\\in \\operatorname{GL}_\\infty (\\mathbb {F}_p\\llbracket T\\rrbracket )$ such that $\\bar{Q}^{\\prime }\\bar{M}\\bar{Q}^{\\prime -1}$ is block upper triangular of the shape $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){N_{11} & N_{12}\\\\\\ 0\\ \\ & N_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Let $Q^{\\prime }\\in \\mathrm {M}_\\infty (R)$ be a special lift of $\\bar{Q}^{\\prime }$ .",
"By Lemma REF , $Q^{\\prime }$ is invertible; and by Corollary REF , $Q^{\\prime }$ is $\\underline{\\lambda }$ -stable.",
"Thus we may assume that $\\bar{M}$ is block upper triangular with the prescribed shape.",
"We first take an infinite subsequence $\\underline{s}^{\\prime }:=(s_1^{\\prime },\\dots )$ of $\\Omega $ such that $s_1^{\\prime }>\\ell _{\\bar{M}}$ as in Lemma REF .",
"By Proposition REF with $M:=M_{[s_1^{\\prime }]}\\in \\mathrm {M}_{s_1^{\\prime }}(R)$ , we get a matrix $X_1\\in \\mathrm {M}_{(s_1^{\\prime }-s_1)\\times s_1}(R)$ such that $H(X_1,1)\\ge \\lambda _{s_1^{\\prime }}-\\lambda _{s_1}$ .",
"If we set $Y_1:=\\begin{bmatrix}I_{s_1}&0\\\\X_1&I_{s_1^{\\prime }-s_1}\\end{bmatrix}$ , then $Y_1$ is $\\underline{\\lambda }^{[s_1^{\\prime }]}$ -stable and $Y_1M_{[s_1^{\\prime }]}Y_1^{-1}$ is of the form $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_1^{\\prime } & B_1^{\\prime } \\\\\\ 0 \\ & D_1^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Let $P_1:=\\begin{bmatrix}Y_1&0\\\\0&I_\\infty \\end{bmatrix}=\\begin{bmatrix}I_{s_1}&0&0\\\\X_1&I_{s_1^{\\prime }-s_1}&0\\\\0&0&I_\\infty \\end{bmatrix}.$ Then $P_1$ is $\\underline{\\lambda }$ -stable and $M_1:=P_1MP_1^{-1}$ is of the form $ \\begin{bmatrix}A_1^{\\prime }&B_1^{\\prime }&\\ast \\\\0&D_1^{\\prime }&\\ast \\\\\\ast &\\ast &\\ast \\end{bmatrix}.$ Now we consider the matrix $(M_1)_{[s_2^{\\prime }]}\\in \\mathrm {M}_{s_2^{\\prime }}(R)$ which satisfies all the hypotheses in Proposition REF with $\\alpha =\\lambda _{s_2^{\\prime }}$ .",
"Moreover, if we write $(M_1)_{[s_2^{\\prime }]}$ in the form $(M_1)_{[s_2^{\\prime }]}=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A & B \\\\C & D \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ then $H(C,1)\\ge \\lambda _{s_2^{\\prime }}$ .",
"Hence there exists $X_2\\in \\mathrm {M}_{(s_2^{\\prime }-s_1)\\times s_1}(\\mathfrak {m})$ such that if we set $Y_2:=\\begin{bmatrix}I_{s_1}&0\\\\X_2&I_{s_2^{\\prime }-s_1}\\end{bmatrix}$ , then $Y_2$ is $\\underline{\\lambda }^{[s_2^{\\prime }]}$ -stable and $Y_2(M_1)_{[s_2^{\\prime }]}Y_2^{-1}$ is of the form $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_2^{\\prime } & B_2^{\\prime } \\\\\\ 0 \\ & D_2^{\\prime } \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$ Moreover, we have $H(X_2,1)\\ge \\lambda _{s_2^{\\prime }}-\\lambda _{s_1}$ .",
"Let $P_2:= \\begin{bmatrix}Y_2&0\\\\0&I_\\infty \\end{bmatrix} = \\begin{bmatrix}I_{s_1}&0&0\\\\X_2&I_{s_2^{\\prime }-s_1}&0\\\\0&0&I_\\infty \\end{bmatrix}.$ Then $P_2$ is $\\underline{\\lambda }$ -stable and $P_2M_1P_2^{-1}=(P_2P_1)M(P_2P_1)^{-1}$ is of the form $\\begin{bmatrix}A_2^{\\prime }&B_2^{\\prime }&\\ast \\\\0&D_2^{\\prime }&\\ast \\\\\\ast &\\ast &\\ast \\end{bmatrix}$ .",
"We repeat this iteration and get a sequence of matrices $\\left(P_k= \\begin{bmatrix}I_{s_1}&0\\\\P_k^{\\prime }&I_\\infty \\end{bmatrix}\\;\\Big |\\;k\\in \\mathbb {N}\\right)\\subset \\operatorname{GL}_\\infty (R)$ such that $P_k$ is $\\underline{\\lambda }$ -stable and $H(P_k^{\\prime },1)\\ge \\lambda _{s_k^{\\prime }}-\\lambda _{s_1}$ .",
"Moreover, $(P_k\\dots P_1)M(P_k\\dots P_1)^{-1}$ is of the form $ \\begin{bmatrix}A_k^{\\prime }&B_k^{\\prime }&\\ast \\\\0&D_k^{\\prime }&\\ast \\\\\\ast &\\ast &\\ast \\end{bmatrix}$ .",
"The infinite series $\\sum \\limits _{i=1}^\\infty P_i^{\\prime }$ converges to a matrix $P^{\\prime }$ and hence the product $P_k\\dots P_1$ converges to $P= \\begin{bmatrix}I_{s_1}&0\\\\P^{\\prime }&I_\\infty \\end{bmatrix}$ .",
"By our construction, $P$ is $\\underline{\\lambda }$ -stable and $PMP^{-1}$ is of the form $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){N_{11} & N_{12} \\\\0 & N_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle s_1};\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle s_1};\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ].$"
],
[
"Notations",
" Let $F$ be a totally real field of degree $g$ over $\\mathbb {Q}$ and $\\mathcal {O}_F$ be its ring of integers.",
"Let $I:=\\operatorname{Hom}(F,\\bar{\\mathbb {Q}})$ denote the set of embeddings of $F$ into $\\bar{\\mathbb {Q}}$ .",
"Fix an odd prime $p$ which splits completely in $F$ .",
"We set $\\mathcal {O}_p:=\\mathcal {O}_F\\otimes _{\\mathbb {Z}}\\mathbb {Z}_p$ and $F_p:=F\\otimes _{\\mathbb {Q}}\\mathbb {Q}_p$ .",
"For every $i\\in I$ , let $\\mathfrak {p}_i$ be the prime of $\\mathcal {O}_F$ induced by the composite $i_p=\\iota _p\\circ i:F\\rightarrow \\bar{\\mathbb {Q}}_p$ .",
"Let $F_{\\mathfrak {p}_i}$ (resp.",
"$\\mathcal {O}_{\\mathfrak {p}_i}$ ) denote the completion of $F$ (resp.",
"$\\mathcal {O}_F$ ) at $\\mathfrak {p}_i$ .",
"For $\\alpha \\in F_p$ and $i\\in I$ , we denote by $\\alpha _i$ the $i$ -component of $\\alpha $ under the natural isomorphism $F_p\\cong \\prod \\limits _{i\\in I}F_{\\mathfrak {p}_i}$ .",
"Since $p$ splits in $F$ , the embedding $\\mathbb {Q}_p\\rightarrow F_{\\mathfrak {p}_i}$ is an isomorphism $ \\mathbb {Q}_p\\cong F_{\\mathfrak {p}_i}$ and hence get $\\mathbb {Z}_p\\cong \\mathcal {O}_{\\mathfrak {p}_i}$ .",
"For every $i\\in I$ , we fix a uniformizer $\\pi _i$ of $\\mathcal {O}_{\\mathfrak {p}_i}$ and let $\\pi :=\\prod \\limits _{i\\in I}\\pi _i\\in \\mathcal {O}_p$ .",
"For every $i\\in I$ , we use $\\Delta _i$ to denote the torsion subgroup of $\\mathcal {O}_{\\mathfrak {p}_i}^\\times $ .",
"Hence we have a decomposition of the multiplicative group $\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\cong \\Delta _i\\times (1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i})$ .",
"Let $A_{F,f}$ (resp.",
"$A_{F,f}^{(p)}$ ) denote the ring of finite adeles (resp.",
"finite prime-to-$p$ adeles) of $F$ .",
"We use $\\textrm {Nm}_{F/\\mathbb {Q}}:F^\\times \\rightarrow \\mathbb {Q}^\\times $ to denote the norm map.",
"For simplicity, the norm map $F_p^\\times \\rightarrow \\mathbb {Q}_p^\\times $ will also be denoted by $\\textrm {Nm}_{F/\\mathbb {Q}}$ .",
"Following [5], we define $\\mathcal {N}:=\\lbrace |x|_p\\;|\\;x\\in \\bar{\\mathbb {Q}}_p^\\times \\rbrace \\cap (0,1]=p^\\mathbb {Q}\\cap (0,1]$ and $\\mathcal {N}^\\times :=\\mathcal {N}\\setminus \\lbrace 1 \\rbrace =p^\\mathbb {Q}\\cap (0,1)$ .",
"We use $\\mathbb {A}^1$ (resp.",
"$\\mathbb {G}_m$ ) to denote the rigid analytification of the affine line (resp.",
"affine line with zero removed).",
"For $r=(r_i)_{i\\in I}\\in \\mathcal {N}^I$ and $\\alpha =(\\alpha _i)_{i\\in I}\\in \\mathcal {O}_p$ , we define $B(\\alpha ,r)$ to be the $\\mathbb {Q}_p$ -polydisc such that $B(\\alpha ,r)(\\mathbb {C}_p):=\\lbrace (x_i)_{i\\in I}\\in \\mathbb {C}_p^I\\;|\\; |x_i-i_p(\\alpha _i)|_p\\le r_i\\text{~for all~} i\\in I \\rbrace $ , and $\\mathbf {B}_r:=\\prod \\limits _{\\alpha \\in \\mathcal {O}_p}B(\\alpha ,r)$ .",
"If $r\\in (\\mathcal {N}^\\times )^I$ , we define $\\mathbf {B}_r^\\times :=\\prod \\limits _{\\alpha \\in \\mathcal {O}_p^\\times }B(\\alpha ,r)$ .",
"Let $K$ be a complete field extension of $\\mathbb {Q}_p$ and $X:=\\mathrm {Sp}(A)$ be a $K$ -affinoid space.",
"By [5], there is a bijection $\\iota :\\mathcal {O}(X)^\\times \\cong \\operatorname{Hom}_{\\mathbb {Q}_p\\text{-rigid space~}}(X,\\mathbb {G}_m)$ .",
"By [5], for any continuous character $\\chi :\\mathcal {O}_p^\\times \\rightarrow A^\\times $ , there exists at least one $r\\in (\\mathcal {N}^\\times )^I$ that satisfies the following: There is a unique map of $K$ -rigid spaces $\\beta _r:\\mathbf {B}_r^\\times \\times X\\rightarrow \\mathbb {G}_m$ such that for every $\\alpha \\in \\mathcal {O}_p^\\times $ we have $ \\iota \\circ \\chi (\\alpha )=\\beta _r(\\alpha ,-)$ , where $\\chi (\\alpha )\\in A^\\times =\\mathcal {O}(X)^\\times $ and $\\beta _r(\\alpha ,-):X\\rightarrow \\mathbb {G}_m$ is obtained by evaluating $\\beta _r$ at $\\alpha \\in \\mathbf {B}_{r}^\\times (\\mathbb {Q}_p)$ .",
"We call $\\beta _r$ the $r$ -thickening of $\\chi $ .",
"Let $\\chi :\\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ be a continuous character.",
"For every $i\\in I$ , we denote by $\\chi _i$ the $i$ -component of $\\chi $ , i.e.",
"$\\chi _i$ is the composite $\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\hookrightarrow \\mathcal {O}_p^\\times \\xrightarrow{}\\mathbb {C}_p^\\times $ .",
"We define the coordinates $T:=(T_i)_{i\\in I}$ of $\\chi $ by $T_i:=T_{\\chi _i}= \\chi _i(\\exp (\\pi _i))-1$ for all $i\\in I$ .",
"For $r=(r_i)_{i\\in I}\\in (\\mathcal {N}^\\times )^I$ , the character $\\chi $ admits the $r$ -thickening if and only if $v_p(T_i)>\\frac{pr_i}{p-1}$ for all $i\\in I$ .",
"Let $\\iota _F:\\mathcal {O}_F^\\times \\hookrightarrow \\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times , x\\mapsto (x,x^2)$ be an embedding.",
"We always regard $\\mathcal {O}_F^\\times $ as a subgroup of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ by $\\iota _F$ .",
"A weight is defined to be a continuous group homomorphism $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ for some affinoid $\\mathbb {Q}_p$ -space $X=\\mathrm {Sp}(A)$ , such that $\\ker (\\kappa )$ contains a subgroup of $\\mathcal {O}_F^\\times $ of finite index.",
"For such a weight $\\kappa $ , we define $r(\\kappa )$ to be the largest element $r$ in $(\\mathcal {N}^\\times )^I$ (with the obvious partial order), such that the continuous homomorphism $n:\\mathcal {O}_p^\\times \\rightarrow \\mathcal {O}(X)^\\times $ admits the $r$ -thickening.",
"Let $D/F$ be a totally definite quaternion algebra over $F$ with discriminant $\\mathfrak {d}$ .",
"Assume that $(p,\\mathfrak {d})=1$ .",
"Let $D_f:=D\\otimes _F A_{F,f}$ and $D_f^{(p)}:=D\\otimes _F A_{F,f}^{(p)}$ .",
"Fix a maximal order $\\mathcal {O}_D$ of $D$ .",
"For each finite place $v$ of $F$ prime to $\\mathfrak {d}$ , we fix an isomorphism $\\mathcal {O}_D\\otimes _{\\mathcal {O}_F}\\mathcal {O}_{F,v}\\cong \\mathrm {M}_2(\\mathcal {O}_{F,v})$ .",
"In particular, we have an isomorphism $\\mathcal {O}_D\\otimes _{\\mathcal {O}_F}\\mathcal {O}_p\\cong \\mathrm {M}_2(\\mathcal {O}_p)$ .",
"For an ideal $\\mathfrak {n}$ of $\\mathcal {O}_F$ prime to $\\mathfrak {d}$ , we denote by $U_0^{D,(p)}(\\mathfrak {n})$ and $U_1^{D,(p)}(\\mathfrak {n})$ the subgroups of $(\\mathcal {O}_D\\otimes \\hat{\\mathbb {Z}}^{(p)})^\\times $ consisting of matrices which are congruent to $\\begin{bmatrix}\\ast &\\ast \\\\0&\\ast \\end{bmatrix}$ and $\\begin{bmatrix}\\ast &\\ast \\\\0&1\\end{bmatrix}$ $\\mathfrak {}\\mod {\\mathfrak {n}}$ , respectively.",
"For a $\\mathbb {Q}_p$ -Banach algebra $A$ with norm $|\\cdot |$ , we define a subring $A\\lbrace \\!\\lbrace X\\rbrace \\!\\rbrace $ of $A\\llbracket X \\rrbracket $ by $A\\lbrace \\!\\lbrace X\\rbrace \\!\\rbrace :=\\left\\lbrace \\sum _{n\\ge 0}c_nX^n\\in A\\llbracket X \\rrbracket \\;\\Bigg |\\; \\lim _{n\\rightarrow \\infty }|c_n|R^n=0, \\text{~for any~} R\\in \\mathbb {R}_{>0} \\right\\rbrace .$"
],
[
"Subgroups of $\\operatorname{GL}_2(F_p)$",
" For a commutative ring $R$ with unit 1, we define several subgroups of $\\operatorname{GL}_2(R)$ by $B(R):=\\begin{bmatrix}R^\\times &R\\\\0&R^\\times \\end{bmatrix}, \\ T(R):=\\begin{bmatrix}R^\\times &0\\\\0&R^\\times \\end{bmatrix}, \\ N(R):=\\begin{bmatrix}1&R\\\\0&1\\end{bmatrix}\\text{~and~}D(R):=\\left\\lbrace \\begin{bmatrix}\\alpha &0\\\\0&\\alpha \\end{bmatrix}\\Big |\\ \\alpha \\in R^\\times \\right\\rbrace .$ For a subgroup $H$ of $R^\\times $ , we set $T(H):=\\begin{bmatrix}H^\\times &0\\\\0&H^\\times \\end{bmatrix}\\text{~and~} D(H):=\\left\\lbrace \\begin{bmatrix}\\alpha &0\\\\0&\\alpha \\end{bmatrix}\\ \\Big |\\ \\alpha \\in H \\right\\rbrace .$ Fix $i\\in I$ and a positive integer $t_i$ .",
"We define the following subgroups of $\\operatorname{GL}_2(\\mathcal {O}_{\\mathfrak {p}_i})$ : $\\mathrm {Iw}_{\\pi _i^{t_i}}:=\\begin{bmatrix}\\mathcal {O}_{\\mathfrak {p}_i}^\\times &\\mathcal {O}_{\\mathfrak {p}_i}\\\\\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}&\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\end{bmatrix},\\ \\bar{N}(\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}):=\\begin{bmatrix}1&0\\\\\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}&1\\end{bmatrix} \\text{~and~}\\bar{B}(\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}):=\\begin{bmatrix}\\mathcal {O}_{\\mathfrak {p}_i}^\\times &0\\\\\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}&\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\end{bmatrix}.$ Given $t=(t_i)_{i\\in I}\\in \\mathbb {N}^I$ , we put $\\mathrm {Iw}_{\\pi ^t}:=\\prod \\limits _{i\\in I}\\mathrm {Iw}_{\\pi _i^{t_i}}$ and similarly define $\\bar{N}(\\pi ^t\\mathcal {O}_p),\\ \\bar{B}(\\pi ^t\\mathcal {O}_p)$ .",
"Note that every $g\\in \\mathrm {Iw}_{\\pi ^t}$ can be represented by $(g_i)_{i\\in I}$ , with $g_i\\in \\mathrm {Iw}_{\\pi _i^{m_i}}$ , and every $g_i\\in \\mathrm {Iw}_{\\pi _i^{t_i}}$ can be viewed as an element in $\\mathrm {Iw}_{\\pi ^t}$ whose $j$ -component is the identity matrix for all $j\\ne i$ .",
"All the groups we define above are profinite and considered as topological groups endowed with the profinite topology.",
"For $i\\in I$ and $t_i\\in \\mathbb {N}$ , we will identify $\\mathcal {O}_{\\mathfrak {p}_i}$ with $\\bar{N}(\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i})$ via the isomorphism $\\bar{n}_i:\\mathcal {O}_{\\mathfrak {p}_i}\\rightarrow \\bar{N}(\\pi _i^{t_i}\\mathcal {O}_{\\mathfrak {p}_i}),\\ z\\mapsto \\begin{bmatrix}1&0\\\\\\\\ \\pi _i^{t_i} z&1\\end{bmatrix}$ .",
"For $t\\in \\mathbb {N}^I$ , we will identify $\\mathcal {O}_p$ with $\\bar{N}(\\pi ^t\\mathcal {O}_p)$ via the isomorphism $\\bar{n}:\\mathcal {O}_p\\rightarrow \\bar{N}(\\pi ^t\\mathcal {O}_p),\\ z\\mapsto \\begin{bmatrix}1&0\\\\\\\\ \\pi ^t z&1\\end{bmatrix}$ .",
"For $i\\in I$ , we denote by $\\mathrm {Iw}_{\\pi _i,1}:= \\begin{bmatrix}1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i} &\\mathcal {O}_{\\mathfrak {p}_i}\\\\\\\\ \\pi _i\\mathcal {O}_{\\mathfrak {p}_i} z&1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}\\end{bmatrix}$ the the pro-$p$ -subgroup of $\\mathrm {Iw}_{\\pi _i}$ .",
"Then the map $T(\\Delta _i)\\times \\mathrm {Iw}_{\\pi ,1}\\rightarrow \\mathrm {Iw}_{\\pi }$ , $(t,g)\\mapsto tg$ is a bijection.",
"We put $\\mathrm {Iw}_{\\pi ,1}:=\\prod \\limits _{i\\in I}\\mathrm {Iw}_{\\pi _i,1}$ .",
"The Iwasawa decomposition is the following bijection: $N(\\mathcal {O}_p)\\times T(\\mathcal {O}_p)\\times \\bar{N}(\\pi ^t\\mathcal {O}_p)\\rightarrow \\mathrm {Iw}_{\\pi ^t},\\ (N,T,\\bar{N})\\mapsto NT\\bar{N}.$ For $i\\in I$ and $t_i\\in \\mathbb {N}$ , we define an anti-involution $\\ast $ on $\\mathrm {Iw}_{\\pi _i^{t_i}}$ by $g=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}\\in \\mathrm {Iw}_{\\pi _i^{t_i}}\\mapsto g^\\ast =\\begin{bmatrix}a&c/\\pi _i^{t_i}\\\\\\pi _i^{t_i} b&d\\end{bmatrix}.$ Here “anti-involution” means that the map $g\\mapsto g^\\ast $ satisfies $(g^\\ast )^\\ast =g$ and $(gh)^\\ast =h^\\ast g^\\ast $ for all $g,h\\in \\mathrm {Iw}_{\\pi _i^{t_i}}$ .",
"We use similar formula to define anti-involutions on $\\mathrm {Iw}_{\\pi ^t}$ for $t\\in \\mathbb {N}^I$ .",
"For $t=(t_i)_{i\\in I}\\in \\mathbb {N}^I$ , we set $\\mathbf {M}_{\\pi ^t}:=\\left\\lbrace \\gamma =(\\gamma _i)_{i\\in I}\\in \\mathrm {M}_2(\\mathcal {O}_p)\\;\\Bigg |\\;\\text{~if~} \\gamma _i= \\begin{bmatrix}a_i&b_i\\\\c_i&d_i\\end{bmatrix} \\text{~then~} \\det (\\gamma _i)\\ne 0, \\pi _i^{t_i}|c_i,\\pi _i\\nmid d_i \\right\\rbrace .$ Then $\\mathbf {M}_{\\pi ^t}$ contains $\\mathrm {Iw}_{\\pi ^t}$ and is a monoid under multiplication.",
"The involution $\\ast $ can be extended to $\\mathbf {M}_{\\pi ^t}$ by the same formula.",
"For $i\\in I$ and $t_i\\in \\mathbb {N}$ , we define the monoid $\\mathbf {M}_{\\pi _i^{t_i}}\\subset \\mathrm {M}_2(\\mathcal {O}_{\\mathfrak {p}_i})$ in a similar way."
],
[
"Induced representations",
"Let $A$ be a topological ring in which $p$ is topologically nilpotent and $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ be a continuous character.",
"Convention 3.3.1 Given the character $\\kappa $ as above, we always extend $\\nu $ to a continuous homomorphism $F_p^\\times \\rightarrow A^\\times $ by requiring $\\nu (\\pi _i)=1$ for all $i\\in I$ .",
"The character $\\kappa $ induces a character $\\kappa _T:T(\\mathcal {O}_p)\\rightarrow A^\\times ,\\ \\begin{bmatrix}a&0\\\\0&d\\end{bmatrix}\\mapsto n(d)\\cdot \\nu (ad)$ .",
"Since $T(\\mathcal {O}_p)$ is a quotient of $B(\\mathcal {O}_p)$ (resp.",
"$\\bar{B}(\\pi \\mathcal {O}_p)$ ), this character $\\kappa _T$ extends to a character $\\kappa _{B}$ (resp.",
"$\\kappa _{\\bar{B}}$ ) of $B(\\mathcal {O}_p)$ (resp.",
"$\\bar{B}(\\pi \\mathcal {O}_p)$ ).",
"Fix $t\\in \\mathbb {N}^I$ .",
"Consider the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi ^t}}(\\kappa _B):= \\Big \\lbrace f\\in \\mathcal {C}(\\mathrm {Iw}_{\\pi ^t},A)\\;\\Big |\\; f(b g)=\\kappa _B(b)f(g) \\text{~for all~} b\\in B(\\mathcal {O}_p),\\ g\\in \\mathrm {Iw}_{\\pi ^t} \\Big \\rbrace .$ The group $\\mathrm {Iw}_{\\pi ^t}$ acts right on $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi ^t}}(\\kappa _B)$ by $f\\circ m(g)=f(gm^\\ast )$ .",
"By Iwasawa decomposition, we obtain the following bijection $\\begin{split}\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi ^t}}(\\kappa _B)&\\rightarrow \\mathcal {C}(\\mathcal {O}_p,A),\\\\f&\\mapsto h(z):= f(\\bar{n}(z))=f\\left(\\begin{bmatrix}1&0\\\\\\pi ^t z&1\\end{bmatrix}\\right).\\end{split}$ The right action of $\\mathrm {Iw}_{\\pi ^t}$ on $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi ^t}}(\\kappa _B)$ induces a right action on $\\mathcal {C}(\\mathcal {O}_p,A)$ , which can be written in the following explicit formula: $h\\circ m(z)=n(cz+d)\\nu (ad-bc)h\\left(\\frac{az+b}{cz+d}\\right), \\text{~for~} m=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}.$ Using the exact formula in (REF ), we can extend this action to the monoid $\\mathbf {M}_{\\pi ^t}$ ."
],
[
"Space of $p$ -adic automorphic forms",
"We will recall Buzzard's construction of overconvergent automorphic forms in [5].",
"Let $A$ be a $\\mathbb {Q}_p$ -affinoid algebra and $X:=\\mathrm {Sp}(A)$ be the corresponding affinoid space.",
"Let $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ be a weight.",
"We write $r(\\kappa ):=(p^{-m_{\\kappa ,i}})_{i\\in I}$ .",
"Fix $r=(r_i)_{i\\in I}=(p^{-m_i})_{i\\in I}\\in \\mathcal {N}^I$ .",
"Define $\\mathcal {A}_{\\kappa ,r}$ to be the $\\mathbb {Q}_p$ -Banach algebra $\\mathcal {O}(\\mathbf {B}_r\\times X)=\\mathcal {O}_{\\mathbf {B}_r}\\hat{\\otimes }_{\\mathbb {Q}_p}A$ .",
"Since $\\mathcal {O}_p$ is Zariski dense in $\\mathbf {B}_r$ , we obtain an embedding $\\mathcal {A}_{\\kappa ,r}\\hookrightarrow \\mathcal {C}(\\mathcal {O}_p,A)$ and hence can view $\\mathcal {A}_{\\kappa ,r}$ as a subspace of $\\mathcal {C}(\\mathcal {O}_p,A)$ via this embedding.",
"Definition 3.4.1 We call $t\\in \\mathbb {N}^I$ that is good for $(\\kappa ,r)$ if $m_i+t_i\\ge m_{\\kappa ,i}$ for all $i\\in I$ .",
"Fix $t\\in \\mathbb {N}^I$ that is good for $(\\kappa ,r)$ .",
"It is easy to check that the right action of $\\mathbf {M}_{\\pi ^t}$ on $\\mathcal {C}(\\mathcal {O}_p,A)$ leaves the subspace $\\mathcal {A}_{\\kappa ,r}$ stable, and hence (REF ) defines a right action of the monoid $\\mathbf {M}_{\\pi ^t}$ on $\\mathcal {A}_{\\kappa ,r}$ .",
"Fix an open compact subgroup $K^p\\subset D_f^{(p),\\times }$ and $t\\in \\mathbb {N}^I$ which is good for $(\\kappa ,r)$ .",
"As in [5], we define the space of $r$ -overconvergent automorphic forms of weight $\\kappa $ and level $K^p\\mathrm {Iw}_{\\pi ^t}$ to be the $A$ -module $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)=\\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {A}_{\\kappa ,r}\\;|\\;\\phi (gu)=\\phi (g)\\circ u,\\text{~for all~} g\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi ^t} \\rbrace .$"
],
[
"Hecke operators",
"Let $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ , $r=(r_i)_{i\\in I}\\in \\mathcal {N}^I$ , and $t\\in \\mathbb {N}^I$ be the same as the previous section.",
"Fix an open compact subgroup $K^p$ of $D_f^{(p),\\times }$ .",
"In the rest of this paper, we will make the following convention on the choice of $K^p$ .",
"Convention 3.5.1 $K^p$ is of the form $U_0^{(p)}(\\mathfrak {n})$ or $U_1^{(p)}(\\mathfrak {n})$ for some ideal $\\mathfrak {n}$ of $\\mathcal {O}_F$ prime to $p\\mathfrak {d}$ .",
"Let $K=K^p\\mathrm {Iw}_{\\pi ^t}$ be the open compact subgroup of $ D_f^\\times $ .",
"Now we recall the definition of Hecke operators on the space $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)$ .",
"Let $v$ be a finite place of $F$ where $D$ splits and fix a uniformizer $\\pi _v$ of $F_v$ .",
"In particular, when $v=i\\in I$ is over $p$ , we choose $\\pi _v=\\pi _i$ as before.",
"Let $\\eta _v=\\left[ \\begin{array}{cc}\\pi _v&0\\\\0&1\\end{array}\\right]\\in \\operatorname{GL}_2(F_v)$ which is viewed as an element in $D_f^\\times $ whose components away from $v$ are the identity element.",
"We fix a double coset decomposition $K\\eta _v K=\\bigsqcup _j Kx_j,$ with $x_j\\in D_f^\\times $ .",
"We use $x_{j,p}$ to denote the $v$ -component of $x_j$ .",
"The Hecke operator $U_v$ on $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)$ is defined by $U_{v}(\\phi ):=\\sum _j\\phi |_{x_j}$ where $\\phi |_{x_j}(x):=\\phi (xx_j^{-1})\\circ x_{j,p}$ for $\\phi \\in S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r).$ When $v=i\\in I$ , the operator $U_i$ depends on the choice of the uniformizer $\\pi _i$ , and we will write $U_{\\pi _i}$ for $U_i$ .",
"For later computations, we give a more explicit expression of the $U_{\\pi _i}$ operator.",
"We fix a double coset decomposition $\\mathrm {Iw}_{\\pi ^t}\\eta _i\\mathrm {Iw}_{\\pi ^t}=\\bigsqcup _{j=0}^{p-1}\\mathrm {Iw}_{\\pi ^t}v_{i,j},$ with $v_{i,j}=\\begin{bmatrix}\\pi _i&0\\\\j\\pi _i^{t_i}&1\\end{bmatrix}\\in \\mathbf {M}_{\\pi _i^{t_i}}\\subset \\mathbf {M}_{\\pi ^t}$ for $j=0,\\dots ,p-1$ .",
"The $U_{\\pi _i}$ -operator can be described by $U_{\\pi _i}(\\phi )=\\sum _j\\phi |_{v_{i,j}} \\text{~with~} \\phi |_{v_{i,j}}(x)=\\phi (xv_{i,j}^{-1})\\circ v_{i,j}.$ When $v$ is prime to $\\mathfrak {n}p$ , we view $\\pi _v$ as a central element in $D_f^\\times $ .",
"Choose a double coset decomposition $K\\pi _v K=\\bigsqcup _j Ky_j,$ with $y_j\\in D_f^\\times $ .",
"The Hecke operator $S_v$ on $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)$ is defined by $S_{v}(\\phi )=\\sum _j\\phi |_{y_j}.$ The Hecke operators $U_v$ 's and $S_v$ 's for all possible $v$ 's commute with each other.",
"In particular, we define $U_{\\pi }=\\prod \\limits _{i\\in I}U_{\\pi _i}$ .",
"By [5], the operator $U_{\\pi }$ acting on $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)$ is compact.",
"Thus, its characteristic power series is well-defined by $\\mathrm {Char}(U_{\\pi };S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)):=\\det (\\mathrm {I}-XU_{\\pi }|_{S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)})\\in A\\lbrace \\!\\lbrace X\\rbrace \\!\\rbrace .$ We make two remarks on the characteristic power series $\\mathrm {Char}(U_{\\pi };S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r))$ .",
"Remark 3.5.2 By [5], $\\mathrm {Char}(U_{\\pi };S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r))$ is independent of $r$ (as long as $t$ is good for the pair $(\\kappa ,r)$ ).",
"By [5], there is a canonical Hecke equivariant isomorphism $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi ^t},r)\\cong S_\\kappa ^D(K^p\\mathrm {Iw}_\\pi ,r^{\\prime })$ for some explicit $r^{\\prime }\\in (\\mathcal {N}^\\times )^I$ .",
"It is a well-known phenomenon that the characteristic power series does not see the higher $\\mathrm {Iw}_{\\pi ^t}$ -structure.",
"For this reason, we can and will only work over the space $S_\\kappa ^D(K^p\\mathrm {Iw}_\\pi ,r)$ , i.e.",
"$t=(1,\\dots ,1)$ ."
],
[
"The eigenvariety datum for $D$",
"In this section, we recall the construction of the spectral varieties and eigenvarieties associated to $D$ as constructed in [5].",
"We will follow [12] to define an eigenvariety datum as we will use Hansen's interpolation theorem to translate our results to Hilbert modular eigenvarieties (see § below).",
"We start with the definition of the weight space.",
"As in [1], the weight space $\\mathcal {W}$ is defined to be the rigid analytic space over $\\mathbb {Q}_p$ associated to the complete group algebra $\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ .",
"A closed point of $\\mathcal {W}$ is a continuous character $\\chi =(\\nu ,\\mu ):\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ .",
"Remark 3.6.1 It will be helpful to compare the weight space constructed in [5] with the weight space constructed above.",
"First we recall its construction in [5].",
"Let $G\\subset \\mathcal {O}_F^\\times $ be a subgroup of finite index.",
"Recall that we regard $G$ as a subgroup of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ via the embedding $\\iota _F:\\mathcal {O}_F^\\times \\rightarrow \\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ , $x\\mapsto (x,x^2)$ .",
"We denote by $\\Gamma _G$ the quotient of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ by the closure of $G$ .",
"By [5], the functor which sends a $\\mathbb {Q}_p$ -rigid space $U$ to the group of continuous group homomorphism $\\Gamma _G\\rightarrow \\mathcal {O}(U)^\\times $ is represented by a $\\mathbb {Q}_p$ -rigid space $\\mathcal {X}_{\\Gamma _G}$ (actually $\\mathcal {X}_{\\Gamma _G}$ is isomorphic to the product of an open unit polydisc and a finite rigid space over $\\mathbb {Q}_p$ ).",
"The weight space $\\mathcal {W}^{\\mathrm {full}}$ is the direct limit $\\varinjlim \\limits _G \\mathcal {X}_{\\Gamma _G} $ as $G$ varies over the set of subgroups of finite index of $\\mathcal {O}_F^\\times $ .",
"If $G_1\\subset G_2\\subset \\mathcal {O}_F^\\times $ are two subgroups of $\\mathcal {O}_F^\\times $ of finite indices, the natural homomorphism $\\Gamma _{G_1}\\rightarrow \\Gamma _{G_2}$ is surjective with finite kernel, and hence the corresponding map $\\mathcal {X}_{\\Gamma _{G_2}}\\rightarrow \\mathcal {X}_{\\Gamma _{G_1}}$ is a closed immersion and geometrically identifies $\\mathcal {X}_{\\Gamma _{G_2}}$ with a union of components of $\\mathcal {X}_{\\Gamma _{G_1}}$ .",
"Therefore, $\\lbrace \\mathcal {X}_{\\Gamma _{G}}|G \\text{~is a subgroup of~} \\mathcal {O}_F^\\times \\text{~of finite index} \\rbrace $ forms an admissible cover of the weight space $\\mathcal {W}$ .",
"It follows that the dimension of $\\mathcal {W}^{\\mathrm {full}}$ is $g+1+\\delta $ , where $\\delta $ is the Leopoldt defect for $(F,p)$ .",
"Now let's explain the relation between the two weight spaces $\\mathcal {W}$ and $\\mathcal {W}^{\\mathrm {full}}$ .",
"Define a continuous homomorphism $\\phi _\\eta :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ , $(\\alpha ,\\beta )\\mapsto (\\alpha ^{-2}\\beta ,\\textrm {Nm}_{F/\\mathbb {Q}}(\\alpha ))$ .",
"The kernel $\\ker (\\phi _\\eta )$ contains the group $G=\\mathcal {O}_F^{\\times ,+}$ and hence $\\phi _\\eta $ induces a map $\\eta :\\mathcal {W}\\rightarrow \\mathcal {X}_{\\Gamma _{G}}$ of rigid analytic spaces over $\\mathbb {Q}_p$ and hence a map $\\eta :\\mathcal {W}\\rightarrow \\mathcal {W}^{\\mathrm {full}}$ .",
"Since $\\dim \\mathcal {W}=g+1$ , if the Leopoldt's conjecture holds for $(F,p)$ , the map $\\eta :\\mathcal {W}\\rightarrow \\mathcal {W}^{\\mathrm {full}}$ would be an immersion which identifies $\\mathcal {W}$ with a set of connected components of $\\mathcal {W}^{\\mathrm {full}}$ .",
"There are two reasons why we use the weight space $\\mathcal {W}$ instead of $\\mathcal {W}^{\\mathrm {full}}$ : first, we do not even know the dimension of the weight space $\\mathcal {W}^{\\mathrm {full}}$ without assuming the Leopoldt's conjecture for $(F,p)$ and it is rather complicated to describe the connected components of $\\mathcal {W}^{\\mathrm {full}}$ and classical weights on every component; second, as an application we will use $p$ -adic Jacquet-Langlands correspondence to translate our results to Hilbert modular eigenvarieties as constructed in [1], and the weight space $\\mathcal {W}$ is the one considered there.",
"Since the action of the monoid $\\mathbf {M}_{\\pi }$ on the spaces $\\mathcal {A}_{\\kappa ,r}$ depends on the characters of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ , it is useful to have an explicit expression of the map $\\mathcal {W}\\rightarrow \\mathcal {W}^{\\mathrm {full}}$ in term of characters.",
"Let $A$ be an affinoid $\\mathbb {Q}_p$ -algebra and $\\chi =(\\nu ,\\mu ):\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ be a continuous character.",
"Then the composite $\\kappa =\\chi \\circ \\phi _\\eta :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ is of the form $\\kappa =(n,\\nu )$ , where $n:\\mathcal {O}_p^\\times \\rightarrow A^\\times $ is the character defined by $\\alpha \\mapsto \\nu (\\alpha )^{-2}\\cdot (\\mu \\circ \\textrm {Nm}_{F/\\mathbb {Q}}(\\alpha ))$ .",
"The character $\\kappa $ is called the character of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ associated to $\\chi $.",
"Convention 3.6.2 In the rest of this paper, a weight $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ is always the character associated to some character $\\chi $ of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ as constructed above.",
"We fix an open compact subgroup $K^p$ of $D_f^{(p),\\times }$ as before.",
"For an affinoid subdomain $U=\\mathrm {Sp}(A)\\subset \\mathcal {W}$ , we denote by $\\chi _U:\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ the universal character and $\\kappa _U:\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ the character associated to $\\chi _U$ .",
"For simplicity, we use $f_U(X)$ to denote $\\mathrm {Char}(U_\\pi ,S_{\\kappa _U}^D(K^p\\mathrm {Iw}_{\\pi },r(\\kappa _U)))\\in A\\lbrace \\!\\lbrace X\\rbrace \\!\\rbrace $ .",
"The spectral variety $\\mathrm {wt}:\\mathcal {Z}_D\\rightarrow \\mathcal {W}$ is defined to be the Fredholm hypersurface of $\\mathcal {W}\\times \\mathbb {A}^1$ associated to the compact operator $U_\\pi $ .",
"More precisely, for every affinoid $U$ of $\\mathcal {W}$ , $\\mathrm {wt}^{-1}(U)\\subset U\\times \\mathbb {A}^1$ is the zero locus of the characteristic power series $f_U(X)$ .",
"We use $\\mathrm {wt}:\\mathcal {Z}_D\\rightarrow \\mathcal {W}$ (resp.",
"$a_\\pi ^{-1}:\\mathcal {Z}_D\\rightarrow \\mathbb {A}^1$ ) to denote the first (resp.",
"second) projection.",
"Definition 3.6.3 An affinoid open subdomain $Y\\subset \\mathcal {Z}_D$ is called slope adapted if there exist $h\\in \\mathbb {Q}$ and an affinoid $U=\\mathrm {Sp}(A)\\subset \\mathcal {W}$ , such that $Y=A\\langle p^hX \\rangle /(f_U(X))$ is an affinoid open subset of $\\mathcal {Z}_D$ and the natural map $Y\\rightarrow U$ is finite and flat.",
"For such an slope adapted affinoid $Y$ , the characteristic power series $f_U(X)$ admits a slope $\\le h$ -factorization $f_U(X)=Q(X)\\cdot R(X)$ and $\\mathcal {O}(Y)\\cong A[X]/(Q(X))$ .",
"More precisely, we have the following characterization of $Q(X)$ and $R(X)$ : $Q(X)$ is a polynomial whose leading coefficient is a unit and the slopes of the Newton polygon of $Q(X)$ are all $\\le h$ ; and $R(X)$ is an entire power series and the slopes of the Newton polygon of $R(X)$ are all $>h$ .",
"Moreover, it follows from [5] that the space $S_{\\kappa _U}^D(K^p\\mathrm {Iw}_{\\pi },r(\\kappa _U))$ admits a slope $\\le h$ -decomposition $S_{\\kappa _U}^D(K^p\\mathrm {Iw}_{\\pi },r(\\kappa _U))=S_{\\kappa _U}^{D,\\le h}\\oplus S_{\\kappa _U}^{D,>h}$ in the sense of [12].",
"In particular, $S_{\\kappa _U}^{D,\\le h}$ is an $\\mathcal {O}(Y)\\cong A[X]/(Q(X))$ -module via the map $X\\mapsto U_\\pi ^{-1}$ .",
"Recall that by [12], the collection of slope adapted affinoids forms an admissible cover of $\\mathcal {Z}_D$ .",
"Hence the association $Y\\mapsto S_{\\kappa _U}^{D,\\le h}$ defines a coherent sheaf $\\mathcal {M}_D$ on $\\mathcal {Z}_D$ .",
"Let $\\mathbf {T}$ be the Hecke algebra over $\\mathbb {Q}_p$ generated by the Hecke operators $U_v$ 's and $S_v$ 's for all finite places $v$ of $F$ not dividing $p\\mathfrak {n}\\mathfrak {d}$ , and all the $U_{\\pi _i}$ 's for all $i\\in I$ defined in §REF and $\\psi :\\mathbf {T}\\rightarrow \\operatorname{End}_{\\mathcal {Z}_D}(\\mathcal {M}_D)$ be the natural homomorphism of $\\mathbb {Q}_p$ -algebras.",
"The tuple $\\mathfrak {D}=(\\mathcal {W},\\mathcal {Z}_D,\\mathcal {M}_D,\\mathbf {T},\\psi )$ is an eigenvariety datum in the sense of [12].",
"We use $\\mathcal {X}_D$ to denote the associated eigenvariety together with the finite morphism $\\pi :\\mathcal {X}_D\\rightarrow \\mathcal {Z}_D$ and a morphism $w:\\mathcal {X}_D\\rightarrow \\mathcal {W}$ .",
"It follows from [12] that every point $x$ of $\\mathcal {X}_D$ lying over $z\\in \\mathcal {Z}_D$ corresponds to a generalized eigenspace for the action of $\\mathbf {T}$ on $\\mathcal {M}_D(z)$ .",
"In particular, we use $a_i(x)$ to denote the eigenvalue of the $U_{\\pi _i}$ -operator for all $i\\in I$ .",
"Let $\\mathcal {W}^\\ast $ be the rigid analytic space associated to $\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\rrbracket $ .",
"The homomorphism $\\phi _\\rho :\\mathcal {O}_p^\\times \\rightarrow \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ , $\\alpha \\mapsto (\\alpha ^{-2},\\textrm {Nm}_{F/\\mathbb {Q}}(\\alpha ))$ induces a map of rigid spaces $\\rho :\\mathcal {W}\\rightarrow \\mathcal {W}^\\ast $ .",
"Explicitly $\\rho $ maps a weight $(\\nu ,\\mu )$ of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ to the character $n$ of $\\mathcal {O}_p^\\times $ defined above.",
"Each closed point of $\\mathcal {W}^\\ast $ corresponds to a continuous character $n:\\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ .",
"The coordinates $T^\\ast _i$ for $i\\in I$ of $n$ defined in §REF form a set of parameters on $\\mathcal {W}^\\ast $ .",
"Let $T_i:=\\phi _\\rho (T_i^\\ast )$ and $T:=[1,\\exp (p)]-1\\in \\mathbb {Z}_p\\llbracket \\mathcal {O}_p\\times \\mathbb {Z}_p\\rrbracket $ .",
"The set $\\lbrace (T_i)_{i\\in I}, T \\rbrace $ forms a full set of parameters on the weight space $\\mathcal {W}$ .",
"Let $J$ be a nonempty subset of $I$ and $r_J:=(r_j)_{j\\in J}\\in (0,1)^J$ .",
"We denote by $\\mathcal {W}^{>r_J}$ the subspace of $\\mathcal {W}$ where $|T_j|_p>r_j$ for all $j\\in J$ .",
"Set $\\mathcal {Z}_D^{>r_J}:=\\mathrm {wt}^{-1}(\\mathcal {W}^{>r_J})$ and $\\mathcal {X}_D^{>r_J}:=w^{-1}(\\mathcal {W}^{>r_J})$ .",
"Remark 3.6.4 We can use a similar trick as in [20] to replace the open compact subgroup $K^p$ be a sufficiently small one.",
"For an ideal $\\mathfrak {n}^{\\prime }$ of $\\mathcal {O}_F$ prime to $p\\delta _D$ , let $K^{\\prime p}=K^p\\cap U_0^{(p)}(\\mathfrak {n}^{\\prime })$ or $K^p\\cap U_1^{(p)}(\\mathfrak {n}^{\\prime })$ .",
"Then the spectral variety (resp.",
"eigenvariety) for $K^p$ is a union of connected components of the spectral variety (resp.",
"eigenvariety) for $K^{\\prime p}$ .",
"Hence it suffices to prove the theorem for sufficiently small $K^p$ ."
],
[
"Integral model of the space of $p$ -adic automorphic forms",
"Let $A$ be a topological ring in which $p$ is topologically nilpotent and $\\chi =(\\nu ,\\mu ):\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ be a continuous character.",
"Let $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ be the associated character of $\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times $ .",
"Recall that from $\\kappa $ , we can define a character $\\kappa _T$ (resp.",
"$\\kappa _B$ , $\\kappa _{\\bar{B}}$ ) of $T(\\mathcal {O}_p)$ (resp.",
"$B(\\mathcal {O}_p)$ , $\\bar{B}(\\pi \\mathcal {O}_p)$ ) as in §REF .",
"For any $i\\in I$ , we use $\\kappa _i=(n_i,\\nu _i):\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\times \\mathcal {O}_{\\mathfrak {p}_i}^\\times \\rightarrow A^\\times $ to denote the $i$ -component of $\\kappa $ .",
"We extend the character $\\nu $ to a character $\\nu :F_p^\\times \\rightarrow A^\\times $ as before.",
"Similar with [20], we make the following definition.",
"Definition 3.7.1 We define the space of integral $p$ -adic automorphic forms for $D$ to be $S_{\\kappa ,I}^D(K^p,A) &:= \\lbrace \\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_\\pi }(\\kappa )| \\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_\\pi \\rbrace \\\\&=\\lbrace \\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathcal {O}_p,A)| \\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_\\pi \\rbrace .$ Remark 3.7.2 For any $t\\in \\mathbb {N}^I$ , we define the space of integral $p$ -adic automorphic forms of level $K^p\\mathrm {Iw}_{\\pi ^t}$ by $S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^t},A) := \\lbrace \\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathcal {O}_p,A)| \\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi ^t} \\rbrace .$ This definition gives no generalization of Definition REF .",
"In fact, for $t\\in \\mathbb {N}^I$ and $s\\in \\mathbb {Z}_{\\ge 0}^I$ , we have an Hecke equivariant isomorphism between the spaces $S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^t},A)$ and $S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t+s}},A)$ .",
"This fact is well known to experts.",
"For completeness, we give a sketch of the construction of the isomorphisms and refer [5], especially Proposition $11.1$ for more details.",
"For $\\phi \\in S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^t},A)$ , we define $\\psi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathcal {O}_p,A)$ by $\\psi (x)=\\phi \\left(x\\begin{bmatrix}\\pi ^{-s}&0\\\\0&1\\end{bmatrix}\\right)\\circ \\begin{bmatrix}\\pi ^{s}&0\\\\0&1\\end{bmatrix}.$ One can check $\\psi \\in S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t+s}},A)$ and we obtain a map $\\iota _1:S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t}},A)\\rightarrow S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t+s}},A)$ .",
"Conversely, given $\\psi \\in S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t+s}},A)$ , we define $\\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathcal {O}_p,A)$ by $\\phi (x)(z^{\\prime }):=\\psi \\left(x\\begin{bmatrix}\\pi ^{s}&\\alpha \\\\0&1\\end{bmatrix}\\right)(z), \\text{~with~} z^{\\prime }=\\pi ^sz+\\alpha \\text{~for some ~} z\\textrm {~and~}\\alpha \\in \\mathcal {O}_p.$ One can verify that this definition is independent of the expression of $z^{\\prime }=\\pi ^sz+\\alpha \\in \\mathcal {O}_p$ and $\\phi \\in S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t}},A)$ .",
"So we obtain another map $\\iota _2:S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t+s}},A)\\rightarrow S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^{t}},A)$ .",
"The two maps $\\iota _1$ and $\\iota _2$ establish the desired Hecke equivariant isomorphism.",
"For a better interpretation of Definition REF below, it is useful to explain the relation between the space of integral $p$ -adic automorphic forms for $D$ and the completed homology group of the Shimura varieties associated to the algebraic group $\\operatorname{Res}_{F/\\mathbb {Q}}D^\\times $ .",
"Recall the completed homology group (of degree 0) is defined as $\\tilde{\\mathrm {H}}_0:= \\varprojlim _{K_p}\\mathrm {H}_0(D^\\times \\setminus D_f^\\times /K^pK_p,\\mathbb {Z}_p)=\\varprojlim _{K_p}\\mathbb {Z}_p[D^\\times \\setminus D_f^\\times /K^pK_p],$ where in the inverse limit, $K_p$ ranges over all open compact subgroups of $\\operatorname{GL}_2(\\mathcal {O}_p)$ .",
"For each $K_p$ , the group $\\mathrm {H}_0(D^\\times \\setminus D_f^\\times /K^pK_p,\\mathbb {Z}_p)$ is endowed with the $p$ -adic topology and $\\tilde{\\mathrm {H}}_0$ is endowed with the inverse limit topology.",
"The complete homology group $\\tilde{\\mathrm {H}}_0$ carries a natural right action of $\\operatorname{GL}_2(F_p)$ induced by the right translation of $\\operatorname{GL}_2(F_p)$ on $D_f^\\times $ .",
"It follows from [11] that $\\tilde{\\mathrm {H}}_0$ is a finitely generated $\\mathbb {Z}_p\\llbracket K_p\\rrbracket $ -module for any (nonempty) open compact subgroup $K_p$ of $\\operatorname{GL}_2(F_p)$ .",
"Proposition 3.7.3 Assume that $A$ is linearly topologized, i.e.",
"$0\\in A$ has a fundamental system of neighborhoods consisting of ideals.",
"There is a natural isomorphism of $A$ -modules: $S_{\\kappa ,I}^D(K^p,A)\\rightarrow \\operatorname{Hom}_{cts,\\bar{B}(\\pi \\mathcal {O}_p)}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}})).$ This result is well known to experts.",
"We only give a sketch of the construction of the isomorphism here.",
"By definition, the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi }}(\\kappa )$ is a subspace of $\\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ .",
"The right action of $\\mathrm {Iw}_{\\pi }$ on $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi }}(\\kappa )$ can be extended to $\\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ by the same formula: $f\\circ u(g)=f(gu^\\ast )$ .",
"It is worthwhile to remark here that the action of the monoid $\\mathbf {M}_1$ cannot be extended to $\\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ .",
"Given a map $\\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ , with the property that $\\phi (xu)=\\phi (x)\\circ u$ , for all $x\\in D_f^\\times $ and $u\\in \\mathrm {Iw}_{\\pi }$ .",
"We define a map $\\tilde{f}:D^\\times \\setminus D_f^\\times /K^p\\rightarrow A$ by $\\tilde{f}(x)=\\phi (x)(I_2)$ , where $I_2\\in \\mathrm {Iw}_{\\pi }$ is the identity matrix.",
"Since $\\phi (x)\\in \\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ , for all $x\\in D_f^\\times $ , $\\tilde{f}$ induces a continuous map $f:\\tilde{\\mathrm {H}}_0\\rightarrow A$ .",
"Moreover, when $\\phi \\in S_{\\kappa ,I}^D(K^p)$ , it is straightforward to check that $f\\in \\operatorname{Hom}_{cts,\\bar{B}(\\pi \\mathcal {O}_p)}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}}))$ .",
"Conversely, given a continuous map $f:\\tilde{\\mathrm {H}}_0\\rightarrow A$ , we view $f$ as a map $D^\\times \\setminus D_f^\\times /K^p\\rightarrow A$ , such that $f \\mod {I}$ is locally constant on cosets of $\\operatorname{GL}_2(F_p)$ for all nonempty open ideals $I$ of $A$ as $A$ is linearly topologized.",
"Define $\\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ by $\\phi (x)(g)=f(xg^\\ast )$ , for $x\\in D_f^\\times $ , $g\\in \\mathrm {Iw}_{\\pi }$ .",
"It is easy to check that $\\phi (xu)=\\phi (x)\\circ u$ , for $x\\in D_f^\\times $ and $u\\in \\mathrm {Iw}_{\\pi }$ , and if $f\\in \\operatorname{Hom}_{cts,\\bar{B}(\\pi \\mathcal {O}_p)}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}}))$ , then $\\phi (x)\\in \\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi }}(\\kappa )$ , for all $x\\in D_f^\\times $ .",
"Remark 3.7.4 Conceptually, Proposition REF follows from the tautological isomorphism $S_{\\kappa ,I}^D(K^p,A)=\\operatorname{Hom}_{\\mathrm {Iw}_p}(D^\\times \\setminus D_f^\\times /K^p,\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_\\pi }(\\kappa ))\\cong \\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_\\pi \\rrbracket }(\\tilde{\\mathrm {H}}_0,\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_\\pi }(\\kappa )))$ and Frobenius reciprocity.",
"However, the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_\\pi }(\\kappa )$ is slightly different from the standard definition.",
"This is why we give a concrete proof here.",
"Inspired by Proposition REF , if we replace the group $\\bar{B}(\\pi \\mathcal {O}_p)$ by $\\bar{B}(\\pi _i\\mathcal {O}_{\\mathfrak {p}_i})$ for some $i\\in I$ , we obtain the space $\\operatorname{Hom}_{\\bar{B}(\\pi _i\\mathcal {O}_{\\mathfrak {p}_i})}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}}))$ that contains $S_{\\kappa ,I}^D(K^p,A)$ .",
"Elements in $\\operatorname{Hom}_{\\bar{B}(\\pi _i\\mathcal {O}_{\\mathfrak {p}_i})}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}}))$ looks like automorphic forms at the place $i\\in I$ , and are continuous functions on the complete homology $\\tilde{\\mathrm {H}}_0$ at all other places $i^{\\prime }\\ne i$ .",
"For our argument, we need a generalization of the space $\\operatorname{Hom}_{\\bar{B}(\\pi _i\\mathcal {O}_{\\mathfrak {p}_i})}(\\tilde{\\mathrm {H}}_0, A(\\kappa _{\\bar{B}}))$ .",
"First we introduce some notations.",
"Notation 3.7.5 Let $J$ be a subset of $I$ .",
"We denote by $J^c$ the complement of $J$ in $I$ .",
"Let $\\mathcal {O}_{p,J}:=\\prod \\limits _{j\\in J}\\mathcal {O}_{\\mathfrak {p}_j}$ , $\\pi _J:=\\prod \\limits _{j\\in J}\\pi _j\\in \\mathcal {O}_{p,J}$ and $\\mathrm {Iw}_{\\pi ,J}:=\\prod \\limits _{j\\in J}\\mathrm {Iw}_{\\pi _j}$ which is regarded as a subgroup of $\\mathrm {Iw}_{\\pi }$ whose $j^{\\prime }$ -component is the identity matrix for all $j^{\\prime }\\notin J$ .",
"Similar to (1), we define subgroup $B(\\mathcal {O}_{p,J})$ (resp.",
"$T(\\mathcal {O}_{p,J})$ , $\\bar{B}(\\pi _J\\mathcal {O}_{p,J})$ , $\\bar{N}(\\pi _J\\mathcal {O}_{p,J})$ , $D(\\mathcal {O}_{p,J})$ ) of $B(\\mathcal {O}_p)$ (resp.",
"$T(\\mathcal {O}_p)$ , $\\bar{B}(\\pi \\mathcal {O}_p)$ , $\\bar{N}(\\pi \\mathcal {O}_p)$ , $D(\\mathcal {O}_p)$ ) and the monoid $\\mathbf {M}_{\\pi ,J}\\subset \\mathrm {M}_2(\\mathcal {O}_{p,J})$ .",
"Fix $j\\in I$ .",
"We set $P_j^{\\prime }:=\\left\\lbrace g\\in \\mathrm {Iw}_{\\pi _j,1}| \\det (g)=1 \\right\\rbrace $ and denote by $P_j$ the subgroup of $\\mathrm {Iw}_{\\pi _j}$ generated by $\\begin{bmatrix}1&0\\\\0&\\Delta _j\\end{bmatrix}\\subset T(\\mathcal {O}_{\\mathfrak {p}_j})$ and $P_j^{\\prime }$ .",
"The map $D(\\mathcal {O}_{\\mathfrak {p}_j})\\times P_j\\rightarrow \\mathrm {Iw}_{\\pi _j}$ , $(d,g)\\mapsto dg$ is an isomorphism of groups (here we use the assumption that $p>2$ ).",
"$P_j^{\\prime }$ is the pro-$p$ normal subgroup of $P_j$ and $P_j/P_j^{\\prime }\\cong \\Delta _j$ .",
"Let $P_J:=\\prod \\limits _{j\\in J}P_j$ and $P_J^{\\prime }:=\\prod \\limits _{j\\in J}P_j^{\\prime }$ .",
"We use $\\chi _J:=(\\nu _J,\\mu ):\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ to denote the restriction to the $J$ -component of a continuous homomorphism $\\chi =(\\nu ,\\mu ):\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ .",
"In the rest of this section, we fix a nonempty subset $J$ of $I$ and a continuous character $\\chi _J=(\\nu _J,\\mu ):\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ .",
"We define a character $\\kappa _{B,J}:B(\\mathcal {O}_{p,J})\\times D(\\mathcal {O}_{p,J^c})\\rightarrow A^\\times $ by $\\kappa _{B,J}\\left( \\begin{bmatrix}a_j&b_j\\\\0&d_j\\end{bmatrix}\\right):=\\nu _j(a_j/d_j)\\mu (\\textrm {Nm}_{F/\\mathbb {Q}}(d_j)),\\text{~for~} j\\in J,$ and $\\kappa _{B,J}\\left( \\begin{bmatrix}a_j&0\\\\0&a_j\\end{bmatrix}\\right):=\\mu (\\textrm {Nm}_{F/\\mathbb {Q}}(a_j)),\\text{~for~} j\\notin J.$ We remark that if $\\chi _J$ is the restriction of a continuous character $\\chi :\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ , then $\\kappa _{B,J}$ is the restriction of the character $\\kappa _B:B(\\mathcal {O}_p)\\rightarrow A^\\times $ , where $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ is the character associated to $\\chi $ and $\\kappa _B$ is the character defined in §REF .",
"Definition 3.7.6 Under the above notations, we set $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A):=\\mathrm {Ind}_{B(\\mathcal {O}_{p,J})\\times D(\\mathcal {O}_{p,J^c})}^{\\mathrm {Iw}_{\\pi }}(\\kappa _{B,J})$ .",
"In particular, when $J=I$ and $\\chi =\\chi _J$ is a character of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ , $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ is the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi }}(\\kappa _B)$ defined in §REF .",
"Remark 3.7.7 By definition, $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ is an $A$ -submodule of the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_{p,J})}^{\\mathrm {Iw}_{\\pi }}(\\kappa _J)\\subset \\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ .",
"Recall that we have defined a right action of $\\mathrm {Iw}_{\\pi }$ on $\\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ via the formula $f\\circ u(g)=f(gu^\\ast )$ for $f\\in \\mathcal {C}(\\mathrm {Iw}_{\\pi },A)$ and $u\\in \\mathrm {Iw}_{\\pi }$ .",
"This action leaves the two spaces $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ and $\\mathrm {Ind}_{B(\\mathcal {O}_{p,J})}^{\\mathrm {Iw}_{\\pi }}(\\kappa _J)$ stable.",
"Under the Iwasawa decomposition, $\\mathrm {Ind}_{B(\\mathcal {O}_{p,J})}^{\\mathrm {Iw}_{\\pi }}(\\kappa _J)$ can be identified with $\\mathcal {C}(\\mathcal {O}_{p,J}\\times \\mathrm {Iw}_{\\pi ,J^c},A)$ , and the $\\mathrm {Iw}_{\\pi }$ -action on the latter space is given by the following explicit formula: $h(z_J,g_{J^c})\\circ u=\\big (\\prod _{j\\in J}n_j(c_jz_j+d_j)v_j(\\det (u_j)) \\big ) h(z_J^{\\prime },g_{J^c}u_{J^c}^\\ast ),$ where $z_J=(z_j)_{j\\in J}\\in \\mathcal {O}_{p,J}$ , $g_{J^c}=(g_{j^{\\prime }})_{j^{\\prime }\\in J^c}\\in \\mathrm {Iw}_{\\pi ,J^c}$ and $z_J^{\\prime }=(\\frac{a_jz_j+b_j}{c_jz_j+d_j})_{j\\in J}\\in \\mathcal {O}_{p,J}$ .",
"Using this formula, we can extend this action to the monoid $\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ .",
"This action also leaves $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ stable.",
"Under the Iwasawa decomposition and the bijection $D(\\mathcal {O}_{\\mathfrak {p}_j})\\times P_j\\rightarrow \\mathrm {Iw}_{\\pi _j}$ for all $j\\in J^c$ , we can identify $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ with $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},A)$ .",
"For $\\alpha \\in \\mathcal {O}_p^\\times $ , the diagonal matrix $\\mathrm {Diag}(\\alpha ):= \\begin{bmatrix}\\alpha &0\\\\0&\\alpha \\end{bmatrix} \\in D(\\mathcal {O}_p)\\subset \\mathrm {Iw}_{\\pi }$ with $\\alpha $ on the diagonal acts on $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ via multiplication by $\\mu (\\textrm {Nm}_{F/\\mathbb {Q}}(\\alpha ))$ .",
"This fact will be used in §REF .",
"Definition 3.7.8 Given a nonempty subset $J$ of $I$ and a continuous character $\\chi _J:\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ , we define the space of generalized integral $p$ -adic automorphic forms for $D$ to be $S_{\\kappa ,J}^D(K^p,A):=\\lbrace \\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)| \\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_\\pi \\rbrace .$ For $j\\in J$ , we can use the exact formula (REF ) to define the $U_{\\pi _j}$ -operator on $S_{\\kappa ,J}^D(K^p,A)$ .",
"Remark 3.7.9 The reason we use the notation $S_{\\kappa ,J}^D(K^p,A)$ to denote the space of generalized integral $p$ -adic automorphic forms is that we want to make it compatible with Definition REF , i.e.",
"when $J=I$ , $S_{\\kappa ,I}^D(K^p,A)$ defined above coincides with the space defined in Definition REF .",
"But we point out that the definition of $S_{\\kappa ,J}^D(K^p,A)$ only depends on the character $\\chi _J$ of $\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times $ , even though it may be the restriction of a character $\\chi $ of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ .",
"Remark 3.7.10 Let $J_1\\subset J_2$ be two nonempty subsets of $I$ and $\\chi _{J_2}:\\mathcal {O}_{p,J_2}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ be a continuous character.",
"Let $\\chi _{J_1}$ be the restriction of $\\chi _{J_2}$ to $\\mathcal {O}_{p,J_1}^\\times \\times \\mathbb {Z}_p^\\times $ .",
"For later argument, it is useful to explain the relation between the two spaces $S_{\\kappa ,J_1}^D(K^p,A)$ and $S_{\\kappa ,J_2}^D(K^p,A)$ .",
"Since $B(\\mathcal {O}_{p,J_1})\\times D(\\mathcal {O}_{p,J_1^c})$ is a subgroup of $B(\\mathcal {O}_{p,J_2})\\times D(\\mathcal {O}_{p,J_2^c})$ , we have a natural injection $\\mathcal {C}_{\\chi _{J_2}}(\\mathrm {Iw}_\\pi ,A)\\hookrightarrow \\mathcal {C}_{\\chi _{J_1}}(\\mathrm {Iw}_\\pi ,A)$ , which is equivariant under the action of the monoid $\\mathbf {M}_{\\pi ,J_1}\\times \\mathrm {Iw}_{\\pi ,J_1^c}$ .",
"This induces an injection $S_{\\kappa ,J_2}^D(K^p,A)\\rightarrow S_{\\kappa ,J_1}^D(K^p,A)$ .",
"This map is compatible with the $U_{\\pi _j}$ -operator on these spaces for all $j\\in J_1$ .",
"We can define a right action of $B(\\mathcal {O}_{p,J^c})$ on $ \\mathcal {C}(\\mathcal {O}_{p,J}\\times \\mathrm {Iw}_{\\pi ,J^c}, A)$ by $(h\\cdot b_{J^c})(z_J,g_{J^c}):=h(z_J,b_{J^c}g_{J^c})$ for $b_{J^c}\\in B(\\mathcal {O}_{p,J^c})$ and $h\\in \\mathcal {C}(\\mathcal {O}_{p,J}\\times \\mathrm {Iw}_{\\pi ,J^c}, A)$ .",
"This action leaves $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ stable as $D(\\mathcal {O}_{p,J^c})$ is central in $\\mathrm {Iw}_{\\pi ,J^c}$ and hence induces a right action of $B(\\mathcal {O}_{p,J^c})$ on $S_{\\kappa ,J}^D(K^p,A)$ .",
"Set $J_3:=J_2\\setminus J_1$ .",
"Then the injection $\\mathcal {C}_{\\chi _{J_2}}(\\mathrm {Iw}_\\pi ,A)\\hookrightarrow \\mathcal {C}_{\\chi _{J_1}}(\\mathrm {Iw}_\\pi ,A)$ (resp.",
"$S_{\\kappa ,J_2}^D(K^p,A)\\rightarrow S_{\\kappa ,J_1}^D(K^p,A)$ ) identifies the first space with the subspace of the latter space on which the Borel subgroup $B(\\mathcal {O}_{p,J_3})$ acts via the character $\\kappa _{B,J_2}$ defined above."
],
[
"Spaces of classical automorphic forms",
"There are three goals in this section.",
"Recall the definitions of classical weights and spaces of classical automorphic forms as defined in [5].",
"Prove an Atkin-Lehner duality result (see Proposition REF below).",
"Explain how to realize some spaces of classical automorphic forms as subspaces of the space of generalized integral $p$ -adic automorphic forms.",
"Let $\\nu =(\\nu _i)_{i\\in I}\\in \\mathbb {Z}^I$ and $\\mu \\in \\mathbb {Z}$ , such that $n=(n_i:= \\mu -2\\nu _i)_{i\\in I}\\in \\mathbb {N}^I$ .",
"Define a character $\\chi :\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow \\mathbb {Q}_p^\\times $ , $(\\alpha ,\\beta )\\mapsto \\beta ^r\\prod \\limits _{i\\in I}i_p(\\alpha _i)^{\\nu _i}$ .",
"The associated character $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow \\mathbb {Q}_p^\\times $ is given by $\\kappa (\\alpha ,\\beta )=\\prod \\limits _{i\\in I}i_p(\\alpha _i)^{n_i}i_p(\\beta _i)^{\\nu _i}$ .",
"We call such a weight algebraic.",
"Let $L$ be a finite extension of $\\mathbb {Q}_p$ .",
"A weight $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ is called locally algebraic, or classical, if $\\kappa $ decomposes as $\\kappa =\\kappa _{alg}\\kappa _{fin}$ , where $\\kappa _{alg}$ (resp.",
"$\\kappa _{fin}=\\psi =(\\psi _1,\\psi _2):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ ) is an algebraic weight (resp.",
"a finite character).",
"Hence a locally algebraic weight can be represented by a triple $((n_i)_{i\\in I}\\in \\mathbb {N}^I,(\\nu _i)_{i\\in I}\\in \\mathbb {Z}^I,\\psi =(\\psi _1,\\psi _2))$ with the property that $n+2\\nu \\in \\mathbb {Z}$ and the character $\\psi _1\\cdot \\psi _2^2:\\mathcal {O}_p^\\times \\rightarrow L^\\times $ factors through the norm map $\\textrm {Nm}_{F/\\mathbb {Q}}:\\mathcal {O}_p^\\times \\rightarrow \\mathbb {Z}_p^\\times $ (see Convention REF ).",
"Fix such a locally algebraic weight $\\kappa =(n=(n_i)_{i\\in I},\\nu =(\\nu _i)_{i\\in I},\\psi )$ .",
"Let $r(\\kappa ):=(p^{-m_{\\kappa ,i}})_{i\\in I}$ and $t:=(t_i)_{i\\in I}\\in \\mathbb {N}^I$ such that $\\psi :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ factors through $(\\mathcal {O}_p/\\pi ^t)^\\times \\times (\\mathcal {O}_p/\\pi ^t)^\\times $ .",
"It is straightforward to verify that $t_i\\ge m_{\\kappa ,i}$ for all $i\\in I$ .",
"In particular, $t$ is good for $(\\kappa ,r)$ for all $r\\in \\mathcal {N}^I$ .",
"From the integer $\\nu _i$ and the character $\\psi _{2,i}:\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\rightarrow L^\\times $ , we define another character $\\tau _i:\\mathcal {O}_{\\mathfrak {p}_i}^\\times \\rightarrow L^\\times $ , $\\beta _i\\mapsto i_p(\\beta _i)^{\\nu _i}\\psi _{2,i}(\\beta _i)$ and extend it to a character of $F_{\\mathfrak {p}_i}^\\times $ by setting $\\tau _i(\\pi _i)=1$ .",
"We put $L_{\\kappa }$ to be the $L$ -vector space with basis $\\Sigma _\\kappa :=\\left\\lbrace \\prod \\limits _{i\\in I}Z_i^{l_i}\\;\\Big |\\;0\\le l_i\\le n_i, \\text{~for all~} i\\right\\rbrace $ , where $Z_i$ 's are independent indeterminates.",
"The space $L_{\\kappa }$ carries a right action of $\\mathbf {M}_{\\pi ^t}$ by $\\left(\\prod \\limits _{i\\in I}Z_i^{l_i}\\right)\\circ \\gamma =\\prod _{i\\in I}\\psi _{1,i}(d_i)\\tau _i(\\det (\\gamma _i))(a_iZ_i+b_i)^{l_i}(c_iZ_i+d_i)^{n_i-l_i},$ for $\\gamma =(\\gamma _i)_{i\\in I}\\in M_{\\pi ^t}$ with $\\gamma _i=\\left[ \\begin{array}{cc}a_i&b_i\\\\c_i&d_i\\end{array}\\right]$ for $i\\in I$ .",
"Let $\\mathcal {O}_L$ be the ring of integers of $L$ .",
"We denote by $L_{\\kappa ,\\mathcal {O}_L}$ the $\\mathcal {O}_L$ -lattice of $L_\\kappa $ spanned by the polynomials in $\\Sigma _\\kappa $ .",
"Since the character $\\kappa $ takes values in $\\mathcal {O}_L^\\times $ , the action of the monoid $\\mathbf {M}_{\\pi ^t}$ on $L_\\kappa $ leaves $L_{\\kappa ,\\mathcal {O}_L}$ stable.",
"Fix an open compact subgroup $K^p$ of $D_f^{(p),\\times }$ and let $K=K^p\\mathrm {Iw}_{\\pi ^t}$ be the subgroup of $D_f^\\times $ .",
"Definition 3.8.1 Under the above notations, define $k=n+2\\in \\mathbb {Z}_{\\ge 2}^I$ and $w=n+\\nu +1\\in \\mathbb {Z}^I$ .",
"The space of classical automorphic forms of weight $(k,w)$ , level $K$ and character $\\psi $ for $D$ is defined by $S_{k,w}^D(K,\\psi ):=\\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow L_\\kappa |\\phi (xu)=\\phi (x)\\circ u,\\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi ^t} \\rbrace .$ We also define $S_{k,w}^D(K,\\psi ,\\mathcal {O}_L):=\\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow L_{\\kappa ,\\mathcal {O}_L}|\\phi (xu)=\\phi (x)\\circ u,\\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi ^t} \\rbrace .$ Note that $S_{k,w}^D(K,\\psi ,\\mathcal {O}_L)$ is an $\\mathcal {O}_L$ -lattice of $S_{k,w}^D(K,\\psi )$ and stable under the $U_{\\pi _i}$ -operators for all $i\\in I$ .",
"Remark 3.8.2 Convention REF on the weight $\\kappa =(n,\\nu ):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ makes our definition of Hecke operators on the spaces of classical automorphic forms slightly different from the `usual' one (see [15] for example).",
"To be more precise, we assume that the character $\\psi $ above is trivial for simplicity.",
"We will define a second action of the monoid $\\mathbf {M}_{\\pi ^t}$ on $L_\\kappa $ below.",
"This action coincides with the one that we defined above when restricting to $\\mathrm {Iw}_{\\pi ^t}$ .",
"In particular, it defines the same spaces of classical automorphic forms.",
"However, this two actions differ by a power of $\\pi $ as an action of the monoid $\\mathbf {M}_{\\pi ^t}$ , and hence define different Hecke operators.",
"If we use $U_{\\pi _i,cl}$ to denote the Hecke operator defined by the second action $\\Vert _\\gamma $ , then it is related with our Hecke operator $U_{\\pi _i}$ defined in §REF via the equality $U_{\\pi _i,cl}=\\pi _i^{\\nu _i}U_{\\pi _i}$ .",
"The reason that we renormalize the Hecke operators is that the quantity $\\pi _i^{\\nu _i}$ does not vary analytically with $\\nu $ .",
"We keep the notations as before.",
"We define another right action of the monoid $\\mathbf {M}_{\\pi ^t}$ on $L_{\\kappa } $ by $\\left(\\prod _{i\\in I}Z_i^{l_i}\\right)\\Big \\Vert _\\gamma :=\\prod _{i\\in I}(\\det (\\gamma _i))^{\\nu _i}((a_iZ_i+b_i)^{l_i}(c_iZ_i+d_i)^{n_i-l_i})$ for $\\gamma =(\\gamma _i)_{i\\in I}\\in \\mathbf {M}_{\\pi ^t}$ and $\\gamma _i=\\left[ \\begin{array}{cc}a_i&b_i\\\\c_i&d_i\\end{array}\\right]$ for all $i\\in I$ .",
"Then the space of classical automorphic forms can be described by $ S_{k,w}^D(K,\\psi )=\\Big \\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow L_{\\kappa }\\;\\Big |\\;\\phi (xu)=\\psi _1(d)\\psi _2(\\det (u))\\phi (x)\\Vert _u,\\\\\\text{~for~} x\\in D_f^\\times ,u=\\left[ \\begin{array}{cc}a&b\\\\c&d\\end{array}\\right]\\in \\mathrm {Iw}_{\\pi ^t} \\Big \\rbrace .$ Note that since $\\psi _2$ factors through $(\\mathcal {O}_p/\\pi ^t)^\\times $ , we have $\\psi _2(\\det (u))=\\psi _2(ad)$ .",
"Fix $j\\in I$ .",
"Define another finite character $\\psi ^{\\prime }=(\\psi _1^{\\prime },\\psi _2^{\\prime }):\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ by setting: $\\psi _{1,i}^{\\prime }:={\\left\\lbrace \\begin{array}{ll}\\psi _{1,i}, &\\mbox{if }i\\ne j\\\\\\psi _{1,i}^{-1}, &\\mbox{if }i=j\\end{array}\\right.}",
"\\textrm {~and~}\\psi _{2,i}^{\\prime }:={\\left\\lbrace \\begin{array}{ll}\\psi _{2,i}, &\\mbox{if }i\\ne j\\\\\\psi _{1,i}\\psi _{2,i}, &\\mbox{if }i=j\\end{array}\\right.",
"}.$ In other words, if we regard $\\psi _{1,j}$ as a character of $\\mathcal {O}_p^\\times $ via the natural projection $\\mathcal {O}_p^\\times \\rightarrow \\mathcal {O}_{\\mathfrak {p}_j}^\\times $ , we have $\\psi _1^{\\prime }=\\psi _1\\psi _{1,j}^{-2}$ and $\\psi _2^{\\prime }=\\psi _2\\psi _{1,j}$ .",
"The Atkin-Lehner map is defined by $\\mathrm {AL}_j:S_{k,w}^D(K,\\psi )\\rightarrow S_{k,w}^D(K,\\psi ^{\\prime }), \\phi \\mapsto \\phi (\\bullet v_j^{-1})\\Vert _{v_j},$ where $v_j=\\left[ \\begin{array}{cc}0&1\\\\\\pi _j^{t_j}&0\\end{array}\\right]\\in \\mathbf {M}_{\\pi _j^{t_j}}$ and we view $v_j$ as an element of $\\mathbf {M}_{\\pi ^t}$ whose $i$ -component is the identity, for all $i\\ne j$ .",
"First we verify that the map $\\mathrm {AL}_j$ is well-defined.",
"Given $\\phi \\in S_{k,w}^D(U,\\psi )$ , $x\\in D_f^\\times $ and $u_j=\\left[ \\begin{array}{cc}a_j&b_j\\\\c_j&d_j\\end{array}\\right]\\in \\mathrm {Iw}_{\\pi _j^{t_j}}$ , we set $\\tilde{u}_j:= v_ju_jv_j^{-1}=\\left[ \\begin{array}{cc}d_j&\\pi _j^{-t_j}c_j\\\\\\pi _j^{t_j}b_j&a_j\\end{array}\\right]$ .",
"By a direct computation, we have $\\begin{split}\\mathrm {AL}_j(\\phi )(xu_j)&=\\phi (xu_jv_j^{-1})\\Vert _{v_j}=\\phi (xv_j^{-1}\\tilde{u}_j)\\Vert _{v_j}=(\\psi _{1,j}(a_j)\\psi _{2,j}(\\det (\\tilde{u}_j))\\phi (xv_j^{-1})\\Vert _{\\tilde{u}_j})\\Vert _{v_j}\\\\&=\\psi _{1,j}(a_j)\\psi _{2,j}(\\det (\\tilde{u}_j))\\phi (xv_j^{-1})\\Vert _{\\tilde{u}_jv_j}=\\psi _{1,j}(a_j)\\psi _{2,j}(\\det (\\tilde{u}_j))(\\phi (xv_j^{-1})\\Vert _{v_j})\\Vert _{u_j}\\\\&=\\psi _{1,j}^{-1}(d_j)\\psi _{1,j}(a_jd_j)\\psi _{2,j}(\\det (u_j))\\mathrm {AL}_j(\\phi )(x)\\Vert _{u_j}=\\psi _{1,j}^{\\prime }(d_j)\\psi _{2,j}^{\\prime }(\\det (u_j))\\mathrm {AL}_j(\\phi )(x)\\Vert _{u_j}.\\end{split}$ Thus, we know that $\\mathrm {AL}_j(\\phi )\\in S_{k,w}^D(K,\\psi ^{\\prime })$ and $\\mathrm {AL}_j$ is well-defined.",
"It is clear that $\\mathrm {AL}_j$ is an $L$ -linear isomorphism and induces an $\\mathcal {O}_L$ -linear isomorphism between $ S_{k,w}^D(K,\\psi ,\\mathcal {O}_L)$ and $ S_{k,w}^D(K,\\psi ^{\\prime },\\mathcal {O}_L)$ .",
"Definition 3.8.3 The locally algebraic weight $\\kappa ^{\\prime }$ corresponding to the triple $(n,v,\\psi ^{\\prime })$ is called the $j$ -Atkin-Lehner dual weight of $\\kappa $ .",
"Proposition 3.8.4 Assume that the finite character $\\psi _{1,j}:\\mathcal {O}_{\\mathfrak {p}_j}^\\times \\rightarrow L^\\times $ has conductor $t_j\\ge 2$ .",
"For $\\phi \\in S_{k,w}^D(K,\\psi )$ , we have $U_{\\pi _j}\\circ \\mathrm {AL}_j\\circ U_{\\pi _j}(\\phi )=p\\pi _j^{n_j}\\mathrm {AL}_j\\circ S_{\\pi _j}(\\phi )$ , where $S_{\\pi _j}:S_{k,w}^D(K,\\psi )\\rightarrow S_{k,w}^D(K,\\psi )$ is the automorphism defined by $\\phi \\mapsto \\phi \\left(\\bullet \\left[ \\begin{array}{cc}\\pi _j^{-1}&0\\\\0&\\pi _j^{-1}\\end{array}\\right]\\right)$ .",
"First we make a remark on notations.",
"In this proposition, we have two locally algebraic weights $\\kappa =(n,v,\\psi )$ and $\\kappa ^{\\prime }=(n,v,\\psi ^{\\prime })$ , and hence two different actions of the monoid $\\mathbf {M}_{\\pi ^t}$ on the $L$ -vector space $L_{\\kappa }=L_{\\kappa ^{\\prime }}$ .",
"However, for $v_{j,l}=\\left[ \\begin{array}{cc}\\pi _j&0\\\\l\\pi _j^{t_j}&1\\end{array}\\right]$ , $l=0,\\dots ,p-1$ , the right action of $v_{j,l}$ on $L_{\\kappa }=L_{\\kappa ^{\\prime }}$ is given by $h\\circ v_{j,l}=h\\Vert _{v_{j,l}}$ , no matter which weights $\\kappa $ or $\\kappa ^{\\prime }$ that we consider.",
"Hence for $\\phi \\in S_{k,w}^D(K,\\psi )$ or $S_{k,w}^D(K,\\psi ^{\\prime })$ , we always have $U_{\\pi _j}(\\phi )(x)=\\sum \\limits _{l=0}^{p-1}\\phi (xv_{j,l}^{-1})\\Vert _{v_{j,l}}$ .",
"For $x\\in D_f^\\times $ and $\\phi \\in S_{k,w}^D(K,\\psi )$ , we have $U_{\\pi _j}\\circ \\mathrm {AL}_j\\circ U_{\\pi _j}(\\phi )(x)=\\sum \\limits _{l,m=0}^{p-1}\\phi (xv_{j,m}^{-1}v_j^{-1}v_{j,l}^{-1})\\Vert _{v_{j,l}v_jv_{j,m}}=\\sum \\limits _{l,m=0}^{p-1}\\phi (xv_j^{-1}(v_jv_{j,m}^{-1}v_j^{-1}v_{j,l}^{-1}))\\Vert _{v_{j,l}v_jv_{j,m}}.$ By a direct computation, we have $\\begin{split}v_jv_{j,m}^{-1}v_j^{-1}v_{j,l}^{-1}= \\begin{bmatrix}1&-m\\pi _j^{-1}\\\\0 &\\pi _j^{-1}\\end{bmatrix}\\begin{bmatrix}\\pi _j^{-1}&0\\\\-l\\pi _j^{t_j-1} &1\\end{bmatrix}=\\begin{bmatrix}\\pi _j^{-1}&0\\\\-l\\pi _j^{t_j-2} &\\pi _j^{-1}\\end{bmatrix} \\begin{bmatrix}1+lm\\pi _j^{t_j-1}&-m\\\\ml^2\\pi _j^{2t_j-2} &1-lm\\pi _j^{t_j-1}\\end{bmatrix}.\\end{split}$ From the hypothesis $t_j\\ge 2$ , the matrix $ \\begin{bmatrix}1+lm\\pi _j^{t_j-1}&-m\\\\ml^2\\pi _j^{2t_j-2} &1-lm\\pi _j^{t_j-1}\\end{bmatrix}$ belongs to $\\mathrm {Iw}_{\\pi _j^{t_j}}$ .",
"Hence, we have $\\phi (xv_j^{-1}(v_jv_{j,m}^{-1}v_j^{-1}v_{j,l}^{-1}))=\\psi _{1,j}(1-lm\\pi _j^{t_j-1})\\phi \\left( xv_j^{-1} \\begin{bmatrix}\\pi _j^{-1}&0\\\\-l\\pi _j^{t_j-2} &\\pi _j^{-1}\\end{bmatrix} \\right)\\Bigg \\Vert _{ \\left[{\\begin{matrix}1+lm\\pi _j^{t_j-1}&-m\\\\ml^2\\pi _j^{2t_j-2} &1-lm\\pi _j^{t_j-1}\\end{matrix}}\\right]},$ and consequently $U_{\\pi _j}\\circ \\mathrm {AL}_j\\circ U_{\\pi _j}(\\phi )(x)\\\\=\\sum \\limits _{l,m=0}^{p-1}\\psi _{1,j}(1-lm\\pi _j^{t_j-1})\\phi \\left( xv_j^{-1} \\begin{bmatrix}\\pi _j^{-1}&0\\\\-l\\pi _j^{t_j-2} &\\pi _j^{-1}\\end{bmatrix} \\right)\\Bigg \\Vert _{\\left[{\\begin{matrix}1+lm\\pi _j^{t_j-1}&-m\\\\ml^2\\pi _j^{2t_j-2} &1-lm\\pi _j^{t_ij-1}\\end{matrix}}\\right]v_{j,l}v_jv_{j,m}}.$ Since $\\psi _{1,j}$ is a finite character of conductor $\\pi _j^{t_j}$ , we have $\\sum _{m=0}^{p-1}\\psi _{1,j}(1-lm\\pi _j^{t_j-1})={\\left\\lbrace \\begin{array}{ll}p, & \\text{~for~} l=0, \\\\0, & \\text{otherwise},\\end{array}\\right.",
"}$ and hence $U_{\\pi _j}\\circ \\mathrm {AL}_j\\circ U_{\\pi _j}(\\phi )(x)=p\\phi \\left( xv_j^{-1}\\left[ \\begin{array}{cc}\\pi _j^{-1}&0\\\\0 &\\pi _j^{-1}\\end{array}\\right] \\right)\\Bigg \\Vert _{\\left[ \\begin{array}{cc}\\pi _j&0\\\\0 &\\pi _j\\end{array}\\right]v_j}=p\\pi _j^{n_j}\\mathrm {AL}_j\\circ S_{\\pi _j}(\\phi )(x).$ Proposition 3.8.5 The slopes of the $U_{\\pi _j}$ operator on the two spaces $S_{k,w}^D(K,\\psi )$ and $S_{k,w}^D(K,\\psi ^{\\prime })$ can be paired such that the slopes in each pair sum to $n_i+1=k_i-1$ .",
"We will give the proof of this proposition after introducing the following lemma.",
"Lemma 3.8.6 Let $L$ be a field with a valuation $v_L(\\cdot )$ and $\\mathcal {O}_L$ be its valuation ring.",
"Consider three matrices $A,B\\in \\mathrm {M}_n(L)$ and $U\\in \\operatorname{GL}_n(\\mathcal {O}_L)$ , such that $U$ commutes with $B$ and $AB=\\alpha U$ for some $\\alpha \\in \\mathcal {O}_L$ with $v_L(\\alpha )=k$ .",
"Then one can pair the slopes of the Newton polygons of $A$ and $B$ , such that the slopes in each pair sum to $k$ .",
"Replacing $L$ by its algebraic closure $\\bar{L}$ and extending $v_L(\\cdot )$ to $\\bar{L}$ , we can assume that $L$ is algebraically closed.",
"We regard $A,B,U$ as $L$ -linear operators on the space $V=L^n$ .",
"Then $V$ has the decomposition $V=\\bigoplus \\limits _{\\lambda \\text{~eigenvalue of~}U}V_\\lambda $ , where $V_\\lambda =\\ker ((U-\\lambda I_n)^n)$ .",
"Since $AB=\\alpha U$ and $U$ commutes with $B$ , the three matrices $A,B,U$ all commute with the others.",
"Hence $A,B$ stabilizes the spaces $V_\\lambda $ 's.",
"Hence we may assume that $V=\\ker ((U-\\lambda I_n)^n)$ for some $\\lambda \\in L$ .",
"For the same reason, we can also assume that $V=\\ker ((A-\\lambda _A I_n)^n)$ for some $\\lambda _A\\in L$ .",
"Since $U\\in \\operatorname{GL}_n(\\mathcal {O}_L)$ , we have $\\lambda \\in \\mathcal {O}_L^\\times $ and hence $v_L(\\lambda )=0$ .",
"Let $\\lambda _B=a\\lambda \\lambda _A^{-1}$ .",
"Then $B-\\lambda _BI_n=\\alpha A^{-1}U-\\alpha \\lambda \\lambda _A^{-1}=\\alpha \\lambda _A^{-1}A^{-1}(\\lambda _AU-\\lambda A)=\\alpha \\lambda _A^{-1}A^{-1}(\\lambda _A(U-\\lambda I_n)+\\lambda (\\lambda _AI_n-A))$ .",
"Therefore $B-\\lambda _BI_n$ is a nilpotent operator on $V$ .",
"The eigenvalues of $B$ are $\\lambda _B$ with multiplicity $n$ .",
"Since $\\lambda _A\\lambda _B=\\alpha \\lambda $ , we have $v_L(\\lambda _A)+v_L(\\lambda _B)=v_L(\\alpha )+v_L(\\lambda )=k$ .",
"We choose an $\\mathcal {O}_L$ -basis $\\Omega $ of $S_{k,w}^D(K,\\psi ,\\mathcal {O}_L)$ which can be viewed as an $L$ -basis of $S_{k,w}^D(K,\\psi )$ .",
"We denote by $A\\in \\mathrm {M}_n(L)$ (resp.",
"$B\\in \\mathrm {M}_n(L)$ , resp.",
"$U\\in \\operatorname{GL}_n(\\mathcal {O}_L)$ ) the matrix corresponding to the map $\\mathrm {AL}_j^{-1}\\circ U_{\\pi _j}\\circ \\mathrm {AL}_j: S_{k,w}^D(K,\\psi )\\rightarrow S_{k,w}^D(K,\\psi )$ (resp.",
"$U_{\\pi _j}:S_{k,w}^D(K,\\psi )\\rightarrow S_{k,w}^D(K,\\psi )$ , resp.",
"$S_{\\pi _j}:S_{k,w}^D(K,\\psi )\\rightarrow S_{k,w}^D(K,\\psi )$ ).",
"Notice that the operator $S_{\\pi _j}$ preserves $S_{k,w}^D(K,\\psi ,\\mathcal {O}_L)$ and commutes with $U_{\\pi _j}$ as $\\left[ \\begin{array}{cc}\\pi _j^{-1}&0\\\\0 &\\pi _j^{-1}\\end{array}\\right]$ is central in $\\operatorname{GL}_2(F_{\\mathfrak {p}_j})$ .",
"Now the proposition follows from Proposition REF and Lemma REF .",
"Fix a nonempty subset $J$ of $I$ and a locally algebraic weight $\\kappa $ corresponding to the triple $((n_i)_{i\\in I}\\in \\mathbb {Z}_{\\ge 0}^I,(\\nu _i)_{i\\in I}\\in \\mathbb {Z}^I,\\psi =(\\psi _1,\\psi _2))$ with the following property: $n_j=\\nu _j=0$ and $\\psi _{1,j}|_{1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}}$ , $\\psi _{2,j}|_{1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}}$ are trivial, for all $j\\notin J$ .",
"Equivalently, the character $\\kappa _{J^c}$ equals to $\\psi _{J^c}=(\\psi _{1,J^c},\\psi _{2,J^c}):\\mathcal {O}_{p,J^c}^\\times \\times \\mathcal {O}_{p,J^c}^\\times \\rightarrow L^\\times $ and factors through $(\\mathcal {O}_{p,J^c}/\\pi _{J^c})^\\times \\times (\\mathcal {O}_{p,J^c}/\\pi _{J^c})^\\times $ .",
"The character $\\kappa _{J^c,T}:T(\\mathcal {O}_{p,J^c})\\rightarrow L^\\times $ is trivial on $T(1+\\pi _{J^c}\\mathcal {O}_{p,J^c})$ under the decomposition $T(\\mathcal {O}_{p,J^c})=T(\\mathcal {O}_{p,J^c}/\\pi _{J^c})\\times T(1+\\pi _{J^c}\\mathcal {O}_{p,J^c})$ .",
"Since $\\mathrm {Iw}_{\\pi ,1,J^c}$ is a normal subgroup of $\\mathrm {Iw}_{\\pi ,J^c}$ with quotient $\\mathrm {Iw}_{\\pi ,J^c}/\\mathrm {Iw}_{\\pi ,1,J^c}\\cong T(\\mathcal {O}_{p,J^c}/\\pi _{J^c})$ , the character $\\kappa _{T,J^c}:T(\\mathcal {O}_{p,J^c})\\rightarrow L^\\times $ induces a character $\\alpha _{J^c}:\\mathrm {Iw}_{\\pi ,J^c}\\rightarrow L^\\times $ .",
"As $p$ splits in $F$ , $\\alpha _{J^c}$ takes values in $\\Delta \\subset \\mathbb {Z}_p^\\times $ .",
"Moreover, if the locally algebraic weight $\\kappa $ is the character associated to a point $\\chi \\in \\mathcal {W}(L)$ , then the character $\\alpha _{J^c}$ only depends on $\\omega :=\\chi |_H$ (or equivalently, the component that $\\chi $ belongs to), where $H$ is the torsion subgroup of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ and the set $J$ .",
"We choose $t=(t_i)_{i\\in I}\\in \\mathbb {N}^I$ such that the character $\\psi $ factors through $(\\mathcal {O}_p/\\pi ^t)^\\times \\times (\\mathcal {O}_p/\\pi ^t)^\\times $ .",
"In particular, we take $t_j=1$ for all $j\\notin J$ .",
"Definition 3.8.7 We call $\\alpha _{J^c}$ the character associated to the character $\\omega \\in H^\\vee $ and the set $J$ .",
"For the locally algebraic weight $\\kappa $ we consider above, let $k,w\\in \\mathbb {Z}^I$ be defined as in Definition REF .",
"We can regard the $L$ -vector space $L_{\\kappa }$ as polynomial functions on $\\mathcal {O}_p=\\prod \\limits _{i\\in I}\\mathcal {O}_{\\mathfrak {p}_i}$ and hence obtain an embedding $L_{n,v}\\rightarrow \\mathcal {C}(\\mathcal {O}_p,L)$ , which is equivariant under the action of the monoid $\\mathbf {M}_{\\pi ^t}$ .",
"So we have an embedding $S_{k,w}^D(K,\\psi )\\rightarrow S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^t},L)$ .",
"Composing with the isomorphism $S_{\\kappa ,I}^D(K^p\\mathrm {Iw}_{\\pi ^t},L)\\cong S_{\\kappa ,I}^D(K^p,L)$ established in Remark REF and the embedding $S_{\\kappa ,I}^D(K^p,L)\\hookrightarrow S_{\\kappa ,J}^D(K^p,L)$ , we obtain an embedding $S_{k,w}^D(K,\\psi )\\rightarrow S_{\\kappa ,J}^D(K^p,L)$ .",
"Then we have the following characterization of the image of this embedding.",
"Proposition 3.8.8 Under the above assumptions on the weight $\\kappa $ , the image of the embedding $S_{k,w}^D(K,\\psi )\\rightarrow S_{\\kappa ,J}^D(K^p,L)$ lies in the subspace of $S_{\\kappa ,J}^D(K^p,L)$ on which the group $\\mathrm {Iw}_{\\pi ,J^c}$ acts via the character $\\alpha _{J^c}:\\mathrm {Iw}_{\\pi ,J^c}\\rightarrow \\mathbb {Z}_p^\\times $ associated to $\\omega \\in H^\\vee $ and the set $J$ .",
"Fix a locally algebraic weight $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow L^\\times $ that corresponds to the triple $(n,\\nu ,\\psi )$ and we choose $t=(t_i)_{i\\in I}\\in \\mathbb {N}^I$ such that $\\psi $ factors through $(\\mathcal {O}_p/\\pi ^t)^\\times \\times (\\mathcal {O}_p/\\pi ^t)^\\times $ as before.",
"Let $K=K^p\\mathrm {Iw}_{\\pi ^t}$ be an open compact subgroup of $D_f^\\times $ .",
"Fix $i\\in I$ .",
"Define $n^{\\prime },\\nu ^{\\prime }\\in \\mathbb {Z}^I$ as follows: $n_j^{\\prime }={\\left\\lbrace \\begin{array}{ll}n_j, &\\mbox{if }j\\ne i\\\\n_j, &\\mbox{if }j=i\\end{array}\\right.}",
"\\text{~and~}\\nu _j^{\\prime }={\\left\\lbrace \\begin{array}{ll}\\nu _j, &\\mbox{if }j\\ne i\\\\\\nu _j+n_j-1, &\\mbox{if }j=i\\end{array}\\right.",
"}.$ Let $\\kappa ^{\\prime }$ be the locally algebraic weight that corresponds to the triple $(n^{\\prime },\\nu ^{\\prime },\\psi )$ .",
"Fix $r\\in \\mathcal {N}^I$ and we denote by $z_j$ the coordinate of the $j$ -component of the polydisc $\\mathbf {B}_r$ for all $j\\in I$ .",
"Under the above notations, the differential operator $(\\frac{d}{dz_i})^{n_i+1}:\\mathcal {A}_{\\kappa ,r}\\rightarrow \\mathcal {A}_{\\kappa ^{\\prime },r}$ induces an operator $\\theta _i:S_{\\kappa }^D(K,r)\\rightarrow S_{\\kappa ^{\\prime }}^D(K,r)$ .",
"Moreover, this map is equivariant for the $U_{\\pi _i}$ -operator on the source and the $\\pi _i^{n_i+1}U_{\\pi _i}$ -operator on the target.",
"When $F=\\mathbb {Q}$ , this is proven in [6].",
"The proof of the general case is similar.",
"At the end of this section, we record the following classicality result that characterizes the image of $S_{k,w}^D(K,\\psi )$ in $S_{\\kappa }^D(K,r)$ .",
"Proposition 3.8.9 Fix $\\phi \\in S_{\\kappa }^D(K,r)$ .",
"Then $\\phi \\in S_{k,w}^D(K,\\psi )$ if and only if $\\theta _i(\\phi )=0$ for all $i\\in I$ .",
"If $\\phi $ is an eigenform for the $U_{\\pi _i}$ -operator with nonzero eigenvalue $\\lambda _i$ such that $v_p(\\lambda _i)<n_i+1$ for all $i\\in I$ , then $\\phi \\in S_{k,w}^D(K,\\psi )$ .",
"Part $(2)$ follows from [2].",
"Part $(1)$ follows from the proof given there."
],
[
"Explicit expression of $p$ -adic automorphic forms",
"Let $A$ be a topological ring and $\\kappa :\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow A^\\times $ as before.",
"Fix a double coset decomposition $D_f^\\times =\\bigsqcup \\limits _{k=0}^{s-1}D^\\times \\gamma _kK^p\\mathrm {Iw}_{\\pi }$ with $\\gamma _k\\in D_f^\\times $ .",
"We have an isomorphism of $A$ -modules: $S_{\\kappa ,I}^D(K^p)\\xrightarrow{}\\bigoplus _{k=0}^{s-1}\\mathcal {C}(\\mathcal {O}_p,A)^{\\Gamma _k}, \\phi \\mapsto (\\phi (\\gamma _k))_{k=0,\\dots ,s-1},$ where $\\Gamma _k=\\gamma _k^{-1}D^\\times \\gamma _k\\cap K^p\\mathrm {Iw}_{\\pi }$ and $\\Gamma _k$ acts on $\\mathcal {C}(\\mathcal {O}_p,A)$ via its $\\operatorname{GL}_2(F_p)$ -component.",
"We embed $\\mathcal {O}_F^\\times $ diagonally to $D_f^\\times $ and hence view $\\mathcal {O}_F^\\times $ as a subgroup of $D_f^\\times $ .",
"From [15] and the assumption that $D/F$ is totally definite, we can choose $K^p$ small enough such that $\\Gamma _k\\subset \\mathcal {O}_F^{\\times ,+}$ , for all $k=0,\\dots ,s-1$ .",
"The subgroup $K^p$ with this property is called neat.",
"For a neat $K^p$ , we have the isomorphism $S_{\\kappa ,I}^D(K^p)\\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}(\\mathcal {O}_p,A)$ .",
"Similarly we have explicit descriptions of the spaces of $p$ -adic overconvergent automorphic forms by evaluating the functions at $\\gamma _k$ 's, for $k=0,\\dots ,s-1$ : $S_\\kappa ^D(K^p\\mathrm {Iw}_{\\pi },r)\\xrightarrow{} \\bigoplus \\limits _{k=0}^{s-1}\\mathcal {A}_{\\kappa ,r}$ , where $A$ a $\\mathbb {Q}_p$ -affinoid algebra and $(1,\\dots , 1)$ good for $(\\kappa ,r)$ .",
"Convention 3.9.1 In the rest of this paper, we always assume that $K^p$ is neat.",
"In general, we fix a nonempty subset $J$ of $I$ and a continuous character $\\chi _J=(\\nu _J,\\mu ):\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ .",
"As observed in Remark REF , for any $\\alpha \\in \\mathcal {O}_F^\\times $ , the matrix $\\mathrm {Diag}(\\alpha )\\in T(\\mathcal {O}_p)$ acts on $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ via multiplication by $\\mu (\\textrm {Nm}_{F/\\mathbb {Q}}(\\alpha ))$ .",
"So we have an isomorphism of $A$ -modules: $S_{\\kappa ,J}^D(K^p,A)\\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ ."
],
[
"Notations",
" We label the elements in $I=\\operatorname{Hom}(F,\\bar{\\mathbb {Q}})$ by $I=\\lbrace i_1,\\dots , i_g\\rbrace $ .",
"For $1\\le l\\le g-1$ , let $J_l=\\lbrace i_1,\\dots , i_l \\rbrace \\subset I$ .",
"Let $H$ be the torsion subgroup of $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ .",
"Hence $\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\cong H\\times ((1+\\pi \\mathcal {O}_p)\\times (1+p\\mathbb {Z}_p))$ and $\\mathcal {W}\\cong \\prod \\limits _{\\omega \\in H^\\vee }\\mathcal {W}_\\omega $ , where $H^\\vee $ is the character group of $H$ , and $\\mathcal {W}_\\omega $ is isomorphic to the $(g+1)$ -dimensional open unit polydisc.",
"The homomorphism $\\phi _\\rho :\\mathcal {O}_p^\\times \\rightarrow \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times $ induces a continuous homomorphism $\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\rrbracket \\rightarrow \\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ , which is still denoted by $\\phi _\\rho $ .",
"For each $i\\in I$ , define $T_i:=\\phi _\\rho ([\\exp (\\pi _i)]-1)\\in \\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ and $T:=[1,\\exp (p)]-1$ .",
"Then $\\lbrace (T_i)_{i\\in I},T \\rbrace $ forms a full set of parameters of the weight space $\\mathcal {W}$ .",
"We denote by $\\Lambda $ the complete group ring $\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ , and put $\\mathfrak {m}_\\Lambda :=(p,(T_i)_{i\\in I})\\subset \\Lambda $ .",
"For every character $\\omega \\in H^\\vee $ , let $\\Lambda _\\omega :=\\Lambda \\otimes _{\\mathbb {Z}_p[H],\\omega }\\mathbb {Z}_p$ .",
"Under the above notations, we have $\\Lambda _\\omega =\\mathbb {Z}_p\\llbracket (T_i)_{i\\in I},T \\rrbracket $ .",
"For a nonempty subset $J$ of $I$ , we set $\\Lambda _J:=\\mathbb {Z}_p\\llbracket \\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ .",
"We use $H_J\\subset H$ to denote the torsion subgroup of $\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times $ .",
"Under the isomorphism $\\Lambda _J\\cong \\mathbb {Z}_p[H_J]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket (T_j)_{j\\in J}, T \\rrbracket $ , we set $\\Lambda _J^{>1/p}:=\\mathbb {Z}_p[H_J]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket (T_j,\\frac{p}{T_j})_{j\\in J},T \\rrbracket $ and $\\mathfrak {m}_{\\Lambda _J^{>1/p}}$ be the ideal of $\\Lambda _J^{>1/p}$ generated by the elements $(T_j)_{j\\in J}$ .",
"Note that since $p=T_j\\cdot \\frac{p}{T_j}$ in $\\Lambda _J^{>1/p}$ , we have $p\\in \\mathfrak {m}_{\\Lambda _J^{>1/p}}$ .",
"When $J=I$ , we write $\\Lambda ^{>1/p}$ (resp.",
"$\\mathfrak {m}_{\\Lambda ^{>1/p}}$ ) for $\\Lambda _J^{>1/p}$ (resp.",
"$\\mathfrak {m}_{\\Lambda _J^{>1/p}}$ ) for simplicity.",
"In particular, we have $\\mathfrak {m}_{\\Lambda ^{>1/p}}=\\mathfrak {m}_\\Lambda \\cdot \\Lambda ^{>1/p}$ , and $\\mathfrak {m}_{\\Lambda ^{>1/p}}$ is generated by $(T_i)_{i\\in I}$ in $\\Lambda ^{>1/p}$ .",
"We also have $p\\in \\mathfrak {m}_{\\Lambda ^{>1/p}}$ ."
],
[
"Explicit expression of $U_{\\pi _j}$ -operators on the space of {{formula:b252d76e-0945-45e7-b74c-f17fe2b4cad5}} -adic automorphic forms",
"First we give an explicit expression of the $U_{\\pi _j}$ -operator on the space of (generalized) integral $p$ -adic automorphic forms.",
"This is a generalization of [20].",
"Proposition 4.2.1 Let $J$ be a nonempty subset of $I$ and $\\chi _J:\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ be a continuous character.",
"Fix $j\\in J$ .",
"Under the isomorphism $S_{\\kappa ,J}^D(K^p,A)\\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ , the $U_{\\pi _j}$ -operator on this space can be described by the following commutative diagram: $@=1.3cm{S_{\\kappa ,J}^D(K^p,A) [r]^{\\cong } [d]^{U_{\\pi _j}} & \\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)[d]^{\\mathcal {U}_j} \\\\S_{\\kappa ,J}^D(K^p,A) [r]^{\\cong } & \\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A).",
"}$ Here the right vertical map $\\mathcal {U}_j$ in the above diagram is given by an $s\\times s$ matrix with the following descriptions: Each entry of $\\mathcal {U}_j$ is a sum of operators of the form $\\circ \\delta _p$ , where $\\delta _p=(\\delta _i)_{i\\in I}\\in \\mathrm {M}_2(\\mathcal {O}_p)$ have the property that $\\delta _i\\in \\mathrm {Iw}_{\\pi _i}$ for all $i\\ne j$ , and $\\delta _j$ belongs to $\\left( \\begin{array}{cc}\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}&\\mathcal {O}_{\\mathfrak {p}_j}\\\\\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}&\\mathcal {O}_{\\mathfrak {p}_j}^\\times \\end{array}\\right)\\subset M_{\\pi ,j}$ , where $\\circ \\delta _p$ is the right action of the monoid $\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ on the spaces $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ defined in §REF .",
"There are exactly $p$ such operators appearing in each row and column of $\\mathcal {U}_j$ .",
"The proof is almost identical with that of [20] and we only give a sketch here.",
"For every $l=0,\\dots ,p-1$ and $k=0,\\dots ,s-1$ , we can write $\\gamma _kv_{j,l}^{-1}$ uniquely as $\\delta _{l,k}\\gamma _{\\alpha _{l,k}}u_{l,k}$ ， for $\\delta _{l,k}\\in D^\\times $ , $\\alpha _{l,k}\\in \\lbrace 0,\\dots ,s-1 \\rbrace $ and $u_{l,k}\\in K^p\\mathrm {Iw}_\\pi $ .",
"Then $\\begin{split}U_{\\pi _j}(\\phi )(\\gamma _k)&=\\sum _{l=0}^{p-1}\\phi (\\gamma _{\\alpha _{l,k}}u_{l,k})\\circ v_{j,l}=\\sum _{l=0}^{p-1}(\\phi (\\gamma _{\\alpha _{l,k}})\\circ u_{l,k,p})\\circ v_{j,l}\\\\&=\\sum _{l=0}^{p-1}\\phi (\\gamma _{\\alpha _{l,k}})\\circ (u_{l,k,p}v_{j,l}),\\end{split}$ where $u_{l,k,p}\\in \\mathrm {Iw}_{\\pi }$ is the $\\operatorname{GL}_2(F_p)$ -component of $u_{l,k}$ .",
"Let $\\delta _{l,k,p}=u_{l,k,p}v_{j,l}$ for all $l$ 's and $k$ 's.",
"It is straightforward to check that $\\delta _{l,k,p}$ 's satisfy the stated property."
],
[
"p-adic analysis",
"It follows from Proposition REF that to study the $U_{\\pi _j}$ -operators for $j\\in I$ , it is crucial to understand the action of the monoid $\\mathbf {M}_\\pi \\subset \\operatorname{GL}_2(F_p)$ on the space $\\mathcal {C}(\\mathcal {O}_p,A)$ .",
"When $F=\\mathbb {Q}$ (and hence $\\mathcal {O}_p=\\mathbb {Z}_p$ ) and $\\kappa :\\mathbb {Z}_p^\\times \\rightarrow \\Lambda ^\\times =(\\mathbb {Z}_p\\llbracket \\mathbb {Z}_p^\\times \\rrbracket )^\\times $ is the universal character, this question has been studied carefully by Liu-Wan-Xiao in [20].",
"We will recall their results in this section and temporarily adopt their notations.",
"For $F=\\mathbb {Q}$ , the monoid $\\mathbf {M}_\\pi $ becomes $\\left\\lbrace \\left[ \\begin{array}{cc}a& b\\\\c&d\\end{array}\\right]\\in \\mathrm {M}_2(\\mathbb {Z}_p)\\;\\Big |\\; p|c, p\\nmid d \\text{~and~} ad-bc\\ne 0 \\right\\rbrace $ .",
"The Iwasawa algebra $\\Lambda $ decomposes as $\\mathbb {Z}_p\\llbracket \\mathbb {Z}_p^\\times \\rrbracket \\cong \\mathbb {Z}_p[\\Delta ]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket T \\rrbracket $ , where $\\Delta \\subset \\mathbb {Z}_p^\\times $ is the torsion subgroup, $T=[\\exp (p)]-1$ .",
"Denote $\\mathfrak {m}_{\\Lambda }:=(p,T)\\subset \\Lambda $ .",
"The space $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda )$ admits an orthonormal basis $\\lbrace e_0=1,e_1=z,e_2=\\binom{z}{2},\\dots \\rbrace $ , which is called the Mahler basis (we refer [20] for more details).",
"It also carries a right action of the monoid $\\mathbf {M}_{\\pi }$ defined by $h\\circ \\delta (z):=\\chi (cz+d)h(\\frac{az+b}{cz+d}) \\text{~for~} h\\in \\mathcal {C}(\\mathbb {Z}_p,\\Lambda ) \\text{~and~} \\delta \\in \\mathbf {M}_{\\pi }.$ For $\\delta _p= \\left[\\begin{array}{cc}a& b\\\\c&d\\end{array}\\right]\\in \\mathbf {M}_\\pi $ , we use $P(\\delta _p)=(P_{m,n}(\\delta _p))_{m,n\\ge 0}$ to denote the infinite matrix for the action of $\\delta _p$ on $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda )$ with respect to the Mahler basis, i.e.",
"$P_{m,n}(\\delta _p)$ is the coefficient of $\\binom{z}{m}$ of the function $\\binom{z}{n}\\circ \\delta _p$ .",
"Then we have the following estimation.",
"Proposition 4.3.1 ([20], Proposition 3.24) When $\\delta _p= \\left[ \\begin{array}{cc}a& b\\\\c&d\\end{array}\\right]\\in \\left[ \\begin{array}{cc}p\\mathbb {Z}_p& \\mathbb {Z}_p\\\\p\\mathbb {Z}_p& \\mathbb {Z}_p^\\times \\end{array}\\right]$ , the coefficient $P_{m,n}(\\delta _p)$ belongs to $\\mathfrak {m}_{\\Lambda }^{\\max \\lbrace m-\\lfloor \\frac{n}{p} \\rfloor ,0 \\rbrace }$ .",
"When $\\delta _p= \\left[ \\begin{array}{cc}a& b\\\\c&d\\end{array}\\right]\\in \\mathbf {M}_\\pi $ , the coefficient $P_{m,n}(\\delta _p)$ belongs to $\\mathfrak {m}_{\\Lambda }^{\\max \\lbrace m-n,0 \\rbrace }$ ."
],
[
"Orthonormalizable spaces and compact operators",
"In this section we recall the notions of orthonormal basis and compact operator defined in [20] and apply the theory to the spaces of (generalized) integral $p$ -adic automorphic forms.",
"Definition 4.4.1 Let $R$ be a complete noetherian ring with ideal of definition $\\mathfrak {m}_R$ .",
"A topological $R$ -module $M$ is called orthonormalizable if it is isomorphic to the topological $R$ -module: $\\hat{\\oplus }_{i\\in \\mathbb {Z}_{\\ge 0}} Re_i:= \\varprojlim _{n}(\\oplus _{i\\in \\mathbb {Z}_{\\ge 0}}(R/\\mathfrak {m}_R^n)e_i).$ More precisely, $R$ is orthonormalizable if there exists $\\lbrace e_i|i\\in \\mathbb {Z}_{\\ge 0} \\rbrace \\subset M$ , such that every element $m$ in $M$ can be written uniquely as $m=\\sum \\limits _{i\\in \\mathbb {Z}_{\\ge 0}}a_ie_i$ , with $a_i\\in R$ and $\\lim \\limits _{i\\rightarrow \\infty }a_i=0$ in $R$ .",
"The set $\\lbrace e_i|i\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ is called an orthonormal basis of $M$ .",
"Let $M$ be an orthonormalizable $R$ -module.",
"Let $U:M\\rightarrow M$ be a continuous $R$ -linear operator on $M$ .",
"$U$ is called compact if the induced operator on $M/\\mathfrak {m}_R^nM$ has finitely generated image for all $n\\in \\mathbb {Z}_{\\ge 0}$ .",
"Keep the notations as in Definition REF .",
"We fix an orthonormal basis $\\lbrace e_m|m\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $M$ , and let $P\\in \\mathrm {M}_\\infty (R)$ be the matrix associated to $U$ under this basis.",
"We define the characteristic power series of the $U$ -operator by $\\mathrm {char}(U,M):= \\det (\\mathrm {I}_\\infty -XP)=\\lim \\limits _{n\\rightarrow \\infty }\\det (\\mathrm {I}_\\infty -X(P\\mathfrak {}\\mod {\\mathfrak {m}}_R^n))\\in R\\llbracket X \\rrbracket $ .",
"The formal power series $\\mathrm {char}(U,M)$ is well defined and it does not depend on the choice of the orthonormal basis.",
"We will refer to [20] for more details.",
"We apply the above notions to the spaces of integral $p$ -adic automorphic forms and their Hecke operators.",
"Let $\\chi ^{\\prime }:\\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow (\\Lambda ^{>1/p})^\\times $ be the universal character and $\\kappa ^{\\prime }:\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow (\\Lambda ^{>1/p})^\\times $ be the associated character.",
"Recall that we can identify the induced representation $\\mathrm {Ind}_{B(\\mathcal {O}_p)}^{\\mathrm {Iw}_{\\pi }}(\\kappa ^{\\prime }_B)$ with the space $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})$ .",
"The latter space admits an orthonormal basis (as a topological $\\Lambda ^{>1/p}$ -module) $\\lbrace e_m=\\prod \\limits _{i\\in I}\\binom{z_i}{m_i}| m=(m_i)_{i\\in I}\\in \\mathbb {Z}_{\\ge 0}^I \\rbrace $ , where $z_i$ is the coordinate of the $i$ th component of $\\mathcal {O}_p=\\prod \\limits _{i\\in I}\\mathcal {O}_{\\mathfrak {p}_i}$ .",
"Similar to [20], we consider a closed subspace $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}=\\hat{\\oplus }_{m\\in \\mathbb {Z}_{\\ge 0}^I}\\Lambda ^{>1/p}e_m^{\\prime },$ where $e_m^{\\prime }=\\prod \\limits _{i\\in I}T_i^{m_i}\\binom{z_i}{m_i}$ .",
"We claim that the space $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ is stable under the action of the monoid $\\mathbf {M}_\\pi $ .",
"In fact, for any $i\\in I$ and $\\delta _i\\in \\mathbf {M}_{\\pi _i}$ , by Proposition REF $(2)$ and $p=T_i\\cdot \\frac{p}{T_i}$ in $\\Lambda ^{>1/p}$ , we have $\\binom{z_i}{m_i}\\circ \\delta _i=\\sum \\limits _{n_i\\ge 0}a_{n_i,m_i}\\binom{z_i}{n_i}$ with $a_{n_i,m_i}\\in T_i^{\\max \\lbrace n_i-m_i,0\\rbrace }\\Lambda ^{>1/p}$ , and hence $(T_i^{m_i}\\binom{z_i}{m_i})\\circ \\delta _i=\\sum \\limits _{n_i\\ge 0}b_{n_i,m_i}T_i^{n_i}\\binom{z_i}{n_i}$ with $b_{n_i,m_i}=T_i^{m_i-n_i}a_{n_i,m_i}\\in \\Lambda ^{>1/p}$ .",
"This implies that $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ is stable under the action of $\\mathbf {M}_{\\pi _i}$ .",
"Combined with $\\mathbf {M}_\\pi =\\prod \\limits _{i\\in I}\\mathbf {M}_{\\pi _i}$ , this proves our claim.",
"The space of integral 1-convergent automorphic forms are defined by $S_{\\kappa ,I}^{D,I}(K^p,\\Lambda ^{>1/p}):= \\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}|\\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi } \\rbrace $ We have an explicit expression of $S_{\\kappa ,I}^{D,I}(K^p)$ as before: $S_{\\kappa ,I}^{D,I}(K^p)\\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ .",
"Hence the topological $\\Lambda ^{>1/p}$ -module $S_{\\kappa ,I}^{D,I}(K^p)$ has an orthonormal basis $\\lbrace e_{k,m}^{\\prime }|k=0,\\dots ,s-1,m\\in \\mathbb {Z}_{\\ge 0}^I \\rbrace $ , such that $\\lbrace e_{k,m}^{\\prime }| m\\in \\mathbb {Z}_{\\ge 0}^I \\rbrace $ is the orthonormal basis of the $k$ -th direct summand in $\\bigoplus \\limits _{k=0}^{s-1}\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ defined above, for every $k=0,\\dots ,s-1$ .",
"I claim that the $U_{\\pi }$ -operator on the space $S_{\\kappa ,I}^{D,I}(K^p)$ is compact.",
"In fact, by Proposition REF $(1)$ , for $\\delta _i\\in \\left[ \\begin{array}{cc}\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}& \\mathcal {O}_{\\mathfrak {p}_i}\\\\\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}& \\mathcal {O}_{\\mathfrak {p}_i}^\\times \\end{array}\\right]$ and $m_i\\in \\mathbb {Z}_{\\ge 0}$ , we have $\\binom{z_i}{m_i}\\circ \\delta _i=\\sum \\limits _{n_i\\ge 0}a_{n_i,m_i}\\binom{z_i}{n_i}$ with $a_{n_i,m_i}\\in T_i^{\\max \\lbrace n_i-\\lfloor \\frac{m_i}{p} \\rfloor ,0\\rbrace }\\Lambda ^{>1/p}$ .",
"Therefore $(T_i^{m_i}\\binom{z_i}{m_i})\\circ \\delta _i=\\sum \\limits _{n_i\\ge 0}b_{n_i,m_i}T_i^{n_i}\\binom{z_i}{n_i}$ , with $b_{n_i,m_i}=T_i^{m_i-n_i}a_{n_i,m_i}\\in T_i^{m_i-\\lfloor \\frac{m_i}{p} \\rfloor }\\Lambda ^{>1/p}\\subset \\mathfrak {m}_{\\Lambda ^{>1/p}}^{m_i-\\lfloor \\frac{m_i}{p} \\rfloor }$ .",
"Hence for $\\delta _p=(\\delta _i)\\in \\left[ \\begin{array}{cc}\\pi \\mathcal {O}_p& \\mathcal {O}_p\\\\\\pi \\mathcal {O}_p& \\mathcal {O}_p^\\times \\end{array}\\right]$ and $m=(m_i)\\in \\mathbb {Z}_{\\ge 0}^I$ , we have $e_m^{\\prime }\\circ \\delta _p=\\left(\\prod _{i\\in I}T_i^{m_i}\\binom{z_i}{m_i}\\right)\\circ \\delta _p=\\prod _{i\\in I}\\left(T_i^{m_i}\\binom{z_i}{m_i}\\right)\\circ \\delta _i=\\sum _{n\\in \\mathbb {Z}_{\\ge 0}^I}b_{n,m}e_n^{\\prime },$ with $b_{n,m}\\in \\mathfrak {m}_{\\Lambda ^{>1/p}}^{\\lambda _m}$ , where $\\lambda _m=\\sum \\limits _{i\\in I}(m_i-\\lfloor \\frac{m_i}{p} \\rfloor )$ .",
"It follows from the above estimation and the explicit expression of the $U_{\\pi }$ -operator on $S_{\\kappa ,I}^{D,I}(K^p)$ that $U_\\pi $ is compact.",
"Remark 4.4.2 Fix a point $x\\in \\mathcal {W}^{>1/p}(\\mathbb {C}_p)$ which corresponds to a continuous homomorphism $\\chi :\\Lambda ^{>1/p}\\rightarrow \\mathbb {C}_p$ .",
"Let $V:= \\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}\\hat{\\otimes }_{\\Lambda ^{>1/p},\\chi }\\mathbb {C}_p$ be the specialization of $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ at $x$ .",
"By [10], there exist $r<r^{\\prime }$ in $\\mathcal {N}^I$ (with the obvious partial order) that depend on the valuations $v_p(\\chi (T_i))$ 's for all $i\\in I$ such that $\\mathcal {A}_{\\kappa ,r}\\subset V\\subset \\mathcal {A}_{\\kappa ,r^{\\prime }}$ , and hence $S_{\\kappa }^D(K^p\\mathrm {Iw}_{\\pi },r)\\subset S_{\\kappa ,I}^{D,I}(K^p,\\Lambda ^{>1/p})\\hat{\\otimes }_{\\Lambda ^{>1/p},\\chi }\\mathbb {C}_p\\subset S_{\\kappa }^D(K^p\\mathrm {Iw}_{\\pi },r^{\\prime })$ .",
"Therefore the space $S_{\\kappa ,I}^{D,I}(K^p,\\Lambda ^{>1/p})\\hat{\\otimes }_{\\Lambda ^{>1/p},\\chi }\\mathbb {C}_p$ contains all the finite $U_{\\pi }$ -slope systems of Hecke eigenvalues."
],
[
"Continuous functions and distribution algebras",
"Let $G$ be a profinite group endowed with the profinite topology.",
"Let $A$ be a $\\mathbb {Z}_p$ -algebra and $I$ be an ideal of $A$ .",
"We assume that $A$ is endowed with the $I$ -adic topology and is complete under this topology.",
"The typical example we are interested is that $A=\\Lambda _J^{>1/p}$ equipped with the $\\mathfrak {m}_{\\Lambda _J^{>1/p}}$ -adic topology for a nonempty subset $J$ of $I$ .",
"On the set $\\mathcal {C}(G,A)$ , we give it the uniform topology, i.e.",
"for any $f\\in \\mathcal {C}(G,A)$ , it has a basis $\\lbrace U_n \\rbrace $ of open neighborhoods as $U_n=\\lbrace g\\in \\mathcal {C}(G,A)\\;|\\;f(x)-g(x)\\in I^n\\text{~for all~} x\\in G \\rbrace $ .",
"Let $A\\llbracket G \\rrbracket :=\\varprojlim \\limits _{U\\subset G}A[G/U]$ be the complete group ring of $G$ over $A$ , where $U$ ranges over all the open normal subgroups of $G$ .",
"For each $U$ , the free $A$ -module $A[G/U]$ is endowed with the product topology and $A\\llbracket G \\rrbracket $ is endowed with the inverse limit topology.",
"We use $\\mathcal {D}(G,A)$ to denote the set $\\operatorname{Hom}_{A}(\\mathcal {C}(G,A),A)$ of continuous $A$ -linear maps from $\\mathcal {C}(G,A)$ to $A$ (here continuity follows from $A$ -linearity).",
"Then we have the following.",
"Proposition 4.5.1 There is a natural isomorphism of $A$ -modules $A\\llbracket G \\rrbracket \\cong \\mathcal {D}(G,A)$ .",
"Under this isomorphism the topology on $A\\llbracket G \\rrbracket $ corresponds to the weak topology on $\\mathcal {D}(G,A)$ .",
"The proof is almost identical to the proof of [18], which handles the case when $A$ is the unit ball of a Banach-Tate $\\mathbb {Z}_p$ -algebra.",
"For completeness, we give a sketch of proof here.",
"For any nonempty open normal subgroup $U$ of $G$ , we have an isomorphism of $A$ -modules $A[G/U]\\xrightarrow{}\\operatorname{Hom}_A(\\mathcal {C}(G/U,A),A)$ , which is also a homeomorphism.",
"The projection $G\\rightarrow G/U$ induces a natural $A$ -module homomorphism $\\mathcal {D}(G,A)\\rightarrow \\mathcal {D}(G/U,A)$ and hence a map $\\mathcal {D}(G,A)\\rightarrow \\varprojlim \\limits _U \\mathcal {D}(G/U,A)$ .",
"If we use $\\mathcal {C}_{sm}(G,A)$ to denote the $A$ -submodule of $\\mathcal {C}(G,A)$ consisting of locally constant functions, the map $\\mathcal {D}(G,A)\\rightarrow \\varprojlim \\limits _U \\mathcal {D}(G/U,A)\\cong A\\llbracket G\\rrbracket $ is the natural homomorphism $\\operatorname{Hom}_A(\\mathcal {C}(G,A),A)\\rightarrow \\operatorname{Hom}_A(\\mathcal {C}_{sm}(G,A),A)$ induced by the inclusion map $\\mathcal {C}_{sm}(G,A)\\rightarrow \\mathcal {C}(G,A)$ .",
"Since $\\mathcal {C}_{sm}(G,A)$ is dense in $\\mathcal {C}(G,A)$ , it is clear that the map $\\operatorname{Hom}_A(\\mathcal {C}(G,A),A)\\rightarrow \\operatorname{Hom}_A(\\mathcal {C}_{sm}(G,A),A)$ is an isomorphism of $A$ -modules as well as a homeomorphism.",
"Remark 4.5.2 We can also define a convolution product on $\\mathcal {D}(G,A)$ so that the isomorphism in Proposition REF is an isomorphism between $A$ -algebras.",
"But we do not need this fact and refer to [18] to details (in a slightly different setting).",
"For latter discussion, we recall a similar result as that of Proposition REF discussed in [18].",
"Let $A$ be a Banach-Tate $\\mathbb {Z}_p$ -algebra with unit ball $A_0$ and a multiplicative pseudo-uniformizer $\\varpi $ in the sense of [18].",
"For a profinite group $G$ , we define $A_0\\llbracket G\\rrbracket =\\varprojlim _{U\\subset G}A_0[G/U]$ and $A\\llbracket G\\rrbracket =A_0\\llbracket G\\rrbracket [\\frac{1}{\\varpi }]$ .",
"It follows from [18] that there is a natural $A$ -Banach algebra isomorphism $A\\llbracket G\\rrbracket \\xrightarrow{}\\mathcal {D}(G,A)$ .",
"It restricts to an $A_0$ -algebra isomorphism $A_0\\llbracket G\\rrbracket \\xrightarrow{}\\mathcal {D}(G,A_0)$ that identifies the inverse star topology on the source with the weak star topology on the target.",
"In particular, when $L$ is a closed subfield of $\\mathbb {C}_p$ , the $L$ -Banach space $L\\llbracket G\\rrbracket $ is defined and isomorphic to $\\mathcal {D}(G,L)$ .",
"Let $H $ be a group or more generally a monoid.",
"Suppose that we have a right action of $H$ on $\\mathcal {C}(G,A)$ .",
"The above isomorphism induces a left action of $H$ on $A\\llbracket G \\rrbracket $ by requiring that $h\\circ \\mu (f)=\\mu (f\\circ h)$ , for $h\\in H$ , $\\mu \\in \\mathcal {D}(G,A)$ and $f\\in \\mathcal {C}(G,A)$ .",
"We apply the above results to the spaces of (generalized) integral $p$ -adic automorphic forms.",
"Fix a nonempty subset $J$ of $I$ .",
"Recall that in §REF , we have an isomorphism $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)\\cong \\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},A)$ .",
"Combined with Proposition REF , it gives us an $A$ -linear isomorphism $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket \\cong \\operatorname{Hom}_A(\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A),A)$ .",
"In §REF , we have defined a right action of the monoid $\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ on $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,A)$ .",
"Therefore, we get a left action of $\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ on $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ .",
"Under the isomorphism $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket \\cong A\\llbracket \\mathcal {O}_{p,J} \\rrbracket \\hat{\\otimes }_A A\\llbracket P_{J^c} \\rrbracket $ , the monoid $\\mathbf {M}_{\\pi ,J}$ acts on $A\\llbracket \\mathcal {O}_{p,J} \\rrbracket $ and the group $\\mathrm {Iw}_{\\pi ,J^c}$ acts on $A\\llbracket P_{J^c} \\rrbracket $ .",
"We remark that the latter action has an explicit expression: for $u\\in \\mathrm {Iw}_{\\pi ,J^c}$ and $x\\in A\\llbracket P_{J^c} \\rrbracket $ , the left action of $u$ on $x$ is given by $u\\circ x=\\kappa _{B,J}(u^\\ast _2)x\\cdot u^\\ast _1$ , where $u_1^\\ast $ (resp.",
"$u_2^\\ast $ ) is the $P_{J^c}$ -component (resp.",
"$D(\\mathcal {O}_{p,J^c})$ -component) of $u^\\ast \\in \\mathrm {Iw}_{\\pi ,J^c}$ under the decomposition $\\mathrm {Iw}_{\\pi ,J^c}=P_{J^c}\\times D(\\mathcal {O}_{p,J^c})$ , $\\kappa _{B,J}$ is the character defined in §REF , and $x\\cdot u_1^\\ast $ is the product in the complete group algebra $A\\llbracket P_{J^c} \\rrbracket $ .",
"Convention 4.5.3 We always view the complete group algebra $A\\llbracket P_{J^c}\\rrbracket $ as a left $A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -module induced by the left multiplication of $P^{\\prime }_{J^c}$ on $P_{J^c}$ .",
"As $P^{\\prime }_{J^c}$ is a normal subgroup of $P_{J^c}$ with quotient $P_{J^c}/P^{\\prime }_{J^c}\\cong \\Delta _{J^c}$ , $A\\llbracket P_{J^c}\\rrbracket $ is a free $A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ of rank $|\\Delta _{J^c}|$ .",
"Under the isomorphism $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket \\cong A\\llbracket \\mathcal {O}_{p,J} \\rrbracket \\hat{\\otimes }_A A\\llbracket P_{J^c} \\rrbracket $ , the group algebra $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ is also endowed with a left $A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -module structure.",
"The left action of $\\mathrm {Iw}_{\\pi ,J^c}$ on $A\\llbracket P_{J^c}^{\\prime }\\rrbracket $ and $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ defined above are $A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -linear.",
"On the other hand, although $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket $ has a (commutative) ring structure, the left action of $\\mathrm {Iw}_{\\pi ,J}$ on $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket $ defined above is not compatible with this ring structure.",
"On the dual side, this is equivalent to the fact that the right action of $\\mathrm {Iw}_{\\pi ,J}$ on the space $\\mathcal {C}(\\mathcal {O}_{p,J},A)$ does not commute with the obvious translation action of $\\mathcal {O}_{p,J}$ on $\\mathcal {C}(\\mathcal {O}_{p,J},A)$ .",
"For this reason we only view $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ as a left $A\\llbracket P^{\\prime }_{J^c} \\rrbracket $ -module in this paper.",
"Lemma 4.5.4 Let $M$ be a finitely generated right $\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_\\pi \\rrbracket $ -module and $N$ be a right $A\\llbracket \\mathrm {Iw}_\\pi \\rrbracket $ -module.",
"Let $\\langle \\cdot ,\\cdot \\rangle _N:N\\times \\operatorname{Hom}_A(N,A)\\rightarrow A$ be the natural bilinear map.",
"We endow $\\operatorname{Hom}_A(N,A)$ with a left action of $\\mathrm {Iw}_{\\pi }$ as before, i.e.",
"we have $\\langle n\\cdot g , l \\rangle _N=\\langle n, g\\cdot l \\rangle _N$ for all $n\\in N$ , $l\\in \\operatorname{Hom}_A(N,A)$ and $g\\in \\mathrm {Iw}_{\\pi }$ .",
"Under the above notations, the map $\\begin{split}M\\times \\operatorname{Hom}_A(N,A) &\\rightarrow \\operatorname{Hom}_A(\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N),A)\\\\(m,l)&\\mapsto F_{(m,l)},\\end{split}$ where $F_{(m,l)}:\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N)\\rightarrow A$ is defined by $F_{(m,l)}(\\varphi ):=\\langle \\varphi (m) , l \\rangle _N$ for any $\\varphi \\in \\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N)$ , induces an $A$ -linear isomorphism $\\iota _{M,N}:M\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }\\operatorname{Hom}_A(N,A)\\xrightarrow{}\\operatorname{Hom}_A(\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N),A).$ First we remark that since $M$ is a right $\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket $ and $\\operatorname{Hom}_A(N,A)$ carries a left $\\mathrm {Iw}_{\\pi }$ -action, the completed tensor product $M\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }\\operatorname{Hom}_A(N,A)$ is meaningful.",
"It is straightforward to verify that the map $M\\times \\operatorname{Hom}_A(N,A) \\rightarrow \\operatorname{Hom}_A(\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N),A)$ induces an $A$ -linear homomorphism $\\iota _{M,N}:M\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }\\operatorname{Hom}_A(N,A)\\rightarrow \\operatorname{Hom}_A(\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N),A)$ and $\\iota _{M,N}$ is an isomorphism when $M$ is free.",
"In the general case, we choose a resolution $F_1\\rightarrow F_0\\rightarrow M\\rightarrow 0$ of $M$ with $F_0$ , $F_1$ finite free $\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket $ -module.",
"We apply the functors $M\\rightarrow M\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }\\operatorname{Hom}_A(N,A)$ and $M\\rightarrow \\operatorname{Hom}_A(\\operatorname{Hom}_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket }(M,N),A)$ (with $N$ fixed) to the resolution $F_1\\rightarrow F_0\\rightarrow M\\rightarrow 0$ .",
"A simple diagram chasing shows that $\\iota _{M,N}$ is an isomorphism.",
"We denote by $S_{\\kappa ,I}^D(K^p,A)^\\vee $ the $A$ -linear dual of the space of $p$ -adic automorphic forms $S_{\\kappa ,I}^D(K^p,A)$ , and for any subset $J\\subset I$ and a continuous character $\\chi _J:\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow A^\\times $ , we define $S_{\\kappa ,J}^D(K^p,A)^\\vee $ to be the $A$ -linear dual of the space $S_{\\kappa ,J}^D(K^p,A)$ .",
"From the tautological isomorphism (REF ) and Lemma REF , we have the following expression of these spaces: $S_{\\kappa ,I}^D(K^p,A)^\\vee =\\tilde{\\mathrm {H}}_0\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket } A\\llbracket \\mathcal {O}_p\\rrbracket \\text{~and~} S_{\\kappa ,J}^D(K^p,A)^\\vee = \\tilde{\\mathrm {H}}_0\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket } A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket ,$ where $\\tilde{\\mathrm {H}}_0$ is the completed homology group defined in §REF .",
"We can translate the results we developed for the spaces $S_{\\kappa ,I}^D(K^p,A)$ and $S_{\\kappa ,J}^D(K^p,A)$ to their $A$ -linear duals.",
"We summarize these results as follows: For $j\\in J$ , we can define the $U_{\\pi _j}$ -operator on $S_{\\kappa ,I}^D(K^p,A)^\\vee $ and $S_{\\kappa ,J}^D(K^p,A)^\\vee $ by $U_{\\pi _j}(h\\hat{\\otimes } \\mu ):=\\sum _{l=0}^{p-1} (h\\cdot v_{j,l}^{-1})\\hat{\\otimes } (v_{j,l}\\cdot \\mu ), \\text{~for~} h\\hat{\\otimes }\\mu \\in \\tilde{\\mathrm {H}}_0\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket } A\\llbracket \\mathcal {O}_p\\rrbracket \\text{~or ~} \\tilde{\\mathrm {H}}_0\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket } A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket .$ Under the natural $A$ -bilinear pairing $\\langle \\cdot ,\\cdot \\rangle :S_{\\kappa ,I}^D(K^p,A)\\times S_{\\kappa ,I}^D(K^p,A)^\\vee \\rightarrow A$ (resp.",
"$\\langle \\cdot , \\cdot \\rangle _J:S_{\\kappa ,J}^D(K^p,A)\\times S_{\\kappa ,J}^D(K^p,A)^\\vee \\rightarrow A$ ), we have $\\langle U_{\\pi _j}(\\phi ),\\psi \\rangle =\\langle \\phi ,U_{\\pi _j}(\\psi ) \\rangle $ (resp.",
"$\\langle U_{\\pi _j}(\\phi ),\\psi \\rangle _J=\\langle \\phi ,U_{\\pi _j}(\\psi ) \\rangle _J$ ).",
"The right action of $B(\\mathcal {O}_{p,J^c})$ on $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_{\\pi },A)$ defined in Remark REF induces a left action of $B(\\mathcal {O}_{p,J^c})$ on $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ .",
"Moreover $A\\llbracket \\mathcal {O}_p \\rrbracket $ can be identified with the coinvariant of $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket $ under this action.",
"For any two subsets $J_1\\subset J_2$ of $I$ , let $J_3=J_2\\setminus J_1$ .",
"We have a natural surjective $A$ -linear map $S_{\\kappa ,J_1}^D(K^p,A)^\\vee \\rightarrow S_{\\kappa ,J_2}^D(K^p,A)^\\vee $ , which identifies the latter space with the $B(\\mathcal {O}_{p,J_3})$ -coinvariants of the first space.",
"Moreover, it is compatible with the $U_{\\pi _j}$ -operators on these two spaces for all $j\\in J_1$ .",
"Let $\\gamma _k$ for $k=0,\\dots , s-1$ be the elements of $D_f^\\times $ defined in §REF , which are viewed as elements in $\\tilde{\\mathrm {H}}_0$ .",
"By (REF ), we write $S_{\\kappa ,I}^D(K^p,A)^\\vee $ and $S_{\\kappa ,J}^D(K^p,A)^\\vee $ explicitly by $\\bigoplus \\limits _{k=0}^{s-1}A\\llbracket \\mathcal {O}_p \\rrbracket \\xrightarrow{}S_{\\kappa ,I}^D(K^p,A)^\\vee , \\ (\\mu _k)_{k=0,\\dots ,s-1}\\mapsto \\sum \\limits _{k=0}^{k-1}\\gamma _k\\hat{\\otimes } \\mu _k$ and $\\bigoplus \\limits _{k=0}^{s-1}A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket \\xrightarrow{} S_{\\kappa ,J}^D(K^p,A)^\\vee ,\\ (\\mu _k)_{k=0,\\dots ,s-1}\\mapsto \\sum \\limits _{k=0}^{k-1}\\gamma _k\\hat{\\otimes } \\mu _k.$ The left $A\\llbracket P_{J^c}^{\\prime }\\rrbracket $ -module structure on $A\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket $ defined in Convention REF induces a left $A\\llbracket P_{J^c}^{\\prime }\\rrbracket $ -module structure on the space $S_{\\kappa ,J}^{D}(K^p,A)^\\vee $ .",
"The $U_{\\pi _j}$ -operator on $S_{\\kappa ,J}^{D}(K^p,A)^\\vee $ is $A\\llbracket P_{J^c}^{\\prime }\\rrbracket $ -linear for all $j\\in J$ .",
"Recall that in §REF , we define a closed subspace $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ of $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})$ , which is closed under the action of the monoid $\\mathbf {M}_\\pi $ .",
"We will generalize this construction to define a closed subspace $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},\\Lambda _J^{>1/p})^{J^{\\prime }-mod}$ of $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},\\Lambda _J^{>1/p})$ for any nonempty subsets $J^{\\prime }\\subset J$ of $I$ .",
"We cannot define this space directly as there is no Mahler basis for the space $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},\\Lambda _J^{>1/p})$ .",
"We will pass to the dual side to make the definition and then dual back.",
"First we explain the construction in the case $F=\\mathbb {Q}$ .",
"Then $\\mathcal {O}_p=\\mathbb {Z}_p$ and we assume $\\Lambda ^{>1/p}=\\mathbb {Z}_p\\llbracket T,\\frac{p}{T} \\rrbracket $ .",
"It follows from Amice transformation that the $\\Lambda ^{>1/p}$ -dual space of $\\mathcal {O}(\\mathbb {Z}_p,\\Lambda ^{>1/p})$ is isomorphic to $\\Lambda ^{>1/p}\\llbracket X \\rrbracket $ and under this isomorphism, the dual basis of the Mahler basis $\\lbrace e_m=\\binom{z}{m}\\;|\\;m\\ge 0 \\rbrace $ is given by $\\lbrace X^m\\;|\\;m\\ge 0 \\rbrace $ .",
"Combining it with Proposition REF , we have an isomorphism $\\Lambda ^{>1/p}\\llbracket \\mathbb {Z}_p \\rrbracket \\xrightarrow{}\\Lambda ^{>1/p}\\llbracket X \\rrbracket $ , under which $[1]-1$ corresponds to $X$ .",
"We use $\\Lambda ^{>1/p}\\llbracket X\\rrbracket ^{mod}$ to denote the $\\Lambda ^{>1/p}$ -dual space of the closed subspace $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda ^{>1/p})^{mod}$ .",
"Then as a $\\Lambda ^{>1/p}$ -module, it is isomorphic to $\\Lambda ^{>1/p}\\llbracket X^{\\prime }\\rrbracket $ , and under this isomorphism the dual basis of the modified Mahler basis $\\lbrace e_m^{\\prime }=T^m\\binom{z}{m}\\;|\\;m\\ge 0 \\rbrace $ corresponds to $\\lbrace X^{\\prime m}\\;|\\;m\\ge 0 \\rbrace $ .",
"The $\\Lambda ^{>1/p}$ -dual of the inclusion $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda ^{>1/p})^{mod}\\rightarrow \\mathcal {C}(\\mathbb {Z}_p,\\Lambda ^{>1/p})$ is the $\\Lambda ^{>1/p}$ -algebra homomorphism $\\Lambda ^{>1/p}\\llbracket X\\rrbracket \\rightarrow \\Lambda ^{>1/p}\\llbracket X^{\\prime } \\rrbracket $ which sends $X$ to $TX^{\\prime }$ .",
"Moreover, as the right action of the monoid $\\mathbf {M}_\\pi $ on $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda ^{>1/p})$ keeps the subspace $\\mathcal {C}(\\mathbb {Z}_p,\\Lambda ^{>1/p})^{mod}$ stable, it induces a left action of $\\mathbf {M}_\\pi $ on $\\Lambda ^{>1/p}\\llbracket X^{\\prime } \\rrbracket $ , and the map $\\Lambda ^{>1/p}\\llbracket X \\rrbracket \\rightarrow \\Lambda ^{>1/p}\\llbracket X^{\\prime } \\rrbracket $ defined above is compatible with the $\\mathbf {M}_\\pi $ -actions on these two spaces.",
"Now we consider the general case.",
"Fix two nonempty subsets $J^{\\prime }\\subset J$ of $I$ .",
"We define an isomorphism of $\\Lambda _J^{>1/p}$ -algebras $\\iota _J:\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J} \\rrbracket \\rightarrow \\Lambda _J^{>1/p}\\llbracket (X_j)_{j\\in J}\\rrbracket ,\\ [1_j]-1\\mapsto X_j$ for all $j\\in J$ , where $1_j\\in \\mathcal {O}_{p,J}=\\prod \\limits _{j\\in J}\\mathcal {O}_{\\mathfrak {p}_j}$ is the element with $j$ -component 1 and $i$ -component 0 for all $i\\ne j$ .",
"We identify $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket $ with $\\Lambda _J^{>1/p}\\llbracket (X_j)_{j\\in J}\\rrbracket $ via $\\iota _J$ .",
"We put $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J} \\rrbracket ^{J^{\\prime }-mod}:=\\Lambda _J^{>1/p}\\llbracket (X_j^{\\prime })_{j\\in J^{\\prime }}, (X_j)_{j\\in J\\setminus J^{\\prime }} \\rrbracket $ and define an embedding of $\\Lambda _J^{>1/p}$ -algebras $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket \\rightarrow \\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}$ by $X_j\\mapsto T_jX_j^{\\prime }$ for $j\\in J^{\\prime }$ and $X_j\\mapsto X_j$ for $j\\in J\\setminus J^{\\prime }$ .",
"Then we put $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket ^{J^{\\prime }-mod}:=\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}\\hat{\\otimes }_{\\Lambda _J^{>1/p}}\\Lambda _J^{>1/p}\\llbracket P_{J^c}\\rrbracket $ .",
"Under the isomorphism $ \\mathcal {O}_{\\mathfrak {p}_j}\\cong \\mathbb {Z}_p$ , we have $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket ^{j-mod}\\cong \\Lambda _J^{>1/p}\\llbracket \\mathbb {Z}_p\\rrbracket ^{mod}$ defined above and hence we can endow $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket ^{j-mod}$ a left action of the monoid $\\mathbf {M}_{\\pi _j}$ .",
"Under the isomorphism $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}\\cong \\left(\\hat{\\otimes }_{j\\in J^{\\prime }}\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket ^{j-mod} \\right)\\hat{\\otimes }_{\\Lambda _J^{>1/p}}\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J\\setminus J^{\\prime }}\\rrbracket $ , we get a left action of the monoid $\\mathbf {M}_{\\pi ,J}$ on $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}$ .",
"Then the embedding $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket \\hookrightarrow \\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}$ (resp.",
"$\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c} \\rrbracket \\rightarrow \\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket ^{J^{\\prime }-mod}$ ) is equivariant under the actions of the monoid $\\mathbf {M}_{\\pi ,J}$ (resp.",
"$\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ ) on these two spaces.",
"Define $\\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,\\Lambda _J^{>1/p})^{J^{\\prime }-mod}$ to be the $\\Lambda _J^{>1/p}$ -dual space of $\\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket ^{J^{\\prime }-mod}$ , which carries a right action of the monoid $\\mathbf {M}_{\\pi ,J}\\times \\mathrm {Iw}_{\\pi ,J^c}$ .",
"Similar with the definition of $S_{\\kappa ,I}^{D,I}(K^p,\\Lambda _J^{>1/p})$ , we set $ S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})\\\\:=\\lbrace \\phi : D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}_{\\chi _J}(\\mathrm {Iw}_\\pi ,\\Lambda _J^{>1/p})^{J^{\\prime }-mod}|\\phi (xu)= \\phi (x)\\circ u \\text{~for all~} x\\in D_f^\\times ,u\\in \\mathrm {Iw}_{\\pi } \\rbrace ,$ and $S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})^\\vee $ to be its $\\Lambda _J^{>1/p}$ -linear dual.",
"Hence, we have $S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})^\\vee =\\tilde{\\mathrm {H}}_0\\hat{\\otimes }_{\\mathbb {Z}_p\\llbracket \\mathrm {Iw}_{\\pi }\\rrbracket } \\Lambda _J^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket ^{J^{\\prime }-mod}.$ Remark 4.5.5 Since our notation above is somewhat complicated, we make a remark to explain its meaning.",
"In the notation $S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})$ , the subscript $J$ under $S$ means that we consider generalized $p$ -adic automorphic forms which behave like automorphic forms at places in $J$ and behave like dual of completed homology groups at places in $I\\setminus J$ .",
"The superscript $J^{\\prime }$ means that we `modify' the basis at places in $J^{\\prime }$ , or more precisely, we consider the closed subspace $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},\\Lambda _J^{>1/p})^{J^{\\prime }-mod}$ of $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c},\\Lambda _J^{>1/p})$ defined before.",
"In particular, we always have $J^{\\prime }\\subset J$ when writing $S_{\\kappa ,I}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})$ , and if $J^{\\prime \\prime }\\subset J^{\\prime }\\subset J$ are three nonempty subsets of $I$ , we have a natural map $S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})\\rightarrow S_{\\kappa ,J}^{D,J^{\\prime \\prime }}(K^p,\\Lambda _J^{>1/p})$ , which is equivariant under the $U_{\\pi _j}$ -operator, for all $j\\in J$ .",
"Fix a nonempty subset $J$ of $I$ and let $A=\\Lambda _J$ or $\\Lambda _J^{>1/p}$ .",
"For $j\\in J$ , under the isomorphism $\\mathbb {Z}_p\\cong \\mathcal {O}_{\\mathfrak {p}_j}$ , we can identify $\\mathcal {C}(\\mathbb {Z}_p,A)$ with $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_j},A)$ .",
"So it is meaningful to talk about the Mahler basis (resp.",
"modified Mahler basis) of the space $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_j},A)$ (resp.",
"$\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_j},A)^{mod}$ ).",
"We define the Mahler basis of $A\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket $ to be the dual basis of the Mahler basis of $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_j},A)$ .",
"Under the isomorphism $A\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket \\cong A\\llbracket X_j\\rrbracket $ constructed as above, the Mahler basis of $A\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket $ is given by $\\lbrace X_j^m|m\\ge 0 \\rbrace $ .",
"In general, from the expression $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket =\\hat{\\otimes }_{j\\in J}A\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket $ , we can talk about the Mahler basis of $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket $ .",
"Under the isomorphism $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket \\cong A\\llbracket (X_j)_{j\\in J}\\rrbracket $ , the Mahler basis is given by $\\left\\lbrace \\prod \\limits _{j\\in J} X_j^{n_j}|n_j\\ge 0 \\right\\rbrace $ .",
"Since $A\\llbracket \\mathcal {O}_{p,J}\\times P^{\\prime }_{J^c}\\rrbracket =A\\llbracket \\mathcal {O}_{p,J}\\rrbracket \\hat{\\otimes }_A A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ , the Mahler basis of $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket $ becomes a basis of the left $A\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -module $A\\llbracket \\mathcal {O}_{p,J}\\times P^{\\prime }_{J^c}\\rrbracket $ , which is called the Mahler basis of $A\\llbracket \\mathcal {O}_{p,J}\\times P^{\\prime }_{J^c}\\rrbracket $ .",
"We make similar definitions for the modified Mahler basis of the spaces $A\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\rrbracket ^{mod}$ , $A\\llbracket \\mathcal {O}_{p,J}\\rrbracket ^{J^{\\prime }-mod}$ and $A\\llbracket \\mathcal {O}_{p,J}\\times P^{\\prime }_{J^c}\\rrbracket ^{J^{\\prime }-mod}$ .",
"Strictly speaking, it is crucial for us to work with the $\\Lambda _J^{>1/p}$ -dual spaces of $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c}, \\Lambda _J^{>1/p})$ and $\\mathcal {C}(\\mathcal {O}_{p,J}\\times P_{J^c}, \\Lambda _J^{>1/p})^{J^{\\prime }-mod}$ .",
"To explain the problem, we assume $J=I$ for simplicity.",
"Recall that $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ is the subspace of $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})$ spanned by the orthonormal basis $\\lbrace e_m^{\\prime }|m\\in \\mathbb {Z}_{\\ge 0}^I \\rbrace $ , i.e.",
"any element in $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ is of the form $\\sum \\limits _{m\\in \\mathbb {Z}_{\\ge 0}^I}a_me_m^{\\prime }$ with $a_m\\in \\Lambda ^{>1/p}$ and $a_m\\rightarrow 0$ as $|m|:= \\sum \\limits _{i\\in I}m_i\\rightarrow \\infty $ under the $\\mathfrak {m}_{\\Lambda ^{>1/p}}$ -adic topology.",
"In particular, $\\mathcal {C}(\\mathcal {O}_p,\\Lambda ^{>1/p})^{mod}$ is not isomorphic to the direct product $\\prod \\limits _{m\\in \\mathbb {Z}_{\\ge 0}^I}\\Lambda ^{>1/p}$ , while its $\\Lambda ^{>1/p}$ -dual is.",
"The computation in §REF actually shows that there is a well-defined action of the monoid $\\mathbf {M}_{\\pi }$ on $\\Lambda ^{>1/p}\\llbracket \\mathcal {O}_p\\rrbracket ^{I-mod}$ , such that the inclusion $\\Lambda ^{>1/p}\\llbracket \\mathcal {O}_p\\rrbracket \\rightarrow \\Lambda ^{>1/p}\\llbracket \\mathcal {O}_p\\rrbracket ^{I-mod}$ is $\\mathbf {M}_{\\pi }$ -equivariant.",
"Hence the spaces $\\Lambda ^{>1/p}\\llbracket \\mathcal {O}_{p,J}\\times P_{J^c}\\rrbracket ^{J^{\\prime }-mod}$ for various subsets $J^{\\prime }\\subset J\\subset I$ are the correct objects to consider.",
"Since $S_{\\kappa ,I}^D(K^p,\\Lambda ^{>1/p})$ and its dual space $S_{\\kappa ,I}^D(K^p,\\Lambda ^{>1/p})^\\vee $ share the same Hecke eigensystem, we can work with the space $S_{\\kappa ,I}^D(K^p,\\Lambda ^{>1/p})^\\vee $ and embedded into the space $S_{\\kappa ,I}^{D,I}(K^p,\\Lambda ^{>1/p})^\\vee $ to study the $U_{\\pi _i}$ -eigenvalues.",
"In the rest of our proof of Theorem REF , we will insist to work with the dual spaces $S_{\\kappa ,J}^{D,J^{\\prime }}(K^p,\\Lambda _J^{>1/p})^\\vee $ of generalized $p$ -adic automorphic forms."
],
[
"A filtration on the space of $p$ -adic automorphic forms with respect to a {{formula:85f22b87-37dc-4282-95f8-b0b6be090a1a}} -operator",
"Throughout this section, we take $J=\\lbrace j \\rbrace \\subset I$ consisting of a single place $j$ in $I$ , and we use $R$ to denote the ring $\\Lambda _J^{>1/p}=\\mathbb {Z}_p[H_J]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket T_j,\\frac{p}{T_j},T\\rrbracket $ .",
"In particular we have $\\mathcal {O}_{p,J}=\\mathcal {O}_{\\mathfrak {p}_j}$ .",
"Let $\\chi _J:\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times =\\mathcal {O}_{\\mathfrak {p}_j}^\\times \\times \\mathbb {Z}_p^\\times \\rightarrow R^\\times $ be the universal character.",
"Recall that the space $S_{\\kappa ,J}^D(K^p,R)^\\vee $ is endowed with a left $R\\llbracket P^{\\prime }_{J^c} \\rrbracket $ -module structure.",
"Moreover, the $U_{\\pi _j}$ -operator and the explicit expression $S_{\\kappa ,J}^D(K^p,R)^\\vee \\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P_{J^c} \\rrbracket $ are both $R\\llbracket P^{\\prime }_{J^c} \\rrbracket $ -linear.",
"We have a `dual' result of Proposition REF : under the isomorphism $S_{\\kappa ,J}^D(K^p,R)^\\vee \\xrightarrow{}\\bigoplus \\limits _{k=0}^{s-1}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P_{J^c} \\rrbracket $ , the $U_{\\pi _j}$ -operator can be described by the following commutative diagram: $@=1.3cm{S_{\\kappa ,J}^D(K^p,R)^\\vee [r]^{\\cong } [d]^{U_{\\pi _j}} & \\bigoplus \\limits _{k=0}^{s-1}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P_{J^c} \\rrbracket [d]^{\\mathcal {U}_j^\\vee } \\\\S_{\\kappa ,J}^D(K^p,R)^\\vee [r]^{\\cong } & \\bigoplus \\limits _{k=0}^{s-1}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P_{J^c} \\rrbracket .",
"}$ Here the right vertical maps $\\mathcal {U}_j^\\vee $ is the `transpose' of the map $\\mathcal {U}_j$ in Proposition REF .",
"More explicitly, $\\mathcal {U}_j^\\vee $ is given by a $t\\times t$ matrix with the following descriptions: Each entry of $\\mathcal {U}_j^\\vee $ is a sum of operators of the form $\\delta _p\\circ $ , where $\\delta _p=(\\delta _i)_{i\\in I}\\in \\mathrm {M}_2(\\mathcal {O}_p)$ have the same property as in Proposition REF .",
"There are exactly $p$ such operators appearing in each row and column of $\\mathcal {U}_j^\\vee $ .",
"Notice that $R\\llbracket P_{J^c}\\rrbracket $ is a free left $R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -module of rank $|\\Delta _{J^c}|$ .",
"If we fix a set of representatives in $P_{J^c}$ of the quotient $P_{J^c}/P^{\\prime }_{J^c}\\cong \\Delta _{J^c}$ , we obtain an $R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -linear isomorphism $\\llbracket P_{J^c}\\rrbracket \\cong \\bigoplus \\limits _{k=0}^{|\\Delta _{J^c}|}R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ .",
"This induces an isomorphism $R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P_{J^c}\\rrbracket \\cong \\bigoplus \\limits _{k=1}^{|\\Delta _{J^c}|}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P^{\\prime }_{J^c}\\rrbracket $ and we have an isomorphism $S_{\\kappa , J}^D(K^p,R)^\\vee \\xrightarrow{}\\bigoplus \\limits _{k=1}^SR\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P^{\\prime }_{J^c}\\rrbracket $ of left $R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -modules, where $S=s\\cdot |\\Delta _{J^c}|$ .",
"Similarly, we have an $R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -linear isomorphism $S_{\\kappa , J}^{D,J}(K^p,R)^\\vee \\xrightarrow{}\\bigoplus \\limits _{k=1}^SR\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P^{\\prime }_{J^c}\\rrbracket ^{J-mod}$ .",
"Under the above isomorphisms, we choose a basis $\\left\\lbrace f_{k,m}|1\\le k\\le S,m\\in \\mathbb {Z}_{\\ge 0}\\right\\rbrace $ of $S_{\\kappa , J}^{D}(K^p,R)^\\vee $ as a left $R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ -module, such that $\\lbrace f_{k,m}|m\\in \\mathbb {Z}_{\\ge 0}\\rbrace $ is the Mahler basis of the $k$ -th direct factor of $\\bigoplus \\limits _{k=0}^{s-1}R\\llbracket \\mathcal {O}_{\\mathfrak {p}_j}\\times P^{\\prime }_{J^c}\\rrbracket $ for all $1\\le k\\le S$ .",
"We choose a basis $\\lbrace f_{k,m}^{mod}|1\\le k\\le S,m\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $S_{\\kappa , J}^{D,J}(K^p,R)^\\vee $ in a similar way.",
"Notice that for any $\\delta _i\\in \\mathbf {M}_{\\pi _i}$ , if we use $P(\\delta _i)=(P_{m,n}(\\delta _i))_{m,n\\ge 0}$ to denote the infinite matrix for the action of $\\delta _i$ on $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_i}, A)$ under the Mahler basis, then the matrix for the action of $\\delta _i$ on $A\\llbracket \\mathcal {O}_{\\mathfrak {p}_i}\\rrbracket $ under the Mahler basis is the transpose of the matrix $P(\\delta _i)$ .",
"We use $N=(N_{m,n})_{m,n\\ge 0}$ (resp.",
"$M=(M_{m,n})_{m,n\\ge 0}$ ) to denote the matrix in $\\mathrm {M}_\\infty (R\\llbracket P^{\\prime }_{J^c}\\rrbracket )$ which corresponds to the $U_{\\pi _j}$ -operator on $S_{\\kappa ,J}^D(K^p,R)^\\vee $ (resp.",
"$S_{\\kappa ,J}^{D,J}(K^p,R)^\\vee $ ) under the basis we choose above.",
"It follows from Propositions REF , REF and the above remark that we have $N_{m,n}\\in (T_j)^{\\max \\lbrace \\lfloor \\frac{n}{S} \\rfloor -\\lfloor \\frac{m}{pS} \\rfloor ,0 \\rbrace }$ .",
"On the other hand, from the construction of the Mahler basis, the matrix $M$ is the conjugation of $N$ by the infinite diagonal matrix with diagonal entries $\\underbrace{1,1,\\dots ,1}_S,\\underbrace{T_j,T_j,\\dots ,T_j}_{S}, \\underbrace{T^2_j,T^2_j,\\dots ,T^2_j}_{S},\\dots $ Define two sequences $\\underline{\\lambda }$ , $\\underline{\\mu }$ of integers as $\\lambda _n=\\lfloor \\frac{n}{S} \\rfloor -\\lfloor \\frac{n}{pS} \\rfloor $ and $\\mu _0=0$ , $\\mu _{n+1}-\\mu _n=\\lambda _n$ for all $n\\in \\mathbb {Z}_{\\ge 0}$ .",
"The above computation implies that the matrix $M\\in \\mathrm {M}_\\infty (R\\llbracket P^{\\prime }_{J^c}\\rrbracket )$ is $\\underline{\\lambda }$ -Hodge bounded with respect to the element $T_j\\in R\\subset R\\llbracket P^{\\prime }_{J^c}\\rrbracket $ .",
"In the following discussion, we fix a character $\\omega =(\\eta _j,\\eta ):H_J=\\Delta _j \\times \\Delta \\rightarrow \\mathbb {Z}_p^\\times $ .",
"Denote $R_{\\omega _J}=R\\otimes _{\\mathbb {Z}_p[H_J],\\omega _J}\\mathbb {Z}_p$ , which is naturally isomorphic to $\\mathbb {Z}_p\\llbracket T_j,\\frac{p}{T_j},T\\rrbracket $ .",
"We construct a chain of ring homomorphisms as follows: Let $R\\rightarrow R_{\\omega _J}$ and $R\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket $ be the natural homomorphisms.",
"Let $R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega _J}$ be the reduction map modulo the augmentation ideal of the complete group algebra $R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket $ .",
"Under the isomorphism $R_{\\omega _J}\\cong \\mathbb {Z}_p\\llbracket T_j,\\frac{p}{T_j},T\\rrbracket $ , let $R_{\\omega _J}\\rightarrow R_j:= \\mathbb {Z}_p\\llbracket T_j,\\frac{p}{T_j}\\rrbracket $ be the reduction map modulo the ideal generated by $T$ .",
"Let $R_j\\rightarrow \\mathbb {F}_p\\llbracket T_j\\rrbracket $ be the reduction map modulo the ideal generated by $\\frac{p}{T_j}$ .",
"We apply the above homomorphisms $R\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega _J}\\rightarrow R_j\\rightarrow \\mathbb {F}_p\\llbracket T_j\\rrbracket $ to entries of the matrix $M$ , and obtain matrices $M_{R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket }\\in \\mathrm {M}_\\infty (R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket )$ , $M_{R_{\\omega _J}}\\in \\mathrm {M}_\\infty (R_{\\omega _J})$ , $M_{R_j}\\in \\mathrm {M}_\\infty (R_j)$ and $\\bar{M}\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T_j\\rrbracket )$ .",
"All these matrices are $\\underline{\\lambda }$ -Hodge bounded with respect to the element $T_j$ as the matrix $M$ is so.",
"Fix $l\\in 2\\mathbb {Z}_{\\ge 0}$ .",
"For every character $\\omega _{J^c}:\\Delta _{J^c}\\rightarrow \\mathbb {Z}_p^\\times $ , we obtain a character $\\omega =(\\omega _J,\\omega _{J^c}):H\\rightarrow \\mathbb {Z}_p^\\times $ .",
"We construct a point $\\chi _l\\in \\mathcal {W}(\\mathbb {C}_p)$ whose associated character $\\kappa _l:\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ is locally algebraic and corresponds to the triple $(n\\in \\mathbb {Z}_{\\ge 0}^I,\\nu \\in \\mathbb {Z}^I,\\psi =(\\psi _1,\\psi _2))$ defined as follows.",
"Define $\\nu :=(\\nu _i)_{i\\in I}$ by $\\nu _j:=-\\frac{l}{2}$ and $\\nu _i:=0$ for all $i\\ne j$ , and set $n:=-2\\nu \\in \\mathbb {Z}_{\\ge 0}^I$ .",
"Define two finite characters $\\psi _1,\\psi _2:\\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ with the following properties: $\\psi _1|_{1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}}$ and $\\psi _2|_{1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}}$ are trivial for all $i\\ne j$ ; $\\psi _2|_{1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}}$ is a nontrivial character which factors through $(1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j})/(1+\\pi _j^2\\mathcal {O}_{\\mathfrak {p}_j})$ ; $\\psi _1|_{1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}}=(\\psi _2|_{1+\\pi _j\\mathcal {O}_{\\mathfrak {p}_j}})^{-2}$ ; and the characters $\\psi _1|_{\\Delta _p}$ and $\\psi _2|_{\\Delta _p}$ are uniquely determined by the condition that the point $\\chi _l$ belongs to the component $\\mathcal {W}_\\omega $ of $\\mathcal {W}$ .",
"It is straightforward to verify that $T_{i,\\chi _l}=0$ for all $i\\ne j$ , and $v_p(T_{j,\\chi _l})=\\frac{1}{p-1}\\in (0,1)$ .",
"Note that the $\\mathrm {Iw}_{\\pi ,1,J^c}$ -coinvariant space of $S_{\\kappa ,J}^D(K^p,R_{\\omega _J})^\\vee $ (resp.",
"$S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee $ ) is given by $S_{\\kappa ,J}^D(K^p,R_{\\omega _J})^\\vee \\otimes _{R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket }R_j$ (resp.",
"$S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee \\otimes _{R_{\\omega _J}\\llbracket P^{\\prime }_{J^c}\\rrbracket }R_j$ ), and this space admits an action of $\\mathrm {Iw}_{\\pi ,J^c}/\\mathrm {Iw}_{\\pi ,1,J^c}\\cong \\Delta _{J^c}$ .",
"Define a homomorphism $\\tau _l:R_j\\rightarrow \\mathbb {C}_p$ of $\\mathbb {Z}_p$ -algebras with $\\tau _l(T_j)=T_{j,\\chi _l}$ and we use $M_{\\tau _l}\\in \\mathrm {M}_\\infty (\\mathbb {C}_p)$ to denote the infinite matrix obtained by applying $\\tau _l$ to entries of $M_{R_j}$ .",
"Now we consider another character $\\omega _J^{\\prime }=(\\omega _1^{-1},\\omega _2):H_J\\rightarrow \\mathbb {Z}_p^\\times $ , and obtain another character $\\omega ^{\\prime }=(\\omega _J^{\\prime },\\omega _{J^c}):H\\rightarrow \\mathbb {Z}_p^\\times $ .",
"We construct another point $\\chi _l^{\\prime }\\in \\mathcal {W}(\\mathbb {C}_p)$ whose associated character $\\kappa _l^{\\prime }:\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ corresponds to the triple $(n,\\nu ,\\psi ^{\\prime }=(\\psi _1^{-1},\\psi _2))$ .",
"It belongs to the weight disc $\\mathcal {W}_{\\omega ^{\\prime }}$ .",
"Applying the homomorphisms $R\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega ^{\\prime }_J}\\llbracket P^{\\prime }_{J^c}\\rrbracket \\rightarrow R_{\\omega ^{\\prime }_J}\\rightarrow R_j$ , we get matrices $M^{\\prime }_{R_{\\omega ^{\\prime }_J}\\llbracket P^{\\prime }_{J^c}\\rrbracket }$ , $M^{\\prime }_{R_{\\omega ^{\\prime }_J}}$ and $M^{\\prime }_{R_j}$ as before.",
"Applying the homomorphism $\\tau _l$ to entries of $M^{\\prime }_{R_j}$ , we get a matrix $M^{\\prime }_{\\tau _l}\\in \\mathrm {M}_\\infty (\\mathbb {C}_p)$ .",
"Define $t\\in \\mathbb {N}^I$ by $t_j=2$ and $t_i=1$ for all $i\\ne j$ .",
"Since we have $|\\Delta _{J^c}|$ choices of the characters $\\omega _{J^c}:\\Delta _{J^c}\\rightarrow \\mathbb {Z}_p^\\times $ , and $\\dim S_{k,w}^D(K^p\\mathrm {Iw}_{\\pi ^t},\\psi )=sp(l+1)$ , it follows from Proposition REF that there are $sp(l+1)|\\Delta _{J^c}|=Sp(l+1)$ pairs of the slopes of the Newton polygons of $M_{\\tau _l}$ and $M^{\\prime }_{\\tau _l}$ , such that the slopes in each pair sum to $l+1$ .",
"Hence the total sum of these slopes is $Sp(l+1)^2$ .",
"Since the matrix $M_{R_j}\\in \\mathrm {M}_\\infty (R_j)$ is $\\underline{\\lambda }$ -Hodge bounded, it follows from the construction of the matrix $M_{\\tau _l}$ that the sum of the first $Sp(l+1)$ slopes of $M_{\\tau _l}$ is at least $v_p(T_{j,\\chi _l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k$ .",
"For the same reason, the first $Sp(l+1)$ slopes of $M_{\\tau ^{\\prime }_l}$ is at least $v_p(T_{j,\\chi ^{\\prime }_l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k$ .",
"As $v_p(T_{j,\\chi _l})=v_p(T_{j,\\chi ^{\\prime }_l})=\\frac{1}{p-1}$ , we have $Sp(l+1)^2=v_p(T_{j,\\chi _l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k+v_p(T_{j,\\chi ^{\\prime }_l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k$ .",
"Therefore the first $Sp(l+1)$ slopes of the matrix $M_{\\tau _l}$ (resp.",
"$M_{\\tau ^{\\prime }_l}$ ) sum to exactly $v_p(T_{j,\\chi _l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k=\\frac{1}{2}Sp(l+1)^2$ (resp.",
"$v_p(T_{j,\\chi ^{\\prime }_l})\\sum \\limits _{k=0}^{Sp(l+1)-1}\\lambda _k=\\frac{1}{2}Sp(l+1)^2$ ).",
"Let $\\mathrm {char}(M_{R_j}):=\\sum \\limits _{n\\ge 0}c_nX^n\\in \\mathbb {Z}_p\\llbracket T_j\\rrbracket \\llbracket X \\rrbracket \\subset R_j\\llbracket X\\rrbracket $ be the characteristic power series of $M_{R_j}$ , with $c_n\\in \\mathbb {Z}_p\\llbracket T_j \\rrbracket $ .",
"The above discussion implies that the Newton polygon of $\\sum \\limits _{n\\ge 0}c_n(T_{j,\\chi _l})X^n$ passes through $(Sp(l+1),\\mu _{Sp(l+1)}v_p(T_{j,\\chi _l}))$ for all $l\\in 2\\mathbb {Z}_{\\ge 0}$ .",
"For simplicity, we denote $n_{l+1}=Sp(l+1)$ for $l\\in 2\\mathbb {Z}_{\\ge 0}$ .",
"Now we can run the exact argument in the proof of [20].",
"We give a summary of their results which will be used in our argument: Write $c_n(T_j)=\\sum \\limits _{m\\ge 0}b_{n,m}T_j^m$ , with $b_{n,m}\\in \\mathbb {Z}_p$ .",
"Then $v_p(b_{n,m})\\ge \\max \\lbrace \\mu _n-m,0 \\rbrace $ .",
"In particular, for $t_j\\in \\mathbb {C}_p$ with $v_p(t_j)\\in (0,1)$ , we have $v_p(c_n(t_j))\\ge \\mu _nv_p(t_j)$ for all $n\\in \\mathbb {Z}_{\\ge 0}$ .",
"The equality holds if and only if $b_{n,\\mu _n}\\in \\mathbb {Z}_p^\\times $ .",
"For any $l\\in 2\\mathbb {Z}_{\\ge 0}$ , there exist two integers $n_{l+1}^-\\in [n_{l+1}-S,n_{l+1}]$ and $n_{l+1}^+\\in [n_{l+1},n_{l+1}+S]$ , such that for any $t_j\\in \\mathbb {C}_p$ with $v_p(t_j)\\in (0,1)$ , we have the following.",
"If $n_{l+1}^-\\ne n_{l+1}^+$ , then the points $(n_{l+1}^-,\\mu _{n_{l+1}^-}v_p(t_j))$ and $(n_{l+1}^+,\\mu _{n_{l+1}^+}v_p(t_j))$ are two consecutive vertices of the Newton polygon of $\\sum \\limits _{n\\ge 0}c_n(t_j)X^n$ .",
"Moreover, the line segment connecting these two vertices has slope $(l+1)(p-1)v_p(t_j)$ ; If $n_{l+1}^-=n_{l+1}=n_{l+1}^+$ , then $(n_{l+1},\\mu _{n_{l+1}v_p(t_j)})$ is a vertex of the Newton polygon of $\\sum \\limits _{n\\ge 0}c_n(t_j)X^n$ .",
"Now we consider the matrix $\\bar{M}\\in \\mathrm {M}_{\\infty }(\\mathbb {F}_p\\llbracket T_j \\rrbracket )$ .",
"Its characteristic power series $\\mathrm {char}(\\bar{M})=\\sum \\limits _{n\\ge 0}d_n(T_j)X^n$ can be obtained by applying the homomorphism $R_j\\rightarrow \\mathbb {F}_p\\llbracket T_j \\rrbracket $ to the coefficients of the power series $\\mathrm {char}(M_{R_j})=\\sum \\limits _{n\\ge 0}c_n(T_j)X^n$ .",
"We define the lower bound polygon of $\\mathrm {char}(\\bar{M})$ to be the lower convex hull of the points $(n,\\mu _n)$ , for all $n\\in \\mathbb {Z}_{\\ge 0}$ .",
"Since $\\bar{M}$ is $\\underline{\\lambda }$ -Hodge bounded, the Newton polygon of $\\mathrm {char}(\\bar{M})$ always lies on or above its lower bound polygon.",
"We observe the following two facts: The kernel of the composite $\\mathbb {Z}_p\\llbracket T_j \\rrbracket \\rightarrow R_j\\rightarrow \\mathbb {F}_p\\llbracket T_j \\rrbracket $ is the principal ideal generated by $p$ .",
"The $x$ -coordinates of the line segment of the lower bound polygon with slope $(l+1)(p-1)$ belong to the interval $[n_{l+1}-S,n_{l+1}+S]$ .",
"Combined the results from [20], these facts imply the following.",
"If $n_{l+1}^-\\ne n_{l+1}^+$ , the points $(n_{l+1}^-,v_{T_j}(d_{n_{l+1}^-}(T_j)))$ and $(n_{l+1}^+,v_{T_i}(d_{n_{l+1}^+}(T_j)))$ lie on the lower bound polygon and the slope of the line segment connecting these two vertices has slope $\\lambda _{n_{l+1}}=(l+1)(p-1)$ ; if $n_{l+1}^-=n_{l+1}=n_{l+1}^+$ , $(n_{l+1},v_{T_j}(d_{n_{l+1}}(T_j)))$ lies on the lower bound polygon.",
"In both cases, the slopes of the Newton polygon of $\\mathrm {char}(\\bar{M})$ on the left of the point $(n_{l+1}^-,v_{T_j}(d_{n_{l+1}^-}(T_j)))$ belong to $[0,\\lambda _{n_{l+1}})$ , and the slopes of the Newton polygon of $\\mathrm {char}(\\bar{M})$ on the right of the point $(n_{l+1}^+,v_{T_j}(d_{n_{l+1}^+}(T_j)))$ belong to $(\\lambda _{n_{l+1}},\\infty )$ .",
"In particular, $(n_{l+1}^-,v_{T_j}(d_{n_{l+1}^-}(T_j)))$ and $(n_{l+1}^+,v_{T_j}(d_{n_{l+1}^+}(T_j)))$ are touching vertices of the Newton polygon of $\\mathrm {char}(\\bar{M})=\\sum \\limits _{n\\ge 0}d_n(T_j)X^n$ .",
"Notice that $(0,0)$ is always a touching vertex of $\\bar{M}$ .",
"The above construction of the numbers $n_{l+1}^+$ 's also applies to the point $(0,0)$ .",
"It follows that we can find an integer $n_0^+\\in [0,S]$ , such that $(n_0^+,v_{T_j}(d_{n_0^+}(T_j)))$ is a touching vertex of the Newton polygon of $\\mathrm {char}(\\bar{M})$ , and the slopes of this Newton polygon on the left of the point $(n_0^+,v_{T_j}(d_{n_0^+}(T_j)))$ are all 0.",
"We denote $n_0^-=0$ for simplicity.",
"Let $\\Omega _{\\omega _J}:=\\lbrace n_0^-,n_0^+ \\rbrace \\cup \\lbrace n_l^-,n_l^+|l\\in 1+2\\mathbb {Z}_{\\ge 0} \\rbrace =\\lbrace n_0^-\\le n_0^+\\le n_1^-\\le n_1^+\\le n_3^-\\le \\dots \\rbrace $ .",
"We can apply Theorem REF to the matrix $M\\in \\mathrm {M}_{\\infty }(R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket )$ and the set $\\Omega _{\\omega _J}$ , and get the following.",
"Proposition 4.6.1 For every character $\\omega _J:H_J\\rightarrow \\mathbb {Z}_p^\\times $ , there exist a basis $\\lbrace v_m|m\\in \\mathbb {N}\\rbrace $ and a filtration $\\lbrace \\tilde{V}_\\alpha |\\alpha \\in \\Omega _{\\omega _J} \\rbrace $ of the left $R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket $ -module $S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee $ with the following properties.",
"If we use $N\\in \\mathrm {M}_\\infty (R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket )$ to denote the matrix corresponding to the $U_{\\pi _j}$ -operator on $S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee $ under the basis $\\lbrace v_m|m\\in \\mathbb {N}\\rbrace $ , the matrix $N$ is $\\underline{\\lambda }$ -Hodge bounded with respect to the element $T_j\\in R_{\\omega _J}$ .",
"The set $\\lbrace \\tilde{V}_\\alpha |\\alpha \\in \\Omega _{\\omega _J} \\rbrace $ is an increasing filtration of $S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee $ ; more precisely, we have $(0)=\\tilde{V}_{n_0^-}\\subset \\tilde{V}_{n_0^+}\\subset \\tilde{V}_{n_1^-}\\subset \\tilde{V}_{n_1^+}\\subset \\dots $ .",
"For every $\\alpha \\in \\Omega _{\\omega _J}$ , $\\tilde{V}_\\alpha $ is a free $R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket $ -submodule of $S_{\\kappa ,J}^{D,J}(K^p,R_{\\omega _J})^\\vee $ of finite rank $d_\\alpha \\ge 0$ , and $\\lbrace v_m|m=1,\\dots , d_\\alpha \\rbrace $ is a basis of $\\tilde{V}_\\alpha $ .",
"For every $\\alpha \\in \\Omega _{\\omega _J}$ , $\\tilde{V}_\\alpha $ is stable under the $U_{\\pi _j}$ -operator.",
"We use $R_\\alpha \\in \\mathrm {M}_{d_\\alpha }(R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket )$ to denote the matrix corresponding to the $U_{\\pi _j}$ -operator on $\\tilde{V}_\\alpha $ under the basis $\\lbrace v_m|m=1,\\dots ,d_\\alpha \\rbrace $ .",
"In particular, the matrix $N$ can be written in the blockwise form: $N=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_\\alpha & B_\\alpha \\\\0\\ \\ & D_\\alpha \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle d_\\alpha };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\alpha };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in \\mathrm {M}_{\\infty }(R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket ).$ Let $\\alpha \\le \\beta $ be two consecutive elements in $\\Omega _{\\omega _J}$ .",
"The matrix $A_\\beta $ can be written in the block uppertriangular forms: $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{11} & A_{12} \\\\\\ 0\\ \\ & A_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle d_\\alpha };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\alpha };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]$ with $A_{11}=A_\\alpha $ .",
"We use $\\bar{A}_{22}\\in \\mathrm {M}_d(\\mathbb {F}_p\\llbracket T_j \\rrbracket )$ to denote the matrix obtained by applying the homomorphism $R_{\\omega _J}\\llbracket P_{J^c}^{\\prime }\\rrbracket \\rightarrow \\mathbb {F}_p\\llbracket T_j \\rrbracket $ to entries of $A_{22}$ , where $d=d_\\beta -d_\\alpha $ .",
"Then the slopes of the Newton polygon of $\\bar{A}_{22}$ (with respect to the $T_j$ -adic valuation) belong to $\\lbrace \\lambda _l \\rbrace $ if $\\alpha =n_l^-$ , $\\beta =n_l^+$ , and belong to $(\\lambda _l,\\lambda _{l^{\\prime }})$ if $\\alpha =n_l^+$ , $\\beta =n_{l^{\\prime }}^-$ ."
],
[
"Proof of the main theorem",
"Notation 5.0.2 In this section, we write $I=\\lbrace j_1,j_2,\\dots , j_g \\rbrace $ .",
"For $1\\le l\\le g$ , we denote by $J_l$ the subset $\\lbrace j_1,\\dots , j_l\\rbrace $ of $I$ .",
"We will drop the letter $J$ or $j$ in the previous notations for simplicity.",
"More precisely, for $1\\le l\\le g$ , we make the following.",
"Let $\\mathfrak {p}_l$ be the prime of $F$ corresponding to the embedding $j_l\\in I$ and $\\pi _l=\\pi _{j_l}$ be the uniformizer of $\\mathcal {O}_{\\mathfrak {p}_l}$ we fixed before.",
"Let $H_l:=H_{J_l}$ denote the torsion subgroup of $\\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times $ and $\\delta _l$ be the torsion subgroup of $\\mathcal {O}_{\\mathfrak {p}_l}^\\times $ .",
"Let $\\Lambda _l$ and $R_l$ denote the ring $\\Lambda _{J_l}=\\mathbb {Z}_p\\llbracket \\mathcal {O}_{p,J}^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ and $\\Lambda _{J_l}^{>1/p}$ , respectively.",
"Let $T_l\\in \\Lambda _l$ be the element $T_{j_l}$ defined in §REF .",
"In particular, we can write $\\Lambda _l=\\mathbb {Z}_p[H_l]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket (T_m)_{1\\le m\\le l}, T\\rrbracket $ and $R_l=\\mathbb {Z}_p[H_l]\\otimes _{\\mathbb {Z}_p}\\mathbb {Z}_p\\llbracket (T_m,\\frac{p}{T_m})_{1\\le m\\le l}, T\\rrbracket $ ."
],
[
"Properties of the filtration on the space of integral $p$ -adic automorphic forms",
"We apply the construction in §REF to the set $J_1$ consisting of a single element $j_1$ .",
"To be more precise, we fix a character $\\omega _{1}=(\\eta _1,\\eta ):H_{1}=\\Delta _1\\times \\Delta \\rightarrow \\mathbb {Z}_p^\\times $ .",
"Let $R_{1,\\omega _{1}}=R_1\\otimes _{\\mathbb {Z}_p[H_{1}],\\omega _{1}}\\mathbb {Z}_p$ , which is isomorphic to $\\mathbb {Z}_p\\llbracket T_{1},\\frac{p}{T_{1}},T\\rrbracket $ and $R_{2,\\omega _2}=R_2\\otimes _{\\mathbb {Z}_p[H_{1}],\\omega _{1}}\\mathbb {Z}_p$ .",
"We apply Proposition REF to the space $S_{\\kappa ,J_1}^{D,J_1}(K^p,R_{1,\\omega _{1}})^\\vee $ and the $U_{\\pi _{1}}$ -operator on it.",
"Then we get a set $\\Omega _{\\omega _{1}}=\\lbrace n_l^-,n_l^+|l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} \\rbrace $ , a basis $\\lbrace v_m|m\\in \\mathbb {N}\\rbrace $ and a filtration $\\lbrace \\tilde{V}_\\alpha |\\alpha \\in \\Omega _{\\omega _{1}}\\rbrace $ of the space $S_{\\kappa ,J_1}^{D,J_1}(K^p,R_{1,\\omega _{1}})^\\vee $ .",
"By Remark REF , we have a surjective map $S_{\\kappa ,J_1}^{D,J_1}(K^p,R_{2,\\omega _{1}})^\\vee \\rightarrow S_{\\kappa ,J_2}^{D,J_1}(K^p,R_{2,\\omega _{1}})^\\vee $ , which identifies the latter space as the $B(\\mathcal {O}_{\\mathfrak {p}_{2}})$ -coinvariants of the first space.",
"For $\\alpha \\in \\Omega _{\\omega _{1}}$ , we use $V_\\alpha $ to denote the image of $\\tilde{V}_\\alpha \\otimes _{R_{1,\\omega _{1}}}R_{2,\\omega _{2}}$ under this map.",
"Proposition 5.1.1 For all $\\alpha \\in \\Omega _{\\omega _{1}}$ , the space $V_{\\alpha }$ is stable under the $U_{\\pi _{2}}$ -operator on $S_{\\kappa ,J_2}^{D,J_1}(K^p,R_{2,\\omega _{1}})^\\vee $ .",
"Throughout the proof, we use $R_1^{\\prime }$ and $R_2^{\\prime }$ to denote the rings $R_{1,\\omega _{1}}$ and $R_{2,\\omega _{2}}$ , respectively.",
"Let $N=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_\\alpha & B_\\alpha \\\\0\\ \\ & D_\\alpha \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle d_\\alpha };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\alpha };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]\\in \\mathrm {M}_{\\infty }(R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c} \\rrbracket )$ be the infinite matrix as in Proposition REF .",
"We apply the homomorphisms $R^{\\prime }_1\\llbracket P^{\\prime }_{J_1^c}\\rrbracket \\rightarrow R_1^{\\prime }\\rightarrow \\mathbb {F}_p\\llbracket T_1\\rrbracket $ to the matrix $A_\\alpha \\in \\mathrm {M}_{d_\\alpha }(R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ , and obtain matrices $A_{\\alpha ,R_1^{\\prime }}\\mathrm {M}_{d_\\alpha }(R_1^{\\prime })$ and $\\bar{A}_{\\alpha }\\in \\mathrm {M}_{d_\\alpha }(\\mathbb {F}_p\\llbracket T_1\\rrbracket )$ .",
"We use $f(X)=\\det (X\\cdot I_{d_{\\alpha }}-A_{\\alpha ,R_1^{\\prime }})=X^{d_\\alpha }+a_{d_\\alpha -1}X^{d_\\alpha -1}+\\cdot +a_0\\in R_1^{\\prime }[X]$ to denote the characteristic polynomial of $A_{\\alpha ,R_1^{\\prime }}$ .",
"Since the matrix $\\bar{A}_\\alpha $ is strictly $\\underline{\\lambda }^{[d_\\alpha ]}$ -Hodge bounded, we can write $a_0=T_1^{n_\\alpha }\\cdot a_0^{\\prime }$ , where $n_\\alpha =\\sum \\limits _{k=0}^{d_\\alpha -1}\\lambda _k$ and $a_0^{\\prime }$ is a unit in $R_1^{\\prime }$ .",
"As $N$ is $\\underline{\\lambda }$ -Hodge bounded and $\\lim \\limits _{k\\rightarrow \\infty }\\lambda _k=\\infty $ , we can find an element $\\beta \\ge \\alpha $ in $\\Omega _{\\omega _1}$ with the following property: if we write the matrix $N$ in the block uppertriangular form $N=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_\\beta & B_\\beta \\\\0\\ \\ & D_\\beta \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle d_\\beta };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\beta };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]$ , then $D_\\beta \\in \\mathrm {M}_\\infty (T_1^{n_\\alpha +1}R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c} \\rrbracket )$ .",
"We write $f(D_\\beta )=a_0I_\\infty +a_1D_\\beta +\\dots +D_\\beta ^{d_\\alpha }=T_1^{n_\\alpha }(a_0^{\\prime }I_\\infty +(T_1^{-n_\\alpha }D_\\beta )(a_1+\\dots +D_\\beta ^{d_\\alpha -1}))$ .",
"Since $T_1^{-n_\\alpha }D_\\beta \\in \\mathrm {M}_\\infty ( T_1 R^{\\prime }_1\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ and $a_0^{\\prime }$ is a unit in $R_1^{\\prime }$ , the matrix $D_\\beta ^{\\prime }=a_0^{\\prime }I_\\infty +(T_1^{-n_\\alpha }D_\\beta )(a_1+\\dots +D_\\beta ^{d_\\alpha -1})$ belongs to $\\operatorname{GL}_\\infty (R^{\\prime }_1\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ by Lemma REF .",
"By Proposition REF , the matrix $A_\\beta \\in \\mathrm {M}_{d_\\beta }(R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ is of the block uppertriangular form $\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{11} & A_{12} \\\\\\ 0 \\ \\ & A_{22} \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=0.5ex]m-2-1.south east);[dotted](m-1-1.south west) -- (m-1-2.south east);\\node [above,text depth=1pt] at ([xshift=2.5ex]m-1-1.north) {\\scriptstyle d_\\alpha };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\alpha };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ]$ with $A_{11}=A_\\alpha $ .",
"Denote $d=d_\\beta -d_\\alpha $ and $g(X)=\\det (X\\cdot I_d-A_{22,R_1^{\\prime }})\\in R_1^{\\prime }[X]$ for the characteristic polynomial of $A_{22,R_1^{\\prime }}$ .",
"It follows from Proposition REF that the slopes of the Newton polygon of $\\bar{A}_{11}\\in \\mathrm {M}_{d_{\\alpha }}(\\mathbb {F}_p\\llbracket T_1 \\rrbracket )$ are strictly less than the slopes of the Newton polygon of $\\bar{A}_{22}\\in \\mathrm {M}_{d^{\\prime }}(\\mathbb {F}_p\\llbracket T_1 \\rrbracket )$ (under the $T_1$ -adic valuation).",
"We use $\\bar{f}(X)$ (resp.",
"$\\bar{g}(X)$ ) to denote the image of $f(X)$ (resp.",
"$g(X)$ ) under the homomorphism $R_1^{\\prime }\\rightarrow \\mathbb {F}_p\\llbracket T_1 \\rrbracket $ .",
"Then $\\bar{f}(X)$ and $\\bar{g}(X)$ are coprime in $\\mathbb {F}_p(\\!(T_1)\\!",
")[X]$ .",
"By [22], $f$ and $g$ are strictly coprime in $R_1^{\\prime }[\\frac{1}{T_1}][X]$ .",
"So we can find $u(X)$ ,$v(X)\\in R_1^{\\prime }[X]$ and $m_\\alpha \\in \\mathbb {Z}_{\\ge 0}$ , such that $f(X)u(X)+g(X)v(X)=T_1^{m_\\alpha }$ in $R_1^{\\prime }[X].$ By Hamilton-Cayley's Theorem (over the commutative ring $R_1^{\\prime }$ ), we have $f(A_{11,R_1^{\\prime }})=0$ and $g(A_{22,R_1^{\\prime }})=0$ .",
"Hence we have $f(A_{22,R_1^{\\prime }})u(A_{22,R_1^{\\prime }})=T_1^{m_\\alpha }I_{d}$ in $\\mathrm {M}_d(R_1^{\\prime })$ .",
"We use $\\mathcal {I}_1\\subset R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ and $\\mathcal {I}_2\\subset R_2^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ to denote the augmentation ideal of the corresponding complete group algebras, respectively.",
"In summary of the above discussion, the matrix $N\\in \\mathrm {M}_\\infty (R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ can be written in the following block uppertriangular form: $N=\\quad \\Biggl [\\hspace{-2.77771pt}\\begin{tikzpicture}[baseline=-.65ex][matrix of math nodes,column sep=1ex,] (m){A_{11} & \\ \\ast \\ & \\ast \\\\0 & A_{22} & \\ast \\\\0 & \\ 0\\ \\ & D_\\beta \\\\};[dotted]([xshift=0.5ex]m-1-1.north east) -- ([xshift=2ex]m-3-1.south east);[dotted]([xshift=0.5ex]m-1-2.north east) -- ([xshift=0.5ex]m-3-2.south east);[dotted](m-1-1.south west) -- (m-1-3.south east);[dotted](m-2-1.south west) -- (m-2-3.south east);\\node [above,text depth=1pt] at ([xshift=3.5ex]m-1-1.north) {\\scriptstyle d_\\alpha };\\node [above,text depth=1pt] at ([xshift=3ex]m-1-2.north) {\\scriptstyle d_\\beta };\\node [left,overlay] at ([xshift=-1.2ex,yshift=-2ex]m-1-1.west) {\\scriptstyle d_\\alpha };\\node [left,overlay] at ([xshift=-3ex,yshift=-2ex]m-2-1.west) {\\scriptstyle d_\\beta };\\end{tikzpicture}\\hspace{-2.77771pt}\\Biggr ],$ such that the matrix $f(N)=\\left[ \\begin{array}{ccc}f(A_{11})& \\ast &\\ast \\\\0&f(A_{22})&\\ast \\\\0&0&f(D_\\beta )\\end{array} \\right]$ satisfies the following properties: $f(A_{11})\\in \\mathrm {M}_{d_\\alpha }(\\mathcal {I}_1)$ ， $f(A_{22})u(A_{22})-T_1^{m_\\alpha }\\in \\mathrm {M}_d(\\mathcal {I}_1)$ ， $f(D_\\beta )=T_1^{n_\\alpha }D^{\\prime }_\\beta $ , for some $D^{\\prime }_\\beta \\in \\operatorname{GL}_\\infty (R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ .",
"Under the basis $\\lbrace v_m|m\\in \\mathbb {N}\\rbrace $ , we have an $R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ -linear isomorphism $S_{\\kappa _1,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee \\cong \\prod \\limits _{m=1}^\\infty R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ .",
"For any positive integer $k$ , we denote by $\\mathfrak {R}_k$ the quotient $R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket /\\mathcal {I}_1^k$ .",
"The above isomorphism induces an isomorphism $S_{\\kappa _1,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee \\otimes _{R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket } \\mathfrak {R}_k \\cong \\prod \\limits _{m=1}^\\infty \\mathfrak {R}_k.$ Since the $U_{\\pi _1}$ -operator on $S_{\\kappa _1,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee $ is $R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ -linear, it induces an $\\mathfrak {R}_k$ -linear map on $S_{\\kappa _1,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee \\otimes _{R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket } \\mathfrak {R}_k$ .",
"Under the above isomorphism, this operator corresponds to the infinite matrix $N_{\\mathfrak {R}_k}$ , and hence $f(U_{\\pi _1})$ corresponds to the matrix $f(N)_{\\mathfrak {R}_k}$ .",
"From the above discussion, we have the following facts for $f(N)_{\\mathfrak {R}_k}$ .",
"The submatrix $f(A_{11})_{\\mathfrak {R}_k}$ is nilpotent as $f(A_{11})\\in \\mathrm {M}_{d_\\alpha }(\\mathcal {I}_1)$ ; more precisely, we have $f(A_{11})_{\\mathfrak {R}_k}^k=0$ .",
"There exists a matrix $B_k\\in \\mathrm {M}_d(\\mathfrak {R}_k)$ , such that $f(A_{22})_{\\mathfrak {R}_k}\\cdot B_k=B_k\\cdot f(A_{22})_{\\mathfrak {R}_k}=T_1^{m_\\alpha k}\\cdot I_d$ ; in fact, if we write $f(A_{22})u(A_{22})=T_1^{m_\\alpha }\\cdot I_d+E$ , with $E\\in \\mathrm {M}_d(\\mathcal {I}_1)$ , we can choose the matrix $B_k$ to be $u(A_{22})\\left(\\sum \\limits _{l=0}^{k-1} T_1^{m_\\alpha l}E^{k-1-l}\\right)$ (viewed as a matrix in $\\mathrm {M}_d(\\mathfrak {R}_k)$ in the obvious way).",
"$f(D_\\beta )_{\\mathfrak {R}_k}=T_1^{n_\\alpha }D^{\\prime }_{\\beta ,\\mathfrak {R}_k}$ with $D^{\\prime }_{\\beta ,\\mathfrak {R}_k}\\in \\operatorname{GL}_\\infty (\\mathfrak {R}_k)$ .",
"So we have $\\tilde{V}_\\alpha /\\mathcal {I}_1^k=\\varinjlim \\limits _{m} \\ker \\left(f(U_{\\pi _1})^m: S_{\\kappa ,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee \\otimes _{R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket } \\mathfrak {R}_k\\rightarrow S_{\\kappa ,J_1}^{D,J_1}(K^p,R_1^{\\prime })^\\vee \\otimes _{R_1^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket } \\mathfrak {R}_k\\right).$ Since we have a surjection $S_{\\kappa ,J_1}^{D,J_1}(K^p,R_2^{\\prime })^\\vee \\rightarrow S_{\\kappa ,J_2}^{D,J_1}(K^p,R_2^{\\prime })^\\vee $ , and $T_1$ is not a zero divisor in $R_2^{\\prime }\\llbracket P^{\\prime }_{J_1^c}\\rrbracket /\\mathcal {I}_2^k$ , for all $k>0$ , we have $V_\\alpha /\\mathcal {I}_2^k=\\varinjlim \\limits _{m} \\ker (f(U_{\\pi _1})^m: S_{\\kappa ,J_2}^{D,J_1}(K^p,R_2^{\\prime })^\\vee /\\mathcal {I}_2^k\\rightarrow S_{\\kappa ,J_2}^{D,J_1}(K^p,R_2^{\\prime })^\\vee /\\mathcal {I}_2^k.$ Since the operator $U_{\\pi _2}$ commutes with $U_{\\pi _1}$ , it also commutes with $f(U_{\\pi _1})$ .",
"So it stabilizes $V_\\alpha /\\mathcal {I}_2^k$ for all $k\\ge 0$ .",
"Since $V_\\alpha =\\varprojlim \\limits _k V_\\alpha /\\mathcal {I}_2^k$ , we see that $U_{\\pi _2}$ stabilizes $V_\\alpha $ .",
"Remark 5.1.2 The proof of Proposition REF is a little lengthy and technical.",
"So it would be helpful to explain the intuition of our argument.",
"We fix a continuous homomorphism $\\chi :R_{2,\\omega _{2}}\\rightarrow \\mathbb {C}_p$ .",
"We will show in §5.3 below that $V_\\alpha \\otimes _{R_{2,\\omega _2},\\chi }\\mathbb {C}_p$ is the subspace of $S_{\\kappa ,J_2}^{D,J_1}(K^p,R_{2,\\omega _{2}})^\\vee \\otimes _{R_{2,\\omega _{2}},\\chi }\\mathbb {C}_p$ on which the slopes of the $U_{\\pi _1}$ -operator belong to the interval $[0,\\lambda _l\\cdot v_p(\\chi (T_1)))$ (resp.",
"$[0,\\lambda _l\\cdot v_p(\\chi (T_1))]$ ) if $\\alpha =n_l^-$ (resp.",
"$\\alpha =n_l^+$ ).",
"Proposition REF follows essentially from this characterization of the spaces $V_\\alpha $ 's and the fact that the operators $U_{\\pi _1}$ and $U_{\\pi _2}$ commutes with each other."
],
[
"A filtration on the space of $p$ -adic automorphic forms with respect to all {{formula:842cc49a-78e9-4560-9424-a7350629042b}} -operators",
"In this section, we fix a character $\\eta _{2}:\\Delta _{2}\\rightarrow \\mathbb {Z}_p^\\times $ .",
"Together with $\\omega _{1}:H_{1}\\rightarrow \\mathbb {Z}_p^\\times $ , it gives a character $\\omega _{2}:H_{2}\\rightarrow \\mathbb {Z}_p^\\times $ .",
"Denote $R_{2,\\omega _{2}}=R_2\\otimes _{\\mathbb {Z}_p[H_{2}],\\omega _{2}}\\mathbb {Z}_p$ and $V_{\\alpha ,\\omega _{2}}=V_\\alpha \\otimes _{\\mathbb {Z}_p[H_{2}],\\omega _{2}}\\mathbb {Z}_p$ for all $\\alpha \\in \\Omega _{\\omega _{1}}$ .",
"The set $\\lbrace V_{\\alpha ,\\omega _{2}}|\\alpha \\in \\Omega _{\\omega _{1}} \\rbrace $ is a filtration of $S_{\\kappa ,J_2}^{D,J_1}(K^p,R_{2,\\omega _{2}})^\\vee $ , and by Proposition REF , this filtration is stable under the $U_{\\pi _2}$ -operator.",
"This filtration induces a filtration $\\lbrace V^{mod}_{\\alpha ,\\omega _{2}}|\\alpha \\in \\Omega _{\\omega _{1}} \\rbrace $ of $S_{\\kappa ,J_2}^{D,J_2}(K^p, R_{2,\\omega _{2}})^\\vee $ .",
"If $\\gamma \\le \\alpha $ are two consecutive elements in $\\Omega _{\\omega _{1}}$ , we set $W_{\\alpha ,\\Omega _{\\omega _{2}}}:=V_{\\alpha ,\\omega _{2}}/V_{\\gamma ,\\omega _{2}}$ and $W^{mod}_{\\alpha ,\\Omega _{\\omega _{2}}}:=V^{mod}_{\\alpha ,\\omega _{2}}/V^{mod}_{\\gamma ,\\omega _{2}}$ .",
"Hence， $W_{\\alpha ,\\Omega _{\\omega _{2}}}$ also carries an action of the $U_{\\pi _2}$ -operator.",
"Under the basis $\\lbrace v_m|m=1,\\dots , d_\\alpha \\rbrace $ of $\\tilde{V}_\\alpha $ , we have an isomorphism $\\tilde{V}_\\alpha \\cong \\prod \\limits _{k=1}^{d_\\alpha }R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ , which induces an isomorphism $V_{\\alpha ,\\omega _{2}}\\cong \\prod \\limits _{k=1}^{d_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket $ .",
"So we obtain an isomorphism $W_{\\alpha ,\\omega _{2}}\\cong \\prod \\limits _{k=1}^{r_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket $ , where $r_\\alpha =d_\\alpha -d_\\gamma $ under the above notations.",
"We also have isomorphisms $V_{\\alpha ,\\omega _{2}}^{mod}\\cong \\prod \\limits _{k=1}^{d_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ and $W_{\\alpha ,\\omega _{2}}^{mod}\\cong \\prod \\limits _{k=1}^{r_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ .",
"Since $V_{\\alpha ,\\omega _{2}}$ is stable under the $U_{\\pi _1}$ and $U_{\\pi _2}$ -operators, a similar computation in §REF shows that $V_{\\alpha ,\\omega _{2}}^{mod}$ is also stable under the $U_{\\pi _1}$ and $U_{\\pi _2}$ -operators.",
"Hence we have well defined $U_{\\pi _1}$ and $U_{\\pi _2}$ -operators on $W_{\\alpha ,\\omega _{2}}^{mod}$ for all $\\alpha $ .",
"Fix $\\alpha \\in \\Omega _{\\omega _{1}}$ in the following discussion.",
"We will construct a filtration of the graded piece $W_{\\alpha ,\\omega _{2}}^{mod}$ adapted to the $U_{\\pi _2}$ -operator based on a similar idea we used in §REF .",
"Under the above isomorphisms, we choose a basis $\\lbrace f_{k,m}|1\\le k\\le r_\\alpha , m\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $W_{\\alpha ,\\omega _{2}}$ as a left $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c} \\rrbracket $ -module, such that $\\lbrace f_{k,m}|m\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ is the Mahler basis of the $k$ -th direct factor of $\\prod \\limits _{k=1}^{r_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket $ for all $1\\le k\\le r_\\alpha $ .",
"We choose a basis $\\lbrace f_{k,m}^{mod}|1\\le k\\le r_\\alpha ,m\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $W_{\\alpha ,\\omega _{2}}^{mod}$ in a similar way.",
"We use $N=(N_{m,n})_{m,n\\ge 0}$ (resp.",
"$M=(M_{m,n})_{m,n\\ge 0}$ ) to denote the matrix in $\\mathrm {M}_\\infty (R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket )$ which corresponds to the $U_{\\pi _2}$ -operator on $W_{\\alpha ,\\omega _{2}}$ (resp.",
"$W_{\\alpha ,\\omega _{2}}^{mod}$ ) under the basis we choose above.",
"It follows from Propositions REF and REF that $N_{m,n}\\in (T_{2})^{\\max \\lbrace \\lfloor \\frac{n}{r_\\alpha }\\rfloor -\\lfloor \\frac{m}{pr_\\alpha }\\rfloor ,0 \\rbrace }$ for all $m,n\\in \\mathbb {Z}_{\\ge 0}$ .",
"Notice that $M$ is the conjugation of $N$ by the diagonal matrix with diagonal entries $\\underbrace{1,1,\\dots ,1}_{r_\\alpha },\\underbrace{T_{2},T_{2},\\dots ,T_{2}}_{r_\\alpha }, \\underbrace{T^2_{2},T^2_{2},\\dots ,T^2_{2}}_{r_\\alpha },\\dots $ Define two sequence of integers $\\underline{\\lambda }, \\underline{\\mu }$ as $\\lambda _n=\\lfloor \\frac{n}{r_\\alpha }\\rfloor -\\lfloor \\frac{n}{pr_\\alpha }\\rfloor $ , $\\mu _0=0$ , $\\mu _{n+1}-\\mu _n=\\lambda _n$ for all $n\\in \\mathbb {Z}_{\\ge 0}$ .",
"Then the matrix $M$ is $\\underline{\\lambda }$ -Hodge bounded with respect to $T_{j_2}\\in R_{2,\\omega _{J_2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ .",
"Let $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket \\rightarrow R_{2,\\omega _{2}}$ be the reduction map modulo the augmentation ideal of the group ring $R_{2,\\omega _{J_2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ .",
"Under the isomorphism $R_{2,\\omega _{2}}\\cong \\mathbb {Z}_p\\llbracket T_{1},T_{2},\\frac{p}{T_{1}},\\frac{p}{T_{2}},T\\rrbracket $ , let $R_{2,\\omega _{2}}\\rightarrow \\mathbb {F}_p\\llbracket T_{2}\\rrbracket $ be the reduction map modulo the ideal generated by $T_{1},\\frac{p}{T_{1}}, \\frac{p}{T_{2}}$ and $T$ .",
"Applying the above two homomorphisms to the entries of the matrix $M$ , we obtain two matrices $M_{R_{2,\\omega _{2}}}\\in \\mathrm {M}_\\infty (R_{2,\\omega _{2}})$ and $\\bar{M}\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T_2\\rrbracket )$ .",
"Both of them are $\\underline{\\lambda }$ -Hodge bounded with respect to $T_2$ .",
"Fix an integer $l_1\\in 2\\mathbb {Z}_{\\ge 0}$ and a finite character $\\varepsilon _1:1+\\pi _1\\mathcal {O}_{\\mathfrak {p}_1}\\rightarrow \\mathbb {C}_p^\\times $ with the following properties: $l_1+1>\\frac{\\lambda _l}{p-1}$ , where $\\alpha =n_l^-$ or $n_l^+$ ; and $\\varepsilon _1$ is nontrivial and factors through $(1+\\pi _1\\mathcal {O}_{\\mathfrak {p}_1})/(1+\\pi ^2_1\\mathcal {O}_{\\mathfrak {p}_1})$ .",
"For every character $\\omega _{J_2^c}:\\Delta _{J_2^c}\\rightarrow \\mathbb {Z}_p^\\times $ , we obtain a character $\\omega =(\\omega _{2},\\omega _{J_2^c}):H\\rightarrow \\mathbb {Z}_p^\\times $ .",
"Given the above datum, for any integer $l_2\\in 2\\mathbb {Z}_{\\ge 0}$ and any nontrivial character $\\varepsilon _2:1+\\pi _2\\mathcal {O}_{\\mathfrak {p}_2}\\rightarrow \\mathbb {C}_p^\\times $ that factors through $1+\\pi ^2_2\\mathcal {O}_{\\mathfrak {p}_2}$ , we construct a point $\\chi _2\\in \\mathcal {W}(\\mathbb {C}_p)$ such that its associated character $\\kappa _{l_2}:\\mathcal {O}_p^\\times \\times \\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ is locally algebraic and corresponds to the triple $(n\\in \\mathbb {Z}_{\\ge 0}^I,\\nu \\in \\mathbb {Z}^I,\\psi =(\\psi _1,\\psi _2))$ defined as follows.",
"Define $\\nu :=(\\nu _i)_{i\\in I}$ by $\\nu _i={\\left\\lbrace \\begin{array}{ll}-l_1/2, &\\textrm {for~} i=j_1, \\\\-l_2/2, &\\textrm {for~} i=j_2, \\\\0, & \\text{otherwise.}\\end{array}\\right.",
"}.$ and $n:=-2\\nu \\in \\mathbb {Z}_{\\ge 0}^I$ .",
"Let $\\psi _1,\\psi _2:\\mathcal {O}_p^\\times \\rightarrow \\mathbb {C}_p^\\times $ be two finite characters with the following properties: $\\psi _1|_{1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}}$ and $\\psi _2|_{1+\\pi _i\\mathcal {O}_{\\mathfrak {p}_i}}$ are trivial for all $i\\ne j_1,j_2$ , $\\psi _2|_{1+\\pi _1\\mathcal {O}_{\\mathfrak {p}_1}}=\\varepsilon _{1}$ , $\\psi _2|_{1+\\pi _2\\mathcal {O}_{\\mathfrak {p}_2}}=\\varepsilon _{2}$ , and $\\psi _1|_{1+\\pi _1\\mathcal {O}_{\\mathfrak {p}_1}}=\\varepsilon _{1}^{-2}$ , $\\psi _1|_{1+\\pi _2\\mathcal {O}_{\\mathfrak {p}_2}}=\\varepsilon _{2}^{-2}$ ; the characters $\\psi _1|_{\\Delta _p}$ and $\\psi _2|_{\\Delta _p}$ are determined by the condition that the point $\\chi _{l_2}$ belongs to the component $\\mathcal {W}_\\omega $ of $\\mathcal {W}$ .",
"Under the above construction, we have $T_{i,\\chi _{l_2}}=0$ for all $i\\ne j_1,j_2$ and $v_p(T_{i,\\chi _{l_2}})=\\frac{1}{p-1}$ for $i=j_1,j_2$ .",
"Remark 5.2.1 The construction of the point $\\chi _2\\in \\mathcal {W}(\\mathbb {C}_p)$ depends on two even integers $l_1,l_2$ and two finite characters $\\varepsilon _1,\\varepsilon _2$ .",
"In the following paragraphs, we will construct a filtration of each graded piece $W_{\\alpha ,\\omega _2}^{mod}$ adapted to the $U_{\\pi _2}$ -operator.",
"Therefore the integer $l_2$ and the finite character $\\varepsilon _2$ play similar roles as those of $l$ and $\\psi _2$ appearing in the construction in §$4.6$ .",
"On the other hand, since we work over the ring $R_{2,\\omega _2}\\cong \\mathbb {Z}_p\\llbracket T_1,\\frac{p}{T_1},T_2,\\frac{p}{T_2},T\\rrbracket $ , we do have a restriction on the $T_1$ -parameter of the locally algebraic weight $\\chi _2\\in \\mathcal {W}(\\mathbb {C}_p)$ , that is, $v_p(T_{1,\\chi _2})\\in (0,1)$ .",
"The additional assumption $l_1+1>\\frac{\\lambda _l}{p-1}$ is to guarantee that the space $V_{\\alpha ,\\omega _2}^{mod}$ lands into the space of classical automorphic forms after specializing to the point $\\chi _2\\in \\mathcal {W}(\\mathbb {C}_p)$ .",
"As we will see in the argument below, the choices of $l_1$ and $\\varepsilon _{1}$ do not affect the filtrations adapted to the $U_{\\pi _2}$ -operator as long as they satisfy the above assumptions.",
"The point $\\chi _{l_2}\\in \\mathcal {W}(\\mathbb {C}_p)$ defines a homomorphism $\\tau _{l_2}:R_{2,\\omega _{2}}\\rightarrow \\mathbb {C}_p$ .",
"We apply this homomorphism to entries of the matrix $M_{R_{2,\\omega _{2}}}$ and get a matrix $M_{\\tau _{l_2}}\\in \\mathrm {M}_\\infty (\\mathbb {C}_p)$ .",
"Let $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket \\rightarrow \\mathbb {C}_p$ be the composite of the homomorphisms $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket \\rightarrow R_{2,\\omega _{2}}$ and $\\tau _{l_2}:R_{2,\\omega _{2}}\\rightarrow \\mathbb {C}_p$ .",
"Denote $\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}=R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}\\otimes _{R_{2,\\omega _{2}},\\tau _{l_2}}\\mathbb {C}_p$ .",
"From the explicit expression $V_{\\alpha ,\\omega _{2}}^{mod}\\cong \\prod \\limits _{k=1}^{d_\\alpha } R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ , we have an isomorphism $V_{\\alpha ,\\omega _{2}}^{mod}\\otimes _{R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket }\\mathbb {C}_p\\cong \\prod \\limits _{k=1}^{d_\\alpha }\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}$ .",
"Under the isomorphism $R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}\\cong R_{2,\\omega _{2}}\\llbracket X_2^{\\prime }\\rrbracket $ , the space $\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}$ admits a quotient $\\mathbb {C}_p\\llbracket X_2^{\\prime }\\rrbracket ^{\\deg \\le l_2}$ consisting of polynomials of $X_2^{\\prime }$ of degree $\\le l_2$ .",
"This quotient is stable under the action of the monoid $\\mathbf {M}_{\\pi _2}$ .",
"Recall that $V_\\alpha ^{mod}$ is a $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ -submodule of $S_{\\kappa ,J_2}^{D,J_2}(K^p,R_{2,\\omega _{2}})^\\vee $ , and we have an $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ -linear isomorphism $V_\\alpha ^{mod}\\cong \\prod \\limits _{k=1}^{d_\\alpha }R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ .",
"We put $V_{\\alpha ,\\mathbb {C}_p}^{mod}:=V_{\\alpha }^{mod}\\otimes _{R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket }\\mathbb {C}_p$ and obtain an isomorphism $V_{\\alpha ,\\mathbb {C}_p}^{mod}\\cong \\prod \\limits _{k=1}^{d_\\alpha }\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}$ .",
"Given the locally algebraic weight $\\kappa _{l_2}$ defined above, we define a subspace $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_2},\\mathbb {C}_p)^{cl}$ of $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_2},\\mathbb {C}_p)$ consisting of functions $f$ with the property that $f|_{a+\\pi _2\\mathcal {O}_{\\mathfrak {p}_2}}$ is a polynomial function of degree less or equal to $l_2$ , for all $a\\in \\mathcal {O}_{\\mathfrak {p}_2}$ .",
"Then we have $\\dim _{\\mathbb {C}_p}\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_2},\\mathbb {C}_p)^{cl}=p(l_2+1)$ .",
"We use $\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{cl}$ to denote the $\\mathbb {C}_p$ -dual of $\\mathcal {C}(\\mathcal {O}_{\\mathfrak {p}_2},\\mathbb {C}_p)^{cl}$ .",
"Then $\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{cl}$ is a quotient of $\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{j_2-mod}$ and hence $V_{\\alpha ,\\mathbb {C}_p}^{mod}$ admits a quotient $V_{\\alpha ,\\mathbb {C}_p}^{cl}:=\\prod \\limits _{k=1}^{d_\\alpha }\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{cl}$ .",
"Moreover, this quotient also carries an action of the $U_{\\pi _1}$ and $U_{\\pi _2}$ -operators induced from these operators on $V_{\\alpha ,\\mathbb {C}_p}^{mod}$ .",
"We define $W_{\\alpha ,\\mathbb {C}_p}^{cl}=V_{\\alpha ,\\mathbb {C}_p}^{cl}/V_{\\gamma ,\\mathbb {C}_p}^{cl}$ , which is isomorphic to $\\prod \\limits _{k=1}^{r_\\alpha }\\mathbb {C}_p\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\rrbracket ^{cl}$ .",
"We use $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }$ to denote the $\\mathbb {C}_p$ -dual of $V_{\\alpha ,\\mathbb {C}_p}^{cl}$ .",
"From the above construction, we see that there exists $r\\in \\mathcal {N}^I$ , such that $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }$ lies in the following subspace $\\lbrace \\phi :D^\\times \\setminus D_f^\\times /K^p\\rightarrow \\mathcal {C}^{r,an}(\\mathcal {O}_{\\mathfrak {p}_1}\\times \\mathcal {O}_{\\mathfrak {p}_2},\\mathbb {C}_p)|\\phi (xu)=\\phi (x)\\circ u, \\text{~for all~} x\\in D_f^\\times , u\\in \\mathrm {Iw}_{\\pi ^t} \\rbrace $ of $S_{\\kappa _l,I}^D(K^p,\\mathbb {C}_p)$ with the additional properties: The slopes of $U_{\\pi _1}$ -operator on $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }$ is less or equal to $\\lambda _l\\cdot v_p(\\chi _{l_2}(T_j))=\\frac{\\lambda _l}{p-1}< l_2+1$ ; and $\\phi (x)|_{\\lbrace x_1\\rbrace \\times \\mathcal {O}_{\\mathfrak {p}_2}}$ is a polynomial function of degree less or equal to $l_2$ , for all $x_1\\in \\mathcal {O}_{\\mathfrak {p}_1}$ .",
"Fix a locally algebraic weight $\\kappa _{l_2}$ as above.",
"For $1\\le l\\le g$ , we have defined an operator $\\theta _{i_l}:S_{\\kappa _{l_2}}^D(K^p,r)\\rightarrow S_{\\kappa ^{\\prime }_{l_2}}^D(K^p,r)$ in §REF .",
"It follows from the first property and the fact that the $U_{\\pi _1}$ -slopes on the space $S_{\\kappa ^{\\prime }_{l_2}}^D(K^p,r)$ are all nonnegative that $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }\\subset \\ker (\\theta _{i_1})$ .",
"It follows from the second property and the definition of $\\theta _{i_l}$ 's that $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }\\subset \\ker (\\theta _{i_l})$ for $l=2,\\dots ,g$ .",
"By Proposition REF , we conclude that $V_{\\alpha ,\\mathbb {C}_p}^{cl,\\vee }\\subset \\bigoplus \\limits _{\\omega _{J_2^c}:\\Delta _{J_2^c}\\rightarrow \\mathbb {Z}_p^\\times }S_{k,w}^D(K^p,\\psi )$ .",
"We make the same construction starting with another character $\\omega _{2}^{\\prime }=(\\eta _{j_1}^{-1},\\eta _{j_2}^{-1},\\eta ):H_{2}\\rightarrow \\mathbb {Z}_p^\\times $ .",
"We obtain $\\underline{\\lambda }$ -Hodge bounded matrices $M^{\\prime }_{R_{2,\\omega _{J_2}^{\\prime }}}\\in \\mathrm {M}_\\infty (R_{2,\\omega _{J_2}^{\\prime }})$ , $\\bar{M}^{\\prime }\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T_j\\rrbracket )$ , and for every $l_2\\in 2\\mathbb {Z}_{\\ge 0}$ , we get a matrix $M^{\\prime }_{\\tau _{l_2}}\\in \\mathrm {M}_\\infty (\\mathbb {C}_p)$ .",
"We also have spaces $(V_{\\alpha ,\\mathbb {C}_p}^{mod})^{\\prime }$ , $(V_{\\alpha ,\\mathbb {C}_p}^{cl})^{\\prime }$ and $(W_{\\alpha ,\\mathbb {C}_p}^{cl})^{\\prime }$ .",
"It follows from Proposition REF that the $U_{\\pi _2}$ -slopes on the spaces $W_{\\alpha ,\\mathbb {C}_p}^{cl}$ and $(W_{\\alpha ,\\mathbb {C}_p}^{cl})^{\\prime }$ can be paired such that the slopes in each pair sum to $l_2-1$ .",
"In other words, there exist $r_\\alpha p(l_2+1)$ pairs of the slopes of the Newton polygons of the matrices $M_{\\tau _{l_2}}$ and $M^{\\prime }_{\\tau _{l_2}}$ , such that the slopes in each pair sum to $l_2-1$ .",
"It follows from a similar argument in §$4.6$ that the first $r_\\alpha p(l_2+1)$ slopes of the matrix $M_{\\tau _{l_2}}$ sum to $\\frac{1}{2}r_\\alpha p(l_2+1)$ .",
"We use $\\mathrm {char}(M_{R_{2,\\omega _{2}}})=\\sum \\limits _{n\\ge 0}c_nX^n\\in R_{2,\\omega _{2}}\\llbracket X\\rrbracket $ to denote the characteristic power series of the matrix $M_{R_{2,\\omega _{2}}}$ with $c_n(\\underline{T})\\in R_{1,\\omega _{1}}\\llbracket T_2\\rrbracket \\subset R_{2,\\omega _{2}}$ and $\\underline{T}=(T_1,T_2,T)$ .",
"For all $l_2\\in 2\\mathbb {Z}_{\\ge 0}$ , the Newton polygon of $\\sum \\limits _{n\\ge 0}c_n(\\chi _{l_2}(\\underline{T}))X^n$ passes through the point $(r_\\alpha p(l_2+1), \\mu _{r_\\alpha p (l_2+1)}v_p(\\chi _{l_2}(T_2)))$ .",
"Write $c_n(\\underline{T})=\\sum \\limits _{m\\ge 0}b_{n,m}T_2^{m}$ with $b_{n,m}\\in R_{1,\\omega _{1}}$ .",
"Recall that $R_{1,\\omega _{1}}$ is isomorphic to $\\mathbb {Z}_p\\llbracket T_1,\\frac{p}{T_1},T\\rrbracket $ .",
"In particular, $R_{1,\\omega _{1}}$ is a local ring and we use $\\mathfrak {m}_{1,\\omega _{1}}$ to denote its maximal ideal.",
"For $b(T_1,T)\\in R_{1,\\omega _{1}}$ , the following statements are equivalent: $b(T_1,T)$ is a unit in $R_{1,\\omega _{1}}$ ; $b(T_1,T)\\notin \\mathfrak {m}_{1,\\omega _{1}}$ ; there exist $t_1,t\\in \\mathbb {C}_p$ with $|t_1|_p\\in (\\frac{1}{p},1)$ and $|t|_p<1$ , such that $|b(t_1,t)|_p=1$ , i.e.",
"$b(t_1,t)$ is a $p$ -adic unit.",
"Now we can run the same argument in §REF to the matrix $M\\in \\mathrm {M}_\\infty (R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket )$ , $M_{R_{2,\\omega _{2}}}\\in \\mathrm {M}_\\infty (R_{2,\\omega _{2}})$ and $\\bar{M}\\in \\mathrm {M}_\\infty (\\mathbb {F}_p\\llbracket T_2\\rrbracket )$ .",
"We obtain a set $\\Omega _{\\omega _{2},\\alpha _1}=\\left\\lbrace n_{\\omega _{2},\\alpha _1,l}^-, n_{\\omega _{2},\\alpha _1,l}^+\\Big | l\\in \\lbrace 0\\rbrace \\bigcup 1+2\\mathbb {Z}_{\\ge 0} \\right\\rbrace ,$ and a filtration $\\lbrace \\tilde{V}_{\\alpha _1,\\alpha _2}|\\alpha _2\\in \\Omega _{\\omega _{2},\\alpha _1} \\rbrace $ of the graded piece $W_{\\alpha _1,\\omega _{2}}^{mod}$ as a left $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ -module.",
"Fix a character $\\omega :H\\rightarrow \\mathbb {Z}_p^\\times $ .",
"For $1\\le l\\le g$ , we use $\\omega _{l}:H_{l}\\rightarrow \\mathbb {Z}_p^\\times $ to denote the restriction of $\\omega $ to $H_{l}$ .",
"Now we apply the above construction inductively to the places $j_3,\\dots , j_g$ .",
"In summary, we get the following datum: There exists a set $\\Omega _{\\omega _1}=\\lbrace n_{\\omega _{1},l}^-, n_{\\omega _{1},l}^+|l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} \\rbrace $ and a filtration $\\lbrace F_{\\alpha _1}|\\alpha _1\\in \\Omega _{\\omega _1} \\rbrace $ of $S_{\\kappa ,I}^{D,I}(K^p,\\Lambda _\\omega ^{>1/p})^\\vee $ consisting of free $\\Lambda _\\omega ^{>1/p}$ -modules.",
"We use $\\lbrace G_{\\alpha _1}|\\alpha _1\\in \\Omega _{\\omega _1} \\rbrace $ to denote the graded pieces of this filtration, i.e.",
"$G_{\\alpha _1}=F_{\\alpha _1}/F_{\\gamma _1}$ , if $\\gamma _1\\le \\alpha _1$ are two consecutive elements in $\\Omega _{1}$ (we denote $G_{n_{\\omega _{1},0}^-}=F_{n_{\\omega _{1},0}^-}=(0)$ ).",
"For any $\\alpha _1\\in \\Omega _{\\omega _{1}}$ , there exist a set $\\Omega _{\\omega _{2},\\alpha _1}=\\lbrace n_{\\omega _{2},\\alpha _1,l}^-, n_{\\omega _{2},\\alpha _1,l}^+|l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} \\rbrace $ and a filtration $\\lbrace F_{\\alpha _1,\\alpha _2}|\\alpha _2\\in \\Omega _{\\omega _{2},\\alpha _1} \\rbrace $ of $G_{\\alpha _1}$ consisting of free $\\Lambda _\\omega ^{>1/p}$ -modules; define $G_{\\alpha _1,\\alpha _2}=F_{\\alpha _1,\\alpha _2}/F_{\\alpha _1,\\gamma _2}$ if $\\gamma _2\\le \\alpha _2$ are two consecutive elements in $\\Omega _{\\omega _{2},\\alpha _1}$ .",
"For $2\\le k\\le g-1$ , given the set $\\Omega _{\\omega _{k},\\alpha _1,\\dots ,\\alpha _{k-1}}$ and the space $G_{\\alpha _1,\\dots , \\alpha _k}$ , there exist a set $\\Omega _{\\omega _{k+1},\\alpha _1,\\dots ,\\alpha _k}=\\lbrace n_{\\omega _{k+1},\\alpha _1,\\dots ,\\alpha _k,l}^-, n_{\\omega _{k+1},\\alpha _1,\\dots ,\\alpha _k,l}^+|l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} \\rbrace $ and a filtration $\\lbrace F_{\\alpha _1,\\dots ,\\alpha _{k+1}}|\\alpha _{k+1}\\in \\Omega _{\\omega _{k+1},\\alpha _1,\\dots ,\\alpha _k} \\rbrace $ of $G_{\\alpha _1,\\dots ,\\alpha _k}$ consisting of free $\\Lambda _\\omega ^{>1/p}$ -modules; define $G_{\\alpha _1,\\dots ,\\alpha _{k+1}}=F_{\\alpha _1,\\dots ,\\alpha _{k+1}}/F_{\\alpha _1,\\dots ,\\gamma _{k+1}}$ if $\\gamma _{k+1}\\le \\alpha _{k+1}$ are two consecutive elements in $\\Omega _{\\omega _{k+1},\\alpha _1,\\dots ,\\alpha _k}$ .",
"Moreover, we have the following properties of the above filtrations: all the filtrations are stable under the $U_{\\pi _i}$ -operators, for all $i\\in I$ ; and the graded pieces $G_{\\alpha _1,\\dots ,\\alpha _g}$ obtained above are free $\\Lambda _\\omega ^{>1/p}$ -modules of finite rank (possibly 0)."
],
[
"Conclusion",
"For any $x\\in \\mathcal {W}^{>1/p}_\\omega (\\mathbb {C}_p)$ , we use $\\chi _x:\\Lambda _\\omega ^{>1/p}\\rightarrow \\mathbb {C}_p$ to denote the corresponding homomorphism.",
"For any free $\\Lambda _\\omega ^{>1/p}$ -module $G_{\\alpha _1,\\dots , \\alpha _g}$ we obtained in the previous section, we denote $G_{\\alpha _1,\\dots ,\\alpha _g}(x)=G_{\\alpha _1,\\dots , \\alpha _g}\\otimes _{\\Lambda _\\omega ^{>1/p},\\chi _x}\\mathbb {C}_p$ .",
"Proposition 5.3.1 For all $1\\le k\\le g$ , if $\\alpha _k=n_{\\omega _{J_{k}},\\alpha _1\\dots ,\\alpha _{k-1},l}^-$ for some $l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} $ , the slopes of the $U_{\\pi _k}$ -operator on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ are all equal to $(p-1)v_p(\\chi _x(T_{i_k}))\\cdot l$ ; if $\\alpha _k=n_{\\omega _{J_{k}},\\alpha _1\\dots ,\\alpha _{k-1},l}^+$ for some $l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0} $ , the slopes of the $U_{\\pi _k}$ -operator on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ all belong to the interval $(p-1)v_p(\\chi _x(T_{i_k}))\\cdot (l,l+2)$ if $l\\ne 0$ , and the interval $(p-1)v_p(\\chi _x(T_{i_k}))\\cdot (0,1)$ if $l=0$ .",
"We give a proof for $l=1$ .",
"The argument for other $l$ 's is similar.",
"Recall that by Proposition REF , we have a filtration $\\lbrace \\tilde{V}_{\\alpha _1}|\\alpha _1\\in \\Omega _{\\omega _{1}} \\rbrace $ of the space $S_{\\kappa _1,J_1}^{D,J_1}(K^p,R_{1,\\omega _{1}})^\\vee $ .",
"We have an isomorphism $\\tilde{V}_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{d_{\\alpha _1}}R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ of left $R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ -modules for all $\\alpha _1\\in \\Omega _{\\omega _{1}}$ .",
"These isomorphisms induces isomorphisms $\\tilde{W}_{\\alpha _1}:= \\tilde{V}_{\\gamma _1}/\\tilde{V}_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{r_{\\alpha _1}}R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ .",
"Fix $\\alpha _1\\in \\Omega _{\\omega _{1}}$ and let $\\gamma _1\\in \\Omega _{\\omega _{1}}$ such that $\\alpha _1\\le \\gamma _1$ are two consecutive elements in $\\Omega _{\\omega _{1}}$ .",
"Under the above isomorphisms, the $U_{\\pi _1}$ -operator on $\\tilde{W}_{\\alpha _1}$ corresponds to a matrix $U_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ .",
"This matrix is strictly $\\lambda ^{(\\alpha _1,\\gamma _1]}$ -Hodge bounded with respect to the element $T_1\\in R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ .",
"First we assume that $\\alpha _1=n_{\\omega _{1},l}^-$ for some $l\\in \\lbrace 0\\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0}$ .",
"We can assume that $\\alpha _1\\ne \\gamma _1$ , or otherwise $G_{\\alpha _1,\\dots ,\\alpha _g}=0$ and the proposition is trivial in this case.",
"From the construction in Proposition REF , we see that $\\lambda _{\\alpha _1+1}=\\lambda _{\\gamma _1}=(p-1)l$ .",
"Hence there exists an invertible matrix $U^{\\prime }_{\\alpha _1}\\in \\operatorname{GL}_{r_{\\alpha _1}}(R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ , such that $U_{\\alpha _1}=T_1^{(p-1)l}U^{\\prime }_{\\alpha _1}$ .",
"Under the isomorphism $\\tilde{W}_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{r_{\\alpha _1}}R_{1,\\omega _{J_1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ , the matrix $U^{\\prime }_{\\alpha _1}$ defines an invertible $R_{1,\\omega _{J_1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ -linear operator $U^{\\prime }_{\\pi _1}$ on $\\tilde{W}_{\\alpha _1}$ .",
"Recall that $V_{\\alpha _1}$ is the image of $\\tilde{V}\\otimes _{R_{1,\\omega _{1}}}R_{2,\\omega _{2}}$ under the map $S_{\\kappa ,J_1}^{D,J_1}(K^p,R_{2,\\omega _{2}})^\\vee \\rightarrow S_{\\kappa ,J_2}^{D,J_1}(K^p,R_{2,\\omega _{2}})^\\vee $ .",
"Hence we have isomorphisms $V_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{d_{\\alpha _1}}R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket $ and $V^{mod}_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{d_{\\alpha _1}}R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ .",
"It induces isomorphisms $W_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{r_{\\alpha _1}}R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket $ and $W^{mod}_{\\alpha _1}\\cong \\prod \\limits _{k=1}^{r_{\\alpha _1}}R_{2,\\omega _{2}}\\llbracket \\mathcal {O}_{\\mathfrak {p}_2}\\times P^{\\prime }_{J_2^c}\\rrbracket ^{j_2-mod}$ .",
"Under these isomorphisms, the matrix $U^{\\prime }_{\\alpha _1}$ defines invertible operators $U^{\\prime }_{\\pi _1}$ on $W_{\\alpha _1}$ and $W_{\\alpha _1}^{mod}$ , but notice that these operators are only $R_{2,\\omega _{2}}\\llbracket P^{\\prime }_{J_2^c}\\rrbracket $ -linear.",
"Moreover, the operator $U_{\\pi _1}^{\\prime }$ preserves the filtrations $\\lbrace W_{\\alpha _1,\\alpha _2}^{mod}|\\alpha _2\\in \\Omega _{\\omega _{2},\\alpha _1} \\rbrace $ as the operator $U_{\\pi _1}$ does.",
"We apply the above argument inductively to the places $i_2,\\dots , i_g$ and conclude that there is an operator $U_{\\pi _1}^{\\prime }$ on the graded pieces $G_{\\alpha _1,\\dots , \\alpha _g}$ such that $U_{\\pi _1}^{\\prime }$ is invertible and $\\Lambda _\\omega ^{>1/p}$ -linear, and satisfies $U_{\\pi _1}=T_1^{(p-1)l}U^{\\prime }_{\\pi _1}$ .",
"Notice that the homomorphism $\\chi _x:\\Lambda _\\omega ^{>1/p}\\rightarrow \\mathbb {C}_p$ factors through $\\mathcal {O}_{\\mathbb {C}_p}$ and hence $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ admits a lattice $L_{\\alpha _1,\\dots ,\\alpha _g}(x)=G_{\\alpha _1,\\dots ,\\alpha _g}\\otimes _{\\Lambda _\\omega ^{>1/p},\\chi _x}\\mathcal {O}_{\\mathbb {C}_p}$ .",
"Both the operators $U_{\\pi _1}$ and $U_{\\pi _1}^{\\prime }$ preserve this lattice.",
"Since $U_{\\pi _1}^{\\prime }$ is invertible on $L_{\\alpha _1,\\dots , \\alpha _g}(x)$ , the slopes of $U^{\\prime }_{\\pi _1}$ are all 0.",
"So the slopes of $U_{\\pi _1}$ on $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ are all equal to $(p-1)v_p(\\chi _x((T_1)))\\cdot l$ .",
"Now we assume that $\\alpha _1=n_{\\omega _{1},l}^+$ for some $l\\in \\lbrace 0 \\rbrace \\cup 1+2\\mathbb {Z}_{\\ge 0}$ .",
"As in the first case, we can find a matrix $U^{\\prime }_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ , such that $U_{\\alpha _1}=T_1^{(p-1)l}U_{\\alpha _1}^{\\prime }$ , but this matrix is not invertible in general.",
"It follows that there exists a $\\Lambda _\\omega ^{>1/p}$ -linear operator $U^{\\prime }_{\\pi _1}$ on $G_{\\alpha _1,\\dots ,\\alpha _g}$ , such that $U_{\\pi _1}=T_1^{(p-1)l}U_{\\pi _1}^{\\prime }$ .",
"Since $U^{\\prime }_{\\pi _1}$ preserves the lattice $L_{\\alpha _1,\\dots ,\\alpha _g}(x)$ , the slopes of $U_{\\pi _1}^{\\prime }$ on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ are all nonnegative.",
"So the slopes of $U_{\\pi _1}$ on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ are all no less than $(p-1)v_p(\\chi _x((T_1)))\\cdot l$ .",
"Recall that $R_{1,\\omega _{1}}$ is isomorphic to $\\mathbb {Z}_p\\llbracket T_1,\\frac{p}{T_1},T\\rrbracket $ .",
"In particular, it is a local ring with residue field $\\mathbb {F}_p$ .",
"Hence we have a natural surjective map $R_{1,\\omega _{1}}\\rightarrow \\mathbb {F}_p$ .",
"We apply the homomorphisms $R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket \\rightarrow \\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ and $\\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket \\rightarrow \\mathbb {F}_p$ to entries of the matrix $U^{\\prime }_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(R_{1,\\omega _{1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ , and get matrices $U^{\\prime }_{\\alpha _1,\\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket }\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ and $U^{\\prime }_{\\alpha _1,\\mathbb {F}_p}\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathbb {F}_p)$ .",
"By Proposition REF , the slopes of the matrix $\\bar{U}_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathbb {F}_p\\llbracket T_1\\rrbracket )$ (with respect to the $T_1$ -adic valuation) all belong to the interval $(\\lambda _{\\alpha _1},\\lambda _{\\gamma _1})=((p-1)l,(p-1)(l+2))$ .",
"So the slopes of $\\bar{U}^{\\prime }_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathbb {F}_p\\llbracket T_1\\rrbracket )$ are all positive.",
"If we use $f(X)\\in \\mathbb {F}_p[X]$ to denote the characteristic polynomial of the matrix $U^{\\prime }_{\\alpha _1,\\mathbb {F}_p}$ , then we have $f(X)=X^{r_{\\alpha _1}}$ .",
"We use $\\mathcal {I}\\subset \\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket $ to denote the augmentation ideal of this complete group ring.",
"It follows from Hamilton-Cayley Theorem that $f(U^{\\prime }_{\\alpha _1,\\mathbb {F}_p})=(U^{\\prime }_{\\alpha _1,\\mathbb {F}_p})^{r_{\\alpha _1}}=0$ , and hence $(U^{\\prime }_{\\alpha _1,\\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket })^{r_{\\alpha _1}}\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathcal {I})$ .",
"Define $\\bar{L}_{\\alpha _1,\\dots ,\\alpha _g}(x)=L_{\\alpha _1,\\dots ,\\alpha _g}(x)\\otimes _{\\mathcal {O}_{\\mathbb {C}_p}}\\bar{\\mathbb {F}}_p$ .",
"It is a finite $\\bar{\\mathbb {F}}_p$ -vector space and $\\mathrm {Iw}_{\\pi }$ acts on it continuously.",
"It follows that the action of $\\mathcal {I}$ on $\\bar{L}_{\\alpha _1,\\dots ,\\alpha _g}(x)$ is nilpotent and hence $U^{\\prime }_{\\alpha _1,\\mathbb {F}_p\\llbracket P^{\\prime }_{J_1^c}\\rrbracket }$ is a nilpotent operator on $\\bar{L}_{\\alpha _1,\\dots ,\\alpha _g}(x)$ .",
"So the slopes of the $U^{\\prime }_{\\pi _1}$ -operator on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ are all positive, and we conclude that the slopes of the $U^{\\prime }_{\\pi _1}$ -operator on $G_{\\alpha _1,\\dots ,\\alpha _g}(x)$ are all strictly larger than $(p-1)v_p(\\chi _x((T_1)))\\cdot l$ .",
"Since the matrix $U_{\\alpha _1}$ is strictly $\\lambda ^{(\\alpha _1,\\gamma _1]}$ -Hodge bounded, we can find a matrix $V_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(R_{1,\\omega _{J_1}}\\llbracket P^{\\prime }_{J_1^c}\\rrbracket )$ , such that $U_{\\alpha _1}V_{\\alpha _1}=V_{\\alpha _1}U_{\\alpha _1}=T_1^{(p-1)(l+2)}\\cdot I_{r_{\\alpha _1}}$ .",
"The matrix $V_{\\alpha _1}$ induces a $\\Lambda _\\omega ^{>1/p}$ -linear operator $V_{\\pi _1}$ on $G_{\\alpha _1,\\dots , \\alpha _g}$ and $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ .",
"It preserves the lattice $L_{\\alpha _1,\\dots , \\alpha _g}(x)$ and satisfies $U_{\\pi _1}V_{\\pi _1}=V_{\\pi _1}U_{\\pi _1}=T_1^{(p-1)(l+2)}\\cdot \\mathrm {Id}$ .",
"It follows from Lemma REF that the slopes of $U_{\\pi _1}$ and $V_{\\pi _1}$ on $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ can be paired such that the slopes in each pair sum to $(p-1)v_p(\\chi _x((T_1)))\\cdot (l+2)$ .",
"We have seen that the slopes of $\\bar{U}_{\\alpha _1}\\in \\mathrm {M}_{r_{\\alpha _1}}(\\mathbb {F}_p\\llbracket T_1\\rrbracket )$ all belong to the interval $((p-1)l,(p-1)(l+2))$ .",
"Hence the slopes of $\\bar{V}_{\\alpha _1}$ are all positive.",
"By a similar argument as above, we can show that the slopes of $V_{\\pi _1}$ on $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ are all positive.",
"It follows that the slopes of $U_{\\pi _1}$ on $G_{\\alpha _1,\\dots , \\alpha _g}(x)$ are all strictly less than $(p-1)v_p(\\chi _x((T_1)))\\cdot (l+2)$ .",
"Now we are ready to prove our main theorem REF .",
"Let $X=\\mathrm {Sp}(A)\\subset \\mathcal {W}_\\omega ^{>1/p}$ be an affinoid subdomain that corresponds to the continuous homomorphism $\\chi :\\Lambda _\\omega ^{>1/p}\\rightarrow A$ .",
"For $l\\in \\lbrace 1,\\dots , g \\rbrace $ and a set of indices $\\lbrace \\alpha _k| k=1,\\dots , l \\rbrace $ with $\\alpha _1\\in \\Omega _{\\omega _{1}}$ , $\\alpha _{k+1}\\in \\Omega _{\\omega _{k+1},\\alpha _1,\\dots , \\alpha _k}$ for $k=1,\\dots , l-1$ , we define $F_{\\alpha _1,\\dots ,\\alpha _l,A}:= F_{\\alpha _1,\\dots ,\\alpha _l}\\hat{\\otimes }_{\\Lambda _\\omega ^{>1/p},\\chi }A$ and $G_{\\alpha _1,\\dots ,\\alpha _l,A}:= G_{\\alpha _1,\\dots ,\\alpha _l}\\hat{\\otimes }_{\\Lambda _\\omega ^{>1/p},\\chi }A$ .",
"Let $S_A:= S_{\\kappa ,I}^{D,I}(K^p,\\Lambda _\\omega ^{>1/p})\\hat{\\otimes }_{\\Lambda _\\omega ^{>1/p},\\chi }A$ and $\\psi _A:\\mathbf {T}\\rightarrow \\operatorname{End}_A(S_A)$ be the natural homomorphism, where $\\mathbf {T}$ is the (abstract) $\\mathbb {Q}_p$ -Hecke algebra defined in §REF .",
"Let $\\mathbf {T}_A$ be the image of $\\psi _A$ .",
"Under the above notations and by our previous construction in §REF , we obtain a filtration $\\lbrace F_{\\alpha _1}|\\alpha _1\\in \\Omega _{\\omega _{1}}\\rbrace $ of the $A$ -Banach module $S_A$ and the Hecke operators on $S_A$ all preserve this filtration.",
"It follows from Proposition REF that the induced homomorphism $\\mathbf {T}_A\\rightarrow \\prod \\limits _{\\alpha _1\\in \\Omega _{\\omega _{1}}}\\operatorname{End}_A(G_{\\alpha _1})$ is injective.",
"We inductively run the above argument, and obtain an injective homomorphism $\\mathbf {T}_A\\rightarrow \\prod \\limits _{\\alpha _1,\\dots ,\\alpha _g}\\operatorname{End}_A(G_{\\alpha _1,\\dots ,\\alpha _g})$ .",
"The main theorem REF follows from the injectivity of this map, the construction of the eigenvariety $\\mathcal {X}_D$ , and Proposition REF ."
],
[
"p-adic Langlands Functoriality",
"Definition 6.1.1 An eigenvariety datum is a tuple $\\mathfrak {D}=(\\mathcal {W},\\mathcal {Z},\\mathcal {M},\\mathbf {T},\\psi )$ , where $\\mathcal {W}$ is a separated reduced equidimensional, relatively factorial (see [12] for the precise definition) rigid analytic space, $\\mathcal {Z}\\subset \\mathcal {W}\\times \\mathbb {A}^1$ is a Fredholm hypersurface, $\\mathcal {M}$ is a coherent analytic sheaf over $\\mathcal {Z}$ , $\\mathbf {T}$ is a commutative $\\mathbb {Q}_p$ -algebra and $\\psi $ is a $\\mathbb {Q}_p$ -algebra homomorphism $\\psi :\\mathbf {T}\\rightarrow \\operatorname{End}_{\\mathcal {O}_\\mathcal {Z}}(\\mathcal {M})$ .",
"Remark 6.1.2 In §REF , we construct a quasi-coherent sheaf $\\mathcal {M}$ on $\\mathcal {W}$ coming from the spaces of of overconvergent automorphic forms for $D$ .",
"We can also construct a coherent sheaf $\\mathcal {M}^\\ast $ over the spectral variety $\\mathcal {Z}_D$ coming from the spaces $S_{\\kappa }^D(K^p\\mathrm {Iw}_{\\pi },r)$ 's.",
"The construction need a special admissible cover of $\\mathcal {Z}_D$ which consists of slope adapted affinoids of $\\mathcal {Z}_D$ .",
"We refer to [5] and [12] for the details of the construction.",
"Given an eigenvariety datum $\\mathfrak {D}$ as above, we use $\\mathcal {X}=\\mathcal {X}(\\mathfrak {D})$ to denote the eigenvariety associated to $\\mathfrak {D}$ , and we have a finite morphism $\\pi :\\mathcal {X}\\rightarrow \\mathcal {Z}$ and a morphism $w:\\mathcal {X}\\rightarrow \\mathcal {W}$ .",
"Recall that in [12] , the core $\\mathcal {X}^\\circ $ of $\\mathcal {W}$ is defined to be the union of $\\dim \\mathcal {W}$ -dimensional irreducible components of the nilreduction $\\mathcal {X}^{\\mathrm {red}}$ , and $\\mathcal {X}$ is unmixed if $\\mathcal {X}^\\circ \\cong \\mathcal {X}$ .",
"Remark 6.1.3 The eigenvarieties we will consider in this section are the eigenvarieties $\\mathcal {X}_D$ constructed in §REF and Hilbert modular eigenvarieties $\\mathcal {X}_{\\operatorname{GL}_{2/F}}$ constructed in [1].",
"These eigenvarieties are unmixed by [2].",
"Hence we assume that all the eigenvarieties are unmixed in the rest of this paper.",
"Given a point $z\\in \\mathcal {Z}$ and any $T\\in \\mathbf {T}$ , we write $D(T,X)(z)\\in k(z)[X]$ for the characteristic polynomial $\\det (1-\\psi (T)X)|_{\\mathcal {M}(z)}$ .",
"Now we can state Hansen's interpolation theorem ([12]), which is the main tool to translate our results to Hilbert modular eigenvarieties.",
"Theorem 6.1.4 Given two eigenvariety data $\\mathfrak {D}_i=(\\mathcal {W}_i,\\mathcal {Z}_i,\\mathcal {M}_i,\\mathbf {T}_i,\\psi _i)$ , let $\\mathcal {X}_i=\\mathcal {X}(\\mathfrak {D}_i)$ be the associated eigenvariety for $i=1,2$ .",
"Suppose that we are given the following additional data: a closed immersion $\\jmath :\\mathcal {W}_1\\rightarrow \\mathcal {W}_2$ and we use $j$ to denote the closed immersion $\\jmath \\times \\mathrm {id}:\\mathcal {W}_1\\times \\mathbb {A}^1\\rightarrow \\mathcal {W}_2\\times \\mathbb {A}^1$ ; a homomorphism of $\\mathbb {Q}_p$ -algebras $\\sigma :\\mathbf {T}_2\\rightarrow \\mathbf {T}_1$ ; a very Zariski dense set $\\mathcal {Z}_1^{\\mathrm {cl}}\\subset \\mathcal {Z}_1$ with $j(\\mathcal {Z}_1^{\\mathrm {cl}})\\subset \\mathcal {Z}_2$ , such that $D(\\sigma (T),X)(z)$ divides $D(T,X)(j(z))$ in $k(z)[X]$ for all $z\\in \\mathcal {Z}_1^{\\mathrm {cl}}$ and all $T\\in \\mathbf {T}_2$ .",
"Then there exists a morphism $i:\\mathcal {X}_1\\rightarrow \\mathcal {X}_2$ with the commutative diagrams ${\\mathcal {X}_1 [r]^{i} [d]^{w_1} & \\mathcal {X}_2 [d]^{w_2} \\\\\\mathcal {W}_1 [r]^{\\jmath } & \\mathcal {W}_2} ,{\\mathcal {O}(\\mathcal {X}_2) [r]^{i^\\ast } & \\mathcal {O}(\\mathcal {X}_1) \\\\\\mathbf {T}_2 [u]^{\\phi _2} [r]^{\\sigma } & \\mathbf {T}_1 [u]^{\\phi _1}}.$ Moreover, $i$ is a composite of a finite morphism followed by a closed immersion.",
"Remark 6.1.5 When $\\mathcal {W}_1=\\mathcal {W}_2=\\mathcal {W}$ , $\\mathbf {T}_1=\\mathbf {T}_2=\\mathbf {T}$ and $\\jmath :\\mathcal {W}_1\\rightarrow \\mathcal {W}_2$ and $\\sigma :\\mathbf {T}_2\\rightarrow \\mathbf {T}_1$ are the identity maps, the map $i:\\mathcal {X}_1\\rightarrow \\mathcal {X}_2$ in the above theorem is a closed immersion."
],
[
"Application to Hilbert modular eigenvarieties",
"Recall that $F$ is a totally real field in which $p$ splits and $D$ is a totally definite quaternion algebra over $F$ with discriminant $\\mathfrak {d}$ .",
"We assume that $(p,\\mathfrak {d})=1$ .",
"Fix a prime ideal $\\mathfrak {n}$ of $F$ p prime to $\\mathfrak {d}p$ .",
"Set $U_1(\\mathfrak {n}):=\\left\\lbrace \\gamma \\in \\operatorname{GL}_2(\\mathcal {O}_F\\otimes \\hat{\\mathbb {Z}})|\\gamma \\equiv \\left( \\begin{array}{cc}\\ast &\\ast \\\\0&1\\end{array}\\right)\\mathfrak {}\\mod {\\mathfrak {n}}\\right\\rbrace .$ When $\\mathfrak {n}$ is prime to $\\mathfrak {d}$ , we set $U^D_1(\\mathfrak {n}):=\\left\\lbrace \\gamma \\in (\\mathcal {O}_D\\otimes \\hat{\\mathbb {Z}})^\\times |\\gamma \\equiv \\left( \\begin{array}{cc}\\ast &\\ast \\\\0&1\\end{array}\\right)\\mathfrak {}\\mod {\\mathfrak {n}}\\right\\rbrace .$ Let $\\mathfrak {D}_1=(\\mathcal {W},\\mathcal {Z}_D,\\mathcal {M}_D,\\mathbf {T},\\psi _D)$ be the eigenvariety datum associated to the spaces of overconvergent automorphic forms of tame level $U^D_1(\\mathfrak {n})$ as constructed in §REF .",
"We use $\\mathcal {X}_D(U_1(\\mathfrak {n}))$ to denote the corresponding eigenvariety.",
"Let $\\mathfrak {D}_2=(\\mathcal {W},\\mathcal {Z},\\mathcal {M},\\mathbf {T},\\psi )$ be the eigenvariety datum associated to the spaces of overconvergent cuspidal Hilbert modular forms of tame level $U_1(\\mathfrak {n})$ as constructed in [1] $§5$ and let $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(U_1(\\mathfrak {n}))$ be the corresponding eigenvariety.",
"We define $\\mathcal {Z}_D^{cl}\\subset \\mathcal {Z}_D(\\mathbb {C}_p)\\subset (\\mathcal {W}\\times \\mathbb {A}^1)(\\mathbb {C}_p)$ to be the set of points $z=(\\chi ,\\alpha ^{-1})$ consisting of a classical weight $\\chi \\in \\mathcal {W}(\\mathbb {C}_p)$ corresponding to $(v=(v_i)\\in \\mathbb {Z}^I,r\\in \\mathbb {Z})$ and $\\alpha \\in \\mathbb {C}_p^\\ast $ with $v_p(\\alpha )<\\min \\limits _{i\\in I}\\lbrace n_i+1 \\rbrace $ .",
"Then it follows from the proof of [8] or the proof of [2] that the set $\\mathcal {Z}_D^{cl}$ is very Zariski dense in $\\mathcal {Z}_D$ .",
"From the classicality theorem for overconvergent automorphic forms for $D$ ([26]) and for overconvergent Hilbert modular forms ([25]), we have $\\mathcal {M}_D(z)=S_{k,w}^D(U_1^D(\\mathfrak {n}\\pi ))^{U_{\\pi }=\\alpha }, \\text{~and~} \\mathcal {M}(z)=S_{k,w}(U_1(\\mathfrak {n}\\mathfrak {d}\\pi ))^{U_\\pi =\\alpha }.$ Now we can apply Theorem REF to the eigenvariety data $\\mathfrak {D}_1$ and $\\mathfrak {D}_2$ together with the additional data $\\mathrm {id}:\\mathcal {W}\\rightarrow \\mathcal {W}$ and $\\mathrm {id}:\\mathbf {T}\\rightarrow \\mathbf {T}$ , and get the following theorem.",
"Theorem 6.2.1 ([2]) There is a closed immersion $i_D:\\mathcal {X}_D(\\mathfrak {n})\\rightarrow \\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n}\\mathfrak {d})$ interpolating the Jacquet-Langlands correspondence on non-critical classical points.",
"Moreover, when $g=[F:\\mathbb {Q}]$ is even, one can choose $D$ with $\\mathfrak {d}=1$ so that $i_D$ is an isomorphism.",
"When $g=[F:\\mathbb {Q}]$ is even, we can choose $D$ with $\\mathfrak {d}=1$ and use the above theorem to translate our main theorem REF to the Hilbert modular eigenvarieties.",
"When $g$ is odd, the discriminant $\\mathfrak {d}$ cannot be 1 and the immersion $i_D$ in Theorem REF is not surjective.",
"We need more work to get the desired results for Hilbert modular eigenvarieties.",
"We choose a quadratic extension $F^{\\prime }/F$ , such that $F^{\\prime }$ is totally real and $p$ splits completely in $F^{\\prime }$ .",
"Let $\\mathfrak {D}_3=(\\mathcal {W}^{\\prime },\\mathcal {Z}^{\\prime },\\mathcal {M}^{\\prime },\\mathbf {T}^{\\prime },\\psi ^{\\prime })$ be the eigenvariety datum associated to the spaces of overconvergent Hilbert modular forms of tame level $U^{\\prime }_1(\\mathfrak {n}^{\\prime })$ , where $\\mathfrak {n}^{\\prime }=\\mathfrak {n}\\mathcal {O}_{F^{\\prime }}$ and $U^{\\prime }_1(\\mathfrak {n}^{\\prime })=\\left\\lbrace \\gamma \\in \\operatorname{GL}_2(\\mathcal {O}_{G^{\\prime }}\\otimes \\hat{\\mathbb {Z}})|\\gamma \\equiv \\left( \\begin{array}{cc}\\ast &\\ast \\\\0&1\\end{array}\\right)\\mathfrak {}\\mod {\\mathfrak {n}}^{\\prime } \\right\\rbrace .$ Let $\\mathcal {O}_p^{\\prime }:=\\mathcal {O}_{F^{\\prime }}\\otimes \\mathbb {Z}_p$ .",
"The norm map $\\textrm {Nm}_{F^{\\prime }/F}:\\mathcal {O}_p^{\\prime \\times }\\rightarrow \\mathcal {O}_p^\\times $ induces a continuous homomorphism of the completed group rings: $\\phi :\\mathbb {Z}_p\\llbracket \\mathcal {O}_p^{\\prime \\times }\\times \\mathbb {Z}_p^\\times \\rrbracket \\rightarrow \\mathbb {Z}_p\\llbracket \\mathcal {O}_p^\\times \\times \\mathbb {Z}_p^\\times \\rrbracket $ , and hence gives a closed immersion $\\jmath :\\mathcal {W}\\rightarrow \\mathcal {W}^{\\prime }$ .",
"Define a homomorphism $\\sigma :\\mathbf {T}^{\\prime }\\rightarrow \\mathbf {T}$ of $\\mathbb {Q}_p$ -algebras as follows: for a prime $\\mathfrak {l}$ of $F$ , define $\\sigma (T_{\\mathfrak {l}_1})=\\sigma (T_{\\mathfrak {l}_2})=T_\\mathfrak {l}$ , if $\\mathfrak {l}$ splits as $\\mathfrak {l}\\mathcal {O}_{F^{\\prime }}=\\mathfrak {l}_1\\cdot \\mathfrak {l}_2$ in $F^{\\prime }$ ; $\\sigma (T_{\\mathfrak {l}^{\\prime }})=T_\\mathfrak {l}^2-2lS_\\mathfrak {l}$ , if $\\mathfrak {l}$ is inert in $\\mathcal {O}_{F^{\\prime }}$ and set $\\mathfrak {l}^{\\prime }=\\mathfrak {l}\\mathcal {O}_{F^{\\prime }}$ ; $\\sigma (T_{\\mathfrak {l}^{\\prime }})=T_\\mathfrak {l}$ , if $\\mathfrak {l}$ is ramified in $F^{\\prime }$ , and let $\\mathfrak {l}^{\\prime }$ be the unique prime of $F^{\\prime }$ over $\\mathfrak {l}$ .",
"Applying Theorem REF to the above datum, we have the following theorem.",
"Theorem 6.2.2 There is a morphism $i_{F^{\\prime }/F}:\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})\\rightarrow \\mathcal {X}_{\\operatorname{GL}_{2/F^{\\prime }}}(\\mathfrak {n}^{\\prime })$ interpolating the quadratic base change from $F$ to $F^{\\prime }$ on non-critical classical points.",
"To apply the above theorem to Hilbert modular eigenvarieties, we need to make some explicit computations.",
"Let $I^{\\prime }=\\operatorname{Hom}(F,\\bar{\\mathbb {Q}})=\\operatorname{Hom}(F,\\bar{\\mathbb {Q}}_p)$ .",
"We identify the set $I$ (resp.",
"$I^{\\prime }$ ) with the set of primes of $F$ (resp.",
"$F^{\\prime }$ ) over $p$ .",
"Since $p$ splits completely in $F^{\\prime }$ , every prime $i$ in $I$ splits into two primes in $I^{\\prime }$ , and we use $i_1^{\\prime }$ and $i_2^{\\prime }$ to denote the two primes.",
"Under the natural map $\\mathcal {O}_{\\mathfrak {p}_i}\\rightarrow \\mathcal {O}^{\\prime }_{\\mathfrak {p}_{i^{\\prime }_j}}$ , the image $\\pi _{i^{\\prime }_j}$ of $\\pi _i$ is a uniformizer of $\\mathcal {O}^{\\prime }_{\\mathfrak {p}_{i^{\\prime }_j}}$ , for $j=1,2$ .",
"As in §REF , we can use these $\\pi _{i^{\\prime }_j}$ 's for all $i\\in I$ and $j=1,2$ to define a full set $\\lbrace (T_{i^{\\prime }_j})_{i\\in I,j=1,2}, T \\rbrace $ of parameters on the weight space $\\mathcal {W}^{\\prime }$ .",
"Under these notations, the closed immersion $\\jmath :\\mathcal {W}\\rightarrow \\mathcal {W}^{\\prime }$ defined above can be described in term of parameters: if $x\\in \\mathcal {W}(\\mathbb {C}_p)$ has parameters $((w_i)_{i\\in I},w_0)$ , and $\\jmath (x)\\in \\mathcal {W}^{\\prime }(\\mathbb {C}_p)$ has parameters $((w_{i^{\\prime }})_{i^{\\prime }\\in I^{\\prime }},w_p^{\\prime })$ , then $w_{i_1^{\\prime }}=w_{i_2^{\\prime }}=w_i$ for all $i\\in I$ and $w_0=w_0^{\\prime }$ .",
"In particular, for any $r=(r_i)_{i\\in I}\\in \\mathcal {N}^I$ , if we define $r^{\\prime }=(r_{i^{\\prime }})_{i^{\\prime }\\in I^{\\prime }}\\in \\mathcal {N}^{I^{\\prime }}$ by $r_{i_1^{\\prime }}=r_{i_2^{\\prime }}=r_i$ for all $i\\in I$ , then $\\jmath ^{-1}(\\mathcal {W}^{\\prime >r^{\\prime }})=\\mathcal {W}^{>r}$ .",
"For any closed point $x^{\\prime }$ of $\\mathcal {X}_{\\operatorname{GL}_{2/F^{\\prime }}}(\\mathfrak {n}^{\\prime })$ , recall that we use $a_{i^{\\prime }}(x)$ to denote its corresponding $U_{\\pi _{i^{\\prime }}}$ -eigenvalue for all $i^{\\prime }\\in I^{\\prime }$ .",
"From the explicit description of the homomorphism $\\sigma :\\mathbf {T}^{\\prime }\\rightarrow \\mathbf {T}$ , we have $a_{i_1^{\\prime }}(i_{F^{\\prime }/F}(x))=a_{i_2^{\\prime }}(i_{F^{\\prime }/F}(x))=a_i(x)$ for any closed point $x$ of $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})$ .",
"Combining Theorems REF , REF and REF , we have the following description of the boundary behavior of Hilbert modular eigenvarieties: Theorem 6.2.3 We use $\\Sigma $ to denote the subset $\\lbrace 0 \\rbrace \\bigcup \\lbrace 1+2k|k\\in \\mathbb {Z}_{\\ge 0} \\rbrace $ of $\\mathbb {Z}$ .",
"The eigenvariety $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})^{>1/p}$ is a disjoint union $\\mathcal {X}_{\\operatorname{GL}_{2/F}}(\\mathfrak {n})^{>1/p}=\\bigsqcup _{l\\in \\Sigma ^I, \\sigma \\in \\lbrace \\pm \\rbrace ^I}\\mathcal {X}_{l,\\sigma }$ of (possibly empty) rigid analytic spaces which are finite over $\\mathcal {W}^{>1/p}$ via $w$ , such that for each closed point $x\\in \\mathcal {X}_{l,\\sigma }(\\mathbb {C}_p)$ with $l=(l_i)_{i\\in I}\\in \\Sigma ^I$ and $\\sigma =(\\sigma _i)_{i\\in I}\\in \\lbrace \\pm \\rbrace ^I$ , we have ${\\left\\lbrace \\begin{array}{ll}v_p(a_i(x)) = (p-1)v_p(T_{i,w(x)})\\cdot l_i,&\\textrm {if}\\ \\sigma _i=-;\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (l_i,l_i+2),&\\textrm {if}\\ \\sigma _i=+ \\textrm {~and~} l_i\\ne 0;\\\\v_p(a_i(x)) \\in (p-1)v_p(T_{i,w(x)})\\cdot (0,1),&\\textrm {if}\\ \\sigma _i=+ \\textrm {~and~} l_i=0,\\end{array}\\right.",
"}$ for all $i\\in I$ ."
]
] | 2005.14267 |
[
[
"Composition Estimation via Shrinkage"
],
[
"Abstract In this note, we explore a simple approach to composition estimation, using penalized likelihood density estimation on a nominal discrete domain.",
"Practical issues such as smoothing parameter selection and the use of prior information are investigated in simulations, and a theoretical analysis is attempted.",
"The method has been implemented in a pair of R functions for use by practitioners."
],
[
"Introduction",
"A composition refers to the proportions of a set of parts that make up a whole.",
"For example, the relative abundance of bacterial genera in some microbiome can be represented by a vector of proportions summing up to 1.",
"The analysis of compositional data [1] found applications in many fields including metagenomics (cf.",
"xing:17).",
"Our task here is to estimate composition using empirical data.",
"lhz:20 had excellent discussions of the challenges posed by composition estimation and the many approaches attempted in the literature.",
"Samples of microbiome are probed for the number of occurrences of bacterial genera, say, and a central issue is the prevalence of zero counts due to limitations of the probing technologies.",
"Zero proportions are not among acceptable answers, and the challenge is to estimate the proportions associated with zero counts in some reasoned manner.",
"Given multiple samples believed to have similar composition patterns, one may form an empirical composition matrix with columns adding up to 1 but containing entries of zeros, then share information among columns to produce estimates with desirable properties; lhz:20 proposed and illustrated an approach doing just that, using as estimates some low-rank approximation of the empirical composition matrix.",
"In this note, we explore a simple approach to composition estimation.",
"We take as input a matrix of raw counts, possibly with many 0's, and return a composition matrix of positive entries with the columns summing up to 1.",
"The method is actually designed to work with one column, with information shared by other columns helping to improve performance; prior information from other sources, if available, could be equally helpful.",
"We shall employ penalized likelihood density estimation on a nominal discrete domain for the task.",
"The method has long been developed and used with success in practice, but primarily on continuous domains.",
"We now take a serious look at how it performs on a discrete domain.",
"We shall first set the notation and outline the method, then investigate important practical issues such as smoothing parameter selection and the effective use of prior information.",
"A pair of R functions are available for use by practitioners.",
"Some theoretical analysis is also attempted to better understand how the method works."
],
[
"Method",
"Consider a multinomial sample $\\mathbf {k}\\sim \\text{Multinomial}(n;\\mathbf {p})$ , where $\\mathbf {p}=(p_{1},\\dots ,p_{m})^{T}$ , $\\sum _{y}p_{y}=1$ , $\\mathbf {k}=(k_{1},\\dots ,k_{m})^{T}$ , $\\sum _{y}k_{y}=n$ .",
"Our task is to estimate $\\mathbf {p}$ .",
"When $m$ is large and some of the $p_{y}$ 's are small, $k_{y}=0$ is not uncommon, and the simple maximum likelihood estimate $\\hat{p}_{y}=k_{y}/n$ is undesirable.",
"The estimation of $\\mathbf {p}$ can be cast as the estimation of probability density $f(y)$ on domain $\\mathcal {Y}=\\lbrace 1,\\dots ,m\\rbrace $ .",
"An approach is via the penalized likelihood method, minimizing $-\\frac{1}{n}\\sum _{y}\\Big \\lbrace k_{y}\\eta (y)-\\log \\int _{\\mathcal {Y}}e^{\\eta (y)}\\Big \\rbrace +\\frac{\\lambda }{2}J(\\eta ),$ where $f(y)=e^{\\eta (y)}/\\int _{\\mathcal {Y}}e^{\\eta (y)}$ , $J(\\eta )$ is a roughness penalty, and the smoothing parameter $\\lambda $ controls the trade-off between goodness-of-fit and smoothness.",
"See, e.g., silver:82 and gq:93.",
"With $y$ nominal, a standard choice for $J(\\eta )$ is $\\sum _{y}\\big (\\eta (y)-\\bar{\\eta }\\big )^{2}$ , where $\\bar{\\eta }=m^{-1}\\sum _{y}\\eta (y)$ , which ensures invariance with respect to permutations of elements in $\\mathcal {Y}$ .",
"This is estimation by shrinkage.",
"Through the specification of $\\int _{\\mathcal {Y}}g(y)$ , one may incorporate possible prior information concerning $p_{y}$ .",
"With $\\int _{\\mathcal {Y}}g(y)=\\sum _{y}w_{y}g(y)$ , the density $f(y)$ is relative to the base measure $\\lbrace w_{y}\\rbrace $ on $\\mathcal {Y}$ , $p_{y}=w_{y}e^{\\eta (y)}/\\sum _{x}w_{x}e^{\\eta (x)}$ .",
"Absent prior information, one simply sets $w_{y}\\propto 1$ , but if there is reason to suggest that $(p_{y}/\\tilde{p}_{y})$ 's are near a constant for some $\\tilde{p}_{y}$ , say, one may want to use $w_{y}\\propto \\tilde{p}_{y}$ .",
"A nearly uniform density (relative to a base measure) is easier to estimate.",
"With multiple samples $\\mathbf {k}_{x}\\sim \\text{Multinomial}(n_{x};\\mathbf {p}_{x})$ , $\\mathbf {p}_{x}=(p_{x,1},\\dots ,p_{x,m})^{T}$ , $\\sum _{y}p_{x,y}=1$ , $\\mathbf {k}_{x}=(k_{x,1},\\dots ,k_{x,m})^{T}$ , $\\sum _{y}k_{x,y}=n_{x}$ , where $\\mathbf {p}_{x}$ 's are believed to be close to each other, one may use the collapsed data $\\mathbf {k}=\\sum _{x}\\mathbf {k}_{x}$ to estimate a $\\tilde{p}_{y}$ with $w_{y}\\propto 1$ , then use $w_{y}\\propto \\tilde{p}_{y}$ for the estimation of individual $\\mathbf {p}_{x}$ 's."
],
[
"Density Estimation on Discrete Domain: Practice",
"We now look at how (REF ) performs on a discrete domain.",
"Some theoretical analysis is to be found in Section 5.",
"Computation is straightforward.",
"Our primary interests here are two fold, to check on the effectiveness of smoothing parameter selection by cross-validation, and to verify the benefit of using $w_{y}\\propto \\tilde{p}_{y}$ when prior information is available.",
"The cross-validation technique for the selection of $\\lambda $ in (REF ) as developed in gw:02 proved to be effective on continuous domains, and plays a central role behind the ssden facility in the R package gss [5]; technical details are to be found in gu:13 (gu:13, Sect.",
"7.3).",
"We now explore its performance on a discrete domain via simple simulation.",
"Take $m=100$ .",
"We first generate $Z_{y}\\sim {N}(0,1)$ , then produce 50 sets of $Z_{x,y}=Z_{y}+z_{x,y}$ where $z_{x,y}\\sim {N}(0,1/4)$ .",
"One then has 50 $\\mathbf {p}_{x}$ 's with $p_{x,y}\\propto {e}^{Z_{x,y}}$ .",
"Now split $N=10000=\\sum _{x}n_{x}$ total size to these 50 multinomial distributions in a random but slightly uneven fashion, with $n_{x}$ ranging from 104 to 314.",
"With the collapsed data of size 10k, one has a cross-validated estimate $\\tilde{p}_{y}$ .",
"For each of the 50 samples, four estimates were calculated, a pair each with $w_{y}\\propto 1$ and $w_{y}\\propto \\tilde{p}_{y}$ .",
"For each pair, one estimate was with $\\lambda $ minimizing the cross-validation score (as implemented in ssden) at $\\lambda _{v}$ , another with $\\lambda $ minimizing the Kullback-Leibler divergence $L(\\lambda )=\\text{KL}(\\mathbf {p},\\hat{\\mathbf {p}})=\\sum _{y}p_{y}\\log {p}_{y}/\\hat{p}_{y}$ at $\\lambda _{o}$ ; the subscript $x$ is omitted in the formula, but the 50 samples were generated from 50 $\\mathbf {p}_{x}$ 's with generally different sizes $n_{x}$ .",
"Ratios of $L(\\lambda _{o})/L(\\lambda _{v})$ are summarized in the boxplots in the left half of the left frame in Figure REF .",
"Figure: Density Estimation on Discrete Domain: Left: Relativeefficacy L(λ o )/L(λ v )L(\\lambda _{o})/L(\\lambda _{v}) with w y ∝1w_{y}\\propto 1 (widerboxes) and w y ∝p ˜ y w_{y}\\propto \\tilde{p}_{y} (thinner boxes).",
"Center andRight: 𝐩\\mathbf {p} (solid) and cross-validated 𝐩 ^\\hat{\\mathbf {p}} (faded)with w y ∝1w_{y}\\propto 1 (center) and w y ∝p ˜ y w_{y}\\propto \\tilde{p}_{y}(right); n=192n=192.The sizes $n_{x}\\in [103,314]$ are very low for $m=100$ , yet even with $w_{y}\\propto 1$ , $L(\\lambda _{o})$ 's were in the range $(0.080,0.257)$ with the median at $0.138$ ; the bahavior of cross-validation was dichotomous, succeeding 28 times and failing 22.",
"The method is capable, only if one can tune it properly by selecting the right $\\lambda $ in practice.",
"Using $w_{y}\\propto \\tilde{p}_{y}$ , $L(\\lambda _{o})$ 's moved to the range $(0.042,0.100)$ with the median at $0.064$ , and cross-validation performed reliably; $L(\\lambda _{v})$ 's had a range $(0.044,0.153)$ with the median at $0.080$ .",
"Repeating the exercise with a more reasonable total size $N=25k$ , for $n_{x}\\in [252,761]$ , parallel boxplots are shown in the right half of the frame; the range of $L(\\lambda _{v})$ 's for $w_{y}\\propto \\tilde{p}_{y}$ are now $(0.033,0.087)$ , with the median at $0.047$ .",
"The cross-validated estimates using a sample of $n=192$ are shown in the center and right frames in Figure 1, with $p_{y}$ 's sorted and plotted in solid.",
"The estimates $\\hat{p}_{y}$ 's in faded, vertically aligned with the $p_{y}$ 's, are invariant to permutations of $y$ , but the sorting cuts down on clutter.",
"There are 39 $k_{y}=0$ in the sample.",
"The $w_{y}\\propto 1$ fit has $\\text{KL}(\\mathbf {p},\\hat{\\mathbf {p}})=1.244$ (really bad); the layers of $\\hat{p}_{y}$ 's from below in the center frame correspond to $k_{y}=0,1,\\dots $ .",
"The $w_{y}\\propto \\tilde{p}_{y}$ fit has $\\text{KL}(\\mathbf {p},\\hat{\\mathbf {p}})=0.108$ , compared to $\\text{KL}(\\mathbf {p},\\tilde{\\mathbf {p}})=0.126$ ."
],
[
"Computation and Software",
"On $\\mathcal {Y}=\\lbrace 1,\\dots ,m\\rbrace $ , basis functions (i.e., the likes of unit vectors) are largely independent of each other, so one needs to entertain $m-1$ coefficients numerically, one each for all but one $y\\in \\mathcal {Y}$ ; the one less is due to a side condition needed on $\\eta (y)$ to ensure a one-to-one mapping $f(y)=e^{\\eta (y)}/\\int _{\\mathcal {Y}}e^{\\eta (y)}$ .",
"When $m$ is large, the $O(m^{3})$ execution time could be demanding.",
"With $w_{y}\\propto 1$ , $\\hat{\\eta }_{y}$ 's associated with $k_{y}=0$ are all equal by symmetry, as seen in the center frame of Figure REF , so one may use that as the baseline, and exclude bases associated with $k_{y}=0$ to save some computation.",
"A pair of R functions have been added to the gss package to facilitate the practical use of the proposed method.",
"One may use sscomp(x,wt) to perform density estimation on a nominal discrete domain, which takes $k_{y}$ in x; $k_{y}=0$ does need to be included, as the length of x gives $m$ .",
"The default $w_{y}\\propto 1$ can be overridden via wt.",
"The function is simply a stripped down version of ssden, tailor-made for use in the current setting.",
"It returns the cross-validated estimate $\\hat{\\mathbf {p}}$ as an $m\\times 1$ matrix.",
"With data in a matrix of $k_{x,y}$ 's of $m$ rows, one may use sscomp2(x), which collapses the columns to estimate $\\tilde{\\mathbf {p}}$ using sscomp with the default $w_{y}\\propto 1$ , calls sscomp with $w_{y}\\propto \\tilde{p}_{y}$ to estimate $\\mathbf {p}_{x}$ for each column, then returns the $\\hat{\\mathbf {p}}_{x}$ 's in a matrix matching the input.",
"Working with the log density $\\eta (y)$ , one guarantees $p_{y}>0$ , however small they might be.",
"The domain $\\mathcal {Y}$ enters (REF ) via $\\int _{\\mathcal {Y}}e^{\\eta (y)}$ , so is part of data in the setting.",
"As seen above in the empirical study of Section 3, and also later in the theoretical analysis of Section 5, $w_{y}$ 's mimicking the “shape” of $p_{y}$ 's allow (REF ) to perform better.",
"This resembles the mechanism behind importance sampling.",
"With a single sample, one may use other sources of prior information in lieu of $\\tilde{\\mathbf {p}}$ , if available.",
"While $k_{y}$ 's should be non-negative integers according to our multinomial formulation, the algorithm and the R code work with any non-negative numbers.",
"The same ratios $k_{y}/n$ represent the same empirical pattern reflecting $p_{y}$ , but a larger $n=\\sum _{y}k_{y}$ makes cross-validation to pick a smaller $\\lambda $ , tilting $\\hat{p}_{y}$ closer to $k_{y}/n$ ."
],
[
"Density Estimation on Discrete Domain: Theory",
"We now attempt some theoretical analysis concerning density estimation via (REF ) on a nominal discrete domain.",
"The theory developed in the literature on continuous domains (silver:82; coxosu:90; gq:93) does not apply here, as conditions concerning eigenvalues of the quadratic roughness functional $J(\\eta )$ no longer hold.",
"For a one-to-one mapping $p_{y}=w_{y}e^{\\eta _{y}}/\\sum _{x}w_{x}e^{\\eta _{x}}$ , we impose a side condition $\\sum _{y}\\eta _{y}=0$ .",
"It follows that $J(\\eta )=\\sum _{y}\\eta _{y}^{2}$ , and (REF ) can be written as $-\\frac{1}{n}\\sum _{y}k_{y}\\eta _{y}+\\log \\sum _{y}w_{y}e^{\\eta _{y}}+\\frac{\\lambda }{2}\\sum _{y}\\eta _{y}^{2}.$"
],
[
"Linear Approximation",
"Denoting by $p_{0,y}=w_{y}e^{\\eta _{0,y}}/\\sum _{x}w_{x}e^{\\eta _{0,x}}$ the true probabilities, substituting $\\log \\sum _{y}w_{y}e^{\\eta _{y}}$ in (REF ) by its quadratic approximation at $\\eta _{0,y}$ , and dropping terms not involving $\\eta _{y}$ , one has $-\\sum _{y}(\\tilde{k}_{y}-p_{0,y})\\eta _{y}+\\frac{1}{2}\\Big \\lbrace \\sum _{y}p_{0,y}(\\eta _{y}-\\eta _{0,y})^{2}-\\Big (\\sum _{y}p_{0,y}(\\eta _{y}-\\eta _{0,y})\\Big )^{2}\\Big \\rbrace +\\frac{\\lambda }{2}\\sum _{y}\\eta _{y}^{2}\\\\=-(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})^{T}\\eta +\\frac{1}{2}(\\eta -\\eta _{0})^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\eta -\\eta _{0})+\\frac{\\lambda }{2}\\eta ^{T}\\eta ,$ where $\\tilde{k}_{y}=k_{y}/n$ , $P_{0}=\\text{diag}(p_{0,1},\\dots ,p_{0,m})$ ; for the quadratic approximation, write $A(\\alpha )=\\log \\sum _{y}w_{y}e^{\\alpha (\\eta _{y}-\\eta _{0,y})+\\eta _{0,y}}$ , differentiate with respect to $\\alpha $ , then $A(1)\\approx {A}(0)+A^{\\prime }(0)+\\frac{1}{2}A^{\\prime \\prime }(0)$ .",
"We shall first analyze the minimizer of (REF ), which is linear in $\\tilde{k}_{y}$ , then bridge it with the minimizer of (REF ).",
"Differentiating (REF ) with respect to $\\eta $ and setting the gradient to 0, the minimizer $\\tilde{\\eta }$ satisfies $(\\lambda {I}+P-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})\\tilde{\\eta }=(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})+(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})\\eta _{0}.$ It follows that $\\tilde{\\eta }-\\eta _{0}=(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}\\big ((\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})-\\lambda \\eta _{0}\\big ).$ Write $B(\\alpha )=\\sum _{y}h_{y}w_{y}e^{\\alpha (\\eta _{1,y}-\\eta _{0,y})+\\eta _{0,y}}/\\sum _{y}w_{y}e^{\\alpha (\\eta _{1,y}-\\eta _{0,y})+\\eta _{0,y}}$ .",
"It is clear that $B(1)-B(0)=\\mathbf {h}^{T}(\\mathbf {p}_{1}-\\mathbf {p}_{0})$ , where $p_{1,y}=w_{y}e^{\\eta _{1,y}}/\\sum _{x}w_{x}e^{\\eta _{1,x}}$ .",
"A Taylor approximation $B(1)\\approx {B}(0)+B^{\\prime }(0)$ yields $\\mathbf {h}^{T}(\\mathbf {p}_{1}-\\mathbf {p}_{0})\\approx \\mathbf {h}^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\eta _{1}-\\eta _{0})$ , so $(\\mathbf {p}_{1}-\\mathbf {p}_{0})^{T}(\\eta _{1}-\\eta _{0})\\approx (\\eta _{1}-\\eta _{0})^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\eta _{1}-\\eta _{0})=V(\\eta _{1}-\\eta _{0});$ the left-hand-side is $\\text{KL}(\\mathbf {p}_{0},\\mathbf {p}_{1})+\\text{KL}(\\mathbf {p}_{1},\\mathbf {p}_{0})$ , the symmetrized Kullback-Leibler, and the right-hand-side quadratic proxy is a weighted mean square error of log probability, with $p_{0,y}$ 's as the weights.",
"Note that $V(\\eta +C)=V(\\eta )$ , invariant to the side condition imposed on $\\eta $ .",
"We shall try to bound $(\\lambda {J}+V)(\\tilde{\\eta }-\\eta _{0})=(\\tilde{\\eta }-\\eta _{0})^{T}(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\tilde{\\eta }-\\eta _{0}).$ By Cauchy-Schwartz, one can treat the bias term involving $\\eta _{0}$ and the variance term involving $(\\tilde{\\mathbf {k}}_{r}-\\mathbf {p}_{0})$ separately.",
"For the variance term, note that $E\\big [(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})^{T}\\big ]=n^{-1}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0})$ , so $E\\big [(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})^{T}(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})\\big ]\\\\=\\frac{1}{n}\\text{tr}\\big ((\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})\\big )=\\frac{1}{n}\\sum _{y}\\frac{\\rho _{y}}{\\lambda +\\rho _{y}}<\\frac{1}{n\\lambda },$ where $\\rho _{y}$ 's are the eigenvalues of $P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T}$ , $\\sum _{y}\\rho _{y}<1$ .",
"For the bias term, $\\lambda ^{2}\\eta _{0}^{T}(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\lambda {I}+P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})^{-1}\\eta _{0}\\le \\lambda \\,\\eta _{0}^{T}\\eta _{0}.$ Putting thing together, one has $(\\lambda {J}+V)(\\tilde{\\eta }-\\eta _{0})=O_{p}\\big (\\lambda \\,\\eta _{0}^{T}\\eta _{0}+(n\\lambda )^{-1}\\big ).$ When $(w_{y}/p_{0,y})$ 's are near a constant, $\\eta _{0}^{T}\\eta _{0}$ is small.",
"These bounds may not be sharp, but in case they are, the optimal rate also depends on how $\\eta _{0}^{T}\\eta _{0}$ grows with $m$ .",
"If $\\eta _{0}^{T}\\eta _{0}\\asymp {m}$ , then the optimal rate is $(\\lambda {J}+V)(\\tilde{\\eta }-\\eta _{0})=O_{p}\\big (\\sqrt{m/n}\\big )$ achieved at $\\lambda \\asymp (mn)^{-1/2}$ ; $m$ is allowed to grow with $n$ , but the growth rate should be slower than $O(n)$ to guarantee consistency.",
"Set $\\eta _{y}=\\hat{\\eta }_{y}+\\alpha {h}_{y}$ in (REF ), where $\\hat{\\eta }_{y}$ 's minimize (REF ) and $h_{y}$ 's are arbitrary.",
"Differentiating with respect to $\\alpha $ and setting the derivative at $\\alpha =0$ to 0, one has $-\\frac{1}{n}\\sum _{y}k_{y}h_{y}+\\sum _{y}\\hat{p}_{y}h_{y}+\\lambda \\sum _{y}\\hat{\\eta }_{y}h_{y}=-\\tilde{\\mathbf {k}}^{T}\\mathbf {h}+\\hat{\\mathbf {p}}^{T}\\mathbf {h}+\\lambda \\,\\hat{\\eta }^{T}\\mathbf {h}=0,$ where $\\hat{p}_{y}=w_{y}e^{\\hat{\\eta }_{y}}/\\sum _{x}w_{x}e^{\\hat{\\eta }_{x}}$ .",
"Setting $\\eta =\\tilde{\\eta }+\\alpha \\mathbf {h}$ in (REF ), differentiating with respect to $\\alpha $ , then setting the derivative at $\\alpha =0$ to 0, one has $-(\\tilde{\\mathbf {k}}-\\mathbf {p}_{0})^{T}\\mathbf {h}+(\\tilde{\\eta }-\\eta _{0})^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})\\mathbf {h}+\\lambda \\tilde{\\eta }^{T}\\mathbf {h}=0.$ Subtracting (REF ) from (REF ) and setting $\\mathbf {h}=\\hat{\\eta }-\\tilde{\\eta }$ , some algebra yields $\\lambda (\\hat{\\eta }-\\tilde{\\eta })^{T}(\\hat{\\eta }-\\tilde{\\eta })+(\\hat{\\mathbf {p}}-\\tilde{\\mathbf {p}})^{T}(\\hat{\\eta }-\\tilde{\\eta })\\\\=(\\tilde{\\eta }-\\eta _{0})^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\hat{\\eta }-\\tilde{\\eta })-(\\tilde{\\mathbf {p}}-\\mathbf {p}_{0})^{T}(\\hat{\\eta }-\\tilde{\\eta }).$ Using the mean value theorem in the arguments leading to (REF ), $(\\hat{\\mathbf {p}}-\\tilde{\\mathbf {p}})^{T}(\\hat{\\eta }-\\tilde{\\eta })=(\\hat{\\eta }-\\tilde{\\eta })^{T}(P_{1}-\\mathbf {p}_{1}\\mathbf {p}_{1}^{T})(\\hat{\\eta }-\\tilde{\\eta })$ where $P_{1}=\\text{diag}(p_{1,1},\\dots ,p_{1,m})$ , $\\mathbf {p}_{1}=(p_{1,1},\\dots ,p_{1,m})^{T}$ correspond to a convex combination $\\eta _{1}$ of $\\hat{\\eta }$ and $\\tilde{\\eta }$ .",
"Likewise, $(\\tilde{\\mathbf {p}}-\\mathbf {p}_{0})^{T}(\\hat{\\eta }-\\tilde{\\eta })=(\\tilde{\\eta }-\\eta _{0})^{T}(P_{2}-\\mathbf {p}_{2}\\mathbf {p}_{2}^{T})(\\hat{\\eta }-\\tilde{\\eta }).$ Assuming $\\mathbf {a}^{T}(P_{i}-\\mathbf {p}_{i}\\mathbf {p}_{i}^{T})\\mathbf {b}$ , $i=1,2$ be bounded by multiples of $\\mathbf {a}^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})\\mathbf {b}$ from below and above, one has from (REF ), $(\\lambda {J}+c_{1}V)(\\hat{\\eta }-\\tilde{\\eta })\\le {c}_{2}\\big |(\\tilde{\\eta }-\\eta _{0})^{T}(P_{0}-\\mathbf {p}_{0}\\mathbf {p}_{0}^{T})(\\hat{\\eta }-\\tilde{\\eta })\\big |$ for some $0<c_{1}, c_{2}<\\infty $ .",
"By Cauchy-Schwartz, $V(\\hat{\\eta }-\\tilde{\\eta })\\le (c_{2}/c_{1})^{2}V(\\tilde{\\eta }-\\eta _{0})$ , and in turn $\\lambda {J}(\\hat{\\eta }-\\tilde{\\eta })\\le {c}_{3}V(\\tilde{\\eta }-\\eta _{0})$ for some $0<c_{3}<\\infty $ .",
"Simple manipulation further propagates the rate of (REF ) to $(\\lambda {J}+V)(\\hat{\\eta }-\\eta _{0})$ ."
]
] | 2005.13988 |
[
[
"A new intrinsic metric and quasiregular maps"
],
[
"Abstract We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics.",
"We also prove distortion results for this metric under quasiregular maps."
],
[
"Introduction",
"Distance functions specific to a domain $G \\subset \\mathbb {R}^n, n\\ge 2\\,,$ or, as we call them, intrinsic metrics, are some of the key notions of geometric function theory and are currently studied by many authors.",
"See for instance the recent monographs [8], [9], [10], [15] and papers [4], [6], [7], [11], [12], [13].",
"In [8] intrinsic metrics are used as a powerful tool to analyse the properties of quasidisks and [10] provides a survey of some recent progress in the field.",
"A list of twelve metrics recurrent in geometric function theory is given in [15].",
"In the classical case $n=2$ one can define the hyperbolic metric of a simply connected domain by use of a conformal mapping given by the Riemann mapping theorem and the hyperbolic metric of the unit disk [3].",
"This metric is conformally invariant and therefore a most useful tool.",
"For dimensions $n\\ge 3,$ by Liouville's theorem, conformal mappings $f \\colon D \\rightarrow D^{\\prime }$ of domains $D, D^{\\prime } \\subset \\mathbb {R}^n$ are of the form $f= g | D$ where $g$ is a Möbius transformation [9], [10], and therefore there is no counterpart of the Riemann mapping theorem, and the planar procedure is not applicable.",
"This state of affairs led many researchers to look for generalized hyperbolic geometries and metrics which share at least some but not all properties of the hyperbolic metric [10].",
"For instance, the quasihyperbolic and distance ratio metrics studied in [4], [8], [9], [10] do not enjoy the full conformal invariance property for any dimension $n\\ge 2,$ both are invariant under similarity transformations only.",
"Here we study a function recently used as a tool by O. Dovgoshey, P. Hariri, and M. Vuorinen [6] and show that this function satisfies the triangle inequality and, indeed, defines an intrinsic metric of a domain.",
"Moreover, we compare it to the distance ratio metric and find two-sided bounds for it.",
"Finally, we study the behavior of this metric under quasiconformal mappings.",
"For a proper nonempty open subset $D \\subset {\\mathbb {R}}^n\\,$ and for all $x,y\\in D$ , the distance ratio metric $j_D$ is defined as $j_D(x,y)=\\log \\left( 1+\\frac{|x-y|}{\\min \\lbrace d_{D}(x),d_{D}(y) \\rbrace } \\right)\\,.$ For a proof of the triangle inequality, see [8], [1].",
"If there is no danger of confusion, we write $d_D(x)=d(x)= d(x,\\partial D)=\\operatorname{dist}(x, \\partial D)\\,.$ In this paper our goal is to prove that the expression (REF ) studied in [6] for $ (X,\\rho )=(D,j_D) $ is, in fact, a metric.",
"We also prove several upper and lower bounds for this new metric.",
"Theorem 1.1 Let $(X,\\rho )$ be a metric space and for $x,y\\in X, c>0\\,,$ let $ W(x,y):=\\log \\Big (1+2c\\sinh \\frac{\\rho (x,y)}{2}\\Big )\\,.$ If $c\\ge 1$ , then $W$ is a metric on $X$ .",
"Moreover, if $\\rho =j_{{\\mathbb {B}^2}}$ where ${\\mathbb {B}^2}$ is the unit disk, then $W$ is a metric on $\\mathbb {B}^2$ if and only if $c\\ge 1\\,.$ Theorem 1.3 Let $G$ be a proper subdomain of $\\mathbb {R}^n\\,.$ The following inequality holds for all $x,y \\in G$ $\\frac{j_G(x,y)}{2} \\le \\log \\Big (1+ 2 \\sinh \\frac{j_G(x,y)}{2}\\Big ) \\le \\min \\left\\lbrace j_G(x,y), \\frac{j_G(x,y)}{2} + \\log \\frac{5}{4} \\right\\rbrace \\,.$ Figure: The graphs of the functions y=t 2,y=log(1+2sinht 2) y=\\frac{t}{2}, y=\\log (1+2\\sinh \\frac{t}{2}), andy=min{t,t 2+log5 4} y=\\min \\lbrace t,\\frac{t}{2}+\\log \\frac{5}{4}\\rbrace in Theorem .We conclude our paper by studying the behavior of the metric of Theorem REF under quasiregular mappings defined on the unit disk and prove the following result, which is based on a recent sharp version of the Schwarz lemma for quasiregular mappings for $n=2$ [17].",
"Theorem 1.4 Let $f\\colon \\mathbb {B}^{2}\\rightarrow \\mathbb {B}^{2}$ be a non-constant $K$ -quasiregular mapping, where $K\\ge 1$ .",
"Denote by $\\rho =\\rho _{\\mathbb {B}^{2}}$ the hyperbolic metric and let $W_{\\lambda }\\left( x,y\\right)=\\log \\left( 1+2\\lambda \\sinh \\frac{\\rho \\left( x,y\\right) }{2}\\right) $ , where $ \\lambda \\ge 1$ and let $c(K)$ be the constant in Theorem REF.",
"For all $x,y\\in \\mathbb {B}^{2}$ $W_{\\lambda }\\left( f\\left( x\\right) ,f\\left( y\\right) \\right)\\le 2\\lambda c(K)\\max \\left\\lbrace W_{\\lambda }\\left( x,y\\right) ^{1/K},W_{\\lambda }\\left(x,y\\right) \\right\\rbrace .$"
],
[
"Preliminaries",
"We recall the definition of the hyperbolic distance $\\rho _{\\mathbb {B}^n}(x,y)$ between two points $x,y \\in \\mathbb {B}^n = \\lbrace x \\in \\mathbb {R}^n: |x|<1 \\rbrace $ [2]: $ \\tanh {\\frac{\\rho _{\\mathbb {B}^n}(x,y)}{2}}=\\frac{|x-y|}{\\sqrt{|x-y|^2+(1-|x|^2)(1-|y|^2)}}\\,.$ One of the main properties of the hyperbolic metric is its invariance under a Möbius self-mapping $T_a\\colon \\mathbb {B}^n \\rightarrow \\mathbb {B}^n\\,, $ with $T_a(a) = 0\\,, |a|<1\\,,$ of the unit ball $\\mathbb {B}^n\\,.",
"$ In other words, the mapping $T_a$ is an isometry.",
"By [2] we have for $ x,y\\in \\mathbb {H}^n = \\lbrace z \\in \\mathbb {R}^n: z_n>0\\rbrace $ $\\cosh {\\rho _{\\mathbb {H}^n}(x,y)}=1+\\frac{|x-y|^2}{2x_ny_n}\\,.$ For $ D \\in \\lbrace \\mathbb {B}^n, \\mathbb {H}^n \\rbrace $ and all $x, y \\in D$ we have by [10] $ j_D(x,y) \\le \\rho _D(x,y) \\le 2 j_D(x,y) \\,.$ By means of the Riemann mapping theorem one can extend the definition of the hyperbolic metric to the case of simply connected plane domains [3]."
],
[
"A new metric",
"Theorem 3.1 [6] Let $D$ be a nonempty open set in a metric space $(X, \\rho )$ and let $\\partial D \\ne \\varnothing $ .",
"Then the function $h_{D,c}(x,y) = \\log \\left(1+c\\frac{\\rho (x,y)}{\\sqrt{d_D(x)d_D(y)}}\\right)\\,,$ is a metric for every $c \\ge 2$ .",
"The constant 2 is best possible here.",
"This metric is listed in [5] and it has found some applications in [14].",
"Proposition 3.2 [6] For $c, t>0$ , let $ F_c(t)=\\log \\left(1+2c \\sinh {\\frac{t}{2}} \\right).$ Then the double inequality $\\frac{c}{2(1+c)}t< F_c(t) < ct$ holds for $c\\ge \\frac{1}{2}$ and $t>0\\, .",
"$ Lemma 3.3 [6] Let $D$ be a proper subdomain of $\\mathbb {R}^n$ .",
"Then for $c>0$ and $x,y \\in D$ $\\log \\Big (1+2 c \\sinh \\frac{j_D(x,y)}{2}\\Big )\\le h_{D,c}(x,y) \\le c j_D(x,y)\\,.$ We will now prove that the expression on the left hand side of the inequality of Lemma REF satisfies the triangle inequality and for that purpose we need the following refined form of Proposition REF for $c \\ge 1\\,.$ This refined result and some of the lower bounds that will be proved below for the function $F_c$ in Proposition REF , also lead to improved constants in some of the results of [6].",
"Lemma 3.4 The function ${F_c(t)}/{t}$ is decreasing from $(0,\\infty ) $ onto $(1/2,c)\\,$ if and only if $c\\ge 1\\,.$ Let $w(t) = 1+2c\\sinh (t/2)\\,.$ Differentiation yields $\\bigg (\\frac{F_c(t)}{t}\\bigg )^{\\prime }=\\frac{1}{t^{2}} g(t)\\,,\\quad g(t):=\\bigg ( \\frac{ct\\cosh (t/2)}{w(t)}-\\log (w(t))\\bigg ) \\,,$ $g^{\\prime }( t ) =\\frac{ct}{2\\left( w(t)\\right) ^{2}}\\bigg ( \\sinh \\Big (\\frac{t}{2}\\Big ) -2c\\bigg ) \\,.$ The equation $\\sinh \\left( \\frac{t}{2}\\right) =2c$ has the unique solution $t_{1}=2\\log \\left( 2c+\\sqrt{4c^{2}+1}\\right) >0\\,.$ We have $g^{\\prime }(t)<0$ for $0<t<t_{1}$ and $g^{\\prime }(t)>0$ for $ t>t_{1}$ .",
"Then $g$ is strictly decreasing on $\\left[ 0,t_{1}\\right] $ and strictly increasing on $[t_{1},\\infty )$ .",
"Note that $g\\left( t\\right) <g\\left( 0\\right) =0$ for $0<t\\le t_{1}$ .",
"Assume that the limit $L(c):=\\underset{t\\rightarrow \\infty }{\\lim }g\\left(t\\right) \\,$ is finite.",
"Then: if $L\\left( c\\right) \\le 0$ , then $g\\left( t\\right) <L\\left( c\\right)\\le 0$ for $t>t_{1}$ .",
"In this case, $g\\left( t\\right) <0$ for all $t>0$ .",
"It follows that ${F_c(t)}/{t}$ is strictly decreasing.",
"if $L\\left( c\\right) >0$ , then there exist a unique point $t_{2}>t_{1}$ such that $g\\left( t_{2}\\right) =0$ .",
"In this case $g\\left( t\\right) >0$ for all $t>t_{2}$ .",
"It follows that ${F_c(t)}/{t}$ is strictly decreasing on $ \\left[ 0,t_{2}\\right] $ and strictly increasing on $[t_{2},\\infty )$ .",
"Now we compute $L(c)\\,.$ Setting $s=t/2$ we see that $L(c)=\\lim _{s\\rightarrow \\infty }\\left( \\frac{2cs\\cosh (s)}{w(s)}-\\log (w(s))\\right)\\,.$ Now we use the change of variable $\\frac{1}{\\sinh \\left( s\\right) }=u$ when $s>0\\,.$ Then $1+2c\\sinh (s)=\\frac{u+2c}{u}$ and $\\cosh (s)=\\sqrt{1+\\frac{1}{u^{2}}}=\\frac{\\sqrt{u^{2}+1}}{u}$ .",
"Moreover, $\\sinh \\left( s\\right) =\\frac{1}{u}>0$ implies $s=\\log \\left( \\frac{1}{u}+\\sqrt{1+\\frac{1}{u^{2}}}\\right) \\,.",
"$ Write $ v=2c\\dfrac{\\sqrt{u^2+1}}{u+2c} $ .",
"It follows that $L(c)&=\\lim _{u\\rightarrow 0}\\left( v\\log \\bigg ( \\frac{1}{u}+\\sqrt{1+\\frac{1}{u^{2}}}\\,\\bigg )-\\log \\Big (1+\\frac{2c}{u}\\Big )\\right) \\\\&=\\lim _{u\\rightarrow 0}\\left[ v\\log \\left( 1+\\sqrt{u^{2}+1}\\right) -\\log \\left( u+2c\\right)+\\left( 1-v\\right) \\log u\\right] \\text{,} $ where $\\lim _{u\\rightarrow 0}v\\log \\left( 1+\\sqrt{u^{2}+1}\\right)=\\log 2\\,, \\quad \\lim _{u\\rightarrow 0}\\log \\left( u+2c\\right)=\\log \\left( 2c\\right) \\,.$ The limit $ \\lim _{u\\rightarrow 0}\\left( 1-v\\right) \\log u$ has the indeterminate form $0\\cdot \\infty $ .",
"But $\\underset{u\\rightarrow 0}{\\lim }\\left( 1-v\\right) \\log u&=\\underset{u\\rightarrow 0}{\\lim }\\left( \\frac{\\left(u+2c\\right) ^{2}-4c^{2}\\left( u^{2}+1\\right) }{\\left( u+2c\\right) \\left(u+2c+2c\\sqrt{u^{2}+1}\\right) }\\right) \\log u \\\\&=\\frac{1}{8c^{2}}\\lim _{u\\rightarrow 0}\\Big (\\left(1-4c^{2}\\right) u^{2}\\log u+4cu\\log u\\Big ) \\text{.}",
"$ Since $\\underset{u\\rightarrow 0}{\\lim }u\\log u=$ $\\underset{u\\rightarrow 0}{\\lim }u^{2}\\log u=0$ , (REF ) and (REF ) imply $L(c) =\\log 2-\\log \\left( 2c\\right) $ .",
"We have $L\\left( c\\right) \\le 0$ if $c\\ge 1$ and $L\\left( c\\right) >0$ if $0<c<1$ .",
"In conclusion, ${F_c(t)}/{t}$ is strictly decreasing on $\\left( 0,\\infty \\right) $ if and only if $c\\ge 1$ and its limit values at 0 and $\\infty $ follow easily.",
"3.5 Proof of Theorem REF .",
"The proof for $c \\ge 1$ follows readily from Lemma REF and a general property of metrics [1].",
"The well-known fact that $j_G(x,y)$ is a metric is recorded e.g.",
"in [1].",
"We next show that for $c \\in (0,1)$ the function $W(x,y) = \\log \\Big (1 + 2 c \\, \\sinh \\frac{j_{{\\mathbb {B}^2}}(x,y)}{2}\\Big )$ fails to satisfy the triangle inequality in the unit disk ${{\\mathbb {B}^2}}\\,.$ Write $E(x,y) = 1 + \\frac{|x-y|}{\\min \\lbrace 1-|x|, 1-|y|\\rbrace } \\,.$ The inequality $W\\left( x,z\\right) >W\\left(x,y\\right) +W\\left( y,z\\right) $ is equivalent to $\\frac{E\\left( x,z\\right) -1}{\\sqrt{E\\left( x,z\\right) }}>\\frac{E\\left(x,y\\right) -1}{\\sqrt{E\\left( x,y\\right) }}+\\frac{E\\left( y,z\\right) -1}{\\sqrt{E\\left( y,z\\right) }}+c\\frac{E\\left( x,y\\right) -1}{\\sqrt{E\\left(x,y\\right) }}\\frac{E\\left( y,z\\right) -1}{\\sqrt{E\\left( y,z\\right) }}\\text{.",
"}$ Assume that $0<y<z<1$ and $x=-y$ .",
"In this case, (REF ) writes as $&\\frac{z+y}{\\sqrt{\\left( 1+y\\right) \\left( 1-z\\right) }} \\\\& \\ >\\frac{2y}{\\sqrt{\\left( 1-y\\right) \\left( 1+y\\right) }}+\\frac{z-y}{\\sqrt{\\left( 1-y\\right)\\left( 1-z\\right) }}+c\\frac{2y\\left( z-y\\right) }{\\left( 1-y\\right)\\sqrt{\\left( 1+y\\right) \\left( 1-z\\right) }}\\text{,}$ which is equivalent to $H\\left( y,z\\right) :=p\\left( y\\right) z+q\\left( y\\right) \\sqrt{1-z}-r\\left(y\\right) <0\\text{,}$ where $p\\left( y\\right):=\\frac{1}{\\sqrt{1-y}}+\\frac{2cy}{\\left( 1-y\\right)\\sqrt{1+y}}-\\frac{1}{\\sqrt{1+y}}\\,,\\quad q\\left( y\\right):=\\frac{2y}{\\sqrt{\\left( 1-y\\right) \\left( 1+y\\right) }}$ $r\\left( y\\right):=y\\left(\\frac{1}{\\sqrt{1-y}}+\\frac{2cy}{\\left( 1-y\\right) \\sqrt{1+y}}+\\frac{1}{\\sqrt{1+y}}\\right) \\,.$ Note that $\\underset{z\\nearrow 1}{\\lim }H\\left( y,z\\right) =p\\left( y\\right)-r\\left( y\\right) =\\sqrt{1-y}+\\frac{2cy}{\\sqrt{1+y}}-\\sqrt{1+y}$ and that $\\underset{y\\nearrow 1}{\\lim }\\left( p\\left( y\\right)-r\\left( y\\right)\\right)=\\sqrt{2}\\left( c-1\\right) <0$ .",
"Take $0<a<1$ such that $p\\left(a\\right) -r\\left( a\\right) <0$ .",
"Since $\\underset{z\\nearrow 1}{\\lim }H\\left(a,z\\right)=p\\left( a\\right) -r\\left( a\\right) <0$ , we may choose $a<b<1$ such that $H\\left( a,b\\right) <0$ .",
"The latter inequality implies $W\\left(-a,b\\right) >W\\left( -a,a\\right) +W\\left( a,b\\right) \\,.$ $\\Box $ Our next result refines, for $(2c-1)t>1\\,,$ the upper bound in Proposition REF .",
"Because Lemma REF will not be used and its proof is straightforward and tedious, its proof is placed in an appendix at the end of the paper.",
"Lemma 3.7 The inequality $F_c(t)\\le \\frac{1}{2}\\frac{t^2+(2c+1)t}{t+1},$ holds for $ c>0 $ and $ t>0 $ .",
"The next result refines the lower bound of Proposition REF for $c\\ge 1, t\\ge 0\\,$ and the upper bound of Lemma REF for each $c>0$ and large enough $t\\,.$ Lemma 3.9 For $t\\ge 0$ , the following inequalities hold $\\frac{t}{t+1}\\log c+\\frac{t}{2}\\le F_c(t)\\,,\\text{ }c\\ge 1\\,, \\qquad \\mathrm {(1)}$ $F_c(t)\\le \\log \\Big (1+\\frac{1}{4c}\\Big )+\\frac{t}{2},\\text{ }c>0\\,.",
"\\qquad \\mathrm {(2)}$ Equality holds in (1) if and only if $t=0$ , respectively in (2) if and only if $c\\ge \\frac{1}{2}$ and $t=2\\log (2c)\\,.$ (REF ) For each fixed $t>0$ , we consider the expression $L_{t}(c)=1+c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})-c^{\\frac{t}{t+1}}e^{\\frac{t}{2}}$ as a function of $c\\,.$ The derivative is $L_{t}^{\\prime }(c)=e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}-\\frac{t}{t+1}c^{-\\frac{1}{t+1}}e^{\\frac{t}{2}},$ and the equation $L_{t}^{\\prime }(c)=0$ has the unique solution $c=c_{0}:=\\Big (\\frac{t}{t+1}\\frac{1}{1-e^{-t}}\\Big )^{t+1}.",
"$ The inequality $e^{t}>t+1$ for $t>0\\,,$ implies that $c_{0}<1$ , and hence $ L_{t}^{\\prime }(c)>0$ holds for $c\\ge 1$ .",
"Since $L_{t}(1)=1-e^{-\\frac{t}{2}}>0$ , we have $L_{t}(c)>0$ for $c\\ge 1$ .",
"Hence the following inequality holds for $c\\ge 1$ and $t>0$ , $c^{\\frac{t}{t+1}}e^{\\frac{t}{2}}<1+c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}).$ Considering the logarithms of both sides, we have the assertion.",
"(REF ) By the arithmetic-geometric mean inequality, $2\\le x+\\frac{1}{x}$ holds for all $x>0$ , hence $2-2ce^{-\\frac{t}{2}}\\le \\frac{1}{2c}e^{\\frac{t}{2}}\\,.$ Adding $2ce^{\\frac{t}{2}}$ to the both sides of this inequality, dividing by 2 and taking the logarithm, we obtain (REF ).",
"Equality holds in (REF ) if and only if $2ce^{-\\frac{t}{2}}=1$ , i.e.",
"$t=2\\log \\left( 2c\\right) $ .",
"Figure: The graphs of the functions y=l(c,t),y=F c (t) y=l(c,t), y=F_c(t), andy=u(c,t) y=u(c,t) in Lemma .Lemma 3.13 Let $l(c,t)=\\left\\lbrace \\begin{array}{ll}\\dfrac{t}{t+1}\\log c+\\dfrac{t}{2}\\quad & \\mbox{if \\ } c\\ge 1,\\ t\\ge 0 \\\\\\dfrac{ct}{2} & \\mbox{if \\ } 0<c\\le 1,\\ t\\ge 0\\end{array}\\right.",
"$ and $u(c,t)=\\min \\Big \\lbrace \\log \\Big (c+\\frac{1}{4c}\\Big )+\\frac{t}{2},\\ \\ \\log (1+ct)+c(e^{t}-1)\\Big \\rbrace .",
"$ Then, for $t>0$ and $c>0$ , the following inequalities hold $l(c,t)< F_c(t) \\le u(c,t).",
"$ The upper bound is attained, for $t>0$ and $c>0,$ if and only if $c>\\frac{1}{2}$ and $t=2\\log (2c).$ We first prove the lower bounds.",
"By Lemma REF it is enough to prove the case $0<c\\le 1.$ For each fixed $t>0$ , let $f_{t}(c)=1+c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})-e^{\\frac{ct}{2}}.$ Then $f_{t}(0)=0$ and $f_{t}^{\\prime }(c)=e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}-\\frac{t}{2}e^{\\frac{ct}{2}}$ is decreasing on the real axis.",
"Because $f_{t}^{\\prime }(0)=e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}-\\frac{t}{2}>0$ , the sign of $f_{t}^{\\prime }(1)=e^{\\frac{t}{2}}\\left( 1-\\frac{t}{2}-e^{-t}\\right) $ depends on $t$ .",
"The equation $1-\\frac{t}{2}-e^{-t}=0$ has a unique positive solution $t_{0}\\in \\left(1,2\\right) $ and we have $f_{t}^{\\prime }(1)>0$ for $0<t<t_{0}$ and $f_{t}^{\\prime }(1)<0$ for $t>t_{0}$ .",
"If $0<t\\le t_{0}$ , it follows that $f_{t}$ is increasing on $\\left[ 0,1\\right] $ , therefore $f_{t}(c)>0$ for $0<c\\le 1$ .",
"If $t>t_{0}$ , there is a unique positive critical point $c=c_{0}$ as a solution of $f_{t}^{\\prime }(c)=0\\,.$ The function $f_{t}$ attains the maximum at $c=c_{0}$ .",
"Since $f_{t}(0)=0$ and $f_{t}(1)=1-e^{-\\frac{t}{2}}>0$ , we have $f_{t}(c)>0$ for $0<c\\le 1$ .",
"Therefore $e^{\\frac{ct}{2}}<1+2c\\sinh \\frac{t}{2}$ holds for $0<c\\le 1$ which yields the desired inequality.",
"We now prove the upper bound.",
"By Lemma REF it is enough to prove that $\\log \\Big (1+2c\\sinh \\frac{t}{2}\\Big )<\\log (1+ct)+c(e^{t}-1)$ holds for $c>0$ and $t>0.$ For each fixed $t>0\\,,$ let $g_{t}(c)=(1+ct)e^{c(e^{t}-1)}-1-c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})\\,.$ Then, $g_{t}(c)$ is increasing on $(0,\\infty )$ , because $g_{t}^{\\prime }(c)=e^{c(e^{t}-1)}\\big ((1+ct)(e^{t}-1)+t\\big )-(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})$ is clearly increasing and $g_{t}^{\\prime }(0)=e^{t}-1+t-(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})=(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})(e^{\\frac{t}{2}}-1)+t>0\\,.$ Therefore, $g_{t}(c)>0$ holds as $g_{t}(0)=0$ .",
"Hence, we have $1+c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})<(1+ct)e^{c(e^{t}-1)}\\,,$ which yields the assertion.",
"Moreover, for $t>0$ , $c>0$ , the equation $\\log \\Big (1+2c\\sinh \\dfrac{t}{2}\\Big )=u(c,t)$ leads to $\\log \\Big (1+2c\\sinh \\dfrac{t}{2}\\Big )=\\log \\Big (c+\\frac{1}{4c}\\Big )+\\frac{t}{2}<\\log (1+ct)+c(e^{t}-1),$ which holds if and only if $c>\\frac{1}{2}$ and $t=2\\log (2c)$ .",
"3.17 Proof of Theorem REF .",
"The upper bound follows from Lemmas REF , REF and the lower bound from Lemma REF .",
"$\\Box $ The above results readily give the following theorem.",
"Theorem 3.18 For points $x,y \\in G,$ and a number $ c \\ge 1\\,,$ we have $L \\, j_G(x,y) \\le W(x,y) \\le U\\, j_G(x,y)$ where $W$ is the metric $W(x,y)= \\log \\Big ( 1+ 2c \\sinh \\frac{j_G(x,y)}{2}\\Big )$ and $L= \\frac{1}{2} +\\frac{\\log (c)}{1+ j_G(x,y)}\\,, \\quad U= \\frac{j_G(x,y)+(2c+1)}{2(1+ j_G(x,y))} \\,.$ Proposition 3.19 Let $\\left( G,\\rho _{G}\\right) $ and $(D,\\rho _{D})$ be two metric spaces and let $\\omega _{G,c}=\\log \\left( 1+2c\\sinh \\frac{\\rho _{G}}{2}\\right) $ and $\\omega _{D,c}=\\log \\left( 1+2c\\sinh \\frac{\\rho _{D}}{2}\\right) $ , where $c\\ge 1$ .",
"If $f\\colon (G,\\rho _{G})\\rightarrow (D,\\rho _{D})$ is an $L$ -Lipschitz function, then $f\\colon (G,\\omega _{G,c})\\rightarrow (D,\\omega _{D,c})$ is $L^{\\prime }$ -Lipschitz, where $L^{\\prime }=L$ if $L\\ge 1$ and $L^{\\prime }=cL$ if $L>0$ .",
"Conversely, if $f\\colon (G,\\omega _{G,c})\\rightarrow (D,\\omega _{D,c})$ is an $L^{\\prime }$ -Lipschitz function with $L^{\\prime }>0 $ , then $f\\colon (G,\\rho _{G})\\rightarrow (D,\\rho _{D})$ is an $2cL^{\\prime }$ -Lipschitz function.",
"Denote $F_{c}(t)=\\log \\big (1+2c\\sinh \\left( \\frac{t}{2}\\right) \\big )$ as in Lemma REF .",
"By Theorem REF , $\\omega _{G,c}=F_{c}\\circ \\rho _{G}$ and $\\omega _{D,c}=F_{c}\\circ \\rho _{D}$ are metrics on $G$ and $D$ , respectively.",
"Fix distinct points $x,y\\in G\\,.$ Assume that $f\\colon (G,\\rho _{G})\\rightarrow (D,\\rho _{D})$ is $L$ -Lipschitz.",
"We have to prove that $F_{c}\\big (\\rho _{D}(f(x),f(y))\\big )\\le L^{\\prime }F_{c}\\big ( \\rho _{G}(x,y)\\big ),$ where $L^{\\prime }=L$ if $L\\ge 1$ and $L^{\\prime }=cL$ whenever $L>0$ .",
"Since $f\\colon (G,\\rho _{G})\\rightarrow (D,\\rho _{D})$ is $L$ -Lipschitz and $F_{c}$ is increasing, $F_{c}\\big (\\rho _{D}(f(x),f(y))\\big )\\le F_{c}\\big ( L\\rho _{G}(x,y)\\big ) \\,.$ Assume first that $L\\ge 1$ .",
"By Lemma REF , $\\frac{F_{c}(t)}{t}$ is decreasing on $\\left( 0,\\infty \\right) $ , therefore $F_{c}(Lt)\\le LF_{c}(t)$ for all $t>0$ , as $L\\ge 1$ .",
"Then $F_{c}\\big ( L\\rho _{G}(x,y)\\big ) \\le LF_{c}\\big ( \\rho _{G}(x,y)\\big ) $ .",
"The latter inequality and (REF ) imply (REF ) with $L^{\\prime }=L$ .",
"By Lemma REF , $\\frac{t}{2}\\le F_{c}(t)\\le ct$ for all $t\\ge 0$ , if $c\\ge 1$ .",
"For all $L>0$ , (REF ) implies $F_{c}\\big (\\rho _{D}(f(x),f(y))\\big )\\le F_{c}\\big ( L\\rho _{G}(x,y)\\big ) \\le cL\\rho _{G}\\left( x,y\\right) .$ Now assume that (REF ) holds.",
"Then $\\rho _{D}(f(x),f(y))&\\le 2F_{c}\\big (\\rho _{D}(f(x),f(y))\\big )\\le L^{\\prime }F_{c}\\big ( \\rho _{G}(x,y)\\big ) \\\\&\\le 2cL^{\\prime }\\rho _{G}(x,y).$"
],
[
"Metrics and quasiregular maps",
"If $D\\in \\lbrace \\mathbb {B}^{n},\\mathbb {H}^{n}\\rbrace $ and $\\rho _{D}$ is the hyperbolic metric on $D$ , then the metric defined on $D$ by $W_{c}(x,y)=\\log \\left( 1+2c\\sinh \\frac{\\rho _{D}\\left( x,y\\right) }{2}\\right) $ , where $c\\ge 1$ , is invariant under Möbius self maps of $D$ , due to the Möbius invariance of the hyperbolic metric.",
"We also recall some notation about special functions and the fundamental distortion result of quasiregular maps, a variant of the Schwarz lemma for these maps, see [10].",
"For $r\\in (0,1)$ and $K>0$ , we define the distortion function $\\varphi _K(r)=\\mu ^{-1}(\\mu (r)/K),$ where $\\mu (r)$ is the modulus of the planar Grötzsch ring, a decreasing homeomorphism $\\mu \\colon (0,1)\\rightarrow (0,\\infty )\\,,$ see [1], [10].",
"Theorem 4.1 Let $G_1$ and $G_2$ be simply-connected domains in $\\mathbb {R}^2$ and let $f\\colon G_1\\rightarrow G_2=f(G_1)$ be a $K$ -quasiregular mapping.",
"Then for all $x,y\\in G_1$ $\\rho _{G_2}(f(x),f(y))\\le c(K)\\max \\lbrace \\rho _{G_1}(x,y),\\rho _{G_1}(x,y)^{1\\slash K}\\rbrace $ where $c(K)$ is as in [10], [17].",
"Remark 4.2 By [10], $K\\le c(K)\\le \\log (2(1+\\sqrt{1-1\\slash e^2}))(K-1)+K$ and, in particular, $c(K)\\rightarrow 1$ , when $K\\rightarrow 1$ .",
"4.3 Proof of Theorem REF .",
"By Theorem REF [10], $\\rho \\left( f\\left( x\\right) ,f\\left( y\\right) \\right) \\le c(K)\\max \\left\\lbrace \\rho \\left( x,y\\right) ^{1/K},\\rho \\left( x,y\\right) \\right\\rbrace .$ According to Lemma REF , $\\frac{t}{2}<\\log \\left( 1+2\\lambda \\sinh \\frac{t}{2}\\right)<\\lambda t \\text{\\,\\, for every } \\,\\, t\\in \\left( 0,\\infty \\right) \\text{.",
"}$ Then for all $x,y\\in \\mathbb {B}^{2}$ $W_{\\lambda }\\left( f\\left( x\\right) ,f\\left( y\\right) \\right)&\\le \\lambda \\rho (f(x),f(y))\\le \\lambda c(K)\\max \\left\\lbrace \\rho \\left(x,y\\right) ^{1/K},\\rho \\left( x,y\\right) \\right\\rbrace \\\\&\\le \\lambda c(K)\\max \\left\\lbrace 2^{1/K}W_{\\lambda }\\left( x,y\\right)^{1/K},2W_{\\lambda }\\left( x,y\\right) \\right\\rbrace $ and (REF ) follows.",
"$\\Box $"
],
[
"Appendix",
"We give here the proof of Lemma REF .",
"We shall apply the inequality $1-e^{-t}=\\frac{e^t-1}{e^t}> \\frac{t}{t+1}\\,$ which easily follows from $ e^{t}-(t+1)>0 \\,, t> 0\\,.$ 5.2 Proof of Lemma REF .",
"The right hand side of (REF ) can be written as $ \\dfrac{1}{2} t+\\dfrac{ct}{t+1} $ .",
"Taking the exponential function of both sides of (REF ), we need to check that for each fixed $ t >0 $ the following inequality holds, $E_t(c):=e^{\\frac{t}{2}}\\cdot e^{\\frac{ct}{t+1}}-1-c(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})>0, \\quad (c>0).$ First, we remark that $ E_t(0) =e^{\\frac{t}{2}}-1> 0 $ holds for $ t>0 $ .",
"Next, we will show that $ E^{\\prime }_t(c) >0 $ holds for $ c>0 $ .",
"It is clear that the derivative $E^{\\prime }_t(c)=e^{\\frac{t}{2}}\\Big (\\frac{t}{t+1}e^{\\frac{t}{t+1}c}-1+e^{-t}\\Big )$ is an increasing function with respect to $ c $ , and satisfies $ \\lim _{c\\rightarrow \\infty }E^{\\prime }_t(c)=\\infty $ .",
"From (REF ), we have $E^{\\prime }_t(0)=e^{\\frac{t}{2}}\\Big (\\frac{t}{t+1}-(1-e^{-t})\\Big )<0.$ Hence, $ E^{\\prime }_t(c)=0 $ has the unique positive root $ c_0 $ , and $ E_t(c) $ attains the minimum at $ c=c_0 $ .",
"Moreover, the root $ c_0 $ is given by the formula $c_0=\\frac{t+1}{t}\\log \\Big (\\frac{t+1}{t}(1-e^{-t})\\Big ).$ Then, $E_t(c_0)=\\frac{t+1}{t}\\Big (1-\\log \\big (\\frac{t+1}{t}(1-e^{-t})\\big )\\Big )(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})-1.$ Therefore, we need to show that $C(t):=\\Big (1-\\log \\big (\\frac{t+1}{t}(1-e^{-t})\\big )\\Big )(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})-\\frac{t}{t+1}>0 \\quad (t>0).$ By using (REF ), we have $C(t) &\\ge \\Big (1-\\log \\big (\\frac{t+1}{t}(1-e^{-t})\\big )\\Big )(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}})-\\frac{e^{t}-1}{e^t}\\\\&= \\Big (1-\\log \\big (\\frac{t+1}{t}(1-e^{-t})\\big )-e^{-\\frac{t}{2}}\\Big )(e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}).$ Since $e^{\\frac{t}{2}}-e^{-\\frac{t}{2}} > 0 $ , we have to check that $\\widetilde{C}(t)&:=1-\\log \\Big (\\frac{t+1}{t}(1-e^{-t})\\Big )-e^{-\\frac{t}{2}}\\\\&=1-\\log (t+1)+\\log t-\\log (e^{t}-1)+t-e^{-\\frac{t}{2}}>0\\,.$ Thus $ \\widetilde{C}(0)=1-\\log 1-e^0=0 $ (note that $ \\displaystyle \\lim _{t\\rightarrow 0}\\dfrac{1-e^{-t}}{t}=1 $ ), and $\\widetilde{C}^{\\prime }\\left( t\\right)& =\\frac{1}{t}-\\frac{1}{t+1}-\\frac{e^{t}}{e^{t}-1}+1+\\frac{1}{2}e^{-\\frac{t}{2}}\\\\&=\\frac{2\\left( e^{t}-1\\right)-2t\\left( t+1\\right) +t(t+1)\\left( e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}\\right) }{2t\\left( t+1\\right) \\left( e^{t}-1\\right)}.$ Applying $ 2t(t+1)(e^t-1)>0 $ , we need to check that $\\widehat{C}(t):=2\\left( e^{t}-1\\right) -2t\\left( t+1\\right)+t(t+1)\\left( e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}\\right)>0.$ Then, $\\widehat{C}(0)=0 $ and $\\widehat{C}^{\\prime }(t)=2e^{t}-2\\left( 2t+1\\right)+\\left(2t+1\\right) \\left( e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}\\right)+\\frac{t\\left(t+1\\right) }{2}\\left( e^{\\frac{t}{2}}+e^{-\\frac{t}{2}}\\right).$ Similarly, $ \\widehat{C}^{\\prime }(0)=0 $ and $\\widehat{C}^{\\prime \\prime }(t)= &2\\left( e^{t}-1\\right)+\\left( e^{\\frac{t}{2}}+e^{-\\frac{t}{2}}-2\\right)+2t\\left( e^{\\frac{t}{2}}+e^{-\\frac{t}{2}}\\right)\\\\&+\\frac{t\\left( t+1\\right)+8}{4}\\left( e^{\\frac{t}{2}}-e^{-\\frac{t}{2}}\\right).$ Therefore, $ \\widehat{C}^{\\prime \\prime }(t)>0 $ , since the coefficient of each term is positive for $ t>0 $ .",
"Then, $ \\widehat{C}^{\\prime }(t)>0 $ , since $ \\widehat{C}^{\\prime }(t) $ is increasing function with $ \\widehat{C}^{\\prime }(0)=0 $ .",
"Similarly, $ \\widehat{C}(t)>0 $ .",
"Hence $\\widetilde{C}\\left( t\\right)>\\underset{s\\searrow 0}{\\lim }\\widetilde{C}\\left( s\\right) =0$ for all $t>0$ .",
"In conclusion, we have $ C(t)>0 $ for each $ t>0 $ .",
"Finally, we have $ E_t(c)>0 \\ (c>0) $ for each $ t>0 $ , and we have the assertion.",
"$\\Box $ Acknowledgments.",
"This work was partially supported by JSPS KAKENHI Grant Number 19K03531 and by JSPS Grant BR171101."
]
] | 2005.14035 |
[
[
"Noether Currents for Eulerian Variational Principles in Non Barotropic\n Magnetohydrodynamics and Topological Conservations Laws"
],
[
"Abstract We derive a Noether current for the Eulerian variational principle of ideal non-barotropic magnetohydrodynamics (MHD).",
"It was shown previously that ideal non-barotropic MHD is mathematically equivalent to a five function field theory with an induced geometrical structure in the case that field lines cover surfaces and this theory can be described using a variational principle.",
"Here we use various symmetries of the flow to derive topological constants of motion through the derived Noether current and discuss their implication for non-barotropic MHD."
],
[
"Introduction",
"Variational principles for MHD were introduced by previous authors both in Lagrangian and Eulerian form.",
"Vladimirov and Moffatt [1] in a series of papers have discussed an Eulerian variational principle for incompressible MHD.",
"However, their variational principle contained three more functions in addition to the seven variables which appear in the standard equations of incompressible MHD which are the magnetic field $\\vec{B}$ the velocity field $\\vec{v}$ and the pressure $P$ .",
"Yahalom & Lynden-Bell [2] obtained an Eulerian Lagrangian principle for barotropic MHD which will depend on only six functions.",
"The variational derivative of this Lagrangian produced all the equations needed to describe barotropic MHD without any additional constraints.",
"Yahalom [3] have shown that for the barotropic case four functions will suffice.",
"Moreover, it was shown that the cuts of some of those functions [4] are topological local conserved quantities.",
"Previous work was concerned only with barotropic magnetohydrodynamics.",
"Variational principles of non barotropic magnetohydrodynamics can be found in the work of Bekenstein & Oron [5] in terms of 15 functions and V.A.",
"Kats [6] in terms of 20 functions.",
"Morrison [7] has suggested a Hamiltonian approach but this also depends on 8 canonical variables (see table 2 [7]).",
"It was shown that this number can be somewhat reduced.",
"In [8], [9] it was demonstrated that only five functions will suffice to describe non barotropic magnetohydrodynamics, and that the reduced lagrangian has a distinct geometrical structure including an induced metric.",
"The theorem of Noether dictates that for every continuous symmetry group of an Action the system must possess a conservation law.",
"For example time translation symmetry results in the conservation of energy, while spatial translation symmetry results in the conservation of linear momentum and rotation symmetry in the conservation of angular momentum to list some well known examples.",
"But sometimes the conservation law is discovered without reference to the Noether theorem by using the equations of the system.",
"In that case one is tempted to inquire what is the hidden symmetry associated with this conservation law and what is the simplest way to represent it.",
"The concept of metage as a label for fluid elements along a vortex line in ideal fluids was first introduced by Lynden-Bell & Katz [10].",
"A translation group of this label was found to be connected to the conservation of Moffat's [1] helicity by Yahalom [11] using a Lagrangian variational principle.",
"The concept of metage was later generalized by Yahalom & Lynden-Bell [2] for barotropic MHD, but now as a label for fluid elements along magnetic field lines which are comoving with the flow in the case of ideal MHD.",
"Yahalom & Lynden-Bell [2] has also shown that the translation group of the magnetic metage is connected to Woltjer [12], [13] conservation of cross helicity for barotropic MHD.",
"Recently the concept of metage was generalized also for non barotropic MHD in which magnetic field lines lie on entropy surfaces [14].",
"This was later generalized by dropping the entropy condition on magnetic field lines [15].",
"In those papers the metage translation symmetry group was used to generate a non-barotropic cross helicity generalization using a Lagrangian variational principle.",
"Cross Helicity was first described by Woltjer [12], [13] and is give by: $ H_{C} \\equiv \\int \\vec{B}\\cdot \\vec{v}d^{3} x,$ in which the integral is taken over the entire flow domain.",
"$H_{C}$ is conserved for barotropic or incompressible MHD and is given a topological interpretation in terms of the knottiness of magnetic and flow field lines.",
"Both conservation laws for the helicity in the fluid dynamics case and the barotropic MHD case were shown to originate from a relabelling symmetry through the Noether theorem [2], [11], [17], [18].",
"Webb et al.",
"[20] have generalized the idea of relabelling symmetry to non-barotropic MHD and derived their generalized cross helicity conservation law by using Noether's theorem but without using the simple representation which is connected with the metage variable.",
"The conservation law deduction involves a divergence symmetry of the action.",
"These conservation laws were written as Eulerian conservation laws of the form $D_t+\\vec{\\nabla }\\cdot \\vec{F} = 0$ where D is the conserved density and F is the conserved flux.",
"Webb et al.",
"[22] discuss the cross helicity conservation law for non-barotropic MHD in a multi-symplectic formulation of MHD.",
"Webb et al.",
"[19], [20] emphasize that the generalized cross helicity conservation law, in MHD and the generalized helicity conservation law in non-barotropic fluids are non-local in the sense that they depend on the auxiliary nonlocal variable $\\sigma $ , which depends on the Lagrangian time integral of the temperature $T(x, t)$ .",
"Notice that a potential vorticity conservation equation for non-barotropic MHD is derived by Webb, G. M. and Mace, R.L.",
"[23] by using Noether's second theorem.",
"Recently the non-barotropic cross helicity was generalized using additional label translation symmetry groups ($\\chi $ and $\\eta $ translations) [25], this led to additional topological conservation laws the $\\chi $ and $\\eta $ cross helicities.",
"Previous analysis depended on Lagrangian variational principles and their Noether currents.",
"Here we introduce a novel approach based on an Eulerian variational principle.",
"We derive the Noether current of the Eulerian variational principle and show how this can be used to derive topological conservation laws using label symmetries.",
"The plan of this paper is as follows: First we introduce the standard notations and equations of non-barotropic magnetohydrodynamics.",
"Next we introduce the Eulerian variational principle suitable for the non-barotropic case.",
"This is followed by a derivations of the Noether Current and finally we use the Noether current to obtain the generalized non- barotropic cross helicities.",
"Implication for non-barotropic MHD dynamics of the topological conservation laws are discussed."
],
[
"Standard formulation of ideal non-barotropic magnetohydrodynamics",
"The standard set of equations solved for non-barotropic magnetohydrodynamics are given below: $\\frac{\\partial {\\vec{B}}}{\\partial t} = \\vec{\\nabla }\\times (\\vec{v} \\times \\vec{B})$ $\\vec{\\nabla }\\cdot \\vec{B} =0$ $\\frac{\\partial {\\rho }}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\vec{v} ) = 0$ $\\rho \\frac{d \\vec{v}}{d t}=\\rho (\\frac{\\partial \\vec{v}}{\\partial t}+(\\vec{v} \\cdot \\vec{\\nabla })\\vec{v}) = -\\vec{\\nabla }p (\\rho ,s) +\\frac{(\\vec{\\nabla }\\times \\vec{B}) \\times \\vec{B}}{4 \\pi }$ $\\frac{d s}{d t}=0$ The following notations are utilized: $\\frac{\\partial }{\\partial t}$ is the temporal derivative, $\\frac{d}{d t}$ is the temporal material derivative and $\\vec{\\nabla }$ has its standard meaning in vector calculus.",
"$\\rho $ is the fluid density and $s$ is the specific entropy.",
"Finally $p (\\rho ,s)$ is the pressure which depends on the density and entropy (the non-barotropic case).",
"The justification for those equations and the conditions under which they apply can be found in standard books on magnetohydrodynamics (see for example [16]).",
"The number of independent variables for which one needs to solve is eight ($\\vec{v},\\vec{B},\\rho ,s$ ) and the number of equations (REF ,REF ,REF ,REF ) is also eight.",
"Notice that equation (REF ) is a condition on the initial $\\vec{B}$ field and is satisfied automatically for any other time due to equation (REF ).",
"We will find it useful to introduce the following thermodynamic equations for later use: $d \\varepsilon &=& T ds - p d \\frac{1}{\\rho } = T ds + \\frac{p}{\\rho ^2} d \\rho \\nonumber \\\\& & \\frac{\\partial \\varepsilon }{\\partial s}_{\\rho } = T, \\qquad \\frac{\\partial \\varepsilon }{\\partial \\rho }_{s} = \\frac{p}{\\rho ^2}\\nonumber \\\\w &=& \\varepsilon + \\frac{p}{\\rho }= \\varepsilon + \\frac{\\partial \\varepsilon }{\\partial \\rho } \\rho = \\frac{\\partial (\\rho \\varepsilon )}{\\partial \\rho }\\nonumber \\\\dw &=& d\\varepsilon + d(\\frac{p}{\\rho }) = T ds + \\frac{1}{\\rho } dp$ in the above: $\\varepsilon $ is the specific internal energy, $T$ is the temperature and $w$ is the specific enthalpy.",
"A special case of equation of state is the polytropic equation of state [26]: $p =K \\rho ^{\\gamma }$ $K$ and $\\gamma $ may depend on the specific entropy $s$ .",
"Hence: $\\frac{\\partial \\varepsilon }{\\partial \\rho } = K \\rho ^{\\gamma -2} \\Rightarrow \\varepsilon = \\frac{K}{\\gamma -1} \\rho ^{\\gamma -1}= \\frac{p}{\\rho (\\gamma -1)} \\Rightarrow \\rho \\varepsilon = \\frac{p}{\\gamma -1}$ the last identity is up to a function dependent on $s$ ."
],
[
"Variational principle of non-barotropic magnetohydrodynamics",
"In the following section we will generalize the approach of [2] for the non-barotropic case [8], [9].",
"Consider the action: $A & \\equiv & \\int {\\cal L} d^3 x dt,\\nonumber \\\\{\\cal L} & \\equiv & {\\cal L}_1 + {\\cal L}_2,\\nonumber \\\\{\\cal L}_1 & \\equiv & \\rho (\\frac{1}{2} \\vec{v}^2 - \\varepsilon (\\rho ,s)) + \\frac{\\vec{B}^2}{8 \\pi },\\nonumber \\\\{\\cal L}_2 & \\equiv & \\nu [\\frac{\\partial {\\rho }}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\vec{v} )]- \\rho \\alpha \\frac{d \\chi }{dt} - \\rho \\beta \\frac{d \\eta }{dt} - \\rho \\sigma \\frac{d s}{dt}\\nonumber \\\\&-& \\frac{\\vec{B}}{4 \\pi } \\cdot \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta .$ In the specific case of a polytropic equation of state we have according to equation (REF ): ${\\cal L}_1 = \\frac{1}{2} \\rho \\vec{v}^2 - \\frac{p}{\\gamma -1} + \\frac{\\vec{B}^2}{8 \\pi }.$ Obviously $\\nu ,\\alpha ,\\beta ,\\sigma $ are Lagrange multipliers which were inserted in such a way that the variational principle will yield the following equations: $& & \\frac{\\partial {\\rho }}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\vec{v} ) = 0,\\nonumber \\\\& & \\rho \\frac{d \\chi }{dt} = 0, \\rho \\frac{d \\eta }{dt} = 0, \\rho \\frac{d s}{dt} = 0.$ It is not assumed that $\\nu ,\\alpha ,\\beta ,\\sigma $ are single valued.",
"Provided $\\rho $ is not null those are just the continuity equation (REF ), entropy conservation and the conditions that Sakurai's functions are comoving.",
"Taking the variational derivative with respect to $\\vec{B}$ we see that: $\\vec{B} = \\hat{\\vec{B}} \\equiv \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta .$ Hence $\\vec{B}$ is in Sakurai's form [27] and satisfies equation (REF ).",
"It can be easily shown that provided that $\\vec{B}$ is in the form given in equation (REF ), and equations (REF ) are satisfied, then also equation (REF ) is satisfied.",
"We notice that the specific form of the magnetic field given in equation (REF ) appear under different names in the literature.",
"The functions $\\chi $ and $\\eta $ are sometimes denoted \"Euler potentials\", \"Clebsch variables\" and also \"flux representation functions\" [28].",
"Equation (REF ) imply that the magnetic field lines lie on surfaces, the lines may be surface filling but not volume filling.",
"For the time being we have showed that all the equations of non-barotropic magnetohydrodynamics can be obtained from the above variational principle except Euler's equations.",
"We will now show that Euler's equations can be derived from the above variational principle as well.",
"Let us take an arbitrary variational derivative of the above action with respect to $\\vec{v}$ , this will result in: $\\delta _{\\vec{v}} A &=& \\hspace{-5.69046pt} \\int dt \\lbrace \\int d^3 x dt \\rho \\delta \\vec{v} \\cdot [\\vec{v} - \\vec{\\nabla }\\nu - \\alpha \\vec{\\nabla }\\chi - \\beta \\vec{\\nabla }\\eta \\nonumber \\\\&-& \\sigma \\vec{\\nabla }s]+ \\oint d \\vec{S} \\cdot \\delta \\vec{v} \\rho \\nu + \\int d \\vec{\\Sigma }\\cdot \\delta \\vec{v} \\rho [\\nu ]\\rbrace .$ The integral $\\oint d \\vec{S} \\cdot \\delta \\vec{v} \\rho \\nu $ vanishes in many physical scenarios.",
"In the case of astrophysical flows this integral will vanish since $\\rho =0$ on the flow boundary, in the case of a fluid contained in a vessel no flux boundary conditions $\\delta \\vec{v} \\cdot \\hat{n} =0$ are induced ($\\hat{n}$ is a unit vector normal to the boundary).",
"The surface integral $\\int d \\vec{\\Sigma }$ on the cut of $\\nu $ vanishes in the case that $\\nu $ is single valued and $[\\nu ]=0$ .",
"In the case that $\\nu $ is not single valued only a Kutta type velocity perturbation [32] in which the velocity perturbation is parallel to the cut will cause the cut integral to vanish.",
"Provided that the surface integrals do vanish and that $\\delta _{\\vec{v}} A =0$ for an arbitrary velocity perturbation we see that $\\vec{v}$ must have the following form: $\\vec{v} = \\hat{\\vec{v}} \\equiv \\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s.$ The above equation is reminiscent of Clebsch representation in non magnetic fluids.",
"A similar expression was obtained by Morrison [7] using an Hamiltonian formalism but in which the $s$ terms is replaced by $\\psi $ which is conjugate to $s$ .",
"Let us now take the variational derivative with respect to the density $\\rho $ we obtain: $\\delta _{\\rho } A & = & \\int d^3 x dt \\delta \\rho [\\frac{1}{2} \\vec{v}^2 - w - \\frac{\\partial {\\nu }}{\\partial t} - \\vec{v} \\cdot \\vec{\\nabla }\\nu ]\\nonumber \\\\& + & \\int dt \\oint d \\vec{S} \\cdot \\vec{v} \\delta \\rho \\nu +\\int dt \\int d \\vec{\\Sigma }\\cdot \\vec{v} \\delta \\rho [\\nu ]\\nonumber \\\\&+& \\int d^3 x \\nu \\delta \\rho |^{t_1}_{t_0}.$ In which $ w= \\frac{\\partial (\\varepsilon \\rho )}{\\partial \\rho }$ is the specific enthalpy.",
"Hence provided that $\\oint d \\vec{S} \\cdot \\vec{v} \\delta \\rho \\nu $ vanishes on the boundary of the domain and $ \\int d \\vec{\\Sigma }\\cdot \\vec{v} \\delta \\rho [\\nu ]$ vanishes on the cut of $\\nu $ in the case that $\\nu $ is not single valuedWhich entails either a Kutta type condition for the velocity or a vanishing density perturbation on the cut.",
"and in initial and final times the following equation must be satisfied: $\\frac{d \\nu }{d t} = \\frac{1}{2} \\vec{v}^2 - w, \\qquad $ Finally we have to calculate the variation with respect to both $\\chi $ and $\\eta $ this will lead us to the following results: $\\delta _{\\chi } A \\hspace{-11.38092pt} & = & \\hspace{-11.38092pt} \\int d^3 x dt \\delta \\chi [\\frac{\\partial {(\\rho \\alpha )}}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\alpha \\vec{v})-\\vec{\\nabla }\\eta \\cdot \\vec{J}]\\nonumber \\\\&+& \\int dt \\oint d \\vec{S} \\cdot [\\frac{\\vec{B}}{4 \\pi } \\times \\vec{\\nabla }\\eta - \\vec{v} \\rho \\alpha ]\\delta \\chi \\nonumber \\\\& + & \\int dt \\int d \\vec{\\Sigma }\\cdot [\\frac{\\vec{B}}{4 \\pi } \\times \\vec{\\nabla }\\eta - \\vec{v} \\rho \\alpha ][\\delta \\chi ]\\nonumber \\\\&-& \\int d^3 x \\rho \\alpha \\delta \\chi |^{t_1}_{t_0},$ $\\delta _{\\eta } A \\hspace{-11.38092pt} & = & \\hspace{-11.38092pt} \\int d^3 x dt \\delta \\eta [\\frac{\\partial {(\\rho \\beta )}}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\beta \\vec{v})+\\vec{\\nabla }\\chi \\cdot \\vec{J}]\\nonumber \\\\&+& \\int dt \\oint d \\vec{S} \\cdot [\\vec{\\nabla }\\chi \\times \\frac{\\vec{B}}{4 \\pi } - \\vec{v} \\rho \\beta ]\\delta \\eta \\nonumber \\\\& + & \\int dt \\int d \\vec{\\Sigma }\\cdot [\\vec{\\nabla }\\chi \\times \\frac{\\vec{B}}{4 \\pi } - \\vec{v} \\rho \\beta ][\\delta \\eta ]\\nonumber \\\\&-& \\int d^3 x \\rho \\beta \\delta \\eta |^{t_1}_{t_0}.$ Provided that the correct temporal and boundary conditions are met with respect to the variations $\\delta \\chi $ and $\\delta \\eta $ on the domain boundary and on the cuts in the case that some (or all) of the relevant functions are non single valued.",
"we obtain the following set of equations: $\\frac{d \\alpha }{dt} = \\frac{\\vec{\\nabla }\\eta \\cdot \\vec{J}}{\\rho }, \\qquad \\frac{d \\beta }{dt} = -\\frac{\\vec{\\nabla }\\chi \\cdot \\vec{J}}{\\rho },$ in which the continuity equation (REF ) was taken into account.",
"By correct temporal conditions we mean that both $\\delta \\eta $ and $\\delta \\chi $ vanish at initial and final times.",
"As for boundary conditions which are sufficient to make the boundary term vanish on can consider the case that the boundary is at infinity and both $\\vec{B}$ and $\\rho $ vanish.",
"Another possibility is that the boundary is impermeable and perfectly conducting.",
"A sufficient condition for the integral over the \"cuts\" to vanish is to use variations $\\delta \\eta $ and $\\delta \\chi $ which are single valued.",
"It can be shown that $\\chi $ can always be taken to be single valued, hence taking $\\delta \\chi $ to be single valued is no restriction at all.",
"In some topologies $\\eta $ is not single valued and in those cases a single valued restriction on $\\delta \\eta $ is sufficient to make the cut term null.",
"Finally we take a variational derivative with respect to the entropy $s$ : $\\delta _{s} A \\hspace{-11.38092pt} & = & \\hspace{-11.38092pt} \\int d^3 x dt \\delta s[\\frac{\\partial {(\\rho \\sigma )}}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\sigma \\vec{v})- \\rho T]\\nonumber \\\\&+& \\int dt \\oint d \\vec{S} \\cdot \\rho \\sigma \\vec{v} \\delta s- \\int d^3 x \\rho \\sigma \\delta s |^{t_1}_{t_0},$ in which the temperature is $T=\\frac{\\partial \\varepsilon }{\\partial s}$ .",
"We notice that according to equation (REF ) $\\sigma $ is single valued and hence no cuts are needed.",
"Taking into account the continuity equation (REF ) we obtain for locations in which the density $\\rho $ is not null the result: $\\frac{d \\sigma }{dt} =T,$ provided that $\\delta _{s} A$ vanished for an arbitrary $\\delta s$ ."
],
[
"Euler's equations",
"We shall now show that a velocity field given by equation (REF ), such that the equations for $\\alpha , \\beta , \\chi , \\eta , \\nu , \\sigma , s$ satisfy the corresponding equations (REF ,REF ,REF ,REF ) must satisfy Euler's equations.",
"Let us calculate the material derivative of $\\vec{v}$ : $\\frac{d\\vec{v}}{dt} &=& \\frac{d\\vec{\\nabla }\\nu }{dt} + \\frac{d\\alpha }{dt} \\vec{\\nabla }\\chi +\\alpha \\frac{d\\vec{\\nabla }\\chi }{dt} +\\frac{d\\beta }{dt} \\vec{\\nabla }\\eta + \\beta \\frac{d\\vec{\\nabla }\\eta }{dt}\\nonumber \\\\&+& \\frac{d\\sigma }{dt} \\vec{\\nabla }s + \\sigma \\frac{d\\vec{\\nabla }s}{dt}$ It can be easily shown that: $\\frac{d\\vec{\\nabla }\\nu }{dt} & = & \\vec{\\nabla }\\frac{d \\nu }{dt}- \\vec{\\nabla }v_k \\frac{\\partial \\nu }{\\partial x_k}= \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2 - w)- \\vec{\\nabla }v_k \\frac{\\partial \\nu }{\\partial x_k},\\nonumber \\\\\\frac{d\\vec{\\nabla }\\eta }{dt} & = & \\vec{\\nabla }\\frac{d \\eta }{dt}- \\vec{\\nabla }v_k \\frac{\\partial \\eta }{\\partial x_k}= - \\vec{\\nabla }v_k \\frac{\\partial \\eta }{\\partial x_k},\\nonumber \\\\\\frac{d\\vec{\\nabla }\\chi }{dt} & = & \\vec{\\nabla }\\frac{d \\chi }{dt}- \\vec{\\nabla }v_k \\frac{\\partial \\chi }{\\partial x_k}= - \\vec{\\nabla }v_k \\frac{\\partial \\chi }{\\partial x_k},\\nonumber \\\\\\frac{d\\vec{\\nabla }s}{dt} & = & \\vec{\\nabla }\\frac{d s}{dt}- \\vec{\\nabla }v_k \\frac{\\partial s}{\\partial x_k}= - \\vec{\\nabla }v_k \\frac{\\partial s}{\\partial x_k}.$ In which $x_k$ is a Cartesian coordinate and a summation convention is assumed.",
"Inserting the result from equations (REF ,REF ) into equation (REF ) yields: $\\frac{d\\vec{v}}{dt} &=& - \\vec{\\nabla }v_k (\\frac{\\partial \\nu }{\\partial x_k} + \\alpha \\frac{\\partial \\chi }{\\partial x_k} +\\beta \\frac{\\partial \\eta }{\\partial x_k} + \\sigma \\frac{\\partial s}{\\partial x_k})\\nonumber \\\\&+& \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2 - w)+ T \\vec{\\nabla }s\\nonumber \\\\&+& \\frac{1}{\\rho } ((\\vec{\\nabla }\\eta \\cdot \\vec{J})\\vec{\\nabla }\\chi - (\\vec{\\nabla }\\chi \\cdot \\vec{J})\\vec{\\nabla }\\eta )\\nonumber \\\\&=& - \\vec{\\nabla }v_k v_k + \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2 - w) + T \\vec{\\nabla }s\\nonumber \\\\&+& \\frac{1}{\\rho } \\vec{J} \\times (\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta )\\nonumber \\\\&=& - \\frac{\\vec{\\nabla }p}{\\rho } + \\frac{1}{\\rho } \\vec{J} \\times \\vec{B}.$ In which we have used both equation (REF ) and equation (REF ) in the above derivation.",
"This of course proves that the non-barotropic Euler equations can be derived from the action given in equation (REF ) and hence all the equations of non-barotropic magnetohydrodynamics can be derived from the above action without restricting the variations in any way except on the relevant boundaries and cuts."
],
[
"Local non-barotropic cross helicity",
"The function $\\nu $ , whose material derivative is given in (REF ), can be multiple valued because only its gradient appears in the velocity (REF ).",
"However, the discontinuity, $[\\nu ]$ , of $\\nu $ is conserved, $\\dfrac{d{[\\nu ]}}{dt} =0$ since the terms on the right-hand side of (REF ) describe physical quantities and hence are single valued.",
"A similar equation also holds for barotropic fluid dynamics and barotropic MHD [2], [3], [4].",
"We now substitute the expressions for ${\\vec{B}}$ and ${\\vec{v}}$ given by (REF ) and (REF ) respectively into the formula $H_{C} \\equiv \\int \\vec{B}\\cdot \\vec{v}{d}^{3} x$ for the cross helicity (see (REF )) to obtain $ H_{C} = \\int {d}\\Phi [\\nu ] + \\int {d}\\Phi \\oint \\sigma d s,$ where the closed line integral taken along a magnetic field line.",
"Furthermore, ${d}\\Phi =\\vec{B}\\cdot d \\vec{S}=(\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta ) \\cdot d \\vec{S}={d}\\chi {d}\\eta $ is a magnetic flux element which is co-moving as governed by (REF ) and ${d}\\vec{S}$ is an infinitesimal area element.",
"Although the cross helicity is not conserved for non-barotropic flows, inspection of the right-hand side of (REF ) reveals that it is made of a sum of two terms.",
"One term is conserved, as both ${d}\\Phi $ and $[\\nu ]$ are co-moving, and the other is not.",
"This suggests the following definition for the non-barotropic cross helicity $H_{CNB} \\equiv \\int {d}\\Phi [\\nu ] = H_{C} - \\int {d}\\Phi \\oint \\sigma {d}s .$ It can be written in the more conventional form: $ H_{CNB} = \\int \\vec{B} \\cdot \\vec{v}_t {d}^{3} x$ in which the topological velocity field is defined as $\\vec{v}_t = \\vec{v} - \\sigma \\vec{\\nabla }s .$ It should be noted that $H_{CNB}$ is conserved even for an MHD not satisfying the Sakurai topological constraint given in (REF ), provided that we have a field $\\sigma $ satisfying the equation $\\dfrac{d{\\sigma }}{dt} = T$ .",
"This can be verified by direct derivation using only the equation of motion and the sigma equation.",
"Thus the non-barotropic cross helicity conservation law, $\\dfrac{{{d}}{H_{CNB}}}{{{d}}{t}} = 0 ,$ is more general than the variational principle described by (REF ) as follows from a direct computation using (REF ) and (REF )–(REF ).",
"Also note that, for a constant specific entropy $s$ , we obtain $H_{CNB}=H_{C}$ and the non-barotropic cross helicity reduces to the standard barotropic cross helicity.",
"The local form of equation (REF ) describing the evolution of $H_{CNB}$ per unit volume was described by [19], [20].",
"To conclude we introduce also a local topological conservation law in the spirit of [4] which is the non-barotropic cross helicity per unit of magnetic flux.",
"This quantity which is equal to the discontinuity, $[\\nu ]$ , of $\\nu $ is conserved and can be written as a sum of the barotropic cross helicity per unit flux and the closed line integral of $s {d}\\sigma $ along a magnetic field line, namely: $ [\\nu ]= \\dfrac{{{d}}{H_{CNB}}}{{{d}}{\\Phi }} = \\dfrac{{{d}}{H_{C}}}{{{d}}{\\Phi }} + \\oint s {d}\\sigma .$"
],
[
"Simplified action",
"The reader of this paper might argue here that the paper is misleading.",
"The author has declared that he is going to present a simplified action for non-barotropic magnetohydrodynamics instead he added six more functions $\\alpha ,\\beta ,\\chi ,\\-\\eta ,\\nu ,\\sigma $ to the standard set $\\vec{B},\\vec{v},\\rho ,s$ .",
"In the following I will show that this is not so and the action given in equation (REF ) in a form suitable for a pedagogic presentation can indeed be simplified.",
"It is easy to show that the Lagrangian density appearing in equation (REF ) can be written in the form: ${\\cal L} & = & -\\rho [\\frac{\\partial {\\nu }}{\\partial t} + \\alpha \\frac{\\partial {\\chi }}{\\partial t}+ \\beta \\frac{\\partial {\\eta }}{\\partial t}+ \\sigma \\frac{\\partial {s}}{\\partial t}+\\varepsilon (\\rho ,s)]\\nonumber \\\\&+&\\frac{1}{2}\\rho [(\\vec{v}-\\hat{\\vec{v}})^2-(\\hat{\\vec{v}})^2]\\nonumber \\\\& + & \\frac{1}{8 \\pi } [(\\vec{B}-\\hat{\\vec{B}})^2-(\\hat{\\vec{B}})^2]+\\frac{\\partial {(\\nu \\rho )}}{\\partial t} + \\vec{\\nabla }\\cdot (\\nu \\rho \\vec{v} )$ In which $\\hat{\\vec{v}}$ is a shorthand notation for $\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi +\\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s $ (see equation (REF )) and $\\hat{\\vec{B}}$ is a shorthand notation for $\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta $ (see equation (REF )).",
"Thus ${\\cal L}$ has four contributions: ${\\cal L} & = & \\hat{\\cal L} + {\\cal L}_{\\vec{v}}+ {\\cal L}_{\\vec{B}}+{\\cal L}_{boundary},\\nonumber \\\\\\hat{\\cal L} \\hspace{34.14322pt} &\\hspace{-71.13188pt} \\equiv & \\hspace{-42.67912pt} -\\rho [\\frac{\\partial {\\nu }}{\\partial t}+ \\alpha \\frac{\\partial {\\chi }}{\\partial t}+ \\beta \\frac{\\partial {\\eta }}{\\partial t}+ \\sigma \\frac{\\partial {s}}{\\partial t}+\\varepsilon (\\rho ,s)\\nonumber \\\\&+& \\frac{1}{2} (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s )^2 ]\\nonumber \\\\&-&\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta )^2\\nonumber \\\\{\\cal L}_{\\vec{v}} &\\equiv & \\frac{1}{2}\\rho (\\vec{v}-\\hat{\\vec{v}})^2,\\nonumber \\\\{\\cal L}_{\\vec{B}} &\\equiv & \\frac{1}{8 \\pi } (\\vec{B}-\\hat{\\vec{B}})^2,\\nonumber \\\\{\\cal L}_{boundary} &\\equiv & \\frac{\\partial {(\\nu \\rho )}}{\\partial t} + \\vec{\\nabla }\\cdot (\\nu \\rho \\vec{v} ).$ The only term containing $\\vec{v}$ is${\\cal L}_{boundary}$ also depends on $\\vec{v}$ but being a boundary term in space and time it does not contribute to the derived equations ${\\cal L}_{\\vec{v}}$ , it can easily be seen that this term will lead, after we nullify the variational derivative with respect to $\\vec{v}$ , to equation (REF ) but will otherwise have no contribution to other variational derivatives.",
"Similarly the only term containing $\\vec{B}$ is ${\\cal L}_{\\vec{B}}$ and it can easily be seen that this term will lead, after we nullify the variational derivative, to equation (REF ) but will have no contribution to other variational derivatives.",
"Also notice that the term ${\\cal L}_{boundary}$ contains only complete partial derivatives and thus can not contribute to the equations although it can change the boundary conditions.",
"Hence we see that equations (REF ), equation (REF ), equations (REF ) and equation (REF ) can be derived using the Lagrangian density: $& & \\hat{\\cal L}[\\alpha ,\\beta ,\\chi ,\\eta ,\\nu ,\\rho ,\\sigma ,s] = -\\rho [\\frac{\\partial {\\nu }}{\\partial t} + \\alpha \\frac{\\partial {\\chi }}{\\partial t}+ \\beta \\frac{\\partial {\\eta }}{\\partial t}\\nonumber \\\\&+& \\sigma \\frac{\\partial {s}}{\\partial t}+\\ \\varepsilon (\\rho ,s) + \\frac{1}{2} (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s )^2 ]\\nonumber \\\\&-&\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta )^2$ in which $\\hat{\\vec{v}}$ replaces $\\vec{v}$ and $\\hat{\\vec{B}}$ replaces $\\vec{B}$ in the relevant equations.",
"Furthermore, after integrating the eight equations (REF ,REF ,REF ,REF ) we can insert the potentials $\\alpha ,\\beta ,\\chi ,\\eta ,\\nu ,\\sigma ,s$ into equations (REF ) and (REF ) to obtain the physical quantities $\\vec{v}$ and $\\vec{B}$ .",
"Hence, the general non-barotropic magnetohydrodynamic problem is reduced from eight equations (REF ,REF ,REF ,REF ) and the additional constraint (REF ) to a problem of eight first order (in the temporal derivative) unconstrained equations.",
"Moreover, the entire set of equations can be derived from the Lagrangian density $\\hat{\\cal L}$ ."
],
[
"Elimination of Variables",
"Let us now look at the three last three equations of (REF ) [8], [9].",
"Those describe three comoving quantities which can be written in terms of the generalized Clebsch form given in equation (REF ) as follows: $& & \\frac{\\partial \\chi }{\\partial t} + (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s)\\cdot \\vec{\\nabla }\\chi = 0\\nonumber \\\\& & \\frac{\\partial \\eta }{\\partial t} + (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s)\\cdot \\vec{\\nabla }\\eta = 0\\nonumber \\\\& & \\frac{\\partial s}{\\partial t} + (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s)\\cdot \\vec{\\nabla }s = 0$ Those are algebraic equations for $\\alpha , \\beta , \\sigma $ , which can be solved such that $\\alpha , \\beta , \\sigma $ can be written as functionals of $\\chi ,\\eta ,\\nu ,s$ , resulting eventually in the description of non-barotropic magnetohydrodynamics in terms of five functions: $\\nu ,\\rho ,\\chi ,\\eta ,s$ .",
"Let us introduce the notation: $\\alpha _i \\equiv (\\alpha , \\beta , \\sigma ), \\chi _i\\equiv (\\chi ,\\eta ,s),k_i \\equiv -\\frac{\\partial \\chi _i}{\\partial t} - \\vec{\\nabla }\\nu \\cdot \\vec{\\nabla }\\chi _i$ $ i\\in (1,2,3)$ .",
"In terms of the above notation equation (REF ) takes the form: $k_i =\\alpha _j \\vec{\\nabla }\\chi _i \\cdot \\vec{\\nabla }\\chi _j, \\qquad j\\in (1,2,3)$ in which the Einstein summation convention is assumed.",
"Let us define the matrix: $A_{ij} \\equiv \\vec{\\nabla }\\chi _i \\cdot \\vec{\\nabla }\\chi _j$ obviously this matrix is symmetric since $A_{ij}=A_{ji}$ .",
"Hence equation (REF ) takes the form: $k_i = A_{ij} \\alpha _j, \\qquad j\\in (1,2,3)$ Provided that the matrix $A_{ij}$ is not singular it has an inverse $A^{-1}_{ij}$ which can be written as: $A^{-1}_{ij}&=&\\left|A\\right|^{-1}\\cdot \\nonumber \\\\& & \\hspace{-56.9055pt}\\left(\\begin{array}{ccc}A_{22} A_{33}-A_{23}^2 & A_{13} A_{23}-A_{12} A_{33} & A_{12} A_{23}-A_{13} A_{22} \\\\A_{13} A_{23}-A_{12} A_{33} & A_{11} A_{33}-A_{13}^2 & A_{12} A_{13}-A_{11} A_{23} \\\\A_{12} A_{23}-A_{13} A_{22} & A_{12} A_{13}-A_{11} A_{23} & A_{11} A_{22}-A_{12}^2\\end{array}\\right)$ In which the determinant $\\left|A\\right|$ is given by the following equation: $\\left|A\\right| &=&A_{11} A_{22} A_{33}-A_{11} A_{23}^2-A_{22} A_{13}^2\\nonumber \\\\&-&A_{33} A_{12}^2 +2 A_{12} A_{13} A_{23}$ In terms of the above equations the $\\alpha _i$ 's can be calculated as functionals of $\\chi _i,\\nu $ as follows: $\\alpha _i [\\chi _i,\\nu ]= A^{-1}_{ij} k_j.$ The velocity equation (REF ) can now be written as: $\\vec{v} &=& \\vec{\\nabla }\\nu + \\alpha _i \\vec{\\nabla }\\chi _i= \\vec{\\nabla }\\nu + A^{-1}_{ij} k_j \\vec{\\nabla }\\chi _i\\nonumber \\\\&=& \\vec{\\nabla }\\nu - A^{-1}_{ij}\\vec{\\nabla }\\chi _i (\\frac{\\partial \\chi _j}{\\partial t} + \\vec{\\nabla }\\nu \\cdot \\vec{\\nabla }\\chi _j).$ Provided that the $\\chi _i$ is a coordinate basis in three dimensions, we may write: $\\vec{\\nabla }\\nu = \\vec{\\nabla }\\chi _n \\frac{\\partial \\nu }{\\partial \\chi _n}, \\qquad n\\in (1,2,3).$ Inserting equation (REF ) into equation (REF ) we obtain: $\\vec{v} = - A^{-1}_{ij}\\vec{\\nabla }\\chi _i \\frac{\\partial \\chi _j}{\\partial t}$ in the above $\\delta _{in}$ is a Kronecker delta.",
"Thus the velocity $\\vec{v} [\\chi _i]$ is a functional of $\\chi _i$ only and is independent of $\\nu $ ."
],
[
"Lagrangian Density and Variational Analysis",
"Let us now rewrite the Lagrangian density $\\hat{\\cal L}[\\chi _i,\\nu ,\\rho ]$ given in equation (REF ) in terms of the new variables: $& & \\hat{\\cal L}[\\chi _i,\\nu ,\\rho ] = -\\rho [\\frac{\\partial {\\nu }}{\\partial t} + \\alpha _k [\\chi _i,\\nu ] \\frac{\\partial {\\chi _k}}{\\partial t}\\nonumber \\\\&+& \\ \\varepsilon (\\rho ,\\chi _3) + \\frac{1}{2} \\vec{v} [\\chi _i]^2 ]-\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2)^2$ Let us calculate the variational derivative of $\\hat{\\cal L}[\\chi _i,\\nu ,\\rho ]$ with respect to $\\chi _i$ this will result in: $\\delta _{\\chi _i}\\hat{\\cal L} &\\hspace{-11.38092pt}=&\\hspace{-11.38092pt} -\\rho [ \\delta _{\\chi _i} \\alpha _k \\frac{\\partial {\\chi _k}}{\\partial t} +\\alpha _{\\underline{i}} \\frac{\\partial \\delta \\chi _{\\underline{i}}}{\\partial t}+\\ \\delta _{\\chi _i} \\varepsilon (\\rho ,\\chi _3) + \\delta _{\\chi _i}\\vec{v} \\cdot \\vec{v} ]\\nonumber \\\\&-& \\frac{ \\vec{B}}{4 \\pi } \\cdot \\delta _{\\chi _i} (\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2)$ in which the summation convention is not applied if the index is underlined.",
"However, due to equation (REF ) we may write: $\\delta _{\\chi _i}\\vec{v}= \\delta _{\\chi _i} \\alpha _k \\vec{\\nabla }\\chi _k + \\alpha _{\\underline{i}} \\vec{\\nabla }\\delta \\chi _{\\underline{i}}.$ Inserting equation (REF ) into equation (REF ) and rearranging the terms we obtain: $\\delta _{\\chi _i}\\hat{\\cal L} &=& -\\rho [ \\delta _{\\chi _i} \\alpha _k (\\frac{\\partial {\\chi _k}}{\\partial t}+ \\vec{v} \\cdot \\vec{\\nabla }\\chi _k )+\\alpha _{\\underline{i}} (\\frac{\\partial \\delta \\chi _{\\underline{i}}}{\\partial t}\\nonumber \\\\&+& \\vec{v} \\cdot \\vec{\\nabla }\\delta \\chi _{\\underline{i}}) +\\ \\delta _{\\chi _i} \\varepsilon (\\rho ,\\chi _3) ]\\nonumber \\\\&-& \\frac{ \\vec{B}}{4 \\pi } \\cdot \\delta _{\\chi _i} (\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2).$ Now by construction $\\vec{v}$ satisfies equation (REF ) and hence $\\frac{\\partial {\\chi _k}}{\\partial t}+ \\vec{v} \\cdot \\vec{\\nabla }\\chi _k = 0$ , this leads to: $\\delta _{\\chi _i}\\hat{\\cal L} = -\\rho \\left[ \\alpha _{\\underline{i}} \\frac{d \\delta \\chi _{\\underline{i}}}{d t}+ \\delta _{\\chi _i} \\varepsilon (\\rho ,\\chi _3) \\right] - \\frac{ \\vec{B}}{4 \\pi } \\cdot \\delta _{\\chi _i} (\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2).$ From now on the derivation proceeds as in equations (REF ,REF ,REF ) resulting in equations (REF ,REF ) and will not be repeated.",
"The difference is that now $\\alpha , \\beta $ and $\\sigma $ are not independent quantities, rather they depend through equation (REF ) on the derivatives of $\\chi _i,\\nu $ .",
"Thus, equations (REF ,REF ,REF ) are not first order equations in time but are second order equations.",
"Now let us calculate the variational derivative with respect to $\\nu $ this will result in the expression: $\\delta _{\\nu } \\hat{\\cal L} = -\\rho [ \\frac{\\partial {\\delta \\nu }}{\\partial t} + \\delta _{\\nu } \\alpha _n \\frac{\\partial {\\chi _n}}{\\partial t}]$ However, $\\delta _{\\nu } \\alpha _k$ can be calculated from equation (REF ): $\\delta _{\\nu } \\alpha _n = A^{-1}_{nj} \\delta _{\\nu } k_j = - A^{-1}_{nj} \\vec{\\nabla }\\delta \\nu \\cdot \\vec{\\nabla }\\chi _j$ Inserting the above equation into equation (REF ): $\\delta _{\\nu } \\hat{\\cal L} &=& -\\rho [ \\frac{\\partial {\\delta \\nu }}{\\partial t} - A^{-1}_{nj} \\vec{\\nabla }\\chi _j\\frac{\\partial {\\chi _n}}{\\partial t} \\cdot \\vec{\\nabla }\\delta \\nu ] =\\nonumber \\\\&-& \\rho [ \\frac{\\partial {\\delta \\nu }}{\\partial t} + \\vec{v} \\cdot \\vec{\\nabla }\\delta \\nu ]=-\\rho \\frac{d{\\delta \\nu }}{d t}$ The above equation can be put to the form: $\\delta _{\\nu } \\hat{\\cal L} = \\delta \\nu [\\frac{\\partial {\\rho }}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\vec{v} )]-\\frac{\\partial {(\\rho \\delta \\nu )}}{\\partial t}- \\vec{\\nabla }\\cdot (\\rho \\vec{v} \\delta \\nu )$ This obviously leads to the continuity equation (REF ) and some boundary terms in space and time.",
"The variational derivative with respect to $\\rho $ is trivial and the analysis is identical to the one in equation (REF ) leading to equation (REF ).",
"To conclude this subsection let us summarize the equations of non-barotropic magnetohydrodynamics: $\\frac{d \\nu }{d t} &=& \\frac{1}{2} \\vec{v}^2 - w,\\frac{\\partial {\\rho }}{\\partial t} + \\vec{\\nabla }\\cdot (\\rho \\vec{v} ) = 0,\\nonumber \\\\\\frac{d \\sigma }{dt} &=& T,\\frac{d \\alpha }{dt} = \\frac{\\vec{\\nabla }\\eta \\cdot \\vec{J}}{\\rho },\\frac{d \\beta }{dt} = -\\frac{\\vec{\\nabla }\\chi \\cdot \\vec{J}}{\\rho }$ in which $\\alpha ,\\beta ,\\sigma ,\\vec{v}$ are functionals of $\\chi ,\\eta ,s,\\nu $ as described above.",
"It is easy to show as in equation (REF ) that those variational equations are equivalent to the physical equations."
],
[
"Lagrangian Density in Explicit Form",
"Let us put the Lagrangian density of equation (REF ) in a slightly more explicit form.",
"First us look at the term $\\vec{v}^2$ : $\\vec{v}^2 &\\hspace{-11.38092pt}=& \\hspace{-11.38092pt}A^{-1}_{ij}\\vec{\\nabla }\\chi _i \\frac{\\partial \\chi _j}{\\partial t} A^{-1}_{mn}\\vec{\\nabla }\\chi _m \\frac{\\partial \\chi _n}{\\partial t}\\nonumber \\\\&=& A^{-1}_{ij} A^{-1}_{mn} A_{im} \\frac{\\partial \\chi _j}{\\partial t} \\frac{\\partial \\chi _n}{\\partial t}= A^{-1}_{jn} \\frac{\\partial \\chi _j}{\\partial t} \\frac{\\partial \\chi _n}{\\partial t}$ in the above we use equation (REF ) and equation (REF ).",
"Next let us look at the expression: $\\alpha _k [\\chi _i,\\nu ] \\frac{\\partial {\\chi _k}}{\\partial t}&\\hspace{-19.91684pt} =& \\hspace{-19.91684pt} A^{-1}_{kj} k_j \\frac{\\partial {\\chi _k}}{\\partial t}=-(\\frac{\\partial \\chi _j}{\\partial t} + \\vec{\\nabla }\\nu \\cdot \\vec{\\nabla }\\chi _j)A^{-1}_{kj} \\frac{\\partial {\\chi _k}}{\\partial t}\\nonumber \\\\&=& -A^{-1}_{jk} \\frac{\\partial \\chi _j}{\\partial t} \\frac{\\partial \\chi _k}{\\partial t}- \\frac{\\partial \\nu }{\\partial \\chi _m} \\frac{\\partial {\\chi _m}}{\\partial t}$ Inserting equation (REF ) and equation (REF ) into equation (REF ) leads to a Lagrangian density of a more standard quadratic form: $\\hat{\\cal L}[\\chi _i,\\nu ,\\rho ] &=& \\rho [\\frac{1}{2} A^{-1}_{jn} \\frac{\\partial \\chi _j}{\\partial t} \\frac{\\partial \\chi _n}{\\partial t}+\\frac{\\partial \\nu }{\\partial \\chi _m} \\frac{\\partial {\\chi _m}}{\\partial t}-\\frac{\\partial {\\nu }}{\\partial t}\\nonumber \\\\&-& \\varepsilon (\\rho ,\\chi _3)]-\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2)^2.$ We now define the metric $g_{jn} = A^{-1}_{jn}$ and obtain the geometrical Lagrangian: $\\hat{\\cal L}[\\chi _i,\\nu ,\\rho ] &=& \\rho [\\frac{1}{2} g_{jn} \\frac{\\partial \\chi _j}{\\partial t} \\frac{\\partial \\chi _n}{\\partial t}+\\frac{\\partial \\nu }{\\partial \\chi _m} \\frac{\\partial {\\chi _m}}{\\partial t}-\\frac{\\partial {\\nu }}{\\partial t}\\nonumber \\\\&-& \\varepsilon (\\rho ,\\chi _3)]-\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi _1 \\times \\vec{\\nabla }\\chi _2)^2.$ The Lagrangian is thus composed of a geometric kinetic term which is quadratic in the temporal derivatives, a \"gyroscopic\" terms which is linear in the temporal derivative and a potential term which is independent of the temporal derivative."
],
[
"Noether Current",
"Let us assume that all the equations of motion and boundary conditions of non barotropic MHD are satisfied.",
"In this case we have according to equation (REF ): $\\delta A = \\int _{t_1}^{t_2} dt \\int d^3 x \\delta \\hat{\\cal L}= -\\left.",
"\\int d^3 x \\rho \\left[ \\delta \\nu + \\alpha \\delta \\chi + \\beta \\delta \\eta + \\sigma \\delta s \\right] \\right|_{t_1}^{t_2}.$ For the current purpose it does not matter if $\\alpha , \\beta $ and $\\sigma $ are independent variational variables or depend on other variational variables through equation (REF ).",
"Now suppose that the variations $\\delta \\nu , \\delta \\chi , \\delta \\eta , \\delta s$ are symmetry variations such that $\\delta A = 0$ .",
"In that case one obtains a conserved Noether current: $\\delta J = - \\int d^3 x \\rho \\left[ \\delta \\nu + \\alpha \\delta \\chi + \\beta \\delta \\eta + \\sigma \\delta s \\right].$ As the variations in the specific entropy $s$ will generally vary the specific internal energy term in the lagrangian we do not expect non trivial entropy symmetry transformation and the action will only be invariant for $\\delta s = 0$ , hence: $\\delta J = - \\int d^3 x \\rho \\left[ \\delta \\nu + \\alpha \\delta \\chi + \\beta \\delta \\eta \\right].$"
],
[
"Lagrangian and Eulerian variations",
"The value of a function $f$ can be modified by evaluating it at a different point in space, the difference between the new and old values would be: $f (\\vec{x} + \\vec{\\xi }) - f (\\vec{x}) = \\vec{\\xi }\\cdot \\vec{\\nabla }f$ in which $\\vec{x}$ is a coordinate vector and $\\vec{\\xi }$ is a displacement vector, the equality is correct to first order in $\\vec{\\xi }$ .",
"Alternatively we can modify the value of a function by changing it to a different function $f^{\\prime }$ , in this case the difference between the new and old values would be: $\\delta f = f^{\\prime } (\\vec{x}) - f (\\vec{x})$ for a small $\\delta f$ this just the standard variation of variational analysis or an Eulerian variation.",
"Finally we can do both, in the last case the difference between the new and old values would be: $\\Delta f = f^{\\prime } (\\vec{x} + \\vec{\\xi }) - f (\\vec{x})$ Hence: $\\Delta f = f^{\\prime } (\\vec{x} + \\vec{\\xi }) - f (\\vec{x} + \\vec{\\xi }) + f (\\vec{x} + \\vec{\\xi }) - f (\\vec{x}).$ Keeping only first order terms we obtain: $\\Delta f = \\delta f + \\vec{\\xi }\\cdot \\vec{\\nabla }f \\Rightarrow \\delta f = \\Delta f - \\vec{\\xi }\\cdot \\vec{\\nabla }f.$ in which $\\Delta $ is a Lagrangian variation.",
"Now suppose that a specific function is connected to a fluid element in such a way that its value in space is determined only by fluid element location.",
"And suppose that the fluid element is displaced as dictated by the flow.",
"Such a function would be denoted a label of the flow and its material derivative would vanish.",
"Moreover, for a label: $f^{\\prime } (\\vec{x} + \\vec{\\xi }) = f (\\vec{x}) \\Rightarrow \\Delta f = 0$ in order to change the value of a label in a certain point in space the fluid element must be displaced and another (with a different label value) must take its place.",
"If follows from equation (REF ) that for a label: $\\delta f = - \\vec{\\xi }\\cdot \\vec{\\nabla }f.$ Now suppose we have a set of three labels $\\tilde{\\chi }_i$ such that: $\\delta \\tilde{\\chi }_i= - \\vec{\\xi }\\cdot \\vec{\\nabla }\\tilde{\\chi }_i = - \\xi _k \\frac{\\partial \\tilde{\\chi }_i}{\\partial x_k},$ in which we use the Einstein summation convention and $x_k$ are Cartesian coordinates.",
"The inverse of the matrix $\\frac{\\partial \\tilde{\\chi }_i}{\\partial x_k}$ is $\\frac{\\partial x_k}{\\partial \\tilde{\\chi }_i}$ as: $\\frac{\\partial \\tilde{\\chi }_i}{\\partial x_k} \\frac{\\partial x_j}{\\partial \\tilde{\\chi }_i} = \\delta _k^j,$ $\\delta _k^j$ is Kronecker's delta.",
"It thus follows that one can calculate the displacement vector $\\vec{\\xi }$ as follows: $\\xi _k = - \\frac{\\partial x_k}{\\partial \\tilde{\\chi }_i} \\delta \\tilde{\\chi }_i \\Rightarrow \\vec{\\xi }= -\\frac{\\partial \\vec{r}}{\\partial \\tilde{\\chi }_i} \\delta \\tilde{\\chi }_i$"
],
[
"Noether current for label symmetries",
"We now study the form of the Noether current equation (REF ) for the case of label symmetry transformations.",
"It is clear from equation (REF ) that $\\chi ,\\eta $ can be taken to be labels.",
"Hence we can write the conserved Noether current defined in equation (REF ) as: $\\delta J = - \\int d^3 x \\rho \\left[ \\Delta \\nu - \\vec{\\xi }\\cdot \\vec{\\nabla }\\nu - \\alpha \\vec{\\xi }\\cdot \\vec{\\nabla }\\chi - \\beta \\vec{\\xi }\\cdot \\vec{\\nabla }\\eta \\right].$ Or using equation (REF ) as: $\\delta J = \\int d^3 x \\rho \\left[\\vec{\\xi }\\cdot (\\vec{v} - \\sigma \\vec{\\nabla }s) - \\Delta \\nu \\right].$ We use the topological velocity defined in equation (REF ): $\\vec{v}_t = \\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta $ and write: $\\delta J = \\int d^3 x \\rho \\left[\\vec{\\xi }\\cdot \\vec{v}_t - \\Delta \\nu \\right].$ Suppose now that we are considering label symmetry transformations with the infinitesimal form: $\\tilde{\\chi }_i + \\delta \\tilde{\\chi }_i$ this type of transformations will induce a transformation on other functions ($\\nu $ as an example) which could be thought of as functions of labels a transformation of the form: $\\delta \\nu = \\nu (\\tilde{\\chi }_i + \\delta \\tilde{\\chi }_i) - \\nu (\\tilde{\\chi }_i ) = \\delta \\tilde{\\chi }_i \\partial _{\\tilde{\\chi }_i} \\nu =- \\vec{\\xi }\\cdot \\vec{\\nabla }\\tilde{\\chi }_i \\partial _{\\tilde{\\chi }_i} \\nu = - \\vec{\\xi }\\cdot \\vec{\\nabla }\\nu .$ Hence by equation (REF ): $\\Delta \\nu = 0.$ it follows that for an induced infinitesimal label transformation any function will transform as a label.",
"From equation (REF ) it follows that the Noether current will take the following form for a symmetry label transformation: $\\delta J = \\int d^3 x \\rho \\vec{\\xi }\\cdot \\vec{v}_t.$ This Noether current form is identical to equation (47) of [15] and equation (14) of [25], which were derive from a Lagrangian variational principle.",
"We note, however, that this form is limited to the case of label transformations and the general form given in equation (REF ) allows us to exploit larger symmetry groups.",
"Next we will study some symmetry transformations of the action $A$ , in order to do this we shall first introduce the Load and Metage quantities."
],
[
"Load and Metage",
"The following section follows closely a similar section in [2], [14], [15], [24].",
"Consider a thin tube surrounding a magnetic field line, the magnetic flux contained within the tube is: $\\Delta \\Phi =\\int \\vec{B} \\cdot d \\vec{S}$ and the mass contained with the tube is: $\\Delta M = \\int \\rho d\\vec{l} \\cdot d \\vec{S},$ in which $dl$ is a length element along the tube.",
"Since the magnetic field lines move with the flow by virtue of equation (REF ) and equation (REF ) both the quantities $\\Delta \\Phi $ and $\\Delta M$ are conserved and since the tube is thin we may define the conserved magnetic load: $\\lambda = \\frac{\\Delta M}{\\Delta \\Phi } = \\oint \\frac{\\rho }{B}dl,$ in which the above integral is performed along the field line.",
"Obviously the parts of the line which go out of the flow to regions in which $\\rho =0$ have a null contribution to the integral.",
"Notice that $\\lambda $ is a single valued function that can be measured in principle.",
"Since $\\lambda $ is conserved it satisfies the equation: $\\frac{d \\lambda }{d t} = 0.$ This can be viewed as a manifestation of the frozen-in law of $\\frac{B}{\\rho }$ .",
"By construction surfaces of constant magnetic load move with the flow and contain magnetic field lines.",
"Hence the gradient to such surfaces must be orthogonal to the field line: $\\vec{\\nabla }\\lambda \\cdot \\vec{B} = 0.$ Now consider an arbitrary comoving point on the magnetic field line and denote it by $i$ , and consider an additional comoving point on the magnetic field line and denote it by $r$ .",
"The integral: $\\mu (r) = \\int _i^r \\frac{\\rho }{B}dl + \\mu (i),$ is also a conserved quantity which we may denote following Lynden-Bell & Katz [10] as the magnetic metage.",
"$\\mu (i)$ is an arbitrary number which can be chosen differently for each magnetic line.",
"By construction: $\\frac{d \\mu }{d t} = 0.$ This can be viewed as another manifestation of the frozen-in law of $\\frac{B}{\\rho }$ .",
"Also it is easy to see that by differentiating along the magnetic field line we obtain: $\\vec{\\nabla }\\mu \\cdot \\vec{B} = \\rho .$ Notice that $\\mu $ will be generally a non single valued function, we will show later in this paper that symmetry to translations in $\\mu $ ; will generate through the Noether theorem the conservation of the magnetic cross helicity.",
"At this point we have two comoving coordinates of flow, namely $\\lambda ,\\mu $ obviously in a three dimensional flow we also have a third coordinate.",
"However, before defining the third coordinate we will find it useful to work not directly with $\\lambda $ but with a function of $\\lambda $ .",
"Now consider the magnetic flux within a surface of constant load $\\Phi (\\lambda )$ .",
"The magnetic flux is a conserved quantity and depends only on the load $\\lambda $ of the surrounding surface.",
"Now we define the quantity: $\\chi = \\frac{\\Phi (\\lambda )}{2\\pi }.$ Obviously $\\chi $ satisfies the equations: $\\frac{d \\chi }{d t} = 0, \\qquad \\vec{B} \\cdot \\vec{\\nabla }\\chi = 0.$ Let us now define an additional comoving coordinate $\\eta ^{*}$ since $\\vec{\\nabla }\\mu $ is not orthogonal to the $\\vec{B}$ lines we can choose $\\vec{\\nabla }\\eta ^{*}$ to be orthogonal to the $\\vec{B}$ lines and not be in the direction of the $\\vec{\\nabla }\\chi $ lines, that is we choose $\\eta ^{*}$ not to depend only on $\\chi $ .",
"Since both $\\vec{\\nabla }\\eta ^{*}$ and $\\vec{\\nabla }\\chi $ are orthogonal to $\\vec{B}$ , $\\vec{B}$ must take the form: $\\vec{B} = A \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta ^{*}.$ However, using equation (REF ) we have: $\\vec{\\nabla }\\cdot \\vec{B} = \\vec{\\nabla }A \\cdot (\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta ^{*})=0.$ Which implies that $A$ is a function of $\\chi ,\\eta ^{*}$ .",
"Now we can define a new comoving function $\\eta $ such that: $\\eta = \\int _0^{\\eta ^{*}}A(\\chi ,\\eta ^{^{\\prime }*})d\\eta ^{^{\\prime }*}, \\qquad \\frac{d \\eta }{d t} = 0.$ In terms of this function we obtain the Sakurai (Euler potentials) presentation: $\\vec{B} = \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta .$ And the density is now given by the Jacobian: $\\rho = \\vec{\\nabla }\\mu \\cdot (\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta )=\\frac{\\partial (\\chi ,\\eta ,\\mu )}{\\partial (x,y,z)}.$ It can easily be shown using the fact that the labels are comoving that the above forms of $\\vec{B}$ and $\\rho $ satisfy equation (REF ), equation (REF ) and equation (REF ) automatically.",
"We can now write a Lagrangian density in terms of the labels, in which $\\rho $ is no longer an independent variational variable but rather a quantity dependent on $\\mu $ through equation (REF ).",
"The Lagrangian density of equation (REF ) takes the form: $& & \\hspace{-34.14322pt}\\hat{\\cal L}[\\alpha ,\\beta ,\\chi ,\\eta ,\\mu ,\\nu ,\\sigma ,s] = -\\frac{\\partial (\\chi ,\\eta ,\\mu )}{\\partial (x,y,z)} \\left[\\frac{\\partial {\\nu }}{\\partial t} + \\alpha \\frac{\\partial {\\chi }}{\\partial t} + \\beta \\frac{\\partial {\\eta }}{\\partial t} + \\sigma \\frac{\\partial {s}}{\\partial t}+ \\varepsilon (\\frac{\\partial (\\chi ,\\eta ,\\mu )}{\\partial (x,y,z)},s) \\right.\\nonumber \\\\&+& \\left.",
"\\frac{1}{2} (\\vec{\\nabla }\\nu + \\alpha \\vec{\\nabla }\\chi + \\beta \\vec{\\nabla }\\eta + \\sigma \\vec{\\nabla }s )^2 \\right]-\\frac{1}{8 \\pi }(\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta )^2$ Notice however, that $\\eta $ is defined in a non unique way since one can redefine $\\eta $ for example by performing the following transformation: $\\eta \\rightarrow \\eta + f(\\chi )$ in which $f(\\chi )$ is an arbitrary function.",
"The comoving coordinates $\\chi ,\\eta $ serve as labels of the magnetic field lines.",
"Moreover the magnetic flux can be calculated as: $\\Phi = \\int \\vec{B} \\cdot d \\vec{S} = \\int d \\chi d \\eta .$ In the case that the surface integral is performed inside a load contour we obtain: $\\Phi (\\lambda ) = \\int _{\\lambda } d \\chi d \\eta = \\chi \\int _{\\lambda } d \\eta =\\left\\lbrace \\begin{array}{c}\\chi [\\eta ] \\\\\\chi (\\eta _{max}-\\eta _{min}) \\\\\\end{array}\\right.$ There are two cases involved; in one case the load surfaces are topological cylinders; in this case $\\eta $ is not single valued and hence we obtain the upper value for $\\Phi (\\lambda )$ .",
"In a second case the load surfaces are topological spheres; in this case $\\eta $ is single valued and has minimal $\\eta _{min}$ and maximal $\\eta _{max}$ values.",
"Hence the lower value of $\\Phi (\\lambda )$ is obtained.",
"For example in some cases $\\eta $ is identical to twice the latitude angle $\\theta $ .",
"In those cases $\\eta _{min}=0$ (value at the \"north pole\") and $\\eta _{max}= 2 \\pi $ (value at the \"south pole\").",
"Comparing the above equation with equation (REF ) we derive that $\\eta $ can be either single valued or not single valued and that its discontinuity across its cut in the non single valued case is $[\\eta ] =2 \\pi $ .",
"The triplet $\\chi ,\\eta ,\\mu $ will suffice to label any fluid element in three dimensions.",
"But for a non-barotropic flow there is also another possible label $s$ which is comoving according to equation (REF ).",
"The question then arises of the relation of this label to the previous three.",
"As one needs to make a choice regarding the preferred set of labels it seems that the physical ones are $\\chi ,\\eta ,s$ in which we use the surfaces on which the magnetic fields lie and the entropy, each label has an obvious physical interpretation.",
"In this case we must look at $\\mu $ as a function of $\\chi ,\\eta ,s$ .",
"If the magnetic field lines lie on entropy surface then $\\mu $ regains its status as an independent label.",
"The density can now be written as: $\\rho = \\frac{\\partial \\mu }{ \\partial s} \\frac{\\partial (\\chi ,\\eta ,s)}{\\partial (x,y,z)}.$ Now as $\\mu $ can be defined for each magnetic field line separately according to equation (REF ) it is obvious that such a choice exist in which $\\mu $ is a function of $s$ only.",
"One may also think of the entropy $s$ as a functions $\\chi ,\\eta ,\\mu $ .",
"However, if one change $\\mu $ in this case this generally entails a change in $s$ and the symmetry described in equation (REF ) is lost in the Action.",
"In what follows we shall ignore the status of $s$ as a label and consider it as a variational variable which only attains a status of a label at the variational extremum."
],
[
"The labelling symmetry group and its subgroups",
"It is obvious that the choice of fluid labels is quite arbitrary.",
"However, when enforcing the $\\chi , \\eta , \\mu $ coordinate system satisfying equation (REF ) the choice is restricted to $\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu }$ such that: ${\\partial (\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu }) \\over \\partial (\\chi , \\eta , \\mu )} = 1.$ Moreover the Euler potential magnetic field representation: $\\vec{B} = \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\eta ,$ reduces the choice further to: ${\\partial (\\tilde{\\chi }, \\tilde{\\eta }) \\over \\partial (\\chi , \\eta )} = 1.$ We further notice that in the Eulerian variation principle approach the label symmetry cannot be realized unless it is coupled to transformation of other variational variables that is the label transformation induces transformation on $\\alpha $ and $\\beta $ as follows: $\\tilde{\\alpha }&=& \\alpha \\partial _{\\tilde{\\chi }} \\chi + \\beta \\partial _{\\tilde{\\chi }} \\eta \\nonumber \\\\\\tilde{\\beta }&=& \\alpha \\partial _{\\tilde{\\eta }} \\chi + \\beta \\partial _{\\tilde{\\eta }} \\eta $ From equation (REF ) it follows that: $\\hat{\\cal L}[\\alpha ,\\beta ,\\chi ,\\eta ,\\mu ,\\nu ,\\sigma ,s] - \\hat{\\cal L}[\\tilde{\\alpha }, \\tilde{\\beta },\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu },\\nu ,\\sigma ,s] = 0$ hence the label transformation is a symmetry transformation.",
"Now suppose that we consider $\\nu $ as a function of the labels: $\\nu (x,y,z,t) = \\bar{\\nu }(\\chi ,\\eta ,\\mu ,t)$ in that case replacing $(\\chi ,\\eta ,\\mu ) \\rightarrow (\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu })$ in the above equation will yield a different function $\\tilde{\\nu }$ of the coordinates such that: $\\bar{\\nu }(\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu },t) = \\tilde{\\nu }(x,y,z,t)$ From this point of view which we adopt in the current paper, symmetry can be only achieved if: $\\int d^3 x \\left[\\hat{\\cal L}[\\alpha ,\\beta ,\\chi ,\\eta ,\\mu ,\\bar{\\nu }(\\chi ,\\eta ,\\mu ,t),\\sigma ,s] -\\hat{\\cal L}[\\tilde{\\alpha }, \\tilde{\\beta },\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu },\\bar{\\nu }(\\tilde{\\chi }, \\tilde{\\eta }, \\tilde{\\mu },t),\\sigma ,s]\\right] = 0$"
],
[
"Metage translations",
"In what follows we consider the transformation (see also equation (REF )): $\\tilde{\\chi }= \\chi , \\tilde{\\eta }= \\eta , \\tilde{\\mu }= \\mu + a (\\chi ,\\eta )$ Hence $a$ is a label displacement which may be different for each magnetic field line, as the field line is closed one need not worry about edge difficulties.",
"This transformation satisfies trivially the conditions (REF ,REF ).",
"If we take the infinitesimal symmetry transformation $\\delta \\mu = a, \\delta \\chi =\\delta \\eta =0$ we can calculate the associated fluid element displacement with this relabelling using equation (REF ) and equation (REF ).",
"$\\vec{\\xi }= -{\\partial \\vec{r} \\over \\partial \\mu }\\delta \\mu = -\\delta \\mu \\frac{\\vec{B}}{\\rho }.$ Inserting equation (REF ) into equation (REF ) we obtain the conservation law: $\\delta J =\\int d^3 x \\rho \\vec{v}_t \\cdot \\vec{\\xi }= -\\int d^3 x \\delta \\mu \\vec{v}_t \\cdot \\vec{B}$ In the simplest case we may take $\\delta \\mu $ to be a small constant, hence: $\\delta J = - \\delta \\mu \\int d^3 x \\vec{v}_t \\cdot \\vec{B} = - \\delta \\mu H_{CNB}$ Where $H_{CNB}$ is the non barotropic global cross helicity [19], [29], [30] defined as: $H_{CNB} \\equiv \\int d^3 x \\vec{v}_t \\cdot \\vec{B}.$ We thus obtain the conservation of non-barotropic cross helicity using the Noether theorem and the symmetry group of metage translations.",
"Of course one can perform a different translation on each magnetic field line, in this case one obtains: $\\delta J = -\\int d^3 x \\delta \\mu \\vec{v}_t \\cdot \\vec{B} =-\\int d \\chi d \\eta \\delta \\mu \\oint _{\\chi ,\\eta } d\\mu \\rho ^{-1} \\vec{v}_t \\cdot \\vec{B}$ Now since $\\delta \\mu $ is an arbitrary (small) function of $\\chi ,\\eta $ it follows that: $I = \\oint _{\\chi ,\\eta } d\\mu \\rho ^{-1} \\vec{v}_t \\cdot \\vec{B}$ is a conserved quantity for each magnetic field line.",
"Along a magnetic field line the following equations hold: $d\\mu = \\vec{\\nabla }\\mu \\cdot d \\vec{r} = \\vec{\\nabla }\\mu \\cdot \\hat{B} dr = \\frac{\\rho }{B} dr$ in the above $\\hat{B}$ is an unit vector in the magnetic field direction an equation (REF ) is used.",
"Inserting equation (REF ) into equation (REF ) we obtain: $I = \\oint _{\\chi ,\\eta } dr \\vec{v}_t \\cdot \\hat{B} = \\oint _{\\chi ,\\eta } d \\vec{r} \\cdot \\vec{v}_t.$ which is just the circulation of the topological velocity along the magnetic field lines.",
"This quantity can be written in terms of the generalized Clebsch representation of the velocity equation (REF ) as: $I = \\oint _{\\chi ,\\eta } d \\vec{r} \\cdot \\vec{v}_t = \\oint _{\\chi ,\\eta } d \\vec{r} \\cdot \\vec{\\nabla }\\nu = [\\nu ].$ $[\\nu ]$ is the discontinuity of $\\nu $ .",
"This was shown to be equal to the amount of non barotropic cross helicity per unit of magnetic flux in equation (REF ) [29], [30].",
"$I=[\\nu ]= \\frac{dH_{CNB}}{d \\Phi }.$"
],
[
"Transformations of magnetic surfaces",
"Consider the following transformations: $\\tilde{\\eta } = \\eta + \\delta \\eta (\\chi ,\\eta ), \\qquad \\tilde{\\chi } = \\chi + \\delta \\chi (\\chi ,\\eta ), \\qquad \\tilde{\\mu } = \\mu $ in which $\\delta \\eta ,\\delta \\chi $ are considered small in some sense.",
"Inserting the above quantities into equation (REF ) and keeping only first order terms we arrive at: $\\partial _{\\eta } \\delta \\eta + \\partial _{\\chi } \\delta \\chi = 0.$ This equation can be solved as follows: $\\delta \\eta = \\partial _{\\chi } \\delta f, \\qquad \\delta \\chi = -\\partial _{\\eta } \\delta f,$ in which $\\delta f= \\delta f (\\chi ,\\eta )$ is an arbitrary small function.",
"In this case we obtain a particle displacements of the form: $\\vec{\\xi }&=& -{\\partial \\vec{r} \\over \\partial \\chi }\\delta \\chi -{\\partial \\vec{r} \\over \\partial \\eta }\\delta \\eta =-\\frac{1}{\\rho } \\left( \\vec{\\nabla }\\eta \\times \\vec{\\nabla }\\mu \\ \\delta \\chi + \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\chi \\ \\delta \\eta \\right)\\nonumber \\\\&=& \\frac{\\vec{\\nabla }\\mu }{\\rho } \\times ( \\vec{\\nabla }\\eta \\delta \\chi - \\vec{\\nabla }\\chi \\delta \\eta )$ A special case that satisfies equation (REF ) is the case of a constant $\\delta \\chi $ and $\\delta \\eta $ , those two independent displacements lead to two new topological conservation laws: $\\delta J_\\chi &=& \\delta \\chi \\int d^3 x \\vec{v} _t \\cdot \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta = \\delta \\chi \\ H_{CNB \\chi },\\nonumber \\\\\\delta J_\\eta &=& \\delta \\eta \\int d^3 x \\vec{v} _t \\cdot \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu = \\delta \\eta \\ H_{CNB \\eta }.$ Where the new non barotropic global cross helicities are defined as: $H_{CNB \\chi } \\equiv \\int d^3 x \\vec{v} _t \\cdot \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta , \\quad H_{CNB \\eta } \\equiv \\int d^3 x \\vec{v} _t \\cdot \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu $ We will find it useful to introduce the abstract \"magnetic fields\" as follows: $\\vec{B}_\\chi \\equiv \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta , \\qquad \\vec{B}_\\eta \\equiv \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu $ In terms of which we obtain the new helicities in a more conventional form: $H_{CNB \\chi } = \\int d^3 x \\vec{v} _t \\cdot \\vec{B}_\\chi , \\qquad H_{CNB \\eta } = \\int d^3 x \\vec{v} _t \\cdot \\vec{B}_\\eta $ It is more plausible that those symmetries and conservation laws hold for magnetic field lines which lie on topological torii.",
"In this case $\\eta $ is non single valued [2] and thus the translation in this direction resembles moving fluid elements along closed loops.",
"Both those helicities suffer a topological interpretation in terms of the knottiness of the abstract magnetic field lines and the flow lines.",
"Finally we remark that for barotropic MHD $\\vec{v}_t$ can be replaced with $\\vec{v}$ ."
],
[
"Direct Derivation",
"Before continuing to discuss the possible applications of the topological constants of motion, we shall demonstrate that the generalized cross helicities are indeed constant without relying on the Noether theorem."
],
[
"Direct derivation of the constancy of non barotropic cross helicity",
"Taking the temporal derivative of the non barotropic cross helicity given in equation (REF ) we obtain: $\\frac{d H_{CNB}}{dt} = \\int d^3 x \\left[ \\partial _t \\vec{v}_t \\cdot \\vec{B} + \\vec{v}_t \\cdot \\partial _t \\vec{B} \\right].$ in the above $\\frac{d }{dt}$ is an ordinary temporal derivative, and we use the notation: $\\partial _t \\equiv \\frac{\\partial }{\\partial t}$ .",
"Using equation (REF ) it follows that: $\\vec{v}_t \\cdot \\partial _t \\vec{B} = \\vec{v}_t \\cdot \\vec{\\nabla }\\times (\\vec{v} \\times \\vec{B})=\\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}) \\times \\vec{v}_t \\right) + (\\vec{v} \\times \\vec{B}) \\cdot \\vec{\\omega }_t$ in which we have used a standard identity of vector analysis and the definition: $\\vec{\\omega }_t = \\vec{\\nabla }\\times \\vec{v}_t = \\vec{\\omega }- \\vec{\\nabla }\\sigma \\times \\vec{\\nabla }s$ $\\vec{v}_t$ is defined in equation (REF ).",
"Next we calculate: $\\partial _t \\vec{v}_t \\cdot \\vec{B} = \\vec{B} \\cdot \\partial _t \\left(\\vec{v} - \\sigma \\vec{\\nabla }s \\right) = \\vec{B} \\cdot \\left( \\partial _t \\vec{v} - \\partial _t \\sigma \\vec{\\nabla }s- \\sigma \\vec{\\nabla }\\partial _t s \\right)$ Taking into account Euler equation (REF ) and the standard thermodynamic identities of equation (REF ): $\\partial _t \\vec{v} &=& -(\\vec{v} \\cdot \\vec{\\nabla })\\vec{v} - \\frac{1}{\\rho }\\vec{\\nabla }p (\\rho ,s) +\\frac{(\\vec{\\nabla }\\times \\vec{B}) \\times \\vec{B}}{4 \\pi \\rho }\\nonumber \\\\&=& \\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2) - \\vec{\\nabla }w + T \\vec{\\nabla }s +\\frac{(\\vec{\\nabla }\\times \\vec{B}) \\times \\vec{B}}{4 \\pi \\rho }$ Hence: $\\vec{B} \\cdot \\partial _t \\vec{v} = \\vec{B} \\cdot \\left(\\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w) + T \\vec{\\nabla }s \\right).$ Taking into account equation (REF ) it follows that: $- \\partial _t \\sigma \\vec{\\nabla }s = (\\vec{v} \\cdot \\vec{\\nabla }\\sigma - T) \\vec{\\nabla }s .$ And taking into account equation (REF ) it follows that: $- \\sigma \\vec{\\nabla }\\partial _t s = \\sigma \\vec{\\nabla }(\\vec{v} \\cdot \\vec{\\nabla }s).$ Inserting equation (REF ), equation (REF ) and equation (REF ) into equation (REF ) it follows that: $\\partial _t \\vec{v}_t \\cdot \\vec{B} = \\vec{B} \\cdot \\left( \\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w)+ (\\vec{v} \\cdot \\vec{\\nabla }\\sigma ) \\vec{\\nabla }s+ \\sigma \\vec{\\nabla }(\\vec{v} \\cdot \\vec{\\nabla }s) \\right)$ Hence: $\\partial _t \\vec{v}_t \\cdot \\vec{B}&=&\\vec{B} \\cdot \\left( \\vec{v} \\times \\vec{\\omega }+ \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right.\\nonumber \\\\&+& \\left.",
"(\\vec{v} \\cdot \\vec{\\nabla }\\sigma ) \\vec{\\nabla }s - (\\vec{v} \\cdot \\vec{\\nabla }s) \\vec{\\nabla }\\sigma \\right)\\nonumber \\\\&& \\hspace{-56.9055pt} = \\vec{B} \\cdot \\left( \\vec{v} \\times \\vec{\\omega }+ \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right)+ (\\vec{\\nabla }\\sigma \\times \\vec{\\nabla }s) \\times \\vec{v} \\right)\\nonumber \\\\& = & \\vec{B} \\cdot \\left( \\vec{v} \\times \\vec{\\omega }_t + \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right)$ Combining equation (REF ) with equation (REF ) we arrive at the result: $\\partial _t \\vec{v}_t \\cdot \\vec{B} + \\vec{v}_t \\cdot \\partial _t \\vec{B} &=& \\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}) \\times \\vec{v}_t \\right)+ \\vec{B} \\cdot \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right)\\nonumber \\\\&=& \\vec{\\nabla }\\cdot \\left[ (\\vec{v} \\times \\vec{B}) \\times \\vec{v}_t+ \\vec{B} \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right]$ in which we take into account equation (REF ).",
"Inserting equation (REF ) into equation (REF ) and using Gauss theorem we obtain a surface integral: $\\frac{d H_{CNB}}{dt} = \\oint d \\vec{S} \\cdot \\left[ (\\vec{v} \\times \\vec{B}) \\times \\vec{v}_t+ \\vec{B} \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right].$ The surface integral encapsulates the volume for which the non barotropic cross helicity is calculated.",
"If the surface is taken at infinity the magnetic fields vanish and thus: $\\frac{d H_{CNB}}{dt} = 0$ which means that $H_{CNB}$ is a constant of motion.",
"We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach.",
"However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced."
],
[
"Direct derivation of the constancy of non barotropic $\\chi $ cross helicity",
"Taking the temporal derivative of the non barotropic $\\chi $ cross helicity given in equation (REF ) we obtain: $\\frac{d H_{CNB\\chi }}{dt} = \\int d^3 x \\left[ \\partial _t \\vec{v}_t \\cdot \\vec{B}_\\chi + \\vec{v}_t \\cdot \\partial _t \\vec{B}_\\chi \\right].$ Let us calculate $\\partial _t \\vec{B}_\\chi $ where $\\vec{B}_\\chi $ is defined in equation (REF ): $\\vec{B}_\\chi = \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta .$ It follows that: $\\partial _t \\vec{B}_\\chi = \\vec{\\nabla }\\partial _t \\mu \\times \\vec{\\nabla }\\eta + \\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\partial _t \\eta .$ Using equation (REF ) and equation (REF ) we obtain: $\\partial _t \\vec{B}_\\chi &=& \\vec{\\nabla }(-\\vec{v} \\cdot \\vec{\\nabla }\\mu ) \\times \\vec{\\nabla }\\eta + \\vec{\\nabla }\\mu \\times \\vec{\\nabla }(-\\vec{v} \\cdot \\vec{\\nabla }\\eta )\\nonumber \\\\&=& \\vec{\\nabla }\\times \\left(\\vec{\\nabla }\\mu (\\vec{v} \\cdot \\vec{\\nabla }\\eta ) - \\vec{\\nabla }\\eta (\\vec{v} \\cdot \\vec{\\nabla }\\mu )\\right)\\nonumber \\\\&=& \\vec{\\nabla }\\times \\left(\\vec{v} \\times (\\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta ) \\right) =\\vec{\\nabla }\\times \\left(\\vec{v} \\times \\vec{B}_\\chi \\right)$ in which we used standard vector analysis identities.",
"It thus follows that: $\\vec{v}_t \\cdot \\partial _t \\vec{B}_\\chi = \\vec{v}_t \\cdot \\vec{\\nabla }\\times (\\vec{v} \\times \\vec{B}_\\chi )=\\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}_\\chi ) \\times \\vec{v}_t \\right) + (\\vec{v} \\times \\vec{B}_\\chi ) \\cdot \\vec{\\omega }_t$ Next we calculate: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\chi = \\vec{B}_\\chi \\cdot \\partial _t \\left(\\vec{v} - \\sigma \\vec{\\nabla }s \\right)= \\vec{B}_\\chi \\cdot \\left( \\partial _t \\vec{v} - \\partial _t \\sigma \\vec{\\nabla }s - \\sigma \\vec{\\nabla }\\partial _t s \\right)$ Taking into account equation (REF ): $\\vec{B}_\\chi \\cdot \\partial _t \\vec{v} = \\vec{B}_\\chi \\cdot \\left(\\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w) + T \\vec{\\nabla }s \\right) + \\vec{B}_\\chi \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B}$ in which the current density is given by: $\\vec{J} = \\frac{\\vec{\\nabla }\\times \\vec{B}}{4 \\pi } \\quad \\Rightarrow \\vec{\\nabla }\\cdot \\vec{J} = 0.$ Now: $\\vec{B}_\\chi \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B} = \\frac{1}{\\rho } \\vec{J} \\cdot \\vec{B} \\times \\vec{B}_\\chi .$ However: $\\vec{B} \\times \\vec{B}_\\chi = \\vec{B} \\times (\\vec{\\nabla }\\mu \\times \\vec{\\nabla }\\eta ) =\\vec{\\nabla }\\mu (\\vec{B} \\cdot \\vec{\\nabla }\\eta ) - \\vec{\\nabla }\\eta (\\vec{B} \\cdot \\vec{\\nabla }\\mu ) = - \\rho \\vec{\\nabla }\\eta .$ It thus follows that: $\\vec{B}_\\chi \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B} = - \\vec{J} \\cdot \\vec{\\nabla }\\eta = - \\vec{\\nabla }\\cdot (\\vec{J} \\eta ) .$ in which we used equation (REF ) and equation (REF ).",
"Inserting equation (REF ) into equation (REF ) will yield: $\\vec{B}_\\chi \\cdot \\partial _t \\vec{v} = \\vec{B}_\\chi \\cdot \\left(\\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w) + T \\vec{\\nabla }s \\right) - \\vec{\\nabla }\\cdot (\\vec{J} \\eta )$ Inserting equation (REF ), equation (REF ) and equation (REF ) into equation (REF ) it follows that: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\chi = \\vec{B}_\\chi \\cdot \\left( \\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w)+ (\\vec{v} \\cdot \\vec{\\nabla }\\sigma ) \\vec{\\nabla }s+ \\sigma \\vec{\\nabla }(\\vec{v} \\cdot \\vec{\\nabla }s) \\right)- \\vec{\\nabla }\\cdot (\\vec{J} \\eta )$ Hence: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\chi = \\vec{B}_\\chi \\cdot \\left( \\vec{v} \\times \\vec{\\omega }_t + \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right)- \\vec{\\nabla }\\cdot (\\vec{J} \\eta )$ Combining equation (REF ) with equation (REF ) we arrive at the result: $& & \\partial _t \\vec{v}_t \\cdot \\vec{B}_\\chi + \\vec{v}_t \\cdot \\partial _t \\vec{B}_\\chi \\nonumber \\\\&=& \\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}_\\chi ) \\times \\vec{v}_t - \\vec{J} \\eta \\right)+ \\vec{B}_\\chi \\cdot \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right)\\nonumber \\\\&=& \\vec{\\nabla }\\cdot \\left[ (\\vec{v} \\times \\vec{B}_\\chi ) \\times \\vec{v}_t+ \\vec{B}_\\chi \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) - \\vec{J} \\eta \\right]$ in which we take into account equation (REF ).",
"Inserting equation (REF ) into equation (REF ) and using Gauss theorem we obtain a surface integral: $\\frac{d H_{CNB \\chi }}{dt} &=& \\oint d \\vec{S} \\cdot \\left[ (\\vec{v} \\times \\vec{B}_\\chi ) \\times \\vec{v}_t+ \\vec{B}_\\chi \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) - \\vec{J} \\eta \\right]\\nonumber \\\\&-& \\int d \\vec{\\Sigma }\\cdot \\vec{J} [\\eta ] .$ The surface integral encapsulates the volume for which the $\\chi $ non barotropic cross helicity is calculated and an additional surface integral is performed along the cut of $\\eta $ , in case that $\\eta $ is not single valued (see equation (REF )).",
"If the surface is taken at infinity the magnetic fields and current densities vanish and thus: $\\frac{d H_{CNB \\chi }}{dt} = - \\int d \\vec{\\Sigma }\\cdot \\vec{J} [\\eta ]$ hence for spherical topologies of magnetic field lines or for a current density $\\vec{J}$ parallel to the cut we obtain: $\\frac{d H_{CNB \\chi }}{dt} = 0.$ which means that $H_{CNB \\chi }$ is a constant of motion.",
"We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach.",
"However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced."
],
[
"Direct derivation of the constancy of non barotropic $\\eta $ cross helicity",
"Taking the temporal derivative of the non barotropic $\\eta $ cross helicity given in equation (REF ) we obtain: $\\frac{d H_{CNB\\eta }}{dt} = \\int d^3 x \\left[ \\partial _t \\vec{v}_t \\cdot \\vec{B}_\\eta + \\vec{v}_t \\cdot \\partial _t \\vec{B}_\\eta \\right].$ Let us calculate $\\partial _t \\vec{B}_\\eta $ where $\\vec{B}_\\eta $ is defined in equation (REF ): $\\vec{B}_\\eta = \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu .$ It follows that: $\\partial _t \\vec{B}_\\eta = \\vec{\\nabla }\\partial _t \\chi \\times \\vec{\\nabla }\\mu + \\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\partial _t \\mu .$ Using equation (REF ) and equation (REF ) we obtain: $\\partial _t \\vec{B}_\\eta &=& \\vec{\\nabla }(-\\vec{v} \\cdot \\vec{\\nabla }\\chi ) \\times \\vec{\\nabla }\\mu + \\vec{\\nabla }\\chi \\times \\vec{\\nabla }(-\\vec{v} \\cdot \\vec{\\nabla }\\mu )\\nonumber \\\\&=& \\vec{\\nabla }\\times \\left(\\vec{\\nabla }\\chi (\\vec{v} \\cdot \\vec{\\nabla }\\mu ) - \\vec{\\nabla }\\mu (\\vec{v} \\cdot \\vec{\\nabla }\\chi )\\right)\\nonumber \\\\&=& \\vec{\\nabla }\\times \\left(\\vec{v} \\times (\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu ) \\right) =\\vec{\\nabla }\\times \\left(\\vec{v} \\times \\vec{B}_\\eta \\right)$ in which we used standard vector analysis identities.",
"It thus follows that: $\\vec{v}_t \\cdot \\partial _t \\vec{B}_\\eta = \\vec{v}_t \\cdot \\vec{\\nabla }\\times (\\vec{v} \\times \\vec{B}_\\eta )=\\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}_\\eta ) \\times \\vec{v}_t \\right) + (\\vec{v} \\times \\vec{B}_\\eta ) \\cdot \\vec{\\omega }_t$ Next we calculate: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\eta = \\vec{B}_\\eta \\cdot \\partial _t \\left(\\vec{v} - \\sigma \\vec{\\nabla }s \\right)= \\vec{B}_\\eta \\cdot \\left( \\partial _t \\vec{v} - \\partial _t \\sigma \\vec{\\nabla }s - \\sigma \\vec{\\nabla }\\partial _t s \\right)$ Taking into account equation (REF ): $\\vec{B}_\\eta \\cdot \\partial _t \\vec{v} = \\vec{B}_\\eta \\cdot \\left(\\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w) + T \\vec{\\nabla }s \\right) + \\vec{B}_\\eta \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B}.$ Now: $\\vec{B}_\\eta \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B} = \\frac{1}{\\rho } \\vec{J} \\cdot \\vec{B} \\times \\vec{B}_\\eta .$ However: $\\vec{B} \\times \\vec{B}_\\eta = \\vec{B} \\times (\\vec{\\nabla }\\chi \\times \\vec{\\nabla }\\mu ) =\\vec{\\nabla }\\chi (\\vec{B} \\cdot \\vec{\\nabla }\\mu ) - \\vec{\\nabla }\\mu (\\vec{B} \\cdot \\vec{\\nabla }\\chi ) = \\rho \\vec{\\nabla }\\chi .$ It thus follows that: $\\vec{B}_\\eta \\cdot \\frac{1}{\\rho }\\vec{J} \\times \\vec{B} = \\vec{J} \\cdot \\vec{\\nabla }\\chi = \\vec{\\nabla }\\cdot (\\vec{J} \\chi ) .$ in which we used equation (REF ) and equation (REF ).",
"Inserting equation (REF ) into equation (REF ) will yield: $\\vec{B}_\\eta \\cdot \\partial _t \\vec{v} = \\vec{B}_\\eta \\cdot \\left(\\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w) + T \\vec{\\nabla }s \\right) + \\vec{\\nabla }\\cdot (\\vec{J} \\chi )$ Inserting equation (REF ), equation (REF ) and equation (REF ) into equation (REF ) it follows that: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\eta = \\vec{B}_\\eta \\cdot \\left( \\vec{v} \\times \\vec{\\omega }- \\vec{\\nabla }(\\frac{1}{2} \\vec{v}^2+w)+ (\\vec{v} \\cdot \\vec{\\nabla }\\sigma ) \\vec{\\nabla }s+ \\sigma \\vec{\\nabla }(\\vec{v} \\cdot \\vec{\\nabla }s) \\right)+ \\vec{\\nabla }\\cdot (\\vec{J} \\chi )$ Hence: $\\partial _t \\vec{v}_t \\cdot \\vec{B}_\\eta = \\vec{B}_\\eta \\cdot \\left( \\vec{v} \\times \\vec{\\omega }_t + \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) \\right)+ \\vec{\\nabla }\\cdot (\\vec{J} \\chi )$ Combining equation (REF ) with equation (REF ) we arrive at the result: $& & \\partial _t \\vec{v}_t \\cdot \\vec{B}_\\eta + \\vec{v}_t \\cdot \\partial _t \\vec{B}_\\eta \\nonumber \\\\&=& \\vec{\\nabla }\\cdot \\left( (\\vec{v} \\times \\vec{B}_\\eta ) \\times \\vec{v}_t + \\vec{J} \\chi \\right)+ \\vec{B}_\\eta \\cdot \\vec{\\nabla }\\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right)\\nonumber \\\\&=& \\vec{\\nabla }\\cdot \\left[ (\\vec{v} \\times \\vec{B}_\\eta ) \\times \\vec{v}_t+ \\vec{B}_\\eta \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) + \\vec{J} \\chi \\right]$ in which we take into account equation (REF ).",
"Inserting equation (REF ) into equation (REF ) and using Gauss theorem we obtain a surface integral: $\\frac{d H_{CNB \\eta }}{dt} = \\oint d \\vec{S} \\cdot \\left[ (\\vec{v} \\times \\vec{B}_\\eta ) \\times \\vec{v}_t+ \\vec{B}_\\eta \\left(\\sigma (\\vec{v} \\cdot \\vec{\\nabla }s)-\\frac{1}{2} \\vec{v}^2-w\\right) + \\vec{J} \\chi \\right].$ The surface integral encapsulates the volume for which the $\\eta $ non barotropic cross helicity is calculated.",
"If the surface is taken at infinity the magnetic fields and current densities vanish and thus: $\\frac{d H_{CNB \\eta }}{dt} = 0.$ which means that $H_{CNB \\eta }$ is a constant of motion.",
"We notice the complexity of the direct derivation with respect to the elegance and simplicity of the Noether theorem approach.",
"However, obtaining the same result using different methods strengthens our confidence that no mathematical error was accidentally introduced."
],
[
"Possible Application",
"In his important review paper \"Physics of magnetically confined plasmas\" A. H. Boozer [31] states that: \"A spiky current profile causes a rapid dissipation of energy relative to magnetic helicity.",
"If the evolution of a magnetic field is rapid, then it must be at constant helicity.\"",
"Usually topological conservation laws are used in order to deduce lower bounds on the \"energy\" of the flow.",
"Those bounds are only approximate in non ideal flows but due to their topological nature simulations show that they are approximately conserved even when the \"energy\" is not.",
"For example it is easy to show that the \"energy\" is bounded from below by the non-barotropic cross helicity as follows (see [24]): $H_{CNB} = \\int \\vec{B}\\cdot \\vec{v_t}d^{3} x \\le \\frac{1}{2}\\int \\left( \\vec{B}^2 + \\vec{v_t}^2 \\right) d^{3} x,$ $H_{CNB} = \\int \\vec{B}\\cdot \\vec{v_t}d^{3} x \\le \\sqrt{\\int \\vec{v_t}^2 d^{3} x}\\sqrt{\\int \\vec{B}^2 d^{3} x},$ the second equation is a result of the Cauchy-Schwartz inequality.",
"In this sense a configuration with a highly complicated topology is more stable since its energy is bounded from below.",
"It is a simple thing to show that similar bounds occur also for the $\\chi $ and $\\eta $ helicities: $H_{CNB \\chi } = \\int \\vec{B_\\chi }\\cdot \\vec{v_t}d^{3} x \\le \\frac{1}{2}\\int \\left( \\vec{B_\\chi }^2 + \\vec{v_t}^2 \\right) d^{3} x,$ $H_{CNB \\chi } = \\int \\vec{B_\\chi }\\cdot \\vec{v_t}d^{3} x \\le \\sqrt{\\int \\vec{v_t}^2 d^{3} x}\\sqrt{\\int \\vec{B_\\chi }^2 d^{3} x},$ $H_{CNB \\eta } = \\int \\vec{B_\\eta }\\cdot \\vec{v_t}d^{3} x \\le \\frac{1}{2}\\int \\left( \\vec{B_\\eta }^2 + \\vec{v_t}^2 \\right) d^{3} x,$ $H_{CNB \\eta } = \\int \\vec{B_\\eta }\\cdot \\vec{v_t}d^{3} x \\le \\sqrt{\\int \\vec{v_t}^2 d^{3} x}\\sqrt{\\int \\vec{B_\\eta }^2 d^{3} x},$ Hence the kinetic energy is bounded by three differen bounds and so it the \"total\" energy.",
"The importance of each of those bounds is dependent on the flow."
],
[
"Conclusion",
"We have derived a Noether current from an Eulerian variational principle on non-barotropic MHD, this was shown to lead to to the conservation of non-barotropic cross helicity.",
"The connection of the translation symmetry groups of labels to both the global non barotropic cross helicity conservation law and the conservation law of circulations of topological velocity along magnetic field lines was elucidated.",
"The latter were shown to be equivalent to the amount of non barotropic cross helicity per unit of magnetic flux [29], [30], [24].",
"Further more we have shown that two additional cross helicity conservation laws exist the $\\chi $ and $\\eta $ cross helicities.",
"Those lead to new bounds on MHD flows in addition to the bounds of the standard non-barotropic cross helicity discussed in [30] for ideal non-barotropic MHD.",
"The importance of constants of motion for stability analysis is also discussed in [39].",
"The significance of those constraints for non-ideal MHD and for plasma physics in general remains to be studied in future works.",
"It is shown that non-barotropic MHD can be derived from a variational principle of five functions.",
"The formalism is given in a Lagrangian presentation with a geometrical structure.",
"Possible applications include stability analysis of stationary MHD configurations and its possible utilization for developing efficient numerical schemes for integrating the MHD equations.",
"It may be more efficient to incorporate the developed formalism in the framework of an existing code instead of developing a new code from scratch.",
"Possible existing codes are described in [33], [34], [35].",
"Applications of this study may be useful to both linear and non-linear stability analysis of known barotropic MHD configurations [36], [37], [38], [39], [40], [41].",
"As for designing efficient numerical schemes for integrating the equations of fluid dynamics and MHD one may follow the approach described in [42], [43], [32], [44].",
"Another possible application of the variational method is in deducing new analytic solutions for the MHD equations.",
"Although the equations are notoriously difficult to solve being both partial differential equations and nonlinear, possible solutions can be found in terms of variational variables.",
"An example for this approach is the self gravitating torus described in [45].",
"One can use continuous symmetries which appear in the variational Lagrangian to derive new conservation laws through the Noether theorem.",
"An example for such derivation which still lacks physical interpretation can be found in [46].",
"It may be that the Lagrangian derived in [3] has a larger symmetry group.",
"And of course one anticipates a different symmetry structure for the non-barotropic case.",
"Topological invariants have always been informative, and there are such invariants in MHD flows.",
"For example the two helicities have long been useful in research into the problem of hydrogen fusion, and in various astrophysical scenarios.",
"In previous works [2], [4], [11] connections between helicities with symmetries of the barotropic fluid equations were made.",
"The Noether current here derived may help us to identify and characterize as yet unknown topological invariants in MHD .",
"Acknowledgement This research was supported by the U.S. Department of Energy (DE-AC02-09CH11466)."
]
] | 2005.14005 |
[
[
"AstroSat observations of GRO J2058+42 during the 2019 outburst"
],
[
"Abstract We present results from AstroSat observation of the recent outburst of GRO J2058+42, an X-ray pulsar in a Be-binary system.",
"The source was observed on April 10, 2019 by LAXPC and SXT instruments on AstroSat during its declining phase of the latest giant outburst.",
"Light curves showed a strong pulsation of the pulsar with a period of 194.2201 \\pm 0.0016 s, and a spin-up rate of (1.65\\pm0.06)\\times10^{-11} Hz s^{-1}.",
"Intermittent flaring was detected in light curves between 3--80 keV energy band with increase in intensity by up to 1.8 times its average intensity.",
"Pulse profiles obtained between 3--80 keV energy band of the pulsar showed strong dependence on energy.",
"A broad peak was observed in the power density spectrum of the source consistently during AstroSat observations with its peak oscillation frequency of 0.090 Hz along with its higher harmonics, which may be due to quasi-periodic oscillations, a commonly observed phenomenon in transient X-ray pulsars, during their outburst.",
"AstroSat observation also detected cyclotron absorption features in its spectrum corresponding to (9.7--14.4) keV, (19.3--23.8) keV and (37.8--43.1) keV.",
"The pulse phase resolved spectroscopy of the source showed phase dependent variation in its energy and relative strength of these features.",
"The spectrum was well fitted with an absorbed black-body, a Fermi Dirac cut-off model and alternatively with an absorbed CompTT model.",
"Both these models were combined with a Fe-line and three Gaussian absorption lines to account for observed cyclotron resonance scattering features in the spectrum."
],
[
"Introduction",
"Many Be-binary systems were observed earlier during outbursts which offered interesting results [8], [66].",
"Bright X-ray outbursts are observed in Be-binaries most likely during periastron passage of its neutron star through a circumstellar disk of its companion.",
"Depending on amount of matter released from its companion and the geometry of the binary system, a rare as well as regular outbursts are observed during its each binary orbit [55], [56], [57].",
"The pulse characteristics of some of these were studied in detail during their outburst activities such as EXO 2030+375 [59], Cepheus X-4 [45], XTE J1946+274 [61].",
"Studies on pulse characteristics offer information on the pulsar geometry and underlying mechanism for its emitted pulse profile.",
"The shape of the emitted pulse depends on modes of accretion inflows, source luminosity, geometry of accretion column and configuration of its magnetic field with respect to an observer's line of sight [59].",
"Therefore, such studies offer understanding of its pulsar system and disk-magnetospheric interaction during the process of mass accretion, which affects its emitted radiation.",
"Quasi-periodic oscillations (QPO) were detected from many Be-binaries earlier, such as A0535+262 [20], [21], EXO2030+375 [2], 4U0115+63 [70], [28], [16], V0032+53 [65].",
"Studies of QPOs offer rich information about accretion torque onto the neutron star, thermodynamic properties of the inner accretion disk and electrodynamics of disk-magnetospheric interaction of the neutron star.",
"Details on sources with observed QPOs, their frequencies and other features along with pulsar spin frequencies etc.",
"are given in tabular form by [15], [25], [46].",
"Some of these transient Be-binary pulsars such as A0535+26 (50 mHz, 9.7 mHz), 1A 1118-61 (92 mHz, 2.5 mHz), XTE J1858+034 (110 mHz, 4.53 mHz), EXO 2030+375 (200 mHz, 24 mHz), SWIFT J1626.6-5156 (1000 mHz, 65 mHz), XTE J0111.2-7317 (1270 mHz, 32 mHz) as well as a persistent Be-binary, X-per (54 mHz, 1.2 mHz) and an OB-type binary, 4U 1907+09 (69 mHz, 2.27 mHz) showed higher QPO frequency than their respective spin-frequency as mentioned in order inside parenthesis [15].",
"These cover an interestingly wide range of QPO frequencies between 50–1270 mHz for these pulsars.",
"Studies of cyclotron absorption features, if present in the spectrum, enable us to determine the strength of the surface magnetic field of the neutron star and offers an insight into the line producing region, structure of accretion column and its geometry [72].",
"The cyclotron absorption features thus provides an important diagnostic probe for detailed studies of the neutron star binaries since its discovery in the spectrum of Her X-1 [79].",
"There are several reports on the detection of cyclotron absorption features in the spectrum of many Be-binaries starting at a lower energy from $\\sim 10$ keV [14], [31] to a higher energy at $\\sim 100$ keV [38].",
"Detailed compilation of such sources and studies are reported by [72] and [40].",
"It is observed from detailed studies that some sources show a wide variation in its cyclotron-line energy with respect to its pulse-phase, source luminosity and with time, such as Vela X-1, Cen X-3 and Her X-1 [72].",
"These interesting properties help in understanding the nature of these sources and also offer an insight into their underlying physical properties governing such changes.",
"The high mass X-ray binary GRO J2058+42 is a transient X-ray pulsar which was first discovered by BATSE on board CGRO during its giant outburst in September–October, 1995.",
"The outburst lasted for about 46 days which peaked at 300 mCrab intensity [87].",
"The spin period of its neutron star decreased from 198 s to 196 s during its 46 days outburst [87].",
"This outburst was subsequently followed by 5 more bursts of lower intensity of about 15 mCrab and of lesser duration of 15 days observed at an interval of about 110 days.",
"Additional shorter outbursts with peak intensity of about 8 mCrab were detected by BATSE, halfway between the first four outbursts of 15-20 mCrab (20-50 keV) intensity.",
"During early 1998, two outbursts of lower intensity were observed with PCA and HEXTE on-board RXTE [88].",
"There were, however, no reports of any giant outburst from the source.",
"Thus, GRO J2058+42 showed a very rare giant outburst activity, only twice so far, since its discovery.",
"The first was detected by BATSE [87], [88] and the most recent one was detected by Swift-BAT [4], [32] and Fermi-GBM [41] in March 2019.",
"GRO J2058+42 was suggested earlier to be a high mass X-ray binary due to its observed properties.",
"It was subsequently confirmed having a Be-star companion through optical observations [67].",
"A very limited details are known so far about this source, due to its rare intense outbursts.",
"AstroSat observed the source on April 10, 2019 during the declining phase of the second giant outburst, for studies of some of the above described properties of a typical Be-binary pulsar.",
"In this work, we present results from our studies on spectral and timing properties of GRO J2058+42 using data from AstroSat observation.",
"Incidentally, there were no reports on detection of any cyclotron absorption features from the earlier outbursts of GRO J2058+42.",
"We, therefore in particular, wish to check for possible detection of cyclotron absorption features in the spectrum and presence of QPOs.",
"After this manuscript was submitted, [44] have reported detection of cyclotron absorption feature using NuSTAR observation of the source during the same outburst.",
"The rest of the paper is organized as follows: Section 2 describes the observations and data analysis including software tools used, Section 3 describes the results and its implications are discussed in Section 4.",
"Finally, Section 5 gives the conclusions from this study.",
"Figure: Intensity variation in the 15–50 keV energy band as observed by Swift-BATduring source outburst is shown by black points while the continuous lineshows a fit to the light curve with cubic B-splines using 32 uniformly distributed knots.The time covered by AstroSat Observation is marked by two vertical lines."
],
[
"Observations and Data Analysis ",
"GRO J2058+42 had its rare and long outburst during March–May, 2019 as its latest event, as seen in the Swift-BAT intensity curve shown in Figure REF .",
"The light curve was downloaded from Swift-BAT light curves Archivehttps://swift.gsfc.nasa.gov/results/transients/index.html and relevant details about these curves are described by [37].",
"As there was a data gap close to the peak of the outburst, the peak of the outburst was approximately determined by fitting a cubic B-spline with 32 uniformly distributed knots to the data while data points with large errors were dropped.",
"It enabled us to determine peak intensity and its corresponding time which was found to be at around MJD 58574.8 with its peak intensity reaching to about 256 mCrab (Swift-BAT 0.0564 cts cm$^{-2}$ s$^{-1}$ ) in the 15–50 keV energy band.",
"When we compared this giant outburst with the earlier outburst, it turns out that, the main flare of this giant outburst lasted for about a similar duration of 46 days [87] and was followed by a relatively weaker and a much narrower secondary flare as seen in Figure REF .",
"There is also a small burst about 110 days before the main burst as seen in the swift-BAT light curve (Figure REF ).",
"CGRO–BATSE observed the earlier giant outburst with a peak intensity of about 300 mCrab [87], [88], comparable to that of its latest outburst which was also followed by a secondary flare observed after about 52 days.",
"AstroSat observed the source on April 10, 2019 during the declining phase of the outburst as marked in the Figure REF from MJD 58583.11 to MJD 58584.67.",
"It was about 8-days after the peak of the outburst, when source intensity had decreased to about 170 mCrab, which is about 66% of its peak intensity as obtained above.",
"The LAXPC was used as the primary instrument for this observation with a total effective exposure of 57 ks.",
"Details of LAXPC instruments on board AstroSat and its operation modes are described in detail by [3].",
"During the observation only one LAXPC detector, i.e., LAXPC20 was working properly.",
"LAXPCs were operated in its event-analysis mode.",
"LAXPC10 was operating at a very low gain and it barely detected pulsation in the source.",
"LAXPC20 detected an average source count-rate of $225.6 \\pm 0.1$ cts s$^{-1}$ in the 6–70 keV energy band.",
"The SXT was operated in its photon counting mode during the whole observation, with an effective exposure of about 21 ks and detected an average count-rate of $3.78\\pm 0.01$ cts s$^{-1}$ in the 0.8–7 keV energy band.",
"The SXT instrument along with its operational modes are described in detail by [69].",
"Thus useful data from the AstroSat SXT and LAXPC20 were used for timing and spectroscopic analysis, covering a total energy band of 0.8–80 keV.",
"The AstroSat observation was made under a Target of Opportunity (TOO) proposal with the observation ID of 20190410_T03_098T01_9000002836.",
"AstroSat data from the proposal are available from the AstroSat Data Archivehttps://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp.",
"LAXPC data were analyzed using LAXPC pipeline software version 3.0.",
"The LAXPC software can be downloaded from the LAXPC sitehttp::www.tifr.res.in/astrosat_laxpc/LaxpcSoft.html.",
"The software takes level-1 data files as input and also utilizes calibration and background files to generate level-2 data products, like light curve and spectrum.",
"The background files are available with the software, which also gives a suitable recommendation for selection of a background and associated response files which may be used for data product generation and analysis.",
"The background observation during April 19, 2019 was used to estimate the contribution from the source.",
"Events were extracted from all main anodes from all the layers of LAXPC20 for this analysis.",
"The pipeline software was executed with default values.",
"However, selection of appropriate energy channels were made for deriving light curves for a particular energy band.",
"The pipeline software also corrects for any shift in the gain based on calibration data and generates corrected spectrum and background subtracted light curves of the source within its set energy band.",
"The SXT data analysis pipeline software version AS1SXTLevel2-1.4b was used for a set of default parameters in this analysis.",
"Spectral response file and auxiliary response files of the SXT including background files are offered along with data analysis tools and can be downloaded from the SXT payload operation center sitehttp::www.tifr.res.in/astrosat_sxt/index.html.",
"Updated CALDB files are directly linked to the pipeline processing software.",
"All photon events were extracted by including a circular area covering a region of interest of 6 arc minutes radius with respect to the source center.",
"All grades between 0–12 were considered for selection of photon events.",
"The pipeline analysis software takes level-1 data as its input along with other relevant files for the analysis of events in steps to finally produce clean and calibrated event lists.",
"The final data products such as spectrum, light curve and image are produced by applying various default filters and appropriate screening of the data.",
"From calibrated clean events, one can produce spectrum and light curves in different energy bands covering 0.8–7.0 keV.",
"The `Xselect' tool version 2.4c was used for screening the data.",
"The solar system barycentric corrections were applied to the time series for both SXT and LAXPC data to correct for arrival time delays of events prior to its detailed timing analysis, using a tool `as1bary'http://astrosat-ssc.iucaa.in/?q=data_and_analysis, developed by the AstroSat science support cell.",
"Heasoft version 6.14 was used for analysis of this data.",
"The standard software XRONOS 5.22 was used for timing analysis; lcurve for generation of combined light curves, efsearch for determination of spin-period of the pulsar, powspec for computing power density spectrum etc.. Standard spectral analysis tool XSPEC version 12.8.1 was used for fitting spectral data with combination of appropriate models as described below.",
"Results obtained from these analysis are presented in the next section."
],
[
"X-ray light curve and folded pulse profiles",
"Extracted light curve of GRO J2058+42 showed regular and strong pulsations along with variation in its intensity.",
"A light curve segment of LAXPC20 with 10 s binning is shown in the left panel of Figure REF .",
"Source intensity variation between the 3–80 keV energy band for the complete observation duration of LAXPC20 is shown in the right panel of Figure REF , obtained by binning the data with its established spin-period.",
"A straight-line over the binned light curve shows its average intensity.",
"Intensity variations were clearly seen for the complete duration of AstroSat observation.",
"Observed source count rate varied from 230 cts s$^{-1}$ to about 520 cts s$^{-1}$ with an average count rate of 294 cts s$^{-1}$ .",
"Intermittent flaring and dips were also seen in the light curve which is a typical intrinsic property of a Be-binary.",
"During the flares, intensity increased by up to 1.8 times the average intensity.",
"The pulsar spin period was derived after applying corrections to pulse arrival time delays with respect to solar system barycenter.",
"The average pulsar period was initially derived with epoch folding and verified using Lomb-Scargle periodogram, on full AstroSat observation data.",
"Spin-period of the pulsar was found to be $194.180\\pm 0.001$ s with 1-$\\sigma $ confidence limit, during AstroSat observation.",
"Total duration of the observation time was then divided into 5 intervals to estimate spin-up rate of the pulsar during AstroSat observation.",
"The pulsar spin-period was determined for each data interval separately along with its estimated error.",
"These periods were plotted with respect to mid-value of respective interval in MJD.",
"An average spin-up rate of $(1.71\\pm 0.14)\\times 10^{-11}$ Hz s$^{-1}$ was derived by fitting a straight line to these 5 data points, along with its error estimated with 90% confidence limit.",
"Then starting with these initially measured values, the spin period at the beginning of observation at $t=t_0$ and its time derivative were accurately determined by correcting the phase using $\\phi (t)=\\phi _0+\\nu _0(t-t_0)+\\dot{\\nu }{(t-t_0)^2\\over 2},$ where $\\phi $ is the phase (in range 0–1 over the period), $\\phi _0$ is the phase at initial time, $t_0$ , $\\nu _0$ is the spin frequency at initial time and $\\dot{\\nu }$ is its time derivative.",
"The fit was performed by fitting a periodic signal with 20 harmonics of the basic period using Eq.",
"REF , $c(t)=c_0+\\sum _{k=1}^N\\Big (a_k\\sin (2\\pi k\\phi )+b_k\\cos (2\\pi k\\phi )\\Big ),$ where $c(t)$ is the observed count rate, $N$ is the number of harmonics included in the fit and $c_0,a_k,b_k$ are the coefficients fitted, apart from $\\nu _0$ and $\\dot{\\nu }$ .",
"The light curve was obtained with a time-bin of 1 s. The best fit values for $\\nu _0$ and $\\dot{\\nu }$ were determined by varying both parameters to minimize $\\chi ^2$ deviation of the light curve from the model (Eq.",
"REF ).",
"Various values of $N$ were tried and $N=20$ was found to be adequate to account for all pulsation signal.",
"In principle, it is possible to use the F-test to decide the required value of $N$ , which gave a value of $N=14$ with a probability threshold of 0.05, but some components after $k=15$ were also significant.",
"Beyond $N=20$ , the next few components were not found to be significant.",
"That is why this value was adopted.",
"To find the errors in the fitted values of $\\nu $ and $\\dot{\\nu }$ , we performed a Monte Carlo simulation by perturbing $c(t)$ and repeating the process for 4000 different realization of noise to find the distribution of parameter values and using this distribution, we found the 90% confidence limits for the parameters.",
"The same program was used to generate the pulse-profiles as well as the GTI intervals for different phase intervals using the fitted values of parameters.",
"The GTI values were then used to calculate the spectra for different phase intervals for phase resolved spectroscopy.",
"The same program was also used to generate the light curve after subtracting the pulse profile as defined by Eq.",
"REF to filter out the contribution of coherent pulsation in the resulting power density spectrum.",
"The fitted value of the period at $t=t_0$ corresponding to MJD 58583.10868148 was found to be $194.2201\\pm 0.0016$ s and $\\dot{\\nu }=(1.65\\pm 0.06)\\times 10^{-11}$ Hz s$^{-1}$ , where the error bars denote the 90% confidence limits.",
"The orbit of the binary system is not known and the orbital motion can also contribute to the period variation during the observation.",
"However, the Fermi-GBM observations during this outbursthttps://gammaray.msfc.nasa.gov/gbm/science/pulsars/lightcurves/groj2058.html suggests that the period variation was restricted to the duration of the giant outburst, and the change in ${\\dot{\\nu }}$ during the outburst showed a good correlation with intensity, with a correlation coefficient of about 0.93.",
"This implies that the variation in period is largely due to accretion during the outburst.",
"Figure: Light curves derived from LAXPC20 in the 3–80 keV energy band are shown, as a segment with 10 s binning(left panel) and also for the entire AstroSat observation with a binning of spin-period of the pulsar (right panel).The reference time t 0 t_0 is MJD 58583.10868148 for both these figures.",
"The horizontal lineon the right panel shows the average source count rate of 294 counts s -1 ^{-1}.Figure: Folded pulse profiles derived from LAXPC20 at different energy bandscovering 3–80 keV.",
"The vertical lines in the lowest panel mark the four divisions ofpulse period used in phase resolved studies.Figure: Count rate and RMS pulse fraction derived from LAXPC20 at different energy bandscovering 3–80 keV.Light curves were extracted corresponding to different energy bands and respective pulse profiles were derived by folding corresponding light curves in 64 bins with its derived spin-period of the neutron star at the beginning of the epoch and its derivative.",
"Two cycles of all such pulse-profiles are shown in Figure REF for clarity.",
"Folded pulse-profiles showed variation with respect to different energy bands.",
"The average pulse-profile covering full 3–80 keV energy band is also shown at the bottom panel of the Figure REF , for reference and relative comparison.",
"We noticed that pulse profiles at lower energies showed pronounced and multiple-pulses which gradually merged into a double-pulse and then in a single asymmetric broad-pulse at higher energies.",
"The RMS pulse fraction measurements and source intensity with respect to energy is shown in Figure REF .",
"RMS pulse fraction showed energy dependent variation, which showed increase from about $20\\%$ to $30\\%$ between 3–20 keV while at higher energies it is nearly constant.",
"Thus pulse profiles showed remarkable variability in shape and its pulse-fraction with respect to considered energy bands between 3–80 keV.",
"Pulse-profiles derived from recent AstroSat observations can be compared with earlier RXTE observation at much lower source intensity of about 10 mCrab.",
"There is a general similarity in shape but structures are seen much more clearly than those reported from November 28, 1996 RXTE observation between 2–60 keV during its earlier lower intensity outburst [87].",
"For the source intensity of about 17 mCrab, the RMS pulse fractions were measured at different energy bands between 2–20 keV from the RXTE PCA observations of February 4, 1998.",
"These were estimated as $11.7 \\pm 2.6 \\%$ (2–4 keV), $19.1 \\pm 4.1\\%$ (4–9 keV) and $27.2 \\pm 5.9 \\%$ (9–20 keV).",
"This can be compared with the recent observation by AstroSat LAXPC20, which gives $15.6 \\pm 3.3 \\%$ (3–4 keV), $22.0 \\pm 4.7 \\%$ (4–9 keV) and $28.6\\pm 6.1 \\%$ (9–20 keV) respectively.",
"The RXTE/PCA and AstroSat/LAXPC20 measurements show that these are comparable within respective error limits, although source intensity was an order of magnitude higher during AstroSat observation.",
"Thus, the pulsar does not show drastic change in shape of pulse profiles and pulse-fractions with change in source luminosity in this range.",
"Figure: Power density spectrum derived from LAXPC20 data between 3–30 keVcovering the observation duration is shown.A prominent QPO feature is observed at 0.09 Hz along with its 2 harmonicswhen signal from coherent spin-frequency of neutron star wasremoved from the time series."
],
[
"Power density spectrum and detection of a QPO feature",
"The source light curve along with its background between 3–30 keV energy band with a bin time of 1.0 second was used to derive power density spectrum.",
"The XRONOS tool `powspec' was used for this purpose.",
"The normalization parameter value of $-2$ was selected so that power density spectra were normalized such that their integral gave the squared RMS fractional variability.",
"Therefore, the power was expressed in the units of $\\mathrm {(RMS)}^2\\, \\mathrm {Hz}^{-1}$ and the expected white noise level was subtracted, to obtain the RMS fractional variability of the time series.",
"For such a normalized power density spectrum where a QPO profile was modeled by a Lorentzian, the strength of a QPO signal was described by its fractional root-mean-squared (RMS) amplitude, which is proportional to its integrated power which contributes to its over all power density spectrum and often expressed in percent [84].",
"The integrated power can be computed from the area under the Lorentzian profile.",
"The amplitude of the Lorentzian multiplied by its FWHM and $\\pi $ /2 would determine the area under the profile in units of $\\mathrm {RMS}^2$ , hence its square-root would give the RMS amplitude of the QPO signal.",
"The power density spectrum was therefore, derived considering above normalization and a total of 1024 bins per interval and total a 73 intervals in a frame were considered for the purpose of obtaining the power density spectrum.",
"However, out of a total of 73 intervals, 16 intervals with less than $50\\%$ data, were rejected for a default window selection.",
"The data gaps were padded with zeros as its default option.",
"A geometric re-binning of 1.04 was applied for generation of power density spectrum to improve on statistical fluctuation at higher frequencies.",
"ccccc 2 Power density spectrum model parameters Parameter Value Amplitude F-TEST Detection (RMS $\\%$ ) FAP Significance ($\\sigma $ ) Power Index $-0.78\\pm 0.08$ Normalization (K) (at 1 Hz) $0.010\\pm 0.001$ QPO-$f_{0}$ Centroid LC (Hz) $0.090\\pm 0.003$ $ 12.1 \\pm 1.0$ $2.7 \\times 10^{-11}$ 6.7 Sigma LW (Hz) $0.081\\pm 0.012$ Normalization LN $0.116 \\pm 0.008$ QPO-$f_{1}$ Centroid LC (Hz) $0.183 \\pm 0.004$ $5.2 \\pm 0.9$ $1.4 \\times 10^{-7}$ 5.3 Sigma LW (Hz) $0.048 \\pm 0.015$ Normalization LN $0.036\\pm 0.006$ QPO-$f_{2}$ Centroid LC (Hz) $0.280 \\pm 0.017$ $ 4.2 \\pm 1.5$ $2.2 \\times 10^{-4}$ 3.7 Sigma LW (Hz) $0.11 \\pm 0.07$ Normalization LN $0.010 \\pm 0.003$ $\\chi ^2_{{red}}$ (dof) 1.08 (62) Errors with 90% confidence range for each parameter.",
"FAP-False alarm probability.",
"The average power density spectrum derived as above showed spin frequency along with its several harmonics.",
"It also showed a broad peak around 0.09 Hz and its 2 harmonics, which were identified as a QPO.",
"The QPO signal was found to be better in the 3–30 keV energy band, compared to a higher energy band, hence power spectra are presented for this energy band.",
"For confirming the presence of QPO, we removed the contribution of coherent pulsation signal by subtracting the fitted pulse profile with 20 harmonics (Eq.",
"REF ) from the time series data as described in section 3.1.",
"Power density spectrum was then derived from this time series.",
"The power density spectrum showed a clear detection of a QPO feature at $0.090 \\pm 0.003$ Hz along with its 2 harmonics.",
"As the pulsation signal was removed from the time series data, power density spectrum did not show any coherent pulsations as was seen earlier.",
"However, the continuum defined by the power law-index were found to be consistent within the 90% error limit, when coherent pulsation signal was present (power-index $= -0.89 \\pm {0.09}$ ) and when pulsation signal was removed, (power-index $= -0.78 \\pm {0.07}$ ).",
"The observed QPO and its 2 higher harmonics defined by Lorentzian function were also found consistent within their parameter uncertainty.",
"The power density spectrum was modeled using a power-law in combination with 3 Lorentzian functions to model QPO and its two higher harmonics.",
"The model parameter values and QPO-frequency and its corresponding 2 harmonics as derived from the fitted model are listed in the Table-1.",
"This is to mention here that since many intervals of data were averaged for generating the power density spectrum, therefore, $\\chi ^2$ test for the fitted model does not introduce significant biases as it is well known that in this case the power density spectrum has distribution close to the normal distribution [5].",
"The model was fitted and parameter values were established from the power density spectrum when its y-axis was normalized and expressed in the units of $\\mathrm {(RMS)}^2\\, \\mathrm {Hz}^{-1}$ and frequency along the x-axis.",
"However, the power density spectrum is shown in Figure REF with y-axis multiplied by corresponding frequency to show prominence of the QPO and its 2 harmonics.",
"The Lorentzian function $L(f)$ used in the model is defined as: $L(f) = \\frac{\\mathrm {LN}}{1 + \\left(\\frac{2 (f-\\mathrm {LC})}{\\mathrm {LW}}\\right)^2} \\;,$ where parameters LN, LC and LW represent its peak amplitude, centroid frequency and its full width at half maximum, respectively.",
"An alternate model, broken-power-law along with 3 Lorentzian functions was also fitted to the same power density spectrum for comparison.",
"The reduced ${\\chi ^2}$ value of 1.12 was found for the 60 degrees of freedom (dof), as compared with power-law model along with 3 Lorentzian functions which yields the reduced ${\\chi ^2}$ of 1.08 for 62 dof.",
"Thus, power-law model resulted in a better fit, compared to broken power-law model.",
"A break frequency of $0.31 \\pm 0.01$ Hz was obtained from broken-power law model when fitted to the power density spectrum.",
"The break frequency is much higher than the observed QPO frequency of 0.09 Hz and hence it cannot mimic the QPO.",
"Thus, RMS amplitudes determined from the averaged power density spectrum, by fitting a power-law and 3 Lorentzian for QPO and its 2 harmonics are also given in Table-1.",
"The statistical significance of detected QPOs was established using the F-test.",
"This was done by determining the difference in $\\chi ^2$ with and without the inclusion of a Lorentzian function, used for modeling detected QPO in the power density spectrum.",
"Similar approach was followed for its other 2 harmonics, individually.",
"The false alarm probability (FAP) and detection significance of QPO expressed in terms of $\\sigma $ for an equivalent normal distribution is computed and listed in Table-1 for QPO and its 2 harmonics.",
"It turns out that the QPO frequency and the two harmonics are significant.",
"It is observed that apart from the first harmonic the other two peaks are rather broad.",
"Figure: Ratio of spectral counts of the source at different energies with respect toCrab spectral counts is shown corresponding to 4 different pulse-phases of the pulsar (asmarked in Figure 3) derived from LAXPC20 spectral data between 4–60 keV energy band.The continuous line shows the fit with a combination model of a polynomialof degree 3 in energy and Gaussians to define dips in the ratio."
],
[
"Spectrum and detection of cyclotron absorption features",
"RXTE derived the first X-ray spectrum of the source during the earlier outburst of 1996 covering energy band between 2.7–50 keV.",
"The spectrum was modeled adequately well with absorbed thermal bremsstrahlung (Phabs*bremss).",
"However, there was, no report on any detection of cyclotron absorption feature from RXTE observations.",
"Therefore, to check for the presence of a cyclotron absorption feature in the spectrum we took the ratio of counts in the spectrum of source to that in the Crab spectrum which was observed by LAXPC20 of AstroSat and the results for the phase-resolved spectra are shown in Figure REF .",
"The overall pulse phase was divided into 4 intervals covering structures of the folded pulse profile as shown by vertical lines on 3–80 keV pulse profile in Figure REF .",
"The phase intervals considered are 0.00–0.18 (phase-1), 0.18–0.43 (phase-2), 0.43–0.70 (phase-3) and 0.70–1.00 (phase-4).",
"The ratio of spectral counts plotted against energy with respect to Crab was fitted with a polynomial of degree 3 along with combination of Gaussians to define corresponding features in the ratio curve.",
"The first phase shows two features around 10 keV and 20 keV and an insignificant feature around 38 keV.",
"The feature around 38 keV is more prominent during other phases as well as in the phase averaged case, while the lower two features are not significant during the last two phases.",
"This is borne out by the results of fitting the spectrum.",
"The variation in properties of features with phase effectively rule out instrumental effects or uncertainties in instrumental response as these effects would be independent of phase.",
"cccc 4 Spectral parameters for phase averaged spectrum Parameter Units Model 1 (FDCUT) Model 2 (CompTT) Phabs(nH) 10$^{22}$ cm$^{-2}$ $0.869^{+0.045}_{-0.046}$ $0.650^{+0.048}_{-0.046}$ bbody(kT) keV $0.83^{+0.04}_{-0.05}$ - bbody(Norm) ph keV$^{-1}$ cm$^{-2}$ s$^{-1}$ $0.0036^{+0.0008}_{-0.0007}$ - power-law(PI) - $1.04^{+0.03}_{-0.03}$ - $E_{cut}$ keV $34.6^{+1.2}_{-2.1}$ - $E_{fold}$ keV $10.95^{+0.27}_{-0.43}$ - power-law(Norm) ph keV$^{-1}$ cm$^{-2}$ s$^{-1}$ $0.10^{+0.01}_{-0.01}$ - at 1 keV CompTT(Redshift) - - 0 (fixed) CompTT(T0) keV - $0.51^{+0.02}_{-0.03}$ CompTT(kT) keV - $8.22^{+0.10}_{-0.10}$ CompTT(tau) - - $5.21^{+0.12}_{-0.12}$ CompTT(approx) - - 0.2 (fixed) CompTT(Norm) - - $0.051^{+0.003}_{-0.002}$ gauss(Line E) keV 6.5 (fixed) 6.5 (fixed) gauss(Sigma) keV 0.24 (fixed) 0.24 (fixed) gauss(Norm) $ph~cm^{-2}~s^{-1}$ $1.8\\times 10^{-4}$ $2.3\\times 10^{-4}$ gabs(Line E1) keV $10.95^{+0.74}_{-0.81}$ $10.81^{+0.48}_{-0.49}$ gabs(Sigma) keV $3.74^{+0.72}_{-0.53}$ $3.20^{+0.39}_{-0.32}$ gabs(Strength) - $1.13^{+0.29}_{-0.29}$ $1.56^{+0.28}_{-0.29}$ gabs(Line E2) keV $21.71^{+0.81}_{-0.83}$ $20.84^{+0.54}_{-0.55}$ gabs(Sigma) keV $3.61^{+1.28}_{-1.32}$ $3.69^{+0.47}_{-0.40}$ gabs(Strength) - $0.90^{+0.48}_{-0.44}$ $1.42^{+0.27}_{-0.29}$ gabs(Line E3) keV $38.73^{+0.90}_{-0.81}$ $38.38^{+1.14}_{-1.08}$ gabs(Sigma) keV $2.64^{+1.73}_{-1.33}$ $2.56^{+1.12}_{-0.77}$ gabs(Strength) - $1.02^{+0.55}_{-0.33}$ $0.96^{+0.21}_{-0.20}$ Flux(3–80 keV) erg cm$^{-2}$ s$^{-1}$ $4.33\\times 10^{-9}$ $4.33\\times 10^{-9}$ $\\chi ^2$ - 879.79 877.36 $\\chi ^2_{{red}}$ (dof) - 1.264(696) 1.257(698) 3cSignificance of cyclotron absorption features from F-TEST: E1 FAP - $1.68 \\times 10^{-2}$ $1.79 \\times 10^{-4}$ significance ($\\sigma $ ) - 2.39 3.75 E2 FAP - $2.57 \\times 10^{-4}$ $1.72 \\times 10^{-4}$ significance ($\\sigma $ ) - 3.66 3.76 E3 FAP - $1.37 \\times 10^{-7}$ $1.55 \\times 10^{-7}$ significance ($\\sigma $ ) - 5.27 5.25 3cSignificance of cyclotron absorption features from NONE-ZERO LINE DEPTH: E1 FAP - $1.19 \\times 10^{-3}$ $3.14 \\times 10^{-6}$ significance ($\\sigma $ ) - 3.24 4.66 E2 FAP - $4.57 \\times 10^{-6}$ $5.93 \\times 10^{-6}$ significance ($\\sigma $ ) - 4.49 4.36 E3 FAP - $5.89 \\times 10^{-13} $ $2.27 \\times 10^{-10}$ significance ($\\sigma $ ) - 7.20 6.34 Errors with 90% confidence range for each parameter.",
"False alarm probability (FAP).",
"Figure: Spectrum fitted with combined model; an absorbed Fermi-Dirac cutoff modelwith a black-body a Gaussian emission line and 3 Gaussian absorption lines (Model 1, left panel), andan absorbed CompTT with a Gaussian emission line and 3 Gaussian absorption lines(Model 2, right panel) are shown.Figure: Phase resolved spectra fitted with combined model; an absorbed CompTT witha Gaussian emission line and3 Gaussian absorptions lines.The graph below shows the residue to the fit in units of chi.cccccc 6 Spectral parameters for phase resolved spectra Parameter Units Phase 1 Phase 2 Phase 3 Phase 4 Phabs(nH) 10$^{22}$ cm$^{-2}$ $0.83^{+0.11}_{-0.11}$ $0.81^{+0.19}_{-0.12}$ $0.65^{+0.10}_{-0.08}$ $0.64^{+0.09}_{-0.08}$ CompTT(Redshift) - 0 (fixed) 0 (fixed) 0 (fixed) 0 (fixed) CompTT(T0) keV $0.41^{+0.05}_{-0.05}$ $0.42^{+0.06}_{-0.11}$ $0.53^{+0.04}_{-0.05}$ $0.47^{+0.04}_{-0.04}$ CompTT(kT) keV $10.03^{+0.22}_{-0.22}$ $8.56^{+0.10}_{-0.11}$ $8.45^{+0.15}_{-0.26}$ $8.36^{+0.15}_{-0.18}$ CompTT(tau) - $4.60^{+0.17}_{-0.16}$ $6.17^{+0.16}_{-0.15}$ $4.91^{+0.27}_{-0.13}$ $4.99^{+0.11}_{-0.23}$ CompTT(approx) - 0.2 (fixed) 0.2 (fixed) 0.2 (fixed) 0.2 (fixed) CompTT(Norm) - $0.037^{+0.003}_{-0.003}$ $0.055^{+0.006}_{-0.003}$ $0.047^{+0.003}_{-0.002}$ $0.042^{+0.007}_{-0.002}$ gauss(Line E) keV 6.5 (fixed) 6.5 (fixed) 6.5 (fixed) 6.5 (fixed) gauss(Sigma) keV 0.24 (fixed) 0.24 (fixed) 0.24 (fixed) 0.24 (fixed) gauss(Norm) $ph~cm^{-2}~s^{-1}$ $2.10\\times 10^{-4}$ $2.20\\times 10^{-4}$ $2.88\\times 10^{-3}$ $2.88\\times 10^{-3}$ gabs(Line E1) keV $10.26^{+0.51}_{-0.52}$ $13.17^{+1.18}_{-0.99}$ $12.44^{+1.55}_{-1.18}$ $12.05^{+1.00}_{-0.89}$ gabs(Sigma) keV $3.59^{+0.57}_{-0.48}$ $2.02^{+1.30}_{-1.01}$ $1.49^{+1.72}_{-0.84}$ $1.28^{+1.26}_{-0.75}$ gabs(Strength) - $2.14^{+0.58}_{-0.52}$ $0.47^{+0.20}_{-0.26}$ $0.20^{+0.26}_{-0.12}$ $0.19^{+0.11}_{-0.13}$ gabs(Line E2) keV $19.72^{+0.39}_{-0.40}$ $22.20^{+0.70}_{-0.78}$ $22.68^{+1.13}_{-1.15}$ $22.11^{+1.27}_{-1.33}$ gabs(Sigma) keV $3.66^{+0.58}_{-0.53}$ $3.68^{+0.61}_{-0.50}$ $2.15^{+1.41}_{-0.89}$ $1.89^{+1.81}_{-1.21}$ gabs(Strength) - $2.21^{+0.57}_{-0.44}$ $1.21^{+0.60}_{-0.46}$ $0.37^{+0.62}_{-0.17}$ $0.27^{+0.17}_{-0.16}$ gabs(Line E3) keV $38.38 (fixed)$ $40.07^{+1.15}_{-1.11}$ $38.54^{+0.70}_{-0.71}$ $41.77^{+1.35}_{-1.90}$ gabs(Sigma) keV $2.2 (fixed) $ $4.60^{+0.87}_{-0.73}$ $2.41^{+1.11}_{-1.08}$ $5.33^{+1.45}_{-1.29}$ gabs(Strength) - $0.059^{+0.21}_{-0.059}$ $1.69^{+0.35}_{-0.36}$ $1.15^{+0.24}_{-0.23}$ $2.39^{+0.98}_{-0.75}$ $\\chi ^2$ - 663.48 528.95 526.29 625.61 $\\chi ^2_{{red}}$ (dof) - 1.211(548) 0.969(546) 0.966(545) 1.148(545) $\\chi ^2_{{diff}}$ (with 3 gabs removed) - 224.0 68.76 66.54 62.79 FAP - $6.08 \\times 10^{-30}$ $5.42 \\times 10^{-11}$ $1.24 \\times 10^{-10}$ $3.67 \\times 10^{-8}$ significance ($\\sigma $ ) - 11.37 6.55 6.43 5.51 4lSignificance test of cyclotron line using F-TEST: E1 FAP - $9.67 \\times 10^{-13}$ $4.13 \\times 10^{-4}$ $9.29 \\times 10^{-2}$ $9.83 \\times 10^{-3}$ significance ($\\sigma $ ) - 7.14 3.53 1.68 2.58 E2 FAP - $2.22 \\times 10^{-7}$ $1.58 \\times 10^{-6}$ $2.51 \\times 10^{-3}$ $1.02 \\times 10^{-1}$ significance ($\\sigma $ ) - 5.18 4.80 3.02 1.64 E3 FAP - $9.9 \\times 10^{-1}$ $7.26 \\times 10^{-13}$ $1.56 \\times 10^{-9}$ $6.11 \\times 10^{-9}$ significance ($\\sigma $ ) - 0.01 7.17 6.04 5.81 4lSignificance test of cyclotron line using NON-ZERO LINE DEPTH: E1 FAP - $4.05 \\times 10^{-17}$ $1.34 \\times 10^{-4}$ $5.13 \\times 10^{-2}$ $3.81 \\times 10^{-3}$ significance ($\\sigma $ ) - 8.41 3.82 1.95 2.89 E2 FAP - $1.76 \\times 10^{-9} $ $3.89 \\times 10^{-7}$ $1.24 \\times 10^{-3}$ $2.27 \\times 10^{-2}$ significance ($\\sigma $ ) - 5.73 4.96 3.23 2.21 E3 FAP - $9.26 \\times 10^{-1}$ $1.38 \\times 10^{-13}$ $5.70 \\times 10^{-14}$ $3.01 \\times 10^{-11}$ significance ($\\sigma $ ) - 0.09 7.40 7.08 6.65 Errors with 90% confidence range for each parameter.",
"False alarm probability (FAP).",
"We then derived X-ray spectrum covering a total energy band from 0.8–70 keV with AstroSat, combining SXT and LAXPC20 spectral data.",
"The combined spectrum was fitted with two different models.",
"The first (Model–1) is an absorbed Fermi-Dirac cutoff model, FDCUT [77] combined with a black body (bbody), a Gaussian for an iron emission line and 3 Gaussian absorption features (gabs) which were introduced to account for presence of cyclotron absorption features in the spectrum to improve the spectral fit.",
"We also tried an alternative model (Model–2) an absorbed CompTT model [78] combined with an iron emission line and 3 absorption features as described above for comparison and measurement of the centroid energy of the cyclotron absorption features.",
"The spectral parameters derived from these two models are tabulated in Table 2 for phase-averaged spectrum, while Table 3 shows the results of fitting the second model to phase-resolved spectra.",
"Spectra along with the fitted models are shown for phase-averaged and phase-resolved spectra in Figures REF and REF , respectively.",
"The Figure REF shows phase dependent variations in spectral residues indicating relative intensities of 3 absorption features individually when the optical depth of corresponding line energy is made zero.",
"In addition, the residue of the fitted model is also presented when all the 3 gabs components, associated with cyclotron resonance scattering features, were removed.",
"This depicts phase dependent variations of relative intensities of cyclotron absorption features for 4 different phases.",
"The difference in $\\chi ^2$ values without the 3 gabs components were established for all the 4 phases and given in the Table 3.",
"The difference in $\\chi ^2$ values clearly showed overall significance of presence of cyclotron absorption features in the spectrum.",
"A systematic error of $2.5\\%$ was added to account for uncertainties in the response of the instruments.",
"Relative difference in the normalization between instruments was accounted by introducing a constant multiplicative factors for the two instruments and by fixing it for LAXPC20 at 1.0 during the fit.",
"Thus the normalization factor of SXT was found to be 0.31 when a circular field of view of a diameter of 40 arc minutes was selected.",
"This is due to a fixed offset between the pointing axis of the two instruments.",
"The region of interest of SXT, however, was restricted to 12 arc-minutes in diameter to accumulate events mainly from region where source photons dominate, to allow better constrain in the lower energy region of the overall spectrum.",
"With this restricted region of interest, the normalization was found to be 0.18 for SXT with respect to the LAXPC20 frozen at unity.",
"Both the spectral models offered a reasonably good fit to the spectral data and measurements of respective energy of cyclotron absorption features are found to be consistent within their error limits.",
"Considering the observed flux of $4.3\\times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ in the 3–80 keV energy band and using source distance of 9 kpc [67], the X-ray luminosity is determined to be $4.2 \\times 10^{37}$ ergs s$^{-1}$ during AstroSat observation.",
"To estimate the statistical significance of the cyclotron absorption features which were detected, we initially tried first two approaches described by [7] and then cross verified these using Monte Carlo technique as a re-verification for a few cases, as described below.",
"Figure: Phase resolved spectra residues showing relative intensitiesof 3 absorption lines (top 3 panels of each figure) when individual optical depth is made zero andfollowed by fitted residue when all the three absorption features were removed from the model (bottom panel of each figure)for all the 4 different phases.",
"The combined spectrum was fitted with Model 2.Figure: Results of Δχ 2 \\Delta \\chi ^2 distribution derived from Monte Carlo simulations of spectra for LAXPC20 for testingsignificance of cyclotron absorption features corresponding to lines at ∼22\\sim 22 keV (left panel)and ∼40\\sim 40 keV (right panel) are shown for pulse phase-2.",
"The observed values of Δχ 2 \\Delta \\chi ^2 correspondingto above two lines are shown by vertical arrows.",
"The first approach was to use the F-test, which uses the reduction in $\\chi ^2$ value when the three parameters defining the cyclotron absorption feature using gabs (multiplicative model) were included.",
"Based on the improvement in $\\chi ^2$ value by adding the feature, one could estimate its significance and false alarm probability.",
"However, this is to specifically mention that the F-test used here is additive and has certain limitations for multiplicative models such as gabs-model [58], [62].",
"This test was applied for an initial assessment and for rough estimates of the significance of 3 individual additional components, defined by gabs-model.",
"In all cases, the significance of each feature was calculated independently and we have calculated the probability of the signal being false and converted that to the significance in terms of equivalent value of $\\sigma $ in the normal distribution.",
"The estimated significance by this F-test for the absorption features in the spectrum are tabulated for phase averaged and 4 different pulse phases in Tables 2 and 3, respectively.",
"In the second approach, we kept the centroid energy fixed for a cyclotron absorption feature and stepped through a grid of values of line-depth and width in steps to study the change in $\\chi ^2$ as a function of these two parameters, using the XSPEC steppar command.",
"The change in the minimum $\\chi ^2$ value corresponding to zero line-depth was noted.",
"Thus, using the minimum difference in $\\chi ^2$ required to get zero line-depth, we could calculate the false alarm probability and the significance of the line.",
"The results are shown in Table 2 for phase averaged spectrum and in Table 3 for four different pulse-phases of the pulsar.",
"These estimates of significance of cyclotron absorption features are consistent with the estimates using option (1).",
"We also assessed the significance of cyclotron absorption features independently using Monte Carlo simulation.",
"This technique has been used to establish significance of cyclotron absorption features in the spectrum for many sources [6], [7], [9].",
"We used LAXPC20 data fitted with the CompTT continuum model along with Fe-line and 3 Gaussian absorption features which were introduced to model observed cyclotron absorption features.",
"Only LAXPC20 data were considered to improve on the speed of simulation and convergence to the fit.",
"We fixed the Hydrogen column density ($N_{H}= 0.81\\times 10^{22}$ cm$^{-2}$ ) as derived from fit to the combined SXT and LAXPC20 spectral data.",
"We then used XSPEC script `simftest' for simulating 1000 spectra.",
"The script does the fit with and without cyclotron absorption feature for each of the simulated data.",
"From these simulations, one can estimate the change in $\\chi ^2$ between a model with and without the cyclotron absorption feature and find the distribution of $\\Delta \\chi ^2$ values.",
"These simulations were done to re-confirm the significance results for Phase-2 spectrum, corresponding to its $\\sim $ 22 keV and $\\sim $ 40 keV cyclotron absorption features, independently.",
"The change in chi-squared ($\\Delta \\chi ^2$ ) distribution were plotted.",
"Since the cyclotron absorption feature is defined by three free parameters, we expect the simulated $\\Delta \\chi ^2$ distribution to follow a $\\chi ^2$ distribution with three degrees of freedom.",
"The observed distribution was found to be consistent with the $\\chi ^2$ distribution as can be seen in the Figure REF .",
"In these two cases maximum values of $\\Delta \\chi ^2$ obtained in simulation was 14 (22 keV line) and 10 (40 keV line) which are much lower than the observed values of 25.5 (22 keV line) and 36.7 (40 keV line) in the LAXPC20 data.",
"By extrapolating the distribution we estimate that a very large number of simulations would be required to get observed value of $\\Delta \\chi ^2$ in one of them by chance.",
"This could be $10^5$ –$10^6$ simulations required for $\\Delta \\chi ^2\\approx 26$ and $10^{8}$ –$10^{9}$ for $\\Delta \\chi ^2\\approx 37$ .",
"Therefore, it is not feasible to perform these many simulations to achieve desired values.",
"The significance of individual features for LAXPC20 data for phase-2, corresponding to these values are $4.38\\sigma $ ($\\mathrm {FAP}=1.2\\times 10^{-5}$ ) for 22 keV line and $5.4\\sigma $ ($\\mathrm {FAP}=5.32 \\times 10^{-8}$ ) for 40 keV line.",
"These simulations therefore confirmed presence of these features with high significance in LAXPC20 spectrum which detected cyclotron absorption features from the source in its sensitive energy band.",
"It thus also confirms that the significance established from the first two methods as shown in the Table 3 are consistent with the results obtained by the Monte Carlo simulation technique as shown here as a test case.",
"We have tried a few more cases and in all of these the significance is consistent with those obtained using options (1) and (2).",
"Our results for phase-1 are in agreement with [44], whereby the first two features around 10 keV and 20 keV were detected with high significance and no other higher energy features were detected.",
"The mean energy, width and depth of the features estimated by us is also consistent with those by [44].",
"Additionally, we detected absorption feature around 38–40 keV for the rest of the 3 phases with high significance.",
"However, we did not detect presence of any cyclotron absorption feature around 30 keV as reported by [44].",
"This was verified using phase-1 spectrum, where a third gabs model was introduced with frozen values of its energy and width as reported by [44] and its optical depth was allowed to vary.",
"The additional gabs-model fitted to the spectral data could not detect any significant optical depth corresponding to the energy $\\sim $ 30 keV.",
"The individual spectrum, Figures 4 & 5 of [44], do not appear to show significant feature around 30 keV.",
"The feature seen around 38–40 keV in LAXPC spectrum, could possibly be either the third harmonic (if the primary absorption feature is around 12 keV) or the fourth harmonic (if the primary feature is around 10 keV).",
"In the latter case, we may have missed the third harmonic as it could possibly be very weak and hence not detectable.",
"Even higher harmonics could be present, but due to lower sensitivity at high energies it is difficult to say anything definitely about them.",
"We thus detected a cyclotron absorption feature around 38–40 keV in the other 3 phases, which is contrary to the results of [44].",
"This detection was possible due to relatively higher effective area of the LAXPC20 [3] by more than an order of magnitude around 38–40 keV compared to FM-modules of the NuSTAR [26], [10].",
"This is evident from their Figure 4 and 5, [44], where errors are relatively large around 38 keV and hence it is likely that 38 keV feature was not detected by the NuSTAR.",
"We detected 38 keV absorption feature with high significance as reported in Table-2, even in the phase averaged spectrum.",
"We also noticed that the spectral ratio with respect to Crab spectrum (Figure REF ) clearly showed depression around 38–40 keV, confirming presence of this absorption feature in the spectrum and the variation in its shape with phase.",
"This rules out the possibility of occurrence of absorption feature due to inaccuracy of instrument response or the modeled background."
],
[
"Power Spectrum and QPO",
"Many accretion powered pulsars show milli-Hertz quasi periodic oscillations (QPOs).",
"X-ray pulsars with high mass companions such as X Per [76], 4U 1907+09 [29], [46], XTE J1858+034 [60], A 0535+26 [20], V 0332+53 [75] and X 0115+63 [70] showed QPOs with their respective peak oscillation frequencies in the range 50–110 mHz.",
"QPOs in X-ray pulsars offer an important diagnostic tool to probe; accretion flows in these binaries, conditions in the inner region of its accretion disk, properties of accretion torques exerted on its neutron stars and disk-magnetospheric coupling and their interaction [24], [25].",
"Now let us consider two main models and their applicability to high mass X-ray binary GRO J2058+42.",
"These are namely: Keplerian frequency and Beat frequency model [82].",
"The Keplerian frequency model, where QPOs are produced due to inhomogeneities at the inner edge of the Keplerian disk, that modulate the X-ray flux at Keplerian frequency, expressed as $\\nu _{\\mbox{QPO}}=\\nu _{k}$ [81].",
"Whereas in the case of Beat frequency model, the accretion flow on to the neutron star is modulated at the Beat frequency between Keplerian frequency at the inner edge of the accretion disk and the neutron star spin frequency, $\\nu _{\\mbox{QPO}}=\\nu _{k} - \\nu _{s}$ [1].",
"In the case of GRO J2058+42, the detected QPO frequency $ \\nu _{QPO}= 9.0 \\times 10^{-2}$ Hz is much higher than the spin frequency of its neutron star, $5.15 \\times 10^{-3} $ Hz.",
"It is, therefore, not possible to differentiate between a Beat frequency and a Keplerian frequency model in the case of GRO J2058+42.",
"We can obtain radius of the inner edge of the accretion disk $r_{\\mbox{QPO}}$ using expression for Keplerian orbital motion, $r_{\\mbox{QPO}}= \\left({\\frac{GM}{4\\pi ^2}} \\right)^{1/3} \\nu _{k}^{-2/3} \\approx 8.0 \\times 10^{8} \\quad \\mbox{cm},$ where $M$ is $1.4 M_{\\odot }$ for a neutron star and $G$ is the gravitational constant and $\\nu _{k}$ its Keplerian frequency.",
"Alternatively, using mass accretion rate and strength of magnetic field of neutron star one can also derive radius of the inner edge of the Keplerian disk, which is equivalent to the magnetospheric radius $r_{m}$ , of the neutron star as expressed by the equation 6.18, page 158 of [22] $r_{m} \\simeq 5.2~\\mu _{30}^{4/7} {\\dot{M}_{16}}^{-2/7} {m_{x}}^{-1/7}\\times 10^{8} \\, \\mbox{cm} = 2.4 \\times 10^{8} \\, \\mbox{cm},$ where $m_{x}= M_{x}/{M_{\\odot }} = 1.4$ , $\\mu _{30}$ is magnetic moment of the neutron star expressed in units of $10^{30}$ G cm$^{3}$ and $\\dot{M}_{16}$ is its mass accretion rate expressed in units of $10^{16}$ g s$^{-1}$ .",
"These are derived from observed values of neutron star magnetic field and source luminosity respectively as detected by AstroSat observation.",
"The co-rotation radius $r_{co}$ , of an X-ray pulsar can be defined where the spin angular velocity of neutron star is equal to the Keplerian angular velocity of matter.",
"It can be derived by equating the Keplerian velocity to the co-rotating Keplerian velocity, $r_{co} = 1.7 \\times 10^{8} P^{2/3} \\left(\\frac{M}{1.4 M_\\odot } \\right)^{1/3} \\, \\mbox{cm},$ where $P$ is the spin period of the neutron star.",
"Using estimated pulse period from AstroSat observation and assuming a neutron star mass of $M = 1.4 M_{\\odot }$ , one can obtain the co-rotation radius $r_{co}$ for GRO J2058+42 as $5.7 \\times 10^{9}$ cm (Equation-REF ).",
"It is therefore evident that the disk radius $r_{\\mbox{QPO}}$ is about an order of magnitude smaller than the co-rotation radius $r_{co}$ .",
"It suggests, therefore, that formation of such a transient disk is possible between magnetosphere and co-rotation radius of the neutron star.",
"Estimated values of the radius of the inner accretion disk derived by Keplerian orbital motion is ${8.0\\times 10^{8}}$ cm (Equation-REF ) and using the accretion torque theory, it is determined to be ${2.4 \\times 10^{8}}$ cm (Equation-REF ).",
"These values are comparable considering the level of approximations involved, and uncertainties in mass, radius and distance of the source as well as in the geometry of the magnetic field.",
"It favors however, formation of a transient accretion disk around the neutron star, which could possibly explain the cause of observed oscillations at 0.090 Hz in the X-ray flux due to either Beat frequency or Keplerian frequency modulations as discussed above.",
"Such properties are quite recurrent in many cases of Be-binary pulsars as mentioned above and also observed in this case, for the first time by LAXPC on-board AstroSat.",
"The neutron star magnetospheric radius established for some cases [49], [15] using measured strength of magnetic field, source distance and its luminosity, assuming the canonical mass and radius of a typical neutron star (Equation-REF ) was found to be very close to inner radius of the accretion disk, as determined from its observed QPO frequency (Equation-REF ) for some sources.",
"Hence, for such cases, it was possible to establish consistency with spectroscopic measured value of their magnetic field strength.",
"The same appears to be true for GRO J2058+42 under uncertainty involved in measured parameters and standard assumptions.",
"Formation of a transient disk may supply necessary spin-up torque to the neutron star, if the disk rotates in the same direction as the pulsar spin.",
"The expected torque ($N_{\\mbox{char}}$ ) on to the neutron star due to transfer of angular momentum of the accreted mass from such a transient disk can be calculated using the expression below, assuming, transfer of angular momentum of all accreted mass from such a disk with a radius $r_{QPO}$ to its neutron star [29], [46] $N_{\\mbox{char}} = \\eta {\\dot{M}}(GM{r_{QPO}})^{1/2},$ where $\\dot{M}$ is the mass accretion rate and $\\eta $ is the duty cycle for the applied torque.",
"If we assume that all the potential energy of the accreted mass liberated during the outburst is transformed into radiation then the luminosity can be expressed as, $L_{37} = 1.33 {\\dot{M}_{17}} (M/M_{\\odot }){R_{6}}^{-1} \\quad \\mbox{erg s}^{-1},$ where $L_{37}$ is the luminosity in units of $10^{37}$ erg s$^{-1}$ and $R_6$ is the radius in units of $10^6$ cm.",
"The mass accretion rate can be calculated from the observed luminosity, which is derived by measurements of flux from spectral model for AstroSat spectrum (Table 2), and known distance of the star.",
"The estimated value of $\\dot{M}$ during the AstroSat observation was $2.3 \\times 10^{17}$ g s$^{-1}$ .",
"Using these parameters, the value of $N_{char}$ is found to be $\\eta ~(9.0 \\times 10^{34})$ g cm$^{2}$ s$^{-2}$ .",
"The observed torque $N_{0}$ of the pulsar can be expressed in terms of the moment of inertia, $I= 10^{45}$ g cm$^{2}$ (for a typical neutron star, having mass as 1.4${M_{\\odot }}$ and radius of 10 km) and $\\dot{\\nu }$ the observed rate of change of frequency of the neutron star as $N_{0} = 2{\\pi }I{\\dot{\\nu }}.$ Now, if we consider the value of $\\eta \\approx 1.0$ , representing that the transient accretion disk was present and torque was applied for almost the whole duration of observation.",
"Then, by equating the above two torques (Eqs.",
"REF and REF ), one can estimate an average spin-up rate of the pulsar as $\\dot{\\nu } = 1.43 \\times 10^{-11}$ Hz s$^{-1}$ or $\\dot{P} = -5.39 \\times 10^{-7}$ s s$^{-1}$ .",
"This is comparable to the observed value of $\\dot{\\nu }$ during AstroSat observation.",
"Considering the value of $\\dot{\\nu }=1.65\\times 10^{-11}$ Hz s$^{-1}$ obtained during the AstroSat observation and assuming that the spin-up rate is proportional to the luminosity during the recent outburst we could integrate $\\dot{\\nu }$ over the entire outburst to calculate the net change in spin frequency or the period.",
"Taking into account the observed period of 194.22 s during the beginning of AstroSat observation we could calculate that the period decreased from about 195.7 s before outburst ($\\mbox{MJD}=58550$ ) to 193.4 s after this latest long outburst of 2019 ($\\mbox{MJD}=58624$ ).",
"This change in period is comparable to the measured period change from 195.6 to 193.5 by Fermi-GBM during this latest outburst.",
"This is also comparable to the period change during earlier outburst observed by BATSE in 1995.",
"Thus, the formation and presence of such a transient accretion disks around neutron star during the source outburst, could cause significant change in the pulsar period in short duration of 46 days."
],
[
"Pulse Phase-averaged Spectrum",
"The X-ray spectrum of the source was first derived by RXTE during the earlier outburst of 1996.",
"The RXTE PCA and HEXTE data were used covering 2.7–25 keV and 11–50 keV respectively [88].",
"The spectral data were fitted with an absorbed thermal bremsstrahlung model (Phabs*bremss) adequately well, particularly at higher intensities during its outburst.",
"The source flux dropped sharply above 20 keV in most cases, therefore, other models such as a power law with a high-energy cutoff, were not used with RXTE data.",
"It was found that the absorption term, $N_{H}$ was nearly constant with its best-fit values in the range (4.6–5.4)$\\times 10^{22}$ cm$^{-2}$ .",
"The temperature, kT, was found to increase with the source intensity, with best-fit values varying from $10.3 \\pm 0.5$ keV at $2.9\\times 10^{-11}$ ergs cm$^{-2}$ s$^{-1}$ to $22.2\\pm 0.4$ keV at $2.6\\times 10^{-10}$ ergs cm$^{-2}$ s$^{-1}$ [88].",
"There was, however, no report on any cyclotron absorption feature from RXTE observations.",
"This was missed very likely due to observations at much lower intensity of the source during low intensity outbursts, which was an order of magnitude lower than the latest outburst.",
"AstroSat data enabled us to derive X-ray spectrum in 0.8–70 keV energy band as shown in Figure 7.",
"Spectrum was fitted reasonably well using two models.",
"The first model was defined as an absorbed Fermi-Dirac cutoff model along with a black-body, a Fe-emission line and 3 Gaussian absorption lines were introduced to model cyclotron scattering features and its two higher harmonics as observed in the spectrum.",
"The CompTT model was used as the second model in combination with a Fe-emission line and 3 Gaussian absorptions lines as defined in the first model.",
"The CompTT-model is generally used for neutron star based low mass X-ray binaries such as Z-type and atoll sources with relatively lower magnetic field ($<10^{12}$ G) of the neutron star [18].",
"However, the model could also successfully define spectrum of some of the pulsars in Be-binaries for example Cep X-4 [30], 4U 1907+09 [83] and GRO J2058+42 [44].",
"The derived parameters from the two models are tabulated in Table 2.",
"The first model estimated a black-body temperature of $0.83 \\pm 0.04$ keV and detected presence of cyclotron resonance scattering feature and its harmonics.",
"The Comptonization model on the other hand enabled us to determine input photon Wien-temperature of $0.52 \\pm 0.02$ keV, the plasma temperature of $8.22 \\pm 0.10$ keV and plasma optical depth of $5.21 \\pm 0.12$ for the phase-averaged spectra.",
"The Wien-temperature, as per CompTT-model, originates away from the neutron star surface and closer to the inner accretion disk i.e., at the outer transition layer, hence Wien-temperature is always found to be relatively lower than the neutron star black-body temperature as it originates close to inner transition layer i.e., near the surface of the neutron star [18].",
"As per the CompTT model bulk Comptonization occurs in the innermost part of the transition-layer region, while thermal Comptonization is dominant in the outer transition layer and presumably within some extended region located above the accretion disk.",
"Similar deviations were also observed in the case of Cep X-4 fitted with CompTT-model and black-body combined with FDCUT-model [30].",
"However, the centroid energy of cyclotron absorption feature and its detected harmonics are found to be consistent within errors for the two models (Table 2).",
"Some of the accretion powered X-ray pulsars showed additional features in emission between 10–20 keV energy band and more rarely in absorption between 8–10 keV in their respective residuals when fitted with a variety of continuum models [12].",
"[12] have argued that such features may be caused by inadequacies of continuum model rather than cyclotron resonance features.",
"For example, it was observed that a single emission line at around 14 keV can fit two features around 10 and 20 keV for Vela X-1 [34], Her X-1 [11], Cep X-4 [42] and an absorption line between 8–10 keV can fit the features for 4U 1907+09, 4U 1538-52, 4U 0352+309 as in figure 6 of [12].",
"In the case of GRO J2058+42, an introduction of a single Gaussian emission line in this range of energy could not appropriately fit the spectrum.",
"Referring to the results described in Section 3.3 using ratios of spectral counts with respect to Crab spectrum derived for 4 different phases of the pulsar (Figure 6), clearly indicates presence of prominent depressions around 10 keV and 20 keV for phase-1 in particular, and its presence at other phases as well.",
"Therefore, it confirms the presence of such absorption features in the spectral data associated with a physical origin and not due to any discrepancies of the continuum model as discussed above.",
"Additionally, it also excludes the possibility of any uncertainty in the response matrix of the detector as the response matrix was not used to deconvolve the spectrum to calculate these ratios.",
"Relative significance of these absorption features were subsequently estimated after modeling the data and results are shown in Table 2 for the phase averaged case and in Table 3 for 4 different pulse phases.",
"These observations strongly favor presence and detection of these absorption features in their respective spectra of the pulsar.",
"The Be-binary pulsar 4U 0115+63 showed interesting features when its measured pulsed fraction was plotted against energy starting from 5 keV onwards.",
"It showed gradual increase in pulsed fraction with energy along with a sharp localized decrease only around 22 keV.",
"This localized decrease was attributed to cyclotron resonant scattering at the second Landau level.",
"No such localized decrease was observed for its higher harmonics likely due to low signal to noise ratio.",
"Such decrease was also not observed corresponding to its fundamental line, despite its high count rate, as it could be due to competing effects such as photon spawning and cyclotron emission [19].",
"The pulse fractions measured at different energies between 3–80 keV from AstroSat observation of GRO J2058+42 (Figure REF ) did not show any localized decrease in its pulse-fraction corresponding to fundamental ($\\sim 10$ keV) or even for its observed higher harmonics ($\\sim 20$ keV, $\\sim 38$ keV).",
"These could possibly be due to similar reasons mentioned as above for 4U 0115+63 for fundamental and higher harmonics or the resultant decrease in pulse fraction is small and it could not be detected within the error limits.",
"This is very similar to another Be-binary V 0332+53, where no such localized change in pulsed-fraction was observed corresponding to its cyclotron line energies, despite their prominent optical depths [80].",
"The pulse fraction for GRO J2058+42, however, showed a gradual increase with energy and then became nearly constant at higher energies above 20 keV as seen in Figure 4."
],
[
"Pulse Phase-resolved Spectra",
"Cyclotron resonant scattering features are produced due to interaction of photons with electrons quantized in the Landau states formed in the accretion column in the presence of a strong magnetic field of a neutron star in accreting X-ray binaries [68].",
"It can also be produced by reflection of X-rays from the atmosphere of the neutron star [64].",
"These interactions results in absorption features in their spectrum at a particular energy and produce complex shape of the absorption features due to complex scattering cross-sections.",
"The line energies and their shapes change depending on the environment of the line-forming region typically located close to the neutron star, strength of magnetic field of the neutron star, nature of the accretion column, source luminosity and pulse-phase of accreting X-ray pulsars [50], [51], [52].",
"The pulse phase-resolved spectra corresponding to 4 different phases of the accreting pulsar GRO J2058+42 are presented in Fig.",
"REF .",
"AstroSat detected a cyclotron absorption feature and its two harmonics in its pulse phase-resolved spectra, which were identified in the phased-averaged spectrum.",
"Spectral parameters derived from the fitted CompTT-model for 4 different pulse phases are given in Table 3.",
"It is noticed that the relative strengths and shape of these absorption features change with pulse-phase as shown in Figure 9 for clarity.",
"The line energies are found to vary within the range of (9.7–14.4) keV, (19.3–23.8) keV and (37.3–43.1) keV respectively for the observed cyclotron scattering features and its higher harmonics.",
"Their respective detection significance along with false alarm probability (FAP) with respect to pulse-phases are also given in Table 3.",
"We noticed a higher value of centroid energy of the fundamental cyclotron line energy at 13.17 for phase-2, but it was found to be consistent with phase-3 and Phase-4 within its associated larger uncertainty.",
"Overall pulse phase-resolved spectra of GRO J2058+42 showed consistency with respect to cyclotron line energies and continuum except for the phase-1 (Table 3).",
"The phase averaged spectrum (Figure 7), fitted using CompTT-model for GRO J2058+52 determined the cyclotron fundamental line energy along with the harmonics from AstroSat observation.",
"Interestingly, the observed ratios between its fundamental and the two harmonics are found to cover a range of $1.93\\pm 0.06$ and $3.6 \\pm 0.12$ respectively with a 1-$\\sigma $ error limit.",
"This indicates non-harmonic ratio, observed for higher harmonics of cyclotron resonance scattering feature with respect to its fundamental line energy.",
"However, looking at the results from pulse phase resolved spectra in Table 3, it is clear that the position of the spectral features as well as their strengths showed variation with phase but when compared with phase averaged spectrum it may give misleading results.",
"For example, the mean position of the first feature is different in Phase-1 as compared to other phases and its strength is highest in Phase-1.",
"Thus the phase averaged spectrum is likely to be biased towards the lower value in Phase-1.",
"On the other hand, the third feature around 38 keV is not seen in Phase-1, while it is significant in other phases and the phase averaged spectrum will reflect this value.",
"If we consider the ratios between energies of different features in phase resolved spectra (Table 3), the ratios for the first two features are $1.69\\pm 0.16$ , $1.82\\pm 0.14$ and $1.84\\pm 0.11$ corresponding to phase-2, phase-3 and phase-4 respectively.",
"Similarly, the ratio between the third and the first feature, ignoring Phase-1 where the third feature is not significant, are $3.04\\pm 0.17$ , $3.10\\pm 0.23$ and $3.47\\pm 0.20$ for the three phases respectively.",
"Within the error bars these ratios are consistent with those expected for harmonics.",
"Hence, there is no evidence for non-harmonic ratios in GRO J2058+42.",
"However, non-harmonic ratios in cyclotron absorption features were observed for some of the pulsars, for example V 0332+53 [63], [36], [48], Her X-1 [17] and Cep X-4 [86].",
"For some cases, marginal deviation in harmonicity could be explained by employing relativistic approximation of photon-electron scattering [43], but for large deviations it could possibly be explained due to influence of non-dipole magnetic field on the line forming region when the field strength increases with height [51], [52].",
"The variation in cyclotron features with pulse phase could be due to superposition of contributions from a number of lines formed at different heights of a line-forming region in a cylindrical accretion column geometry, influenced by a large gradient of its magnetic field.",
"This could result in the observed variation of the centroid energy and shape depending on visibility of accretion column with respect to line of sight depending on system orientation during spin of the neutron star.",
"This could occur either in the accretion column or in the neutron star atmosphere for an anisotropic injection of energy with an emission peak exiting in a particular direction [53].",
"In such cases, cyclotron scattering features could be observed only for the interval depending on its pulse-phase as seen for pulse phase-1 for GRO J2058+42.",
"There could be other possibilities where the accretion column itself could partially be eclipsed by the neutron star itself for certain pulse-phases, such that only a portion of the accretion column could be visible to the observer [47].",
"This may cause dispersion in magnetic field strength across the visible portion of the accretion column and may cause resultant change in the line-energy and its shape with pulse-phase.",
"This suggest that the observed phase dependent changes in the cyclotron line energy and its shape for GRO J2058+42 during AstroSat observation, could likely be due to changes in the geometry of the line producing region with respect to line of sight during its spin, for a stable luminosity of the source.",
"We find from Table 3 that photo electric absorption and input soft photon temperature CompTT(T0) for all the 4 phases were almost constant within 90% error limit.",
"The plasma temperature CompTT(kT) for phase-1 was, however, found to be relatively higher by $\\sim 1.5$ keV as compared to an average of the remaining 3 phases at 8.5 keV.",
"The plasma optical depth for phase-2 was found to be higher at 6.17 compared to its average value of 4.8 for the other 3 phases.",
"These measurements indicated that for the phase-1, the difference in plasma temperature and possibly local change in configuration of its magnetic field could be responsible for the observed change in its continuum and line parameters as compared to the other 3 phases.",
"The observed ratio of the spectral counts with respect to Crab for phase-1 also clearly showed that there are significant changes in the spectrum relative to other phases (Figure REF ).",
"Therefore, the evident differences in its continuum and line parameters indicate that for this narrow phase-1, emission probably comes mainly from a different region and hence emission components are different with respect to the rest of the three phases.",
"For example, the radiation could originate from one column instead of another, from its visible higher column height as opposed to near the stellar surface.",
"The stellar surface area associated with a relatively hotter plasma accumulation may also be suggestive of a scenario where a magnetically more intense area produces a hotter region contributing a substantial variation in the spectral shape due to the interaction of accelerated high energy particles which could up-scatter soft photons giving rise to the Comptonization spectrum with change in its spectral shape.",
"The recent work of [54] could explain more favorably the observed change in spectral continuum as well as decrease in the centroid energy of the cyclotron lines in general and in particular that of the phase-1.",
"The velocity of bulk motion of in-falling plasma in the cyclotron line forming regions plays a dominant role.",
"Line forming regions are located near the walls of the cylindrical accretion column whereas spectral continuum are formed above and around the accumulated mound close to neutron star surface.",
"Therefore, relative location of these two regions where continuum and cyclotron-lines are formed with respect to a line of sight of the observer, one could see the observed change in the spectral continuum as well as a decrease in centroid energy of the cyclotron-line during certain pulse-phase of the pulsar depending on optimal conditions such as velocity of the bulk motion of the in-falling matter, effect of gravitational bending of the emitted radiation and suitable variation of the local magnetic field [54].",
"The fundamental centroid energy of the cyclotron absorption feature showed overall variation of $\\sim $ 30% with pulse-phase measured for GRO J2058+42.",
"There are several other sources, namely Cen X-3 [13], [73], Vela X-1 [39], [34], 4U 0115+63 [27], GX 301–2 [35], [74], and Her X-1 [85], [71], [33], which showed variation in their fundamental energy of about 10–30% over the pulse-phase.",
"The cyclotron absorption features of GRO J2058+42 clearly showed comparable variation in cyclotron line-energy and shape with variation in its pulse-phase."
],
[
"Determination of the Strength of Magnetic Field of the Pulsar",
"The strength of the surface magnetic field B of a neutron star can be obtained using $E_{c} \\simeq 11.6 n \\bigg ({\\frac{1}{1+z}} \\bigg ) \\bigg ({\\frac{B}{10^{12}G}} \\bigg )\\; \\mbox{keV}.$ We assume that the observed cyclotron absorption feature at $E_{c}$ is associated with its fundamental line ($n = 1$ ), and considering a gravitational redshift of $z = 0.3$ at the surface of a typical neutron star having a mass of $1.4 M_{\\odot }$ and a radius of 10 km.",
"From the measurements of the centroid energy at $10.81^{+0.48}_{-0.49}$ keV, for the phase averaged case, we could thus establish the strength of the magnetic field of the pulsar as $(1.21^{+0.05}_{-0.06}) \\times 10^{12}$ G. Considering the variation in energy of first cyclotron line with pulse phase we can get a range of (9.7–14.4) keV after including the error bars.",
"This would translate to a variation in the magnetic field strength of (1.1–1.6)$\\times 10^{12}$ G over the pulse phase of the pulsar.",
"The strength of the magnetic field derived for GRO J2058+42 from its spectrum is comparable to the magnitude of those measured for other X-ray pulsars, such as KS 1947+300 (12.2 keV) [23], Swift J1626.3–5156 (10 keV) [14] and 4U 0115+63 (12 keV) [31], [19] where respective energy values of their fundamental cyclotron absorption features are shown within parenthesis."
],
[
"Conclusion",
"AstroSat observed the Be-binary pulsar GRO J2058+42 on April 10, 2019 during its latest long outburst between March–May 2019, using both LAXPC and SXT instruments for a total exposure of 57 ks.",
"The source intensity during this observation had declined to 170 mCrab, about 66% of its peak intensity of 256 mCrab.",
"AstroSat observation showed a clear detection of strong pulsation from the source.",
"The spin-period of the pulsar was determined as $194.2201\\pm 0.0016$ s and its average spin-up rate was $\\dot{\\nu } = (1.65\\pm 0.06) \\times 10^{-11}$ Hz s$^{-1}$ corresponding to MJD 58583.10868148.",
"Pulse-profiles derived at different energy bands covering 3–80 keV, showed pronounced and multiple pulse structure, which showed a variation in shape and pulse-fraction with energy.",
"The RMS pulse fraction varied from about 20% to 30% between 3–20 keV, beyond which it was approximately constant.",
"Pulse-profiles derived from recent AstroSat observations were found to be similar in shape to those reported from RXTE observations, at a relatively lower source intensity by an order of magnitude.",
"Thus, the source did not show drastic change in its pulse shape and pulse-fraction within the observed variation of source luminosity.",
"A broad QPO feature corresponding to a frequency of 0.090 Hz was detected for the first time from AstroSat observations.",
"This provided an evidence for the formation of a transient accretion disk around the neutron star.",
"The QPO, therefore, offered us a tool to probe inner region of the accretion disk and to quantify the transfer of angular momentum through such accretion disk.",
"This enabled us to estimate the torque applied to the neutron star by mass transfer from a transient accretion disk.",
"This resulted in the prediction and determination of observed change in spin-period of the pulsar by 2.3 seconds during the recent 46 days outburst.",
"Similar change in spin-period was observed by BATSE during the earlier outburst in 1995 and also by the Fermi-GBM during this latest outburst.",
"This therefore, favors a scenario where pulsar could be effectively spun-up by such a magnitude, in a short span of 46 days.",
"The centroid energy of cyclotron line and its harmonics were found to be consistent within errors for the two models used for spectral fits.",
"The line energies are found to vary within the range of (9.7–14.4) keV, (19.3–23.8) keV and (37.8–43.1) keV respectively for the observed 3 absorption features.",
"The detection of cyclotron line lead to determination of strength of its strong magnetic field of the neutron star.",
"Therefore, using AstroSat observation, we could establish the strength of magnetic field of the pulsar as (1.1–1.6)$\\times 10^{12}$ G. Further observations are required to study spectral variation with source luminosity, variation of magnetic field strength during its long outburst duration and with time, to probe the nature of its accretion column and geometry where cyclotron lines are produced.",
"We gratefully acknowledge all the support received from the Indian Space Research Organization (ISRO) for successful realization of AstroSat mission from the initial phase of instrument building, tests and qualifications to software developments and mission operations.",
"We also acknowledge the support received from the LAXPC Payload Operation Center (POC), TIFR, Mumbai for the release of verified data, calibration data products and pipeline processing tools.",
"This work has also utilized the data from the Soft X-ray Telescope (SXT) and hence thankfully acknowledge SXT POC at TIFR for releasing the data through the ISSDC data archive center and providing the necessary software tools.",
"We acknowledge software engineers Sanket Kotak, Ashutosh Bajpai and Harshal Pawar, for their vital services in software development activities and timely completion of the full SXT pipeline processing chain along with relevant documentation.",
"We also acknowledge the support of the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.02-0200.",
"This research has also made use of data obtained from Swift-BAT and RXTE through the High Energy Astrophysics Science Archive Research Center On-line Services, provided by the NASA/Goddard Space Flight Center, we acknowledge their vital support.",
"We also thank Fermi-GBM team of NASA/MSFC for sharing monitoring data of the pulsar period measurements of this source during its outburst.",
"We also acknowledge generous support of NASA's HEASARC for offering all the useful software and tools for analysis of Astronomical data.",
"We thank an anonymous Referee for critical comments, which have improved the manuscript significantly.",
"Astrosat"
]
] | 2005.14044 |
[
[
"Inflation with non-canonical scalar fields revisited"
],
[
"Abstract We revisit inflation with non-canonical scalar fields by applying deformed-steepness exponential potentials.",
"We show that the resulting scenario can lead to inflationary observables, and in particular to scalar spectral index and tensor-to-scalar ratio, in remarkable agreement with observations.",
"Additionally, a significant advantage of the scenario is that the required parameter values, such as the non-canonicality exponent and scale, as well as the potential exponent and scale, do not need to acquire unnatural values and hence can accept a theoretical justification.",
"Hence, we obtain a significant improvement with respect to alternative schemes, and we present distinct correlations between the model parameters that better fit the data, which can be tested in future probes.",
"This combination of observational efficiency and theoretical justification makes the scenario at hand a good candidate for the description of inflation."
],
[
"Introduction",
"Inflation is now a crucial part of the Standard Model of Cosmology [1], [2], [3], [4], [5].",
"Its solution to the horizon and flatness problems, together with the predictions for an almost scale invariant perturbation spectral index, have been confirmed by measurements of the cosmic microwave background (CMB) radiation.",
"Nevertheless, the specific mechanism that triggers the inflationary epoch is one of the most outstanding issues in contemporary particle physics and cosmology.",
"As a result, the building of theoretical models that explain this early accelerating expansion of the universe has exploded in recent years.",
"The first main class of mechanisms that can lead to successful inflation is based on the introduction of a scalar field, while the second main class is obtained through gravitational modifications (for reviews see [6], [7], [8], [9], [10]).",
"Consequently, inflation-related observations have provided significant insight to both modified gravity [11], [12], [13], [14], [15], as well as to particle physics model building.",
"The literature on the latter is very extensive, particularly within the framework of supersymmetry [16], [17], [18], [19], supergravity [20], [21], [22], [23], theories of extra dimensions such as superstring and brane theories [24], [25], [26], and technicolor too [27].",
"Detailed lists of references on different theoretical constructions can be found in [6], [7], [10].",
"In trying to understand the above issues (often in the framework of a single theory) several problems have been encountered, including fine-tuning issues (tiny dimensionless constants) and large predictions for tensor fluctuations.",
"In this respect, theories of scalar fields with non-canonical kinetic terms, as expected in supergravity and superstring theories, including the $k$ -inflation subclass [28], [29], [30], were found to have significant advantages.",
"These theories arise commonly in the framework of supergravity and string compactifications, which typically contain a large number of light scalar fields $X$ (moduli), whose dynamics are governed by a non-trivial moduli space metric $G_{ij}$ .",
"As long as the moduli space metric is not flat, we generically expect non-canonical kinetic terms.",
"Such effects could, but need not, be suppressed by the high scale of the corresponding Ultra-Violet physics (e.g.",
"moduli masses, string scale), but they can still have significant cosmological consequences through the dynamics of the dilaton and moduli fields.",
"Among their many advantages, non-canonical scalars satisfy in a more natural way the slow-roll conditions of inflation, since the additional effective friction terms in the equations of motion of the inflaton slow down the scalar field for potentials which would otherwise be too steep.",
"Hence, the resulting tensor-to-scalar ratio is significantly reduced [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48].",
"Moreover, models with non-canonical kinetic terms often allow for the kinetic term to play the role of dark matter and the potential terms to generate dark energy and inflation [49], [50], [51], [52].",
"Additionally, note that in the inflation realization in the context of Galileon and Horndeski theories, the role of the non-canonical kinetic term is also crucial [53], [54], [55], [56], [57], [58].",
"The form of the non-canonical terms can vary significantly, since there are many plausible models, including different ways to achieve compactification.",
"The recent cosmological data, however, together with the requirement to avoid fine-tuning and unnatural solutions, severely constrain the available possibilities.",
"On the other hand, an alternative way to improve the inflationary observables is by introducing an extra parameter as an exponent in the known potential forms, and thus affecting their steepness.",
"In this way the dynamics of the scalar field can be additionally deformed, offering an alternative way to bring the tensor-to-scalar ratio to lower values without ruining the necessary spectral index [59], [60], [61], [62], [63], [64], [65], [66].",
"One possible disadvantage of the above inflationary models, namely those with non-canonical terms and those with extra steepness parameter in the potential, is that the parameter values needed for acceptable observables are unnatural and hard to be justified from the field-theoretical point of view.",
"In particular, the non-canonical exponents need to be large, or the mass and potential parameters take trans-Planckian values.",
"Hence, in this work, we are interested in studying a combination of the above models, specifically introducing a scalar field with non-canonical kinetic terms on top of a deformed-steepness potential with an extra parameter.",
"As we will show, this enhances the range of solutions and leads to very satisfactory observables, for natural sets of model parameters that we proceed to identify and classify."
],
[
"Non-canonical inflation with deformed-steepness potentials",
"In this section we present the scenario of non-canonical inflation with deformed-steepness potentials.",
"We will focus on the usual non-canonical Lagrangian, which is well justified theoretically, and takes the form [28], [29], [30], [68], [67], [35], $\\mathcal {L}(\\phi , X)=X \\left(\\frac{X}{M^4} \\right)^{\\alpha -1}-V(\\phi ),$ where $X=\\frac{1}{2} \\partial _{\\mu } \\phi \\partial ^{\\mu } \\phi $ is the kinetic energy of the scalar field, and thus the action of the scenario reads $\\mathcal {S}=\\int d^{4}x \\sqrt{-g} \\left[ M^{2}_{pl} \\frac{R}{2} +X\\left(\\frac{X}{M^4} \\right)^{\\alpha -1}-V(\\phi ) \\right].$ The parameter $M$ has dimensions of mass and determines the scale in which the non-canonical effects become significant, while $M_{pl}$ is the Planck mass.",
"Concerning the potential, in this work we will consider the deformed-steepness potential that was introduced in [59], [61], namely $V(\\phi )=V_{0}\\exp ^{-\\lambda \\phi ^{n}/M^{n}_{pl}},$ with $V_0$ and $\\lambda $ the usual potential parameters and $n$ the new exponent parameter that determines the deformed-steepness.",
"We consider a homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) metric of the form $ds^{2}=-dt^{2}+a^{2}(t)\\delta _{ij}dx^{i}dx^{j}\\,,$ where $a(t)$ is the scale factor.",
"Variation of the action (REF ) in terms of the metric gives the following Friedmann equations $H^2=\\frac{1}{3 M^{2}_{pl}} \\left[(2\\alpha -1)X \\left(\\frac{X}{M^4}\\right)^{\\alpha -1}+V_{0}\\exp ^{-\\lambda \\phi ^{n}/M^{n}_{pl}} \\right]$ $\\dot{H}=- \\frac{1}{M^{2}_{pl}} \\alpha X\\left(\\frac{X}{M^4} \\right)^{\\alpha -1},$ where $H=\\frac{\\dot{a}}{a}$ is the Hubble parameter.",
"Additionally, variation in terms of the scalar field leads to the Klein-Gordon equation $\\ddot{\\phi }+\\frac{3H\\dot{\\phi }}{2\\alpha -1}-\\frac{\\lambda n\\phi ^{n-1}V_{0}\\exp ^{-\\lambda \\phi ^{n}/M^{n}_{pl}}}{\\alpha (2\\alpha -1)M^{n}_{pl}}\\left(\\frac{2M^4}{\\dot{\\phi ^2}} \\right)^{\\alpha -1}=0.$ Note that one can write the above equation in the form of the usual conservation equation $\\dot{\\rho }_{\\phi }+3H(\\rho _\\phi +p_\\phi )$ , using the definitions $&&\\rho _{\\phi } = (2\\alpha -1)X \\left(\\frac{X}{M^4}\\right)^{\\alpha -1}+V_{0}\\exp ^{-\\lambda \\phi ^{n}/M^{n}_{pl}}\\nonumber \\\\ \\nonumber \\\\&&p_{\\phi } = X \\left(\\frac{X}{M^4} \\right)^{\\alpha -1}-V_{0}\\exp ^{-\\lambda \\phi ^{n}/M^{n}_{pl}}.$ In every inflationary scenario the important quantities are the inflation-related observables, namely the scalar spectral index of the curvature perturbations $n_\\mathrm {s}$ and its running $\\alpha _\\mathrm {s} \\equiv dn_\\mathrm {s}/d\\ln k$ , with $k$ the measure of the wave number $\\vec{k}$ , the tensor spectral index $n_\\mathrm {T}$ and its running, as well as the tensor-to-scalar ratio $r$ .",
"In a given scenario these quantities depend on the model parameters, and hence confrontation with observational data can lead to constraints on these model parameters.",
"In order to extract the relations for the inflation-related observables, a detailed and thorough perturbation analysis is needed.",
"In the simple case of canonical fields minimally coupled to gravity, and introducing the slow-roll parameters, full perturbation analysis indicates that the inflationary observables can be expressed solely in terms of the scalar potential and its derivatives [69], [7], [10].",
"However, in the case where non-canonical terms or forms of non-minimally coupling are present, as well as in the case where the potential itself is absent (as for instance in modified gravity inflation), one should instead introduce the Hubble slow-roll parameters $\\epsilon _n$ (with $n$ positive integer), defined as [70], [71], [10], [72] $\\epsilon _{n+1}\\equiv \\frac{d\\ln |\\epsilon _n|}{dN},$ where $N\\equiv \\ln (a/a_{ini})$ is the e-folding number, and $\\epsilon _0\\equiv H_{ini}/H$ , where $a_{ini}$ is the initial scale factor with $H_{ini}$ the corresponding Hubble parameter (as usual inflation ends when $\\epsilon _1=1$ ).",
"Thus, the first three $\\epsilon _n$ are found to be $&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\epsilon _1\\equiv -\\frac{\\dot{H}}{H^2},\\\\&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\epsilon _2 \\equiv \\frac{\\ddot{H}}{H\\dot{H}}-\\frac{2\\dot{H}}{H^2},\\\\&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\epsilon _3 \\equiv \\left(\\ddot{H}H-2\\dot{H}^2\\right)^{-1}\\left[\\frac{H\\dot{H}\\dddot{H}-\\ddot{H}(\\dot{H}^2+H\\ddot{H}) }{ H\\dot{ H } }-\\frac{2\\dot{H}}{H^2}(H\\ddot{H}-2\\dot{H}^2)\\right].$ With these definitions, the basic inflationary observables are given as [10] $r &\\approx &16\\epsilon _1 ,\\\\n_\\mathrm {s} &\\approx & 1-2\\epsilon _1 - \\epsilon _2 , \\\\\\alpha _\\mathrm {s} &\\approx & -2 \\epsilon _1 \\epsilon _2 - \\epsilon _2 \\epsilon _3 ,\\\\n_\\mathrm {T} &\\approx & -2\\epsilon _1 .$ In the scenario of non-canonical inflation with deformed-steepness potentials, described by equations (REF )-(REF ), the dynamics, i.e.",
"the Hubble function, is determined by the parameters $\\alpha $ and $M$ related to “non-canonicality”, by the standard potential parameters $V_0$ and $\\lambda $ , alongside the deformed-steepness parameter $n$ .",
"Hence, we deduce that the above inflationary observables (REF )-() will be determined by these model parameters too.",
"In the next section we analyze in detail the effect of each parameter on the inflationary observables, and we will show which combinations can bring the predictions deep inside the observational contours."
],
[
"Results",
"In this section, we investigate the inflationary observables in the scenario of non-canonical inflation with deformed-steepness potentials.",
"In particular, we desire to see how the scalar spectral index $n_\\mathrm {s}$ and the tensor-to-scalar ratio $r$ are affected by the model parameters.",
"Since the involved equations (REF )-(REF ), the slow-roll parameters (REF )-() and the observables expressions (REF )-() are in general too complicated to admit analytical solutions, we investigate them numerically.",
"Specifically, for a given set of parameter values we impose the conditions for $\\phi $ , $\\dot{\\phi }$ and $H$ corresponding to small $\\epsilon _i$ .",
"We evolve the system and we determine the end of inflation by demanding $\\epsilon _{1}=1$ (cases of eternal inflation are considered non-physical), and thus by imposing the desired e-folding number $N$ we extract the time at the beginning of inflation.",
"Hence, we can use the corresponding Hubble parameter to calculate the inflationary observables corresponding to the given parameter values and the imposed e-folding number $N$ .",
"We start our investigation by examining the effect of the non-canonical parameter $\\alpha $ and the deformed-steepness parameter $n$ .",
"Therefore, we fix $M$ and $V_0$ at theoretically motivated values and we calculate $n_\\mathrm {s}$ and $r$ for various combinations of $\\alpha $ and $n$ , adjusting suitably only the value of $\\lambda $ , and for the e-folding number $N$ taking, as usual, the values 50, 60 and 70.",
"In Table REF we summarise the obtained observable predictions.",
"Additionally, in order to present the information in a more transparent way that allows comparison with observational data, in Fig.",
"REF we depict the results of Table REF on top of the 1$\\sigma $ and 2$\\sigma $ contours of the Planck 2018 data [73].",
"Table: Predictions for the scalar spectral index n s n_\\mathrm {s} andthe tensor-to-scalar ratio rr for the scenario of non-canonical inflation withdeformed-steepness potential, for various combinations of α\\alpha and nn,adjusting the values of λ\\lambda , and for e-folding number NN equal to 50, 60and 70.",
"For this Table we fix M=10 -4 M pl M=10^{-4} M_{pl} and V 0 =10 16 V_{0}=10^{16}(GeV) 4 (GeV)^4, with M pl =10 18 M_{pl}=10^{18} GeVGeV.Figure: 1σ\\sigma (yellow) and 2σ\\sigma (light yellow) contours for Planck 2018results (Planck +TT+lowP+TT+lowP) , on then s -rn_{\\mathrm {s}}-r plane.Furthermore, wedepict the predictions of Table , of the scenario at hand forvarious values of thethe non-canonicalparameter α\\alpha and the deformed-steepness parameter nn, adjusting thevalues of λ\\lambda , and keepingfixed M=10 -4 M pl M=10^{-4} M_{pl} and V 0 =10 16 V_{0}= 10^{16}(GeV) 4 (GeV)^4, with M pl =10 18 M_{pl}=10^{18} GeV.",
"In every line the first (black) pointcorresponds to e-folding number N=50N=50, the middle (red) point to N=60N=60, andthe third (green) to N=70N=70.Upper left panel: Black - solid for n=4n=4, λ=10 -13 \\lambda =10^{-13}, blue -dashed for n=5n=5, λ=10 -12 \\lambda =10^{-12}, green - dotted for n=6n=6,λ=10 -11 \\lambda =10^{-11}, red - dashed-dotted for n=7n=7, λ=10 -11 \\lambda =10^{-11} , magenta- dashed-dotted-dotted for n=8n=8, λ=10 -11 \\lambda =10^{-11}.Upper right panel: Black - solid for n=5n=5, λ=3·10 -9 \\lambda =3\\cdot 10^{-9},blue - dashedfor n=6n=6, λ=10 -7 \\lambda =10^{-7}, green - dotted for n=7n=7, λ=10 -6 \\lambda =10^{-6},red - dashed-dotted for n=8n=8, λ=10 -5 \\lambda =10^{-5}.Lower panel:Black - solid for n=6n=6, λ=3·10 -6 \\lambda =3\\cdot 10^{-6},blue - dashedfor n=7n=7, λ=10 -4 \\lambda =10^{-4}, green - dotted for n=8n=8, λ=10 -2 \\lambda =10^{-2}.A general observation is that the predictions of the scenario at hand lie well inside the 1$\\sigma $ region of the Planck 2018 data, without the need to use large values for the non-canonical parameter $\\alpha $ or the deformed-steepness parameter $n$ , which was indeed the main motivation behind the present work.",
"Additionally, the predictions of the scenario are better, compared to the simple non-canonical models, as well as to the simple deformed-steepness models.",
"Concerning the specific features, we find the following: For any given set of model-parameters, increasing the e-folding values $N$ leads to increased $n_{\\mathrm {s}}$ and decreased $r$ , as is usual in the majority of inflationary scenarios.",
"Now, for a given $\\alpha $ , as $n$ increases both $n_{\\mathrm {s}}$ and $r$ also increase.",
"On the other hand, for a given $n$ , as $\\alpha $ increases there is no particular tendency for $n_{\\mathrm {s}}$ and $r$ .",
"However, for larger $\\alpha $ , such as $\\alpha =7$ , the effect of $n$ is less significant and the different curves actually coincide.",
"Moreover, as $\\alpha $ grows, higher values of $\\lambda $ can be accommodated, whereas for values of $\\alpha $ closer to the canonical case, $\\lambda $ needs to be reduced significantly if we desire to obtain observables inside the data contours.",
"Similarly, as $n$ increases $\\lambda $ needs to acquire higher values too.",
"We proceed by investigating the effect on the observables of the parameters $M$ and $V_0$ , which determine the scale of non-canonicality and of the potential, respectively, keeping in mind that the scale of inflation in theoretically motivated constructions can be anywhere from just below the unification scale (mostly Grand Unification), to energies as low as the scales within reach of the LHC (see e.g.",
"[74]), with many possibilities in between, that can be linked i.e.",
"to different stages of symmetry breaking.",
"Without loss of generality we fix $\\alpha =3$ , $n=6$ and $\\lambda =10^{-11}$ , and we calculate $n_\\mathrm {s}$ and $r$ for e-folding number $N$ being as usual 50, 60 and 70.",
"We first additionally fix $M$ and change $V_0$ , then we fix $V_0$ and change $M$ , and finally we change both $M$ and $V_0$ .",
"We summarise the obtained observable predictions in Table REF .",
"Furthermore, in order to present the results in a more transparent way, in Fig.",
"REF we depict the results of Table REF on top of the 1$\\sigma $ and 2$\\sigma $ contours of the Planck 2018 data [73].",
"Table: Predictions for the scalar spectral index n s n_\\mathrm {s} andthe tensor-to-scalar ratio rr for the scenario of non-canonical inflation withdeformed-steepness potential, for fixed α=3\\alpha =3, n=6n=6, andλ=10 -11 \\lambda =10^{-11}, for various combinations of MM and V 0 V_0, and for e-foldingnumber NN being50, 60 and 70.",
"For the upper panel we additionally fix M=10 -4 M pl M=10^{-4} M_{pl}, withM pl =10 18 M_{pl}=10^{18} GeVGeV, while for the middle panel we additionally fixV 0 =10 9 V_{0}=10^{9} (GeV) 4 (GeV)^4.Figure: 1σ\\sigma (yellow) and 2σ\\sigma (light yellow) contours for Planck 2018results (Planck +TT+lowP+TT+lowP) , on then s -rn_{\\mathrm {s}}-r plane.Moreover, wedepict the predictions of the upper panels of Table , of thescenario at hand forfor fixed α=3\\alpha =3, n=6n=6, andλ=10 -11 \\lambda =10^{-11}, forvarious combinations of MM and V 0 V_0.",
"In every line the first (black) pointcorresponds to e-folding number N=50N=50, the middle (red) point to N=60N=60, andthe third (green) to N=70N=70.Upper left panel: Fixed M=10 -4 M pl M=10^{-4} M_{pl}.",
"Black - solid forV 0 =10 15 V_{0}=10^{15} (GeV) 4 (GeV)^4, blue -dashed for V 0 =10 16 V_{0}=10^{16} (GeV) 4 (GeV)^4, green - dotted for V 0 =10 17 V_{0}=10^{17}(GeV) 4 (GeV)^4.Upper right panel:Fixed V 0 =10 9 V_{0}=10^{9} (GeV) 4 (GeV)^4.",
"Black - solid forM=10 -3 M pl M=10^{-3} M_{pl}, blue -dashed for M=5·10 -4 M pl M=5\\cdot 10^{-4} M_{pl}, green - dotted for M=10 -4 M pl M=10^{-4} M_{pl}.Lower panel: Black - solid forM=5·10 -3 M pl M=5\\cdot 10^{-3} M_{pl}, V 0 =10 3 V_{0}=10^{3} (GeV) 4 (GeV)^4 blue -dashed for M=10 -3 M pl M=10^{-3} M_{pl}, V 0 =10 8 V_{0}=10^{8} (GeV) 4 (GeV)^4, green - dotted forM=10 -5 M pl M=10^{-5} M_{pl}, V 0 =10 18 V_{0}=10^{18} (GeV) 4 (GeV)^4.The main observation is that the predictions of the scenario lie well inside the 1$\\sigma $ region of observational data.",
"Now, for fixed $M$ , increasing $V_0$ leads to lower values of $r$ and $n_{\\mathrm {s}}$ ; moreover, the variation of $r$ is much faster than that of $n_{\\mathrm {s}}$ .",
"Additionally, for fixed $V_0$ , increasing $M$ leads also to lower values of $r$ and $n_{\\mathrm {s}}$ ; nevertheless the change in $n_{\\mathrm {s}}$ is strongly affected by the change in $M$ (since $M$ appears in powers of four in the equations) that it can easily be led outside the observational contours.",
"Finally, in the case where both $M$ and $V_0$ are allowed to vary, we find that for increasing $M$ we need to significantly decrease $V_0$ in order to remain inside the observational contours.",
"This was expected, since in the scalar-field equation (REF ) these two parameters appear multiplied.",
"Nevertheless, this is not a trivial result, since $M$ is related to the non-canonicality scale while $V_0$ to the potential scale.",
"From the above analysis we deduce that non-canonical kinetic terms combined with deformed-steepness potentials can provide inflationary predictions in very good agreement with observations, compared to simple non-canonical models [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48] as well as to canonical models with deformed-steepness potentials [59], [60], [61], [62], [63], [64], [65], [66].",
"An additional significant advantage is that the above combination allows to achieve good predictions without the need to use unnaturally large values for $\\alpha $ or $n$ , or unnaturally tuned values for the non-canonicality and potential scales $M$ and $V_0$ , as well as for the potential exponent $\\lambda $ .",
"In particular, we see that $M$ and $V_0$ remain in reasonable sub-Planckian regions, with values that can be easily predicted and accepted from field theoretical point of view.",
"This combination of observational efficiency and theoretical justification is a significant advantage of the scenario at hand."
],
[
"Conclusions",
"In this work we revisited inflation with non-canonical scalar fields by applying deformed-steepness exponential potentials.",
"Non-canonical kinetic terms can arise naturally in models of supergravity and superstrings, while exponential potentials have remarkable properties, as they greatly facilitate slow roll and result to scaling behaviour at large scales.",
"As we have shown, the resulting scenario can lead to inflationary observables, and in particular to scalar spectral index of the curvature perturbations $n_\\mathrm {s}$ and tensor-to-scalar ratio $r$ , in remarkable agreement with the observations of Planck 2018, being well inside the 1$\\sigma $ region.",
"Apart from observational predictability, a significant additional advantage of the proposed scenario arises from the theoretical point of view.",
"In particular, in order to obtain acceptable observables, in simple non-canonical models one needs to use relatively large non-canonical exponent $\\alpha $ or ranges of values for the non-canonicality scale $M$ , while in canonical models with deformed-steepness potentials relatively large values of the extra exponent $n$ need to be imposed, and, hence, these models cannot be well-justified theoretically.",
"On the other hand, in the scenario of the present work the exponents $\\alpha $ and $n$ are small, as well as the non-canonicality and potential scales $M$ and $V_0$ remain in reasonable sub-Planckian regions.",
"Our analysis revealed that, for a given $\\alpha $ , as $n$ increases both $n_{\\mathrm {s}}$ and $r$ increase too, while on the other hand, for a given $n$ , as $\\alpha $ increases there is no particular tendency for $n_{\\mathrm {s}}$ and $r$ .",
"Moreover, as $\\alpha $ grows, higher values of $\\lambda $ can be accommodated, whereas for $\\alpha $ values closer to the canonical case, $\\lambda $ needs to be reduced significantly in order to obtain observables inside the data contours.",
"Similarly, as $n$ increases, $\\lambda $ needs to acquire higher values too.",
"Additionally, for fixed $M$ , increasing $V_0$ leads to lower values of $r$ and $n_{\\mathrm {s}}$ , while for fixed $V_0$ , increasing $M$ leads to lower values of $r$ and $n_{\\mathrm {s}}$ too.",
"In summary, we showed that revisiting non-canonical inflation models by applying potentials with deformed steepness, increases the observational predictability, bringing the scalar spectral index and the tensor-to-scalar ratio more deeply into the observational contours, offering a better theoretical justification for the required parameters.",
"This combination of observational efficiency and theoretical justification is a significant advantage of the scenario at hand, and hence, non-canonical models with deformed-steepness potential need to be further explored, as additional observational data will be coming forward."
]
] | 2005.14069 |
[
[
"Variational regularisation for inverse problems with imperfect forward\n operators and general noise models"
],
[
"Abstract We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice.",
"We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals.",
"Both for a-priori and a-posteriori parameter choice rules, we obtain convergence rates of the regularized solutions in terms of Bregman distances.",
"Our results apply to fidelity terms such as Wasserstein distances, f-divergences, norms, as well as sums and infimal convolutions of those."
],
[
"[figure]skip=12pt compat=newest plot coordinates/math parser=false external [prefix=pics/] =1"
]
] | 2005.14131 |
[
[
"Empathic AI Painter: A Computational Creativity System with Embodied\n Conversational Interaction"
],
[
"Abstract There is a growing recognition that artists use valuable ways to understand and work with cognitive and perceptual mechanisms to convey desired experiences and narrative in their created artworks (DiPaola et al., 2010; Zeki, 2001).",
"This paper documents our attempt to computationally model the creative process of a portrait painter, who relies on understanding human traits (i.e., personality and emotions) to inform their art.",
"Our system includes an empathic conversational interaction component to capture the dominant personality category of the user and a generative AI Portraiture system that uses this categorization to create a personalized stylization of the user's portrait.",
"This paper includes the description of our systems and the real-time interaction results obtained during the demonstration session of the NeurIPS 2019 Conference."
],
[
"Introduction",
"Representation of human traits has always been of great value in the creation of artworks.",
"Even though science and art seem to have very different toolsets and goals, these differences could be seen as blurred where an artist seems to follow scientific knowledge to inform their craft and where science studies art to understand the human condition.",
"Using well-established insights from psychological research on personality, emotion, and other human traits to inform, create and critique artwork is not a new idea [15].",
"Traditional portrait artists use knowledge on human vision and perception to create a painterly portrait of a live or photographed sitter [7].",
"To create a portrait, a human portrait painter would not only set up the room/lighting, position the sitter, interview the sitter to understand/capture their physical and mental characteristics, but also try to convey their painting style in the trajectory of their painting career as well as strive for some personal and universal (e.g., emotional and cultural concepts) truth of the current world they live in.",
"An artist has a palette of choices of themes, brush style, colour plan, edge and line plan, abstraction style, and emotional narrative at their disposal to create the final painting that balances these goals.",
"The discipline of computational creativity and generative art opens up new possibilities for modeling scientific knowledge to emulate this process and advance our understanding of human creativity further.",
"Our research uses a host of artificial intelligence techniques as an attempt to begin to understand and emulate this creative process.",
"The system we created, the Empathic AI Painter, is aimed to examine new ways of balancing or blending different aesthetic, conceptual, abstraction concerns at a semantic level.",
"We showcased our work at the demo session of the NeurIPS 2019 Conference in Vancouver, Canada.",
"This paper includes a detailed description of the research, the process and the demonstration, followed by some discussion on the results obtained and future work."
],
[
"System Description",
"The Empathic Painter System is designed to emulate the interaction of a live portrait artist with a human (the ‘sitter’ in art terms).",
"It attempts to understand the traits (i.e., emotion and personality) to emphatically create a unique portrait of the sitter through selecting the right color palette, style, and abstraction techniques that match the sitter's emotion and personality traits.",
"Our system incorporates two distinct components implemented as a two-stage process: in the first step, it aims to capture the characteristics of the sitter; second, using these traits to generate an artistic representation of their portrait during the second stage of the process (see fig:System).",
"Figure: The two components of the system: conversational interaction and generative portrait stylization.",
"The Big-5 categorical mapping between these components is used to create a personality-based stylized portrait of the sitter.The first stage, capturing the personality-based characteristics of the sitter, is achieved during the conversational interaction of the sitter with our embodied conversational agent that uses empathic interaction methods.",
"We are using the M-Path conversational agent system [20] that was previously developed by our research group.",
"For this demonstration, we modified the M-Path system to perform an interview based on the Big-5 personality questionnaire (see sec:big5 and sec:eca) to categorize the sitter in one of the well-established personality dimensions.",
"This information is used to map the personality traits to a particular artistic style.",
"This mapping is passed to our Generative AI Portrait Stylization system at the second stage, to generate an artistic portrait of the sitter (see sec:genart).",
"Figure: An example setup of the interaction with the M-Path system, where the 3D embodied conversational agent conducts Big-5 questionnaire with the participant using a webcam and microphone.The interaction scenario used during the demonstration included several steps.",
"First, the source portrait of the sitter is taken in a controlled lighting condition, and a unique ID is assigned while obtaining consent for participation and usage of the portrait.",
"Then, the sitter is informed about the M-Path system and given instructions on how to interact with the conversational agent.",
"The sitter then initiates the interaction until a successful conversation is achieved, and the conversational agent informs the sitter that the interaction is over.",
"M-Path system uses the information gathered during the interaction to classify the personality dimension of the sitter as a category.",
"This category is then passed to the Generative AI Portraiture system to generate a personalized portrait style of the sitter.",
"The generated portraits are displayed on a large monitor for all the participants and the demonstration crowd to see and evaluate.",
"fig:ECA shows an example interaction setting with the M-Path system.",
"Details on both stages and the mapping process are described in the following sections."
],
[
"Big-5 Personality Mapping",
"The five-factor model of personality (FFM), also referred to as the \"Big-5 Personality Model,\" is constructed as an empirical generalization to capture the covariation of widely-accepted personality traits between individuals [14].",
"It is constructed as an attempt to categorize the personality variations along five dimensions: extraversion, openness, conscientiousness, neuroticism, and agreeableness.",
"Each personality dimension describes a broad domain of psychological functioning that is composed from a set of more specific and narrow traits [24], [1].",
"Extraversion describes the extent to which people are assertive, dominant, energetic, active, talkative, and enthusiastic.",
"Openness is a personality dimension that characterizes someone who is intellectually curious and tends to seek new experiences and explore novel ideas, which can be described as creative, innovative, imaginative, reflective, untraditional, cultured, curious, original, broad-minded, intelligent, and artistically sensitive.",
"Conscientiousness indicates an individual’s degree of organization, persistence, hard work, and motivation in the pursuit of goal accomplishment.",
"People on the high end of this dimension are thought to be organized, plan oriented, and determined.",
"Neuroticism represents individual differences in adjustment and emotional stability, which is also called Emotional Stability, Stability, or Emotionality.",
"Individuals with high Neuroticism scores tend to experience a number of negative emotions including anxiety, hostility, depression, self-consciousness, impulsiveness, vulnerability, anger, embarrassment, and worrisome.",
"Lastly, agreeableness has been interpreted as likability, friendliness, social conformity or compliance [1].",
"Individuals with high Agreeableness scores can be characterized as trusting, forgiving, caring, altruistic, gullible, courteous, flexible, good-natured, cooperative, soft-hearted, and tolerant.",
"Several independent researchers used factor analysis of verbal descriptors of human behavior to define this model [11].",
"The instrument used in this paper is based on the shortened version of the Revised NEO Personality Inventory (NEO-PI-R) by [4].",
"The original questionnaire consists of 120 statements, 25 statements for each dimension that takes about 45 minutes to finish.",
"For the online demonstration purposes, we only used one statement per each dimension where the whole conversational interaction takes less than 5 minutes to complete.",
"Each question is further modified to fit the conversation setup in the conference environment (see tab:questions).",
"Table: The questions used for the personality dimensions.The answers to these questions are then evaluated based on its polarity (i.e., positive, neutral, negative) and mapped onto two-factor dimensions for personality adjectives.",
"The mapping model we used in this paper is the Abridged Big Five Circumplex Model (AB5C), where the facets of the Big Five dimensions are mapped as combinations of two factors [10].",
"AB5C mapping includes descriptive personality terms for each of the 90 resulting combinations, where the most distinctive trait of an individual is used to select the column, and the next most distinctive trait selects the row.",
"The traits could be either positive or negative.",
"The mapping from our Big-5 traits to our Generative AI portrait styles was given to art experts in our research group, who independently matched the styles into the Big-5 categories and came to an agreement.",
"We planned a study and mechanism to refine and verify these results using both an art and psychology students.",
"However, given the compressed timescale of the demonstration, we were not able to schedule for the demo.",
"We are looking to complete such a study in the future."
],
[
"Empathic Conversational Avatar",
"The initial point of interaction for our system is the empathic conversational agent, M-Path [20], which was developed using a framework that is based on a computational model of empathy [21].",
"M-Path is implemented in an embodied human-like avatar that is capable of initiating and maintaining an emotional conversation, based on the predetermined goal of the dialogueThe conversational system is shared for open-source usage and can be reached at https://github.com/onyalcin/M-PATH.",
"The interaction scenario includes a face-to-face conversation with a human interaction partner, similar to a video-conference setting with audio-visual input and output modalities.",
"The agent processes the real-time inputs from the interaction partner in terms of their affective and linguistic properties, to be able to generate empathic verbal and non-verbal response behavior (see fig:mpath).",
"The overall goal of the dialogue is to complete the modified Big-5 questionnaire, to be able to assign a personality category to the conversation partner and send this information to be further processed in the generative art system.",
"Figure: Our Empathic Conversational Agent system and example captured avatar behavior.In the following sections, we will give a detailed description of the empathic conversational system based on its input processing, decision making and output generation capabilities.",
"These mechanisms are controlled by three distinct modules within the system: perceptual module, behavior controller and behavior manager."
],
[
"Perceptual Module",
"M-Path system processes the participants' input in its perceptual module.",
"This module gathers the audio and video input signals via a microphone and a video camera when the conversation partner is speaking.",
"During the demo session, this process was triggered with the aid of the push-to-talk system, where users provide their input via pressing and holding a button.",
"When the user starts speaking using this mechanism, M-Path enters to the “listening” state.",
"During this state, the speech and facial expressions are processed in real-time for speech and emotional understanding.",
"The video input is processed in the facial emotion recognition module, with an OpenCV face-recognition algorithm [3] to detect the face.",
"After detecting the face, the emotions are recognized in 6 basic emotion categories (i.e., anger, disgust, fear, joy, sadness, surprise and contempt), which is categorized by a CNN model trained over CK+ Dataset [13].",
"The speech input from the microphone is first sent to the speech-to-text module, which uses Google Speech-to-Text (STT) service to get streaming speech recognition [9].",
"The sentiment analysis component uses the text received from the STT service to evaluate the response in terms of polarity (positive-neutral-negative) value of the text.",
"In the live-demo, we used the SO-CAL Sentiment Analyzer [18], which was re-trained over the NRC-Canada lexicon [16].",
"The text of the speech is later sent to the decision-making component to generate conversational responses.",
"This processing continues until the speech of the conversation partner is finished, which concludes the listening state.",
"This processed information is then sent to the decision-making component as the agent enters the “thinking” state."
],
[
"Behavior Controller",
"The behavior controller module is responsible for generating empathic and goal-directed verbal and non-verbal responses during all the cycles of the conversation: listening, thinking, and speaking.",
"This is achieved by processing the user response and emotion information, both of which were perceived during the listening act.",
"The conversation cycle starts with the user’s initiation with a greeting and ends after the agent receives satisfactory responses to the questions of the personality survey.",
"The listening, thinking and speaking state of the agent is looped in sequence until a successful categorization of the user is reached.",
"During the listening stage, the agent shows non-verbal affect matching response and backchanneling behavior.",
"Affect matching is created with a facial expression that matches the user's facial expressions in real-time, which are chosen by the processes of empathy mechanisms.",
"Backchanneling is created with nodding behavior that is triggered by the detection of the pauses while conversing with the user.",
"Both of these behaviors are combined to generate natural and empathic listening behavior.",
"Previous work showed that the empathic responses generated during both the listening and speaking stages were indeed increasing the perception of the agent as empathic [22].",
"After the conversation with the participant is concluded, the final result received from the STT engine is passed to the Dialogue Manager (DM) along with the overall sentiment of the user, and ultimately sent to the Empathy Mechanisms (EM) component.",
"The goal of the DM is to complete the modified Big-5 personality questionnaire in order to assign a personality category.",
"The goal of the EM is to make sure that the DM generates empathic responses while reaching its own goal.",
"DM collects the appropriate emotional response decided by the EM to create an emotionally appropriate verbal reaction to the user's response, which is followed by a survey-related coping response, and finally the next survey question.",
"Our system uses the scikit-learn library [17] in Python for the TF-IDF vectorizer model, while using NLTK Lemmatizer [12].",
"The second model is generated by fine-tuning the pre-trained language representation model BERT [5] for the classification of user responses based on sentiment and the Big-5 questionnaire answers.",
"The answers to the Big-5 questionnaire are collected to select the most dominant personality dimensions of the user based on the polarity and their assigned probability values.",
"The Big-5 mapping mentioned in the sec:big5 is then used to select a category for the user along with its adjectives.",
"This categorization is then sent to the generative art cycle to create a personalized portrait of the user.",
"After each appropriate response is generated by the dialogue manager component, it is then sent to the behavior manager to be performed by the embodied conversational agent during the speaking state."
],
[
"Behavior Manager",
"In order to create natural conversational behavior, M-Path continuously generates non-verbal and/or verbal behaviors in all states of the dialogue.",
"A combination of facial expressions, body gestures, head gestures, posture and lip movements are synchronized with the speech of the agent and sent as a BML (Behavior Markup Language) message to the Smartbody character animation platform [19] to display the generated behaviors.",
"The stylistic rendering of the portraits is generated by the generative art component of our system.",
"Here, the portrait of the sitter goes through three main phases of processing (see fig:subfigex2).",
"In the first phase, the original portrait fig:orig of the sitter image is pre-processed by using the first AI-processing tool that segments the background from the foreground [2], which will be later used to stylized the portrait in a focused way.",
"Then, the light and color balance of the face is achieved to create a Rembrandt lighting effect, where one side of the face is dramatically shown fig:pro1.",
"Figure: Process flow of our Generative AI Portraiture system from raw source to final portrait.The next phase uses this image and the personality category as inputs to our modified-Deep Dream (mDD) system with successive passes on the image to create the baseline style fig:pro2.",
"While most DD systems use pre-trained networks with object recognition data such as ImageNet, we have implemented our own modified DD system (mDD) and train new models with artistic paintings and drawings as training data.",
"We now have amassed a dataset of 160,000 labeled and categorized paintings from 3000 artists for a total size of 67 gigabytes of artistic visual data (one of the largest in an AI research group).",
"However, we have discovered that even with such a large dataset, most artists make under 200 paintings in their lifetime (except Picasso), which might not be rigorous and large enough for an advanced CNN training for art styles.",
"In order to overcome this issue, we developed a method that we call “hierarchical tight style and tile” [8].",
"Figure: Example styles created by the system that maps a variety of personality categories shown with the adjectives.In the last phase, the source image created from the previous phase is further refined using the personality category, where our ePainterly system uses a combination of Deep Style techniques as a surface texture manipulator, as well as a series of Non-Photorealistic Rendering (NPR) techniques such as particle systems, color palette manipulation and stroke engine techniques fig:pro3.",
"This iterative process refines and completes the finished portrait style, and the final result is displayed in an online gallery system for viewing (see fig:pro4).",
"Our ePainterly module is an extension of our former painting system Painterly [6] that models the cognitive processes of artists based on years of research in this area.",
"The NPR subclass of stroke-based rendering is used as the final part of our ePainterly process to realize the internal mDD models with stroke-based output.",
"As it can be seen in fig:subfigex2, the aesthetic advantages of this additional step include reducing noise artifacts from the generated mDD output, cohesive stroke-based clustering, and a better distributed color space.",
"The Empathic AI Painter was introduced during the demo session of the NeurIPS 2019 Conference in Vancouver, Canada.",
"A total of 42 participants assigned for testing the system, where 26 of them completed both the portrait-taking and interaction scenario with the system during the 3-hour demonstration session.",
"Each conversational interaction with the M-Path system lasted about 5 minutes.",
"After the demonstration, we evaluated the performance of M-Path interaction individually.",
"On average, 84.72% of the speech input from the sitter was recognized correctly, where 82.51% of them had a correct categorization as answers to the personality questionnaire.",
"The 26 participants with a complete interaction showed 17 distinct personality categories.",
"A large number of participants contacted us to receive high-resolution versions of their finished portraits.",
"Anecdotally, we heard from the expert conference crowd that the portraits being generated seemed to depict the participants' perceived personality quite well, although we intend to validate this feedback in future studies.",
"We believe our approach has the potential to create better AI generative art systems that can resonate with human traits, which is closer to the creative process of human artists.",
"We have begun the next phase of this project in three major areas: 1) move beyond the Big-5 model to include affective models for portrait art depiction and evaluation, 2) plan user studies to determine aspects of cognitive reception of portraits and to validate different user personality models, 3) incorporate our telerobot system to automate the source photo taking and other aspects of our process that have current human intervention."
]
] | 2005.14223 |
[
[
"Attention: to Better Stand on the Shoulders of Giants"
],
[
"Abstract Science of science (SciSci) is an emerging discipline wherein science is used to study the structure and evolution of science itself using large data sets.",
"The increasing availability of digital data on scholarly outcomes offers unprecedented opportunities to explore SciSci.",
"In the progress of science, the previously discovered knowledge principally inspires new scientific ideas, and citation is a reasonably good reflection of this cumulative nature of scientific research.",
"The researches that choose potentially influential references will have a lead over the emerging publications.",
"Although the peer review process is the mainly reliable way of predicting a paper's future impact, the ability to foresee the lasting impact based on citation records is increasingly essential in the scientific impact analysis in the era of big data.",
"This paper develops an attention mechanism for the long-term scientific impact prediction and validates the method based on a real large-scale citation data set.",
"The results break conventional thinking.",
"Instead of accurately simulating the original power-law distribution, emphasizing the limited attention can better stand on the shoulders of giants."
],
[
"Introduction",
"The increasing availability of digital data on scholarly outcomes offers unprecedented opportunities to explore the science of science (SciSci) [7].",
"Based on the empirical analysis of big data, SciSci provides a quantitative understanding of scientific discovery, creativity, and practice.",
"It discovers that the previously discovered knowledge mainly inspires new scientific ideas, and citation is a relatively good reflection of this cumulative nature of scientific research.",
"Citation count, which has been used to evaluate the quality and influence of scientific work for a long time, stands out from many quantification measure metrics of scientific impact.",
"With the rapid evolution of scientific research, there is a huge volume of literature published every year, and this situation is expected to remain within the foreseeable future.",
"Fig.",
"REF shows the statistics on AMiner [21], which is a large literature database in Computer Science.",
"Fig.",
"REF visualizes the explosive increase on the volume of publications in the past years from 1990 to 2015.",
"It shows that the literature quantity assumes the exponential order to grow.",
"Useful scientific research requires reviewing the previous researches.",
"It is not wise, nor possible, for researchers to track all existing related work due to the enormous volume of the existing publications.",
"In general, researchers follow or cite merely a small proportion of high-quality publications.",
"SciSci provides several quantification methods for scientific impact measurement in article-level, author-level, and journal-level.",
"Much SciSci work has been done on the evaluation metrics for the quality and influence of scientific work, including citation count, h-index [9], and impact factor [8].",
"One of the most basic quantification measure metrics of scientific impact is citation count.",
"It measures the number of received citations for an article.",
"Many other essential evaluation criteria of authors (e.g., h-index) and journals (e.g., Impact Factor) are calculated based on citation count.",
"Figure: The citation Distribution.A lot of SciSci researchers have focused on the characterization of scientific impact, such as the universal citation distributions [17], the characteristics of citation networks [10], [15], [11], and the growth pattern of scientific impact [6].",
"The results reveal the regularity of scientific progress that a few research papers attract the vast majority of citations [3], long-distance interdisciplinarity leads to higher scientific impact [12], [27].",
"Fig.",
"REF illustrates the citation distribution (the number of papers vs. citation counts) of about two million papers in AMiner.",
"It is natural to find that not all publications attract equal attention in academia.",
"A few research papers accumulate the vast majority of citations, and most of the other papers attract only a few citations [3].",
"The citation distribution follows the power-law distribution.",
"A small number of scholarly outcomes are more likely to attract scientists' attention than the others accounting for a vast majority.",
"For the ever-growing literature quantity, it is significative to forecast which paper is more likely to attract more attention in academia.",
"The fact is that the current citation count and the derived metrics can only capture the past accomplishment.",
"They lack the predictive power to quantify future impact [1].",
"Predicting an individual paper's citation count over time is significant, but (arguably) very difficult.",
"To predict the citation count of individual items within a complex evolving system, current models are falling into two main paradigms.",
"One formulates the citation count over time as time series, and then makes predictions by either exploiting temporal correlations [20], or fitting these time series with certain classes of designed functions [14], [2], including the regression models [26], the counting process [22], the point process, the Poisson process [23], Reinforced Poisson Process (RPP) [18], self-excited Hawkes Process [16], RPP with self-excited Hawkes Process [24].",
"The designed functions consider various factors.",
"The other prevalent line utilize Deep Neural Network (DNN) based models to solve the scientific impact prediction problem.",
"Recently, Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN), have received considerable attention from both the academia and the industry.",
"RNN has been proven to perform particularly well on temporal data series [19].",
"Due to the vanishing gradient problem, RNN always fails to handle the temporal contingencies present in the input/output sequences spanning long intervals [4].",
"The networks with loops in RNN allow information to persist in a long time.",
"Long short-term memory (LSTM) is proven to be capable of learning long-term dependencies.",
"RNN with LSTM units performs rather well in handling long-term temporal data series [25].",
"All the existing methods try to tune the citation distribution exactly as the power-law distribution.",
"However, this paper argues that the effectiveness of quantifying long-term scientific impact is fundamentally limited in this line of thinking.",
"This paper proposes to put more attention on some specific items.",
"The authors validate the proposed line of thinking on a real large-scale citation dataset.",
"Extensive experiment results demonstrate that the proposed method possesses remarkable power at predicting the long-term scientific citation.",
"The most important contribution is that this paper is the first to change the line of thinking in quantifying the long-term scientific impact.",
"Instead of simulating the power-law distribution, researchers need to pay more attention to the limited attention to better stand on the shoulders of giants."
],
[
"Problem Formulation",
"The basic evaluation metric for scientific impact is citation count.",
"The received citation count of an individual paper $d$ during time period $[0, T]$ is characterized by a time-stamped sequence $\\lbrace n_{d}^{t} \\rbrace _{t=0}^{T}$ , where $n_{d}^{t}$ represents the number of citation counts received by paper $d$ at time $t$ , $n_{d}^{t}$ is an integer greater than or equal to zero.",
"In the context of giving the historical citation records, the goal is to model the future citation count and predict it over an arbitrary time.",
"Scientific Impact.",
"Given the literature corpus $D$ , $\\mathrm {card}(D)$ $=M$ , the scientific impact of a literature article $d\\in D$ at time $t$ is defined as its citation counts $n_{d}^{t}$ : $\\begin{aligned}citing_d^t &= \\lbrace \\tilde{d} \\in D, \\tilde{d}\\ne d:\\tilde{d}^t~cites~d\\rbrace ,\\\\n_{d}^{t} &= \\mathrm {card}(citing_d^t).\\end{aligned}$ The underlying assumption of the citation count here is the accumulated citations, which make it possible to quantify citations for different items at different times.",
"The long-term scientific impact of individual item $d$ can be formalized as the following time series $\\lbrace n_{d}^{0},\\cdots , n_{d}^{t},\\cdots , n_{d}^{T}\\rbrace $ .",
"Without loss of generality, the number of accumulated citation count increase over time.",
"And then, we have $0=n_{d}^{0}\\le \\cdots \\le n_{d}^{t}\\le \\cdots \\le n_{d}^{T}=N_d$ .",
"The scientific impact prediction problem can be formalized as follows.",
"Input: For each paper $d$ , the input is $\\lbrace (x_{d}^{0},n_{d}^{0}),\\cdots ,$ $(x_{d}^{t},n_{d}^{t}),\\cdots \\rbrace $ $\\in \\mathbb {N}^K \\times \\mathbb {N}$ , where $\\vec{X}=\\lbrace x_{d}^{0},\\cdots ,x_{d}^{t},\\cdots \\rbrace $ , and $x_{d}^{t}$ is expressed as a $K$ -dimensional feature vector, and $n_{d}^{t}$ is the citation counts of paper $d$ at time $t$ .",
"Learning: The goal of citation count prediction is to learn a predictive function $f$ ($\\mathbb {N}^K \\rightarrow \\mathbb {N}$ ) to predict the citation counts of an article $d$ after a given time period $t$ .",
"Formally, we have $f(d|\\vec{X}, t)\\rightarrow \\hat{n}_d^{t},$ where $\\hat{n}_d^{t}$ is the predicted citation count and $n_{d}^{t}$ is the actual one.",
"Prediction: Based on the learned prediction function, we can predict the citation count of a paper for the next years, for example, the citation count of paper $d$ at time $t$ is given by $f(d|\\vec{X}, t)$ .",
"Figure: The attention model."
],
[
"Scientific Impact Prediction",
"As the most efficient scientific impact prediction method found so far, RNN has already achieved compelling performance in predicting the scientific impact.",
"This paper embeds the RNN with LSTM units as a baseline and then emphasize highly cited papers in the proposed attention mechanism.",
"Although many other fields have used the attention mechanism, the proposed method gives the new insight about long-term quantifying scientific impact.",
"Instead of adapting citation distribution to a power-law distribution, the findings in this paper provide a new line of thinking for the SciSci research."
],
[
"Deep Learning Attention Mechanism",
"Given a time-stamped sequence $\\lbrace n_{d}^{t} \\rbrace _{t=0}^{T}$ , a $K$ -dimensional feature vector $\\vec{X}=\\lbrace x_{d}^{0},\\cdots ,x_{d}^{t},\\cdots ,x_{d}^{T}\\rbrace $ needs to be designed as input.",
"The input space of every item with popularity records $\\lbrace (x^{0},n^{0}),$ $\\cdots ,(x^{t},n^{t}),\\cdots ,(x^{T},n^{T})\\rbrace $ reflects the intrinsic quality of the item.",
"Fig.",
"REF gives an overview of the model architecture.",
"There are two key components in the architecture: the RNN with LSTM units and the attention model.",
"As illustrated in Fig.",
"REF , it arranges the LSTM units in the form of RNN with $L$ layers.",
"In the deep neural network, the parameter $L$ depends on the input scale.",
"RNN is famous for its popularity and well-known capability for efficient time series learning [25].",
"The LSTM units capture the long-range dependency in long-term scientific impact quantification.",
"The RNN with LSTM Units.",
"The LSTM units are arranged in the form of RNN, as illustrated in Fig.",
"REF .",
"There are four major components in a standard LSTM unit, including a memory cell, a forget gate $\\Gamma _f$ , an input gate $\\Gamma _i$ , and an output gate $\\Gamma _o$ .",
"The gates are responsible for information processing and storage over arbitrary time intervals.",
"Usually, the outputs of these gates are between 0 and 1.",
"A new study gives suggestions to push the output values of the gates towards 0 or 1.",
"By doing so, the gates are mostly open or closed, instead of in a middle state [13].",
"This paper arranges the LSTM units in the form of RNN.",
"In this way, introducing the memory cell will solve the vanishing gradient problem.",
"Thus, it can store information for either short or long periods in the LSTM unit.",
"Intuitively, the input gate controls the extent to which a new value flows into the memory cell.",
"A function of the inputs passes through the input gate and is added to the cell state to update it.",
"The following formula for the input gate is used: $\\Gamma _{i}^{t}=\\sigma \\left( W_i\\left[ h^{t-1},x^{t} \\right] +b_{i}\\right),$ where matrix $W_i$ collects the weights of the input and recurrent connections.",
"The symbol $\\sigma $ represents the Sigmoid function.",
"The values of the vector $\\Gamma _{i}^{t}$ are between 0 and 1.",
"If one of the values of $\\Gamma _{i}^{t}$ is 0 (or close to 0), it means that this input gate is closed and no new information is allowed into the memory cell at time $t$ .",
"If one of the values is 1, the input gate is open for new coming value at time $t$ .",
"Otherwise, the gate is in the state of half-open half-clearance.",
"The forget gate controls the extent to which a value remains in the memory cell.",
"It provides a way to get rid of the previously stored memory value.",
"Here is the formulation of the forget gate: $\\Gamma _{f}^{t}=\\sigma \\left( W_f\\left[ h^{t-1},x^{t} \\right] +b_{f}\\right),$ where $W_f$ is the weight matrix that governs the behavior of the forget gate.",
"Similar to $\\Gamma _{i}^{t}$ , $\\Gamma _{f}^{t}$ is also a vector of values between 0 and 1.",
"If one of the values of $\\Gamma _{f}^{t}$ is 0 (or close to 0), it means that the memory cell should remove that piece of information in the corresponding component in the cell.",
"If one of the values is 1, the corresponding information will be kept.",
"Remembering information for long periods of time is practically the default behavior of LSTM.",
"The long-term accumulative influence is formulated as follows: $c^{t}=\\Gamma _{f}^{t}\\ast c^{t-1}+\\Gamma _{i}^{t}\\ast \\tilde{c}^{t},$ where $\\ast $ denotes the Hadamard product (the element-wise multiplication of matrices), $\\tilde{c}^{t}$ is calculated as follows: $\\tilde{c}^{t}=\\tanh \\left( W_c\\left[ h^{t-1},x^{t} \\right] +b_{c}\\right).$ That is, the information in memory cell consists of two parts: the retained old information $\\Gamma _{f}^{t}\\ast c^{t-1}$ (controlled by the forget gate), and the new coming information $\\Gamma _{i}^{t}\\ast \\tilde{c}^{t}$ (controlled by the input gate).",
"The output gate controls the extent to which the value in the cell is used to compute the output activation of the LSTM unit.",
"The following output function is used: $\\Gamma _{o}^{t}=\\sigma \\left( W_o\\left[ h^{t-1},x^{t} \\right] +b_{o}\\right).$ The weight matrices and bias vector parameters are needed to be learned during training.",
"This paper updates the current working state as the following formula: $h^{t}=\\Gamma _{o}^{t}\\ast \\tanh \\left( c^{t}\\right).$ The items stored in the current working state have an advantage in reading over those stored in long-term memory.",
"In the time series modeling of scientific impact, the recent items stored in the short-term working state have an advantage over those stored in the long-term memory.",
"The next step introduces the attention mechanism based on $h^{t}$ .",
"The Attention Model.",
"The artificial attention mechanism, inspired by the attention behavior in neuroscience, has been applied in deep learning for speech recognition, translation, and visual identification of object.Broadly, attention mechanisms are components of prediction systems that allow the system to focus on different subsets of the input sequentially.",
"It aims to capture the critical points and focuses on the relevant parts more than the remote parts as a human does.",
"More specifically, the content-based attention generates attention distribution.",
"Only part of a subset of the input information is focused.",
"The attention function needs to be differentiable, so that everywhere of the input is focused, just to different extents.",
"The deep learning attention mechanism used in this paper works as follows: given an input $\\vec{X}=\\lbrace x_{d}^{0},\\cdots ,x_{d}^{t},\\cdots , x_{d}^{T}\\rbrace $ , the aforementioned LSTM units generate $\\vec{h}=\\lbrace h_1, \\cdots , h_t, \\cdots , h_T\\rbrace $ to represent the hidden patterns of the input.",
"The output is the summary of the $h_t$ focusing on information linked to the input.",
"In this formulation, attention produces a fixed-length embedding of the input sequence by computing an adaptive weighted average of the state sequence $\\vec{h}$ .",
"The graphical representation of the attention model is shown in Fig.",
"REF .",
"The input $\\vec{X}$ and the hidden layer $\\vec{h}$ of LSTM network (a RNN composed of LSTM units) are the input of the attention model.",
"Then, it computes the following formula: $a^{t}=\\tanh \\left( W_a\\left[ x^{t},h^{t} \\right]\\right),$ where $W_a$ is the weight matrix.",
"An important remark here is that each $a^t$ is computed independently without looking at the other $x^{t^{\\prime }}$ for $t^{\\prime }\\ne t$ .",
"Then, each $a^t$ is linked to a Softmax layer, which function is given by: $\\alpha ^t=\\frac{e^{a^t}}{\\sum _{t}e^{a^t}}, \\text{for}~t=1,\\cdots ,T$ where $\\sum _{t}\\alpha ^t=1$ , the $\\alpha ^t$ is the softmax of the $a^t$ projected on a learned direction.",
"The output is a weighted arithmetic mean of the input, and the weights reflects the relevance of $\\vec{h}$ and the input.",
"It is calculated as the following formula: $O=\\sum _{t}\\alpha ^t x_t.$ Finally, the popularity of item $d$ at time $t$ is given by the prediction $f(d|\\vec{X}, t)=O$ .",
"Table: The performance of various models on the data set."
],
[
"Key Factor in Quantifying Long-term Impact",
"As widely acknowledged, the citation distribution follows the power-law distribution.",
"This finding leads the way of research in this domain.",
"Researchers try to simulate the citation distribution as the power-law distribution.",
"This paper changes the line of thinking.",
"Although the number of research papers has exploded, the reading time of scientists has not.",
"The attention shifts toward the top 1% over time [3].",
"Even though the citation distribution follows the power-law distribution, attention is also vital in quantifying the long-term scientific impact.",
"In the fact of limited attention, Matthew effect dominates in quantifying the long-term scientific impact.",
"The experiments will confirm it.",
"The citation count captures the inherent differences between papers, accounting for the perceived novelty and importance of a paper.",
"The “rich-get-richer\" phenomenon summarizes the Matthew effect of accumulated advantage, i.e., previously accumulated attention triggers more subsequent attentions [5].",
"It is in fact that the highly popular items are more visible and more likely to be viewed than others.",
"The proposed model emphasizes highly cited papers under limited attention.",
"The memory cell in the LSTM unit considers the long-term dependencies.",
"As shown in Eq.",
"(REF ), previously accumulated attention stored in the long-term memory triggers more subsequent attention.",
"What is more, the attention model, which focuses on the most popular part of the time series as Eq.",
"(REF ) does, also emphasizes the Matthew effect."
],
[
"Experiments",
"This section demonstrates the effectiveness of putting particular emphasis on the vital factor in quantifying the long-term scientific impact."
],
[
"Dataset",
"The authors extract the data from an academic search and mining platform called AMiner and construct a real large-scale scholarly datasethttps://www.aminer.cn/data.",
"The full graph of citation network contained in this dataset has about 2 million vertices (papers) and 8 million edges (citations).",
"In detail, the dataset is composed of $2,092,356$ digitalized papers spanning from 1936 to 2016 (for more than 80 years), and $8,024,869$ citations between them.",
"By convention, the authors eliminate those papers with less than 5 citations during the first 5 years after publication and only retain the remaining papers as the training data.",
"As a result, $143,902$ papers published in 1956 to 2015 are retained."
],
[
"Baseline Models and Evaluation Metrics",
"To compare the predictive performance of the proposed attention model against other models, we introduce several published models that have been used to predict scientific impact.",
"Specifically, the comparison methods in the experiments are LR, CART, SVR (the three basic machine learning methods used in [26]), RPP [23], [18], and RNN [25].",
"The advantage of deep learning is the utilization of various features.",
"For the sake of fairness, the authors only use the citation count records and the same feature used in [23], [18].",
"This paper uses two basic metrics for scientific impact evaluation: Mean Absolute Percentage Error (MAPE) and Accuracy (ACC).",
"Let $n_{d}^{t}$ be the observed citations of paper $d$ up to time $t$ , and $\\hat{n}_{d}^{t}$ be the predicted one.",
"The MAPE measures the average deviation between the predicted and observed citations over all papers.",
"For a dataset of $M$ papers, the MAPE is given by: $\\mathrm {MAPE} = \\frac{1}{M}\\sum _{d=1}^{M}\\left| \\frac{\\hat{n}_{d}^{t}-n_{d}^{t}}{n_{d}^{t}} \\right|.$ ACC measures the fraction of papers correctly predicted under a given error tolerance $\\epsilon $ .",
"Specifically, the accuracy of citation prediction over $M$ papers is defined as: $\\mathrm {ACC} = \\frac{1}{M}\\sum _{d=1}^{M}\\mathrm {I}\\left[\\left| \\frac{\\hat{n}_{d}^{t}-n_{d}^{t}}{n_{d}^{t}} \\right|\\le \\epsilon \\right],$ where $\\mathrm {I}[\\theta ]$ is an indicator function which returns 1 if the statement $\\theta $ is true, otherwise returns 0.",
"We find that our method always outperforms regardless the value of $\\epsilon $ .",
"In this paper, we set $\\epsilon = 0.3$ ."
],
[
"Model Setting",
"The experiment results show that the longer the duration of the training set, the better the long-term prediction performance.",
"This paper sets the training period as 5 years and then predict the citation counts for each paper from the $1^\\mathrm {st}$ to $5^\\mathrm {th}$ after the training period.",
"For example, $t=1$ means that the first observation year after the training period.",
"In the experiments, the features with positive contributions are the citation history, the h-index of the paper author, and the level of the publication journal.",
"For the convenience of performance comparison, the input feature used here is the citation history for every sub-window of length 10 years.",
"The value of the parameter $L$ is 2.",
"The loss function used here is MAPE.",
"Adadelta is the gradient descent optimization algorithm.",
"The attention layer is fully connected and uses tanh activation.",
"Figure: ATT-A-LT (DLAM)."
],
[
"Results",
"As shown in the Table.",
"REF , the proposed model exhibits the best performance in terms of ACC in all the situations of $t=1$ , 2, 3, 4, and 5.",
"It means that the DLAM consistently achieves the higher accuracy than other models across different observation time.",
"What is more, the proposed model also exhibits the best performance in terms of MAPE in all the situations mentioned above.",
"That is, the proposed model achieves higher accuracy and lowers error rates simultaneously.",
"In the experiments, all the models used for comparison achieve acceptable low error rates, except RPP.",
"RPP can avoid this problem with prior [18], which incorporates conjugate prior for the fitness parameter.",
"However, the RPP with prior does not improve the ACC performance.",
"Overall, the proposed model also outperforms than RPP with prior.",
"Compared to the other methods in terms of ACC and MAPE, the proposed model increases with the number of years after the training period.",
"Compare to RNN (the most efficient method certified in recent works), the proposed model achieves a few performance improvements about $1.65\\%$ in terms of MAPE, and about $2.13\\%$ in terms of ACC, when $t=1$ .",
"However, in the situation of $t=5$ , the proposed model achieves significant performance improvement about $24.31\\%$ in terms of MAPE, and about $16.7\\%$ in terms of ACC.",
"In other words, the proposed model shows much superiority than other models in scientific impact prediction, especially in the long-term situation."
],
[
"Further Exploration",
" Effectiveness of the attention mechanism.",
"The authors remove the attention module of the proposed model to verify the effectiveness of the attention mechanism.",
"The remainder is RNN with LSTM units (labeled as LT-CCP), which is proven to be useful in long-term citation count prediction.",
"In the next step, we add the attention mechanism in two different ways.",
"Firstly, we add the attention module before the RNN module, which is labeled as ATT-B-LT (attention before LT-CCP).",
"In a second way, we add the attention module after the RNN module, which is labeled as ATT-A-LT (attention after LT-CCP).",
"As shown in Fig.",
"REF and Fig.",
"REF , the ACC is increased, and the corresponding MAPE is decreased.",
"Both ATT-B-LT and ATT-A-LT perform better than LT-CCP in terms of MAPE and ACC.",
"Introducing the attention module improves the ability of scientific impact prediction.",
"The effectiveness of the attention mechanism is verified.",
"In addition, we can see that the ATT-A-LT performs better than ATT-B-LT.",
"It indicates that the deep learning model can learn the implicit features underlying the citation records, which provide a further boost in the performance.",
"Analysis of the citation distribution.",
"We illustrate the actual and the predicted citations distribution of LT-CCP (RNN with LSTM), ATT-B-LT and ATT-A-LT (DLAM) when $t=5$ in Fig.",
"REF , Fig.",
"REF and Fig.",
"REF respectively.",
"The LT-CCP (RNN with LSTM) illustrated in Fig.",
"REF shows the best simulation of the power-law distribution.",
"But the ATT-B-LT shown in Fig.",
"REF and the ATT-A-LT (DLAM) shown in Fig.",
"REF present bad simulation of the power-law distribution.",
"The results show that LT-CCP (RNN with LSTM) matches very well with that of real citations, but the ATT-B-LT and the ATT-A-LT (DLAM) don't.",
"Usually, it is believed that more similar to the power-law distribution, the whole result is more better.",
"At first glance, it seems that LT-CCP (RNN with LSTM) performs the best.",
"However, the first thought is wrong.",
"As verified in Fig.",
"REF and Fig.",
"REF , the LT-CCP (RNN with LSTM) performs the worst.",
"In fact, the LT-CCP only has better fitting effect on the papers with little citation counts.",
"On the contrary, the ATT-B-LT and ATT-A-LT (DLAM) have better fitting effect on the highly cited papers.",
"The methods with attention mechanism achieve better overall performance.",
"It is more accordant with practical prediction requirements that a few papers occupy vast number of citations.",
"It further proves the effectiveness of the attention model.",
"The experimental results indicate that we need to change the fixed pattern of thinking in quantifying long-term scientific impact."
],
[
"Conclusion",
"Scientific impact evaluation is always a key point in decision making concerning with recruitment and funding in the scientific community.",
"Based on big data based empirical analysis, science of science provides quantitative understanding of the scientific impact.",
"In this paper, the authors introduce the attention mechanism in long-term scientific impact prediction, and verify its effectiveness.",
"More importantly, this paper provides us great insights in understanding the key factor in quantifying long-term scientific impact.",
"Usually, it is believed that the citation distribution is more similar to the power-law distribution, the whole result is more better.",
"However, the experimental results in the paper discredit this conclusion.",
"In the future research work, we need to change the fixed pattern of thinking in quantifying long-term scientific impact, and make better use of limited attention to better stand on the shoulders of giants."
],
[
"Acknowledgments",
"The work is supported by the National Natural Science Foundation of China (NSFC) under Grant No.",
"61806111 and No.",
"61806109, and NSFC for Distinguished Young Scholar under Grant No.",
"61825602."
]
] | 2005.14256 |
[
[
"Reduced order modeling of fluid flows: Machine learning, Kolmogorov\n barrier, closure modeling, and partitioning"
],
[
"Abstract In this paper, we put forth a long short-term memory (LSTM) nudging framework for the enhancement of reduced order models (ROMs) of fluid flows utilizing noisy measurements.",
"We build on the fact that in a realistic application, there are uncertainties in initial conditions, boundary conditions, model parameters, and/or field measurements.",
"Moreover, conventional nonlinear ROMs based on Galerkin projection (GROMs) suffer from imperfection and solution instabilities due to the modal truncation, especially for advection-dominated flows with slow decay in the Kolmogorov width.",
"In the presented LSTM-Nudge approach, we fuse forecasts from a combination of imperfect GROM and uncertain state estimates, with sparse Eulerian sensor measurements to provide more reliable predictions in a dynamical data assimilation framework.",
"We illustrate the idea with the viscous Burgers problem, as a benchmark test bed with quadratic nonlinearity and Laplacian dissipation.",
"We investigate the effects of measurements noise and state estimate uncertainty on the performance of the LSTM-Nudge behavior.",
"We also demonstrate that it can sufficiently handle different levels of temporal and spatial measurement sparsity.",
"This first step in our assessment of the proposed model shows that the LSTM nudging could represent a viable realtime predictive tool in emerging digital twin systems."
],
[
"Introduction",
"Reduced order modeling (ROM) is a family of protocols that aim at representing the system's dynamics of interest with minimal computational burden [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].",
"Standard approaches usually consist of two major steps; (1) tailor a low-order subspace, where the flow trajectory can be sufficiently approximated to live, (2) build a surrogate model to cheaply evolve this trajectory in time.",
"For the former, modal decomposition techniques have shown substantial success in extracting the physically important features and underlying patterns of the flow.",
"Examples include proper orthogonal decomposition (POD) [19], [20], [11], [21], [22], balanced proper orthogonal decomposition (BPOD) [23], [24], [25], [26], spectral proper orthogonal decomposition (SPOD) [27], [28], [12], [29], [30], and dynamic mode decomposition (DMD) [31], [32], [33], [34], [35], [36], [37], [38].",
"Of particular interest, POD has gained historical recognition in fluid dynamics community, representing a set of data with minimal number of basis functions or modes while preserving as much energy as possible [39], [20], [11].",
"In particular, POD generates a set of hierarchically arranged modes, sorted by their respective contribution to the total variance of information in the data.",
"In fluid flow applications, with velocity field data, this information corresponds to the flow's kinetic energy.",
"As a mathematical measure of the system's reducibility and the quality of a constructed (linear) subspace, the Kolmogorov $n$ -width [40] is a classical concept from approximation theory that quantifies the worst-case scenario error that might arise from the projection of solution trajectory onto an optimal subspace.",
"Mathematically, it is defined as follows [41], [42], [43], $d_n({\\cal {M}}) := \\inf _{{\\cal {S}}_n} \\sup _{q \\in {\\cal {M}}} \\inf _{w\\in {\\cal {S}}_n} \\Vert q - w \\Vert ,$ where ${\\cal {S}}_n$ is a linear $n$ -dimensional subspace, and ${\\cal {M}}$ is the solution manifold.",
"The first infimum is taken over all possible $n$ -dimensional subspaces, $q$ is a state on the solution manifold ${\\cal {M}}$ , while the last infimum sweeps all corresponding states that live in ${\\cal {S}}_n$ .",
"In other words, $d_n({\\cal {M}})$ estimates the largest error that might arise from approximating the solution manifold using the best-possible $n$ -dimensional linear subspace.",
"Assuming an orthogonal projection of $q$ onto ${\\cal {S}}_n$ is possible, then the previous relation reduces to $d_n({\\cal {M}}) := \\inf _{{\\cal {S}}_n} \\sup _{q \\in {\\cal {M}}} \\Vert q- \\Pi _{{\\cal {S}}_n} q \\Vert ,$ where $\\Pi _{{\\cal {S}}_n}$ is the orthogonal projector onto ${\\cal {S}}_n$ .",
"Using information about the decay rate of $d_n({\\cal {M}})$ with increasing $n$ , the system's reducibility and the quality of a reduced order approximation can be judged.",
"Unfortunately, most of the fluid systems of practical relevance exhibit a slow decay of the Kolmogorov width, hindering reasonable approximation of the flow dynamics using a linear subspace.",
"This has been denoted as the “Kolmogorov barrier” in ROM context.",
"Recently, efforts have been devoted to break-up or bypass this barrier by building more representative and concise subspaces.",
"This can achieved either by partitioning techniques with the aim of localizing the resulting basis functions [44], [45], [46], [47], [48], [49], [50], [51], [52], [53] and preventing modal deformation, or constructing nonlinear latent subspaces using auto-encoders [54], [55], [56], [57], [58], [59].",
"Building surrogate models to evolve on reduced manifolds has been traditionally categorized into two groups; physics-based and data-based.",
"Physics-based models rely on the governing equations from first principles, where the full order model (FOM) operators are projected (e.g., using Galerkin-type techniques) onto the reduced subspace to structure a reduced order model (ROM).",
"Those are favorable because of their interpretability and generalizability, as well as the existence of robust techniques for stability and uncertainty analysis.",
"However, those are usually expensive to represent for turbulent and advection-dominated flows, with a slow decay of the Kolmogorov $n$ -width, necessitating an increase in the number of modes (or degrees of freedom) to be retained in ROM.",
"Otherwise, modal truncation results in a Galerkin-based ROM (GROM) that might be eventually unstable.",
"In this regard, closure techniques have been introduced to stabilize GROMs and account for the effect of truncated modes on the retained modes' dynamics [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82].",
"On the other hand, data-based models solely depend on archival data (thus called nonintrusive) to learn the underlying relations that govern the dynamical evolution of the system.",
"Nonintrusive ROMs have benefited from the widespread machine learning (ML) tools to build stable and accurate models, compared to their GROM counterparts.",
"In particular, (deep) neural networks have been extensively utilized to emulate the dynamical evolution of ROMs [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93].",
"However, those often lack human interpretability and generalizability, and can even become prohibitively “data-hungry”.",
"More recently, there has been a momentum in the research community to establish hybrid frameworks, that exploit machine learning algorithms and abundant data streams along with physical models to maximize their benefits [94], [95], [96], [97], [98], [75], [76], [99], [100], [101], [102].",
"It has been shown that such hybridization can provide models that are superior to their individual components.",
"Likewise, physics-informed machine learning is also gaining tremendous popularity, using human-knowledge and physical intuition to constrain the neural network predictions [103], [104], [105], [106].",
"Along similar lines, in this paper, we propose a hybrid framework that blends live measurement streams with physical models in order to achieve better predictions.",
"Moreover, we suppose that both components (i.e., the physics-based model and data) are imperfect, thus avoiding biases in predictions.",
"The physics-based ROMs (e.g., GROMs) are inherently imperfect due to the modal truncation and intrinsic nonlinearity.",
"We also perturb the initial conditions to further mimic erroneous state estimates in practice.",
"Meanwhile, we realize that, more often than not, sensor signals are noisy.",
"Thus, we utilize recurrent neural networks, namely the long short-term memory (LSTM) variant, to combine the possibly defective model prediction with noisy measurements to “nudge” the model's solution towards the true states.",
"Nudging is a data assimilation (DA) technique, which is a well-established family of predictive tools in geosciences, especially numerical weather forecasts [107], [108], [109], [110], [111].",
"It works by relaxing the model state toward observations by adding correction (or nudging) terms, proportional to the difference between observations and model state, known as innovation in DA context (e.g., [112], [113]).",
"Usually, this proportionality is assumed to be linear, and the proportionality constants (or weights) are empirically tuned.",
"Here, we use a simplistic LSTM architecture to generalize this relation to consider nonlinear mappings among the innovation and nudging terms.",
"We apply the proposed framework (called LSTM-Nudge in the paper) for the reduced order modeling of the one-dimensional viscous Burgers equation as a starting benchmark problem for advection-dominated fluid flows with quadratic nonlinearity.",
"We test the performance of LSTM-Nudge with various levels of measurement noises, initial field perturbations, and sensors signals sparsity.",
"Therefore, the hybrid modeling approach presented in this paper illustrates a novel way of combining the best of both the physics-driven and the data-driven modeling approaches.",
"Despite being shown in the context of a relatively simple dynamical system, this approach would be ideal for accurate and realtime modeling of complex systems, and therefore can be considered as a viable enabler for emerging digital twin technologies [114], [115], [116]."
],
[
"Reduced Order Modeling",
"In this section, we present the reduced order formulations adopted in this study.",
"In particular, we utilize proper orthogonal decomposition (POD) as a data-driven tool to extract the flow's coherent structures and build a reduced order subspace that best approximates the flow fields of interest.",
"Then, we adopt a Galerkin approach to project the full order model operators onto that reduced space to build a “'physics-constrained” reduced order model."
],
[
"Governing equation",
"Here, we consider the one-dimensional (1D) viscous Burgers equation as a test bed.",
"It represents a simple form of the Navier-Stokes equations in 1D setting with similar quadratic nonlinear interactions and Laplacian dissipation.",
"It is therefore considered as a standard benchmark for the analysis of nonlinear advection-diffusion problems.",
"The evolution of the velocity field $u(x, t)$ , in a dimensionless form, is given by $\\dfrac{\\partial u}{\\partial t} + u \\dfrac{\\partial u}{\\partial x} = \\dfrac{1}{\\text{Re}} \\dfrac{\\partial ^2 u}{\\partial x^2}, $ where $\\text{Re}$ is the dimensionless Reynolds number, defined as the ratio of inertial effects to viscous effects.",
"In dimensionless form, the reciprocal of Reynolds number can be denoted as the dimensionless kinematic viscosity $\\nu $ .",
"Therefore, Eq.",
"REF can be rewritten as below, $\\dfrac{\\partial u}{\\partial t} + u \\dfrac{\\partial u}{\\partial x} = \\nu \\dfrac{\\partial ^2 u}{\\partial x^2}.",
"$"
],
[
"Proper orthogonal decomposition",
"The first step for building a projection-based reduced order model is to design a low-order subspace that is capable of capturing the essential features of the system of interest.",
"In the fluid dynamics community, proper orthogonal decomposition (POD) is one of the most popular techniques in this regard.",
"Starting from a collection of system's realizations (called snapshots), POD provides a systematic algorithm to construct a set of orthonormal basis functions (called POD modes) that best describes that collection of snapshot data (in the $\\ell _2$ sense).",
"More importantly, those bases are sorted based on their contributions to the system's total energy, making the modal selection a straightforward process.",
"This is a significant advantage compared to other modal decomposition techniques like dynamic mode decomposition, where further sorting and selection criterion has to be carefully defined [117], [118], [119], [120].",
"Usually, the method of snapshots [19] is followed in practice to perform POD efficiently and economically, especially for high dimensional systems.",
"However, we adopt the singular value decomposition (SVD) based approach here for the sake of simplicity and brevity of presentation.",
"Suppose we have a collection of $N$ system realizations, denoted as $ u(x_i,t_n)$ for $i=1,2,\\dots , M$ , and $n=1,2,\\dots , N$ , where $M$ is the number of spatial locations and $N$ is the number of snapshots.",
"Thus, we can build a snapshot matrix $\\mathbf {A} \\in \\mathbb {R}^{M \\times N}$ as follows, $\\mathbf {A} = \\begin{bmatrix}u(x_1,t_1) & u(x_1,t_2) & \\dots & u(x_1,t_{N}) \\\\u(x_2,t_1) & u(x_2,t_2) & \\dots & u(x_2,t_{N}) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\u(x_{M},t_1) & u(x_{M},t_2) & \\dots & u(x_{M},t_{N}) \\\\\\end{bmatrix}.$ Then, a thin (reduced) SVD is performed on $\\mathbf {A}$ in the following form, $ \\mathbf {A} = \\mathbf {U} \\mathbf {\\Sigma } \\mathbf {V}^T,$ where $\\mathbf {U} \\in \\mathbb {R}^{M \\times N}$ is a matrix with orthonormal columns, called the left singular vectors of $\\mathbf {A}$ and represent the spatial basis, while the columns of $\\mathbf {V} \\in \\mathbb {R}^{N \\times N}$ are the right singular vectors of $\\mathbf {A}$ , representing the temporal basis.",
"The singular values of $\\mathbf {A}$ are stored in descending order as the entries of the diagonal matrix $\\mathbf {\\Sigma } \\in \\mathbb {R}^{N \\times N}$ .",
"Thus, Eq.",
"REF can be expanded as, $\\mathbf {A} = \\begin{bmatrix}U_1(x_1) & U_2(x_1) & \\dots & U_{N}(x_1) \\\\U_1(x_2) & U_2(x_2) & \\dots & U_{N}(x_2) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\U_1(x_{M}) & U_2(x_{M}) & \\dots & U_{N}(x_{M})\\end{bmatrix}\\begin{bmatrix}\\sigma _1 & & & \\\\& \\sigma _2 & & \\\\& & \\ddots & \\\\& & & \\sigma _{N}\\end{bmatrix}\\begin{bmatrix}V_1(t_1) & V_2(t_1) & \\dots & V_{N}(t_1) \\\\V_1(t_2) & V_2(t_2) & \\dots & V_{N}(t_2) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\V_1(t_{N}) & V_2(t_{N}) & \\dots & V_{N}(t_{N})\\end{bmatrix}^{T},$ where $\\sigma _1 \\ge \\sigma _2 \\ge \\dots \\sigma _{N} \\ge 0$ .",
"For dimensionality reduction purposes, only the first $R$ columns of $\\mathbf {U}$ , the first $R$ columns of $\\mathbf {V}$ , and the upper-left $R\\times R$ sub-matrix of $\\mathbf {\\Sigma }$ are considered, corresponding to the largest $R$ singular values.",
"Specifically, the first $R$ columns of $\\mathbf {U}$ represent the most effective $R$ POD modes, denoted as $\\lbrace \\phi _k\\rbrace _{k=1}^{R}$ in the rest of the manuscript.",
"The velocity field $u(x,t)$ is thus approximated as a linear superposition of the contributions of the first $R$ modes, which can be mathematically expressed as $ u(x,t) = \\sum _{k=1}^{R} a_k(t) \\phi _k(x),$ where $\\phi _k(x)$ are the spatial modes, $a_k(t)$ are the time-dependent modal coefficients (also known as generalized coordinates), and $R$ is the number of retained modes in ROM approximation (i.e., ROM dimension)."
],
[
"Galerkin projection",
"After constructing a set of POD basis functions, an orthogonal Galerkin projection can be performed to obtain the Galerkin-based ROM (GROM).",
"To do so, the ROM approximation (Eq.",
"REF ) is substituted into the governing equation (Eq.",
"REF ).",
"Noting that the POD bases are only spatial functions (i.e., independent of time) and the modal coefficients are independent of space, we get the following, $\\left(\\sum _{i=1}^{R} \\dfrac{\\partial a_i}{\\partial t} \\phi _i \\right) + \\left(\\sum _{i=1}^{R} a_i \\phi _i \\right) \\left(\\sum _{i=1}^{R} a_i \\dfrac{\\partial \\phi _i }{\\partial x} \\right) = \\nu \\left(\\sum _{i=1}^{R} a_i \\dfrac{\\partial ^2 \\phi _i}{\\partial x^2} \\right).$ We note that the POD basis functions are orthonormal by construction as $\\langle \\phi _i ; \\phi _j \\rangle ={\\left\\lbrace \\begin{array}{ll}1 &\\quad \\text{if } i = j \\\\0 &\\quad \\text{otherwise,}\\end{array}\\right.",
"}$ where the angle parentheses $\\langle \\bullet ; \\bullet \\rangle $ stand for the standard inner product in Euclidean space (i.e., dot product).",
"Then, an inner product with an arbitrary basis function $\\phi _k$ can be conducted.",
"Utilizing the orthonormality property of the basis function to simplify ROM derivation, we get the following set of ordinary differential equations (ODEs) representing the tensorial form of GROM, $\\dfrac{\\text{d}a_k}{\\text{d}t} &= \\nu \\sum _{i=1}^{R} \\mathfrak {L}_{i,k} a_i + \\sum _{i=1}^{R} \\sum _{j=1}^{R} \\mathfrak {N}_{i,j,k} a_i a_j, $ where $\\mathfrak {L}$ and $\\mathfrak {N}$ are the matrix and tensor of predetermined model coefficients corresponding to linear and nonlinear terms, respectively.",
"Those are precomputed during an offline stage as $\\mathfrak {L}_{i,k} & = \\big \\langle \\dfrac{\\partial ^2 \\phi _i }{\\partial x^2} ; \\phi _k \\big \\rangle , \\\\\\mathfrak {N}_{i,j,k} &= \\big \\langle -\\phi _i \\dfrac{\\partial \\phi _j}{\\partial x}; \\phi _k \\big \\rangle .$ Equation REF can be rewritten in a compact form as $ \\dot{\\mathbf {a}} = \\mathbf {f}(\\mathbf {a}),$ where $\\mathbf {a} = [a_1, a_2, \\dots , a_R]^T$ , and the (continuous-time) model map $\\mathbf {f}$ is defined as follows, $\\mathbf {f} =\\begin{bmatrix}\\nu \\sum _{i=1}^{R} \\mathfrak {L}_{i,1} a_i + \\sum _{i=1}^{R} \\sum _{j=1}^{R} \\mathfrak {N}_{i,j,1} a_i a_j \\\\\\nu \\sum _{i=1}^{R} \\mathfrak {L}_{i,2} a_i + \\sum _{i=1}^{R} \\sum _{j=1}^{R} \\mathfrak {N}_{i,j,2} a_i a_j \\\\\\vdots \\\\\\nu \\sum _{i=1}^{R} \\mathfrak {L}_{i,R} a_i + \\sum _{i=1}^{R} \\sum _{j=1}^{R} \\mathfrak {N}_{i,j,R} a_i a_j\\end{bmatrix}.$ Alternatively, Eq.",
"REF can be used in a discrete-time version as $ \\mathbf {a}^{n+1} = \\mathbf {M}(\\mathbf {a}^n),$ where $\\mathbf {M}(\\cdot )$ is the discrete-time map obtained by any suitable temporal integration technique.",
"Here, we use the fourth-order Runge-Kutta (RK4) method as follows, $\\mathbf {a}^{n+1} &= \\mathbf {a}^n + \\dfrac{\\Delta t}{6} (\\mathbf {g}_1 + 2\\mathbf {g}_2 + 2\\mathbf {g}_3 + \\mathbf {g}_4),$ where $\\mathbf {g}_1 &= \\mathbf {f}(\\mathbf {a}^n), \\\\\\mathbf {g}_2 &= \\mathbf {f}(\\mathbf {a}^n + \\dfrac{\\Delta t}{2} \\cdot \\mathbf {g}_1), \\\\\\mathbf {g}_3 &= \\mathbf {f}(\\mathbf {a}^n + \\dfrac{\\Delta t}{2} \\cdot \\mathbf {g}_2), \\\\\\mathbf {g}_4 &= \\mathbf {f}(\\mathbf {a}^n + \\Delta t \\cdot \\mathbf {g}_3).$ Thus the discrete-time map defining the transition from time $t_n$ to time $t_{n+1}$ is written as $\\mathbf {M}(\\mathbf {a}^n) = \\mathbf {a}^n + \\dfrac{\\Delta t}{6} (\\mathbf {g}_1 + 2\\mathbf {g}_2 + 2\\mathbf {g}_3 + \\mathbf {g}_4).$ Due to the quadratic nonlinearity in the governing equation, and consequently the GROM, the online computational cost of solving Eq.",
"REF is $O(R^3)$ (i.e., it scales cubically with the number of retained modes).",
"Therefore, this has to be kept as low as possible for the feasible implementation of ROM in applications that require near realtime responses (e.g., active control).",
"However, this is often not an easy task for systems with slow decay of the Kolmogorov n-width.",
"Examples include advection-dominated flows with strong nonlinear interactions between wide range of modes.",
"Consequently, the resulting GROM is an intrinsically imperfect model.",
"That is even with the true initial conditions, and absence of numerical errors, the GROM might give inaccurate or false predictions.",
"Indeed, Carlberg et al.",
"[121] showed that GROM becomes unstable for long time intervals.",
"Moreover, in most realistic cases, proper specification of the initial state, boundary conditions, and/or model parameters is rarely attainable.",
"This uncertainty in problem definition, in conjunction with model imperfection, poses challenges for accurate predictions.",
"In this study, we put forth a nudging-based methodology that fuses prior model forecast (using imperfect initial condition specification and imperfect model) with the available Eulerian sensor measurements to provide a more accurate posterior prediction.",
"Relating our setting to realistic applications, we build our framework on the assumption that measurements are usually noisy and sparse both in space and time.",
"Nudging has a prestigious history in data assimilation, being a simple and unbiased approach [112].",
"The idea behind nudging is to penalize the dynamical model evolution with the discrepancy between the model's predictions and observations.",
"In other words, the forward model given in Eq.",
"REF is supplied with a nudging (or correction) term rewritten in the following form, $\\mathbf {a}^{n+1} = \\mathbf {M}(\\mathbf {a}^n) + \\mathbf {G}(\\mathbf {z}^{n+1}-h(\\mathbf {a}^{n+1})),$ where $\\mathbf {G}$ is called the nudging (gain) matrix and $\\mathbf {z}$ is the set of measurements (observations), while $h(\\cdot )$ is a mapping from model space to observation space.",
"For example, $h(\\cdot )$ can be a reconstruction map, from ROM space to FOM space.",
"In other words, $h(\\mathbf {a})$ represents the “model prediction of the measured quantity”, while $\\mathbf {z}$ is the “actual” observations.",
"Given the simplicity of Eq.",
"REF , the specification/definition of the gain matrix $\\mathbf {G}$ is not as straightforward [122], [123], [124], [125].",
"In the proposed framework, we utilize recurrent neural networks, namely the long short-term memory (LSTM) variant, to define a nudging map.",
"In particular, Eq.",
"REF implies that each component of $\\mathbf {a}^{n+1}$ (i.e., $a_1,a_2\\dots ,a_R$ ) is corrected using a linear superposition of the the components of $\\mathbf {z}^{n+1}-h(\\mathbf {a}^{n+1})$ , weighted by the gain matrix.",
"Here, we relax this linearity assumption and generalize it to a possibly nonlinear mapping $\\mathbf {C}(\\mathbf {a}, \\mathbf {z})$ as, $\\mathbf {a}^{n+1} = \\mathbf {M}(\\mathbf {a}^n) + \\mathbf {C}(\\mathbf {a}_b^{n+1}, \\mathbf {z}^{n+1}),$ where the map $\\mathbf {C}(\\mathbf {a}, \\mathbf {z})$ is learnt (or fit) using an LSTM neural network, and $\\mathbf {a}_b^{n+1}$ is the prior model prediction computed using imperfect model and/or imperfect initial conditions (called background in data assimilation terminology), defined as $\\mathbf {a}_b^{n+1} = \\mathbf {M}(\\mathbf {a}^n)$ .",
"Thus, Eq.",
"REF can be rewritten as follows, $\\mathbf {a}^{n+1} = \\mathbf {a}_b^{n+1} + \\mathbf {C}(\\mathbf {a}_b^{n+1}, \\mathbf {z}^{n+1}).$ In order to learn the map $\\mathbf {C}(\\mathbf {a}_b,\\mathbf {z})$ , we consider the case with an imperfect model, defective initial conditions, and noisy observations.",
"Moreover, we suppose sensors are sparse in space, and measurement signals are sparse in time too.",
"Specifically, we use sensors located at a few equally-spaced grid points, but a generalization to off-grid sensor placement is possible.",
"Also, we assume sensors send measurement signals every $\\tau $ time units.",
"To mimic sensor measurements and noisy initial conditions, we run a twin experiment as follows, Solve the FOM equation (i.e., Eq.",
"REF ) and sample true field data ($u_{true}(x,t_n)$ ) each $\\tau $ time units.",
"In other words, store $u_{true}(x,t_n)$ at $t_n\\in \\lbrace 0,\\tau ,2\\tau ,\\dots T\\rbrace $ where $T$ is the total (maximum) time and $\\tau $ is the time window over which measurements are collected.",
"Define erroneous initial field estimate as $u_{err} (x,t_n) = u_{true}(x,t_n) + \\epsilon _b$ , where $t_n\\in \\lbrace 0,\\tau ,2\\tau ,\\dots T-\\tau \\rbrace $ .",
"$\\epsilon _b$ stands for noise in initial state estimate, assumed as white Gaussian noise with zero mean and covariance matrix $B$ (i.e., $\\epsilon _b \\sim {\\cal {N}}(0,B)$ ).",
"Define sparse and noisy measurements as $\\mathbf {z} = u_{true}(x_{Obs},t_n) + \\epsilon _m $ , for $t_n\\in \\lbrace \\tau ,2\\tau ,\\dots T\\rbrace $ .",
"Similarly, $\\epsilon _m$ stands for the measurements noise, assumed to be white Gaussian noise with zero mean and covariance matrix $Q$ (i.e., $\\epsilon _m \\sim {\\cal {N}}(0,Q)$ ).",
"For LSTM training data, we project the erroneous field estimates (from Step 2) onto the POD basis functions to get the erroneous POD modal coefficients (i.e., $\\mathbf {a}_{err}(t_n)$ , for $t_n\\in \\lbrace 0,\\tau ,2\\tau ,\\dots T-\\tau \\rbrace $ .",
"Then, we integrate those erroneous coefficients for $\\tau $ time units to get the background prediction $\\mathbf {a}_b(t_n)$ , for $t_n\\in \\lbrace \\tau ,2\\tau ,\\dots T\\rbrace $ .",
"Then, we train the LSTM using $\\mathbf {a}_b(t_n)$ and $\\mathbf {z}(t_n)$ as inputs, and set the target as the correction $(\\mathbf {a}_{true}(t_n) - \\mathbf {a}_b(t_n))$ , for $t_n\\in \\lbrace \\tau ,2\\tau ,\\dots T\\rbrace $ .",
"The true modal coefficients ($\\mathbf {a}_{true}$ ) are obtained by projecting the true field data (from Step 1) onto the POD bases, where the projection is defined via the inner product as $a_k(t) = \\langle u(x,t) ; \\phi _k(x) \\rangle $ .",
"In order to enrich the training data set, Step 2 and Step 3 are repeated several times giving an ensemble of erroneous state estimates and noisy measurements at every time instant of interest.",
"Each member of those ensembles represents one training sample.",
"This also assists the LSTM network to handle wider range of noise.",
"We emphasize that the proposed LSTM-Nudge approach not only cures model imperfection (i.e., provides model closure and accounts for any missing physical processes) but also treats uncertainties in initial state estimates.",
"Moreover, the field measurements (i.e., the nudging data) are assumed to be sparse and noisy to mimic real-life situations."
],
[
"Results",
"We test the proposed methodology using the 1D Burgers problem introduced in Sec.",
"REF .",
"In particular, we consider a domain of dimensionless length of one, with a square wave as initial condition defined as, $u(x,0) = {\\left\\lbrace \\begin{array}{ll}1 , &\\quad \\text{if } 0 < x \\le 0.5 \\\\0 , &\\quad \\text{if } 0.5 < x \\le 1.0,\\end{array}\\right.",
"}$ with zero Dirichlet boundary conditions, $u(0,t) = u(1,t) = 0$ .",
"We solve Eq.",
"REF at $\\text{Re} = 10^4$ for $t \\in [0,1]$ .",
"For numerical computations, we use a family of fourth order compact schemes for spatial derivatives [126], and skew-symmetric formulation for the nonlinear term.",
"For the FOM simulation, we use a time step of $10^{-4}$ over a spatial grid of 4096, and for POD basis generation, we collect 100 snapshots (i.e., every 100 FOM time steps).",
"The temporal evolution of the 1D Burgers problem using the described setup is shown in Fig.",
"REF , illustrating the advection of the shock wave over time.",
"Figure: Evolution of the FOM velocity field for the 1D Burgers problem, characterized by a moving shock with initial square wave.For ROM computations, 6 modes are retained in the reduced order approximation (i.e., $R=6$ ) and a time step of $0.01$ is adopted for the temporal integration of GROM equations.",
"In order to implement the LSTM-Nudge approach, we begin at erroneous initial condition defined as $u_{err}(x,0) = u_{true}(x,0) + \\epsilon _b$ , where $u_{true}(x,0)$ is defined with Eq.",
"REF , and $\\epsilon _b$ is a white Gaussian noise with zero mean and covariance matrix $B$ .",
"For simplicity, we assume $B=\\sigma _b^2 \\mathbf {I}$ , where $\\sigma _b$ is the standard deviation in “background” estimate of the initial condition and $\\mathbf {I}$ is the identity matrix.",
"We note that this formulation implies that our estimates of the initial velocity field at different spatial locations are uncorrelated.",
"As nudging field data, we locate sensors to measure the velocity field $u(x,t)$ every 256 grid points (i.e., a total of 17 sensors with $s_{freq} = 256$ , where $s_{freq}$ is the number of spatial steps between sensors locations), and collect measurements every 10 time steps (i.e., each $0.1$ time unit with $t_{freq}=10$ , where $t_{freq}$ is the number of time steps between measurement signals).",
"To account for noisy observations, white Gaussian noise of zero mean and covariance matrix of $Q$ is added to the true velocity field obtained from the FOM simulation.",
"Similar to $B$ , we set $Q=\\sigma _m^2 \\mathbf {I}$ , where $\\sigma _m$ is the standard deviation of measurement noise.",
"This assumes that sensors measurements are not correlated to each other, and all sensors have similar quality (i.e., add similar amounts of noise to the measurements).",
"As a base case, we set $\\sigma _b=1$ , and $\\sigma _m=1$ .",
"The procedure presented in Sec.",
"is applied using the numerical setup described above, and compared against the reference case of GROM with the erroneous initial condition and inherent model imperfections due to modal truncation.",
"In Fig.",
"REF , the temporal evolution of the POD modal coefficients is shown for the true projection, background, and LSTM-Nudge results.",
"The true projection results are obtained by the projection (i.e., via inner product) of the true FOM field at different time instants onto the corresponding basis functions.",
"The background trajectory is the reference solution obtained by standard GROM using the erroneous initial condition, without any closure or corrections.",
"It can be seen that the background trajectory gets off the true trajectory by time as a manifestation of model imperfection.",
"Also, note that the background solution does not begin from the same point as true projection due to the noise in initial condition.",
"On the other hand, the LSTM-Nudge predictions almost perfectly match the true projection solution, implying that the approach is capable of blending noisy observations with a prior estimate to gain more accurate predictions.",
"Figure: Temporal evolution of the POD modal coefficients for the 1D Burgers problem.In order to better visualize the predictive capabilities of the LSTM-Nudge methodology, we compute the reconstructed velocity field using Eq.",
"REF .",
"Moreover, the root mean-squares error ($RMSE$ ) of the reconstructed field with respect to the FOM solution is calculated as a function of time as follows, $ RMSE(t) = \\sqrt{\\dfrac{1}{M}\\sum _{i=1}^{M}{\\bigg (u_{FOM}(x_i,t) - u_{ROM}(x_i,t) \\bigg )^2} },$ where $u_{FOM}$ is the true velocity field obtained from solving the FOM equation, while $u_{ROM}$ is the reduced order approximation computed through true projection, background (reference) solution, or LSTM-Nudge method.",
"The reconstructed velocity field at final time (i.e., at $t=1$ ) is shown in Fig.",
"REF along with the $RMSE$ as a function of time.",
"As described before, the true projection solution is simply the projection of the FOM field onto the reduced POD space, and it represents the optimal solution that can be approximated using a linear subspace spanned by $R$ modes.",
"In order to get rid of those Gibbs oscillations, we would need either a larger number of modes or a more representative subspace (e.g., through partitioning or auto-encoders).",
"Therefore, it is fair to compare our results against the true projection solution, rather than the FOM since we do not address any issues regarding the resolution or representability capabilities of the POD subspace.",
"Figure: Final velocity field (at t=1t=1) [left] and the root mean-squares error [right] for the 1D Burgers problem."
],
[
"Effect of noise",
"Here, we investigate the effect of noise (both in initial condition and measurements) on the performance of the LSTM-Nudge framework.",
"In other words, we study how much noise it can handle sufficiently.",
"For the training phase, the LSTM was trained using noisy data with $\\sigma _b = \\sigma _m = 1.0$ .",
"Now, we test using data with smaller and larger amounts of noise.",
"In particular, we vary $\\sigma _b$ and $\\sigma _m$ between $0.1$ , $1.0$ , and $10.0$ .",
"Readers should be aware that the true velocity field spans between 0 and 1.",
"Thus, a noise with a standard deviation of 10 is an extreme case, corresponding to very cheap sensors.",
"In Fig.",
"REF , we show the root mean-squares error of the reconstructed velocity fields based on true projection, background solution, and LSTM-Nudge predictions using different levels of noise.",
"We can see that the LSTM-Nudge is performing very well, compared to the background solution, and almost matching the true projection results.",
"More importantly, we find that the prediction accuracy is more dependent on measurement noise than noise in the initial condition.",
"For instance, the LSTM-Nudge almost recovers the true state estimate very shortly using adequate measurements (i.e., Fig.",
"REF ).",
"This is even more visualized in the surface plots in Fig.",
"REF .",
"In contrast, the model imperfections cannot be cured well at many time instances using highly noisy observations, even with moderate noise in initial condition (e.g., see Fig.",
"REF ).",
"The situation is worse in Fig.",
"REF , with severe noise in both initial conditions and measurements.",
"Figure: Root mean-squares error in reconstructed velocity field, with different levels of background and measurement noises.The dependence of the framework on measurements noise significantly more than background noise might be attributed to the input features we are using in the LSTM architecture.",
"It includes a combination between the background modal coefficients (obtained from erroneous initial condition) and the observed velocity field at sparse locations.",
"For the modal coefficients, the erroneous initial conditions are first projected onto the POD subspace to obtain the initial (erroneous) modal coefficients, which are then integrated in time for $\\tau $ time units.",
"This projection is known to filter a large amount of added noise, which can be seen as a preprocessing step to reduce the effect of initial condition perturbation.",
"That is why we can barely see a difference between erroneous trajectory and true projection at time zero, except for the extreme case with $\\sigma _b=10$ .",
"On the other hand, observations are fed to the LSTM network as is, without any preprocessing or prior treatment.",
"This makes the predictions more sensitive to the measurement quality.",
"Figure: Surface plots for the spatio-temporal evolution of the 1D Burgers problem with σ b =10\\sigma _b=10, and σ m =1\\sigma _m=1."
],
[
"Effect of measurements sparsity",
"Since the LSTM-Nudge in ROM context is found to be relatively sensitive to sensor quality (observational noise), we study the effect of measurement sparsity as well.",
"In particular, both the temporal sparsity (i.e., frequency of measurement signals) and spatial sparsity (i.e., number of sensors) are explored.",
"For the base case, we are collecting measurements every 10 time steps (i.e., $t_{freq}=10, \\tau =0.1$ ), and sensors are placed every 256 grid points (i.e., $s_{freq}=256$ ).",
"First, we consider the case when measurement signals are only available every 20 time steps (i.e., $t_{freq}=20$ ), with the same spatial sparsity (i.e., 17 sensors).",
"The POD modal coefficients are plotted in Fig.",
"REF as predicted by LSTM-Nudge, compared to the background case (without corrections) and true projection trajectory.",
"We find that the framework is sufficiently able to handle this variation in measurement signal frequency.",
"We note here that we use the same LSTM network, trained using the base case data (i.e., trained with $t_{freq}=10$ and tested for $t_{freq}=20$ ).",
"For all, we use the same level of noise as before (i.e., $\\sigma _b=\\sigma _m=1$ ).",
"Figure: Temporal evolution of the POD modal coefficients for the 1D Burgers problem, when measurements are taken every 20 time steps.We also plot the final velocity field reconstructed through the background solution, and LSTM-Nudge compared to both FOM and true projection in Fig.",
"REF as well as the root mean-squares errors at different times.",
"Although we can see a discrepancy between true projection and LSTM-Nudge at several time instants, we notice a jump in LSTM-Nudge solution towards the true projection almost every $0.2$ time units.",
"This is consistent with the fact that LSTM-Nudge goes into effect every 20 time steps, when measurements become available, implying that the LSTM-Nudge is still capable of sufficiently rectifying the model and state imperfections whenever measurements are received.",
"Figure: Final velocity field (at t=1t=1) [left] as well as the root mean-squares error [right] for the 1D Burgers problem, when measurements are taken every 20 time steps.In order to examine the effect of spatial sparsity (number of sensors), we vary the spatial frequency (i.e., number of grid points between sensors) as $s_{freq} \\in \\lbrace 128, 512, 1024, 2048\\rbrace $ .",
"The first case (i.e., $s_{freq}=128$ ) corresponds to more sensors than base case, while the others correspond to more sparse measurement points (less sensors).",
"From Fig.",
"REF , we can deduce that the effect of number of sensors is minimal in this case, even with very few sensors (e.g., 3 sensors in Fig REF ).",
"However, we should state here that each of those cases requires retraining the LSTM network with the relevant number of measurement points.",
"This is because the LSTM for the base case has an input dimension of 23 (i.e., 6 modal coefficients and 17 measurements), and changing the number of measurements would require a different size of input vector.",
"Although we assume equally-spaced and collocated sensors (i.e., placed exactly on the numerical grid), compressive sensing ideas can be adopted to intelligently locate sensors for optimal performance.",
"Figure: Root mean-squares error in reconstructed velocity field, with different number of sensors located sparsely at grid points."
],
[
"Effect of measured quantity",
"As described in Sec.",
", the input to the LSTM-Nudge framework is a combination of modal coefficients (i.e., GROM state variables), and direct measurements (i.e., velocity field) without constraining any mapping between them.",
"The LSTM has shown brilliant effectiveness learning the map between model state and observations to approximate the required correction/nudging.",
"In this section, we elaborate more on this feature by exploring the performance of the LSTM-Nudge with a different measured quantity.",
"In practice, direct field variable measurement may not be feasible.",
"For example, the dynamics of sea surface temperature can be only inferred from satellite measurement of radiated thermal energy.",
"Defining a map between observable quantity and model (state) variable is not usually straightforward.",
"Hence, utilizing neural networks strengths of discovering underlying patterns and relations to learn such maps is highly desired.",
"Instead of measuring the velocity $u(x,t)$ , we hypothesize that we can only observe the square of this velocity field (i.e., $u^2(x,t)$ ).",
"This is related to the kinetic energy of the flow.",
"We repeat the LSTM training using the new input features (i.e., modal coefficients and square of velocity), and test using the base case parameters (i.e., $\\sigma _b=\\sigma _m=1$ , $t_{freq}=10$ , and $s_{freq}=256$ ).",
"The LSTM-Nudge is found to perform sufficiently well with this new observable quantity, as shown in Fig.",
"REF for the temporal modal coefficients and Fig.",
"REF for the velocity field reconstruction.",
"We also emphasize here that the similar behavior between observing $u(x,t)$ and observing $u^2(x,t)$ might be attributed to the values of the velocity field varying between 0 and 1.",
"Thus, observing either the velocity or its squared value basically has the same pattern and range.",
"For different situations, where the observable has significantly different pattern, the behavior might vary as well.",
"For example, for a sine wave moving between $-1$ and 1, observing the squared value (or the absolute value) would result in measurements between 0 and 1, neglecting the negative part.",
"Figure: Temporal evolution of the POD modal coefficients for the 1D Burgers problem, with u 2 u^2 as the available measurements.Figure: Final velocity field (at t=1t=1) [left] as well as the root mean-squares error [right] for the 1D Burgers problem, with u 2 u^2 as the available measurements."
],
[
"Concluding Remarks",
"In the current study, we have developed a methodology to utilize machine learning to cure model deficiency through online measurement data adopting ideas from dynamic data assimilation.",
"In particular, an LSTM architecture has been trained to nudge prior predictions towards true state values using a combination of background information with sparse and noisy observations.",
"The proposed framework is distinguished from previous studies in the sense that it is built on the assumption that all the computing ingredients are imperfect, including a truncated GROM model, erroneous initial conditions, and defective sensors.",
"We have applied the proposed LSTM-Nudge to the 1D Burgers problem with a moving discontinuity, and investigated the effects of measurement noise and initial condition perturbation on its behavior.",
"Although the framework works sufficiently well for a wide range of noise and perturbation, numerical experiments have indicated relatively more dependence of performance on measurement quality (noise).",
"Meanwhile, we have found that sensors sparsity has minimal effects on results.",
"We emphasize that the proposed framework represents one way of hybridizing human knowledge, physics-based models, measurement information, and data-driven tools to maximize their benefits rather than discarding any of them.",
"This might represent a viable key enabler for the emerging digital twin applications.",
"Nonetheless, the scalability of the approach has yet to be tested using more complex and higher-dimensional problems."
],
[
"Acknowledgments",
"This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290.",
"O.S.",
"gratefully acknowledges their support.",
"Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government.",
"Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.",
"Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.",
"The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof."
]
] | 2005.14246 |
[
[
"Variance Reduction in Simulation of Multiclass Processing Networks"
],
[
"Abstract We use simulation to estimate the steady-state performance of a stable multiclass queueing network.",
"Standard estimators have been seen to perform poorly when the network is heavily loaded.",
"We introduce two new simulation estimators.",
"The first provides substantial variance reductions in moderately-loaded networks at very little additional computational cost.",
"The second estimator provides substantial variance reductions in heavy traffic, again for a small additional computational cost.",
"Both methods employ the variance reduction method of control variates, and differ in terms of how the control variates are constructed."
],
[
"Abstract",
"We use simulation to estimate the steady-state performance of a stable multiclass queueing network.",
"Standard estimators have been seen to perform poorly when the network is heavily loaded.",
"We introduce two new simulation estimators.",
"The first provides substantial variance reductions in moderately-loaded networks at very little additional computational cost.",
"The second estimator provides substantial variance reductions in heavy traffic, again for a small additional computational cost.",
"Both methods employ the variance reduction method of control variates, and differ in terms of how the control variates are constructed."
],
[
"Introduction",
"Owing to a mistake in the editorial process, this paper was accepted for publication but never actually appeared.",
"At the request of a friend I am posting it on arXiv.",
"A multiclass queueing network is a network of service stations through which multiple classes of customers move.",
"Each customer class can have different service-time characteristics at a single service station.",
"Multiclass queueing networks are of great interest in a large variety of applications [4], [29], [8], [15] because of their tremendous modeling flexibility.",
"Perhaps the most common reason for modeling a system using a multiclass queueing network is to try to determine a suitable operating policy for the network.",
"An operating policy is a policy that determines which customers should be worked on at which times.",
"For example, if there are multiple customer classes at a single service station, then which class should the station work on?",
"In order to make comparisons between operating policies, one must define a suitable performance measure, such as expected steady-state work in process, or expected steady-state throughput, etc.",
"For broad classes of networks one can compute certain performance measures analytically [23], [3], [24], [17], [18], or one can turn to numerical computation [42], [40], [10], [44].",
"In general, however, these approaches are either infeasible, or intractable due to the high complexity of network models.",
"Some results are available in the form of bounds on performance measures through the construction of linear programs [28], [5], [27], [43], [38].",
"Unfortunately, these bounds are often quite loose, and so it can be difficult to compare operating policies based on such bounds alone.",
"It is natural then, to turn to simulation.",
"Given that one is going to simulate many different operating policies, it is important that any simulation return relatively accurate answers as quickly as possible.",
"This suggests the need for variance-reduction techniques that can increase the accuracy in simulation results for a given computational budget.",
"Another reason for desiring efficient simulation techniques is that the network under consideration is often moderately to heavily loaded, in the sense that some of the resources of the network are close to full utilization.",
"It has been noted that, in such settings, simulation can take a tremendously long time to return precise estimates of performance [47], [2].",
"So we are strongly motivated to seek special variance-reduction techniques for multiclass networks.",
"In this paper we develop two such variance-reduction techniques.",
"Both are based on the approximating martingale process method [20], [19], which is a specialization of the method of control variates; see, for example, [31] for an introduction to control variates.",
"This paper is an outgrowth of [21].",
"The methods introduced there and here have since seen further development in [22] and [7].",
"The theoretical results on the order of the variance constants given here have been considerably extended in [35], [36].",
"These papers also describe further insights on network behavior that have been uncovered since the work presented in this paper was completed.",
"Furthermore, [25] and [26] have since introduced a new family of variance reduction techniques that may lead to even greater variance reductions than those seen in this paper.",
"Although our presentation concentrates on the estimation of the mean steady-state number of customers (of all classes) in the system, our methods may be tailored to the steady-state estimation of any linear function of the individual customer-class populations.",
"In particular, we can also estimate, for instance, the mean steady-state number of customers of a particular class present in the system.",
"In Section we describe our model of a multiclass-queueing system.",
"We also review Poisson's equation and explain its importance in our context.",
"In particular, we wish to approximate the solution to Poisson's equation to construct efficient simulators.",
"In Section , we explore quadratic forms as approximations to the solution to Poisson's equation.",
"Computational results are given for the resulting simulation estimator, which we term the quadratic estimator.",
"The quadratic estimator significantly outperforms a more standard simulation estimator in lightly to moderately loaded networks.",
"In heavily loaded networks, the difference between the performance of the two estimators closes, although the quadratic estimator still provides variance reductions that would significantly reduce the computational effort involved in exploring a class of operating policies.",
"In Section , we explore alternative approximations to the solution to Poisson's equation based on the concept of a fluid limit.",
"The resulting simulation estimator, the fluid estimator, yields significant variance reductions in heavily loaded networks, and modest variance reductions in less heavily loaded networks.",
"For a given simulation run-length, it is slightly more expensive to compute than the standard estimator, so that the issue of variance reduction versus computational effort needs to be considered [16].",
"We discuss the choice of simulation estimator for a given network in Section ."
],
[
"Multiclass Queueing Networks",
"Consider a system consisting of $d$ stations (or machines) and $\\ell $ classes of customers (or jobs).",
"Class $i$ customers require service at station $s(i)$ .",
"Upon completion of service at station $s(i)$ , a class $i$ customer becomes a class $j$ customer with probability $R_{ij}$ , and exits the system with probability $R_{i0} \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}1 - \\sum _{j=1}^\\ell R_{ij}.$ The service times for class $i$ customers are assumed to form an i.i.d.",
"(independent and identically distributed) sequence of exponentially distributed r.v.",
"'s (random variables) with mean $\\mu _i^{-1}$ .",
"Class $i$ customers arrive exogenously to station $s(i)$ according to a Poisson process with rate $\\lambda _i$ (which may be zero).",
"The Poisson-arrival processes and the service-time processes are mutually independent.",
"We let $\\mu $ denote the $\\ell $ -dimensional vector of service rates, and $\\lambda $ the $d$ -dimensional vector of arrival rates.",
"Unless otherwise stated, all vectors are assumed to be column vectors.",
"We require that the routing matrix $R = (R_{ij}: 1 \\le i, j\\le \\ell )$ be transient, so that the following inverse exists: $(I-R)^{-1} = \\sum _{k=0}^\\infty R^k .$ This ensures that all customers that enter the system will eventually leave, and we can also be assured that there is a unique solution $\\gamma \\ge 0$ to the traffic equations, $0 = \\lambda -\\gamma + R^{\\prime } \\gamma ,$ where $R^{\\prime }$ denotes the transpose of the matrix $R$ .",
"We assume throughout that $\\gamma _i>0$ for all $i$ .",
"Figure: A multiclass network with d=2d=2 stations and ℓ=3\\ell =3 customer classes.Figure REF illustrates an example with $d=2$ stations, $\\ell = 3$ customer classes, and a single exogenous arrival process (two of the arrival rates are zero).",
"Customer classes 1 and 3 are served at Station 1, so that $s(1) = s(3) = 1$ .",
"Similarly $s(2) = 2$ .",
"The routing values $R_{ij}$ are zero except for $R_{12} = R_{23} = 1$ .",
"Let $X_i(t)$ denote the number of class $i$ customers present in the system at time $t$ , and let $X(t) = (X_i(t): 1 \\le i \\le \\ell )$ be the vector of customer populations at time $t$ .",
"Let $V_i(t)$ denote the fraction of station $s(i)$ 's effort allocated to serving customers of class $i$ at time $t$ , and let $V(t)$ denote the corresponding vector quantity.",
"We must have $V_i(t) \\ge 0$ for all $i$ and $t$ , and $\\sum _{i: s(i) = s} V_i(t) \\le 1$ for all $s$ and $t$ , i.e., a station $s$ can never allocate negative effort, or more than unit effort, to the classes $\\lbrace i: s(i) = s\\rbrace $ that are served at the station.",
"For the network in Figure REF for example, suppose that at time $t$ Machine 1 is serving Class-3 jobs, and Machine 2 is empty.",
"Then $V(t) = (0, 0, 1)^{\\prime }$ .",
"We require the operating policy adopted by stations to be stationary, non-idling, and $0-1$ .",
"We next define and explain each of these terms.",
"By stationary, we mean that $V(t)$ is a deterministic function of $X(t)$ , so that the workload allocations depend only on the current customer-class levels.",
"This allows the modeling of preemptive priority policies, for example, but precludes the modeling of policies such as FIFO which rely on additional information such as the order in which customers arrive to a station.",
"By non-idling, we mean that if $\\sum _{i: s(i) = s} X_i(t) > 0$ , then $\\sum _{i: s(i) = s} V_i(t)=1$ , i.e., if there are customers present at station $s$ at time $t$ , then the station allocates all of its effort at that time.",
"By $0-1$ , we mean that at any given station, at most one class receives service at any given time.",
"This assumption is applied only for notational convenience.",
"The estimators we derive may also be applied to networks controlled by randomized or processor-sharing policies.",
"With the above structure in place, we may conclude that $X = (X(t): t \\ge 0)$ is a time-homogeneous continuous-time Markov chain.",
"However, we prefer to work in discrete time because it simplifies the analysis.",
"So we first rescale time so that $e^{\\prime } (\\lambda + \\mu ) = 1$ , where $e$ denotes a vector of ones.",
"Then we uniformize; see, e.g., [41].",
"Uniformization is a process that allows us to study the continuous-time process $X$ through a related discrete-time process $Y = (Y(n): n \\ge 0)$ .",
"Define $\\tau _0 = 0$ and let the times $\\lbrace \\tau _n: n \\ge 1\\rbrace $ correspond to epochs when either arrivals, real service completions, or virtual service completions occur in the uniformized process.",
"For $n \\ge 0$ , let $Y(n) =X(\\tau _n)$ and $W(n) = V(\\tau _n)$ .",
"Then the process $Y= (Y(n): n \\ge 0)$ is a discrete-time Markov chain evolving on a (countable) state space $S$ that is a subset of $\\lbrace 0, 1, 2, ...\\rbrace ^\\ell $ .",
"Example 1: For the M/M/1 queue, $Y$ is a Markov chain on $\\lbrace 0, 1, 2, \\ldots \\rbrace $ with transition matrix $P$ , where $P_{ij} = (\\mu +\\lambda )^{-1} \\left\\lbrace \\begin{array}{ll}\\lambda & \\mbox{if $j = i+1$,} \\\\\\mu & \\mbox{if $j=\\max (i-1, 0)$, and} \\\\0 & \\mbox{otherwise.",
"}\\end{array}\\right.$ Our goal is to estimate $\\alpha $ , the steady-state mean number of customers in the system, i.e., the steady-state mean of $|Y(0)|\\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}e^{\\prime }Y(0)$ (i.e.",
"$|\\,\\cdot \\,|$ is the $L_1$ norm).",
"To ensure (among other things) that $\\alpha $ exists and is finite, we make a certain assumption (A) below.",
"The assumption (A) is known as a Lyapunov, or Foster-Lyapunov, condition.",
"Intuitively, the function $V$ represents energy, and (REF ) indicates that there is an expected loss in energy for states $y$ that are “large.” This then ensures that energy never gets too large, and so the chain remains stable.",
"For any function $V$ on $ \\mbox{I\\hspace{-2.27621pt}R}_+^\\ell $ we define $PV(y) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\operatorname{\\mathbb {E}}_yV(Y(1)), \\qquad \\Delta _V(y) = PV\\, (y) - V(y), \\qquad \\qquad y\\in S,$ where $\\operatorname{\\mathbb {E}}_y(\\cdot ) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\operatorname{\\mathbb {E}}(\\cdot \\, | \\, Y(0) = y)$ .",
"Intuitively, $PV(y)$ $(\\Delta _V(y))$ represents the expected energy (expected change in energy) one step from now, assuming that the chain is currently in state $y$ .",
"(A) There exists a function $V: \\mbox{I\\hspace{-2.27621pt}R}_+^\\ell \\rightarrow \\mbox{I\\hspace{-2.27621pt}R}_+$ satisfying $V$ is equivalent to a quintic in the sense that for some $\\delta < 1$ , $\\delta (| y| ^5 + 1) \\le V(y) \\le \\delta ^{-1} (| y| ^5 + 1) \\mbox{;and}$ for some $\\eta > 0$ , and all $y$ , $ \\Delta _V(y)=PV\\, (y) - V(y) \\le -| y| ^4 + \\eta .$ Continuation of Example : For the M/M/1 queue, we may take $V(y) = b_0 y^5$ , with $b_0$ a sufficiently large constant, and then (A) is satisfied as long as $\\rho \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\lambda / \\mu < 1$ .",
"In general, this condition will be satisfied if the stability-linear program of [27] admits a solution, generating a co-positive $\\ell \\times \\ell $ matrix $Q$ .",
"In this case, the function $V$ may be taken as $V(y) = (y^{\\prime } Q y)^{5/2}$ .",
"Alternatively, if a fluid model is stable, and we define $\\kappa =\\inf \\lbrace n \\ge 1: Y(n) = 0\\rbrace $ , then the function $V(y) = \\operatorname{\\mathbb {E}}_y \\sum _{k=0}^\\kappa | Y(k)| ^4 $ is bounded as in Condition 1 [11], and this function is known to satisfy Condition 2 [37].",
"Under the assumption (A), the chain $Y$ possesses a unique stationary distribution $\\pi $ , and $\\operatorname{\\mathbb {E}}_\\pi | Y(0)| ^4 \\le \\eta < \\infty $ [37], where $\\operatorname{\\mathbb {E}}_\\pi (\\,\\cdot \\,) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\int _S \\operatorname{\\mathbb {E}}(\\,\\cdot \\,| Y(0) = y)\\pi (dy).$ Hence, in particular, the steady-state mean number of customers $\\alpha = \\operatorname{\\mathbb {E}}_\\pi |Y(0)| <\\infty $ .",
"The Lyapunov condition (A) is stronger than is strictly necessary to ensure that $\\alpha = \\operatorname{\\mathbb {E}}_\\pi |Y(0)|$ is finite.",
"A “tighter” requirement is that there is a function $h: \\mbox{I\\hspace{-2.27621pt}R}_+^\\ell \\rightarrow \\mbox{I\\hspace{-2.27621pt}R}_+$ satisfying $\\Delta _h(y) = Ph(y) - h(y) \\le -|y| + \\eta $ for all $y \\in S$ and some constant $\\eta > 0$ .",
"Given such a function, Theorem 14.3.7 of [37] allows us to conclude that $\\alpha = E_\\pi |Y(0)| \\le \\eta $ .",
"One might then ask whether the inequality in (REF ) can be made an equality, thereby yielding a tighter upper bound $\\eta $ on $\\alpha $ .",
"In such a case we would have $\\Delta _{h^*}(y) = Ph^*(y) - h^*(y) = -|y| + \\eta .$ This equation is known as Poisson's equation.",
"If $h^*$ is a solution to Poisson's equation, then it is easy to see that $h^* + c$ is also a solution for any constant $c$ .",
"In fact, Proposition 17.4.1 of [37] shows that any two $\\pi $ -integrable solutions to Poisson's equation must differ by an additive constant, in the sense that for all $y$ in a set $A$ with $\\pi (A) = 1$ , $h_1(y) - \\pi (h_1) = h_2(y) -\\pi (h_2)$ where, for a real-valued function $g:S \\rightarrow \\mbox{I\\hspace{-2.27621pt}R}$ , we denote $\\pi (g) = \\int _Sg(y) \\pi (dy)$ .",
"One may estimate $\\alpha $ using $\\alpha (n) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}|\\bar{Y}(n)|$ , where $\\bar{Y}(n) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}n^{-1} \\sum _{i=0}^{n-1} Y(i)$ , the mean number of customers in the system up to time $n$ .",
"The following result is a special case of Theorem 17.0.1 of [37], and shows that the estimator $\\alpha (n)$ is consistent, and satisfies a central limit theorem.",
"Theorem 1 Suppose that (A) holds.",
"Then $\\alpha (n) \\rightarrow \\alpha $ almost surely (a.s.), and furthermore, $n^{1/2}(\\alpha (n) - \\alpha ) \\mbox{ $\\Rightarrow $ }\\sigma N(0, 1),$ as $n \\rightarrow \\infty $ .",
"The time-average variance constant (TAVC) $\\sigma ^2$ is given by $\\sigma ^2 = \\operatorname{\\mathbb {E}}[h(Y)(|Y| - \\operatorname{\\mathbb {E}}|Y|)] - \\operatorname{Var}|Y|$ , where $Y$ is distributed according to the stationary distribution $\\pi $ , and $h$ solves Poisson's equation; see (REF ).",
"As noted in the introduction, it has been observed that simulation can take a very long time to yield accurate answers for heavily loaded networks [2], [47].",
"This problem is exhibited in our framework through the TAVC.",
"Our simulation experiments indicate that the TAVC grows rapidly as the network becomes heavily loaded.",
"Our next result lends further weight to these observations.",
"Consider a multiclass queueing system consisting of a single station and (possibly) multiple customer classes.",
"Suppose that the service rates $\\mu $ and arrival rates $\\lambda $ are such that $e^{\\prime } {\\cal M}^{-1} (I-R^{\\prime })^{-1} \\lambda = 1,$ where ${\\cal M} =$ diag$(\\mu )$ .",
"(This corresponds to a situation where the resources of the network are exactly matched by the demand.)",
"Now consider a family of queueing systems indexed by $\\rho \\in (0, 1)$ , where the $\\rho $ th system has arrival rate vector $\\lambda (\\rho ) = \\rho \\lambda $ .",
"For the sake of clarity, we occasionally suppress dependence on $\\rho $ .",
"For a given vector of buffer levels $y$ , let $f(y) = d^{\\prime }y$ be a measure of the work in the system, where $d^{\\prime } = e^{\\prime } {\\cal M}^{-1} (I - R^{\\prime })^{-1} = e^{\\prime }Q,$ and $Q = {\\cal M}^{-1} (I-R^{\\prime })^{-1}$ .",
"Intuitively, $f(y)$ measures that total expected amount of processing required to completely serve all of the customers presently in the system as given by $y$ .",
"Let $\\sigma ^2_f(\\rho )$ be the TAVC associated with the estimator $f({\\bar{Y}}(n)) = d^{\\prime } {\\bar{Y}}(n)$ , and let $\\sigma ^2_i(\\rho )$ be the TAVC associated with ${\\bar{Y}}_i(n)$ $(i = 1, \\ldots , \\ell )$ .",
"If $\\Sigma _\\rho $ denotes the time-average covariance matrix of $Y$ , then $\\sigma ^2_f(\\rho ) = d^{\\prime }\\Sigma _\\rho d$ and $\\sigma ^2_i(\\rho ) = e_i^{\\prime }\\Sigma _\\rho e_i$ , where $e_i$ denotes the $i$ th basis vector.",
"The proof of the following result may be found in the appendix.",
"Theorem 2 Consider the family of multiclass queueing systems above under any non-idling work-allocation policy.",
"Then the following are true for each $\\rho < 1$ .",
"Assumption (A) holds with the function $b_\\rho f^5$ for some sufficiently large constant $b_\\rho $ , and the TAVCs $\\sigma ^2_f(\\rho )$ and $\\sigma ^2_i(\\rho )$ $(i = 1, \\ldots , \\ell )$ are finite.",
"The solution to Poisson's equation for the estimator $f({\\bar{Y}}(n))$ is given by $h(y; \\rho ) = \\frac{f^2(y)}{2(1-\\rho )} + c_\\rho ^{\\prime } y,$ where the vector $c_\\rho $ is of the order $(1-\\rho )^{-1}$ .",
"Furthermore, there exist constants $A, B > 0$ (independent of $\\rho $ ) such that for $\\rho $ sufficiently close to 1, $\\frac{A}{(1-\\rho )^4} \\le \\sigma ^2_f(\\rho ) \\le \\frac{B}{(1-\\rho )^4},$ and finally $\\mbox{trace} \\,\\Sigma _\\rho = \\sum _{i=1}^\\ell \\sigma ^2_i(\\rho ) \\ge \\frac{A^{\\prime }}{(1-\\rho )^4}$ for some constant $A^{\\prime }$ (again, independent of $\\rho $ ).",
"Theorem REF shows that trace$\\,\\Sigma _\\rho $ is of the order $(1-\\rho )^{-4}$ as $\\rho \\rightarrow 1$ , and this suggests, although it does not necessarily prove, that the TAVC $e^{\\prime } \\Sigma _\\rho e$ of the standard estimator $\\alpha (n)$ is of the same order (see also [35] where this precise order is verified for the TAVC in a diffusion model.)",
"We have already mentioned that our simulation experiments also indicate that $\\alpha (n)$ has high variance in heavily congested networks.",
"Therefore, there is strong motivation for identifying alternative estimators to $\\alpha (n)$ that can improve performance in heavy traffic.",
"The key to these estimators is Poisson's equation.",
"If $h^*$ is $\\pi $ -integrable, then by taking expectations with respect to $\\pi $ in (REF ), we see that $\\alpha = \\eta $ .",
"Therefore, if the solution to Poisson's equation is known, then so is $\\alpha $ .",
"In general then, we cannot expect to know the solution to Poisson's equation.",
"But what if an approximation is known?",
"If the approximation $h$ is $\\pi $ -integrable, then $\\pi (\\Delta _h)= 0$ , i.e., $\\Delta _h(Y_n)$ has steady-state mean 0, and so one might consider using $\\Delta _h$ in building a simulation control variate for estimating $\\alpha $ .",
"In particular, one might consider using the controlled estimator $\\alpha _c(n) = \\alpha (n) + \\frac{\\beta }{n} \\sum _{i=0}^{n-1} \\Delta _h(Y(i)),$ where $\\beta $ is an adjustable constant.",
"If $h = h^*$ and we take $\\beta = 1$ , then $\\alpha _c(n) = \\alpha $ , and we obtain a zero-variance estimator of $\\alpha $ .",
"In general, we can expect useful variance reductions using the estimator $\\alpha _c(n)$ provided that $h$ is a suitable approximation to the solution to Poisson's equation.",
"We will discuss the choice of the constant $\\beta $ later.",
"For more details on this approach to variance reduction see [20], [19], [22], [7] and [35].",
"So how should one go about determining an approximation to the solution $h^*$ to Poisson's equation?",
"It is known that for any `reasonable' policy, the function $h^*$ is equivalent to a quadratic, in the sense of (REF ) (see [27], [33], [34], [35] and Theorem REF below).",
"So it is reasonable to search for a quadratic function $h$ that approximately solves (REF )."
],
[
"A Quadratic Approximation",
"We begin this section by demonstrating the general ideas of the approach on the stable M/M/1 queue.",
"Continuation of Example : Recall that the solution to Poisson's equation is equivalent to a quadratic.",
"So it is reasonable to approximate the solution $h^*$ to Poisson's equation by a quadratic function.",
"In the linear case $h(y) = y$ , we have that $\\Delta _h(y) = \\lambda -\\mu w$ , where $w = \\mathbb {I}(y > 0)$ ($\\mathbb {I}(\\cdot )$ is the indicator function that is 1 if its argument is true and 0 otherwise) represents the only non-idling policy: The server works when customers are present, and idles when customers are not present.",
"Taking expectations with respect to $\\pi $ , we see that the expected fraction of time that the server is working is $\\operatorname{\\mathbb {E}}_\\pi W(0) = \\lambda / \\mu $ , and this is the result expected by work conservation.",
"Taking the pure quadratic $h(y) = a y^2$ , we have $ \\Delta _h(y) = 2a y(\\lambda - \\mu ) + a \\lambda + a \\mu w.$ We now choose $a = (\\mu -\\lambda )^{-1} / 2$ to ensure that the coefficient of $y$ is $-1$ .",
"We could then use the right-hand side of (REF ) as a control variate as in ().",
"However, it is instructive (and useful) to adopt a slightly different approach where we avoid estimation of the variable $w$ , and replace it by its known steady-state mean $\\operatorname{\\mathbb {E}}_\\pi W(0)=\\pi (w)=\\lambda /\\mu $ .",
"We may then use the controlled estimator $\\bar{Y}(n) + \\beta (-\\bar{Y}(n) + 2\\lambda a)$ .",
"This estimator is in fact equal to the estimator $\\bar{Y}(n)+\\beta \\Delta _{h^*}(\\bar{Y}(n))$ , with $h^*(y) = a(y^2 + y)$ , the solution to Poisson's equation.",
"Hence if $\\beta = 1$ , this estimator has zero variance.",
"We turn now to the general case.",
"We assume throughout this section that the assumption (A) is in place, so that the network is stable, and all steady-state expectations that we use exist.",
"A similar development for reentrant lines may be found in [21].",
"Since the solution to Poisson's equation is equivalent to a quadratic, it is reasonable to use $h(y) = y^{\\prime } Q y + q^{\\prime }y$ , for some symmetric matrix $Q$ and vector $q$ .",
"This approach was originally proposed in [27], based on the prior approaches to bounding network performance presented in [28], [5] and [13].",
"We first consider the linear part of the function, and then turn to the quadratic terms.",
"Consider the function $h_j(y) = y_j$ for some $j$ .",
"Then, letting $w$ denote the work allocation vector corresponding to the vector $y$ , we have $\\Delta _{h_j}(y) = \\lambda _j - \\mu _j w_j + \\sum _i \\mu _i w_i R_{ij}.$ Denote by $\\bar{x}$ the vector $(\\mu _i \\operatorname{\\mathbb {E}}_\\pi W_i(0): 1 \\le i \\le \\ell )$ .",
"Taking expectations with respect to $\\pi $ in (REF ) and writing the equations (one for each $j$ ) in vector form, we obtain $0 = \\lambda -\\bar{x}+ R^{\\prime } \\bar{x}.$ Since the solution to (REF ) is unique, we conclude that $\\operatorname{\\mathbb {E}}_\\pi W_j(0) = \\gamma _j / \\mu _j$ .",
"Thus by considering linear functions we have shown that the usual traffic conditions hold.",
"Now consider the function $h_{jk}(y) = y_j y_k$ for $j \\ne k$ .",
"Then, $\\Delta _{h_{jk}}(y) & = & \\lambda _j y_k + \\lambda _k y_j - \\mu _j w_jy_k - \\mu _k w_k y_j \\nonumber \\\\& + & \\sum _i \\mu _i w_i (R_{ij} y_k + R_{ik} y_j) -\\mu _j w_j R_{jk} - \\mu _k w_k R_{kj}.",
"$ Notice that (REF ) is a nonlinear expression in $y$ , due to the presence of the $w$ terms.",
"We would prefer to work with linear expressions.",
"To this end, introduce the variables $Z_{ij}(n) =W_i(n)Y_j(n)$ , and let ${\\bar{z}}_{ij} = \\operatorname{\\mathbb {E}}_\\pi W_i(0) Y_j(0)$ for $1 \\le i, j \\le \\ell $ .",
"Under our assumptions on the policy it follows that $Z(n)$ is a fixed, deterministic function of $Y(n)$ .",
"Furthermore, let ${\\bar{y}}_j = \\operatorname{\\mathbb {E}}_\\pi Y_j(0)$ .",
"Taking expectations with respect to $\\pi $ in (REF ), we obtain $ 0 = \\lambda _j {\\bar{y}}_k + \\lambda _k {\\bar{y}}_j - \\mu _j {\\bar{z}}_{jk} -\\mu _k{\\bar{z}}_{kj} + \\sum _i \\mu _i (R_{ij} {\\bar{z}}_{ik} + R_{ik} {\\bar{z}}_{ij}) - \\gamma _jR_{jk} - \\gamma _k R_{kj}.$ Now, the non-idling condition implies that whenever $y_j > 0$ so that work is present at station $s(j)$ , $\\sum _{i: s(i) = s(j)} w_i = 1,$ so that station $j$ allocates all of its effort.",
"Hence, for any value of $y_j$ , including 0, $y_j = \\sum _{i:s(i) = s(j)} w_i y_j,$ and consequently, $ {\\bar{y}}_j = \\sum _{i:s(i) = s(j)} {\\bar{z}}_{ij}.$ Therefore, if we let ${\\bar{z}}$ be a column vector containing the ${\\bar{z}}_{jk}$ 's, then the expression (REF ) may be written as $u_{jk}^{\\prime } {\\bar{z}}= c_{jk}$ , for a suitably defined column vector $u_{jk}$ and constant $c_{jk}$ .",
"By considering the function $h_{jj}(y) = y_j^2$ , we obtain $0 = 2 \\gamma _j + 2 \\lambda _j {\\bar{y}}_j - 2 \\mu _j {\\bar{z}}_{jj} + 2 \\sum _i\\mu _i R_{ij} {\\bar{z}}_{ij},$ for $j = 1, \\ldots , \\ell $ , and again these equations can be written as $u_{jj}^{\\prime } {\\bar{z}}= c_{jj}$ for suitably defined $u_{jj}$ and $c_{jj}$ .",
"We obtain one equation for each $1 \\le j \\le k \\le \\ell $ , so that in all there are $\\ell (\\ell +1) / 2$ equations of the form $u_{jk}^{\\prime } {\\bar{z}}= c_{jk}$ .",
"If we now write the vectors $\\lbrace u_{jk} \\rbrace $ as columns in a matrix $U$ , and the values $\\lbrace c_{jk} \\rbrace $ in a vector $c$ , these equations can be written as $U^{\\prime } {\\bar{z}}= c$ .",
"The matrix $U$ has $\\ell ^2$ rows, and $\\ell (\\ell +1) / 2$ columns.",
"Although we began with expressions involving the Markov chain $Y$ , we are now working with the Markov chain $Z = (Z(n): n \\ge 0)$ .",
"Therefore, it is useful to express the function $|y|$ as $p^{\\prime }z$ , for some vector $p$ .",
"In particular, $p_{ij} = 1$ if $s(i) = s(j)$ , and 0 otherwise.",
"In view of (REF ), the estimator $\\alpha (n)$ may be written as $p^{\\prime }{\\bar{Z}}(n)$ , where ${\\bar{Z}}(n) = n^{-1} \\sum _{i=0}^{n-1} Z(i)$ .",
"So define the quadratic estimator as $\\alpha _q(n) & \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}& p^{\\prime } {\\bar{Z}}(n) + \\beta \\nu ^{\\prime } (U^{\\prime } {\\bar{Z}}(n) - c)\\nonumber \\\\& = & (p + \\beta U \\nu )^{\\prime } {\\bar{Z}}(n) - \\beta \\nu ^{\\prime } c, $ where $\\nu $ is a vector of coefficients.",
"This is again of the form $\\alpha _q(n) = |\\bar{Y}(n)| +\\beta \\Delta _h (\\bar{Y}(n) ),$ where $h$ is a quadratic, $h(y) = y^{\\prime }Qy + \\zeta ^{\\prime }y$ for some matrix $Q$ and vector $\\zeta $ .",
"However in the case of networks, we can only hope that $h$ approximately solves Poisson's equation, in the sense that $Ph(y) - h(y) = - d(y)+ \\alpha _d$ , with $d(\\,\\cdot \\,)\\approx |\\,\\cdot \\,|$ .",
"Let us assume (for now) that $\\beta = 1$ .",
"The variance of (REF ) is then given by $ (p + U \\nu )^{\\prime } \\Lambda _n (p + U\\nu ),$ where $\\Lambda _n$ is the covariance matrix of ${\\bar{Z}}(n)$ .",
"But under appropriate initial conditions and assuming (A) holds, $\\Lambda _n \\sim \\Lambda / n$ , where $\\Lambda $ is the time-average covariance matrix for ${\\bar{Z}}(n)$ .",
"Then (REF ) is asymptotically given by $ n^{-1} \\Vert p + U \\nu \\Vert _\\Lambda ^2 \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}n^{-1} (p + U \\nu )^{\\prime } \\Lambda (p +U\\nu ).$ Standard control variate methodology suggests that one could estimate the covariance matrix $\\Lambda $ (or $U^{\\prime } \\Lambda U$ ), and then choose $\\nu $ to minimize the vector norm (REF ) [31].",
"However, as cautioned in [31], there is the danger of a variance increase associated with the additional estimation of the covariance matrix.",
"The “loss factor” is discussed in [30] and [39] for terminating simulations, and in [32] for steady-state simulation.",
"The loss factor can become an issue when many control variables are used (as is potentially the case here).",
"Rather than attempt to determine an optimal selection of control variables from those at our disposal, we instead choose to avoid the issue altogether by preselecting $\\nu $ , and then using a standard approach to select the single parameter $\\beta $ .",
"See [21] for further discussion related to this point.",
"From (REF ), it is “optimal” to choose $\\nu $ to minimize $\\Vert U \\nu + p\\Vert _\\Lambda $ .",
"Since $\\Lambda $ is unknown prior to the simulation, we instead choose $\\nu $ to minimize $\\Vert U\\nu + p\\Vert _i$ for some $L_i$ norm $\\Vert \\cdot \\Vert $ .",
"This problem can be solved using linear programming if $i$ is chosen to be 1 or $\\infty $ , or using least-squares methods if $i = 2$ .",
"In addition, for some workload policies, for example preemptive priority policies, it is known that ${\\bar{z}}_{ij} = 0$ for some $i$ and $j$ .",
"In this case, we may modify the norm used in the minimization slightly to ignore the cost coefficient of ${\\bar{z}}_{ij}$ .",
"See [21] for further discussion of this point, and [27] for the related concept of auxiliary constraints.",
"It remains to explain how $\\beta $ is selected.",
"Given a response $X$ , a control variable $C$ , and the form of the controlled estimator $X +\\beta C$ , it is well known that the value of $\\beta $ that minimizes the controlled variance is $\\beta ^* = -\\operatorname{Cov}(X, C) / \\operatorname{Var}C$ .",
"In our case, the response $X$ is $p^{\\prime } {\\bar{Z}}(n)$ and the control $C$ is $\\nu ^{\\prime }(U^{\\prime }{\\bar{Z}}(n) - c)$ .",
"One may use any reasonable approach to estimate $\\beta ^*$ from the simulation.",
"In particular, [32] discusses how this may be done in a steady-state context using both regenerative and batch-means approaches.",
"The process $Z$ is regenerative with regeneration times defined by the hitting times of the state 0, but the regenerative cycles can be expected to be very long.",
"So we suggest instead using a batch-means approach to estimating $\\beta ^*$ .",
"The required calculations are taken from [32], and are summarized in the appendix.",
"Theorem REF , also in the appendix, gives the relevant asymptotic theory for the quadratic estimator.",
"Basically, under the assumption (A), the quadratic estimator converges in probability, and is asymptotically $t$ -distributed when suitably normalized.",
"The convergence mode being “in probability” results from the fact that the estimator for $\\beta ^*$ is only weakly consistent.",
"If however, a strongly consistent estimator for $\\beta $ were used, or if $\\beta $ were chosen to be a constant (e.g., 1), then the quadratic estimator would be strongly consistent.",
"In summary, to estimate $\\alpha $ : Choose $\\nu $ to minimize $\\Vert U \\nu + p\\Vert $ for some suitable norm.",
"Simulate the Markov chain $Z$ up until time $n$ .",
"(This amounts to simulating $Y$ since $Z$ is a deterministic function of $Y$ .)",
"Compute $\\beta (n)$ , the estimate of $\\beta ^*$ .",
"Compute the estimator $\\alpha _q(n) = p^{\\prime } {\\bar{Z}}(n) + \\beta (n)\\nu ^{\\prime }(U^{\\prime } {\\bar{Z}}(n) - c)$ .",
"Simulation results for the quadratic estimator and three separate queueing networks are given in [21].",
"Simulation results for the network in Figure REF operating under the FBFS (first buffer, first-served) preemptive priority policy are given in Table REF .",
"We minimized $\\Vert U \\nu + p\\Vert _2$ to determine $\\nu $ , and ignored the additional information that ${\\bar{z}}_{31} = 0$ under the chosen service policy.",
"The results are representative of all the other networks we experimented with.",
"Table: Simulation Results for the Two-Station Three-BufferExample (to two significant figures).We took $\\mu _1 = \\mu _3 = 22$ , $\\mu _2 =10$ , and then chose $\\lambda $ to ensure that $\\rho _2 \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\lambda /\\mu _2$ was as specified in the table.",
"Before conducting the experiments, time was rescaled so that $\\lambda + \\sum _i \\mu _i = 1$ .",
"Results for the standard estimator are provided in the first two columns, and those for the quadratic estimator are in the next two columns.",
"The simulations were run for 100,000 time steps using 20 batches, and we repeated the experiments 200 times to obtain an estimate of the error in the estimators.",
"For each estimator we supply the estimated mean and variance (over the 200 runs).",
"The column labeled “Reduction” gives the ratio of the observed variances.",
"The results for $\\rho = 0.99$ are subject to some suspicion, owing to the fact that our batch means exhibited correlation.",
"We see that for lightly to moderately loaded systems, variance reduction factors on the order of between 120 and 7 are observed.",
"The largest variance reduction occurs in light traffic, while smaller variance reductions are obtained in moderately loaded systems.",
"These variance reductions are certainly useful, and come at very little additional computational cost, since the only real additional computational cost in computing the estimator $\\alpha _q(n)$ as opposed to the standard estimator $\\alpha (n)$ is the solution of the optimization problem to choose $\\nu $ .",
"But this problem is solved once only before the simulation begins, and takes a (very) small amount of time to solve relative to the computational effort devoted to the simulation.",
"As discussed earlier, we are more interested in the performance of these simulation estimators in moderately to heavily loaded systems.",
"The results in Table REF suggest that our estimator is less effective (relative to the standard estimator) in heavy traffic.",
"In particular, for (very) heavily loaded systems, only modest variance reductions are seen.",
"It is conceivable that the disappointing performance of the controlled estimator in heavy traffic is due to the heuristic of choosing the multipliers $\\nu $ prior to the simulation, as opposed to attempting to make an “optimal” choice.",
"To test this idea, we estimated the maximum possible variance reduction using the estimator $\\alpha _q(n)$ .",
"For several networks, and for various traffic loadings, we estimated the time-average covariance matrix $\\Lambda $ .",
"As discussed earlier, the maximum possible variance reduction (asymptotically) is obtained by selecting $\\nu $ to minimize $\\Vert U\\nu + p\\Vert _\\Lambda $ .",
"The results for the network in Figure REF are presented in Table REF .",
"Table: The best possible performance for the quadratic estimator inthe two-station three-buffer example (2 significant figures).The columns headed “Standard” and “Quadratic” give an estimate of the asymptotic variance for the standard and best possible quadratic estimators respectively.",
"The column headed “Reduction” gives the ratio of these two values.",
"The simulation run lengths required to get reasonable estimates of $\\Lambda $ for $\\rho _2 > 0.9$ were infeasibly large, and so we omitted these values.",
"Comparing these results with those of Table REF , we see that the potential variance reductions appear to decrease as congestion increases.",
"However, substantial variance reductions may yet be possible with a carefully chosen weighting vector $\\nu $ .",
"In view of the “loss factor,” the question of how best to choose $\\nu $ to achieve greater variance reduction using the quadratic estimator is an interesting open question.",
"In the absence of a more-effective candidate than the one we have suggested, it would seem that a different approach to generating control variates is warranted for heavily loaded systems."
],
[
"Fluid Models and Stability",
"To motivate our second approach to generating control variates for the simulation of multiclass queueing networks, we consider an alternative expression [37] for the solution $h^*$ to Poisson's equation (REF ), namely $ h^*(y) = \\operatorname{\\mathbb {E}}_y \\sum _{k=0}^{\\kappa } (|Y(k)| - \\alpha ),$ where $\\kappa = \\inf \\lbrace n \\ge 0: Y(n) = 0\\rbrace $ .",
"Although it is difficult to compute (REF ) exactly, we can certainly approximate it through the use of fluid models.",
"As before, we introduce the key ideas through the M/M/1 queue.",
"Continuation of Example : Consider a uniformized (and discrete-time) process $Y =(Y(n): n \\ge 0)$ describing the queue length in an M/M/1 queue.",
"We have the recursion $Y(n+1) = (Y(n) + I(n+1))^+,$ for $n \\ge 0$ , where $I = (I(n): n \\ge 1)$ is a Bernoulli, i.i.d.",
"process: $\\lambda = P(I(n) = 1)$ is the arrival rate, and $\\mu = P(I(n)= -1)$ is the service rate.",
"Time has been normalized so that $\\lambda + \\mu = 1$ .",
"Figure: (a) A sample path YY of the M/M/1 queue withρ=λ/μ=0.9\\rho =\\lambda /\\mu = 0.9, and Y(0)=400Y(0)=400.",
"(b) A solutionto the differential equation φ ˙=𝕀(φ>0)(λ-μ)\\dot{\\phi }=\\mathbb {I}(\\phi > 0)(\\lambda -\\mu )starting from the same initial condition.To construct an approximation to the solution to Poisson's equation, first note that in heavy traffic, the network will typically be somewhat congested, and so we are primarily concerned with “large” states.",
"So it may pay to consider the process starting from a large initial condition.",
"In the left-hand side of Figure REF we see one such simulation.",
"One approach to computing the solution to Poisson's equation is to compute (REF ).",
"While this is easy for the M/M/1 queue, such computation can be formidable for more complex network models.",
"However, consider the right-hand side of Figure REF which shows a sample path of the deterministic fluid, or leaky bucket model.",
"This satisfies the differential equation $\\dot{\\phi }=\\mathbb {I}(\\phi > 0)(\\lambda - \\mu )$ , where $\\mathbb {I}(\\cdot )$ is the indicator function that is 1 if its argument is true and 0 otherwise.",
"The behavior of the two processes looks similar when viewed on this large spatial/temporal scale.",
"It appears that a good approximation is $h^*(x)\\approx $ $\\begin{array}{rcl}\\displaystyle h(x) &\\Delta \\over =&\\displaystyle \\int _0^\\infty \\phi (t)\\, dt , \\quad \\phi (0)=x,\\strut \\\\\\displaystyle & =&\\displaystyle \\frac{1}{2}\\frac{x^2}{\\mu -\\lambda }.\\end{array}$ This is the same approximation arrived at in Section .",
"Of course, the M/M/1 queue is a very special case of a multiclass queueing network, and so it is worthwhile investigating this approximation more carefully before adopting it wholesale.",
"We return now to the case of a general multiclass queueing network.",
"The dynamics of the process $Y$ can be described by a random linear system after a slight extension of the previous definitions.",
"Define a sequence of i.i.d.",
"random matrices $\\lbrace I(n): n \\ge 1\\rbrace $ on $\\lbrace 0,1\\rbrace ^{(\\ell + 1)^2}$ , with $\\operatorname{\\mathbb {P}}\\lbrace \\sum _j \\sum _k I_{j, k}(n) = 1\\rbrace =1$ , and $\\operatorname{\\mathbb {E}}[I_{j, k}(n)]=\\mu _j R_{jk}$ , where $\\mu _j$ denotes the service rate for class $j$ customers.",
"Note that exactly one element of $I(n)$ is positive for each $n$ .",
"These random variables indicate which event in the uniformized process $Y = (Y(n): n \\ge 0)$ is to occur.",
"The variable $I_{jk}(n) = 1$ if and only if a class $j$ job completes service and moves to station $k$ .",
"It is convenient to capture the exogenous arrival processes within the same framework.",
"An exogenous arrival is indicated by $j = 0$ , and a departure from the system is indicated by $k = 0$ .",
"For $j=0$ , let $\\mu _0 \\Delta \\over =\\sum _{k=1}^\\ell \\lambda _k$ denote the overall arrival rate of customers to the system.",
"Define $R_{0, 0} = 0$ , and for $1 \\le k \\le \\ell $ , $R_{0, k} = \\lambda _k / \\mu _0$ .",
"Thus, we pool all customer arrivals into one stream with rate $\\mu _0$ , and an arriving customer is allocated to one of the $\\ell $ classes according to the appropriate probability.",
"For $1 \\le j \\le \\ell $ , let $W_j(n) = 1$ if station $s(j)$ is allocating its entire effort to customers of class $j$ at time $n$ , and 0 otherwise.",
"As before we require that $W_j(n)$ is a deterministic function of $Y(n)$ for all $n \\ge 0$ and all $j = 1, \\ldots , \\ell $ .",
"We define $W_0(n) = 1$ for all $n$ , indicating that the exogenous arrival process is always active.",
"For $1\\le k \\le \\ell $ , let $e^k$ denote the $k$ th basis vector in $\\mathbb {R}^\\ell $ , and set $e^0 \\Delta \\over =0$ .",
"The random linear system can then be defined as $Y(n+1) = Y(n) +\\sum _{j=0}^\\ell \\sum _{k=0}^\\ell I_{j,k}(n+1) [-e^j + e^k ] W_j(n),$ where the state process $Y$ denotes the vector of customer classes in the system as before.",
"To define the fluid model associated with this network we suppose that the initial condition is large so that $m=|Y(0)|\\gg 1$ .",
"We then construct a continuous time process $\\phi ^y (t)$ as follows: If $tm$ is an integer, we set $\\phi ^y (t) = \\frac{1 }{ m} Y (m t),$ where $Y(0) = y$ and $|y| = m$ .",
"For all other $t \\ge 0$ , we define $\\phi ^y(t)$ by linear interpolation, so that it is continuous and piecewise linear in $t$ .",
"Note that $|\\phi ^y(0)|=1$ , and that $\\phi ^y$ is Lipschitz continuous (see, e.g., [1] for a definition of Lipschitz continuity.)",
"The collection of all “fluid limits” is defined by ${\\cal L}\\Delta \\over =\\bigcap _{m=1}^\\infty \\overline{ \\lbrace \\phi ^y : |y| > m \\rbrace }$ where the overbar denotes weak closure.",
"The set ${\\cal L}$ depends upon the particular policy chosen, and for many policies such as preemptive priority policies, it is a family of purely deterministic functions.",
"Any process $\\phi \\in {\\cal L}$ evolves on the state space $\\mathbb {R}_+^\\ell $ and, for a wide class of scheduling policies, satisfies a differential equation of the form $\\frac{d}{dt} \\phi (t) = \\sum _{j=0}^\\ell \\sum _{k=0}^\\ell \\mu _j R_{jk}[-e^j+e^k] u_j(t)$ where the function $u(\\cdot )$ is analogous to the discrete control, and satisfies similar constraints (see the M/M/1 queue model described earlier, or [9] and [12] for more general examples).",
"In many cases the differential equation (REF ) admits a unique solution, from any initial condition, even though typically in practice the control $u$ is a discontinuous function of the state $\\phi $ (consider again any priority policy).",
"It is now known that stability of (REF ) is closely connected with the stability of the fluid model [9], [27], [11].",
"The fluid model ${\\cal L}$ is called $L_p$ -stable if $\\lim _{t \\rightarrow \\infty } \\ \\sup _{\\phi \\in {\\cal L}} \\ \\operatorname{\\mathbb {E}}[ | \\phi (t) |^p] = 0.$ Let $T_0$ denote the first hitting time $\\inf \\lbrace t\\ge 0 : \\phi (t) =0\\rbrace $ .",
"It is shown in [33] that $\\sup _{\\phi \\in {\\cal L}}\\operatorname{\\mathbb {E}}[T_0]<\\infty $ when the model is $L_2$ -stable.",
"Hence, when ${\\cal L}$ is non-random, $L_2$ -stability is equivalent to stability in the sense of [9]: There is some time $T$ such that $\\phi (t)=0$ for $t\\ge T$ , $\\phi \\in {\\cal L}$ .",
"For example, in the M/M/1 queue with $\\lambda < \\mu $ , the queue eventually hits 0 as seen in Figure REF .",
"The following result is a minor generalization of results from [27], [11].",
"Its proof is omitted.",
"Theorem 3 The following two stability criteria are equivalent for the network under any non-idling policy, and any $p\\ge 2$ .",
"(i) There is a function $V$ , and a constant $b<\\infty $ satisfying $P V(y) - V(y) \\le -|y|^{p-1} + b$ where for some $\\delta >0$ , $\\delta (1+ |y|^p) \\le V(y) \\le \\delta ^{-1}(1+ |y|^p),\\qquad y\\in S.$ (ii) The fluid model ${\\cal L}$ is $L_p$ -stable.",
"Thus, $L_p$ stability can be verified through the Lyapunov condition (A).",
"Using this result it is possible to show that the solution to Poisson's equation is asymptotically equal to a value function for the associated fluid model, provided that the fluid model is $L_2$ -stable.",
"It can be shown that many policies for the fluid model are piecewise constant on a finite set of cones in $\\mbox{I\\hspace{-2.27621pt}R}_+^\\ell $ .",
"This is certainly the case for buffer priority policies, and also holds for $L_1$ optimal policies (for a discussion see [46]).",
"It then follows that for such policies the fluid value function $V$ is piecewise quadratic.",
"The proof of the following result appears in the appendix.",
"Theorem 4 Suppose that for a given non-idling policy $w$ , the fluid model ${\\cal L}$ is $L_2$ -stable and non-random.",
"Suppose moreover that limits are unique, in the sense that $\\phi _1(0)\\ne \\phi _2(0)$ for any two distinct $\\phi _i\\in {\\cal L}$ .",
"Then a solution $h^*$ to Poisson's equation exists, and $\\limsup _{ |y| \\rightarrow \\infty } \\Bigl | \\frac{h^*(y)}{ V(y)} - 1\\Bigr | =0,$ where $V(y) = |y|^2 \\int _0^\\infty |\\phi (t)| \\, dt, \\qquad \\phi (0)= \\frac{y}{|y|}.$ Hence, under the conditions of Theorem REF , a solution $h$ to Poisson's equation is intimately related to the fluid value function $V$ .",
"This result then strongly motivates the use of a fluid value function as an approximation for the solution to Poisson's equation.",
"Another way to motivate the fluid approximation is to note that (from (REF )) $Y(n+1) &=& Y(n) + \\sum _{j, k} \\mu _j R_{jk}(-e^j + e^k) W_j(n)\\nonumber \\\\& + & \\sum _{j, k} (I_{jk}(n+1) - \\mu _j R_{jk})(-e^j + e^k) W_j(n)\\nonumber \\\\& = & Y(n) + B W(n) + D(n+1) \\nonumber \\\\& = & Y(0) + \\sum _{i=0}^n B W(i) + M(n+1), $ where $B$ is an $(\\ell +1) \\times (\\ell + 1)$ matrix.",
"The process $M(\\cdot )$ is a vector-valued martingale with respect to the natural filtration, and $D(i)$ is the martingale difference $M(i) - M(i-1)$ .",
"(See, e.g., [41] for an introduction to martingales.)",
"It is straightforward to check that $E D(i)^{\\prime } D(i)$ is bounded in $i$ (by $b$ say), so that $E M(n)^{\\prime }M(n)\\le b n$ for all $n \\ge 0$ .",
"Hence, the network is essentially a deterministic fluid model with a `disturbance' $M$ .",
"When the initial condition $Y(0)$ is large, then the state dominates this disturbance, and hence the network behavior appears deterministic.",
"We therefore have strong motivation for approximating the solution to Poisson's equation $h^*$ by $V$ , where $V(y) \\mbox{$\\,\\stackrel{\\bigtriangleup }{=}\\,$}\\int _0^\\infty \\phi (t)\\, dt,$ and $\\phi $ solves the differential equation (REF ) with $\\phi (0) = y$ .",
"The fluid estimator of $\\alpha $ is then $ \\alpha _f(n) = |{\\bar{Y}}(n)| + \\frac{\\beta }{n} \\sum _{k=0}^{n-1}\\Delta _V(Y(k)).$ The parameter $\\beta $ is again a constant that may be chosen to attempt to minimize the variance of the fluid estimator.",
"We use the methodology outlined in the appendix to estimate the optimal $\\beta $ .",
"Consequently, the asymptotic results for the quadratic estimator also apply here, namely that under the assumption (A), the fluid estimator is weakly consistent, and is asymptotically $t$ -distributed when suitably normalized.",
"Clearly, to implement the fluid estimator we need to be able to compute $\\Delta _V$ .",
"For any function $V$ , we have that $PV(y) & = & \\operatorname{\\mathbb {E}}_y V(Y(1)) \\\\& = & \\sum _{j=0}^\\ell \\sum _{k=0}^\\ell \\mu _j R_{jk} V(y - e^j +e^k) W_j(y),$ so that it suffices to be able to compute $V(y)$ for $y \\in S$ .",
"Solving the differential equation (REF ) to find $\\phi $ is not difficult when the fluid control $u$ is piecewise constant on a finite set of cones in $\\mbox{I\\hspace{-2.27621pt}R}_+^\\ell $ since in this case $\\phi $ is piecewise linear.",
"Integrating $\\phi $ to find the fluid value function $V$ is then straightforward.",
"In other words, for a specific model, some preliminary work has to be done to give code that can compute $V$ , but this is usually not a difficult step.",
"Algorithms for computation or approximation of $V$ are described in [14] and [7].",
"It should be apparent from the above discussion that computing the control $\\Delta _V$ for the estimator (REF ) may be moderately time-consuming (computationally speaking) relative to the time taken to simply simulate the process $Y$ .",
"Be that as it may, it is certainly the case that the time taken to compute the control is relatively insensitive to the congestion in the system.",
"In Table REF we present simulation results for the fluid estimator on the network of Figure REF .",
"(Similar results were obtained for all other multiclass queueing networks that we tried.)",
"The entries in Table REF have the same interpretation as those in Table REF .",
"In particular, the column headed “Reduction” represents the variance reduction factor over the standard estimator.",
"Table: Simulation results for the two-station three-bufferexample (2 significant figures).",
"The interpretation of the values given is the same as in Table .The best value of $\\beta $ was found to be close to unity in each of the simulations, particularly at high loads where it was found to be within $\\pm 5$ % of unity.",
"Observe that for low traffic intensities, the fluid estimator yields reasonable variance reductions over the standard estimator.",
"However, because it is more expensive to compute than the standard estimator, these results are not particularly encouraging.",
"But as the system becomes more and more congested, the fluid estimator yields large variance reductions over the standard estimator, meaning that the extra computational effort per iteration is certainly worthwhile.",
"For very high traffic intensities, the fluid estimator significantly outperforms both the standard estimator and the quadratic estimator, and so we have achieved our goal of deriving an estimator that can be effective in heavy traffic."
],
[
"Conclusions",
"We have given two simulation estimators for estimating a linear function of the steady-state customer class population.",
"The quadratic estimator produces very useful variance reductions in light to moderate traffic at very little additional computational cost.",
"We recommend that it be used in simulations of such lightly loaded networks.",
"The quadratic estimator is less effective in simulations of heavily loaded networks, but could potentially provide useful variance reductions in this regime if a better choice of weighting vector $\\nu $ can be employed.",
"The fluid estimator provides modest variance reduction in light to moderate traffic, but appears to be very effective in heavy traffic.",
"There is an additional computational overhead in computing the fluid estimator, but this overhead is (roughly) independent of the load on the network.",
"Hence we may conclude that in heavily loaded systems, the fluid estimator should yield significant computational improvements, and should therefore be used.",
"One might conclude from the above discussion that the quadratic and fluid estimators could be combined using the method of multiple control variates to yield a single “combined” estimator.",
"However, we believe that it is unlikely that a combined estimator would yield significant improvements over the use of either the quadratic estimator (in light traffic) or the fluid estimator (in heavy traffic).",
"In light traffic, we expect that the additional reductions in variance would be negated by the increased computational effort.",
"And in heavy traffic we expect that any additional variance reduction would be modest, owing to the weaker performance of the quadratic estimator in this regime.",
"Finally, we note that [45] explores bounded perturbations of the fluid value function in approximate dynamic programming.",
"Related techniques are considered in current research to refine the fluid estimator."
],
[
"Proof of Theorem ",
"It is straightforward to show that $ \\Delta _{f^2}(y) = -2(1-\\rho )f(y) + e^{\\prime }Q [L + Q^{-1}W - {\\cal M}WR +\\mbox{diag}(e^{\\prime }{\\cal M}WR) ]Q^{\\prime }e,$ where $W$ is the diagonal matrix containing the work allocation vector $w$ corresponding to $y$ , and $L =$ diag$(\\lambda )$ .",
"Furthermore, for $m \\ge 3$ , $ \\Delta _{f^m}(y) = -m(1-\\rho ) f^{m-1}(y) + \\mbox{lower order terms}.$ It follows from (REF ) with $m = 5$ that (A) holds.",
"This implies (Theorem 17.5.3 of [37]) that the TAVC's are finite, and furthermore, that $\\sigma ^2_f(\\rho ) = \\lim _{n \\rightarrow \\infty } n \\operatorname{Var}(d^{\\prime } {\\bar{Y}}(n)), \\mbox{ and}$ $\\sigma ^2_i(\\rho ) = \\lim _{n \\rightarrow \\infty } n \\operatorname{Var}({\\bar{Y}}_i(n)).$ The fact that $h$ is of the form given in the theorem follows from (REF ).",
"Observe that the function $f^2 / (2(1-\\rho ))$ is “almost” the solution to Poisson's equation.",
"It needs to be adjusted slightly to remove the terms in the RHS of (REF ) involving the work allocation vector $w$ .",
"These terms are of the form $f^{\\prime }w$ , where the coefficients in $f$ are bounded in $\\rho $ .",
"Therefore, the solution to Poisson's equation is as given in the theorem.",
"So then the TAVC $\\sigma ^2_f(\\rho )$ is given by $ \\operatorname{\\mathbb {E}}h(Y)(f(Y) - d^{\\prime }{\\bar{y}}) = \\frac{\\operatorname{Cov}(f^2(Y), f(Y))}{2(1-\\rho )}- \\operatorname{\\mathbb {E}}(c_\\rho ^{\\prime }Y \\, (f(Y) - d^{\\prime }{\\bar{y}})),$ where $Y$ is distributed according to the stationary distribution $\\pi $ and ${\\bar{y}}= \\operatorname{\\mathbb {E}}Y$ .",
"Since $\\operatorname{\\mathbb {E}}\\Delta _{f^m}(Y) = 0$ for $m = 0, \\ldots , 4$ , it follows from (REF ) that $\\operatorname{\\mathbb {E}}f(Y)$ is of the order $(1-\\rho )^{-1}$ as $\\rho \\rightarrow 1$ .",
"Then, by induction using (REF ), $\\operatorname{\\mathbb {E}}f^m(Y)$ is of the order $(1-\\rho )^{-m}$ as $\\rho \\rightarrow 1$ for $m = 1, \\ldots , 4$ .",
"The second term on the right-hand side of (REF ) is therefore of the order $(1-\\rho )^{-3}$ as $\\rho \\rightarrow 1$ .",
"As for the first term, we have the easily proved inequality that for any non-negative r.v.",
"$X$ with $\\operatorname{\\mathbb {E}}X^3 < \\infty $ , $\\operatorname{Cov}(X^2, X) \\ge \\frac{\\operatorname{Var}X}{\\operatorname{\\mathbb {E}}X^2} \\operatorname{\\mathbb {E}}X^3.$ Applying this inequality to the first term on the right-hand side of (REF ), and noting that $\\operatorname{Var}f(Y)$ is of the same order as $\\operatorname{\\mathbb {E}}f(Y)^2$ as $\\rho \\rightarrow 1$ , we obtain the required result that $\\sigma ^2_f(\\rho )$ is of the order $(1-\\rho )^{-4}$ as $\\rho \\rightarrow 1$ .",
"The last statement of the theorem follows from the fact that $\\sigma ^2_f(\\rho ) & = & \\lim _{n \\rightarrow \\infty } n \\operatorname{Var}(d^{\\prime } {\\bar{Y}}(n)) \\\\& = & \\lim _{n \\rightarrow \\infty } n \\operatorname{Var}(\\sum _{i=1}^\\ell d_i {\\bar{Y}}_i(n)) \\\\& \\le & \\lim _{n \\rightarrow \\infty } \\ell n \\sum _{i=1}^\\ell \\operatorname{Var}(d_i \\bar{Y}_i(n)) \\\\& = & \\ell \\sum _{i=1}^\\ell d_i^2 \\sigma ^2_i(\\rho ).$ We repeat formulae from [32] for estimating the control variate parameter $\\beta $ that is used in both the quadratic estimator (REF ) and the fluid estimator (REF ) via the batch means method of simulation output analysis.",
"To encapsulate both estimators and avoid repetition, we give the formulae for the case where a real-valued stochastic process $X =(X(n): n \\ge 0)$ is simulated, and a real-valued control $C = (C(n): n\\ge 0)$ is recorded.",
"Let the $b$ batches each consist of $m$ observations, so that the simulation run-length $n=mb$ .",
"If $X_i$ and $C_i$ are the $i$ th batch means of the process and control respectively, then for $0 \\le i \\le b-1$ , we have $X_i = \\frac{1}{m} \\sum _{j=im}^{(i+1)m - 1} X(j) \\mbox{ and }C_i = \\frac{1}{m} \\sum _{j=im}^{(i+1)m - 1} C(j).$ Let ${\\bar{X}}_n$ and ${\\bar{C}}_n$ denote the overall (sample) means of the process and control respectively.",
"Define $V_{XX}(n) & = & \\frac{1}{b-1} \\sum _{i=0}^{b-1} (X_i - {\\bar{X}}_n)^2, \\\\V_{CC}(n) & = & \\frac{1}{b-1} \\sum _{i=0}^{b-1} (C_i - {\\bar{C}}_n)^2, \\mbox{and}\\\\V_{XC}(n) & = & \\frac{1}{b-1} \\sum _{i=0}^{b-1} (X_i - {\\bar{X}}_n)(C_i -{\\bar{C}}_n).$ Define $\\beta = -V_{XC} / V_{CC}$ , and let $\\alpha _n = {\\bar{X}}_n + \\beta {\\bar{C}}_n$ be the controlled estimator.",
"Finally, let $R^2(n) = \\frac{b-1}{b-2}\\left(V_{XX}(n) -\\frac{V_{XC}(n)^2}{V_{CC}(n)} \\right)$ and $ S^2(n) = R^2(n) \\left( \\frac{1}{b} + \\frac{1}{b-1} \\frac{{\\bar{C}}_n^2}{V_{CC}} \\right).$ Using the above computational process, we can construct the quadratic estimator $\\alpha _q(n)$ and the fluid estimator $\\alpha _f(n)$ .",
"The following result describes the asymptotic behavior of these estimators.",
"Theorem 5 (Loh 1994) Under the assumption (A), the estimators $\\alpha _q(n)$ and $\\alpha _f(n)$ converge in probability to $\\alpha $ , and for $j = q, f$ , $\\frac{\\alpha _j(n) - \\alpha }{S_j(n)} \\mbox{ $\\Rightarrow $ }T_{b-2},$ where $T_{b-2}$ has the Student's $t$ -distribution with $b-2$ degrees of freedom, and $S_j(n)$ is defined in the obvious way through (REF )."
],
[
"The second result is proved in Section 1.3.2 of [32].",
"In addition, Proposition 1.5 of [32] shows that $nS_j^2(n)$ converges in distribution to a finite-valued random variable as $n\\rightarrow \\infty $ so that the first result follows."
],
[
"Proof of Theorem ",
"First note that for any $\\phi \\in {\\cal L}$ we have $|\\phi (0)|=1$ , and under the assumptions of the theorem there is a $T>0$ such that $V(y) = |y|^2 \\int _0^T |\\phi (t)| \\, dt, \\qquad \\phi (0)=\\frac{y}{|y|} ,\\quad \\phi \\in {\\cal L}.$ We shall fix such a $T$ throughout the proof.",
"From Theorem REF , a Lyapunov function exists that satisfies (REF ), which is equivalent to a quadratic in the sense of (REF ).",
"It follows that $\\pi (c) <\\infty $ , where $c(y) = |y|$ , and that a solution to Poisson's equation exists which is bounded from above by a quadratic, and uniformly bounded from below [37].",
"We can then take the solution to Poisson's equation and iterate as follows: $P^nh = h - \\sum _{i=0}^{n-1} P^i\\bar{c}$ , where $\\bar{c}(y) = |y| -\\alpha $ .",
"Let $m=|y|$ and take $n = [m T] =$ the integer part of $mT$ to give $\\frac{\\operatorname{\\mathbb {E}}_{ y}[h(Y( m T))]}{ m^2}= \\frac{h( y)}{ m^2}- \\frac{\\operatorname{\\mathbb {E}}_{ y}\\Bigl [ \\sum _{i=0}^{ [m T]-1}\\frac{|Y(i)|}{m}\\Bigr ]}{m}- \\frac{T\\alpha }{ m}.$ Since $h$ is bounded above by a quadratic, there is a $K<\\infty $ such that $\\left| \\frac{[h(Y( m T))]}{ m^2} \\right|\\le K\\Bigl (1+ \\frac{ |Y( m T)|^2}{ m^2}\\Bigr )$ The random variable on the right hand side is uniformly bounded by $K(1+1/m+ T)^2$ for all initial $y$ since at most one customer can arrive during each time slot.",
"It then follows from weak convergence (see, e.g., [6]) and the definition of $T$ that $\\limsup _{|y|\\rightarrow \\infty } \\operatorname{\\mathbb {E}}_y \\left| \\frac{[h(Y( m T))]}{ m^2}\\right|=0$ Moreover, again by Lipschitz continuity of the fluid model we have $\\limsup _{|y|\\rightarrow \\infty }\\left| \\operatorname{\\mathbb {E}}_{ y}\\left[\\frac{1}{ m}\\sum _{i=0}^{ [m T]-1} \\frac{|Y(i)|}{ m}\\right] - V(\\frac{y}{m})\\right| =0.$ Putting these results together we see that $\\limsup _{|y|\\rightarrow \\infty }\\left| \\frac{h(y)}{m^2} -V(\\frac{y}{m})\\right| = 0,$ proving the result."
]
] | 2005.14179 |
[
[
"Testing Lambert$W$ equation of state with observational Hubble parameter\n data"
],
[
"Abstract In this paper, we investigate the possibility that the Universe is driven by a single dark fluid described by a Lambert $W$ equation of state parameter, $w_{eff}$, which is essentially dependent on two parameters $\\vartheta_{1}$ and $\\vartheta_{2}$ which need to be fixed from observations.",
"We obtain the constraints on these parameters using the latest 51 data points of $H(z)$ measurements, spanning the redshift range $0.07\\leq z \\leq 2.36$.",
"The present study shows that the Universe is indeed undergoing an accelerated expansion phase following the decelerated one at the transition redshift, $z_{t}=0.77\\pm0.03$ ($1\\sigma$) and is well consistent with the recent observations.",
"We also find that at low redshifts, $w_{eff}$ evolves only in the quintessence regime ($-1<w_{eff}<-\\frac{1}{3}$) within $1\\sigma$ confidence level.",
"Its present value is found to be $-0.96\\pm0.02$ ($1\\sigma$).",
"The fact that the present value of $w_{eff}$ is very close to the Cosmological Constant $\\Lambda$ implies that our proposed equation of state parameter might serve as a unification of dark matter and dark energy.",
"Furthermore, we compare the evolution of $H(z)$ for the model under consideration with that of the $\\Lambda$CDM model.",
"Finally, we observe that for the best-fit case, the differences between the two models are negligible at $z\\sim 0.67$."
],
[
"Introduction",
"In 1998, two independent teams of cosmologists, the High-Z Supernova Search Team founded by B.P.",
"Schmidt [1] and led by A. Riess et al.",
"[2] and the Supernova Cosmology Project led by S. Perlmutter et al.",
"[3] analyzed observational evidences from Supernovae Type Ia (SNIa), cosmic microwave background (CMB) radiation, baryon acoustic oscillations (BAO), large scale structure (LSS) of spacetime, and weak lensing and established that our Universe is presently exhibiting a phase of accelerated expansion.",
"The whole astronomical community was startled at this discovery because contemporary theoretical Cosmology had predicted a decelerated, matter dominated universe.",
"Due to this unexpected result, cosmologists were forced to modify the standard model of Cosmology so that this new observational result could be incorporated into the theory.",
"To this effect, most cosmologists took either of the following two ways— The domain of the stress-energy tensor, $T_{\\mu \\nu }$ , was extended to include a dark energy component, a fluid with exotic properties such as a huge negative pressure.",
"These type of fluid models later came to be known as modified matter models.",
"The geometric part of the Einstein's field equations was modified to obtain a gravity theory different from General Relativity.",
"These type of models later came to be known as modified gravity models.",
"For an extensive review on the two approaches, one may see Refs.",
"[4], [5], [7], [8], [9], [10], [6].",
"One must note that the the $\\Lambda $ -Cold-Dark-Matter ($\\Lambda $ CDM) model, regarded as the simplest modified matter model, is identified as the standard model in 21st century cosmology.",
"This model consists of a tiny cosmological constant ($\\Lambda $ ) which acts as dark energy (the dominant component) and cold dark matter in the form of dust.",
"These two entities together make up almost 96% of the energy budget of the Universe.",
"However, the cosmological constant is plagued with several problems, particularly, the fine tuning and the cosmic coincidence problems.",
"In addition to this, recent local measurement of Hubble constant $H_{0}$ by Hubble Space Telescope [11] is in disagreement with the $\\Lambda $ CDM cosmology.",
"These have prompted cosmologists to devise alternative dark energy models with the assumption that the cosmological constant problem is solved in such a way that $\\Lambda $ vanishes completely.",
"On the other hand, dark energy is usually characterized by an dynamical effective equation of state (EoS) parameter.",
"A large number of functional forms for the EoS parameter have been studied to account for this unknown component.",
"For reviews on the various dark energy candidates, one can refer to [4], [5], [6].",
"The proposed candidates for the EoS parameter for dark energy are constrained with different observational data sets in order to check the viability of a particular model.",
"Although these models fit the observational data sets quite well, yet all of these models have their own demerits.",
"Hence the study of cosmic acceleration will continue for the foreseeable future.",
"In this context, very recently Saha and Bamba [12] have introduced a new fluid which deals with a special mathematical function, known as the Lambert $W$ function.",
"The functional form of the proposed Lambert $W$ EoS parameter is not very simple and straight forward.",
"As we shall see, the EoS parameter of this fluid has two free parameters, $\\vartheta _1$ and $\\vartheta _2$ .",
"Using some motivated choices of these parameters, they have shown that this new fluid can, in principle, explain the evolutionary stages of the Universe.",
"This fact has motivated us to study the model more deeply in order to have a better understanding of the effect of this special function in the observational point of view.",
"In the present work, we wish to constrain these free parameters ($\\vartheta _1$ and $\\vartheta _2$ ) using the latest 51 data points of $H(z)$ measurements, spanning the redshift range $0.07\\le z \\le 2.36$ .",
"We would like to emphasize that this work represents the first observational study on this new fluid.",
"Using the best fit values of $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ , we then reconstruct the evolutions of the effective EoS parameter and the deceleration parameter for the Lambert $W$ model.",
"We also study the evolutions of the Hubble parameter and the distance modulus for the present model and the standard $\\Lambda $ CDM model and compare that with the observational datasets.",
"The structure of this paper is as follows.",
"In section , we introduce the basic properties of the Lambert $W$ function.",
"We briefly describe the Lambert $W$ cosmological model in section .",
"In section , we discuss the dataset and method used in this work along with the results obtained from the analysis of observational data.",
"Finally, in section , we present our conclusions.",
"Throughout the text, the symbol dot indicates derivative with respect to the cosmic time."
],
[
"The Lambert $W$ Function",
"We now turn our attention towards the Lambert $W$ function which holds a central place in the present work.",
"The Lambert $W$ function, also sometimes referred to as the “omega function\" or the “product logarithm\", is defined mathematically as the multivalued inverse of the function $xe^{x}$ , i.e., $ \\text{Lambert}W(y) \\cdot e^{\\text{Lambert}W(y)}=y.$ Eq.",
"(REF ) has two real solutions if $-\\frac{1}{e} \\le y < 0$ , which correspond to two real branches of $\\text{Lambert}W$Note here that $W(y)$ at $y=-e^{-1},0,1$ can be computed as $-1,0,0.567143$ respectively.",
"These three values might prove useful for our work.",
"[13].",
"However, infinitely many solutions of Eq.",
"(REF ) can be obtained with imaginary values of $y$ which shall correspond to infinitely many imaginary branches [14], [15].",
"Euler [17] is often credited with the earliest mention of Eq.",
"(REF ) but Euler himself credited Lambert for his earlier work on the transcendental equation of the form [18] $ x^m-x^n=(m-n)\\nu x^{m+n},$ where $m,n,\\nu $ are constants.",
"As a matter of fact, Lambert initially obtained a series solution in $p$ of the trinomial equation [14] $ x=p+x^{\\alpha }\\qquad \\mathrm {(2^{\\prime })}$ and later extended the series to find powers of $x$ as well [18], [19].",
"Euler [17] used the substitution $x^{-n}$ for $x$ and setting $\\alpha = mn$ and $p=(m-n)\\nu $ .",
"to transform Eq.",
"(REF ) into the more symmetrical form given in Eq.",
"(REF ).",
"We can compute the $n^{\\text{th}}$ derivatives of the Lambert $W$Henceforth, we shall write Lambert $W$ simply as $W$ .",
"Anyone interested in the history behind the choice of the letter $``W\"$ is referred to the article by Hayes [20].",
"function as $W^{n}(y)=\\frac{W^{n-1}(y)}{y^n[1+W(y)]^{2n-1}}\\varphi _{k=1}^{n} \\delta _{kn}W^{k}(y),~~~~y \\ne -\\frac{1}{e},$ where $\\delta _{kn}$ is the number triangle $ \\begin{array}{ccccc}\\phantom{+}1 & & & &\\\\-2 & -1 & & & \\\\\\phantom{+}9 & \\phantom{+}8 & \\phantom{+}2 & & \\\\-64 & -79 & -36 & -6 & \\\\\\phantom{+}625 & \\phantom{+}974 & \\phantom{+}622 & \\phantom{+}192 & \\phantom{+}24\\end{array} .$ Particularly, the first order derivative of $W(y)$ can be evaluated as $W^{\\prime }(y) &=& \\frac{W(y)}{y[1+W(y)]}, ~\\mbox{if~} y \\ne 0 \\nonumber \\\\&=& \\frac{e^{-W(y)}}{1+W(y)}.$ The antiderivative of $W(y)$ is given as $\\int W(y)\\text{d}y=y\\left[W(y)-1+\\frac{1}{W(y)}\\right]+C,$ where $C$ is the arbitrary constant of integration.",
"We have defined and outlined the basic properties of the Lambert $W$ function in the last two paragraphs.",
"Many other mathematical properties of this special function can be found in Refs.",
"[13], [16], [21], [22].",
"It is remarkable to know that numerous real-life applications of the Lambert $W$ function can be found in Mathematics, Physics, and Computer Science.",
"For an extensive discussion on some of such applications, one may see the article by Corless [14].",
"In General Relativity, the Lambert $W$ function is employed in finding solutions to the (1+1)-gravity problem [23] and in finding inverse of Regge-Finkelstein coordinates [24].",
"Motivated by the above facts, in the next section, we explore its implications in studying the cosmic history of the Universe."
],
[
"Cosmology with the Lambert $W$ Function",
"Saha and Bamba [12] recently studied the Lambert $W$ function in the context of Cosmology.",
"They were the first to propose a novel equation of state (EoS) parameter which incorporates this function in a special fashion.",
"It is worthwhile to mention here that the Lambert $W$ function appears while deriving solutions of the continuity equation in the gravitational particle creation scenario [25].",
"This served as a motivation for them to study the evolutionary history of the Universe with the Lambert $W$ function.",
"They assumed a spatially flat Friedmann-Robertson-Walker (FRW) universe as the spacetime metric and considered a perfect fluid having an effective EoS [12] $ w_{\\mathrm {eff}} = \\frac{P}{\\rho } = \\left[\\vartheta _\\mathrm {1}\\text{ln}\\left\\lbrace W\\left(\\frac{a}{a_0}\\right)\\right\\rbrace +\\vartheta _\\mathrm {2}\\left\\lbrace W\\left(\\frac{a}{a_0}\\right)\\right\\rbrace ^3\\right],$ where $P$ and $\\rho $ are the pressure and energy density of the cosmic fluid, respectively.",
"$a_{0}$ is the value of the scale factor, $a$ , at the present epoch, while $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ are dimensionless parameters which must be fixed from observational data.",
"The proposed EoS looks phenomenological and seems to be a bit speculative at first, but, theoretically, this EoS has been predicted to smoothly describe the evolutionary history of the Universe [12].",
"An important advantage of the form of EoS assumed in equation (REF ) is that it is independent of any prior assumption about the nature of dark energy.",
"The expression for the energy density, $\\rho $ , can be obtained from the continuity equation as ($\\rho _{0}$ is a positive constant) [12] $\\rho = \\rho _{0} \\text{exp}\\left[-3\\left\\lbrace \\text{ln}[W(a)][\\vartheta _\\mathrm {1} W(a)+\\vartheta _\\mathrm {1} +1]+W(a)(1-\\vartheta _\\mathrm {1})+\\frac{\\vartheta _\\mathrm {2}}{12}W(a)^3[4+3W(a)]\\right\\rbrace \\right],$ while the deceleration parameter $q$ , in terms of redshift $z$ , is given by [12] $q=-\\frac{\\ddot{a}}{aH^2}=\\frac{3}{2}\\left\\lbrace 1+\\vartheta _\\mathrm {1}\\text{ln}\\left[W\\left(\\frac{1}{1+z}\\right)\\right]+\\vartheta _\\mathrm {2}W\\left(\\frac{1}{1+z}\\right)^3\\right\\rbrace -1.$ where, $z=\\frac{1}{a}-1$ .",
"Note that the functional form of $q(z)$ depends crucially on the values of the model parameters $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ .",
"In the next section, we first constrain these parameters using the observational data and with the best fit values obtained, we then try to reconstruct the functional dependence of $q(z)$ ."
],
[
"Observational Constraints on the Lambert $W$ EoS",
"The cosmological model, discussed in the present context, has been confronted with latest cosmological observations.",
"The observational data, used to constraint the model parameters are briefly discussed in the following.",
"It is well known that the observational Hubble parameter dataset (OHD) is one of the most robust probes to analyze different cosmological models for its model independent nature.",
"Recently, a plethora of papers have been published, for example, [26], [27], [28], [29], [30], [31], [32], which determine the dynamical characteristics of many cosmological models.",
"In this work, we consider the latest 51 data points of $H(z)$ measurements in the redshift range $0.07 \\le z \\le 2.36$ , obtained from different surveys [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48] and the corresponding $H(z)$ values are given in [32].",
"Among them 31 data points calculated from the differential age method (i.e., cosmic chronometers technique), however, 20 data points of this sample are calculated from the BAO measurements under different fiducial cosmologies based on the standard $\\Lambda $ CDM model.",
"Although some of the data points from the BAO measurements are being correlated, however, we assume here that they are independent measurements.",
"On the other hand, the cosmic chronometers method [49] offers to directly measure the expansion rate of the universe (i.e., $H(z)$ ) using spectroscopic dating of passively-evolving galaxy to compare their ages, providing $H(z)$ measurements that are model-independent.",
"Note that these data points (31 points) constitute the majority of our $H(z)$ sample.",
"Additionally, the latest SH0ES measurement of the Hubble constant $H_{0} = 74.03 \\pm 1.42$ km/s/Mpc (at 68% CL) [11], denoted as R19, is also included in the analysis.",
"We use the above sample to constrain the free parameters of the model as given in equation (REF ), and search for an alternative solution to the accelerated expansion of the Universe.",
"The $\\chi ^{2}$ function for this dataset is defined as $\\chi ^2= \\sum ^{N}_{i=1}\\frac{[{H}_{obs}(z_{i}) - {H}_{th}(z_{i},\\theta _p)]^2}{\\sigma ^2_{H}(z_{i})}$ where $N$ stands for the number of the observational Hubble parameter ${H}_{obs}(z_{i})$ at $z_{i}$ and $\\sigma _{H}(z_{i})$ represents the error associated with the $i^{th}$ data point.",
"Also, ${H}_{th}(z_{i}, \\theta _p)$ stands for the theoretical Hubble parameter for a given model depending on model parameters $\\theta _1$ , $\\theta _2$ ... $\\theta _p$ .",
"One can now use the maximum likelihood method and take the likelihood function as ${\\cal L}={\\rm exp}\\left[-\\frac{\\chi ^2}{2}\\right]$ The best-fit corresponds to the free parameters for which $\\chi ^2$ function is minimized (say, $\\chi ^{2}_{min}$ ).",
"In this work, we have minimized $\\chi ^{2}$ with respect to the parameters $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ to calculate their best-fit values.",
"In what follows, we discuss the results obtained from the statistical analysis of the above mentioned datasets.",
"Results: Figure REF shows the $1\\sigma $ and $2\\sigma $ confidence level confidence contours on the set of parameters ($\\vartheta _\\mathrm {1}$ , $\\vartheta _\\mathrm {2}$ ) and the marginalized likelihood function of the present model obtained in the combined analysis with the combinations of the datasets OHD and R19.",
"The best-fit values for the model parameters are obtained as $\\vartheta _\\mathrm {1}=-0.166\\pm 0.104$ ($1\\sigma $ ) and $\\vartheta _\\mathrm {2}=-4.746\\pm 0.479$ ($1\\sigma $ ) with $\\chi ^{2}_{min}=36.853$ .",
"Figure: Marginalized posterior distribution of the set of parameters (ϑ 1 \\vartheta _\\mathrm {1}, ϑ 2 \\vartheta _\\mathrm {2}) and corresponding 2D confidence contours obtained from the χ 2 \\chi ^{2} analysis for the present model utilizing the joint OHD+R19 dataset.",
"The red point represents the best-fit values of the parameter pair (ϑ 1 ,ϑ 2 \\vartheta _{1},\\vartheta _{2}).Figure REF shows the evolution of the effective EoS parameter $w_{eff}(z)$ within $1\\sigma $ error region for our model.",
"The reconstruction of $w_{eff}(z)$ has been done by the joint (OHD+R19) dataset.",
"It has been observed from figure REF that $w_{eff}(z)$ tends to zero at high redshift for any values of $\\vartheta _{1}$ and $\\vartheta _{2}$ and thus it become indistinguishable from the dark matter component at high redshift.",
"On the other hand, $w_{eff}(z)$ enters in the quintessence regime ($-1 < w_{eff} <-\\frac{1}{3}$ , within $1\\sigma $ confidence level) at relatively low redshifts and its present value is found to be $-0.96\\pm 0.02$ ($1\\sigma $ ).",
"Therefore, the functional form of $w_{eff}(z)$ , as given in equation (REF ), can easily accommodate both the phases of the Universe, i.e., early matter dominated era and late-time dark energy dominated era.",
"It is also evident that the present value of $w_{eff}$ is very close to the Cosmological Constant $\\Lambda $ .",
"This implies that our proposed EoS might serve as a unification of dark matter and dark energy.",
"Figure: Reconstructed w eff w_{eff} as a function of redshift zz.",
"In this plot, the red curve corresponds the evolution of q(z)q(z) for the best-fit case and the gray shaded region indicates 1σ1\\sigma error region.",
"Here, the horizontal green line is for w eff =-1 3w_{eff}=-\\frac{1}{3}.Similarly, the evolution of the deceleration parameter $q(z)$ within $1\\sigma $ confidence level is shown in figure REF .",
"The redshift at which $q$ changes sign from positive to negative corresponds to the onset of late-time cosmic acceleration.",
"The redshift around which the transition from the decelerating ($q>0$ ) expansion to the accelerating ($q<0$ ) expansion occurs is found to be $0.77\\pm 0.03$ ($1\\sigma $ ).",
"The results are in good agreement with the measured transition redshift $z_{t}$ based on the OHD (cosmic chronometer) dataset [27], [28], [29] including the standard $\\Lambda $ CDM prediction ($z_{t}\\approx 0.7$ ).",
"On the other hand, in figure REF , we have plotted the evolution of the normalized energy density $\\rho /\\rho _0$ in the logarithmic scale against the normalized scale factor $a/a_0$ .",
"We have assumed $a_0=1$ without any loss of generality and considered the best-fit values, $\\vartheta _\\mathrm {1}=-0.166$ $\\&$ $\\vartheta _\\mathrm {2}=-4.746$ , of the coefficients in the Lambert $W$ EoS parameter.",
"The figure clearly shows that the Lambert $W$ EoS exhibits a transition of a dark matter dominated era to a dark energy dominated era with the evolution of the Universe.",
"Interestingly enough, though, it will allow the Universe to transit from the dark energy era to a future dark matter dominated era.",
"Furthermore, in the left panel of figure REF , we have shown the evolution of the Hubble parameter $H(z)$ within $2\\sigma $ confidence level for our model and have compared that with the latest 51 points of $H(z)$ dataset [32] as well as the flat $\\Lambda $ CDM model.",
"From this figure, we have observed that the model is well consistent with the OHD+R19 dataset against redshift parameter.",
"In the right panel of figure REF , we observed that for the best-fit case, the relative difference $\\bigtriangleup E$ is close to 0.98$\\%$ at $z\\sim 0.5$ , while the differences between the present and $\\Lambda $ CDM models are negligible around $z\\sim 0.67$ .",
"It has also been observed that $H_{{\\rm Lambert}W}(z)<H_{\\Lambda CDM}(z)$ at high redshifts, whereas $H_{{\\rm Lambert}W}(z)>H_{\\Lambda CDM}(z)$ at relatively low redshifts.",
"Finally, the best fit of distance modulus $\\mu (z)$ as a function of $z$ for the present model and the 580 points of Union 2.1 compilation [51] Supernovae Type Ia (SNIa) datasets are plotted in figure REF .",
"The evolution of $\\mu (z)$ for the standard $\\Lambda $ CDM model is also shown in figure REF for model comparision.",
"From figure REF , we have found that the Lambert $W$ model reproduces the observed values of $\\mu (z)$ quite effectively.",
"Furthermore, we have checked that the nature of the evolution of $\\mu (z)$ is hardly affected by a small change in the values of $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ within $1\\sigma $ confidence limit.",
"Figure: Plot of the deceleration parameter qq as a function of redshift zz is shown in 1σ1\\sigma error region (gray) by considering OHD+R19 dataset.",
"Here,the red curve represents the corresponding evolution of qq for the best-fit case and the horizontal line stands for q(z)=0q(z)=0.Figure: This figure shows the evolution of the normalized energy density ρ/ρ 0 \\rho /\\rho _0 in the logarithmic scale against the normalized scale factor a/a 0 a/a_0.",
"We have assumed a 0 =1a_0=1 without any loss of generality and considered the best-fit values, ϑ 1 =-0.166\\vartheta _\\mathrm {1}=-0.166 &\\& ϑ 2 =-4.746\\vartheta _\\mathrm {2}=-4.746, of the coefficients in the Lambert WW EoS parameter.Figure: Left panel: The evolution of H(z)H(z) within 1σ1\\sigma (gray) and 2σ2\\sigma (dashed) confidence levels are shown for the present model by considering theOHD+R19 dataset.",
"In this plot, the dots correspond to the 51 H(z)H(z) data points, whereas the green shaded contour (1σ1\\sigma ) indicates the corresponding evolution of H(z)H(z) for a flat Λ\\Lambda CDM model .",
"For each model, the solid curve inside the shaded region corresponds the evolution of H(z)H(z) for the best-fit case.",
"Right Panel: The corresponding relative difference △H(%)=100×[H Lambert W (z)-H ΛCDM (z)]/H ΛCDM (z)\\bigtriangleup H(\\%)=100\\times [H_{{\\rm Lambert}W}(z)-H_{\\Lambda CDM}(z)]/H_{\\Lambda CDM}(z).Figure: This figure shows the Error bar plot of 580 points of Union 2.1 compilation Supernovae Type Ia data sets (red dots) together with the presented model shown in solid black line with ϑ 1 =-0.166\\vartheta _\\mathrm {1}=-0.166 &\\& ϑ 2 =-4.746\\vartheta _\\mathrm {2}=-4.746.",
"The standard Λ\\Lambda CDM model is also shown in solid green line for model comparision.",
"Here, μ(z)\\mu (z) denotes distance modulus, which is thedifference between the apparent magnitude and the absolute magnitude of the observed supernova, is given by μ(z)=5 log 10 (d L Mpc )+25\\mu (z)=5{\\rm log}_{10}(\\frac{d_{L}}{\\rm Mpc})+25, where d L d_{L} is the luminosity distance."
],
[
"Conclusions",
"In summary, we have investigated the possibility that the Universe is driven by a single dark fluid described by a Lambert $W$ EoS parameter which is dependent on two free parameters $\\vartheta _{1}$ and $\\vartheta _{2}$ .",
"We have then fixed the values of $\\vartheta _{1}$ and $\\vartheta _{2}$ from the analysis of recent observational datasets.",
"As discussed in the previous section, the measurements of Hubble parameter at different redshift from the differential age of galaxies and the BAO methods are incorporated in the present analysis.",
"Also, the latest measurement of $H_{0}$ from [11] is also taken into account.",
"The best fit values of ($\\vartheta _\\mathrm {1}$ , $\\vartheta _\\mathrm {2}$ ) for the combined OHD+R19 dataset are obtained as $\\vartheta _\\mathrm {1}=-0.166$ and $\\vartheta _\\mathrm {2}=-4.746$ .",
"Furthermore, using the best fit values of $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ , we have reconstructed the evolutions of $w_{eff}(z)$ , $q(z)$ and $H(z)$ for the present model.",
"The main results of our study are summarized as follows.",
"We have found that the effective EoS parameter $w_{eff}(z)$ can easily accommodate both the phases of cosmic evolution, i.e., early matter (dust) dominated phase and late-time dark energy dominated phase.",
"Additionally, we have observed that $w_{eff}$ remains in the quintessence regime.",
"Its present value has been found to be $-0.96\\pm 0.02$ ($1\\sigma $ ).",
"This shows that the present value of $w_{eff}$ is very close to the cosmological constant $\\Lambda $ .",
"Thus, our proposed EoS parameter might serve as a unification of dark matter and dark energy.",
"It has also been found that the deceleration parameter $q$ undergoes a smooth transition from a decelerated ($q>0$ ) to an accelerated ($q<0$ ) phase of expansion at the redshift $z_{t}=0.77\\pm 0.03$ ($1\\sigma $ ).",
"This result is in good agreement with the measured $z_{t}$ based on the cosmic chronometer dataset [27], [28], [29] including the standard $\\Lambda $ CDM prediction ($z_{t}\\approx 0.7$ ).",
"It is worth emphasizing that the evolution scenarios of $w_{eff}(z)$ and $q(z)$ are necessary to explain both the observed growth of structures at the early epoch and the late-time cosmic acceleration measurements.",
"Finally, we have shown the evolution of $H(z)$ within $2\\sigma $ confidence level for our model and have compared that with the latest 51 points of $H(z)$ dataset [32] as well as the $\\Lambda $ CDM model.",
"For the best-fit case, we observed that the relative difference $\\bigtriangleup E$ is close to 0.98$\\%$ at $z\\sim 0.5$ , while the differences between the two models are negligible around $z\\sim 0.67$ .",
"We have also found that the present model reproduces the observed values of the distance modulus $\\mu (z)$ quite effectively (see figure REF ).",
"In this context, it desreves to mention here that in a recent work [52], a group of authors including us, have studied the effect on the growth of perturbations for the Lambert $W$ dark energy model.",
"We have performed the analysis for two different cosmological scenarios.",
"In the first case, we have considered the universe to be filled with two different fluid components, namely, the LambertW dark energy component and the baryonic matter component, while in the second case, we have considered that there is a single fluid component in the universe whose EoS parameter is described by the Lambert $W$ function.",
"We have then compared the growth rates of the Lambert $W$ model with that for a $\\Lambda $ CDM model as well as the CPL model.",
"It has been observed that the presence of Lambert $W$ dynamical dark energy sector changes the growth rate and affects the matter fluctuations in the Universe to a great extent.",
"We conclude that the Lambert $W$ EoS parameter provides some interesting consequences in the cosmological perspective, and thus it can be a candidate for the description of nature.",
"However, it is natural to extend the present work with addition of other datasets from Supernovae Type Ia, BAO and CMB probes in order to constrain the new parameters $\\vartheta _\\mathrm {1}$ and $\\vartheta _\\mathrm {2}$ more precisely.",
"The present analysis is one preliminary step towards that direction."
]
] | 2005.14061 |
[
[
"Language (Technology) is Power: A Critical Survey of \"Bias\" in NLP"
],
[
"Abstract We survey 146 papers analyzing \"bias\" in NLP systems, finding that their motivations are often vague, inconsistent, and lacking in normative reasoning, despite the fact that analyzing \"bias\" is an inherently normative process.",
"We further find that these papers' proposed quantitative techniques for measuring or mitigating \"bias\" are poorly matched to their motivations and do not engage with the relevant literature outside of NLP.",
"Based on these findings, we describe the beginnings of a path forward by proposing three recommendations that should guide work analyzing \"bias\" in NLP systems.",
"These recommendations rest on a greater recognition of the relationships between language and social hierarchies, encouraging researchers and practitioners to articulate their conceptualizations of \"bias\"---i.e., what kinds of system behaviors are harmful, in what ways, to whom, and why, as well as the normative reasoning underlying these statements---and to center work around the lived experiences of members of communities affected by NLP systems, while interrogating and reimagining the power relations between technologists and such communities."
],
[
"Allocational harms",
", , , ."
],
[
"Stereotyping",
", , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ."
],
[
"Other representational harms",
", , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ."
],
[
"Questionable correlations",
", , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ."
],
[
"Vague/unstated",
"None."
],
[
"Surveys, frameworks, and meta-analyses",
", , , , , , , , , , , , , , , , , , , ."
]
] | 2005.14050 |
[
[
"Low regularity of non-$L^2(R^n)$ local solutions to gMHD-alpha systems"
],
[
"Abstract The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations.",
"Recently it has become common to study generalizations of fluids-based differential equations.",
"Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$\\alpha$) system, which differs from the original MHD system by including an additional non-linear terms (indexed by $\\alpha$), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form $-|\\xi|^\\gamma / g(|\\xi|)$.",
"In a paper by Pennington, the problem was considered with initial data in the Sobolev space $H^{s,2}(\\mathbb{R}^n)$ with $n \\geq 3$.",
"Here we consider the problem with initial data in $H^{s,p}(\\mathbb{R}^n)$ with $n \\geq 3$ and $p > 2$.",
"Our goal is to minimize the regularity required for obtaining uniqueness of a solution."
],
[
"Introduction",
"This paper is concerned with the generalized Magneto-Hydrodynamic alpha (gMHD-$\\alpha $ ) system of equations, reported below in its full generality: $& \\partial _t v + (u \\cdot \\nabla ) v + \\sum _{i = 1}^n v_i \\nabla u_i - \\nu _1 \\mathcal {L}_1 v + \\frac{1}{2} \\nabla \\left|B \\right|^2 = - \\nabla p + (B \\cdot \\nabla )B, \\\\& \\partial _t B + (u \\cdot \\nabla ) B - (B \\cdot \\nabla ) u - \\nu _2 \\mathcal {L}_2 B = 0, \\\\& v = (1 - \\alpha ^2 \\mathcal {L}_3) u, \\\\& \\operatorname{div}u = \\operatorname{div}B = 0, \\\\& u(0,x) = u_0(x), \\quad B(0,x) = B_0(x), \\quad x \\in \\mathbb {R}^n.",
"$ Since these equations govern the motion of fluids subject to a magnetic field, (almost) each term has a specific physical meaning: in the above, $u$ is the fluid velocity, $B$ the magnetic field, and $p$ the scalar-valued pressure of the fluid; for the constants, $\\nu _1 > 0$ is the fluid viscosity, $\\nu _2 > 0$ the magnetic diffusion, and $\\alpha > 0$ a constant coming from varying the Hamiltonian that originally gave rise to the standard MHD equations (see [4]).",
"Finally, the $\\mathcal {L}_i$ terms are Fourier multipliers with symbol $-\\left|\\xi \\right|^{\\gamma _i} / g_i(\\left|\\xi \\right|)$ , where $g_i$ is a positive scalar function and $\\gamma _i > 0$ .",
"The standard MHD system is the special case obtained when setting $\\alpha = 0$ , $g_1 = g_2 = 1$ , and $\\gamma _1 = \\gamma _2 = 2$ , so that $v = u$ and $\\mathcal {L}_1 = \\mathcal {L}_2 = \\Delta $ .",
"The existence of a global solution up to initial conditions is a classic result, at least in two dimensions.",
"Unfortunately, the presence of nonlinear terms makes the MHD equations particularly complex to solve in arbitrary dimensions, and so a common strategy has been to study modified versions of them.",
"One modification, termed Lagrangian Averaged MHD-$\\alpha $ after the Lagrangian Averaged Navier-Stokes equation, is obtained from Equations (REF )-() by setting $\\mathcal {L}_i = \\Delta $ for $i = 1,2,3$ ($\\gamma _i = 2$ and $g_i = 1$ ).",
"Linshiz and Titi proved the existence of a global solution for smooth initial data in three dimension [4].",
"Another version is obtained by setting $\\alpha = 0$ and $g_1 = g_2 = 1$ and leaving the $\\gamma _i$ 's unspecified.",
"Zhao and Zhu used these generalized operators to guarantee a global solution to Equations (REF )-() in the case of $n = 3$ , provided that $g_1 = g_2 = g_3 = 1$ , $\\gamma _1 = \\gamma _2 = n/2$ , and $\\gamma _3 = 2$ [14].",
"The first incorporation of a non-constant value for any $g_i$ appeared in [9], where Tao proved the existence of a unique global solution to the generalized Navier-Stokes equation ($B_0 = \\alpha = 0$ ) when $\\gamma _1 = n/2 + 1$ and when $g_1$ is a radial non-decreasing function bounded below satisfying $ \\int _1^\\infty \\frac{ds}{s g_1(s)^4} = \\infty ,$ the prototypical example of which is essentially a logarithm.",
"Wu obtained a similar result for the generalized MHD system in [12], specifically showing that there is a unique global solution provided that $u_0, B_0 \\in H^{r,2}(\\mathbb {R}^n)$ with $r > n/2 + 1$ ; $\\gamma _1 \\ge n/2 + 1$ , $\\gamma _2 > 0$ , and $\\gamma _1 + \\gamma _2 \\ge 1$ ; and $g_1, g_2$ are non-decreasing, bounded below by 1, and satisfy $ \\int _1^\\infty \\frac{ds}{s (g_1(s) + g_2(s))^2} = \\infty .$ This work was ultimately extended to the gMHD-$\\alpha $ system in [13], where Yamazaki obtained a unique global solution in three dimensions provided that $\\gamma _1 + \\gamma _2 + \\gamma _3 \\ge 5$ , $\\min \\lbrace \\gamma _1, \\gamma _3\\rbrace > \\gamma _2 > 0$ , $\\gamma _3 + 2 \\gamma _1 > 3$ , and the $g_i$ satisfy $ \\int _1^\\infty \\frac{ds}{s g_1(s)^2 g_2(s) g_3(s)^2} = \\infty .$ In [8], one of the authors considered a generalization of the equations in [14] with the incorporation of non-constant $g_i$ , $i = 1, 2, 3$ , while still leaving $\\mathcal {L}_3 = \\Delta $ , and guaranteed a unique global solution.",
"In this paper we will extend those results for the case of $\\gamma _3 \\ne 2$ and non-constant $g_3$ .",
"We will particularly focus on the case of low-regularity initial data in a non-$L^2(\\mathbb {R}^n)$ setting to then obtain, in the future, global $L^p(\\mathbb {R}^n)$ solutions using an interpolation technique, the details of which can be found in [5] and [1].",
"The rest of the paper is organized as follows.",
"Section 2 is devoted to explaining the notation we will use and some supporting results necessary for the algorithm.",
"Section 3 contains the main result (Theorem REF ) of this paper and its proof.",
"We end this section with two important spacial cases of Theorem REF .",
"Theorem 1 Let $g_1, g_2, g_3 : [0, \\infty ) \\rightarrow \\mathbb {R}$ be non-decreasing functions bounded below by 1 satisfying $ g_i^{(k)}(s) \\le Cs^{-k}$ for $i = 1,2,3$ and $0 \\le k \\le n/2+1$ .",
"Moreover, assume $0 \\le \\gamma _3 \\le 1$ and $p, q \\ge n$ with $2p > q$ .",
"Then, for any divergence-free $u_0 \\in L^p(\\mathbb {R}^n)$ and $B_0 \\in L^q(\\mathbb {R}^n)$ , there exists a unique local solution $(u,B)$ to the generalized MHD-$\\alpha $ system (REF )-() provided that $\\gamma _1^- & > 6 - \\gamma _3, \\\\\\gamma _2^- & > 1 + \\frac{n}{p}.$ The condition (REF ) is a modification of the condition in the Mikhlin multiplier thoerem that is necessary for supporting estimates in Proposition (REF ) and (REF ).",
"The functions that satisfy it are still essentially logarithms, the same type of functions that satisfy (REF )-(REF ).",
"Theorem 2 Let $\\gamma _3^- - 1 \\le \\frac{n}{2p} \\le \\gamma _3^-$ , $\\frac{n}{2q} - 1 + \\gamma _3^- \\le \\frac{n}{2p}$ , and let $p, q \\ge n$ with $q < 3p/2$ .",
"Moreover, assume that $g_1, g_2, g_3$ satisfy the inequality (REF ).",
"Then, for each divergence-free $u_0 \\in H^{n/2p,p}(\\mathbb {R}^n)$ and $B_0 \\in H^{n/2q,q}(\\mathbb {R}^n)$ , there exists a unique local solution $(u,B)$ to the generalized MHD system from Equations (REF )-() provided that $\\gamma _1^- & > 6 - \\gamma _3^- - \\frac{n}{p}, \\\\\\gamma _2^- & > 1 + \\frac{n}{2p}.$ Note that, in the statement of Theorem REF and REF and in what follows, we use $x^- = x - \\varepsilon $ for some positive $\\varepsilon $ , i.e.",
"$x^-$ denotes a number arbitrarily close to, but strictly smaller than, $x$ ."
],
[
"Notation and supporting facts",
"We let $H^{r,p}(\\mathbb {R}^n)$ be the usual Sobolev space, and we write $\\left\\Vert f \\right\\Vert _{r,p}$ to mean $\\left\\Vert f \\right\\Vert _{H^{r,p}(\\mathbb {R}^n)}$ and $\\left\\Vert f \\right\\Vert _{p}$ for $\\left\\Vert f \\right\\Vert _{L^p(\\mathbb {R}^n)}$ .",
"Due to the nature of the procedure we will use, we require that the solutions live in an auxiliary continuous-in-time space $C^T_{a;r,p}(\\mathbb {R}^n)$ defined by $C^T_{a;r,p}(\\mathbb {R}^n) := \\left\\lbrace f \\in C((0,T), H^{r,p}(\\mathbb {R}^n)) : \\left\\Vert f \\right\\Vert _{a;r,p} < \\infty \\right\\rbrace ,$ where $T > 0$ , $a \\ge 0$ , $C(X,Y)$ is the space of continuous maps $X \\rightarrow Y$ , and $\\left\\Vert f \\right\\Vert _{a;r,p} := \\sup _{(0,T)} t^a \\left\\Vert f(t) \\right\\Vert _{r,p}.$ Finally, we denote by $\\dot{C}^T_{a;r,p}(\\mathbb {R}^n)$ the subspace of $C^T_{a;r,p}(\\mathbb {R}^n)$ consisting of functions $f$ such that $\\lim _{t \\rightarrow 0^+} t^{a} f(t) = 0$ and by $BC(X,Y) \\subset C(X,Y)$ the subspace of bounded continuous maps $X \\rightarrow Y$ .",
"The following are some supporting propositions that we will use throughout the paper.",
"This first one is a product estimate, the proof of which can be found in Chapter 2 of [10]: Proposition 1 If $r \\ge 0$ and $1 < p \\le \\infty $ , then $\\left\\Vert fg \\right\\Vert _{r,p} \\le C \\left( \\left\\Vert f \\right\\Vert _{p_1} \\left\\Vert g \\right\\Vert _{r,p_2} + \\left\\Vert f \\right\\Vert _{r, q_1} \\left\\Vert g \\right\\Vert _{q_2} \\right),$ where $\\frac{1}{p} = \\frac{1}{p_1} + \\frac{1}{p_2} = \\frac{1}{q_1} + \\frac{1}{q_2}$ and $p_1, p_2, q_1, q_2 \\in [1, \\infty ]$ .",
"The following is a useful Sobolev embedding which is a straightforward extension of a result from Chapter 13 of [11]: Proposition 2 Let $s \\ge r$ and $(s-r)p < n$ .",
"Then $\\left\\Vert f \\right\\Vert _{r,q} \\le C \\left\\Vert f \\right\\Vert _{s,p}$ provided that $\\frac{1}{q} - \\frac{r}{n} = \\frac{1}{p} - \\frac{s}{n}.$ Our next result follows from a simple calculus exercise: Proposition 3 If $0 < a, b \\in \\mathbb {R}$ , then $\\sup _{t \\in [0,T]} \\int _0^t (t-s)^{-a} s^{-b} \\, ds \\le CT^{1 - a - b},$ provided that $a + b < 1$ .",
"Our final two propositions consist of an estimate for the semigroup $e^{t\\mathcal {L}_i}$ analogous to similar results for the heat kernel $e^{t\\Delta }$ and an estimate for the operator $(1 - \\mathcal {L}_i)^{-1}$ .",
"The proofs of both propositions can be found in [7].",
"We recall that $x^-$ is a number arbitrarily close to, but strictly smaller than, $x$ .",
"Proposition 4 Let $1 < p_1 \\le p_2 < \\infty $ , $r_1 \\le r_2$ , $g(x)$ be a non-decreasing function bounded below by 1 satisfying $\\left|g^{(k)}(x) \\right| \\le C \\left|x \\right|^{-k}$ for all $1 \\le k \\le n/2 + 1$ .",
"Then $e^{t\\mathcal {L}_i} : H^{r_1,p_1}(\\mathbb {R}^n) \\rightarrow H^{r_2, p_2}(\\mathbb {R}^n)$ and $\\left\\Vert e^{t\\mathcal {L}_i} f \\right\\Vert _{r_2, p_2} \\le t^{-(r_2 - r_1 + n/p_1 - n/p_2)/\\gamma _i^-} \\left\\Vert f \\right\\Vert _{r_1, p_1}.$ Note that it is this proposition which necessitates the requirements on the $g_i$ 's.",
"Proposition 5 Let $1 < p < \\infty $ , $r \\in \\mathbb {R}$ , $g(x)$ be a non-decreasing function bounded below by 1 satisfying $\\left|g^{(k)}(x) \\right| \\le C \\left|x \\right|^{-k}$ for all $1 \\le k \\le n/2 + 1$ .",
"Then $\\left\\Vert (1 - \\mathcal {L}_i)^{-1} f \\right\\Vert _{r,p} \\le C \\left\\Vert f \\right\\Vert _{r - \\gamma _i^-, p}.$"
],
[
"Main Theorem and Proof",
"In this section we state the most general form of the theorem and then proceed with the proof.",
"Theorem 3 Let $g_1, g_2, g_3 : [0, \\infty ) \\rightarrow \\mathbb {R}$ be non-decreasing functions bounded below by 1 satisfying $g_i^{(k)}(s) \\le Cs^{-k}$ for $i = 1,2,3$ and $0 \\le k \\le n/2+1$ .",
"Let $r_0, r_1, r_2 \\ge 0$ and let $p_0, p_1, p_2 \\ge n$ with $p_0 \\le p_1$ and $p_2 < 2p_0$ .",
"Moreover, assume that $\\gamma _3^- - 1 \\le r_0 \\le \\gamma _3^- \\le r_1, \\\\r_2 - 1 + \\gamma _3^- \\le r_0, \\\\r_2 \\le r_0 < \\frac{n}{p_1}, \\\\2r_1 \\ge \\max \\left\\lbrace 2, 1 + \\gamma _3^- - \\frac{n}{p_0} + \\frac{2n}{p_1} \\right\\rbrace , \\\\r_2 < \\min \\left\\lbrace \\frac{n}{p_2}, \\frac{2n}{p_2} - \\frac{n}{p_0} \\right\\rbrace .$ Then, for any divergence-free $u_0 \\in H^{r_0,p_0}(\\mathbb {R}^n)$ and $B_0 \\in H^{r_2,p_2}(\\mathbb {R}^n)$ , there exists a unique local solution $(u,B)$ to the generalized MHD-$\\alpha $ system (REF )-() provided that $\\gamma _1^- & > 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1}, \\\\\\gamma _1^- & > 1 - 2r_2 + r_1 - \\gamma _3^- - \\frac{n}{p_1} + \\frac{2n}{p_2}, \\\\\\gamma _2^- & > 1 - r_0 + \\frac{n}{p_0}.$ Note: Theorem REF may be recovered by setting $r_0 = r_2 = 0$ , $r_1 = 2$ , $p := p_0 = p_1$ , and $q := p_2$ , while Theorem REF may be recovered by setting $r_0 = n/(2p)$ , $r_1 = 2$ , $r_2 = n/(2q)$ , $p := p_0 = p_1$ , and $q := p_2$ .",
"For the sake of clarity and to highlight some technical details, the proof will be divided in subsections.",
"We first write the generalized MHD-$\\alpha $ system in a more helpful form.",
"Without loss of generality, we set $\\alpha = \\nu _1 = \\nu _2 = 1$ .",
"We pass to divergence-free vector fields by applying the Hodge operator $P$ to Equations (REF ) and () (more information about the Hodge operator can be found in Chapter 11 of [3]), and then we apply $(1 - \\mathcal {L}_3)^{-1}$ to Equation (REF ).",
"By noting that $P$ , $(1 - \\mathcal {L}_3)^{-1}$ , and $\\partial _t$ all commute since they are Fourier multipliers, we obtain $\\partial _t u + P(1 - \\mathcal {L}_3)^{-1} \\left((u \\cdot \\nabla ) v + \\sum _{i = 1}^n v_i \\nabla u_i - (B \\cdot \\nabla ) B \\right) - \\mathcal {L}_1 u = P \\left( -\\nabla p - \\frac{1}{2} \\nabla \\left|B \\right|^2 \\right) = 0.$ An application of the divergence-free condition in Equation () allows us to rewrite the terms of the form $(x \\cdot \\nabla ) y$ as $\\operatorname{div}(x \\otimes y)$ .",
"Note that $x \\otimes y$ is the matrix whose $(i,j)$ entry is $x_i y_j$ , so that the product estimate in Proposition REF applies to $x \\otimes y$ .",
"We then have the following system: $& \\partial _t u + P(1 - \\mathcal {L}_3)^{-1} \\left(\\operatorname{div}(u \\otimes v) + \\sum _{i = 1}^n v_i \\nabla u_i - \\operatorname{div}(B \\otimes B) \\right) - \\mathcal {L}_1 u = 0, \\\\& \\partial _t B + P \\left(\\operatorname{div}(u \\otimes B) - \\operatorname{div}(B \\otimes u)\\right) - \\mathcal {L}_2 B = 0, \\\\& v = (1 - \\mathcal {L}_3) u, \\\\& \\operatorname{div}u = \\operatorname{div}B = 0, \\\\& u(x,0) = u_0(x), \\quad B(0,x) = B_0(x), \\quad x \\in \\mathbb {R}^n.$ An application of Duhamel's principle shows that $(u,B)$ is a solution to the system if and only if $(u,B)$ is a fixed point of the map $\\Phi (u,B) := (\\Phi _1(u,B), \\Phi _2(u,B))$ defined by $\\Phi _1(u,B) & := e^{t\\mathcal {L}_1} u_0 - \\int _0^t e^{(t-s) \\mathcal {L}_1} \\left( W_1 (u,v) + W_2(u,v) - W_1 (B,B)\\right) ds, \\\\\\Phi _2(u,B) & := e^{t\\mathcal {L}_2} B_0 - \\int _0^t e^{(t-s)\\mathcal {L}_2} \\left( W_3 (u,B) - W_3(B,u) \\right) ds,$ where $W_1(x,y) & = P (1-\\mathcal {L}_3)^{-1} \\operatorname{div}(x \\otimes y), \\\\W_2(x,y) & = P (1-\\mathcal {L}_3)^{-1} \\left( \\sum _{i = 1}^n y_i \\nabla x_i \\right), \\\\W_3(x,y) & = P \\operatorname{div}(x \\otimes y).$ By the contraction mapping theorem, it suffices to show that $\\Phi $ is a contraction on the space $X_{T,M} \\times Y_{T,M}$ , where $X_{T,M} := \\bigg \\lbrace f \\in BC\\left([0,T), H^{r_0,p_0}(\\mathbb {R}^n)\\right) \\cap \\dot{C}_{a_1, r_1, p_1}(\\mathbb {R}^n) \\\\\\text{ and } \\sup _{(0,T)} \\left( \\left\\Vert f(t) - e^{t\\mathcal {L}_1}u_0 \\right\\Vert _{r_0, p_0} + \\left\\Vert f(t) \\right\\Vert _{a_1; r_1, p_1} \\right) < M \\bigg \\rbrace $ and $Y_{T,M} := \\bigg \\lbrace f \\in BC([0,T), H^{r_2,p_2}(\\mathbb {R}^n)) \\text{ and }\\sup _{(0,T)} \\left\\Vert f(t) - e^{t\\mathcal {L}_2}B_0 \\right\\Vert _{r_2, p_2} < M \\bigg \\rbrace $ for some $0 < T < 1$ and $M > 0$ .",
"Following the methods in [6], [2], we will complete the proof by showing that $I_1 & = \\sup _{(0,T)} t^{a_1} \\left\\Vert e^{t\\mathcal {L}_1} u_0 \\right\\Vert _{r_1,p_1} < M/4, \\\\I_2 & = \\sup _{(0,T)} \\left\\Vert \\int _0^t e^{(t-s) \\mathcal {L}_1} \\left( W_1 (u,v) + W_2(u,v) - W_1 (B,B)\\right) ds \\right\\Vert _{r_0,p_0} < M/4, \\\\I_3 & = \\sup _{(0,T)} t^{a_1} \\left\\Vert \\int _0^t e^{(t-s) \\mathcal {L}_1} \\left( W_1 (u,v) + W_2(u,v) - W_1 (B,B)\\right) ds \\right\\Vert _{r_1,p_1} < M/4, \\\\I_4 & = \\sup _{(0,T)} \\left\\Vert \\int _0^t e^{(t-s) \\mathcal {L}_2} \\left( W_3 (u,B) - W_3(B,u) \\right) ds \\right\\Vert _{r_2,p_2} < M/4.\\\\$ We start with $I_1$ .",
"If $\\varphi $ is in the Schwartz space, we have that $I_1 & = \\sup _{(0,T)} t^{a_1} \\left\\Vert e^{t\\mathcal {L}_1} (u_0 - \\varphi + \\varphi ) \\right\\Vert _{r_1,p_1} \\\\& \\le \\sup _{(0,T)} t^{a_1} \\left\\Vert e^{t\\mathcal {L}_1} (u_0 - \\varphi ) \\right\\Vert _{r_1, p_1} + \\sup _{(0,T)} t^{a_1} \\left\\Vert e^{t\\mathcal {L}_1} \\varphi \\right\\Vert _{r_1, p_1} \\\\& \\le \\sup _{(0,T)} t^{a_1} t^{-a_1} \\left\\Vert u_0 - \\varphi \\right\\Vert _{r_0, p_0} + \\sup _{(0,T)} t^{a_1} \\left\\Vert \\varphi \\right\\Vert _{r_1, p_1} \\\\& \\le \\left\\Vert u_0 - \\varphi \\right\\Vert _{r_0, p_0} + T^{a_1} \\left\\Vert \\varphi \\right\\Vert _{r_1, p_1},$ provided that (by Proposition REF ) $0 \\le a_1 = \\frac{r_1 - r_0 + \\frac{n}{p_0} - \\frac{n}{p_1}}{\\gamma _1^-} < 1$ and $p_0 \\le p_1$ .",
"We can choose $\\varphi $ so that $\\left\\Vert u_0 - \\varphi \\right\\Vert _{r_0, p_0}$ is arbitrarily small, and then we can choose $T$ small enough to reduce $T^{a_1} \\left\\Vert \\varphi \\right\\Vert _{r_1, p_1}$ so that the sum of the two is bounded by $M/4$ .",
"subsection2 .5plus.7-.5em $I_2$ and $I_3$ .",
"Minkowski's inequality gives us $I_2 & \\le J_1 + J_2 + J_3, \\\\I_3 & \\le K_1 + K_2 + K_3,$ where $J_1 & := \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (u,v) \\right\\Vert _{r_0,p_0} ds, & K_1 & := \\sup _{(0,T)} t^{a_1} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (u,v) \\right\\Vert _{r_1,p_1} ds, \\\\J_2 & := \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_2 (u,v) \\right\\Vert _{r_0,p_0} ds, & K_2 & := \\sup _{(0,T)} t^{a_1} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_2 (u,v) \\right\\Vert _{r_1,p_1} ds, \\\\J_3 & := \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (B,B) \\right\\Vert _{r_0,p_0} ds, & K_3 & := \\sup _{(0,T)} t^{a_1} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (B,B) \\right\\Vert _{r_1,p_1} ds.$ We will show that each term is bounded above by $C M^2 T^k$ for various values of $k > 0$ , which, since $T < 1$ , will imply $I_2, I_3 < M/4$ provided that $CM^2 < M/4$ .",
"We begin our algorithm with $J_1$ and $K_1$ , showing the details of the calculations and highlighting the choices of parameters.",
"The argument for the other two pairs of integrals is very similar, and the details will be omitted.",
"subsection2 .5plus.7-.5em $J_1$ and $K_1$ By Proposition REF , $J_1$ is bounded by $J_1 \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^-} \\left\\Vert W_1 (u,v) \\right\\Vert _{\\gamma _3^- - 1,\\pi _1}ds$ provided that $\\gamma _3^- - 1 \\le r_0$ and where $\\pi _1$ is an intermediate parameter that will be specified later.",
"Now we work towards bounding $W_1(u,v)$ , and an application of Proposition REF gives us $\\left\\Vert W_1 (u,v) \\right\\Vert _{\\gamma _3^- - 1,\\pi _1} = \\left\\Vert P(1 - \\mathcal {L}_3)^{-1} \\operatorname{div}(u \\otimes v) \\right\\Vert _{\\gamma _3^- - 1,\\pi _1} \\le C \\left\\Vert u \\otimes v \\right\\Vert _{\\pi _1}.$ We chose a regularity of $\\gamma _3^- - 1$ when performing the semigroup estimate in order to end up with a product in zero regularity, so that we can apply Holder's inequality: by Proposition REF , if $ \\frac{1}{\\pi _1} = \\frac{1}{p^{\\prime }} + \\frac{1}{p_1},$ we have $\\left\\Vert u \\otimes v \\right\\Vert _{\\pi _1} \\le C \\left\\Vert u \\right\\Vert _{p^{\\prime }} \\left\\Vert v \\right\\Vert _{p_1} \\le C \\left\\Vert u \\right\\Vert _{p^{\\prime }} \\left\\Vert u \\right\\Vert _{\\gamma _3^-, p_1}.$ Note that Equation (REF ) specifies the required value of $\\pi _1$ .",
"We obtain $\\left\\Vert u \\right\\Vert _{p^{\\prime }} \\le \\left\\Vert u \\right\\Vert _{r_0,p_0}$ by Proposition REF if $ r_0 < \\frac{n}{p_0} \\text{ and } \\frac{1}{p^{\\prime }} = \\frac{1}{p_0} - \\frac{r_0}{n};$ we also get $\\left\\Vert u \\right\\Vert _{\\gamma _3^-,p_1} \\le \\left\\Vert u \\right\\Vert _{r_1, p_1}$ by requiring that $r_1 \\ge \\gamma _3^-$ .",
"Combining the two bounds gives us $ \\left\\Vert W_1(u,v) \\right\\Vert _{\\gamma _3^- - 1, \\pi _1} \\le C \\left\\Vert u \\otimes v \\right\\Vert _{\\pi _1} \\le C \\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert u \\right\\Vert _{r_1, p_1}.$ Note that Equations (REF ) and (REF ) give us $\\frac{1}{\\pi _1} = \\frac{1}{p_0} + \\frac{1}{p_1} - \\frac{r_0}{n}.$ With this new bound on $W_1(u,v)$ , we come back to $J_1$ and see that $J_1 & \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^-} \\left\\Vert W_1 (u,v) \\right\\Vert _{\\gamma _3^- - 1,\\pi _1}ds \\\\& \\le C \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^-} \\left\\Vert u \\right\\Vert _{r_0, p_0} \\left\\Vert u \\right\\Vert _{r_1, p_1} ds \\\\& = C \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^-} s^{-a_1} \\left\\Vert u \\right\\Vert _{r_0, p_0} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1, p_1} ds \\\\& \\le C \\left\\Vert u \\right\\Vert _{0; r_0, p_0} \\left\\Vert u \\right\\Vert _{a_1; r_1, p_1} \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^-} s^{-a_1} ds \\\\& < CM^2 T^{1 - (r_0 - (\\gamma _3^- - 1) + n/{\\pi _1} - n/p_0)/\\gamma _1^- - a_1},$ where the last inequality holds by Proposition REF if $ \\gamma _1^- & > r_0 - (\\gamma _3^- - 1) + \\frac{n}{\\pi _1} - \\frac{n}{p_0} + \\gamma _1 a_1 \\\\& = r_0 - (\\gamma _3^- - 1) + n \\left( \\frac{1}{p_0} + \\frac{1}{p_1} - \\frac{r_0}{n} \\right) - \\frac{n}{p_0} + r_1 - r_0 + \\frac{n}{p_0} - \\frac{n}{p_1} \\\\& = 1 - r_0 + r_1 - \\gamma _3^- + \\frac{n}{p_0}.$ We further note that the requirement in (REF ) also guarantees that the exponent on $T$ is positive, as desired.",
"We now turn our attention to $K_1$ .",
"Proposition REF guarantees that, if $p_1 \\ge \\pi _1^{\\prime }$ , $K_1 \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_1)/\\gamma _1^-} \\left\\Vert W_1 (u,v) \\right\\Vert _{r_1 - r_0 + \\gamma _3^- - 1,\\pi _1^{\\prime }} ds.$ This time, we chose $r_1 - r_0 + \\gamma _3^- - 1$ in order to match the previous “jump” in regularity from $r_0$ to $\\gamma _3^- - 1$ .",
"Propositions REF and REF give us $\\left\\Vert W_1 (u,v) \\right\\Vert _{r_1 - r_0 + \\gamma _3^- - 1,\\pi _1^{\\prime }} & = \\left\\Vert P(1 - \\mathcal {L}_3)^{-1} \\operatorname{div}(u \\otimes v) \\right\\Vert _{r_1 - r_0 + \\gamma _3^- - 1,\\pi _1^{\\prime }} \\\\& \\le C \\left\\Vert u \\otimes v \\right\\Vert _{r_1 - r_0, \\pi _1^{\\prime }} \\\\& \\le C \\left( \\left\\Vert u \\right\\Vert _{r_1 - r_0, p^{\\prime }} \\left\\Vert v \\right\\Vert _{p^{\\prime \\prime }} + \\left\\Vert v \\right\\Vert _{r_1 - r_0, q^{\\prime }} \\left\\Vert u \\right\\Vert _{q^{\\prime \\prime }} \\right),$ provided that $\\frac{1}{\\pi _1^{\\prime }} = \\frac{1}{p^{\\prime }} + \\frac{1}{p^{\\prime \\prime }} = \\frac{1}{q^{\\prime }} + \\frac{1}{q^{\\prime \\prime }}.$ Four applications of Proposition REF lead us to the following bounds: $& \\left\\Vert u \\right\\Vert _{r_1 - r_0, p^{\\prime }} \\le \\left\\Vert u \\right\\Vert _{r_1,p_1} & \\text{if } r_0 < \\frac{n}{p_1} \\text{ and } \\frac{1}{p^{\\prime }} = \\frac{1}{p_1} - \\frac{r_0}{n}, \\\\& \\left\\Vert v \\right\\Vert _{p^{\\prime \\prime }} \\le \\left\\Vert v \\right\\Vert _{r_0 - \\gamma _3^-, p_0} & \\text{if } r_0 < \\frac{n}{p_0} + \\gamma _3^- \\text{ and } \\frac{1}{p^{\\prime \\prime }} = \\frac{1}{p_0} - \\frac{r_0 - \\gamma _3^-}{n}, \\\\& \\left\\Vert v \\right\\Vert _{r_1 - r_0, q^{\\prime }} \\le \\left\\Vert v \\right\\Vert _{r_1 - \\gamma _3^-, p_1} & \\text{if } r_0 < \\frac{n}{p_0} + \\gamma _3^- \\text{ and } \\frac{1}{q^{\\prime }} = \\frac{1}{p_1} - \\frac{r_0 - \\gamma _3^-}{n}, \\\\& \\left\\Vert u \\right\\Vert _{q^{\\prime \\prime }} \\le \\left\\Vert u \\right\\Vert _{r_0,p_0} & \\text{if } r_0 < \\frac{n}{p_0} \\text{ and } \\frac{1}{q^{\\prime \\prime }} = \\frac{1}{p_0} - \\frac{r_0}{n}.",
"$ Combining the parameters specified by (REF )-(), we obtain $\\frac{1}{\\pi _1^{\\prime }} = \\frac{1}{p_0} + \\frac{1}{p_1} - \\frac{2r_0 - \\gamma _3^-}{n}$ and $\\left\\Vert W_1(u,v) \\right\\Vert _{r_1 - r_0 + \\gamma _3^- - 1,\\pi _1^{\\prime }} \\le C \\left\\Vert u \\otimes v \\right\\Vert _{r_1 - r_0, \\pi _1^{\\prime }} \\le C \\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert u \\right\\Vert _{r_1,p_1}.$ Moreover, the integrability requirement from Proposition REF necessitates $\\frac{1}{p_1} \\le \\frac{1}{\\pi _1^{\\prime }} = \\frac{1}{p_0} + \\frac{1}{p_1} - \\frac{2r_0 - \\gamma _3^-}{n},$ and so $ r_0 \\le \\frac{1}{2} \\left( \\frac{n}{p_0} + \\gamma _3^- \\right).$ We can finally plug this bound into $K_1$ : $K_1 & \\le \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_1)/\\gamma _1^-} \\left\\Vert W_1 (u,v) \\right\\Vert _{r_1 - r_0 + \\gamma _3^- - 1,\\pi _1^{\\prime }} ds \\\\& \\le C \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_1)/\\gamma _1^-} \\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert u \\right\\Vert _{r_1,p_1} ds \\\\& = C \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_1)/\\gamma _1^-} s^{-a_1} \\left\\Vert u \\right\\Vert _{r_0,p_0} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} ds \\\\& \\le C \\left\\Vert u \\right\\Vert _{0;r_0,p_0} \\left\\Vert u \\right\\Vert _{a_1; r_1,p_1} \\sup _{(0,T)} t^{a_1}\\int _0^t (t-s)^{-(r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_1)/\\gamma _1^-} s^{-a_1} ds \\\\& < CM^2 T^{1 - (r_1 - (r_1 - r_0 + \\gamma _3^- - 1) + n/{\\pi _1^{\\prime }} - n/p_0)/\\gamma _1^-},$ where the last inequality follows by Proposition REF if $\\gamma _1^- & > r_1 - \\left( r_1 - r_0 + \\gamma _3^- - 1 \\right) + \\frac{n}{\\pi _1^{\\prime }} - \\frac{n}{p_1} + \\gamma _1^- a_1 \\\\& = r_0 - \\gamma _3^-+ 1 + n \\left( \\frac{1}{p_0} + \\frac{1}{p_1} - \\frac{2r_0 - \\gamma _3^-}{n} \\right) - \\frac{n}{p_1} + r_1 - r_0 + \\frac{n}{p_0} - \\frac{n}{p_1} \\\\& = 1 - 2r_0 + r_1 + \\frac{2n}{p_0} - \\frac{n}{p_1} .$ Note that, once again, the requirement that Proposition REF hold is sufficient to guarantee that the exponent on $T$ be positive.",
"To summarize, here is the list of inequalities needed to obtain the desired bounds on $J_1$ and $K_1$ : $& r_0 \\le \\gamma _3^- & \\text{assumption}, \\\\& \\gamma _3^- - 1 \\le r_0 & \\text{semigroup estimate for $J_1$}, \\\\& r_1 - r_0 + \\gamma _3^- - 1 \\le r_1 & \\text{semigroup estimate for $K_1$} \\\\& r_0 < \\frac{n}{p_0} & (\\ref {J_1-p^{\\prime }}), \\\\& r_0 < \\frac{n}{p_1} & (\\ref {K_1-p^{\\prime }}), \\\\& r_0 \\le \\frac{1}{2} \\left( \\frac{n}{p_0} + \\gamma _3^- \\right) & (\\ref {extra-cond}), \\\\& r_0 < \\frac{n}{p_0} + \\gamma _3^- & (\\ref {J_2-p^{\\prime \\prime }}), \\\\& r_1 \\ge \\gamma _3^- & \\text{bound on $\\left\\Vert u \\right\\Vert _{\\gamma _3^-,p_1}$ in $J_1$}, \\\\& \\gamma _1^- > 1 - r_0 + r_1 - \\gamma _3^- + \\frac{n}{p_0} & (\\ref {J_1-gamma_1}), \\\\& \\gamma _1^- > 1 - 2r_0 + r_1 + \\frac{2n}{p_0} - \\frac{n}{p_1} & (\\ref {K_1-gamma_1-new}).$ After some obvious simplifications and after noting that (REF ) implies (REF ) since $\\underbrace{\\left( 1 - 2r_0 + r_1 + \\frac{2n}{p_0} - \\frac{n}{p_1} \\right)}_{\\text{RHS of (\\ref {K_1-gamma_1-new})}} - \\underbrace{\\left( 1 - r_0 + r_1 - \\gamma _3^- + \\frac{n}{p_0} \\right)}_{\\text{RHS of (\\ref {J_1-gamma_1})}} = \\gamma _3^- - r_0 + \\frac{n}{p_0} - \\frac{n}{p_1} \\ge 0,$ the list reduces to $\\gamma _3^- - 1 \\le r_0 \\le \\gamma _3^- \\le r_1, \\\\r_0 < \\frac{n}{p_1}, \\\\\\gamma _1^- > 1 - 2r_0 + r_1 + \\frac{2n}{p_0} - \\frac{n}{p_1}.$ subsection2 .5plus.7-.5em $J_2$ and $K_2$ .",
"We have $J_2 = \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_2 (u,v) \\right\\Vert _{r_0,p_0} ds \\le \\sup _{(0,T)} \\int _0^t \\left\\Vert W_2 (u,v) \\right\\Vert _{r_0, p_0} ds.$ We now work towards bounding $W_2(u,v)$ .",
"We immediately see, thanks to Proposition REF , that $\\left\\Vert W_2(u,v) \\right\\Vert _{r_0, p_0} & = \\left\\Vert P (1 - \\mathcal {L}_3)^{-1} \\sum _{i = 1}^n v_i \\nabla u_i \\right\\Vert _{r_0,p_0} \\\\& \\le C \\left\\Vert \\sum _{i = 1}^n v_i \\nabla u_i \\right\\Vert _{r_0 - \\gamma _3^-,p_0} \\\\& \\le C \\sum _{i = 1}^n \\left\\Vert v_i \\nabla u_i \\right\\Vert _{p_0}$ since $r_0 \\le \\gamma _3^-$ .",
"Now the product estimate is nothing more than Holder's inequality, so if $\\frac{1}{p_0} = \\frac{1}{p^{\\prime }} + \\frac{1}{p^{\\prime \\prime }}$ we obtain $\\left\\Vert W_2(u,v) \\right\\Vert _{r_0, p_0} & \\le C \\sum _{i = 1}^n \\left\\Vert v_i \\right\\Vert _{p^{\\prime }} \\left\\Vert \\nabla u_i \\right\\Vert _{p^{\\prime \\prime }} \\\\& \\le C \\sum _{i = 1}^n \\left\\Vert v \\right\\Vert _{p^{\\prime }} \\left\\Vert \\nabla u \\right\\Vert _{p^{\\prime \\prime }} \\\\& \\le C \\left\\Vert u \\right\\Vert _{\\gamma _3^-, p^{\\prime }} \\left\\Vert u \\right\\Vert _{1, p^{\\prime \\prime }}.$ By Proposition REF we have that $\\left\\Vert u \\right\\Vert _{\\gamma _3^-, p^{\\prime }} \\le C \\left\\Vert u \\right\\Vert _{\\gamma _3^- + \\beta , p_1} \\le C \\left\\Vert u \\right\\Vert _{r_1,p_1} \\text{ and } \\left\\Vert u \\right\\Vert _{1, p^{\\prime \\prime }} \\le C \\left\\Vert u \\right\\Vert _{r_1,p_1},$ where the first set of inequalities requires that $ 0 \\le \\beta < \\frac{n}{p_1}, \\quad \\frac{1}{p^{\\prime }} = \\frac{1}{p_1} - \\frac{\\beta }{n}, \\quad r_1 \\ge \\gamma _3^- + \\beta ,$ and the second inequality requires that $ \\frac{1}{p^{\\prime \\prime }} = \\frac{1}{p_1} - \\frac{r_1 - 1}{n} \\text{ and } r_1 \\ge 1.$ We finally obtain $\\left\\Vert W_2(u,v) \\right\\Vert _{r_0, p_0} \\le C \\left\\Vert u \\right\\Vert _{\\gamma _3^-, p^{\\prime }} \\left\\Vert u \\right\\Vert _{1, p^{\\prime \\prime }} \\le C \\left\\Vert u \\right\\Vert _{r_1, p_1}^2.$ We pause here to note that, without the presence of the space $\\dot{C}_{a_1;r_1,p_1}(\\mathbb {R}^n)$ in the definition of $X_{T,M}$ , we would not be able to bound this $W_2$ term.",
"Returning to $J_2$ , we have $J_2 & \\le \\sup _{(0,T)} \\int _0^t \\left\\Vert W_2 (u,v) \\right\\Vert _{r_0, \\pi _2} ds \\\\& \\le C \\sup _{(0,T)} \\int _0^t \\left\\Vert u \\right\\Vert _{r_1,p_1}^2 ds \\\\& = C \\sup _{(0,T)} \\int _0^t s^{-2a_1} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} ds \\\\& \\le C \\left\\Vert u \\right\\Vert _{a_1;r_1,p_1}^2 \\sup _{(0,T)} \\int _0^t s^{-2a_1} ds \\\\& < CM^2 T^{1 - 2a_1},$ provided $2a_1 > 1$ , which is equivalent to $ \\gamma _1^- > 2r_1 - 2r_0 + \\frac{2n}{p_0} - \\frac{2n}{p_1}$ and we recall that $\\frac{n}{p_0} = \\frac{n}{p^{\\prime }} + \\frac{n}{p^{\\prime \\prime }} = \\frac{2n}{p_1} - \\beta - r_1 + 1.$ We choose $\\beta $ to be exactly $ \\beta = 1 - r_1 - \\frac{n}{p_0} + \\frac{2n}{p_1},$ and so the two requirements in (REF ) become $ r_1 \\ge 1 - \\frac{n}{p_0} + \\frac{n}{p_1} \\text{ and } 2r_1 \\ge 1 + \\gamma _3^- - \\frac{n}{p_0} + \\frac{2n}{p_1}.$ Turning to $K_2$ , noting that we go down to $\\gamma _3^-$ instead of $r_0$ , we have $K_2 & = \\sup _{(0,T)} t^{a_1} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_2 (u,v) \\right\\Vert _{r_1,p_1} ds \\\\& \\le \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - \\gamma _3^- + n/{p_0} - n/p_1)/\\gamma _1^-} \\left\\Vert W_2 (u,v) \\right\\Vert _{\\gamma _3^-,p_0}ds \\\\& \\le C \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - \\gamma _3^- + n/{p_0} - n/p_1)/\\gamma _1^-} \\left\\Vert u \\right\\Vert _{r_1,p_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} ds \\\\& = C \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - \\gamma _3^- + n/{p_0} - n/p_1)/\\gamma _1^-} s^{-2a_1} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} s^{a_1} \\left\\Vert u \\right\\Vert _{r_1,p_1} ds \\\\& \\le C \\left\\Vert u \\right\\Vert _{a_1;r_0,p_0}^2 \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - \\gamma _3^- + n/{p_0} - n/p_1)/\\gamma _1^-} s^{-2a_1} ds \\\\& < CM^2 T^{1 - (r_1 - \\gamma _3^- + n/\\pi _2 - n/p_1)/\\gamma _1^- - a_1},$ where, by Proposition REF , the last inequality holds if $ \\gamma _1^- & > r_1 - \\gamma _3^- + \\frac{n}{p_0} - \\frac{n}{p_1} + 2\\gamma _1^-a_1 \\\\& = 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1}.$ Here is a summary of the inequalities needed to obtain the required bounds on $J_2$ and $K_2$ : $& r_1 \\ge 1 & (\\ref {J_2-p^{\\prime \\prime }}), \\\\& r_1 \\ge 1 - \\frac{n}{p_0} + \\frac{n}{p_1} & (\\ref {J_2-p^{\\prime }-new}), \\\\& 2r_1 \\ge 1 + \\gamma _3^- - \\frac{n}{p_0} + \\frac{2n}{p_1} & (\\ref {J_2-p^{\\prime }-new}), \\\\& \\gamma _1^- > 2r_1 - 2r_0 + \\frac{2n}{p_0} - \\frac{2n}{p_1} & (\\ref {J_2-gamma_1-new}), \\\\& \\gamma _1^- > 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1} & (\\ref {K_2-gamma_1}).$ By noting that $\\underbrace{\\left(3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1} \\right)}_{\\text{RHS of (\\ref {K_2-gamma_1})}} - \\underbrace{\\left( 2r_1 - 2r_0 + \\frac{2n}{p_0} - \\frac{2n}{p_1}\\right)}_{\\text{RHS of (\\ref {J_2-gamma_1-new})}} = r_1 - \\gamma _3^- + \\frac{n}{p_0} - \\frac{n}{p_1} \\ge 0$ we conclude that (REF ) implies (REF ), and so the list reduces to $2r_1 \\ge \\max \\left\\lbrace 2, 1 + \\gamma _3^- - \\frac{n}{p_0} + \\frac{2n}{p_1} \\right\\rbrace , \\\\\\gamma _1^- > 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1}.$ subsection2 .5plus.7-.5em $J_3$ and $K_3$ .",
"Provided that $r_2 - 1 + \\gamma _3^- \\le r_0$ and $\\frac{1}{\\pi _3} \\ge \\frac{1}{p_0}$ , Proposition REF gives us $J_3 & = \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (B,B) \\right\\Vert _{r_0,p_0} ds \\\\& \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (r_2 - 1 + \\gamma _3^-) + n/{\\pi _3} - n/p_0)/\\gamma _1^-} \\left\\Vert W_1 (B,B) \\right\\Vert _{r_2 - 1 + \\gamma _3^-,\\pi _3} ds.$ Once again, applying Propositions REF and REF gets us $\\left\\Vert W_1(B,B) \\right\\Vert _{r_2 - 1 + \\gamma _3^-, \\pi _3} & = \\left\\Vert P (1 - \\mathcal {L}_3)^{-1} \\operatorname{div}(B \\otimes B) \\right\\Vert _{r_2 - 1 + \\gamma _3^-, \\pi _3} \\\\& \\le C \\left\\Vert B \\otimes B \\right\\Vert _{r_2, \\pi _3} \\\\& \\le C \\left\\Vert B \\right\\Vert _{r_2, p_2} \\left\\Vert B \\right\\Vert _{p^{\\prime }},$ where the product estimate requires that $r_2 \\ge 0$ and $\\frac{1}{\\pi _3} = \\frac{1}{p_2} + \\frac{1}{p^{\\prime }}.$ Provided that $ r_2 < \\frac{n}{p_2} \\text{ and } \\frac{1}{p^{\\prime }} = \\frac{1}{p_2} - \\frac{r_2}{n},$ which combines with the previous equation to give $\\frac{1}{\\pi _3} = \\frac{2}{p_2} - \\frac{r_2}{n},$ we can bound $\\left\\Vert B \\right\\Vert _{p^{\\prime }}$ by $\\left\\Vert B \\right\\Vert _{r_2,p_2}$ thanks to Proposition REF .",
"Thus, $\\left\\Vert W_1(B,B) \\right\\Vert _{r_2-1+\\gamma _3^-,\\pi _3} \\le C \\left\\Vert B \\right\\Vert _{r_2,p_2}^2.$ Moreover, Proposition (REF ) requires that $\\frac{1}{p_0} \\le \\frac{1}{\\pi _3} = \\frac{2}{p_2} - \\frac{r_2}{n},$ which can be restated as $ r_2 \\le \\frac{2n}{p_2} - \\frac{n}{p_0}.$ Plugging the bound for $W_1(B,B)$ back into the integral gives us $J_3 & \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (r_2 - 1 + \\gamma _3^-) + n/{\\pi _3} - n/p_0)/\\gamma _1^-} \\left\\Vert W_1 (B,B) \\right\\Vert _{r_2 - 1 + \\gamma _3^-,\\pi _3} ds \\\\& \\le C \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (r_2 - 1 + \\gamma _3^-) + n/{\\pi _3} - n/p_0)/\\gamma _1^-} \\left\\Vert B \\right\\Vert _{r_2,p_2}^2 ds \\\\& \\le C \\left\\Vert B \\right\\Vert _{0;r_2,p_2}^2 \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_0 - (r_2 - 1 + \\gamma _3^-) + n/{\\pi _3} - n/p_0)/\\gamma _1^-} ds \\\\& < C M^2 T^{1 - (r_0 - (r_2 - 1 + \\gamma _3^-) + n/{\\pi _3} - n/p_0)/\\gamma _1^-},$ where once again the last inequality holds if $\\gamma _1^- & > r_0 - (r_2 - 1 + \\gamma _3^-) + \\frac{n}{\\pi _3} - \\frac{n}{p_0} \\\\& = r_0 - 2r_2 - \\gamma _3^- + 1 + \\frac{2n}{p_2} - \\frac{n}{p_0}.",
"$ The same bounds for $W_1(B,B)$ work in the case of $K_3$ , so that $K_3 & = \\sup _{(0,T)} t^{a_1} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_1} W_1 (B,B) \\right\\Vert _{r_1,p_1} ds \\\\& \\le \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_2 - 1 + \\gamma _3^-) + n/\\pi _3 - n/p_1)/\\gamma _1^-} \\left\\Vert W_1 (B,B) \\right\\Vert _{r_2 - 1 + \\gamma _3^-,\\pi _3} ds \\\\& \\le C \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_2 - 1 + \\gamma _3^-) + n/\\pi _3 - n/p_1)/\\gamma _1^-} \\left\\Vert B \\right\\Vert _{r_2,p_2}^2 ds \\\\& \\le C \\left\\Vert B \\right\\Vert _{0;r_2,p_2}^2 \\sup _{(0,T)} t^{a_1} \\int _0^t (t-s)^{-(r_1 - (r_2 - 1 + \\gamma _3^-) + n/\\pi _3 - n/p_1)/\\gamma _1^-} ds \\\\& < C M^2 T^{1 - (r_1 - (r_2 - 1 + \\gamma _3^-) + n/\\pi _3 - n/p_1)/\\gamma _1^- + a_1},$ which holds provided that $ \\gamma _1^- & > r_1 - (r_2 - 1 + \\gamma _3^-) + \\frac{n}{\\pi _3} - \\frac{n}{p_1} \\\\& = r_1 - 2r_2 - \\gamma _3^- + 1 - \\frac{n}{p_1} + \\frac{2n}{p_2}.$ What follows is the list of inequalities needed to bound $J_3$ and $K_3$ as desired: $& r_2 - 1 + \\gamma _3^- \\le r_0 & \\text{semigroup estimate for $J_3$}, \\\\& r_2 - 1 + \\gamma _3^- \\le r_1 & \\text{semigroup estimate for $K_3$}, \\\\& r_2 \\ge 0 & \\text{product estimate}, \\\\& r_2 < \\min \\left\\lbrace \\frac{n}{p_2}, \\frac{2n}{p_2} - \\frac{n}{p_0} \\right\\rbrace & (\\ref {J_3-p^{\\prime }})-(\\ref {r_2-extra}), \\\\& \\gamma _1^- > r_0- 2r_2 - \\gamma _3^- + 1 - \\frac{n}{p_0} + \\frac{2n}{p_2} & (\\ref {J_3-gamma_1}), \\\\& \\gamma _1^- > r_1 - 2r_2 - \\gamma _3^- + 1 + \\frac{2n}{p_2} - \\frac{n}{p_1} & (\\ref {K_3-gamma_1}).$ We see that (REF ) suffices for (REF ) since $\\underbrace{\\left( r_1 - 2r_2 - \\gamma _3^- + 1 + \\frac{2n}{p_2} - \\frac{n}{p_1} \\right)}_{\\text{RHS of (\\ref {K_3-gamma_1})}} - \\underbrace{\\left( r_0- 2r_2 - \\gamma _3^- + 1 - \\frac{n}{p_0} + \\frac{2n}{p_2} \\right)}_{\\text{RHS of (\\ref {J_3-gamma_1})}} = r_1 - r_0 + \\frac{n}{p_0} - \\frac{n}{p_1} \\ge 0,$ and so the list reduces to $r_2 - 1 + \\gamma _3^- \\le r_0, \\\\0 \\le r_2 < \\min \\left\\lbrace \\frac{n}{p_2}, \\frac{2n}{p_2} - \\frac{n}{p_0} \\right\\rbrace , \\\\\\gamma _1^- > r_1 - 2r_2 - \\gamma _3^- + 1 + \\frac{2n}{p_2} - \\frac{n}{p_1}.$ subsection2 .5plus.7-.5em Bounding $I_4$ .",
"Applying Minkowski's inequality to $I_4$ gives $I_4 \\le L_1 + L_2,$ where $L_1 & := \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_2} W_3(u,B) ds \\right\\Vert _{r_2,p_2} \\\\L_2 & := \\sup _{(0,T)} \\int _0^t \\left\\Vert e^{(t-s) \\mathcal {L}_2} W_3(B,u) ds \\right\\Vert _{r_2,p_2}$ We can immediately note that, since $W_3$ is not symmetric, $L_1 \\ne L_2$ , but our techniques will give the same bound for each.",
"So, we set $L := L_1$ and proceed to bound only $L_1$ .",
"Proposition REF gives us $L \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(r_2 - (r_2 - 1) + n/\\pi _4 - n/p_2)/\\gamma _2^-} \\left\\Vert W_3(u,B) \\right\\Vert _{r_2 - 1, \\pi _4} ds,$ provided that $\\frac{1}{\\pi _4} \\ge \\frac{1}{p_2}$ .",
"Continuing with $W_3(u,B)$ , we obtain $\\left\\Vert W_3(u,B) \\right\\Vert _{r_2 - 1, \\pi _4} = \\left\\Vert P \\operatorname{div}(u \\otimes B) \\right\\Vert _{r_2 - 1, \\pi _4} \\le C \\left\\Vert u \\otimes B \\right\\Vert _{r_2, \\pi _4};$ an application of Proposition REF gives us $\\left\\Vert u \\otimes B \\right\\Vert _{r_2 , \\pi _4} \\le C \\left( \\left\\Vert u \\right\\Vert _{r_2,p^{\\prime }} \\left\\Vert B \\right\\Vert _{p^{\\prime \\prime }} + \\left\\Vert B \\right\\Vert _{r_2,p_2} \\left\\Vert u \\right\\Vert _{q^{\\prime \\prime }} \\right)$ as long as $\\frac{1}{\\pi _4} = \\frac{1}{p^{\\prime }} + \\frac{1}{p^{\\prime \\prime }} = \\frac{1}{p_2} + \\frac{1}{q^{\\prime \\prime }}.$ We want to bound $\\left\\Vert u \\otimes B \\right\\Vert _{r_2, \\pi _4}$ by $\\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert B \\right\\Vert _{r_2,p_2}$ , which requires three applications of Proposition REF .",
"First, we obtain $\\left\\Vert u \\right\\Vert _{r_2,p^{\\prime }} \\le \\left\\Vert u \\right\\Vert _{r_0,p_0}$ if $ 0 \\le r_0 - r_2 < \\frac{n}{p_0} \\text{ and } \\frac{1}{p^{\\prime }} - \\frac{r_2}{n} = \\frac{1}{p_0} - \\frac{r_0}{n}.$ We further get $\\left\\Vert u \\right\\Vert _{q^{\\prime \\prime }} \\le \\left\\Vert u \\right\\Vert _{r_0,p_0}$ provided that $ r_0 < \\frac{n}{p_0} \\text{ and } \\frac{1}{q^{\\prime \\prime }} = \\frac{1}{p_0} - \\frac{r_0}{n}.$ The last embedding, $\\left\\Vert B \\right\\Vert _{p^{\\prime \\prime }} \\le \\left\\Vert B \\right\\Vert _{r_2,p_2}$ , requires $ r_2 < \\frac{n}{p_2} \\text{ and } \\frac{1}{p^{\\prime \\prime }} = \\frac{1}{p_2} - \\frac{r_2}{n}.$ Combining Equations (REF )-(REF ) together gives us $\\frac{1}{\\pi _4} = \\frac{1}{p_0} + \\frac{1}{p_2} - \\frac{r_0}{n},$ which is required to satisfy $ \\frac{1}{p_2} \\le \\frac{1}{\\pi _4} = \\frac{1}{p_0} + \\frac{1}{p_2} - \\frac{r_0}{n} \\Rightarrow r_0 \\le \\frac{n}{p_0}.$ This is the bound we were looking for: $\\left\\Vert W_3(u,B) \\right\\Vert _{r_2 - 1, \\pi _4} \\le C \\left( \\left\\Vert u \\right\\Vert _{r_2,p^{\\prime }} \\left\\Vert B \\right\\Vert _{p^{\\prime \\prime }} + \\left\\Vert B \\right\\Vert _{r_2,q^{\\prime }} \\left\\Vert u \\right\\Vert _{q^{\\prime \\prime }} \\right) \\le C \\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert B \\right\\Vert _{r_2,p_2}.$ We can plug the above into $L$ and obtain $L & \\le \\sup _{(0,T)} \\int _0^t (t-s)^{-(1 + n/\\pi _4 - n/p_2)/\\gamma _2^-} \\left\\Vert W_3(u,B) \\right\\Vert _{r_2 - 1, \\pi _4} ds \\\\& \\le C \\sup _{(0,T)} \\int _0^t (t-s)^{-(1 + n/\\pi _4 - n/p_2)/\\gamma _2^-} \\left\\Vert u \\right\\Vert _{r_0,p_0} \\left\\Vert B \\right\\Vert _{r_2,p_2} ds \\\\& \\le C \\left\\Vert u \\right\\Vert _{0;r_0,p_0} \\left\\Vert B \\right\\Vert _{0;r_2,p_2} \\sup _{(0,T)} \\int _0^t (t-s)^{-(1 + n/\\pi _4 - n/p_2)/\\gamma _2^-} ds \\\\& \\le C M^2 T^{1 - (1 + n/\\pi _4 - n/p_2)/\\gamma _2^-},$ which holds if $ \\gamma _2^- & > 1 + \\frac{n}{\\pi _4} - \\frac{n}{p_2} \\\\& = 1 - r_0 + \\frac{n}{p_0}.$ The list of inequalities necessary to bound $L$ is thus $& 0 \\le r_0 - r_2 < \\frac{n}{p_0} & (\\ref {L-p^{\\prime }}), \\\\& r_2 < \\frac{n}{p_2} & (\\ref {L-p^{\\prime \\prime }}), \\\\& r_0 \\le \\frac{n}{p_0} & (\\ref {extra-cond-2}), \\\\& \\gamma _2^- > 1 - r_0 + \\frac{n}{p_0} & (\\ref {L-gamma_2}).$ subsection2 .5plus.7-.5em Wrapping up On one final note, we point out that since $\\underbrace{\\left( 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1} \\right)}_{\\text{RHS of (\\ref {K_2-gamma_1})}} - \\underbrace{\\left( 1 - 2r_0 + r_1 + \\frac{2n}{p_0} - \\frac{n}{p_1} \\right)}_{\\text{RHS of (\\ref {K_1-gamma_1-new})}} = 2r_1 - 1 - \\gamma _3^- + \\frac{n}{p_0} - \\frac{2n}{p_1} \\ge 0$ we have that (REF ) implies (REF ), and so the following is the definitive list containing all the inequalities needed for $I_2, I_3,$ and $I_4$ : $\\gamma _3^- - 1 \\le r_0 \\le \\gamma _3^- \\le r_1, \\\\r_2 - 1 + \\gamma _3^- \\le r_0, \\\\r_2 \\le r_0 < \\frac{n}{p_1}, \\\\2r_1 \\ge \\max \\left\\lbrace 2, 1 + \\gamma _3^- - \\frac{n}{p_0} + \\frac{2n}{p_1} \\right\\rbrace , \\\\r_2 < \\min \\left\\lbrace \\frac{n}{p_2}, \\frac{2n}{p_2} - \\frac{n}{p_0} \\right\\rbrace , \\\\\\gamma _1^- > 3r_1 - 2r_0 - \\gamma _3^- + \\frac{3n}{p_0} - \\frac{3n}{p_1}, \\\\\\gamma _1^- > 1 - 2r_2 + r_1 - \\gamma _3^- - \\frac{n}{p_1} + \\frac{2n}{p_2}, \\\\\\gamma _2^- > 1 - r_0 + \\frac{n}{p_0}.$ The above inequalities coincide with those in Theorem REF , and so we are done."
]
] | 2005.14130 |
[
[
"Antiferromagnetism mediated by heavy electrons: singlet shielding vs\n \"high-order\" RKKY"
],
[
"Abstract Can the antiferromagnetic (AF) order be induced via the local moments' hybridization with the heavy electrons instead of conduction electrons?",
"We address this intriguingly fundamental question via a prototypical model to describe the interplay between local moments and heavy electrons.",
"We discover that the AF order can be mediated via the heavy electrons through the mechanism of \"high-order\" Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction.",
"Moreover, the induced AF order can coexist with its metallicity in a finite regime of the phase diagram, which competes with and ultimately destroys the AF order concurrently with the breakdown of heavy electrons.",
"The potential relevance to the heavy fermion compound Ce$_3$(Pt/Pd)In$_{11}$ is discussed."
],
[
"Introduction",
"The conventional Kondo/Anderson lattice model (KLM/PAM) describes the competition of antiferromagnetism and Kondo screening as a fundamental model of heavy fermion physics [1], [3], [4], [5], [6], [2].",
"Generally, there are two well-known exchange mechanism responsible for the formation of the antiferromagnetic phase, i.e.",
"(1) the superexchange interaction between localized electrons via their hybridization with the conduction band and (2) the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction originating from the scattering of the conduction electrons from two localized moments [7].",
"The former always favors the antiferromagnetic order between localized moment but is strongly suppressed if the conduction electron band approaches to half filled [8]; while the latter's sign and magnitude vary with the distance between localized moments and also the conduction band filling [9].",
"Take the conventional PAM as an example, it is well known that at small hybridization between the conduction and localized electrons, the indirect RKKY interaction induces the antiferromagnetic ground state, which competes with the paramagnetic spin liquid ground state formed by Kondo screening the local electrons by the conduction band at large hybridization.",
"The common thread between the aforementioned two exchange mechanism relies on the conduction electrons.",
"It is natural to ask whether or not the heavy electrons can similarly act as a “glue” for the antiferromagnetic (AF) order between local moments.",
"Until now, surprisingly, there has been few studies on this fundamental question despite that multiple 4f orbitals was considered in the context of Cerium volume collapse considering the inherent 4f electronic correlations [10].",
"We point out that the AF order mediated by heavy electrons is not only an abstract theoretical question but also relevant to recent discovery of the microscopic coexistence between AF and superconductivity in a particular family of heavy fermion compounds Ce$_3$ (Pt/Pd)In$_{11}$ harboring two inequivalent Ce sites [11], [12], [13], [14], where the most fascinating scenario proposed for their coexistence claims that the Ce(1) sublattice is fully Kondo screened and responsible to the superconducting state while the Ce(2) sublattice forms the magnetic ordering.",
"Therefore, one intrinsic problem is the interplay between Ce(1) and Ce(2) sublattices, particularly the possible Ce(2) magnetic order induced by Ce(1) sublattice.",
"Motivated by these experimental progress, in this proof of principle study, we explore the possibility of AF order mediated by heavy electrons via a prototypical model to describe the interplay between the local moments and heavy electrons.",
"Specifically, we discover that the hybridization between the local moments and heavy electrons can indeed induce AF order between the local moments through the so-called “high-order” RKKY interaction that resembles the conventional one mediated via the conduction electrons in standard PAM/KLM."
],
[
"Model and methodology",
"To illustrate our findings, we adopt the simplest and prototypical model consisting of two distinct localized $f$ -orbitals together with the conduction electrons on two-dimensional square lattice, which reads in the half-filled form: ${\\cal H} &=& - t \\sum \\limits _{\\langle ij \\rangle \\sigma }(c^{\\dagger }_{i\\sigma }c_{j\\sigma }^{\\vphantom{dagger}}+c^{\\dagger }_{j\\sigma }c_{i\\sigma }^{\\vphantom{dagger}})- \\mu \\sum \\limits _{i\\sigma } (n^{c}_{i\\sigma }+ n^{f_1}_{i\\sigma } +n^{f_2}_{i\\sigma }) \\nonumber \\\\&+& V \\sum \\limits _{i \\sigma } (c^{\\dagger }_{i\\sigma }f_{1i\\sigma }^{\\vphantom{dagger}}+ f^{\\dagger }_{1i\\sigma }c_{i\\sigma }^{\\vphantom{dagger}})+ t_{\\perp } \\sum \\limits _{i \\sigma } (f^{\\dagger }_{1i\\sigma }f_{2i\\sigma }^{\\vphantom{dagger}}+ f^{\\dagger }_{2i\\sigma }f_{1i\\sigma }^{\\vphantom{dagger}}) \\nonumber \\\\&+& U \\sum \\limits _{mi} (n^{f_m}_{i\\uparrow }-\\frac{1}{2}) (n^{f_m}_{i\\downarrow }-\\frac{1}{2})$ where $c^{\\dagger }_{i\\sigma }(c_{i\\sigma }^{\\vphantom{dagger}})$ and $f^{\\dagger }_{mi\\sigma }(f_{mi\\sigma }^{\\vphantom{dagger}})$ with $m=1,2$ are creation(destruction) operators for conduction and two local $f_{1,2}$ electrons on site $i$ with spin $\\sigma $ .",
"$n^{c,f_m}_{i\\sigma }$ are the associated number operators.",
"The chemical potential $\\mu $ can be tuned for a desired average occupancy of three orbitals.",
"The hopping $t=1$ between conduction electrons on nearest neighbor sites $\\langle ij \\rangle $ sets the energy scale.",
"$U$ is the local repulsive interaction in the $f_{1,2}$ orbital.",
"Note that in this work we only consider the case that the $f_{1,2}$ orbitals share an identical $U$ for simplicity although generally they can differ.",
"The two remaining control parameters are two distinct hybridizations, namely $V$ between $c$ -$f_1$ and $t_\\perp $ between $f_1$ -$f_2$ .",
"Before proceeding, we remark that in the heavy fermion compounds with multiple crystallographic inequivalent local moment sites, such as Ce$_3$ (Pt/Pd)In$_{11}$ [11], [12], [13], [14], the local environment of two Ce ions are different leading to distinct Kondo interaction strengths with the conduction electrons.",
"Here our focus is on the sole effects of the heavy electrons from $c$ -$f_1$ Kondo singlets on the additional $f_2$ local moments.",
"Thus, we neglect the $c$ -$f_2$ hybridization to avoid the additional Kondo screening from $c$ electrons and associated complexity.",
"The more realistic modelling of the heavy fermion compounds is left for future investigation.",
"In addition, to explicitly investigate the AF order without charge fluctuation, we stick on the half-filled systems by setting $\\mu =0$ so that the $c$ and $f_{1,2}$ orbitals are individually half-filled, which also ensures that the superexchange interaction between local moments is suppressed [8] as discussed before so that only the RKKY-type interaction can take the role of mediating the AF order.",
"To gather some initial insights of this model, it is worthwhile elaborating on some limiting cases.",
"In the absence of Hubbard interaction $U$ , the three-orbital unit cell gives rise to three energy bands such that the system hosts a metallic ground state for any finite $V, t_{\\perp }$ at half-filling.",
"As discussed later, turning on $U$ opens the orbital-selective spectral gap.",
"In the extreme case of $t_{\\perp } \\ll V$ , the system separates into conventional PAM plus additional individual local moments; in contrary, if $t_{\\perp } \\gg V$ , the system becomes a conduction band plus individual strongly bound dimers.",
"To fully take into account all the energy scales on the equal footing, we use the well established numerical technique of finite temperature determinant Quantum Monte Carlo (DQMC) [15] to explore the physics of Eq.",
"REF .",
"As a celebrated computational method, DQMC provides an approximation-free solution in the presence of strong correlations.",
"Besides, finite size scaling can facilitate the extraction of the AF order parameter reliably so that all the quantities throughout the paper are extracted values in the thermodynamic limit.",
"Throughout the paper, we concentrate on the characteristic intermediate coupling strength $U=4.0t$ , where it has been widely believed that the critical $c$ -$f$ hybridization strength separating the Kondo singlet and antiferromagnetic insulating ground states in PAM is $V_c \\sim 1.0t$ [16], [6].",
"Because the major purpose of this work is the AF order induced by heavy electrons, we only explore the systems with $V/t \\ge 1.2$ such that the $c$ -$f_1$ subsystem is readily within the heavy electron regime.",
"Besides, all the physical quantities are obtained through finite-size scaling in lattice sizes as large as $N=10 \\times 10$ at lowest temperature $T=0.025t$ with periodic boundary.",
"Figure: Tentative phase diagram of Eq.",
"at half-filling at lowest simulated temperature T=0.025tT=0.025t.",
"The gray and cyan regimes exhibit the f 2 f_2-AF order mediated via heavy electrons (cc-f 1 f_1).",
"The red dashed line highlights the position with maximal AF structure factor.",
"See text for details.Our major findings are illustrated in the tentative phase diagram Figure REF and summarized as follows: Singlet shielding (SS): $f_2$ local moments are effectively standing alone and shielded from the heavy electrons ($c$ -$f_1$ singlets); $f_2$ -AF$^I$: $f_2$ -AF ordered insulator via “high-order” RKKY coexisting with $c$ -$f_1$ heavy electrons; $f_2$ -AF$^M$: $f_2$ -AF order with metallic feature coexisting with partially broken heavy electrons (metallic $c$ electrons); Orbital-selective metallicity: $f_2$ -AF disappears due to broken heavy electrons; both $c$ and $f_2$ exhibit metallicity while $f_1$ remains insulating.",
"Some remarks follow in order.",
"First of all, in all phases, the $f_1$ orbital exhibits the stable insulating behavior even though $c$ and $f_2$ exhibit metallic behavior at large $t_\\perp $ .",
"Besides, $f_1$ orbital does not host the AF order unless at relatively small $V/t=1.2, 1.6$ due to the combined effects of proximity effect from $f_2$ -AF and weakened $c$ -$f_1$ Kondo screening.",
"Secondly, there is a crossover (blue dashed) line separating the $f_2$ -AF$^I$ and $f_2$ -AF$^M$ phases, which is supported by the strong metallic tendency of $c$ and $f_2$ orbitals that competes with and ultimately destroys the $f_2$ -AF order.",
"Finally, at sufficiently large $t_\\perp $ , $c$ -$f_2$ -M phase will be replaced by strongly coupled $f_1$ -$f_2$ dimers together with the free conduction electrons.",
"In Fig.",
"REF , in addition, the red dashed line highlights the $t_{\\perp }$ at which the $f_2$ -AF structure factor reaches its maximum.",
"Note that the $f_2$ -AF rapidly turns on upon its emergence and gradually disappears.",
"In what follows, we start providing the concrete numerical evidence to support the phase diagram in detail."
],
[
"Magnetic properties",
"We first illustrate our findings of the induced $f_2$ -AF order mediated via heavy electrons ($c$ -$f_1$ singlets) by the magnetic properties.",
"The AF order is manifested by the AF structure factor of the $f_{1,2}$ local moments $S^{f_m}_{AF}(V_1,V_2)=\\frac{1}{N} \\sum _{ij} e^{-i \\mathbf {q} \\cdot (\\mathbf {R_i}-\\mathbf {R_j})} \\langle (n^{f_m}_{i\\uparrow }-n^{f_m}_{i\\downarrow }) (n^{f_m}_{j\\uparrow }-n^{f_m}_{j\\downarrow }) \\rangle $ with $m=1,2$ at $\\mathbf {q}=(\\pi ,\\pi )$ , where $\\mathbf {R_i}$ denotes the coordinates of site $i$ and $N$ is the lattice size.",
"Figure REF (a) shows the finite-size scaling of $S^{f_m}_{AF}/N$ at fixed $V/t=2.0$ with dashed/solid lines denoting $m=1,2$ respectively.",
"Clearly, at small $t_{\\perp }$ , the $c$ -$f_1$ singlets shields the additional $f_2$ local moments and the two subsystems are effectively separated so that both show absence of AF order.",
"With increasing $t_{\\perp }$ , the $f_2$ -AF order emerges while $f_1$ local moments remain forming the Kondo singlets with the conduction electrons, which indicates that the $f_2$ -AF order is not induced by the proximity effect from a “$f_1$ -AF” order liberated by the $t_{\\perp }$ hybridization.",
"In the standard PAM ($t_{\\perp }=0$ ), the RKKY interaction scales as $\\sim J^{2}/W$ with $J\\sim V^{2}/U$ and $W$ the conduction bandwidth.",
"Here we claim that $f_2$ -AF is realized through a mechanism of “high-order” RKKY interaction with modified $J\\sim V^2t_{\\perp }^2/U$ via an indirect $c$ -$f_2$ hybridization, which competes with the Kondo screening scaling as $\\sim W e^{-W/J}$ .",
"As expected, further increasing $t_{\\perp }$ leads to the gradual diminish of $f_2$ -AF due to the strong $f_1$ -$f_2$ hybridization, which finally results in individual strongly bound dimers.",
"To make further progress, Fig.",
"REF (b) demonstrates the evolution of extrapolated $S^{f_m}_{AF}/N$ with $t_{\\perp }$ for diverse $V$ , where the general peak structure of $f_2$ -AF order (solid lines) and the absence of $f_1$ -AF order in most cases (dashed lines) can be seen.",
"Additionally, the $f_2$ -AF rapidly turns on upon its emergence and gradually disappears.",
"It is natural that stronger $V$ requires larger critical $t_{\\perp }$ to overcome the $c$ -$f_1$ Kondo screening to partially liberate the conduction electrons for its essential role in mediating the “high-order” RKKY that induces $f_2$ -AF order.",
"In contrary, only the systems of “light” heavy electrons with relatively small Kondo screening, e.g.",
"$V/t=1.2, 1.6$ (blue and orange dashed lines) clearly exhibit the $f_1$ -AF order whose maximum are concomitant with that of $f_2$ -AF.",
"This observation indicates the feedback among $c$ -$f_1$ -$f_2$ orbitals: (a) $t_{\\perp }$ tends to break the heavy electrons to liberate the $c$ -electrons; (b) $c$ -electrons mediate the $f_2$ -AF order via “high-order” RKKY; (c) the induced $f_2$ -AF order has proximity effect to induce the potential $f_1$ -AF order unfavored by heavy electrons.",
"Certainly, the partially liberated $c$ -electrons can also mediate the $f_1$ -AF order to some extent, although their combined effects quickly decay with enlarging the $c$ -$f_1$ hybridization $V$ .",
"Figure: (a) Local spin correlations C ab C^{ab} between orbitals: C cf 2 C^{cf_2} (full symbols), C cf 1 C^{cf_1} (half-filled symbols), and C f 1 f 2 C^{f_1f_2} (unfilled symbols) (b) local moments 〈m 2 〉\\langle m^2 \\rangle of f 1 f_{1} (dashed lines) and f 2 f_2 (solid lines) versus t ⊥ t_{\\perp } for diverse VV at T=0.025tT=0.025t.To further support our scenario of “high-order” RKKY, we resort to the local spin correlations $C^{ab}=\\langle (n^a_{\\uparrow }-n^a_{\\downarrow }) (n^b_{\\uparrow }-n^b_{\\downarrow }) \\rangle $ between three orbitals $a,b=c,f_1,f_2$ in Figure REF (a).",
"Apparently, $C^{cf_1}$ ($C^{f_1f_2}$ ) decreases (increases) in magnitude with turning on $t_{\\perp }$ .",
"Nonetheless, the striking difference shows up in the indirect $C^{cf_2}$ correlation, which exhibits a nontrivial peak, whose position is consistent with the maximal $S^{f_2}_{AF}$ shown in Fig.",
"REF .",
"This strongly indicates that the “heavy” $c$ electrons dressed by $f_1$ local moments are mediating the $f_2$ -AF order in an indirect “high-order” manner.",
"More careful comparison reveals that this common peak occurs at the specific $t_{\\perp }$ where $C^{cf_1} \\approx C^{f_1f_2}$ and also changes most rapidly.",
"This observation implies that the homogeneous $C^{cf_1}$ and $C^{f_1f_2}$ spin correlations are favored for enhancing $C^{cf_2}$ and in turn strengthening the “high-order” RKKY interaction to mediate the $f_2$ -AF order.",
"In addition, the rapid evolution of $C^{cf_1}$ and $C^{f_1f_2}$ in this regime reflects the crucial delicate balance between $C^{cf_1}$ and $C^{f_1f_2}$ .",
"Furthermore, all these observations vividly implies the vital role of the heavy electrons ($c$ -$f_1$ singlets) in mediating the $f_2$ -AF order.",
"Note that $C^{cf_1}$ gradually vanishes at large $t_{\\perp }$ denoting the breakdown of heavy electrons, where the $f_2$ -AF order disappears concurrently.",
"The essential physics of our model can be also described in the viewpoint of the competition and balance between $t_{\\perp }$ and $V$ , which can be explored by investigating another indicator of the magnetic properties, namely the local moments $\\langle m^2 \\rangle $ of $f_{1,2}$ .",
"Fig.",
"REF (b) illustrates its behavior of $f_1$ (dashed lines) and $f_2$ (solid lines).",
"Naturally, the $f_2$ local moment decreases with $t_{\\perp }$ , which is most rapidly in the regime where the $f_2$ -AF order emerges.",
"Nevertheless, the $f_1$ local moment does not vary much but only possesses a bump in the regime with the maximal $f_2$ -AF order, which can be traced to its quantum fluctuation subject to two-fold hybridization with $c$ and $f_2$ .",
"This provides further evidence on the steadily frozen behavior of $f_1$ orbital, whose major role is to dress the $c$ electrons."
],
[
"Spectral properties",
"At this stage, we have mainly focussed on the SS and $f_2$ -AF$^I$ regimes at moderate $t_{\\perp }$ in the phase diagram Fig.",
"REF .",
"To further understand the $f_2$ -AF$^M$ and $c$ -$f_2$ -M regimes, we have to rely on the spectral properties.",
"The specific question we want to address is the fate of the $f_2$ -AF order at large $t_{\\perp }\\ge V$ .",
"To this aim, we examined the single-particle orbital-dependent local density of states (DOS) $N^{a}(\\omega )$ with $a=c,f_1,f_2$ relating to the local imaginary-time Green's function $G^{a}(\\tau )=- \\sum \\limits _{{\\bf j}} \\langle a^{\\phantom{\\dagger }}_{{\\bf j}}(\\tau ) a^{\\dagger }_{{\\bf j}\\pm }(0) \\rangle $ via $G^{a}(\\tau )= \\int _{-\\infty }^{\\infty }\\ d\\omega \\frac{e^{-\\omega \\tau }}{e^{-\\beta \\omega }+1}\\ N^{a}(\\omega ) $ To avoid the ambiguity from analytical continuation such as maximum entropy method[17], we resort to the approximate formula $N^a(\\omega =0) \\approx \\beta G^a(\\tau =\\beta /2)/\\pi $ assuming that the temperature is much lower than the energy scale on which there are structures in DOS[18].",
"As shown in Figure REF , the dominant feature associated with both the $f_2$ -AF$^M$ and $c$ -$f_2$ -M phases at large $t_{\\perp }\\ge V$ is the metallic behavior of both $c$ and $f_2$ orbitals while $f_1$ orbital is readily insulating.",
"The stronger metallicity of the conduction electron can be easily understood as the consequence of significantly weakened $c$ -$f_1$ spin correlation (Fig.",
"REF ) that liberates the $c$ electrons so that $\\beta G^{c}(\\beta /2)$ keeps growing with $t_{\\perp }$ .",
"Strikingly, the comparison with Fig.",
"REF demonstrates that the metallicity of $f_2$ starts within the phase with $f_2$ -AF order (cyan regime in Fig.",
"REF ).",
"Therefore, the $f_2$ -AF$^M$ phase displays the coexistence and competition of the metallicity and AF order of $f_2$ orbital.",
"In fact, the metallicity participates in destroying the $f_2$ -AF order.",
"The distinct difference between $f_1$ and $f_2$ reflects the more freedom of $f_2$ despite of the gradually stronger $f_1$ -$f_2$ binding, which is consistent with the smoother variation of $f_1$ local moment with $t_{\\perp }$ in Fig.",
"REF (b).",
"Apparently, we confirm that $\\beta G^{f_2}(\\beta /2)$ finally vanishes at sufficiently large $t_{\\perp }$ , e.g.",
"at $V/t=1.2$ (blue solid line), where the system becomes strongly bound $f_1$ -$f_2$ dimers plus nearly free conduction electrons."
],
[
"Conclusion",
"In conclusion, as a proof of principle study, we have addressed the fundamental question of whether or not the antiferromagnetic (AF) order can be induced via the local moments' hybridization with the heavy electrons instead of conduction electrons.",
"We provided strong numerical evidence to confirm its possibility via a prototypical model through determinant QMC simulations.",
"In particular, we claim that this AF order mediated by heavy electrons is realized by a so-called “high-order” RKKY interaction that resembles the conventional RKKY mediated via the conduction electrons in standard PAM/KLM.",
"We emphasize that the induced AF order only emerges if the heavy electrons are present, whose breakdown coincides with the disappearance of the ordering.",
"Moreover, we further prove that the induced AF order can coexist with its metallicity in a finite regime of the phase diagram, which competes with and ultimately destroys the AF order.",
"As our motivation partly came from the potential relevance to the heavy fermion compound Ce$_3$ (Pt/Pd)In$_{11}$ [11], [12], [13], [14], we remark that the three orbitals $c,f_1,f_2$ in our prototypical model can be used to mimic Pt/Pd, Ce(1), and Ce(2) separately of Ce$_3$ (Pt/Pd)In$_{11}$ [19].",
"Our findings implies that the experimentally observed magnetic ordering of Ce(2) ($f_2$ orbital) can indeed coexist microscopically with the fully Kondo screened Ce(1) ($f_1$ orbital) and in fact the Ce(1) plays a significant role in forming the AF order of Ce(2) sublattice.",
"To some extent, however, our model has intrinsic limitation due to its neglecting of conduction electron reservoir from In sites because it has been shown that the strong hybridization with the out of plane In plays an important role in other Ce-based compounds, such as Ce-115 materials [20].",
"Therefore, it is requisite to explore the more appropriate models for the potential connection of our findings reported here to the realistic materials, which is left for future investigation.",
"Another fascinating theoretical question regards on the reverse role of Ce(2) on the superconductivity claimed experimentally to be responsible by Ce(1) [14].",
"Besides, the thorough understanding and realization of the proposed “high-order” RKKY interaction in other contexts would be highly interesting.",
"We acknowledge Richard Scalettar and Jeroen Custers for fruitful discussion in the initial stage.",
"This work was funded by the Stewart Blusson Quantum Matter Institute at University of British Columbia, and by the Natural Sciences and Engineering Research Council of Canada."
]
] | 2005.14148 |
[
[
"Visible lattice points along curves"
],
[
"Abstract This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously.",
"By proposing the concept of level of visibility, we are able to analyze more carefully about both the \"visible\" points and the \"invisible\" points in the definition of previous research.",
"We prove asymptotic formulas for the number of lattice points in different levels of visibility."
],
[
"Background",
"A lattice point $(m,n)\\in \\mathbb {N}\\times \\mathbb {N}$ is said to be visible to the lattice point $(u,v)\\in \\mathbb {N}\\times \\mathbb {N}$ along lines if there are no other integer lattice points on the straight line segment joining $(m, n)$ and $(u, v)$ .",
"In 1883, it was showed by Sylvester [15] that the proportion of lattice points that are visible to the origin $(0,0)$ is $1/\\zeta (2)=6/\\pi ^2\\approx 0.60793$ , where $\\zeta (s)$ is the Riemann zeta function.",
"Since then, the study of the distribution of visible lattice points continues to intrigue mathematicians till now.",
"For example, one may refer to Adhikari-Granville [1], Baker [2], Boca-Cobeli-Zaharescu [3], Chaubey-Tamazyan-Zaharescu [4], Chen [5], Huxley-Nowak [10] for part of related works and some generalizations in recent years.",
"In 2018, Goins, Harris, Kubik and Mbirika [7] considered the integer lattice points in the plane which are visible to the origin $(0,0)$ along curves $y=r x^k$ with $k\\in \\mathbb {N}$ fixed and some $r\\in \\mathbb {Q}$ .",
"They showed that the proportion of such integer lattice points is $1/\\zeta (k+1)$ .",
"In the same year, Harris and Omar [8] futher considered the case of rational exponent $k$ .",
"Recently, Benedetti, Estupián and Harris [6] studied the proportion of visible lattice points to the origin along such curves in higher dimensional space.",
"All the above results are concerned about the lattice points visible to only one base point.",
"It is natural to consider the distribution of lattice points which are visible to more base points simultaneously.",
"For the case of visibility along straight lines, the earliest work originates from Rearick in 1960s.",
"In his Ph.D. thesis, Rearick [13] first showed that the density of integer lattice points in the plane which are jointly visible along straight lines to $N$ ($N=2$ or 3) base points is $\\prod _p\\left(1-N/p^2\\right)$ , where the base points are mutually visible in pairs and the product is over all the primes.",
"Then in [14], he generalized this result to lattice points in higher dimensional space and larger $N$ .",
"The joint visibility of lattice points along curves has not been considered yet.",
"In this paper, we focus on this topic and we also propose the concept of level of visibility.",
"Level-1 visibility matches the definition of \"$k$ -visible\" in [7].",
"We use higher level of visibility to analyze more carefully about the \"invisible\" points along certain curves.",
"We give asymptotic formulas for the number of lattice points which are visible to a set of $N$ base points along certain curves in different levels of visibility."
],
[
"Our results",
"For any positive integer $k$ and integer lattice points $(u,v),(m,n)\\in \\mathbb {N}\\times \\mathbb {N}$ , let $r\\in \\mathbb {Q}$ be given by $n-v=r(m-u)^k$ and $\\mathcal {C}$ be the curve $y-v=r(x-u)^k$ .",
"If there is no integer lattice points lying on the segment of $\\mathcal {C}$ between points $(m,n)$ and $(u,v)$ , we say $(m,n)$ is (Level-1) $k$ -visible to $(u,v)$ .",
"Further, if there is at most one integer lattice points lying on the segment of $\\mathcal {C}$ between points $(m,n)$ and $(u,v)$ , we say point $(m,n)$ is Level-2 $k$ -visible to $(u,v)$ .",
"One can see that $k$ -visibility is mutual.",
"Precisely, if a point $(m,n)$ is Level-1 or Level-2 $k$ -visible to the point $(u,v)$ along the curve $y-v=r(x-u)^k$ , then $(u,v)$ is also Level-1 or Level-2 $k$ -visible to $(m,n)$ , respectively, along the curve $y-n=(-1)^{k+1}r(x-m)^k$ .",
"Throughout this paper, we always assume $S$ is a given set of integer lattice points in the plane.",
"We say an integer lattice point $(m,n)$ is Level-1 $k$ -visible to $S$ if it belongs to the set $V_k^1(S):=\\lbrace (m,n)\\in \\mathbb {N}\\times \\mathbb {N}: (m,n) ~\\text{is}~k\\text{-visible to every point in}~ S \\rbrace .$ Similarly, we say a point $(m,n)\\in \\mathbb {N}\\times \\mathbb {N}$ is Level-2 $k$ -visible to $S$ if it belongs to the set $V_k^2(S):=\\lbrace (m,n)\\in \\mathbb {N}\\times \\mathbb {N}: (m,n) ~\\text{is Level-2}~k\\text{-visible to every point in}~ S \\rbrace .$ One may define higher Level $k$ -visible points to $S$ this way.",
"But in this paper, we focus on Level-1 and Level-2 $k$ -visible points.",
"For $x\\ge 2$ , we consider visible lattice points along curves in the square $[1,x]\\times [1,x]$ .",
"Denote $N^1_k(S,x):=&\\#\\lbrace (m,n)\\in V_k^1(S): m,n\\le x\\rbrace ,$ and $N^2_k(S,x):=&\\#\\lbrace (m,n)\\in V_k^2(S): m,n\\le x\\rbrace .$ An important case is that the points of $S$ are pairwise $k$ -visible to each other.",
"The cardinality of such $S$ can't be too large.",
"In fact, we have $\\#S\\le 2^{k+1}$ by Proposition REF in the next section.",
"For such type of $S$ , we obtain the following asymptotic formulas for $N^1_k(S,x)$ and $N_k^2(S,x)$ .",
"Theorem 1.1 Assume the elements of $S$ are pairwise $k$ -visible to each other and $N=\\#S< 2^{k+1}$ .",
"For any $k\\ge 2$ , we have $N^1_k(S, x)=x^2\\prod _p \\Bigg (1-\\frac{N}{p^{k+1}} \\Bigg )+E_1(x),$ where $p$ runs over all primes, and $ E_1(x)={\\left\\lbrace \\begin{array}{ll}O_k(x\\log ^N x),& \\text{if}~1\\le N\\le k;\\\\O_{k,\\varepsilon }(x^{2-\\frac{2k}{N+k}+\\varepsilon }), &\\text{if}~k<N<2^{k+1}.\\end{array}\\right.}",
"$ Remark 1.",
"If $N=\\#S=2^{k+1}$ , by Proposition REF , there is no lattice point outside $S$ which is (Level-1) $k$ -visible to all elements of $S$ .",
"By the above Theorem, the density of (Level-1) $k$ -visible points to every elements of $S$ is $\\prod _p \\left(1-N/p^{k+1}\\right)$ .",
"For $k=1$ , it is done by the work in [14], where the author studied the visible points along straight lines.",
"The special case $N=1$ for $k\\ge 2$ in Theorem REF covers the result in [7], where only the main term was given.",
"We also give asymptotic formulas for Level-2 $k$ -visible points.",
"Note that such set actually includes some \"invisible\" points in the definition of previous research.",
"We are able to analyze more carefully about these \"invisible\" points.",
"Theorem 1.2 Assume the elements of $S$ are pairwise $k$ -visible to each other and $N=\\#S\\le 2^{k+1}$ .",
"For any $k\\ge 1$ , we have $N_k^2(S, x)= N_k^1(S, x)+x^2 \\frac{N}{2^{k+1}}\\left(1-\\frac{1}{2^{k+1}}\\right)\\prod _{p>2~{\\rm prime}} \\bigg ( 1-\\frac{N}{p^{k+1}}\\bigg )+E_2(x),$ where $ E_2(x)={\\left\\lbrace \\begin{array}{ll}O_k(x\\log ^N x),& \\text{if}~1\\le N\\le k;\\\\O_{k,\\varepsilon }(x^{2-\\frac{2k}{N+k}+\\varepsilon }), &\\text{if}~k<N\\le 2^{k+1}.\\end{array}\\right.}",
"$ Remark 2.",
"Note that when $N=2^{k+1}$ , $N_k^1(S, x)=0$ , there is no Level-1 $k$ -visible points to $S$ .",
"But there are still positive proportion of lattice points in the plane which are Level-2 $k$ -visible to $S$ .",
"For the special case $N=1$ and $k=1$ , our problem is the same as the so-called \"primitive lattice problem\" inside a square.",
"Nowak [12], Zhai [17] and Wu [16] have studied the number of primitive lattice points inside a circle.",
"Primitive lattice points in general planar domains have also been studied by Hensley [9], Huxley and Nowak [10] and Baker [2] etc.",
"Assuming the Riemann hypothesis(RH), they continuously improved the error term of the concerned asymptotic formulas by estimating certain exponential sums.",
"One may wonder how much we can do to improve the estimates of $E_1(x)$ and $E_2(x)$ by similar argument under RH.",
"However, we do not focus on pursuing the best possible error term in this paper.",
"Taking $S=\\lbrace (0,0), (1,1)\\rbrace $ , we did numerical calculations for densities of Level-1 and Level-2 $k$ -visible points for $x=10000$ and $k=2, 3, \\ldots , 9$ (See Table REF and Figure REF below).",
"We see that the numerical results match the theoretical predictions very well.",
"Table: Densities of kk-visible points to set of two elementsFigure: Densities of kk-visible points to set of two elementsWe also calculate the case when $S=\\lbrace (0,0), (1,2), (2,1) \\rbrace $ , and we get the following data for densities of Level-1 and Level-2 $k$ -visible points to $S$ (See Table REF and Figure REF ).",
"Table: Densities of kk-visible points to set of three elementsFigure: Densities of kk-visible points to set of three elementsNotations.",
"We use $\\mathbb {Z}$ to denote the set of integers; $\\mathbb {N}$ to denote the set of positive integers; $\\mathbb {Q}$ to denote the set of rational numbers; $\\# S$ to denote the cardinality of a set $S$ .",
"As usual, we use the expressions $f=O(g)$ or $f\\ll g$ to mean $|f|\\le Cg$ for some constant $C>0$ .",
"In the case when this constant $C>0$ may depend on some parameters $\\rho $ , we write $f=O_\\rho (g)$ or $f\\ll _\\rho g$ ."
],
[
"Preliminaries",
"We define the degree-$k$ greatest common divisor of $m$ , $n\\in \\mathbb {Z}$ as ${{\\rm gcd}_k }(m,n):=\\max \\lbrace d\\in \\mathbb {N}: d\\mid m,\\ d^k\\mid n \\rbrace .$ Proposition 2.1 For any integer $k\\ge 1$ , assume any two distinct elements $(u_i, v_i)$ , $(u_j, v_j)\\in S$ are $k$ -visible to each other, then we have $\\#S\\le 2^{k+1}$ .",
"To see this, we consider the map $\\lambda : S\\rightarrow \\widetilde{S}:=\\lbrace (u\\bmod 2, v\\bmod 2^{k}) : (u, v)\\in S \\rbrace .$ The size of the image $\\widetilde{S}$ is at most $2^{k+1}$ .",
"If $S$ has more than $2^{k+1}$ points, there must be two distinct elements which map to the same element in $\\widetilde{S}$ , say $\\lambda ((u_1, v_1) )=\\lambda ((u_2, v_2)).$ Thus we have $2\\mid (u_2-u_1) ~\\text{and}~ 2^k\\mid (v_2-v_1),$ and hence ${{\\rm gcd}_k }(u_2-u_1, v_2-v_1)\\ge 2$ , which contradicts our assumption on $S$ .",
"By the definition of $k$ -visible points and elementary argument, we get the following lemma.",
"One may refer to [7] (Proposition 3) for similar argument.",
"Here we omit the proof.",
"Lemma 2.2 For any $k\\ge 1$ , if $m-u\\ne 0$ and $n-v\\ne 0$ , we have Point $(m,n)$ is $k$ -visible to point $(u,v)$ if and only if ${{\\rm gcd}_k }(m-u,n-v)=1$ .",
"There exists exactly one integer point lying on the segment of the curve $y-v=r(x-u)^k$ joining $(u,v)$ and $(m,n)$ for some $r\\in \\mathbb {Q}$ if and only if ${{\\rm gcd}_k }(m-u,n-v)=2$ .",
"We also need the following well-known result for $l$ -fold divisor function $\\tau _l(n)=\\sum _{d_1\\cdots d_l=n} 1$ .",
"Lemma 2.3 ([11], formula (1.80)) Let $l\\ge 2$ be an integer.",
"For any $x\\ge 2$ , we have $\\sum \\limits _{n\\le x}\\tau _l(n)\\ll _{l} x\\log ^{l-1} x.$"
],
[
"Proof of Theorem ",
"Given a set $S$ , if we shift $S$ such that it contains the origin, the error occurs to our counting function is $O_{S}(x)$ .",
"Thus, we may assume $(0,0)\\in S$ .",
"Denote the elements of $S$ as $(u_j, v_j),\\ 0\\le j\\le N-1$ with $(u_0,v_0)=(0,0)$ .",
"By Proposition REF , the contribution of points $(m, n)$ with $m=u_j$ or $n=v_{j^{\\prime }}$ for some $j, j^{\\prime }$ is $O(|S|x)=O_k(x)$ .",
"Hence, we only need to estimate the contribution of points $(m, n)$ with $m\\ne u_j$ and $n\\ne v_{j^{\\prime }}$ for all $0\\le j, j^{\\prime }\\le N-1$ .",
"Throughout all our proofs, we implicitly assume the input of ${{\\rm gcd}_k }(*, *)$ has no zero coordinates unless otherwise specified.",
"By Lemma REF we have $N^1_k(S, x)&=\\sum _{\\begin{array}{c}m, n\\le x\\\\{{\\rm gcd}_k }(m-u_j, n-v_j)=1\\\\ m\\ne u_j, n\\ne v_j\\\\0\\le j\\le N-1\\end{array}} 1+O_k(x)=: \\widetilde{N}^1_k(S,x)+O_k(x).$ Applying the formula $\\sum _{d|n}\\mu (d)={\\left\\lbrace \\begin{array}{ll}1,\\ {\\rm if}\\ n=1;\\\\0,\\ {\\rm otherwise},\\end{array}\\right.",
"}$ where $\\mu $ is the Möbius function, we write $\\widetilde{N}^1_k(S,x)=\\sum _{m, n\\le x}\\sum \\limits _{\\begin{array}{c}d_j|\\gcd _k(m-u_j,n-v_j)\\\\0\\le j\\le N-1\\end{array}}\\mu (d_0)\\cdots \\mu (d_{N-1}).$ Let $D>0$ be a parameter to be chosen later.",
"Divide the sum over $d_0,\\cdots ,d_{N-1}$ into two parts: $d_0\\cdots d_{N-1}\\le D$ and $d_0\\cdots d_{N-1}> D$ , and denote their contributions to $\\widetilde{N}^1_k(S,x)$ by $\\sum _{\\le }$ and $\\sum _{>}$ respectively.",
"Then we have $\\widetilde{N}^1_k(S,x)={\\sum }_{\\le }+{\\sum }_{>}.$ For $\\sum _{\\le }$ , we change the order of the summation and obtain ${\\sum }_{\\le }=\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\end{array}}\\mu (d_0)\\cdots \\mu (d_{N-1})\\Bigg (\\sum _{\\begin{array}{c}m,n\\le x\\\\ d_j\\mid m-u_j,d_j^k\\mid n-v_j\\\\ 0\\le j\\le N-1\\end{array}}1\\Bigg ).$ Note that $(u_0,v_0)=(0,0)$ , then for any given $d_0$ , $\\cdots $ , $d_{N-1}$ , the inner sum over $m,n$ in the above formula actually equals $\\Bigg (\\sum _{\\begin{array}{c}s\\le x/d_0\\\\sd_0\\equiv u_j(\\bmod d_j)\\\\1\\le j\\le N-1\\end{array}}1\\Bigg )\\Bigg (\\sum _{\\begin{array}{c}t\\le x/d_0^k\\\\td_0^k\\equiv v_j(\\bmod d_{j}^k)\\\\1\\le j\\le N-1\\end{array}}1\\Bigg ).$ Since the points in $S$ are mutually $k$ -visible, then by Lemma REF we have ${{\\rm gcd}_k }(u_l-u_j, v_l-v_j)=1,\\ \\text{for}\\ (u_l, v_l), (u_j, v_j)\\in S,\\ 0\\le j\\ne l\\le N-1.$ This implies $\\gcd (d_j, d_l)=1 \\ \\text{for}\\ 0\\le j\\ne l\\le N-1.$ It then follows that ${\\sum }_{\\le }=&\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\\\ \\gcd (d_j, d_l)=1, ~\\forall \\ 0\\le j\\ne l\\le N-1 \\end{array}}\\mu (d_0)\\cdots \\mu (d_{N-1})\\\\& \\left(\\frac{x}{d_0\\cdots d_{N-1}}+O(1)\\right)\\left(\\frac{x}{d_0^k\\cdots d_{N-1}^k}+O(1)\\right).$ Then by Lemma REF we obtain ${\\sum }_{\\le }=&x^2\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\\\ \\gcd (d_j, d_l)=1, ~\\forall \\ 0\\le j\\ne l\\le N-1 \\end{array}}\\frac{\\mu (d_0)\\cdots \\mu (d_{N-1})}{d_0^{k+1}\\cdots d_{N-1}^{k+1}}+O\\Bigg (x\\sum _{d_0\\cdots d_{N-1}\\le D} \\frac{1}{d_0\\cdots d_{N-1}} \\Bigg )\\nonumber \\\\&+O\\Bigg (x\\sum _{d_0\\cdots d_{N-1}\\le D} \\frac{1}{d_0^k\\cdots d^k_{N-1}} \\Bigg )+O\\Bigg ( \\sum _{d_0\\cdots d_{N-1}\\le D} 1 \\Bigg ) \\nonumber \\\\=& x^2\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\\\ \\gcd (d_j, d_l)=1, ~\\forall \\ 0\\le j\\ne l\\le N-1 \\end{array}}\\frac{\\mu (d_0)\\cdots \\mu (d_{N-1})}{d_0^{k+1}\\cdots d_{N-1}^{k+1}}+O_k\\Big (D\\log ^{N-1}D+x\\log ^{N}D\\Big ).$ Writing $n=d_0\\cdots d_{N-1}$ , we then have ${\\sum }_{\\le }=x^2\\sum _{n\\le D}\\frac{\\mu (n)\\tau _{N}(n)}{n^{k+1}}+O_k(D\\log ^{N-1}D+x\\log ^N D).$ Using Lemma REF , we obtain ${\\sum }_{\\le }=x^2\\sum _{n=1}^{\\infty }\\frac{\\mu (n)\\tau _{N}(n)}{n^{k+1}}+O_k \\big ({x^2D^{-k}\\log ^{N-1} D}+D\\log ^{N-1}D+x\\log ^N D\\big ).$ i) If $1\\le N\\le k$ , then we choose $D=x$ .",
"In this case, since each $d_j\\le x^{1/k}$ , $d_0\\cdots d_{N-1}\\le x^{N/k}\\le x$ .",
"Thus the second sum $\\sum _{>}$ in (REF ) is empty since we already exclude zero inputs of ${{\\rm gcd}_k }(*,*)$ in the beginning of the proof.",
"Inserting (REF ) into (REF ) yields (REF ) in Theorem REF with $E_1(x)\\ll _k x\\log ^{N}x ~\\text{for}~1\\le N\\le k.$ ii) If $k<N<2^{k+1}$ , we need to make another choice for $D$ , and deal with ${\\sum }_{>}$ more carefully.",
"Taking absolute value of $\\mu (d)$ , we obtain ${\\sum }_{>}=\\sum _{m, n\\le x}\\sum \\limits _{\\begin{array}{c}d_0\\cdots d_{N-1}>D\\\\d_j|\\gcd _k(m-u_j,n-v_j)\\\\0\\le j\\le N-1\\end{array}}\\mu (d_0)\\cdots \\mu (d_{N-1})\\ll \\sum _{m, n\\le x}\\sum \\limits _{\\begin{array}{c}d_0\\cdots d_{N-1}>D\\\\d_j|\\gcd _k(m-u_j,n-v_j)\\\\0\\le j\\le N-1\\end{array}}1,$ which implies ${\\sum }_{>}\\ll \\sum _{\\begin{array}{c}m, n\\le x\\\\ \\prod \\limits _{0\\le j\\le N-1}\\gcd _{k}(m-u_j,n-v_j)>D\\end{array}}\\tau \\big ({{\\rm gcd}_k }(m-u_0,n-v_0)\\big )\\cdots \\tau \\big ({{\\rm gcd}_k }(m-u_{N-1},n-v_{N-1})\\big )$ Using the bounds $\\tau (n)\\ll _{\\varepsilon }n^{\\varepsilon }$ for any $\\varepsilon >0$ , $N<2^{k+1}$ and ${{\\rm gcd}_k }(m-u_j, n-v_j)\\le x^{1/k}$ , we have ${\\sum }_{>}\\ll _{k, \\varepsilon } x^{\\varepsilon }\\sum _{\\begin{array}{c}m, n\\le x\\\\ \\prod \\limits _{0\\le j\\le N-1}{{\\rm gcd}_k }(m-u_j,n-v_j)>D\\end{array}}1.$ Since $\\prod \\limits _{0\\le j\\le N-1}\\gcd _{k}(m-u_j,n-v_j)>D$ implies ${{\\rm gcd}_k }(m-u_{j^*},n-v_{j^*})>D^{1/N}$ for some $j^*\\in \\lbrace 0,\\cdots ,N-1\\rbrace $ , we obtain ${\\sum }_{>}\\ll _{k, \\varepsilon } x^{\\varepsilon }\\sum \\limits _{0\\le j\\le N-1}\\sum _{\\begin{array}{c}m, n\\le x\\\\ {{\\rm gcd}_k }(m-u_j,n-v_j)>D^{1/N}\\end{array}}1.$ By the definition of ${{\\rm gcd}_k }$ , we have ${\\sum }_{>}\\ll _{k, \\varepsilon } x^{\\varepsilon }\\sum \\limits _{0\\le j\\le N-1}\\sum \\limits _{D^{1/N}<d\\le x^{1/k}}\\sum _{\\begin{array}{c}m, n\\le x\\\\d\\mid m-u_j\\\\d^k\\mid n-v_j\\end{array}}1.$ It follows that ${\\sum }_{>}\\ll _{k, \\varepsilon } x^{2+\\varepsilon }\\sum \\limits _{0\\le j\\le N-1}\\sum \\limits _{D^{1/N}<d\\le x^{1/k}}\\frac{1}{d^{1+k}}\\ll _{k, \\epsilon } x^{2+\\varepsilon }D^{-k/N}.$ Collecting all the above gives $N^1_k(S, x)=x^2\\prod _p \\Bigg (1-\\frac{N}{p^{k+1}} \\Bigg )+O_{k,\\epsilon }\\big (D\\log ^{N-1}D+x^{2+\\varepsilon }D^{-k/N}+x\\log ^N x\\big ).$ Taking $D=x^{\\frac{2N}{N+k}}$ yields (REF ) with $E_{1}(x)\\ll _{k, \\epsilon } x^{2-\\frac{2k}{N+k}+\\varepsilon } ~\\text{for}~k<N<2^{k+1}.$"
],
[
"Proof of Theorem ",
"In this section, we also assume ${{\\rm gcd}_k }(*,*)$ does not take any input with zero coordinates.",
"If elements of $S$ are pairwise $k$ -visible to each other, then for any $(m,n)\\in V_k^2(S)$ , there exists at most one $(u,v)\\in S$ such that ${{\\rm gcd}_k }(m-u, n-v)=2$ .",
"Indeed, suppose ${{\\rm gcd}_k }(m-u_1, n-v_1)={{\\rm gcd}_k }(m-u_2, n-v_2)=2$ for some $(u_1, v_1)\\ne (u_2, v_2)\\in S$ , then we have $2|(m-u_1), ~2|(m-u_2), ~2^k|(n-v_1), ~2^k|(n-v_2).$ Thus, $2|(u_2-u_1)$ and $2^k|(v_2-v_1)$ , which contradicts the assumption ${{\\rm gcd}_k }(u_2-u_1, v_2-v_1)=1$ .",
"By the above argument, we write $N_k^2(S, x)=N_k^1(S, x)+\\sum _{0\\le l\\le N-1}\\sum _{\\begin{array}{c} m, n\\le x\\\\ {{\\rm gcd}_k }(m-u_l, n-v_l)=2\\\\ {{\\rm gcd}_k }(m-u_j, n-v_j)=1\\\\ j\\ne l\\end{array}}1+O_k(x).$ Without loss of generality, we may assume $(u_0, v_0)=(0,0)$ .",
"We only need to estimate the inner sum of the second term in (REF ) for $l=0$ , other cases are similar.",
"Denote $I(x):=\\sum _{\\begin{array}{c}m, n\\le x\\\\ {{\\rm gcd}_k }(m, n)=2\\\\ {{\\rm gcd}_k }(m-u_j, n-v_j)=1\\\\ 1\\le j\\le N-1 \\end{array} } 1.$ We have $I(x)=\\sum _{\\begin{array}{c}m, n\\le x\\\\2\\mid m,2^k\\mid n\\\\ {{\\rm gcd}_k }(m/2, n/2^{k})=1\\\\ {{\\rm gcd}_k }(m-u_j, n-v_j)=1\\\\ 1\\le j\\le N-1 \\end{array} } 1=\\sum _{\\begin{array}{c}m,n\\le x\\\\2\\mid m,2^k\\mid n\\end{array}}\\sum _{\\begin{array}{c}d_0\\mid {{\\rm gcd}_k }(m/2, n/2^k)\\\\ d_j\\mid {{\\rm gcd}_k }(m-u_j, n-v_j)\\\\ 1\\le j\\le N-1\\end{array}} \\mu (d_0)\\cdots \\mu (d_{N-1}).$ By changing the order of summation and making the substitutions $m=2d_0s$ and $n=(2d_0)^kt$ , we obtain $I(x)=\\sum _{d_0, \\cdots , d_{N-1}\\le x^{1/k}}\\mu (d_0)\\cdots \\mu (d_{N-1})\\sum _{\\begin{array}{c}s\\le x/(2d_0), t\\le x/(2d_0)^k\\\\2d_0 s\\equiv u_j(\\bmod d_j) \\\\ (2d_0)^k t\\equiv v_j(\\bmod d_j^k) \\\\1\\le j\\le N-1 \\end{array} }1.$ In order to get estimates of $I(x)$ , we need to analyze the conditions in the inner sum.",
"Fix $d_0, \\cdots , d_{N-1}$ , in order for those congruence equations having solutions, we need $\\gcd (2d_0, d_j)\\mid u_j, ~\\gcd ((2d_0)^k, d_j^k)\\mid v_j$ for $1\\le j\\le N-1$ .",
"Since points $(u_j,v_j)$ are $k$ -visible to point $(u_0,v_0)$ , then Lemma REF gives ${{\\rm gcd}_k }(u_j, v_j)=1$ .",
"It follows that $\\gcd (2d_0, d_j)=1$ for $1\\le j\\le N-1$ .",
"Moreover, in order for those congruence equations having solutions, we also need the following equations $d_{j_1}l_{1}-d_{j_2}l_{2}= u_{j_2}-u_{j_1}, ~d^k_{j_1}t_{1}-d^k_{j_2}t_{2}= v_{j_2}-v_{j_1}$ have solutions for any $d_{j_1}$ and $d_{j_2}$ with $1\\le j_1\\ne j_2\\le N-1$ .",
"This implies $\\gcd (d_{j_1},d_{j_2})\\mid u_{j_2}-u_{j_1},\\ \\gcd (d_{j_1}^k,d_{j_2}^k)\\mid v_{j_2}-v_{j_1}$ for $1\\le j_1\\ne j_2\\le N-1$ .",
"By the assumption of pairwise $k$ -visibility of elements of $S$ , we have ${{\\rm gcd}_k }(u_{j_2}-u_{j_1}, v_{j_2}-v_{j_1})=1$ , and thus $\\gcd (d_{j_1}, d_{j_2})=1$ for any $1\\le j_1\\ne j_2\\le N-1$ .",
"As what we did in Section , we divide the sum over $d_0,\\cdots ,d_{N-1}$ into two parts according to $d_0\\cdots d_{N-1}\\le D$ or not.",
"Denote them by $I_{\\le }$ and $I_{>}$ respectively, then $I(x)=I_{\\le }+I_{>}.$ For $I_{\\le }$ , we have $I_{\\le }=\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\\\ \\gcd (d_{j_1}, d_{j_2})=1, \\forall j_1\\ne j_2\\\\\\gcd (2, d_j)=1, 1\\le j\\le N-1\\end{array}} \\mu (d_0)\\cdots \\mu (d_{N-1})\\left(\\frac{x}{2d_0\\cdots d_{N-1}}+O(1)\\right)\\left(\\frac{x}{2^kd_0^k\\cdots d_{N-1}^k}+O(1)\\right),$ and by Lemma REF , we get $I_{\\le }=\\frac{x^2}{2^{k+1}} \\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}\\le D\\\\ \\gcd (d_{j_1}, d_{j_2})=1,\\forall j_1\\ne j_2\\\\\\gcd (2, d_j)=1, 1\\le j\\le N-1\\end{array}}\\frac{\\mu (d_0)\\cdots \\mu (d_{N-1})}{d_0^{k+1}\\cdots d_{N-1}^{k+1}}+O_k\\left(x\\log ^N x+D\\log ^{N-1} D\\right).$ Making the substitution $n=d_0\\cdots d_{N-1}$ , we obtain $I_{\\le }=\\frac{x^2}{2^{k+1}} \\sum _{n\\le D} \\frac{\\mu (n)}{n^{k+1}}h(n)+O_k\\left(x\\log ^N x+D\\log ^{N-1} D\\right).$ where $h(n)=\\sum \\limits _{\\begin{array}{c}n=d_0\\cdots d_{N-1}\\\\ d_1,\\cdots , d_{N-1} ~\\text{odd} \\end{array}} 1.$ Extending the sum over $n$ and using the bound $h(n)\\le \\tau _{N}(n)$ , and by Lemma REF , we derive $I_{\\le }=\\frac{x^2}{2^{k+1}} \\sum _{n=1}^{\\infty } \\frac{\\mu (n)}{n^{k+1}}h(n)+O_k\\left(x^{2}D^{-k}\\log ^{N-1} D+x\\log ^N x+D\\log ^{N-1} D\\right).$ Note that $h(n)$ is multiplicative with $h(2)=1$ and $h(p)=N$ for $p> 2$ prime.",
"Thus $\\sum _{n=1}^{\\infty } \\frac{\\mu (n)}{n^{k+1}}h(n)=\\bigg (1-\\frac{1}{2^{k+1}} \\bigg ) \\prod _{p>2} \\bigg (1-\\frac{N}{p^{k+1}} \\bigg ).$ i) If $1\\le N\\le k$ , then we choose $D=x$ .",
"In this case, since each $d_j\\le x^{1/k}$ , $d_0\\cdots d_{N-1}\\le x^{N/k}\\le x$ .",
"Thus the second term $I_{>}$ in (REF ) is empty.",
"Inserting (REF ) into (REF ) yields (REF ) in Theorem REF with $E_1(x)\\ll _k x\\log ^{N}x ~\\text{for}~1\\le N\\le k.$ ii) If $k<N\\le 2^{k+1}$ , we need to make another choice for $D$ and deal with $I_{>}$ .",
"By similar argument as before, we obtain $I_{>}\\ll \\sum _{\\begin{array}{c}m,n\\le x\\\\2\\mid m,2^k\\mid n\\end{array}}\\sum _{\\begin{array}{c}d_0\\cdots d_{N-1}>D\\\\d_0\\mid {{\\rm gcd}_k }(m/2, n/2^k)\\\\ d_j\\mid {{\\rm gcd}_k }(m-u_j, n-v_j)\\\\ 1\\le j\\le N-1\\end{array}} 1,$ which gives $I_{>}\\ll \\sum \\limits _{\\begin{array}{c}m,n\\le x\\\\2\\mid m,2^k\\mid n\\\\ \\prod \\limits _{0\\le j\\le N-1}{{\\rm gcd}_k }(m-u_j,n-v_j)>D\\end{array}}\\tau ({{\\rm gcd}_k }(m/2,n/2^k))\\prod \\limits _{1\\le j\\le N-1}\\tau ({{\\rm gcd}_k }(m-u_j,n-v_j)).$ Using the bound $\\tau (n)\\ll _{\\varepsilon } n^{\\varepsilon }$ for any $\\varepsilon >0$ , by a similar argument as in the proof of Theorem REF , we obtain $I_{>}\\ll _{\\varepsilon } x^{\\varepsilon }\\sum \\limits _{\\begin{array}{c}m,n\\le x\\\\ \\prod \\limits _{0\\le j\\le N-1}{{\\rm gcd}_k }(m-u_j,n-v_j)>D\\end{array}}1\\ll _{\\varepsilon } x^{2+\\varepsilon }D^{-k/N}.$ Hence, combining all the estimates and taking $D=x^{\\frac{2N}{N+k}}$ yields $I(x)=\\frac{x^2}{2^{k+1}} \\bigg (1-\\frac{1}{2^{k+1}} \\bigg ) \\prod _{p>2} \\bigg (1-\\frac{N}{p^{k+1}} \\bigg ) +O_{k,\\epsilon }(x^{2-\\frac{2k}{N+k}+\\varepsilon }+x\\log ^N x).$ Plugging this into (REF ), we obtain (REF ) in Theorem REF with $E_{2}(x)\\ll _{k,\\varepsilon } x^{2-\\frac{2k}{N+k}+\\varepsilon } ~\\text{for}~k<N\\le 2^{k+1}.$ Acknowledgements.",
"The first author is partially supported by Shandong Provincial Natural Science Foundation (Grant No.",
"ZR2019BA028).",
"The second author is partially supported by the Humboldt Professorship of Professor Harald Helfgott.",
"Both authors thank the anonymous referee for valuable suggestions."
]
] | 2005.13994 |
[
[
"DQM Tools and Techniques of the SND Detector"
],
[
"Abstract SND detector operates at the VEPP-2000 collider (BINP, Novosibirsk).",
"To improve events selection for physical analysis and facilitate online detector control we developed new data quality monitoring (DQM) system.",
"The system includes online and reprocess control modules, automatic decision making scripts, interactive (web based) and program (python) access to various quality estimates.",
"This access is implemented with node.js server with data in RDBMS MySQL.",
"We describe here general system logics, its components and some implementation details."
],
[
"Introduction",
"The SND detector [1], [2], [3] operates at the VEPP-2000 collider [4] since 2008.",
"It produces hundreds gigabytes of stored raw data per day [5].",
"The data are complemented with dozens megabytes of metadata, facility conditions (beam energy, luminosity, crate temperatures, etc.)",
"and additional statistics (histograms etc.)",
"that could be used in reconstruction, processing and system control.",
"An important part of experiment software is a data quality monitoring (DQM) system.",
"This is necessary to obtain the meaningful data for physical analysis and to control the detector state.",
"Recently DQM software was seriously redesigned.",
"We present here the new system and its first usage experience.",
"The data quality metadata are generated at every stage of data collecting and reprocessing.",
"The new DQM system includes software tools that show data acquisition summary and histograms; collect quality data from automated scripts and users input; have hierarchical quality model; support several parameter sets for different stages of data processing; provide quality information getter UI's and API's."
],
[
"Estimating Run Quality",
"The minimal collection of events for analysis is referred to here and below as “run”.",
"This is also the minimal unit for the data quality estimation.",
"Parameters to monitor are defined by the detector subsystem experts with optional scripts which assign data quality marks to runs according to configuration.",
"These marks could be “bad”, “user has to decide”, “in doubt”, “good”, “no data”.",
"They could be also assigned manually by dedicated persons which are usually operators, run coordinator and detector subsystems experts."
],
[
"The Data Acquisition Stage",
"At this stage an operator monitors the experiment data quality right after the data acquisition.",
"A large set of histograms (e.g.",
"drift chamber layers statistics, calorimeter energies distribution) becomes available minutes/hours later after the data are processed by a high level trigger and recorded.",
"Our DQM system then launches predefined scripts which assign quality marks where possible, and then displays the histograms and automatic quality marks to an operator.",
"The operator shall check them, assign the quality marks which were not set automatically, and, if necessary, correct automatic marks.",
"Not all important quality parameters yet covered with automatic decision scripts.",
"The DQM system requires that an operator fills the gaps.",
"In order to help one to do it without special knowledge, the interface displays reference histograms.",
"Subsystem experts also may leave comments about their decision like for example “the histogram has to have two peaks”.",
"Having checked all the parameters an operator could either proceed with other activities or report a problem to a run coordinator the same day it appeared.",
"During the data collection dedicated person (the run coordinator) makes sure that the data collection goes smoothly.",
"This person keeps an eye on the operators checking quality data.",
"The DQM system provides a day summary and a month view for this purpose in addition to the individual run view.",
"Interactive DQM interface is implemented as a web application.",
"So the run coordinator and the experts can remotely discuss the quality and make sure the detector works fine.",
"A list of good runs could be exported also for prompt calibration programs to use.",
"Checks performed at this stage: Check the run validity: enough time, enough events, good collider currents etc.",
"Check the detector subsystems: calorimeter, tracking system, aerogel counters, muon system, trigger electronics etc."
],
[
"The Reprocessing Stage",
"The second stage of the data quality control is done when data has been reprocessed for analysis.",
"At that time we have more information, including that available only after completion the experiment.",
"Reprocessing software applies proper and final calibration (conditions) data and produces new meta-statistics (histograms, counters, averages) to check.",
"These meta-data are analysed later with related scripts based on configuration.",
"Data preparation requires creative approach and immersion for several days or even weeks.",
"The person who deals with this task have more general view than individual runs.",
"It could be necessary to make quality decisions based on run ranges or run sets at once.",
"Now the DQM system provides a single point of storing, discussing and retrieving quality information.",
"The data experts can easily access the first stage quality data or investigate some data mysteries with the detector subsystem experts.",
"Having done that, we can export a list of good runs for processing.",
"Checks of the stage could include check the run validity enough run time, enough events, good event number ratio for $e^{+}e^{-} \\rightarrow e^{+}e^{-}$ , $e^{+}e^{-} \\rightarrow \\gamma \\gamma $ ; check the subsystems, examine specific runs in detail."
],
[
"Interaction With Users",
"An operator, a physicist or subsystem expert can interact with the DQM system using three interfaces: web interface, program getter access and DQM scripts respectively.",
"In principle direct access to the DQM database is also possible, but this way is generally discouraged for any use other than system administration and development."
],
[
"Web Interface",
"This is the main way of manual interaction.",
"It allows users to compare actual numbers and histograms with reference ones, to assign their quality marks and leave comments, to select runs by several quality criteria and to view or edit their quality data.",
"The interface provides different views for operators and experts.",
"The expert views are optimized for investigating quality data (figure REF , on the left).",
"They contain forms for filtering runs by quality, run list view and particular run summary with parameters accessible by a mouse click.",
"The operator views are optimized to check limited set of parameters for each run (figure REF , on the right).",
"It displays actual and reference representation (histograms, averages etc.)",
"of pre-defined parameters set.",
"An operator can monitor the run log and walk back and forth at the run quality view."
],
[
"Program Getter Access",
"Data quality information is also available using program getter access from Python scripts.",
"This language is chosen because embedded Python interpreter is used for configuration of experiment data processing framework [6].",
"The interaction (figure REF ) is relatively simple.",
"The retrieved data (overall or per-system quality marks) could be used to filter qualified runs in automated calibration software or in analysis.",
"Figure: Examples of using DQM python getter.The program getter can either interact with web interface (figure REF , single run example), or load cached integral quality information (figure REF , multiple runs example) from the database.",
"The single mode could be used for triggering quality estimation when login credentials are provided.",
"The multiple mode is faster.",
"However it may result in missing some information if some cache entries are expired or don't exist."
],
[
"DQM Scripts",
"Automated data quality estimation is performed by executing special scripts.",
"A script (like at figure REF ) is a ROOT [7] macro that accepts several parameters like run number, histogram, file path, etc.",
"The script shall analyze the data and assign quality marks to related subsystem for one run.",
"It also can set parameters titles/comments, choose custom histograms/numbers to show or even hide them.",
"These actions are performed using a simple C++ API.",
"The system provides the infrastructure.",
"Usually detector subsystem experts create their scripts based on their understanding.",
"These scripts are executed by a server when an authorized user accesses a web page containing run quality data.",
"Having executed the scripts a user can review the results and correct some data if necessary at any time.",
"The output produced by scripts is cached, so they are executed when accessing for the first time.",
"Cache invalidation is available for experts."
],
[
"Implementation Details",
"The described software is implemented as an SND information system [8] component.",
"The server part is integrated to the Node.js [9] application (in JavaScript).",
"The system uses SND databases running under MySQL RDBMS and node-mysql for accessing them.",
"Please refer to figure REF for more details.",
"The histograms mentioned before are served by another Node.js application that uses JSROOT [10] both at server and client sides for reading histogram files and rendering histograms.",
"The server can use also old CERNLIB HBOOK files converting them by h2root utility."
],
[
"Applying to Experiment Data",
"Having put the new software into production, we implemented DQM scripts for several subsystems (calorimeter, muon system, trigger electronics).",
"The new interface was used at the data acquisition stage in 2019.",
"At the same time, the data collected in the previous (2018) year were marked up during the reprocessing stage.",
"More than five thousands runs (including cosmic ones) containing 3.1 billion stored events were checked.",
"This check resulted in detecting 23% runs having bad quality (cosmic, short, test or erroneous runs), 62% good ones and 14% ones with tolerable quality.",
"The results are shown at the figure REF .",
"Please note that bad runs tend to be significantly shorter in terms of time, events count and integral luminosity, however they occupy the same area on the figure.",
"Figure: Integral quality data for RHO2018 experiment.",
"Each cell represents a run (green, yellow and red backgrounds for good, tolerable and bad quality respectively)."
],
[
"Conclusion",
"A new version of the SND experiment DQM system provides a framework for automatic and manual data quality estimation, interactive (web) and program user interfaces.",
"SND data quality is monitored in data acquisition and processing stages with different goals and assumptions.",
"The new DQM system was put into production in 2019.",
"The data acquisition configuration was used during data taking of 2019 and 2020.",
"The reprocessing configuration was used for data of 2018 and 2019.",
"This work is partly supported by the RFBR grant 18-02-00382."
]
] | 2005.14143 |
[
[
"Higher R\\'edei reciprocity and integral points on conics"
],
[
"Abstract Fix an integer $l$ such that $|l|$ is a prime $3$ modulo $4$.",
"Let $d > 0$ be a squarefree integer and let $N_d(x, y)$ be the principal binary quadratic form of $\\mathbb{Q}(\\sqrt{d})$.",
"Building on a breakthrough of Alexander Smith, we give an asymptotic formula for the solubility of $N_d(x, y) = l$ in integers $x$ and $y$ as $d$ varies among squarefree integers divisible by $l$.",
"As a corollary we give, in case $l > 0$, an asymptotic formula for the event that the Hasse Unit Index of the field $\\mathbb{Q}(\\sqrt{-l}, \\sqrt{d})$ is $2$ as $d$ varies over all positive squarefree integers.",
"We also improve the results of Fouvry and Kl\\\"uners and recent results of Chan, Milovic and the authors on the solubility of the negative Pell equation.",
"Our main new tool is a generalization of a classical reciprocity law due to R\\'edei."
],
[
"Introduction",
"The study of integral points on conics goes back to at least the ancient Greeks.",
"Much later significant progress was made by the Indian mathematicians Brahmagupta and Bhaskara II around the years 650 and 1150 respectively.",
"Brahmagupta was able to solve the Pell equation $x^2 - dy^2 = 1 \\text{ in } x, y \\in \\mathbb {Z}$ in special cases, while Bhaskara II was the first to give a method to solve the Pell equation in full generality.",
"Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4 or let $l = -1$ .",
"For a squarefree integer $d > 0$ , we define $N_d(x, y) =\\left\\lbrace \\begin{array}{ll}x^2 + xy - \\frac{d - 1}{4}y^2 & \\mbox{if } d \\equiv 1 \\bmod 4 \\\\x^2 - dy^2 & \\mbox{otherwise,}\\end{array}\\right.$ which is the principal binary quadratic form of $\\mathbb {Q}(\\sqrt{d})$ .",
"In this paper we look at the equation $N_d(x, y) = l \\text{ in } x, y \\in \\mathbb {Z}$ with $d$ squarefree.",
"Unlike equation (REF ) it is not always possible to find $x, y \\in \\mathbb {Z}$ that satisfy the above equation.",
"We denote by $H(K)$ the narrow Hilbert class field of a number field $K$ , which is the maximal abelian extension of $K$ that is unramified at all finite places, while the ordinary Hilbert class field must also be unramified at the infinite places.",
"If $l = -1$ , equation (REF ) is soluble if and only if the narrow and ordinary Hilbert class fields of $\\mathbb {Q}(\\sqrt{d})$ coincide.",
"Instead, if $l > 0$ is a prime 3 modulo 4, then equation (REF ) is soluble if and only if there is an ideal in $\\mathbb {Q}(\\sqrt{d})$ with norm $l$ and trivial Artin symbol in the narrow Hilbert class field of $\\mathbb {Q}(\\sqrt{d})$ .",
"We will now focus on $l$ a prime 3 modulo 4, and shall later discuss the very classical case $l = -1$ known as the negative Pell equation.",
"Given $d$ , there exists an algorithm to compute the Hilbert class field of $\\mathbb {Q}(\\sqrt{d})$ both in the narrow and ordinary sense.",
"Hence it is possible to decide given $l$ and $d$ whether equation (REF ) is soluble.",
"In fact, for a fixed squarefree integer $d$ , an appeal to the Chebotarev Density Theorem gives an asymptotic for the number of primes $l$ such that equation (REF ) is soluble.",
"In this paper we ask the opposite question.",
"Instead of fixing $d$ , we shall treat $l$ as fixed and vary $d$ .",
"Equivalently, we ask how often there is some ideal with norm $l$ and trivial Artin symbol in $H(\\mathbb {Q}(\\sqrt{d}))$ as $d$ varies.",
"Unfortunately, the distribution of the Hilbert class field as $d$ varies is not well understood at the moment.",
"In fact, the only proven results for the distribution of $\\text{Cl}(K)$ with $K$ imaginary quadratic are Davenport–Heilbronn [6] on 3-torsion, Fouvry–Klüners [9], [10] on 4-torsion, based on earlier work of Heath-Brown [16] on 2-Selmer groups, and Smith [32], [33] on respectively 8-torsion and $2^\\infty $ -torsion.",
"Heuristically, we understand the situation much better due to the seminal work of Cohen and Lenstra [3], which was later extended by Gerth [14].",
"Therefore we restrict our attention to only those squarefree integers $d$ that are divisible by $l$ .",
"In this case we know that $l$ ramifies in $\\mathbb {Q}(\\sqrt{d})$ .",
"Gauss genus theory states that the ramified primes generate $\\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2]$ , and that there is precisely one non-trivial relation between them.",
"Here $\\text{Cl}$ denotes the narrow class group.",
"In particular we see that $\\mathfrak {l} \\in \\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2]$ , where $\\mathfrak {l}$ is the unique ideal above $l$ .",
"In this case equation (REF ) is soluble if and only if $\\mathfrak {l}$ is the relation in $\\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2]$ .",
"Hence we need to study the distribution of $\\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2^\\infty ]$ , and this naturally brings the methods of Smith [33] into play.",
"Note that for equation (REF ) to be soluble, it is necessary that it is soluble over $\\mathbb {Q}$ .",
"Or formulated differently, $\\mathfrak {l}$ must split in the genus field of $\\mathbb {Q}(\\sqrt{d})$ , which is by definition the maximal subextension of $H(\\mathbb {Q}(\\sqrt{d}))$ that is abelian over $\\mathbb {Q}$ .",
"By the Hasse–Minkowski theorem it is easy to determine necessary and sufficient conditions on $d$ for the solubility of equation (REF ) over $\\mathbb {Q}$ .",
"With this in mind we can state our first main theorem after introducing the following quantities $\\eta _k := \\prod _{j = 1}^k (1 - 2^{-j}) \\text{ with } k \\in \\mathbb {Z}_{\\ge 0} \\cup \\lbrace \\infty \\rbrace , \\quad \\gamma := \\sum _{j = 0}^\\infty \\frac{2^{-j^2} \\eta _\\infty \\eta _j^{-2}}{2^{j + 1} - 1}$ and with $R \\in \\lbrace \\mathbb {Z}, \\mathbb {Q}\\rbrace $ and $l$ any integer $S_{R, X, l} := \\lbrace 0 < d < X : d \\textup { squarefree}, \\ l \\mid d, N_d(x, y) = l \\textup { is soluble with } x, y \\in R\\rbrace .$ Theorem 1.1 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"Then we have $\\lim _{X \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, X, l}|}{|S_{\\mathbb {Q}, X, l}|} = \\gamma .$ We remark that $\\gamma $ has a very natural interpretation.",
"Informally speaking, the quantity $2^{-j^2} \\eta _\\infty \\eta _j^{-2}$ represents the probability that the 4–rank of a random element in the set $S_{\\mathbb {Q}, X, l}$ is equal to $j$ .",
"This will be made precise in Theorem REF .",
"Note that if the 4–rank of $\\text{Cl}(K)$ is $j$ , we have a natural generating set, coming from Gauss genus theory, of size $j + 1$ for $2 \\text{Cl}(K)[4]$ .",
"Furthermore, Gauss genus theory says that there is exactly one relation between the generators.",
"Hence $1/(2^{j + 1} - 1)$ represents the probability that the ideal above $l$ is the relation, if one thinks of the relation as being “random”.",
"This is very much in spirit of Stevenhagen's conjecture [34] on the solubility of the negative Pell equation.",
"Although we shall not prove it, our techniques readily give the distribution of $2\\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2^\\infty ]$ as $d$ varies in $S_{\\mathbb {Q}, \\infty , l}$ .",
"By classical techniques one can give an asymptotic formula for $|S_{\\mathbb {Q}, X, l}|$ ; this requires only slight modifications of [25], see also [27].",
"Indeed, we have $|S_{\\mathbb {Q}, X, l}| \\sim \\frac{1}{\\sqrt{\\pi }} \\cdot \\frac{C(l) \\cdot \\delta (l)}{|l|} \\cdot \\frac{X}{\\sqrt{\\log X}},$ where $C(l) = \\lim _{s \\rightarrow 1} \\left(\\sqrt{s - 1} \\cdot \\prod _{\\begin{array}{c}p \\text{ odd} \\\\ (l/p) = 1\\end{array}} \\left(1 + \\frac{1}{p^s}\\right)\\right), \\quad \\delta (l)=\\left\\lbrace \\begin{array}{ll}3/2 & \\mbox{if } l \\equiv 1 \\bmod 8 \\\\3/4 & \\mbox{if } l \\equiv 3 \\bmod 8 \\\\1 & \\mbox{if } l \\equiv 5 \\bmod 8 \\\\3/4 & \\mbox{if } l \\equiv 7 \\bmod 8.\\end{array}\\right.$ This yields the following corollary of Theorem REF .",
"Corollary 1.2 Take $l$ to be an integer such that $|l|$ is a prime 3 modulo 4.",
"Then $|S_{\\mathbb {Z}, X, l}| \\sim \\frac{\\gamma }{\\sqrt{\\pi }} \\cdot \\frac{C(l) \\cdot \\delta (l)}{|l|} \\cdot \\frac{X}{\\sqrt{\\log X}}.$ Earlier work was done by Milovic [24], who showed that $S_{\\mathbb {Z}, X, \\pm 2}$ has the same order of magnitude as $S_{\\mathbb {Q}, X, \\pm 2}$ .",
"It is plausible that our methods can be adapted to the case $l = \\pm 2$ as well.",
"An immediate application is the following result.",
"For a biquadratic field $\\mathbb {Q}(\\sqrt{a}, \\sqrt{b})$ , the Hasse Unit Index is defined to be $H_{a, b} := \\left[\\mathcal {O}_{\\mathbb {Q}(\\sqrt{a}, \\sqrt{b})}^\\ast : \\mathcal {O}_{\\mathbb {Q}(\\sqrt{a})}^\\ast \\mathcal {O}_{\\mathbb {Q}(\\sqrt{b})}^\\ast \\mathcal {O}_{\\mathbb {Q}(\\sqrt{ab})}^\\ast \\right].$ If the biquadratic field is totally complex, then it is known that $H_{a, b} \\in \\lbrace 1, 2\\rbrace $ , see for example the work of Lemmermeyer [23].",
"Our next theorem determines the distribution of the Hasse Unit Index in many cases.",
"Corollary 1.3 Let $l > 3$ be a prime 3 modulo 4.",
"Then we have $|\\lbrace 0 < d < X \\textup { squarefree} : H_{-l, d} = 2\\rbrace | \\sim |S_{\\mathbb {Z}, X, l}| + |S_{\\mathbb {Z}, X, -l}| \\sim \\\\\\left(\\frac{\\gamma }{\\sqrt{\\pi }} \\cdot \\frac{C(l) \\cdot \\delta (l)}{l} +\\frac{\\gamma }{\\sqrt{\\pi }} \\cdot \\frac{C(-l) \\cdot \\delta (-l)}{l}\\right) \\cdot \\frac{X}{\\sqrt{\\log X}}.$ From a more geometric perspective, Theorem REF counts how often there exists an integral point in a family of conics.",
"As such, it is natural to view this result from the perspective of the integral Brauer–Manin obstruction.",
"The seminal work [4] was the first to systematically study the integral Brauer–Manin obstruction.",
"We shall now return to the case $l = -1$ .",
"In this case Stevenhagen [34] heuristically predicted how often equation (REF ) is soluble.",
"His heuristical framework can be adjusted to also predict the constant appearing in Theorem REF , and we shall do so in Appendix .",
"The first major result towards Stevenhagen's conjecture is that of Fouvry–Klüners [11].",
"They showed that $\\alpha - o(1) \\le \\frac{|S_{\\mathbb {Z}, X, -1}|}{|S_{\\mathbb {Q}, X, -1}|} \\le \\frac{2}{3} + o(1)$ as $X \\rightarrow \\infty $ .",
"Here $\\alpha $ is known as Stevenhagen's constant and equals $\\alpha := \\prod _{j = 1}^\\infty (1 + 2^{-j})^{-1} \\approx 0.4194.$ Fouvry–Klüners [12] later improved the lower bound to $\\frac{5}{4}\\alpha $ , which was recently improved to $\\beta \\alpha $ in [2], where $\\beta := \\sum _{n = 0}^\\infty 2^{-n(n + 3)/2} \\approx 1.2832, \\quad \\beta \\alpha \\approx 0.53823.$ In this paper we improve both the upper and lower bounds.",
"Theorem 1.4 We have $0.54302 - o(1) \\le \\frac{|S_{\\mathbb {Z}, X, -1}|}{|S_{\\mathbb {Q}, X, -1}|} \\le 0.59944 + o(1)$ as $X \\rightarrow \\infty $ .",
"In Theorem REF we have only given the first five decimals.",
"The full constants can be found in Subsection REF including a detailed explanation where the improvement comes from.",
"Proving the full Stevenhagen conjecture seems to be very hard.",
"The reason for this is that the algebraic results in Smith break down due to the fact that all odd prime divisors are 1 modulo 4 in this family.",
"For a more elaborate discussion on this topic, see Subsection REF or [2].",
"It is also for this reason that we restrict our attention to $|l| \\equiv 3 \\bmod 4$ .",
"Theorem REF and Theorem REF make crucial use of a generalization of a reciprocity law due to Rédei [31].",
"This generalization is proven in Section .",
"An extensive treatment of the classical Rédei reciprocity law can be found in Corsman [5], and was one of the main ingredients in Smith's work on 4-Selmer groups and 8-torsion of class groups [32].",
"Corsman's and Smith's formulations of the Rédei reciprocity law are not correct as stated, and this flaw was discovered and corrected by Stevenhagen [35].",
"We will now roughly explain how we make use of our new reciprocity law.",
"Following Smith's method, we need to prove equidistribution of $\\text{Frob}_{K_{x_1, \\dots , x_m, y}/\\mathbb {Q}}(l)$ as we vary $y$ , where $K_{x_1, \\dots , x_m, y}$ is a completely explicit field depending only on $x_1, \\dots , x_m$ and $y$ .",
"Our reciprocity law implies that under suitable conditions $\\text{Frob}_{K_{x_1, \\dots , x_m, y}/\\mathbb {Q}}(l) = \\text{Frob}_{K_{x_1, \\dots , x_m, l}/\\mathbb {Q}}(y).$ This allows us to apply the Chebotarev Density Theorem to obtain the desired equidistribution.",
"In the case $m = 1$ , the fields $K_{x_1, y}$ are constructed by Rédei, and one recovers the Rédei reciprocity law.",
"In the case $m = 2$ , the field $K_{x_1, x_2, y}$ first appears in Amano [1] for special values of $x_1$ , $x_2$ and $y$ , while the fields $K_{x_1, \\dots , x_m, y}$ are constructed in full generality by Smith [33].",
"In the language of Smith, these fields are the field of definition of certain maps from $G_\\mathbb {Q}$ to $\\mathbb {F}_2$ that Smith calls $\\phi _{x_1, \\dots , x_m, y}$ or simply $\\phi _{\\bar{x}}$ .",
"The field of definition is an unramified multiquadratic extension of a multiquadratic extension of $\\mathbb {Q}$ .",
"As such, they are intimately related to the 2-torsion of the class groups of multiquadratic fields.",
"This connection is explored in recent work of the authors [20].",
"We finish the introduction by mentioning some other important results related to class groups.",
"A lot of attention has recently be given to providing non-trivial upper bounds for $\\text{Cl}(K)[l]$ for a fixed prime $l$ .",
"This was initiated by Pierce [28], [29] for $l = 3$ and continued by Ellenberg and Venkatesh [8], Ellenberg, Pierce and Wood [7], Frei and Widmer [13], Pierce, Turnage-Butterbaugh and Wood [30].",
"Instead of studying class groups of quadratic extensions of $\\mathbb {Q}$ , one can study the distribution of class groups in the family of degree $l$ cyclic extensions of $\\mathbb {Q}$ .",
"This was explored by Gerth [15] and Klys [18], whose work was later generalized by the authors [19] using the Smith method [33].",
"It is natural to wonder if the methods in this paper can also be used to study norm forms coming from degree $l$ cyclic extensions."
],
[
"Acknowledgements",
"We are most grateful to Alexander Smith for explaining his work to us on several occasions.",
"Peter Stevenhagen kindly explained his proof of Rédei reciprocity to us, which inspired us to prove a more general version of the Rédei reciprocity law.",
"We thank Vladimir Mitankin for showing us a useful reference and we thank Stephanie Chan and Djordjo Milovic for our many insightful conversations about negative Pell.",
"Both authors are grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support."
],
[
"Algebraic criteria",
"In this section we collect the algebraic lemmas that link our theorems to questions about the narrow class group.",
"These lemmas are valid for arbitrary non-zero integers $l$ , and we shall only later restrict to $l$ with $|l| \\equiv 3 \\bmod 4$ a prime.",
"For a non-zero integer $l$ , we define $\\text{sign}(l) = 0$ if $l > 0$ and $\\text{sign}(l) = 1$ if $l < 0$ .",
"Lemma 2.1 Let $l$ be a non-zero integer and let $d > 0$ be a squarefree integer.",
"Then there are $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = l$ if and only if there is an integral ideal $I$ of $\\mathcal {O}_{\\mathbb {Q}(\\sqrt{d})}$ with norm $|l|$ such that $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ has trivial Artin symbol in $H(\\mathbb {Q}(\\sqrt{d}))$ .",
"Suppose that there are $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = l$ .",
"In case $d \\equiv 1 \\bmod 4$ , we look at the ideal $I = (x + y\\frac{\\sqrt{d} + 1}{2})$ .",
"It has norm $|l|$ , and furthermore the element $x + y\\frac{\\sqrt{d} + 1}{2}$ has norm $l$ .",
"Then $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ is a principal ideal that has an element with positive norm.",
"This implies that $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ is a principal ideal with a totally positive generator, and hence it has trivial Artin symbol in $H(\\mathbb {Q}(\\sqrt{d}))$ .",
"In case $d \\lnot \\equiv 1 \\bmod 4$ , we use a similar argument with the ideal $I = (x + y\\sqrt{d})$ .",
"For the other direction suppose that there is an integral ideal $I$ of $\\mathcal {O}_{\\mathbb {Q}(\\sqrt{d})}$ with norm $|l|$ and $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ has trivial Artin symbol in $H(\\mathbb {Q}(\\sqrt{d}))$ .",
"Then $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ is a principal ideal with a totally positive generator $\\alpha $ , so $N_{\\mathbb {Q}(\\sqrt{d})/\\mathbb {Q}}(\\alpha ) = d^{\\text{sign}(l)} |l|$ .",
"Hence we have $I = \\left(\\frac{\\alpha }{\\sqrt{d}^{\\text{sign}(l)}}\\right) \\quad \\text{and} \\quad N_{\\mathbb {Q}(\\sqrt{d})/\\mathbb {Q}}\\left(\\frac{\\alpha }{\\sqrt{d}^{\\text{sign}(l)}}\\right) = l.$ Expanding $\\alpha /\\sqrt{d}^{\\text{sign}(l)}$ as $x + y\\frac{\\sqrt{d} + 1}{2}$ if $d \\equiv 1 \\bmod 4$ and $x + y\\sqrt{d}$ otherwise, we get the desired $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = l$ .",
"In case that $l \\mid d$ , we see that every prime dividing $l$ ramifies in $\\mathbb {Q}(\\sqrt{d})$ .",
"Hence there is exactly one ideal $\\mathfrak {l}$ of $\\mathbb {Q}(\\sqrt{d})$ with norm $|l|$ .",
"Furthermore, since $\\mathfrak {l} \\in \\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2]$ , we see that it is enough to demand that $\\mathfrak {l}$ has trivial Artin symbol in the narrow $2^\\infty $ -Hilbert class field of $\\mathbb {Q}(\\sqrt{d})$ , denoted $H_2(\\mathbb {Q}(\\sqrt{d}))$ , which is the maximal abelian extension of $\\mathbb {Q}(\\sqrt{d})$ that is unramified at all finite places and has degree a power of 2.",
"This yields the following criterion.",
"Lemma 2.2 Take a non-zero integer $l$ and take a squarefree integer $d > 0$ divisible by $l$ .",
"Then there exist $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = l$ if and only if there is an integral ideal $I$ of $\\mathcal {O}_{\\mathbb {Q}(\\sqrt{d})}$ with norm $|l|$ such that $I \\cdot (\\sqrt{d})^{\\textup {sign}(l)}$ has trivial Artin symbol in $H_2(\\mathbb {Q}(\\sqrt{d}))$ .",
"It is a well-known result that there are $x, y \\in \\mathbb {Z}$ such that $N_d(x, y) = -1$ if and only if there are $x, y \\in \\mathbb {Z}$ such that $x^2 - dy^2 = -1$ , see for example [34].",
"In this way we can also apply the above lemma to study the negative Pell equation.",
"Our final lemma allows us to deduce Corollary REF directly from Theorem REF .",
"Lemma 2.3 Suppose that $l > 3$ is an odd squarefree integer and let $d > 0$ be a squarefree integer with $d \\ne l$ and $d \\ne 3l$ .",
"Then we have $H_{-l, d} = 2$ if and only if $l \\mid d$ and there are $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = l$ or $N_d(x, y) = -l$ .",
"By our assumptions on $l$ and $d$ we have that the roots of unity of $\\mathbb {Q}(\\sqrt{-l}, \\sqrt{d})$ are $\\lbrace \\pm 1\\rbrace $ .",
"Let $\\epsilon $ be the fundamental unit of $\\mathbb {Q}(\\sqrt{d})$ .",
"Then Kubota's work [22] shows that $H_{-l, d} = 2$ if and only if $\\pm \\epsilon $ is a square in $\\mathbb {Q}(\\sqrt{-l}, \\sqrt{d})$ .",
"By Kummer theory this is equivalent to $\\epsilon = \\pm lz^2$ for some $z \\in \\mathbb {Q}(\\sqrt{d})^\\ast $ .",
"This is in turn equivalent to the requirements that every prime dividing $l$ must ramify in $\\mathbb {Q}(\\sqrt{d})$ , and furthermore that the unique ideal with norm $|l|$ is principal in $\\mathbb {Q}(\\sqrt{d})$ .",
"These last two conditions are equivalent to $l \\mid d$ and the existence of $x, y \\in \\mathbb {Z}$ with $N_d(x, y) = \\pm l$ ."
],
[
"Higher Rédei reciprocity",
"This section contains the main algebraic innovation of this paper, which is a generalization of the classical Rédei reciprocity law (in turn a generalization of quadratic reciprocity).",
"Fix an algebraic closure $\\overline{\\mathbb {Q}}$ of $\\mathbb {Q}$ for the rest of the paper.",
"All our number fields are implicitly taken inside this fixed algebraic closure $\\overline{\\mathbb {Q}}$ .",
"If $K$ is a number field, we define $G_K := \\mathrm {Gal}(\\overline{\\mathbb {Q}}/K)$ .",
"Throughout, we view $\\mathbb {F}_2$ as a discrete $G_\\mathbb {Q}$ -module with trivial action.",
"If $\\phi : G_\\mathbb {Q}\\rightarrow X$ is a continuous map with $X$ a discrete topological space, we define $L(\\phi )$ to be the smallest Galois extension $K$ of $\\mathbb {Q}$ through which $\\phi $ factors via the canonical projection map $G_\\mathbb {Q}\\rightarrow \\mathrm {Gal}(K/\\mathbb {Q})$ .",
"This is well-defined by [19].",
"For us an unramified extension $L/K$ shall always mean unramified at all finite places of $K$ ."
],
[
"Statement of the reciprocity law",
"Let $n \\in \\mathbb {Z}_{\\ge 1}$ and let $A \\subseteq \\Gamma _{\\mathbb {F}_2}(\\mathbb {Q}):=\\text{Hom}_{\\text{top.gr.",
"}}(G_{\\mathbb {Q}}, \\mathbb {F}_2)$ with $|A| = n$ .",
"Let $\\chi _1,\\chi _2$ be two distinct elements of $\\Gamma _{\\mathbb {F}_2}(\\mathbb {Q})-A$ .",
"Write $A_1:=A \\cup \\lbrace \\chi _1\\rbrace , A_2:=A \\cup \\lbrace \\chi _2\\rbrace .$ For a finite extension $L/\\mathbb {Q}$ , we denote by $\\text{Ram}(L/\\mathbb {Q})$ the set of places of $\\mathbb {Q}$ that ramify in $L/\\mathbb {Q}$ .",
"Furthermore, for a collection of characters $T \\subseteq \\Gamma _{\\mathbb {F}_2}(\\mathbb {Q})$ , we denote by $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in T})$ the corresponding multiquadratic extension of $\\mathbb {Q}$ .",
"We assume that as $\\chi $ varies in $A_1 \\cup A_2$ the $n + 2$ sets $\\text{Ram}(\\mathbb {Q}(\\chi )/\\mathbb {Q})$ are non-empty and pairwise disjoint.",
"This in particular implies that $A_1 \\cup A_2$ is a set of $n + 2$ linearly independent characters over $\\mathbb {F}_2$ .",
"We now recall a definition from [20].",
"Definition 3.1 Let $X \\subseteq \\Gamma _{\\mathbb {F}_2}(\\mathbb {Q})$ be linearly independent and let $\\chi _0 \\in X$ .",
"An expansion map with support $X$ and pointer $\\chi _0$ is a continuous group homomorphism $\\psi : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2[\\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }] \\rtimes \\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }$ such that $\\pi _\\chi \\circ \\psi = \\chi $ for every $\\chi \\in X - \\lbrace \\chi _0\\rbrace $ , where $\\pi _\\chi : \\mathbb {F}_2[\\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }] \\rtimes \\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace } \\rightarrow \\mathbb {F}_2$ is the natural projection, and $\\pi \\circ \\psi = \\chi _0$ , where $\\pi : \\mathbb {F}_2[\\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }] \\rightarrow \\mathbb {F}_2$ is the unique non-trivial character that sends the subgroup $\\lbrace 0\\rbrace \\rtimes \\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }$ to 0.",
"Note that an expansion map is automatically surjective.",
"There is another characterization of expansion maps that we give now, first given in Section 3.3 of [20].",
"We have an isomorphism $\\mathbb {F}_2[\\mathbb {F}_2^{X - \\lbrace \\chi _0\\rbrace }] \\cong \\mathbb {F}_2[\\lbrace t_x\\rbrace _{x \\in X - \\lbrace \\chi _0\\rbrace }]/(\\lbrace t_x^2\\rbrace _{x \\in X - \\lbrace \\chi _0\\rbrace })$ by sending $t_x$ to $1 \\cdot \\text{id} + 1 \\cdot e_x$ , where $e_x$ is the vector that is 1 exactly on the $x$ -th coordinate.",
"Note that the squarefree monomials $t_Y := \\prod _{y \\in Y} t_y$ give a basis of $\\mathbb {F}_2[\\lbrace t_x\\rbrace _{x \\in X - \\lbrace \\chi _0\\rbrace }]/(\\lbrace t_x^2\\rbrace _{x \\in X - \\lbrace \\chi _0\\rbrace })$ , as $Y$ varies through the subsets of $X - \\lbrace \\chi _0\\rbrace $ .",
"Therefore, projection on monomials gives rise to continuous 1-cochains $\\phi _Y(\\psi ) : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ for every $Y \\subseteq X - \\lbrace \\chi _0\\rbrace $ .",
"Together they allow us to reconstruct $\\psi $ by the formula $\\psi (g) = \\left(\\sum _{Y \\subseteq X - \\lbrace \\chi _0\\rbrace } \\phi _Y(\\psi )(g) t_Y, \\lbrace \\chi (g)\\rbrace _{\\chi \\in X - \\lbrace \\chi _0\\rbrace }\\right).$ Now define $\\chi _S := \\prod _{\\chi \\in S} \\chi $ , where the product is taken in $\\mathbb {F}_2$ .",
"From equation (REF ) and the composition law for the semidirect product we deduce that $(d\\phi _Y(\\psi ))(g_1, g_2) = \\sum _{\\emptyset \\subsetneq S \\subseteq Y} \\chi _S(g_1) \\phi _{Y - S}(\\psi )(g_2),$ where $d$ is the operator that sends $\\text{Map}(G_\\mathbb {Q}, \\mathbb {F}_2)$ to $\\text{Map}(G_\\mathbb {Q}\\times G_\\mathbb {Q}, \\mathbb {F}_2)$ with the rule $(d\\phi )(g_1, g_2) = \\phi (g_1) + \\phi (g_2) + \\phi (g_1g_2).$ Equation (REF ) is simply equation (2.2) of Smith [33].",
"Conversely, if we are given a system of maps $\\lbrace \\phi _Y\\rbrace _{Y \\subseteq X - \\lbrace \\chi _0\\rbrace }$ satisfying equation (REF ) and $\\phi _\\emptyset = \\chi _0$ , we get an expansion map $\\psi $ with support $X$ and pointer $\\chi _0$ .",
"Now suppose that we have two expansion maps $\\psi _1, \\psi _2: G_{\\mathbb {Q}} \\twoheadrightarrow \\mathbb {F}_2[\\mathbb {F}_2^A] \\rtimes \\mathbb {F}_2^A,$ with supports $A_1,A_2$ and pointers $\\chi _1,\\chi _2$ respectively.",
"Let $\\lbrace \\phi _{1, B}\\rbrace _{B \\subseteq A}, \\lbrace \\phi _{2, B}\\rbrace _{B \\subseteq A}$ be the corresponding system of continuous 1-cochains from $G_{\\mathbb {Q}}$ to $\\mathbb {F}_2$ with $\\phi _{1, \\emptyset } = \\chi _1, \\phi _{2, \\emptyset } = \\chi _2$ .",
"Note that $L(\\psi _1), L(\\psi _2)$ are central $\\mathbb {F}_2$ -extensions of $M(\\psi _1) := \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\prod _{B \\subsetneq A}L(\\phi _{1, B})/ \\mathbb {Q}, \\quad M(\\psi _2) := \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\prod _{B \\subsetneq A}L(\\phi _{2, B})/ \\mathbb {Q}.$ We need one more definition before stating our reciprocity law.",
"Definition 3.2 Let $(A_1, A_2, \\psi _1, \\psi _2)$ be a 4-tuple as above.",
"We say $(A_1, A_2, \\psi _1, \\psi _2)$ is Rédei admissible if the extensions $L(\\psi _1)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A_1}), L(\\psi _2)/ \\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A_2})$ are unramified; each place of $\\textup {Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})$ splits completely in $M(\\psi _2)/\\mathbb {Q}$ and similarly each place of $\\textup {Ram}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ splits completely in $M(\\psi _1)/\\mathbb {Q}$ ; if the infinite place $\\infty $ of $\\mathbb {Q}$ splits completely in $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A_1 \\cup A_2})$ , then $\\infty $ splits completely in $M(\\psi _1)M(\\psi _2)/\\mathbb {Q}$ as well.",
"We call $(\\chi _1,\\chi _2)$ the pointer vector of the 4-tuple and we call $A$ the base set of the 4-tuple.",
"Let now $(A_1,A_2,\\psi _1,\\psi _2)$ be a Rédei admissible 4-tuple, as above, with pointer vector $(\\chi _1,\\chi _2)$ .",
"Then it follows by the definition that each place $v$ in $\\text{Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})$ is unramified in $L(\\psi _2)/\\mathbb {Q}$ and furthermore the consequently defined Artin class $\\text{Art}(v, L(\\psi _2)/\\mathbb {Q})$ lands in $\\text{Gal}(L(\\psi _2)/M(\\psi _2))$ , which is a central subgroup of $\\text{Gal}(L(\\psi _2)/\\mathbb {Q})$ of size equal to 2 and hence can uniquely be identified with $\\mathbb {F}_2$ .",
"We conclude that $\\text{Art}(v, L(\\psi _2)/\\mathbb {Q})$ is a well-defined element of $\\mathbb {F}_2$ .",
"Symmetrically, the same holds if we swap the role of 1 and 2.",
"Finally, for a quadratic extension $\\mathbb {Q}(\\sqrt{d})/\\mathbb {Q}$ , we put $\\widetilde{\\text{Ram}}(\\mathbb {Q}(\\sqrt{d})/\\mathbb {Q})$ to be the set of places in $\\text{Ram}(\\mathbb {Q}(\\sqrt{d})/\\mathbb {Q})$ with the only exception of $(2)$ , which is excluded in case $d$ has even 2-adic valuation.",
"We can now state the reciprocity law.",
"Theorem 3.3 Let $(A_1,A_2,\\psi _1,\\psi _2)$ be a Rédei admissible 4-tuple with pointer vector $(\\chi _1,\\chi _2)$ .",
"Then we have $\\sum _{v \\in \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})}\\emph {Art}(v, L(\\psi _2)/\\mathbb {Q})=\\sum _{v^{\\prime } \\in \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})}\\emph {Art}(v^{\\prime }, L(\\psi _1)/\\mathbb {Q}).$"
],
[
"Proof of Theorem ",
"Let $(A_1,A_2,\\psi _1,\\psi _2)$ be a Rédei admissible 4-tuple with pointer vector $(\\chi _1,\\chi _2)$ and base set $A$ .",
"We start by observing that equation (REF ) implies that, for $i \\in \\lbrace 1,2\\rbrace $ , the set of cochains $\\lbrace \\phi _{i, B}\\rbrace _{B \\subseteq A}$ , when restricted to $G_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})}$ , becomes a set of quadratic characters.",
"By abuse of notation we shall use the same symbols for such quadratic characters.",
"Furthermore, it is clear from the definition of an expansion map that, for each $i \\in \\lbrace 1,2\\rbrace $ , the character $\\phi _{i, A}$ generates a rank 1 free module over the ring $\\mathbb {F}_2[\\text{Gal}(\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})/\\mathbb {Q})]$ and that the corresponding Galois extension of $\\mathbb {Q}$ given by this module is precisely $L(\\psi _i)/\\mathbb {Q}$ .",
"For each $B \\subseteq A$ we denote by $\\alpha _{i, B} \\in \\frac{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in B})^{*}}{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in B})^{*2}}$ the unique element, provided by Kummer theory, corresponding to $\\phi _{i, B}$ .",
"We have the following fact.",
"The reader with some familiarity with Rédei symbols, as for instance treated in [35], will recognize that this fact is a generalization of the connection between Rédei fields and solution sets of certain attached conics, see [35].",
"Proposition 3.4 Let $i \\in \\lbrace 1, 2\\rbrace $ .",
"For each $B \\subseteq A$ we have that $N_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})/\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in B})}(\\alpha _{i, A})=\\alpha _{i, B},$ as elements of $\\frac{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in B})^{*}}{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in B})^{*2}}$ .",
"From the recursive formula (REF ) we see that it suffices to show the proposition when $B$ is obtained from $A$ by deleting a single element $a \\in A$ : the full proposition then follows by applying this repeatedly.",
"Hence we need to show that, for an element $a \\in A$ , we have $N_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})/\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A-\\lbrace a\\rbrace })}(\\alpha _{i, A})=\\alpha _{i, A-\\lbrace a\\rbrace } \\text{ in } \\frac{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A-\\lbrace a\\rbrace })^{*}}{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A-\\lbrace a\\rbrace })^{*2}}.$ By Kummer theory, this is equivalent to showing that the co-restriction of the character $\\phi _{i, A}$ from $G_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})}$ to $G_{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A-\\lbrace a\\rbrace })}$ equals the character $\\phi _{i, A-\\lbrace a\\rbrace }$ .",
"Let us recall the following basic fact.",
"Let $G_1 \\subseteq G_2$ be a continuous index 2 inclusion of profinite groups, and let $\\chi :G_1 \\rightarrow \\mathbb {F}_2, \\chi ^{\\prime }:G_2 \\rightarrow \\mathbb {F}_2$ be two continuous characters.",
"Then the co-restriction of $\\chi $ to $G_2$ equals $\\chi ^{\\prime }$ if and only $\\chi (\\sigma ^2) = \\chi ^{\\prime }(\\sigma )$ and $\\chi (\\sigma \\tau \\sigma ^{-1}) + \\chi (\\tau ) = \\chi ^{\\prime }(\\tau )$ for each $\\sigma \\in G_2 - G_1$ , $\\tau \\in G_1$ .",
"The second relation only implies that the co-restriction of $\\chi $ equals $\\chi ^{\\prime }$ as characters of the index 2 subgroup $G_1$ .",
"This leaves two possibilities for the character from the larger group $G_2$ : these two possibilities constitute a single coset under the subgroup generated by the character $\\epsilon : G_2 \\twoheadrightarrow \\frac{G_2}{G_1}=\\mathbb {F}_2$ .",
"In other words the second relation forces the co-restriction of $\\chi $ to be in the set $\\lbrace \\chi ^{\\prime }, \\chi ^{\\prime } + \\epsilon \\rbrace $ .",
"This ambiguity is resolved by the first relation.",
"The second relation in our case follows precisely by the definition of expansion maps as coordinates of monomials in a semidirect product.",
"In this language the fact that the monomial $t_{A-\\lbrace a\\rbrace }$ maps under multiplication by the variable $t_a$ to the monomial $t_A$ , translates precisely to the dual fact that the character $\\phi _{i, A}$ norms to $\\phi _{i, A - \\lbrace a\\rbrace }$ , when viewed as characters of the group $G_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})}$ , which plays the role of $G_1$ .",
"We now check the first relation, which forces the norm relation to hold as characters of the larger group $G_{\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A-\\lbrace a\\rbrace })}$ , playing here the role of $G_2$ .",
"To this end, pick $\\sigma $ as above, and plug in $(\\sigma ,\\sigma )$ in equation (REF ): the left hand side gives $\\phi _{i, A}(\\sigma ^2)$ , which is the quantity we are after, while the right hand side gives $\\phi _{i, A - \\lbrace a\\rbrace }(\\sigma )$ .",
"This establishes the desired conclusion.",
"Remark 1 One could ask whether, as in the case of ordinary Rédei fields [35], it is possible to give a converse of the above Proposition REF .",
"This is also discussed in [20].",
"We hope to return to this topic in future work.",
"Let $\\Omega $ be the set of all places of $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ .",
"An immediate consequence of Proposition REF is the following crucial fact.",
"Corollary 3.5 Let $i \\in \\lbrace 1, 2\\rbrace $ .",
"Let $v \\in \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _i)/\\mathbb {Q})$ be a finite place.",
"Then the number of elements $w$ of $\\Omega $ lying above $v$ and with $w(\\alpha _{i, A}) \\equiv 1 \\bmod 2$ has odd cardinality.",
"Suppose that $\\infty \\in \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _i)/\\mathbb {Q})$ .",
"Then the number of embeddings $\\sigma : \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\rightarrow \\mathbb {R}$ with $\\sigma (\\alpha _{i, A}) < 0$ has odd cardinality.",
"We shall explain the argument for part $(a)$ , part $(b)$ can be obtained in the same way.",
"Recall that for each $i \\in \\lbrace 1,2\\rbrace $ we have that $\\phi _{i, \\emptyset }=\\chi _i$ .",
"Therefore Proposition REF , applied to $B:=\\emptyset $ yields $N_{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})/\\mathbb {Q}}(\\alpha _{i, A}) = \\alpha _{i, \\emptyset },$ which gives $\\sum _{w \\in \\Omega : w \\mid v} w(\\alpha _{i, A}) \\equiv v(\\alpha _{i, \\emptyset }) \\bmod 2.$ On the other hand by definition $\\mathbb {Q}(\\chi _i)=\\mathbb {Q}(\\sqrt{\\alpha _{i, \\emptyset }})$ .",
"By the definition of $\\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _i)/\\mathbb {Q})$ , this quadratic extension locally at $v$ is obtained by adding the square root of a uniformizer.",
"It follows that $v(\\alpha _{i, \\emptyset }) \\equiv 1 \\bmod 2$ .",
"Hence the desired conclusion follows immediately from equation (REF ).",
"We shall make use of the following general lemma.",
"For a local field $K$ we denote by $(-,-)_K$ the Hilbert pairing on $\\frac{K^{*}}{K^{*2}}$ .",
"If $v$ denotes a place of a number field $L$ , we shall denote, by abuse of notation, by $(-,-)_v$ the pairing $(-,-)_{L_v}$ : in our context the choice of $L$ will be clear so the abuse of notation will not cause ambiguities.",
"We recall the following fundamental fact.",
"Lemma 3.6 Let $p$ be a rational prime and let $K/\\mathbb {Q}_p$ be a finite extension.",
"Let $\\alpha $ be the unique class of $\\frac{K^{*}}{K^{*2}}$ giving the unique unramified quadratic extension $K(\\sqrt{\\alpha })/K$ .",
"Then the linear functional $(\\alpha ,-)_K: \\frac{K^{*}}{K^{*2}} \\rightarrow \\mathbb {F}_2$ equals $[v_K(-)]_{\\emph {mod} \\ 2}$ .",
"Let $\\pi $ be a uniformizer in $K$ .",
"The local Artin map for $K$ will send $\\pi $ to the generator of $\\text{Gal}(K(\\sqrt{\\alpha })/K)$ , which shows that the Hilbert symbol $(\\alpha , \\pi )_K$ is non-trivial.",
"Since this holds for all uniformizers, one immediately obtains the desired conclusion.",
"We need one final ingredient.",
"Let $w$ be a real place of a number field $L$ , i.e.",
"a place corresponding to an embedding $\\sigma :L \\rightarrow \\mathbb {R}$ .",
"We put $w(\\alpha )$ to be 0 if $\\sigma (\\alpha )>0$ and 1 otherwise.",
"Proposition 3.7 We have the following facts.",
"Let $v$ be a finite place outside $\\emph {Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\emph {Ram}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ .",
"Then $(\\alpha _{1, A}, \\alpha _{2, A})_w=0$ for each element $w$ of $\\Omega $ lying above $v$ .",
"Suppose that the infinite place is outside $\\emph {Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\emph {Ram}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ .",
"Then the value of $(\\alpha _{1, A}, \\alpha _{2, A})_w$ is the same for all $w \\in \\Omega $ lying above $\\infty $ .",
"Suppose $(2) \\in \\emph {Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\emph {Ram}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ , $(2) \\notin \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ .",
"Then $(\\alpha _{1, A}, \\alpha _{2, A})_w=0$ for each element $w$ of $\\Omega $ lying above $(2)$ .",
"Let $\\lbrace 1,2\\rbrace =\\lbrace i,j\\rbrace $ and let $v$ be in $\\widetilde{\\emph {Ram}}(\\mathbb {Q}(\\chi _i)/\\mathbb {Q})$ .",
"Then $(\\alpha _{1, A}, \\alpha _{2, A})_w=w(\\alpha _{i, A}) \\cdot \\emph {Art}(v,L(\\psi _j)/\\mathbb {Q})$ for each element $w$ of $\\Omega $ lying above $v$ , where the product is taken in $\\mathbb {F}_2$ .",
"[Proof of Proposition REF part (a.1)] By definition, the extension $L(\\psi _1)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A_1})$ and the extension $L(\\psi _2)/\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A_2})$ are unramified above all finite places.",
"It follows that for each $v$ as in the statement we must have that $L(\\psi _1)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ and $L(\\psi _2)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ are unramified at any place $w \\in \\Omega $ above $v$ .",
"Now, since the fields $L(\\psi _1), L(\\psi _2)$ are respectively equal to the Galois closure (over $\\mathbb {Q}$ ) of the quadratic extensions $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})(\\sqrt{\\alpha _{1, A}})/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}), \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})(\\sqrt{\\alpha _{2, A}})/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ , it follows that the classes of $\\alpha _{1, A}, \\alpha _{2, A}$ are unramified classes in $\\frac{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})_w^{*}}{\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})_w^{*2}}$ .",
"But, thanks to Lemma REF , the Hilbert symbol between two unramified classes is always trivial and so we obtain the desired conclusion.",
"[Proof of Proposition REF part (a.2)] In this case $\\infty $ splits completely in $M(\\psi _1)M(\\psi _2)$ .",
"Since $\\alpha _{1, A}, \\alpha _{2, A}$ are $G_{\\mathbb {Q}}$ -invariants of, respectively, $\\frac{M(\\psi _1)^{*}}{M(\\psi _1)^{*2}}$ and $\\frac{M(\\psi _2)^{*}}{M(\\psi _2)^{*2}}$ , it follows that $\\sigma (\\alpha _{i, A})$ has constantly the same sign as we vary $\\sigma : \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\rightarrow \\mathbb {R}$ for $i \\in \\lbrace 1, 2\\rbrace $ .",
"Indeed, the conjugates of $\\alpha _{i, A}$ are equal to $\\alpha _{i, A}$ times a square in $M(\\psi _i)$ , therefore $\\sigma (\\alpha _{i, A})$ changes by the square of a real number, which is positive.",
"Hence the conclusion follows at once, since the Hilbert symbol in the local field $\\mathbb {R}$ is entirely determined by the sign of the two entries.",
"[Proof of Proposition REF part (b)] Suppose, without loss of generality, that $(2)$ ramifies in $\\mathbb {Q}(\\chi _1)$ , and thus not in $\\mathbb {Q}(\\chi _2)$ .",
"By assumption we additionally know that $\\mathbb {Q}(\\chi _1)/\\mathbb {Q}$ is obtained, locally at 2, by adding the square root of a unit.",
"We then claim that for each place $w$ as in the statement, we must have that $w(\\alpha _{1, A})$ is even.",
"If not we would have that the extension $L(\\psi _1)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A_1})$ ramifies at the unique place above $w$ in $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A_1})$ , which would contradict that this extension is unramified at all finite places.",
"On the other hand we know that $L(\\psi _2)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ is unramified at $w$ : this follows from the fact that $\\mathbb {Q}(\\chi _2)/\\mathbb {Q}$ is unramified at 2 and the reasoning in part (a.1).",
"Therefore we obtain the desired conclusion as an immediate consequence of Lemma REF .",
"[Proof of Proposition REF part (c)] Let us firstly suppose that $v$ is a finite place and let $w \\in \\Omega $ be above $v$ .",
"We claim that $L(\\psi _j)/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ is unramified at $w$ .",
"It is certainly true that any $w^{\\prime }$ above $w$ in $\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A_j})/\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ will be unramified in $L(\\psi _j)/\\mathbb {Q}(\\lbrace \\chi ^{\\prime }\\rbrace _{\\chi ^{\\prime } \\in A_j})$ , since this last extension is unramified at all finite places.",
"But then our claim follows immediately from the fact that $v$ is unramified in $\\mathbb {Q}(\\chi _j)/\\mathbb {Q}$ , as we already argued in part $(a.1)$ .",
"Furthermore, we know that $v$ splits completely in $M(\\psi _j)/\\mathbb {Q}$ and $\\alpha _{j, A}$ is a $G_{\\mathbb {Q}}$ -invariant class in $\\frac{M(\\psi _j)^{*}}{M(\\psi _j)^{*2}}$ .",
"It follows that among all $2^n$ embeddings of $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\rightarrow \\mathbb {Q}_v$ , we have that $\\alpha _{j, A}$ always lands in the same unramified class of $\\frac{\\mathbb {Q}_v^{*}}{\\mathbb {Q}_v^{*2}}$ .",
"Recalling that $L(\\psi _j) = M(\\psi _j)\\left(\\sqrt{\\alpha _{j, A}}\\right),$ we see that this class is trivial if and only if $\\text{Art}(v,L(\\psi _j)/\\mathbb {Q})$ is trivial.",
"On the other hand, Lemma REF tells us that $(\\alpha _{i, A}, \\alpha _{j, A}) = w(\\alpha _{i, A})$ in case $\\text{Art}(v, L(\\psi _j)/\\mathbb {Q})$ is non-trivial and equals 0 in case $\\text{Art}(v, L(\\psi _j)/\\mathbb {Q})$ is trivial.",
"This is precisely the desired conclusion.",
"The case that $v$ equals the place $\\infty $ of $\\mathbb {Q}$ goes as follows.",
"The formula is correct as soon as $w(\\alpha _{i, A})=0$ , because the Hilbert symbol on the reals vanishes as soon as one of the two entries is positive.",
"But we know that $\\infty $ splits completely in $M(\\psi _j)/\\mathbb {Q}$ and that $L(\\psi _j)$ is obtained by adding the square root of $\\alpha _{j, A}$ .",
"Hence, recalling one more time that $\\alpha _{j, A}$ is a $G_{\\mathbb {Q}}$ -invariant class in $\\frac{M(\\psi _j)^{*}}{M(\\psi _j)^{*2}}$ , we have that the sign of $\\sigma (\\alpha _{j, A})$ is independent of the embedding $\\sigma : \\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A}) \\rightarrow \\mathbb {R}$ , and it is positive if and only if $\\text{Art}(\\infty , L(\\psi _j)/\\mathbb {Q})$ is trivial.",
"Hence the desired conclusion holds also when $w(\\alpha _{i, A}) = 1$ .",
"This ends the proof.",
"We are now ready to prove the main result of this section.",
"[Proof of Theorem REF ] Recall that $\\Omega $ denotes the set of all places of $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})$ .",
"Hilbert's reciprocity law yields $\\sum _{w \\in \\Omega } (\\alpha _{1, A}, \\alpha _{2, A})_w = 0.$ Thanks to Proposition REF part $(a.1)$ and part $(b)$ we obtain that each of the Hilbert symbols with $w$ lying above a finite place outside of $\\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ is 0.",
"From Proposition REF part $(a.2)$ we deduce that if $\\infty $ is outside $\\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ (which happens if and only if it is outside $\\text{Ram}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q}) \\cup \\text{Ram}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ ), then the total contribution coming from all the places above $\\infty $ in $\\Omega $ equals $2^n$ times the same number, and, hence, since $n \\ge 1$ , we get 0.",
"Let now $v$ be a place in $\\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})$ .",
"Then thanks to Proposition REF , part $(c)$ , we obtain that for each $w$ above $v$ in $\\mathbb {Q}(\\lbrace \\chi \\rbrace _{\\chi \\in A})/\\mathbb {Q}$ we have $(\\alpha _{1, A}, \\alpha _{2, A})_w = w(\\alpha _{1, A}) \\cdot \\text{Art}(v,L(\\psi _2)/\\mathbb {Q})).$ Therefore Corollary REF allows us to conclude that $\\sum _{w \\in \\Omega : w \\mid v} (\\alpha _{1, A}, \\alpha _{2, A})_w = \\text{Art}(v,L(\\psi _2)/\\mathbb {Q}))$ and similarly $\\sum _{w \\in \\Omega : w \\mid v} (\\alpha _{1, A}, \\alpha _{2, A})_w = \\text{Art}(v,L(\\psi _1)/\\mathbb {Q}))$ for a place $v \\in \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})$ .",
"Hence in total equation (REF ) becomes $\\sum _{v \\in \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})} \\text{Art}(v,L(\\psi _2)/\\mathbb {Q}) + \\sum _{v^{\\prime } \\in \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})}\\text{Art}(v^{\\prime },L(\\psi _1)/\\mathbb {Q}) = 0,$ which can be rewritten as $\\sum _{v \\in \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _1)/\\mathbb {Q})} \\text{Art}(v,L(\\psi _2)/\\mathbb {Q}) = \\sum _{v^{\\prime } \\in \\widetilde{\\text{Ram}}(\\mathbb {Q}(\\chi _2)/\\mathbb {Q})}\\text{Art}(v^{\\prime },L(\\psi _1)/\\mathbb {Q}),$ and this is precisely the desired conclusion."
],
[
"A reflection principle",
"The material in this section is based directly on [33], and therefore we shall go over it rather quickly.",
"We start by introducing some important notation, which will be similar to Smith's notation [33] and [19].",
"Then we shall develop the theory of expansions.",
"Once this is done, we introduce the notions of minimality and agreement.",
"The section ends with two reflection principles, which relate the class group structure of different fields.",
"This will serve as the algebraic input for our analytic machinery."
],
[
"Notation",
"In this paper $X$ will always be a product set $X_1 \\times \\dots \\times X_r$ , where each $X_i$ is a finite, non-empty set of primes intersecting trivially with all the other $X_j$ .",
"This allows us to identify $(x_1, \\dots , x_r) \\in X$ with the squarefree integer $x_1 \\cdot \\ldots \\cdot x_r$ , and we shall often do so implicitly.",
"For $a \\in \\mathbb {Z}_{\\ge 0}$ , we will write $[a]$ for the set $\\lbrace 1, \\dots , a\\rbrace $ .",
"If $S \\subseteq [r]$ , we define $\\overline{X}_S := \\prod _{i \\in S} (X_i \\times X_i) \\times \\prod _{i \\notin [r] - S} X_i,$ and we let $\\pi _i$ be the projection to $X_i \\times X_i$ if $i \\in S$ and to $X_i$ if $i \\notin S$ .",
"The natural projection maps from $X_i \\times X_i$ to $X_i$ are denoted by $\\text{pr}_1$ and $\\text{pr}_2$ .",
"For two subsets $S, S_0 \\subseteq [r]$ , we let $\\pi _{S, S_0}$ be the projection map from $\\overline{X}_S$ to $\\prod _{i \\in S \\cap S_0} (X_i \\times X_i) \\times \\prod _{i \\in ([r] - S) \\cap S_0} X_i$ given by $\\pi _i$ on each $i \\in S_0$ .",
"The set $S$ shall often be clear from context, in case we will simply write $\\pi _{S_0}$ for $\\pi _{S, S_0}$ .",
"Finally, take some $\\bar{x} \\in \\overline{X}_S$ and $T \\subseteq S \\subseteq [r]$ .",
"Then we define $\\bar{x}(T)$ to be the following multiset $\\lbrace \\bar{y} \\in \\overline{X}_T : \\pi _{[r] - (S - T)}(\\bar{y}) = \\pi _{[r] - (S - T)}(\\bar{x}) \\text{ and } \\forall i \\in S - T \\exists j \\in [2] : \\pi _i(\\bar{y}) = \\text{pr}_j(\\pi _i(\\bar{x}))\\rbrace $ with the multiplicity of $\\bar{y} \\in \\bar{x}(T)$ being $\\prod _{i \\in S - T} \\left|\\left\\lbrace j \\in [2] : \\pi _i(\\bar{y}) = \\text{pr}_j(\\pi _i(\\bar{x}))\\right\\rbrace \\right|.$ With these notations, we can now start deriving a reflection principle, which will be almost identical to [33].",
"To do so, we will need to introduce expansions, governing expansions, minimality and agreement."
],
[
"Expansions",
"In this subsection, we shall quickly recall the facts about expansions that we will need.",
"These are treated more elaborately in [33], [19].",
"The paper [20] is entirely devoted to a careful study of the properties of expansions.",
"We start by defining pre-expansions and expansions.",
"Definition 4.1 For any integer $x$ , we let $\\chi _x : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ be the character corresponding to $\\mathbb {Q}(\\sqrt{x})$ .",
"Let $X := X_1 \\times \\dots \\times X_r$ with $|X_i| = 2$ for $i \\in [r]$ .",
"For a subset $U \\subseteq [r]$ , we declare $\\chi _U : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ to be $\\chi _U(\\sigma ) := \\prod _{i \\in U} \\chi _{\\textup {pr}_1(\\pi _i(x)) \\cdot \\textup {pr}_2(\\pi _i(x))}(\\sigma ).$ A pre-expansion for $X$ is a sequence $\\lbrace \\phi _T\\rbrace _{T \\subsetneq [r]}$ where each $\\phi _T: G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ is a continuous 1-cochain satisfying $(d\\phi _T)(\\sigma , \\tau ) = \\sum _{\\emptyset \\ne U \\subseteq T} \\chi _U(\\sigma ) \\phi _{T - U}(\\tau ).$ Recall that $d\\phi _T(\\sigma , \\tau ) := \\phi _T(\\sigma ) + \\phi _T(\\tau ) + \\phi _T(\\sigma \\tau )$ .",
"For the remainder of the paper, we shall always assume that $\\phi _\\emptyset $ is linearly independent from the space of characters spanned by $\\lbrace \\chi _{\\lbrace i\\rbrace }\\rbrace _{i \\in [r]}$ .",
"We say that a pre-expansion is promising if for every $i \\in [r]$ , every prime $p \\in X_i$ splits completely in $L(\\phi _{[r] - \\lbrace i\\rbrace })$ .",
"Furthermore, a pre-expansion is said to be good if $L(\\phi _T)$ is an unramified extension of $L(\\chi _T \\cdot \\phi _\\emptyset )$ for all $T \\subsetneq [r]$ .",
"An expansion for $X$ is a sequence $\\lbrace \\phi _T\\rbrace _{T \\subseteq [r]}$ satisfying the recursive equation (REF ) for each $T \\subseteq [r]$ .",
"An expansion is good if $L(\\phi _T)$ is an unramified extension of $L(\\chi _T \\cdot \\phi _\\emptyset )$ for all $T \\subseteq [r]$ .",
"Finally, we say that $X$ is cooperative if for each distinct $i, j \\in [r]$ , we have that the character $\\chi _{\\lbrace i\\rbrace }$ is locally trivial at 2 and at each prime $p \\in X_j$ .",
"The following result is the key result regarding expansions.",
"It is a rephrasing of [33], see also [19] for a similar statement.",
"Informally, it shows that a pre-expansion can be completed to a good expansion under favorable circumstances.",
"Proposition 4.2 Let $X := X_1 \\times \\dots \\times X_r$ with $|X_i| = 2$ for $i \\in [r]$ .",
"Suppose that $X$ is cooperative and let $\\lbrace \\phi _T\\rbrace _{T \\subsetneq [r]}$ be a promising, good pre-expansion for $X$ .",
"Then there is a continuous map $\\phi _{[r]} : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ such that $(d\\phi _{[r]})(\\sigma , \\tau ) = \\sum _{\\emptyset \\ne U \\subseteq [r]} \\chi _U(\\sigma ) \\phi _{[r] - U}(\\tau )$ with $L(\\phi _{[r]})$ an unramified extension of $L(\\chi _{[r]} \\cdot \\phi _\\emptyset )$ .",
"In Section we work with many expansions simultaneously.",
"For a box $X = X_1 \\times \\dots \\times X_r$ , $S \\subseteq [r]$ and $\\bar{x} \\in \\overline{X}_S$ , we shall use the shorthand $\\phi _{\\bar{x}, a}$ for an expansion map $\\phi _S$ associated to the box $\\prod _{i \\in S} \\pi _i(\\bar{x})$ with $\\phi _\\emptyset = \\chi _a$ .",
"Observe that $\\phi _{\\bar{x}, a}$ only depends on $a$ and the sets $\\pi _i(\\bar{x})$ with $i \\in S$ .",
"In case that $\\text{pr}_1(\\pi _i(\\bar{x})) \\ne \\text{pr}_2(\\pi _i(\\bar{x}))$ for all $i \\in S$ , we can naturally view each $\\pi _i(\\bar{x})$ as a set with two primes as required for Definition REF .",
"If instead $\\text{pr}_1(\\pi _i(\\bar{x})) = \\text{pr}_2(\\pi _i(\\bar{x}))$ for some $i \\in S$ , we set $\\phi _{\\bar{x}, a}$ to be zero."
],
[
"Governing expansions",
"Since we have to work with many expansions simultaneously in the final section, we abstract the essential properties in the notion of governing expansions.",
"Definition 4.3 Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ and let $a \\in \\mathbb {Z}$ be squarefree.",
"We say that there exists a governing expansion $\\mathfrak {G}$ on $(X, S, a)$ if we have for all $T \\subseteq S$ and all $\\bar{x} \\in \\overline{X}_T$ a good expansion $\\phi _{\\bar{x}, a}$ satisfying $(d\\phi _{\\bar{x}, a})(\\sigma , \\tau ) = \\sum _{\\emptyset \\subsetneq T^{\\prime } \\subseteq T} \\chi _{T^{\\prime }}(\\sigma ) \\phi _{\\pi _{T - T^{\\prime }}(\\bar{x}), a}(\\tau );$ take $T \\subseteq S$ , $i \\in T$ and $\\bar{x}_0, \\bar{x}_1, \\bar{x}_2 \\in \\overline{X}_T$ .",
"Suppose that $\\pi _{T - \\lbrace i\\rbrace }(\\bar{x}_0) = \\pi _{T - \\lbrace i\\rbrace }(\\bar{x}_1) = \\pi _{T - \\lbrace i\\rbrace }(\\bar{x}_2)$ and that there are primes $p_0, p_1, p_2$ satisfying $\\textup {pr}_j(\\pi _i(\\bar{x}_k)) = p_{k + j - 1}$ for all $j \\in \\lbrace 1, 2\\rbrace $ and $k \\in \\lbrace 0, 1, 2\\rbrace $ , where the indices are taken modulo 3.",
"Then we have $\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} = \\phi _{\\bar{x}_2, a}.$ These conditions are rather stringent, and typically there does not exist a governing expansion $\\mathfrak {G}$ on $(X, S, a)$ .",
"To construct governing expansions, we introduce additive systems.",
"Definition 4.4 Let $X := X_1 \\times \\dots \\times X_r$ .",
"An additive system $\\mathfrak {A}$ on $X$ is a tuple $(\\overline{Y}_S, \\overline{Y}_S^\\circ , F_S, A_S)_{S \\subseteq [r]}$ satisfying for each $S \\subseteq [r]$ , we have that $A_S$ is a finite $\\mathbb {F}_2$ vector space, $\\overline{Y}_S$ and $\\overline{Y}_S^\\circ $ are sets satisfying $\\overline{Y}_S^\\circ \\subseteq \\overline{Y}_S \\subseteq \\overline{X}_S$ and $F_S : \\overline{Y}_S \\rightarrow A_S$ is a function such that $\\overline{Y}_S^\\circ := \\lbrace \\bar{y} \\in \\overline{Y}_S : F_S(\\bar{y}) = 0\\rbrace ;$ we have for all non-empty $S \\subseteq [r]$ that $\\overline{Y}_S = \\lbrace \\bar{x} \\in \\overline{X}_S : \\bar{x}(T) \\subseteq \\overline{Y}_T^\\circ \\textup { for all } T \\subsetneq S\\rbrace .$ Here we view $\\bar{x}(T)$ as a set by forgetting the multiplicities; take $i \\in S \\subseteq [r]$ and take $\\bar{x}_0, \\bar{x}_1, \\bar{x}_2 \\in \\overline{Y}_S$ satisfying $\\pi _{S - \\lbrace i\\rbrace }(\\bar{x}_0) = \\pi _{S - \\lbrace i\\rbrace }(\\bar{x}_1) = \\pi _{S - \\lbrace i\\rbrace }(\\bar{x}_2)$ such that there are primes $p_0, p_1, p_2$ with $\\textup {pr}_j(\\pi _i(\\bar{x}_k)) = p_{k + j - 1}$ for all $j \\in \\lbrace 1, 2\\rbrace $ and all $k \\in \\lbrace 0, 1, 2\\rbrace $ , where the indices are taken modulo 3.",
"Then we have $F_S(\\bar{x}_0) + F_S(\\bar{x}_1) = F_S(\\bar{x}_2).$ We will sometimes write $\\overline{Y}_S(\\mathfrak {A})$ , $\\overline{Y}_S^\\circ (\\mathfrak {A})$ , $F_S(\\mathfrak {A})$ and $A_S(\\mathfrak {A})$ to stress that this data is associated to the additive system $\\mathfrak {A}$ .",
"We remark that equation (REF ) implies that $F_S(\\bar{x}) = 0$ in case $\\text{pr}_1(\\pi _i(\\bar{x})) = \\text{pr}_2(\\pi _i(\\bar{x}))$ .",
"We will now construct an additive system that will help us find governing expansions.",
"Lemma 4.5 Let $r \\ge 2$ be an integer and let $X := X_1 \\times \\dots \\times X_r$ be such that $X_j$ contains only odd primes.",
"Take an odd squarefree integer $a = q_1 \\cdot \\ldots \\cdot q_t$ such that $\\left(\\frac{a}{p}\\right) = 1 \\textup { and } \\left(\\frac{p}{q_i}\\right) = 1 \\textup { for all } i \\in [t]$ for all $p \\in X_j$ with $j \\in [r]$ .",
"Let $\\Omega $ be a set of places of $\\mathbb {Q}$ disjoint from the $X_j$ and $q_i$ .",
"We assume that every $v \\in \\Omega $ splits in $\\mathbb {Q}(\\sqrt{a})$ .",
"Let $W \\subseteq X$ be a subset such that for all $w_1, w_2 \\in W$ , for all distinct $i, j \\in [r]$ and for all $v \\in \\Omega $ $\\left(\\frac{\\pi _i(w_1)}{\\pi _j(w_1)}\\right) = \\left(\\frac{\\pi _i(w_2)}{\\pi _j(w_2)}\\right), \\quad \\pi _i(w_1) \\pi _i(w_2) \\equiv 1 \\bmod 8, \\quad \\pi _i(w_1) \\pi _i(w_2) \\equiv \\square \\bmod v.$ Then there exists an additive system $\\mathfrak {A}$ on $X$ such that we have $\\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A}) = W$ ; we have $|A_S(\\mathfrak {A})| \\le 2^{r - 1 + |\\Omega |}$ for all $S \\subseteq [r]$ ; suppose that $Z := Z_1 \\times \\dots \\times Z_r$ satisfies $Z_i \\subseteq X_i$ and suppose that $\\overline{Z}_{[r]} \\subseteq \\overline{Y}_{[r]}(\\mathfrak {A}).$ Then there exists a governing expansion $\\mathfrak {G}$ on $(Z, [r], a)$ such that every $v \\in \\Omega $ splits completely in $\\phi _{\\bar{z}, a}$ for $\\bar{z} \\in \\overline{Z}_{[r]}$ .",
"Note that an additive system $\\mathfrak {A}$ is uniquely specified by the maps $F_S(\\mathfrak {A})$ and the set $\\overline{Y}_\\emptyset (\\mathfrak {A})$ .",
"We take $\\overline{Y}_\\emptyset (\\mathfrak {A}) = W$ and we will inductively construct the maps $F_S(\\mathfrak {A})$ .",
"If $S = \\emptyset $ , we take $F_\\emptyset (\\mathfrak {A})$ to be the zero map.",
"Now suppose that $S = \\lbrace i\\rbrace $ .",
"For $\\bar{x} \\in \\overline{Y}_i(\\mathfrak {A})$ , Proposition REF and equation (REF ) imply that there is a good expansion $\\phi _{\\bar{x}, a}$ .",
"Define $K := \\mathbb {Q}\\left(\\left\\lbrace \\sqrt{q_1}, \\dots , \\sqrt{q_t}\\right\\rbrace \\cup \\left\\lbrace \\sqrt{p} : p \\in X_j \\text{ for some } j \\in [r]\\right\\rbrace \\right)$ and let $M$ be the narrow Hilbert class field of $K$ .",
"For every prime $p$ ramifying in $K$ , we choose once and for all an inertia subgroup $I_p$ of $\\mathrm {Gal}(M/\\mathbb {Q})$ , which has size 2, and we denote by $\\sigma _p$ the non-trivial element of $I_p$ .",
"Then by twisting $\\phi _{\\bar{x}, a}$ with characters, we can ensure that $\\phi _{\\bar{x}, a}(\\sigma _p) = 0$ for all $p$ ramifying in $K$ .",
"Observe that such a twist is then still a good expansion.",
"Furthermore, with this choice of $\\phi _{\\bar{x}, a}$ we claim that equation (REF ) holds.",
"Indeed, suppose that $\\bar{x}_0, \\bar{x}_1, \\bar{x}_2 \\in \\overline{Y}_i(\\mathfrak {A})$ satisfy the assumptions for equation (REF ).",
"Then we have $d\\left(\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} + \\phi _{\\bar{x}_2, a}\\right) = 0.$ Since $\\phi _{\\bar{x}_j, a}$ is a good expansion for $j \\in \\lbrace 0, 1, 2\\rbrace $ , this shows that $\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} + \\phi _{\\bar{x}_2, a}$ is a quadratic character with field of definition inside $\\mathrm {Gal}(M/\\mathbb {Q})$ .",
"Then equation (REF ) shows that $\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} + \\phi _{\\bar{x}_2, a}$ is the trivial character, proving the claim.",
"Then we define $F_i(\\mathfrak {A})$ by sending $\\bar{x} \\in \\overline{Y}_i(\\mathfrak {A})$ to $\\phi _{\\bar{x}, a}(\\text{Frob } k)$ as $k$ runs through $\\pi _j(\\bar{x})$ for $j \\in [r] - \\lbrace i\\rbrace $ and $\\Omega $ .",
"We observe that $\\text{Frob}(k)$ splits completely in the field $M(\\phi _{\\bar{x}, a})$ by the assumption $\\bar{x} \\in \\overline{Y}_i(\\mathfrak {A})$ and the assumptions on $W$ .",
"Then equation (REF ) follows from equation (REF ).",
"Now we proceed inductively.",
"We see that Proposition REF implies that there is a good expansion $\\phi _{\\bar{x}, a}$ for $\\bar{x} \\in \\overline{Y}_S(\\mathfrak {A})$ .",
"As before, we twist $\\phi _{\\bar{x}, a}$ to guarantee that $\\phi _{\\bar{x}, a}(\\sigma _p) = 0.$ Again we have that $d\\left(\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} + \\phi _{\\bar{x}_2, a}\\right) = 0,$ which follows from the fact that equation (REF ) holds for all the $\\phi _{\\pi _T(\\bar{x}_j), a}$ for $T \\subsetneq S$ by the induction hypothesis.",
"From this, we deduce just like before that $\\phi _{\\bar{x}_0, a} + \\phi _{\\bar{x}_1, a} + \\phi _{\\bar{x}_2, a} = 0.$ We define $F_S(\\mathfrak {A})$ by sending $\\bar{x} \\in \\overline{Y}_S(\\mathfrak {A})$ to $\\phi _{\\bar{x}, a}(\\text{Frob } k)$ as $k$ runs through $\\pi _j(\\bar{x})$ for $j \\in [r] - S$ and $\\Omega $ .",
"This defines our additive system $\\mathfrak {A}$ , which indeed satisfies the listed properties."
],
[
"Raw cocycles and minimality",
"Let $N := \\mathbb {Q}_2/\\mathbb {Z}_2$ , which we endow with the discrete topology.",
"We view $N$ as a $G_\\mathbb {Q}$ -module with trivial action.",
"For any $x \\in X$ , we let $N(x)$ be the $G_\\mathbb {Q}$ -module $N$ twisted with the action of $\\mathbb {Q}(\\sqrt{x})$ , i.e.",
"$\\sigma \\cdot _x n = n$ if $\\chi _x(\\sigma ) = 0$ and $\\sigma \\cdot _x n = -n$ if $\\chi _x(\\sigma ) = 1$ .",
"Definition 4.6 We define $\\textup {Cocy}(G_\\mathbb {Q}, N(x)[2^k])$ to be the set of continuous 1-cocycles $\\psi $ such that $\\mathbb {Q}(\\sqrt{x}) L(\\psi )/\\mathbb {Q}(\\sqrt{x})$ is unramified.",
"Remark 2 If $k > 1$ , then $L(\\psi )$ automatically contains $\\mathbb {Q}(\\sqrt{x})$ .",
"However, this need not be the case if $k = 1$ .",
"By definition of $\\textup {Cocy}(G_\\mathbb {Q}, N(x)[2^k])$ , restriction of cocycles induces an isomorphism $\\text{Cocy}(G_\\mathbb {Q}, N(x)[2^k]) \\cong \\text{Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}), N(x)[2^k]).$ Of fundamental importance is the split exact sequence $0 \\rightarrow N(x)[2^k] \\rightarrow \\textup {Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}), N(x)[2^k]) \\rightarrow \\text{Cl}^\\vee (\\mathbb {Q}(\\sqrt{x}))[2^k] \\rightarrow 0.$ Fix an element $\\sigma \\in \\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q})$ projecting non-trivially in $\\text{Gal}(\\mathbb {Q}(\\sqrt{x})/\\mathbb {Q})$ .",
"The first map is given by sending $n$ to the unique cocycle that sends $\\sigma $ to $n$ and sends the group $\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}(\\sqrt{x}))$ to zero, while the second map is simply restriction of cocycles.",
"The exact sequence is split, since all the groups appearing are killed by $2^k$ and $N(x)[2^k] \\cong \\mathbb {Z}/2^k\\mathbb {Z}$ as abelian groups.",
"This allows us to work with cocycles instead of the class group.",
"If $x$ is a squarefree integer, we have a natural map $f: \\lbrace d \\text{ squarefree} : d \\mid \\Delta _{\\mathbb {Q}(\\sqrt{x})}\\rbrace \\rightarrow \\text{Cl}(\\mathbb {Q}(\\sqrt{x}))[2],$ where $\\Delta _K$ denotes the discriminant of $K$ .",
"We now define the $m$ -th Artin pairing $\\text{Art}_{m, x} : f^{-1}(2^{m - 1} \\text{Cl}(\\mathbb {Q}(\\sqrt{x}))[2^m]) \\times 2^{m - 1} \\text{Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}), N(x)[2^m]) \\rightarrow \\mathbb {F}_2$ by sending $(b, \\chi )$ to $\\psi (\\text{Frob } \\mathfrak {b})$ , where $\\mathfrak {b}$ is the unique ideal of norm $b$ and furthermore $\\psi \\in \\text{Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}), N(x)[2^m])$ is any lift of $\\chi $ satisfying $2^{m - 1} \\psi = \\chi $ .",
"The left kernel of this pairing is $f^{-1}(2^m \\text{Cl}(\\mathbb {Q}(\\sqrt{x}))[2^{m + 1}])$ and the right kernel of this pairing is $2^m \\text{Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x}))/\\mathbb {Q}), N(x)[2^{m + 1}])$ .",
"We introduce the notion of a raw cocycle and minimality.",
"Definition 4.7 A raw cocycle for $x$ is a sequence $\\lbrace \\psi _i\\rbrace _{i = 0}^k$ satisfying $\\psi _i \\in \\textup {Cocy}(G_\\mathbb {Q}, N(x)[2^k])$ and $2\\psi _i = \\psi _{i - 1}$ for all $1 \\le i \\le k$ .",
"We call $k$ the order of the raw cocycle.",
"Definition 4.8 Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ and let $\\bar{x} \\in \\overline{X}_S$ .",
"Suppose that we are given for each $x \\in \\bar{x}(\\emptyset )$ a raw cocycle $\\lbrace \\psi _{i, x}\\rbrace _{i = 0}^{|S|}$ .",
"We say that the set of raw cocycles is minimal at $\\bar{x}$ if $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{|T|, y} = 0$ for all $T \\subseteq S$ and all $\\bar{y} \\in \\bar{x}(T)$ .",
"Note that the sum here has to be taken with multiplicities.",
"We have all the necessary notation to state our first reflection principle, which is directly based on part (i) of Theorem 2.8 of Smith [33].",
"Theorem 4.9 Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ with $|S| \\ge 2$ and let $\\bar{x} \\in \\overline{X}_S$ .",
"Take $x_0 \\in \\bar{x}(\\emptyset )$ .",
"Let $\\lbrace \\psi _{i, x}\\rbrace _{i = 0}^{|S|}$ be a raw cocycle for all $x$ in $\\bar{x}(\\emptyset )$ except $x_0$ .",
"We assume that for all $T \\subseteq S$ and all $\\bar{y} \\in \\bar{x}(T)$ not containing $x_0$ , we have that $\\lbrace \\psi _{i, y}\\rbrace _{i = 0}^{|T|}$ is minimal at $\\bar{y}$ .",
"Then there is a raw cocycle $\\lbrace \\psi _{i, x_0}\\rbrace _{i = 0}^{|S|}$ such that $\\psi _{1, x_0} = \\psi _{1, x}$ for all $x \\in \\bar{x}(\\emptyset )$ .",
"Additionally, suppose that there is an integer $b$ such that $b \\in f^{-1}(2^{m - 1} \\textup {Cl}(\\mathbb {Q}(\\sqrt{x}))[2^m])$ for all $x \\in \\bar{x}(\\emptyset )$ .",
"Also suppose that for every $i \\in S$ we have that $\\textup {pr}_1(\\pi _i(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _i(\\bar{x}))$ is a square locally at 2 and at all primes in $\\pi _j(\\bar{x})$ with $i \\ne j$ .",
"Then we also have $\\sum _{x \\in \\bar{x}(\\emptyset )} \\textup {Art}_{|S|, x}(b, \\psi _{1, x}) = 0.$ We note that minimality implies that $\\psi _{1, x} = \\psi _{1, x^{\\prime }}$ for all $x, x^{\\prime } \\in \\bar{x}(\\emptyset )$ not equal to $x_0$ .",
"For the first part, we put $\\psi _{|S|, x_0} := - \\sum _{\\begin{array}{c}x \\in \\bar{x}(\\emptyset ) \\\\ x \\ne x_0\\end{array}} \\psi _{|S|, x}.$ Then it is easily seen that $2^{|S| - 1} \\psi _{|S|, x_0} = \\psi _{1, x}$ for any $x \\in \\bar{x}(\\emptyset )$ .",
"Smith [33] shows that $\\psi _{|S|, x_0}$ is a cocycle from $G_\\mathbb {Q}$ to $N(x_0)[2^k]$ .",
"It remains to deal with the ramification locus of $L(\\psi _{|S|, x_0})$ .",
"We first claim that $2\\psi _{|S|, x_0} \\in \\text{Cocy}(\\text{Gal}(H(\\mathbb {Q}(\\sqrt{x_0}))/\\mathbb {Q}), N(x_0)[2^{|S| - 1}])$ .",
"But observe that the minimality assumptions and equation (REF ) imply that $2\\psi _{|S|, x_0} = - \\sum _{\\begin{array}{c}x \\in \\bar{x}(\\emptyset ) - \\bar{y}(\\emptyset ) \\\\ x \\ne x_0\\end{array}} \\psi _{|S| - 1, x}$ for any $\\bar{y} \\in \\bar{x}(T)$ not containing $x_0$ such that $|T| = |S| - 1$ .",
"From this we immediately deduce the claim, since for any prime $p$ not dividing $\\Delta _{\\mathbb {Q}(\\sqrt{x_0})}$ we can take $\\bar{y}$ such that $p$ is unramified in every $L(\\psi _{|S| - 1, x})$ for $x \\in \\bar{x}(\\emptyset ) - \\bar{y}(\\emptyset )$ .",
"Since $L(\\psi _{|S|, x_0})/L(2\\psi _{|S|, x_0})$ is a central $\\mathbb {F}_2$ -extension, we see that $L(\\psi _{|S|, x_0})/L(2\\psi _{|S|, x_0})$ can be made unramified over $\\mathbb {Q}$ at all primes, except those that ramify in $L(2\\psi _{|S|, x_0})$ , by twisting with a character $\\chi $ , see for example [19].",
"But from the shape of $\\psi _{|S|, x_0}$ , it follows that the primes that already ramify in $L(2\\psi _{|S|, x_0})$ can not ramify further in $L(\\psi _{|S|, x_0})/L(2\\psi _{|S|, x_0})$ .",
"Indeed, from equation (REF ) it follows that the ramification degree of any prime $p$ in $L(\\psi _{|S|, x_0})$ is at most 2.",
"Therefore $L(\\psi _{|S|, x_0} + \\chi )$ is an unramified extension of $\\mathbb {Q}(\\sqrt{x_0})$ for some character $\\chi $ , proving the first part.",
"For the second part, recall that $\\text{Art}_{|S|, x}(b, \\psi _{1, x})$ does not depend on the choice of the lift $\\psi $ with $2^{|S| - 1} \\psi = \\psi _{1, x}$ , and hence we may choose the lift $\\psi _{|S|, x_0} + \\chi $ for $\\psi _{1, x_0}$ .",
"By definition we have that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\text{Art}_{|S|, x}(b, \\psi _{1, x}) = \\sum _{x \\in \\bar{x}(\\emptyset )} \\psi _{|S|, x}(\\text{Frob } \\mathfrak {b}).$ Now locally at any prime $p$ dividing $b$ , we know that $\\mathbb {Q}_p(\\sqrt{x}) = \\mathbb {Q}_p(\\sqrt{x^{\\prime }})$ for all $x, x^{\\prime } \\in \\overline{x}(\\emptyset )$ by our assumptions.",
"Since $\\psi _{|S|, x}$ becomes a character when restricted to $\\mathbb {Q}_p(\\sqrt{x})$ , the relation $\\sum _{x \\in \\bar{x}(\\emptyset )} \\psi _{|S|, x} = \\chi $ yields $\\sum _{x \\in \\bar{x}(\\emptyset )} \\psi _{|S|, x}(\\text{Frob } \\mathfrak {b}) = \\chi (\\text{Frob } \\mathfrak {b}).$ We claim that the last expression is trivial.",
"Indeed $\\mathfrak {b}$ is in $2\\text{Cl}(\\mathbb {Q}(\\sqrt{x}))[4]$ for all $x \\in \\bar{x}(\\emptyset )$ , and therefore pairs trivially with any character that is unramified outside the union of the primes dividing $\\Delta _{\\mathbb {Q}(\\sqrt{x})}$ as $x$ ranges through $\\bar{x}(\\emptyset )$ .",
"But equation (REF ) shows that the character $\\chi $ is of this shape, concluding the proof of our theorem."
],
[
"Agreement",
"Our second reflection principle is based on the notion of agreement.",
"Definition 4.10 Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ , let $i_a \\in S$ and let $\\bar{x} \\in \\overline{X}_S$ be given.",
"Take for each $x \\in \\bar{x}(\\emptyset )$ a raw cocycle $\\lbrace \\psi _{i, x}\\rbrace _{i = 0}^{|S|}$ .",
"We further assume that we have an expansion $\\phi _{\\pi _{S - \\lbrace i_a\\rbrace }(\\bar{x}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))}$ .",
"We say that the set of raw cocycles agrees with the expansion at $\\bar{x}$ if $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{|T|, y}=\\left\\lbrace \\begin{array}{ll}\\phi _{\\pi _{T - \\lbrace i_a\\rbrace }(\\bar{y}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))} & \\mbox{if } i_a \\in T \\\\0 & \\mbox{if } i_a \\notin T\\end{array}\\right.$ for all $T \\subseteq S$ and all $\\bar{y} \\in \\bar{x}(T)$ .",
"We are now ready to state a reflection principle that is very similar to part (ii) of [33].",
"Theorem 4.11 Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ with $|S| \\ge 2$ , let $i_a \\in S$ and let $\\bar{x} \\in \\overline{X}_S$ be given.",
"Take $x_0 \\in \\bar{x}(\\emptyset )$ .",
"Let $\\lbrace \\psi _{i, x}\\rbrace _{i = 0}^{|S|}$ be a raw cocycle for all $x$ in $\\bar{x}(\\emptyset )$ except $x_0$ such that there is a character $\\chi $ with the property $\\psi _{1, x} = \\chi + \\chi _{\\pi _{i_a}(x)} \\textup { for all } x.$ Assume that there is a good expansion $\\phi _{\\pi _{S - \\lbrace i_a\\rbrace }(\\bar{x}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))}$ .",
"We further assume that for all $T \\subseteq S$ and all $\\bar{y} \\in \\bar{x}(T)$ not containing $x_0$ , we have that $\\lbrace \\psi _{i, y}\\rbrace _{i = 0}^{|T|}$ agrees with the expansion at $\\bar{y}$ .",
"Then there is a raw cocycle $\\lbrace \\psi _{i, x_0}\\rbrace _{i = 0}^{|S|}$ such that $\\psi _{1, x_0} = \\psi _{1, x}$ for all $x \\in \\bar{x}(\\emptyset )$ with $\\pi _{i_a}(x_0) = \\pi _{i_a}(x)$ .",
"Further suppose that for every $i \\in S$ , $\\textup {pr}_1(\\pi _i(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _i(\\bar{x}))$ is a square locally at 2 and at all primes in $\\pi _j(\\bar{x})$ with $i \\ne j$ .",
"Moreover, assume that there exists an integer $b$ satisfying $b \\in f^{-1}(2^{m - 1} \\textup {Cl}(\\mathbb {Q}(\\sqrt{x}))[2^m])$ for all $x \\in \\bar{x}(\\emptyset )$ .",
"Then we have $\\sum _{x \\in \\bar{x}(\\emptyset )} \\textup {Art}_{|S|, x}(b, \\psi _{1, x}) = \\sum _{p \\mid b} \\phi _{\\pi _{S - \\lbrace i_a\\rbrace }(\\bar{x}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))}(\\textup {Frob } p).$ If instead $x/b \\in f^{-1}(2^{m - 1} \\textup {Cl}(\\mathbb {Q}(\\sqrt{x}))[2^m])$ for all $x \\in \\bar{x}(\\emptyset )$ $\\sum _{x \\in \\bar{x}(\\emptyset )} \\textup {Art}_{|S|, x}(x/b, \\psi _{1, x}) = \\sum _{p \\mid b \\infty } \\phi _{\\pi _{S - \\lbrace i_a\\rbrace }(\\bar{x}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))}(\\textup {Frob } p).$ This can be proven in the same way as Theorem REF .",
"Remark 3 Write $\\phi = \\phi _{\\pi _{S - \\lbrace i_a\\rbrace }(\\bar{x}), \\textup {pr}_1(\\pi _{i_a}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_a}(\\bar{x}))}$ .",
"The agreement assumptions imply that $p$ splits completely in $M(\\phi )$ , so that $\\textup {Frob}(p)$ lands in $\\mathrm {Gal}(L(\\phi )/M(\\phi )) \\cong \\mathbb {F}_2$ ."
],
[
"Analytic prerequisites",
"Throughout the paper our implied constants may depend on $l$ .",
"We shall not record this dependence.",
"The material in this section is rather similar to [33] and [33], but there is one major hurdle to overcome.",
"Indeed, Smith does not prove the analogue of Corollary 6.11 for class groups.",
"To do so, one needs to make the Markov chain analysis of Gerth effective.",
"Another significant complicating factor is that the 4-rank distribution in our family is different due to the fact that $N_d(x, y) = l$ is soluble over $\\mathbb {Q}$ by assumption.",
"This requires some changes to be made to the Markov chains appearing in Gerth [14].",
"These problems are dealt with in our companion paper [21]."
],
[
"Combinatorial results",
"In Section we will make essential use of the following two combinatorial results first proven in Smith [33] with slightly different notation.",
"Let $X := X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ and let $Z \\subseteq X$ with $|\\pi _{[r] - S}(Z)| = 1$ .",
"We define the $\\mathbb {F}_2$ -vector spaces $V := \\text{Map}(Z, \\mathbb {F}_2), \\quad W := \\text{Map}(\\text{Cube}_S(Z), \\mathbb {F}_2),$ where $\\text{Cube}_S(Z)$ is the set of $\\bar{x} \\in \\overline{X}_S$ such that $\\bar{x}(\\emptyset ) \\subseteq Z$ .",
"We define a linear map $d : V \\rightarrow W$ , not to be confused with the map $d$ on 1-cochains, by $dF(\\bar{x}) = \\sum _{x \\in \\bar{x}(\\emptyset )} F(x),$ where the sum has to be taken with multiplicities.",
"This has the effect that $dF(\\bar{x}) = 0$ as soon as there exists some $i \\in S$ with $\\text{pr}_1(\\pi _i(\\bar{x})) = \\text{pr}_2(\\pi _i(\\bar{x}))$ just like in Smith [33].",
"Define ${G}_S(Z)$ to be the image of $d$ .",
"Lemma 5.1 Let $X = X_1 \\times \\dots \\times X_r$ , let $S \\subseteq [r]$ and suppose that $|X_i| = 1$ for $i \\in [r] - S$ .",
"We have that $\\dim _{\\mathbb {F}_2} {G}_S(X) = \\prod _{i \\in S} (|X_i| - 1).$ See [19].",
"Recall the definition of an additive system given in Definition REF .",
"Given an additive system $\\mathfrak {A}$ , we set $C(\\mathfrak {A}) := \\bigcap _{i \\in S} \\left\\lbrace \\bar{x} \\in \\overline{X}_S : \\bar{x}(S - \\lbrace i\\rbrace ) \\cap \\overline{Y}_{S - \\lbrace i\\rbrace }^\\circ (\\mathfrak {A}) \\ne \\emptyset \\right\\rbrace .$ We call an additive system $\\mathfrak {A}$ on $X$ $(a, S)$ -acceptable if $|A_T(\\mathfrak {A})| \\le a$ for all subsets $T$ of $S$ ; $\\bar{x} \\in C(\\mathfrak {A})$ implies $\\bar{x}(\\emptyset ) \\subseteq \\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})$ .",
"Proposition 5.2 There exists an absolute constant $A > 0$ such that the following holds.",
"Let $r > 0$ be an integer, let $X_1, \\dots , X_r$ be finite non-empty sets and let $X$ be their product.",
"Take $S \\subseteq [r]$ with $|S| \\ge 2$ , $|\\pi _{[r] - S}(X)| = 1$ and put $n := \\textup {min}_{i \\in S} |X_i|$ .",
"Let $a \\ge 2$ and $\\epsilon > 0$ be given.",
"Assume that $\\epsilon < a^{-1}$ and $\\log n \\ge A \\cdot 6^{|S|} \\cdot \\log \\epsilon ^{-1}.$ Then there exists $g \\in {G}_S(X)$ such that for all $(a, S)$ -acceptable additive systems $\\mathfrak {A}$ on $X$ and for all $F: \\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A}) \\rightarrow \\mathbb {F}_2$ satisfying $d F(\\bar{x}) = g(\\bar{x})$ for all $\\bar{x} \\in C(\\mathfrak {A})$ , we have $\\frac{|\\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})|}{2} - |X| \\cdot \\epsilon \\le |F^{-1}(0)| \\le \\frac{|\\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})|}{2} + |X| \\cdot \\epsilon .$ This is Proposition 4.4 in Smith [33] and reproven in a slightly more general setting in Proposition 8.7 of [19]."
],
[
"Prime divisors",
"Let $l$ be an integer such that $|l|$ is prime and congruent to 3 modulo 4 or let $l = -1$ .",
"In Theorem REF we are only interested in those squarefree integers $d > 0$ with the properties $l \\mid d, \\quad p \\mid \\frac{d}{l} &\\text{ implies } \\left(\\frac{l}{p}\\right) = 1 \\text{ or } p = 2 \\\\&\\left(\\frac{-d/l}{|l|}\\right) = 1.$ Indeed, this is equivalent to $l \\mid d$ and the solubility of the equation $x^2 - dy^2 = l \\text{ in } x, y \\in \\mathbb {Q}.$ We remark that equation () is equivalent to a set of congruence conditions for $d$ modulo 8.",
"Hence we need to insert congruence conditions in Section 5 of Smith [33].",
"This has already been done in Section 10 of [19] for squarefree integers $d$ such that $p \\mid d$ implies $p \\equiv 0,1 \\bmod l$ , and in Section 4 of [2] with completely different techniques for squarefree integers $d$ such that $p \\mid d$ implies $p \\equiv 1, 2 \\bmod 4$ .",
"Both these techniques are straightforward to generalize to obtain the following results, here we shall follow [33].",
"Define $S(N, l) := \\lbrace 1 \\le d < N : d \\text{ squarefree and satisfies equation (\\ref {ed}) and (\\ref {ed2})}\\rbrace $ and $S_r(N, l) := \\lbrace d \\in S(N, l) : \\omega (d) = r\\rbrace .$ We list the distinct prime divisors of $d$ as $p_1 < p_2 < \\dots < p_r$ .",
"With these notations we can state our next theorem.",
"Theorem 5.3 Fix an integer $l$ such that $|l|$ is prime and congruent to 3 modulo 4 or $l = -1$ .",
"Let $N$ be a real number and put $\\mu := \\frac{1}{2} \\log \\log N$ .",
"Then there are absolute constants $A_1, A_2 > 0$ such that $\\frac{A_1N}{\\log N} \\cdot \\frac{\\mu ^{r - 1}}{(r - 1)!}",
"\\le |S_r(N, l)| \\le \\frac{A_2N}{\\log N} \\cdot \\frac{\\mu ^{r - 1}}{(r - 1)!",
"}$ for all $1 \\le r \\le 200\\mu $ and $\\frac{|\\lbrace d \\in S(N, l) : |\\omega (d) - \\mu | > \\mu ^{2/3}\\rbrace |}{|S(N, l)|} \\ll \\exp \\left(-\\frac{1}{3}\\mu ^{1/3}\\right).$ Now assume that $r$ is such that $|r - \\mu | \\le \\mu ^{2/3}$ and take $D_1 > 3$ and $C_0 > 1$ .",
"In this case we have the bound $1 - \\frac{|\\lbrace d \\in S_r(N, l) : 2D_1 < p_i < p_{i + 1}/2 \\textup { for all } p_i > D_1\\rbrace |}{|S_r(N, l)|} \\ll \\frac{1}{\\log D_1} + \\frac{1}{(\\log N)^{1/4}};$ the bound $1 - \\frac{\\left|\\left\\lbrace d \\in S_r(N, l) : \\left|\\frac{1}{2} \\log \\log p_i - i\\right| < C_0^{1/5} \\max (i, C_0)^{4/5} \\textup { for all } i < \\frac{1}{3}r\\right\\rbrace \\right|}{|S_r(N, l)|} \\ll \\exp (-kC_0)$ for some absolute constant $k$ ; the bound $\\frac{\\left|\\left\\lbrace d \\in S_r(N, l) : \\frac{\\log p_i}{\\log \\log p_i} \\le (\\log \\log \\log N)^{1/2} \\cdot \\sum _{j = 1}^{i - 1} \\log p_j \\textup { for all } \\frac{1}{2}r^{1/2} < i < \\frac{1}{2}r\\right\\rbrace \\right|}{|S_r(N, l)|} \\\\\\ll \\exp \\left(-(\\log \\log \\log N)^{1/4}\\right).$ Condition (REF ) is incorporated in Smith's argument just as in [19] by inserting a congruence condition in the definition of $F(x)$ [33].",
"To deal with the congruence condition (), we simply impose further congruence conditions on the primes for each invertible residue class in $(\\mathbb {Z}/8\\mathbb {Z})^\\ast $ and then sum up the contributions.",
"More explicitly, we define for a congruence class $a \\in (\\mathbb {Z}/8\\mathbb {Z})^\\ast $ the sum $F_a(x) = \\sum _{\\begin{array}{c}p \\le x \\\\ (l/p) = 1 \\\\ p \\equiv a_i \\bmod 8\\end{array}} \\frac{1}{p}$ so that there exist constants $B_a$ , $A$ , $c > 0$ such that $\\left|F_a(x) - \\frac{\\log \\log x}{8} - B_a\\right| \\le A \\cdot e^{-c \\sqrt{\\log x}}.$ Let $(a_1, \\dots , a_r) \\in {(\\mathbb {Z}/8\\mathbb {Z})^\\ast }^r$ be a vector.",
"Then for $T$ any set of tuples of primes $(p_1, \\dots , p_r)$ of length $r$ with $p_i \\equiv a_i \\bmod 8$ and $(l/p) = 1$ , we define the grid $\\text{Grid}(T) = \\bigcup _{(p_1, \\dots , p_r) \\in T} \\prod _{1 \\le i \\le r} \\left[8 \\cdot \\left(F_{a_i}(p_i) - \\frac{1}{p_i} - B_{a_i}\\right), 8 \\cdot \\left(F_{a_i}(p_i) - B_{a_i}\\right)\\right].$ Following the proof of Smith [33], we compare the quantity $\\sum _{\\begin{array}{c}p_1 \\cdot \\ldots \\cdot p_r < N \\\\ (l/p) = 1 \\\\ p_i \\equiv a_i \\bmod 8\\end{array}} \\frac{8^r}{p_1 \\cdot \\ldots \\cdot p_r}$ against the integral $I_r$ for each vector $(a_1, \\dots , a_r) \\in {(\\mathbb {Z}/8\\mathbb {Z})^\\ast }^r$ .",
"Then equation (REF ) follows from a version of [33] for the set of squarefree integers $d$ satisfying equation (REF ) and the condition $p_i \\equiv a_i \\bmod 8$ after summing over all possible vectors $(a_1, \\dots , a_r) \\in {(\\mathbb {Z}/8\\mathbb {Z})^\\ast }^r$ such that $x^2 - a_1 \\cdot \\ldots \\cdot a_ry^2 = l$ is soluble in $\\mathbb {Q}_2$ .",
"The assertion (REF ) is deduced from equation (REF ), from standard bounds on the tails of the Poisson distribution and from a good bound for the number of integers with more than $100 \\log \\log N$ prime divisors.",
"Such a bound follows immediately when one computes the average of $\\tau (n)$ .",
"The claims (i), (ii) and (iii) are a straightforward generalization of the material in Section 5 of Smith [33]."
],
[
"Graph theory",
"In our final section we shall use some results in graph theory that we prove here.",
"This subsection may be skipped on first reading.",
"Definition 5.4 We say that an undirected graph $G$ on $n$ vertices is $(\\kappa , m)$ -bad if there is an induced subgraph with at least $\\kappa $ vertices and no clique of size $m$ .",
"Lemma 5.5 Let $\\kappa $ be an integer with $\\kappa \\ge n / \\log n$ and let $m$ be an integer satisfying $3 \\le m \\le (\\log \\kappa )^{1/4}.$ Then there is an absolute constant $C$ such that for all $n > C$ there are no more than $2^{\\binom{n}{2}} \\cdot 2^{- \\frac{1}{4m}\\binom{\\kappa }{2}}$ $(\\kappa , m)$ -bad graphs $G$ .",
"Theorem 1 of [26] states that under our conditions the number of graphs on $n$ vertices without a clique of size $m$ is at most $2^{\\left(1 - \\frac{1}{m - 1}\\right) \\cdot \\binom{n}{2} + o(n^2/m)}.$ Hence the number of bad graphs $G$ is bounded by $2^n \\cdot 2^{\\binom{n}{2} - \\binom{\\kappa }{2}} \\cdot 2^{\\left(1 - \\frac{1}{m - 1}\\right) \\binom{\\kappa }{2} + o(\\kappa ^2/m)}.$ For sufficiently large $n$ , we have that $2^n \\cdot 2^{\\left(- \\frac{1}{m - 1}\\right) \\binom{\\kappa }{2} + o(\\kappa ^2/m)} \\le 2^{- \\frac{1}{4m}\\binom{\\kappa }{2}}$ and this proves the lemma.",
"Lemma 5.6 Let $V$ , $W$ be sets and let $m$ be an integer.",
"There are at most $\\binom{|V|}{m} \\cdot \\left(1 - \\frac{1}{2^m}\\right)^{|W|} \\cdot 2^{|V| \\cdot |W|}$ functions $f: V \\times W \\rightarrow \\mathbb {F}_2$ with the property that there are pairwise distinct $v_1, \\ldots , v_m \\in V$ such that for all $w \\in W$ there is some $i \\in [m]$ with $f(v_i, w) = 0$ .",
"We start by fixing a subset $\\lbrace v_1, \\ldots , v_m\\rbrace \\subseteq V$ of cardinality $m$ and some $w \\in W$ .",
"Then the probability that $f(v_i, w) = 0$ for some $i \\in [m]$ is $1 - \\frac{1}{2^m}.$ This probability is independent as we vary $w$ , hence the probability that for all $w \\in W$ there is some $i \\in [m]$ with $f(v_i, w) = 0$ is given by $\\left(1 - \\frac{1}{2^m}\\right)^{|W|}.$ Since there are at most $\\binom{|V|}{m}$ subsets of cardinality $m$ , the probability, that there are pairwise distinct $v_1, \\ldots , v_m \\in V$ such that for all $w \\in W$ there is some $i \\in [m]$ with $f(v_i, w) = 0$ , is at most $\\binom{|V|}{m} \\cdot \\left(1 - \\frac{1}{2^m}\\right)^{|W|}.$ This implies the lemma."
],
[
"Equidistribution of Legendre symbol matrices",
"In this subsection we state several equidistribution results pertaining to matrices of Legendre symbols.",
"These results are straightforward modifications of the material in [2], [33].",
"We start with two definitions.",
"Definition 5.7 Suppose that $l$ is a non-zero integer.",
"A prebox is a pair $(X, P)$ satisfying $P$ consists entirely of prime numbers such that the images of $P$ , $l$ and $-1$ are linearly independent in $\\frac{\\mathbb {Q}^\\ast }{\\mathbb {Q}^{\\ast 2}}$ ; $X = X_1 \\times \\dots \\times X_r$ , where each $X_i$ consists entirely of prime numbers with $X_i \\cap P = \\emptyset $ ; there exists a sequence of real numbers $0 < s_1 < t_1 < s_2 < t_2 < \\dots < s_r < t_r$ such that every prime $p \\in X_i$ satisfies $s_i < p < t_i$ and $(l/p) = 1$ .",
"Define the (potentially infinite) sequence $d_1, d_2, \\ldots $ as in Definition 6.2 of Smith [33].",
"Then we have $d_i^2 < |d_{i + 1}|$ .",
"We say that $(X, P)$ is Siegel-less above $t$ if for all $x \\in X$ we have that $d_i \\mid lx \\prod _{p \\in P} p$ implies $|d_i| < t$ .",
"Definition 5.8 Write $A \\sqcup B$ for the disjoint union of two sets $A$ and $B$ .",
"Let $(X, P)$ be a prebox.",
"Put $M := \\lbrace (i, j) : 1 \\le i < j \\le r\\rbrace , \\quad M_P := [r] \\times (P \\sqcup \\lbrace -1\\rbrace ).$ Let $\\mathcal {M} \\subseteq M$ and let $\\mathcal {N} \\subseteq M_P$ .",
"Given a map $a : \\mathcal {M} \\sqcup \\mathcal {N} \\rightarrow \\lbrace \\pm 1\\rbrace $ , we define $X(a)$ to be the subset of tuples $(x_1, \\ldots , x_r) \\in X$ with $\\left(\\frac{x_i}{x_j}\\right) = a(i, j) \\textup { for all } (i, j) \\in \\mathcal {M}, \\quad \\left(\\frac{p}{x_i}\\right) = a(i, p) \\textup { for all } (i, p) \\in \\mathcal {N}.$ Ideally we would like to show that every $X(a)$ is of the expected size.",
"Although we are not able to prove this in full generality, we will prove slightly weaker results that still suffice for our application.",
"If $S \\subseteq [r]$ , $Q \\in \\prod _{i \\in S} X_i$ and $j \\in S$ , we write $X_j(a, Q) := \\left\\lbrace x_j \\in X_j : \\left(\\frac{\\pi _i(Q)}{x_j}\\right) = a(i, j) \\textup { for } i \\in S \\textup { and } \\left(\\frac{p}{x_j}\\right) = a(j, p) \\textup { for } p \\in P\\right\\rbrace .$ Here we use the convention that for $i > j$ $a(i, j) := a(j ,i) \\cdot (-1)^{\\frac{a(i, -1) - 1}{2} \\cdot \\frac{a(j, -1) - 1}{2}}.$ We also define $X(a, Q)$ to be the subset of $x \\in X(a)$ with $\\pi _S(x) = Q$ .",
"Proposition 5.9 Let $l$ be a non-zero integer.",
"For every choice of positive constants $c_1, \\dots , c_8$ satisfying $c_3 > 1$ , $c_5 > 3$ and $\\frac{1}{8} > c_8 + \\frac{c_7 \\log 2}{2} + \\frac{1}{c_1} + \\frac{c_2c_4}{2},$ there exists a constant $A$ such that the following holds.",
"Let $A < t < s_1$ and suppose that $(X, P)$ is a prebox that is Siegel-less above $t$ .",
"Let $\\mathcal {M} \\subseteq M$ and let $\\mathcal {N} \\subseteq M_P$ .",
"Let $1 \\le k \\le r$ be an integer such that $(i, p) \\in \\mathcal {M}$ implies $i > k$ .",
"Furthermore, if $i > k$ , we assume that $X_i$ equals the set of primes in $s_i < q < t_i$ satisfying $\\left(\\frac{l}{q}\\right) = 1 \\textup { and } \\left(\\frac{p}{q}\\right) = a(i, p) \\textup { for all } (i, p) \\in \\mathcal {N}.$ Finally assume that (i) $p \\in P$ implies $p < s_1$ and $|P| \\le \\log t_i - i$ for all $1 \\le i \\le r$ ; (ii) $\\log t_k < t_1^{c_2}$ and if $k < r$ , we assume that $\\log t_{k + 1} > \\max ((\\log t_1)^{c_5}, t^{c_6})$ ; (iii) we assume that for all $1 \\le i \\le r$ $|X_i| \\ge \\frac{2^{c_3i} \\cdot t_i}{(\\log t_i)^{c_4}};$ (iv) $r^{c_1} < t_1$ ; (v) putting $j_i := i - 1 + \\lfloor c_7 \\log t_i \\rfloor $ , we assume that $j_1 > k$ .",
"Furthermore, $j_i \\le r$ implies $(\\log t_i)^{c_5} < \\log t_{j_i}.$ Then we have for all $a : \\mathcal {M} \\sqcup \\mathcal {N} \\rightarrow \\lbrace \\pm 1\\rbrace $ $\\left||X(a)| - \\frac{|X|}{2^{|\\mathcal {M}|}}\\right| \\le t_1^{-c_8} \\cdot \\frac{|X|}{2^{|\\mathcal {M}|}}.$ Now additionally assume that $\\mathcal {M} = M$ and $\\mathcal {N}$ contains $(i, p)$ for every $i > k$ and every $p \\in P \\sqcup \\lbrace -1\\rbrace $ .",
"Furthermore, suppose that $U, V \\subseteq [r]$ are disjoint subsets such that $U \\cup V = [r^{\\prime }]$ and suppose that $u \\in U$ implies $u > k$ and $\\log \\log s_u > \\frac{1}{5} \\max (r, \\log \\log t_r).$ We say that $Q \\in \\pi _V(X)$ is poor if there is some $u \\in U$ such that $\\left||X_u(a, Q)| - \\frac{|X_u|}{2^{|V|}}\\right| > \\frac{|V| \\cdot |X_u|}{t_1^{\\frac{1}{c_1} + c_8}}.$ Then we also have $\\sum _{Q \\in \\pi _V(X) \\textup { poor}} |X(a, Q)| \\le \\frac{r \\cdot |X|}{t_1^{c_8} \\cdot 2^{|\\mathcal {M}|}}.$ This is a straightforward generalization of [33] and [2].",
"Our next proposition deals with the small primes but at the cost of introducing permutations.",
"We define $\\mathcal {P}(k_2)$ to be the set of permutations $\\sigma : [r] \\rightarrow [r]$ that fix every $k_2 < i \\le r$ .",
"Furthermore, if $a: M \\sqcup M_P \\rightarrow \\lbrace \\pm 1\\rbrace $ , we define $\\sigma (a)$ to be $\\sigma (a)(i, j) = a(\\sigma (i), \\sigma (j)), \\quad \\sigma (a)(i, p) = a(\\sigma (i), p).$ Proposition 5.10 Let $l$ be a non-zero integer.",
"For every choice of positive constants $c_1, \\dots , c_{12}$ , satisfying $c_3 > 1$ , $c_5 > 3$ and $\\frac{1}{8} > c_8 + \\frac{c_7 \\log 2}{2} + \\frac{1}{c_1} + \\frac{c_2c_4}{2}, \\quad c_{10} \\log 2 + 2c_{11} + c_{12} < 1 \\quad \\textup {and} \\quad c_{12} + c_{11} < c_9,$ there exists a constant $A$ such that the following holds.",
"Let $A < t$ and suppose that $(X, P)$ is a prebox that is Siegel-less above $t$ such that $X_i$ equals the set of primes $p$ in the interval $(s_i, t_i)$ satisfying $(l/p) = 1$ .",
"Let $k_0, k_1, k_2$ be integers such that $0 \\le k_0 < k_1 < k_2 \\le r$ .",
"We assume that (i) $p \\in P$ implies $p < s_{k_0 + 1}$ and $|P| \\le \\log t_i - i$ for all $i > k_0$ ; (ii) $\\log t_{k_1} < t_{k_0 + 1}^{c_2}$ and $\\log t_{k_1 + 1} > \\max ((\\log t_{k_0 + 1})^{c_5}, t^{c_6})$ ; (iii) for all $i > k_0$ $|X_i| \\ge \\frac{2^{|P| + c_3i} \\cdot k_2^{c_9} \\cdot t_i}{(\\log t_i)^{c_4}};$ (iv) $r^{c_1} < t_{k_0 + 1}$ ; (v) we assume that $k_1 - k_0 < c_7 \\log t_{k_0 + 1}$ .",
"Furthermore, $i > k_0$ and $i - 1 + \\lfloor c_7 \\log t_i\\rfloor \\le j \\le r$ implies $(\\log t_i)^{c_5} < \\log t_j;$ (vi) $k_2 > A$ and $s_{k_0 + 1} > t$ ; (vii) $c_{10} \\log k_2 > |P| + k_0$ and $c_{11} \\log k_2 > \\log k_1$ .",
"Then we have $\\sum _{a \\in \\mathbb {F}_2^{M \\sqcup M_P}} \\left|2^{-|M \\sqcup M_P|} \\cdot k_2!",
"\\cdot |X| - \\sum _{\\sigma \\in \\mathcal {P}(k_2)} |X(\\sigma (a))|\\right| \\le \\left(k_2^{-c_{12}} + t_{k_0 + 1}^{-c_8}\\right) \\cdot k_2!",
"\\cdot |X|.$ This is a straightforward generalization of [33]."
],
[
"Boxes",
"We now define boxes.",
"Boxes are product spaces of the shape $X := X_1 \\times \\dots \\times X_r$ , where $X_1, \\dots , X_r$ are “nice” sets of primes.",
"Then we state an important proposition that allows us to transition from squarefree integers to boxes.",
"Definition 5.11 Let $l$ be such that $|l|$ is a prime 3 modulo 4 or $l = -1$ .",
"Suppose that $D_1 > \\max (100, |l|)$ is a real number and let $1 \\le k \\le r$ be integers.",
"Let $\\mathfrak {t} := (p_1, \\dots , p_k, t_{k + 1}, \\dots , t_r)$ be a tuple satisfying the following properties the $p_i$ are prime numbers satisfying $p_1 < \\dots < p_k < D_1$ and the $t_j$ are real numbers with $D_1 < t_{k + 1} < \\dots < t_r$ ; we have $|l| \\in \\lbrace p_1, \\dots , p_k\\rbrace $ and we have for all $i = 1, \\dots , k$ that $\\gcd (2l, p_i) > 1$ or $\\left(\\frac{l}{p_i}\\right) = 1$ .",
"To $\\mathfrak {t}$ we associate a box $X := X_1 \\times \\dots \\times X_r$ as follows; we set $X_i := \\lbrace p_i\\rbrace $ for $1 \\le i \\le k$ , while for $i > k$ we let $X_i$ be the set of prime numbers $p$ with $\\left(\\frac{l}{p}\\right) = 1$ in the interval $\\left(t_i, \\left(1 + \\frac{1}{e^{i - k} \\log D_1}\\right) \\cdot t_i\\right).$ Note that for $l = -1$ this is the same definition as Definition 5.11 in [2].",
"Furthermore, if $l \\ne -1$ , we can turn any box into a prebox by removing $\\lbrace |l|\\rbrace $ and taking $P = \\emptyset $ .",
"We define $S^\\ast (N, l) := \\lbrace 1 \\le d < N : d \\text{ squarefree and satisfies equation (\\ref {ed})}\\rbrace $ and $S_r^\\ast (N, l) := \\lbrace d \\in S^\\ast (N, l) : \\omega (d) = r\\rbrace .$ Then there is a natural injective map $i: X \\rightarrow S_r^\\ast (\\infty , l)$ , which is a superset of $S_r(\\infty , l)$ .",
"Hence it makes sense to speak of the intersection $i(X) \\cap V$ for $V$ a subset of $S_r(\\infty , l)$ .",
"We can now state our analogue of Proposition 6.9 in Smith [33].",
"Proposition 5.12 Take $l$ to be an integer such that $|l|$ is a prime 3 modulo 4 or $l = -1$ .",
"Let $N \\ge D_1 > \\max (100, |l|)$ and $\\log \\log N \\ge 2 \\log \\log D_1$ .",
"Take any $r$ satisfying equation (REF ).",
"Let $V, W$ be subsets of $S_r(N, l)$ with the additional requirement that $W \\subseteq \\lbrace d \\in S_r(N, l) : 2D_1 < p_i < p_{i + 1}/2 \\textup { for all } p_i > D_1\\rbrace .$ Take any $\\epsilon > 0$ with $|W| > (1 - \\epsilon ) \\cdot |S_r(N, l)|.$ Assume that there exists a real number $\\delta > 0$ such that for all boxes $X$ with $i(X) \\subseteq S_r^\\ast (N, l)$ and $i(X) \\cap W \\ne \\emptyset $ we have $(\\delta - \\epsilon ) \\cdot |i(X) \\cap S_r(N, l)| \\le |i(X) \\cap V| \\le (\\delta + \\epsilon ) \\cdot |i(X) \\cap S_r(N, l)|.$ Then $|V| = \\delta \\cdot |S_r(N, l)| + O\\left(\\left(\\epsilon + \\frac{1}{\\log D_1}\\right) \\cdot |S_r(N, l)|\\right).$ This is a straightforward adaptation of Proposition 6.9 in Smith [33].",
"When we apply Proposition 6.3 and Theorem 6.4 of Smith [33], we need to ensure the Siegel-less condition, i.e.",
"we need to avoid all boxes $X$ such that there are $x \\in X$ and some $i$ with $|d_i| > D_1$ and $d_i \\mid x$ .",
"To do so, we shall add the union of all such boxes $X$ to $W$ .",
"Therefore it is important to show that this union is small, and this is exactly what the following proposition does.",
"Proposition 5.13 Let $l$ be an integer such that $|l|$ is a prime or $l = -1$ .",
"Take $N$ and $r$ satisfying equation (REF ).",
"Also take $N \\ge D_1 > \\max (100, |l|)$ with $\\log \\log N \\ge 2 \\log \\log D_1$ .",
"Let $f_1, f_2, \\ldots $ be any sequence of squarefree integers greater than $D_1$ satisfying $f_i^2 < f_{i + 1}$ .",
"Define $W_i := \\lbrace d \\in S_r(N, l) : \\textup {there is a box } X \\textup { with } d \\in X \\textup { and } f_i \\mid x \\textup { for some } x \\in X\\rbrace .$ Then we have $\\left|\\bigcup _{i = 1}^\\infty W_i\\right| \\ll \\frac{|S_r(N, l)|}{\\log D_1}.$ This is a small generalization of Theorem 5.13 in [2], which is based on Proposition 6.10 in Smith [33]."
],
[
"Rédei matrices",
"The previous subsections provide us with enough tools to deal with the 4-rank distribution in our family of discriminants.",
"We only handle the case where $l$ is such that $|l|$ is a prime 3 modulo 4.",
"The analogous results for $l = -1$ can be found in [2].",
"We now define the Rédei matrix associated to a squarefree integer $d > 0$ .",
"Definition 5.14 Let $d > 0$ be a squarefree integer and suppose that $\\Delta _{\\mathbb {Q}(\\sqrt{d})}$ has $t$ prime divisors, say $p_1, \\dots , p_t$ .",
"We can uniquely decompose $\\chi _d$ as $\\chi _d = \\sum _{i = 1}^t \\chi _i,$ where $\\chi _i: G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ has conductor a power of $p_i$ .",
"In case $p_i \\ne 2$ , we have $\\chi _i = \\chi _{p_i^\\ast }$ , where $p_i^\\ast $ has the same absolute value as $p_i$ and is 1 modulo 4.",
"When $p_i = 2$ , we have $\\chi _i \\in \\lbrace \\chi _{-4}, \\chi _{-8}, \\chi _8\\rbrace $ .",
"The Rédei matrix $R(d)$ is a $t \\times t$ matrix with entry $(i, j)$ equal to $\\chi _j(\\textup {Frob } p_i) \\text{ if } i \\ne j, \\quad \\sum _{k \\ne i} \\chi _k(\\textup {Frob } p_i) \\text{ if } i = j,$ so the sum of every row is zero.",
"It is a classical fact that $\\textup {rk}_4 \\ \\textup {Cl}(\\mathbb {Q}(\\sqrt{d})) = t - 1 - \\textup {rk} \\ R(d).$ One of the pleasant properties of $X(a)$ is that all $x \\in X(a)$ have the same Rédei matrix, and hence the same 4-rank.",
"There are several constraints for the possible shapes of the Rédei matrix.",
"First of all, there is quadratic reciprocity that relates the entry $(i, j)$ with $(j, i)$ .",
"Second of all, if $d \\in S(N, l)$ , then there are further constraints coming from equation (REF ) and equation ().",
"We will now indicate what conditions this forces on $a$ .",
"Definition 5.15 Let $X$ be a box corresponding to $\\mathbf {t} = (p_1, \\dots , p_k, t_{k + 1}, \\dots , t_r)$ and let $\\tilde{j}$ be the index for which $X_{\\tilde{j}} = \\lbrace |l|\\rbrace $ .",
"We define $\\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ to be the set of maps from $M \\sqcup M_\\emptyset $ to $\\lbrace \\pm 1\\rbrace $ .",
"Put $\\widetilde{\\textup {Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ to be the subset of $\\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ satisfying if $X_1 \\ne \\lbrace 2\\rbrace $ and $l > 0$ , then $a(i, \\tilde{j}) = a(i, -1)$ for all $i < \\tilde{j}$ , $a(\\tilde{j}, i) = 1$ for all $i > \\tilde{j}$ and $\\prod _{i = 1}^r a(i, -1) = 1;$ if $X_1 \\ne \\lbrace 2\\rbrace $ and $l < 0$ , then $a(i, \\tilde{j}) = 1$ for all $i < \\tilde{j}$ , $a(\\tilde{j}, i) = a(i, -1)$ for all $i > \\tilde{j}$ ; if $X_1 = \\lbrace 2\\rbrace $ and $l > 0$ , then $a(i, \\tilde{j}) = a(i, -1)$ for all $2 \\le i < \\tilde{j}$ , $a(\\tilde{j}, i) = 1$ for all $i > \\tilde{j}$ and $\\prod _{i = 1}^r a(i, -1) = \\left(\\frac{2}{|l|}\\right);$ if $X_1 = \\lbrace 2\\rbrace $ and $l < 0$ , then $a(i, \\tilde{j}) = 1$ for all $2 \\le i < \\tilde{j}$ , $a(\\tilde{j}, i) = a(i, -1)$ for all $i > \\tilde{j}$ and $l \\equiv 1 \\bmod 8$ .",
"We will now describe exactly the kind of boxes that we will be working with for the rest of the paper.",
"Definition 5.16 Let $X$ be a box and let $N$ be a real number.",
"Put $D_1 := e^{(\\log \\log N)^{\\frac{1}{10}}}, \\quad C_0 := \\frac{\\log \\log \\log N}{100}, \\quad C_0^{\\prime } := \\sqrt{\\log \\log \\log N}.$ We let $W$ be the largest subset of $S_r(N, l)$ satisfying the requirement $W \\cap W_i = \\emptyset $ for all $i \\ge 1$ , where $W_i$ is the set as constructed in Proposition REF ; the requirement $W \\subseteq \\lbrace d \\in S_r(N, l) : 2D_1 < p_i < p_{i + 1}/2 \\textup { for all } p_i > D_1\\rbrace ;$ and the requirement $W \\subseteq \\left\\lbrace d \\in S_r(N, l) : \\left|\\frac{1}{2} \\log \\log p_i - i\\right| < C_0^{1/5} \\max (i, C_0)^{4/5}\\right\\rbrace .$ We say that $X$ is $N$ -decent if $r$ satisfies equation (REF ), $i(X) \\subseteq S_r^\\ast (N, l)$ and $i(X) \\cap W \\ne \\emptyset $ .",
"Now let $W^{\\prime }$ be the largest subset of $W$ satisfying the requirement $W^{\\prime } \\subseteq \\left\\lbrace d \\in S_r(N, l) : \\left|\\frac{1}{2} \\log \\log p_i - i\\right| < C_0^{\\prime 1/5} \\max (i, C_0^{\\prime })^{4/5}\\right\\rbrace ;$ and the requirement that for every $d \\in W^{\\prime }$ there is some $i$ with $\\frac{1}{2}r^{1/2} < i < \\frac{1}{2}$ and $\\frac{\\log p_i}{\\log \\log p_i} > (\\log \\log \\log N)^{1/2} \\cdot \\sum _{j = 1}^{i - 1} \\log p_j.$ We say that $X$ is $N$ -good if $X$ is $N$ -decent and $i(X) \\cap W^{\\prime } \\ne \\emptyset $ .",
"The main point of Definition REF is that we can apply the results in Subsection REF to these boxes provided that $N$ is sufficiently large.",
"Let $P(m, n, j)$ be the probability that a randomly chosen $m \\times n$ matrix with coefficients in $\\mathbb {F}_2$ has right kernel of rank $j$ .",
"Then we have the explicit formula $P(m, n, j) = \\frac{1}{2^{nm}} \\prod _{i = 0}^{n - j - 1} \\frac{(2^m - 2^i)(2^n - 2^i)}{2^{n - j} - 2^i},$ which we will use throughout the paper.",
"For the remainder of this paper, $\\iota $ denotes the unique group isomorphism between $\\lbrace \\pm 1\\rbrace $ and $\\mathbb {F}_2$ .",
"To prove the next theorem, it suffices to work with $N$ -decent boxes $X$ , while we will work with $N$ -good boxes in Section .",
"Theorem 5.17 Let $l$ be such that $|l|$ is a prime 3 modulo 4.",
"Then we have for all $k \\ge 0$ $\\left|\\lim _{s \\rightarrow \\infty } P(s, s, k) \\cdot |S(N, l)| - \\left|\\left\\lbrace d \\in S(N, l) : \\textup {rk}_4 \\ \\textup {Cl}(\\mathbb {Q}(\\sqrt{d})) = k\\right\\rbrace \\right|\\right| = O\\left(\\frac{|S(N, l)|}{(\\log \\log N)^c}\\right)$ for some absolute constant $c > 0$ .",
"Let $r$ be an integer satisfying equation (REF ).",
"Since one easily bounds the differences $\\left|\\lim _{s \\rightarrow \\infty } P(s, s, k) - P(r - 1, r - 1, k)\\right|,$ we may work with $P(r - 1, r - 1, k)$ instead.",
"We now follow the proof of Smith [33].",
"The first step is to reduce to $N$ -decent $X$ , for which we use our Theorem REF , Proposition REF and Proposition REF .",
"Now let $X = X_1 \\times \\dots \\times X_r$ be an $N$ -decent box, so in particular $i(X) \\subseteq S_r^\\ast (N, l)$ .",
"It suffices to prove that $\\left|P(r - 1, r - 1, k) \\cdot |i(X) \\cap S_r(N, l)| - \\left|\\left\\lbrace d \\in i(X) \\cap S_r(N, l) : \\textup {rk}_4 \\ \\textup {Cl}(\\mathbb {Q}(\\sqrt{d})) = k\\right\\rbrace \\right|\\right| \\\\= O\\left(\\frac{|X|}{(\\log \\log N)^c}\\right)$ for some absolute constant $c > 0$ , since $|X| \\ll |i(X) \\cap S_r(N, l)|$ .",
"We now apply Proposition REF to the box $X^{\\prime }$ with $X_{\\tilde{j}}$ removed.",
"Then we obtain an absolute constant $c^{\\prime }$ such that $\\sum _{a \\in \\widetilde{\\text{Map}}(M^{\\prime } \\sqcup M^{\\prime }_\\emptyset , \\lbrace \\pm 1\\rbrace )} \\left| 2^{-|M^{\\prime }| - |M^{\\prime }_\\emptyset |} \\cdot (r - 1)!",
"\\cdot |X^{\\prime }| - \\sum _{\\sigma \\in \\mathcal {P}(r - 1)} |X^{\\prime }(\\sigma (a))| \\right| \\le \\frac{(r - 1)!",
"\\cdot |X^{\\prime }|}{(\\log \\log N)^{c^{\\prime }}},$ where $M^{\\prime } = \\lbrace (i, j) : 1 \\le i < j \\le r - 1\\rbrace $ and $M^{\\prime }_\\emptyset = [r - 1] \\times \\lbrace -1\\rbrace $ .",
"Let $S$ be the set of permutations of $[r]$ that fix $\\tilde{j}$ .",
"Then equation (REF ) implies $\\sum _{a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)} \\left| 2^{-|M^{\\prime }| - |M^{\\prime }_\\emptyset |} \\cdot (r - 1)!",
"\\cdot |X| - \\sum _{\\sigma \\in S} |X(\\sigma (a))| \\right| \\le \\frac{(r - 1)!",
"\\cdot |X|}{(\\log \\log N)^{c^{\\prime }}}.$ Note that if $a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ , then so is $\\sigma (a)$ for any permutation $\\sigma \\in S$ .",
"Also observe that $a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ implies $i(X(a)) \\subseteq S_r(N, l)$ .",
"Furthermore, if $i(X(a)) \\cap S_r(N, l) \\ne \\emptyset $ , then we certainly have $a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ .",
"Set $Q(X, k, l) := \\frac{|\\lbrace a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l) : \\dim _{\\mathbb {F}_2} \\text{ker}(A) = k + 1\\rbrace |}{|\\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)|}.$ Here $A$ is the Rédei matrix associated to $a$ in the obvious way.",
"Note that the the matrix $A^{\\prime }$ associated to $\\sigma (a)$ has the same rank as the matrix $A$ associated to $a$ .",
"Then, because of equation (REF ), it is enough to show that there is an absolute constant $c > 0$ such that $\\left|P(r - 1 , r - 1, k) - Q(X, k, l)\\right| = O\\left(\\frac{1}{(\\log \\log N)^c}\\right)$ for every $r$ satisfying equation (REF ).",
"But this follows from [21]."
],
[
"Proof of main theorems",
"The aim of this section is to prove Theorem REF and Theorem REF .",
"We start with the proof of Theorem REF .",
"The proof of Theorem REF is almost identical and we shall only indicate the necessary changes in Subsection REF ."
],
[
"Proof of Theorem ",
"Let $D_{l, k}(n)$ be the set of squarefree integers $d$ divisible by $l$ such that $\\text{rk}_{2^k} \\text{Cl}(\\mathbb {Q}(\\sqrt{d})) = n$ and furthermore $\\mathfrak {l} \\in 2^{k - 1} \\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2^k]$ if $l > 0$ and $\\mathfrak {l}(\\sqrt{d}) \\in 2^{k - 1} \\text{Cl}(\\mathbb {Q}(\\sqrt{d}))[2^k]$ if $l < 0$ , where $\\mathfrak {l}$ is the unique ideal with norm $l$ .",
"Then we have the decomposition $\\bigcup _{n = 0}^\\infty D_{l, 2}(n) = S(\\infty , l).$ Our next theorem is very much in spirit of the heuristical assumptions that led to Stevenhagen's conjecture [34].",
"Theorem REF is an immediate consequence of Theorem REF , the material in Appendix and the theorem below.",
"Theorem 6.1 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"There are $c, A, N_0 > 0$ such that for all integers $N > N_0$ , all integers $m \\ge 2$ and all sequences of integers $n_2 \\ge \\dots \\ge n_m \\ge n_{m + 1} \\ge 0$ $\\left|\\left|[N] \\cap \\bigcap _{i = 2}^{m + 1} D_{l, i}(n_i)\\right| - \\frac{P(n_m, n_m, n_{m + 1})}{2^{n_m}} \\cdot \\left|[N] \\cap \\bigcap _{i = 2}^m D_{l, i}(n_i)\\right|\\right| \\le \\frac{A \\cdot |S(N, l)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m^2 6^m}}}.$ To prove Theorem REF , our first step is to reduce to boxes with some nice properties.",
"Definition REF precisely pinpoints the boxes for which we will prove the desired equidistribution.",
"We will now state a proposition and prove that the proposition implies Theorem REF , so that it remains to prove the proposition.",
"Proposition 6.2 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"There are $c, A, N_0 > 0$ such that for all integers $N > N_0$ , all integers $m \\ge 2$ , all sequences of integers $n_2 \\ge \\dots \\ge n_m \\ge n_{m + 1} \\ge 0$ and all $N$ -good boxes $X$ $\\left|\\left|i(X) \\cap \\bigcap _{i = 2}^{m + 1} D_{l, i}(n_i)\\right| - \\frac{P(n_m, n_m, n_{m + 1})}{2^{n_m}} \\cdot \\left|i(X) \\cap \\bigcap _{i = 2}^m D_{l, i}(n_i)\\right|\\right| \\le \\\\\\frac{A \\cdot |i(X) \\cap S_r(N, l)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m^2 6^m}}}.$ [Proof that Proposition REF implies Theorem REF ] Due to equation (REF ) we may restrict to $S_r(N, l)$ with $r$ satisfying equation (REF ).",
"Let $D_1$ and $W$ be as in Definition REF .",
"Part (i), (ii) and (iii) of Theorem REF give upper bounds for the complements of the sets appearing in equation (REF ), equation (REF ) and equation (REF ) respectively.",
"Furthermore, Proposition REF shows that most $d \\in W$ are outside the union of the $W_i$ .",
"Therefore we see that there is an absolute constant $C > 0$ with $|W| > \\left(1 - \\frac{C}{\\exp \\left(\\left(\\log \\log \\log N\\right)^{1/4}\\right)}\\right) \\cdot |S_r(N, l)|.$ We now apply Proposition REF two times; in both case with our $D_1$ and $W$ , and $V _1 := [N] \\cap \\bigcap _{i = 2}^{m + 1} D_{l, i}(n_i), \\quad V_2 := [N] \\cap \\bigcap _{i = 2}^m D_{l, i}(n_i)$ respectively.",
"Theorem REF and Proposition REF ensure that equation (REF ) is satisfied.",
"Then we get $V_1 = \\lim _{s \\rightarrow \\infty } P(s, s, n_2) \\cdot \\prod _{i = 2}^m \\frac{P(n_i, n_i, n_{i + 1})}{2^{n_i}} \\cdot |S_r(N, l)| + O\\left(\\frac{|S_r(N, l)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m^2 6^m}}}\\right)$ and $V_2 = \\lim _{s \\rightarrow \\infty } P(s, s, n_2) \\cdot \\prod _{i = 2}^{m - 1} \\frac{P(n_i, n_i, n_{i + 1})}{2^{n_i}} \\cdot |S_r(N, l)| + O\\left(\\frac{|S_r(N, l)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m^2 6^m}}}\\right).$ This quickly implies Theorem REF .",
"Our next goal is to fix the first Rédei matrix.",
"In other words, we split $X$ into the union $X(a)$ with $a$ running over all maps from $M \\sqcup M_\\emptyset $ to $\\lbrace \\pm 1\\rbrace $ .",
"Smith's method does not prove equidistribution for all $a$ , but only for most $a$ .",
"This prompts our next definition.",
"Definition 6.3 Let $X$ be a $N$ -good box and let $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ .",
"Set $r^{\\prime }(a, X) :=\\left\\lbrace \\begin{array}{ll}r & \\mbox{\\textup {if} } X_1 = \\lbrace 2\\rbrace \\textup { or } \\prod _{i = 1}^r a(i, -1) = 1 \\\\r + 1 & \\mbox{\\textup {otherwise.",
"}}\\end{array}\\right.$ Recall that we associated a $r^{\\prime }(a, X) \\times r^{\\prime }(a, X)$ matrix $A$ with coefficients in $\\mathbb {F}_2$ to $a$ during the proof of Theorem REF , which is simply the Rédei matrix of $x$ for any choice of $x \\in X(a)$ .",
"Let $V$ be the vector space $\\mathbb {F}_2^{r^{\\prime }(a, X)}$ .",
"We define $D_{a, 2} := \\lbrace v \\in V : v^TA = 0\\rbrace , \\quad D_{a, 2}^\\vee := \\lbrace v \\in V : Av = 0\\rbrace .$ Put $n_{\\textup {max}} := \\left\\lfloor \\sqrt{\\frac{c^{\\prime }}{m^2 6^m} \\log \\log \\log \\log \\log N} \\right\\rfloor , \\quad n_{a, 2} := \\dim _{\\mathbb {F}_2} D_{a, 2} - 1,$ where $c^{\\prime }$ is a constant specified later.",
"Let $X$ be a $N$ -good box and let $\\tilde{j}$ be the index such that $X_{\\tilde{j}} = \\lbrace |l|\\rbrace $ .",
"We define the vectors $R := (1, 1, \\dots , 1) \\in D_{a, 2}^\\vee $ and $C :=\\left\\lbrace \\begin{array}{ll}(1, \\dots , 1) & \\mbox{\\textup {if} } X_1 = \\lbrace 2\\rbrace \\textup { or } \\prod _{i = 1}^r a(i, -1) = 1 \\\\(0, 1, 1, 1, \\dots , 1) & \\mbox{\\textup {otherwise.",
"}}\\end{array}\\right.$ We next define the vector $L :=\\left\\lbrace \\begin{array}{ll}(0, \\dots , 0, 1, 0, \\dots , 0) \\in D_{a, 2} & \\mbox{\\textup {if} } l > 0 \\\\(0, \\dots , 0, 1, 0, \\dots , 0) + C \\in D_{a, 2} & \\mbox{\\textup {if} } l < 0,\\end{array}\\right.$ where $(0, \\dots , 0, 1, 0, \\dots , 0)$ has a 1 exactly on the $\\tilde{j}$ -th position.",
"Since $l \\mid d$ , the solubility of $x^2 - dy^2 = l$ in $x, y \\in \\mathbb {Q}$ is precisely equivalent to $L$ being in $D_{a, 2}$ .",
"We fix a choice of an index $i$ satisfying equation (REF ) and we call it $k_{\\textup {gap}}$ .",
"Then we say that $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ is $(N, m, X)$ -acceptable if the following conditions are satisfied $n_{a, 2} \\le n_{\\textup {max}}$ ; we have $a \\in \\widetilde{\\textup {Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ , see Definition REF ; we have for all $j > k$ $|X_j(a, Q)| \\ge \\frac{|X_j|}{(\\log t_{k + 1})^{100}},$ where $Q$ is the unique point in $X_1 \\times \\dots \\times X_k$ ; putting $S_{\\textup {pre}, 4} := \\left\\lbrace i \\in [r] : \\frac{k_{\\textup {gap}}}{2} \\le i < k_{\\textup {gap}} \\textup { and } a(i, -1) = 1\\right\\rbrace , \\quad \\alpha _{\\textup {pre}} := \\left|S_{\\textup {pre}, 4}\\right|$ and $S_{\\textup {post}, 4} := \\left\\lbrace i \\in [r] : k_{\\textup {gap}} \\le i \\le 2k_{\\textup {gap}} \\textup { and } a(i, -1) = 1\\right\\rbrace , \\quad \\alpha _{\\textup {post}} := \\left|S_{\\textup {post}, 4}\\right|,$ we have $\\left|\\alpha _{\\textup {pre}} - \\frac{k_{\\textup {gap}}}{4}\\right| \\le \\frac{k_{\\textup {gap}}}{\\log \\log \\log \\log N}, \\quad \\left|\\alpha _{\\textup {post}} - \\frac{k_{\\textup {gap}}}{2}\\right| \\le \\frac{k_{\\textup {gap}}}{\\log \\log \\log \\log N}$ and we further have for all $T_1 \\in D_{a, 2}$ , $T_2 \\in D_{a, 2}^\\vee $ such that $T_1 \\notin \\langle L \\rangle $ or $T_2 \\notin \\langle R \\rangle $ $\\left|\\left|\\left\\lbrace i \\in S_{\\textup {pre}, 4} : \\pi _i(T_1 + T_2) = 0\\right\\rbrace \\right| - \\frac{\\alpha _{\\textup {pre}}}{2}\\right| \\le \\frac{\\alpha _{\\textup {pre}}}{\\log \\log \\log \\log N}$ and $\\left|\\left|\\left\\lbrace i \\in S_{\\textup {post}, 4} : \\pi _i(T_1 + T_2) = 0\\right\\rbrace \\right| - \\frac{\\alpha _{\\textup {post}}}{2}\\right| \\le \\frac{\\alpha _{\\textup {post}}}{\\log \\log \\log \\log N}.$ Let us explain the second condition.",
"Note that given $a \\in \\text{Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ , $i(X(a))$ is entirely contained in $S_r(N, l)$ or completely disjoint from $S_r(N, l)$ .",
"Since we only care about the intersection $i(X) \\cap S_r(N, l)$ , we only restrict to those $a$ with $i(X(a)) \\subseteq S_r(N, l)$ , and this is exactly what the second condition does.",
"The importance of the fourth condition will be explained after our next two definitions.",
"Once we have fixed the first Rédei matrix, we are ready to study all the higher Rédei matrices.",
"In fact, we can prove equidistribution of all the higher Rédei matrices.",
"We formalize this as follows.",
"Definition 6.4 Let $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ and $m \\in \\mathbb {Z}_{\\ge 2}$ be given.",
"Choose filtrations of vector spaces $D_{a, 2} \\supseteq \\dots \\supseteq D_{a, m}, \\quad D_{a, 2}^\\vee \\supseteq \\dots \\supseteq D_{a, m}^\\vee $ with $L \\in D_{a, m}$ and $R \\in D_{a, m}^\\vee $ .",
"Define for $2 \\le i \\le m$ $n_{a, i} := \\dim _{\\mathbb {F}_2} D_{a, i} - 1.$ If $\\textup {Art}_{a, i} : D_{a, i} \\times D_{a, i}^\\vee \\rightarrow \\mathbb {F}_2$ are bilinear pairings for $2 \\le i \\le m$ , we call the set $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m}$ a sequence of Artin pairings if for every $2 \\le i < m$ the left kernel of $\\textup {Art}_{a, i}$ is $D_{a, i + 1}$ and the right kernel of $\\textup {Art}_{a, i}$ is $D_{a, i + 1}^\\vee $ .",
"We say that a bilinear pairing $\\textup {Art}_{a, i} : D_{a, i} \\times D_{a, i}^\\vee \\rightarrow \\mathbb {F}_2$ is valid if $L$ and $R$ are respectively in the left and right kernel.",
"We call a sequence of Artin pairings valid if every element of the sequence is.",
"Let $X$ be an $N$ -good box, let $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ be $(N, m, X)$ -acceptable and also let $d \\in i(X(a))$ .",
"We can naturally associate an infinite sequence of Artin pairings to $d$ as follows.",
"Write the prime divisors of the discriminant of $\\mathbb {Q}(\\sqrt{d})$ as $p_1, \\dots , p_{r^{\\prime }(a, X)}$ with $p_1 < \\dots < p_{r^{\\prime }(a, X)}$ .",
"By construction, we have that for each $v \\in D_{a, 2}$ that the unique ideal in $\\mathbb {Q}(\\sqrt{d})$ with norm $\\prod _{i = 1}^{r^{\\prime }(a, X)} p_i^{\\pi _i(v)}$ is in $2 \\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[4]$ .",
"Similarly we have for each $v \\in D_{a, 2}^\\vee $ that the character $\\sum _{i = 1}^{r^{\\prime }(a, X)} \\pi _i(v) \\chi _i$ is in $2 \\textup {Cl}^\\vee (\\mathbb {Q}(\\sqrt{d}))[4]$ , where $\\chi _i$ is as in Definition REF .",
"In other words, we have natural epimorphisms $D_{a, 2} \\rightarrow 2 \\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[4] \\textup { and } D_{a, 2}^\\vee \\rightarrow 2 \\textup {Cl}^\\vee (\\mathbb {Q}(\\sqrt{d}))[4].$ Now we de declare $D_{a, i, d}$ and $D_{a, i, d}^\\vee $ to be the inverse image of respectively $2^{i - 1} \\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[2^i]$ and $2^{i - 1} \\textup {Cl}^\\vee (\\mathbb {Q}(\\sqrt{d}))[2^i]$ under these maps.",
"Furthermore, we let $\\textup {Art}_{a, i, d}$ be the natural pairing $2^{i - 1} \\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[2^i] \\times 2^{i - 1} \\textup {Cl}^\\vee (\\mathbb {Q}(\\sqrt{d}))[2^i] \\rightarrow \\mathbb {F}_2$ pulled back to $D_{a, i, d}$ and $D_{a, i, d}^\\vee $ .",
"This gives an infinite sequence of Artin pairings $\\textup {Art}_{a, i, d}$ for every $d$ .",
"Furthermore, the sequence is valid if and only if equation (REF ) is soluble.",
"Finally, we define for a sequence of Artin pairings $X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m}) := \\lbrace d \\in X(a) : \\textup {Art}_{a, i, d} = \\textup {Art}_{a, i} \\textup { for } 2 \\le i \\le m\\rbrace .$ For $d \\in i(X(a))$ we have a natural isomorphism between $D_{a, 2}$ and $f^{-1}(2\\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[4])$ .",
"Similarly, we have a natural isomorphism between $D_{a, 2}^\\vee $ and $2\\textup {Cocy}(\\textup {Gal}(H(\\mathbb {Q}(\\sqrt{d}))/\\mathbb {Q}))[4]$ .",
"Furthermore, the resulting Artin pairings are compatible.",
"This is also true for $D_{a, m}$ and $f^{-1}(2^{m - 1}\\textup {Cl}(\\mathbb {Q}(\\sqrt{d}))[2^m])$ provided that $\\text{Art}_{a, i, d}$ and $\\text{Art}_{a, i}$ are equal for $2 \\le i < m$ , and the same holds on the dual side.",
"Take a non-trivial character $F: \\text{Mat}(n_m + 1, n_m, \\mathbb {F}_2) \\rightarrow \\mathbb {F}_2$ .",
"Our goal is to prove equidistribution of $F(\\text{Art}_{a, m, d})$ , where we view $\\text{Art}_{a, m, d}$ as a matrix using a fixed basis.",
"When we prove equidistribution of $F$ , we shall only vary a very limited set of indices in $[r]$ and fix one choice of prime outside those indices.",
"We make this precise in our next definition, where we shall also fix the basis used to identify $\\text{Art}_{a, m, d}$ with a matrix.",
"Definition 6.5 Let $a \\in \\widetilde{\\textup {Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ and let $m \\in \\mathbb {Z}_{\\ge 2}$ be an integer.",
"We fix a basis $v_1, \\ldots , v_{n_{a, 2}}, L$ for $D_{a, 2}$ and a basis $w_1, \\ldots , w_{n_{a, 2}}, R$ for $D_{a, 2}^\\vee $ for the rest of the paper in such a way that $v_1, \\ldots , v_{n_{a, i}}, L$ is a basis for $D_{a, i}$ and $w_1, \\ldots , w_{n_{a, i}}, R$ is a basis for $D_{a, i}^\\vee $ for $2 \\le i \\le m$ .",
"We can decompose any $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ as $F = \\prod _{\\begin{array}{c}1 \\le j_1 \\le n_{a, m} + 1 \\\\ 1 \\le j_2 \\le n_{a, m}\\end{array}} E_{j_1, j_2}^{c_{j_1, j_2}(F)},$ where $c_{j_1, j_2}(F) \\in \\mathbb {F}_2$ and $E_{j_1, j_2}$ is the map that sends a matrix to the coefficient in the entry $(j_1, j_2)$ viewed as an element of $\\lbrace \\pm 1\\rbrace $ via $\\iota ^{-1}$ .",
"We say that $S \\subseteq [r]$ is a set of variable indices for a non-zero character $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ if there are integers $1 \\le j_1 \\le n_{a, m} + 1$ and $1 \\le j_2 \\le n_{a, m}$ and integers $i_1(j_1, j_2), i_2(j_1, j_2)$ with $c_{j_1, j_2}(F) \\ne 0$ , $i_2(j_1, j_2) \\in S_{\\textup {post}, 4} \\cap S$ and $S - \\lbrace i_2(j_1, j_2)\\rbrace \\subseteq S_{\\textup {pre}, 4}$ such that in case $j_1 = n_{a, m} + 1$ , we have $c_{j_1, j_2}(F) = 0$ for all $1 \\le j_1, j_2 \\le n_{a, m}$ , $|S| = m$ , $i_1(j_1, j_2) = \\tilde{j}$ and $S - \\lbrace i_1(j_1, j_2), i_2(j_1, j_2)\\rbrace \\subseteq \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(v_i) = 0\\rbrace \\cap \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(w_i) = 0\\rbrace $ and $i_2(j_1, j_2) \\in \\lbrace j \\in [r] : \\pi _j(w_{j_2}) = 1\\rbrace \\cap \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(v_i) = 0\\rbrace \\cap \\bigcap _{\\begin{array}{c}i = 1 \\\\ i \\ne j_2\\end{array}}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(w_i) = 0\\rbrace $ and $a(i, j) = 1$ for all distinct $i, j \\in S$ .",
"in case $j_1 \\ne n_{a, m} + 1$ , we have $|S| = m + 1$ , $i_1(j_1, j_2) \\in S$ and $S - \\lbrace i_1(j_1, j_2), i_2(j_1, j_2)\\rbrace \\subseteq \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(v_i) = 0\\rbrace \\cap \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(w_i) = 0\\rbrace $ and $i_1(j_1, j_2) \\in \\lbrace j \\in [r] : \\pi _j(w_{j_2}) = 1\\rbrace \\cap \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(v_i) = 0\\rbrace \\cap \\bigcap _{\\begin{array}{c}i = 1 \\\\ i \\ne j_2\\end{array}}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(w_i) = 0\\rbrace $ and $i_2(j_1, j_2) \\in \\lbrace j \\in [r] : \\pi _j(v_{j_1}) = 1\\rbrace \\cap \\bigcap _{\\begin{array}{c}i = 1 \\\\ i \\ne j_1\\end{array}}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(v_i) = 0\\rbrace \\cap \\bigcap _{i = 1}^{n_{a, 2}} \\lbrace j \\in [r] : \\pi _j(w_i) = 0\\rbrace .$ In simple words, the last row of $\\text{Art}_{a, m, d}$ corresponds to the Artin pairing with $l$ .",
"This is exactly the case $j_1 = n_{a, m} + 1$ in the above definition.",
"To prove equidistribution of these entries, we will use our higher Rédei reciprocity law.",
"It is for this reason that in this case our choice of variable indices is different than Smith's choice [33], while if $j_1 \\le n_{a, m}$ we make exactly the same choice as Smith.",
"In case $j_1 = n_{a, m} + 1$ , it will be essential that we restrict our variable indices to always be 1 modulo 4 and also satisfy $a(i, j) = 1$ for all distinct $i, j \\in S$ .",
"Indeed, this is needed to apply our higher Rédei reciprocity law.",
"In case $j_1 \\ne n_{a, m} + 1$ neither of these facts will be important, just as in Smith's original work.",
"It is a non-trivial task to show that we can find variable indices for most $a$ .",
"It is here that the material in Subsection REF and the fourth condition in Definition REF turn out to be crucial.",
"To find our variable indices, we start with a lemma.",
"Lemma 6.6 Suppose that $a \\in \\widetilde{\\textup {Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ satisfies equation (REF ).",
"Assume that $v_1, \\ldots , v_d, L \\in D_{a, 2}$ and $v_{d + 1}, \\ldots , v_e, R \\in D_{a, 2}^\\vee $ are linearly independent.",
"Then we have for all $\\mathbf {v} \\in \\mathbb {F}_2^e$ the estimate $\\left|\\left|\\left\\lbrace i \\in S_{\\textup {pre}, 4} : \\pi _i(v_j) = \\pi _j(\\mathbf {v}) \\textup { for all } 1 \\le j \\le e\\right\\rbrace \\right| - \\frac{\\alpha _{\\textup {pre}}}{2^e}\\right| \\le \\frac{100^e \\cdot k_{\\textup {gap}}}{\\log \\log \\log \\log N}.$ This is a small adjustment of Lemma 13.7 in [19].",
"We stress that the term generic in [19] is an unfortunate clash of terminology, and refers to $a$ satisfying the natural analog of our equation (REF ).",
"We have a completely similar result for the range $k_{\\text{gap}} < i \\le 2k_{\\text{gap}}$ using equation (REF ).",
"Let us now construct a graph $G$ associated to $a$ .",
"The set of vertices of $G$ is $S_{\\text{pre}, 4}$ .",
"Furthermore, for distinct $i, j \\in S_{\\text{pre, 4}}$ there is an edge between $i$ and $j$ if and only if $a(i, j) = 1$ .",
"Note that this is independent of the order of $i$ and $j$ , since $a(i, -1) = 1$ for all $i \\in S_{\\text{pre, 4}}$ .",
"Hence $G$ is an undirected graph.",
"To find our variable indices, we certainly need to be able to find a clique in $G$ of size $m$ .",
"This will follow, for most $G$ , from Lemma REF .",
"The final index $i_2(j_1, j_2)$ is then found using Lemma REF .",
"Definition 6.7 We say that $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ is very $(N, m, X)$ -acceptable if $a$ is $(N, m, X)$ -acceptable, the graph $G$ associated to $a$ is not $(n/\\log n, m)$ -bad and furthermore the function $f: S_{\\textup {pre}, 4} \\times S_{\\textup {post}, 4} \\rightarrow \\mathbb {F}_2$ given by $f(s, t) = \\iota (a(s, t))$ is such that for all pairwise distinct $v_1, \\ldots , v_m \\in S_{\\textup {pre}, 4}$ there is some $w \\in S_{\\textup {post}, 4}$ such that for all $i \\in [m]$ we have $f(v_i, w) = 0$ .",
"This brings us to our next reduction step.",
"Proposition 6.8 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"There are $c, A, N_0 > 0$ such that for all integers $N > N_0$ , all integers $m \\ge 2$ , all sequences of integers $n_2 \\ge \\dots \\ge n_m \\ge 0$ , all $N$ -good boxes $X$ , all very $(N, m, X)$ -acceptable $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ , all sequences of valid Artin pairings $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}$ with $n_{a, i} = n_i$ for $2 \\le i \\le m$ and an Artin pairing $\\textup {Art}_{a, m} : D_{a, m} \\times D_{a, m}^\\vee \\rightarrow \\mathbb {F}_2$ with $R$ in the right kernel $\\left||X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})| - 2^{-n_m(n_m + 1)} \\cdot |X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m})|\\right| \\le \\frac{A \\cdot |X(a)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m 6^m}}}.$ Remark 4 We do not need to assume that $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}$ is valid, but it suffices for our purposes and avoids some casework later on.",
"[Proof that Proposition REF implies Proposition REF ] We observe that $i(X(a)) \\cap S_r(N, l) \\ne \\emptyset $ implies $a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ .",
"Hence we can bound $\\left|\\left|i(X) \\cap \\bigcap _{i = 2}^{m + 1} D_{l, i}(n_i)\\right| - \\frac{P(n_m, n_m, n_{m + 1})}{2^{n_m}} \\cdot \\left|i(X) \\cap \\bigcap _{i = 2}^m D_{l, i}(n_i)\\right|\\right| \\le \\\\\\sum _{a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)} \\hspace{-2.84544pt} \\left|\\left|i(X(a)) \\cap \\bigcap _{i = 2}^{m + 1} D_{l, i}(n_i)\\right| - \\frac{P(n_m, n_m, n_{m + 1})}{2^{n_m}} \\cdot \\left|i(X(a)) \\cap \\bigcap _{i = 2}^m D_{l, i}(n_i)\\right|\\right|.$ We split this sum over the very $(N, m, X)$ -acceptable $a \\in \\text{Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ and the remaining $a$ .",
"For the very $(N, m, X)$ -acceptable $a$ we may apply Proposition REF by further splitting the sum over all possible sequences of valid Artin pairings and an Artin pairing $D_{a, m} \\times D_{a, m}^\\vee \\rightarrow \\mathbb {F}_2$ with $R$ in the right kernel.",
"Note that in the set of bilinear pairings $D_{a, m} \\times D_{a, m}^\\vee \\rightarrow \\mathbb {F}_2$ with $R$ in the right kernel, there are precisely $2^{n_m(n_m + 1)} \\cdot \\frac{P(n_m, n_m, n_{m + 1})}{2^{n_m}}$ such that the left kernel has dimension $n_{m + 1} + 1$ and $L$ is in the left kernel.",
"There are at most $2^{mn_{\\text{max}}^2}$ sequences of Artin pairings, so we stay within the error term of Proposition REF provided that we take the constant $c^{\\prime }$ in the definition of $n_{\\text{max}}$ smaller than the constant $c$ guaranteed by Proposition REF .",
"Hence it suffices to bound $\\sum _{\\begin{array}{c}a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l) \\\\ a \\text{ not very } (N, m, X)\\text{-acceptable}\\end{array}} |i(X(a))|.$ We first tackle those $a$ for which $n_{a, 2} > n_{\\text{max}}$ .",
"These $a$ can easily be dealt with using equation (REF ) for $k \\le n_{\\text{max}}$ inducing an error of size $O\\left(\\frac{|i(X) \\cap S_r(N, l)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m^2 6^m}}}\\right)$ for some absolute constant $c > 0$ .",
"We will now dispatch those $a$ that fail equation (REF ).",
"We declare two maps $a, a^{\\prime } \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ to be equivalent at some integer $i > k$ , written as $a \\sim _i a^{\\prime }$ if $a(j, i) = a^{\\prime }(j, i) \\text{ for all } 1 \\le j \\le k \\text{ and } a(i, -1) = a^{\\prime }(i, -1).$ Observe that if $a$ fails equation (REF ), then so does any $a^{\\prime }$ with $a \\sim _i a^{\\prime }$ .",
"We call an equivalence class bad if there exists some $a$ in the equivalence classes failing equation (REF ).",
"In a given bad equivalence class we clearly have the bound $\\left|\\bigcup _{a^{\\prime } : a \\sim _i a^{\\prime }} X(a^{\\prime })\\right| \\le \\frac{|X|}{(\\log t_{k + 1})^{100}}.$ A simple computation shows that we stay within the error term of Proposition REF when we sum over all $i$ and all bad equivalence classes.",
"We still have to deal with those $a$ failing equation (REF ), equation (REF ) or equation (REF ).",
"To do so, we will use the ideas introduced in [32].",
"Call $a$ generic if $D_{a, 2} \\cap D_{a, 2}^\\vee = \\lbrace 0\\rbrace $ , where we view $D_{a, 2}$ and $D_{a, 2}^\\vee $ as subspaces of $V$ .",
"Let us now suppose that $r = r^{\\prime }(a, X)$ , the other case can be dealt with similarly.",
"Take a non-zero vector $v \\in \\mathbb {F}_2^r$ with $\\lambda $ ones with $v \\ne L$ and $v \\ne R$ .",
"We claim that $\\frac{|\\lbrace a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l) : v \\in D_{a, 2} \\cap D_{a, 2}^\\vee , r = r^{\\prime }(a, X)\\rbrace |}{|\\lbrace a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l) : r = r^{\\prime }(a, X)\\rbrace |} = O(2^{-r - \\lambda }).$ We have that the proportion of $a$ with $v \\in D_{a, 2}$ is equal to $O(2^{-r})$ .",
"Furthermore, the condition that also $v \\in D_{a, 2}^\\vee $ implies that for every $i$ with $\\pi _i(v) = 1$ we have $a(i, -1) = 1$ .",
"These are $O(2^{-\\lambda })$ independent extra conditions giving a total of $O(2^{-r-\\lambda })$ .",
"This establishes the claim.",
"For the case that $v = L$ or $v = R$ , we make fundamental use of the fact that $|l|$ is equivalent to 3 modulo 4 to show that the above proportion is still $O(2^{-r})$ .",
"Summing over all non-zero vectors $v \\in V$ then gives that the proportion of $a \\in \\widetilde{\\text{Map}}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace , \\tilde{j}, l)$ , which are not generic, is bounded by $O\\left(\\sum _{\\lambda = 1}^r 2^{-r - \\lambda } \\binom{r}{\\lambda }\\right) = O\\left(0.75^r\\right).$ Take some $v, w \\in V$ .",
"Recall that the proportion of $a$ with $v \\in D_{a, 2}$ is bounded by $O(2^{-r})$ provided that $v \\notin \\langle L \\rangle $ .",
"Similarly, the proportion of $a$ with $w \\in D_{a, 2}^\\vee $ is bounded by $O(2^{-r})$ if $w \\notin \\langle R \\rangle $ .",
"Finally, if $a$ is generic, the proportion of $a$ with $(v, w) \\in D_{a, 2} \\times D_{a, 2}^\\vee $ is bounded by $O(4^{-r})$ as long as $v \\notin \\langle L \\rangle $ and $w \\notin \\langle R \\rangle $ .",
"But Hoeffding's inequality yields that the proportion of $(v, w) \\in V \\times V$ satisfying $\\left|\\left|\\left\\lbrace i \\in S_{\\textup {pre}, 4} : \\pi _i(v + w) = 0\\right\\rbrace \\right| - \\frac{\\alpha _{\\textup {pre}}}{2}\\right| > \\frac{\\alpha _{\\textup {pre}}}{\\log \\log \\log \\log N}$ is at most $O\\left(\\exp \\left(-(\\log \\log \\log \\log N)^{-2} \\cdot \\alpha _{\\textup {pre}}\\right)\\right).$ From the last two observations we quickly deduce that the proportion of generic $a$ for which equation (REF ) fails is also bounded by $O\\left(\\exp \\left(-(\\log \\log \\log \\log N)^{-2} \\cdot \\alpha _{\\textup {pre}}\\right)\\right),$ and a similar argument applies for the proportion of $a$ failing equation (REF ).",
"For the proportion of $a$ failing equation (REF ) it is even easier to get an upper bound.",
"We have now found an upper bound for the proportion of $a$ failing equation (REF ), equation (REF ) or equation (REF ).",
"From Lemma REF and Lemma REF we directly deduce an upper bound for the proportion of $a$ that are $(N, m, X)$ -acceptable but not very $(N, m, X)$ -acceptable.",
"To finish the proof, we merely need to bound the union of $X(a)$ over these $a$ .",
"This follows from Proposition REF .",
"We remark that we can always find variable indices as in Definition REF if $a$ is very $(N, m, X)$ -acceptable and $N$ is sufficiently large.",
"This is a simple computation once we use that $m < \\log \\log \\log \\log \\log \\log N,$ since otherwise Theorem REF is trivial.",
"We now have all the required setup for our next proposition, where we fix one prime for all indices smaller than $k_{\\textup {gap}}$ except the variable indices.",
"Proposition 6.9 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"There are $c, A, N_0 > 0$ such that for all integers $N > N_0$ , all integers $m \\ge 2$ , all $N$ -good boxes $X$ , all very $(N, m, X)$ -acceptable $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ , all sequences of valid Artin pairings $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}$ , all non-zero multiplicative characters $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ , all sets of variable indices $S$ for $F$ and all $Q \\in \\prod _{i \\in [k_{\\textup {gap}}] - S} X_i$ such that $|X_j(a, Q)| \\ge 4^{-k_{\\textup {gap}}} \\cdot |X_j|$ for all $j \\in S$ , we have $\\left|\\sum _{x \\in X(a, Q, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})} F\\left(\\textup {Art}_{a, m, i(x)}\\right)\\right| \\le \\frac{A \\cdot |X(a, Q)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m 6^m}}}.$ Here $X(a, Q, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})$ is defined as the subset of $x \\in X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})$ with $\\pi _i(x)$ equal to $\\pi _i(Q)$ for $i \\in [k_{\\textup {gap}}] - S$ .",
"[Proof that Proposition REF implies Proposition REF ] The proof is almost identical to the one given in [2].",
"We have to show that equation (REF ) is typically satisfied.",
"We apply Proposition REF to $(X_{k + 1}(a, Q^{\\prime }) \\times \\dots \\times X_r(a, Q^{\\prime }), Q^{\\prime }),$ where $Q^{\\prime }$ is the unique element of $X_1 \\times \\dots \\times X_k$ .",
"Crucially, all the required conditions for Proposition REF are satisfied due to equation (REF ), completing our reduction step.",
"It is time for our final reduction step.",
"If $c_{j_1, j_2}(F) \\ne 0$ for some $1 \\le j_1, j_2 \\le n_{a, m}$ , Smith's method applies without any significant changes.",
"If however $c_{j_1, j_2}(F) = 0$ for all $1 \\le j_1, j_2 \\le n_{a, m}$ , Smith's method breaks down.",
"It is here that we make essential use of our generalized Rédei reciprocity law.",
"We shall now add the algebraic structure needed to apply our reflection principles.",
"The required equidistribution will then be a consequence of the Chebotarev Density Theorem and Proposition REF .",
"From now on we shall make heavy use of the notation introduced in Subsections REF and REF .",
"Definition 6.10 Take a $N$ -good box $X$ , a very $(N, m, X)$ -acceptable $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ and a non-zero multiplicative character $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ .",
"Let $S$ be a set of variable indices for $F$ .",
"Fix a choice of $j_1$ and $j_2$ with $c_{j_1, j_2}(F) \\ne 0$ as in Definition REF .",
"Put $S^{\\prime } := S \\cap [k_{\\textup {gap}}]$ .",
"For each $i \\in S^{\\prime }$ , let $Z_i$ be subsets of $X_i$ with cardinality $M_{\\textup {box}} := \\left\\lfloor (\\log \\log \\log \\log N)^{\\frac{1}{5(m + 1)}} \\right\\rfloor .$ Note that $M_{\\textup {box}} \\ge 2$ for $N$ greater than an absolute constant by equation (REF ).",
"Put $Z := \\prod _{i \\in S^{\\prime }} Z_i, \\quad Z^{\\prime } := \\prod _{i \\in S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace } Z_i.$ If $j_1 \\le n_{a, m}$ , we say that $Z$ is well-governed for $(a, F)$ if for every distinct $a_1, a_2 \\in Z_{i_1(j_1, j_2)}$ there is a governing expansion $\\mathfrak {G}$ on $(Z^{\\prime }, S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace , a_1a_2)$ .",
"Put $M_\\circ (Z) := \\prod _{\\begin{array}{c}a_1, a_2 \\in Z_{i_1(j_1, j_2)} \\\\ a_1 \\ne a_2\\end{array}} \\prod _{T \\subsetneq S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace } \\prod _{\\bar{x} \\in \\overline{Y}_T} L(\\phi _{\\bar{x}, a_1a_2})$ and $M(Z) := \\prod _{\\begin{array}{c}a_1, a_2 \\in Z_{i_1(j_1, j_2)} \\\\ a_1 \\ne a_2\\end{array}} \\prod _{\\bar{x} \\in \\overline{X}_{S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }} L(\\phi _{\\bar{x}, a_1a_2}).$ If $j_1 = n_{a, m} + 1$ , we say that $Z$ is well-governed for $(a, F)$ if there is a governing expansion $\\mathfrak {G}_l$ on $(Z, S^{\\prime }, l)$ and furthermore for each $j \\in S^{\\prime }$ and each $q \\in X_j$ there is a governing expansion $\\mathfrak {G}_{j, q}$ on $(\\prod _{i \\in S^{\\prime } - \\lbrace j\\rbrace } X_i, S^{\\prime } - \\lbrace j\\rbrace , q)$ .",
"Put $M_\\circ (Z) := \\prod _{T \\subsetneq S^{\\prime }} \\prod _{\\bar{x} \\in \\overline{X}_T} L(\\phi _{\\bar{x}, l}) \\prod _{j \\in S^{\\prime }} \\prod _{q \\in X_j} \\prod _{\\bar{x} \\in \\overline{X}_{S^{\\prime } - \\lbrace j\\rbrace }} L(\\phi _{\\bar{x}, q})$ and $M(Z) := M_\\circ (Z) \\prod _{\\bar{x} \\in \\overline{X}_{S^{\\prime }}} L(\\phi _{\\bar{x}, l}).$ so $M(Z)$ is a central Galois extension of $M_\\circ (Z)$ in both cases.",
"Take some $Q \\in \\prod _{i \\in [k_{\\textup {gap}}] - S^{\\prime }} X_i$ .",
"Then we define, for $i > k_{\\textup {gap}}$ , $X_i(a, Q, M_\\circ (Z))$ to be the subset of primes $p \\in X_i$ such that $p$ splits completely in $M_\\circ (Z)$ , $p \\in X_i(a, Q)$ and $\\left(\\frac{z}{p}\\right) = a(j, i) \\textup { for all } j \\in S^{\\prime } \\textup { and all } z \\in Z_j.$ Note that these conditions are equivalent to $\\textup {Frob}_p$ being equal to a given central element in the Galois group of the compositum of $M_\\circ (Z)$ and $\\mathbb {Q}(\\sqrt{x})$ with $x$ running through $-1$ , the prime divisors of $Q$ and the primes in $Z_j$ for $j \\in S^{\\prime }$ .",
"We let $\\widetilde{Z} := Q \\times Z \\times \\prod _{i > k_{\\textup {gap}}} X_i(a, Q, M_\\circ (Z)).$ We call $\\widetilde{Z}$ a satisfactory product space for $(X, a, F, Q)$ if $Z$ is well-governed for $(a, F)$ , the primes in $Q$ split completely in $M_\\circ (Z)$ and if we have for all $i < j$ with $i, j \\in S^{\\prime }$ , all $z_i \\in Z_i$ and all $z_j \\in Z_j$ $\\left(\\frac{z_i}{z_j}\\right) = a(i, j) = 1$ and $Z_i \\subseteq X_i(a, Q)$ .",
"Once we added the necessary algebraic structure to our box, we can construct a suitable additive system $\\mathfrak {A}$ to which we apply Proposition REF .",
"This is the goal of the next lemma, which provides the critical link between our algebraic results and Proposition REF .",
"Lemma 6.11 Let a very $(N, m, X)$ -acceptable $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ , a sequence of valid Artin pairings $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}$ , a non-zero multiplicative character $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ and a set of variable indices $S$ for $F$ be given.",
"Take $\\widetilde{Z}$ to be a satisfactory product space for $(X, a, F, Q)$ .",
"Then there is a $(2^{n_{\\textup {max}}(n_{\\textup {max}} + m + 2)}, S)$ -acceptable additive system $\\mathfrak {A}$ with $\\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A}) = \\widetilde{Z} \\cap X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})$ such that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (F(\\textup {Art}_{a, m, x})) = \\phi _{\\pi _{S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{x}), c}(\\textup {Frob}(p_1) \\cdot \\textup {Frob}(p_2))$ for all $\\bar{x} \\in C(\\mathfrak {A})$ , where $(p_1, p_2) := \\pi _{i_2(j_1, j_2)}(\\bar{x})$ .",
"Here $c$ equals $\\textup {pr}_1(\\pi _{i_1(j_1, j_2)}(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _{i_1(j_1, j_2)}(\\bar{x}))$ if $j_1 \\le n_{a, m}$ and equals $l$ otherwise.",
"We shall proceed to explicitly construct $\\mathfrak {A}$ by induction.",
"We start by introducing some notation.",
"Let $w \\in D_{a, 2}^\\vee $ be one of the chosen basis vectors and let $x \\in X(a)$ be given.",
"A raw cocycle for $(x, w)$ is a sequence $\\lbrace \\psi _{x, w, i}\\rbrace _{i = 0}^k$ of maximal length with $\\psi _{x, w, i} \\in \\text{Cocy}(G_\\mathbb {Q}, N(x)[2^i])$ , $2\\psi _{x, w, i + 1} = \\psi _{x, w, i}$ , $L(\\psi _{x, w, i})/\\mathbb {Q}(\\sqrt{x})$ unramified and $\\psi _{x, w, 1} = \\sum _{i = 1}^{r^{\\prime }(a, X)} \\pi _i(w) \\chi _i$ with $\\chi _i$ as in Definition REF .",
"A raw cocycle $\\mathfrak {R}(w)$ for $(X(a), w)$ is a choice of raw cocycle for every $(x, w)$ with $x \\in X(a)$ .",
"Recall that $i_1(j_1, j_2)$ , $i_2(j_1, j_2)$ , $j_1$ and $j_2$ are the integers associated to our set of variable indices $S$ as in Definition REF .",
"Set $\\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A}) := \\widetilde{Z} \\cap X(a, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}).$ If $j_1 = n_{a, m} + 1$ , we claim that there is a governing expansion $\\mathfrak {G}^{\\prime }$ on $(Z, S^{\\prime }, q_1q_2)$ for any two distinct elements $q_1, q_2 \\in X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ .",
"To prove the claim, take any prime $q$ in $X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ .",
"We shall prove by induction on $T \\subseteq S^{\\prime }$ that there is a governing expansion $\\mathfrak {G}^{\\prime }(T)$ on $\\left(\\prod _{i \\in T} Z_i, T, q\\right).$ Once this is proven, the claim follows upon taking $T = S^{\\prime }$ and using equation (REF ).",
"In the base case $T = \\emptyset $ our inductive statement is clear.",
"Now suppose that $|T| > 0$ .",
"By Proposition REF and induction, it suffices to show that for every $i \\in T$ , every $p \\in Z_i$ and every $\\bar{x} \\in \\overline{X}_{T - \\lbrace i\\rbrace }$ , we have that $p$ splits completely in $L(\\phi _{\\bar{x}, q})$ .",
"We have already checked, earlier in our induction, that $p$ splits completely in $\\prod _{j \\in T - \\lbrace i\\rbrace } L(\\phi _{\\pi _{T - \\lbrace j\\rbrace }(\\bar{x}), q}).$ By Theorem REF the splitting of $p$ in $L(\\phi _{\\bar{x}, q})$ is equivalent to $q$ splitting completely in $L(\\phi _{\\bar{x}, p})$ , which is ensured by the fact that $q \\in X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ .",
"Note that $p$ and $q$ are 1 modulo 4, so that the sets $\\text{Ram}(\\mathbb {Q}(\\chi _p)/\\mathbb {Q})$ and $\\text{Ram}(\\mathbb {Q}(\\chi _q)/\\mathbb {Q})$ are disjoint.",
"This establishes the claim.",
"Let $K$ be the field obtained by adjoining $\\sqrt{l}$ and all the $\\sqrt{z_i}$ and $\\sqrt{q}$ to $\\mathbb {Q}$ , where $z_i \\in Z_i$ for some $i \\in S^{\\prime }$ and $q \\in X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ .",
"Take $M$ to be the narrow Hilbert class field of $K$ .",
"For each prime $p$ ramifying in $M$ , any inertia subgroup at $p$ has size 2 and hence precisely one non-trivial element.",
"Choose such an element $\\sigma _p$ for every $p$ that ramifies in $M$ .",
"First suppose that $j_1 \\le n_{a, m}$ .",
"To shorten our formulas, we define for $\\bar{x} \\in \\overline{X}_S$ and $i \\in S$ $\\text{prp}(\\bar{x}, i) = \\textup {pr}_1(\\pi _i(\\bar{x})) \\cdot \\textup {pr}_2(\\pi _i(\\bar{x})).$ Then we can always choose our maps $\\phi _{\\bar{x}, \\text{prp}(\\bar{x}, i_1(j_1, j_2))} : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ in such a way that $\\phi _{\\bar{x}, \\text{prp}(\\bar{x}, i_1(j_1, j_2))}(\\sigma _p) = 0$ for all $p$ , where $\\bar{x}$ runs over all elements of $\\overline{Z}_T$ with $T \\subseteq S^{\\prime }$ not containing $i_1(j_1, j_2)$ .",
"Proposition 2.3 of Smith [33] shows that with this choice the $\\phi $ maps are additive, in other words equation (REF ) holds.",
"Let $T \\subsetneq S$ .",
"We shall construct our maps $F_{T^{\\prime }}$ with $T^{\\prime } \\subseteq T$ in such a way that $\\overline{Y}_T(\\mathfrak {A})$ is precisely the set of cubes $\\bar{x}$ satisfying $\\bar{x}(\\emptyset ) \\subseteq \\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})$ and the following properties we have for all $T^{\\prime } \\subsetneq T$ with $i_2(j_1, j_2) \\notin T^{\\prime }$ , all $\\bar{y} \\in \\bar{x}(T^{\\prime })$ and all $j \\ne j_2$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_j, |T^{\\prime }|} = 0;$ we have for all $T^{\\prime } \\subsetneq T$ with $i_2(j_1, j_2) \\notin T^{\\prime }$ and all $\\bar{y} \\in \\bar{x}(T^{\\prime })$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_{j_2}, |T^{\\prime }|} = \\left\\lbrace \\begin{array}{ll}\\phi _{\\pi _{T^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{y}), \\text{prp}(\\bar{x}, i_1(j_1, j_2))} & \\mbox{if } i_1(j_1, j_2) \\in T^{\\prime }\\\\0 & \\mbox{if } i_1(j_1, j_2) \\notin T^{\\prime };\\end{array}\\right.$ we have for all $T^{\\prime } \\subsetneq T$ with $i_2(j_1, j_2) \\notin T^{\\prime }$ , $\\bar{y} \\in \\bar{x}(T^{\\prime })$ , all $j$ and $i \\in S - T^{\\prime }$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_j, |T^{\\prime }| + 1}(\\sigma _{\\pi _i(\\bar{x})}) = 0.$ Now suppose that $j_1 = n_{a, m} + 1$ .",
"We now choose our maps $\\phi _{\\bar{x}, q_1q_2} : G_\\mathbb {Q}\\rightarrow \\mathbb {F}_2$ in such a way that $\\phi _{\\bar{x}, q_1q_2}(\\sigma _p) = 0$ for all $p$ , for all $\\bar{x} \\in \\overline{Z}_T$ with $T \\subseteq S^{\\prime }$ and all $q_1, q_2 \\in X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ .",
"Let $T \\subseteq S$ .",
"In this case we construct our maps $F_{T^{\\prime }}(\\mathfrak {A})$ such that $\\overline{Y}_T(\\mathfrak {A})$ equals the cubes $\\bar{x}$ with $\\bar{x}(\\emptyset ) \\subseteq \\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})$ and we have for all $T^{\\prime } \\subsetneq T$ , all $\\bar{y} \\in \\bar{x}(T^{\\prime })$ and all $j \\ne j_2$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_j, |T^{\\prime }|} = 0;$ we have for all $T^{\\prime } \\subsetneq T$ and all $\\bar{y} \\in \\bar{x}(T^{\\prime })$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_{j_2}, |T^{\\prime }|} = \\left\\lbrace \\begin{array}{ll}\\phi _{\\pi _{T^{\\prime } - \\lbrace i_2(j_1, j_2)\\rbrace }(\\bar{y}), \\text{prp}(\\bar{x}, i_2(j_1, j_2))} & \\mbox{if } i_2(j_1, j_2) \\in T^{\\prime }\\\\0 & \\mbox{if } i_2(j_1, j_2) \\notin T^{\\prime };\\end{array}\\right.$ we have for all $T^{\\prime } \\subsetneq T$ , $\\bar{y} \\in \\bar{x}(T^{\\prime })$ , all $j$ and $i \\in S - T^{\\prime }$ $\\sum _{y \\in \\bar{y}(\\emptyset )} \\psi _{y, w_j, |T^{\\prime }| + 1}(\\sigma _{\\pi _i(\\bar{x})}) = 0.$ Let us prove by induction that $\\overline{Y}_T(\\mathfrak {A})$ is as claimed.",
"We shall construct the map $F_T(\\mathfrak {A})$ during the induction.",
"Until otherwise stated, we shall treat the case $j_1 \\le n_{a, m}$ .",
"At the end we indicate the modifications necessary to deal with the case $j_1 = n_{a, m} + 1$ .",
"Take $\\bar{x} \\in \\overline{Y}_T(\\mathfrak {A})$ .",
"If $i_2(j_1, j_2) \\in T$ or $T = S - \\lbrace i_2(j_1, j_2)\\rbrace $ , we simply let $F_T$ be the zero map.",
"Henceforth we will assume that $i_2(j_1, j_2) \\notin T$ and $|T| < |S| - 1$ .",
"Then we define $\\psi _j :=\\left\\lbrace \\begin{array}{ll}\\sum \\limits _{x \\in \\overline{x}(\\emptyset )} \\psi _{x, w_j, |T|} & \\mbox{if } j \\ne j_2 \\text{ or } i_1(j_1, j_2) \\notin T \\\\\\phi _{\\pi _{T - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{x}), \\text{prp}(\\bar{x}, i_1(j_1, j_2))} + \\sum \\limits _{x \\in \\overline{x}(\\emptyset )} \\psi _{x, w_{j_2}, |T|} & \\mbox{otherwise.",
"}\\end{array}\\right.$ In the former case Proposition 2.5 of Smith [33] implies that $\\psi _j$ is a quadratic character of $G_\\mathbb {Q}$ , while in the latter case Proposition 2.6 of Smith [33] demonstrates that $\\psi _j$ is a quadratic character.",
"Take a point $x \\in \\bar{x}(\\emptyset )$ .",
"We claim that $\\psi _j$ is an unramified character of $\\mathbb {Q}(\\sqrt{x})$ .",
"If $p = \\pi _i(\\bar{x})$ with $i \\notin T$ , this is clear.",
"So suppose that $i \\in T$ and write $\\pi _i(\\bar{x}) = \\lbrace p_1, p_2\\rbrace $ with $p_1 = \\pi _i(x)$ .",
"It is clear that $\\psi _j$ does not ramify at $p_1$ , so it suffices to show that $\\psi _j$ does not ramify at $p_2$ .",
"Let $\\bar{y}_k \\in \\bar{x}(T - \\lbrace i\\rbrace )$ be the cube with $\\pi _i(\\bar{y}_k) = p_k$ .",
"Then we have $\\psi _j(\\sigma _{p_2}) = \\sum _{x \\in \\overline{x}(\\emptyset )} \\psi _{x, w_j, |T|}(\\sigma _{p_2}) = \\sum _{y \\in \\overline{y}_1(\\emptyset )} \\psi _{y, w_j, |T|}(\\sigma _{p_2}) + \\sum _{y \\in \\overline{y}_2(\\emptyset )} \\psi _{y, w_j, |T|}(\\sigma _{p_2}) = 0 + 0 = 0.$ The first sum is clearly zero, since all the $\\psi _{y, w_j, |T|}$ with $y \\in \\bar{y}_1(\\emptyset )$ are unramified at $p_2$ .",
"Furthermore, the second sum is zero by equation (REF ) with $T^{\\prime } := T - \\lbrace i\\rbrace $ .",
"This proves our claim.",
"Next we claim that $\\pi _i(\\bar{x})$ splits completely in $L(\\psi _j)$ for all $i \\notin T$ .",
"But indeed, we even have that $\\pi _i(\\bar{x})$ has residue field degree 1 in every $\\psi _{x, w_j, |T|}$ for $x \\in \\bar{x}(\\emptyset )$ because $2\\psi _{x, w_j, |T| + 1} = \\psi _{x, w_j, |T|}$ .",
"Pick some $x \\in \\bar{x}(\\emptyset )$ and let $p \\in \\pi _i(x)$ for some $i \\in T$ .",
"It is straightforward to deduce from $\\bar{x}(\\emptyset ) \\subseteq X(a)$ that $\\psi _j|_{G_{\\mathbb {Q}(\\sqrt{x})}}(\\text{Frob}(\\mathfrak {p}))$ does not depend on $x$ , where $\\mathfrak {p}$ is the unique ideal above $p$ in $\\mathbb {Q}(\\sqrt{x})$ .",
"From this, it becomes clear, from the additivity of $\\psi _j$ , that this defines an additive map $F_{T, j, 1}$ to $\\mathbb {F}_2^{|T|}$ .",
"It follows from Lemma REF that there exists a set $A \\subseteq [r]$ and a bijection $f: [n_{a, 2} + 1] \\rightarrow A$ such that $A \\cap S = \\emptyset $ and $\\pi _{f(i)}(w_k) = \\delta _{i, k}$ for $1 \\le i, k \\le n_{a, 2}$ and furthermore $\\pi _{f(n_{a, 2} + 1)}(w_k) = 0$ for all $1 \\le k \\le n_{a, 2}$ .",
"Then we define an additive map $F_{T, j, 2}$ to $\\mathbb {F}_2^{n_{a, 2} + 1}$ by $(\\psi _j(\\sigma _{\\pi _i(x)}))_{i \\in A}.$ Finally, we define an additive map $F_{T, j, 3}$ to $\\mathbb {F}_2^{|S| - |T|}$ by sending $\\bar{x}$ to $\\left(\\sum _{x \\in \\bar{x}(\\emptyset )} \\psi _{x, w_j, |T| + 1}(\\sigma _{\\pi _i(\\bar{x})})\\right)_{i \\in S - T}.$ We define our map $F_T(\\mathfrak {A})$ to be $(F_{T, j, 1}, F_{T, j, 2}, F_{T, j, 3})_{1 \\le j \\le n_{a, 2}}$ .",
"Note that the maps $F_{T, j, 1}$ and $F_{T, j, 2}$ encode precisely when $\\psi _j = 0$ .",
"From this it becomes obvious that $\\overline{Y}_T(\\mathfrak {A})$ has the claimed shape.",
"Our next task is to verify that our additive system is $(2^{n_{\\text{max}}(n_{\\text{max}} + m + 2)}, S)$ -acceptable.",
"For the first requirement, this is obvious from the construction of $F_T$ above and the inequality $n_{a, 2} \\le n_{\\text{max}}$ .",
"We still need to deal with the second requirement.",
"Take $\\bar{x} \\in C(\\mathfrak {A})$ .",
"If there is some $i \\in S$ such that $|\\bar{x}(S - \\lbrace i\\rbrace ) \\cap \\overline{Y}_{S - \\lbrace i\\rbrace }^\\circ (\\mathfrak {A})| = 2,$ then we are done.",
"Henceforth we assume that $|\\bar{x}(S - \\lbrace i\\rbrace ) \\cap \\overline{Y}_{S - \\lbrace i\\rbrace }^\\circ (\\mathfrak {A})| = 1$ for all $i \\in S$ and let $x_0$ be the unique element in $\\bar{x}(\\emptyset )$ outside $\\bar{x}(S - \\lbrace i\\rbrace ) \\cap \\overline{Y}_{S - \\lbrace i\\rbrace }^\\circ (\\mathfrak {A})$ for all $i \\in S$ .",
"Then we need to prove that $x_0 \\in \\overline{Y}_\\emptyset ^\\circ (\\mathfrak {A})$ .",
"Clearly, $x_0 \\in \\widetilde{Z} \\cap X(a)$ .",
"Take an integer $2 \\le m^{\\prime } \\le m - 1$ , integers $1 \\le j_1^{\\prime } \\le n_{a, m^{\\prime }} + 1$ and $1 \\le j_2^{\\prime } \\le n_{a, m^{\\prime }}$ .",
"It suffices to prove that $\\iota (E_{j_1^{\\prime }, j_2^{\\prime }}(\\text{Art}_{a, m^{\\prime }, x_0})) = \\iota (E_{j_1^{\\prime }, j_2^{\\prime }}(\\text{Art}_{a, m^{\\prime }})).$ Choose a subset $T$ of $S$ of size $m^{\\prime }$ not containing $i_1(j_1, j_2)$ and $i_2(j_1, j_2)$ .",
"Then the above identity follows from Theorem REF applied to any cube in $\\bar{x}(T)$ containing $x_0$ .",
"We still need to prove equation (REF ).",
"Recall that $j_1 \\le n_{a, m}$ .",
"Take some indices $(j_3, j_4)$ with $(j_3, j_4) \\ne (j_1, j_2)$ .",
"We claim that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_3, j_4}(\\textup {Art}_{a, m, x})) = 0.$ First suppose that $j_3 \\le n_{a, m}$ .",
"Then this follows from two applications of Theorem REF .",
"In case $j_3 = n_{a, m} + 1$ we apply Theorem REF twice to obtain $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_3, j_4}(\\textup {Art}_{a, m, x})) = 0.$ Here we use equation (REF ), if $l > 0$ , and equation (REF ), if $l < 0$ .",
"We deduce that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_1, j_2}(\\textup {Art}_{a, m, x})) = \\phi _{\\pi _{S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{x}), \\text{prp}(\\bar{x}, i_1(j_1, j_2))}(\\text{Frob}(p_1) \\cdot \\text{Frob}(p_2)).$ Adding these identities together yields equation (REF ).",
"This proves the lemma for $j_1 \\le n_{a, m}$ .",
"It remains to indicate the necessary changes in case $j_1 = n_{a, m} + 1$ .",
"In this case we let $F_T$ be the zero map if $T = S$ .",
"Otherwise we define $\\psi _j :=\\left\\lbrace \\begin{array}{ll}\\sum \\limits _{x \\in \\overline{x}(\\emptyset )} \\psi _{x, w_j, |T|} & \\mbox{if } j \\ne j_2 \\text{ or } i_2(j_1, j_2) \\notin T \\\\\\phi _{\\pi _{T - \\lbrace i_2(j_1, j_2)\\rbrace }(\\bar{x}), \\text{prp}(\\bar{x}, i_2(j_1, j_2))} + \\sum \\limits _{x \\in \\overline{x}(\\emptyset )} \\psi _{x, w_{j_2}, |T|} & \\mbox{otherwise.",
"}\\end{array}\\right.$ Now we proceed by defining the maps $F_{T, j, i}$ just as in the case $j_1 \\le n_{a, m}$ .",
"Then we see that $\\mathfrak {A}$ is certainly $(2^{n_{\\text{max}}(n_{\\text{max}} + m + 2)}, S)$ -acceptable.",
"Now we have for all $(j_3, j_4)$ with $j_3 \\le n_{a, m}$ $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_3, j_4}^{c_{j_3, j_4}(F)}(\\textup {Art}_{a, m, x})) = 0$ simply because $c_{j_3, j_4}(F) = 0$ by our choice of variable indices.",
"Furthermore, Theorem REF shows that for all $(j_3, j_4)$ with $j_3 = n_{a, m + 1}$ and $j_2 \\ne j_4$ $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_3, j_4}(\\textup {Art}_{a, m, x})) = 0.$ Finally, Theorem REF and Theorem REF imply that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (E_{j_1, j_2}(\\textup {Art}_{a, m, x})) &= \\phi _{\\pi _{S^{\\prime }}(\\bar{x}), \\text{prp}(\\bar{x}, i_2(j_1, j_2))}(\\text{Frob}(l)) \\\\&= \\phi _{\\pi _{S^{\\prime }}(\\bar{x}), l}(\\textup {Frob}(p_1) \\cdot \\textup {Frob}(p_2))$ with $(p_1, p_2) := \\pi _{i_2(j_1, j_2)}(\\bar{x})$ .",
"Here $\\text{Frob}(l)$ is to be interpreted as $\\text{Frob}(|l|) \\cdot \\text{Frob}(\\infty )$ if $l < 0$ .",
"Hence we conclude that $\\sum _{x \\in \\bar{x}(\\emptyset )} \\iota (F(\\textup {Art}_{a, m, x})) = \\phi _{\\pi _{S^{\\prime }}(\\bar{x}), l}(\\textup {Frob}(p_1) \\cdot \\textup {Frob}(p_2)),$ which completes the proof of our lemma because $i_1(j_1, j_2) = \\tilde{j} \\notin S^{\\prime }$ in this case.",
"Proposition 6.12 Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"There are $c, A, N_0 > 0$ such that for all integers $N > N_0$ , all integers $m \\ge 2$ , all $N$ -good boxes $X$ , all very $(N, m, X)$ -acceptable $a \\in \\textup {Map}(M \\sqcup M_\\emptyset , \\lbrace \\pm 1\\rbrace )$ , all sequences of valid Artin pairings $\\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1}$ , all non-zero multiplicative characters $F : \\textup {Mat}(n_{a, m} + 1, n_{a, m}, \\mathbb {F}_2) \\rightarrow \\lbrace \\pm 1\\rbrace $ , all sets of variable indices $S$ for $F$ , all $Q \\in \\prod _{i \\in [k_{\\textup {gap}}] - S} X_i$ and all satisfactory product spaces $\\widetilde{Z}$ for $(X, a, F, Q)$ $\\left|\\sum _{x \\in \\widetilde{Z} \\cap X(a, Q, \\lbrace \\textup {Art}_{a, i}\\rbrace _{2 \\le i \\le m - 1})} F\\left(\\textup {Art}_{a, m, i(x)}\\right)\\right| \\le \\frac{A \\cdot |\\widetilde{Z} \\cap X(a, Q)|}{(\\log \\log \\log \\log N)^{\\frac{c}{m 6^m}}}.$ [Proof that Proposition REF implies Proposition REF ] The proof is very similar to the proof of Proposition 7.5 implies Proposition 7.4 in Smith [33].",
"We only indicate the necessary changes here.",
"There is a small gap in Smith's argument, namely when he applies the Chebotarev Density Theorem on page 81.",
"Indeed, Smith does not argue why there are no Siegel zeroes.",
"Fortunately, this can be easily overcome by an appeal to the classical result of Heilbronn [17] and the fact that our box $X$ is Siegel-less.",
"We need to construct an additive system $\\mathfrak {A}^{\\prime }$ on $S^{\\prime }$ that guarantees the existence of the governing expansions $\\mathfrak {G}$ , $\\mathfrak {G}_l$ and $\\mathfrak {G}_{j, q}$ .",
"This is done in Lemma REF and Proposition 3.3 of Smith [33], but it is essential here that $a(i, -1) = 1$ for all $i \\in S^{\\prime }$ and $a(i, j) = 1$ for all distinct $i, j \\in S^{\\prime }$ to ensure the validity of equation (REF ).",
"Now let $Z$ and $Z^{\\prime }$ be well-governed for $(a, F)$ and suppose that $Z \\cap Z^{\\prime } = \\lbrace x\\rbrace $ .",
"Let $K$ be the field obtained by adjoining $\\sqrt{p}$ to $\\mathbb {Q}$ where $p$ runs over all the prime divisors of $x$ .",
"Then, for Smith's reduction step to work, we need to prove that $[KM_\\circ (Z) M_\\circ (Z^{\\prime }) : K] = [KM_\\circ (Z) : K]^2 = [KM_\\circ (Z^{\\prime }) : K]^2,$ which follows from Proposition 2.4 of Smith [33].",
"[Proof of Proposition REF ] This proof is similar to the proof of Proposition 7.5 in Smith [33] except that one needs to use the additive system constructed in Lemma REF instead of the one constructed in Section 3 of Smith [33].",
"We will now give all the details.",
"Take $\\sigma \\in \\mathrm {Gal}(M(Z)/M_\\circ (Z))$ and define $X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z), \\sigma )$ to be the subset of $p \\in X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ that map to $\\sigma $ under Frobenius.",
"By [33] we have an isomorphism $\\mathrm {Gal}(M(Z)/M_\\circ (Z)) \\cong {G}_{S^{\\prime }}(Z)$ by sending $\\sigma $ to the map $\\bar{x} \\mapsto \\left\\lbrace \\begin{array}{ll}\\phi _{\\bar{x}, l}(\\sigma ) & \\mbox{if } j_1 = n_{a, m} + 1 \\\\\\phi _{\\pi _{S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{x}), \\text{prp}(\\bar{x}, i_1(j_1, j_2))}(\\sigma ) & \\mbox{otherwise.",
"}\\end{array}\\right.$ The Chebotarev Density Theorem and Lemma REF imply that $|X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z), \\sigma )| = \\frac{|X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))|}{2^{(M_{\\text{box}} - 1)^{|S^{\\prime }|}}} \\cdot \\left(1 + O\\left(e^{-2k_{\\text{gap}}}\\right)\\right).$ Then it follows from Proposition REF that for almost all choices of $Q^{\\prime } \\in \\prod _{i \\in [r] - [k_{\\text{gap}}] - \\lbrace i_2(j_1, j_2)\\rbrace } X_i(a, Q, M_\\circ (Z)) \\quad \\text{ with } \\quad \\left(\\frac{\\pi _i(Q^{\\prime })}{\\pi _j(Q^{\\prime })}\\right) = a(i, j),$ we have $|X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z), \\sigma )| = \\frac{|X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))|}{2^{(M_{\\text{box}} - 1)^{|S^{\\prime }|}}} \\cdot \\left(1 + O\\left(e^{-k_{\\text{gap}}}\\right)\\right)$ for each $\\sigma $ , where $X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))$ is the subset of $X_{i_2(j_1, j_2)}(a, Q, M_\\circ (Z))$ projecting to $Q^{\\prime }$ , and similarly for $X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z), \\sigma )$ .",
"We now apply Proposition REF to the space $Z \\times [M_{\\text{box}}]$ with $\\epsilon = \\frac{1}{(\\log \\log \\log \\log N)^{\\frac{c}{(m + 1)6^m}}}$ for some sufficiently small constant $c$ .",
"Let $g_0 \\in {G}_S(Z \\times [M_{\\text{box}}])$ be the function guaranteed by Proposition REF .",
"If we pick primes $x_1, \\dots , x_{M_{\\text{box}}} \\in X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))$ , then we have an obvious isomorphism $\\varphi : Z \\times [M_{\\text{box}}] \\cong \\lbrace Q\\rbrace \\times \\lbrace Q^{\\prime }\\rbrace \\times Z \\times \\lbrace x_1, \\dots , x_{M_{\\text{box}}}\\rbrace .$ To the primes $x_1, \\dots , x_{M_{\\text{box}}} \\in X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))$ , we can associate a function $g_{x_1, \\dots , x_{M_{\\text{box}}}} \\in {G}_S(Z \\times [M_{\\text{box}}])$ by setting $(\\bar{z}, (i, j)) \\mapsto \\phi _{\\bar{z}}(\\text{Frob } x_i) + \\phi _{\\bar{z}}(\\text{Frob } x_j),$ where $\\phi _{\\bar{z}}$ is $\\phi _{\\bar{z}, l}$ or $\\phi _{\\pi _{S^{\\prime } - \\lbrace i_1(j_1, j_2)\\rbrace }(\\bar{z}), \\text{prp}(\\bar{z}, i_1(j_1, j_2))}$ depending on the value of $j_1$ .",
"In case $g = g_0$ , we get the desired oscillation from Proposition REF applied to the function $F(\\text{Art}_{a, m, i(x)})$ pulled back to $Z \\times [M_{\\text{box}}]$ via $\\varphi $ and the additive system $\\mathfrak {A}$ from Lemma REF also pulled back to $Z \\times [M_{\\text{box}}]$ via $\\varphi $ .",
"It remains to split the set $X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))$ in blocks of size $M_{\\text{box}}$ (and a small remainder) such that we have $g_{x_1, \\dots , x_{M_{\\text{box}}}} = g_0$ for almost every block.",
"For this we we claim that given $\\text{Frob}(x_1)$ , there is a unique choice of $\\text{Frob}(x_2), \\dots , \\text{Frob}(x_{M_{\\text{box}}})$ such that $g_{x_1, \\dots , x_{M_{\\text{box}}}} = g_0,$ and furthermore $\\text{Frob}(x_2), \\dots , \\text{Frob}(x_{M_{\\text{box}}})$ are linear functions of $\\text{Frob}(x_1)$ .",
"Once we establish the claim, we use equation (REF ) to partition $X_{i_2(j_1, j_2)}(a, Q \\times Q^{\\prime }, M_\\circ (Z))$ in the desired way.",
"To prove the claim, we remark that there is an isomorphism between ${G}_S(Z \\times [M_{\\text{box}}])$ and the sets of maps $g$ from $[M_{\\text{box}}] \\times [M_{\\text{box}}]$ to ${G}_{S^{\\prime }}(Z)$ satisfying $g(i, j) + g(j, k) = g(i, k).$ Hence, thinking of $g_0$ as a map from $[M_{\\text{box}}] \\times [M_{\\text{box}}]$ to ${G}_{S^{\\prime }}(Z)$ , we see that for any $1 < j \\le M_{\\text{box}}$ $\\phi _{\\bar{z}}(\\text{Frob } x_1) + \\phi _{\\bar{z}}(\\text{Frob } x_j) = g_0(1, j) \\in {G}_{S^{\\prime }}(Z),$ which uniquely specifies $\\text{Frob}(x_j)$ as linear function of $\\text{Frob}(x_1)$ and $g_0$ by equation (REF ).",
"Finally, we see that with this choice of $\\text{Frob}(x_2), \\dots , \\text{Frob}(x_{M_{\\text{box}}})$ , we also have for all $i, j \\in [M_{\\text{box}}]$ $\\phi _{\\bar{z}}(\\text{Frob } x_i) + \\phi _{\\bar{z}}(\\text{Frob } x_j) = g_0(i, j)$ so that $g_{x_1, \\dots , x_{M_{\\text{box}}}} = g_0$ as desired."
],
[
"Proof of Theorem ",
"The goal of this subsection is to sketch the proof of Theorem REF .",
"The major issue is that there is no choice of variable indices as in Definition REF for such discriminants.",
"This comes from the fact that the Rédei matrix is symmetric in this case, and hence the left and right kernels coincide.",
"For this reason one can not apply the reflection principles from Section , and Smith's method breaks down.",
"Nevertheless, one can still make inroads for this problem.",
"The 8-rank can be attacked using classical Rédei symbols.",
"In this case one is actually able to prove analogues of the results in Section by making substantial use of the classical Rédei reciprocity law.",
"The key feature here is that the Rédei symbol is fully symmetric in all its entries, while this is not the case for higher Rédei symbols.",
"This is the approach taken in [2], which leads to the following result.",
"We write $\\text{ClOrd}$ for the ordinary class group.",
"Theorem 6.13 Define for any $n \\ge m \\ge 0$ $\\alpha := \\prod _{j = 1}^\\infty (1 + 2^{-j})^{-1}, \\quad f(n, m) := \\frac{\\alpha \\cdot P(n, n, m)}{2^n \\cdot \\prod _{j = 1}^n (2^j - 1)} .$ Put $A_{n, m}(X) := \\lbrace d \\in S_{\\mathbb {Q}, X, -1}:\\textup {rk}_4 \\textup {Cl}(\\mathbb {Q}(\\sqrt{d})) = \\textup {rk}_4 \\textup {ClOrd}(\\mathbb {Q}(\\sqrt{d})) = n \\textup { and } \\textup {rk}_8 \\textup {Cl}(\\mathbb {Q}(\\sqrt{d})) = m\\rbrace .$ Then we have $\\lim _{X \\rightarrow \\infty } \\frac{|A_{n, m}(X)|}{|S_{\\mathbb {Q}, X, -1}|} = f(n, m).$ This is [2] except we order by discriminants instead of radicands in [2].",
"It is straightforward to adjust the proof to account for this new ordering.",
"Note that $\\alpha /\\prod _{j = 1}^n (2^j - 1)$ is the probability that the 4-rank is $n$ in the family $S_{\\mathbb {Q}, \\infty , -1}$ .",
"The term $P(n, n, m)/2^n$ is already familiar, since it also appeared in the previous subsection.",
"We improve on this result in two rather distinct ways.",
"Firstly, we still have equation (REF ) from Theorem REF .",
"This allows us to deal with the Artin pairing of $(\\sqrt{d})$ and therefore we can detect whether the 8-rank of the narrow and ordinary class group are different.",
"Crucially, Theorem REF ensures that we have the correct 8-rank distribution for the narrow class group.",
"This allows us to substantially improve the known upper bounds for the solubility of the negative Pell equations.",
"Secondly, the pairing $\\text{Art}_2$ need no longer be symmetric.",
"In case that the left and right kernel intersect trivially, we can use the ideas from the previous subsection.",
"This gives only minor improvements to the known upper and lower bounds.",
"The reason for this is that most discriminants have a small 4-rank, while this trick is especially effective when the 4-rank gets large.",
"We believe that any further progress will require substantial new ideas.",
"The remainder of this subsection is devoted to computing the improvement.",
"We start by giving precise statements for the results mentioned in the previous two paragraphs.",
"Theorem 6.14 We have for all $n \\ge m \\ge 0$ $\\lim _{X \\rightarrow \\infty } \\frac{|\\lbrace d \\in A_{n, m}(X) : \\textup {rk}_8 \\textup {ClOrd}(\\mathbb {Q}(\\sqrt{d})) = m - 1\\rbrace |}{|S_{\\mathbb {Q}, X, -1}|} = f(n, m) \\cdot \\frac{2^m - 1}{2^m}.$ Let $g(n, m)$ be the probability that a uniformly chosen $n \\times n$ matrix with coefficients in $\\mathbb {F}_2$ is such that the left and right kernel intersect trivially, given that the kernel has dimension $m$ .",
"Theorem 6.15 We have for all $n \\ge m \\ge 0$ $\\lim _{X \\rightarrow \\infty } \\frac{|\\lbrace d \\in A_{n, m}(X) : \\textup {LKer}(\\textup {Art}_{2, d}) \\cap \\textup {RKer}(\\textup {Art}_{2, d}) = \\langle (1, \\dots , 1) \\rangle \\rbrace |}{|S_{\\mathbb {Q}, X, -1}|} = f(n, m) \\cdot g(n, m).$ Here $\\textup {LKer}$ and $\\textup {RKer}$ denote respectively the left and right kernel.",
"Note that these are both naturally subspaces of $\\mathbb {F}_2^r$ , where $r$ is the number of prime divisors of $d$ .",
"Hence it makes sense to intersect them.",
"This result would follow immediately once one proves that $\\text{Art}_{2, d}$ is a random matrix.",
"But this is precisely what is done in the proof of [2].",
"Remark 5 It is not very hard to give an explicit formula for $g(n, m)$ .",
"To do so, we will compute the probability that the left and right kernel intersect trivially, and the right kernel is some given subspace $W$ of dimension $m$ .",
"We will see soon that this probability is independent on the choice of subspace $W$ and hence equals $g(n, m)$ .",
"Indeed, after right multiplying $A$ by an invertible matrix $X$ (and left multiplying by the transpose of $X$ ), it suffices to deal with the case that the right kernel is $\\lbrace e_1, \\dots , e_m\\rbrace $ , where the $e_i$ are standard basis vectors.",
"For such matrices, it is straightforward to count how many have do not have a non-zero vector in the span of $\\lbrace e_1, \\dots , e_m\\rbrace $ in the left kernel.",
"This leads to the explicit formula $g(n, m) = \\frac{P(n - m, m, 0) \\cdot \\prod _{j = 0}^{n - 2m - 1} (1 - 2^{j - n + m})}{P(n, n - m, 0)}.$ From Theorem REF we deduce that the proportion of squarefree integers $d$ for which negative Pell is soluble is at most $\\frac{2}{3} - \\sum _{n \\ge m \\ge 1} f(n, m) \\cdot \\frac{2^m - 1}{2^m},$ where the constant $\\frac{2}{3}$ comes from squarefree integers $d$ for which the 4-rank of the narrow and ordinary class group are different.",
"These are already dealt with in the work of Fouvry–Klüners [11].",
"We still need to compute the further improvement coming from Theorem REF and the methods in the previous subsection.",
"To compute the improvement to the upper and lower bounds that we get, first observe that we must restrict to squarefree integers $d$ for which the 4-ranks and 8-ranks of the narrow and ordinary class group coincide, and for which the 8-rank is at least 1.",
"Indeed, these are precisely the squarefree integers $d$ that do not fall under the purview of the theorems in [11], [12], Theorem REF or Theorem REF .",
"Furthermore, we must obviously restrict to squarefree integers such that $\\text{Art}_{2, d}$ has left and right kernel that intersect trivially.",
"The total proportion of such squarefree integers $d$ is $\\sum _{n \\ge m \\ge 1} \\frac{f(n, m) \\cdot g(n, m)}{2^m}.$ From the material in Appendix we see that $\\sum _{n \\ge m \\ge 1} \\frac{f(n, m) \\cdot g(n, m)}{2^m} \\cdot \\frac{2^m}{2^{m + 1} - 1}$ are such that negative Pell is soluble and $\\sum _{n \\ge m \\ge 1} \\frac{f(n, m) \\cdot g(n, m)}{2^m} \\cdot \\frac{2^m - 1}{2^{m + 1} - 1}$ are such that negative Pell is not soluble.",
"This yields Theorem REF after a numerical computation."
],
[
"Stevenhagen's conjecture revisited",
"Let $l$ be an integer such that $|l|$ is a prime 3 modulo 4.",
"Define for any integer $n \\ge 0$ the quantities $\\text{Pr}_{l, 2}(n) := \\lim _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n)|}{|[N] \\cap D_{l, 2}(n)|},$ where $D_{l, k}(n)$ is defined at the beginning of Section .",
"Let us first prove that the limit exists.",
"To do so, we look at $\\liminf _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n)|}{|[N] \\cap D_{l, 2}(n)|} \\text{ and } \\limsup _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n)|}{|[N] \\cap D_{l, 2}(n)|}.$ Theorem REF gives increasingly better lower bounds for $\\liminf $ , and increasingly better upper bounds for $\\limsup $ .",
"We conclude that the $\\liminf $ and $\\limsup $ are equal, and hence the limit exists.",
"From the Markov chain behavior in Theorem REF , we also see that $\\text{Pr}_{l, 3}(m, n) := \\lim _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(m) \\cap D_{l, 3}(n)|}{|[N] \\cap D_{l, 2}(m) \\cap D_{l, 3}(n)|}$ exists and equals $\\text{Pr}_{l, 2}(n)$ for every $m \\ge n$ .",
"Then we deduce from the identity $\\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n)|}{|[N] \\cap D_{l, 2}(n)|} = \\sum _{i = 0}^n \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n) \\cap D_{l, 3}(i)|}{|[N] \\cap D_{l, 2}(n) \\cap D_{l, 3}(i)|} \\cdot \\frac{|[N] \\cap D_{l, 2}(n) \\cap D_{l, 3}(i)|}{|[N] \\cap D_{l, 2}(n)|}$ by taking $N \\rightarrow \\infty $ that $\\text{Pr}_{l, 2}(n) = \\sum _{i = 0}^n \\text{Pr}_{l, 3}(n, i) \\cdot \\frac{P(n, n, i)}{2^n} = \\sum _{i = 0}^n \\text{Pr}_{l, 2}(i) \\cdot \\frac{P(n, n, i)}{2^n}.$ We claim that $\\frac{1}{2^{n + 1} - 1} = \\sum _{i = 0}^n \\frac{1}{2^{i + 1} - 1} \\cdot \\frac{P(n, n, i)}{2^n}.$ Let us first show that the claim implies Theorem REF .",
"Since we clearly have $\\text{Pr}_{l, 2}(0) = 1$ , the claim and equation (REF ) imply that $\\text{Pr}_{l, 2}(n) = \\frac{1}{2^{n + 1} - 1}.$ Now consider the decomposition $\\frac{|S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} = \\sum _{n = 0}^\\infty \\frac{|S_{\\mathbb {Z}, N, l} \\cap D_{l, 2}(n)|}{|[N] \\cap D_{l, 2}(n)|} \\cdot \\frac{|[N] \\cap D_{l, 2}(n)|}{|S_{\\mathbb {Q}, N, l}|}.$ Then equation (REF ), Theorem REF and Fatou's lemma imply $\\limsup _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} \\ge \\liminf _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} \\ge \\sum _{n = 0}^\\infty \\frac{2^{-n^2} \\eta _\\infty \\eta _n^{-2}}{2^{n + 1} - 1}.$ Similarly, we get $\\limsup _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Q}, N, l} \\setminus S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} \\ge \\liminf _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Q}, N, l} \\setminus S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} \\ge \\sum _{n = 0}^\\infty \\frac{2^{-n^2} \\eta _\\infty \\eta _n^{-2} \\cdot (2^{n + 1} - 2)}{2^{n + 1} - 1}.$ But it is a classical fact that $\\sum _{n = 0}^\\infty 2^{-n^2} \\eta _\\infty \\eta _n^{-2} = 1.$ Therefore equation (REF ) and equation (REF ) imply that $\\liminf _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} = \\limsup _{N \\rightarrow \\infty } \\frac{|S_{\\mathbb {Z}, N, l}|}{|S_{\\mathbb {Q}, N, l}|} = \\sum _{n = 0}^\\infty \\frac{2^{-n^2} \\eta _\\infty \\eta _n^{-2}}{2^{n + 1} - 1},$ and Theorem REF follows.",
"It remains to prove the claim.",
"Look at the probability space of pairs $(T, U)$ with the uniform measure, where $T$ is a surjective linear map $\\mathbb {F}_2^{[n + 1]} \\rightarrow \\mathbb {F}_2^{[n]}$ and $U$ is a pairing $\\mathbb {F}_2^{[n + 1]} \\times \\mathbb {F}_2^{[n]} \\rightarrow \\mathbb {F}_2$ .",
"All our probabilities will be with respect to this probability space.",
"Now fix a non-zero element $x \\in \\mathbb {F}_2^{[n + 1]}$ .",
"Note that $\\mathbb {P}(x \\in \\text{ker}(T)) = \\frac{1}{2^{n + 1} - 1}.$ We write $\\text{leftker}(U)$ for the set of vectors $v \\in \\mathbb {F}_2^{[n + 1]}$ such that $U(v, w) = 0$ for all $w \\in \\mathbb {F}_2^{[n]}$ .",
"Write $A_{i, x}$ for the event that $x \\in \\text{leftker}(U) \\text{ and } \\text{dim}_{\\mathbb {F}_2} \\text{leftker}(U) = i + 1$ .",
"Then we have $\\mathbb {P}(x \\in \\text{ker}(T)) = \\sum _{i = 0}^n \\mathbb {P}(x \\in \\text{ker}(T) | A_{i, x}) \\cdot \\mathbb {P}(A_{i, x}).$ Observe that $\\mathbb {P}(A_{i, x}) = P(n, n, i)/2^n$ .",
"Next we have $\\mathbb {P}(x \\in \\text{ker}(T) | A_{i, x}) = \\sum _{V} \\mathbb {P}(x \\in \\text{ker}(T) | A_{i, x}, T(\\text{leftker}(U)) = V) \\cdot \\mathbb {P}(T(\\text{leftker}(U)) = V | A_{i, x}),$ where the sum is over $i$ -dimensional subspaces $V$ of $\\mathbb {F}_2^{[n]}$ .",
"But we have $\\mathbb {P}(x \\in \\text{ker}(T) | A_{i, x}, T(\\text{leftker}(U)) = V) = \\frac{1}{2^{i + 1} - 1},$ since restricting $T$ to $\\text{leftker}(U)$ gives a random surjective linear map $T^{\\prime }$ from $\\text{leftker}(U)$ to $V$ , with $x \\in \\text{leftker}(U)$ .",
"Hence we conclude that $\\mathbb {P}(x \\in \\text{ker}(T) | A_{i, x}) = \\frac{1}{2^{i + 1} - 1}.$ Inserting this in equation (REF ) and (REF ), we obtain the desired identity."
]
] | 2005.14157 |
[
[
"The unexpected narrowness of eccentric debris rings: a sign of\n eccentricity during the protoplanetary disc phase"
],
[
"Abstract This paper shows that the eccentric debris rings seen around the stars Fomalhaut and HD 202628 are narrower than expected in the standard eccentric planet perturbation scenario (sometimes referred to as \"pericenter glow\").",
"The standard scenario posits an initially circular and narrow belt of planetesimals at semi-major axis $a$, whose eccentricity is increased to $e_f$ after the gas disc has dispersed by secular perturbations from an eccentric planet, resulting in a belt of width $2ae_f$.",
"In a minor modification of this scenario, narrower belts can arise if the planetesimals are initially eccentric, which could result from earlier planet perturbations during the gas-rich protoplanetary disc phase.",
"However, a primordial eccentricity could alternatively be caused by instabilities that increase the disc eccentricity, without the need for any planets.",
"Whether these scenarios produce detectable eccentric rings within protoplanetary discs is unclear, but they nevertheless predict that narrow eccentric planetesimal rings should exist before the gas in protoplanetary discs is dispersed.",
"PDS 70 is noted as a system hosting an asymmetric protoplanetary disc that may be a progenitor of eccentric debris ring systems."
],
[
"Introduction",
"Debris discs are the circumstellar discs that are seen to orbit main-sequence stars, including our Sun.",
"The dust that is observed derives from a parent population of planetesimals; around other stars this replenishment is inferred because the lifetime of the dust in the presence of radiation and stellar wind forces is typically shorter than the stellar age [1], while in the Solar system the connection is more direct because both the dust and parent bodies are detectable [2], [3].",
"In the Solar system the structure of the asteroid and Edgeworth-Kuiper belts is shaped by planets.",
"Indeed, these small body populations have arguably contributed far more per unit mass to our understanding of Solar system history than the planets themselves.",
"Perhaps the most famous example is the inference of Neptune's outward migration from Pluto's capture into the 2:3 mean-motion resonance [4].",
"One key challenge for debris disc science is the successful application of similar concepts to other stars.",
"While a future aspiration is to unravel the histories of other planetary systems [5], one past and present goal is to correctly infer the presence of as-yet undetectable planets via their gravitational influence [6], [7].",
"Because these perturbations tend to act on timescales that are longer than dust lifetimes, the general expectation is that the planetary influence is imprinted on the parent planetesimal population and inherited by the collisional fragments that are observed.",
"Connecting disc structures, of which myriad are seen, to planets, has proven hard, primarily because detecting the putative planets is hard.",
"Indeed, it is normally impossible to rule out the proposed planets, partly because they commonly lie somewhere along a locus in mass–semi-major axis parameter space rather than in specific locations, but primarily because their masses can be far too small for detection.",
"The single successful example of prediction and subsequent detection is for the edge-on disc $\\beta $ Pictoris, whose warp was attributed to an inclined planet [8] that was subsequently discovered by direct imaging [9], [10].",
"While $\\beta $ Pic b was predicted based on the long-term “secular” perturbations from a misaligned planet, warps in discs are normally hard to detect because most systems do not have the optimal edge-on geometry.",
"In-plane perturbations that result in azimuthally dependent structures are more generally detectable, typically manifesting as a significant eccentricity, which may be imaged directly or inferred from a brightness asymmetry at lower spatial resolution.",
"The first detection of an eccentric debris disc was for HR 4796 [11], [12], where a brightness asymmetry (the so-called “pericenter glow”) from low resolution mid-infrared imaging was attributed to an unseen planet [6], [13].",
"Despite a high sensitivity to asymmetry (due to the exponential dependence of flux density on dust temperature), mid-IR pericenter glow is rarely the method by which asymmetric discs are identified, simply because few discs are bright enough to be imaged near 10$\\mu $ m (both in absolute terms, and relative to the host star).",
"Instead, scattered light images have proven far more successful and yielded a veritable zoo of structures [14], [15], [16], [17].",
"While many of these images suggest the influence of unseen planets [7], a major limitation is that these images trace small dust.",
"This dust is subject to strong radiation and stellar wind forces, and possibly gas drag which in addition to opening the possibility of entirely different sculpting scenarios [18], [19], makes connecting the observed structure to the orbits of the underlying parent planetesimals difficult.",
"Ideally, inferences of unseen planets would be made at longer wavelengths, where the typical grain sizes are large enough to be immune to non-gravitational perturbations, and the observed structure more reasonably assumed to be representative of the planetesimal orbits.",
"Until recently, mm-wave observation of debris discs was limited to photometry and marginally resolved imaging [20], [21].",
"However, the unprecedented sensitivity and resolution of the Atacama Large Millimetre/Submillimetre Array (ALMA) means that well-resolved debris disc images are now of sufficient quality to be confronted with dynamical and collisional models.",
"Indeed, after considering the motivation in more detail (section ) this paper shows that ALMA images of the debris discs in the Fomalhaut and HD 202628 systems have a narrower radial width than is expected based on the secular perturbation scenario originally devised to explain pericenter glow for HR 4796 (section ).",
"Possible origins, including the possibility that these eccentric discs are actually a relic from the protoplanetary disc phase, are discussed in section , and concluding remarks made in section ."
],
[
"Motivation and expectations",
"The basic idea that undepins this work is that the width of any debris ring provides information on the eccentricities of the objects that are observed.",
"An axisymmetric debris ring's width could be entirely explained by orbits of a single semi-major axis $a$ and non-zero eccentricity, as long as the pericenter angles are uniformly distributed in azimuth.",
"The eccentricities could however be lower, because some (or all) of the ring width could also arise from a range of semi-major axes.",
"Thus, in most cases one expects to be able to derive an upper limit on the eccentricities.",
"Essentially the same argument applies to eccentric debris rings, but the constraint is a bit more complex because the pericenter angles must have a preferred direction to break the symmetry.",
"To understand this constraint and why it is useful, this section briefly describes the secular perturbation mechanism that is supposed to be the origin of eccentric debris rings , and how particle eccentricities and pericenters can be related to observable disc structure.",
"These ideas form the basis for the model used below.",
"Figure: Illustration of how secular perturbations produce aneccentric debris ring, with eωe\\omega -space in the left panel, and particleorbits viewed from above in the right panel.",
"Particles thatinitially have zero eccentricity precess anticlockwise around theforced eccentricity (here e f =0.2e_f=0.2, ω=0\\omega =0 is marked by theblue + symbol).",
"A precessing particle starting at the blue dot(i.e.",
"with e p =0.2e_p=0.2) would pass through the magenta dot, then theblack dot, then red, then blue, etc..",
"The orbits for these fourpoints in the precession cycle are shown in the right panel, wherethe star is marked by the blue + symbol.",
"In particular, note thatthe black orbit is eccentric while the blue one is circular.",
"The sumof many such orbits distributed in a circle around e f e_f produces aneccentric ring (see upper right panel in Figure ).Secular (long-term) perturbations from an eccentric planet cause disc particles' pericenter angles to precess, during which their eccentricities vary in a systematic way.",
"As applied to debris discs, this scenario assumes that it is the large and long-lived planetesimals that are perturbed onto eccentric orbits, and that smaller bodies inherit these eccentric orbits when they are created.",
"This orbital evolution is best visualised in eccentricity–pericenter phase space (here called $e\\omega $ -space, but $hk$ -space is also used), where the distance from the origin is the eccentricity, and the anti-clockwise angle from the positive x-axis the pericenter direction relative to some reference direction (normally the ascending node).",
"The planet imparts a “forced” eccentricity $e_f$ on test particles, whose magnitude is approximately the planet/particle semi-major axis ratio (assuming an interior planet) multiplied by the planet eccentricity, and whose pericenter direction is the same as the planet's.",
"A test particle with the forced eccentricity stays on this orbit, but particles elsewhere in $e\\omega $ -space move continuously in an anticlockwise circle around the forced eccentricity (i.e.",
"their eccentricity changes as they precess).",
"The precession rate depends on the strength of the planet's perturbation, via the planet mass and planet/planetesimal semi-major axis ratio.",
"An illustrative example for a particle with zero initial eccentricity is given in Figure REF , which shows the particle and the corresponding orbits at four points in the precession cycle.",
"The difference between a particle's actual eccentricity and the forced eccentricity is termed the “free” or “proper” eccentricity $e_p$ , which is also the radius of the circle drawn out by the particle.",
"The motion in $e\\omega $ -space is such that the particle's orbit is most eccentric only when the pericenter angle is near zero (i.e.",
"near the black dot); this behaviour is the key to how secular perturbations produce an eccentric ring.",
"Figure: Single-planet secular perturbations with different initialeccentricities.",
"In each panel the left plot shows theeωe\\omega -space (initial=orange, final=blue), and the right image shows theresulting eccentric disc image, with the star marked by the blue +(assuming face-on geometry, the stated range of semi-major axes, andincluding a 1/r1/\\sqrt{r} weighting to account for the radiallyvarying surface brightness at long wavelengths).",
"The line in theright image shows the radial profile along the pericenterdirection.",
"The forced ecentricity is the same in all panels.",
"The topleft panel shows a narrow ring, while the other three show how thedisc width can arise from small initial eccentricities (top right),a large initial eccentricity dispersion about the forcedeccentricity (bottom right), or a range of semi-major axes (bottomleft).",
"Only a range of semi-major axes yields an azimuthally varyingring width.Thus, given an initial population of particles at some common location in $e\\omega $ -space, but with a finite range of semi-major axes, the long term evolution due to secular perturbations from an eccentric planet is that these particles are spread around a circle of radius $e_p$ that encloses the forced eccentricity.",
"The time taken for particles to spread out depends on their differential precession rates, and thus the range of semi-major axes.",
"The case of a Gaussian distribution of near-zero initial eccentricities with dispersionIn this work the word “dispersion” is used to describe the width of a Gaussian distribution, while “range” refers to the full range of values covered by some parameter.",
"So in the upper right panel of Figure REF , the initial eccentricity dispersion is small, but the range of final eccentricities is large.",
"Below, both Gaussian and top-hat semi-major axis distributions are used, in which case “range” is used to describe the width of either distribution.",
"$\\sigma _{e,p}=0.01$ is shown in the upper right panel of Figure REF , with the initial eccentricities as orange dots, and the final eccentricities after many precession cycles as blue dots (the semi-major axes are distributed as a narrow Gaussian with dispersion $\\sigma _a/a=0.01$ ).",
"In this case, $e_p \\approx e_f$ , the full range of eccentricities is $\\approx $$2e_f$ , and the resulting debris ring has a uniform width of $\\approx $$2ae_f$ (looking back at Figure REF may help make these results clear).",
"Alternatively, if the initial eccentricities were very close to $e_f$ , as shown in the upper left panel of Figure REF , the final eccentricities are also near $e_f$ , and the resulting debris ring appears very narrow (i.e.",
"approximately follows a single eccentric orbit).",
"Naively, one expects that the initial planetesimal eccentricities are small, the result of damping during the gas-rich protoplanetary disc phase, so the former of these two cases seems more physically plausible [22].",
"That is, with this model one expects eccentric debris rings to have widths of $\\approx $$2ae_f$ , and narrower widths would imply non-zero initial eccentricities that are shifted towards the forced eccentricity.",
"Further variations on this model are of course possible; the semi-major axis distribution can be made wide enough to dominate the belt width, in which case the width varies as a function of azimuth (e.g.",
"with $\\sigma _a/a=0.1$ and $e_p=0$ , see the lower left panel of Figure REF ).",
"The dispersion of the initial eccentricity distribution can also be varied; as $\\sigma _{e,p}$ increases the radial disc profile acquires a lower level “halo” that is not present when the dispersion is small (compare the lower right and upper left panels in Figure REF ).",
"Thus, $e_p$ changes the width, and $\\sigma _{e,p}$ the radial concentration of an eccentric ring, and the semi-major axis distribution provides a further means to change the width, but in a way that is azimuthally-dependent.",
"While alternative parameterisations could also describe an eccentric disc, it could not achieve the same flexibility with fewer parameters, and the advantage here is that the model is physically connected to the initial conditions via the assumed secular perturbation scenario.",
"Of course, whether these parameters are all needed depends on the data in hand, which is considered below.",
"Figure: Single-planet secular perturbations with different semi-majoraxis distributions.",
"Conventions are as in Figure .",
"Theleft panel is similar to the lower left panel in Figure, but with a non-zero proper eccentricity.",
"The rightpanel has the same eccentricity parameters, but the semi-major axisdistribution is uniform in the range a±δ a /2a \\pm \\delta _a/2.",
"Thedouble-ring structure in the right panel can be thought of as thesuperposition of two rings like the top right panel of Figure with semi-major axes a-δ a /2a-\\delta _a/2 anda+δ a /2a+\\delta _a/2.",
"As can be seen from the radial profiles, near thepeak brightness the disc is narrower at apocenter than pericenter(in contrast to all other models).Whether the range of semi-major axes should be small, or be distributed in a particular way is unclear.",
"To briefly explore this aspect, Figure REF shows two examples that have the same eccentricity distributions; the left panel uses a Gaussian distribution of semi-major axes centered at $a$ with dispersion $\\sigma _a$ , while the right uses a uniform distribution between $a \\pm \\delta _a/2$ .",
"In terms of the resulting disc images, the left panel in Figure REF is similar to the lower left panel in Figure REF , with some additional width contributed by an increased range of eccentricities.",
"The right panel in Figure REF is similar to the upper right panel in Figure REF , but has two bright rings near the middle.",
"These arise because this disc is essentially a superposition of two such rings, one at $a-\\delta _a/2$ and the other at $a+\\delta _a/2$ .",
"The outer of the bright rings is the outer edge of the component with $a-\\delta _a/2$ , and the inner ring the inner edge of the component with $a+\\delta _a/2$ .",
"Perhaps importantly, the distance between these rings is not constant with azimuth, and is actually smaller near apocenter, in contrast to all other models.",
"Given an appropriate disc configuration, observations that moderately resolve the disc width might be able to detect or rule out such a variation.",
"Note however that this example is highly idealised, with sharp edges in the semi-major axis distribution, and eccentricities that do not vary with semi-major axis."
],
[
"Previous work",
"Lee & Chiang [7] developed a similar model, which was extended to incorporate radiation forces on small grains, and showed that many of the unusual scattered light morphologies observed for bright debris discs can be explained by highly eccentric parent belts ($e_f \\sim 0.6$ ).",
"Their work assumed a very small proper eccentricity of 0.02, thus implicitly assuming either that the parent belt was initially very eccentric, or that particles damp to the forced eccentricity before they fragment to small enough sizes that they are strongly perturbed by radiation forces.",
"It seems unlikely that their reproduction of scattered light structures would appear as compelling with $e_f=e_p$ , as the parent belts would be predicted to be much wider (the effect on smaller particles is less clear, most likely the structures would become less distinct).",
"To take a specific example, Esposito et al.",
"[23] model the HD 61005 disc and find $e_f=0.21$ and $e_p=0.08$ , where the small proper eccentricity is required to reproduce the relatively narrow parent belt.",
"Thus, it seems probable that if highly eccentric belts do explain the range of scattered light structures, those belts should have proper eccentricities that are smaller than is expected.",
"However, whether the range of observed debris disc structures can be “unified” remains unclear, as these models predict that asymmetric structure seen in scattered light should in at least some cases be accompanied by asymmetric structure in mm-wave observations.",
"Few such systems have been observed by ALMA, but HD 15115 provides a first test.",
"The scattered light structure is highly asymmetric and can be explained with an eccentric belt with a pericenter direction in the sky plane [7], which predicts that the mm-wave emission should also appear asymmetric.",
"However, MacGregor et al.",
"[24] find that the structure is consistent with being symmetric, suggesting that in this case a highly eccentric parent belt is not the reason for the asymmetric scattered light structure.",
"The potential for such stark differences shows that connecting observations with the underlying planetesimal belt structure is more easily done with thermal emission at longer wavelengths, such as probed by ALMA."
],
[
"The ring widths of Fomalhaut and HD 202628",
"Given the motivation above, several eccentric systems are well-suited to quantifying the ring width.",
"Here Fomalhaut and HD 202628 are singled out as the best cases, because their debris rings are obviously eccentric and narrow, and because extant ALMA observations have sufficient spatial resolution to constrain the ring width to better than the expected $2ae_f$ .",
"That is, the null hypothesis is that the ring widths are consistent with that expected from secular perturbations and initially near-zero eccentricities.",
"The assumption that the dust grains detected with ALMA share the same orbits as the planetesimals from which they are derived will be revisited below.",
"Clean images of these discs are shown in Figure REF , primarily to aid the interpretation of the residual images shown in Figures REF and REF , The Fomalhaut observations were at 1.3 mm (band 6) and have a spatial resolution of $1.6 \\times 1.2$ arcsec resolution ($12 \\times 9$ au at the 7.7 pc distance of Fomalhaut).",
"The large angular size relative to the primary beam means that seven pointings were required to cover the disc adequately.",
"The HD 202628 observations were also at 1.3 mm and use a single pointing.",
"The spatial resolution is $0.9 \\times 0.8$ arcsec, which is $21 \\times 19$ au at the 23.8 pc distance of HD 202628.",
"The reader is referred to the papers cited below for more observational details.",
"The HD 202628 ring was modelled by Faramaz et al.",
"[25], and with the assumption of constant surface density (justified by a marginally radially resolved image) the width was found to be 22 au.",
"Given their best-fit eccentricity of 0.09 the expected width of $2ae_f = 28$ au is larger than observed, suggesting that the proper eccentricity is lower than the forced eccentricity.",
"The ring is at best marginally resolved radially and not detected with a high signal to noise ratio, but the star is detected at the expected location which significantly improves the constraints on the ring geometry.",
"Their model was simplified in that it used an offset circular ring; here the data are modelled again using the more complex eccentric ring model.",
"Given the data this is not necessarily a superior approach, but has the benefit that ring parameters directly related to the secular perturbation scenario are derived explicitly, and their (joint) posterior distributions quantified.",
"Figure: Radial surface brightness profile along the disc major axisfor Fomalhaut.",
"The blue line shows the data, and the grey lines showGaussian profiles with FWHM 1.2 arcsec (i.e.",
"the spatial resolution)scaled to each peak.",
"The measured (unconvolved) disc FWHM in eachansa are 2.6 (SE) and 2.1 (NW) arcsec, or 20 and 16 au.The width of the Fomalhaut ring has also been suggested to be narrower than expected by White et al.",
"[26] and MacGregor et al.",
"[27], with these works finding full-width at half-maxima (FWHM) of 13-13.5 au.",
"This width is to be compared to the expectation of $2ae_f = 33$ au.",
"However, the significance of this narrowness is not particularly clear for two reasons.",
"Figure 5 of MacGregor et al.",
"[27] suggests that the disc has a FWHM along the major axis of $\\sim $ 5 arcsec (39 au), but the contours in their Figure 1 show that the disc is at most only $\\sim $ 2 beams (i.e.",
"$\\sim $ 2 arcsec) wide where it is detected (suggesting an axis labelling/scaling error).",
"Figure REF shows the radial profile for a naturally weighted image with a 1 arcsec wide swath through the star and along the disc major axis.",
"The disc widths along the disc major axis are 16 and 20 au, with the NW ansa being narrower, and accounting for the beam width of about 1.2 arcsec these widths are consistent with a true disc FWHM of about 13 au (i.e.",
"much less than 33 au).",
"A second issue is that the modelling in MacGregor et al.",
"[27] finds a proper eccentricity ($0.06 \\pm 0.04$ ) that is consistent with the forced eccentricity (which was $0.12 \\pm 0.01$ ).",
"This result therefore suggests that the disc can be consistent with a width of 33 au (4.3 arcsec) at 2$\\sigma $ , which is clearly incompatible with the data (e.g.",
"from Figure REF here, or the contours in their Figure 1).",
"The probable reason lies with the details of the modelling, which generated disc images by generating N particles on eccentric orbits (i.e.",
"as in Figures REF and REF ).",
"In that work $N= 10^4$ particles were used, but in the course of this work at least 10$^6$ particles were found to be necessary.",
"With small $N$ , shot noise renders two model images with identical parameters sufficiently different that their $\\chi ^2$ can also be very different.",
"This leads to issues with convergence of model fitting, including low acceptance fractions in Markov chain Monte Carlo (MCMC) fitting, and inflated uncertainties.",
"This issue appears to have affected the uncertainties more than the best-fit parameters (which largely agree with the results here), although some differences are discussed below.",
"Finally, MacGregor et al.",
"[27] used a uniform range of semi-major axes and in fact yields the double inner-ring structure discussed above, which can be seen (albeit indistinctly) in their Figure 2.",
"The radial profile in Figure REF does indeed show that the disc is narrower towards the NW ansa (which is near the apocenter), and whether this model provides a good explanation for the data is explored below."
],
[
"The model",
"A slightly more complex version of the secular perturbation model used by MacGregor et al.",
"[27] was used to model the ALMA data for Fomalhaut and HD 202628.",
"Bascially, $N$ particles are generated that sample distributions of eccentricity, inclination, semi-major axis, and true anomaly.",
"These are then binned into an 2d array for the desired viewing geometry to create an image.",
"The fundamental assumption is that the particles have evolved to their current state due to secular perturbations (i.e.",
"are spread evenly around the forced pericenter in $e\\omega $ -space).",
"This assumption means that the models are created analytically, with no need for $n$ -body simulations.",
"From this assumption, and similar secular timescales for inclination evolution, it also follows that the particle nodes are also spread randomly, meaning that any initial misalignment ($i_f$ ) of the particles with the planet's orbital plane causes the disc to have a height $2i_f$ .",
"Viewed edge-on the disc appears brighter at the upper and lower surfaces, because their sinusoidal vertical motions means that particles spend more time there (i.e.",
"similar to the limb-brightening effect seen in the top right panel of Figure REF ).",
"While the modelling here does not resolve the scale height of either disc, a finite vertical extent is included to include any effect on the other parameters.",
"The model parameters are as follows: i) $x_0$ and $y_0$ allow for any shift of the star+disc position from the observation phase center, ii) sky geometry position angle $\\Omega $ , inclination $i$ , and forced pericenter angle $\\omega $ (measured from $\\Omega $ ), iii) total disc and star flux densities $F$ and $F_{\\rm star}$ , iv) disc semi-major axis $a_0$ and Gaussian width $\\sigma _a$ or full width $\\delta _a$ , v) forced eccentricity $e_f$ , proper eccentricity $e_p$ (radius of the circle in $e\\omega $ -space), and proper eccentricity Gaussian dispersion $\\sigma _{e,p}$ , vi) forced inclination $i_f$ and Gaussian inclination dispersion $\\sigma _{i,p}$ , and vii) a data re-weighting parameter $f_w$ (see below).",
"A total of 15 basic parameters are included, and three more are included in the model for HD 202628 for the position and flux of a bright point source that could otherwise influence the model results.",
"The important difference compared to the model used by MacGregor et al.",
"[27] is the inclusion of a proper eccentricity dispersion, which as shown in Figure REF provides an additional means to introduce a finite disc width.",
"Given that the Fomalhaut ring appears narrower at apocenter than pericenter, thus disfavouring a range of semi-major axes, alternative means of generating the disc width are potentially important.",
"Practically, model images are created with the following steps: i) the eccentricity parameters $e_p$ and $\\sigma _{e,p}$ are used to create $N$ initial particles in a 2-d Gaussian in $e\\omega $ -space.",
"As these will be precessed into a circle around $e_f$ , $e_p$ is taken to be the distance from $e_f$ towards the origin in $e\\omega $ -space (these are the orange dots in Figures REF and REF ), ii) these particles are then distributed evenly around the forced eccentricity, giving a set of $N$ eccentricities and pericenters (i.e.",
"the blue dots in the same figures), iii) $N$ inclinations are generated with a Gaussian distribution of dispersion $\\sigma _{i,p}$ centered on $i_f$ , and these are given randomly chosen ascending nodes, iv) $N$ random mean anomalies are generated, and using the eccentricity of each particle these are converted into true anomalies, v) using the eccentricities and true anomalies, the radius of each particle from the star is calculated, and using the true anomaly, pericenter, and node, the angular particle locations are calculated.",
"These steps generate an initial 3-d particle model of the disc viewed from above, with the forced pericenter measured anticlockwise from the positive $x$ direction.",
"The final steps are vi) to apply two rotations to move these particles to the observed geometry; first the $y$ coordinates are multiplied by $\\cos {i}$ to incline the disc, and then the $x,y$ coordinates are rotated by $\\Omega +\\pi $ to place the ascending node at the observed position angle, and vii) finally, the image is generated by binning all particles into a 2-d grid, including a $1/\\sqrt{r}$ weighting to account for decreasing temperature with radius.",
"Model images computed using $N=10^7$ are compared with the ALMA visibilities using the GALARIO [28] software (which returns a $\\chi ^2$ value for a given model image).",
"Absolute uncertainties are first estimated with the CASA statwt task, which derives weights ($=1/\\sigma ^2$ ) based on the variance of the visibilities.",
"The parameter space is explored using MultiNest [29] and emcee [30], with the log likelihood of a given model: $\\ln \\mathcal {L} = -\\frac{1}{2} \\left( \\chi ^2 f_w + \\sum _i^M 2 \\ln \\frac{2 \\pi }{w_if_w} \\right) \\, ,$ where there are $M$ complex visibilities $V$ , with weights $w$The factor $f_w$ , which the visibility weights are multiplied by, is included because CASA statwt is not guaranteed to produce accurate absolute uncertainties.",
"The log likelihood function used in the fitting is therefore derived by assuming that the data are distributed as a Gaussian about the model with the correct normalisation.",
"The factor two in the summation of equation (REF ) arises because each visibility is assumed to have two independent degrees of freedom (i.e.",
"amplitude and phase)..",
"While both methods give similar results, the potential advantage of using MultiNest is that it explores all parameter space within given ranges, meaning that similarly well-fitting solutions in the parameter space can be identified (though no multi-modal solutions were found here).",
"Only the Multinest results are reported here, which used 75 live points to search a broad parameter space around previously found best-fit parameters.",
"All parameters have uniform priors in linear space."
],
[
"Fomalhaut",
"The same seven-pointing data of the Fomalhaut disc that were presented in MacGregor et al.",
"[27] were modelled as described above.",
"The data were calibrated using the observatory-supplied script, with the only additional steps being spectral averaging to a single channel per spectral window (spw), 30-second time averaging, and re-weighting of all visibilities using the CASA 5.6 statwt task.",
"The signal to noise ratio of the disc detection is sufficiently high and the disc sufficiently large that the finite ALMA bandwidth must be accounted for (i.e.",
"the baselines in $uv$ space vary sufficiently across channels that a single frequency cannot be assumed across all channels).",
"The $\\chi ^2$ are therefore computed using visibilities that assume the average frequency for each spectral window, for each of the seven pointings, and the 28 $\\chi ^2$ values summed.",
"The precision of the pointings was found to be good enough relative to the resolution that no per-pointing offset parameters were necessary.",
"Figure: Residual maps for the best-fit Uniform simple (left) and full(right) models of the Fomalhaut ring.",
"The white contour shows thelocation of the ring, and black contours highlight parts of theimage above 3 times the noise level of 13μ\\mu Jy beam -1 ^{-1}.Four slightly different models were fitted to the Fomalhaut data; two “full” models that use all parameters noted above, with “Gaussian” and “Uniform” semi-major axis distributions, and two “simple” models with the same two semi-major axis distributions, and where the eccentricity and inclination dispersion parameters, and the forced inclination, are set near zero (i.e.",
"vertically flat models that look similar to the upper right or lower left panels in Figure REF ).",
"From the radial profile in Figure REF it is clear that a purely Gaussian radial distribution is a poor description of the data, and this is confirmed by a poor fit with the Gaussian simple model, so this model is not discussed further.",
"The Gaussian full model however yields a very good fit, because the additional radial extent allowed by the eccentricity dispersion can account for surface brightness interior and exterior to the brightest part of the ring.",
"Both Uniform models provide good fits, though the full model does a much better job of reproducing the low surface brightness immediately interior and exterior to the ring.",
"A test using the Schwarz criterionAlso known as the Bayesian Information Criterion, ${\\rm BIC} = \\chi ^2+k \\ln n$ , where k is the number of parameters and n the degrees of freedom (here $k$ is 12 or 15 for the simple and full models, and $n \\approx 1.7 \\times 10^6$ for Fomalhaut).",
"Smaller BIC numbers are better, with a decrease of more than about 6 indicating that additional parameters are warranted given the data.",
"[31] finds that while the second “penalty” term increases by 45 for the full models relative to the simple uniform model, the $\\chi ^2$ decreases by 240.",
"Such a large decrease indicates that for Fomalhaut the inclusion of these extra parameters is justified by the improvement in the fit.",
"Table: Partial list of best fit model parameters,with the ring radius and width in units of arcseconds andinclinations in radians (full posterior parameter distributionsare shown in Figures–).",
"Uncertainties are1σ\\sigma , and upper limits are 3σ\\sigma .The modelling results are summarised in Table REF and Figure REF which show the (dirty) residual map for the Uniform models (the Gaussian full model residuals are indistinguishable to the Uniform full model, Figures REF –REF show the posterior parameter distributions for most parameters).",
"All models find the same disc position angle of $\\Omega = 156.4 \\pm 0.1^\\circ $ and inclination $i = 66.6 \\pm 0.1^\\circ $ .",
"Figure REF shows that the models are a good representation of the data, with the only large residual in the NW ansa.",
"This residual noise lies within the ring and has negative residuals interior and exterior, suggesting that the ring is narrower than the model at this location.",
"The Uniform simple model, which is essentially the same as that used by MacGregor et al.",
"[27], finds very similar results, but with the expected smaller uncertainties.",
"All models find that the pericenter location is located $41 \\pm 1^\\circ $ anti-clockwise from the SE disc ansa (i.e.",
"the ascending node).",
"This value is most likely different to the $23 \\pm 4^\\circ $ found by MacGregor et al.",
"[27] for the reasons discussed earlier.",
"Figure: Best-fit Uniform full model to Fomalhaut, shown face-on withpericenter along the positive x axis.",
"Conventions are as in Figure.",
"The ring has a narrow bright component whose width isset by e p e_p, and is surrounded by a lower surface brightness“halo” whose width is set by σ e,p \\sigma _{e,p}.Less obvious from Figure REF is that a greater eccentricity dispersion via the $\\sigma _{e,p}$ parameter improves the fit; low level ($\\sim $ 1$\\sigma $ ) residuals just interior/exterior to the brightest part of the belt remain for the Uniform simple model (left panel), but are reduced for the full models (right panel).",
"The narrower disc width near apocenter means that narrow semi-major axis distributions are preferred, and thus for the full models the disc width is instead generated by a combination of non-zero proper eccentricity and eccentricity dispersion.",
"The best-fit model is shown at high resolution with the $e\\omega $ -space in Figure REF , which shows that the model is composed of a narrow top-hat-like “core” that is produced by the relatively small typical proper eccentricity $e_p$ , and a “halo” that is produced by the dispersion in proper eccentricity $\\sigma _{e,p}$ .",
"The semi-major axis distribution does not contribute to the structure, so the Gaussian full model looks the same.",
"While it was suggested earlier than the Uniform model may be better able to produce a disc that is narrower near apocenter through a quirk of combining a range of semi-major axes and proper eccentricities, this possibility is not borne out by the modelling, primarily because the best models do not require a range of semi-major axes.",
"Overall, these models show that the disc is consistent with an eccentric ring model, but only if the proper eccentricities are smaller than the forced eccentricity.",
"While the constraints on the model parameters appear very small, this is at least in part because of the restricted nature of the models themselves, which for example are relatively inflexible in terms of semi-major axis distributions.",
"While the conclusion of small proper eccentricities is robust from both empirical and modelling approaches, further insight into the azimuthal dependence of the ring width can only be gained with higher resolution imaging."
],
[
"HD 202628",
"The same modelling procedure was applied to HD 202628.",
"The 12m array data from Faramaz et al.",
"[25] were obtained from the ALMA archive and calibrated using the observatory-supplied script (the ACA data have insufficient resolution to be valuable here, and are dominated by a background source, so were not included).",
"The CASA statwt command was used to reweight the data, and all visibilities were exported and modelled assuming a single wavelength.",
"Compared to Fomalhaut this disc is smaller and detected at lower s/n, so the assumption of a single wavelength was found to be adequate.",
"The same four models were fitted, and all four are able to reproduce the data well and have similar $\\chi ^2$ values (i.e.",
"including the additional parameters in the full models is not well justified given the data).",
"Figure: Residual map for the best-fit Gaussian full model of theHD 202628 ring.",
"The white contour shows the location of the ring,and black contours highlight parts of the image above 3 times thenoise level of 6μ\\mu Jy beam -1 ^{-1}.",
"A bright point source has beensubtracted on the NE part of the disc, while three others were notincluded in the model.",
"All are assumed to be unrelated to the disc.The results of the modelling are summarised in Table REF , and Figures REF , which shows the (dirty) residual map for the Gaussian full model (Figures REF –REF show the posterior parameter distributions).",
"All models find the same disc position angle of $\\Omega = 130.7 \\pm 0.1^\\circ $ and inclination $i = 57.4 \\pm 0.1^\\circ $ .",
"The best-fit parameters are in general agreement with Faramaz et al.",
"[25], though with a slightly forced larger eccentricity (0.12 here vs. their 0.09).",
"Whether the pericenter directions agree is unclear; their Figure 3 suggests that pericenter lies near the NW ansa, but Figure 10 puts it more directly West of the stellar position.",
"The latter appears consistent with the value of 143$^\\circ $ from the SE ansa found here so seems more likely to be correct.",
"These differences may be related to the difference in eccentricity, and will be addressed in the near future (Faramaz et al., in preparation).",
"As with Fomalhaut, the residuals show that the model is a good representation of the data, though achieving this is less of a challenge for HD 202628 given the much lower s/n.",
"No significant residuals that are clearly related to the disc remain, though imperfect subtraction of the bright point source NW of the star is apparent.",
"While the eccentricity and disc width parameters $\\sigma _a$ and $e_p$ are given as upper limits, this is based on their 1-dimensional posterior distributions; inspection of the 2-d distributions finds that these two parameters cannot simultaneously be zero, implying that the ring is radially resolved (albeit marginally).",
"As with Fomalhaut, the best fit model finds that the proper eccentricity is significantly lower than the forced eccentricity,This significance is better verified directly from the posterior parameter distributions in the appendix, as the values in Table REF do not reflect asymmetric uncertainty ranges, nor that the 3$\\sigma $ limit for a parameter is not necessarily 3 times the 1$\\sigma $ uncertainty.",
"again implying that the ring cannot be modelled as a secularly perturbed ring whose eccentricities were initally near zero.",
"The lower s/n and spatial resolution means that the constraints on the parameters are not as strong, and higher spatial resolution data would be needed to obtain any insight into how the ring width varies as a function of azimuth."
],
[
"Eccentric ring origins - why so narrow?",
"Having established that the Fomalhaut and HD 202628 debris rings are too narrow to be explained by secularly perturbed particles that started with near-zero initial eccentricities, several possibile origins are discussed.",
"Figure REF is shown to frame this discussion, which shows the eccentricity and relative width of selected debris discs [35].",
"Only discs with ALMA-measured widths and eccentricities are shown, since these provide a quantity that is not affected by small dust grain dynamics (the eccentricity for HR 4796 was not detected by ALMA, so relies on scattered light observation, but is included because the structure at both wavelengths is consistent).",
"While other discs have also been observed with ALMA, these are wider than those shown, and are not known to have significant eccentricity.",
"The possible importance of Figure REF is that the narrower discs are also those that are most eccentric.",
"Fomalhaut and HD 202628 both lie above the dashed line, indicating that their width is narrower than expected given their forced eccentricities.",
"HR 4796 is a marginal case – Kennedy et al.",
"[34] note that the disc is only marginally resolved, and could be narrower (i.e.",
"further to the left of the dashed line).",
"Empirically, an important goal seems to be to fill out Figure REF , in particular for discs whose widths are $\\lesssim $ 30% of their semi-major axis; if the trend holds then it may be reasonably expected that the mechanism that radially concentrates planetesimals also excites a forced eccentricity.",
"Figure REF may eventually reveal a demarcation between narrow/eccentric and wide/circular disc systems, which could indicate whether the concentration mechanism operated in a given system.",
"The possible origins of narrow eccentric debris rings may be grouped into two main types of scenarios, depending on whether the $\\sim $ mm-sized dust observed by ALMA does, or does not, trace the planetesimal population.",
"In the former case the narrow rings are explained by either modifying the secular perturbation scenario to incorporate non-zero initial eccentricities, or perhaps discarding it entirely, while in the latter case secular perturbations may be retained, but the orbits of bodies changes as a function of their size (i.e.",
"particle free eccentricities \"damp\" towards the forced eccentricity as they move down the collisional cascade).",
"These scenarios, plus several others, are discussed below."
],
[
"Eccentric initial conditions",
"The simplest explanation for the narrow debris rings is that the (perhaps naive) expectation of small initial eccentricities in the secular perturbation scenario is not met.",
"That is, the initial conditions in $e\\omega $ -space may look more like the left two panels in Figure REF than the upper right panel, though the initial eccentricities need not be at the location of the forced eccentricity.",
"In this case there are two main scenarios to consider.",
"One possibility is that the planetesimals acquire significant coherent eccentricities (i.e.",
"with a preferred pericenter direction) prior to the evolution that is assumed in a standard secular perturbation scenario.",
"For example, a single planet may exert some influence on planetesimals prior to dispersal of the gas disc, but this influence is diminished in a way that simply decreases the magnitude of the forced eccentricity [36].",
"Thus, when the gas is removed the planetesimal eccentricities are already shifted somewhat towards the forced eccentricity, and as they precess under the influence of the same planet, they trace out a smaller circle in $e\\omega $ -space, resulting in a narrower debris ring.",
"To also provide the tentative trend in Figure REF , this process should also radially concentrate planetesimals, perhaps by trapping them in a gas pressure maximum exterior to the planet [37], or trapping dust in a narrow ring which then goes on to form said planetesimals via the streaming instability [38].",
"It is of course possible that the planetesimals acquire all of their eccentricity before the gas disc is dispersed, in which case no further eccentricity evolution via secular perturbations is necessary.",
"In this case, there is not necessarily a need for any planet, as it may be that the initial eccentricitity excitation is related to planet-free gas dynamics or dust/gas interaction, For example, Lyra & Kuchner [39] show that narrow eccentric rings could form in planet-free discs with high dust/gas ratios, which could occur in local gas pressure maxima, or more globally as the density drops during gas disc dispersal.",
"How might evidence for these possible scenarios be sought?",
"Most simply, they both predict that planetesimals should form and acquire non-zero eccentricites during the gas-rich phase of protoplanetary disc evolution.",
"One system showing a possible asymmetry is PDS 70, which hosts a protoplanetary disc and at least one interior planet [40], [41].",
"The dust component is fairly narrow and shows both a brightness asymmetry [42], and a stellocentric offset that places this ansa farther from the star [43].",
"While these combined properties might be interpreted as apocenter glow [44], the disc is likely optically thick and the origin of the asymmetry may not be so simple.",
"Nevertheless, PDS 70 is a possible precursor to systems such as Fomalhaut and HD 202628, and thus circumstantial evidence that eccentric rings can exist before the debris phase.",
"A possible difference between the single and no-planet scenarios is that in the presence of an eccentric planet the eccentricity of debris rings should increase with time.",
"Immediately after gas disc dispersal secular perturbations have yet to act to their full effect, so debris rings only posess their initial eccentricity, but later acquire their full eccentricity, which is larger.",
"A young disc such as that around HD 4796, with eccentricity $\\approx $ 0.06 [45] is suggestive, but there are too few discs with well-constrained eccentricities and widths for this test to be possible at present."
],
[
"Eccentricity damping",
"While the prior discussion assumed that the observed mm-wave disc structure is a true reflection of the underlying planetesimal orbits, this assumption might be violated.",
"Specifically, if particle free eccentricities decrease as they become smaller, a debris ring could appear narrower at mm wavelengths than the underlying planetesimal belt.",
"A possible way to damp orbital eccentricities in a debris disc is in collisions, if the random velocities of post-collision fragments tend to be smaller than the targets'.",
"Because this damping decreases the relative orbital velocities, the effect in an eccentric disc with a preferred pericenter direction is to drive down the free eccentricity.",
"This scenario was explored analytically by Pan & Schlichting [46], and in general finds the expected decrease in velocities for smaller objects.",
"This scenario however essentially relies on destructive collisions being between objects of similar size, as the orbit of the center of mass has a lower random velocity than either of the two bodies.",
"If the typical destructor is much smaller than the target body, then the post-collision fragments will tend to retain their original velocities, and the damping is inefficient.Picture two people throwing pumpkins towards each other at high velocity, neither is likely to be splattered with fragments (most will fall to the ground below the collision).",
"However, if one throws a much smaller object, such as a pebble, the pebble thrower will be hit by the fragmented pumpkin (which has much more momentum).",
"A significant size difference is generally expected for destructive collisions, because the size distribution is always such that smaller objects are much more common, and therefore the impactor that destroys a larger object is usually the smallest one that can do so [47].",
"In this case, orbital velocities do not decrease significantly with object size, and the entire size distribution inherits eccentricities from the largest bodies.",
"This picture lacks some nuance, and for example ignores the effect of non-destructive collisions, but collisional damping seems unlikely to be significant, and if anything eccentricities seem more likely to increase in collisions.",
"For example, in their simulations of impacts with 100km-diameter targets, Jutzi et al.",
"[48] find that the largest fragments, which dominate the mass, tend to have low velocities relative to the target center of mass, so the approximation that all bodies share the same orbit as the collision center of mass [49] is reasonable.",
"A potentially attractive aspect for collisional damping is that it could be observationally testable.",
"If free eccentricities tend to decrease with particle size in the mm to cm size regime, observations may be able to measure different debris ring widths as a function of wavelength.",
"While the discussion above suggests that significant damping is unlikely, modelling and observations focussing on this specific size range would be valuable.",
"A side effect of damping is that the relative velocities between smaller particles is also smaller, leading to longer collisional lifetimes and a steeper size distribution than would be expected if all objects share the same random velocities [46].",
"The steeper size distribution in turn means that for a given observed disc brightness (which is proportional to the surface area of small grains), the inferred total mass (which is dominated by large bodies) can be much smaller.",
"Damping therefore provides a possible resolution of the debris disc “mass problem” [50], in which disc masses in systems such as HR 4796 are inferred to be implausibly large [34]."
],
[
"A single event",
"A third way to explain the narrow debris rings is to discard the standard debris disc paradigm entirely (i.e.",
"the idea of dust derived from an underlying population of planetesimals undergoing continuous collisions in a pseudo-steady state).",
"That is, to consider transient and/or stochastic events as a way to produce a population of objects on similar eccentric orbits.",
"The scenario for creating a narrow debris ring is then based on the breakup of a single large body, which must have been destroyed in such a way that the velocity dispersion of fragments is small (i.e.",
"similar to described above).",
"The scenario may therefore be very similar to that explored by Jackson et al.",
"[51], where debris is released from a large body in a collision.",
"This scenario tends to produce non-axisymmetric dust density distributions, as more dust is concentrated at the spatial location of the original dust production event (and more radially distributed on the opposite side of the star).",
"However, for a small velocity dispersion the asymmetry will be smaller, and if there are other perturbing bodies in the system, the asymmetry is decreased as the fragments' orbits precess.",
"In some ways this scenario is similar to a normal debris ring scenario, in that once created, a population of fragments evolves in the same way, and is indistinguishable from a planetesimal population with similar orbits.",
"It is therefore very hard to rule out that narrow debris rings (eccentric or otherwise) originate from planetesimals that are themselves a family of fragments.Another difference noted by Cataldi et al.",
"[52] is that if CO gas is also liberated by the collisional evolution, then very different levels of atomic carbon (which results from rapid photodissociation of CO) may result.",
"As carbon is not necessarily easily removed from the system, it builds up over time, so a system where the observed debris is the result of a more recent event should have lower levels of carbon than one that has been evolving for longer.",
"A basic objection however is why only a single debris ring should be detected around a given star, and not a series of near-concentric rings that arise from similar events.",
"A more serious issue is that inferred masses for debris discs can be of order tens of Earth masses [47], and it therefore seems unlikely that this mass in solid fragments can be created from a single event."
],
[
"Shepherding panets",
"Boley et al.",
"[53] proposed that the width of the Fomalhaut debris ring is constrained by a pair of shepherding planets, by analogy with Uranus' $\\epsilon $ ring and Saturn's F ring.",
"In these cases the moons were inferred to exist prior to their discovery because the rings should rapidly spread radially, and some external force is required to confine them [54].",
"As discussed above, collisions in debris discs are such that significant spreading of narrow rings is not necessarily expected.",
"This picture does not however mean that shepherding moons do not exist, just that they are not required by the analogy that narrow debris rings appear similar to narrow planetary rings.",
"As discussed at the outset, a problem with planet-interaction scenarios is that the putative planets are often not detectable.",
"In the case of Fomalhaut, the inferred shepherding planet masses were several Earth masses [53], meaning that their detection at $\\sim $ 140 au from the star is currently impossible.",
"An avenue that has not yet been explored is whether shepherding planets should induce detectable azimuthal structure in the ring edges near the planets [55]; such a study would require simulations, and no doubt higher spatial resolution imaging.",
"In the meantime, the shepherding planet scenario is considered more complicated than is required by the data; while shepherding requires two eccentric planets, the secular perturbation scenario only requires one."
],
[
"Massive debris ring",
"The secular perturbation theory as applied to debris discs usually assumes that the belt mass can be ignored; the planet perturbs the belt, but the belt does not perturb the planet (i.e.",
"cause it to precess).",
"In the case of eccentric debris rings this assumption seems justified, as a precessing planet would cause the forced eccentricity to move in $e\\omega $ -space over time, and differing precession rates for ring particles at different semi-major axes would cause the ring to quickly lose the observed coherence.",
"However, this issue might be circumvented with shepherding satellites as discussed above, or if the ring has sufficient mass that self-gravity forces all particle pericenters to remain similar.",
"If such a coherence-maintaining mechanism could operate, then it is possible that the debris ring was initially near-circular, and gained an observable eccentricity through mutual interaction with an initially eccentric planet.",
"While these ideas were developed for planetary rings in the Solar system , whether they apply here is not clear, so this scenario requires further work.",
"This paper shows that the eccentric Fomalhaut and HD 202628 debris rings are narrower than expected, based on a secular perturbation model and the expectation of near-zero initial eccentricities.",
"In the case of Fomalhaut, this narrowness is clear from simply measuring the radial profile as observed by ALMA (Figure REF ), but for HD 202628 the s/n is per beam is lower so the conclusion drawn from several similar eccentric ring models.",
"What does this narrowness mean?",
"The most likely implication is that in a secular perturbation scenario the planetesimals did not initially have near-circular orbits.",
"This prior excitation could be a consequence of planetary perturbations within the gas-rich protoplanetary disc, but if other processes can produce coherently eccentric planetesimal orbits, may not require a planet at all.",
"In either case however, a clear prediction is that eccentric rings should exist within protoplanetary discs.",
"Whether these rings are observable is less clear, but PDS 70, host to a protoplanetary disc that shows both geometric and brightness asymmetry, and at least one planet, is singled out as a possible progenitor of systems such as Fomalhaut and HD 202628.",
"Making and extending such links would be valuable to understand whether debris discs or their direct progenitors exist within gas-rich protoplanetary discs.",
"An alternative possibility is that the planetesimal belt is wider than observed with ALMA, but that objects are damped as they fragment to smaller sizes.",
"This possibility is disfavoured, but may be testable if mm to cm-size grains have different enough eccentricities.",
"Observational tests of collisional damping may also be relevant to the debris disc mass problem.",
"A plot of belt forced eccentricity against relative radial width (Figure REF ) provides circumstantial evidence of a trend that the narrowest debris rings are also the most eccentric.",
"However, a lack of eccentric systems limits the numbers in this Figure, and more systems will need to be characterised to explore this possible relation further.",
"For a start, HR 4796 is identified as possibly being narrower than expected, and should be imaged at higher spatial resolution.",
"This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.00966.S, ADS/JAO.ALMA#2016.1.00515.S, which are available from the ALMA archive: http://almascience.nrao.edu/aq/.",
"Relevant code for this research work is stored in GitHub: https://github.com/drgmk/eccentric-width and have been archived within the Zenodo repository: https://doi.org/10.5281/zenodo.3832148.",
"GMK declares no competing interests.",
"GMK is supported by the Royal Society as a Royal Society University Research Fellow.",
"Thanks to both referees for constructive reports, Jane Huang for noting the possibly eccentric structure of the PDS 70 disc, to Luca Matrà for advice/discussions on modelling ALMA data, to Meredith MacGregor and Virginie Faramaz for discussions on their prior work on Fomalhaut and HD 202628, and to Mark Wyatt for the reminder that planetesimal belts have mass.",
"ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.",
"The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ."
]
] | 2005.14200 |
[
[
"A Hermite-Gaussian Based Radial Velocity Estimation Method"
],
[
"Abstract As the first successful technique used to detect exoplanets orbiting distant stars, the Radial Velocity Method aims to detect a periodic Doppler shift in a star's spectrum.",
"We introduce a new, mathematically rigorous, approach to detect such a signal that accounts for functional relationships of neighboring wavelengths, minimizes the role of wavelength interpolation, accounts for heteroskedastic noise, and easily allows for statistical inference.",
"Using Hermite-Gaussian functions, we show that the problem of detecting a Doppler shift in the spectrum can be reduced to linear regression in many settings.",
"A simulation study demonstrates that the proposed method is able to accurately estimate an individual spectrum's radial velocity with precision below 0.3 m/s.",
"Furthermore, the new method outperforms the traditional Cross-Correlation Function approach by reducing the root mean squared error up to 15 cm/s.",
"The proposed method is also demonstrated on a new set of observations from the EXtreme PREcision Spectrometer (EXPRES) for the star 51 Pegasi, and successfully recovers estimates that agree well with previous studies of this planetary system.",
"Data and Python3 code associated with this work can be found at https://github.com/parkerholzer/hgrv_method.",
"The method is also implemented in the open source R package rvmethod."
],
[
"?",
"??",
"?",
"??",
"?",
"??",
"?",
"??[label=e1]???",
"e1 ?",
"?",
"?"
]
] | 2005.14083 |
[
[
"Fast flavor conversions in supernovae: the rise of mu-tau neutrinos"
],
[
"Abstract Neutrinos in a core-collapse supernova can undergo fast flavor conversions with a possible impact on the explosion mechanism and nucleosynthesis.",
"We perform the first non-linear simulations of fast conversions in the presence of three neutrino flavors.",
"The recent supernova simulations with muon production call for such an analysis, as they relax the standard $\\nu_{\\mu,\\tau}=\\bar{\\nu}_{\\mu,\\tau}$ (two-flavor) assumption.",
"Our results show the significance of muon and tau lepton number angular distributions, together with the traditional electron lepton number ones.",
"Indeed, our three-flavor results are potentially very different from two-flavor ones.",
"These results strengthen the need to further investigate the occurrence of fast conversions in supernova simulation data, including the degeneracy breaking of mu and tau neutrinos."
],
[
" MPP-2020-79 Fast flavor conversions in supernovae: the rise of mu-tau neutrinos Francesco Capozzi [email protected] Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany Madhurima Chakraborty [email protected] Indian Institute of Technology, Guwahati, Assam-781039, India Sovan Chakraborty [email protected] Indian Institute of Technology, Guwahati, Assam-781039, India Manibrata Sen [email protected] Northwestern University, Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA Department of Physics, University of California Berkeley, Berkeley, California 94720, USA Neutrinos in a core-collapse supernova can undergo fast flavor conversions with a possible impact on the explosion mechanism and nucleosynthesis.",
"We perform the first non-linear simulations of fast conversions in the presence of three neutrino flavors.",
"The recent supernova simulations with muon production call for such an analysis, as they relax the standard $\\nu _{\\mu ,\\tau }=\\bar{\\nu }_{\\mu ,\\tau }$ (two-flavor) assumption.",
"Our results show the significance of muon and tau lepton number angular distributions, together with the traditional electron lepton number ones.",
"Indeed, our three-flavor results are potentially very different from two-flavor ones.",
"These results strengthen the need to further investigate the occurrence of fast conversions in supernova simulation data, including the degeneracy breaking of mu and tau neutrinos.",
"Introduction.",
"– Neutrinos streaming out of a supernova (SN) encounter a large density of ambient neutrinos and antineutrinos.",
"Forward-scattering off this surrounding neutrino bath has been shown to cause self-induced effects, leading to collective flavor oscillations, where neutrinos with different oscillation frequencies change their flavors in a coherent fashion (see [1], [2], [3], [4] for a comprehensive review).",
"Recently, it was shown that under certain conditions, these collective oscillations can be “fast\", growing with the extremely large neutrino density $n_\\nu $ , while being independent of the neutrino mass-squared difference $\\Delta m^2$ or the neutrino energy $E$ [5], [6], [7].",
"These fast flavor conversions (FFC) can occur close to the region of neutrino free-streaming, and can grow as large as $10^5$ times faster than the vacuum oscillations.",
"A host of studies [8], [9], [5], [6], [7], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] in the past few years suggest that a necessary, though not sufficient, condition for the existence of these fast instabilities is the presence of a zero-crossing in the angular distribution of the neutrino electron lepton number (ELN), i.e., the difference between the electron neutrino and the antineutrino angular emission spectra should go through a zero for some emission angle.",
"Though neutrinos of other flavors are also present in the SN environment, ELN is considered to be the only driving quantity, as these studies are performed assuming an effective two-flavor scenario ($e$ and $x$ flavor, where $x=\\mu ,\\,\\tau $ or a linear combination of both).",
"This is due to the fact that in the absence of muons and taus in traditional SN simulations, the heavy lepton neutrinos have identical microphysics and similar number density ($n_{\\nu _\\mu }=n_{\\nu _\\tau }=n_{\\bar{\\nu }_\\mu }=n_{\\bar{\\nu }_\\tau }=n_{\\nu _x}$ ) for all emission angles.",
"Crossings are naively expected to occur near the neutrino free-streaming zone and require a comprehensive study of the interplay of collisions and fast conversions, thereby necessitating the use of quantum kinetic equations [21], [22], [23], [24].",
"Recently, dedicated studies have performed a thorough scan of existing data from SN hydrodynamical simulations.",
"In [25] it was shown that crossings in the ELN, sometimes related to the lepton number emission self-sustained asymmetry (LESA) [26], [27], are present both in the convective layer of the proto-neutron star (PNS) and outside the neutrino decoupling region.",
"Such results have been confirmed with further insights from [18].",
"Furthermore, in [28], it was claimed that coherent neutrino-nucleus scattering in the pre-shock region can lead to a tiny zero crossing in the backward direction, leading to FFC at a distance of $\\mathcal {O}(100)\\,{\\rm km}$ from the PNS.",
"The presence of these recently discovered FFCs well inside a SN can lead to paradigm changes in our understanding of the explosion dynamics, requiring a more detailed analysis of the different approximations going into the studies.",
"In particular, the above results are based on the effective two-flavor setup.",
"This set up has severe limitations as the $\\nu _x$ and $\\bar{\\nu }_x$ flux would differ naturally if nucleon recoil effects, implying different neutral-current scattering cross sections, are taken into account.",
"Moreover, it would be also natural to expect that the high temperatures during the accretion phase would create muons in the nascent neutron stars.",
"In fact, [29] has shown that the addition of muons can enhance neutrino energy deposition to the stalled shockwave, leading to a successful explosion.",
"Muon production in a SN [29] can create differences in the heavy lepton flavor neutrino fluxes, thereby necessitating the inclusion of three-flavor effects.",
"The oscillation treatment of such a scenario has been pointed out in [30] and recently, [31] has done a detailed analysis of fast conversions to three neutrino flavors.",
"Using a linear stability analysis, it demonstrated the possibility of altering the instability growth rates obtained in the standard two-flavor setup.",
"Going a step forward with respect to current literature, we, for the first time, perform a fully non-linear computation of FFCs in the presence of three neutrino flavors.",
"Motivated by the difference in the heavy lepton flavor neutrino fluxes observed in [29], we propose simple toy models and demonstrate that it is not only the ELN, but also $\\mu \\,{\\rm LN}$ and $\\tau \\,{\\rm LN}$ that drives the onset of these rapid conversions.",
"In fact, it is the difference of any of the two-flavor lepton numbers that goes into the evolution equations [31].",
"This emphasizes the incompleteness of a two-flavor analysis, especially while analyzing the presence of FFC in the pre-shock region, where tiny crossings in the ELN can get erased by the $\\mu {\\rm LN}$ and or $\\tau {\\rm LN}$ .",
"Our results suggest the importance of such a fully non-linear three-flavor study of FFC with detailed hydrodynamic SN simulations, including muon production, to gauge its impact on SN dynamics.",
"Equations of motion.",
"– The dynamics of the neutrino occupation number matrices $\\rho _{{\\bf p}, {\\bf x},t}$ for momentum ${\\bf p}$ at position ${\\bf x}$ and time $t$ is governed by the equations of motion (EoMs) [32] $i\\left(\\partial _t + {\\bf v}_{\\bf p} \\cdot \\nabla _{\\bf x}\\right) \\rho _{{\\bf p}, {\\bf x},t}= [\\Omega _{{\\bf p}, {\\bf x},t}, \\rho _{{\\bf p}, {\\bf x},t}]\\,\\ ,$ where the left-hand-side accounts for time and spatial dependence of $\\rho _{{\\bf p}, {\\bf x},t}$ .",
"The Hamiltonian $\\Omega _{{\\bf p}, {\\bf x},t}$ on the right-hand-side consists of the vacuum term $(\\Delta m^2/2\\,E)$ , the Mikheyev-Smirnov-Wolfenstein matter potential depending on the background charged lepton number density and the $\\nu -\\nu $ self-interaction potential, $\\int d^3{\\bf q}/(2\\pi )^3 (1-{\\bf v}_{\\bf p}\\cdot {\\bf v}_{\\bf q})({\\rho }_{{\\bf q}, {\\bf x},t}-{\\bar{\\rho }}_{{\\bf q}, {\\bf x},t})$ [2] In the context of fast oscillations, the vacuum term does not play a major role, except for seeding the oscillations.",
"For a two-flavor FFC computation with both spatial and temporal evolution, one can neglect the matter term due to background electron density under certain circumstances [14] (see [33] for a detailed analysis on the role of background matter), whereas in a three-flavor setup with both electrons and muons in the background, this is not trivial [34], [35].",
"However, our numerical examples only deal with time evolution and the matter terms can be safely neglected [7].",
"Thus, only the non-linear term governs the dynamics of our setup.",
"Note that there can be radiative corrections in this term as well, due to the presence of three-flavors, however, these are small and can be dropped [35].",
"What enters the EoMs is the difference in the occupation numbers of the different flavors integrated over energy, dubbed in literature as the corresponding neutrino flavor lepton number.",
"In the two-flavor scenario, assuming the non-electron flavor angular distributions are identical for both the particles and the antiparticles, one defines the electron lepton number (ELN) as [10] $G^e_{\\bf v} = \\sqrt{2} G_F \\int _{0}^{\\infty }\\frac{dE\\,E^2}{2 \\pi ^2}\\left[\\rho _{ee}(E,{\\bf v})-\\bar{\\rho }_{ee}(E,{\\bf v})\\right] \\,\\,.$ However, 2-D simulations [29] show that this picture is not entirely accurate; the inclusion of muons does create an appreciable flux hierarchy between the $\\nu _\\mu $ and $\\bar{\\nu }_\\mu $ in the accretion phase.",
"This is primarily because muons can be pair-produced from electrons, and can participate in $\\beta $ -processes to create $\\nu _\\mu $ and $\\bar{\\nu }_\\mu $ .",
"The relative neutron-to-proton ratio governs the extent of muonization $(\\nu _\\mu -\\bar{\\nu }_\\mu )$ in the PNS.",
"In contrast, due to the high value of $m_\\tau $ , the $\\tau $ density remains small throughout, although a tiny asymmetry may arise due to different scattering cross-sections with nucleons.",
"Thus, within the three-flavor formalism [30], [31], one needs to define the corresponding muon lepton number $(\\mu \\,{\\rm LN})$ and tau lepton number $(\\tau \\,{\\rm LN})$ as $G^\\mu _{\\bf v} &=&\\sqrt{2} G_F \\int _{0}^{\\infty }\\frac{dE\\,E^2}{2 \\pi ^2}\\left[\\rho _{\\mu \\mu }(E,{\\bf v})-\\bar{\\rho }_{\\mu \\mu }(E,{\\bf v})\\right] \\,\\,,\\\\G^\\tau _{\\bf v} &=&\\sqrt{2} G_F \\int _{0}^{\\infty }\\frac{dE\\,E^2}{2 \\pi ^2}\\left[\\rho _{\\tau \\tau }(E,{\\bf v})-\\bar{\\rho }_{\\tau \\tau }(E,{\\bf v})\\right] \\,\\,.$ The EoM is sensitive to the difference of these lepton numbers via the following quantities, $G^{e\\mu }_{\\bf v}&=&G^e_{\\bf v}-G^\\mu _{\\bf v} \\,\\,,\\\\G^{e\\tau }_{\\bf v}&=&G^e_{\\bf v}-G^\\tau _{\\bf v}\\,,\\\\G^{\\mu \\tau }_{\\bf v}&=&G^\\mu _{\\bf v}-G^\\tau _{\\bf v}\\,.$ Note that $G^{e\\mu }_{\\bf v}$ reduces to the ELN in the two-flavor scenario, where $n_{\\nu _\\mu }=n_{\\bar{\\nu }_\\mu }$ .",
"There exists three off-diagonal elements of the occupation number matrix: $\\rho _{e\\mu },\\,\\rho _{e\\tau },\\,\\rho _{\\mu \\tau }$ , which might undergo an exponential growth (i.e.",
"an instability), when a crossing is created in one of the angular distributions.",
"Therefore, a crossing in the ELN is not enough, since one needs to consider $G^{e\\mu },\\, G^{e\\tau }$ or $G^{\\mu \\tau }$ [31].",
"This has important consequences for FFC.",
"For instance, a tiny crossing in the ELN can be erased by a negative $\\mu \\,{\\rm LN}$ .",
"Analogously, the absence of a crossing in the ELN can be compensated by a positive $\\mu \\,{\\rm LN}$ .",
"Similar arguments hold for the other sectors.",
"As a result, claiming the presence of fast oscillations, focusing only on tiny ELN crossings, as done for example in [28], might lead to incomplete conclusions.",
"In what follows, we consider simple toy models to demonstrate this important point.",
"Numerical examples – We assume a spatially homogeneous flavor composition, and only consider the time evolution in Eq.",
"REF .",
"We assume that the initial neutrino angular distributions are axially symmetric around the $z-$ axis, i.e., they only depend on the zenith angle.",
"Nevertheless, we do consider the azimuthal angles in our calculations, so that also axially breaking instabilities are allowed to develop [36].",
"We take $\\mu =\\sqrt{2}G_Fn_\\nu =4\\times 10^5$ km$^{-1}$ , which is a typical value in the neutrino decoupling region.",
"Concerning the vacuum term, we use as oscillation frequencies $\\Delta m^2_{31}/(2E)=0.5$ km$^{-1}$ and $\\Delta m^2_{21}/(2E)=0.01$ km$^{-1}$ , whereas we set the mixing angles to be $\\theta _{12}=\\theta _{13}=\\theta _{23}=10^{-3}$ .",
"In this way, we mimic the suppression of $\\theta _{ij}$ induced by the large potentials.",
"Moreover, as we focus only on the time evolution we neglect the matter terms, $\\lambda _e =\\lambda _{\\mu } =\\lambda _{\\tau } =0$ .",
"In the following, we consider four toy cases of $G^{e\\mu }_{\\bf v}$ , $G^{e\\tau }_{\\bf v}$ and $G^{\\mu \\tau }_{\\bf v}$ , highlighting the fact that the effective two-flavor formalism leads to different conclusions.",
"Note that current state of the art SN simulations [29] can only provide angular moments of the neutrino distribution, from which one can construct realistic angle-dependence of the neutrino flavor intensity but only near the neutrinosphere.",
"However, these distributions are not very reliable at larger radii [37].",
"We leave to future work the assessment of which lepton number angular distributions are realized in nature.",
"Figure: The upper panels show the angular distribution of the fluxes and the effective lepton numbers for different flavors.",
"The effective lepton number panel (G 𝐯 αβ G^{\\alpha \\beta }_{\\bf v} vs 𝐯\\bf v) shows both the three-flavor (solid lines) and the two-flavor ELN (dashed line).",
"The lower panel shows the evolution of the angle-averaged off-diagonal elements 〈ρ αβ 〉\\left|\\langle \\rho _{\\alpha \\beta }\\rangle \\right| of the occupation number matrix with time (in ns).",
"The three-flavor evolution (solid lines) has no instability, while the effective two-flavor evolution, assuming ν x =ν ¯ x \\nu _x=\\bar{\\nu }_x, shows large exponential growth (dashed line).Firstly, we consider the scenario in Fig.REF , where each of $G^{e}_{\\bf v}$ , $G^{\\tau }_{\\bf v}$ and $G^{\\mu }_{\\bf v}$ have crossings as can be interpreted from the angular distribution of the fluxes (uppermost panel).",
"We show the $G^{e}_{\\bf v}$ crossing by the dashed lines in the $`G^{\\alpha \\beta }_{\\bf v}$ vs $\\bf v$ ' panel.",
"Thus, in the two-flavor scenario ($n_{\\nu _\\mu }=n_{\\bar{\\nu }_\\mu }$ ), there is an exponential growth in $\\rho _{e\\mu }$ (dashed line, lower panel), due to this crossing in $G^{e}_{\\bf v}$ .",
"However, it is evident that no crossing persists in $G^{e\\mu }_{\\bf v},\\,G^{e\\tau }_{\\bf v}$ or $G^{\\mu \\tau }_{\\bf v}$ .",
"Consequently, the time evolution of the off-diagonal elements of $|\\langle \\rho _{\\alpha \\beta }\\rangle |$ (solid lines, lower panel) does not show any exponential growth.",
"Here, we have defined $\\left|\\langle \\rho _{\\alpha \\beta }\\rangle \\right|=\\left|\\int d{\\bf v}\\rho _{\\alpha \\beta }({\\bf v})\\right|\\,.$ This simple but crucial example clearly demonstrates that some of the crossings found in [18], [25], [28] might disappear once the corresponding hydrodynamical simulation include the full three-flavor neutrino transport, including the production of muons.",
"Figure: Panels represent same as in Fig. .",
"However, due to large crossings in G 𝐯 eμ ,G 𝐯 eτ G^{e\\mu }_{\\bf v},\\,G^{e\\tau }_{\\bf v}, substantial flavor conversions are seen in all the sectors.",
"As a second example, consider Fig.",
"REF (upper panels), where the ELN ($G^{e}_{\\bf v}$ ) has a regular crossing (dashed line), but there are none in the $\\mu \\,{\\rm LN}$ or the $\\tau \\,{\\rm LN}$ .",
"The spectra are designed such that the flavor lepton number difference $G^{e\\mu }_{\\bf v},\\,G^{e\\tau }_{\\bf v}$ exhibit deep crossings (solid lines).",
"On the other hand, the crossing in $G^{\\mu \\tau }_{\\bf v}$ is extremely shallow.",
"The naive two-flavor intuition is that there should be exponential growths in the $e-\\mu $ and $e-\\tau $ sector.",
"However, the non-linear, coupled nature of the problem intertwines the growths in all the sectors.",
"This is borne by the lower panel, where substantial flavor conversion is seen in all the sectors, though with different growth rates.",
"Note that the growth in $\\rho _{\\mu \\tau }$ is inherently a non-linear effect, and will not be captured by a linear stability analysis.",
"The corresponding two-flavor evolution of $\\rho _{e\\mu }$ is also shown for comparison.",
"Clearly, the two-flavor evolution is very different, with a larger onset time and growth rate.",
"Figure: Panels represent same as in Fig. .",
"Crossing only in G 𝐯 μτ G^{\\mu \\tau }_{\\bf v} causes an exponential growth in μ-τ\\mu -\\tau sector (lower panel).",
"The two-flavor evolution shows no instability.The third case, shown in Fig.",
"REF , presents a crossing only in $G^{\\mu \\tau }_{\\bf v}$ .",
"There exists a reasonable asymmetry (upper panels) between $\\nu _\\mu $ and $\\nu _\\tau $ (and between their antiparticles as well) to generate an exponential growth in the $\\mu -\\tau $ sector.",
"This is what is seen in the lower panel, where the $\\mu -\\tau $ sector experiences a flavor instability, while the other two do not.",
"Indeed, the two-flavor analysis (dashed line) also does not exhibit any instability due to the lack of a crossing in the ELN ($G^{e}_{\\bf v}$ ).",
"This example advances the hypothesis that those regions where no ELN crossing was found in [18], [25], [28] might, in reality, have fast instabilities once the differences between $\\nu _\\mu $ and $\\bar{\\nu }_\\mu $ are taken into account.",
"Another comment is in order: the amplitude of the exponential growth in our toy model is not enough to cause substantial flavor conversions.",
"However, the background conditions for these solutions may dynamically change in a realistic SN environment, or if spatial evolution is taken into account, and may result in flavor conversions.",
"Figure: Panels represent same as in Fig. .",
"However, shallow crossings are present in G 𝐯 eμ ,G 𝐯 eτ G^{e\\mu }_{\\bf v},\\,G^{e\\tau }_{\\bf v} in the backward direction (upper panels), leading to exponential growths in 〈ρ αβ 〉\\left|\\langle \\rho _{\\alpha \\beta }\\rangle \\right| (lower panel) for the three-flavor setup.",
"The two-flavor evolution shows no instability.As a final example, we consider a scenario in Fig.",
"REF where shallow crossings are present in $G^{e\\mu }_{\\bf v},\\,G^{e\\tau }_{\\bf v}$ in the backward direction, whereas $G^{\\mu \\tau }_{\\bf v}$ shows a significant crossing in the forward direction as well.",
"To contrast with the two-flavor examples, the setup is constructed such that the ELN ($G^{e}_{\\bf v}$ ) also has a shallow crossing but in the forward direction only.",
"We find that such shallow crossings readily lead to an instability in the three-flavor case, whereas the two-flavor setup shows no instability (lower panel).",
"This example is motivated from [28], where it was pointed out that shallow crossings in the backward directions can lead to a fast instability.",
"In [28], such backward crossings were associated with residual coherent scattering on heavy nuclei, which is slightly enhanced for $\\bar{\\nu }_e$ with respect to $\\nu _e$ , because of their larger average energy.",
"Our toy model advances the hypothesis that the existing differences between $\\nu _\\mu $ and $\\bar{\\nu }_\\mu $ could (at least in principle) be the real cause of these crossings.",
"Similarly, the non negligible muon lepton number can also erase a potential shallow (backward) crossing in the ELN.",
"Our examples clearly establish the importance of detailed three-flavor treatments to assess whether such possibilities are indeed realized in nature.",
"We leave this task to future work.",
"Discussion and conclusions.",
"– Fast neutrino flavor conversion near the SN core can lead to a paradigm change in our understanding of flavor evolution of supernova neutrinos.",
"Within a two-flavor formalism, these ultra-rapid flavor conversions are believed to occur mainly when the ELN, i.e., the difference between the $\\nu _e$ and $\\bar{\\nu }_e$ angular spectra, exhibit a zero-crossing.",
"Such flavor mixing can have a drastic impact on the shockwave revival, as well as nucleosynthesis.",
"Hence, it is crucial to appreciate these flavor conversions, using a complete three-flavor analysis.",
"We have performed, for the first time, a completely non-linear, three-flavor treatment of fast flavor conversions of neutrinos.",
"We find that the inclusion of three-flavors can significantly alter our understanding of the conditions for fast conversions.",
"Using simple toy spectra, we demonstrate that it is not the ELN, or correspondingly, the $\\mu $ LN, and $\\tau $ LN, but rather their differences that govern these fast modes.",
"The examples studied in this paper clearly show that three-flavor evolution can result in instabilities that are not captured by a two-flavor study; conversely, it can also wash out the instabilities predicted by a two-flavor study.",
"These results also show that the linear stability analysis cannot capture all the instability signatures of the full non-linear analysis, as expected.",
"Our findings further indicate caution against claiming the presence of fast conversions for shallow crossings in the ELN, because such crossings can easily be nullified by an opposite crossing in the $\\mu $ LN.",
"This is particularly relevant in the shock region, where a significant population of $\\nu _\\mu ,\\nu _\\tau $ can be expected in the accretion phase.",
"This motivates the necessity of including muons in a dedicated analysis of fast-flavor conversions to gauge their impact on supernova dynamics.",
"Acknowledgments –We would like to thank Georg Raffelt, Alessandro Mirizzi, Basudeb Dasgupta and Sajad Abbar for useful comments on the manuscript.",
"MS acknowledges support from the National Science Foundation, Grant PHY-1630782, and to the Heising-Simons Foundation, Grant 2017-228.",
"The work of F.C.",
"is supported by the Deutsche Forschungsgemeinschaft through Grants SFB-1258 “Neutrinos and Dark Matter in Astro- and Particle Physics (NDM)” and EXC 2094 “ORIGINS: From the Origin of the Universe to the First Building Blocks of Life”.",
"SC acknowledges the support of the Max Planck India Mobility Grant from the Max Planck Society.",
"MC and SC also recived funding/support from the European Union’s Horizon 2020 research and innovation programme through the InvisiblesPlus RISE under the Marie Skłodowska-Curie grant agreement No 690575 and through the Elusives ITN under the Marie Skłodowska -Curie grant agreement No 674896."
]
] | 2005.14204 |
[
[
"FCN+RL: A Fully Convolutional Network followed by Refinement Layers to\n Offline Handwritten Signature Segmentation"
],
[
"Abstract Although secular, handwritten signature is one of the most reliable biometric methods used by most countries.",
"In the last ten years, the application of technology for verification of handwritten signatures has evolved strongly, including forensic aspects.",
"Some factors, such as the complexity of the background and the small size of the region of interest - signature pixels - increase the difficulty of the targeting task.",
"Other factors that make it challenging are the various variations present in handwritten signatures such as location, type of ink, color and type of pen, and the type of stroke.",
"In this work, we propose an approach to locate and extract the pixels of handwritten signatures on identification documents, without any prior information on the location of the signatures.",
"The technique used is based on a fully convolutional encoder-decoder network combined with a block of refinement layers for the alpha channel of the predicted image.",
"The experimental results demonstrate that the technique outputs a clean signature with higher fidelity in the lines than the traditional approaches and preservation of the pertinent characteristics to the signer's spelling.",
"To evaluate the quality of our proposal, we use the following image similarity metrics: SSIM, SIFT, and Dice Coefficient.",
"The qualitative and quantitative results show a significant improvement in comparison with the baseline system."
],
[
"Introduction",
"Handwritten signature is a biometric authentication method widely used for personal documents and legal contract validations.",
"Besides, experts in forensic analysis examine handwritten signatures to certify the authenticity of the writing and reveal possible fraud, which in some cases can mean high-value financial losses.",
"One possible approach to signature authentication is through human operators who must compare the signature present in the document with the signature of the original subscriber.",
"However, this approach can be expensive and time-consuming, given the amount of data accumulated in institutions that use a handwritten signature as a means of identification and authentication [1].",
"Several approaches have been developed in the field of machine learning and statistical methods to perform the signature detection and verification tasks automatically.",
"Among these approaches, we can mention techniques based on Artificial Neural Networks (ANN) [2], Hidden Markov Models (HMM) [3], Support Vector Machines (SVM) [4], and Fuzzy Logic [5].",
"Among the neural network techniques, it is important to mention Faster Region-based Convolutional Neural Networks (RCNN) [6] and YOLOv2 [7].",
"Both models were adapted for logos and signature localization in noisy documents [8].",
"Many of the techniques that address signature verification use public databases [9][10][11][12].",
"However, such bases as GPDS [13], and MCYT [14] present images with a light background and dark signatures.",
"This characteristic does not present an environment of great complexity for segmenting the signature pixels.",
"Besides, due to the insertion of mobile devices and their growing popularity, several commercial and banking applications, for example, use images captured by smartphones for transactions, payments, account opening, and copies of documents [15][16].",
"On the other hand, document images captured by smartphone cameras are usually presented with distortions and background noise.",
"Therefore, treating these images in such a way that only handwritten signatures can be extracted for analysis of their characteristics becomes a challenging task in image processing.",
"These images do not always present the desired features or the expected quality, negatively influencing the process of recognition and classification of these handwritten signatures.",
"The situation may become more critical if the image of the source document has unwanted characteristics, such as imperfections, backgrounds, printed text, shape, and variations in size.",
"Another condition that can affect the quality of the attributes of handwritten signatures occurs when the image presents some distortion, such as perspective, inclination, scale, or unexpected resolution, all of them when scanning photos.",
"All these interference can also harm the verification systems of handwritten signatures with the increase of false positives or false negatives in the classification process.",
"In this work, we propose an approach to the pixel-level segmentation of handwritten signatures on images.",
"Our model has been trained with ID document images with the same characteristics and interference that can arise in a real-world scenario.",
"With this, our model will be able to get around the problems presented during the capture of signature images in different identification documents in noisy environments.",
"Our proposal will also enable the acquisition of signatures with greater fidelity in the strokes regardless of the types of pen, ink, background, preserving the graphic characteristics.",
"Another contribution is that the preservation of the characteristics of the signature features will also make it possible to carry out graphotechnical analyzes.",
"These features are used by forensic experts and may be applied in future systems for verifying handwritten signatures with a bias in forensic science.",
"We use a Fully Convolutional Network (FCN)[17] for signature segmentation on identity document images with refinement layers for the alpha channel of the image.",
"The remainder of this paper is organized as follows: The Section presents the Related Works; in Section we describe the Proposed system; Section presents the Material and Methods; in Section we present the results and analysis of our proposal; and finally, the Section depict the conclusions obtained from this work."
],
[
"Related Works",
"Handwritten signatures can be analyzed by online verification systems when an analysis is carried out during its production.",
"In the most recent work by [18], an online handwritten signature verification system based on a critical segment is proposed.",
"The system identifies and exploits the segments that remain unchanged in the signatures to capture the intrinsic behavior in the signature incorporated in the signatures of each signatory.",
"Another way to perform the signature analysis is offline when the signature has already been produced by the signatory.",
"In [19], a model based on Convolutional Neural Networks inspired by the architecture of Inception V1 is presented to learn about the characteristics existing in genuine signatures and forged signatures.",
"The study uses offline subscriptions to public databases such as CEDAR.",
"Our work is focused on offline subscriptions.",
"Studies have been carried out to investigate the performance of the Deep Learning algorithms from literature facing the task of signature and logo detection.",
"The deep learning-based object detectors, namely, Faster R-CNN, ZF, VGG16, $VGG_M$ and YOLOv2 where examined for this task.",
"The proposed approach detects Signatures and Logos simultaneously [8].",
"Mainly, in that study, the authors worked to detect signatures rather than segmentation of signature traits.",
"Thus, bounding boxes were generated around the detected signatures and logos.",
"The dataset used was the Tobacco-800[20], which has a clear background and is composed of scanned documents comprising printed text, signatures, and logos.",
"Other scientific papers have presented methods for the stroke-based extraction of signatures from document images.",
"The proposed approach in [1] is based on an FCN trained to learn, map, and extract the handwritten signatures from documents.",
"Although the proposal achieves good results, the network architecture requires a fixed size (512 x 512) of the input images [1].",
"In [21], a similar approach is used for signature extraction in identification documents.",
"For this, the authors used an optimized U-net network with less trainable parameters and input nodes than [1].",
"To increase the generalization of the model, the authors applied the data augmentation technique in the database, generating greater image diversity during training.",
"The model proposed in [21] achieved higher rates than [1], despite having fewer parameters.",
"To compose the structure of the first stage of our FCN+RL model, we selected an approach similar to the one proposed by [21].",
"The proposal of [21] presents a segmentation model at the stroke level, with promising results for the objectives of the first stage of our system proposed in this work.",
"In fact, stroke pixel integrity is of great importance to the offline signature verification process.",
"Maintaining this integrity to the maximum can increase confidence in Handwritten Signature Verification (HSV) systems, especially if more technical approaches, such as graphoscopy in forensic science, are used.",
"In this sense, proposed methods have been presented for feature extraction using Deep Convolutional Neural Network combined with the SVM classifier to writer-independent (WI) handwritten signature verification systems.",
"The proposed approach described in [22] outperformed other WI-HSV methods from the literature and outperformed writer-dependent methods from literature in some Brazilian dataset.",
"Nevertheless, both works, which reach the state-of-the-art in the HSV task, assume the image signature pixels are available in a clean area.",
"In this paper, we are proposing a robust handwritten signature segmentation method that can be used to detect and extract only the signature pixels in some document.",
"As much more the signature pixels can be extracted without noisy and distortion, better will be the results achieved by any HSV system in the signature verification task."
],
[
"The FCN+RL proposed architecture",
"The handwritten signature segmentation task is performed by an approach using an FCN encoder-decoder network architecture along with the refinement layers (RL) for the alpha channel of the signature image.",
"The FCN is based on the signature segmentation neural network architecture proposed by [21], that uses the FCN U-net architecture [23].",
"This FCN U-net model is then improved by the addition of the RL model, motivated by the results obtained by Xu et al.",
"[24].",
"Fig.",
"REF shows the architecture of our proposed model.",
"Figure: The FCN+RL proposed architecture.",
"In this example, FCN performs segmentation by pixel classification, providing an image with the estimated signature pixels in the foreground.",
"After the first stage, the concatenation between the input image and the image predicted by the FCN encoder-decoder layers is sent for input from the layer block for RL at stage two."
],
[
"The FCN Encoder-Decoder stage",
"The convolutional operations (in blue) are set to $3\\times 3$ of size with ReLU activation function.",
"The max-pooling layers (in red) are set to $2\\times 2$ with stride 2.",
"In the expansive path, there are upstream operations (in green) of size $2\\times 2$ concatenating with the corresponding characteristics of the path of contraction (gray arrows), followed by two convolutions of size $3\\times 3$ followed by ReLU operation.",
"The training is carried out by two stages.",
"First, we train the FCN encoder-decoder layers to learn the signature pixels.",
"At the FCN stage, the $512\\times 512$ image is sent to the network input that follows the contraction layers.",
"Feature maps generated in the contraction layers are cropped and copy applied to concatenate the expansion layers.",
"The expected output result is a binary image containing only the signature pixels predicted by the model.",
"To calculate the error and adjust the weights, a ground-truth image is used.",
"In this way, the model returns the error between the prediction and the ground truth images to fit the weights later.",
"In [24], the authors use the original image and a corresponding trimap of the original images that are concatenated during the train process.",
"These images then proceed to the first convolution layer.",
"The disadvantage of using the trimap in our case is that the trimap image is also needed in the inference process.",
"Therefore, in our model, we ignore the trimap and send only the original image to the input layer.",
"Concatenation is performed between the contraction and expansion layers when training the FCN.",
"Thus, in the inference process, we use only the original image, to keep as much signature information as possible, such as the forensic experts would use in a real-world scenario."
],
[
"The Refinement Layer stage",
"The RL model architecture consists of 4 convolutional layers.",
"A non-linear ReLu layer follows each of the three first layers.",
"Each convolution layer has a $64\\times 3\\times 3$ setting.",
"Convolutional layers for refinement do not have max-pooling layers or upstream layers, so we add a Batch-normalization layer after each of the first three convolutional layers.",
"Before training the refinement layers, the weights of the FCN layers need to be frozen.",
"The RL input image is generated using the concatenation between the original image and the output image predicted in the previous stage.",
"This concatenation extends the alpha channel information and assists in the refinement process for the subscription region.",
"The weight adjustment of RL block is performed using the same ground-truth image and procedure used for training the FCN block.",
"The expected result of the whole process is a binary image with the pixels of handwritten signatures as one and the any other irrelevant pixel as zero.",
"Therefore, this binary image serves as a mask to select the pixels of the handwritten signatures in RGB or gray level from the original input image."
],
[
"Datasets",
"Public datasets of handwritten signatures, such as MCYT and GPDS, do not meet the purpose of this work due to their composition because they do not show the poor image conditions that might impair the classification process.",
"Another challenge for the acquisition of a database is the confidentiality of information, as these are documents with personal information.",
"To overcome this drawbacks, the DSSigDataset-2 database [21] was used for the experiments in this work.",
"The DSSigDataset-2 is made up of $20,000$ document´s images with 200 background samples, and different distortion in the image.",
"The handwritten signatures blended in the document image were selected from the MCYT dataset [25] together with voluntarily generated signatures.",
"Another aspect of the DSSigDataset-2 is the type of pen used, in which different types and different colors were used to avoid possible bias in the learning of the network regarding the color and type of ink.",
"Fig.",
"REF presents an example of an image from the DSSigDataset-2 database.",
"A small area of the target handwritten signature is highlighted to give an idea of the challenge to detect the signature pixels in such conditions.",
"Figure: Example of an image of an identification document in the composition of the training database."
],
[
"Training procedure",
"We split the 20,000 images of DSSigDataset-2 for training, validation, and test using the cross-validation method.",
"We assigned $80\\%$ for training and $15\\%$ for validation, and $5\\%$ for test, which respectively resulted in $16,000$ training images, $3,000$ validation images, and $1,000$ test images taking into account different handwritten signatures in all partitions.",
"Also, since we had random image transformations applied in the document and the background, during the DSSigDataset-2 construction, we assure a complete unbiased dataset.",
"We used the Adam optimizer [26] to minimize the objective function, which was the Dice coefficient (DC) [27], shown in Equation REF .",
"$DC = 2 \\dfrac{|A \\cap B|}{|A|+|B|}$ where $A$ represents the ground-truth image and $B$ represents the segmented image at the network output.",
"Figure: Graph of the evolution of the similarity rate for the evaluation metric with the Dice coefficient in the validation and training sets.Figure: Graph of the evolution of the rate for objective function with the Dice coefficient in the validation and training sets.For training the FCN encoder-decoder (first stage), 10,000 epochs were used.",
"For training the refinement layers, 5,000 epochs were used.",
"Despite this number, our model has already obtained results with rates above 0.80 from epoch 1,000 for the data similarity rate between the output image and the ground-truth.",
"Fig.REF shows the similarity rate between the predicted and the expected data (pixels) for the Dice similarity coefficient (evaluation metric) during the training.",
"Parallel to the increase in the similarity rate, the model showed a loss rate consistent with the results.",
"Fig.REF shows the evolution of the rate for the loss function (objective function)."
],
[
"Results and Discussion",
"We performed several experiments to determine the best possible configuration and to validate the model's ability.",
"We tested different configurations by evaluating the effects on optimization of hyper-parameters.",
"However, we report in this paper the best validated configurations after all the preliminary experiments.",
"To compare the results obtained from the experiments of this work, we implemented the model described in [21] as a baseline system.",
"The reference model was also subjected to training with the DSSigDataset-2 database under the same conditions for the number of epochs and division of training and validation data.",
"Tests applied to both models took place under the same conditions on the test set.",
"Three metrics were used on the predictions of the models covered in this work: Structural Similarity (SSIM) index [28], Scale Invariant Feature Transform (SIFT) [29], and Dice Similarity Coefficient (DSC).",
"Quantitative results for similarity metrics are presented in Table REF .",
"Table: results for similarity metricsThe FCN+RL model presents superior results for the three evaluation metrics.",
"The results show different rates for different parameters.",
"This observation is relevant to the characteristics considered by each metric used.",
"The DSC technique presents lower values because it evaluates pixel-by-pixel of the entire region of the image and not just the morphology of the segmented area.",
"Moreover, our proposed model is much more robust against scale and other distortions in the signature image.",
"The RL block is responsible to filter out irrelevant pixels and filter in some pixels which cannot be detected by the FCN autoencoder block.",
"After performing the tests and applying the similarity metrics, three sets of data were selected with thirty samples for each set.",
"These data were collected from the results of the SSIM, SIFT, and DSC tests and subjected to statistical analyses.",
"First, we applied the Shapiro Wilk test to verify the normality of the data.",
"This test indicates that the data do not follow a normal distribution for the datasets, so we applied the Wilcoxon-Mann-Whitney test.",
"Tables REF and REF shows the results of the normality test and Wilcoxon test, respectively.",
"For the null hypothesis, we consider that the means are the same for both models, and as an alternative hypothesis, we consider the difference in means between the two models.",
"We considered a significance level of 0.05.",
"Table: Normality test resultsTable: Results of the Wilcoxon-Mann-Whitney test.The results of the Wilcoxon test show that the p-value values are lower than the level of significance.",
"Therefore, we reject the null hypothesis.",
"This result grants evidence that the models perform differently for the tests performed in this work.",
"Observing the results of similarity rates between the two models, the statistical tests show evidence that our proposed model is statistically superior to the reference model.",
"Finally, we performed a qualitative assessment of the images predicted by the models.",
"In this evaluation, it was possible to infer about the noise levels that negatively impact the subscription segmentation.",
"Qualitative results show an improvement in the segmentation of handwritten signatures by our FCN+RL model.",
"Fig.",
"REF shows the results for the two models, the reference model [21] and our model FCN+RL.",
"Fig.",
"REF shows the enlarged results of the subscriptions.",
"It is easy to observe the results produced by our FCN+RL model show greater fidelity to the original image, in addition to having less background noise and keeping the fine details of the signatures in both examples."
],
[
"Conclusions",
"Extracting handwritten signatures from images with a complex background and noise interference, such as identification documents, is a complex but promising task for the application of signature verification systems.",
"Achieving the maximum fidelity of a signature's characteristics can have a positive impact on the classification results, including guaranteeing the graphotechnical characteristics used by forensic specialists.",
"In this article, we proposed an approach to an FCN encoder-decoder network with refinement layers using a concatenation of the alpha channel of the region of interest on the original image for the segmentation of handwritten signatures, the FCN+RL.",
"The technique used with the alpha channel opacity over the image in the second stage of the convolution layers, refinement stage, provided an increase in the similarity rate between the predicted images and the ground truth.",
"This refinement reduces the scattering of the region of interest and presents the pixels of the signatures much less sparse and greater preservation of the graphic characteristics of the signatures.",
"With this result, signature verification systems can be used on handwritten signatures extracted from different images of ID documents captured by various computing devices such as smartphones.",
"Catch signatures will be much cleaner and preserved in the segmentation process.",
"In this way, more technical analyzes, such as those used by forensic science, may make use of the graphotechnical characteristics held in the signatures, even submitted in digital systems.",
"The evaluation metrics used were SSIM, SIFT, and the Data Similarity Coefficient.",
"To compare the results obtained with the proposed approach, we replicate the subscription segmentation model proposed by [21] to use as a reference.",
"The best results of our proposed model obtained an improvement rate of more than 40 percentage points about the reference model.",
"The qualitative results show a greater fidelity in the characteristics of the signature and a low level of background noise.",
"In the next works, we intend to present the signature verification models that are being developed by us and will be used for the calculation of graphotechnical analyzes.",
"As possible future works, it is also possible to point out the implementation of a semantic segmentation solution based on the proposed architecture for the application domain relevant to the structure and characteristics of documents.",
"An example is the detection of anomalies in documents forged or created by gross forgeries—interest area of the forensic science, the documentoscopy.",
"In addition to the implementation, as mentioned earlier, instantiating the model to other application domains would allow a way to assess the generalization capacity of the proposed model, as well as to detect possible adjustable points in the architecture."
],
[
"Acknowledgment",
"This study was financed in part by: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, FACEPE, and CNPq - Brazilian research agencies.",
"Figure: Qualitative result of the model used in and our FCN+RL proposed model.Figure: Zoom in the images shown in Fig.",
"."
]
] | 2005.14229 |
[
[
"Human migration and the motion of substance in a channel of a network"
],
[
"Abstract We study the motion of a substance in a channel of a network that consists of chain of nodes of a network (the nodes can be considered as boxes) and edges that connect the nodes and form the way for motion of the substance.",
"The nodes of the channel can have different \"leakage\", i.e., some amount of the substance can leave the channel at a node and the rate of leaving may be different for the different nodes of the channel.",
"In addition the nodes close to the end of the channel for some (construction or other) reason may be more \"attractive\" for the substance in comparison to the nodes around the entry node of the channel.",
"We discuss channels containing infinite or finite number of nodes and obtain the distribution of the substance along the nodes.",
"Two regimes of functioning of the channels are studied: stationary regime and non-stationary regime.",
"The distribution of the substance along the nodes of the channel for the case of stationary regime is a generalization of the Waring distribution (for channel with infinite number of nodes) or generalization of the truncated Waring distribution (for channel with finite number of nodes).",
"In the non-stationary regime of functioning of the channel one observes an exponential increase or exponential decrease of the amount of substance in the nodes.",
"Despite this the asymptotic distribution of the substance among the nodes of the channel in this regime is stationary.",
"The developed theory is applied for a study of the distrribution of migrants in countries that form migration channels."
],
[
"Introduction",
"In the last decades the researchers realized the importance of dynamics of complex systems and this leaded to intensive studies of such systems, especially in the area of social dynamics and population dynamics [1] - [19].",
"In the course of these studies the networks have appeared as important part of the structure of many complex systems [20] - [22].",
"Research on network flows has some of its roots in the studies on transportation problems, e.g., in the developing of minimal cost transportation models.",
"This research topic was established in 1960's especially after the publishing the book of Ford and Fulkerson [23].",
"At the beginning of the research the problems of interest have been, e.g., how by minimal number of individuals to meet a fixed schedule of tasks; minimal cost flow problems; or possible maximal flows in a network.",
"In course of the years the area of problems connected to network flows has increased very much.",
"Today one uses the methodology from the theory of network flows [24], [25] to solve problems connected to: (i) shortest path finding, (ii) just in time scheduling, (iii) facility layout and location, (iv) project management (determining minimum project duration), (v) optimal electronic route guidance in urban traffic networks [26], (vi) self-organizing network flows, (vii) modeling and optimization of scalar flows in networks [27], (viii) memory effects [28], (ix) isoform identification of RNA [29], etc.",
"(just some other examples are [30] - [36]).",
"Below we shall consider a specific network flow problem: motion of a substance through a network channel in presence of possibility for \"leakage\" in the nodes of the channel (loss of substance or usage of a part of substance in some process).",
"In addition we shall assume the existence of possibility that the substance may have preference for some of the nodes of the channel (e.g., the channel may be structured in such a way that the substance tends to concentrate in some of the nodes).",
"This feature will allow us to use the model for study of motions of animals or humans.",
"We note that the discussed model contains also the particular case when there is no preference of the substance with respect to the nodes of the channel.",
"The obvious application of the model is for the flow of some non-living substance through a chanel with usage of part of substance for some industrial process in the nodes of the channel.",
"We shall show that the model has more applications by another illustration: modeling of large human migration flows.",
"The large flows allow continuous modeling as in this case the discrete quantities can be approximated by a continuous ones.",
"This choice of an illustration of the model has been made because of the actuality of the problem of human migration [37].",
"Indeed the study of international migration becomes very actual after the large migration flows directed to Europe in 2015.",
"Much efforts are invested also in the study of internal migration in order to understand this migration and to make projection of the migration flows that may be very important for taking decisions about economic development of regions of a country [38]- [46].",
"Human migration models are of interest also for applied mathematics as they can be classified as probability models (exponential model, Poisson model, multinomial model, Markov chain models of migration [47]- [58]) or deterministic models (e.g., gravity model of migration [59]).",
"Human migration is closely connected to migration networks [60], [61], to ideological struggles [62],[63] and to waves and statistical distributions in population systems [64] -[67].",
"The paper is organized as follows.",
"In Sect.2 we discuss a model for motion of substance in a channel containing an infinite number of nodes.",
"Two regimes of functioning of the channel: stationary regime and non-stationary regime are studied.",
"Statistical distributions of the amount of substance in the nodes of the channel are obtained.",
"A particular case of the distribution for the stationary regime of functioning of the channel is the Waring distribution.",
"Sect.",
"3 is devoted to the case of channel containing finite number of nodes.",
"This case is of interest for the problem of human migration channels.",
"The distribution of substance for stationary regime of functioning of such a channel is given by a distribution that is a generalization of the truncated Waring distribution and a more complicated distribution is obtained for the asymptotic state of the channel functioning in a non-stationary regime.",
"In Sect.",
"4 we discuss application of the obtained mathematical results to a human migration channel of finite length and discuss effects such as the possibility of concentration of migrants in the last node of the channel (the final destination country).",
"Several concluding remarks are summarized in Sect.",
"5."
],
[
"Channel containing infinite number of nodes",
"We consider a channel consisting of a chain of nodes of a network.The nodes are connected by edges and each node is connected only to the two neighboring nodes of the channel exclusive for the first and the last node of the channel that are connected only to the neighboring node.",
"We study a model of the motion of substance through such a channel which is an extension of the model discussed in [68] and [69].",
"We consider each node as a cell (box), i.e., we consider an array of infinite number of cells indexed in succession by non-negative integers.",
"The first cell has index 0 and the last cell has index $N$ (in the case discussed here $N=\\infty $ ).",
"We assume that an amount $x$ of some substance is distributed among the cells and this substance can move from one cell to another cell.",
"Let $x_i$ be the amount of the substance in the $i$ -th cell.",
"Then $x = \\sum \\limits _{i=0}^\\infty x_ i$ The fractions $y_i = x_i/x$ can be considered as probability values of distribution of a discrete random variable $\\zeta $ $y_i = p(\\zeta = i), \\ i=0,1, \\dots $ The content $x_i$ of any cell may change due to the following 3 processes: Some amount $s$ of the substance $x$ enters the system of cells from the external environment through the 0-th cell; Rate $f_i$ from $x_i$ is transferred from the $i$ -th cell into the $i+1$ -th cell; Rate $g_i$ from $x_i$ leaks out the $i$ -th cell into the external environment.",
"We assume that the process of the motion of the substance is continuous in the time.",
"Then the process can be modeled mathematically by the system of ordinary differential equations: $ \\frac{dx_0}{dt} &=& s-f_0-g_0; \\nonumber \\\\\\frac{dx_i}{dt} &=& f_{i-1} -f_i - g_i, \\ i=1,2,\\dots .$ There are two regimes of functioning of the channel: stationary regime and non-stationary regime."
],
[
"Stationary regime of functioning of the channel",
"In the stationary regime of the functioning of the channel $\\frac{dx_i}{dt}=0$ , $i=0,1,\\dots $ .",
"Let us mark the quantities for the stationary case with $^*$ .",
"Then from Eqs.",
"(REF ) one obtains $f_0^*=s^*-g_0^*; \\ \\ f_i^*=f_{i-1}^*-g_i.$ This result can be written also as $f_i^* = s^*- \\sum \\limits _{j=0}^i g_j^*$ Hence for the stationary case the situation in the channel is determined by the quantities $s^*$ and $g_j^*$ , $j=0,1,\\dots $ .",
"In this paper we shall assume the following forms of the amount of the moving substances in Eqs.",
"(REF ) ($\\alpha , \\beta , \\gamma _i, \\sigma $ are constants) $s &=& \\sigma x_0 = \\sigma _0; \\ \\ \\sigma _0 > 0 \\nonumber \\\\f_i &=& (\\alpha _i + \\beta _i i) x_i; \\ \\ \\ \\alpha _i >0, \\ \\beta _i \\ge 0 \\rightarrow \\textrm {cumulative advantage of higher nodes} \\nonumber \\\\g_i &=& \\gamma _i x_i; \\ \\ \\ \\gamma _i \\ge 0 \\rightarrow \\textrm {non-uniform leakagein the nodes}$ The rules (REF ) differ from the rules in [68] in 3 points: $s$ is proportional to the substance in the 0th node (the amount of this substance is $x_0$ ).",
"In [68] $s$ is proportional to the amount $x$ of the substance in the entire channel $x = \\sum \\limits _{i=0}^N x_i$ ; Leakage rate $\\gamma _i$ is different for the different nodes.",
"In [68] the leakage rate is constant and equal to $\\gamma $ for all nodes of the channel (i.e., there is uniform leakage in the nodes).",
"Parameters $\\alpha _i$ and $\\beta _i$ are different for the different cells.",
"In [68] these parameters are the same for all cells of the channel.",
"Substitution of Eqs.",
"(REF ) in Eqs.",
"(REF ) leads to the relationships $\\frac{dx_0}{dt} &=& \\sigma _0 x_0 - \\alpha _0 x_0 - \\gamma _0 x_0; \\nonumber \\\\\\frac{dx_i}{dt} &=& [\\alpha _{i-1} + (i-1) \\beta _{i-1}]x_{i-1} - (\\alpha _i + i \\beta _i + \\gamma _i)x_i; \\ \\ \\ i=1,2,\\dots $ As we shall consider the stationary regime of functioning of the channel then from the first of the Eqs.",
"(REF ) it follows that $\\sigma _0 = \\alpha _0 + \\gamma _0$ .",
"This means that $x_0$ (the amount of the substance in the 0-th cell of the channel) is free parameter.",
"In this case the solution of Eqs.",
"(REF ) is $x_i = x_i^* + \\sum \\limits _{j=0}^i b_{ij} \\exp [-(\\alpha _j + j \\beta _j + \\gamma _j)t]$ where $x_i^*$ is the stationary part of the solution.",
"For $x_i^*$ one obtains the relationship $x_i^* = \\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{\\alpha _i + i \\beta _i + \\gamma _i} x_{i-1}^*, \\ i=1,2,\\dots $ The corresponding relationships for the coefficients $b_{ij}$ are ($i=1,\\dots $ ): $b_{ij} = \\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{(\\alpha _i - \\alpha _j) + (i \\beta _i -j \\beta _j) + (\\gamma _i - \\gamma _j)} b_{i-1,j},\\ j=0,1,\\dots ,i-1$ From Eq.",
"(REF ) one obtains $x_i^* = \\frac{\\prod \\limits _{j=0}^{i-1}[\\alpha _{i-j-1}+(i-j-1)\\beta _{i-j-1}]}{\\prod \\limits _{j=0}^{i-1} \\alpha _{i-j} + (i-j) \\beta _{i-j} + \\gamma _{i-j}} x_0^*$ The form of the corresponding stationary distribution $y_i^* = x_i^*/x^*$ (where $x^*$ is the amount of the substance in all of the cells of the channel) is $y_i^* = \\frac{\\prod \\limits _{j=0}^{i-1}[\\alpha _{i-j-1}+(i-j-1)\\beta _{i-j-1}]}{\\prod \\limits _{j=0}^{i-1} \\alpha _{i-j} + (i-j) \\beta _{i-j} + \\gamma _{i-j}} y_0^*$ (Note that the requirement $\\sum \\limits _{i=1}^{\\infty } y_i^* =1$ has to be satisfied).",
"To the best of our knowledge the distribution presented by Eq.",
"(REF ) was not discussed by other authors.",
"Let us show that this distributions contains as particular cases several famous distributions such as Waring distribution, Zipf distribution, and Yule-Simon distribution.",
"In order to do this we consider the particular case when $\\beta _i \\ne 0$ and write $x_i$ from Eq.",
"(REF ) as follows $x_i^* = \\frac{\\prod \\limits _{j=0}^{i-1} b_{i-j} [k_{i-j-1} + (i-j-1)]}{\\prod \\limits _{j=0}^{i-1} [k_{i-j} + a_{i-j} + (i-j)]} x_0^*$ where $k_i = \\alpha _i/\\beta _i$ ; $a_i = \\gamma _i/\\beta _i$ ; $b_i = \\beta _{i-1}/\\beta _i$ .",
"The form of the corresponding stationary distribution $y_i^* = x_i^*/x^*$ is $y_i^* = \\frac{\\prod \\limits _{j=0}^{i-1} b_{i-j} [k_{i-j-1} + (i-j-1)]}{\\prod \\limits _{j=0}^{i-1} [k_{i-j} + a_{i-j} + (i-j)]} y_0^*$ Let us now consider the particular case where $\\alpha _i = \\alpha $ and $\\beta _i = \\beta $ for $i=0,1,2,\\dots $ .",
"Then from Eqs.",
"(REF ) and (REF ) one obtains $x_i^* = \\frac{[k+(i-1)]!}{(k-1)!",
"\\prod \\limits _{j=1}^i (k+j+a_j)} x_0^*$ where $k = \\alpha /\\beta $ and $a_j=\\gamma _j/\\beta $ .",
"The form of the corresponding stationary distribution $y_i^* = x_i^*/x^*$ is $y_i^* = \\frac{[k+(i-1)]!}{(k-1)!",
"\\prod \\limits _{j=1}^i (k+j+a_j)} y_0^*$ Let us consider the particular case where $a_0 = \\dots = a_N$ .",
"In this case the distribution from Eq.",
"(REF ) is reduced to the distribution: $P(\\zeta = i) &=& P(\\zeta =0) \\frac{(k-1)^{[i]}}{(a+k)^{[i]}}; \\ \\ k^{[i]} = \\frac{(k+i)!}{k!",
"}; \\ i=1, 2, \\dots $ $P(\\zeta =0)=y_0^* = x_0^*/x^*$ is the percentage of substance that is located in the first cell of the channel.",
"Let this percentage be $y_0^* = \\frac{a}{a+k}$ The case described by Eq.",
"(REF ) corresponds to the situation where the amount of substance in the first cell is proportional of the amount of substance in the entire channel (self-reproduction property of the substance).",
"In this case Eq.",
"(REF ) is reduced to the distribution: $P(\\zeta = i) &=& \\frac{a}{a+k} \\frac{(k-1)^{[i]}}{(a+k)^{[i]}}; \\ \\ k^{[i]} = \\frac{(k+i)!}{k!",
"}; \\ i=1, 2, \\dots $ Let us denote $\\rho = a$ and $k=l$ .",
"The distribution (REF ) is exactly the Waring distribution (probability distribution of non-negative integers named after Edward Waring - the 6th Lucasian professor of Mathematics in Cambridge from the 18th century) [70] - [72] $p_l = \\rho \\frac{\\alpha _{(l)}}{(\\rho + \\alpha )_{(l+1)}}; \\ \\alpha _{(l)} = \\alpha (\\alpha +1) \\dots (\\alpha +l-1)$ Waring distribution may be written also as follows $p_0 &=& \\rho \\frac{\\alpha _{(0)}}{(\\rho + \\alpha )_{(1)}} = \\frac{\\rho }{\\alpha + \\rho }\\nonumber \\\\p_l &=& \\frac{\\alpha +(l-1)}{\\alpha + \\rho + l}p_{l-1}.$ The mean $\\mu $ (the expected value) of the Waring distribution is $\\mu = \\frac{\\alpha }{\\rho -1} \\ \\textrm {if} \\ \\rho >1$ The variance of the Waring distribution is $V = \\frac{\\alpha \\rho (\\alpha + \\rho -1)}{(\\rho -1)^2(\\rho - 2)} \\ \\textrm {if} \\ \\rho >2$ $\\rho $ is called the tail parameter as it controls the tail of the Waring distribution.",
"Waring distribution contains various distributions as particular cases.",
"Let $i \\rightarrow \\infty $ Then the Waring distribution is reduced to $p_l \\approx \\frac{1}{l^{(1+\\rho )}}.$ which is the frequency form of the Zipf distribution [73].",
"If $\\alpha \\rightarrow 0$ the Waring distribution is reduced to the Yule-Simon distribution [74] $p(\\zeta = l \\mid \\zeta > 0) = \\rho B(\\rho +1,l)$ where $B$ is the beta-function."
],
[
"Non-stationary regime of functioning of the channel",
"In the nonstationary case $dx_0/dt \\ne 0$ .",
"In this case the solution of the first equation of the system of equations (REF ) is $x_0 = b_{00} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t]$ where $b_{00}$ is a constant of integration.",
"$x_i$ must be obtained by solution of the corresponding Eqs.",
"(REF ).",
"The form of $x_i$ is $x_i = \\sum \\limits _{j=0}^i b_{ij} \\exp [-(\\alpha _j + j \\beta _j + \\gamma _j - \\sigma _j )t]$ The solution of the system of equations (REF ) is (REF ) where $\\sigma _i =0$ , $i=1,\\dots ,$ ($\\sigma _0 = \\sigma $ ): $b_{ij} &=& \\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{(\\alpha _i - \\alpha _j) +(i \\beta _i - j \\beta _j) + (\\gamma _i - \\gamma _j)} b_{i-1,j}; \\ i=1,\\dots ;j=1,\\dots , i-1$ and $b_{ii}$ are determined from the initial conditions in the cells of the channel.",
"The asymptotic solution ($t \\rightarrow \\infty $ ) is $x_i^a = b_{i0} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t]$ This means that the asymptotic distribution $y_i^a = x_i^a/x^a$ is stationary $y_i^a = \\frac{b_{i0}}{\\sum \\limits _{j=0}^\\infty b_{j0}}$ regardless of the fact that the amount of substance in the two cells may increase or decrease exponentially.",
"The explicit form of this distribution is $y_0^a &=& \\frac{1}{1+ \\sum \\limits _{i=1}^{\\infty } \\prod \\limits _{k=1}^i\\frac{\\alpha _{k-1} + (k-1) \\beta _{k-1}}{(\\alpha _k - \\alpha _0) + k \\beta _k +(\\gamma _k - \\gamma _0)}} \\nonumber \\\\y_i^a &=& \\frac{\\prod \\limits _{k=1}^i \\frac{\\alpha _{k-1} + (k-1) \\beta _{k-1}}{(\\alpha _k - \\alpha _0) + k \\beta _k + (\\gamma _k - \\gamma _0)}}{\\sum \\limits _{i=0}^{\\infty } \\prod \\limits _{k=1}^i\\frac{\\alpha _{k-1} + (k-1) \\beta _{k-1}}{(\\alpha _k - \\alpha _0) + k \\beta _k + (\\gamma _k - \\gamma _0)}},i=1,\\dots ,$"
],
[
"Channel containing finite number of nodes",
"Finite size channels are very interesting from the point of view of the applications of the theory, e.g., to migrant flows.",
"Let us consider a channel consisting of $N+1$ nodes (cells) and corresponding edges.",
"The nodes are indexed in succession by non-negative integers, i.e., the first cell has index 0 and the last cell has index $N$ .",
"In this case the total amount of substance in the channel is $x = \\sum \\limits _{i=0}^N x_ i$ The fractions $y_i = x_i/x$ can be considered as probability values of distribution of a discrete random variable $\\zeta $ $y_i = p(\\zeta = i), \\ i=0,1, \\dots , N$ The mathematical model of the finite channel is as follows: $ \\frac{dx_0}{dt} &=& s-f_0-g_0; \\nonumber \\\\\\frac{dx_i}{dt} &=& f_{i-1} -f_i - g_i, \\ i=1,2,\\dots , N-1 \\nonumber \\\\\\frac{dx_N}{dt} &=& f_{N-1} - g_N .$ The relationships for the amount of the moving substances are the same as in the case of infinite channel: $s &=& \\sigma x_0 = \\sigma _0 x_0; \\ \\ \\sigma = \\sigma _0 > 0 \\nonumber \\\\f_i &=& (\\alpha _i + \\beta _i i) x_i; \\ \\ \\ \\alpha _i >0, \\ \\beta _i \\ge 0 \\rightarrow \\textrm {cumulative advantage of higher nodes} \\nonumber \\\\g_i &=& \\gamma _i x_i; \\ \\ \\ \\gamma _i \\ge 0 \\rightarrow \\textrm {non-uniform leakagein the nodes}$ Substitution of Eqs.",
"(REF ) in Eqs.",
"(REF ) leads to the relationships $\\frac{dx_0}{dt} &=& \\sigma _0 x_0 - \\alpha _0 x_0 - \\gamma _0 x_0; \\nonumber \\\\\\frac{dx_i}{dt} &=& [\\alpha _{i-1} + (i-1) \\beta _{i-1}]x_{i-1} - (\\alpha _i + i \\beta _i + \\gamma _i)x_i,\\ i=1,2,\\dots ,N-1\\nonumber \\\\\\frac{dx_N}{dt} &=& [\\alpha _{N-1} + (N-1) \\beta _{N-1}]x_{N-1} - \\gamma _N x_N$"
],
[
"Stationary regime of functioning of the channel",
"In the stationary regime of functioning of the channel $\\frac{dx_0}{dt}=0$ , i.e.",
"$\\sigma _0 = \\alpha _0 + \\gamma _0$ .",
"In this case the system of equations (REF ) has a stationary solution with a free parameter $x_0$ .",
"This solution is $x_i^* &=& \\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{\\alpha _i + i \\beta _i + \\gamma _i} x_{i-1}^*, \\ i=1,2,\\dots , N-1 \\nonumber \\\\x_N^* &=& \\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{\\gamma _N} x_{N-1}^*.$ The solution of Eqs.",
"(REF ) is $x_i = x_i^* + \\sum \\limits _{j=0}^i b_{ij} \\exp [-(\\alpha _j + j \\beta _j + \\gamma _j)t]$ The substitution of Eq.",
"(REF ) in Eqs.",
"(REF ) leads to the following relationships for the coefficients $b_{ij}$ ($\\alpha _N = \\beta _N =0$ as there is no $N+1$ -st node(cell) where the substance can move from the $N$ -th mode (cell)) $b_{ij} &=& \\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{(\\alpha _i - \\alpha _j) + (i \\beta _i - j \\beta _j) + (\\gamma _i - \\gamma _j)} b_{i-1,j}; \\ i=1,\\dots ,N-1\\nonumber \\\\b_{Nj} &=& \\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{\\gamma _N - \\gamma _j - \\alpha _j - j \\beta _j} b_{N-1,j}, \\ j=0, \\dots , N-1$ $b_{ij}$ that are not determined by Eqs.",
"(REF ) may be determined by the initial conditions and in this process $b_{00}$ may be fixed too.",
"In the exponential function in Eq.",
"(REF ) there are no negative coefficients and because of this when $t \\rightarrow \\infty $ the system comes to the stationary solution from Eqs.",
"(REF ).",
"The form of this stationary solution is $x_i^* &=& \\frac{\\prod \\limits _{j=1}^i [\\alpha _{i-j} + (i-j) \\beta _{i-j}] }{\\prod \\limits _{j=1}^i (\\alpha _j + j \\beta _j + \\gamma _j)} x_0^*, \\ i=1,\\dots , N-1 \\nonumber \\\\x_N^* &=& \\frac{\\prod \\limits _{j=1}^N [\\alpha _{N-j} + (N-j) \\beta _{N-j}]}{ \\gamma _N \\prod \\limits _{j=1}^{N-1} (\\alpha _j + j \\beta _j + \\gamma _j)} x_0^*$ Let $x^*=\\sum \\limits _{i=0}^N x_i$ be the total amount of the substance for the case of stationary state of the channel.",
"Then we can consider the distribution $y_i^* = x_i^*/x^*$ .",
"Its form is $y_i^* &=& \\frac{\\prod \\limits _{j=1}^i [\\alpha _{i-j} + (i-j) \\beta _{i-j}] }{\\prod \\limits _{j=1}^i (\\alpha _j + j \\beta _j + \\gamma _j)} y_0^*, \\ i=1,\\dots , N-1 \\nonumber \\\\y_N^* &=& \\frac{\\prod \\limits _{j=1}^N [\\alpha _{N-j} + (N-j) \\beta _{N-j}]}{ \\gamma _N \\prod \\limits _{j=1}^{N-1} (\\alpha _j + j \\beta _j + \\gamma _j)} y_0^*$ To the best of our knowledge the distribution presented by Eq.",
"(REF ) was not discussed by other authors.",
"Let us consider the particular case where $\\alpha _i = \\alpha $ , $\\beta _i = \\beta $ and $\\gamma _i = \\gamma $ and $k = \\alpha /\\beta $ , $a = \\gamma /\\beta $ .",
"In this case the distribution from Eq.",
"(REF ) is reduced to the distribution: $P(\\zeta = i) &=& P(\\zeta =0) \\frac{(k-1)^{[i]}}{(a+k)^{[i]}}; \\ \\ k^{[i]} = \\frac{(k+i)!}{k!",
"}; \\ i=0,\\dots ,N-1 \\nonumber \\\\P(\\zeta = N) &=& \\frac{P(\\zeta =0)}{a} \\frac{(k-1)^{[N]}}{(a+k)^{[N-1]}},$ $P(\\zeta =0)=y_0^* = x_0^*/x^*$ is the percentage of substance that is located in the first node of the channel.",
"Let this percentage be $y_0^* = \\frac{a}{a+k}$ The case described by Eq.",
"(REF ) corresponds to the situation where the amount of substance in the first node is proportional of the amount of substance in the entire channel (self-reproduction property of the substance).",
"In this case Eq.",
"(REF ) is reduced to the truncated Waring distribution: $P(\\zeta = i) &=& \\frac{a}{a+k} \\frac{(k-1)^{[i]}}{(a+k)^{[i]}}; \\ \\ k^{[i]} = \\frac{(k+i)!}{k!",
"}; \\ i=0,\\dots ,N-1 \\nonumber \\\\P(\\zeta = N) &=& \\frac{1}{a+k} \\frac{(k-1)^{[N]}}{(a+k)^{[N-1]}},$ The truncated Waring distribution (REF ) is close to the Waring distribution that was discussed above in the text.",
"A characteristic feature of the truncated Waring distribution is the possibility for accumulation of substance in the last node of the channel (and this concentration can be quite significant) [69].",
"Let us note that that for the case of distribution (REF ) $y_N^* > y_{N-1}^*$ when $k+(N-1) > a_N$ , i.e., the concentration of the substrate in the last node of the channel depends on the situation in the last two nodes (from the parameters $a_{N-1}, \\beta _{n-1}, \\gamma _N$ ).",
"For the case of truncated Waring distribution (REF ) $y_N^* > y_{N-1}^*$ when $k+(N-1) > a$ where $a= \\sigma /\\beta $ ,i.e., the concentration of substance in the last node of the channel depends on the situation in the first node of the channel.",
"This may be important for the case of channels for human migrants."
],
[
"Non-stationary regime of functioning of the channel",
"In the nonstationary case $dx_0/dt \\ne 0$ .",
"In this case the solution of the first equation of the system of equations (REF ) is $x_0 = b_{00} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t]$ where $b_{00}$ is a constant of integration.",
"$x_i$ must be obtained by solution of the corresponding Eqs.",
"(REF ).",
"The form of $x_i$ is $x_i = \\sum \\limits _{j=0}^i b_{ij} \\exp [-(\\alpha _j + j \\beta _j + \\gamma _j - \\sigma _j )t]$ In order to understand the processes in the channel let us consider first the case of channel consisting of two nodes ($N=1$ ).",
"In this case we have to solve the additional equation $\\frac{dx_1}{dt} = \\alpha _0 x_0 - \\gamma _1 x_1$ The solution is $x_1 = \\frac{\\alpha _0}{\\gamma _1 - \\gamma _0 + \\sigma _0 - \\alpha _0} b_{00} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t] + b_{11} \\exp (-\\gamma _1 t)$ $b_{11}$ can be determined from the initial conditions at $t=0$ .",
"The asymptotic form of the obtained solution ($t \\rightarrow \\infty $ ) is $x_0^a &=& b_{00} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t] \\nonumber \\\\x_1^a &=& \\frac{\\alpha _0}{\\gamma _1 - \\gamma _0 + \\sigma _0 - \\alpha _0} b_{00} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t]$ as $\\gamma _1 >0$ .",
"Let us consider the asymptotic distribution $y_i^a = x_i^a/x^a$ where $x^a = \\sum \\limits _{i=0}^N x_i$ .",
"$x_0^a$ and $x_1^a$ depend on $t$ but nevertheless the asymptotic distribution is stationary $y_0^a = \\frac{1}{1+ \\frac{\\alpha _0}{\\gamma _1 - \\gamma _0 + \\sigma _0 - \\alpha _0}}; \\ \\ y_1^a = \\frac{1}{1+ \\frac{\\gamma _1 - \\gamma _0 + \\sigma _0 - \\alpha _0}{\\alpha _0}}$ Thus the distribution of the substance in the channel tend to a stationary asymptotic distribution regardless of the fact that the amount of substance in the two nodes may increase or decrease exponentially.",
"Let us now consider the case of channel containing more than 2 nodes ($N>1$ ).",
"In this case the solution of the system of equations (REF ) is (REF ) where $\\sigma _i =0$ , $i=1,\\dots ,N-1$ ; $b_{ij} &=& \\frac{\\alpha _{i-1} + (i-1)\\beta _{i-1}}{(\\alpha _i - \\alpha _j) +(i\\beta _i - j\\beta _j) +(\\gamma _i - \\gamma _j)} b_{i-1,j}; \\ i=1,\\dots ,N-1\\nonumber \\\\b_{Nj} &=& \\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{ \\gamma _N - \\gamma _j - \\alpha _j - j \\beta _j} b_{N-1,j}, \\ j=0, \\dots , N-1\\nonumber \\\\\\sigma _N &=& \\alpha _N + N \\beta _N,$ and $b_{ii}$ are determined from the initial conditions in the nodes of the channel.",
"The asymptotic solution ($t \\rightarrow \\infty $ ) is $x_i^a = b_{i0} \\exp [(\\sigma _0 - \\alpha _0 - \\gamma _0)t]$ This means that the asymptotic distribution $y_i^a = x_i^a/x^a$ is stationary $y_i^a = \\frac{b_{i0}}{\\sum \\limits _{j=0}^N b_{j0}}$ regardless of the fact that the amount of substance in the two cells many increase or decrease exponentially.",
"The explicit form of this distribution is $y_0^a &=& \\frac{1}{\\Omega } \\nonumber \\\\y_i^a &=& \\frac{\\prod \\limits _{k=1}^i\\frac{\\alpha _{i-k} + (i-k) \\beta _{i-k}}{(-\\alpha _0 + \\alpha _{i-k+1}) +(i-k+1)\\beta _{i-k+1} + (-\\gamma _0 + \\gamma _{i-k+1})}}{\\Omega },\\nonumber \\\\i&=&1,\\dots , N-1 \\nonumber \\\\y_N^a &= & \\frac{\\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{(\\gamma _N - \\gamma _0) - \\alpha _0} \\prod \\limits _{k=1}^{N-1}\\frac{\\alpha _{i-k} + (i-k) \\beta _{i-k}}{(-\\alpha _0 + \\alpha _{i-k+1}) +(i-k+1)\\beta _{i-k+1} + (-\\gamma _0 + \\gamma _{i-k+1})}}{\\Omega }\\nonumber \\\\$ where $\\Omega &=& 1+ \\sum \\limits _{i=1}^{N-1} \\prod \\limits _{k=1}^i\\frac{\\alpha _{i-k} + (i-k) \\beta _{i-k}}{(-\\alpha _0 + \\alpha _{i-k+1}) +(i-k+1)\\beta _{i-k+1} + (-\\gamma _0 + \\gamma _{i-k+1})} + \\nonumber \\\\&& \\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{(\\gamma _N - \\gamma _0) - \\alpha _0} \\prod \\limits _{k=1}^{N-1}\\frac{\\alpha _{i-k} + (i-k) \\beta _{i-k}}{(-\\alpha _0 + \\alpha _{i-k+1}) +(i-k+1)\\beta _{i-k+1} + (-\\gamma _0 + \\gamma _{i-k+1})} \\nonumber \\\\$"
],
[
"Application of obtained results to channels of migration networks",
"The model discussed above can be used for a study of motion of substance through cells of appropriate technological systems.",
"The model can be applied also for investigation of other systems.",
"Below we shall discuss it in connection with channels of human migration for the case when the migration flows are large and continuous approximation of these flows can be used.",
"Let us consider a chain of $N+1$ countries or cities.",
"This chain may be considered as a channel in a migration network.",
"The nodes of this network (corresponding to the countries of the channel for an example) may be considered as boxes (cells).",
"A flow of migrants moves through this migration channel from the country of entrance to the final destination country.",
"The entry country will be the node with label 0 and the final destination country will be the node with label $N$ .",
"Let us have a number $x$ of migrants that are distributed among the countries.",
"Let $x_i$ be the number of migrants in the $i$ -th country.",
"This number can change on the basis of the following three processes: (i) A number $s$ of migrants enter the channel from the external environment through the 0-th node (country of entrance); A rate $f_i$ from $x_i$ is transferred from the $i$ -th country to the $i+1$ -th country; (iii) A rate $g_i$ from $x_i$ change their status (e.g.",
"they are not anymore migrants and may become citizens of the corresponding country, may return home, etc.).",
"The values of $x_i$ can be determined by Eqs.",
"(REF ).",
"The relationships (REF ) mean that: (i) The number of migrants $s$ that enter the channel is proportional of the current number of migrants in the entry country of the channel; (ii) There may be preference for some countries, e.g.",
"migrants may prefer the countries that are around the end of the migration channel (and the final destination country may be the most preferred one); (iii) It is assumed that the conditions along the channel are different with respect to 'leakage' of migrants, e.g.",
"the different rates $\\gamma _i$ of migrants leave the flow of migrants in different countries of the channel.",
"In addition the the transition from country to country may have different grade of difficulty (different $\\alpha _i$ ) and the attractiveness of the countries along the channel may be different for migrants (different $\\beta _i$ ).",
"$\\sigma _0$ is the \"gate\" parameter as it regulates the number of migrants that enter the channel.",
"The parameters $\\gamma _i$ regulate the \"absorption\" of the channel as they reflect the change of the status of some migrants.",
"The large values of $\\gamma _i$ may compensate the value of $\\sigma $ and even may lead to decrease of the number of migrants in the channel.",
"The large values of $\\gamma _i$ may however lead to integration problems connected to migrants.",
"Small values of parameters $\\alpha _i$ mean that the way of the migrants through the channel is more difficult and because of this the migrants tend to concentrate in the entry country (and eventually in the second country of the channel).",
"The countries that are in the second half of the migration channel and especially the final destination country may try to decrease $\\alpha _i$ by agreements that commit the entry country to keep the migrants on its territory.",
"Any increase of $\\alpha _i$ may lead to increase of the proportion of migrants that reach the second half of the migration channel and especially the final destination country.",
"The parameters $\\beta _i$ regulate the attractiveness of the countries along the migration channel.",
"Large values of $\\beta _i$ mean that the remaining countries in the channel and especially the final destination country are very attractive for some reason.",
"This increases the attractiveness of the countries from the second half of the channel (migrants want more to reach these countries as in such a way the distance to the final destination country is smaller).",
"If for some reason $\\beta _i$ are kept at high values a flood of migrants may reach the final destination country which may lead to large logistic and other problems.",
"Let us now consider the case of channel consisting of finite number of nodes and the stationary case.",
"Then from Eq.",
"(REF ) we obtain the relationship $\\frac{y_N^*}{y_{N-1}^*} = \\frac{\\alpha _{N-1} + (N-1) \\beta _{N-1}}{\\gamma _N}$ If $\\frac{y_N^*}{y_{N-1}^*} >1$ there is an effect of concentration of migrants in the final destination country.",
"This happens when $\\alpha _{N-1} + (N-1) \\beta _{N-1} > \\gamma _N$ i.e., if the attractivity of the final destination country is large (large value of $\\beta _{N-1}$ ) and it is relatively easy to cross the border to the final destination country (large $\\alpha _{N-1}$ ) and the probability of change of the status of migrants in the final destination country of the channel are not large enough to compensate for this popularity (value of $\\gamma _N$ is relatively small).",
"In order to avoid the arising of the effect of the concentration of migrants in the final destination country one has to achieve $\\alpha _{N-1} + (N-1)\\beta _{N-1} < \\gamma _N$ This means that one should try to decrease the number of migrants entering the final destination country (to lower the value of $\\alpha _{N-1}$ ); to decrease the attractivity of the final destination country (to lower the value of $\\beta _{N-1}$ ) and to increase the probabilities for change of the status of the migrants in the entry country and/or in the final destination country (to increase the value of $\\gamma _N$ ).",
"A new effect with respect to the theory developed in our previous study [69] is the possibility of accumulation of migrants not only in the final destination country but also in any country of the channel.",
"This can happen (see Eqs.",
"(REF )) when (for some value of $i$ ) $\\frac{\\alpha _{i-1} + (i-1) \\beta _{i-1}}{\\alpha _i + i \\beta _i + \\gamma _i} >1$ Eq.",
"(REF ) shows that such a case of concentration of migrants may happen when the entry of migrants in the $i$ -th country is easy and this country is popular among migrants (large values of $\\alpha _{i-1}$ and $\\beta _{i-1}$ with respect to $\\alpha _i$ and $\\beta _i$ ) and in addition the value of $\\gamma _N$ is small.",
"The change of the total number of migrants in the channel can be obtained by taking the sum of the equations (REF ).",
"The result is $\\frac{dx}{dt}= \\sigma _0 x_0 - \\sum \\limits _{l=0}^N \\gamma _l x_l$ Thus the total number of migrants in the channel may increase fast when many migrants enter the channel (when the value of $\\sigma _0$ is large) and decrease when the probability for change of the status of the migrants increases along the channel (when the values of some of $\\gamma _l$ or the values of all $\\gamma _l$ increase).",
"In the stationary regime of functioning of a finite channel the total number of migrants in the countries of the channel is $x^* = x^*_0 \\left\\lbrace 1 + \\sum \\limits _{i=1}^{N-1}\\frac{\\prod \\limits _{j=1}^i [\\alpha _{i-j} + (i-j) \\beta _{j-1}] }{\\prod \\limits _{j=1}^i (\\alpha _j + j \\beta _j + \\gamma _j)} +\\frac{\\prod \\limits _{j=1}^N [\\alpha _{i-j} + (N-j) \\beta _{j-1}]}{ \\gamma _N \\prod \\limits _{j=1}^{N-1} (\\alpha _j + j \\beta _j + \\gamma _j)}\\right\\rbrace $ Thus the number of migrants in the countries of the channel may be decreased if one manages to decrease the number of migrants in the entry country of the channel (i.e.",
"if one manages to decrease the value of $x^*_0$ ).",
"Finally let us consider the case of channel consisting of finite number of countries and in stationary regime of functioning.",
"Let in addition the motion of migrants be determined by the attractivity of the final destination country at the expense of the easiness of moving through the borders between the countries of the channel (i.e., $\\beta _i >> \\alpha _i$ and parameter $\\alpha _i$ may be neglected in the relationships for all cells except in the numerator for the 0-th cell in Eq.",
"(REF )).",
"Let in addition there be no \"leakage\" (the migrants are very much fascinated by the final destination countries and the change of the statuses along the channel is small, i.e., one can neglect all $\\gamma _i$ except for $\\gamma _N$ which is assumed to be significant).",
"Thus from Eq.",
"(REF ) we obtain $x_1^* = \\frac{\\alpha _0}{\\beta _1} x_0^*; \\ x_i^* = \\frac{(i-1)}{ i} \\frac{\\beta _{i-1}}{\\beta _i} x^*_{i-1}, i=2,..., N-1; \\ x_N^* = \\frac{ (N-1)\\beta _{N - 1}}{\\gamma _N} x_{N-1}^*.$ From the second of equations (REF ) one easily obtains $x_i^* = \\frac{i-k}{i} \\frac{\\beta _{i-k}}{\\beta _i} x^*_{i-k}$ .",
"Then the approximate total number of the migrants in the channel is $x^* &=& x_0^* \\left\\lbrace 1+ \\alpha _0 \\sum \\limits _{l=1}^{N-1}\\frac{1}{l \\beta _l} + \\frac{\\alpha _0}{\\gamma _N} \\right\\rbrace $ and the distribution of the migrants among the countries of the channel is $y_0^* &=& \\frac{1}{1+ \\alpha _0 \\sum \\limits _{l=1}^{N-1}\\frac{1}{l \\beta _l} + \\frac{\\alpha _0}{\\gamma _N}} \\nonumber \\\\y_1 &=& \\frac{\\alpha _0}{\\beta _1 \\left[1+ \\alpha _0 \\sum \\limits _{l=1}^{N-1}\\frac{1}{l \\beta _l} + \\frac{\\alpha _0}{\\gamma _N} \\right]} \\nonumber \\\\y_i^* &=& \\frac{\\alpha _0}{i \\beta _i \\left[ 1+ \\alpha _0 \\sum \\limits _{l=1}^{N-1}\\frac{1}{l \\beta _l} + \\frac{\\alpha _0}{\\gamma _N}\\right]},\\ \\ i=2, \\dots , N-1 \\nonumber \\\\y_N^* &=& \\frac{\\alpha _0}{\\gamma _N \\left[ 1+ \\alpha _0 \\sum \\limits _{l=1}^{N-1}\\frac{1}{l \\beta _l} + \\frac{\\alpha _0}{\\gamma _N}\\right]}$ Let us now denote $\\alpha _0$ as $\\alpha $ and assume that $\\beta _1 = \\dots =\\beta _{N-1} = \\beta $ .",
"Then $x^* &=& x_0^* \\left\\lbrace 1+ \\frac{\\alpha }{\\beta } \\sum \\limits _{l=1}^{N-1}\\frac{1}{l} + \\frac{\\alpha }{\\gamma _N} \\right\\rbrace = x^*_0 \\left\\lbrace 1 + \\frac{\\alpha }{\\beta } H_{N-1} + \\frac{\\alpha }{\\gamma } \\right\\rbrace $ where $H_{N-1}$ is the $N-1$ -th harmonic number.",
"Let us use the approximate relationship for harmonic numbers $H_N = \\ln (N) + C_E + \\frac{1}{2N} - \\frac{1}{12 N^2} + \\frac{1}{120 N^4} - \\dots $ ($C_E$ is the constant of Euler).",
"Then for a migration chanel of finite and not very large length we obtain the relationship $x^* \\approx x^*_0 \\left\\lbrace 1 + \\frac{\\alpha }{\\beta } \\left[ \\ln (N-1) + C_E + \\frac{1}{2(N-1)} \\right] + \\frac{\\alpha }{\\gamma } \\right\\rbrace $ and the approximate distribution of the migrants in the channel $y_i^* = \\frac{x_i^*}{x^*}$ will be $y_0^* &=& \\frac{1}{\\left\\lbrace 1 + \\frac{\\alpha }{\\beta } \\left[ \\ln (N-1) + C_E + \\frac{1}{2(N-1)} \\right] + \\frac{\\alpha }{\\gamma } \\right\\rbrace } \\nonumber \\\\y_1^* &=& \\frac{\\alpha }{\\beta \\left\\lbrace 1 + \\frac{\\alpha }{\\beta } \\left[ \\ln (N-1) + C_E + \\frac{1}{2(N-1)} \\right] + \\frac{\\alpha }{\\gamma } \\right\\rbrace }\\nonumber \\\\y_i^* &=& \\frac{\\alpha }{\\beta i \\left\\lbrace 1 + \\frac{\\alpha }{\\beta } \\left[ \\ln (N-1) + C_E + \\frac{1}{2(N-1)} \\right] + \\frac{\\alpha }{\\gamma } \\right\\rbrace }, i=2,\\dots ,N-1 \\nonumber \\\\y_N^* &=& \\frac{\\alpha }{\\gamma _N \\left\\lbrace 1 + \\frac{\\alpha }{\\beta } \\left[ \\ln (N-1) + C_E + \\frac{1}{2(N-1)} \\right] + \\frac{\\alpha }{\\gamma } \\right\\rbrace }$ which is a version of truncated Zipf distribution.",
"As an application of the above theory we shall discuss a stationary regime of functioning of a migration channel, containing 3 countries.",
"Let the migrants come overseas (e.g., by boats) to the entry country of the channel.",
"The attractive country in the channel is the third country of the channel (the final destination country).",
"The second country of the channel is not attractive for the migrants but in order to reach the final destination country the migrants have to move through its territory.",
"Let the flow of migrants be large.",
"Then we can apply the part of the model described by Eqs.",
"(REF ) and (REF ).",
"The number of the migrants in the three countries of the channel are $x^*_0; \\ \\ x^*_1 = x^*_0 \\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1}; \\ \\ x_2^* = x_0^* \\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1}$ The total number of migrants in the countries of the channel will be $x^* = x_0^* \\left( 1+ \\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1} +\\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1} \\right)$ and the distribution of the migrants along the countries of the channel will be $y_0^* = \\frac{1}{\\left( 1+ \\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1} +\\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1} \\right)}\\nonumber \\\\y_1^* =\\frac{\\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1}}{\\left( 1+ \\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1} +\\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1} \\right)}\\nonumber \\\\y_2^* =\\frac{\\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1}}{\\left( 1+ \\frac{\\alpha _0}{\\alpha _1+\\beta _1+\\gamma _1} +\\frac{\\alpha _1+\\beta _1}{\\gamma _2} \\frac{\\alpha _0}{\\alpha _1+\\beta _1 + \\gamma _1} \\right)}$ Let us discuss two scenarios.",
"In the first scenario there are no measures to decrease the number of migrants in the entry country of the channel and their number remains, e.g., $x_0^* = 300,000$ .",
"let the stationary state of the channel be characterized by values of parameters: $\\alpha _0 = \\alpha _1 = \\beta _1 = \\gamma _1 = \\gamma _2 = 0.0001$ .",
"Then in countries of the channel there are $600,000$ migrants: $300,000$ in the entry country ($50\\%$ ), $100,000$ in the second country of the channel (about $16,7\\%$ ) and $200,000$ in the final destination country (about $33.3\\%$ ).",
"Now let the number of the migrants in the first country of the channel is a large burden for this country, and it decides to increase $\\alpha _0$ (e.g., to ease the border control and to increase the probability that the migrants may move successfully (and mostly illegally) to the second country of the channel.",
"Let the result of such a behaviour be that $\\alpha _0$ increase from $0.0001$ to $0.0002$ .",
"After some time the channel will have new stationary state of operation where $300,000$ ($x_0^*$ remains unchanged) but the number of migrants in the second and third country of the channel will increase to $200,000$ in the second country of the channel and $400,000$ in the final destination country.",
"Then the second country of the channel may decide that the migration burden is too high for it and this country take measures that may lead to increase of $\\alpha _1$ .",
"Let this increase be from $\\alpha _1 =0.0001$ to $\\alpha _1=0.0002$ .",
"We remember that $\\alpha _0=0.0002$ and the other parameters of the channel have values of $0.0001$ .",
"This will have the following effect of the stationary regime of the functioning of the channel: there will be $300,000$ migrants in the entry country of the channel (as no measures are taken to decrease $x_0^*$ ).",
"Then there will be $150,000$ migrants in the second country of the channel (increasing value of $\\alpha _1$ decreased the number of migrants in this country) and there will be $450,000$ migrants in the final destination country (part of the migrants move from the second country of the channel to the final destination country).",
"Let us now the final destination country decides that the migration burden it too large for it and takes measures to decrease its popularity.",
"Let these measures lead to a new value of $\\beta _1$ : $\\beta _1 =0$ .",
"Then the stationary regime of the functioning of the migration channel will be characterized by the following number of migrants in the three countries: $300,000$ migrants in the entry country, $200,000$ migrants in the second country of the channel, and $400,000$ migrants in the final destination country of the channel.",
"Thus the number of migrants in the final destination country will decrease at the expense of the migrants in the second country of the channel.",
"The sole actions of the countries from the scenario 1 above can continue but as we have seen this will not lead to significant decrease of migration burden.",
"Such significant decrease can be realized in the scenario 2: the countries concentrate their efforts in order to decrease $x_0$ keeping the values of the other parameters unchanged.",
"This may have large effect as follows.",
"Let the initial stationary state of the channel be characterized by $x_0^*=300,000$ , and $\\alpha _0 = \\alpha _1 = \\beta _1 = \\gamma _1 =\\gamma _2 = 0.0001$ .",
"Then the number of migrants in the second country of the channel will be $100,000$ and the number of migrants in the final destination country will be $200,000$ .",
"Let now the measures be taken and the number of migrants in the entry country of the channel decreases to $x_0^*=200,000$ .",
"Then the stationary state of the channel will be characterized by about $67,000$ migrants in the second country of the channel and about $133,000$ migrants in the final destination country.",
"If the measures for decreasing $x_0^*$ continue and the number of migrants reduces to $45,000$ then the corresponding numbers of migrants will be $15,000$ in the second country of the channel and $30,000$ in the final destination country.",
"Thus scenario 2 is much more effective from the point of view of decreasing migration burden on the countries of the channel."
],
[
"Concluding remarks",
"Above we have discussed a model for motion of a substance in channels of networks.",
"Two regimes of functioning of the channel are studied: stationary regime of motion of the substance and nonstationary regime of motion of the substance.",
"The main result of the study are the obtained distributions of substance along the cells for the case of stationary regime of motion of the substance.",
"The corresponding distribution for the case of channel containing infinite number of modes is a generalization of the Waring distribution.",
"The distribution obtained for the case of a channel containing finite number of nodes is a generalization of the truncated Waring distribution.",
"In addition to classical application of the model for calculation of motion and distribution of substance in channels of technological systems we have discussed the model also in the context of motion of large amounts of migrants through migration channels, e.g., connecting several countries.",
"Specific characteristic of the discussed model is the possibility for different \"leakage\" of the nodes, i.e., the different probabilities for a change of the status of a migrant in the different nodes (countries) of the channel.",
"For the case of non-stationary regime of functioning of the channel the number of migrants in the channel may increase or decrease exponentially but the asymptotic distribution of the migrants in the countries of the channel is stationary and depends strongly on the situation in the first node (entry country) of the channel.",
"The corresponding distributions of the migrants in the countries of the channel are more complicated in comparison to the distributions for the case of stationary regime of functioning of the channel.",
"In the stationary regime of functioning of a channel consisting of finite number of nodes an effect of concentration of migrants in the final destination country (last node of the channel) may be observed if the final destination country is popular enough.",
"The possibility of different \"leakage\" and different preferences may lead to another concentration effect: the concentration of migrants may happen not only in the last country of the channel but also in the countries that are between the entry country of the channel and final destination country.",
"Let us note finally that if the popularity of the countries close to the final destination country is large and the motion from node to node (from country to country) is not so easy and in addition the migrants are not interested in change of their status in the countries of the channel except for the final destination country, then the distribution of the migrants in the channel is a version of the Zipf distribution."
]
] | 1709.01833 |
[
[
"Optimal Sub-sampling with Influence Functions"
],
[
"Abstract Sub-sampling is a common and often effective method to deal with the computational challenges of large datasets.",
"However, for most statistical models, there is no well-motivated approach for drawing a non-uniform subsample.",
"We show that the concept of an asymptotically linear estimator and the associated influence function leads to optimal sampling procedures for a wide class of popular models.",
"Furthermore, for linear regression models which have well-studied procedures for non-uniform sub-sampling, we show our optimal influence function based method outperforms previous approaches.",
"We empirically show the improved performance of our method on real datasets."
],
[
"Introduction",
"As the amount of data in the world increases, the question arises as to how best to deal with the large datasets.",
"The associated tasks can be varied as they include training models on big data, rapid prototyping when datasets do not fit on a single machine, parameter tuning and model selection, and data exploration and visualization of “important” data points such as outliers.",
"Although a number of data sketching techniques such as random projections exist to reduce the size of the data and the associated computational costs, many can only be applied to some pre-specified task and may require custom code to use.",
"However, sampling methods are a common method for dealing with the problem of size as they provide an exceptionally flexible summarization of the data that can be applied to almost all tasks in a simple, straightforward manner.",
"The simplest sampling method is uniform random sampling.",
"However, it is inefficient as it does not exploit any notion of the importance of a data point for the relevant tasks.",
"We would ideally like to sample the data efficiently, preferentially sampling the data that will accurately approximate the estimates from the full data set while avoiding wasting resources on data that are nearly irrelevant.",
"Although several approaches introduce preferential sampling probabilities such as sampling based on leverage scores [7] or gradients [17], we show that the resulting sampling probabilities are still inefficient and can demonstrate pathological behavior.",
"Furthermore, most methods are derived only for a linear regression model.",
"To the authors' knowledge, there is limited work on sampling for more general machine learning models outside of logistic regression which was studied by [9].",
"In this paper, we examine the problem of finding optimal sub-sampling probabilities for nearly any estimation problem.",
"To this end, we propose using the influence function as a measure of sampling importance.",
"The influence function measures the change in the objective or values of interest due to a single point.",
"It is a particularly general approach as many model and estimators, such as maximum likelihood and M-estimators, can be cast in the framework and can work with non-differentiable objectives.",
"We prove that the regularized version of our sampling design is asymptotically optimal among all regularized designs of the same expected size.",
"This is a substantially stronger result than other results that minimize a loose probabilistic upper bound on the error.",
"Beyond the improved performance of our method, the influence-based approach allows one to fundamentally understand the problem of optimal sub-sampling.",
"Rather than proposing an ad hoc method and analyzing its theoretical properties, the influence function and the notion of asymptotically linear estimators reduces the problem of accurately approximating the estimate from the full data set to the problem of calculating the mean of influence functions.",
"Thus, the problem of finding an optimal sampling design or probabilities for estimating a model can be converted to the more straightforward problem of optimal sampling design for a mean.",
"This design can depend on the specific task on hand.",
"In particular, good sampling designs to estimate the parameters of a model can substantially differ from good sampling designs to optimize the resulting predictions even when the same model is used in both.",
"This fact is also borne out in our experimental results.",
"We explicitly derive sampling probabilities for linear regression, quantile regression, and generalized linear models.",
"In doing so, we are able to separate the “influence” of the residuals $y_i - \\hat{y}_i$ from the “influence” of the regression design or predictors $X$ .",
"As a result, we are able to show that existing approaches often only appropriately exploits one of the two, whereas our method appropriately incorporates both."
],
[
"Related work",
" A number of methods exist for sub-sampling when the relevant task is linear $L_p$ regression or matrix approximation.",
"The dominant approach in the literature for least squares regression is based on statistical leverage scores [13], [7].",
"A number of papers [3], [14], [5], [4] address more general $L_p$ linear regression problems and derive a corresponding leverage for $L_p$ regression.",
"These methods focus on generating sampling designs from the design matrix or predictors $X$ , and make no or limited use of the responses $Y$ .",
"The resulting sampling designs are obtained via a relatively expensive to compute and complex random projection or low distortion embedding.",
"For linear regression models that make use of the responses, the gradient-based approach of [17] and the Uluru algorithm in [6] provide methods that subsample the data based on residuals given a pilot estimate of the coefficients.",
"Although the results for the gradient-based approach only deal with linear least squares regression, the gradient can be computed for any differentiable loss function.",
"We show that the Uluru algorithm is a special case of our optimal sampling procedure when applied to the coefficients of a linear regression problem.",
"Interestingly, although the theoretical results for Uluru deal are stated in terms of the prediction error, we find Uluru is suboptimal in this regime.",
"Outside of linear models, local case-control sampling [9] provides an effective sampling method for logistic regression based on the residuals as well.",
"A number of other other techniques are relevant for fast model fitting on large datasets.",
"Stochastic gradient descent (SGD) employs sampling and is useful in large-scale learning problems [2].",
"Other techniques such as [15] use random projections to speed up the computation in the inner loop of a model fitting algorithm.",
"In these cases, the data size is not reduced and the result is an estimate for the specific model being fit.",
"In contrast, we focus on sampling as a way to reduce computational complexity for a specific model or models while also being able to further use the sample for other purposes, for example, in parameter tuning."
],
[
"Influence functions and asymptotically linear estimators",
" One key idea in this paper is that many parameter estimators can be asymptotically expressed as a mean of influence vectors.",
"This allows us to express the problem of optimal sub-sampling for a statistical model in terms of optimal sub-sampling for a mean.",
"We give a brief overview of the theoretical machinery needed for our method and refer the interested reader to [16] for more details.",
"We consider the class of estimators are plug-in estimators $\\hat{\\theta }(P)$ taking a distribution to a real-valued vector of parameter estimates.",
"This is a highly flexible class of estimators.",
"For example, any M-estimator $\\hat{\\theta }(P) = \\arg \\!\\min _t {\\mathbb {E}}_{P} Loss(t, X)$ is of this form.",
"The argument for the M-estimator is the empirical distribution $\\mathbb {P}_n$ .",
"When it exists, the influence function for this estimator is defined by its Gateaux derivative $\\psi _P(x) &= \\lim _{\\epsilon \\rightarrow 0^+} \\frac{1}{\\epsilon }\\left(\\hat{\\theta }\\left((1-\\epsilon )P + \\epsilon \\delta _x\\right) - \\hat{\\theta }(P)\\right)$ where $\\delta _x$ is the Dirac delta measure at $x$ .",
"It represents the infinitesimal change in the estimate by adding the point $x$ to the sample.",
"The estimator $\\hat{\\theta }$ is an asymptotically linear estimator with influence function $\\psi $ if it satisfies $\\sqrt{n}\\left(\\hat{\\theta } - \\theta \\right) =\\frac{1}{\\sqrt{n}} \\sum _{i=1}^n \\psi (X_i) + o_p(1),$ with ${\\mathbb {E}}\\psi = 0$ and ${\\mathbb {E}}\\psi ^T \\psi < \\infty $ .",
"$o_p(1)$ denotes convergence in probability in some normed space.",
"Asymptotically linear estimators are pervasive in statistical modeling.",
"Under sufficient regularity conditions, the previously mentioned M-estimators and maximum likelihood estimators as well as Z-estimators, non-degenerate U-statistics, and Generalized Method of Moments estimators are asymptotically linear estimators.",
"For maximum likelihood estimation with correctly specified and sufficiently regular models, the influence function can be described in terms of the derivative of the log-likelihood with respect to the parameter, in other words, the score function $s_\\theta (x)$ .",
"The influence function can be related to score by $s_{\\theta }(x) &= \\partial \\ell (\\theta ; x) / \\partial \\theta , \\qquad \\qquad \\psi _{\\theta } = I_{\\theta }^{-1} s_{\\theta }$ where $I_\\theta $ is the Fisher information.",
"Unlike gradient-based sampling, there is no differentiability requirement.",
"Thus, non-differentiable likelihoods such as a double exponential location family or quantile regressions fit in the framework.",
"Furthermore, we find both in theory and practice that gradient-based sampling accounts for the regression design (i.e.",
"the predictors) on an inappropriate scale.",
"As a consequence, even sampling using only the residuals as weights and completely ignoring the regression design often performs better than the full gradient as shown in figure REF .",
"As the influence function asymptotically encodes the effect of a single data point, it is sensible to use it to determine the point's sampling weight.",
"The asymptotic form of an asymptotically linear estimator shows that the problem of sampling for the estimator can be reduced to a problem of estimating a multivariate mean."
],
[
"Linear Least Squares Influence",
"In the context of linear regression, the influence function for the coefficients $\\hat{\\theta }$ is given by $\\psi _{P_\\theta }(x_i, y_i) &= (y_i - x_i^T \\theta ) \\Sigma ^{-1} x_i$ where $\\Sigma = \\frac{1}{n}\\left(X^T X\\right)$ is the empirical second moment matrix and $\\theta $ are the true coefficients.",
"Taking the norm of the influence yields a sampling weight that differs from gradient-based sampling only by the scaling.",
"The weight is proportional to $\\Vert (y_i - \\hat{y}_i)\\Sigma ^{-1} x_i\\Vert $ rather than $\\Vert (y_i - \\hat{y}_i)x_i\\Vert $ .",
"The influence-based scaling is the more sensible of the two.",
"For example, if one wishes to make the $j^{th}$ coordinate more important, $\\theta _j$ must be scaled to be larger and $x_{ij}$ to be smaller.",
"Under gradient-based sampling, a previously important point with a large $x_{ij}$ is perversely made less important as $x_{ij}$ is scaled downwards.",
"Influence-based sampling correctly increases the importance of the point."
],
[
"Influence on Predictions",
"The notion of influence can be extended beyond estimators for parameters.",
"For example, points can be sampled according to their influence on predictions rather than on the coefficients themselves.",
"For many prediction problems, this measure may be more sensible as one neither cares about the exact values of the coefficient nor the scale of the variables if the predictive performance is good.",
"The influence on the prediction is easily derived from the influence on the coefficients.",
"When each prediction $\\hat{y}_i(\\theta )$ is a twice differentiable function of the parameter $\\theta $ , then, with a slight abuse of notation, the influence on the vector of predictions is given by the chain rule $\\psi ^{(pred)}_{P_\\theta }(x_i, y_i) = \\hat{y}^{\\prime }(\\theta ) \\frac{d \\theta }{d \\delta _{(x_i,y_i)}} =\\hat{y}^{\\prime }(\\theta ) \\psi _{P_\\theta }(x_i, y_i).$ In the context of linear least squares regression, the influence is $\\psi ^{(pred)}_{P_\\theta }(x_i, y_i) = X (X^TX)^{-1} x_i (y_i - \\hat{y}_i) = r_i H_{i\\cdot }$ where $r_i$ is the residual error and $H_{\\cdot i}$ is the $i^{th}$ column of the hat matrix.",
"Since the hat matrix $H$ is idempotent and symmetric, it follows that the squared norm $\\Vert H_{i \\cdot } \\Vert ^2 = e_i^T H^T H e_i = e_i^T H e_i = H_{ii}$ .",
"A sensible univariate measure for a sampling weight is thus $|r_i| \\sqrt{H_{ii}}$ .",
"The prediction on the influence provides a strong connection to leverage-based sampling.",
"In the classical statistical setting where the experimenter does not have knowledge of the response $Y_i$ when setting the regression design $X$ , a sensible measure of influence takes the expectation over the unknown $Y_i$ .",
"In this case, one obtains a sampling weight proportional to the root leverage score $\\sqrt{H_{ii}}$ .",
"If only the influence on the prediction $\\hat{y}_i$ is considered, the influence is $\\psi ^{(pred,i)}_{P_\\theta }(x_i, y_i) = x_i (X^TX)^{-1} x_i (y_i - \\hat{y}_i) = r_i H_{ii}$ .",
"Taking an expectation over an unknown $Y_i$ exactly gives back leverage score sampling.",
"Thus, we see that leverage-based sampling throws away two pieces of information that are available in sampling for computational problem, the effect of the regression design on points other than the $i^{th}$ point and more importantly, the information on the response $Y_i$ ."
],
[
"Sampling design and Estimation from samples",
"Reducing computational costs using the influence function requires three components: 1) a method to estimate the influence function, 2) a method to convert the influence function into a sampling design and sampling probabilities, and 3) a method to provide good estimates from the sample.",
"We address these in reverse order."
],
[
"Estimation from samples",
"A good sub-sampling procedure allows one to accurately approximate the estimator $\\hat{\\theta }$ .",
"This is a somewhat different problem from approximating the true parameters.",
"Let $\\tilde{\\theta }$ be an estimator based on a subsample.",
"As $\\tilde{\\theta }$ cannot be expected to improve upon the estimator on the full data $\\hat{\\theta }(\\mathbb {P}_n)$ , it is reasonable to assume that its expectation is close to or equal to $\\hat{\\theta }(\\mathbb {P}_n)$ , and the variance ${\\rm Var}(\\tilde{\\theta }) \\approx {\\mathbb {E}}{\\rm Var}(\\tilde{\\theta } | \\hat{\\theta }- \\theta )$ .",
"In other words, the subsample should be the best sample for approximating the deviation $\\hat{\\theta } - \\theta $ .",
"This deviation has the asymptotically linear form given in equation REF , so that approximating the deviation is asymptotically equivalent to approximating the mean of influences.",
"The quantities of interest such as the objective or estimators that we consider in this paper can be expressed as functions of the empirical distribution $\\mathbb {P}$ .",
"For any subsampling procedure, an unbiased estimate of the true empirical distribution can be obtained by reweighting the sample points by their inverse sampling probability.",
"In the case of mean estimation, the resulting estimator called the Horvitz-Thompson estimator $\\hat{\\mu } = n^{-1} \\sum _i X_i Z_i / \\pi _i$ where $Z_i$ is the indicator that a point was sampled and $\\pi _i = {\\mathbb {E}}Z_i$ is the sampling probability.",
"This can also be applied to obtain an unbiased estimate $\\hat{\\mathbb {P}}$ of the true empirical distribution and resulting parameter estimator $\\hat{\\mathbb {P}} &= \\left(\\sum _{i=1}^n \\frac{1}{\\pi _i}\\right)^{-1}\\sum _{i=1}^n \\frac{Z_i}{\\pi _i} \\delta _{x_i}, \\qquad \\qquad \\tilde{\\theta } = \\hat{\\theta }(\\hat{\\mathbb {P}}) .$"
],
[
"Optimal design",
"The next question is how to choose an appropriate sampling procedure and convert a vector valued influence to a univariate sampling probability.",
"A sample can then be drawn by independently selecting each item according to its sampling probability.",
"This scheme is called Poisson sampling.",
"Oftentimes, there is a sampling objective that should be minimized as well a size budget for the number of samples that can be stored.",
"Consider the objective of minimizing $L_2$ error of the parameters, and suppose the original data points are drawn i.i.d.",
"from some distribution.",
"One has $\\Vert \\hat{\\theta } - \\theta \\Vert ^2 \\approx \\frac{1}{n^2} \\sum _i \\Vert \\psi _i\\Vert ^2 \\propto Trace({\\rm Var}\\, \\psi _i)$ .",
"If each $\\psi _i$ is sampled independently with probability $\\pi _i$ , the optimal probabilities $\\pi _i$ for drawing a sample of expected size $m$ minimizes the objective $\\min V\\left( \\sum _i \\Vert \\psi _i\\Vert Z_i / \\pi _i\\right)$ subject to the constraints $\\sum _i \\pi _i = m$ and $\\pi _i \\le 1$ .",
"Applying the method of Lagrange multipliers gives $\\pi _i \\propto \\Vert \\psi _i\\Vert $ for all $\\pi _i \\ne 1$ .",
"This gives an instance of probability proportional to size (PPS) sampling where the auxiliary measure of size is the norm of the influence function for the parameters.",
"Similarly if the objective is minimizing the $L_2$ error in the predictions, an appropriate sampling scheme uses PPS sampling with size equal to the norm of the influence function of the predictions.",
"We note that the problem of choosing a sampling weight is non-obvious.",
"For least squares regression, the optimal choice for approximating the loss at $\\hat{\\theta }$ is to sample with probability proportional to the squared residual $(y_i - \\hat{y}_i)^2$ .",
"We found this to be a poor choice for approximating the coefficients in our experiments.",
"The reason is that the raw value of the loss is unimportant.",
"Asymptotically, only the gradient and Hessian are relevant.",
"We also note that although past work [7] sampled with probability proportional to the leverage score, our work surprisingly shows that the square root of the leverage may be a more appropriate measure.",
"In other cases, one may be interested in quantities other than the predictions or all the coefficients together.",
"For example, if one wishes to study the effect of gas and electricity pricing on consumption, any reasonable model would adjust for weather effects.",
"In this case, only a subset of coefficients may be of interest while others, like those for weather, are nuisance parameters.",
"In this case, one may use the influence restricted to the coefficients of interest."
],
[
"Regularization",
"Since each sampled point is weighted by its inverse sampling probability, the resulting estimate may have high variance if a sampling probability is too small.",
"The solution in this paper is to add a small amount of regularization to ensure that no sampling probability is too small.",
"This ensures convergence $\\gamma _n(\\hat{\\mathbb {P}} -\\mathbb {P}_n) \\rightsquigarrow T$ to some limit process $T$ under an appropriate scaling $\\gamma _n$ .",
"In deriving an optimal sampling design, this corresponds to adding the convex constraint that $\\alpha \\le \\pi _i \\le 1$ to the optimization in REF .",
"The resulting sampling probabilities have $\\pi _i \\propto \\Vert \\psi (x_i) \\Vert $ if $\\pi _i \\ne 1$ or $\\alpha $ ."
],
[
"Influence function estimation",
"Thus far, we have derived the exact influence function given a population distribution or in finite sample cases, the empirical distribution.",
"In most cases, the influence function depends on the true parameter $\\theta $ .",
"A simple estimate of the influence function uses a pilot estimate $\\theta _0$ as a substitute.",
"The pilot estimate may be readily available or easily obtained.",
"For example, in many machine learning applications, the estimated parameters from one day may be used as a pilot estimate for the next day.",
"If one has no pre-existing pilot estimate, then one can first draw a uniform sample, or even a reasonable convenience sample, from the data to form a pilot that can be used on the remaining data.",
"A second approach requires only the ability to compute the gradient of the log-likelihood, in other words the score function.",
"As described in section , for a sufficiently regular family, the influence function for the maximum likelihood estimate is a rescaling of the score function $s$ by the inverse of the Fisher information $I_\\theta = {\\mathbb {E}}ss^T$ .",
"A simple procedure to estimate the influence function is to 1) start with a pilot estimate $\\theta _0$ , 2) compute an online estimate of the covariance $\\hat{V}_{\\theta _0}$ for the scores at $\\theta _0$ , and 3) rescale the score $\\hat{\\psi }(x) = \\hat{V}_{\\theta _0}^{-1} s_{\\theta _0}(x)$ .",
"In experiments we find that the approximation to the leverage score or non-residual component of the influence can negatively affect the quality of the sample.",
"In such cases, approximating the component with a constant yields the extraordinarily simple—but often effective—sampling design with probability proportional to the residual."
],
[
"Influence function computation",
"For many estimators, computing the influence function requires a matrix inversion or pseudo-inverse.",
"This may be costly to compute.",
"A simple strategy is to replace $V$ with its diagonal, in other words, the Jacobi preconditioner.",
"Another strategy is to apply matrix sketching.",
"These can include fast random projection methods [1] as well as deterministic sketching methods [11].",
"When the influence function is on the predicted values rather than the parameters themselves, section REF shows that the influence of a point can sometimes be expressed as the product of a function of the residual and the leverage of the point.",
"Methods to approximately compute leverage scores [7] may be applied.",
"The simplest leverage score approximation is to assume equal leverage of all points.",
"The resulting sampling weight depends only on the residual."
],
[
"Examples",
"We provide two examples for the influence function in addition to the linear least squares model that is already derived in section .",
"First, we consider the common case of generalized linear model.",
"Second, we consider a non-differentiable quantile regression problem."
],
[
"Generalized Linear Model",
"Generalized linear models (GLMs) are a generalization of the normal linear model to allow for non-normal error distributions such as when responses are discrete valued.",
"They are flexible and commonly used.",
"Logistic, Poisson, and exponential regression models are examples of GLMs.",
"A generalized linear model with a canonical link function has log likelihood given by $\\log p(y_i, x_i | \\theta ) = y_i x_i^T \\theta - A(x_i^T \\theta )$ where $A(t) = \\log \\int _\\Omega exp(y t) dy$ is the log partition function and $\\Omega $ is the set of possible $y$ values.",
"Under smoothness conditions on the error distribution and when the model is correctly specified, the influence function is given by the rescaled score given in equation REF .",
"A straightforward derivation gives that $A^{\\prime }(x_i^T\\theta ) = {\\mathbb {E}}(Y_i | x_i^T\\theta )$ and $A^{\\prime \\prime }(x_i^T\\theta ) = 1/{\\rm Var}(Y_i | x_i^T\\theta )$ .",
"Thus, the score function is $s_{\\theta }(x_i, y_i) = (y_i - \\hat{y}_i) x_i$ and the Fisher information is $(X^TWX)$ where $W$ is a diagonal weight matrix with $W_{ii} = 1/{\\rm Var}_{\\theta } (Y_i | x_i^T \\theta )$ .",
"Thus the influence functions for $\\hat{\\theta }$ and $\\hat{y}$ under correct model specification are given by $\\psi _{\\theta }(x_i, y_i) &= (y_i - \\hat{y}_i) (X^T W X)^{-1} x_i \\\\\\psi ^{(pred)}_\\theta (x_i, y_i) &= (y_i - \\hat{y}_i) W X (X^TWX)^{-1} x_i =r_i H_{i \\cdot }^T.$ where $r_i$ is the residual and $H = X (X^TWX)^{-1} X^T W$ so that $\\hat{\\theta } = H Y$ .",
"Unlike linear least squares regression, the matrix $H$ is non-symmetric so the norm of the influence function cannot be expressed exactly in terms of the diagonal of the hat matrix .",
"For the special case of logistic regression, local case-control sampling provides a sampling method that has both good empirical and theoretical properties.",
"It chooses sampling probabilities proportional to the “surprise” $y_i(1-\\hat{p}_i) + (1-y_i)\\hat{p}_i$ so that a point is likely to be sampled only if it did not match the prediction.",
"The surprise can also be expressed as the absolute value of the residual $|y_i - \\hat{p}_i|$ .",
"Thus, local case-control sampling is equivalent to influence-based sampling under the approximation that there is no essentially no effect due to the regression design.",
"Alternatively, influence-based sampling can be seen as local case-control sampling but with the addition of information about the predictors or regression design $X$ ."
],
[
"Quantile regression",
"Quantile regression [10] provides another useful generalization of linear models.",
"While linear least squares regression focuses on estimating the conditional mean, in some cases, the quantity of interest is not the average but the upper or lower tails of a distribution.",
"For example, a charitable foundation may be interested in predicting conditional quantiles in order to set suggested donation amounts.",
"In the case of median or $L_1$ regression, the true regression coefficients match the least squares coefficients when the error distribution is symmetric; however, median regression enjoys robustness properties that make it less sensitive to outliers.",
"For quantile regression, the loss function is the non-differentiable “check” function $\\ell _\\tau (x) = (1-\\tau ) x 1(x < 0) + \\tau x 1(x \\ge 0)$ rather than the squared residual.",
"When the desired quantile is $\\tau $ and the true conditional quantile is linear, the influence function is given by $\\psi (x_i, y_i) = [\\tau (1-\\tau )]^{-1} V^{-1} x\\, \\rho (y_i - x_i^T \\theta )$ where $\\rho $ is a subgradient of the loss, $\\rho (z) = 1-\\tau $ if $z < 0$ and $\\tau $ if $z > 0$ , and $V = \\int xx^T f(0 | x)dG(x)$ when the $X_i$ are randomly drawn from a distribution $X_i \\sim G$ and the error $Y_i - x_i^T\\theta $ has density $f(\\cdot | x_i)$ .",
"In particular, if the error distribution is independent of the predictors $X$ , $\\Sigma = \\frac{1}{n} X^TX$ is a consistent estimator of $V$ .",
"This gives the following estimated influence functions on the coefficients and predictions $\\hat{\\psi }(x_i, y_i) &= [\\tau (1-\\tau )]^{-1} \\rho (r_i) \\Sigma ^{-1} x_i \\\\\\hat{\\psi }^{(pred)}(x_i, y_i) &= [\\tau (1-\\tau )]^{-1} \\rho (r_i) H_{\\cdot i}.$ We note that this influence function has the same form as the influence function for linear regression in equation REF .",
"The residual $r_i$ in the influence function for linear regression is simply replaced by $\\rho (r_i)$ in the quantile influence function.",
"We will refer to $\\rho (r_i)$ as the \"residual\" for quantile regression."
],
[
"Error analysis",
"Let $\\phi (\\cdot )$ be some real-valued function on distributions in some P-Donsker class $\\mathcal {F}$ .",
"Suppose it is Hadamard differentiable at $P_{\\theta }$ under the uniform norm $\\ell ({\\mathcal {F}})^{\\infty }$ with influence function $\\psi _\\theta $ .",
"Assume values $X_i$ are drawn i.i.d.",
"from $P_{\\theta }$ .",
"Consider the set of measures $Q$ that are mutually absolutely continuous with respect to $P_{\\theta }$ with $\\alpha \\le dQ / dP \\le 1$ almost everywhere and the total measure of $Q$ is some constant $c$ .",
"$Q$ defines an importance sub-sampling distribution which is generated by taking the empirical distribution $\\mathbb {P}_n$ and keeping item $x$ in the empirical distribution with probability $\\pi _x = dQ/dP(x)$ .",
"Let $\\hat{\\mathbb {P}}^Q_n = n^{-1} \\sum _{i=1}^n Z_i \\frac{dP}{dQ}(X_i) \\delta _{X_i} $ be the resulting estimated empirical measure where $Z_i$ indicates $X_i$ is the the subsample and equals 1 with probability $\\frac{dQ}{dP}$ .",
"Since the weights $\\frac{dP}{dQ}$ are bounded and fixed, it follows that $\\sqrt{\\frac{n}{c}}(\\hat{\\mathbb {P}}^Q_n - P) \\rightsquigarrow R^Q$ where $R^Q$ is tight in $\\ell (\\mathcal {F})^{\\infty }$ .",
"The functional delta method [16] gives that $\\sqrt{\\frac{n}{c}}(\\phi (\\hat{\\mathbb {P}}^Q_n) - \\phi (P))\\rightsquigarrow \\int \\psi (x) dR(x)$ .",
"Furthermore, this limit is $Normal(0,V^Q)$ where $V^Q = \\int \\psi (x)^2 \\left(\\frac{dP}{dQ}\\right)^2 dQ(x)$ .",
"It is thus sensible to define the optimal importance sub-sampling measure $Q_{opt}$ to be the one that minimizes the asymptotic variance $V^Q$ .",
"Let $\\hat{\\pi } \\in \\Pi _{\\alpha , n}$ be estimated regularized inclusion probabilities for a sample of expected size $m$ based on the influence function.",
"These probabilities are asymptotically optimal in the sense that the resulting estimates converge to some limit Normal distribution where the variance of that limit distribution is equal to the limit variance under the optimal importance sub-sampling distribution.",
"The proof is deferred to the supplementary material.",
"Theorem 1 Suppose the pilot estimate of the influence function is consistent $\\tilde{\\psi } = \\psi _{\\theta _0} + o_p(1)$ under the uniform norm.",
"As $m,n \\rightarrow \\infty $ with $m /n \\rightarrow c > 0$ , the plug-in estimator $\\phi (\\hat{\\mathbb {P}}_n(\\hat{\\pi }) ) \\rightsquigarrow Normal(0, V^{Q_{opt}})$ ."
],
[
"Experiments",
"We consider two real datasets from the UCI repository, the CASP [12] ($n=45730$ , $d=9$ ) and the Online News Popularity ($n=38644$ , $d=59$ ) datasets, and show the results in Figures REF and REF , respectively.",
"We consider both least squares and quantile regression models.",
"For the Online News Popularity dataset, we removed 4 columns due to collinearity.",
"In each case, we use a 5% random sample of the data to derive a pilot estimate and drew a weighted sub-sample from the remainder.",
"We considered 2 linear regression models, linear least squares and quantile regression.",
"The quality of the fit on a subsample is measured either by squared error in the coefficients $\\Vert \\hat{\\theta } - \\theta _{opt}\\Vert ^2$ or by the loss function corresponding to the optimization.",
"As we wish to make only one pass through the data, we use the same approach as in [6] for computing approximate matrix inverses needed for the influence and leverage rather than the fast Johnstone-Lindenstrauss transforms needed in [7].",
"We also regularize the $d\\times d$ matrix inverse by taking $(V+\\lambda I)^{-1}$ where $\\lambda = Trace(V) / 10d$ .",
"For completeness, we compute the exact leverage scores and show that they still perform worse.",
"Figure: Results of our method on the CASP data set .",
"The left 4 subplots show the performance of methods based on approximations to the influence and leverage while the right 4 are based on the exact values.",
"As expected, our influence-based methods are the top performers.",
"They require 1/31/3 to 1/21/2 the sample size compared to uniform sampling to achieve the same error.",
"This holds for both least squares and quantile regression tasks when their customized influence functions are used.",
"We also find that the gradient-based sampling method performed worse than just using the residuals.",
"This was likely due to the widely ranging scales of each variable as we did not make an additional pass to normalize the data.",
"Furthermore, we note that the approximate-leverage-based method performs better than uniform sampling to minimize prediction error but worse in minimizing error in the coefficients.Figure: Left: Results of our method on the NEWS data set.",
"The results on the NEWS dataset tell a similar story to the CASP dataset but with much larger gains over uniform sampling for least squares regression.",
"This is likely due to the heavy tailed nature of the NEWS dataset compared to CASP.",
"The dataset also exhibits poor approximation of the matrix inverse needed to compute the “influence” of the design, so that simply using the residual with no approximated influence from the regression design performs best for least squares.",
"Right: Sampling patterns for different importance measures for the model Y=5X+1000Z+ϵY = 5 X + 1000 Z + \\epsilon where Z=10 -3 Z = 10^{-3}.",
"Red denotes high inclusion probability while light blue denotes low inclusion probability.",
"The differences between the methods are made more apparent in the supplementary material.Leverage focuses only on the extremes of the design while completely ignoring large residual near the center.",
"The gradient-based design also ignores large residuals near the center though to a much smaller degree.",
"The influence-based design picks an appropriate balance of residual and regression design effects where points near the center are slightly less likely to be sampled given the same residual.",
"The “influence” of the residual dominates over the “influence” of the regression design here."
],
[
"Conclusion",
"We have demonstrated both theoretically and empirically that influence functions yield good and principled sub-sampling procedures for handling big data.",
"They allow one to fundamentally understand and recast the problem as optimal sampling for a mean estimation problem where the mean is taken over influences.",
"In particular, we show that our approach yields the best possible asymptotic variance over all Poisson sampling designs with the same size and regularization.",
"Our approach can be applied to a wide range of statistical and machine learning models.",
"Furthermore, although the full influence often requires a matrix inversion, simple approximations to the influence that take only $O(nd)$ time, such as using only the residual, can perform well while being easy to both compute and implement."
],
[
"Proof",
"The proof consists of two parts.",
"One establishes that sampling with probability proportional to influence leads to the optimal importance sub-sampling measure.",
"This argument is a straightforward application of the same argument in section REF .",
"The second consists of showing that the sample drawn using the estimated influences effectively yields the same sample as one using the exact influences.",
"The consistency of the estimated influence $\\hat{\\psi }$ gives that $\\hat{\\pi } = dQ / dP + o_p(1)$ .",
"For Poisson sampling with , the item $X_i, Y_i$ is in the sample if $U_i < \\pi _i$ for independent $U_i \\sim Uniform(0,1)$ .",
"Let $\\pi _i = dQ/dP(X_i)$ and $\\tilde{\\pi }_i$ be the estimated inclusion probability using the estimated influence.",
"Let $Z_i = 1$ if $U_i < \\pi _i$ and 0 otherwise.",
"Likewise, $\\tilde{Z}_i = 1$ if $U_i <\\tilde{\\pi }_i$ and 0 otherwise.",
"Let $\\epsilon _i = \\tilde{\\pi }_i - \\pi _i$ .",
"The difference $Z_i / \\pi _i - \\tilde{Z}_i / \\tilde{\\pi }_i &=\\frac{Z_i - \\tilde{Z}_i}{\\pi _i} (1 - \\epsilon _i + O(\\epsilon _i^2))$ Taking the numerator, we have $|Z_i - \\tilde{Z}_i| \\sim Bernoulli(\\epsilon _i)$ .",
"And the overall expectation of the absolute value is $O(\\epsilon _i / \\pi _i)$ .",
"Since ${\\rm Var}(Z_i / \\pi _i) = (1-\\pi _i)$ , it follows that the empirical estimate based on the estimated influences $\\sqrt{\\frac{n}{c}}(\\hat{\\mathbb {P}} - P)$ converges to the same limit as that under the optimal $Q_{opt}$ .",
"Hence, $\\sqrt{m}(\\phi (\\hat{\\mathbb {P}}) - \\phi (P))$ and $\\sqrt{m}(\\phi (\\hat{\\mathbb {P}}^{Q_{opt}}) - \\phi (P))$ also converge to the same limit by the functional delta method."
]
] | 1709.01716 |
[
[
"Primordial gravitational waves amplification from causal fluids"
],
[
"Abstract We consider the evolution of the gravitational wave spectrum for super-Hubble modes in interaction with a relativistic fluid, which is regarded as an effective description of fluctuations in a light scalar minimally coupled field, during the earliest epoch of the radiation dominated era after the end of inflation.",
"We obtain the initial conditions for gravitons and fluid from quantum fluctuations at the end of inflation, and assume instantaneous reheating.",
"We model the fluid by using relativistic causal hydrodynamics.",
"There are two dimensionful parameters, the relaxation time $\\tau$ and temperature.",
"In particular we study the interaction between gravitational waves and the non trivial tensor (spin 2) part of the fluid energy-momentum tensor.",
"Our main result is that the new dimensionful parameter $\\tau$ introduces a new relevant scale which distinguishes two kinds of super-Hubble modes.",
"For modes with $H^{-1}<\\lambda<\\tau$ the fluid-graviton interaction increases the amplitude of the primordial gravitational wave spectrum at the electroweak transition by a factor of about $1.3$ with respect to the usual scale invariant spectrum."
],
[
"Introduction",
"In this paper we shall consider the evolution of the primordial gravitational wave background during the early radiation dominated era [1] [2] [3], from reheating after inflation up to the cosmological electroweak transition.",
"We will use second order hydrodynamics [4] [5] as an effective theory for the matter fields, and obtain a linear theory for gravitons consistently coupled to the spin-2 component of the matter energy-momentum tensor.",
"Our motivation in using hydrodynamics as an effective theory comes from the highly successful description of the early evolution of the fireball created in relativistic heavy ion collisions (RHICs) by these methods, even in early stages where it is unlikely that local thermal equilibrium has been established [6] [7].",
"As a matter of fact, our problem bears a significant similarity to RHICs [8].",
"Our main assumption is that among the fundamental fields there is at least one that is not conformally coupled; for simplicity we shall take this to be a light (effectively massless), minimally coupled scalar field with small coupling constants.",
"These fields are commonly related to “axion-like particles” (ALPs) [9] [10].",
"Inflationary expansion brings this field to its De Sitter invariant vacuum state.",
"However, this state is highly squeezed and its quantum fluctuations are much higher than those of the local vacuum state of adiabatic observers.",
"Upon horizon exit, and particularly after reheating, these fluctuations lose quantum coherence and may be treated as classical particles [11] [12] [13] [14] [15] [16] – thus resembling the quark-gluon plasma generated in RHICs.",
"These particles compose our “fluid”.",
"As we have learned from RHICs, the proper treatment of a real relativistic fluids on timescales not much larger than the fluid relaxation time requires the use of “second order” theories rather than the better known Eckart or Landau-Lifshitz formulations [17] [18] [19]; one of the main points of this paper is that this is the relevant framework for our discussion.",
"In second order theories, the viscous part of the energy-momentum tensor, or some other equivalent variable, is considered as an independent degree of freedom following a Cattaneo-Maxwell type dynamical equation [20].",
"This equation, together with the Einstein equations and the relevant conservation laws, completes the fully consistent dynamics we are looking for.",
"During reheating and afterwards, we must distinguish between the physics of modes inside or outside the horizon.",
"Reheating is dominated by the most out of equilibrium phenomenon in the history of our Universe, the sudden conversion of the energy-momentum of the inflaton field into radiation energy-momentum [21] [22] [23] [24] [25].",
"We do not assume our scalar field is decoupled from the rest of matter, and so it partakes of this essentially nonlinear phenomenon.",
"However, the nonlinearities are restricted by causality and therefore they are strong only within the horizon.",
"Outside the horizon the evolution of the graviton-effective fluid system may be described accurately enough by linearized equations.",
"At the most basic level, a gravitational wave presents itself through an anisotropy in the rest frame of the fluid.",
"Ideal hydrodynamics is restricted by the Pascal principle, namely, the state of the ideal fluid is defined solely by the chemical potentials associated to conserved charges (which moreover vanish for a conformal theory) and by the inverse temperature four-vector, and so it is locally isotropic on surfaces perpendicular to this vector.",
"Moreover, for a true equilibrium state, the inverse temperature four-vector must be a (conformal) Killing vector [4], and it may happen that for a given spacetime there are no such vectors.",
"However, in that case hydrodynamics is not built on true equilibria, but only on approximated local equilibria.",
"Any space time will allow for the construction of coordinate systems, such as Riemann normal coordinates [14] [26], which look locally isotropic.",
"Therefore, in the usual approach to hydrodynamics, temperature will be isotropic in the rest frame.",
"The shear tensor, on the other hand, may be anisotropic, but because it is built from derivatives of a vector, it cannot have the symmetry of a spin-2 field.",
"To account for the kind of anisotropy associated to a gravitational wave it is necessary to go beyond the usual framework by considering higher orders or else including from scratch a new spin-2 degree of freedom, as we shall do in the following.",
"For further discussion we refer to [27].",
"Unlike ideal and first order hydrodynamics, there is no universally accepted approach to second order hydrodynamics.",
"However, in the linearized regime we are interested in, most formalisms converge.",
"For simplicity, we shall adopt a divergence-type theory scheme [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] where the conformally invariant fluid is described by a dimensionful parameter $T$ (which becomes the temperature when in equilibrium), the fluid four-velocity $u^{\\mu }$ (which obeys $u_{\\mu }u^{\\mu }=-1$ , we adopt MTW conventions) and a dimensionless, symmetric, traceless and transverse tensor $\\zeta ^{\\mu \\nu }$ ($\\zeta ^{\\mu }_{\\mu }=\\zeta ^{\\mu \\nu }u_{\\nu }=0$ ).",
"We scale this tensor so that in the linearized theory $\\zeta _{\\mu \\nu }=\\Pi _{\\mu \\nu }/\\rho $ , where $\\Pi _{\\mu \\nu }$ is the viscous energy-momentum tensor and $\\rho $ the energy density.",
"For simplicity we shall not consider an explicit coupling of the fluid to other matter fields, the self and gauge interactions of the fluid will appear through the constitutive relations for the fluid, that is its relaxation time $\\tau $ (to be discussed in Section ), and its temperature.",
"Under this approximation the equations of the model are the Einstein equations, energy-momentum conservation, and a Cattaneo-Maxwell equation for $\\zeta _{\\mu \\nu }$ to be provided below.",
"In summary, we assume that at the end of reheating super-Hubble modes are in a state determined by their state at the end of inflation (namely, that reheating is so fast that no significant processing occurs during reheating itself), and then thermalize to the state determined by the dominant cosmic radiation background [38]; this thermalization is well described by linearized hydrodynamics.",
"Moreover, at the relevant temperature scales the fluid may be regarded as composed of massless particles, whereby hydrodynamics becomes conformally invariant [39].",
"The tensor field $\\zeta ^{\\mu \\nu }$ may be decomposed into scalar (spin 0), vector (spin 1) and tensor (spin 2) parts which are decoupled from each other at linear order.",
"Our interest lies in the spin 2 part, which couples directly to the graviton field; for simplicity we shall disregard the scalar and vector sectors, and focus on the spin-2 sector alone.",
"The spin 1 is relevant in scenarios including gauge fields, since it is related to magnetic field generation [40] [41] [42] [43] [44].",
"It is well known that the spin 2 part of the matter energy-momentum tensor may seed a primordial gravitational field [45] [46] [47] [48] [49] [50] [51] [52] [53].",
"In the literature there are several estimates of the gravitational background created by different fields, such as the inflaton [54] [55], the Higgs field [56] [57] [58], primordial density fluctuations [59], scalars and non abelian charged scalars [60], and Fermi fields [61].",
"In principle, the effect on the gravitational wave background may be observed through its impact on the CMB [62].",
"The present work is closest to [63] [64] which considers the gravitational field created out of a spectator field.",
"However, three differences stand out, namely we put the emphasis in achieving a self-consistent dynamics, including the back reaction of the gravitons on the spectator field, we incorporate the thermalization to the dominant radiation background into this picture, and we read the initial conditions for field and gravitons directly off quantum fluctuations of super-Hubble modes just before inflation ends, rather than the Starobinsky-Yokohama equation [65].",
"Let us elaborate on this last point.",
"Under the assumption of instantaneous reheating we may obtain the initial conditions for these equations from the analysis of quantum fluctuations just before reheating.",
"For the graviton field this is conventional, for completitude the main necessary results will be summarized below.",
"For the effective fluid we shall treat $\\zeta ^{\\mu \\nu }$ as a stochastic Gaussian field whose self-correlation is derived from the energy-momentum self correlation of a quantum minimally coupled scalar field during inflation.",
"Of course this is a divergent quantity, but the divergence is associated to short wavelength modes within the horizon; we shall assume a local observer will subtract the correlations corresponding to the instantaneous vacuum state (as defined by adiabatic modes), and associate the remainder with the effective fluid [14] [66] [67].",
"The new dimensionful quantity $\\tau $ (Eq.",
"(REF )) splits the range of super-horizon modes $k\\le H$ , where $H$ is Hubble’s constant during inflation, in two.",
"For modes where $k\\le \\tau ^{-1}$ as well, the fluid relaxation is efficient and there is no substantial effect of the fluid on the gravitons; the energy associated with the spin 2 field is just dissipated into heat.",
"However, when $\\tau ^{-1}\\le k\\le H$ there is some amplification of the primordial gravitational spectrum due to the decay of the spin 2 part of the fluid into gravitons.",
"This means that this mechanism may be the source of a local feature (a step) in the graviton spectrum around $k\\sim \\tau ^{-1}\\ll H$ .",
"We quantify the height of this step by solving the linearized equations from reheating up to the time of the electroweak transition, after which the primordial gravitational wave spectrum is subject to further processing [1].",
"We shall show that given appropriate values of the coupling constant (similar to some axion-like particle models) this step may fall in an observationally relevant range.",
"This is the main result of this paper.",
"The paper is organized as follows.",
"In Section we introduce the framework of divergence type theories from which we extract the causal hydrodynamic equations for the fluid, particularly we derive to linearized order the expression for the energy-momentum tensor and the dynamic equation for the non-equilibrium tensor.",
"In order to deduce the system of fluid-gravitons coupled equations we gather the closure and linearized Einstein's equations in Section .",
"Section provides the initial conditions for gravitons and non-equilibrium variable from quantum fluctuations during inflation.",
"Section is the main part of this paper; here we analyze the solutions of the previous system.",
"We compute the evolution of the primordial gravitational wave spectrum for super-Hubble modes up to the electroweak transition and show that some amplification occurs for modes with $H^{-1}<\\lambda <\\tau $ .",
"Then we study the values of the relaxation time $\\tau $ in Section from quantum field theory for a scalar field with gauge coupling constant $g$ .",
"Finally we conclude with some brief final remarks summarizing the most important results.",
"We add two appendices.",
"Appendix discusses the conformal invariance of fluid equations in the limit of massless particles, and Appendix clarifies some technical tools to calculate the Fourier transform of the noise kernel for scalar fields."
],
[
"Fluid dynamics from divergence-type theory",
"We assume inflation brings every non-conformally coupled matter field into its de Sitter invariant vacuum state, except the inflaton which is slowly-rolling down through its potential.",
"We also assume an instantaneous reheating, so the universe goes from inflation to radiation domination in essentially no time [38].",
"When inflation ends, quantum fluctuations of non-conformally coupled fields become much higher than those of the local vacuum state of adiabatic observers.",
"After inflation, these fluctuations enter in the nonlinear regime and decohere.",
"It therefore becomes adequate to treat them like an effective fluid.",
"In other words, the end of inflation sets the initial conditions for the later evolution of every field in a radiation dominated universe.",
"The proper theoretical framework for the discussion of the further evolution is given by causal relativistic hydrodynamics.",
"We shall follow a dissipative-type theory scheme as derived from kinetic theory for massless scalar particles obeying Bose-Einstein statistics [68].",
"To linearized order we may consider any other relevant approach, such as viscous anisotropic hydrodynamics [7] [69] [70] [71] [72] [73] [74] or theories based on the so-called `Entropy Production Variational Principle' [75] with equivalent results.",
"This approach consists in formulating an ansatz for the one-particle distribution function (1pdf), parametrized by the hydrodynamic variables.",
"Later on the hydrodynamic currents such as the particle number current and the energy-momentum tensor are derived as moments of the parameterized 1pdf, and the corresponding equations as moments of the Boltzmann equation.",
"We assume a perturbed Friedmann-Robertson-Walker Universe with metric $g_{\\mu \\nu }=a^2(\\eta )\\bar{g}_{\\mu \\nu }$ with $a(\\eta )$ the scale factor depending only on conformal time $\\eta $ , and $\\bar{g}_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu }$ , where $\\eta _{\\mu \\nu }$ is the Minkowsky metric (with signature $\\left(-,+,+,+\\right)$ ) and $h_{\\mu \\nu }$ represents the primordial gravitational waves.",
"Upon reheating the inflaton decays into radiation which is left in a state of thermal equilibrium, namely its four-velocity $U^{\\mu }_{rad}=a^{-1}U^{\\mu }$ follows the conformal Killing field of the Friedmann-Robertson-Walker background ($U^{\\mu }=\\left(1,0,0,0\\right)$ ), and its temperature $T_{rad}=a^{-1}\\mathbf {T}$ decays as the inverse radius of the Universe.",
"The spectator field, which is not decoupled from radiation, thermalizes into this state, a process which may be described by linear relaxation equations.",
"Moreover as $p^{\\mu }p_{\\mu }=m^2\\ll T_{rad}^2$ this theory is effectively conformally invariant.",
"This implies the energy-momentum and non-equilibrium tensor (Eq.",
"(REF )) are traceless.",
"Further the Boltzmann equation for massless particles also is conformally invariant and since the procedure of taking moments does not spoil this symmetry every conservation equation is conformally invariant as well.",
"See Appendix for details.",
"Through conformal invariance we are able to eliminate the scale factor $a$ from all equations.",
"As we are interested in the equilibration process of this scalar fluid to the dominant radiation, we analyze linear perturbations around a state thermalized to the dominant radiation equilibrium state.",
"In consequence we consider a linear deviation from a Bose-Einstein equilibrium distribution $f_0=1/\\left(\\exp {\\left(\\beta ^{\\mu }p_{\\mu }\\right)}-1\\right)$ where $\\beta ^{\\mu }=U^{\\mu }/\\mathbf {T}$ .",
"To introduce fluctuations we define the complete 1pdf as $f=\\frac{1}{\\exp {\\left(-\\displaystyle \\frac{u^{\\mu }p_{\\mu }}{T}-\\kappa \\,\\displaystyle \\frac{\\zeta ^{\\mu \\nu }}{T^2}\\,p_{\\mu }p_{\\nu }\\right)}-1},$ where $u^{\\mu }$ , $T$ and $\\zeta _{\\mu \\nu }$ are velocity, temperature and dimensionless non-equilibrium variable of the fluid respectively.",
"The constant in front of $\\zeta _{\\mu \\nu }$ is chosen so that later on we shall obtain $\\zeta ^{\\mu \\nu }=\\Pi ^{\\mu \\nu }/\\rho $ , where $\\Pi ^{\\mu \\nu }$ is the viscous part of the energy-momentum tensor and $\\rho $ the energy density, to linear order.",
"It has the value $\\kappa =\\pi ^4/\\left(2\\,5!\\,\\zeta (5)\\right)$ with $\\zeta (n)$ the Riemann function.",
"For the collision integral we take an Anderson-Witting linear ansatz [76] [77] [78] $I_{col}=\\frac{u_{\\mu }p^{\\mu }}{\\tau }\\left(f-f_0\\right),$ where $\\tau $ is the relaxation time of the fluid.",
"This is an external parameter of the theory, which must be derived from consideration of the fluid particles interactions between themselves and with radiation.",
"We shall discuss this parameter in Section .",
"The idea is to decompose all fields into an (homogeneous) average and a fluctuation, and obtain linearized equations for the fluctuations.",
"From the cosmological principle we assume the background quantities have the FRW symmetry, in particular $\\zeta ^{\\mu \\nu }$ vanishes in the background.",
"Since our purpose is to analyze interactions between the fluid and the gravitons we consider only tensor perturbations.",
"The linearized 1pdf reads $f\\simeq f_0\\left[1+(1+f_0)\\,\\kappa \\,\\frac{\\zeta _{\\mu \\nu }}{\\mathbf {T}^2}\\, p^{\\mu }p^{\\nu }\\right].$ We choose a gauge where $h_{\\mu \\nu }U^{\\nu }=0$ , due to the tensor character of perturbations also ${h^{\\mu }}_{\\mu }=0$ .",
"Since $\\zeta ^{\\mu \\nu }$ is transverse to the four-velocity to linear order we find $U_{\\mu }\\zeta ^{\\mu \\nu }=\\zeta ^{\\mu }_{\\mu }=0$ ."
],
[
"Hydrodynamic equations",
"To deduce the hydrodynamic equations we define the comoving energy-momentum tensor and non-equilibrium tensor as usual [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [79] [68].",
"The fluid energy-momentum tensor reads $\\overline{T}^{\\mu \\nu }=\\int \\bar{Dp}\\,p^{\\mu }p^{\\nu }f,$ and the non-equilibrium current $\\overline{A}^{\\mu \\nu \\lambda }=\\int \\bar{Dp}\\,p^{\\mu }p^{\\nu }p^{\\lambda }f.$ We also need the second moment of the collision integral $\\overline{I}^{\\mu \\nu }=\\int \\bar{Dp}\\,p^{\\mu }p^{\\nu }\\overline{I}_{col}.$ In Eqs.",
"(REF )-(REF ) the invariant relativistic measure is $\\bar{Dp}=\\frac{2\\prod _{\\mu =0}^{4}dp_{\\mu }\\,\\delta (p^2)}{(2\\pi )^3\\sqrt{-\\bar{g}}}\\Theta (p^0).$ The equations are the conservation equation for energy-momentum tensor ${\\overline{T}^{\\mu \\nu }}_{;\\mu }=0$ and the closure equation for non-equilibrium current $\\begin{split}&\\left({S^{\\alpha }}_{\\mu }{S^{\\beta }}_{\\nu }-\\frac{1}{3}S^{\\alpha \\beta }S_{\\mu \\nu }\\right){\\overline{A}^{\\mu \\nu \\lambda }}_{;\\lambda }=\\\\&=\\left({S^{\\alpha }}_{\\mu }{S^{\\beta }}_{\\nu }-\\frac{1}{3}S^{\\alpha \\beta }S_{\\mu \\nu }\\right)\\overline{I}^{\\mu \\nu }\\end{split}$ where $S^{\\mu }_{\\nu }=\\delta ^{\\mu }_{\\nu }+U^{\\mu }U_{\\nu }$ .",
"The relevant integrals were computed in [79].",
"Here we summarize the final expressions ${{\\overline{T}}^{\\mu }}_{\\nu }=\\frac{\\pi ^2}{30}\\mathbf {T}^4\\left(U^{\\mu }U_{\\nu }+\\frac{1}{3}{S^{\\mu }}_{\\nu }+{\\zeta ^{\\mu }}_{\\nu }\\right),$ $\\overline{I}^{\\mu \\nu }=-\\frac{3\\pi ^2}{15}\\,\\frac{\\zeta (6)}{\\zeta (5)}\\,\\frac{1}{\\tau }\\,\\mathbf {T}^5\\zeta ^{\\mu \\nu }$ and $\\begin{split}{{\\overline{A}}^{\\mu \\nu \\lambda }}&=\\frac{12\\zeta (5)}{\\pi ^2}\\mathbf {T}^5\\bigg [U^{\\mu }U^{\\nu }U^{\\lambda }+\\\\&+\\frac{1}{3}\\left(S^{\\mu \\nu }U^{\\lambda }+S^{\\mu \\lambda }U^{\\nu }+S^{\\lambda \\nu }U^{\\mu }\\right)\\bigg ]-\\\\&-\\frac{4\\,\\zeta (5)}{\\pi ^2}\\mathbf {T}^5\\left(U^{\\mu }h^{\\nu \\lambda }+U^{\\nu }h^{\\mu \\lambda }+U^{\\lambda }h^{\\mu \\nu }\\right)+\\\\&+\\frac{3\\pi ^2}{15}\\,\\frac{\\zeta (6)}{\\zeta (5)}\\,\\mathbf {T}^5\\left(\\zeta ^{\\mu \\nu }U^{\\lambda }+\\zeta ^{\\lambda \\nu }U^{\\mu }+\\zeta ^{\\mu \\lambda }U^{\\nu }\\right).\\end{split}$ In order to derive the linearized equations in the following section, we consider a purely spin-2 perturbation ($\\textrm {TT}$ ) of the energy-momentum tensor (REF ) in mixed components and closure equation (REF ) to first order.",
"These expressions are ${{{{\\overline{T}}^{(1)}}^{\\mu }}_{\\nu }}^{\\textrm {TT}}=\\frac{\\pi ^2}{30}\\mathbf {T}^4\\,{\\zeta ^{\\mu }}_{\\nu },$ $b\\,{h^{\\alpha \\beta }}_{,0}+{\\zeta ^{\\alpha \\beta }}_{,0}+\\frac{1}{\\tau }\\zeta ^{\\alpha \\beta }=0.$ respectively, with $b=20\\,\\zeta ^2(5)/\\left(\\pi ^4\\,\\zeta (6)\\right)$ .",
"If we had used a Maxwell-Juttner equilibrium distribution, we would have derived the same equation but with $b=2/9$ .",
"Note the ratio of both $b_{MJ}/b_{BE}\\simeq 1.024$ .",
"In order to relate $\\tau $ with the usual transport coefficients we compute the energy-momentum tensor up to first order in $\\tau $ .",
"For this purpose we may discard the interaction with gravitons taking $h_{\\mu \\nu }=0$ .",
"However it is need to introduce perturbations in temperature $\\delta T$ and velocity $v^{\\mu }$ , in addition to the tensor one $\\zeta ^{\\mu \\nu }$ .",
"Then the energy-momentum tensor reads $\\begin{split}{{{{\\overline{T}}^{(1)}}^{\\mu }}_{\\nu }}=\\frac{\\pi ^2}{30}\\mathbf {T}^4\\,\\bigg [&4\\frac{\\delta T}{\\mathbf {T}}\\left(U^{\\mu }U_{\\nu }+\\frac{1}{3}{S^{\\mu }}_{\\nu }\\right)+\\\\&+\\,\\frac{4}{3}\\left(U^{\\mu }v_{\\nu }+U_{\\nu }v^{\\mu }\\right)+{\\zeta ^{\\mu }}_{\\nu }\\bigg ].\\end{split}$ Including the velocity perturbation Eq.",
"(REF ) becomes ${\\zeta ^{\\alpha \\beta }}_{,0}+\\frac{1}{\\tau }\\zeta ^{\\alpha \\beta }+b\\,\\sigma ^{\\alpha \\beta }=0$ which implies, to first order in $\\tau $ , $\\zeta ^{\\alpha \\beta }=-\\tau \\,b\\,\\sigma ^{\\alpha \\beta }$ .",
"In consequence by simple comparison with the usual viscous energy-momentum tensor, the well-known kinematic viscosity coefficient $\\nu =b\\,\\tau $ ."
],
[
"Fluid-gravitons coupled equations",
"From now on we normalize $H\\eta \\rightarrow \\eta $ , $Hr\\rightarrow r$ , where $H$ is the Hubble constant at the moment of reheating; we also define $\\eta =0$ there and $a\\left(0\\right)=1$ .",
"From the linearized Einstein's equation in mixed components we get ${{G^{(1)}}^{\\mu }}_{\\nu }=\\frac{1}{a^2(\\eta )M_{pl}^2}\\,{{{\\overline{T}}^{(1)}}^{\\mu }}_{\\nu },$ with $M_{pl}$ the reduced Planck mass.",
"We apply tensor projectors to Eq.",
"(REF ) in spatial indexes.",
"It reads ${{{G^{(1)}}^{i}}_{j}}^{\\textrm {TT}}=\\frac{H^2}{2}\\left[-\\eta ^{\\rho \\sigma }\\partial _{\\rho }\\partial _{\\sigma }+2\\frac{a^{\\prime }(\\eta )}{a(\\eta )}\\partial _{\\eta }\\right]h_{ij},$ and for ${{{\\overline{T}}^{(1)i}}_{j}}^{\\textrm {TT}}$ we use Eq.",
"(REF ).",
"Since $h_{ij}$ and $\\zeta _{ij}$ are tensor degrees of freedom we write the following Fourier decomposition for both $h_{ij}(\\mathbf {r},\\eta )=\\sum _{\\lambda =+,\\times }\\int \\frac{d^3k}{(2\\pi )^{3/2}}\\,\\epsilon _{ij}^{\\lambda }(\\hat{k})\\,h^{\\lambda }_{k}(\\eta )\\,e^{i\\mathbf {k}\\mathbf {r}},$ where the conformal wave number is $\\bar{k}_{phys}=Hk$ , $\\lambda =+,\\times $ indicates polarization and the polarization tensors $\\epsilon _{ij}^{\\lambda }(\\hat{k})$ satisfy $\\epsilon _{ij}^{\\lambda }(\\hat{k})\\,\\delta ^{ij}=k^i\\,\\epsilon _{ij}^{\\lambda }(\\hat{k})=0$ and $\\epsilon _{ij}^{\\lambda }(\\hat{k})\\,\\epsilon ^{ij}_{\\lambda ^{\\prime }}(\\hat{k})=\\delta _{\\lambda \\lambda ^{\\prime }}$ .",
"Gathering the expressions above we derive similar equations for either polarization.",
"Dropping the $\\lambda $ index in $h_k$ and $\\zeta _k$ , together with Eq.",
"(REF ), we get the system of equations to linear order for $h_k$ and $\\zeta _k$ ${\\left\\lbrace \\begin{array}{ll}\\left[\\partial ^2_{\\eta }+k^2+2\\displaystyle \\frac{a^{\\prime }(\\eta )}{a(\\eta )}\\partial _{\\eta }\\right]h_k(\\eta )=\\displaystyle \\frac{1}{a^2(\\eta )}\\,K_0\\zeta _k(\\eta ) \\\\\\partial _{\\eta }\\zeta _k(\\eta )+\\displaystyle \\frac{1}{\\tau _0}\\zeta _k(\\eta )=-b\\,\\partial _{\\eta }h_k(\\eta ),\\end{array}\\right.",
"}$ where $K_0=\\pi ^2\\mathbf {T}^4/\\left(15H^2M_{pl}^2\\right)$ and $\\tau _0=H\\tau $ .",
"In the radiation dominated era $a\\left(\\eta \\right)=1+\\eta $ and $H(\\eta )=\\left(1+\\eta \\right)^{-2}$ .",
"We change variables $\\eta \\rightarrow z(\\eta )=k(1+\\eta )$ and $h_k(z)=\\chi _k(z)/z$ , therefore ${\\left\\lbrace \\begin{array}{ll}\\partial _z^2\\chi _k(z)+\\chi _k(z)=\\displaystyle \\frac{K_0\\zeta _k(z)}{z}\\\\\\partial _z\\zeta _k(z)+\\displaystyle \\frac{\\zeta _k(z)}{k\\tau _0}=-b\\,\\partial _z\\left(\\displaystyle \\frac{\\chi _k(z)}{z}\\right).\\end{array}\\right.",
"}$ To solve our problem we need the solution of (REF ) with the appropriate initial conditions for $h_k$ and $\\zeta _k$ , to be discussed in next section.",
"The magnitude of the parameter $K_0$ measures the interaction strength between the tensor degrees of freedom $\\zeta _k$ and $h_k$ .",
"Using instantaneous and effective reheating $H^2\\simeq g_*\\frac{\\pi ^2}{30}\\mathbf {T}^4/3M_{pl}^2$ where $g_*$ is the number of relativistic degrees of freedom at temperature $\\mathbf {T}$ .",
"Since $O(10^2\\textrm { GeV})\\ll \\mathbf {T}\\le M_{pl}$ , then $g_*\\gtrsim 10^2$ and $K_0\\simeq 6\\,\\frac{\\rho _S}{\\rho _{\\gamma }}=\\frac{6}{g_*}\\lesssim 10^{-2}.$"
],
[
"Initial conditions",
"The purpose of this section is to compute the initial conditions for $h_k$ and $\\zeta _k$ at the beginning of the radiation dominated era.",
"To do this we regard them as classical stochastic Gaussian variables with zero mean, whose self correlation matches the Hadamard propagator of the corresponding quantum operators in the Bunch-Davies vacuum at the end of inflation."
],
[
"Gravitons $h$",
"Gravitons are tensor metric perturbations.",
"As we have seen before there are two polarizations $h^+$ and $h^{\\times }$ .",
"As it is well known [80], the amplitude for both can be treated as massless real scalar fields.",
"As usual, to quantize them we use decomposition (REF ) and apply canonical quantization to the auxiliary field $\\chi $ defined by $h_k(\\eta )=\\chi _k(\\eta )/a(\\eta )$ .",
"Explicitly $\\centering \\begin{split}&h_{ij}(\\mathbf {r},\\eta )=\\frac{\\chi _{ij}(\\mathbf {r},\\eta )}{a(\\eta )}=\\\\&=\\sum _{\\lambda =+,\\times }\\int \\frac{d^3k}{(2\\pi )^{3/2}}\\frac{\\epsilon _{ij}^{\\lambda }(\\hat{k})}{a(\\eta )}\\left[\\chi _k^{\\lambda }(\\eta )\\hat{a}_{\\mathbf {k}}+\\chi _k^{\\lambda \\,*}(\\eta )\\hat{a}^{\\dagger }_{\\mathbf {-k}}\\right]e^{i\\mathbf {k}\\mathbf {r}}.\\end{split}$ This field $\\chi $ must be dimensionless as well as $h$ .",
"As before we obtain the same equation for both polarizations of $\\chi _k$ .",
"During inflation $\\chi _k^{\\prime \\prime }+\\left[k^2-\\frac{a^{\\prime \\prime }}{a}\\right]\\chi _k=0.$ During inflation $\\eta \\le 0$ and $a(\\eta )=1/(1-\\eta )$ .",
"We adopt the Bunch-Davies positive frequency solution [81] of (REF ) $\\chi _k^I(\\eta )=\\frac{H}{M_{pl}}\\frac{e^{-ik\\eta }}{\\sqrt{2k}}\\left[1+i\\frac{1}{k(1-\\eta )}\\right].$ Under the scheme of instantaneous reheating our initial conditions for the evolution of Fourier components $h_k$ during radiation dominated Universe $\\eta \\ge 0$ are $h_{\\mathbf {k}}(\\eta =0)=i\\,\\frac{H}{M_{pl}}\\frac{1}{\\sqrt{2k^3}}\\,e_{\\mathbf {k}}+\\frac{H}{M_{pl}}\\frac{1}{\\sqrt{2k}}\\,b_{\\mathbf {k}}$ and $h^{\\prime }_{\\mathbf {k}}(\\eta =0)=-i\\frac{H}{M_{pl}}\\sqrt{\\frac{k}{2}}\\,e_{\\mathbf {k}},$ where $e_{\\mathbf {k}}=\\hat{a}_{\\mathbf {k}}-\\hat{a}^{\\dagger }_{-\\mathbf {k}}$ and $b_{\\mathbf {k}}=\\hat{a}_{\\mathbf {k}}+\\hat{a}^{\\dagger }_{-\\mathbf {k}}$ .",
"Next we assume the Landau prescription $\\langle AB\\rangle _{S}=1/2\\,\\langle 0\\left|\\left\\lbrace A;B\\right\\rbrace \\right|0\\rangle $ to convert quantum expectation values into stochastic ensemble averages [82] [83].",
"In consequence $\\begin{split}&\\langle e_{\\mathbf {k}}e_{\\mathbf {k}^{\\prime }}^*\\rangle _S=\\\\&=\\frac{1}{2} \\langle 0| \\left\\lbrace \\left(a_{\\mathbf {k}}-a_{-\\mathbf {k}}^{\\dagger }\\right);\\left(a_{\\mathbf {k}^{\\prime }}^{\\dagger }-a_{-\\mathbf {k}^{\\prime }}\\right)\\right\\rbrace |0\\rangle _Q=\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime }),\\end{split}$ $\\begin{split}&\\langle b_{\\mathbf {k}}b_{\\mathbf {k}^{\\prime }}^*\\rangle _S=\\\\&=\\frac{1}{2} \\langle 0| \\left\\lbrace \\left(a_{\\mathbf {k}}+a_{-\\mathbf {k}}^{\\dagger }\\right);\\left(a_{\\mathbf {k}^{\\prime }}^{\\dagger }+a_{-\\mathbf {k}^{\\prime }}\\right)\\right\\rbrace |0\\rangle _Q=\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime }),\\end{split}$ $\\begin{split}&\\langle e_{\\mathbf {k}}b_{\\mathbf {k}^{\\prime }}^*\\rangle _S=\\\\&=\\frac{1}{2} \\langle 0| \\left\\lbrace \\left(a_{\\mathbf {k}}-a_{-\\mathbf {k}}^{\\dagger }\\right);\\left(a_{\\mathbf {k}^{\\prime }}^{\\dagger }+a_{-\\mathbf {k}^{\\prime }}\\right)\\right\\rbrace |0\\rangle _Q=0.\\end{split}$ For instance, initial correlation for modes outside the horizon ($k\\ll 1$ ) at $\\eta =0$ develop a scale invariant spectrum, namely $\\langle h_k(\\eta )h^*_{k^{\\prime }}(\\eta ^{\\prime })\\rangle =\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\frac{H^2}{2M_{pl}^2k^3}.$ This case is more complicated because there is no immediate relation between the stochastic non-equilibrium variable $\\zeta $ and some canonical quantum field during inflation.",
"Instead, we write the tensor part of the energy-momentum tensor self correlation for a minimally coupled scalar field during inflation, namely the so-called noise kernel ${{{N^{\\mu }}_{\\nu }}^{\\rho }}_{\\sigma }$ .",
"Then we match it at $\\eta =0$ to the stochastic self correlation function of $\\zeta $ calculated during the radiation dominated era.",
"The noise kernel is defined as ${{{N^{\\mu }}_{\\nu }}^{\\rho }}_{\\sigma }=\\frac{1}{2}\\left[\\langle \\left\\lbrace {T^{\\mu }}_{\\nu }(x),{T^{\\rho }}_{\\sigma }(y)\\right\\rbrace \\rangle -2\\langle {T^{\\mu }}_{\\nu }(x)\\rangle \\langle {T^{\\rho }}_{\\sigma }(y)\\rangle \\right].$ Since we will take the tensor part of the noise kernel, the only possible contribution comes from the kinetic term of the energy-momentum tensor [84].",
"${{{N^{\\mu }}_{\\nu }}^{\\rho }}_{\\sigma }$ was computed in [85].",
"For the massless ($m/H\\ll 1$ ) and large scales ($r\\gg 1$ ) limit at the end of inflation ($\\eta =0$ ), which is our case of interest, [85] obtains the following result for the kinetic term contribution $\\begin{split}&N^{ijkl}(r,\\eta =0)\\simeq \\\\&\\simeq \\frac{H^8}{16\\pi ^4\\,r^4}\\left[\\delta ^{il}\\delta ^{jk}-2\\left(\\delta ^{il}\\hat{r}^j\\hat{r}^k+\\delta ^{jk}\\hat{r}^i\\hat{r}^l\\right)+4\\,\\hat{r}^i\\hat{r}^j\\hat{r}^k\\hat{r}^l\\right]+\\\\&+(k\\leftrightarrow l).\\end{split}$ We disregard a term which becomes constant at large separations, since it does not contribute to the tensor part.",
"In Fourier space we define the projector ${{{\\Lambda ^a}_i}^b}_j$ into tensor part (divergenceless and traceless) like ${{{\\Lambda ^a}_i}^b}_j={M^a}_i{M^b}_j-\\frac{1}{2}M^{ab}M_{ij},$ with ${M^a}_i={\\delta ^a}_i-\\frac{k^ak_i}{k^2}.$ Recalling that $r=\\left|\\mathbf {x}-\\mathbf {x}^{\\prime }\\right|$ , when Fourier transforming we get two different momenta for each spatial point $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ .",
"Due to homogeneity and isotropy, the tensor part of the Fourier transformed noise kernel $N_T^{abcd}$ results $\\begin{split}&{N_{T\\,}}^{abcd}(\\mathbf {k},\\mathbf {k}^{\\prime })=\\\\&={{{\\Lambda ^a}_i}^b}_j{{{\\Lambda ^c}_k}^d}_l\\,\\,\\langle \\frac{1}{2}\\left\\lbrace {\\left.T_{\\mathbf {k}}\\right.^i}_{j}-\\langle {\\left.T_{\\mathbf {k}}\\right.^i}_{j}\\rangle ;{\\left.T^*_{\\mathbf {k}^{\\prime }}\\right.^k}_{l}-\\langle {\\left.T^*_{\\mathbf {k}^{\\prime }}\\right.^k}_{l}\\rangle \\right\\rbrace \\rangle =\\\\&={{{\\Lambda ^a}_i}^b}_j{{{\\Lambda ^c}_k}^d}_l\\,\\,{{{\\left.N\\right.^i}_{j}}^{k}}_{l}(\\mathbf {k},\\mathbf {k}^{\\prime })=\\\\&=\\delta \\left(\\mathbf {k}-\\mathbf {k}^{\\prime }\\right)\\,F(k)\\,\\left[\\Lambda ^{adbc}+\\Lambda ^{acbd}\\right],\\end{split}$ with $F(k)=c\\,H^8\\,k+O(k^2),$ and $c=6911/(12\\,\\pi ^2)$ (see Appendix ).",
"This result provides us the quantum fluctuations from inflation.",
"In order to match it with our fluid non-equilibrium correlation we must to subtract the local vacuum fluctuations.",
"It is possible to show that the pathological behaviour of (REF ) at short distance is caused entirely by the mentioned local vacuum fluctuations.",
"In fact if we calculate the noise kernel using the local fourth order adiabatic vacua at time $\\eta =0$ we obtain the same terms as in (REF ).",
"However computations also show that these vacuum fluctuations are only valid for small scales ($k>1$ ).",
"In consequence, after the subtraction of the local vacuum, the quantum noise kernel for large scales ($k\\ll 1$ ) is (REF ).",
"On the other hand, we analyze the stochastic fluctuations of the fluid energy-momentum tensor in momentum space.",
"We know that the energy-momentum tensor satisfies ${T^{\\mu }}_{\\nu }(\\eta =0)=a^{-2}(\\eta =0){{{\\overline{T}}^{\\,\\mu }}}_{\\nu }={{{\\overline{T}}^{\\,\\mu }}}_{\\nu }$ .",
"From (REF ) and using decomposition (REF ), we arrive ${{{{{\\overline{T}}^{\\,(1)}_k}}^i}_j}^{\\textrm {TT}}=\\frac{\\pi ^2}{30}\\mathbf {T}^4\\sum _{\\lambda =+,\\times }\\epsilon ^{\\lambda }_{ij}(\\hat{k})\\,\\zeta _k^{\\lambda }(\\eta ).$ Setting $\\mathbf {k}=k\\hat{z}$ and $\\zeta ^{\\lambda }_k(\\eta =0)=\\zeta ^{\\lambda }_k$ , the most general choice is ${{{\\overline{T}^{(1)}_k}^i}_j}^{\\textrm {TT}}=\\frac{\\pi ^2}{30}\\mathbf {T}^4\\left(\\begin{array}{ccc}\\zeta ^+_k & \\zeta ^{\\times }_k & 0 \\\\\\zeta ^{\\times }_k & -\\zeta ^+_k & 0 \\\\0 & 0 & 0\\end{array}\\right)^{ij}.$ The projected correlation at time zero is $\\begin{split}&{{{\\Lambda ^a}_i}^b}_j{{{\\Lambda ^c}_k}^d}_l\\,\\langle {{T^{(1)}_k}^i}_j{{T^{(1)*}_{k^{\\prime }}}^k}_l\\rangle ={{{\\Lambda ^a}_i}^b}_j{{{\\Lambda ^c}_k}^d}_l\\,\\langle {{{\\overline{T}}^{(1)}_k}^i}_j{{{\\overline{T}}^{(1)*}_{k^{\\prime }}}^k}_l\\rangle =\\\\&=\\frac{\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\mathbf {T}^8\\pi ^4}{30^2}\\langle \\left(\\begin{array}{ccc}\\zeta ^{+}_k & \\zeta ^{\\times }_k & 0 \\\\\\zeta ^{\\times }_k & -\\zeta ^{+}_k & 0 \\\\0 & 0 & 0\\end{array}\\right)^{ab}\\left(\\begin{array}{ccc}\\zeta ^{+*}_{k^{\\prime }} & \\zeta ^{\\times *}_{k^{\\prime }} & 0 \\\\\\zeta ^{\\times *}_{k^{\\prime }} & -\\zeta ^{+*}_{k^{\\prime }} & 0 \\\\0 & 0 & 0\\end{array}\\right)^{cd}\\rangle \\end{split}$ Terms like $\\langle {T^{i}}_{j}\\rangle \\langle {T^{k}}_{l}\\rangle $ are zero to first order.",
"Just like in the quantum case a $\\delta $ -function appears due to homogeneity.",
"We match stochastic and quantum tensor correlation comparing Eqs.",
"(REF ) and (REF ) in the frame where $\\mathbf {k}=k\\hat{z}$ and the initial time $\\eta =0$ .",
"It results $\\begin{split}\\langle \\zeta ^{\\times }_{\\mathbf {k}}\\zeta ^{\\times *}_{\\mathbf {k}^{\\prime }}\\rangle =\\langle \\zeta ^{+}_{\\mathbf {k}}\\zeta ^{+*}_{\\mathbf {k}^{\\prime }}\\rangle =\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\left[\\,d\\,\\left(\\frac{H}{\\mathbf {T}}\\right)^8\\,k+O(k^2)\\right]\\end{split}$ and $\\langle \\zeta ^{\\times }_{\\mathbf {k}}\\zeta ^{+*}_{\\mathbf {k}^{\\prime }}\\rangle =0,$ with $d=(30/\\pi ^2)^2\\,c$ (crf.",
"Eq.",
"(REF )).",
"As we see both polarizations follow identical equations decoupled from each other.",
"Henceforth we shall drop the polarization label."
],
[
"Tensor mode evolution",
"To study the solutions of the system (REF ) we make a distinction between sub-horizon ($k/a\\left(\\eta \\right)>H\\left(\\eta \\right)$ ) and super-horizon ($k/a\\left(\\eta \\right)<H\\left(\\eta \\right)$ ) modes.",
"Recalling $z=k\\left(1+\\eta \\right)$ the former involve $z>1$ and the latter $z<1$ .",
"Since we only concentrate in super-horizon modes, our analysis would be valid until modes re-enter in the horizon at $z=1$ .",
"Further we consider our model to be valid up to the electroweak transition, where new effects must be considered due to the change in the number of relativistic degrees of freedom.",
"In consequence we will analyze solutions in the limit $k\\rightarrow 0$ and $\\eta $ bounded by the condition $z=k(1+\\eta )<1$ or by the electroweak time, whatever happens first.",
"We only keep the dominant terms in the power series expansion for $k\\ll 1$ valid for super-horizon modes until the electroweak transition.",
"We interpret $K_0$ (Eq.",
"(REF )) as an interaction parameter between gravitons and tensor fluid modes.",
"If $K_0=0$ gravitons decouple from the fluid.",
"We determine its evolution by solving the first equation of (REF ) with the initial conditions (REF )-(REF ).",
"The dominant terms in the limit $k\\ll 1$ are $h_{\\mathbf {k}}(\\eta )=i\\,\\frac{H}{M_{pl}}\\frac{1}{\\sqrt{2k^3}}\\,e_{\\mathbf {k}}+\\frac{H}{M_{pl}}\\frac{1}{\\sqrt{2k}}\\,b_{\\mathbf {k}}+O\\left(\\sqrt{k}\\,\\right).$ So $\\langle h_{\\mathbf {k}}(\\eta )h_{\\mathbf {k}^{\\prime }}^*(\\eta )\\rangle =\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\left[\\frac{H^2}{2M_{pl}^2k^3}+\\frac{H^2}{2M_{pl}^2k}+\\dots \\right]$ Neglecting the second term in (REF ) we obtain the so-called scale invariant spectrum, $\\langle h_{\\mathbf {k}}(\\eta )h_{\\mathbf {k}^{\\prime }}^*(\\eta )\\rangle \\propto \\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })/k^3$ .",
"In the general case with $K_0\\ne 0$ it is enough to consider the two limiting cases of (REF ), namely $k\\tau _0\\ll 1$ and $k\\tau _0\\gg 1$ .",
"Hereafter we assume $1/\\tau \\ll H$ ; we shall discuss in the Section whether this is a realistic hypothesis.",
"We solve the system (REF ) with initial conditions (REF )-(REF ) for gravitons and (REF )-(REF ) for tensor fluid modes.",
"When $k\\tau _0\\ll 1$ ($k\\ll 1/\\tau \\ll H$ in unnormalized units) the fluid modes decay before they can interact meaningfully with gravitons.",
"For these modes with very large wavelengths we recover to leading order the usual scale invariant spectrum, namely the first term in Eq.",
"(REF ).",
"The most interesting case is when $k\\tau _0\\gg 1$ .",
"It means $1/\\tau \\ll k\\ll H$ and enables us to neglect the term $\\zeta _k/(k\\tau _0)$ in equations (REF ).",
"The system takes the form ${\\left\\lbrace \\begin{array}{ll}\\partial _z^2\\chi _k(z)+\\chi _k(z)=\\displaystyle \\frac{K_0\\zeta _k(z)}{z}\\\\\\partial _z\\zeta _k(z)=-b\\,\\partial _z\\left(\\displaystyle \\frac{\\chi _k(z)}{z}\\right).\\end{array}\\right.",
"}$ Then, $\\zeta _k(z)=-b\\,h_k(z)+C_{\\mathbf {k}},$ $C_{\\mathbf {k}}$ will be set by matching the quantum noise kernel spectrum to the correlation $\\langle \\zeta _k(\\eta )\\zeta ^*_{k^{\\prime }}(\\eta )\\rangle $ at initial time $\\eta =0$ .",
"We assume null cross correlation $\\langle \\zeta _kh_{k^{\\prime }}^*\\rangle =0$ , because both variables have different physical origin.",
"In consequence $\\langle C_{\\mathbf {k}} C^*_{\\mathbf {k}^{\\prime }}\\rangle =\\left.\\langle \\zeta _{\\mathbf {k}}\\zeta _{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}+b^2\\left.\\langle h_{\\mathbf {k}}h_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}.$ Using $\\langle \\zeta _kh_{k^{\\prime }}^*\\rangle =0$ explicitly, we get $\\left.\\langle C_{\\mathbf {k}}h_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}=\\left.\\langle h_{\\mathbf {k}}C_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}=b\\left.\\langle h_{\\mathbf {k}}h_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0},$ so, considering initial conditions (REF )-(REF ) and (REF )-(REF ) we derive $\\begin{split}&\\langle C_{\\mathbf {k}} C^*_{\\mathbf {k}^{\\prime }}\\rangle =\\\\&=\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\left[\\,d\\,\\left(\\frac{H}{\\mathbf {T}}\\right)^8\\,k+b^2\\frac{H^2}{2M_{pl}^2k^3}+b^2\\frac{H^2}{2M_{pl}^2k}\\right],\\end{split}$ and $\\begin{split}&\\left.\\langle C_{\\mathbf {k}}h_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}=\\left.\\langle h_{\\mathbf {k}}C_{\\mathbf {k}^{\\prime }}^*\\rangle \\right|_{\\eta =0}=\\\\&=\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\left[b\\,\\frac{H^2}{2M_{pl}^2k^3}+b\\,\\frac{H^2}{2M_{pl}^2k}\\right].\\end{split}$ The equation for $\\chi _k(z)$ reads $\\partial _z^2\\chi _k(z)+\\chi _k(z)+K_0\\,b\\,\\frac{\\chi _k(z)}{z^2}=K_0\\frac{C_{\\mathbf {k}}}{z}.$ Let $\\chi _k=\\sqrt{z}\\,\\psi _k$ and so $h_k=\\psi _k/\\sqrt{z}$ , therefore $\\begin{split}&z^2\\psi ^{\\prime \\prime }_k(z)+z\\psi ^{\\prime }_k(z)+\\left[z^2-\\left(\\frac{1}{4}-b\\,K_0\\right)\\right]\\psi _k(z)=\\\\&=K_0C_{\\mathbf {k}}\\sqrt{z},\\end{split}$ whose solution is $\\begin{split}\\psi _k(z)&=\\overline{C}_{1\\mathbf {k}}J_{\\nu }(z)+\\overline{C}_{2\\mathbf {k}}Y_{\\nu }(z)+\\\\&+\\frac{\\pi }{2}Y_{\\nu }(z)\\int _{z_0}^{z}\\frac{J_{\\nu }(z^{\\prime })}{z^{\\prime }}K_0C\\sqrt{z^{\\prime }}\\,dz^{\\prime }-\\\\&-\\frac{\\pi }{2}J_{\\nu }(z)\\int _{z_0}^{z}\\frac{Y_{\\nu }(z^{\\prime })}{z^{\\prime }}K_0C_{\\mathbf {k}}\\sqrt{z^{\\prime }}\\,dz^{\\prime },\\end{split}$ where $\\nu ^2=1/4-b\\,K_0$ , and $J_{\\nu }(z)$ ($j_{\\nu }(z)$ ) and $Y_{\\nu }(z)$ ($y_{\\nu }(z)$ ) are (spherical) Bessel's functions of first and second kind respectively.",
"The expression for $h_k(z)$ is $\\begin{split}h_k(z)=&\\,C_{1\\mathbf {k}}\\,j_{\\nu -1/2}(z)+C_{2\\mathbf {k}}\\,y_{\\nu -1/2}(z)+ \\\\&+\\frac{\\pi }{2}\\,K_0\\,C_{\\mathbf {k}}\\left[y_{\\nu -1/2}(z)\\int _{z_0}^{z}j_{\\nu -1/2}(z^{\\prime })\\,dz^{\\prime }\\right.-\\\\&-\\left.j_{\\nu -1/2}(z)\\int _{z_0}^{z}y_{\\nu -1/2}(z^{\\prime })\\,dz^{\\prime }\\right],\\end{split}$ Our solution in the limit $k\\ll 1$ is $h_{\\mathbf {k}}(\\eta )=\\,h_{\\mathbf {k}}(0)\\,G_1(\\eta ,\\nu )+\\frac{\\pi }{2}\\,K_0\\,C_{\\mathbf {k}}\\,G_2(\\eta ,\\nu ),$ where $\\begin{split}&G_1(\\eta ,\\nu )=\\\\&=\\frac{\\left(1+\\eta \\right)^{-\\nu -1/2}\\left(-1+2\\nu +\\left(1+\\eta \\right)^{2\\nu }\\left(1+2\\nu \\right)\\right)}{4\\nu },\\end{split}$ $\\begin{split}&G_2(\\eta ,\\nu )=\\\\&=\\frac{\\left(1+\\eta \\right)^{-\\nu -1/2}}{\\nu \\left(4\\nu ^2-1\\right)}\\bigg [-1+2\\nu -4\\nu (1+\\eta )^{\\nu +1/2}+\\\\&+(1+\\eta )^{2\\nu }(1+2\\nu )\\bigg ].\\end{split}$ Thus, equal time self correlation for gravitons reads $\\begin{split}&\\langle h_{\\mathbf {k}}(\\eta )h^*_{\\mathbf {k}^{\\prime }}(\\eta )\\rangle =\\\\&=\\langle h_{\\mathbf {k}}(0)h^*_{\\mathbf {k}^{\\prime }}(0)\\rangle \\left[G_1(\\eta ,\\nu )+b\\frac{\\pi }{2}K_0G_2(\\eta ,\\nu )\\right]^2+\\\\&+\\langle \\zeta _{\\mathbf {k}}(0)\\zeta _{\\mathbf {k}^{\\prime }}^*(0)\\rangle \\left(\\frac{\\pi }{2}K_0\\right)^2{G_2}^2(\\eta ,\\nu ).\\end{split}$ Let us make an ascending series expansion in $K_0$ around zero, recalling $\\nu =\\sqrt{1/4-b\\,K_0}$ , and replace the initial correlations.",
"In that case we obtain to leading order in $k$ and $K_0$ $\\begin{split}&\\langle h_{\\mathbf {k}}(\\eta )h^*_{\\mathbf {k}^{\\prime }}(\\eta )\\rangle =\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\frac{H^2}{2M_{pl}^2k^3}\\times \\\\&\\times \\left[1+b\\,K_0\\left(\\frac{\\pi }{2}-1\\right)\\left(\\log {(1+\\eta )}-\\frac{\\eta }{1+\\eta }\\right)\\right]^2.\\end{split}$ Our description of the spectrum evolution holds up to a certain time $\\eta _{k,max}$ , depending on $k$ , at which either the modes re-enter in the horizon or the electroweak transition takes place.",
"To estimate the electroweak time $\\eta _{EW}$ we use the ratio of the scale factor between the end of inflation and the electroweak transition, which is $a_{EW}/a_{EOI}=T_{EOI}/T_{EW}$ .",
"The typical energy of electroweak transition is $T_{EW}\\simeq 10^2 \\textrm { GeV}$ and $T_{EOI}=T_{\\gamma }=\\mathbf {T}=10^n\\textrm { GeV}$ .",
"Therefore $a_{EW}=1+\\eta _{EW}=10^{n-2}$ and $\\eta _{EW}\\simeq 10^{n-2}$ .",
"On the other hand we may find the conformal time at the re-entry in the horizon $\\eta _{k,re-entry}$ , which depends explicitly on $k$ , from the relation $\\lambda _{phys}(\\eta )=\\lambda _ca(\\eta )$ .",
"It results $\\eta _{k,\\textrm {re-entry}}\\simeq 1/k$ .",
"In Fig.",
"REF we show a scheme to study the evolution of physical wavelengths while the Universe expands and the horizon (Hubble radius) changes.",
"Physical wavelengths evolve proportionally to the scale factor $a$ .",
"Modes re-enter in the horizon when $\\lambda _{phys}(\\eta )=H^{-1}(\\eta )\\rightarrow kH(\\eta )/a(\\eta )\\simeq 1$ , so the smaller the wavenumber the later its entry.",
"In particular at $\\eta =\\eta _{EW}$ one mode with comoving wavenumber $k=k_{EW}$ re-enters the horizon.",
"Therefore the evolution of modes with $k<k_{EW}$ is bounded by $\\eta _{EW}$ .",
"Conversely the time bound for modes whose $k>k_{EW}$ is $\\eta _{k,\\textrm {re-entry}}$ .",
"To finish it is relevant to know what happens with $k=1/\\tau $ .",
"We consider fields whose relaxation time $\\tau $ produces perturbations of cosmological interest, namely perturbations whose wavelength today is at least as long as 1 kpc.",
"In comparison, the mode $k=k_{EW}$ has a wavelength today $\\lambda _{EW,0}\\lesssim 1\\textrm { pc}$ , so we get $\\lambda _{\\tau ,0}\\gg \\lambda _{EW,0}$ , as it is shown in Fig.",
"REF .",
"Therefore $1/\\tau _0\\ll k_{EW}$ .",
"Summarizing, we derive the following time bounds $\\begin{split}&\\eta _{k,max}=\\textrm {min}_k\\left\\lbrace \\eta _{EW},\\eta _{k,\\textrm {re-entry}}\\right\\rbrace =\\\\&={\\left\\lbrace \\begin{array}{ll}\\eta _{k,\\textrm {re-entry}}=\\displaystyle \\frac{1}{k}-1&\\textrm { if }\\,k_{EW}<k<1\\\\\\eta _{EW}=10^{n-2}-1&\\textrm { if }\\,\\displaystyle \\frac{1}{\\tau _0}<k<k_{EW}.\\end{array}\\right.",
"}\\end{split}$ Finally, using these bounds in Eq.",
"(REF ) within the range of comoving unnormalized units $1/\\tau <k<k_{EW}$ , we obtain at $\\eta =\\eta _{EW}$ the gravitational wave spectrum for each polarization $\\langle h_{\\mathbf {k}}(\\eta )h^*_{\\mathbf {k}^{\\prime }}(\\eta )\\rangle \\simeq 1.35\\,\\left(\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\frac{H^2}{2M_{pl}^2k^3}\\right).$ Figure: Physical wavelength vs. scale factor.This is a typical scheme to study evolution of perturbations during the expansion of the Universe.",
"We show distinct events: the end of inflation (EOIEOI), electroweak transition (EWEW), matter-radiation equality (EqEq), recombination (RecRec) and today.",
"The Hubble radius H -1 H^{-1} is represented by the black solid line and its evolution depends on the epoch of domination.",
"λ\\lambda represents the physical wavelength of the perturbations and it scales λ∝a\\lambda \\propto a.",
"We show different wavelengths for the distinct values of the Hubble radius at the moments said.",
"These scales are related with multipoles in the CMB correlation spectrum, for instance l Rec ∼100l_{Rec}\\sim 100.",
"Usually H -1 H^{-1} is the only relevant scale that distinguishes the evolution of perturbations between super-Hubble (λ>H -1 \\lambda >H^{-1}) and sub-Hubble (λ<H -1 \\lambda <H^{-1}) modes.",
"We always concentrate in the former, but here it is important to note that the presence of the new dimensionful parameter τ\\tau (Eq.",
"()) introduces another scale which splits the evolution of super-Hubble modes in two.",
"First, for modes with λ>λ τ ≃τ\\lambda >\\lambda _{\\tau }\\simeq \\tau we recover the usual invariant spectrum.",
"However for modes with H -1 <λ<λ τ H^{-1}<\\lambda <\\lambda _{\\tau } the fluid-graviton interaction produces an energy transfer from the fluid to gravitons and increases the amplitude of the spectrum.",
"We are able to extend our description until the electroweak transition.",
"Thus, shaded zone represents the modes which are amplified with respect to the usual invariant spectrum by a factor of about 1.31.3 at the electroweak time according to Eq.",
"()."
],
[
"Estimates of $\\tau $",
"The main goal of this section is to estimate the relaxation time $\\tau $ of the field we have considered throughout the paper.",
"First we get a feature (step) in the spectrum at comoving wavenumber $k_{\\tau }=1/\\tau $ and comoving wavelength $\\lambda _{\\tau }=2\\pi /k_{\\tau }\\sim \\tau $ .",
"We have set $a(\\eta )=1+\\eta $ and $\\eta =0$ at the end of inflation.",
"For instantaneous reheating, it coincides with the onset of the radiation dominated epoch where $a_{\\gamma }=a(\\eta =0)=1$ .",
"The evolution of physical perturbation wavelengths from the end of inflation until today may be calculated as $\\lambda _{\\tau ,0}=\\lambda _{\\tau }\\,\\frac{a_{0}}{a_{\\gamma }}=2\\pi \\,\\tau \\,\\frac{a_{0}}{a_{\\gamma }},$ with $a_0$ the scale factor today (subscript 0 means today).",
"To compute the ratio $a_0/a_{\\gamma }$ we consider a nearly adiabatic expansion of the Universe in which $a(\\eta )\\propto 1/T_{rad}$ .",
"In consequence $\\frac{a_{0}}{a_{\\gamma }}\\simeq O(1)\\,\\frac{T_{\\gamma }}{T_0}\\simeq 10^{n+14},$ where $T_{\\gamma }=T_{rad}(\\eta =0)=10^n$ GeV is the reheating temperature.",
"Therefore $\\lambda _{\\tau ,0}=\\lambda _{\\tau }\\,10^{n+14}.$ Recall that physical wavelengths of cosmological interest are in the range $\\lambda _{0}\\gtrsim 1\\textrm { kpc}$ .",
"In particular we would like to concentrate on $\\lambda _{0}\\gtrsim 1$ Mpc which implies $\\lambda _{\\tau ,0}\\gtrsim 1$ Mpc.",
"Let us consider a scalar field with a gauge coupling constant $g$ .",
"[14] and [86] show that it is possible to compute the relaxation time $\\tau $ in the Boltzmann equation from quantum field theory.",
"Basically it is given by $\\frac{1}{\\tau }\\sim \\frac{\\textrm {Im}\\left[\\Sigma \\right]}{T},$ where $\\Sigma $ is the self-energy of the field we are considering and $\\textrm {Im}\\left[x\\right]$ takes the imaginary part of $x$ .",
"We could expand $\\textrm {Im}\\left[\\Sigma \\right]$ in Feynman diagrams and prove that the first non-null contribution appears at the two-loop order.",
"We conclude on dimensional grounds that $\\textrm {Im}\\left[\\Sigma \\right]\\sim g^4T^2=\\alpha _g^2 T^2,$ where $\\alpha _g^2=g^4$ represents the fine structure constant of this theory.",
"If we take the reheating temperature $T_{\\gamma }\\sim 10^{16}-10^{15}$ GeV and values of $g\\sim 10^{-6}$ we find that $\\lambda _{\\tau ,0}\\sim 10$ Mpc which lies in the range of cosmological interest.",
"The characteristic multipole $l$ for this scale reads $l\\sim \\pi R_{LSS}/x\\sim 10^3$ , where $R_{LSS}\\simeq 14$ Gpc is the distance to the last scattering surface (LSS) and $x\\simeq 10$ Mpc represents the perturbation wavelength.",
"In addition from the range of reheating temperature $T_{\\gamma }\\sim 10^{16}-10^{15}$ GeV we consider, we estimate a tensor to scalar ratio about $r\\sim 10^{-1}-10^{-5}$ respectively [1].",
"The values of $\\tau $ we are regarding here are consistent with the values for its analogous $\\Gamma _{a\\rightarrow \\gamma \\gamma }^{-1}$ (axion lifetime) in known ALP-models in the literature [87] [88] [89] [90].",
"We assume that the relaxation time $\\tau $ and the thermalization time are of the same order and that hydrodynamics is already valid for earlier times.",
"The validity of applying hydrodynamics in this regime has been discussed by [91] [92] [93] [94] [95] who argue that the hydrodynamic framework is valid at time scales shorter than the corresponding for isotropization and thermalization, driven by a novel dynamical attractor whose details vary according to the theory under consideration.",
"Such attractor solutions show that hydrodynamics displays a new degree of universality far-from-equilibrium regardless of the details of the initial state of the system.",
"In fact, the approach to the dynamical attractor effectively wipes out information about the specific initial condition used for the evolution, before the true equilibrium state and consequently, thermalization, is reached.",
"This process is described as hydrodynamization to distinguish it from ordinary thermalization, and it has been shown by those authors that it develops on shorter time scales than thermalization.",
"In the context of kinetic theory and standard statistical mechanics, thermalization is understood as the development of an isotropic thermal one-particle distribution function.",
"In some particular cases, it is possible to show that even with relative anisotropies of about 50% the hydrodynamic description matches the full solution [96] [97]."
],
[
"Final Remarks",
"When studying the early Universe, particularly just after inflation, it is important to include full interactions between all fields in our description.",
"This may be a daunting challenge.",
"In that way, we propose to treat the fields and its interactions with effective relativistic hydrodynamic theories.",
"Nonetheless we discard ideal fluids in order to incorporate dissipative effects, as we have learned from relativistic heavy ions collisions.",
"Further we go beyond covariant Navier-Stokes theory to avoid known causality and stability issues.",
"Thus our main hypothesis lies in using causal hydrodynamics to obtain an adequate description of the phenomena we are interested in, specially during the very early Universe when almost all the matter fields could be described as a hot plasma.",
"Incorporating these causal theories to model the fields as effective fluids during the very early Universe may bring forth new effects [79].",
"Throughout the paper we have analyzed a simplified case of interaction between a spectator minimally coupled scalar field and the tensor metric perturbations after inflation.",
"Unlike ideal or Navier-Stokes hydrodynamics, this interaction may be present in any causal theory because the tensor part of the dissipative energy-momentum tensor is regarded as a new variable with non-trivial dynamics.",
"Covariant Navier-Stokes equations has no proper tensor degree of freedom, in spite of the fact that the energy-momentum tensor of a quantum scalar field has such a part [49] [83].",
"Causal theories allow us to keep this component of the energy-momentum tensor and thus follow its interaction with the gravitational field.",
"In consequence causal hydrodynamics enables the description of effects that are lost in covariant Navier-Stokes theory.",
"Its importance would be estimated by considering the constitutive parameters.",
"To be concrete we analyze the evolution of gravitational wave spectrum.",
"Usually $H^{-1}$ is the only relevant scale that distinguishes the evolution of perturbations between super-Hubble ($\\lambda >H^{-1}$ ) and sub-Hubble ($\\lambda <H^{-1}$ ) modes, where $\\lambda $ represents the physical wavelength.",
"We always concentrate in the former, but here it is important to note that the presence of the new dimensionful parameter $\\tau $ which provides us the characteristic relaxation time of the fluid dynamics (Eq.",
"(REF )) introduces another scale which splits the evolution of super-Hubble modes in two, as it is shown in Fig.",
"REF .",
"Considering the values of the parameters on previous sections we get that for modes with $\\lambda >\\lambda _{\\tau }\\simeq \\tau $ we recover the usual invariant spectrum.",
"However for modes with $H^{-1}<\\lambda <\\lambda _{\\tau }$ the fluid-graviton interaction produces an energy transfer from the fluid to gravitons and increases the amplitude of the spectrum.",
"We are able to extend our description until the electroweak transition.",
"Thus, shaded zone in Fig.",
"REF represents the modes which are amplified with respect to the usual invariant spectrum by a factor of about $1.3$ at the electroweak time according to Eq.",
"(REF ).",
"Fields at extreme conditions, like highly energetic collisions or very large temperatures in the early Universe, evidence the need for new schemes of description which incorporate interactions and non-ideal processes such as dissipation and thermalization.",
"Causal relativistic hydrodynamic theories are promising candidates to include characteristic effects of these regimes in a consistent framework."
],
[
"Conformal invariance",
"We shall show that the Boltzmann equation for massless particles is conformally invariant, and that conformal invariance is not broken by taking moments.",
"The Boltzmann equation in curved space is $p^{\\mu }\\left[\\frac{\\partial }{\\partial x^{\\mu }}+\\Gamma ^{\\nu }_{\\mu \\rho }p_{\\nu }\\frac{\\partial }{\\partial p_{\\rho }}\\right]f=I_{col}$ We write $g_{\\mu \\nu }=a^2(\\eta )\\bar{g}_{\\mu \\nu }$ .",
"So we split the metric connection $\\Gamma ^{\\nu }_{\\mu \\rho }={\\bar{\\Gamma }}^{\\nu }_{\\mu \\rho }+\\frac{a^{\\prime }}{a}\\,\\gamma ^{\\nu }_{\\mu \\rho },$ where $\\gamma ^{\\nu }_{\\mu \\rho }={\\delta ^{\\nu }}_{\\rho }{\\delta ^{0}}_{\\mu }+{\\delta ^{\\nu }}_{\\mu }{\\delta ^{0}}_{\\rho }-\\bar{g}^{\\nu 0}\\bar{g}_{\\mu \\rho }.$ We also assume that $f\\left( x^{\\mu },p_{\\nu }\\right)$ is invariant and $I_{col}=a^{-2}\\overline{I}_{col}$ .",
"Thus Boltzmann equation reads $\\bar{g}^{\\mu \\sigma }p_{\\sigma }\\left[\\frac{\\partial }{\\partial x^{\\mu }}+\\bar{\\Gamma }^{\\nu }_{\\mu \\rho }p_{\\nu }\\frac{\\partial }{\\partial p_{\\rho }}+\\frac{a^{\\prime }}{a}\\gamma ^{\\nu }_{\\mu \\rho }p_{\\nu }\\frac{\\partial }{\\partial p_{\\rho }}\\right]f=\\overline{I}_{col}$ Conformal invariance follows if we show that $\\bar{g}^{\\mu \\sigma }\\gamma ^{\\nu }_{\\mu \\rho }\\,p_{\\sigma }p_{\\nu }=0$ for a massless theory, namely when $\\bar{g}^{\\mu \\sigma }p_{\\sigma }p_{\\mu }=0$ .",
"Indeed, using (REF ) it is straightforward to show that $\\bar{g}^{\\mu \\sigma }\\gamma ^{\\nu }_{\\mu \\rho }\\,p_{\\sigma }p_{\\nu }=\\bar{g}^{\\mu \\sigma }p_{\\sigma }p_{\\mu }{\\delta ^0}_{\\rho }=0.$ We define the covariant moments of the distribution function as $A^{\\mu _1,\\ldots ,\\mu _n}=\\int Dp\\;p^{\\mu _1}\\ldots p^{\\mu _n}f$ where $Dp=\\frac{2dp_0\\prod _idp_i}{\\left(2\\pi \\right)^3\\sqrt{-g}}\\;\\delta \\left( p^2\\right)\\Theta (p^0) =a^{-2}\\bar{Dp},$ $\\bar{Dp}$ is defined in Eq.",
"(REF ).",
"Then the moments transform as $A^{\\mu _1,\\ldots ,\\mu _n}=a^{-2\\left( n+1\\right) }\\overline{A}^{\\mu _1,\\dots ,\\mu _n}$ and $I^{\\mu _1,\\ldots ,\\mu _n}=\\int Dp\\;p^{\\mu _1}\\ldots p^{\\mu _n}I_{col}=a^{-2\\left( n+2\\right) }\\overline{I}^{\\mu _1,\\ldots ,\\mu _n}$ The covariant equation for the moments reads $A^{\\mu \\mu _1,\\ldots ,\\mu _n}_{;\\mu } =I^{\\mu _1,\\ldots ,\\mu _n}$ and becomes $\\begin{split}&\\overline{A}^{\\mu \\mu _1,\\ldots ,\\mu _n}_{,\\mu }+\\bar{\\Gamma }^{\\mu }_{\\mu \\rho }\\,\\overline{A}^{\\rho \\mu _1,\\ldots ,\\mu _n}+\\sum _{i=1}^{n}\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\dots ,\\mu _n}+\\\\&+\\frac{a^{\\prime }}{a}\\left[-2n\\overline{A}^{0\\mu _1,\\ldots ,\\mu _n}+\\sum _{i=1}^n\\gamma ^{\\mu _i}_{\\mu \\rho }\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}\\right]=\\\\&=\\overline{I}^{\\mu _1,\\ldots ,\\mu _n},\\end{split}$ where $\\left(\\mu _i\\right)$ means that $\\mu _i$ index is excluded.",
"Following we need to show $\\sum _{i=1}^n\\gamma ^{\\mu _i}_{\\mu \\rho }\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=2n\\overline{A}^{0\\mu _1,\\ldots ,\\mu _n}$ given that the moments are totally symmetric and traceless on any pair of indexes.",
"Actually, for each term we have $\\gamma ^{\\mu _i}_{\\mu \\rho }\\,\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=2\\overline{A}^{0\\mu _1,\\ldots ,\\mu _n}$ because if $\\mu _i=0$ this gives $\\begin{split}&\\gamma ^{0}_{\\mu \\rho }\\,\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=\\\\&=2\\overline{A}^{00\\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}-\\bar{g}^{00}{{\\overline{A}^{\\mu }}_{\\mu }}^{\\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=\\\\&=2\\overline{A}^{0\\mu _1,\\ldots ,\\mu _i=0,\\ldots ,\\mu _n}\\end{split}$ and if $\\mu _i=j\\ne 0$ then we get $\\begin{split}&\\gamma ^{j}_{\\mu \\rho }\\,\\overline{A}^{\\mu \\rho \\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=\\\\&=\\overline{A}^{0j\\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}+\\overline{A}^{j0\\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}-\\\\&-\\bar{g}^{j0}{{\\overline{A}^{\\mu }}_{\\mu }}^{\\mu _1,\\ldots \\left(\\mu _i\\right)\\ldots ,\\mu _n}=2\\overline{A}^{0\\mu _1,\\ldots ,\\mu _i=j,\\ldots ,\\mu _n},\\end{split}$ which ends up proving (REF ).",
"We now show that our ansatz for the distribution function and the collision integral is consistent with conformal invariance.",
"Indeed, we take the one-particle distribution function given in (REF ) $f=\\frac{1}{\\displaystyle {\\exp {\\left(-\\beta ^{\\mu }p_{\\mu }-\\kappa \\,\\zeta ^{\\mu \\nu }\\,p_{\\mu }p_{\\nu }/T^2\\right)}}-1}.$ Since $p_{\\mu }$ is invariant we require transformation laws which implies invariance of $\\beta ^{\\mu }$ and $\\zeta ^{\\mu \\nu }/T^2$ .",
"Index disposition matters.",
"From $T=\\mathbf {T}/a$ we arrive to $\\beta _{\\mu }=a^2\\bar{\\beta }_{\\mu }$ , $u^{\\mu }=a^{-1}\\bar{u}^{\\mu }$ and $\\zeta _{\\mu \\nu }=a^2\\bar{\\zeta }_{\\mu \\nu }$ .",
"In addition as $\\tau $ is a scale dimensional parameter we assume that $\\tau =a\\bar{\\tau }$ , thus $I_{col}=\\frac{u^{\\mu }p_{\\mu }}{\\tau }\\left(f-f_0\\right)$ also has the required transformation law."
],
[
"Tensor part of the noise kernel",
"In this appendix we clarify the calculation of tensor part of noise kernel in Fourier space.",
"From Eq.",
"(REF ) we write $\\begin{split}&{{{{N}^{i}}_{j}}^{k}}_{l}(\\mathbf {x},\\mathbf {x}^{\\prime })=\\\\&=\\left[r^ir^jr^kr^l\\,F_1(r)+\\left(\\delta ^{il}r^jr^k+\\delta ^{jk}r^ir^l\\right)F_2(r)\\right.+\\\\&+\\left.\\delta ^{il}\\delta ^{jk}\\,F_3(r)\\right]+(k\\leftrightarrow l),\\end{split}$ with $F_1(r)=\\frac{H^8}{4\\pi ^4r^{8}},\\;F_2(r)=-\\frac{H^8}{8\\pi ^4r^{6}}\\;\\textrm {and}\\;F_3(r)=\\frac{H^8}{16\\pi ^4r^{4}}.$ Thus applying tensor projectors (REF ) to (REF ) in Fourier space we get $\\begin{split}&{N_{T\\,}}^{abcd}(\\mathbf {k},\\mathbf {k}^{\\prime })={{{\\Lambda ^a}_i}^b}_j{{{\\Lambda ^c}_k}^d}_l\\,\\,{{{\\left.N\\right.^i}_{j}}^{k}}_{l}(\\mathbf {k},\\mathbf {k}^{\\prime })=\\\\&=\\delta (\\mathbf {k}-\\mathbf {k}^{\\prime })\\,F(k)\\,\\left[\\Lambda ^{adbc}+\\Lambda ^{acbd}\\right],\\end{split}$ where $F(k)=\\left[\\frac{2F_1^{\\prime \\prime }(k)}{k^2}-\\frac{2F_1^{\\prime }(k)}{k^3}-\\frac{2F_2^{\\prime }(k)}{k}+F_3(k)\\right].$ To compute Fourier transforms $F_i(k)$ we use the following relation $\\int \\,r^{-2n}\\,e^{-i\\,\\mathbf {k}\\cdot \\mathbf {r}}d^3r=\\pi ^{3/2}\\,\\frac{\\Gamma (3/2-n)}{\\Gamma (n)}\\,\\left(\\frac{k^2}{4}\\right)^{n-3/2},$ and finally it results $F(k)=\\frac{6911}{12}\\,\\frac{H^8}{\\pi ^2}\\,k+O(k^2).$ Work supported in part by CONICET and University of Buenos Aires.",
"It is a pleasure to thank A. Kandus, D. López Nacir and G. Pérez-Nadal for discussions."
]
] | 1709.01661 |
[
[
"Radio Weak Lensing Shear Measurement in the Visibility Domain - II.\n Source Extraction"
],
[
"Abstract This paper extends the method introduced in Rivi et al.",
"(2016b) to measure galaxy ellipticities in the visibility domain for radio weak lensing surveys.",
"In that paper we focused on the development and testing of the method for the simple case of individual galaxies located at the phase centre, and proposed to extend it to the realistic case of many sources in the field of view by isolating visibilities of each source with a faceting technique.",
"In this second paper we present a detailed algorithm for source extraction in the visibility domain and show its effectiveness as a function of the source number density by running simulations of SKA1-MID observations in the band 950-1150 MHz and comparing original and measured values of galaxies' ellipticities.",
"Shear measurements from a realistic population of 10^4 galaxies randomly located in a field of view of 1 deg^2 (i.e.",
"the source density expected for the current radio weak lensing survey proposal with SKA1) are also performed.",
"At SNR >= 10, the multiplicative bias is only a factor 1.5 worse than what found when analysing individual sources, and is still comparable to the bias values reported for similar measurement methods at optical wavelengths.",
"The additive bias is unchanged from the case of individual sources, but is significantly larger than typically found in optical surveys.",
"This bias depends on the shape of the uv coverage and we suggest that a uv-plane weighting scheme to produce a more isotropic shape could reduce and control additive bias."
],
[
"Introduction",
"Cosmological or targeted surveys of weak gravitational lensing at radio wavelengths may have a relevant role in the next years, when the Square Kilometre Array (SKA)https://www.skatelescope.org radio telescope will start to operate, providing a density of detected galaxies sufficient for shear measurement and a resolution to reliably measure their shapes.",
"They will also be able to probe to higher redshifts given the different galaxy redshift distributions compared to the optical band [9].",
"Although the galaxy number density in the radio band will be lower than in the optical, the possibility to observe deeper can make radio weak lensing surveys for cosmology measurements competitive with the corresponding optical surveys, as shown in recent forecasts from SKA simulations [20].",
"Moreover, radio observations have the advantage of a deterministic knowledge of the image-plane Point Spread Function (PSF), being the Fourier Transform of the uv coverage, and will provide unique approaches for mitigating intrinsic alignments, such as concurrent measurements of polarization [8] and galaxy rotation velocities [3], [32].",
"Being subject to different observational systematics, cross-correlation with optical observations of the same field will allow suppression of systematic errors on shear measurement from future large surveys [36], [16], [11].",
"This is quite relevant for precision cosmology as these errors may become comparable to, and larger than, the statistical noise.",
"The precursor radio weak lensing survey SuperCLASShttp://www.e-merlin.ac.uk/legacy/projects/superclass.html is already underway and will soon provide data that may be used to test new methods required for accurate galaxy shape measurement in the radio band.",
"A natural approach for such methods is working in the visibility domain where the data originates and the noise is Gaussian, avoiding non-linear data manipulation of the imaging process.",
"SKA simulations have already shown that current imaging methods produce images with structures in the residuals which may dominate the cosmological signal [37].",
"Also cross-correlation analysis using real data images shows that no evidence of correlation is found between the optical and radio intrinsic shape of the matched objects [36], [47].",
"This result suggests the presence of systematics in the procedure adopted for turning the visibility data into images, although a significant percentage of AGN sources in the observed population may be another possible explanation, as well as the astrophysical scatter between optical and radio position angle due to the different emission mechanisms in the two bands.",
"Currently, cosmic shear in the radio band has been successfully detected only working in the visibility domain but obtaining sources position from the image [13].",
"Galaxies' ellipticities from the VLA FIRST survey [2] were measured by decomposing them into shapelets, an orthonormal basis of functions corresponding to perturbations around a circular Gaussian invariant under Fourier transform [12].",
"Since the data size and the number of resolved sources ($\\sim 20-30 \\deg ^{-2}$ ) of each pointing is quite small, a joint fitting of the shapelet coefficients was possible by solving normal equations.",
"Such an approach, computationally convenient, becomes very challenging when dealing with the order of $10^4$ sources per square degree and a very large dataset per pointing (order of PetaBytes), as expected from SKA Phase 1 continuum surveys [9].",
"Moreover shapelets introduce a shear bias as they cannot accurately model steep brightness profiles and highly elliptical galaxy shapes [29].",
"In the companion paper ([40], hereafter Paper I) we presented RadioLensfit, an alternative method working in the visibility domain where model fitting is performed on a single source at a time using an exponential profile as model for the galaxy.",
"It is an adaptation of the optical Bayesian lensfit method [31] to radio data, where model visibilities are defined analytically and the likelihood is marginalised over uninteresting parameters.",
"The method was tested in the simple case of individual galaxy visibilities simulated adopting the SKA1 uv coverage described in Section , and the shear noise bias [38], [30] estimated as a function of the signal-to-noise ratio (SNR).",
"Results compared with requirements [9] for the proposed SKA1 radio weak lensing survey [5] showed that the multiplicative shear bias may need calibration corrections similar to those for optical surveys, while the additive bias have to be controlled by an isotropic sampling of the visibility plane.",
"In this paper we extend this work implementing the method for isolating source visibilities from realistic data, i.e.",
"when many galaxies are in the field of view.",
"We estimate its effectiveness in terms of ellipticity fitting and shear measurement by running SKA1-MID simulations as we did in the previous paper.",
"We finally investigate the effect of the shape of the uv coverage on the additive shear bias.",
"This paper is organised as follows.",
"In Section 2 we summarise RadioLensfit and present the extraction algorithm.",
"In Section 3 details of SKA1 simulations are provided, while in Sections 4 and 5 results for galaxy ellipticity and shear measurements are presented respectively.",
"Finally we discuss the shear additive noise bias in Section 6."
],
[
"Overview of RadioLensfit",
"RadioLensfit is a method for measuring radio galaxy ellipticities in the visibility domain.",
"The idea is to adapt the approach used in the optical case to radio data, i.e.",
"extracting from visibilities and model fitting a single source at a time.",
"Source extraction is difficult in the Fourier domain because signals from all sources in the primary beam are mixed altogether in the visibilities and sources are no longer localised.",
"For this reason a joint analysis with the image domain may be needed: it allows us to identify sources and measure their position and flux with sufficient accuracy.",
"With such information we can also compute a model of the observed sky and use it to approximate the signal from the other galaxies that must be removed when extracting each source.",
"The extraction is completed using a faceting technique that phase shifts the phase centre to the source position and further reduces its signal contamination by averaging visibilities in a coarse grid.",
"Finally the model fitting can be performed as for the simple case of a single galaxy in the primary beam located at the phase centre as summarised in Section REF .",
"This way we can largely reduce the computational time when a huge number of sources are in the field of view (as for SKA) instead of trying a challenging joint fitting of all sources.",
"A detailed algorithm for the fitting of many sources in the primary beam is presented in Section REF ."
],
[
"Galaxy ellipticity fitting",
"In Paper I we introduced this method as an adaptation of lenfit [31] by performing the chi-square fitting of single source visibilities.",
"They are evaluated at the uv points, that are defined by the baselines formed between two antennae projected on the plane orthogonal to the pointing direction.",
"Model visibilities of a star-forming (SF) galaxy are computed analytically as the Fourier transform of the exponential brightness profile (Sérsic index $n=1$ ): $V(u,v) = \\Big ( \\frac{\\lambda _\\textrm {ref}}{\\lambda }\\Big )^\\beta \\frac{S_{\\lambda _\\textrm {ref}}\\mathrm {e}^{2\\pi \\mathrm {i} (u l + v m)}}{\\big (1+4\\pi ^2 \\alpha ^2 |{A}^{-T}\\mathbf {k}|^2\\big )^{3/2}},$ where $\\mathbf {k}=(u,v)^T$ is measured in wavenumber units, $\\beta =-0.7$ is the assumed spectral index for the synchrotron radiation emitted by the galaxy disc, $(l,m)$ and $\\alpha $ are the source position and scalelength respectively, $S_{\\lambda _\\textrm {ref}}$ is the source flux at reference wavelength $\\lambda _\\textrm {ref}$ .",
"The ellipticity parameters $(e_1, e_2)$ are contained in the matrix ${A}$ that linearly transforms the circular exponential profile to an ellipse: ${A} = \\left( \\begin{array}{cc} 1-e_{1} & -e_{2} \\\\ -e_{2} & 1+e_{1} \\end{array} \\right).$ We assume the following ellipticity definition: $\\mathbf {e} = e_1 +\\mathrm {i}e_2 = \\frac{a-b}{a+b}\\mathrm {e}^{2\\mathrm {i}\\theta },$ where $a$ and $b$ are the galaxy major and minor axes respectively, and $\\theta $ is the galaxy orientation.",
"The likelihood is marginalised over non-interesting parameters such as flux, scalelength and position, adopting uniform priors for the flux and position, and a lognormal prior dependent on the flux for the scalelength (see Section ).",
"This way we obtain a likelihood function of only the ellipticity parameters.",
"The galaxy ellipticity measurement is given by the likelihood mean point and 1D standard deviation (defined as the square root of the covariance matrix determinant), obtained after sampling the likelihood with an adaptive grid covering a neighbourhood around the maximum point.",
"In real observations the finite channel bandwidth and sampling time introduce smearing effects that are dependent on the source position.",
"These effects may be approximated analytically and included in the visibilities model [6], [46].",
"For example for frequency smearing, assuming a square bandpass filter in the expression of the smeared visibility presented in [6], we obtain: $\\tilde{V}(u,v)= V(u,v) \\textrm {sinc}[\\pi (ul+vm)\\Delta \\nu /(\\nu _0 c)],$ where uv coordinates are taken at the mid-channel frequency $\\nu _0$ , $\\Delta \\nu $ is the channel bandwidth and $\\text{sinc}(x)=\\sin (x)/x$ .",
"Another option is to make the observation with very tiny frequency channels and sampling time intervals.",
"[19] proposed to use $\\sim $ 30 kHz channel bandwidth and 0.5 s sampling time to make smearing tolerable, but meaning a huge number of uv points.",
"In this case, raw visibilities may be averaged into a single uv grid without jeopardizing ellipticity measurements.",
"In fact, observations from the same pair of antennae at different frequencies (resp.",
"times) correspond to visibilities evaluated at different uv points along a radial (resp.",
"tangential) direction, therefore these visibilities can be treated as the ones evaluated at uv points related to different baselines.",
"Depending on the grid size, data volume and then computational time may be considerably reduced."
],
[
"Source extraction",
"We assume flux and source positions are provided.",
"For example they may be estimated from a cleaned image of the same data that are analysed, or applying MC methods to the visibilities (e.g.",
"using a multimodal nested sampling with a single source model as in [18]).",
"From this information we define an initial sky model where the visibilities of each source $s$ in the field of view are computed according to equation (REF ) with ellipticity $\\mathbf {e}= \\mathbf {0}$ , i.e.",
"circular source, and scalelength provided by the linear relation between the log of the median scalelength $\\alpha _\\mathrm {med}$ and flux density $S$ [39]: $\\ln {[\\alpha _\\mathrm {med}/\\textrm {arcsec}]} = -0.93 +0.33\\ln {[S/\\mu \\textrm {Jy}]}.$ The sky visibilities are obtained adding the model visibilities of each source in the beam: $V_\\mathrm {sky}(u,v) = \\sum _{s=1}^N V_s(u,v).$ Starting from this sky model, the source extraction and fitting procedure is performed according to decreasing flux order, i.e.",
"decreasing SNR, as follows: 1.",
"Given the position of the source $(l,m)$ , remove the corresponding circular source model visibilities from the sky model and then take the difference between the data and the sky model, so that the visibilities of the current source (with a reduced contamination from the others) are isolated.",
"2.",
"Apply faceting [15]: phase shift these visibilities in order to move the phase centre to the location of the source, by multiplying each visibility by the factor $\\exp (-2\\pi \\mathrm {i} (u l + v m))$ , and average them in a coarse grid (facet).",
"This way we reduce the field of view to a small patch around the source, with the advantage of reducing the number of visibilities used for the model fitting and therefore accelerating the computation.",
"On the other hand this procedure limits the maximum wavelength of the Fourier mode that can be measured because of the finite spacing of the facet uv points.",
"3.",
"Use the source visibilities for measuring the corresponding source ellipticity as in Section REF .",
"4.",
"Use the estimated ellipticity to improve model visibilities of the current source and remove them from the data.",
"5.",
"Repeat from step 1 until all sources are fitted.",
"Note that in this algorithm the sky model is improved after each source fitting by replacing circular sources with the elliptical source that has been fitted.",
"Moreover, by ordering the source extraction by decreasing flux, the source fitting is performed with a better approximation of the sky model for sources at low SNR.",
"In the case of ”bad measurements”, the corresponding sources are not removed in the first instance from the data and sky model visibilities, but they are re-fitted at the end of the procedure, when the ellipticities of all the other sources are measured and a better sky model is obtained.",
"These unreliable fits are recognised by a too small standard deviation of the ellipticity likelihood to be realistic.",
"This seems related to errors in the likelihood computation, when the cross-correlation function is not sufficiently smooth to be marginalised over the source position, possibly due to PSF sidelobes or too much noise in the data.",
"Bad measurements are given weight zero in the shear computation."
],
[
"SKA1 Simulations",
"As in Paper I, we simulate SKA-MID eight-hour observations of 1 square degree at declination $\\delta = -30^\\circ $ by using the SKA-MIDSKA-MID latitude is -304948.00 S. Phase 1 antennae configuration provided in [21].",
"This integration time provides a complete circular coverage, i.e.",
"without large gaps (because of the 3 telescope arms), and allow to reach a sensitivity of 10$\\mu $ Jy at 10$\\sigma $ .",
"It would allow a targeted area of $800 \\deg ^2$ to be observed with such sensitivity in 10,000 hours in a forthcoming SKA1 radio weak lensing survey, sufficient for measuring cluster lensing.",
"We choose the following conservative approximation of the frequency bandwidth: 950 - 1190 MHz, as proposed in [4].",
"This seems to be the optimum observation frequency for a weak lensing survey with SKA1-MID, in case only 30% of the full bandwidth of SKA Band 2 is usable (because of RFI problems, other surveys commensality, etc.).",
"Visibilities are sampled every $\\tau _\\textrm {acc} = 60$ s and we consider one large channel because smearing effects are not included on shorter time and bandwidth scales.",
"The observations are simulated by using equations (REF ) and (REF ) and adding an uncorrelated Gaussian noise whose variance is given by [48]: $\\sigma ^2 = \\frac{\\textrm {SEFD}^2}{2\\eta ^2\\Delta \\nu \\tau _\\textrm {acc}},$ where $\\Delta \\nu $ is the frequency channel bandwidth, SEFD = 400 Jy is the system equivalent flux density for SKA-MID dishes and $\\eta =0.9$ is the system efficiency [5].",
"For simplicity we assume that the SKA1-MID core is composed of only SKA dish antennae even if part of it actually contains 64 MeerKAT dishes.",
"SF galaxy populations are generated according to distributions estimated from archival data of VLA radio surveys at 1.4 GHz.",
"As described in [39], we estimated distributions of the flux $S$ and the scalelength $\\alpha $ of sources modelled by an exponential profile by the analysis of faint sources (order of tens $\\mu $ Jy) catalogue of the SWIRE field survey [35].",
"This is the radio catalogue with the largest number of SF galaxies, being related to the deepest survey so far in terms of the radio source density (source flux cut $\\sim $ 15 $\\mu $ Jy), and it contains source size measurements from the imaged data.",
"The flux distribution is fitted by a power law: $p(S) \\propto S^{-1.34}$ .",
"The scalelength is obtained by the relation with the measured full width at half maximumThe FWHM is derived from the Gaussian model fit of the source after PSF deconvolution.",
": FWHM = $2\\alpha \\ln (2)$ , and its distribution is fitted dependently on source flux by a lognormal function with mean $\\mu = \\ln (\\alpha _\\mathrm {med})$ and variance $\\sigma \\sim 0.3$ , where $\\alpha _\\mathrm {med}$ is given by eq.",
"(REF ).",
"The variance value is suitably chosen in the middle of a range well representing scalelength distributions for different flux values.",
"The modulus $e$ of the intrinsic ellipticities is generated according to a function proposed by [31]: $P(e) = \\frac{Ne\\left(1-\\exp \\left[ \\frac{e-e_\\textrm {max}}{c}\\right]\\right)}{(1+e)(e^2+e_0^2)^{1/2}}.$ The parameter values, $c = 0.2298$ and $e_0 = 0.0732$ are obtained from fitting the VLA-COSMOS field data.",
"Although this survey is less deep than the SWIRE, but still detecting $\\mu $ Jy sources (flux cut $\\sim $ 75 $\\mu $ Jy), we rely on a recent re-analysis of the L-band radio visibility data where the level of systematics in the measurement of the galaxy position angle is significantly reduced [47].",
"In fact the previous analysis [42] as well as the one of the VLA-SWIRE were mainly focused on faint source counts.",
"The normalisation factor is $N = 2.595$ .",
"We generate galaxy populations with flux densities ranging between $10\\mu $ Jy and $200\\mu $ Jy.",
"According to our flux distribution we obtain a source number density of 2.7 gal/arcmin$^2$ .",
"To be consistent, we adopt this source number density in our simulation, although more accurate modelling from recent radio continuum surveys suggest that a higher source number density should be detected at such a flux density cut in real observations [14], [25].",
"This is the expected source number density for the proposed 2-yr SKA1 radio weak lensing survey covering 5000 $\\deg ^2$ [9]."
],
[
"Galaxy shape measurement",
"In this section, first we select the facet size to be used in the source extraction by simulating visibilities of individual sources.",
"Then we simulate populations of galaxies located simultaneously in the field of view in order to show the efficacy of our source extraction algorithm as a function of the source number density."
],
[
"Facet size",
"The facet size is affected by the weighting scheme [7] adopted in the gridding phase.",
"For example, natural weighting optimises the sensitivity for detecting weak sources by emphasising the data from short baselines.",
"In this case, a relatively large facet size is expected even for covering a single galaxy because a large contribution to the signal is from long wavelength modes which must be adequately sampled by small facet cells.",
"In effect, the source in the image domain turns to be convolved with a large natural-weighted PSF with a broad low-level plateau (see left panel of Fig.",
"REF ).",
"Uniform weighting will require instead a much smaller facet size because it emphasises data from long baselines, where most of the source shape signal is contained.",
"This is reflected by the small uniform-weighted PSF (see right panel of Fig.",
"REF ).",
"On the other side, the weighting scheme used in the faceting procedure shouldn't affect the model fitting, provided that the measurement uncertainties are propagated correctly in the likelihood computation (see equation (21) of Paper I for the natural case) and model visibilities are consistent with the observed data.",
"This may not be true for measurement methods in the image domain, as shown in [47].",
"Since we are interested in the detection of faint sources for radio weak lensing, we adopt a natural weighting scheme.",
"To minimise the number of sources falling in the same facet, we define a facet size dependent on source flux, as it is related to the size of the source.",
"We split the flux total range of the simulated population, i.e.",
"10-200 $\\mu $ Jy, in 7 bins as shown in Table REF .",
"Facet uv point coordinates have to be re-computed only once per bin as the model fitting is performed according to source flux order.",
"We chose larger bins at large fluxes because the sizes of such sources increase more gradually with the flux compared to the ones with low flux (see equation (REF )).",
"To estimate the facet size for each bin, we simulate raw visibilities of a single galaxy in the primary beam in order to avoid any source contamination effects, and vary the noise added to the visibilities (to have SNR $\\ge 100$ ) in order to see the effect of the source size only.",
"We measure the galaxy ellipticity after averaging visibilities in the facet.",
"The best-fit slopeConsistently with Paper I, we refer to the ellipticity best-fit slope instead of the multiplicative bias when measuring galaxy shapes.",
"This terminology is used to clearly distinguish it from the shear multiplicative bias, which is obtained from the best-fit of shear measurements (each being the weighted average of galaxy ellipticities).",
"for the ellipticity measurements of 1000 galaxies is computed for different facet sizes and source flux ranging between the flux bin bounds.",
"We select the facet size when a fixed best-fit slope threshold of about 0.97 is reached, as listed in Table REF .",
"Note that the fitting for the first ellipticity component is better than the second one because of a slightly anisotropy of the PSF as discussed in Section .",
"The selected facet sizes are consistent with the relation between the uv grid cell size $\\Delta u$ (in units of wavelengths) and the related field of view (in radians): $\\psi = 1/\\Delta u$ [7].",
"We also note that for small sources (low flux) it is actually better to use smaller facets even in the case of a single source in the field of view.",
"This is shown in Fig.",
"REF where the difference between binned measurements and true values of the first ellipticity component of 1000 galaxies, with realistic flux distribution in the range 10-200 $\\mu $ Jy, are plotted both for the case where the facet size is constant and equal to 600 (left panel) and when the facet has a variable size dependent on the source flux (right panel).",
"Similar plots are obtained for the second component.",
"In the latter case we obtained 25 bad measurements (see Section REF ) and the best-fit slopes of the two ellipticity components are $0.9552 \\pm 0.0057$ and $0.9426 \\pm 0.0061$ respectively, whereas for the case of $600 \\times 600$ facet the best-fit slopes for the same galaxy population and noise are $0.9306 \\pm 0.0054$ and $0.9135 \\pm 0.0056$ and the number of bad measurements is 3 times larger.",
"These results are due to the fact that we do not model exactly the primary beam because the model visibilities are directly sampled on the uv facet points.",
"This means that in the image domain the sidelobes of the source model are not suppressed by any apodisation, whereas the gridding of the original uv coverage causes the full image to be apodised by a broad 2D sinc function which has the effect in the data of suppressing background sources that are a long way from the phase centre and the distant sidelobes from the primary source.",
"The grid sampling causes the resulting image domain facet to become a small, but aliased version of the apodised image.",
"The aliased model is an incorrect description of the apodised and aliased data, and the discrepancy will get worse for smaller facets and at large distances from the source.",
"Fig.",
"REF shows that a suitable facet size dependent on the source flux/size may be a trade-off between these two effects.",
"We could improve the model by applying the same gridding operations as in the data (sampling on the original uv coverage and then averaging in the facet), but this will add a large amount of computational time.",
"Our results show that the adopted model approximation is acceptable, provided that the facet sizes are sufficiently large to not affect the significant sidelobes in the image domain.",
"Otherwise we expect the discrepancy between data and model to become severe and the biases may become less robust and hence less calibratable."
],
[
"Dependence on source number density",
"We estimate the efficacy of the source extraction method by measuring the slope of the best-fit line of $10^4$ galaxy ellipticity measurements as a function of the source number density.",
"For each measurement we simulate sources located simultaneously in the field of view according to a uniform distribution.",
"Results are plotted in Fig.",
"REF and show reliable fits, independent of the source density up to 2.8 gal/arcmin$^2$ .",
"In this case the best-fit slopes of the correlation for each ellipticity component are $0.9365 \\pm 0.0017$ and $0.9262 \\pm 0.0017$ respectively and the number of bad measurements is about 1%.",
"At higher densities galaxy ellipticity measurement starts to deteriorate, as residuals of nearby galaxies affect the model fitting, but may still be good enough for shear measurement because of the improved statistics (as shown in Section ).",
"Shape measurements of galaxies may be improved by a joint fitting within facets by applying the Hamiltonian Monte Carlo technique [34].",
"RadioLensfit results used as starting points should reduce the burn-in phase and accelerate convergence.",
"Since the number of sources in the facet will be relatively small this approach becomes more feasible and preliminary results with this method show a better accuracy in the galaxy ellipticity fitting, although requiring a large computational time (Rivi et al., in preparation).",
"Using a single channel, the serial version of RadioLensfit running on an Intel Xeon E5-2650 takes on average about 10 sec/gal computing time for the model fitting.",
"As discussed in Paper I, the shared memory parallelisation with OpenMP allows to exploit all the computational resources when the amount of memory for the source model fitting requires the usage of the full CPU.",
"Its implementation has been optimised by distributing to each thread the likelihood computation and marginalisation over the position parameters for different scalelength values of the model.",
"It doesn't scale linearly with the number of threads because the likelihood mariginalisation over the scalelength parameter is not parallel and there is an overhead for the creation and destruction of OpenMP threads at each iteration of the likelihood maximisation and sampling.",
"This version running on all the eight cores of the CPU takes on average about 2.4 sec/gal.",
"Figure: Best-fit line slope for both galaxy ellipticity components as a function of the source number density."
],
[
"Shear",
"Following Paper I, for shear measurement we simulate galaxy populations as described in Section in a field of view of 1 $\\deg ^2$ .",
"We generate populations free of shape noise [33], [27]: for each ellipticity modulus, 10 equally-spaced galaxy orientations are generated so that the corresponding ellipticity values are distributed uniformly on a circle, and galaxies whose ellipticity is on the same ring are given the same size and flux.",
"We generate sources randomly located according to a uniform distribution.",
"All measurements are performed with facet size dependent on the source flux as defined in Table 1.",
"Figure: Shear components estimated from a galaxy population in 1 deg 2 \\deg ^2 as a function of the source number density for input 𝐠=0\\mathbf {g} = \\mathbf {0}.We measure the reduced shear $\\mathbf {g}=g_1 + \\mathrm {i}g_2$ as the weighted average of the galaxies' ellipticities using weights that approximate the inverse-variance of each ellipticity measurement.",
"Error bars are given by the standard deviation of the shear values estimated from 1000 bootstrap resamples.",
"Fig.",
"REF shows shear measurements as a function of the source number density up to 8 gal/armin$^2$ , when no input shear is applied, i.e.",
"$g_1=g_2=0$ .",
"At high densities the larger number of sources compensates the less accurate galaxy shape fitting (Fig.",
"REF ), still producing shear values consistent with the results obtained at the SKA1 source density corresponding to a population of about $10^4$ galaxies.",
"For this population we measure the shear also for input reduced shear values with amplitude $g=0.04$ and eight different orientations.",
"The input shear $\\mathbf {g}$ action on the intrinsic galaxy ellipticity $\\mathbf {e}^s$ is simulated following [44]: $\\mathbf {e} = \\frac{\\mathbf {e}^s + \\mathbf {g}}{1+\\mathbf {g}^*\\mathbf {e}^s},$ where $\\mathbf {g}^*$ is the shear complex conjugate.",
"We compare results with the optimal case where each galaxy is at the phase centre and the only one contained in the field of view, considering the same galaxy population.",
"Results are plotted in Fig.",
"REF , both for SNR $\\ge 10$ and SNR $\\ge 25$ , where measurements from individual source visibilities are green crosses and the ones from the same population but with all sources simultaneously in the primary beam are black crosses.",
"Clearly error bars (cross arms) are larger at SNR $\\ge 10$ as the galaxy population is dominated by lower flux sources.",
"Figure: Comparison of shear measurements: input values are blue points, measured values from single sources at the phase centre are green crosses, measured values from sources simultaneously in the f.o.v are black crosses.Table: Shear bias components estimated from a realistic population of ∼10 4 \\sim 10^4 galaxies randomly distributed in 1 deg 2 \\deg ^2, corresponding to a source number density of 2.8 gal/arcmin 2 ^2.The measured shear bias, defined as $g_i^m - g_i = m_i g_i + c_i, \\qquad i=1,2,$ is shown in Table 2.",
"At SNR $\\ge 10$ , the multiplicative biases $m_i$ for the two shear components are respectively 1.6 and 1.4 times the ideal case of a single source in the field of view, while additive bias $c_i$ are almost consistent.",
"Selecting galaxies with SNR $\\ge 25$ the population reduces to 5810 sources (i.e.",
"1.6 gal/arcmin$^2$ ).",
"This should affect the bias uncertainty only, as the bias on the shear measurement should not depend on the number of sources.",
"At this regime the $m$ values reduce by a factor two instead of three as in the single source case.",
"This is due to the source signal contamination by residuals of nearby galaxies, which is a new contribution to the shear bias that seems to have an effect on the multiplicative terms only.",
"It may be mitigated by refining ellipticity measurements by joint fitting within larger facets, as explained in Section REF .",
"Note that ”neighbour bias” affects optical lensing measurements also (see [31], [23], [26], [41] and [49]).",
"As discussed in Paper I, the noise bias values exceed SKA1 survey requirementsFor a 2-yr SKA1-MID weak lensing survey over 5000 $\\deg ^2$ and $z_\\mathrm {med} = 1.0$ the requirements for cosmological parameters measurements to be dominated by statistical rather than systematic errors are: multiplicative bias $m < 0.0067$ , additive bias $c < 0.00082$ [9].",
"They are derived using the rules provided in [1].",
"except for the additive component at SNR $\\ge 25$ .",
"However they are comparable to the ones obtained from optical surveys using lensfit [17] and im3shape [49], where a shear calibration correction reduced the multiplicative bias to well below the percent level.",
"Standard approaches in the optical domain derive such calibration by inferring the bias from simulated data matching the observations [10] or parametrising the bias as a function of the observed galaxy properties [24], [23].",
"Recently a self-calibration approach, implemented in the Metacalibration method [22], [45] and used in the analysis of the Dark Energy Surveyhttps://www.darkenergysurvey.org (DES) [49], proved to be the most efficient, being able to recover the input shear in realistic simulations to better than a part in a thousand.",
"It also isotropises the PSF to remove any additive bias.",
"The key idea of the method is to compute the shear estimator response for a shape measurement directly from observed data perturbed with a small known shear.",
"This way all the features present in real data are already included, which are instead extremely difficult to model accurately in external simulations, and it can be applied to any shear measurement method based on averages of galaxy shapes.",
"A similar approach may then be implemented quite easily in the interferometer data analysis, with the advantage that for the additive bias at radio wavelengths we know the PSF much better than at optical wavelengths and we can make the PSF isotropic directly by weighting the uv-plane (as discussed in Section )."
],
[
"Additive bias dependence on the image-plane PSF shape",
"It is well known from weak lensing optical surveys that shear additive bias is dependent on the PSF shape [31].",
"For radio interferometers the PSF is deterministically defined by the uv coverage of the telescope (i.e.",
"antennae locations and pointing direction).",
"For example, if we increase the antenna pointing declination to a larger zenith distance the PSF shape becomes compressed along the y-axis.",
"Figure: PSF ellipticity components versus the zenith distance at the starting of the observation.We measure the additive noise bias for different pointing declinations at various SNR values.",
"Starting from our uv coverage (corresponding to the same declination as the observatory latitude), we simulate the effect on the uv points when the phase centre declination is increased by an angle $\\phi $ .",
"A plot of the image-plane PSF ellipticity components as functions of the zenith distance is given in Fig.",
"REF .",
"The R-squared size of the PSF slightly increases from 14.83 arcsec$^2$ to 15.54 arcsec$^2$ .",
"These values are computed from the quadrupole moments of the image domain as follows [43]: $& \\mathbf {e} = \\frac{Q_{xx}-Q_{yy}+2\\mathrm {i}Q_{xy}}{Q_{xx}+Q_{yy}+2(Q_{xx}Q_{yy}-Q^2_{xy})^{\\frac{1}{2}}}, \\\\& R = Q_{xx}+Q_{yy}.$ We simulate individual source visibilities, to avoid nearby source contamination effects, and assume a constant maximum facet size $1000 \\times 1000$ to ensure that the galaxy convolved with the PSF is contained in the facet even when the PSF becomes highly anisotropic.",
"Figure: Shear additive bias as a function of the zenith distance at different lower signal-to-noise.",
"Facet size fixed at 1000×10001000 \\times 1000.Figure: Shear additive bias at SNR ≥10\\ge 10 as a function of the zenith distance.",
"Left: facet size dependent on source flux according to Table for all declinations.",
"Right: facet size increased with source declination produce similar results as for a constant large facet size.We observe that the additive bias is dependent on source size.",
"In fact measurements at the same SNR obtained by lowering the noise instead of increasing the source flux cut, produce larger bias values, meaning that the additive bias worsens when source sizes decreases.",
"This is consistent with the analysis presented in [28].",
"Noise bias causes a correlation between measured shear and PSF ellipticity even when we correct for the PSF in the model fitting.",
"This becomes noticeable at low SNR, where the first ellipticity component increases significantly towards larger negative values, as the PSF becomes more compressed along the y direction (see left panel of Fig.",
"REF ).",
"At large SNR the additive bias almost disappears independently of the PSF shape (see right panel of Fig.",
"REF ).",
"This is in good agreement with what we should expect.",
"Because of our long integration time the PSF is not isotropic even when the declination equals the observatory latitude, as in the baseline simulation, besides the fact that the distribution of the SKA-MID baselines on the ground are not circularly symmetric.",
"Therefore a large additive bias is still measured at small zenith distances.",
"The PSF anisotropy may be reduced by combining snapshots obtained over a range of hour angles, however this may not be sufficient to reach an additive bias acceptable for weak lensing surveys.",
"To further reduce the noise bias at low SNR we also need to weight the uv plane to ensure that the PSF is more isotropic.",
"A standard technique in radio imaging to improve PSF shape is to use tapering functions [7] to define uv points weights, although a more specific weighting scheme may be required.",
"Moreover, as for the multiplicative bias, we can calibrate the additive shear bias with simulations.",
"This is more feasible than in optical surveys because the PSF is deterministic at radio wavelengths.",
"However, any such calibration would be strongly dependent on distributions of source properties, so isotropising the PSF is a much better option.",
"Note that when using variable size faceting, the facet size must be dependent not only on the source size but also on the PSF shape.",
"In fact as the PSF becomes anisotropic the facet size may become too small relative to the size of the source convolved with the PSF, modifying the effective shape of the source.",
"For example the left panel of Fig.",
"REF shows what happens at SNR $\\ge 10$ when we maintain the same flux dependent facet size (Table REF ) for all pointing declinations: at large zenith distances the shortest baselines occupy smaller uv frequencies and therefore can measure wavelengths longer than the limit imposed by the small facet size used to extract the majority of the galaxy population.",
"If we increase the facet size according to the source declination, e.g.",
"50 cells per side every $10^\\circ $ declination increment, we obtain consistent results with the case of one single large facet (see right panel of Fig.",
"REF )."
],
[
"Conclusions",
"We have extended the presentation of the RadioLensfit method, introduced in Paper I for the simple case of individual galaxies located at the phase centre, to the real case where many galaxies are randomly located in the field of view.",
"This has been done by isolating the visibilities of each source and shifting the phase centre to the source position so that a coarse grid can still be used to reduce nearby galaxy residuals contamination and accelerate ellipticity measurement computation.",
"Source extraction has been performed by removing apart from the data the simulated visibilities of the sky model, but the source of interest, given the positions and fluxes of all sources in the field of view from the image, and down-weighting what remains of nearby source-contamination by averaging visibilities in a coarse grid (facet).",
"For gridding we adopted a natural weighting to maximise sensitivity and estimated the smallest facet size dependent on source flux thresholds in order to minimise the number of nearby galaxies included in the facet.",
"We tested the source extraction procedure, simulating visibilities of SF galaxy populations observed by SKA1-MID in the first 30 per cent of frequency Band 2.",
"We adopted flux and scalelength parameters distributions estimated from the VLA SWIRE catalogue and used the lensfit ellipticity prior with coefficients fitted from a new version of the VLA COSMOS catalogue optimised on shape measurements.",
"We showed the efficacy of our source extraction algorithm as a function of the source number density, obtaining a reliable galaxy ellipticity fitting for the density expected from the current proposal of the SKA1 radio weak lensing survey.",
"Shear measurements from eight-hour observation of one square degree show that the bias due to the extraction procedure mainly affects the multiplicative bias as no significant change has been observed for the additive bias when comparing with the bias obtained for the ideal case of a single source at the phase centre at a time.",
"This bias may be mitigated by a second step in the galaxy ellipticity measurement, where a joint fitting within the facets is performed with HMC, starting from the values obtained with RadioLensfit.",
"However multiplicative noise bias calibration is also required as for the optical domain.",
"We finally observed that because of our uv coverage the PSF is slightly anisotropic even if pointing close to the zenith, therefore we obtain a large additive bias (on average about $0.0068 \\pm 0.0006$ at SNR $\\ge 10$ ).",
"Although a suitable choice of the integration time (split and distributed along a longer period of time) may reduce the PSF anisotropy, a uv weighting scheme may still be required to satisfy weak lensing requirements.",
"It should be optimised to avoid any significant reduction of the signal-to-noise."
],
[
"Acknowledgements",
"We thank Tom Kitching for useful discussions and Ben Tunbridge for providing the distribution parameters of our ellipticity prior, obtained by fitting VLA-COSMOS data as for the prior function presented in his paper.",
"We also thank Sphesihle Makhathini for the support in the simulation of the SKA1-MID uv coverages.",
"MR acknowledges the support of the Science and Technology Facilities Council (STFC) via an SKA grant.",
"LM acknowledges STFC grant ST/N000919/1."
]
] | 1709.01827 |
[
[
"CONE: Community Oriented Network Embedding"
],
[
"Abstract Detecting communities has long been popular in the research on networks.",
"It is usually modeled as an unsupervised clustering problem on graphs, based on heuristic assumptions about community characteristics, such as edge density and node homogeneity.",
"In this work, we doubt the universality of these widely adopted assumptions and compare human labeled communities with machine predicted ones obtained via various mainstream algorithms.",
"Based on supportive results, we argue that communities are defined by various social patterns and unsupervised learning based on heuristics is incapable of capturing all of them.",
"Therefore, we propose to inject supervision into community detection through Community Oriented Network Embedding (CONE), which leverages limited ground-truth communities as examples to learn an embedding model aware of the social patterns underlying them.",
"Specifically, a deep architecture is developed by combining recurrent neural networks with random-walks on graphs towards capturing social patterns directed by ground-truth communities.",
"Generic clustering algorithms on the embeddings of other nodes produced by the learned model then effectively reveals more communities that share similar social patterns with the ground-truth ones."
],
[
"Introduction",
"One of the most popular topics in network research is to identify communities.",
"On the one hand, as networks are growing larger than ever before, it is efficient and necessary to look into smaller sub-networks, which consist of specific groups of interacting objects (i.e., nodes) and their links (i.e., edges).",
"On the other hand, the knowledge of community structures allows us to better understand the status of an object within a group and the relations between it and its peers, so as to enable multiple benefits including the discovery of functionally related objects [1], the study of interactions between modules [2], the inference of missing contents [3], the prediction of unobserved connections [4] and so on.",
"Many algorithms aim to solve this problem [5], [6], [7], [8], [9], [10], [11].",
"However, they are all formulated based on the heuristic assumptions of edge density and node homogeneity, i.e., communities are constructed by densely connected nodes and nodes within one community are all homogeneous in some ways.",
"Most surprisingly, even the ground-truth community labels used for evaluation in many standard datasets are generated by machines rather than humans based on these two assumptions [12].",
"In this work, we doubt the universality of these widely trusted assumptions.",
"To see how these two assumptions do not necessarily hold true, consider the scenario in Figure REF , where two groups of students and professors are circled by dashed lines.",
"Each of the two is naturally a valid community as everyone is affiliated to the same research group.",
"However, they are not always densely connected, as it is possible that students work on their individual projects and hardly meet each other.",
"The community is not homogeneous in a specific way either.",
"Students may have similar age range, salary range and come from the same hometown, but these might be quite different from those of the professors.",
"Therefore, unsupervised community detection algorithms based on edge density and node homogeneity can hardly detect communities like these.",
"Figure: Toy example of communities in a CS departmentAlthough neither of the density and homogeneity characteristics is prominent in the communities in Figure REF , we observe that there are indeed some interesting social patterns.",
"For example, a pattern of star is clearly formed within the communities, and the secretory obviously acts as a bridge between communities.",
"Those patterns are characterized by both link structures and user contents.",
"For instance, the professor as a star center is distinctive in ages from the students around but may possess similar other contents such as research interest and department.",
"The professor is also densely connected with the students around, but the students may not directly work with each other.",
"On the other hand, the secretory as a bridge shares similar contents like department with all others, while also connects to many of them.",
"Nevertheless, she/he belongs to neither of the two groups.",
"Clearly, communities are defined and should be detected by the underlying social patterns.",
"Intuitively, a social pattern as we want to leverage should be formed by users with certain contents into specific local structures, and it should appear often in communities.",
"However, while we can think of some intuitive ones like those in Figure REF , the actual social patterns should be fuzzy, complex and varying across networks.",
"It is impractical to enumerate all possible ones and design an unsupervised algorithm to detect communities of all patterns.",
"Moreover, traditional ways of finding and matching patterns on even small networks are notoriously time-consuming and almost impossible on real-world large social networks [13].",
"To the best of our knowledge, there exists no previous work that aims to leverage such fuzz, complex and varying patterns on networks.",
"In this work, we propose to inject weak supervision into community detection by learning an embedding model that automatically captures fuzzy social patterns.",
"Instead of taking pre-defined assumptions about community characteristics, it is more reliable to learn from the data what the underlying social patterns look like.",
"Specifically, given a network, we propose to leverage a few labeled communities as examples, and automatically explore and memorize their important social patterns.",
"The model can then easily compute the embeddings on other parts of the network or other similar networks, which can then be leveraged for the detection of unlabeled communities there.",
"For example, in Figure REF , it is intuitive to leverage the star pattern learned from one community to detect the other, and the patterns learned from research groups in the CS department can be well leveraged to detect research groups in other departments, shools, etc.",
"While network embedding has attracted intense research attention recently [14], [15], [16], [17], [18], a supervised embedding directed by social patterns is non-trivial and has never been explored.",
"The task is challenging due to the lack of an efficient way to explore and combine complex user contents, network structures and community labels into a unified learning framework that effectively captures social patterns.",
"In this work, we develop an end-to-end deep architecture of Community Oriented Network Embedding (CONE).",
"The key advantages of CONE over some existing network embedding techniques are as follows: Content expressive: to deeply explore and understand high-dimensional noisy contents as signatures of social patterns, we start from content embedding, by decomposing it into multiple nonlinear layers of recurrent neural networks (RNN).",
"Structure aware: to leverage links and capture social patterns with local structures, we design a network regularization layer, which essentially generalizes the embedding from single users to local structures according to high-order random-walk transitions on the graph.",
"Community oriented: to exploit example communities, we connect a softmax supervision layer into an end-to-end framework to directly require users within the same communities and thus forming some special social patterns to be close in the embedded space.",
"In this way, the supervision can be properly back propagated to the content embedding layer via the network regularization layer, which explores their mutual reinforcement and reliably captures prominent social patterns.",
"Out-of-sample: CONE directly learns an embedding model, rather than the embeddings of a specific set of nodes.",
"Therefore, it is able to handle the out-of-sample problem, which addresses the challenges of limited supervision and dynamic network."
],
[
"Motivating Study",
"In this section, we show the deficiency of traditional unsupervised community detection algorithms by quantitively examining the density and homogeneity assumptions they adopt and their consequences.",
"On the contrary, we demonstrate that there are indeed some communities that are neither dense nor homogeneous, but they share some underlying social patterns.",
"We conduct a set of analysis using the Facebook dataset described in [8].",
"This dataset includes 10 ego-networks, consisting of 193 social circles and 4,039 nodes.",
"10 ego users have manually identified all the communities to which their friends belong.",
"The average number of social circles in each ego-network is 19, with an average community size of 22 users.",
"We trust these manually labeled social circles and use them as the ground-truth communities.",
"Let a graph $G=\\lbrace V, E\\rbrace $ represent a network with a vertex set $V$ and an edge set $E$ .",
"A community can be represented as a sub-graph $C=\\lbrace V_C,E_C\\rbrace $ with $V_C \\subset V$ and $E_C \\subset E$ .",
"$\\mathbf {a}_i$ is the content vector of a vertex $v_i \\in V$ .",
"We evaluate the density and homogeneity of both ground-truth communities and communities detected by the state-of-the-art algorithms.",
"Metrics.",
"We use a density metric $D(\\cdot )$ and a homogeneity metric $H(\\cdot )$ defined as follows: $D(C) = \\frac{2|E_C|}{|V_C|(|V_C| - 1)}.$ The value of $D(C)$ ranges from 0 (a graph without edges) to 1 (a complete clique).",
"It is a standard measure of the density of a network widely used in related literature [12].",
"$H(C) = \\frac{\\sum \\limits _{v_i,v_j \\in V_C, v_i\\ne v_j}\\sigma (\\frac{\\mathbf {a}_i^T \\mathbf {a}_j}{avg(\\mathbf {a}_i^T\\mathbf {a}_j)})}{|V_C|(|V_C|-1)},$ where $avg(\\mathbf {a}_i^T \\mathbf {a}_j)$ is the average of content similarity among all pairs of nodes in the whole network $G$ , and $\\sigma (t)= \\frac{1-e^{-t}}{1+e^{-t}}$ is the adjusted half sigmoid function in the range of $[0,1]$ .",
"The value of $H(C)$ also ranges from 0 (none of the nodes share the same contents) to 1 (all nodes share the same contents), while the value increases rapidly as the content similarity in $C$ is small compared with the average content similarity in the whole network $G$ .",
"Compared algorithms.",
"We study three mainstream community detection algorithms within the state-of-the-art: MinCut [7] (based on network modularity), CESNA [10] (based on probabilistic generative model) and InfoMaps [19] (based on information theory).",
"Density and homogeneity analysis.",
"Assumption 1: nodes within the same community are densely connected.",
"Conflicting evidence: In Figure REF , we present $D(C)$ computed on the ground-truth communities as well as the communities detected by the state-of-the-art algorithms in ego-network 686 as an example.",
"It can be observed that ground-truth communities do not always have a high density.",
"In fact, it is possible that there are more low-density communities than high-density ones as in (a).",
"The evidence directly contradicts with Assumption 1.",
"Based on this inaccurate assumption, traditional unsupervised community detection algorithms tend to detect communities with high density.",
"As an example, the distributions of $D(C)$ computed on the detected communities in the same ego-network are presented in Figure REF (b)-(d).",
"It can be observed that most of the detected communities have a high density ($D(C)>0.5$ ), which is inconsistent with the density distribution of the ground-truth communities.",
"It suggests that the inaccurate assumption of edge density indeed leads to unreliable community detection results.",
"Figure: InfoMapFigure: InfoMapsAssumption 2: nodes within the same community are homogeneous in some ways.",
"Conflicting evidence: As an example, the distribution of $H(C)$ computed on the ground-truth communities in ego-network 698 is contradictory to Assumption 2.",
"As shown in Figure REF (a), few communities are indeed homogeneous (e.g., the one with homogeneity higher than $0.8$ ), while the majority are much less homogeneous.",
"Given $\\sigma (1)\\simeq 0.46$ , the results indicate that most communities have a homogeneity similar to that of the whole network.",
"We conduct the same homogeneity analysis on communities detected by the same group of algorithms.",
"As can be seen in Figure REF (c)-(d), since CESNA and InfoMaps assume node homogeneity in communities, they do detect communities of slightly higher homogeneity, which diverge from the ground-truth ones.",
"MinCut does not assume node homogeneity.",
"Figure: Cmt 15 in ego-net 3437Real community social pattern analysis.",
"After showing that nodes within the same communities are not always densely connected or homogeneous in some ways, we look into the structures of ground-truth communities to find evidence for the existence of social patterns, so as to further motivate our novel approach of community detection.",
"In Figure REF , among many patterns and examples, we show two communities that clearly share a co-star pattern, where almost all members connect to two center nodes, while they do not densely connect within themselves.",
"Moreover, the center nodes in both communities indeed have some special contents such as a certain degree in the college or job in the company, which are not shared by other non-center members.",
"(We removed a few isolated noise nodes for clear visualization.)"
],
[
"Overall Framework",
"The analysis in Sec.",
"clearly demonstrates the deficiency of existing unsupervised community detection algorithms, which is a consequence of their falsifiable assumptions of edge density and node homogeneity.",
"As we motivated in Sec.",
", the existence of fuzzy, complex and varying social patterns underlying communities further indicates an urge for developing a novel community detection algorithm that aims at leveraging such patterns.",
"In this work, rather than enumerating and evaluating all important social patterns, we propose to automatically capture them from labeled communities through weak supervision.",
"Instead of exhaustive graph matching, we model the problem as representation learning.",
"Specifically, we do not care about the exact shapes of the patterns, but we intuitively require the users within the same patterns to be close in an embedded space.",
"We formulate our framework as follows.",
"We are given a network modeled by $\\mathcal {G}=\\lbrace \\mathcal {V},\\mathcal {E},\\mathcal {A},\\mathcal {C}\\rbrace $ , where $\\mathcal {V}$ is the set of $n$ users, $\\mathcal {E}$ is the set of observed links among $\\mathcal {V}$ .",
"$\\mathcal {A}$ is the set of observed user contents on $\\mathcal {V}$ , where $\\mathbf {a_i}$ is the set of contents on user $v_i$ .",
"$\\mathcal {C}$ is a set of ground-truth community memberships, where $\\mathbf {c}_k$ is the set of users in community $c_k$ .",
"For learning the model, only a small number of ground-truth communities are required as examples, while others are used for evaluation.",
"To detect more communities, we firstly learn an embedding model that captures social patterns in $\\mathcal {G}$ based on $\\mathcal {A}$ , $\\mathcal {E}$ and $\\mathcal {C}$ .",
"The model should effectively explore various social patterns and transform each user into a $p$ -dimensional vector.",
"In the embedded space, users forming some important social patterns and thus within each example community are close.",
"More importantly, the model should be able to leverage the social patterns and produce embeddings on all users in $\\mathcal {V}$ in a similar way.",
"Therefore, new communities can then be detected through generic clustering algorithms such as $k$ -means on the $p$ -dimensional features.",
"Overlapping communities can also be discovered by algorithms like MOC [20].",
"Figure: The end-to-end deep architecture of Community Oriented Network Embedding (CONE).Figure REF illustrates the overall architecture of our CONE model.",
"We take the input of $\\mathcal {A}$ for all users $\\mathcal {V}$ and compute the content embeddings $\\mathbf {H}$ , which is then regularized by network structures $\\mathcal {E}$ to yield the community oriented embeddings $\\mathbf {S}$ .",
"For representation learning, $\\mathbf {S}$ is then used to generate community predictions $\\mathbf {Y}$ , which is supervised by the ground-truth community labels $\\mathbf {L}$ derived from the example communities in $\\mathcal {C}$ .",
"$\\mathbf {L}$ can be either point-wise, where $l_{ik}=1$ means $v_i$ is a member of community $c_k$ , or pair-wise, where $l_{ij}=1$ means $v_i$ and $v_j$ are in the same community.",
"To detect more communities, we take $\\mathbf {S}$ as the actual output of CONE and apply generic clustering algorithms like $k$ -means on it.",
"To ensure a desirable representation, ideally we should require users within the same example communities indicated by $\\mathbf {L}$ , and thus forming some specific social patterns, to be close in the embedded space.",
"That is, we need to compute the pair-wise loss among all users and our overall high-level objective function should look like the following, $\\mathcal {J}=\\Phi (\\mathbf {Y},\\mathbf {L})=\\sum _{(v_i,v_j), i\\ne j}\\phi (\\hat{y}_{ij}, l_{ij}).$ where $\\Phi $ is a loss function, $\\mathbf {L}$ is the pair-wise community labels and $\\mathbf {Y}$ is the pair-wise prediction.",
"In what follows, we further explain the reasons for our model architecture and how it works in details."
],
[
"Deep Architecture",
"While it is intuitive to automatically learn an embedding model that captures important social patterns, the task is non-trivial and posts some unique challenges: [leftmargin=20pt] Explore high-dimensional noisy user contents.",
"Leverage complex links and local structures.",
"Exploit limited supervision of example communities.",
"To deal with all challenges above, we develop a deep architecture of CONE, which inherently combines RNN with random-walks in an end-to-end supervised learning framework.",
"RNN: deeply explore and understand user contents as signatures of social patterns.",
"Existing network embedding algorithms are insufficient in exploring contents for capturing social patterns.",
"They usually focus on preserving the link structures among users [14], [16], [15], and incorporate user contents as augmented attribute nodes [21], text feature matrices [17] or bag-of-word vectors [18].",
"Thus, the deep semantics within user contents are not fully explored.",
"In our situation, contents are so important that they often become the signatures of social patterns.",
"For example, in a football fans' club, a popular player identified by the contents of his tweets is likely to be the center, surrounded by groups of fans identified by their semantically different tweet contents.",
"However, exploring and understanding user contents in social networks is a non-trivial task, since they can be complex, noisy and high-dimensional.",
"E.g., in text-rich networks like Twitter, contents can be a list of recent tweets; in tag-rich networks like Flickr, they can be a list of frequently used hashtags in a timely order; in networks with explicit attributes like Facebook and LinkedIn, they can be a set of categorical variables like Birthday, School, Occupation with lots of noisy and missing data.",
"The situation reminds us of the task in natural language processing (NLP) of understanding noisy text sequences, where the semantics is usually hidden in ordered tokens of variable lengths.",
"In this work, instead of starting from the links, we start from deeply modeling user contents.",
"To this end, we employ RNN from NLP, which has been proven advantageous over various other methods in understanding text-like sequences, due to its supreme expressiveness within the neural networks to explore, understand and memorize important semantic patterns.",
"The deep learning framework of RNN also provides us with the flexibility to modify the neural network architectures in order to leverage other information like network structures and community supervision.",
"To the best of our knowledge, this is the first work that models user contents as raw texts and successfully applies RNN to network embedding.",
"To leverage RNN, given a user $v_i$ 's textual contents such as a list of recent tweets, we can concatenate them into a single sequence $\\mathbf {s}_i$ , each element of which is the index of the corresponding word after stemming and stop word removing; given $v_i$ 's categorical contents $\\mathbf {a}_i$ such as education and work on Facebook, we transform it into a sequence $\\mathbf {s}_i=\\lbrace j| a_{ij} = 1\\rbrace $ .",
"The sequence $\\mathbf {s}_i$ is then used as the input of RNN.",
"As mentioned in [22], simple RNN would be difficult to train due to the resulting long term dependencies.",
"Therefore, we use the long-short term memory (LSTM) cells [23] instead to embed $\\mathcal {A}$ .",
"The architecture and implementation of LSTM can be found in the public websitehttp://deeplearning.net/tutorial/lstm.html.",
"Upon each input $\\mathbf {s}_i$ , there will be one output from the LSTM cell as a semantic embedding of user contents.",
"To further improve model expressiveness, we use $d$ LSTM cells to output $d$ embedding vectors $\\lbrace \\mathbf {h}_i^1, \\mathbf {h}_i^2,\\ldots , \\mathbf {h}_i^d\\rbrace $ , and apply mean pooling on them, which is commonly used to integrate features and reduce deviation.",
"So we have $\\mathbf {h}_i=h(\\mathbf {s}_i) = \\frac{1}{d}\\sum _{j=1}^d \\mathbf {h}_i^j,$ where $h(\\cdot )$ denotes the overall deep content embedding function.",
"Supposing the embedding size of each LSTM cell is $p$ , we get a content embedding matrix $\\mathbf {H} \\in \\mathbb {R}^{p\\times n}$ to represent the embeddings of $\\mathcal {V}$ , where the $i$ -th column of $\\mathbf {H}$ equals to $\\mathbf {h}_i$ .",
"Note that, while RNN is especially useful for exploring complex text-like sequences, for simple numerical or categorical contents like user attributes, it also makes sense to use simpler models like feedforward neural networks, which can be incorporated into our end-to-end framework in the same way.",
"We will also show the performance of such basic neural networks in our experiments.",
"Random-walk: leverage links and capture social patterns with local structures.",
"The objectives of existing network embedding algorithms are not appropriate for leveraging network structures to capture social patterns.",
"They usually model network structures by sampling a set of paths from the networks and applying a Skipgram-based model [24] to uniformly require the embeddings of nodes that share similar graph context to be similar [14], [15], [17], [18].",
"To capture social patterns, we want the user embedding to be guided by community supervision.",
"Users having similar network context do not necessarily belong to the same communities, and thus should not be required to have similar embeddings without discrimination.",
"Besides, by sampling the networks into paths, structural information beyond paths is not efficiently leveraged.",
"In our situation, the shapes of local network structures are extremely important, since they may well indicate the existence of specific social patterns, like the star and co-star shapes as we mentioned in Sec.",
"and .",
"In this work, we directly leverage local structures to regularize the supervision of example communities.",
"To this end, we armor our RNN-based content embedding with random-walk-based network regularization.",
"It efficiently generalizes the embeddings on single users to their ambient local structures, allowing RNN to explore content embedding under the regularization of network local structures and the guidance of example communities.",
"Consider the RNN-based content embedding.",
"Due to Eq.",
"REF , if $\\mathbf {a}_i$ is similar to $\\mathbf {a}_j$ , then it is likely that the embeddings $\\mathbf {h}_i$ and $\\mathbf {h}_j$ are also close.",
"However, this may not be ideal, because similar users can well form different communities.",
"The key question is, are they also on the same local structure?",
"To account for this, we insert a random-walk-based network regularization layer, which recomputes the embedding of each user w.r.t.",
"her neighbors, according to their distances measured by random-walks of $k$ steps, i.e., $\\mathbf {s}_i=\\sum _{j=1}^n t^k_{ji}\\mathbf {h}_j$ , where $t^k_{ji}$ is the $k$ -step random-walk transition probability from $v_j$ to $v_i$ .",
"In matrix form, a more compact formula is $\\mathbf {S}=\\mathbf {H}\\mathbf {T}^k$ .",
"$t_{ij}=w_{ij}/d_i$ , where $w_{ij}$ is the binary or real-valued weights on edge $e_{ij}$ and $d_i=\\sum _{j}w_{ij}$ .",
"Intuitively, each $v_i$ `collects' the embeddings transmitted from its local neighbors.",
"Compared with $\\mathbf {h}_i$ , $\\mathbf {s}_i$ encodes the local structure around $v_i$ , rather than just the semantic information from $\\mathbf {a}_i$ .",
"As a consequence, only users with similar contents as well as local structures will be embedded as close.",
"Moreover, consider the loss in Eq.",
"REF brought by the supervision of example communities.",
"Without network regularization, the loss on content embeddings can be formulated as $\\mathcal {J}_H=\\sum _{(v_i,v_j),i\\ne j} \\phi (\\hat{y}(h(\\mathbf {a}_i),h(\\mathbf {a}_j)), l_{ij}).$ Under $\\mathcal {J}_H$ , e.g., if there are some professors and students in the same example communities, all professors and students will be required to get similar content embeddings, which should not be the case.",
"Instead, we apply supervision on the regularized embeddings as $\\mathcal {J}_S=\\sum _{(v_i,v_j),i\\ne j} \\phi (\\hat{y}(\\mathbf {s}_i,\\mathbf {s}_j), l_{ij}).$ As a consequence, only professors and students connected in certain local structures indicated by supervision are embedded as close.",
"End-to-end learning: exploit community supervision and the mutual reinforcement between contents and links.",
"As we discussed about Eq.",
"REF before, to directly meet our goal of embedding users in the same example communities to be close, we need to construct a pair-wise loss on each pair of users in the same communities.",
"However, pair-wise loss functions can not be built into our end-to-end embedding framework and permit efficient training of the neural networks.",
"Moreover, converting the community memberships to pair-wise 0-1 labels will lead to significant information loss, especially for overlapping communities.",
"To overcome these difficulties, we find the point-wise softmax prediction with cross entropy loss as a suitable substitute to the exact pair-wise loss [25].",
"While having a different objective, softmax with cross entropy basically ensures that instances predicted with the same label are close in a space that can be viewed as a linear projection of the original embedding space.",
"Therefore, it can be viewed as achieving a similar goal as the direct pair-wise loss.",
"Moreover, softmax is commonly used in end-to-end deep learning frameworks to predict multiple labels, and the cross entropy loss can be efficiently back propagated through neural networks.",
"By allowing the training on multi-label predictions, it is also able to leverage overlapping example communities and distinguish among different communities.",
"Therefore, we incorporate a softmax supervision layer to form our end-to-end deep learning framework, where community supervision can be exploited as a guidance for exploring social patterns, and the mutual reinforcement between user contents and link structures can be leveraged.",
"Our practical objective function based on cross entropy loss is as follows: $\\hat{y}_{ik} = \\frac{\\exp (\\mathbf {w}^T_k\\mathbf {s}_i)}{\\sum _{k^{\\prime }=1}^K\\exp (\\mathbf {w}^T_{k^{\\prime }}\\mathbf {s}_i)},\\quad \\mathcal {J} = \\sum _{i=1}^n\\sum _{k=1}^K l_{ik} log(\\hat{y}_{ik}),$ where $\\mathbf {w}$ 's are the model parameters in the softmax layer and $K$ is the total number of example communities.",
"$\\mathbf {Y}$ and $\\mathbf {L}$ encode the predicted and ground-truth community labels, respectively, where $\\hat{y}_{ik}=1$ means $v_i$ is predicted to be a member of community $c_k$ , and $\\mathbf {L}$ is the point-wise community labels.",
"To optimize Eq.",
"REF , we employ stochastic gradient descent (SGD) with the diagonal variant of AdaGrad from [26].",
"At the $t$ -th step, the parameters $\\Theta $ is updated by: $\\Theta _{t} \\leftarrow \\Theta _{t-1}-\\frac{\\rho }{\\sqrt{\\sum _{i=1}^{t}g_{i}^{2}}}g_{t},$ where $\\rho $ is the initial learning rate and $g_{t}$ is the sub-gradient at time $t$ .",
"Since our network regularization layer is implemented as a matrix multiplication, the gradients can be efficiently back-propagated to the content embedding layer, and the overall embedding framework is end-to-end.",
"When detecting communities in a network, embeddings of users produced by the CONE model learned on labeled data are fed into generic clustering algorithms like $k$ -means.",
"We apply cross-validation to automatically choose the optimal number of communities as done in [10].",
"The efficiency of CONE: CONE is an end-to-end deep learning framework implemented using TensorFlowhttps://www.tensorflow.org/.",
"Only minimum setup is required to run it efficiently on GPU.",
"We will make the code available upon acceptance of the work."
],
[
"Experiments",
"In this section, we evaluate CONE for community detection with extensive experiments on 3 real-world networks."
],
[
"Experimental Settings",
"Datasets.",
"We use three real-world network datasets.",
"The first is the Facebook dataset we used in Sec.",
"for data analysis, which consists of 10 ego-networks with community labels explicitly provided by the ego users [8].",
"The contents in this dataset are well-defined user profiles such as education, work and location, and links are undirected friendships among users.",
"The second is a Flickr dataset collected by [27].",
"The community labels are generated from the groups joined by users.",
"The contents are the tags aggregated on users' posted photos and the links are undirected friendships.",
"The third is a Twitter dataset also collected by [8], which consists of 973 ego-networks.",
"The community labels are generated from friend circles (or lists), i.e., friends put into the same list by the ego user are regarded as within one community.",
"The contents are the hashtags and popular account mentions within users' recent tweets and the links are directed followings.",
"Detailed statistics of the three datasets we use are shown in Table 1.",
"Table: Statistics of 3 real-world network datasets.Compared algorithms.",
"We compare with two groups of algorithms from the state-of-the-art to comprehensively evaluate the performance of CONE.",
"Community detection algorithms.",
"Some algorithms are based on the edge density assumption alone, while others also assume node homogeneity.",
"We compare with this group of algorithms to show the advantage of abandoning these inaccurate assumptions and leveraging the supervision of example communities.",
"[leftmargin=20pt] MinCut [7]: a classic community detection algorithm based on modularity.",
"BigClam [9]: an advanced community detection algorithm solely based on network structure.",
"Circles [8]: a generative model of edges w.r.t.",
"attribute similarity to detect communities.",
"CESNA [10]: a generative model of edges and attributes to detect communities.",
"SGM [28]: a semi-supervised framework incorporating individual labels and pair-wise constraints.",
"Network embedding algorithms.",
"While we find it intuitive to model the problem as representation learning, we compare with the state-of-the-art network embedding algorithms to show that CONE is advantageous in capturing social patterns.",
"The embeddings learned by all algorithms are fed into the same $k$ -means clustering algorithm as CONE to produce community detection results.",
"[leftmargin=15pt] DeepWalk [15]: an embedding algorithm based on truncated random walks that only considers network structures.",
"node2vec [14]: an embedding algorithm based on $2nd$ order random walks that only considers network structures.",
"TADW [17]: an embedding algorithm that generalizes DeepWalk to consider both node attributes and network structures by tensor factorization.",
"PTE [21]: an embedding algorithm that generalizes LINE [16] to consider node attributes, network structures and class labels by graph augmentation.",
"Planetoid [18]: an embedding algorithm that extends DeepWalk to consider node features, network structures and class labels by jointly predicting labels and contexts.",
"The number of communities to detect is tuned via standard 5-fold cross validation for all algorithms.",
"The implementations of compared algorithms are all provided by the original authors.",
"Metrics.",
"Two widely used metrics for evaluating community detection results are used in our experiments.",
"For a detected community $c_i^*$ and a ground-truth community $c_i$ , the F1 similarity and Jaccard similarity are defined as $F1 = \\frac{2 \\cdot precision \\cdot recall}{precision + recall},$ $Jaccard = \\frac{|c_i \\cap c^*_i|}{|c_i \\cup c^*_i|},$ where $precision = \\frac{c_i \\cap c^*_i}{|c^*_i|}$ , $recall = \\frac{c_i \\cap c^*_i}{|c_i|}$ .",
"For a set of ground-truth communities $\\lbrace c^*_i\\rbrace _{i=1}^M$ and a set of detected communities $\\lbrace c_i\\rbrace _{i=1}^N$ , we compute the score as $\\frac{1}{2|C|} \\cdot \\sum _{c_i \\in C} \\max _{c^*_i \\in C^*} eval(c_i, c^*_i) + \\frac{1}{2|C^*|} \\cdot \\sum _{c^*_i \\in C^*} \\max _{c_i \\in C} eval(c_i, c^*_i),$ where $eval(c_i, c^*_i)$ can be either replaced by $F1$ or $Jaccard$ ."
],
[
"Performance Comparison with Baselines",
"We quantitively evaluate CONE against all baselines on community detection.",
"We randomly split the labeled communities into training and testing sets.",
"We use $10\\%$ of the labeled communities as examples to learn the models for CONE, SGM, PTE and Planetoid.",
"All compared algorithms are then evaluated on the rest $90\\%$ labeled communities.",
"To observe significant difference in performance, we split the training and testing sets 10 times and conduct statistical t-tests with $p$ -value 0.01.",
"Table: Performance comparison on 3 datasets.Table REF shows the average F1 and Jaccard scores evaluated on all compared algorithms over the same 10 random splits of training and testing sets.",
"The results all passed our t-tests.",
"The parameters of baselines are all set to the default values as suggested in the original works.",
"Some numbers are a bit different with those in the original work, because we directly use all node contents as input and only evaluate on the testing set.",
"For CONE, we intuitively use two random transitions in the regularization layer to stress network locality and leverage the link structures in close neighborhoods.",
"CONE reaches around 30% relative improvements on the Facebook dataset and more than 10% improvements on the other two datasets, compared with the second-runner among the 10 baselines on both F1 and Jaccard scores, while they have varying performance.",
"This indicates the robustness and general advantages of our proposed approach.",
"Taking a closer look, we observe that the scores on the Facebook dataset look much better than those on the other two.",
"This is due to the high quality of both contents and links in the dataset.",
"Since the Facebook dataset is the only one that has explicit human labels of communities, the large performance improvement on it indicates the advantage of CONE in leveraging user provided community examples to understand specific social patterns.",
"The scores of network embedding algorithms are lower than traditional community detection algorithms on the Facebook dataset, probably because traditional models work better on few and clean contents.",
"However, as the contents become high-dimensional and noisy like in the Flickr and Twitter datasets, network embedding algorithms excel.",
"This indicates the advantage of leveraging embeddings for traditional social network tasks and further proves the efficacy of CONE in dealing with complex user contents.",
"All experiments are done on a local PC with two 2.5 GHz Intel i7 processors and 8GB memory.",
"The runtime of CONE is comparable to all baselines, while it is trivial to run more efficiently on GPU."
],
[
"Model Selection",
"We comprehensively evaluate the performance of CONE with different amounts of supervision and varying neural architecture.",
"Impact of supervision.",
"We study the impact of supervision on the performance of CONE by varying the training and testing set portions.",
"Figure REF shows the results.",
"CONE's advantage of leveraging community examples is significant on all three datasets.",
"As can be observed, CONE efficiently leverages supervision by learning from very small amounts of labeled communities, and the performance converges rapidly as the training set portion reaches around 11% on the Facabook and Flickr datasets.",
"On datasets with large amounts of labeled communities such as Twitter, around 6% of them are enough to learn a good CONE model.",
"Note that the other three supervised algorithms do not leverage community labels effectively, basically because they are not designed to capture social patterns and leverage them for out-of-sample community detection.",
"Impact of architecture.",
"We study the impact of parameters inside CONE by varying the number of random-walk transitions and the size of embeddings.",
"We also substitute the RNN in the content embedding layer with basic fully connected feed forward neural networks to demonstrate the effectiveness of using RNN to deeply explore the high-dimensional noisy contents.",
"Among the basic neural networks we experimented on with varying layers (from 1 to 4) and embedding sizes (the last layer with sizes of 4, 8, 16 and 32), we show the performance of a three-layer basic neural network (embedding sizes are $64\\rightarrow 32\\rightarrow 16$ from bottom up as a common practice [29]) with ReLU as the activation function, which generally yields the best performance.",
"Figure REF shows the results.",
"The number of random-walk transitions has a large impact on the performance of CONE, especially when the number is small.",
"This demonstrates the utility of our novel network reconfiguration layer.",
"As the number of transitions grows larger and the stationary distribution of random walk is approximated, the performance becomes stable.",
"Note that large numbers of transitions do not necessarily lead to optimal performance, because community structures are often well described within small neighborhoods.",
"The number of embedding dimensions does not have a significant impact on the performance with RNN as the content embedding layer, indicating the robustness of CONE inherited from RNN.",
"However, the performance is significantly worse without RNN as the content embedding layer, which justifies our motivation of leveraging RNN to effectively explore the noisy contents in high-dimensional spaces.",
"The runtime of RNN is also significantly shorter than basic neural networks, especially with more layers and larger embedding sizes."
],
[
"Case Study",
"To understand the advantage of CONE, we conduct the same density and homogeneity analysis on our detected communities and present the results on the same ego-networks we showed in Sec. .",
"Comparing Figure REF (a) and REF (b) with Figure REF (a) and REF (a), we observe that CONE indeed finds communities that are similar in distributions of density and homogeneity as the ground-truth, which indicates its effectiveness in capturing the social patterns that essentially define user-described communities.",
"We also looked into the structures of communities found by different algorithms.",
"Still taking the example of the clear co-star pattern, when we took the ground-truth communities in ego-network 1684 as supervision, CONE indeed found a total of 6 co-star communities on other parts of the dataset, specifically, in ego-networks 0, 107, 348 and 3437.",
"We did not observe communities of such a co-star pattern detected by any other algorithms, from all communities they detected through a single execution on the same training and testing sets.",
"Figure: Homogeneity in ego-net 698Figure: TwitterFigure: Twitter"
],
[
"Community detection",
"Traditional community detection algorithms are mostly unsupervised, based on either of the edge density or node homogeneity assumptions, or both.",
"Therefore, the communities they detect are only constructed by densely connected or homogeneous nodes.",
"Among them, many are based on probabilistic generative models.",
"For instance, COCOMP [11] extends LDA [30] by taking community assignments as latent variables.",
"It leverages node homogeneity by assuming that each community is a group of people that use a specific distribution of words.",
"BigClam [9] does not make use of node content at all.",
"It leverages edge density by modeling the assignments of communities solely based on existing links.",
"A variation of BigClam called CESNA [10] is later proposed, with additional consideration of binary-valued node attributes.",
"Another method within the state-of-the-art, Circles [8], also leverages the same assumptions with a slightly different model.",
"Many non-generative algorithms have also been proposed.",
"Among them, some are based on graph metrics such as modularity [7] and maximal clique [31].",
"They are fundamentally leveraging the edge density assumption.",
"As for node homogeneity, some algorithms firstly augment the graph with content links and then do clustering on the augmented graph [32], [33], while some others attempt to find a partition that yields the minimum encoding cost based on information theory [5], [19].",
"They do not scale trivially to networks with high-dimensional noisy contents.",
"As we leverage whole labeled communities as examples to learn the underlying social patterns, CONE is also different from the few semi-supervised community detection algorithms that leverage individual labels and pair-wise constraints as part of some communities [28]."
],
[
"Network embedding",
"We aim to find node embeddings that are suitable for the task of community detection, under the social pattern assumption.",
"Unlike the techniques that compute embeddings based on node proximities in the network [34], [35], [36], we frame the objective as learning a representation that captures social patterns.",
"Moreover, we aim to learn it under the guidance of communities, instead of the unsupervised ways based on relational data in the networks [37], [38] or structural relationships between concepts [39], [40].",
"As discussed in Sec.",
", the most related techniques to ours are the Skipgram-based network embeddings [14], [15], [16].",
"Recent works [21], [18], [17] have extended such techniques to incorporate node attributes and class labels.",
"Their models using augmented nodes, text feature matrices and bag-of-word vectors are ineffective for deeply exploring high-dimensional noisy contents.",
"Moreover, they all use network context as supervision to uniformly require nodes with similar context to have similar embeddings.",
"Our objective is different, where we only require users within the same communities to be embedded as close.",
"To this end, our supervision is directly guided by community examples, and we leverage network structures only as a regularization.",
"Finally, since they sample networks into paths and predict context based on paths, the techniques are inefficient in capturing local network structures beyond paths.",
"Our regularization efficiently generalizes the embedding of single users to their ambient local structures, which are deterministic rather than approximated by paths."
],
[
"Node classification",
"Traditional supervised learning on networks is essentially node classification [41], [42], [35], [43], which uses node attributes as labels and leverages network structures for tasks like content prediction.",
"Our problem is quite different from node classification.",
"Although we utilize supervision to learn the community oriented features, our final goal is to perform unsupervised detection of unlabeled communities.",
"To be more specific, our predictions are new clusters of nodes, for which no predefined categories or labeled data are available at all.",
"As a consequence, node classification algorithms are not applicable to our problem."
],
[
"Conclusion",
"In this paper, we doubt the generality of the two widely trusted assumptions about community characteristics, i.e., edge density and node homogeneity, and we show the deficiency of mainstream algorithms adopting those assumptions through real data analysis.",
"To deal with this deficiency, we propose to leverage the underlying social patterns that define and detect network communities.",
"We design CONE that effectively explores and captures important social patterns under the guidance of example communities through network embedding.",
"Generic clustering algorithms performed on the embeddings can yield reliable community detection results.",
"While CONE was originally designed for leveraging supervision for reliable community detection free from falsifiable assumptions, it can be easily applied to many network learning tasks to coherently leverage node contents, link structures and various kinds of supervision or constraints.",
"As we observe from our experiments, CONE is especially advantageous in dealing with high-dimensional noisy contents, such as sequences of hashtags and even raw texts."
]
] | 1709.01554 |
[
[
"Ample continua in Cartesian products of continua"
],
[
"Abstract We show that the Cartesian product of the arc and a solenoid has the fupcon property, therefore answering a question raised by Illanes.",
"This combined with Illanes' result implies that the product of a Knaster continuum and a solenoid has the fupcon property, therefore answering a question raised by Bellamy and \\L ysko in the affirmative.",
"Finally, we show that a product of two Smith's nonmetric pseudo-arcs has the fupcon property."
],
[
"Introduction",
"The present paper is concerned with the property of having arbitrarily small open neighborhoods for continua in Cartesian products of continua; i.e.",
"given a continuum $M\\subseteq X\\times Y$ we are interested if (*) for every open neighborhood $U$ of $M$ there exists an open and connected set $V$ such that $M\\subseteq V\\subseteq U$ .",
"The property (*) is closely related to the property of being an ampleThe notion of an ample continuum was introduced by Prajs and Whittington in [8].",
"continuum in the product.",
"Recall that $M$ is ample in $X\\times Y$ provided that for each open subset $U\\subseteq X\\times Y$ such that $M\\subseteq U$ , there exists a subcontinuum $L$ of $X\\times Y$ such that $M\\subseteq \\operatorname{int}_{X\\times Y}(L)\\subseteq L\\subseteq U$ .",
"In fact, according to [1], the two properties are equivalent in the class of Kelley continua.",
"Motivation for the study of ample continua comes from fact that in the hyperspace $C(X\\times Y)$ of subcontinua of $X\\times Y$ ample continua are the points where $C(X\\times Y)$ is locally connected.",
"In this context in [1] Bellamy and Łysko studied the fupconThe abbreviation fupcon stands for full projections imply connected open neighborhoods.",
"It was introduced by Illanes in [6].",
"property of Cartesian products.",
"The product of continua $X\\times Y$ has the fupcon property if whenever $M\\subseteq X\\times Y$ is a continuum with full projections onto coordinate spaces (i.e.",
"$\\pi _X(M)=X$ and $\\pi _Y(M)=Y$ ) then $M$ has the property (*), and the notion naturally generalizes to Cartesian products of more than two continua.",
"Bellamy and Łysko showed that arbitrary Cartesian products of Knaster continua and arbitrary Cartesian products of pseudo-arcs have the fupcon property.",
"Furthermore, the property (*) for subcontinua of such products is in fact equivalent to the property of having full projections onto all coordinate spaces.",
"The authors also showed that the diagonal in a Cartesian square $G$ of a compact and connected topological group has the property (*) if and only if $G$ is locally connected, and therefore if $G$ is a solenoid then $G\\times G$ does not have the fupcon property.",
"Important related results on ample diagonals can be found in the recent work of Prajs [9].",
"Motivated by the aforementioned results, Bellamy and Łysko raised the following question.",
"Question 1.",
"(Bellamy&Łysko, [1]) Let $K$ be a Knaster continuum and $S$ be a solenoid.",
"Does $K\\times S$ have the fupcon property?",
"A partial step towards a solution to the above problem was achieved by Illanes, who showed the following.",
"Theorem A.",
"(Illanes, [6]) Let $X$ be a continuum such that $X \\times [0, 1]$ has the fupcon property.",
"Then for each Knaster continuum $K$ , $X \\times K$ has the fupcon property.",
"Consequently, Question 1 was reduced to the following, potentially simpler problem.",
"Question 2.",
"(Illanes, [6]) Let $S$ be a solenoid.",
"Does $[0,1]\\times S$ have the fupcon property?",
"We answer this question in the affirmative, and in turn obtain positive answer to Question 1.",
"Theorem 1.1 Let $S$ be a solenoid.",
"Then $[0,1]\\times S$ has the fupcon property.",
"Theorem 1.2 Let $S$ be a solenoid and $K$ be a Knaster continuum.",
"Then $K\\times S$ has the fupcon property.",
"In 1985 M. Smith [10] constructed a nonmetric pseudo-arc $\\mathcal {M}$ ; i.e.",
"a Hausdorff chainable, homogeneous, hereditary equivalent and hereditary indecomposable continuum.",
"This continuum has been recently used by the first and third author to provide a new counterexample to Wood's Conjecture in the isometric theory of Banach spaces [2].",
"Relying on the result of Bellamy and Łysko that products of metric pseudo-arcs have the fupcon property, we shall show that their result holds also for products of $\\mathcal {M}$ .",
"Theorem 1.3 Let $\\mathcal {M}$ be Smith's nonmetric pseudo-arc.",
"Any Cartesian power of $\\mathcal {M}$ has the fupcon property.",
"Earlier, Lewis showed [7] that for any 1-dimensional continuum $X$ there exists a continuum $X_P$ that admits a continuous decomposition into pseudo-arcs, and whose decomposition space is homeomorphic to $X$ .",
"Recently, Boroński and Smith [3] extended Lewis' result to continuous curves of Smith's nonmetric pseudo-arc.",
"In particular, given any metric 1-dimensional continuum $X$ there exists a continuum $X_\\mathcal {M}$ that admits a continuous decomposition into nonmetric pseudo-arcs, and whose decomposition space is homeomorphic to $X$ .",
"$X_\\mathcal {M}$ can be seen as “$X$ of nonmetric pseudoarcs”.",
"Here we observe that using the method of proof of Theorem REF one obtains the following generalization.",
"Corollary 1.4 Suppose $X$ and $Y$ are metric 1-dimensional continua such that $X_P\\times Y_P$ has the fupcon property.",
"Then $X_\\mathcal {M}\\times Y_\\mathcal {M}$ has the fupcon property."
],
[
"Proofs",
"(of Theorem REF ) We shall assume that $S$ is the 2-adic solenoid, and give a proof for $[1,2]\\times S$ .",
"For other solenoids the proof is analogous.",
"We shall use the following inverse limit representation of $[1,2]\\times S$ : $[1,2]\\times S=\\lim _{\\leftarrow }\\lbrace [0,1]\\times \\mathbb {S}_i, \\operatorname{id}\\times z^2_i\\rbrace ,$ where $z^2_i:\\mathbb {S}_{i+1}\\rightarrow \\mathbb {S}_i$ is the doubling map on the unit circle.",
"For convenience we set $[1,2]\\times \\mathbb {S}_i=\\mathbb {A}_i=\\lbrace (r,\\theta ):1\\le r\\le 2,0\\le \\theta <2\\pi \\rbrace $ in polar coordinates, and $\\tau _i=\\operatorname{id}\\times z^2_i$ .",
"Then $\\tau _i:\\mathbb {A}_{i+1}\\rightarrow \\mathbb {A}_i$ is the 2-fold covering map given by $\\tau _i(r,\\theta )=(r,2\\theta \\mod {2}\\pi )$ for every positive integer $i$ .",
"Let $M\\subseteq [1,2]\\times S$ be a continuum with full projections onto both coordinate spaces.",
"Let $\\Pi _i:[1,2]\\times S\\rightarrow \\mathbb {A}_i$ be the projection.",
"Claim 2.1 $M_i=\\Pi _i(M)$ is essential in $\\mathbb {A}_i$ for every $i$ .",
"(of Claim REF ) Recall that a continuum $C$ is essential in an annulus $\\mathbb {A}$ if it separates the two components of the boundary.",
"First note that $\\mathbb {A}_{i+j}$ is the $2^{j}$ -fold cover of $\\mathbb {A}_i$ with the covering map given by $\\tau _{i,j}=\\tau _i\\circ \\ldots \\circ \\tau _{i+j}.$ In addition, if $\\tilde{\\mathbb {A}}=\\lbrace (r,\\theta ):1\\le r\\le 2,-\\infty <\\theta <\\infty \\rbrace $ is the universal cover, and $\\phi _k:\\tilde{\\mathbb {A}}\\rightarrow \\mathbb {A}_k$ is given by $\\phi _k(r,\\theta )=(r,2^k\\theta \\mod {2}\\pi )$ then $\\phi _i=\\tau _{i,j}\\circ \\phi _{i+j}$ .",
"Figure: Proof of Claim : an inessential continuum M i M_i with full projections in [1,2]×𝕊 i [1,2]\\times \\mathbb {S}_iBy contradiction suppose $M_i$ is inessential in $\\mathbb {A}_i$ .",
"Then $M_i$ is contained in a closed disk $D_i$ .",
"Since $D_i$ is simply connected, for each positive integer $i$ any component of $\\tau ^{-1}_{i}(M_{i})$ is a homeomorphic copy of $M_i$ .",
"In particular $M_{i+j}$ is homeomorphic to $M_i$ for each $j$ .",
"In addition, $\\lim _{j\\rightarrow \\infty }\\operatorname{diam}(M_{i+j})=0.$ Indeed, since any component $\\tilde{M}$ of $\\phi ^{-1}_i(M_i)$ in the universal cover $\\tilde{\\mathbb {A}}$ is bounded, one can take $j$ large enough so that the projection $N_{i+j}=\\phi _{i+j}(\\tilde{M})$ from $\\tilde{\\mathbb {A}}$ onto $\\mathbb {A}_{i+j}$ is as small as desired.",
"In particular, it is true when $N_{i+j}=M_{i+j}$ .",
"Therefore, there exists a $j_o$ such that the projection of $M_{i+j}\\subseteq \\mathbb {A}_{i+j}$ onto $\\mathbb {S}_{i+j}$ is a proper subset of $\\mathbb {S}_{i+j}$ .",
"This implies that $M$ does not have a full projection onto $S$ , resulting in a contradiction and completing the proof of Claim REF .",
"Claim 2.2 $\\tau ^{-1}_i(M_i)=M_{i+1}$ for every $i$ ; i.e.",
"$\\tau ^{-1}_i(M_i)$ is connected for each $i$ .",
"(of Claim REF ) We use a similar argument to that of Example 1 in [4].",
"By contradiction, suppose $\\tau ^{-1}_i(M_i)$ is disconnected.",
"Without loss of generality let us assume that $i=1$ .",
"Then there are two components $M_2$ and $N_2$ of $\\tau ^{-1}_1(M_1)$ , and each of them maps onto $M_1$ .",
"They are homeomorphic, since the map $\\sigma :\\mathbb {A}_i\\rightarrow \\mathbb {A}_i$ given by $\\sigma (r,\\theta )=\\left(r,(\\theta +2\\pi )\\mod {2}\\pi \\right)$ is a homeomorphism with $\\sigma (M_2)=N_2$ .",
"By Claim REF $M_2$ is essential.",
"Since $M_2$ has a full projection onto $[1,2]$ it must connect the two boundary circles and so there is a point $c\\in M_2\\cap N_2$ .",
"This contradiction implies that $M_2=N_2$ and completes the proof of Claim REF .",
"To finish the proof of Theorem REF let $W$ be an open neighborhood of $M$ .",
"Note that since $\\mathbb {A}_1$ is locally connected, $M_1$ has arbitrarily small connected open neighborhoods.",
"So if $W_1=\\Pi _1(W)$ then there exists a connected open neighborhood $U_1$ such that $M_1\\subseteq U_1\\subseteq W_1$ .",
"Reasoning as above in Claim REF , we deduce that $U_2=\\tau ^{-1}_1(U_1)$ is connected, and then proceeding by induction that $U_{i+1}=\\tau ^{-1}_i(U_i)$ we obtain that $U_{i+1}$ is connected for each $i$ .",
"Consequently $U=(\\tau ^{-1}_i(U_i):i=1,2,\\ldots )$ is an open and connected set such that $M\\subseteq U\\subseteq W$ and Theorem REF is proved.",
"(of Theorem REF ) This follows from Illanes' Theorem A, since by Theorem REF $[0,1]\\times S$ has the fupcon property.",
"(of Theorem REF ) For simplicity of notation we prove it for the product of two Smith's pseudo-arcs.",
"The general case is similar thanks to the result of Bellamy and Lysko for arbitrary products of pseudo-arcs.",
"We consider $\\mathcal {M}$ as the following long inverse limit $\\mathcal {M}=\\lim _{\\leftarrow }\\lbrace P_\\alpha ,p^{\\beta }_{\\alpha }:\\alpha <\\beta <\\omega _1\\rbrace ,$ where each $P_\\alpha $ is a metric pseudo-arc, and $p^{\\beta }_{\\alpha }:P_\\beta \\rightarrow P_\\alpha $ is an open, closed and monotone map, such that $(p^{\\beta }_{\\alpha })^{-1}(x)$ is a pseudo-arc contained in $P_\\beta $ for each $x\\in P_\\alpha $ .",
"Consider the Cartesian square of $\\mathcal {M}$ as the following inverse limit.",
"$\\mathcal {M}\\times \\mathcal {M}^{\\prime }=\\lim _{\\leftarrow }\\lbrace P_\\alpha \\times P^{\\prime }_\\alpha ,p^{\\beta }_{\\alpha }\\times q^{\\beta }_{\\alpha }:\\alpha <\\beta <\\omega _1\\rbrace .$ Let $\\Gamma _\\alpha :\\mathcal {M}\\times \\mathcal {M}\\rightarrow P_\\alpha \\times P^{\\prime }_\\alpha $ be given by $\\Gamma _\\alpha ((\\lbrace x_\\alpha \\rbrace _{\\alpha <\\omega _1},\\lbrace y_\\alpha \\rbrace _{\\alpha <\\omega _1}))=(x_\\alpha ,y_\\alpha ).$ Note that $\\Gamma _\\alpha $ is monotone (i.e.",
"pre-images of points are connected), open and closed for each $\\alpha <\\omega _1$ .",
"Let $M\\subseteq \\mathcal {M}\\times \\mathcal {M}$ be a continuum with full projections onto both coordinate spaces, and $W$ be an open set around $M$ .",
"Then the projection of $W$ onto the square of $\\alpha $ th coordinate spaces $W_\\alpha =\\Gamma _\\alpha (W)$ is an open set around the continuum $M_\\alpha =\\Gamma _\\alpha (M)$ .",
"Since $M_\\alpha $ has full projections onto both coordinate spaces $P_\\alpha $ and $P^{\\prime }_\\alpha $ , by Theorem 4.4 in [1], there exists an open and connected set $V_\\alpha $ such that $M_\\alpha \\subseteq V_\\alpha \\subseteq W_\\alpha $ .",
"By Theorem 6.1.29. in [5], p.358, it follows that $V=\\Gamma _{\\alpha }^{-1}(V_\\alpha )$ is an open and connected set, such that $M\\subseteq V\\subseteq W$ .",
"This completes the proof.",
"The proof of Corollary REF is analogous to the one of Theorem REF , and is left to the reader.",
"We conclude with the following questions.",
"Question 3.",
"(Bellamy&Łysko, [1]) Does the product of two nonhomeomorphic solenoids have the fupcon property?",
"Question 4.",
"(Illanes [6])Let $X$ and $Y$ be chainable Kelley continua.",
"Does $X\\times Y$ have the fupcon property?",
"Question 5.",
"Suppose $X$ and $Y$ are 1-dimensional continua such that $X\\times Y$ has the fupcon property.",
"Does the product $X_P\\times Y_P$ have the fupcon property?",
"Question 6.",
"Does the product of $[0,1]$ and pseudo-circle have the fupcon property?",
"Question 7.",
"Does the product of a pseudo-arc and pseudo-circle have the fupcon property?",
"Question 8.",
"Does the product of two pseudo-circles have the fupcon property?"
],
[
"Acknowledgements",
"The first author is grateful to J. Prajs for drawing his attention to the topic of this paper and some informative conversations during the 32nd Summer Conference on Topology and its Applications in June 2017 at the University of Dayton.",
"It was during that conference and the following two weeks when an important part of this collaboration was carried out.",
"It was made possible thanks to the support from the Moravian-Silesian Region of the Czech Republic by MSK grant 01211/2016/RRC “Strengthening international cooperation in science, research and education”.",
"This work was also supported by the NPU II project LQ1602 IT4Innovations excellence in science."
]
] | 1709.01885 |
[
[
"Input-to-State Stability with Respect to Boundary Disturbances for a\n Class of Semi-linear Parabolic Equations"
],
[
"Abstract This paper studies the input-to-state stability (ISS) properties based on the method of Lyapunov functionals for a class of semi-linear parabolic partial differential equations (PDEs) with respect to boundary disturbances.",
"In order to avoid the appearance of time derivatives of the disturbances in ISS estimates, some technical inequalities are first developed, which allow directly dealing with the boundary conditions and establishing the ISS based on the method of Lyapunov functionals.",
"The well-posedness analysis of the considered problem is carried out and the conditions for ISS are derived.",
"Two examples are used to illustrate the application of the developed result."
],
[
"Introduction",
"In the past few years, there has been a considerable effort devoted to extending the input-to-state stability (ISS) theory, which was originally introduced by Sontag for finite-dimensional nonlinear systems [28], [29], to infinite dimensional systems governed by partial differential equations (PDEs).",
"In particular, significant progresses on the establishment of ISS properties with respect to disturbances for different PDEs have been reported in the recent literature [1], [2], [4], [5], [6], [9], [12], [13], [14], [15], [16], [20], [21], [22], [23], [27], [31].",
"It is noticed that the majority of the existing work dealt with disturbances distributed over the domain for which the method of Lyapunov functionals is shown to be a well-suited tool.",
"However, difficulties may be encountered when considering disturbances acting on the boundaries.",
"This is mainly due to the fact that the latter case usually leads to a formulation involving unbounded operators, which may be an obstacle for the construction of Lyapunov functionals as explained in [13], [14], [15].",
"It is shown in [8], [9] that for a class of linear PDEs, the exponential stability plus a certain admissibility implies the ISS with respect to boundary disturbances.",
"However, it may be difficult to characterize the admissibility for nonlinear PDEs.",
"To avoid dealing with unbounded operators, it is proposed in [2] to transform the boundary disturbance to a distributed one, which allows for the application of the well-established tools, in particular the method of Lyapunov functionals.",
"However, as pointed out in [13], [14], [15] the result given in [2] may end up with ISS estimates expressed by boundary disturbances and their time derivatives, which is not strictly in the original form of ISS formulation.",
"To resolve this concern, it is proposed in [13], [14], [15] to derive the ISS property directly from the estimates of the solution to the considered PDEs by using eigenfunction expansions or finite-difference schemes.",
"An advantage of these methods is that they can be applied to a wide range of linear and nonlinear PDEs.",
"Whereas, these methods may involve heavy computations.",
"In a recent work [24], a new method based monotonicity has been introduced for studying the ISS of nonlinear parabolic equations with boundary disturbances.",
"As an application of this method, the ISS properties in $L^p$ -norm ($p>2$ ) for some linear parabolic equations with Dirichlet boundary disturbances have been established.",
"Nevertheless, it is still of great interest to investigate the applicability of the well-established method of Lyapunov functionals to the establishment of ISS properties with respect to boundary disturbances for nonlinear PDEs, including those investigated recently in [13], [14], [15], [24].",
"This motivates the present work.",
"The aim of this work is to establish the aforementioned ISS property for a class of semi-linear parabolic PDEs with Robin (or Neumann) boundary conditions based on the method of Lyapunov functionals.",
"To achieve this objective, we have developed first in Section some technical inequalities (Lemma REF and Lemma REF ) that establish some relationships between the value of a real-valued $C^1$ -function at any point and its norms.",
"This is a key feature that allows dealing directly with the boundary conditions and avoiding the appearance of time derivatives of the disturbance in ISS estimates.",
"The well-posedness of the problem described in Section REF is addressed in Section .",
"A quite standard Lyapunov functional [21] is then used in Section to establish the ISS estimates of the solutions with respect to in-domain and boundary disturbances.",
"Finally, the ISS analysis of two parabolic PDEs are given in Section to illustrate the proposed method.",
"The main contribution of the present work is the derivation of the ISS property of the considered PDEs from a Lyapunov functional using the developed techniques that can be useful in the study of other types of PDEs.",
"Notation.",
"In this paper, $\\mathbb {R}_+$ denotes the set of positive real numbers and $\\mathbb {R}_{\\ge 0} := 0\\cup \\mathbb {R}_+$ .",
"$L^2(a,b)=\\lbrace u:(a,b)\\rightarrow \\mathbb {R}|\\ \\int _{a}^bu^2(x)\\text{d}x<+\\infty \\rbrace $ , which is a Hilbert space endowed with the natural product $\\langle u,v\\rangle = \\int _{a}^bu(x)v(x)\\text{d}x$ .",
"$H^2(a,b)=\\lbrace u:(a,b)\\rightarrow \\mathbb {R}|\\ u\\in L^2(a,b)$ and the derivatives of first order and second order $u_x,u_{xx}\\in L^2(a,b)\\rbrace $ .",
"$C(\\mathbb {R}_{\\ge 0};\\mathbb {R})=\\lbrace u:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}|$ $u$ is continuous on $\\mathbb {R}_{\\ge 0}\\rbrace $ .",
"$C^1([a,b];\\mathbb {R})=\\lbrace u:[a,b]\\rightarrow \\mathbb {R}|\\ u$ and $u_x$ are continuous on $[a,b]\\rbrace $ .",
"$C^2(\\mathbb {R}_{\\ge 0};\\mathbb {R})=\\lbrace u:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}|\\ u$ , $u_x$ and $u_{xx}$ are continuous on $\\mathbb {R}_{\\ge 0}\\rbrace $ .",
"$C^0(\\mathbb {R}_{\\ge 0}\\times (a,b);\\mathbb {H})=\\lbrace u:\\mathbb {R}_{\\ge 0}\\times (a,b)\\rightarrow \\mathbb {R}| \\ u(t,\\cdot )\\rightarrow \\mathbb {H},\\ u$ is continuous (in $t$ ) on $\\mathbb {R}_{\\ge 0}\\rbrace $ , where $\\mathbb {H} $ is some function space.",
"$C^1(\\mathbb {R}_{+}\\times (a,b);\\mathbb {H})=\\lbrace u:\\mathbb {R}_{+}\\times (a,b)\\rightarrow \\mathbb {R}| \\ u(t,\\cdot )\\rightarrow \\mathbb {H},u_t(t,\\cdot )\\rightarrow \\mathbb {H},\\ u$ and $u_t$ are continuous (in $t$ ) on $\\mathbb {R}_{+}\\rbrace $ , where $\\mathbb {H} $ is some function space.",
"Let $\\mathcal {K}=\\lbrace \\gamma : \\mathbb {R}_{\\ge 0} \\rightarrow \\mathbb {R}_{\\ge 0}|\\ \\gamma (0)=0,\\gamma $ is continuous, strictly increasing$\\rbrace $ ; $ \\mathcal {K}_{\\infty }=\\lbrace \\theta \\in \\mathcal {K}|\\ \\lim \\limits _{s\\rightarrow \\infty }\\theta (s)=\\infty \\rbrace $ ; $ \\mathcal {L}=\\lbrace \\gamma : \\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}|\\ \\gamma $ is continuous, strictly decreasing, $\\lim \\limits _{s\\rightarrow \\infty }\\gamma (s)=0\\rbrace $ ; $ \\mathcal {K}\\mathcal {L}=\\lbrace \\beta : \\mathbb {R}_{\\ge 0}\\times \\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}|\\ \\beta (\\cdot ,t)\\in \\mathcal {K}, \\forall t \\in \\mathbb {R}_{\\ge 0}$ , and $\\beta (s,\\cdot )\\in \\mathcal {L}, \\forall s \\in \\mathbb {R}_{\\ge 0}\\rbrace $ .",
"Throughout this paper, we always denote $\\Vert u\\Vert _{L^2(a,b)} $ , or $\\Vert u\\Vert _{L^2(0,1)}$ , by $\\Vert u\\Vert $ for notational simplicity.",
"We consider the following semi-linear 1-$D$ parabolic equation $u_t-\\mu u_{xx}=f(t,x,u,u_x) \\ \\ \\ \\ \\text{in}\\ \\ \\mathbb {R}_{\\ge 0}\\times (0,1)$ with the boundary and initial conditions $&a_1u(t,1)+a_2u_x(t,1)=0,\\\\&b_1u(t,0)+b_2u_x(t,0)=d_1(t),\\\\&u(0,x)=u_0(x),$ where $d_1(t)$ is the disturbance acting on the boundary, $a_1,a_2,b_1$ , $b_2$ are nonnegative constants and $\\mu $ is a positive constant.",
"In (REF ), for the function $f:\\mathbb {R}_{\\ge 0}\\times (0,1)\\times \\mathbb {R}\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ , there exist a continuous function $\\rho :\\mathbb {R}_{\\ge 0}\\times \\mathbb {R}\\rightarrow \\mathbb {R}_{\\ge 0}$ , which is monotonously increasing in the second argument, a constant $\\gamma \\in [1,3)$ , and a constant $\\vartheta \\in (0,1]$ , such that for any $T\\in \\mathbb {R}_{+}$ , there hold $&|f(t,x,u,p)|\\le \\rho (t,|u|)(1+|p|^{\\gamma }),\\\\&|f(s,x,u,p)-f(t,x,u,q)|\\\\&\\;\\;\\;\\;\\;\\;\\; \\le \\rho (0,|u|)(1+|p|^{\\gamma })|s-t|^{\\vartheta },\\\\&|f(t,x,u,p)-f(t,x,u,q)|\\\\&\\;\\;\\;\\;\\;\\;\\; \\le \\rho (t,|u|)(1+|p|^{\\gamma -1}+|q|^{\\gamma -1})|p-q|,\\\\&|f(t,x,u,p)-f(t,x,v,p)| \\\\&\\;\\;\\;\\;\\;\\;\\; \\le \\rho (t,|u|+|v|)(1+|p|^{\\gamma })|u-v|,$ for a.e.",
"$x\\in (0,1)$ and all $ s,t\\in [0,T) , u\\in \\mathbb {R},v\\in \\mathbb {R},p\\in \\mathbb {R}$ ."
],
[
"Preliminaries",
" In the subsequent development, we employ extensively the following inequalities.",
"Young's inequality: For real numbers $a\\ge 0$ , $b\\ge 0$ , and $\\varepsilon >0$ , there holds $ab\\le \\frac{a^2}{2\\varepsilon }+\\frac{\\varepsilon b^2}{2}$ .",
"Gronwall's inequality: Suppose that $y:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}$ is absolutely continuous on $[0,T]$ for any $T>0$ and satisfies for a.e.",
"$t \\ge 0$ the differential inequality $\\frac{\\text{d}y}{\\text{d}t}(t)\\le g(t)y(t)+h(t),$ where $g,h\\in L^1([0,T];\\mathbb {R})$ for any $T > 0$ .",
"Then for all $t \\in \\mathbb {R}_{\\ge 0}$ , there holds $y(t) \\le y(0)e^{\\int _{0}^{t}g(s)\\text{d}s}+\\displaystyle \\int _{0}^{t}h(s) e^{\\int _{s}^{t}g(\\tau )\\text{d}\\tau }\\text{d}s.$ The following inequalities will be used to deal with the items associated with boundary points.",
"They are essential for establishing the ISS property with respect to boundary disturbances without invoking their time derivatives in a priori estimates of the solution.",
"Lemma 1 Suppose that $u\\in C^{1}([a,b];\\mathbb {R})$ , then $u^2(c)\\le \\frac{2}{b-a}\\Vert u\\Vert ^2+(b-a)\\Vert u_x\\Vert ^2,\\ \\ \\ \\ \\forall c\\in [a,b].$ Proof.",
"For each $c\\in [a,b]$ , let $g(x)=\\leavevmode {\\color {black}\\int _{c}^xu^2_z(z)\\text{d}z}$ .",
"Note that $g_x(x)=u^2_x(x)$ .",
"By Cauchy-Schwarz inequality, we have $\\bigg (\\int _{c}^xu_z(z)\\text{d}z\\bigg )^2&\\le \\bigg |(x-c)\\int _{c}^xu^2_z(z)\\text{d}z\\bigg |\\\\&=(x-c)\\int _{c}^xu^2_z(z)\\text{d}z.$ It follows $&\\int _{a}^b\\bigg (\\int _{c}^xu^2_z(z)\\text{d}z\\bigg )^2\\text{d}x \\le \\int _{a}^b(x-c) g(x)\\text{d}x\\\\=&\\bigg [\\frac{(x-c)^2}{2}g(x)\\bigg ]\\bigg |_{x=a}^{x=b}-\\int _{a}^b\\frac{(x-c)^2}{2} u^2_x(x) \\text{d}x\\\\\\le &\\frac{(b-c)^2}{2}\\int _{c}^bu^2_x(x)\\text{d}x- \\frac{(a-c)^2}{2}\\int _{c}^au^2_x(x)\\text{d}x\\\\=&\\frac{(b-c)^2}{2}\\int _{c}^bu^2_x(x)\\text{d}x+ \\frac{(a-c)^2}{2}\\int _{a}^cu^2_x(x)\\text{d}x\\\\\\le &\\frac{(b-a)^2}{2}\\int _{a}^bu^2_x(x)\\text{d}x.$ Note that $u^2(c)\\!=\\!\\bigg (\\!\\!u(x)+\\!\\!\\leavevmode {\\color {black}\\int _{x}^c\\!\\!u^2_z(z)\\text{d}z}\\bigg )^2\\!\\!\\le \\!",
"2u^2(x)+2\\bigg (\\leavevmode {\\color {black}\\int _{x}^c\\!\\!u^2_z(z)\\text{d}z}\\bigg )^2.$ Integrating over $[a,b]$ and noting (REF ), we get $u^2(c)(b-a)\\le 2\\int _{a}^bu^2(x)\\text{d}x+(b-a)^2\\int _{a}^bu^2_x(x)\\text{d}x,$ which yields the claimed result.",
"$\\blacksquare $ Lemma 2 Suppose that $u\\in C^{1}([a,b];\\mathbb {R})$ .",
"(i) If $u(c_0)=0$ for some $c_0\\in [a,b]$ , there holds $\\leavevmode {\\color {black}\\Vert u\\Vert ^2\\le \\frac{(b-a)^2}{2}\\Vert u_x\\Vert ^2.",
"}$ (ii) For any $c\\in [a,b]$ , there holds $\\Vert u\\Vert ^2\\le 2u^2(c)(b-a)+(b-a)^2\\Vert u_x\\Vert ^2.$ Proof.",
"Note that for any $w\\in C^1([0,1]) $ , there holds [18] $\\leavevmode {\\color {black}\\Vert w\\Vert ^2_{L^2(0,1)}\\le w^2(i)+\\frac{1}{2}\\Vert w_x\\Vert ^2_{L^2(0,1)},\\ i=0,1.",
"}$ For any $c\\in [a,b]$ , let $v(x)=u(c-(c-a)x)$ .",
"Then we get $\\Vert u\\Vert ^2_{L^2(a,c)}&=(c-a)\\Vert v\\Vert ^2_{L^2(0,1)}\\\\&\\le (c-a) v^2(0)+ \\frac{c-a}{2}\\Vert v_x\\Vert ^2_{L^2(0,1)}\\\\&=(c-a) u^2(c)+\\frac{(c-a)^2}{2}\\Vert u_x\\Vert ^2_{L^2(a,c)}.$ Similarly, one may get $\\Vert u\\Vert ^2_{L^2(c,b)}\\le (b-c) u^2(c)+\\frac{(b-c)^2}{2}\\Vert u_x\\Vert ^2_{L^2(c,b)}.$ Finally, one has $\\Vert u\\Vert ^2_{L^2(a,b)}&= \\Vert u\\Vert ^2_{L^2(a,c)}+\\Vert u\\Vert ^2_{L^2(c,b)}\\\\&\\le (b-a) u^2(c)+\\frac{(b-a)^2}{2}\\Vert u\\Vert ^2_{L^2(a,b)}.$ Then, we can conclude that (i) and (ii) hold true.",
"$\\blacksquare $"
],
[
"Well-posedness Analysis",
"Consider first the solution to (REF ) with disturbance free boundary conditions: $&a_1u(t,1)+a_2u_x(t,1)=0,\\\\& b_1u(t,0)+b_2u_x(t,0)=0,\\\\&u(0,x)=\\leavevmode {\\color {black}u_0(x)}.$ In this section, we always assume that $&{(a_1+a_2)b_1\\ne a_1b_2},\\ d_{1}\\in C^{2}(\\mathbb {R}_{\\ge 0};\\mathbb {R}), \\\\&u_0\\in \\mathbb {H}^2_{(0)}:=\\lbrace u\\in H^2(0,1); a_1u(1)+a_2u_x(1)= 0, \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; b_1u(0)+b_2u_x(0)=0\\rbrace .$ Moreover, we make the following assumptions.",
"When $a_2b_2= 0$ , we always assume that $\\frac{a_1}{a_2}&\\ge -\\frac{1}{2}, &\\text{if}\\ \\ a_2\\ne 0\\ \\ \\text{and}\\ \\ b_2=0, \\\\\\frac{b_1}{b_2}&\\le \\frac{1}{2}, &\\text{if}\\ \\ a_2=0\\ \\ \\text{and}\\ \\ b_2\\ne 0.$ When $a_2b_2\\ne 0$ , we always assume that there exist $A_1, A_2\\in \\mathbb {R}_{\\ge 0}$ satisfying $A_1+A_2=1 $ such that $\\frac{a_1}{a_2} \\ge 2A_2,\\ \\frac{b_1}{b_2} \\le A_1,\\ A_2-2A_1 \\ge 0,$ or, there exist $B_1,B_2\\in \\mathbb {R}_{\\ge 0}$ satisfying $B_1+B_2=1 $ such that $\\frac{a_1}{a_2} &\\ge -B_2,\\ \\frac{b_1}{b_2}\\le -2B_1,\\ B_1-2B_2\\ge 0.$ Remark 1 Under the above assumptions, we always have $a_1^2+a_2^2>0$ , $b_1^2+b_2^2>0$ and $a_1^2+b_1^2>0$ .",
"Proposition 3 Assume that $u_1\\in \\mathbb {H}^2_{(0)}$ .",
"Then there exists a unique solution $u\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ to (REF ) with boundary-initial conditions ().",
"Proof.",
"Let $D(\\mathcal {A})=\\mathbb {H}^2_{(0)}$ with the operator $\\mathcal {A}: D(\\mathcal {A})\\rightarrow L^2(0,1)$ defined as $\\mathcal {A}u=\\mu u_{xx}$ .",
"For $\\alpha \\in [0,1]$ , let $\\mathcal {H}_{\\alpha }=\\mu (-\\mathcal {A})^{\\alpha } $ and $\\Vert u\\Vert _{\\alpha }=\\Vert (-\\mathcal {A})^{\\alpha }u\\Vert $ .",
"Let $F(t,u)(x)=f(t,x,u,u_x)$ .",
"Then () can be expressed by an abstract evolutionary equation $\\frac{\\text{d}u}{\\text{d}t}=\\mathcal {A}u+F(t,u)$ .",
"The proof is based on the theory of Lipschitz perturbations of linear evolution equations [19] (see also [26]), which consists in two steps: first to prove that $\\mathcal {A}$ is the infinitesimal generator of a $C_0$ -semigroup of contractions on $L^2(0,1)$ ; and second to prove that $F(t,u) $ satisfies local Hölder condition, i.e., for $F:\\mathbb {R}_{\\ge 0}\\times U\\rightarrow L^2(0,1)$ , where $U$ is an open subset of $\\mathcal {H}_{\\alpha }$ , for every $(t,u)\\in U$ , there is a neighborhood $V\\subset U$ and constants $L\\ge 0$ , $0<\\vartheta \\le 1$ (see, e.g., [19], [26]) such that $\\Vert F(t_1,u_1)-F(t_2,u_2)\\Vert \\le &L(|t_1 -t_2|^{\\vartheta }+\\Vert u_1-u_2\\Vert _{\\alpha }),\\\\& \\forall (t_i,u_i)\\in V,i=1,2.$ First, since $\\mathcal {A}$ is a densely defined closed linear operator and self-adjoint, it suffices to prove that $\\mathcal {A}$ is dissipative.",
"Then, the claim that $\\mathcal {A}$ generates a $C_0$ -semigroup follows from Lumer-Phillips theorem (see [26], [10]).",
"Indeed, due to $\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle =& \\int _{0}^1 u_{xx}u\\text{d}x \\\\=& u_{x}(t,1)u(t,1)- u_{x}(t,0)u(t,0)- \\Vert u_x\\Vert ^2,$ we may argue for four cases.",
"(i) $b_2=a_2=0$ .",
"In this case, $u(t,1)=u(t,0)=0$ .",
"$\\mathcal {A}$ is obviously dissipative.",
"(ii) $b_2=0,a_2\\ne 0$ .",
"In this case, $u(t,0)=0$ .",
"It follows $\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle =\\int _{0}^1u_{xx}u\\text{d}x= -\\frac{a_1}{a_2}u^2(t,1)- \\Vert u_x\\Vert ^2.$ For ${\\frac{a_1}{a_2}\\ge -\\frac{1}{2}}$ , $ -\\frac{a_1}{a_2}u^2(t,1)\\le \\frac{1}{2}u^2(t,1)$ .",
"By Lemma REF and Lemma REF , we get $u^2(t,1)\\le 2\\Vert u\\Vert ^2+\\Vert u_x\\Vert ^2\\le 2\\Vert u_x\\Vert ^2.$ Then $\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle =&-\\frac{a_1}{a_2}u^2(t,1)- \\Vert u_x\\Vert ^2 \\\\\\le & \\frac{1}{2}\\times 2\\Vert u_x\\Vert ^2- \\Vert u_x\\Vert ^2= 0.$ (iii) $b_2\\ne 0,a_2= 0$ .",
"In this case, $u(t,1)=0$ .",
"Arguing as in (ii), for $\\frac{b_1}{b_2}\\le \\frac{1}{2}$ , one may get $\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle \\le 0.$ (iv) $b_2\\ne 0,a_2\\ne 0$ .",
"For $\\frac{a_1}{a_2}\\ge 2A_2,\\ \\frac{b_1}{b_2}\\le A_1,\\ A_2-2A_1\\ge 0$ , note that by Lemma REF and Lemma REF there hold $\\begin{split}\\Vert u_x\\Vert ^2\\ge u^2(t,0)-2\\Vert u\\Vert ^2,\\\\\\Vert u_x\\Vert ^2\\ge \\Vert u\\Vert ^2-2u^2(t,1).\\end{split}$ Then we get $&\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle \\\\=& -\\frac{a_1}{a_2}u^2(t,1)+\\frac{b_1}{b_2}u^2(t,0)-A_1\\Vert u_x\\Vert ^2-A_2\\Vert u_x\\Vert ^2\\\\\\le & -\\frac{a_1}{a_2}u^2(t,1)+\\frac{b_1}{b_2}u^2(t,0)-A_2( \\Vert u\\Vert ^2-2u^2(t,1)) \\\\& \\;\\;\\; -A_1( u^2(t,0)-2\\Vert u\\Vert ^2)\\\\=&\\bigg ( 2A_2-\\frac{a_1}{a_2}\\bigg )u^2(t,1)+\\bigg ( \\frac{b_1}{b_2}-A_1\\bigg )u^2(t,0)\\\\& \\;\\;\\; +(2A_1-A_2)\\Vert u\\Vert ^2 \\le 0.$ Similarly, for $\\frac{a_1}{a_2}\\ge -B_2,\\ \\frac{b_1}{b_2}\\le -2B_1,\\ B_1-2B_2\\ge 0$ , one may get $\\frac{1}{\\mu }\\langle \\mathcal {A}u,u\\rangle \\le 0.$ Thus, $\\mathcal {A}$ is a dissipative operator.",
"The second step of proof can be proceeded in the same way as in [19].",
"First, since $\\mathcal {A} $ is a Sturm-Liouville operator [3], [25], all eigenvalues of $\\mathcal {A} $ are real, and form an infinite, increasing sequence $0>\\lambda _1>\\lambda _2>\\cdots >\\lambda _n>\\cdots $ with $\\lim \\limits _{n\\rightarrow \\infty }\\lambda _n=-\\infty $ .",
"Corresponding to each $\\lambda _n\\in \\mathbb {R}, n=1,2,\\ldots $ , there is exactly one eigenfunction $ \\varphi _n \\in D(\\mathcal {A})\\cap C^2([0,1]) $ satisfying $ \\mathcal {A}\\varphi _n=\\lambda _n\\varphi _n$ .",
"The eigenfunctions form an orthonormal basis of $ L^2(0,1)$ .",
"Second, one may proceed exactly as in [19] to show that the norm $ \\Vert u\\Vert +\\Vert u\\Vert _{\\alpha }$ on $ \\mathcal {H}_{\\alpha }$ is equivalent to the norm $\\Vert u\\Vert _{\\alpha }$ and $ \\mathcal {H}_{\\alpha }\\subset W^{1,2\\gamma }(0,1)\\cap L^{\\infty }(0,1)$ for $ \\max \\lbrace \\frac{3}{4},\\frac{5\\gamma -3}{4\\gamma }\\rbrace <\\alpha <1$ .",
"Furthermore, Theorem 12 in [19] holds.",
"Then proceeding exactly as in (4.17)-(4.20) in [26], one may verify that $F(t,u)$ satisfies (REF ).",
"Finally, Theorem 12 in [19] guarantees the existence of a unique classical solution.$\\blacksquare $ Remark 2 Conditions on the constants $a_i, b_i (i=1,2)$ are only required in order to guarantee exponential stability of a semigroup in the proof of Proposition REF .",
"Theorem 4 There exists a unique solution $u\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ of (REF ) satisfying (REF ), (), and ().",
"Proof.",
"Consider first the case where $b_1^2+b_2^2=1$ .",
"Let $g(x)=b_1+b_2x+c_1x^2+c_2x^3$ , where $c_1, c_2\\in \\mathbb {R}$ satisfy $(a_1+2a_2)c_1+(a_1+3a_2)c_2=-a_1b_1-(a_1+a_2)b_2$ .",
"The existence of $c_1, c_2$ is guaranteed by $a_1^2+a_2^2\\ne 0$ .",
"One may check that $a_1g(1)+a_2g_x(1)=b_1g(0)+b_2g_x(0)=0$ due to $b_1^2+b_2^2=1$ .",
"Let $\\tilde{f}(t,x,v,p)=d_{1t}(t)g(x)+\\mu d_1(t)g_{xx}(x)+f(t,x,v+d_1(t)g(x),p+d_1(t)g_x(x))$ .",
"Consider the following equation $&v_t-\\mu v_{xx}=\\tilde{f}(t,x,v,v_x),\\\\&a_1v(t,1)+a_2v_x(t,1)=0,\\\\& b_1v(t,0)+b_2v_x(t,0)=0,\\\\&v(0,x)=v_0(x),$ where $v_0=u_0-d_{1}(0)g(x)\\in \\mathbb {H}^2_{(0)}$ since $u_0\\in \\mathbb {H}^2_{(0)}$ .",
"Note that $|g(x)|\\le |b_1|+|b_2|+|c_1|+|c_2|:=g_0$ , $|g_x|\\le |b_2|+2|c_1|+3|c_2|:=g_1$ and $|g_{xx}|\\le 2|c_1|+6|c_2|:=g_2$ .",
"Let $\\tilde{\\rho }(t,r)=g_0|d_{1t}(t)|+\\mu g_2|d_1(t)|+\\leavevmode {\\color {black}2^\\gamma (1+g_1^\\gamma |d_1(t)|^\\gamma })\\times \\rho (t,r+2g_0|d_1(t)|)$ , which is continuous in $t$ and $r$ .",
"One may verify that $\\tilde{f}(t,x,v,p)$ satisfies the structural conditions (REF ) with $ \\tilde{\\rho }(t,r)$ instead of $\\rho (t,r)$.",
"According to Proposition REF , () has a unique solution $v\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ .",
"Finally, $u=v+d_1(t)g(x)$ is the unique solution of (REF ) and satisfies (REF ), () and ().",
"For $b_1^2+b_2^2\\ne 1$ , we set $\\tilde{b}_i=\\frac{b_i}{\\sqrt{b_1^2+b_2^2}} (i=1,2),\\ \\tilde{d}_1=\\frac{d_1}{\\sqrt{b_1^2+b_2^2}}.$ Then the boundary condition () is equivalent to $ \\tilde{b}_1u(t,0)+\\tilde{b}_2u_x(t,0)=\\tilde{d}_{1}(t),$ where $\\tilde{b}_1^2+\\tilde{b}_2^2= 1 $ .",
"Therefore, (REF ) has a unique solution $u\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ .",
"$\\blacksquare $"
],
[
"Stability Assessment",
"In stability analysis, we choose the energy of the system, $E(t) = \\Vert u(t,\\cdot )\\Vert ^2$ , as the Lyapunov functional candidate.",
"Let $\\mathbb {H}^2_{(0)}$ be defined as in Section .",
"Note that in order to apply Lemma REF and Lemma REF to deal with the terms of $u_x$ on the boundaries, we always assume that $b_2\\ne 0$ (i.e., we consider the problem with Robin (or Neumann) boundary conditions)."
],
[
"The case where the function $f(t,x,u,p)$ is in a general form",
"We assume that there exists $d(t,x) \\in C^1(\\mathbb {R}_{\\ge 0}\\times (0,1); \\mathbb {R})$ such that $f(t,x,u,p)u\\le M_1u^2+(|d(t,x)|+M_2|p|)|u|,$ for a.e.",
"$x\\in (0,1)$ and all $ t\\in \\mathbb {R}_{\\ge 0} , u\\in \\mathbb {R},p\\in \\mathbb {R}$ , and $M_1\\in \\mathbb {R}$ and $M_2\\in \\mathbb {R}_{\\ge 0}$ are constants.",
"Note that $d(t,x)$ can be used to describe the disturbance in the domain.",
"For simplicity, we assume that $|d(x,t)|\\le |d_2(t)|$ for almost all $x\\in (0,1)$ and any $t>0$ , where $d_2\\in C^1(\\mathbb {R}_{\\ge 0}; \\mathbb {R})$ , i.e., we assume that $f(t,x,u,p)u\\le M_1u^2+(|d_2(t)|+M_2|p|)|u|,$ for a.e.",
"$x\\in (0,1)$ and all $ t\\in \\mathbb {R}_{\\ge 0} , u\\in \\mathbb {R},p\\in \\mathbb {R}$ .",
"Definition 1 System (REF ) with (REF ) is said to be input-to-state stable (ISS), or respectively integral input-to-state stable (iISS), w.r.t.",
"the disturbances $d_1(t)$ and $d_2(t)$ , if there exist functions $\\beta \\in \\mathcal {K}\\mathcal {L},\\theta _1,\\theta _2\\in \\mathcal {K}_{\\infty }$ and $ \\gamma _1, \\gamma _2,\\in \\mathcal {K}$ such that the solution of (REF ) with (REF ) satisfies $\\begin{split}\\Vert u(t,\\cdot )\\Vert \\le &\\beta ( \\Vert {u_0}\\Vert ,t)+\\gamma _1(\\Vert d_1\\Vert _{L^{\\infty }(0,t)}) \\\\&+\\gamma _2(\\Vert d_2\\Vert _{L^{\\infty }(0,t)}),\\ \\forall t\\ge 0,\\end{split}$ or respectively $\\begin{split}\\Vert u(t,\\cdot )\\Vert \\le & \\beta ( \\Vert {u_0}\\Vert ,t)+\\theta _1\\bigg (\\int _{0}^{t} \\gamma _1(|d_1(s)|)\\bigg )\\\\&+\\theta _2\\bigg (\\int _{0}^{t} \\gamma _2(|d_2(s)|)\\bigg ),\\ \\forall t\\ge 0.\\end{split}$ Moreover, System (REF ) with (REF ) is said to be exponential input-to-state stable (EISS), or exponential integral input-to-state stable (EiISS), w.r.t.",
"the disturbances $d_1(t)$ and $d_2(t)$ , if there exist $\\beta ^{\\prime }\\in \\mathcal {K}_{\\infty }$ and a constat $\\lambda > 0$ such that $\\beta ( \\Vert {u_0}\\Vert ,t) \\le \\beta ^{\\prime }(\\Vert {u_0}\\Vert )e^{-\\lambda t}$ in (REF ) or (REF ).",
"Remark 3 While the ISS typically refers to norm-estimates for the input/disturbance in the $L^\\infty $ -norm, other norms can also be considered.",
"It should be mentioned that the latter case usually relates to the integration of the input/disturbance and can be defined as the “integral input-to-state stability (iISS)\" (see, e.g., [8]).",
"This property differs from the ISS in the sense that it allows for unbounded inputs that have “finite energy\" [30].",
"There indeed exist many practically relevant systems that are iISS, but not ISS (see, e.g., [11], [23] for more detailed discussions).",
"In order to obtain the stability of the system, we need some additional assumptions on $a_1,a_2,b_1,b_2,M_1$ and $M_2$ .",
"Specifically, if $a_2\\ne 0$ , we make the following assumptions.",
"Assumption 1 Suppose that (REF ) holds.",
"Moreover, suppose that there exist $A^{\\prime }_1,A^{\\prime }_2,A^{\\prime }_3\\in \\mathbb {R}_{\\ge 0}$ satisfying $A^{\\prime }_1+A^{\\prime }_2+A^{\\prime }_3=\\mu $ and $\\frac{a_1}{a_2}\\mu \\ge 2A^{\\prime }_2,\\ \\frac{b_1}{b_2}\\mu < A^{\\prime }_1.$ Assume further that there exists $\\varepsilon _0 \\in \\mathbb {R}_{+}$ such that $M_1+\\frac{\\varepsilon _0M_2}{2}<A^{\\prime }_2-2A^{\\prime }_1,\\ \\frac{M_2}{2\\varepsilon _0}\\le A^{\\prime }_3.$ Assumption 2 Suppose that (REF ) holds.",
"Moreover, suppose there exist $B^{\\prime }_1,B^{\\prime }_2,B^{\\prime }_3\\in \\mathbb {R}_{\\ge 0}$ satisfying $B^{\\prime }_1+B^{\\prime }_2+B^{\\prime }_3=\\mu $ and $\\frac{a_1}{a_2}\\mu \\ge -B^{\\prime }_2,\\ \\frac{b_1}{b_2}\\mu < -2B^{\\prime }_1.$ Assume further that there exists $\\varepsilon _0 \\in \\mathbb {R}_{+}$ such that $M_1+\\frac{\\varepsilon _0M_2}{2}<B^{\\prime }_1-2B^{\\prime }_2,\\ \\frac{M_2}{2\\varepsilon _0}\\le B^{\\prime }_3.$ If $a_2=0$ , we make the following assumption.",
"Assumption 3 We assume that $\\frac{b_1}{b_2}< \\frac{1}{2} $ , which guarantees that there exist $A^{\\prime }_1,A^{\\prime }_2,A^{\\prime }_3\\in \\mathbb {R}_{\\ge 0}$ satisfying $A^{\\prime }_1+A^{\\prime }_2+A^{\\prime }_3=\\mu $ and $\\frac{b_1}{b_2}\\mu < A^{\\prime }_1.$ Assume further that there exists $\\varepsilon _0 \\in \\mathbb {R}_{+}$ such that $M_1+\\frac{\\varepsilon _0M_2}{2}<A^{\\prime }_2-2A^{\\prime }_1,\\ \\frac{M_2}{2\\varepsilon _0}\\le A^{\\prime }_3.$ Theorem 5 Let $u\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ be the unique solution of (REF ), (REF ),() and ().",
"Under Assumption REF , or Assumption REF , or Assumption REF , System (REF ) with (REF ) is EiISS and EISS having the estimates: $\\Vert u(t,\\cdot )\\Vert ^2\\le &\\Vert u(0,\\cdot )\\Vert ^2e^{-C_0t}+C_1\\int _{0}^t |d_1(s)|^2\\text{d}s \\\\& \\;\\;\\;+ C_2\\int _{0}^t|d_2(s)|^2\\text{d}s,$ and $\\Vert u(t,\\cdot )\\Vert ^2\\le &\\Vert u(0,\\cdot )\\Vert ^2e^{-C_0t}+\\Big (1-e^{-C_0t}\\Big ) \\\\& \\times \\Big (C_1\\Vert d_1\\Vert _{L^{\\infty }(0,t)}^2+C_2\\Vert d_2\\Vert _{L^{\\infty }(0,t)}^2\\Big )$ for some positive constants $C_0,C_1,C_2$ .",
"Proof.",
"We prove first the case for $a_2\\ne 0$ under Assumption REF .",
"Multiplying (REF ) with $u$ and integrating over $[0,1]$ , we have $\\int _{0}^1u_tu\\text{d}x&-\\mu \\int _{0}^1u_{xx}u\\text{d}x= \\int _{0}^1f(t,x,u,u_x)u\\text{d}x\\\\\\le & \\int _{0}^1\\big ((|d_2(t)|+M_2|u_x|)|u|+M_1u^2\\big )\\text{d}x := I_1,$ which is $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2-\\mu u_{x}(t,1)u(t,1)+\\mu u_{x}(t,0)u(t,0)+\\mu \\Vert u_x\\Vert ^2\\le I_1.$ By (REF ), () and Young's inequality, it follows $&\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+\\mu \\Vert u_x\\Vert ^2\\\\\\le & I_1-\\frac{1}{b_2}d_1(t)\\mu u(t,0)+\\frac{b_1}{b_2}\\mu u^2(t,0)-\\frac{a_1}{a_2}\\mu u^2(t,1)\\\\\\le & I_1+\\frac{\\mu |d_1(t)|^2}{2\\varepsilon _1b_2^2}+\\bigg (\\frac{b_1}{b_2}+\\frac{\\varepsilon _1}{2}\\bigg )\\mu u^2(t,0)-\\frac{a_1}{a_2}\\mu u^2(t,1).$ By Young's inequality, we have $I_1\\le \\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{\\varepsilon _2}{2}+M_1+\\frac{\\varepsilon _3M_2}{2}\\bigg )\\Vert u\\Vert ^2+\\frac{M_2}{2\\varepsilon _3}\\Vert u_x\\Vert ^2.$ Then we infer from (REF ), (REF ) and (REF ) that $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&+A^{\\prime }_1u^2(t,0)+(A^{\\prime }_2-2A^{\\prime }_1)\\Vert u\\Vert ^2-2A^{\\prime }_2u^2(t,1)\\\\&+A^{\\prime }_3\\Vert u_x\\Vert ^2\\\\=&\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+A^{\\prime }_1(u^2(t,0)-2\\Vert u\\Vert ^2)\\\\&+A^{\\prime }_2(\\Vert u\\Vert ^2-2u^2(t,1) )+A^{\\prime }_3\\Vert u_x\\Vert ^2\\\\\\le &\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+A^{\\prime }_1\\Vert u_x\\Vert ^2+A^{\\prime }_2\\Vert u_x\\Vert ^2+A^{\\prime }_3\\Vert u_x\\Vert ^2\\\\=&\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+\\mu \\Vert u_x\\Vert ^2\\\\\\le &\\frac{\\mu |d_1(t)|^2}{2\\varepsilon _1b_2^2}+\\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{b_1}{b_2}+\\frac{\\varepsilon _1}{2}\\bigg )\\mu u^2(t,0)\\\\&-\\frac{a_1}{a_2}\\mu u^2(t,1) +\\bigg (\\frac{\\varepsilon _2}{2}+M_1+\\frac{\\varepsilon _3M_2}{2}\\bigg )\\Vert u\\Vert ^2 \\\\&+\\frac{M_2}{2\\varepsilon _3}\\Vert u_x\\Vert ^2.$ Recalling (REF ) and (REF ), one may choose $\\varepsilon _3=\\varepsilon _0$ and $\\varepsilon _1, \\varepsilon _2>0$ small enough such that $&C_0:= A^{\\prime }_2-2A^{\\prime }_1-\\bigg (\\frac{\\varepsilon _2}{2}+M_1+\\frac{\\varepsilon _3M_2}{2}\\bigg )>0,\\\\& \\bigg (\\frac{b_1}{b_2}+\\frac{\\varepsilon _1}{2}\\bigg )\\mu \\le A^{\\prime }_1,\\ \\frac{M_2}{2\\varepsilon _3}\\le A^{\\prime }_3.$ Then we have $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&\\le - C_0\\Vert u\\Vert ^2+\\frac{\\leavevmode {\\color {black}\\mu |d_1(t)|^2}}{2\\varepsilon _1b_2^2}+\\frac{|d_2(t)|^2}{2\\varepsilon _2}\\\\&:=-C_0\\Vert u\\Vert ^2+C_1|d_1(t)|^2+C_2|d_2(t)|^2\\\\&\\le -C_0\\Vert u\\Vert ^2+C_1\\Vert d_1\\Vert _{L^{\\infty }(0,t)}^2+C_2\\Vert d_2\\Vert _{L^{\\infty }(0,t)}^2.$ By (REF ) and Gronwall's inequality, we obtain (REF ).",
"By () and Gronwall's inequality, we obtain (REF ).",
"For $a_2\\ne 0$ and under Assumption REF , it suffices to note that by Lemma REF and Lemma REF , we have $\\Vert u_x\\Vert ^2\\ge u^2(t,1)-2\\Vert u\\Vert ^2$ and $\\Vert u_x\\Vert ^2\\ge \\Vert u\\Vert ^2-2u^2(t,0)$ .",
"Then proceeding as above, one may get $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&+(B^{\\prime }_1-2B^{\\prime }_2)\\Vert u\\Vert ^2+B^{\\prime }_3\\Vert u_x\\Vert ^2 -2B^{\\prime }_1 u^2(t,0) \\\\& +B^{\\prime }_2u^2(t,1) \\\\\\le & \\frac{\\mu |d_1(t)|^2}{2\\varepsilon _1b_2^2}+\\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{b_1}{b_2}+\\frac{\\varepsilon _1}{2}\\bigg )\\mu u^2(t,0)\\\\&-\\frac{a_1}{a_2}\\mu u^2(t,1) +\\bigg (\\frac{\\varepsilon _2}{2}+M_1+\\frac{\\varepsilon _3M_2}{2}\\bigg )\\Vert u\\Vert ^2 \\\\&+\\frac{M_2}{2\\varepsilon _3}\\Vert u_x\\Vert ^2.$ The ISS can be established as well.",
"Now for $a_2=0$ , it suffices to note that $u(t,1)=0$ .",
"Under Assumption REF , the ISS can be obtained as above.",
"$\\blacksquare $ Remark 4 It should be noticed that the assumptions that (REF ) (or (REF )) holds in Assumption REF (or Assumption REF ) are only for assuring the existence of a solution.",
"For ISS assessment, it suffices to relax these assumptions to (REF ) and (REF ) (or (REF ) and (REF )), or some other weaker conditions."
],
[
"The case where the function $f(t,x,u,p)$ has a special form",
"In the following part, we assume that $f(t,x,u,p)$ is with the form $f(t,x,u,p)=\\leavevmode {\\color {black}d(t,x)}+ M_1u+M_2p,$ where $M_1, M_2\\in \\mathbb {R}$ are constants, $|d(t,x)|\\le |d_2(t)|$ for a.e.",
"$x\\in (0,1)$ , $d_2\\in C(\\mathbb {R}_{\\ge 0}; \\mathbb {R})$ .",
"As $f(t,x,u,p)$ grows lineally w.r.t.",
"$u$ and $p$ , the conditions given in Assumption REF , Assumption REF , and Assumption REF can be relaxed.",
"For the case where $a_2\\ne 0$ , we make the following assumptions.",
"Assumption 4 Suppose that there exist $A^{\\prime }_1,A^{\\prime }_2\\in \\mathbb {R}_{\\ge 0}$ satisfying $A^{\\prime }_1+A^{\\prime }_2=\\mu $ and $- \\frac{a_1}{a_2}\\mu +\\frac{M_2}{2}\\le -2A^{\\prime }_2,\\ \\frac{b_1}{b_2}\\mu -\\frac{M_2}{2}< A^{\\prime }_1,\\ M_1<A^{\\prime }_2-2A^{\\prime }_1.$ Assumption 5 Suppose that there exist $B^{\\prime }_1,B^{\\prime }_2\\in \\mathbb {R}_{\\ge 0}$ satisfying $B^{\\prime }_1+B^{\\prime }_2=\\mu $ and $-\\frac{a_1}{a_2}\\mu +\\frac{M_2}{2}\\le B^{\\prime }_2,\\ \\frac{b_1}{b_2}\\mu -\\frac{M_2}{2}< -2B^{\\prime }_1,\\ M_1<B^{\\prime }_1-2B^{\\prime }_2.$ For the case where $a_2= 0$ , we make the following assumption.",
"Assumption 6 We assume that there exist $A^{\\prime }_1,A^{\\prime }_2\\in \\mathbb {R}_{\\ge 0}$ satisfying $A^{\\prime }_1+A^{\\prime }_2=\\mu $ such that $\\frac{b_1}{b_2}\\mu -\\frac{M_2}{2}<A_1^{\\prime },\\ M_1<A^{\\prime }_2-2A^{\\prime }_1.$ Theorem 6 Let $u\\in C^{0}(\\mathbb {R}_{\\ge 0}\\times (0,1);\\mathbb {H}^2_{(0)})\\cap C^{1}(\\mathbb {R}_{+}\\times (0,1);L^2(0,1))$ be the unique solution of (REF ), (REF ),() and ().",
"Under Assumption REF , or Assumption REF , or Assumption REF , System (REF ) with (REF ) is EiISS and EISS having the estimates: $\\Vert u(t,\\cdot )\\Vert ^2\\le &\\Vert u(0,\\cdot )\\Vert ^2e^{-C_3t}+C_4\\int _{0}^t |d_1(s)|^2\\text{d}s \\\\& \\;\\;\\;+ C_5\\int _{0}^t|d_2(s)|^2\\text{d}s,$ and $\\Vert u(t,\\cdot )\\Vert ^2\\le &\\Vert u(0,\\cdot )\\Vert ^2e^{-C_3t}+ \\Big (1-e^{-C_0t}\\Big ) \\\\& \\;\\;\\; \\times \\Big (C_4\\Vert d_1\\Vert _{L^{\\infty }(0,t)}^2+C_5\\Vert d_2\\Vert _{L^{\\infty }(0,t)}^2\\Big )$ for some positive constants $C_3,C_4,C_5$ .",
"Proof.",
"We proceed as in Theorem REF and only prove the result under Assumption REF .",
"Multiplying (REF ) with $u$ and integrating over $[0,1]$ , we have $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&-\\mu u_{x}(t,1)u(t,1)+\\mu u_{x}(t,0)u(t,0)+\\mu \\Vert u_x\\Vert ^2\\\\= & \\int _{0}^1f(t,x,u,u_x)u\\text{d}x\\\\\\le & \\int _{0}^1\\big (|d_2(t)||u|+M_1u^2+M_2u_xu\\big )\\text{d}x := I_2.$ Note that $\\int _{0}^1u_xu\\text{d}x=\\frac{1}{2}u^2(t,x)|^{x=1}_{x=0}=\\frac{1}{2}(u^2(t,1)- u^2(t,0))$ .",
"It follows $I_2\\le & \\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\frac{\\varepsilon _2}{2}\\Vert u\\Vert ^2+M_1\\Vert u\\Vert ^2+\\frac{M_2}{2}(u^2(t,1)- u^2(t,0))\\\\=&\\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{\\varepsilon _2}{2}+M_1\\bigg )\\Vert u\\Vert ^2+\\frac{M_2}{2}(u^2(t,1)- u^2(t,0)).$ Then we have $&\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+\\mu \\Vert u_x\\Vert ^2\\\\\\le & I_2-\\frac{1}{b_2}d_1(t)\\mu u(t,0)+\\frac{b_1}{b_2}\\mu u^2(t,0)-\\frac{a_1}{a_2}\\mu u^2(t,1)\\\\\\le & \\frac{\\mu |d_1(t)|^2}{2\\varepsilon _1b_2^2}+\\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{\\varepsilon _2}{2}+M_1\\bigg )\\Vert u\\Vert ^2\\\\&+\\bigg (\\frac{b_1\\mu }{b_2}-\\frac{M_2}{2}+\\frac{\\varepsilon _1\\mu }{2}\\bigg ) u^2(t,0)+\\bigg (\\frac{M_2}{2}-\\frac{a_1}{a_2}\\mu \\bigg )u^2(t,1).$ We get by splitting $\\mu \\Vert u_x\\Vert ^2$ as in (REF ) and using (REF ) $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&+A^{\\prime }_1u^2(t,0)+(A^{\\prime }_2-2A^{\\prime }_1)\\Vert u\\Vert ^2-2A^{\\prime }_2u^2(t,1)\\\\=&\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+A^{\\prime }_1(u^2(t,0)-2\\Vert u\\Vert ^2)\\\\&+A^{\\prime }_2(\\Vert u\\Vert ^2-2u^2(t,1) )\\\\\\le &\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2+\\mu \\Vert u_x\\Vert ^2\\\\\\le &\\frac{\\mu |d_1(t)|^2}{2\\varepsilon _1b_2^2}+\\frac{|d_2(t)|^2}{2\\varepsilon _2}+\\bigg (\\frac{\\varepsilon _2}{2}+M_1\\bigg )\\Vert u\\Vert ^2\\\\&+\\bigg (\\frac{b_1\\mu }{b_2}-\\frac{M_2}{2}+\\frac{\\varepsilon _1\\mu }{2}\\bigg ) u^2(t,0)\\\\&+\\bigg (\\frac{M_2}{2}-\\frac{a_1}{a_2}\\mu \\bigg )u^2(t,1).$ Choosing $ \\varepsilon _1,\\varepsilon _2$ small enough, such that $&\\frac{b_1\\mu }{b_2}-\\frac{M_2}{2}+\\frac{\\varepsilon _1\\mu }{2}\\le A_1^{\\prime },\\\\&\\frac{\\varepsilon _2}{2}+M_1<A_2^{\\prime }-2A_1^{\\prime },\\frac{M_2}{2}-\\frac{a_1}{a_2}\\mu \\le -2A_2^{\\prime }.$ Then we have $\\frac{\\text{d}}{\\text{d}t}\\Vert u\\Vert ^2&\\le -C_3\\Vert u\\Vert ^2+C_4|d_1(t)|^2+C_5|d_2(t)|^2\\\\&\\le -C_3\\Vert u\\Vert ^2+C_4\\Vert d_1\\Vert _{L^{\\infty }(0,t)}^2+C_5\\Vert d_2\\Vert _{L^{\\infty }(0,t)}^2.$ Finally, one may obtain the desired results by Gronwall's inequality.",
"$\\blacksquare $ Remark 5 Note that with Assumption REF , or Assumption REF , or Assumption REF , Proposition REF guarantees that the operator $\\mathcal {A}$ generates an exponentially stable semi-group, when $d_1(t)=0$ .",
"Then, it follows directly from Proposition 4 in [23] or Proposition 2.13 in [8] that System (REF ) with (REF ) is ISS w.r.t.",
"$d_2$ in $L^p$ -norm ($p \\ge 1$ ).",
"For $d_1(t)\\ne 0$ , the ISS w.r.t.",
"disturbances in $L^\\infty $ -norm is obtained in [8], [15].",
"This result is weaker than that obtained in Theorem REF , which gives an ISS w.r.t.",
"disturbances in $L^2$ -norm.",
"It should be mentioned that a strict Lyapunov functional has also been constructed to establish the ISS w.r.t.",
"in-domain disturbances for a semi-linear parabolic PDE with periodic boundary conditions [21]."
],
[
"Illustration Examples",
"Two examples are used to illustrate the developed results."
],
[
"Ginzburg-Landau equations with real coefficients",
"Consider first the Ginzburg-Landau equation with real coefficients (see, e.g., [17]) $u_t= {\\mu u_{xx}+\\alpha u}-\\beta |u|^2u,$ and the generalized Ginzburg-Landau equation with real coefficients (see, e.g., [7]) $u_t= {\\mu u_{xx}+\\alpha u}-\\beta |u|^2u-\\gamma |u|^4u+\\lambda u_x,$ under the Robin (or Neumann) boundary conditions $u(x,1)=0, b_1u(x,0)+b_2u_x(t,0)&=d(t),$ where $\\mu ,\\beta ,\\gamma >0,\\alpha ,\\lambda ,b_1,b_2\\in \\mathbb {R},b_2\\ne 0$ and $d\\in C^{2}(\\mathbb {R}_{\\ge 0};\\mathbb {R})$ .",
"In the above boundary conditions, $a_1=1$ , $a_2=0$ , $d_1(t)=d(t)$ , and $d_2(t)=0$ .",
"In (REF ), $f(t,x,u,p)=\\alpha u-\\beta |u|^2u$ .",
"In (REF ), $f(t,x,u,p)=\\alpha u-\\beta |u|^2u-\\gamma |u|^4u+\\lambda p$ .",
"In both cases, $f(t,x,u,p)$ satisfies the structural conditions (REF ).",
"Assume that $\\frac{b_1}{b_2}\\le \\frac{1}{2}$ , then there exists a unique real solution of (REF ) and (REF ) respectively.",
"Now for (REF ), $f(t,x,u,p)$ satisfies (REF ) with $M_1=\\alpha ,M_2=0.$ If we assume further that $\\frac{b_1}{b_2}< \\frac{1}{3},\\alpha <0,$ and set $A_1^{\\prime }=\\frac{1}{3}\\mu ,A_2^{\\prime }=\\frac{2}{3}\\mu , {A_3^{\\prime }=0}$ , then Assumption REF holds.",
"Therefore (REF ) is ISS.",
"For (REF ), $f(t,x,u,p)$ satisfies (REF ) with $M_1=\\alpha ,M_2=|\\lambda |.$ If we assume further that $\\frac{b_1}{b_2}< \\frac{1}{4},\\alpha +{|\\lambda |}<0,|\\lambda |\\le \\mu ,$ and set $A_1^{\\prime }=\\frac{1}{4}\\mu ,A_2^{\\prime }=\\frac{1}{2}\\mu , {A_3^{\\prime }=\\frac{1}{4}\\mu },\\varepsilon _0=2$ , then Assumption REF holds.",
"Therefore (REF ) is ISS."
],
[
"1-$D$ transport partial differential equation",
"We consider the following 1-$D$ transport PDE: $u_t=\\mu u_{xx}-mu_{x}-nu,$ under the following boundary conditions $\\begin{split}u_x(t,1)&=\\left(\\frac{m}{2\\mu }-a\\right)u(t,1),\\\\u_x(t,0)&=\\left(\\frac{m}{2\\mu }-b\\right)u(t,0)+d(t),\\end{split}$ where $\\mu >0,m\\ge 0, {n,a,b}\\in \\mathbb {R}$ and $d\\in C^{2}(\\mathbb {R}_{\\ge 0};\\mathbb {R})$ .",
"In order to make the manipulations easier, we set $w(t,x)=e^{\\frac{mx}{2\\mu }}u(t,x)$ .",
"We can then transform the PDE (REF ) with boundary conditions to the following problem (see also [13]): $&w_t=\\mu w_{xx}-\\bigg (\\frac{m^2}{4\\mu }+n\\bigg )w,\\\\&w_x(t,1)=-aw(t,1),\\\\&w_x(t,0)=-bw(t,0)+d(t).$ In this case, $&a_1=a,a_2=1,b_1=b,b_2=1,\\\\&d_1(t)=d(t),d_2(t)=0,M_1=-\\bigg (\\frac{m^2}{4\\mu }+n\\bigg ),M_2=0.$ If we assume that $a\\ge \\frac{4}{3},b<\\frac{1}{3},\\frac{m^2}{4\\mu }+n>0,$ and set $ A_1=\\frac{1}{3},A_2=\\frac{2}{3},A_1^{\\prime }=\\frac{1}{3}\\mu ,A_2^{\\prime }=\\frac{2}{3}\\mu $ , then condition (REF ) and Assumption REF hold.",
"If we assume that $a\\ge -\\frac{1}{3},b<-\\frac{4}{3},\\frac{m^2}{4\\mu }+n>0,$ and set $ B_1=\\frac{2}{3},B_2=\\frac{1}{3},B_1^{\\prime }=\\frac{2}{3}\\mu ,B_2^{\\prime }=\\frac{1}{3}\\mu $ , then condition (REF ) and Assumption REF hold.",
"Under the above two assumptions, (REF ) has a unique solution and (REF ) is ISS, and so is (REF ).",
"Remark 6 If it is easy to fix $A^{\\prime }_i, B^{\\prime }_i\\ (i=1,2)$ , one may verify the conditions in Assumption REF and Assumption REF to conclude the ISS of (REF ) directly.",
"It should be noticed that Assumption REF or Assumption REF is not a necessary condition for the ISS.",
"Therefore, the system may be ISS even if Assumption REF or Assumption REF fails (see also Remark REF ).",
"Remark 7 The system (REF ) was considered in [13] under the boundary conditions: Dirichlet boundary conditions: $u(t,1)=0, \\; u(t,0)=d(t).$ Robin (or Neumann) boundary conditions: $u(t,0)=d(t),u_x(t,1)=\\left(\\frac{m}{2\\mu }-a\\right)u(t,1), (a\\ge 0).$ In the above boundary conditions, $b_2=0$ , which is slightly different from (REF ).",
"The ISS property of (REF ) was obtained by Parseval's identity and the expansions of eigenfunctions of the Sturm-Liouville operator under the same assumption that $\\frac{m^2}{4\\mu }+n>0$ .",
"For the system (REF ) ($m=0$ ) with Dirichlet boundary conditions under a boundary state feedback, the ISS in $L^p$ -norm ($p\\in (2,+\\infty )$ ) is established in [24] by the monotonicity-based method."
],
[
"Conclusion",
"This paper demonstrated via the considered semi-linear PDE that the ISS property with respect to Robin (or Neumann) boundary disturbances can be derived from suitable Lyapunov functionals.",
"The obtained results confirmed that the appearance of the derivatives of boundary disturbances in the ISS estimates can be avoided by directly dealing with the boundary conditions with disturbances.",
"Compared to the work reported in [13], [14], [15], the application of Lyapunov functionals in the establishment of a priori estimates of the solution seems to be less computationally demanding.",
"Therefore, it can be expected that the developed techniques may be applicable in the study of ISS properties for a wider class of PDEs.",
"Finally, it should be mentioned that the technique developed in this work cannot deal with the ISS w.r.t.",
"Dirichlet boundary disturbances, which is the case where $b_2 = 0$ in (REF ).",
"To tackle this type of problems, a method is developed in a parallel work [32]."
]
] | 1709.01880 |
[
[
"Factoring in the Chicken McNugget monoid"
],
[
"Abstract Every day, 34 million Chicken McNuggets are sold worldwide.",
"At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces.",
"However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20.",
"The use of these latter three numbers spawned the so-called Chicken McNugget problem, which asks: \"what numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?\"",
"In this paper, we present an accessible introduction to this problem, as well as several related questions whose motivation comes from the theory of non-unique factorization."
],
[
"Introduction",
"Every day, 34 million Chicken McNuggets are sold worldwide [4].",
"At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces.",
"However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20.",
"The following problem spawned from the use of these latter three numbers.",
"The Chicken McNugget Problem What numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?",
"Early references to this problem can be found in [29], [33].",
"Positive integers satisfying the Chicken McNugget Problem are now known as McNugget numbers [24].",
"In particular, if $n$ is a McNugget number, then there is an ordered triple $(a,b,c)$ of nonnegative integers such that $6a + 9b + 20c = n.$ We will call $(a,b,c)$ a McNugget expansion of $n$ (again see [24]).",
"Since both $(3,0,0)$ and $(0,2,0)$ are McNugget expansions of 18, it is clear that McNugget expansions are not unique.",
"This phenomenon will be the central focus of the remainder of this article.",
"If $\\max \\lbrace a,b,c\\rbrace \\ge 8$ in (REF ), then $n \\ge 48$ and hence determining the numbers $x$ with $0 \\le x \\le 48$ that are McNugget numbers can be checked either by hand or your favorite computer algebra system.",
"The only such $x$ 's that are not McNugget numbers are: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43.",
"(The non-McNugget numbers are sequence A065003 in the On-Line Encyclopedia of Integer Sequences [25].)",
"We demonstrate this in Table REF with a chart that offers the McNugget expansions (when they exist) of all numbers $\\le 50$ .",
"Table: The McNugget numbers and their expansions from 0 to 50.What happens with larger values?",
"Table REF has already verified that 44, 45, 46, 47, 48, and 49 are McNugget numbers.",
"Hence, we have a sequence of 6 consecutive McNugget numbers, and by repeatedly adding 6 to these values, we obtain the following.",
"Proposition 1.1 Any $x > 43$ is a McNugget number.",
"Thus, 43 is the largest number of McNuggets that cannot be ordered with packs of 6, 9, and 20.",
"Our aim in this paper is to consider issues related to the multiple occurances of McNugget expansions as seen in Table REF .",
"Such investigations fall under the more general purview of the theory of non-unique factorizations in integral domains and monoids (a good technical reference on this subject is [22]).",
"Using a general context, we show that the McNugget numbers form an additive monoid and discuss some properites shared by the class of additive submonoids of the nonnegative integers.",
"We then define several combinatorial characteristics arising in non-unique factorization theory, and compute their explicit values for the McNugget Monoid.",
"Figure: The 9 piece box.By emphasizing results concerning McNugget numbers, we offer the reader a glimpse into the vast literature surrounding non-unque factorizations.",
"While we stick to the calculation of basic factorization invariants, our results indicate that such computations involve a fair amount of complexity.",
"Many of the results we touch on have appeared in papers authored or co-authored by undergraduates in National Science Foundation Sponsored REU Programs.",
"This is an area that remains rich in open problems, and we hope our discussion here spurs our readers (both young and old) to explore this rewarding subject more deeply."
],
[
"A brief diversion into generality",
"As illustrated above, Chicken McNugget numbers fit into a long studied mathematical concept.",
"Whether called the Postage Stamp Problem [28], the Coin Problem [17], or the Knapsack Problem [23], the idea is as follows.",
"Given a set of $k$ objects with predetermined values $n_1, n_2, \\ldots , n_k$ , what possible values of $n$ can be had from combinations of these objects?",
"Thus, if a value of $n$ can be obtained, then there is an ordered $k$ -tuple of nonnegative integers $(x_1, \\ldots , x_k)$ that satisfies the linear diophatine equation $n = x_1n_1 + x_2n_2 + \\cdots + x_kn_k.$ We view this in a more algebraic manner.",
"Given integers $n_1, \\ldots , n_k > 0$ , set $\\langle n_1, \\ldots , n_k \\rangle = \\lbrace x_1n_1 + \\cdots + x_kn_k \\mid x_1, \\ldots , x_k \\in \\mathbb {N}_0\\rbrace .$ Notice that if $s_1$ and $s_2$ are in $\\langle n_1, \\ldots , n_k\\rangle $ , then $s_1 + s_2$ is also in $\\langle n_1, \\ldots , n_k\\rangle $ .",
"Since $0 \\in \\langle n_1, \\ldots , n_k\\rangle $ and $+$ is an associative operation, the set $\\langle n_1, \\ldots , n_k\\rangle $ under $+$ forms a monoid.",
"Monoids of nonnegative integers under addition, like the one above, are known as numerical monoids, and $n_1,\\ldots ,n_k$ are called generators.",
"We will call the numerical monoid $\\langle 6, 9, 20 \\rangle $ the Chicken McNugget monoid, and denote it by $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Figure: The 20 piece boxSince $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ consists of the same elements as those in $\\langle 6, 9, 20, 27\\rangle $ , it is clear that generating sets are not unique.",
"Using elementary number theory, it is easy to argue that any numerical monoid $\\langle n_1, \\ldots , n_k\\rangle $ does have a unique generating set with minimal cardinality obtained by eliminating those generators $n_i$ that lie in the numerical monoid generated by $\\lbrace n_1, \\ldots , n_k\\rbrace - \\lbrace n_i\\rbrace $ .",
"In this way, it is clear that $\\lbrace 6, 9, 20\\rbrace $ is indeed the minimal generating set of $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"When dealing with a general numerical monoid $\\langle n_1, \\ldots , n_k\\rangle $ , we will assume without loss of generality that the given generating set $\\lbrace n_1, \\ldots , n_k\\rbrace $ is minimal.",
"In view of this broader setting, the Chicken McNugget Problem can be generalized as follows.",
"The Numerical Monoid Problem If $n_1, \\ldots , n_k$ are positive integers, then which nonnegative integers lie in $\\langle n_1, \\ldots , n_k\\rangle $ ?",
"Example 2.1 We have already determined above exactly which nonnegative integers are McNugget numbers.",
"Suppose the Post Office issues stamps in denominations of 4 cents, 7 cents, and 10 cents.",
"What values of postage can be placed on a letter (assuming that as many stamps as necessary can be placed on the envelope)?",
"In particular, we are looking for the elements of $\\langle 4, 7, 10 \\rangle $ .",
"We can again use brute force to find all the solutions to $4a + 7b + 10c = n$ and conclude that 1, 2, 3, 5, 6, 9, and 13 cannot be obtained.",
"Since 14, 15, 16, and 17 can, all postage values larger than 13 are possible.",
"$\\Box $ Let's return to the largest number of McNuggets that can't be ordered (namely, 43) and the companion number 13 obtained in Example REF .",
"The existence of these numbers is no accident.",
"To see this in general, let $n_1,\\ldots , n_k$ be a set of positive integers that are relatively prime.",
"By elementary number theory, there is a set $y_1, \\ldots , y_k$ of (possibly negative) integers such that $1 = y_1n_1 + \\cdots + y_kn_k.$ By choosing an element $V = x_1n_1 + \\cdots + x_kn_k \\in \\langle n_1, \\ldots , n_k\\rangle $ with sufficiently large coefficients (for instance, if each $x_i \\ge n_1|y_i|$ ), we see $V + 1, \\ldots , V + n_1$ all lie in $\\langle n_1, \\ldots , n_k\\rangle $ as well.",
"As such, any integer greater than $V$ can be obtained in $\\langle n_1, \\ldots , n_k\\rangle $ by adding copies of $n_1$ .",
"This motivates the following definition.",
"Definition 2.2 If $n_1, \\ldots , n_k$ are relatively prime positive integers, then the Frobenius number of $\\langle n_1, \\ldots , n_k\\rangle $ , denoted $F(\\langle n_1, \\ldots , n_k\\rangle )$ , is the largest positive integer $n$ such that $n \\notin \\langle n_1, \\ldots , n_k\\rangle $ .",
"We have already shown that $F(\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}) = 43$ and $F(\\langle 4, 7, 10 \\rangle ) = 13$ .",
"A famous result of Sylvester from 1884 [32] states that if $a$ and $b$ are relatively prime, then $F(\\langle a, b \\rangle ) = ab - a - b$ (a nice proof of this can be found in [7]).",
"This is where the fun begins, as strictly speaking no closed formula exists for the Frobenius number of numerical monoids that require 3 or more generators.",
"While there are fast algorithms that can compute $F(\\langle n_1, n_2, n_3 \\rangle )$ (see for instance [19]), at best formulas for $F(\\langle n_1, \\ldots , n_k\\rangle )$ exist only in special cases (you can find one such special case where $F(\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}) = 43$ pops out in [1]).",
"Our purpose is not to compile or expand upon the vast literature behind the Frobenius number; in fact, we direct the reader to the excellent monograph of Ramírez Alfonsín [30] for more background reading on the Diophatine Frobenius Problem."
],
[
"The McNugget factorization toolkit",
"We focus now on the multiple McNugget expansions we saw in Table REF .",
"In particular, notice that there are McNugget numbers that have unique triples associated to them (6, 9, 12, 15, 20, 21, 26, 29, 32, 35, 40, 41, 46, and 49), some that have two (18, 24, 27, 30, 33, 35, 39, 44, 47, and 50), and even some that have three (36, 42, 45, and 48).",
"While the “normal” notion of factoring occurs in systems where multiplication prevails, notice that the ordered triples representing McNugget numbers are actually factorizations of these numbers into “additive” factors of 6, 9, and 20.",
"Let's borrow some terminology from abstract algebra ([20] is a good beginning reference on the topic).",
"Let $x$ and $y\\in \\langle n_1, \\ldots , n_k\\rangle $ .",
"We say that $x$ divides $y$ in $\\langle n_1, \\ldots , n_k\\rangle $ if there exists a $z\\in \\langle n_1, \\ldots , n_k\\rangle $ such that $y=x+z$ .",
"We call a nonzero element $x \\in \\langle n_1, \\ldots , n_k\\rangle $ irreducible if whenever $x = y + z$ , either $y = 0$ or $z = 0$ .",
"(Hence, $x$ is irreducible if its only proper divisors are 0 and itself).",
"Both of these definitions are obtained from the usual “multiplicative” definition by replacing “$\\cdot $ ” with “$+$ ” and 1 with 0.",
"We leave the proof of the following to the reader.",
"Proposition 3.1 If $\\langle n_1, \\ldots , n_k\\rangle $ is a numerical monoid, then its irreducible elements are precisely $n_1,\\ldots ,n_k$ .",
"Related to irreducibility is the notion of prime elements.",
"A nonzero element $x \\in \\langle n_1, \\ldots , n_k\\rangle $ is prime if whenever $x$ divides a sum $y+z$ , then either $x$ divides $y$ or $x$ divides $z$ (this definition is again borrowed from the multiplicative setting).",
"It is easy to check from the definitions that prime elements are always irreducible, but it turns out that in general irreducible elements need not be prime.",
"In fact, the irreducible elements $n_1, \\ldots , n_k$ of a numerical monoid are never prime.",
"To see this, let $n_i$ be an irreducible element and let $T$ be the numerical monoid generated by $\\lbrace n_1, \\ldots , n_k\\rbrace - \\lbrace n_i\\rbrace $ .",
"Although $n_i \\notin T$ , some multiple of $n_i$ must lie in $T$ (take, for instance, $n_2n_i$ ).",
"Let $kn = \\sum _{j\\ne i}x_jn_j$ (for some $k > 1$ ) be the smallest multiple of $n_i$ in $T$ .",
"Then $n$ divides $\\sum _{j \\ne i} x_jn_j$ over $\\langle n_1, \\ldots , n_k\\rangle $ , but by the minimality of $k$ , $n$ does not divide any proper subsum.",
"Thus $n_i$ is not prime.",
"For our purposes, we restate Proposition REF in terms of $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Corollary 3.2 The irreducible elements of the McNugget monoid are 6, 9, and 20.",
"There are no prime elements."
],
[
"The set of factorizations of an element",
"We refer once again to the elements in Table REF with multiple irreducible factorizations.",
"For each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , let $\\mathsf {Z}(x)=\\lbrace (a,b,c)\\,|\\, 6a+9b+20c = x\\rbrace .$ We will refer to $\\mathsf {Z}(x)$ as the complete set of factorizations $x$ in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , and as such, we could relabel columns 2, 4, and 6 of Table REF as “$\\mathsf {Z}(x)$ .” While we will not dwell on general structure problems involving $\\mathsf {Z}(x)$ , we do briefly address one in the next example.",
"Example 3.3 What elements $x$ in the McNugget monoid are uniquely factorable (i.e., $|\\mathsf {Z}(x)| = 1$ )?",
"A quick glance at Table REF yields 14 such nonzero elements (namely, 6, 9, 12, 15, 20, 21, 26, 29, 32, 35, 40, 41, 46, 49).",
"Are there others?",
"We begin by noting in Table REF that $(3,0,0), (0,2,0) \\in \\mathsf {Z}(18) \\qquad \\mbox{and} \\qquad (10,0,0),(0,0,3) \\in \\mathsf {Z}(60).$ This implies that in any factorization in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , 3 copies of 6 can be freely replaced with 2 copies of 9 (this is called a trade).",
"Similarly, 2 copies of 9 can be traded for 3 copies of 6, and 3 copies of 20 can be traded for 10 copies of 6.",
"In particular, for $n = 6a + 9b + 20c \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , if either $a \\ge 3$ , $b \\ge 2$ or $c \\ge 3$ , then $n$ has more than one factorization in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"As such, if $n$ is to have unique factorization, then $0 \\le a \\le 2$ , $0 \\le b \\le 1$ , and $0 \\le c \\le 2$ .",
"This leaves 18 possibilities, and a quick check yields that the 3 missing elements are $52 = (2,0,2)$ , $55 = (1,1,2)$ and $61 = (2,1,2)$ .",
"$\\Box $ .",
"The argument in Example REF easily generalizes – every numerical monoid that requires more than one generator has finitely many elements that factor uniquely – but note that minimal trades need not be as simple as replacing a multiple of one generator with a multiple of another.",
"Indeed, in the numerical monoid $\\langle 5, 7, 9, 11 \\rangle $ , there is a trade $(1,0,0,1), (0,1,1,0) \\in \\mathsf {Z}(16)$ , though 16 is not a multiple of any generator.",
"Determining the “minimal” trades of a numerical monoid, even computationally, is known to be a very hard problem in general [31]."
],
[
"The length set of an element and related invariants",
"Extracting information from the factorizations of numerical monoid elements (or even simply writing them all down) can be a tall order.",
"To this end, combinatorially-flavored factorization invariants are often used, assigning to each element (or to the monoid as a whole) a value measuring its failure to admit unique factorization.",
"We devote the remainder of this paper to examining several factorization invariants, and what they tell us about the McNugget monoid as compared to more general numerical monoids.",
"We begin by considering a set, derived from the set of factorizations, that has been the focus of many papers in the mathematical literature over the past 30 years.",
"If $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ and $(a,b,c) \\in \\mathsf {Z}(x)$ , then the length of the factorization $(a,b,c)$ is denoted by $|(a,b,c)| = a + b + c.$ We have shown earlier that factorizations in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ may not be unique, and a quick look at Table REF shows that their lengths can also differ.",
"For instance, 42 has three different factorizations, with lengths 5, 6 and 7, respectively.",
"Thus, we denote the set of lengths of $x$ in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ by $\\mathcal {L}(x) = \\lbrace |(a,b,c)| : (a,b,c) \\in \\mathsf {Z}(x)\\rbrace .$ In particular, $\\mathcal {L}(42) = \\lbrace 5,6,7\\rbrace $ .",
"Moreover, set $\\ell (x) = \\min \\mathcal {L}(x) \\qquad \\mbox{and} \\qquad L(x) = \\max \\mathcal {L}(x).$ (In our setting, it is easy to argue that $\\mathcal {L}(x)$ must be finite, so the maximum and minimum above are both well defined.)",
"To give the reader a feel for these invariants, in Table 2 we list all the McNugget numbers from 1 to 50 and their associated values $\\mathcal {L}(x)$ , $\\ell (x)$ , and $L(x)$ .",
"Table: The McNugget numbers from 0 to 50 with ℒ(x)\\mathcal {L}(x), ℓ(x)\\ell (x), and L(x)L(x).The following recent result describes the functions $L(x)$ and $\\ell (x)$ for elements $x \\in \\langle n_1, \\ldots , n_k\\rangle $ that are sufficiently large with respect to the generators.",
"Intuitively, Theorem REF says that for “most” elements $x$ , any factorization with maximal length is almost entirely comprised of $n_1$ , so $L(x + n_1)$ is obtained by taking a maximum length factorization for $x$ and adding one additional copy of $n_1$ .",
"In general, the “sufficiently large” hypothesis is needed, since, for example, both $41 = 2 \\cdot 9 + 1 \\cdot 23$ and $50 = 5 \\cdot 10$ are maximum length factorizations in the numerical monoid $\\langle 9, 10, 23 \\rangle $ .",
"Theorem 3.4 ([5]) Suppose $\\langle n_1, \\ldots , n_k\\rangle $ is a numerical monoid.",
"If $x > n_1n_k$ , then $L(x + n_1) = L(x) + 1,$ and if $x > n_{k-1}n_k$ , then $\\ell (x + n_k) = \\ell (x) + 1.$ We will return to this result in Section REF , where we give a closed formula for $L(x)$ and $\\ell (x)$ that holds for all $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Given our definitions to this point, we can now mention perhaps the most heavily studied invariant in the theory of non-unique factorizations.",
"For $x \\in \\langle n_1, \\ldots , n_k\\rangle $ , the ratio $\\rho (x) = \\frac{L(x)}{\\ell (x)},$ is called the elasticity of $x$ , and $\\rho (\\langle n_1, \\ldots , n_k\\rangle ) = \\sup \\lbrace \\rho (x) \\mid x\\in \\langle n_1, \\ldots , n_k\\rangle \\rbrace $ is the elasticity of $\\langle n_1, \\ldots , n_k\\rangle $ .",
"The elasticity of an element $n \\in \\langle n_1, \\ldots , n_k\\rangle $ measures how “spread out” its factorization lengths are; the larger $\\rho (n)$ is, the more spread out $\\mathcal {L}(n)$ is.",
"To this end, the elasticity $\\rho (\\langle n_1, \\ldots , n_k\\rangle )$ encodes the highest such “spread” throughout the entire monoid.",
"For example, if $\\rho (\\langle n_1, \\ldots , n_k\\rangle ) = 2$ , then the maximum factorization length of any element $n \\in \\langle n_1, \\ldots , n_k\\rangle $ is at most twice its minimum factorization length.",
"A formula for the elasticity of a general numerical monoid, given below, was given in [12], and was the result of an undergraduate research project.",
"Theorem 3.5 ([12], Theorem 2.1 and Corollary 2.3) The elasticity of the numerical monoid $\\langle n_1, \\ldots , n_k\\rangle $ is $\\rho (\\langle n_1, \\ldots , n_k\\rangle ) = \\frac{n_k}{n_1}.$ Moreover, $\\rho (n) = \\frac{n_k}{n_1}$ precisely when $n$ is an integer multiple of the least common multiple of $n_1$ and $n_k$ , and for any rational $r < \\frac{n_k}{n_1}$ , there are only finitely many elements $x \\in \\langle n_1, \\ldots , n_k\\rangle $ with $\\rho (x)\\le r$ .",
"The significance of the final statement in Theorem REF is that there are rationals $1 \\le q \\le \\frac{n_k}{n_1}$ that do not lie in the set $\\lbrace \\rho (x) \\mid x \\in \\langle n_1, \\ldots , n_k\\rangle \\rbrace $ and hence $\\lbrace \\rho (x) \\mid x \\in \\langle n_1, \\ldots , n_k\\rangle \\rbrace \\subsetneq \\mathbb {Q} \\cap [1, \\frac{n_k}{n_1}]$ (to use terminology from the literature, numerical monoids are not fully elastic).",
"Figure REF depicts the elasticities of elements of $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ up to $n = 400$ ; indeed, as $n$ increases, the elasticity $\\rho (n)$ appears to converge to $\\frac{10}{3} = \\rho (\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}})$ .",
"In general, the complete image $\\lbrace \\rho (x) \\mid x \\in \\langle n_1, \\ldots , n_k\\rangle \\rbrace $ has been determined by Barron, O'Neill, and Pelayo in another student co-authored paper [5]; we direct the reader there for a thorough mathematical description of Figure REF .",
"Figure: A plot depicting the elasticity function ρ(n)\\rho (n) for n∈[height=1.7ex]hen-311285.pdfn \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}.We close our discussion of elasticity with the following.",
"Corollary 3.6 The elasticity of the McNugget monoid is $\\rho (\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}) = \\frac{10}{3}.$ While a popular invariant to study, the elasticity only tells us about the largest and smallest elements of $\\mathcal {L}(x)$ .",
"Looking at Table 2, it appears that the length sets of the first few McNugget numbers are uniformly constructed (each is of the form $[a,b]\\cap \\mathbb {N}$ for positive integers $a$ and $b$ ).",
"One need not look too much further to break this pattern; the element $60 \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ has $\\mathsf {Z}(60) = \\lbrace (0,0,3), (1,6,0), (4,4,0), (7,2,0), (10,0,0)\\rbrace $ and thus $\\mathcal {L}(60) = \\lbrace 3, 7, 8, 9, 10\\rbrace .$ This behavior motivates the following “finer” factorization invariant.",
"Fix $x \\in \\langle n_1, \\ldots , n_k\\rangle $ , and let $\\mathcal {L}(x) = \\lbrace m_1, \\ldots , m_t\\rbrace $ with $m_1 < m_2 < \\cdots < m_t$ .",
"Define the delta set of $x$ as $\\Delta (x) = \\lbrace m_i - m_{i-1} \\mid 2 \\le i \\le t\\rbrace ,$ and the delta set of $\\langle n_1, \\ldots , n_k\\rangle $ as $\\Delta (\\langle n_1, \\ldots , n_k\\rangle ) = \\bigcup _{x \\in \\langle n_1, \\ldots , n_k\\rangle } \\Delta (x).$ The study of the delta sets of numerical monoids (and more generally, of cancellative commutative monoids) has been an extremely popular topic; many such papers feature results from REU programs (see, for instance, [8], [10], [11], [13], [14], [16]).",
"From Table REF we see that the McNugget numbers from 1 to 50 all have delta set $\\emptyset $ or $\\lbrace 1\\rbrace $ , and we have further showed that $\\Delta (60) = \\lbrace 1,4\\rbrace $ .",
"What is the delta set of $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ and moreover, what possible subsets of this set occur as $\\Delta (x)$ for some $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ ?",
"We will address those questions is Section REF , with the help of a result from [13], stated below as Theorem REF .",
"One of the primary difficulties in determining the set $\\Delta (\\langle n_1, \\ldots , n_k\\rangle )$ is that even though each element's delta set $\\Delta (x)$ is finite, the definition of $\\Delta (\\langle n_1, \\ldots , n_k\\rangle )$ involves the union of infinitely many such sets.",
"The key turns out to be a description of the sequence $\\lbrace \\Delta (x)\\rbrace _{x \\in \\langle n_1, \\ldots , n_k\\rangle }$ for large $x$ (note that this is a sequence of sets, not integers).",
"Baginski conjectured during the writing of [8] that this sequence is eventually periodic, and three years later this was settled in the affirmative, again in an REU project.",
"Theorem 3.7 ([13]) For $x \\in \\langle n_1, \\ldots , n_k\\rangle $ , $\\Delta (x) = \\Delta (x + n_1n_k)$ whenever $x > 2kn_2n_k^2$ .",
"In particular, $\\Delta (\\langle n_1, \\ldots , n_k\\rangle ) = \\bigcup _{x \\in D} \\Delta (x)$ where $D = \\lbrace x \\in \\langle n_1, \\ldots , n_k\\rangle \\mid x \\le 2kn_2n_k^2 + n_1n_k\\rbrace $ is a finite set.",
"Thus $\\Delta (\\langle n_1, \\ldots , n_k\\rangle )$ can be computed in finite time.",
"The bound given in Theorem REF is far from optimal; it is drastically improved in [21], albeit with a much less concise formula.",
"For convenience, we will use the bound given above in our computation of $\\Delta (\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}})$ in Section REF ."
],
[
"Beyond the length set",
"We remarked earlier that no element of a numerical monoid is prime.",
"Let's consider this more closely in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"For instance, since 6 is not prime, there is a sum $x + y$ in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ such that 6 divides $x + y$ , but 6 does not divide $x$ nor does 6 divide $y$ (take, for instance, $x = y = 9$ ).",
"But note that 6 satisfies the following slightly weaker property.",
"Suppose that 6 divides a sum $x_1+ \\cdots + x_t$ where $t > 3$ .",
"Then there is a subsum of at most 3 of the $x_i$ 's that 6 does divide.",
"To see this, notice that if 6 divides any of the $x_i$ 's, then we are done.",
"So suppose it does not.",
"If 9 divides both $x_i$ and $x_j$ , then 6 divides $x_i + x_j$ since 6 divides $9 + 9$ .",
"If no two $x_i$ 's are divisible by 9, then at least 3 $x_i$ 's are divisible by 20, and nearly identical reasoning to the previous case completes the argument.",
"This value of 3 offers some measure as to how far 6 is from being prime, and motivates the following definition.",
"Definition 3.8 Let $\\langle n_1, \\ldots , n_k\\rangle $ be a numerical monoid.",
"For any nonzero $x \\in \\langle n_1, \\ldots , n_k\\rangle $ , define $\\omega (x) = m$ if $m$ is the smallest positive integer such that whenever $x$ divides $x_1 + \\cdots + x_t$ , with $x_i \\in \\langle n_1, \\ldots , n_k\\rangle $ , then there is a set $T \\subset \\lbrace 1,2,\\ldots ,t\\rbrace $ of indices with $|T| \\le m$ such that $x$ divides $\\sum _{i \\in T} x_i$ .",
"Using Definition REF , a prime element would have $\\omega $ -value 1, so $\\omega (x)$ can be interpreted as a measure of how far $x$ is from being prime.",
"In $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , we argued that $\\omega (6) = 3$ ; a similar argument yields $\\omega (9) = 3$ and $\\omega (20) = 10$ .",
"Notice that the computation of $\\omega (x)$ is dependent more on $\\mathsf {Z}(x)$ than $\\mathcal {L}(x)$ , and hence encodes much different information than either $\\rho (x)$ or $\\Delta (x)$ .",
"Let us more closely examine the argument that $\\omega (6) = 3$ .",
"The key is that 6 divides $9 + 9$ and $20 + 20 + 20$ , but does not divide any subsum of either.",
"Indeed, the latter of these expressions yields a lower bound of $\\omega (6) \\ge 3$ , and the given argument implies that equality holds.",
"With this in mind, we give the following equivalent form of Definition REF , which often simplifies the computation of $\\omega (x)$ .",
"Theorem 3.9 ([27]) Suppose $\\langle n_1, \\ldots , n_k\\rangle $ is a numerical monoid and $x \\in \\langle n_1, \\ldots , n_k\\rangle $ .",
"The following conditions are equivalent.",
"$\\omega (x) = m$ .",
"$m$ is the maximum length of a sum $x_1 + \\cdots + x_t$ of irreducible elements in $\\langle n_1, \\ldots , n_k\\rangle $ with the property that (i) $x$ divides $x_1 + \\cdots + x_t$ , and (ii) $x$ does not divide $x_1 + \\cdots + x_{j-1} + x_{j+1} + \\cdots + x_t$ for $1 \\le j \\le t$ .",
"The sum $x_1 + \\cdots + x_t$ alluded to in part (b) above is called a bullet for $x$ .",
"Hence, $20 + 20 + 20$ is a bullet for 6 in $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , and moreover has maximal length.",
"The benefit of Theorem REF is twofold: (i) each $x \\in \\langle n_1, \\ldots , n_k\\rangle $ has only finitely many bullets, and (ii) the list of bullets can be computed in a similar fashion to the set $\\mathsf {Z}(x)$ of factorizations.",
"We refer the reader to [3], [6], both of which give explicit algorithms (again resulting from undergraduate research projects) for computing $\\omega $ -values.",
"Our goal is to completely describe the behavior of the $\\omega $ -function of the McNugget Monoid.",
"We do so in Section REF , using the following result, which is clearly similar in spirit to Theorems REF and REF .",
"Theorem 3.10 ([26]) For $x \\in \\langle n_1, \\ldots , n_k\\rangle $ sufficiently large, $\\omega (x + n_1) = \\omega (x) + 1.$ In particular, this holds for $x > \\frac{F + n_2}{n_2/n_1 - 1}$ where $F = F(\\langle n_1, \\ldots , n_k\\rangle )$ is the Frobenius number.",
"The similarity between Theorems REF and REF is not a coincidence.",
"While $L(x)$ and $\\omega (x)$ are indeed different functions (for instance, $L(6) = 1$ while $\\omega (6) = 3$ ), they are closely related; the $\\omega $ -function can be expressed in terms of max factorization length that is computed when some collections of generators are omitted.",
"We direct the interested reader to [6], where an explicit formula of this form for $\\omega (n)$ is given."
],
[
"Calculations for the Chicken McNugget monoid",
"In the final section of this paper, we give explicit expressions for $L(x)$ , $\\ell (x)$ , $\\Delta (x)$ and $\\omega (x)$ for every $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"The derivation of each such expression makes use of a theoretical result in Section .",
"We note that each of the formulas provided in this section could also be derived in a purely computational manner, using Theorems REF , REF , and REF and the inductive algorithms introduced in [6] (indeed, these computations finish in a reasonably short amount of time using the implementation in the numericalsgps package discussed in Section ).",
"However, several of the following results identify an interesting phenomenon that distinguish $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ from more general numerical monoids (see the discussion preceeding Question REF ), and the arguments that follow give the reader an idea of how theorems involving factorization in numerical monoids can be proven."
],
[
"Calculating factorization lengths",
"Theorem REF states that $L(x + n_1) = L(x) + 1$ and $\\ell (x + n_k) = \\ell (x) + 1$ for sufficiently large $x \\in \\langle n_1, \\ldots , n_k\\rangle $ .",
"but, it was observed during the writing of [5] that for many numerical monoids, the “sufficiently large” requirement is unecessary.",
"As it turns out, one such example is the McNugget monoid $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , which we detail below.",
"Theorem 4.1 For each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , $L(x + 6) = L(x) + 1$ .",
"In particular, if we write $x = 6q + r$ for $q, r \\in \\mathbb {N}$ and $r < 6$ , then $L(x) = \\left\\lbrace \\begin{array}{l@{\\qquad }l}q & \\textnormal {if } r = 0 \\textnormal { or } 3, \\\\q - 5 & \\textnormal {if } r = 1, \\\\q - 2 & \\textnormal {if } r = 2 \\textnormal { or } 5, \\\\q - 4 & \\textnormal {if } r = 4, \\\\\\end{array}\\right.$ for each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Fix $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ and a factorization $(a, b, c)$ of $x$ .",
"If $b > 1$ , then $x$ has another factorization $(a + 3, b - 2, c)$ with length $a + b + c + 1$ .",
"Similarly, if $c \\ge 3$ , then $(a + 10, b, c - 3)$ is also a factorization of $x$ and has length $a + b + c + 7$ .",
"This implies that if $(a, b, c)$ has maximum length among factorizations of $x$ , then $b \\le 1$ and $c \\le 2$ .",
"Upon inspecting Table REF , we see that unless $x \\in \\lbrace 0, 9, 20, 29, 40, 49\\rbrace $ , we must have $a > 0$ .",
"Now, assume $(a,b,c)$ has maximum length among factorizations of $x$ .",
"We claim $(a+1,b,c)$ is a factorization of $x + 6$ with maximum length.",
"From Table REF , we see that since $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , we must have $x + 6 \\notin \\lbrace 0, 9, 20, 29, 40, 49\\rbrace $ , meaning any maximum length factorization of $x+6$ must have the form $(a^{\\prime } + 1, b^{\\prime }, c^{\\prime })$ .",
"This yields a factorization $(a^{\\prime }, b^{\\prime }, c^{\\prime })$ of $x$ , and since $(a,b,c)$ has maximum length, we have $a + b + c \\ge a^{\\prime } + b^{\\prime } + c^{\\prime }$ .",
"As such, $(a+1,b,c)$ is at least as long as $(a^{\\prime } + 1, b^{\\prime }, c^{\\prime })$ , and the claim is proved.",
"Thus, $L(x+6) = a + 1 + b + c = L(x) + 1.$ From here, the given formula for $L(x)$ now follows from the first claim and the values $L(0)$ , $L(9)$ , $L(20)$ , $L(29)$ , $L(40)$ , and $L(49)$ in Table REF .",
"A similar expression can be obtained for $\\ell (x)$ , ableit with 20 cases instead of 6, this time based on the value of $x$ modulo 20.",
"We encourage the reader to adapt the argument above for Theorem REF .",
"Theorem 4.2 For each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , $\\ell (x + 20) = \\ell (x) + 1$ .",
"In particular, if we write $x = 20q + r$ for $q, r \\in \\mathbb {N}$ and $r < 20$ , then $\\ell (x) = \\left\\lbrace \\begin{array}{l@{\\qquad }l}q & \\textnormal {if } r = 0, \\\\q + 1 & \\textnormal {if } r = 6, 9, \\\\q + 2 & \\textnormal {if } r = 1, 4, 7, 12, 15, 18, \\\\q + 3 & \\textnormal {if } r = 2, 5, 10, 13, 16, \\\\q + 4 & \\textnormal {if } r = 8, 11, 14, 19, \\\\q + 5 & \\textnormal {if } r = 3, 17, \\\\\\end{array}\\right.$ for each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Theorems REF and REF together yield a closed form for $\\rho (x)$ that holds for all $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Since $\\text{lcm}(6,20) = 60$ cases are required, we leave the construction of this closed form to the interested reader."
],
[
"Calculating delta sets",
"Unlike maximum and minimum factorization length, $\\Delta (x)$ is periodic for sufficiently large $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"For example, a computer algebra system can be used to check that $\\Delta (91) = \\lbrace 1\\rbrace $ while $\\Delta (211) = \\lbrace 1,2\\rbrace $ .",
"Theorem REF guarantees $\\Delta (x + 120) = \\Delta (x)$ for $x > 21600$ , but some considerable reductions can be made.",
"In particular, we will reduce the period from 120 down to 20, and will show that equality holds for all $x \\ge 92$ (that is to say, 91 is the largest value of $x$ for which $\\Delta (x + 20) \\ne \\Delta (x)$ ).",
"Theorem 4.3 Each $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ with $x \\ge 92$ has $\\Delta (x + 20) = \\Delta (x)$ .",
"Moreover, $\\Delta (x) = \\left\\lbrace \\begin{array}{l@{\\qquad }l}\\lbrace 1\\rbrace & \\textnormal {if } r = 3, 8, 14, 17, \\\\\\lbrace 1, 2\\rbrace & \\textnormal {if } r = 2, 5, 10, 11, 16, 19, \\\\\\lbrace 1, 3\\rbrace & \\textnormal {if } r = 1, 4, 7, 12, 13, 18, \\\\\\lbrace 1, 4\\rbrace & \\textnormal {if } r = 0, 6, 9, 15, \\\\\\end{array}\\right.$ where $x = 20q + r$ for $q, r \\in \\mathbb {N}$ and $r < 20$ .",
"Hence $\\Delta (\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}})=\\lbrace 1,2,3,4\\rbrace $ .",
"We will show that $\\Delta (x + 20) = \\Delta (x)$ for each $x > 103$ .",
"The remaining claims can be verified by extending Table 2 using computer software.",
"Suppose $x > 103$ , fix a factorization $(a,b,c)$ for $x$ , and let $l = a + b + c$ .",
"If $c \\ge 3$ , then $x$ also has factorizations $(a + 10, b, c - 3)$ , $(a + 7, b + 2, c - 3)$ , $(a + 4, b + 4, c - 3)$ , and $(a + 1, b + 6, c - 3)$ , meaning $\\lbrace l, l + 4, l + 5, l + 6, l + 7\\rbrace \\subset \\mathcal {L}(x).$ Alternatively, since $x > 103$ , if $c \\le 2$ , then $6a + 9b \\ge 63$ , and thus $l \\ge a + b + 2 \\ge 9 \\ge \\ell (x) + 4.$ The above arguments imply (i) any gap in successive lengths in $\\mathcal {L}(x)$ occurs between $\\ell (x)$ and $\\ell (x) + 4$ , and (ii) every factorization with length in that interval has at least one copy of 20.",
"As such, $x + 20$ has the same gaps between $\\ell (x + 20)$ and $\\ell (x + 20) + 4$ as $x$ does between $\\ell (x)$ and $\\ell (x) + 4$ , which proves $\\Delta (x + 20) = \\Delta (x)$ for all $x > 103$ .",
"With a slightly more refined argument than the one given above, one can prove without the use of software that $\\Delta (x + 20) = \\Delta (x)$ for all $x \\ge 92$ .",
"We encourage the interested reader to work out such an argument."
],
[
"Calculating $\\omega $ -primality",
"We conclude our study of $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ with an expression for the $\\omega $ -primality of $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ and show (in some sense) how far a McNugget number is from being prime.",
"We proceed in a similar fashion to Theorems REF and REF , showing that with only two exceptions, $\\omega (x + n_1) = \\omega (x) + 1$ for all $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Theorem 4.4 With the exception of $x = 6$ and $x = 12$ , every nonzero $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ satisfies $\\omega (x + 6) = \\omega (x) + 1$ .",
"In particular, we have $\\omega (x) = \\left\\lbrace \\begin{array}{l@{\\qquad }l}q & \\textnormal {if } r = 0, \\\\q + 5 & \\textnormal {if } r = 1, \\\\q + 7 & \\textnormal {if } r = 2, \\\\q + 2 & \\textnormal {if } r = 3, \\\\q + 4 & \\textnormal {if } r = 4, \\\\q + 9 & \\textnormal {if } r = 5, \\\\\\end{array}\\right.$ for each $x \\ne 6, 12$ , where $x = 6q + r$ for $q, r \\in \\mathbb {N}$ and $r < 6$ .",
"Fix $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Following the spirit of the proof of Theorem REF , we begin by proving each $x > 12$ has a maximum length bullet $(a,b,c)$ with $a > 0$ .",
"Indeed, suppose $(0,b,c)$ is a bullet for $x$ for some $b, c \\ge 0$ .",
"The element $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ also has some bullet of the form $(a^{\\prime },0,0)$ , where $a^{\\prime }$ the smallest integer such that $6a^{\\prime } - x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Notice $a^{\\prime } \\ge 3$ since $x > 12$ .",
"We consider several cases.",
"If $c = 0$ , then $9b - x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ but $9b - x - 9 \\notin \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"If $b \\le 3$ , then $a^{\\prime } \\ge b$ .",
"Otherwise, either $9(b-1)$ or $9(b-2)$ is a multiple of 6, and since $9(b-2) - x \\notin \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ as well, we see $a^{\\prime } \\ge \\frac{3}{2}(b-2) + 1 \\ge b$ .",
"If $b = 0$ , then there are two possibilities.",
"If $c \\le 3$ , then $a^{\\prime } \\ge c$ .",
"Otherwise, either $20(c-1)$ , $20(c-2)$ or $20(c-3)$ is a multiple of 6, so we conclude $a^{\\prime } \\ge \\frac{10}{3}(c - 3) + 1 \\ge c$ .",
"If $b, c > 0$ , then $9b + 20c - x - 9, 9b + 20c - x - 20 \\notin \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , so $9b + 20c - x$ is either 0, 6, or 12.",
"This means either $(3,b-1,c)$ , $(2,b-1,c)$ , or $(1,b-1,c)$ is also a bullet for $x$ , respectively.",
"In each case, we have constructed a bullet for $x$ at least as long as $(0,b,c)$ , but with positive first coordinate, so we conclude $x$ has a maximal bullet with nonzero first coordinate.",
"Now, using a similar argument to that given in the proof of Theorem REF , if $(a + 1, b, c)$ is a maximum length bullet for $x + 6$ , then $(a, b, c)$ is a maximum length bullet for $x$ .",
"This implies $\\omega (x + 6) = \\omega (x) + 1$ whenever $x + 6$ has a maximum length bullet with positive first coordinate, which by the above argument holds whenever $x > 12$ .",
"This proves the first claim.",
"The formula for $\\omega (x)$ now follows from the first claim, the computations $\\omega (9) = 3$ and $\\omega (20) = 10$ from Section REF , and analogous computations for $\\omega (15) = 4$ , $\\omega (18) = 3$ , $\\omega (29) = 13$ , $\\omega (40) = 10$ , and $\\omega (49) = 13$ .",
"Figure REF plots $\\omega $ -values of elements of the McNugget monoid $\\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ .",
"Since $\\omega (x + 6) = \\omega (x) + 1$ for large $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , most of the plotted points occur on one of 6 lines with slope $\\frac{1}{6}$ .",
"It is also evident in the plot that $x = 6$ and $x = 12$ are the only exceptions.",
"Figure: A plot depicting the ω\\omega -primality function ω(n)\\omega (n) for n∈[height=1.7ex]hen-311285.pdfn \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}.Although $\\omega (x + 6) = \\omega (x) + 1$ does not hold for every $x \\in \\mathchoice{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.7ex]{hen-311285.pdf}}{\\includegraphics [height=1.5ex]{hen-311285.pdf}}{\\includegraphics [height=1.0ex]{hen-311285.pdf}}$ , there are some numerical monoids for which the “sufficiently large” hypothesis in Theorem REF can be dropped (for instance, any numerical monoids with 2 minimal generators has this property).",
"Hence, we conclude with a problem suitable for attack by undergraduates.",
"Question 4.5 Determine which numerical monoids $\\langle n_1, \\ldots , n_k\\rangle $ satisfy each of the following conditions for all $x$ (i.e., not just sufficiently large $x$ ): $L(x + n_1) = L(x) + 1$ , $\\ell (x + n_k) = \\ell (x) + 1$ , or $\\omega (x + n_1) = \\omega (x) + 1$ ."
],
[
"Appendix: computer software for numerical monoids",
"Many of the computations referenced in this paper can be performed using the numericalsgps package [18] for the computer algebra system GAP.",
"The brief snippet of sample code below demonstrates how the package is used to compute various quantities discussed in this paper.",
"gap> LoadPackage(\"num\"); true gap> McN:=NumericalSemigroup(6,9,20); <Numerical semigroup with 3 generators> gap> FrobeniusNumberOfNumericalSemigroup(McN); 43 gap> 43 in McN; false gap> 44 in McN; true gap> FactorizationsElementWRTNumericalSemigroup(18,McN); [ [ 3, 0, 0 ], [ 0, 2, 0 ] ] gap> OmegaPrimalityOfElementInNumericalSemigroup(6,McN); 3 This only scratches the surface of the extensive functionality offered by the numericalsgps package.",
"We encourage the interested reader to install and experiment with the package; instructions can be found on the official webpage, whose URL is included below.",
"https://www.gap-system.org/Packages/numericalsgps.html"
]
] | 1709.01606 |
[
[
"The Proca Field in Curved Spacetimes and its Zero Mass Limit"
],
[
"Abstract We investigate the classical and quantum Proca field (a massive vector potential) of mass $m>0$ in arbitrary globally hyperbolic spacetimes and in the presence of external sources.",
"We motivate a notion of continuity in the mass for families of observables $\\left\\{O_m\\right\\}_{m>0}$ and we investigate the massless limit $m\\to0$.",
"Our limiting procedure is local and covariant and it does not require a choice of reference state.",
"We find that the limit exists only on a subset of observables, which automatically implements a gauge equivalence on the massless vector potential.",
"For topologically non-trivial spacetimes, one may consider several inequivalent choices of gauge equivalence and our procedure selects the one which is expected from considerations involving the Aharonov-Bohm effect and Gauss' law.",
"We note that the limiting theory does not automatically reproduce Maxwell's equation, but it can be imposed consistently when the external current is conserved.",
"To recover the correct Maxwell dynamics from the limiting procedure would require an additional control on limits of states.",
"We illustrate this only in the classical case, where the dynamics is recovered when the Lorenz constraint remains well behaved in the limit."
],
[
"Introduction",
"Massive vector potentials satisfying Proca's equation are the most straightforward massive generalization of the massless vector potential of electromagnetism.",
"They may be used for an effective description of vector particles in the standard model, such as W- and Z-bosons (who really acquire their mass through the Higgs mechanism), or as a modification of the massless photon.",
"In the latter scenario, the Proca field provides a theoretical framework to study upper bounds on the photon mass.",
"It is important to note, however, that the Proca field does not have a gauge symmetry, unlike the massless vector potential of electromagnetismAn alternative approach to massive electrodynamics due to Stueckelberg preserves the gauge invariance by introducing an extra scalar field, cf.",
"..",
"In this paper we will make a theoretical investigation of the massless limit of the Proca field in curved spacetimes, with special attention to the emergence of the gauge symmetry.",
"In Minkowski space this massless limit is textbook material (cf.",
"), but the corresponding problem in curved spacetimes poses some additional interesting challenges, which we now discuss.",
"Firstly, to define the quantum Proca field we cannot avail ourselves of a vacuum state or a preferred Hilbert space representation for the quantum theory.",
"However, it is well understood how to circumvent this problem using an algebraic approach.",
"On a given spacetime we can then describe the Proca field of mass $m>0$ with an external current $j$ by an abstract $^*$ -algebra ${A}_{m,j}$ .",
"For $j=0$ such a construction has already been given by Furlani , imposing some topological restrictions, and later by Dappiaggi .",
"The methods needed to include non-trivial currents $j$ are also well known in principle, see e.g.",
"or .",
"In this paper we will not pursue the investigation of states and Hilbert space representations, which forms the next step in the description of the quantum theory.",
"Secondly, to define a notion of continuity in the mass, we will need to compare the algebras ${A}_{m,j}$ at different values of $m$ .",
"Once again we cannot resort to preferred vacuum states or Hilbert space representations.",
"Instead we will propose a notion of continuity in the mass for families of observables $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ , which is formulated entirely at the algebraic level.",
"This continuity makes use of the fact that for all $m>0$ the algebras ${A}_{m,j}$ are isomorphic to an algebra of initial data on a Cauchy surface, which is independent of $m$ .",
"We prove that our notion of continuity is independent of the choice of Cauchy surface before we define the massless limit of the Proca field.",
"Thirdly, the gauge freedom of free electromagnetism admits at least three generalisations from Minkowski space to spacetimes with non-trivial topologies.",
"One may use e.g.",
"the field strength tensor $F$ , or equivalence classes of one-forms $A$ , where the pure gauge solutions are either the closed or the exact one-forms.",
"One of us has previously argued that the latter choice is the preferred one in a generally covariant setting, because it allows the correct description of the Aharonov-Bohm effect and Gauss' law .",
"We will show that this choice of gauge equivalence also arises naturally from the limiting procedure, thereby providing an additional justification for it.",
"In Section below we will review the classical and the quantum Proca field in an arbitrary globally hyperbolic spacetime with a fixed mass $m>0$ and external current $j$ .",
"In Section we will then formulate the continuity in the mass and define the zero mass limit.",
"We will show that this limit exists only for a certain sub-algebra of observables and, by choosing this algebra as large as possible, we automatically arrive at the gauge equivalence given by exact forms, as preferred by .",
"In this section we also comment on the fact that the zero mass limit yields a theory that does not automatically include Maxwell's equations.",
"We believe that this is due to the fact that we did not include the behaviour of states in the zero mass limit, and we illustrate this with an argument concerning the classical Proca field.",
"Although it may be possible to include classes of states (e.g.",
"Hadamard states ) and to study their behaviour during a limiting process, we will not pursue this in the present investigation.",
"Section contains our conclusions and a brief outlook.",
"We will use the remainder of this section to introduce some conventions and notations that will be used throughout the paper.",
"We let $(\\mathcal {M},g)$ denote a spacetime, consisting of a smooth, four dimensional manifold $\\mathcal {M}$ , assumed to be Hausdorff, connected, oriented and para-compact, and a Lorentzian metric $g$ , whose signature is chosen to be $(-,+,+,+)$ .",
"We assume that $(\\mathcal {M},g)$ is globally hyperbolic and time-oriented.",
"A generic smooth, space-like Cauchy surface is denoted by $\\Sigma $ , with an induced Riemannian metric $h$ .",
"The Levi-Civita connection on $(\\mathcal {M},g)$ will be denoted by $\\nabla $ and the one on $\\Sigma $ by $\\nabla _{(\\Sigma )}$ .",
"For further standard notations regarding spacetimes (e.g.",
"causal relations and tensor calculus) we refer to .",
"The space of smooth differential forms on $\\mathcal {M}$ of degree $p$ will be denoted by $\\Omega ^p(\\mathcal {M})$ , and the subspace of compactly supported forms by $\\Omega ^p_0(\\mathcal {M})$ .",
"The space of all differential forms is an algebra under the exterior product $\\wedge $ .",
"Using the metric we can define a Hodge $*$ -operation such that $A\\wedge *B=\\frac{1}{p!",
"}A^{\\mu _1\\ldots \\mu _p}B_{\\mu _1\\ldots \\mu _p}d\\mathrm {vol}_g$ , where $d\\mathrm {vol}_g$ is the natural volume form determined by the metric.",
"We may define a pairing on the space of $p$ -forms by $\\langle A,B\\rangle _{\\mathcal {M}} \\int _{\\mathcal {M}} A\\wedge {*B}$ when the support of $A\\wedge {*B}$ is compact.",
"The pairing is symmetric, $\\langle A,B\\rangle _{\\mathcal {M}}=\\langle B,A\\rangle _{\\mathcal {M}}$ , and it defines an inner product on the spaces $\\Omega ^p_0(\\mathcal {M})$ .",
"The co-derivative $\\delta $ is defined in terms of the exterior derivative $d$ by $\\delta (-1)^{s+1+n(p-1)}{*d*}$ when acting on $p$ -forms, where $n$ is the dimension of the manifold ($n=4$ on $\\mathcal {M}$ and $n=3$ on the Cauchy surface $\\Sigma $ ) and $s$ is the number of negative eigenvalues of the metric ($s=1$ on $\\mathcal {M}$ and $s=0$ on $\\Sigma $ ).",
"One may show that $\\delta $ and $d$ are each other's (formal) adjoints under the pairing $\\langle \\hspace{0.20004pt}\\cdot \\hspace{0.20004pt},\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}\\rangle _{\\mathcal {M}}$ .",
"The Laplace-Beltrami operator on $p$ -forms is defined by $\\square =d\\delta +\\delta d$ , which is a normally hyperbolic operator.",
"A form $A$ is called closed when $dA=0$ and exact when $A=dB$ for some differential form $B$ .",
"It will be convenient to denote the space of closed $p$ -forms on $\\mathcal {M}$ by $\\Omega ^p_d(\\mathcal {M})$ and the compactly supported closed $p$ -forms by $\\Omega ^p_{0,d}(\\mathcal {M})$ .",
"Similarly, $A$ is called co-closed when $\\delta A=0$ and co-exact when $A={\\delta B}$ for some differential form $B$ .",
"Once again it will be convenient to denote the space of co-closed forms on $\\mathcal {M}$ by $\\Omega ^p_{\\delta }(\\mathcal {M})$ and the compactly supported co-closed $p$ -forms by $\\Omega ^p_{0,\\delta }(\\mathcal {M})$ .",
"For more details on differential forms we refer the reader to .",
"Let $A, j \\in \\Omega ^1 (\\mathcal {M})$ be smooth one-forms on $\\mathcal {M}$ and $m >0$ a positive constant.",
"We will call $A$ the Proca field, $m$ its mass and $j$ an external current.",
"The Proca equation reads: $\\left( \\delta d + m^2 \\right) A = j \\,.$ Accordingly, the Proca operator is defined as $(\\delta d + m^2)$ .",
"It is well known that the Proca operator is Green-hyperbolic but not normally hyperbolic .",
"However, we can decompose Proca's equation into a wave equation and a Lorenz constraint: $\\left(\\square +m^2 \\right) A &= j + m^{-2} \\, d \\delta j \\,,\\\\\\delta A &= m^{-2} \\delta j \\,,$ which together are equivalent to the Proca equation (REF ) when $m>0$ .",
"Indeed, applying $\\delta $ to (REF ) yields (), and in the presence of this equality, (REF ) and (REF ) are equivalent.",
"Following Dimock , Furlani and Pfenning we parametrise the initial data of differential forms with the following operators: Definition 2.1 Let $i : \\Sigma \\hookrightarrow \\mathcal {M}$ be the inclusion of the Cauchy surface $\\Sigma $ with pullback $i^*$ .",
"The operators $\\rho _{(0)}, \\rho _{(d)}: \\Omega ^p(\\mathcal {M}) \\rightarrow \\Omega ^p(\\Sigma )$ and $\\rho _{(n)}, \\rho _{(\\delta )}: \\Omega ^p(\\mathcal {M}) \\rightarrow \\Omega ^{p-1}(\\Sigma )$ are defined as: $\\rho _{(0)}= i^* \\,,\\quad \\rho _{(d)}= -{*_{(\\Sigma )} i^*} {* d} \\,,\\quad \\rho _{(\\delta )}= i^*\\delta \\quad \\text{and}\\quad \\rho _{(n)}= -{*_{(\\Sigma )} i^* *} \\,.$ Let $A \\in \\Omega ^1(\\mathcal {M})$ .",
"The differential forms ${A_{(0)}}, {A_{(d)}}\\in \\Omega ^1(\\Sigma )$ and ${A_{(n)}}, {A_{(\\delta )}}\\in \\Omega ^0(\\Sigma )$ are defined as: ${A_{(0)}}= \\rho _{(0)}A \\,, \\quad {A_{(d)}}= \\rho _{(d)}A \\,, \\quad {A_{(n)}}= \\rho _{(n)}A \\quad \\textrm {and}\\quad {A_{(\\delta )}}= \\rho _{(\\delta )}A \\,.$ Specifying these differential forms is equivalent to specifying the initial data $A_\\mu $ and $n^\\alpha \\nabla _\\alpha A_\\mu $ on the Cauchy surface $\\Sigma $ with future pointing unit normal vector field $n$ .",
"The wave operator $(\\square + m^2)$ on $p$ -forms has unique advanced $(-)$ and retarded $(+)$ fundamental solutions $E_m^\\pm : \\Omega ^p_0(\\mathcal {M}) \\rightarrow \\Omega ^p(\\mathcal {M})$ with $\\mathrm {supp}\\left(E^\\pm _m F\\right) \\subset J^\\pm (\\mathrm {supp}\\left(F\\right))$ .",
"It is straightforward to show that the fundamental solutions intertwine their action with the interior and exterior derivative, i. e. it holds $E_m^\\pm d = d E_m^\\pm $ and $E_m^\\pm \\delta = \\delta E_m^\\pm $ .",
"The advanced minus retarded fundamental solution is denoted by $E_m = E_m^- - E_m^+$ .",
"With the notion of the fundamental solutions we can state a solution to the wave equation (REF ) in form of the following Theorem 2.2 (Solution of the wave equation) Let ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ and ${A_{(n)}},{A_{(\\delta )}}\\in \\Omega ^0(\\Sigma )$ specify initial data on the Cauchy surface $\\Sigma $ .",
"Let $F \\in \\Omega ^1_0(\\mathcal {M})$ be a test one-form and $\\kappa \\in \\Omega ^1(\\mathcal {M})$ an external source.",
"Then, for any $m\\ge 0$ , $\\langle A, F \\rangle _\\mathcal {M}= \\sum \\limits _\\pm \\langle E_m^\\mp F , \\kappa \\rangle _{J^{\\pm }(\\Sigma )}&- \\langle {A_{(0)}}, \\rho _{(d)}E_m F \\rangle _\\Sigma - \\langle {A_{(\\delta )}}, \\rho _{(n)}E_m F \\rangle _\\Sigma \\\\&+ \\langle {A_{(n)}}, \\rho _{(\\delta )}E_m F \\rangle _\\Sigma + \\langle {A_{(d)}}, \\rho _{(0)}E_m F \\rangle _\\Sigma $ specifies the unique smooth solution $A\\in \\Omega ^1(\\mathcal {M})$ of the wave equation $(\\square + m^2)A= \\kappa $ with the given initial data.",
"Furthermore, the solution depends continuously on the initial data.",
"The proof is a straightforward generalization of the source free case , see e. g. Theorem 2.3 and Lemma 2.4 of .",
"Now that we have solved the wave equation (REF ), we turn to the Lorenz constraint ().",
"Assume that $A\\in \\Omega ^1(\\mathcal {M})$ solves the wave equation, $(\\square +m^2)A = j + m^{-2} d \\delta j$ .",
"We observe $(\\square +m^2) \\delta A&= \\delta (\\square + m^2) A= \\delta \\left( j + m^{-2} d \\delta j \\right) \\\\&= (\\square + m^2 ) m^{-2} \\delta j \\,.$ The solution $A$ to the wave equation therefore yields a Klein-Gordon equation for $\\delta A - m^{-2} \\delta j$ .",
"This ensures that the Lorenz constraint () propagates and, to impose the constraint (and hence obtain a solution to Proca's equation), it suffices to require that the initial data of $\\delta A - m^{-2} \\delta j$ vanish on the Cauchy surface $\\Sigma $ .",
"We will re-express this requirement in terms of constraints on initial data of $A$ , making use of the following two lemmas: Lemma 2.3 Let $\\Sigma $ be a Cauchy surface with unit normal vector field $n$ .",
"For any smooth zero-form $f\\in \\Omega ^0(\\mathcal {M})$ it holds that $\\rho _{(n)}f = 0 \\,, \\quad \\rho _{(\\delta )}f = 0 \\,, \\quad \\rho _{(0)}f = {\\left.\\hspace{0.0pt}f \\vphantom{\\big |} \\right|_{\\Sigma } } \\,, \\quad \\textrm {and}\\quad \\rho _{(d)}f = {\\left.\\hspace{0.0pt}(df)(n) \\vphantom{\\big |} \\right|_{\\Sigma } } = {\\left.\\hspace{0.0pt}\\left(n^\\alpha \\nabla _\\alpha f\\right) \\vphantom{\\big |} \\right|_{\\Sigma } } \\,.$ Therefore, with respect to the Klein Gordon equation, $\\rho _{(0)}f$ and $\\rho _{(d)}f$ specify initial data on $\\Sigma $ .",
"Proof: 2.1 The proof of these identities is straightforward (cf. ).",
"For example, $\\rho _{(d)}f=\\rho _{(n)}df={\\left.\\hspace{0.0pt}n^\\alpha (df)_\\alpha \\vphantom{\\big |} \\right|_{\\Sigma } }$ by , and $(df)_{\\alpha }=\\nabla _\\alpha f$ .",
"Lemma 2.4 (Gaussian Coordinates) Let $\\Sigma $ be a Cauchy surface of $\\mathcal {M}$ with future pointing unit normal vector field $n$ .",
"We can extend $n$ to a neighbourhood of $\\Sigma $ such that $n^\\alpha \\nabla _\\alpha n^\\beta = 0 \\,, \\quad dn = 2 \\nabla _{[\\mu } n_{\\nu ]} = 0 \\,.",
"$ Proof: 2.2 An introduction to Gaussian (normal) coordinates is for example given in or where the first equation of (REF ) is shown to hold by construction.",
"The second Equation of (REF ) can be derived by using Frobenius' theorem (see for example ) as explained in .",
"With this normal vector field we can write the metric $g$ of the spacetime $\\mathcal {M}$ in a neighbourhood of the Cauchy surface as $g{_{\\mu \\nu }} = - n_\\mu n_\\nu + h{_{\\mu \\nu }}$ , where $h$ extends the induced metric on $\\Sigma $ .",
"To state the main result of this section we also introduce fundamental solutions for the Proca operator $(\\delta d + m^2)$ .",
"The Proca operator, being Green hyperbolic, has unique advanced $(-)$ and retarded $(+)$ fundamental solutions $G_m^\\pm : \\Omega ^p_0(\\mathcal {M}) \\rightarrow \\Omega ^p(\\mathcal {M})$ which are given in terms of the fundamental solutions of the wave operator by $G_m^\\pm = (m^{-2}{d\\delta } +1) E_m^\\pm \\,,$ cf.",
".",
"Analogously we define $G_m = (m^{-2}{d\\delta } +1) E_m$ .",
"We then have Theorem 2.5 (Solution of Proca's equation) Let ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ on $\\Sigma $ , $F \\in \\Omega ^1_0(\\mathcal {M})$ a test one-form, $j \\in \\Omega ^1(\\mathcal {M})$ an external source and $m >0$ a mass.",
"Then, $\\langle A, F \\rangle _\\mathcal {M}= \\sum \\limits _\\pm \\langle j , G_m^\\mp F \\rangle _{J^{\\pm }(\\Sigma )}- \\langle {A_{(0)}}, \\rho _{(d)}G_m F \\rangle _\\Sigma + \\langle {A_{(d)}}, \\rho _{(0)}G_m F \\rangle _\\Sigma $ specifies the unique smooth solution of Proca's equation $\\left( \\delta d + m^2 \\right) A = j$ with the given ${A_{(0)}}$ and ${A_{(d)}}$ .",
"Furthermore, the solution depends continuously on these initial data, and we have ${A_{(\\delta )}}= m^{-2} \\rho _{(\\delta )}j \\quad \\text{and} \\quad m^2 \\, {A_{(n)}}= \\rho _{(n)}j + \\delta _{(\\Sigma )}{A_{(d)}}\\,.$ Proof: 2.3 We use the equivalence of the Proca equation (REF ) with the wave equation (REF ) and the vanishing of the initial data of $\\delta A-m^{-2}\\delta j$ .",
"We let $A$ be a solution of the wave equation $(\\square + m^2) A = \\kappa $ with $\\kappa = j + m^{-2} \\, d \\delta j$ .",
"We first show that the specified constraints (REF ) on the initial data are equivalent to the vanishing of the initial data of $\\delta A - m^{-2} \\delta j$ .",
"For this we use Lemma REF and REF .",
"The vanishing of the initial value yields, using the linearity of the pullback and Definition REF : $0 = \\rho _{(0)}\\left( \\delta A - m^{-2} \\delta j\\right)= \\rho _{(\\delta )}A - m^{-2} \\rho _{(\\delta )}j \\,.$ We will calculate the vanishing of the normal derivative in Gaussian normal coordinates and in the end turn back to a coordinate independent notation: $0 = \\rho _{(d)}\\left( \\delta A - m^{-2} \\delta j\\right)= {\\left.\\hspace{0.0pt}\\Big (n^\\alpha \\nabla _\\alpha \\delta A \\Big ) \\vphantom{\\big |} \\right|_{\\Sigma } } - m^{-2} \\rho _{(d)}\\delta j \\,.",
"$ We will take a separate look at the first summand: $n^\\alpha \\nabla _\\alpha \\delta A&= n^\\alpha \\left( d\\delta A \\right)_{\\alpha }= n^\\alpha \\square A_\\alpha - n^\\beta \\left( \\delta d A \\right)_\\beta \\\\&= n^\\alpha \\kappa _\\alpha -{m^2} \\, n^\\mu A_\\mu + 2 n^\\beta \\nabla ^\\nu \\nabla _{[\\nu } A_{\\beta ]} \\\\&= n^\\alpha \\kappa _\\alpha - {m^2} \\, n^\\mu A_\\mu + 2 \\nabla ^\\nu \\left( n^\\beta \\nabla _{[\\nu } A_{\\beta ]} \\right) \\,,$ where we have used that $\\nabla ^\\nu n^\\beta $ is symmetric by Lemma REF .",
"Writing $g{_{\\mu \\nu }} = -n_\\mu n_\\nu + h{_{\\mu \\nu }}$ and using Lemma REF we find: ${\\left.\\hspace{0.0pt}g{^{\\sigma \\nu }} \\nabla _\\sigma \\left( n^\\beta \\nabla _{[\\nu } A_{\\beta ]} \\right) \\vphantom{\\big |} \\right|_{\\Sigma } }&= {\\left.\\hspace{0.0pt}\\left(- n^\\sigma n^\\nu + h{^{\\sigma \\nu }} \\right) \\nabla _\\sigma \\left( n^\\beta \\nabla _{[\\nu } A_{\\beta ]} \\right) \\vphantom{\\big |} \\right|_{\\Sigma } } \\\\&= 0 + {\\left.\\hspace{0.0pt}\\nabla _{(\\Sigma )}^{\\nu } \\left( n^\\beta \\nabla _{[\\nu } A_{\\beta ]} \\right) \\vphantom{\\big |} \\right|_{\\Sigma } } \\,.$ Here we have made use of the identification $ h{^{\\sigma \\nu }} \\nabla _\\sigma B_\\mu = \\nabla _{(\\Sigma )}^\\nu B_\\mu $ for any one-form $B$ tangential to $\\Sigma $ .",
"We identify $2 n^\\beta \\nabla _{[\\nu } A_{\\beta ]} = -2 n^\\beta \\nabla _{[\\beta } A_{\\nu ]} =- A_{(d)\\nu }$ and use $\\delta _{(\\Sigma )} B = -\\nabla _{(\\Sigma )} ^\\alpha B_\\alpha $ to obtain ${\\left.\\hspace{0.0pt}n^\\alpha \\nabla _\\alpha \\delta A \\vphantom{\\big |} \\right|_{\\Sigma } }=\\rho _{(n)}\\kappa - m^2 {A_{(n)}}+ \\delta _{(\\Sigma )} {A_{(d)}}\\,.$ Inserting this into Equation (REF ) and using the definition of the source term $\\kappa = j + m^{-2} {d \\delta j}$ , we find from $\\rho _{(d)}= \\rho _{(n)}d$ that $m^2 {A_{(n)}}= \\rho _{(n)}\\kappa - m^{-2} \\rho _{(d)}\\delta j + \\delta _{(\\Sigma )} {A_{(d)}}= \\rho _{(n)}j + \\delta _{(\\Sigma )} {A_{(d)}}\\,.$ This proves that (REF ) are the required constraints.",
"We now substitute the constraints (REF ) in the formula of Theorem REF and show that we recover Equation (REF ).",
"We find $ \\langle A, F \\rangle _\\mathcal {M}=& \\sum \\limits _\\pm \\langle j + m^{-2} d\\delta j, E_m^\\mp F \\rangle _{J^{\\pm }(\\Sigma )}- \\langle {A_{(0)}}, \\rho _{(d)}E_m F \\rangle _\\Sigma - m^{-2} \\langle \\rho _{(\\delta )}j, \\rho _{(n)}E_m F \\rangle _\\Sigma \\\\& + m^{-2} \\langle \\delta _{(\\Sigma )} {A_{(d)}}, \\rho _{(\\delta )}E_m F \\rangle _\\Sigma + m^{-2} \\langle \\rho _{(n)}j, \\rho _{(\\delta )}E_m F \\rangle _\\Sigma + \\langle {A_{(d)}}, \\rho _{(0)}E_m F \\rangle _\\Sigma \\,.$ Now, for clarity's sake, we take a look at the appearing terms separately.",
"To get rid of the divergence of ${A_{(d)}}$ , we use the formal adjointness of $\\delta $ and $d$ and the commutativity of $d$ with the pullback $i^*$ : $m^{-2}\\langle \\delta _{(\\Sigma )} {A_{(d)}}, \\rho _{(\\delta )}E_m F \\rangle _\\Sigma =& m^{-2}\\langle {A_{(d)}}, d_{(\\Sigma )} i^* \\delta E_m F \\rangle _\\Sigma \\\\=& m^{-2}\\langle {A_{(d)}}, i^* d \\delta E_m F \\rangle _\\Sigma \\\\=& m^{-2}\\langle {A_{(d)}}, \\rho _{(0)}d \\delta E_m F \\rangle _\\Sigma \\,,$ which, together with $\\langle {A_{(d)}}, \\rho _{(0)}E_m F \\rangle _\\Sigma $ , combines to $\\langle {A_{(d)}}, \\rho _{(0)}G_m F \\rangle _\\Sigma $ .",
"Next, we have a look at a part of the sum term and use Stoke's theorem (we get a sign $\\mp $ due to the orientation of $\\Sigma $ with respect to $J^{\\pm }(\\Sigma )$ ) for a partial integration, at the cost of some boundary terms: $\\sum \\limits _\\pm &\\langle d \\delta j , E_m^\\mp F \\rangle _{J^{\\pm }(\\Sigma )} -\\langle j , d \\delta E_m^\\mp F \\rangle _{J^{\\pm }(\\Sigma )}\\nonumber \\\\&= \\sum \\limits _\\pm \\int _{J^{\\pm }(\\Sigma )} d \\delta j \\wedge {* E_m^\\mp } F -d \\delta E_m^\\mp F \\wedge {* j}\\nonumber \\\\&= \\sum \\limits _\\pm \\int _{J^{\\pm }(\\Sigma )} d \\left(\\delta j \\wedge {* E_m^\\mp } F -\\delta E_m^\\mp F \\wedge {* j}\\right)+ \\delta j \\wedge {* \\delta E_m^\\mp } F - \\delta E_m^\\mp F \\wedge {* \\delta j}\\nonumber \\\\&= \\sum \\limits _\\pm \\mp \\int _\\Sigma i^*\\left(\\delta j \\wedge {* E_m^\\mp } F - \\delta E_m^\\mp F \\wedge {* j}\\right)\\nonumber \\\\&= -\\int _\\Sigma i^*\\left(\\delta j \\wedge {* E_m F} - \\delta E_m F \\wedge {* j}\\right).\\nonumber \\\\&= -\\langle i^*\\delta j, *_{(\\Sigma )} {i^* *} E_m F\\rangle _{\\Sigma } + \\langle i^*\\delta E_m F, *_{(\\Sigma )} {i^* *} j\\rangle _{\\Sigma }\\nonumber \\\\&=\\langle \\rho _{(\\delta )}j, \\rho _{(n)}E_m F\\rangle _{\\Sigma } - \\langle \\rho _{(\\delta )}E_m F, \\rho _{(n)}j\\rangle _{\\Sigma } \\,.$ Multiplying this equality by $m^{-2}$ and rearranging, we see that the first, third and fifth terms of (REF ) combine to the first term of (REF ).",
"Finally, we note that in the second term of (REF ), $d^2 =0$ implies $\\rho _{(d)}G_m = - {*_{(\\Sigma )} {i^* *}} d\\left( m^{-2}{d \\delta } +1 \\right) E_m = - {*_{(\\Sigma )}{i^* *}} d E_m = \\rho _{(d)}E_m$ which completes the proof."
],
[
"The quantum Proca field in curved spacetimes",
"The procedure to quantize the Proca field in a generally covariant way in the framework of Brunetti, Fredenhagen and Verch is well understood, see e. g. for the source free case.",
"The modifications needed to account for external currents can be made analogously to (see also ).",
"Throughout this section, the mass $m>0$ is assumed to be fixed.",
"For simplicity we will mostly consider a single fixed spacetime $(\\mathcal {M},g)$ and source $j\\in \\Omega ^1(\\mathcal {M})$ .",
"The quantum Proca field is then described by the following algebra: Definition 2.6 The unital $^*$ -algebra ${A}_{m,j}$ is obtained from the free algebra, generated by $\\mathbb {1}$ and the objects $\\mathcal {A}_{m,j}(F)$ , $F\\in \\Omega _0^1(\\mathcal {M})$ , by factoring out the relations $\\text{(i)}\\; &\\mathcal {A}_{m,j}(c F + c^{\\prime } F^{\\prime }) = c\\,\\mathcal {A}_{m,j}(F) + c^{\\prime } \\mathcal {A}_{m,j}(F^{\\prime }) &\\textrm {linearity,} \\\\\\text{(ii)}\\; &\\mathcal {A}_{m,j}(F)^* = \\mathcal {A}_{m,j}(\\mathchoice{{\\m@th \\displaystyle F}nullfont\\hspace{0.0pt}\\hspace{2.0pt} \\overline{\\hspace{-1.66656pt}\\hspace{0.0pt}\\box \\hspace{0.0pt}\\hspace{-0.55542pt}}\\hspace{0.0pt}}{}{}{}$ nullfont $\\m@th \\textstyle F$ nullfont $\\m@th \\scriptstyle F$ nullfont $\\m@th \\scriptscriptstyle F$ ) hermitian field, (iii) $\\mathcal {A}$ m,j( (d + m2) F ) = j, F $\\mathcal {M}$ 1 equation of motion, (iv) [$\\mathcal {A}$ m,j(F) , $\\mathcal {A}$ m,j(F') ] = $\\mathrm {i}$$\\mathcal {G}_m(F,F^{\\prime })$ 1 commutation relations, for all $c, c^{\\prime } \\in \\mathbb {C}$ and $F,F^{\\prime } \\in \\Omega ^1_0(\\mathcal {M})$ , where we write $\\mathcal {G}_m(F,F^{\\prime }) = \\langle F, G_mF^{\\prime } \\rangle _\\mathcal {M}$ .",
"For our later investigation of the zero mass limit it will be useful to describe the algebra ${A}_{m,j}$ and its topology in more detail in the next few sections."
],
[
"The Borchers-Uhlmann algebra",
"The algebra ${A}_{m,j}$ is obtained as a quotient of the Borchers-Uhlmann algebra (BU-algebra), which is definedHere, $\\otimes $ denotes the algebraic tensor product, without taking any topological completion.",
"as the tensor algebra of the vector space $\\Omega ^1_0(\\mathcal {M})$ , ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\bigoplus _{n=0}^{\\infty }\\big (\\Omega ^1_0(\\mathcal {M})\\big )^{\\otimes n} \\,.$ Elements $f \\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ are tuples $f= \\big (f^{(0)}, f^{(1)}, f^{(2)}, \\dots \\big )$ , where the components $f^{(0)} \\in \\mathbb {C}$ and for $f^{(n)} \\in \\big (\\Omega ^1_0(\\mathcal {M})\\big )^{\\otimes n}$ for $n > 0$ such that only finitely many $f^{(n)}$ 's are non-vanishing.",
"We will call the component $f^{(n)}$ the degree-n-part of $f$ .",
"Addition and scalar multiplication in ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ are defined component-wise, and we can define a (tensor) product and $*$ -operation by defining their degree-$n$ -parts as $(f\\cdot g)^{(n)}(p_1,p_2, \\dots , p_n) &= \\sum _{i+j = n} f^{(i)} (p_1, p_2, \\dots , p_i) g^{(j)} (p_{i+1} , \\dots , p_n) \\,,\\\\(f^*)^{(n)}(p_1, \\dots , p_n) &= \\overline{f^{(n)}(p_n, p_{n-1}, \\dots , p_1)}\\,$ for all elements $f,g$ and $p_i\\in \\mathcal {M}$ .",
"This makes ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ a *-algebra with unit element $\\mathbb {1}_{\\mathcal {BU}(\\Omega ^1_0(\\mathcal {M}))} = (1, 0, 0, \\dots )$ .",
"The BU-algebra can be endowed with a locally convex topology , obtained from the locally convex topology ofFor a construction, see .",
"$\\Omega ^1_0(\\mathcal {M})$ .",
"More precisely, we can view it as a dense sub-algebra of the complete BU-algebra $\\overline{{\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}} \\bigoplus _{n=0}^{\\infty }\\Gamma _0((T^*\\mathcal {M})^{\\boxtimes n}) \\,,$ where $(T^*M)^{\\boxtimes n}$ denotes the $n$ -fold outer product bundle over $\\mathcal {M}^n$ (cf.",
").",
"We note that the multiplication in $\\overline{{\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}}$ is a jointly continuous bilinear map and hence so is the product in ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ .",
"We want to identify smeared quantum fields $\\mathcal {A}_{m,j}(F)$ with elements $(0, F, 0, 0, \\dots )$ , but the BU-algebra ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ incorporates neither any dynamics, nor the desired quantum commutation relations.",
"It will be convenient to implement the Proca equation (in a distributional sense) and the canonical commutation relations (CCR) in a two step procedure.",
"First we divide out the two-sided ideal $\\mathcal {I}_{m,j}^\\mathrm {\\,dyn}$ in ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ that is generated by elements $\\big (-\\langle j, F \\rangle _\\mathcal {M}, (\\delta d + m^2)F,0,0,\\dots \\big ) \\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\,,$ for $F \\in \\Omega ^1_0(\\mathcal {M})$ , to implement the dynamics.",
"That means, by definition, that every $f\\in \\mathcal {I}_{m,j}^\\mathrm {\\,dyn}$ can be written as a finite sum $f = \\sum _i g_i \\cdot \\left(-\\langle j, F_i \\rangle _\\mathcal {M}, (\\delta d + m^2)F_i,0,0,\\dots \\right) \\cdot h_i \\,,$ for some $F_i \\in \\Omega ^1_0(\\mathcal {M})$ and $g_i, h_i \\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ .",
"We define $\\mathcal {BU}_{m,j}^\\mathrm {dyn}{{\\scalebox {1.2}{{{\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}}{\\mathcal {I}_{m,j}^\\mathrm {\\,dyn}}}}} \\,.$ Elements $f \\in \\mathcal {BU}_{m,j}^\\mathrm {dyn}$ are then equivalence classes $ f = \\left[ g \\right]_{m,j}^\\text{dyn}$ where $g \\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ .",
"Now, in the second step, we incorporate the CCR by dividing out the two-sided ideal $\\mathcal {I}_{m,j}^\\mathrm {\\,CCR}$ that is generated by elements $\\big [ \\big (-\\mathrm {i}{F}{F^{\\prime }}, 0 , F \\otimes F^{\\prime } - F^{\\prime } \\otimes F, 0 , 0 , \\dots \\big ) \\big ]_{m,j}^\\text{dyn} \\in \\mathcal {BU}_{m,j}^\\mathrm {dyn}$ to obtain the final field algebra ${A}_{m,j} = {{\\scalebox {1.2}{{\\mathcal {BU}_{m,j}^\\mathrm {dyn}}{\\mathcal {I}_{m,j}^\\mathrm {\\,CCR}}}}} \\,.$ We will sometimes equivalently write ${A}_{m,j} = {{\\scalebox {1.2}{{{\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}}{\\mathcal {I}_{m,j}}}}}$ , where $\\mathcal {I}_{m,j}$ is the two-sided ideal generated by both of the wanted relations.",
"A smeared quantum Proca field is then an element $\\mathcal {A}_{m,j}(F) \\big [\\big (0,F,0,0,\\dots \\big )\\big ]_{m,j} \\in {A}_{m,j} \\,,$ where the equivalence class $[\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}]_{m,j}$ is taken w.r.t.",
"$\\mathcal {I}_{m,j}$ .",
"By construction, the quantum Proca fields fulfill the desired dynamical and commutation relations.",
"We can endow ${A}_{m,j}$ with the locally convex quotient topology obtained from ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ (cf.",
"), which is induced by the semi-norms $q_{m,j, \\alpha }( [f]_{m,j} ) = \\inf \\big \\lbrace p_\\alpha (g) : g \\in [f]_{m,j} \\big \\rbrace $ where $\\left\\lbrace p_\\alpha \\right\\rbrace _\\alpha $ is a family of semi-norms on ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ that induces its topology .",
"Note that the multiplication in ${A}_{m,j}$ is again jointly continuousTo ensure that the quotient space is Hausdorff, we will show below that the ideals $\\mathcal {I}_{m,j}^\\mathrm {\\,dyn}$ and $\\mathcal {I}_{m,j}^\\mathrm {\\,CCR}$ are closed.."
],
[
"Reduction to the current-free case",
"We now show that the algebra ${A}_{m,j}$ with source dependent dynamics is homeomorphic to the algebra ${A}_{m,0}$ with vanishing source, where the subscript 0 indicates that we set $j=0$ .",
"Let us fix a solution $\\varphi $ of the classical source dependent Proca equation, $(\\delta d +m^2) \\varphi = j$ .",
"We may then define a *-algebra-homomorphism $\\Gamma _{\\varphi }$ on ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ which preserves the unit and which is then uniquely determined by its action on homogeneous elements of degree one: $\\Gamma _{\\varphi } :\\big (0,F,0,0,\\dots \\big ) \\mapsto \\big (-\\langle \\varphi , F \\rangle _\\mathcal {M}, F , 0,0,\\dots \\big )$ for all $F \\in \\Omega ^1_0(\\mathcal {M})$ .",
"Theorem 2.7 Let $m > 0$ and $j \\in \\Omega ^1(\\mathcal {M})$ and $\\varphi \\in \\Omega ^1(\\mathcal {M})$ a solution of $(\\delta d +m^2) \\varphi = j$ .",
"Then the map $\\Gamma _{\\varphi }$ is a homeomorphism of ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ which descends to a homeomorphism $\\Psi _{\\varphi } : {A}_{m,0} \\rightarrow {A}_{m,j}$ .",
"Proof: 2.4 The inverse $\\Gamma _{\\varphi }$ is obviously determined by $\\Gamma _{\\varphi }^{-1} : {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}&\\rightarrow {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\\\\\big (0,F,0,0,\\dots \\big ) &\\mapsto \\big (+\\langle \\varphi , F \\rangle _\\mathcal {M}, F , 0,0,\\dots \\big ) \\,$ and both $\\Gamma _{\\varphi }$ and $\\Gamma _{\\varphi }^{-1}$ are continuous on ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ .",
"We now show that $\\Gamma _{\\varphi }$ maps the ideal $\\mathcal {I}_{m,0}$ onto $\\mathcal {I}_{m,j}$ .",
"It suffices to show that the generators of the source-free ideal map under $\\Gamma _{\\varphi }$ to the corresponding generators of the source dependent ideal and vice versa.",
"Let $F\\in \\Omega ^1_0(\\mathcal {M})$ , then $\\Gamma _{\\varphi }&\\Big ( \\big ( 0,(\\delta d +m^2)F , 0 , 0 , \\dots \\big ) \\Big ) \\\\&= \\big ( - \\langle \\varphi , (\\delta d + m^2)F \\rangle _\\mathcal {M},(\\delta d +m^2)F , 0 , 0 , \\dots \\big ) \\\\&= \\big ( - \\langle (\\delta d + m^2) \\varphi , F \\rangle _\\mathcal {M},(\\delta d +m^2)F , 0 , 0 , \\dots \\big ) \\\\&= \\big ( - \\langle j, F \\rangle _\\mathcal {M},(\\delta d +m^2)F , 0 , 0 , \\dots \\big ) \\,,$ so the generators for the dynamics transform in the desired way.",
"For the commutation relations we first decompose: $\\big ( -\\mathrm {i}{F}{F^{\\prime }},0 , F \\otimes F^{\\prime } - F^{\\prime } \\otimes F , 0 ,0, \\dots \\big ) &=\\big ( -\\mathrm {i}{F}{F^{\\prime }}, 0 ,0, \\dots \\big ) \\\\&\\phantom{M}+ \\big ( 0,F,0,0,\\dots \\big )\\cdot \\big ( 0,F^{\\prime },0,0,\\dots \\big ) \\\\&\\phantom{M}-\\big ( 0,F^{\\prime },0,0,\\dots \\big )\\cdot \\big ( 0,F,0,0,\\dots \\big ) $ and therefore obtain $\\Gamma _{\\varphi }\\Big ( \\big ( -\\mathrm {i}{F}{F^{\\prime }},&0 , F \\otimes F^{\\prime } - F^{\\prime } \\otimes F , 0 ,0, \\dots \\big ) \\Big ) \\\\&=\\big ( -\\mathrm {i}{F}{F^{\\prime }}, 0 ,0, \\dots \\big ) \\\\&\\phantom{M}+ \\big ( - \\langle \\varphi ,F \\rangle _\\mathcal {M},F,0,0,\\dots \\big )\\cdot \\big ( - \\langle \\varphi ,F^{\\prime } \\rangle _\\mathcal {M},F^{\\prime },0,0,\\dots \\big ) \\\\&\\phantom{M}-\\big ( - \\langle \\varphi ,F^{\\prime } \\rangle _\\mathcal {M},F^{\\prime },0,0,\\dots \\big )\\cdot \\big ( - \\langle \\varphi ,F \\rangle _\\mathcal {M},F,0,0,\\dots \\big ) \\\\ &= \\big ( -\\mathrm {i}{F}{F^{\\prime }},0 , F \\otimes F^{\\prime } - F^{\\prime } \\otimes F , 0 ,0, \\dots \\big ) \\,.$ It is straightforward to check in a completely analogous fashion that the generators of the source-dependent ideal map under $\\Gamma _{\\varphi }^{-1}$ to the generators of the source-free ideal.",
"In conclusion, we find that $\\Gamma _{\\varphi }(\\mathcal {I}_{m,0})=\\mathcal {I}_{m,j}$ , and diving out the ideals yields the diffeomorphism $\\Psi _{\\varphi }$ .",
"We refer to for more details.",
"Given an observable of the source free theory $\\mathcal {A}_{m,0}(F)$ , we obtain $\\mathcal {A}_{m,j}(F)= \\langle \\varphi , F \\rangle _\\mathcal {M}\\cdot \\mathbb {1}_{{A}_{m,j}} + \\Psi _{\\varphi }\\big (\\mathcal {A}_{m,0}(F)\\big ) \\,.$ Hence, the dynamics and commutation relations for $\\mathcal {A}_{m,0}$ imply those of $\\mathcal {A}_{m,j}$ and vice versa."
],
[
"Initial value-formulation",
"In order to divide out the dynamical ideal $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ in the source-free case it is convenient to make use of an initial value formulation.",
"First, however, we characterise the generators of this ideal: Lemma 2.8 $F\\in \\Omega ^1_0(\\mathcal {M})$ is of the form $F=(\\delta d+m^2)F^{\\prime }$ for some $F^{\\prime }\\in \\Omega ^1_0(\\mathcal {M})$ if and only if $G_mF=0$ .",
"Proof: 2.5 If $F=(\\delta d + m^2)F^{\\prime }$ , then $G_mF=G_m(\\delta d+m^2)F^{\\prime }=0$ .",
"Conversely, if $G_mF=0$ , then $F^{\\prime }=G_m^+F=G_m^-F$ has compact support and $F=(\\delta d+m^2)F^{\\prime }$ .",
"Now let $\\Sigma $ be an arbitrary, fixed Cauchy surface.",
"We will use the short-hand notation $\\mathcal {D}_{0}(\\Sigma )= \\Omega ^1_0(\\Sigma ) \\oplus \\Omega ^1_0(\\Sigma )$ for the space of initial data on $\\Sigma $ .",
"We define the map $\\kappa _m: \\Omega ^1_0(\\mathcal {M}) \\rightarrow \\mathcal {D}_{0}(\\Sigma )\\,,\\quad F \\mapsto (\\rho _{(0)}G_m F , \\rho _{(d)}G_m F) \\,,$ which maps a test one-form $F$ to the solution $G_mF$ of Proca's equation and then to its initial data on $\\Sigma $ (cf.",
"Theorem REF and Definition REF ).",
"In the notation, we omit the dependence of the map on the Cauchy surface.",
"For any value of $m>0$ , $\\kappa _m$ is continuous w.r.t.",
"the direct sum topology on $\\mathcal {D}_{0}(\\Sigma )$ , and hence $\\mathrm {ker}{\\left(\\kappa _m\\right)}$ is closed .",
"By Lemma REF and Theorem REF we have $\\mathrm {ker}{\\left(\\kappa _m\\right)} = \\big \\lbrace F \\in \\Omega ^1_0(\\mathcal {M}) \\mid G_mF=0 \\big \\rbrace = (\\delta d + m^2)\\Omega ^1_0(\\mathcal {M}) \\,.$ By a standard construction , illustrated in Diagram REF , $\\kappa _m$ gives rise to a linear map $\\xi _m : {{\\scalebox {1.2}{{\\Omega ^1_0(\\mathcal {M})}{\\mathrm {ker}{\\left(\\kappa _m\\right)}}}}} \\rightarrow \\mathrm {img}{}(\\kappa _m)$ , which is the unique bijective map such that $\\xi _m([F]_m) = \\kappa _m(F)$ , where $[F]_m$ denotes equivalence classes in the quotient space .",
"We will now show Table: Illustrating the construction of the homeomorphism ξ m \\xi _m of the space of dynamical test one-forms and the space of initial data.Lemma 2.9 $\\xi _m$ is a homeomorphism onto $\\mathcal {D}_{0}(\\Sigma )$ .",
"Proof: 2.6 First we will show that $\\kappa _m$ is surjective, by constructing a map $\\vartheta _m : \\mathcal {D}_{0}(\\Sigma )\\rightarrow \\Omega ^1_0(\\mathcal {M})$ such that $\\xi _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ []m m = id$.$ We choose a fixed $\\chi \\in \\Omega ^0(\\mathcal {M})$ such that $\\chi = 1$ on $J^+(\\Sigma _+)$ and $\\chi \\equiv 0$ on $J^-(\\Sigma _-)$ , where $\\Sigma _\\pm $ are Cauchy surfaces in the future (+) and past (-) of $\\Sigma $ .",
"Now let $(\\varphi ,\\pi ) \\in \\mathcal {D}_{0}(\\Sigma )$ specify initial data on a Cauchy surface $\\Sigma $ .",
"Then, by Theorem REF , there exists a unique solution $A \\in \\Omega ^1(\\mathcal {M})$ to the source free Proca equation $(\\delta d + m^2) A = 0$ with the given data.",
"We note that $\\mathrm {supp}\\left(A\\right) \\subset J\\big ( \\mathrm {supp}\\left(\\varphi \\right) \\cup \\mathrm {supp}\\left(\\pi \\right) \\big )$ (see ) and hence, by defining $\\vartheta _m(\\varphi ,\\pi ) -(\\delta d + m^2) \\chi A,$ we see that $\\vartheta _m(\\varphi ,\\pi )$ is a compactly supported one-form with support contained in the compact set $J\\big ( \\mathrm {supp}\\left(\\varphi \\right) \\cup \\mathrm {supp}\\left(\\pi \\right)\\big ) \\cap J^-(\\Sigma _+) \\cap J^+(\\Sigma _-)$ .",
"We want to show that $\\kappa _m \\vartheta _m(\\varphi ,\\pi ) = (\\varphi ,\\pi )$ .",
"For this we note that the domains of $G_m^\\pm $ can be extended to forms with past (+) resp.",
"future (-) compact supports , .",
"With these extended definitions we find $G_m^+ \\vartheta _m(\\varphi ,\\pi )&= - \\chi A ,\\\\G_m^- \\vartheta _m(\\varphi ,\\pi )&= G_m^- (\\delta d + m^2)(1-\\chi ) A \\\\&= (1-\\chi ) A \\,,$ because $\\vartheta _m(\\varphi ,\\pi )=(\\delta d+m^2)(1-\\chi ) A$ .",
"We therefore find the result $G_m \\vartheta _m(\\varphi ,\\pi )&= (G_m^- - G_m^+)\\vartheta _m(\\varphi ,\\pi ) \\\\&= (1- \\chi ) A + \\chi A = A$ and hence $\\kappa _m\\vartheta _m(\\varphi ,\\pi )=(\\varphi ,\\pi )$ , which completes the proof of surjectivity.",
"That is, we have found $\\mathrm {img}{}(\\kappa _m) = \\mathcal {D}_{0}(\\Sigma )$ .",
"It remains to show that the bijection $\\xi _m$ is a homeomorphism.",
"By construction, $\\xi _m$ is continuous because $\\kappa _m$ is continuous .",
"The inverse is given by $\\xi _m^{-1} = [\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}]_m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m$, where $ m$ is continuous, because $ A$ depends continuously on the initial data $ (,)$.",
"Since $ []m$ is also continuous, so is $ -1m$.",
"This completes the proof.$ We will now generalize these ideas to the algebra ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ in order to implement the dynamics by dividing out the ideal generated by $\\big (0,(\\delta d + m^2)F,0,0,\\dots \\big )$ , where $F\\in \\Omega ^1_0(\\mathcal {M})$ .",
"As we did on the degree-one level, we would like to find a map $K_m : {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\rightarrow \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ such that $\\mathrm {ker}{\\left(K_m\\right)} = \\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ and then show that $\\mathcal {BU}_{m,0}^\\mathrm {dyn}$ is homeomorphic to $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ .",
"We do this by lifting the map $\\kappa _m$ to the BU-algebra: We define $K_m : {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\rightarrow \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big ) $ as a BU-algebra-homomorphism which preserves the units and which is then completely determined by its action on homogeneous degree-one elements: $K_m : {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}&\\rightarrow \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big ) \\\\\\big (0,F,0,0,\\dots \\big ) &\\mapsto \\big (0, \\kappa _m (F),0,0,\\dots \\big )\\,.$ With this map we can, analogously to the degree-one-part, construct a homeomorphism that implements the dynamics.",
"Lemma 2.10 Let $m>0$ and $j=0$ .",
"Then the map $K_m : {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}\\rightarrow \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ descends to a homeomorphism $\\Xi _m : \\mathcal {BU}_{m,0}^\\mathrm {dyn}\\rightarrow \\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )$ with $\\Xi _m \\big ([f]_{m,0}^\\mathrm {dyn}\\big ) = K_m(f)$ where $f \\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ .",
"Proof: 2.7 The surjectivity of $K_m$ follow directly from the surjectivity of $\\kappa _m$ , which was established in the proof of Lemma REF .",
"Because $\\kappa _m$ is continuous, so is $\\kappa _m^{\\otimes n}$ on $\\Gamma _0(T^*\\mathcal {M}^{\\boxtimes n})$ for any $n\\ge 1$ , by Schwartz' Kernels Theorem.",
"Therefore, $\\kappa _m^{\\otimes n}$ is also continuous on the algebraic tensor product $\\big (\\Omega ^1_0(\\mathcal {M})\\big )^{\\otimes n}$ and hence $K_m$ is continuous.",
"It follows that $K_m$ descends to a continuous linear map $\\Xi _m : {{\\scalebox {1.2}{{{\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}}{\\mathrm {ker}{\\left(K_m\\right)}}}}}\\rightarrow \\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )$ (cf.",
").",
"The inclusion $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}\\subset \\mathrm {ker}{\\left(K_m\\right)}$ is obvious from the facts that $K_m$ is an algebra homomorphisms and that the generators of $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ are of the form $\\big (0,F_i,0,0,\\dots \\big )$ with $F_i\\in \\mathrm {ker}{\\left(\\kappa _m\\right)}$ , cf. Equation.",
"(REF ).",
"The non-trivial part is to show the converse inclusion $\\mathrm {ker}{\\left(K_m\\right)} \\subset \\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ .",
"Consider and arbitrary $f = \\big (f^{(0)}, f^{(1)}, f^{(2)}, \\dots , f^{(N)} , 0 , 0 ,\\dots \\big ) \\in \\mathrm {ker}{\\left(K_m\\right)}$ , $f^{(k)} \\in \\left(\\Omega ^1_0(\\mathcal {M})\\right)^{\\otimes k}$ .",
"Because $K_m$ preserves degrees, each homogeneous element $\\big (0,\\dots ,0 , f^{(n)} , 0 ,0, \\dots \\big )$ is in $\\mathrm {ker}{\\left(K_m\\right)}$ , and it suffices to prove that these homogeneous elements are in the ideal $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ .",
"We will show by induction in the degree $n$ that an arbitrary homogeneous element $\\big (0,\\dots ,0, f^{(n)} , 0 ,0, \\dots \\big )$ with $\\kappa _m^{\\otimes n}\\left(f^{(n)}\\right) = 0$ is in the ideal $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ .",
"At degree 0, $\\kappa _m^{\\otimes 0}$ is the identity mapping, so its kernel is trivial.",
"At degree 1, we use the fact that $\\kappa _m(F)=0$ if and only if $\\big (0,F,0,0,\\dots \\big )$ is a generator of $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ (cf.",
"Equation (REF )).",
"We can now make the induction step and assume that the claim holds for homogeneous elements of degree $\\le n$ for some $n\\ge 1$ .",
"Consider a homogeneous element $\\big (0,\\dots ,0,f^{(n+1)},0,0,\\dots \\big )$ where $f^{(n+1)} \\in \\left(\\Omega ^1_0(\\mathcal {M})\\right)^{\\otimes (n+1)}$ such that $\\kappa _m^{\\otimes (n+1)}(f^{(n+1)}) = 0$ .",
"We can write this more explicitly for some $F_i \\in \\Omega ^1_0(\\mathcal {M})$ and some $\\mathcal {F}_i^{(n)}\\in \\big (\\Omega ^1_0(\\mathcal {M})\\big )^{\\otimes n}$ as $\\big (0,\\dots ,0,f^{(n+1)},0,0, \\dots \\big ) = \\big (0,\\dots ,0, \\sum \\limits _{i=1}^{M} F_i \\otimes \\mathcal {F}_i^{(n)},0, 0 , \\dots \\big ) \\,.$ Let $V \\mathrm {span}{\\big \\lbrace F_1,F_2, \\dots , F_M\\big \\rbrace }$ and $W V \\hspace{0.05005pt}\\cap \\hspace{0.09995pt} \\mathrm {ker}{\\left(\\kappa _m\\right)}$ , which define finite dimensional subspaces of $\\Omega ^1_0(\\mathcal {M})$ .",
"We find a basis $\\lbrace \\widetilde{F}_1, \\dots ,\\widetilde{F}_\\mu \\rbrace $ , $\\mu \\le M$ , of $W$ which we can extend to a basis $\\lbrace \\widetilde{F}_1, \\dots ,\\widetilde{F}_M \\rbrace $ of $V$ .",
"With the use of this basis we can re-write $f^{(n+1)} = \\sum \\limits _{i=1}^{M} F_i \\otimes \\mathcal {F}_i^{(n)}&= \\sum \\limits _{i=1}^{\\mu } \\widetilde{F}_i \\otimes \\widetilde{\\mathcal {F}}_i^{(n)} + \\sum \\limits _{i=\\mu + 1}^{M} \\widetilde{F}_i \\otimes \\widetilde{\\mathcal {F}}_i^{(n)} \\\\& X_1^{(n+1)} + X_2^{(n+1)} \\,.$ Here, each $\\widetilde{\\mathcal {F}}_i^{(n)}$ can be constructed as a linear combination of the ${\\mathcal {F}}_i^{(n)}$ 's.",
"We first have a look at $X_1^{(n+1)}$ .",
"We know by construction for $i=1,\\dots ,\\mu $ that $\\kappa _m ( \\widetilde{F_i} ) = 0$ .",
"It follows that $\\big (0,\\ldots ,0,X_1^{(n+1)},0,\\dots \\big ) = \\sum \\limits _{i=1}^{\\mu } \\big (0,\\widetilde{F}_i,0,0, \\dots \\big ) \\otimes \\big (0,\\dots ,0,\\widetilde{\\mathcal {F}}_i^{(n)},0,0, \\dots \\big )$ is in $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ and that $\\kappa _m^{\\otimes (n+1)}(X_1^{(n+1)}) = 0$ .",
"Now we have a closer look at the remaining part $X_2^{(n+1)}$ , which must then also have $\\kappa _m^{\\otimes (n+1)}(X_2^{(n+1)}) = 0$ .",
"However, by construction, it holds $\\mathrm {span}{\\big \\lbrace \\widetilde{F}_{\\mu +1}, \\dots ,\\widetilde{F}_M \\big \\rbrace } \\cap \\mathrm {ker}{\\left(\\kappa _m\\right)} = \\lbrace 0 \\rbrace $ , which implies that the $\\kappa _m(\\widetilde{F_i})$ 's are linearly independent for $i=\\mu +1,\\dots ,M$ .",
"With this, it then follows that we must have $\\kappa _m^{\\otimes n}( \\widetilde{\\mathcal {F}}_i^{(n)}) = 0$ for all $i=\\mu +1,\\dots ,M$ .",
"Since $\\widetilde{\\mathcal {F}}_i^{(n)}$ is of degree $n$ , we can apply the induction hypothesis and find that $(0,\\dots ,0, \\widetilde{\\mathcal {F}}_i^{(n)} , 0 , 0 , \\dots ) \\in \\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ and hence $\\big (0,\\ldots ,0,X_2^{(n+1)},0,\\dots \\big ) = \\sum \\limits _{i=\\mu +1}^{M} \\big (0,\\widetilde{F}_i,0,\\dots \\big ) \\otimes \\big (0,\\dots ,0,\\widetilde{\\mathcal {F}}_i^{(n)},0,\\dots \\big )$ is also in $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ .",
"Hence $\\big (0,\\dots , 0, f^{(n+1)} , 0 , 0 , \\dots \\big ) = \\big (0,\\dots , 0, X_1^{(n+1)} + X_2^{(n+1)} , 0 , 0 , \\dots \\big ) \\in \\mathcal {I}_{m,0}^\\mathrm {\\,dyn}$ which completes the proof by induction.",
"The continuity of $K_m$ and the proof above imply in particular that the ideal $\\mathcal {I}_{m,0}^\\mathrm {\\,dyn}=\\mathrm {ker}{\\left(K_m\\right)}$ is closed."
],
[
"Canonical commutation relations",
"We are left to include the quantum nature of the fields by dividing out the relation that implements the CCR.",
"In $\\mathcal {BU}_{m,0}^\\mathrm {dyn}$ , we need to divide out the two-sided ideal $\\mathcal {I}_{m,0}^\\mathrm {\\,CCR}$ that is generated by elements $\\big (-\\mathrm {i}{F}{F^{\\prime }}, 0 , F \\otimes F^{\\prime } - F^{\\prime } \\otimes F , 0 , 0 , \\dots \\big )$ .",
"For $\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )$ we make use of the following lemma: Lemma 2.11 Let $F,F^{\\prime } \\in \\Omega ^1_0(\\mathcal {M})$ be two test one-forms on $\\mathcal {M}$ and let $(\\varphi ,\\pi )=\\kappa _m(F)$ and $(\\varphi ^{\\prime },\\pi ^{\\prime })=\\kappa _m(F^{\\prime })$ .",
"Then $\\mathcal {G}_m(F,F^{\\prime }) = \\mathcal {G}^{(\\Sigma )}\\big (\\kappa _m(F),\\kappa _m(F^{\\prime })\\big ) \\,,$ where $\\mathcal {G}^{(\\Sigma )}\\big ((\\varphi ,\\pi ),(\\varphi ^{\\prime },\\pi ^{\\prime })\\big )= \\langle \\varphi , \\pi ^{\\prime } \\rangle _\\Sigma - \\langle \\pi , \\varphi ^{\\prime } \\rangle _\\Sigma \\,$ is a symplectic form on the space $\\mathcal {D}_{0}(\\Sigma )$ of initial data, i. e., it is bilinear, anti-symmetric and non-degenerate.",
"Proof: 2.8 It is straightforward to show that $\\mathcal {G}^{(\\Sigma )}$ is a symplectic form.",
"Now let $F,F^{\\prime } \\in \\Omega ^1_0(\\mathcal {M})$ and recall that $G_m F^{\\prime }$ is a solution to the source free Proca equation with initial data $\\kappa _m(F)$ , and similarly for $G_mF^{\\prime }$ .",
"Then, using the definition of $\\mathcal {G}_m(F,F^{\\prime })$ and $\\langle F , G_m^\\pm F^{\\prime } \\rangle _\\mathcal {M}=\\langle G_m^\\mp F , F^{\\prime } \\rangle _\\mathcal {M}$ , $\\mathcal {G}_m(F,F^{\\prime })&= \\langle F , G_m F^{\\prime } \\rangle _\\mathcal {M}= - \\langle G_m F, F^{\\prime } \\rangle _\\mathcal {M}\\,\\\\&= \\langle \\rho _{(0)}G_m F, \\rho _{(d)}G_mF^{\\prime } \\rangle _\\Sigma - \\langle \\rho _{(d)}G_m F, \\rho _{(0)}G_m F^{\\prime } \\rangle _\\Sigma \\\\&= \\mathcal {G}^{(\\Sigma )}\\big ((\\varphi ,\\pi ),(\\varphi ^{\\prime },\\pi ^{\\prime })\\big )$ by Theorem REF with $j=0$ .",
"It follows from this lemma that $\\mathcal {I}_{m,0}^\\mathrm {\\,CCR}$ maps under $\\Xi _m$ to the two-sided ideal ${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\subset \\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )$ that is generated by elements $\\big (-\\mathrm {i}( \\langle \\varphi , \\pi ^{\\prime } \\rangle _\\Sigma - \\langle \\pi , \\varphi ^{\\prime }\\rangle _\\Sigma ), 0 , (\\varphi , \\pi ) \\otimes (\\varphi ^{\\prime }, \\pi ^{\\prime }) - (\\varphi ^{\\prime }, \\pi ^{\\prime }) \\otimes (\\varphi , \\pi ) , 0 , 0 , \\dots \\big )\\,.$ Lemma REF in Appendix shows that the ideal ${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is the kernel of a continuous linear map.",
"This implies in particular that ${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ , and hence also $\\mathcal {I}_{m,0}^\\mathrm {\\,CCR}$ , is closed.",
"With these results the following theorem follows easily.",
"Theorem 2.12 Let $m>0$ and $j=0$ .",
"Then the map $\\Xi _m : \\mathcal {BU}_{m,0}^\\mathrm {dyn}\\rightarrow \\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )$ descends to a homeomorphism $\\Lambda _m : {A}_{m,0}(M) \\rightarrow {\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ .",
"We omit the proof and refer to for the details.",
"The results of this section can be combined with those of Section REF and REF and illustrated as in Diagram REF .",
"Table: A commutative diagram illustrating the various quotients of BU-algebras and their relations.",
"Bi-directional arrows represent homeomorphisms."
],
[
"Locality of the quantum Proca field",
"Finally we consider the quantum Proca field in the generally covariant setting, using a categorical framework as Brunetti, Fredenhagen and Verch .",
"For this purpose we introduce the following Definition 2.13 By an admissible embedding $\\psi : (\\mathcal {M},g_\\mathcal {M}) \\rightarrow (\\mathcal {N},g_\\mathcal {N})$ we mean an orientation and time orientation preserving isometric embedding $\\psi :\\mathcal {M}\\rightarrow \\mathcal {N}$ such that for every $p \\in \\mathcal {M}$ it holds $J_\\mathcal {M}^\\pm (p) = \\psi ^{-1} \\big ( J_\\mathcal {N}^\\pm ( \\psi (p)) \\big )$ .",
"The category $\\mathsf {SpacCurr}$ consists of triples $M=(\\mathcal {M},g_\\mathcal {M},j_\\mathcal {M})$ as objects, where $(\\mathcal {M},g_\\mathcal {M})$ is a (oriented and time-oriented) globally hyperbolic spacetime and $j_\\mathcal {M}\\in \\Omega ^1(\\mathcal {M})$ is a background current, and morphisms $\\psi $ , where $\\psi $ is an admissible embedding such that $\\psi ^* j_\\mathcal {N}= j_\\mathcal {M}$ .",
"The category $\\mathsf {Alg}$ consists of unital $^*$ -algebras as objects and unit preserving $^*$ -algebra-homomorphisms as morphisms.",
"The category $\\mathsf {Alg}^{\\prime }$ is the subcategory of $\\mathsf {Alg}$ consisting of the same objects but only injective morphisms.",
"Definition 2.14 A generally covariant quantum field theory with background source is a covariant functor between the categories $\\mathsf {SpacCurr}$ and $\\mathsf {Alg}$ .",
"The theory is called locally covariant if and only if the range of the functor is contained in $\\mathsf {Alg}^{\\prime }$ .",
"The construction of this functor $\\mathbf {A}_m$ for the Proca field of mass $m>0$ is straightforward: To each $M$ we associate the $^*$ -algebra $\\mathbf {A}_m(M){A}_{m,j}$ constructed on $M$ as above and to any morphism $\\psi :M\\rightarrow N$ we associate the unit preserving $^*$ -algebra-homomorphism $\\mathbf {A}_m(\\psi ) \\equiv \\alpha _\\psi : \\mathbf {A}_m(M) \\rightarrow \\mathbf {A}_m(N)$ , whose action is fully determined by the action on the generators $\\mathcal {A}_{m,M}(F)$ , which we previously denoted by $\\mathcal {A}_{m,j}(F)$ without explicitly referring to the background spacetime $M$ , as $\\alpha _\\psi \\big (\\mathcal {A}_{m,M}(F)\\big ) = \\mathcal {A}_{m,N}(\\psi _*(F)) \\,.$ It is straightforward to show that the above functor is well-defined for all $m>0$ .",
"A detailed verification is given in .",
"We now show that for $m>0$ the functor $\\mathbf {A}_m$ defines a locally covariant QFT, i. e. that the homomorphisms $\\mathbf {A}_m(\\psi ) \\equiv \\alpha _\\psi $ are injective.",
"Theorem 2.15 $\\mathbf {A}_m$ as given above defines a locally covariant QFT of the Proca field, i. e. it is a functor $\\mathbf {A}_m : \\mathsf {SpacCurr}\\rightarrow \\mathsf {Alg}^{\\prime }$ .",
"Proof: 2.9 $\\mathbf {A}_m$ is given as a functor into $\\mathsf {Alg}$ , so it only remains to show that the morphisms $\\mathbf {A}_m(\\psi ) \\equiv \\alpha _\\psi $ are injective.",
"By Lemma REF $\\mathcal {G}^{(\\Sigma )}$ is a symplectic form on $\\mathcal {D}_{0}(\\Sigma )$ and hence the algebra ${\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ is simple (cf.",
").",
"The same is true for the homeomorphic algebra $\\mathbf {A}_m(M)$ (cf.",
"Theorems REF and REF ).",
"Since $\\mathbf {A}_m(M)$ is simple, the homomorphism $\\alpha _\\psi $ has either full or trivial kernel.",
"As $\\alpha _\\psi $ is defined to be unit preserving, it follows that the kernel is trivial and hence $\\alpha _\\psi $ is injective.",
"For the main results of this article we will investigate the zero mass limit of the Proca field a in curved spacetime in both the classical and the quantum case.",
"In Section REF we will formulate the key notion of continuity of the field theory with respect to the mass and establish its basic properties.",
"We then define the massless limit in a general, state independent setup first for the classical Proca field in Section REF and then for the quantum Proca field in Section REF .",
"At given points, we compare our results with the theory of the (quantum) vector potential of electromagnetism in curved spacetimes as studied in , ."
],
[
"Continuity in the mass",
"When defining a notion of continuity of the field theory with respect to the mass, the basic problem is that at different masses the smeared fields $\\mathcal {A}_{m,j}(F)$ are elements of different algebras ${A}_{m,j}$ .",
"Indeed, when constructing ${A}_{m,j}$ as a quotient of the BU-algebra, the ideals that implement the dynamics and the commutation relations both depend on the mass.",
"We therefore need to find a way of comparing the Proca fields at different masses with each other.",
"One could try to solve this using the $C^*$ -Weyl algebra to describe the quantum Proca field and the notion of a continuous field of $C^*$ -algebras depending on the mass parameter (cf. ).",
"This would work very nicely, if the theories were described by a weakly continuous family of (non-degenerate) symplectic forms on a fixed linear space (cf.",
", which generalises ).",
"However, as it turns out, this approach is ill-suited for the problem at hand.",
"Indeed, one would like linear combinations of Weyl operators $W_{m,j}(F_i)=\\mathrm {e}^{\\mathrm {i}\\mathcal {A}_{m,j}(F_i)}$ with fixed test-forms $F_i\\in \\Omega _0^1(\\mathcal {M})$ to depend continuously on the mass, but for $j=0$ the norm of an operator like $W_{m,0}\\big ((\\delta d +m_0^2)F\\big )-1$ , with a fixed $F$ and $m_0$ , can be seen to be discontinuous at $m=m_0$ , where the operator vanishes.",
"A different attempt, which we have hinted at in Section REF , is to use the semi-norms $q_{m, j, \\alpha }\\big ( [f]_{m,j} \\big ) = \\inf \\big \\lbrace p_\\alpha (g) : g \\in [f]_{m,j} \\big \\rbrace $ to define a notion of continuity of the theory with respect to the mass $m$ .",
"We could call a family of operators $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ with $O_m \\in {A}_{m,j}$ continuous if and only if the map $m \\mapsto q_{m, j, \\alpha }\\big (O_m\\big )$ is continuous for all $\\alpha $ with respect to the standard topology in $\\mathbb {R}$ .",
"While this definition seems appropriate at first sight, it is non-trivial to show the desirable property that for a fixed $F\\in \\Omega _0^1(\\mathcal {M})$ the smeared field operators $\\mathcal {A}_{m,j}(F)$ vary continuously with $m$ .",
"Even for $j=0$ and considering only the one-particle level, we were unable to prove this.",
"In this paper we therefore opt for the following solution, which makes use of the Borchers-Uhlmann algebra of initial data.",
"For simplicity we first consider the case $j=0$ and a family of operators $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ with $O_m \\in {A}_{m,0}$ .",
"Since we have found for every mass $m>0$ that ${A}_{m,0}$ is homeomorphic to ${\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ , we can map the family $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ to a family of operators in the single algebra ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , which already carries a topology and hence a notion of continuity.",
"When $j\\ne 0$ we combine this idea with the fact that ${A}_{m,j}$ is homeomorphic to ${A}_{m,0}$ .",
"In this way we arrive at the following notion of continuity.",
"Definition 3.1 (Continuity with respect to the mass) Let $j \\in \\Omega ^1(\\mathcal {M})$ be fixed and let $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ be a family of operators with $O_m\\in {A}_{m,j}$ .",
"We call $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ continuous if and only if the map $\\mathbb {R}_+ &\\rightarrow {{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}} \\,,\\\\m &\\mapsto \\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) (Om) is continuous, where $\\Lambda _{m}$ and $\\Psi _{\\varphi _{m,j}}$ are as defined in Section REF and REF and $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m>0}$ is a family of classical solutions to the inhomogeneous Proca equation $(\\delta d + m^2) \\varphi _{m,j} = j$ which depends continuously on $m$ (i. e. $m\\mapsto \\varphi _{m,j}\\in \\Omega ^1(\\mathcal {M})$ is continuous).",
"Equivalently, identifying $O_m = \\big [\\tilde{O}_m\\big ]_{m,j}$ for some $\\tilde{O}_m \\subset {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ , the family $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ is continuous if and only if the map $\\mathbb {R}_+ &\\rightarrow {{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}} \\,,\\\\m &\\mapsto \\big [ \\big ( K_m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) (Om) ]CCR is continuous, with $K_m$ and $\\Gamma _{\\varphi _{m,j}}$ as defined in Section REF and REF .",
"We now aim to establish some desirable properties of this notion of continuity, most importantly that it is independent of the choice of Cauchy surface $\\Sigma $ and of the choice of the continuous family $\\varphi _{m,j}$ of classical solutions.",
"Our arguments will make essential use of the following result for normally hyperbolic operators: Theorem 3.2 Let $P$ be a normally hyperbolic operator on a real vector bundle $V$ over a globally hyperbolic spacetime $M$ .",
"Let $u_0,u_1\\in \\Gamma (V|_{\\Sigma })$ be initial data on a Cauchy surface $\\Sigma $ and $f\\in \\Gamma _0(V)$ .",
"For $r\\in \\mathbb {R}$ , let $u^{(r)}$ be the unique solution to $(P+r)u^{(r)}=f$ with initial data $u_0,u_1$ on $\\Sigma $ .",
"Then $r\\mapsto u^{(r)}$ is a continuous map from $\\mathbb {R}$ to $\\Gamma (V)$ .",
"Proof: 3.1 It suffices to prove continuity at $r=0$ , after shifting $P$ by a constant.",
"We may write $P=\\nabla ^{\\alpha }\\nabla _{\\alpha }+B$ , where $B$ is a bundle endomorphism .",
"Here, $\\nabla _{\\alpha }$ is a connection on $V$ , which may be extended with the Levi-Civita connection to tensor product bundles of $V$ , $TM$ and their dual bundles.",
"We write for $k=0,1,2,\\ldots $ $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k} \\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}(u^{(r)}-u^{(0)})$ and we note that $(P+r)(u^{(r)}-u^{(0)})=-ru^{(0)}$ and hence $(P+r)v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k} = -r\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}u^{(0)}-(B^{(k)}v^{(k,r)})_{\\alpha _1\\cdots \\alpha _k}+\\sum _{l=0}^{k-1}(C^{(k,l)}v^{(l,r)})_{\\alpha _1\\cdots \\alpha _k} \\,,$ where $B^{(k)}$ and $C^{(k,l)}$ are bundle homomorphisms which involve $B$ and the curvature of $\\nabla $ .",
"It follows that $v^{(k,r)}$ solves an inhomogeneous normally hyperbolic equation with the operator $P+B^{(k)}+r$ and an inhomogeneous term determined by $u^{(0)}$ and $v^{(l,r)}$ with $l<k$ .",
"We now first prove by induction over $k$ that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ as $r\\rightarrow 0$ .",
"For $k=0$ this claim is trivial, because $v^{(0,r)}=u^{(r)}-u^{(0)}$ has vanishing initial data for all $r$ .",
"Now suppose that the claim is true for all $0\\le l\\le k-1$ and consider $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k}$ .",
"Using the unit normal vector field $n$ to $\\Sigma $ we may express $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k}$ as a sum of terms in which all indices are either projected onto the conormal direction or onto the space-like directions cotangent to $\\Sigma $ .",
"If one of the indices is projected onto the space-like directions, then we may commute the derivatives in Equation (REF ) to bring the space-like index to the left.",
"The commutator terms involve the curvature, which is independent of $r$ , and at most $k-2$ derivatives.",
"Hence its initial data vanish as $r\\rightarrow 0$ by the induction hypothesis.",
"Similarly, if the first index is space-like, then the initial data of the term vanish as $r\\rightarrow 0$ by the induction hypothesis, since convergence in $\\Gamma (V|_{\\Sigma })$ entails convergence of all spacelike derivatives.",
"Finally we consider the term where all indices are projected onto the conormal direction.",
"For this term we may use Equation (REF ) to eliminate two normal derivatives in favour of spacelike derivatives and lower order terms.",
"Again the initial data of this term vanish in the limit $r\\rightarrow 0$ by the induction hypothesis.",
"Adding all components together proves that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ as $r\\rightarrow 0$ .",
"At this point our proof uses an energy estimate.",
"To formulate it, we endow the vector bundles $V$ and $TM$ with auxiliary smooth Riemannian metrics, and we denote the corresponding pointwise norms by $\\Vert {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} \\Vert $ .",
"For every compact $K\\subset \\Sigma $ and $L\\subset \\mathbb {R}$ there is a $C>0$ such that for all $r\\in L$ $\\int _{D(K)}*{v^{(r)}}^2\\le C\\int _K \\left(*{ {\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+ *{n^{\\alpha }\\nabla _{\\alpha } {\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\right)+C\\int _{D(K)} *{f^{(r)}}^2 \\,,$ where $D(k)$ is the domain of dependence and $v^{(r)}$ is a solution toAn explicit proof of this estimate is in Appendix .",
"Cf.",
"for an energy estimate of a quite similar form, where the independence of $C$ on $r$ can be established by retracing the steps in the proof.",
"$(P+r)v^{(r)}=f^{(r)}$ .",
"We now apply this result to $T^*M^{\\otimes k}\\otimes V$ instead of $V$ and prove by induction that each $v^{(k,r)}$ converges to 0 in the $L^2$ -sense on every compact set $\\widetilde{K}\\subset M$ .",
"Indeed, $\\widetilde{K}\\subset D(K)$ for some compact $K\\subset \\Sigma $ , so it suffices to apply the above energy estimate to $v^{(k,r)}$ and show that the right-hand side converges to 0.",
"Note that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ , and hence also in the $L^2$ -norm on every compact $K$ .",
"It remains to consider the source term of Equation (REF ), $-r\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}u^{(0)}+\\sum _{l=0}^{k-1}(C^{(k,l)}v^{(l,r)})_{\\alpha _1\\cdots \\alpha _k} \\,.$ Because $u^{(0)}$ is independent of $r$ we see immediately that the first term converges to 0 as $r\\rightarrow 0$ .",
"For $k=0$ the summation vanishes, so the energy estimate proves the desired convergence of $v^{(0,r)}$ .",
"For $k>0$ we use a proof by induction.",
"Assuming that $v^{(l,r)}\\rightarrow 0$ in the $L^2$ -sense as $r\\rightarrow 0$ for all $0\\le l\\le k-1$ , the energy estimate then proves the claim also for $v^{(k,r)}$ .",
"Finally, since $v^{(0,r)}$ and all its derivatives converge to 0 in an $L^2$ sense on every compact set, they also converge in $\\Gamma (V)$ by the Sobolev Embedding Theorem ( ).",
"For us, the following consequence is most relevant: Corollary 3.3 For fixed $F\\in \\Omega ^1_0(\\mathcal {M})$ , the advanced and retarded solutions $E^{\\pm }_mF$ depend continuously on $m\\in \\mathbb {R}$ .",
"Consequently, $E_mF$ is continuous in $m\\in \\mathbb {R}$ and $G_m^\\pm F$ and $G_mF$ are continuous in $m>0$ .",
"Proof: 3.2 We apply Theorem REF to $\\Box +m^2$ with $r=m^2$ .",
"Choosing e. g. $u_0,u_1=0$ and $\\Sigma $ to the past/future of the support of $f$ , we see that $E^{\\pm }_mF$ depend continuously on $m\\in \\mathbb {R}$ , and hence so does $E_mF$ .",
"The continuity of $G^{\\pm }_mF$ and $G_mF$ follows from the formula $G_m^\\pm = (m^{-2}{d\\delta } +1) E_m^\\pm $ as long as $m\\ne 0$ .",
"Let us now return to the continuity of families of observables and verify that it behaves well in the simplest examples.",
"Lemma 3.4 For a fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ and $j\\in \\Omega ^1(\\mathcal {M})$ the family of operators $\\left\\lbrace \\mathcal {A}_{m,j}(F)\\right\\rbrace _{m>0}$ is continuous.",
"Proof: 3.3 We see from the definitions of the maps involved in Definition REF that $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1)($\\mathcal {A}$ m,j(F)) = [( m,j , F $\\mathcal {M}$ , mF,0,0,...)]CCR, where $[{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}]_\\sim ^\\text{CCR}$ is continuous and does not depend on the mass.",
"Because $\\varphi _{m,j}$ depends continuously on $m>0$ , so does $\\langle \\varphi _{m,j} , F \\rangle _{\\mathcal {M}}$ .",
"Furthermore, $G_mF$ is continuous in $m>0$ by Corollary REF and the operators $\\rho _{({\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}})}$ are continuous and independent of $m$ , therefore, the initial data $\\kappa _m F=(\\rho _{(0)}G_m F , \\rho _{(d)}G_m F)$ also depend continuously on $m>0$ .",
"Combining these continuous maps proves the lemma.",
"We have found the desirable property that the quantum fields vary continuously with respect to the mass.",
"Note that this result is in fact independent of the choice of the Cauchy surface $\\Sigma $ , since $\\kappa _m(F)$ is continuous in $m$ for every Cauchy surface.",
"Indeed, we will now show quite generally that the notion of continuity in Definition REF is independent of the choice of the Cauchy surface $\\Sigma $ and of the family of classical solutions $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _m$ .",
"Theorem 3.5 The notion of continuity in Definition REF is independent of the choice of the Cauchy surface $\\Sigma $ and of the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m>0}$ of classical solutions to the inhomogeneous Proca equation.",
"Proof: 3.4 In this proof we will make repeated use of a joint continuity lemma, which we state and prove as Lemma REF in Appendix .",
"This lemma makes use of barrelled locally convex spaces, and we prove in Lemma REF that the complete BU-algebra is such a space.",
"Let $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ be a family of operators with $O_m\\in {A}_{m,j}$ .",
"We first verify the independence of the choice of Cauchy surface.",
"For this we choose two Cauchy surfaces $\\Sigma $ , $\\Sigma ^{\\prime }$ and we consider the family of operators $O^{\\prime }_m \\Psi _{\\varphi _{m,j}}^{-1} (O_m)\\in {A}_{m,0}$ .",
"It then suffices to prove that the continuity of $\\Lambda ^{(\\Sigma )}_m(O^{\\prime }_m)$ implies the continuity of $\\Lambda ^{(\\Sigma ^{\\prime })}_m(O^{\\prime }_m)$ , where we have made the dependence on the Cauchy surfaces explicit.",
"Let us first consider the space of initial data on the Cauchy surface for the wave equation on one-forms, $\\widetilde{\\mathcal {D}}_0(\\Sigma ) \\Omega ^1_0(\\Sigma )\\oplus \\Omega ^1_0(\\Sigma )\\oplus \\Omega ^0_0(\\Sigma )\\oplus \\Omega ^0_0(\\Sigma )$ and its analogue $\\widetilde{\\mathcal {D}}_0(\\Sigma ^{\\prime })$ .",
"For each $m$ we may define a continuous linear map $L_m:\\widetilde{\\mathcal {D}}_0(\\Sigma )\\rightarrow \\widetilde{\\mathcal {D}}_0(\\Sigma ^{\\prime })$ , which propagates the initial data under the wave operator $\\square +m^2$ .",
"By Theorem REF , $L_m$ is weakly continuous.",
"Fixed initial data $\\psi = (\\varphi , \\pi ) \\in \\mathcal {D}_{0}(\\Sigma )$ can be extended to initial data $\\Psi _m\\in \\widetilde{\\mathcal {D}}_0(\\Sigma )$ , using the constraint equations of Theorem REF with $m>0$ and $j=0$ .",
"Note that $\\Psi _m$ depends continuously on the mass $m$ .",
"Because $\\widetilde{\\mathcal {D}}_0(\\Sigma )$ is a barrelled space (cf.",
"the proof of Lemma REF ) we may apply Lemma REF and conclude that $L_m\\Psi _{m^{\\prime }}$ is jointly continuous in $(m,m^{\\prime })$ on $\\mathbb {R}_+\\times \\mathbb {R}_+$ .",
"In particular, $m\\mapsto L_m\\Psi _m$ depends continuously on $m>0$ .",
"Consequently, the map $\\tau _m : \\mathcal {D}_{0}(\\Sigma )\\rightarrow \\mathcal {D}_0(\\Sigma ^{\\prime })$ , which propagates initial data for the Proca field of mass $m$ , is also weakly continuous in $m>0$ .",
"We now extend this result as follows.",
"For $1\\le n\\le N$ we consider the continuous linear map $T^{N,n}_m 1^{\\otimes n-1}\\otimes \\tau _m\\otimes 1^{\\otimes N-n}$ on $\\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes n}\\boxtimes (T^*\\Sigma ^{\\prime }\\oplus T^*\\Sigma ^{\\prime })^{\\boxtimes N-n}\\big )$ , which may be defined using Schwartz' Kernels Theorem.",
"One may extend the proof of Theorem REF and Corollary REF to show that $T^{N,n}_m$ is also weakly continuous in $m>0$ .",
"We then define the map $T^N_m T^{N,1}\\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ TN,2 TN, N$ which is again weakly continuous in $ m>0$, by a repeated application of the joint continuity Lemma \\ref {lem:jointcontinuity}, using the fact that each of the spaces $ 0(($T^*\\Sigma $$T^*\\Sigma $ )n($T^*\\Sigma $ '$T^*\\Sigma $ ')N-n)$ is barrelled.",
"Let us now consider the lift of $ m$ to a continuous linear map $ Tm : $\\mathcal {BU}$ ( $\\mathcal {D}_{0}(\\Sigma )$ )$\\mathcal {BU}$ ( D0('))$ between complete BU-algebras (using sections of the bundle $$T^*\\Sigma $$T^*\\Sigma $$ and its analogue on $ '$).",
"Its action on $ 0(($T^*\\Sigma $$T^*\\Sigma $ )n)$ is simply given by $ TNm$, which shows that $ Tm$ is also weakly continuous in $ m>0$.",
"We note that each $ Tm$ is a homeomorphism and that it maps the ideal $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$$ onto $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}$$.",
"This means that it also maps the closed ideal $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$$ onto $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}$$ and it descends to a homeomorphism $ Tm$ between the quotient algebras $${\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$$ and $${\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big ( \\mathcal {D}_0(\\Sigma ^{\\prime })\\big )}}{\\; \\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}}}}}$$.",
"The weak continuity of $ Tm$ in $ m>0$ implies the weak continuity of $ Tm$ in $ m>0$.$ The complete algebra $\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}$ is barrelled, as shown in Lemma REF , and hence so is the quotient ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ .",
"Furthermore, because the ideal ${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is a closed subspace of $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ (cf.",
"Section REF ), the quotient ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$ is a dense subspace of ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ .",
"On this subspace, $\\widetilde{T}_m$ restricts to $\\Lambda ^{(\\Sigma ^{\\prime })}_{m} \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ ( ()m )-1$.",
"Identifying\\begin{equation}\\Lambda ^{(\\Sigma ^{\\prime })}_m(O^{\\prime }_m) = \\widetilde{T}_m \\Lambda ^{(\\Sigma )}_m(O^{\\prime }_m)\\end{equation}we may therefore use the assumed continuity of $ ()m(O'm)$ in $ m>0$ and the known weak continuity of $ Tm$ together with Lemma \\ref {lem:jointcontinuity} to find that $ m(')m(O'm)$ is continuous in $ m>0$.",
"This proves the independence of the choice of $$.$ We now turn to the independence of the choice of classical solutions.",
"Let $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _m$ and $\\left\\lbrace \\varphi ^{\\prime }_{m,j} \\right\\rbrace _m$ specify continuous families of classical solutions to the inhomogeneous Proca equation and fix a Cauchy surface $\\Sigma $ .",
"We denote the initial data of $\\varphi _{m,j}$ and $\\varphi ^{\\prime }_{m,j}$ by $(\\phi _m,\\pi _m)$ and $(\\phi _m^{\\prime },\\pi _m^{\\prime })$ , respectively.",
"For each $m>0$ we now define an algebra homeomorphism $L_m$ on the BU-algebra $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ by setting stipulating that $L_m$ preserves the unit and acts on homogeneous elements of degree 1 as $L_m\\big (0,(\\alpha ,\\beta ),0,0,\\ldots \\big ) \\big (\\mathcal {G}^{(\\Sigma )}\\big ((\\phi _m-\\phi ^{\\prime }_m,\\pi _m-\\pi ^{\\prime }_m),(\\alpha ,\\beta )\\big ),(\\alpha ,\\beta ),0,0,\\ldots \\big ) \\,.$ We can extend each $L_m$ in a unique way to a homeomorphism of the completed BU-algebra $\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}$ , using Schwartz' Kernels Theorem.",
"We denote the extended operator by the same symbol $L_m$ .",
"The action of $L_m$ on a homogeneous element $\\psi ^{(N)}$ of degree $N$ , i. e. on a section $\\psi ^{(N)}\\in \\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N}\\big )$ , can be written out explicitly as a sum of terms of degrees $\\le N$ .",
"Because $\\phi _m,\\pi _m,\\phi _m^{\\prime }$ and $\\pi _m^{\\prime }$ depend continuously on $m>0$ , so does the section $(\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m)$ and also the sections $\\big ((\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m)\\big )^{\\boxtimes n}$ for eachThis may be shown by induction over $n\\ge 1$ , e. g. using the joint continuity Lemma REF and noting that the linear map $\\gamma \\mapsto ((\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m))\\boxtimes \\gamma $ is weakly continuous in $m>0$ for any section $\\gamma $ of any vector bundle.",
"$n\\ge 1$ .",
"It follows that the components of $L_m\\psi ^{(N)}$ also depend continuously on $m>0$ .",
"Thus we see that $L_m$ is weakly continuous in $m>0$ .",
"Note that $L_m$ preserves the ideal ${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$ (just as in the proof of Theorem REF ), and hence it also preserves the closed ideal $\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}$ .",
"The $L_m$ therefore descend to homeomorphisms $\\tilde{L}_m$ of the quotient algebra ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\ \\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ , and the weak continuity of $L_m$ in $m>0$ implies the weak continuity of $\\tilde{L}_m$ in $m>0$ .",
"We note that $\\tilde{L}_m$ preserves the dense subalgebra ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$ and one may verify directly from Theorem REF and the definitions of the relevant maps that $\\tilde{L}_m$ acts on this subalgebra as $\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ 'm,j-1 'm,j m -1.",
"If $\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 (Om)$ depends continuously on $ m>0$, then so does\\begin{equation}\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}\\end{equation}{\\hbox{$\\scriptscriptstyle \\circ $}}$ 'm,j-1 (Om) = Lm m m,j-1 (Om) by the joint continuity Lemma REF ."
],
[
"The classical case",
"For fixed initial data ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ on a fixed Cauchy surface $\\Sigma $ there is a family of solutions $A_{m,j}$ to the Proca equation of mass $m>0$ with source term $j\\in \\Omega ^1_0(\\mathcal {M})$ .",
"We have seen in Theorem REF that these solutions take the form $\\langle A_{m,j} , F \\rangle _\\mathcal {M}= \\sum \\limits _\\pm \\langle j , G_m^\\mp F \\rangle _{J^\\pm (\\Sigma )} +\\langle {A_{(0)}}, \\rho _{(d)}G_m F \\rangle _\\Sigma - \\langle {A_{(d)}}, \\rho _{(0)}G_m F \\rangle _\\Sigma $ for any fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ .",
"We may think of $F$ as the mathematical representation of an experimental setup which measures the field configuration $A$ through the pairing $\\langle A,F\\rangle _{\\mathcal {M}}$ and we wish to investigate for which $F$ , if any, we can take the limit $m\\rightarrow 0$ in Equation (REF ) above for all choices of $\\Sigma $ and all initial data ${A_{(0)}},{A_{(d)}}$ .",
"Lemma 3.6 For fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ , the limit $m\\rightarrow 0$ of the right-hand side of Equation (REF ) exists for all smooth space-like Cauchy surfaces $\\Sigma $ and all initial data ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ , if and only if $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"Proof: 3.5 Suppose that for a given $F \\in \\Omega ^1_0(\\mathcal {M})$ the right-hand side of Equation (REF ) converges as $m\\rightarrow 0$ for all smooth space-like Cauchy surfaces $\\Sigma $ and all ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ .",
"Because we can vary the initial data arbitrarily and independently, all three terms in Equation (REF ) must converge separately.",
"In particular, $\\lim _{m\\rightarrow 0}\\rho _{(0)}G_mF$ must exist in a distributional sense.",
"Recall that $G_mF=m^{-2}E_md\\delta F+E_mF$ , where the second term is in $\\Omega ^1(\\mathcal {M})$ and depends continuously on $m\\in \\mathbb {R}$ by Corollary REF .",
"It then follows from the same corollary and from the continuity and linearity of $\\rho _{(0)}$ that $\\rho _{(0)}E_0d\\delta F &= \\lim _{m\\rightarrow 0} \\rho _{(0)}E_md\\delta F\\\\&= \\lim _{m\\rightarrow 0} m^2 \\rho _{(0)}\\left(G_mF-E_mF\\right)\\\\&= \\lim _{m\\rightarrow 0} m^2 \\cdot \\left(\\lim _{m\\rightarrow 0}\\rho _{(0)}G_mF - \\rho _{(0)}E_0F\\right)=0,$ where we used the existence of the limit of $\\rho _{(0)}G_mF$ .",
"Because this holds on every Cauchy surface, the one-form $E_0d\\delta F$ must annihilate every space-like vector at every point.",
"Because all tangent vectors are linear combinations of space-like vectors we conclude that $E_0d\\delta F=0$ and hence also $E_0\\delta dF=E_0(\\delta d+d\\delta )F=0$ .",
"We may then define $F^{\\prime } E_0^+\\delta dF=E_0^-\\delta dF$ and $F^{\\prime \\prime } E_0^+d\\delta F=E_0^-d\\delta F$ and note that these have compact supports.",
"Furthermore, since $\\delta $ and $d$ intertwine with $E_0^+$ on forms, $\\delta F^{\\prime }=0=dF^{\\prime \\prime }$ and $F^{\\prime } + F ^{\\prime \\prime } = E_0^+ (d \\delta + \\delta d) F = F \\,.$ Combining this formula with $G_m^\\pm =E_m^\\pm (m^{-2}d\\delta +1)$ we find $G_m^\\pm F &=E_m^\\pm F^{\\prime } + m^{-2}E_m^\\pm (d\\delta +\\delta d+m^2)F^{\\prime \\prime }\\\\&=E_m^\\pm F^{\\prime } + m^{-2}F^{\\prime \\prime }.$ Substituting this in the first term of Equation (REF ) we see that $\\sum \\limits _\\pm \\langle j , G_m^\\mp F \\rangle _{J^\\pm (\\Sigma )}= \\sum \\limits _\\pm \\langle j , E_m^\\pm F^{\\prime } \\rangle _{J^\\pm (\\Sigma )} + m^{-2} \\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}$ must converge as $m\\rightarrow 0$ .",
"The terms in the first sum converge as $m\\rightarrow 0$ by Corollary REF , and hence the last term must also converge.",
"This clearly implies $\\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}=0$ , showing that $F$ must have the stated form.",
"Conversely, when $F=F^{\\prime }+F^{\\prime \\prime }$ with $\\delta F^{\\prime }=0=dF^{\\prime \\prime }$ and $\\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}=0$ , then it follows from Equation (REF ) that $G_mF=E_mF^{\\prime }$ , which has a limit as $m\\rightarrow 0$ .",
"Together with Equation (REF ) and the continuity of $\\rho _{(d)}$ and $\\rho _{(0)}$ it follows that the right-side of Equation (REF ) converges as $m\\rightarrow 0$ .",
"Note that $F^{\\prime }$ and $F^{\\prime \\prime }$ are uniquely determined by $F=F^{\\prime }+F^{\\prime \\prime }$ and $\\delta F^{\\prime }=0=dF$ , because $\\Omega ^1_{0,\\delta }(\\mathcal {M}){\\hspace{1.00006pt}\\cap \\hspace{1.00006pt}}\\Omega ^1_{0,d}(\\mathcal {M})=\\lbrace 0\\rbrace $ .",
"Indeed, if $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ satisfies $d\\tilde{F}=\\delta \\tilde{F}=0$ , then also $\\square \\tilde{F}=0$ and hence $\\tilde{F}=0$ by .",
"For a fixed $m>0$ and $j\\in \\Omega ^1(\\mathcal {M})$ there are $F\\in \\Omega ^1_0(\\mathcal {M})$ which define trivial observables in the sense that $\\langle A_{m,j},F\\rangle _{\\mathcal {M}}=0$ for all field configurations (i. e. for all initial data).",
"The following lemma characterizes them: Lemma 3.7 For fixed $m>0$ and $j\\in \\Omega ^1(\\mathcal {M})$ , $F\\in \\Omega ^1_0(\\mathcal {M})$ defines a trivial observable if and only if $F=(\\delta d+m^2) \\tilde{F}$ for some $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ with $\\langle j,\\tilde{F}\\rangle _{\\mathcal {M}}=0$ .",
"Proof: 3.6 Arguing as in the proof of Lemma REF we see that $F$ defines a trivial observable if and only if $G_mF=0$ and $\\langle j,G_m^+F\\rangle _{\\mathcal {M}}=0$ .",
"The first condition is equivalent to $F=(\\delta d+m^2)\\tilde{F}$ for some $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ by Lemma REF .",
"The second condition then means that $\\langle j,\\tilde{F}\\rangle _{\\mathcal {M}}=0$ .",
"For any fixed $m$ and $j$ one would normally divide out these trivial observables, because they are redundant.",
"For our purposes, however, this is rather awkward, because the space of trivial observables depends on $m$ and $j$ .",
"However, we can remove some of the redundancy in the following way: Theorem 3.8 (Existence of the zero mass limit) Fix $j\\in \\Omega ^1(\\mathcal {M})$ .",
"For $F\\in \\Omega ^1_0(\\mathcal {M})$ , Equation (REF ) admits a massless limit for all initial data on all Cauchy surfaces if and only if there is a $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ such that $F-F^{\\prime }$ is a trivial observable for all $m>0$ .",
"Proof: 3.7 It follows from Lemma REF that $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j, F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"From Equation (REF ) we see that for all $m>0$ it holds $G_mF^{\\prime \\prime }=0$ and $\\langle j,G_m^+F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=m^{-2}\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ , so $F^{\\prime \\prime }=F-F^{\\prime }$ defines a trivial observable for all $m>0$ by Lemma REF and its proof.",
"In other words, for the massless limit it sufficesIt is unclear if there is any remaining redundancy.",
"to consider all co-closed forms $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ .",
"The meaning of this can be quite easily understood under the duality $\\langle {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} , {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} \\rangle _\\mathcal {M}$ .",
"One finds that ${\\scalebox {1.2}{{\\mathcal {D}^1(\\mathcal {M})}{d\\mathcal {D}^{0}(\\mathcal {M})}}}$ is dual to $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ (see ).",
"Here, $\\mathcal {D}^1(\\mathcal {M})$ denotes the set of distributional one-forms (in a physical sense, these are classical vector potentials), so restricting to co-closed test one-forms is equivalent to implementing the gauge equivalence $A \\rightarrow A + d\\chi $ , for $A \\in \\mathcal {D}^1(\\mathcal {M})$ and $ \\chi \\in \\mathcal {D}^0(\\mathcal {M})$ in the theory.",
"This dual relation is easily checked for $A^{\\prime } = A + d\\chi $ and $F \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ $\\langle A^{\\prime }, F \\rangle _{\\mathcal {M}}&= \\langle A, F \\rangle _{\\mathcal {M}} + \\langle d\\chi , F \\rangle _{\\mathcal {M}} \\\\&= \\langle A, F \\rangle _{\\mathcal {M}} + \\langle \\chi , \\delta F \\rangle _{\\mathcal {M}} = \\langle A, F \\rangle _{\\mathcal {M}} \\,.$ This is a nice result, because it elucidates the gauge equivalence in the Maxwell theory.",
"Note that it is a priori unclear how to implement the gauge equivalence in Maxwell's theory on curved spacetimes due to the non-trivial topology.",
"Maxwell's equation $\\delta d A = 0$ suggests that two solutions that differ by a closed one-form give rise to the same configuration, but one can argue that only exact one-forms should be treated as pure gauge solutions, because the Aharonov-Bohm effect does distinguish between configurations that differ by a form that is closed but not exact .",
"It is gratifying to see that we arrive at a gauge equivalence given by the class of exact forms, simply by keeping the set of linear observables as large as possible in the limit, i. e. $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ .",
"Hence, we have already captured one important feature of the Maxwell theory in the massless limit of the Proca theory!",
"It remains to check whether also the dynamics are well behaved in the massless limit."
],
[
"Dynamics and the zero mass limit",
"In the massless limit one may hope to find a vector potential $A_{0,j}$ satisfying Maxwell's equations $\\delta dA_{0,j}=j$ at least in a distributional sense, i. e. $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}=\\langle j , \\delta d F \\rangle _\\mathcal {M}$ for every test one-form $F\\in \\Omega ^1_0(\\mathcal {M})$ .",
"Note that $\\delta dF$ is co-closed, so by Theorem REF we may substitute $\\tilde{F}=\\delta dF$ in the limit $\\langle A_{0,j} , \\tilde{F} \\rangle _\\mathcal {M}&\\lim \\limits _{m \\rightarrow 0} \\langle A_{m,j}, \\tilde{F} \\rangle _\\mathcal {M}\\\\&= \\lim \\limits _{m \\rightarrow 0}\\Big ( \\sum \\limits _\\pm \\langle j , G_m^\\mp \\tilde{F} \\rangle _{J^{\\pm }(\\Sigma )} +\\langle {A_{(0)}}, \\rho _{(d)}G_m \\tilde{F} \\rangle _\\Sigma - \\langle {A_{(d)}}, \\rho _{(0)}G_m \\tilde{F} \\rangle _\\Sigma \\Big ) \\,$ for any given initial data ${A_{(0)}},{A_{(d)}}$ on any Cauchy surface $\\Sigma $ .",
"However, using $\\lim _{m\\rightarrow 0}G_m^{\\pm }\\delta dF&=\\lim _{m\\rightarrow 0}E_m^{\\pm }\\delta dF= E_0^{\\pm }\\delta dF\\\\&=F - E_0^{\\pm }d\\delta F$ we only find $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}&=\\sum \\limits _\\pm \\langle j , F-E_0^{\\mp }d \\delta F\\rangle _{J^\\pm (\\Sigma )} - \\langle {A_{(0)}}, \\rho _{(d)}E_0 d\\delta F \\rangle _\\Sigma + \\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma \\\\&=\\langle j , F\\rangle _{\\mathcal {M}}- \\sum \\limits _\\pm \\langle j , E_0^{\\mp }d \\delta F\\rangle _{J^\\pm (\\Sigma )}+ \\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma ,\\,$ where we used the fact that $\\rho _{(d)}E_m d\\delta F = - *_{(\\Sigma )}i^* * d E_m d \\delta F = 0$ since $d$ and $E_m$ commute.",
"The second term in Equation (REF ) will not vanish in general (e. g. when $dF=0$ but $\\langle j,F\\rangle _{\\mathcal {M}}\\ne 0$ ).",
"Ergo, the fields $A_{0,j}$ defined as the zero mass limit of the Proca field $A_{m,j}$ will not fulfill Maxwell's equation in a distributional sense.",
"While this might seem surprising at first, it is quite easy to understand when we recall how we have found solutions to Proca's equation, using the massive wave equation (REF ) combined with constraint equations on the initial data to ensure that the Lorenz constraint () is fulfilled.",
"Similarly, one solves Maxwell's equation by specifying a solution to the massless wave equation $(\\delta d + d \\delta )A_{0,j} = j$ and restricting the initial data such that the Lorenz constraint $\\delta A_{0,j} = 0$ is fulfilled.",
"The problem in the massless limit lies with the constraints.",
"Recall from Theorem REF that, in order to implement the Lorenz constraint, we have restricted the initial data by ${A_{(\\delta )}}= m^{-2}\\rho _{(\\delta )}j \\,, \\quad \\textrm {and} \\quad {A_{(n)}}= m^{-2}\\left( \\rho _{(n)}j + \\delta _{(\\Sigma )} {A_{(d)}}\\right) \\,.$ It is obvious that, in general, the resulting ${A_{(\\delta )}}$ and ${A_{(n)}}$ diverge in the zero mass limit, so there is no corresponding solution to Maxwell's equations with the same initial data.",
"In order to keep the dynamics in the zero mass limit, we need to make sure that the constraints are well behaved in the limit.",
"Since we do not want the external source or the initial data to be dependent of the mass, we have to require that ${A_{(\\delta )}}$ and ${A_{(n)}}$ vanish, i. e. we need to specifyThe first equation follows from $\\rho _{(\\delta )}j=0$ on all Cauchy surfaces.",
"$\\delta j &= 0 \\,, \\quad \\textrm {and} \\\\\\delta _{(\\Sigma )} {A_{(d)}}&= -\\rho _{(n)}j \\,.$ This corresponds exactly to the constraints on the initial data for the Maxwell equation which implement the Lorenz gauge (cf.",
"Pfenning ).",
"With these constraints, we can now look at the remaining term of $\\langle A_{0,j} , \\delta d F\\rangle _\\mathcal {M}$ in Equation (REF ).",
"We do this separately for the two summands.",
"Using that $d$ commutes with pullbacks and inserting the constraints on the initial data, we find $\\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma &= \\langle {A_{(d)}}, d_{(\\Sigma )} \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma \\\\&= \\langle \\delta _{(\\Sigma )}{A_{(d)}}, \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma \\\\&= -\\langle \\rho _{(n)}j , \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma $ For the first summand $\\sum _\\pm \\langle j , E_0^\\mp d\\delta F \\rangle _{\\Sigma ^\\pm }$ we use the partial integration in Equation (REF ) in the proof of Theorem REF and find, using $m=0$ and the constraint $\\delta j = 0$ as specified above, $\\sum _\\pm \\langle j , E_0^\\mp d\\delta F \\rangle _{J^\\pm (\\Sigma )}&= \\sum _\\pm \\langle d \\delta j , E_0^\\mp F \\rangle _{J^\\pm (\\Sigma )} +\\langle \\rho _{(\\delta )}E_0F,\\rho _{(n)}j\\rangle _\\Sigma - \\langle \\rho _{(\\delta )}j,\\rho _{(n)}E_0F\\rangle _\\Sigma \\\\&= \\langle \\rho _{(0)}E_0\\delta F,\\rho _{(n)}j\\rangle _\\Sigma \\,.$ Using the symmetry of the inner product $\\langle {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}},{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}\\rangle _{\\mathcal {M}}$ we find that the remaining terms of Equation (REF ) cancel when restricting the initial data such that they are well defined in the zero mass limit.",
"We therefore obtain the correct dynamics in that case: $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}&= \\langle j , F \\rangle _\\mathcal {M}- \\lim \\limits _{m \\rightarrow 0}\\Big (\\sum \\limits _\\pm \\langle j , E_m^\\mp d\\delta F \\rangle _{J^\\pm (\\Sigma )}- \\langle {A_{(d)}}, \\rho _{(0)}E_m d\\delta F \\rangle _\\Sigma \\Big ) \\\\&= \\langle j , F \\rangle _\\mathcal {M}\\,.$ In combination with Theorem REF we have thus shown Theorem 3.9 (The zero mass limit of the Proca field) Let $F\\in \\Omega ^1_0(\\mathcal {M})$ be a test one-form and $j \\in \\Omega ^1(\\mathcal {M})$ an external current.",
"Let $A_{m,j}$ be the solution to Proca's equation specified by initial data ${A_{(0)}}, {A_{(d)}}\\in \\Omega ^1_0(\\Sigma )$ via Theorem REF .",
"Defining the zero mass limit $\\langle A_{0,j} , F \\rangle _\\mathcal {M}= \\lim _{m \\rightarrow 0} \\langle A_{m,j}, F \\rangle _\\mathcal {M}$ of the Proca field, the following holds: The limit exists if and only if $F$ is equivalent to an observable $F^{\\prime }$ (for all $m>0$ ) with $\\delta F^{\\prime } = 0$ , effectively implementing the gauge equivalence of the Maxwell theory.",
"The field $A_{0,j}$ is a Maxwell field, that is, it solves Maxwell's equation, if and only if the current is conserved, $\\delta j = 0$ , and $\\rho _{(n)}j = - \\delta _{(\\Sigma )} {A_{(d)}}$ , implementing the Lorenz gauge.",
"Note that the conservation of the external current $\\delta j = 0$ is not required to solve Proca's equation, but it is necessary to solve Maxwell's equations ($\\delta dA=j$ entails $\\delta j=0$ ).",
"It is therefore not surprising that this condition is also necessary to recover the dynamics in the zero mass limit.",
"In analogy to the quantum theory, we may think of the field configuration $A$ as a state, whereas $F$ is an observable.",
"We then see from the theorem that the limits of observables give rise to the gauge equivalence of the classical vector potential, but additional conditions on the limits of states and external currents are needed in order to recover Maxwell's equation."
],
[
"The quantum case",
"In the quantum case we define the observables in the zero mass limit as follows: Definition 3.10 (Zero mass limit theory) For any fixed $j \\in \\Omega ^1(\\mathcal {M})$ and $O\\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ we say that $[O]_{m,j}\\in {A}_{m,j}$ has a zero mass limit if and only if $\\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) ([O]m,j) exists for all Cauchy surfaces $\\Sigma $ and all families $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ of classical solutions to the inhomogeneous equation $(\\delta d + m^2) \\varphi _{m,j} = j$ which depend continuously on $m$ .",
"Here, $\\Lambda _{m}$ and $\\Psi _{\\varphi _{m,j}}$ are as defined in Section REF and REF .",
"We call the zero mass limit trivial if and only if the above limit vanishes for all Cauchy surfaces $\\Sigma $ and all families $\\lbrace \\varphi _{m,j}\\rbrace _{m\\ge 0}$ .",
"If the zero mass limit exists, we denote its equivalence class modulo trivial observables by $[O]_{0,j}$ .",
"Note that we included $m=0$ in the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"This is done for the following reason.",
"Even when $j=0$ we may choose a non-trivial family $\\left\\lbrace \\varphi _{m,0} \\right\\rbrace _{m\\ge 0}$ and due to the isomorphism $\\Psi _{\\varphi _{m,0}}^{-1}$ we are then considering quantum fluctuations around the classical solutions $\\varphi _{m,0}$ .",
"If the quantum field is to converge, it seems reasonable to require that the classical background field $\\varphi _{m,0}$ also converges.",
"For general sources this implies that $\\varphi _{0,j}$ satisfies Maxwell's equations and hence the current must be conserved, $\\delta j=0$ .",
"We can think of the zero mass limit of an operator $O$ as a family of operators in the algebras ${{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , indexed by $\\Sigma $ and by the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"Using the properties of the topological algebras ${{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , it is not hard to see that the operators $O\\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ which have a zero mass limit form a $^*$ -subalgebra of ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ in which the operators with a trivial zero mass limit form an ideal.",
"We are interested in the quotient algebra which we denote by ${A}_{0,j}$ and which is generated by $\\mathbb {1}$ and by homogeneous degree-one elements, which we denote by $\\mathcal {A}_{0,j}(F)$ .",
"These are the massless field operators and we can think of them as the massless limits of the field operators $\\mathcal {A}_{m,j}(F)$ .",
"Our next theorem focuses on these field operators.",
"As our main result we determine for which $F\\in \\Omega ^1_0(\\mathcal {M})$ the limit $\\mathcal {A}_{0,j}(F)$ exists.",
"Theorem 3.11 (Existence of the zero mass limit) For given $j\\in \\Omega ^1_{\\delta }(\\mathcal {M})$ , $\\mathcal {A}_{m,j}(F)$ has a zero mass limit $\\mathcal {A}_{0,j}(F)$ if and only if $F\\in \\Omega ^1_0(\\mathcal {M})$ is of the form $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"The zero mass limit is trivial when $F^{\\prime }=0$ .",
"Proof: 3.8 Note that $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F)) =[(m,j,F$\\mathcal {M}$ ,mF,0,0,...)]CCR .",
"Just as in the last paragraph of the proof of Lemma REF we see that all $F$ of the stated form have a limit $\\lim _{m\\rightarrow 0}G_mF=\\lim _{m\\rightarrow 0}E_mF$ and hence the limit of the initial data $\\lim _{m\\rightarrow 0}\\kappa _mF$ exists on every Cauchy surface.",
"By assumption on the $\\varphi _{m,j}$ , $\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}}$ also has a limit as $m\\rightarrow 0$ .",
"Because $[{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}]_\\sim ^\\text{CCR}$ is continuous and independent of $m$ we see that $\\lim _{m\\rightarrow 0}\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F))$ exists for all $ F$ of the stated form.$ When $F^{\\prime }=0$ , then $F=F^{\\prime \\prime }$ and $G_mF=0$ (cf.",
"the proof of Theorem REF ) and hence $\\kappa _mF$ on every Cauchy surface.",
"Furthermore, $\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}}=m^{-2}\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ by Theorem REF and Equation (REF ).",
"Thus the zero mass limit is trivial.",
"Assume that $\\lim _{m\\rightarrow 0}\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F))$ exists.",
"This means that for each Cauchy surface $$ there is a family of elements $ gm${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$$ such that $ m0 (m,j,F$\\mathcal {M}$ ,mF,0,...)+gm$ exists in $$\\mathcal {BU}$ ($\\mathcal {D}_{0}(\\Sigma )$ )$.",
"Using the projection $ S$ of Lemma \\ref {lem:symmetrization-of-fields}, we have{\\begin{@align}{1}{-1}S\\big (\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )&=\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )\\\\&=S\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )\\,,\\end{@align}}because $ (m,j,F$\\mathcal {M}$ ,mF,0,...)$ is homogeneous of degree 1 and hence symmetric.",
"The continuity of $ S$ then implies that{\\begin{@align}{1}{-1}S\\big (\\lim _{m\\rightarrow 0} \\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )&=\\lim _{m\\rightarrow 0} S\\big (\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )\\\\&=\\lim _{m\\rightarrow 0} \\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big ),\\end{@align}}exists.",
"This implies that both $ m0m,j,F$\\mathcal {M}$$ and $ m0mF$ exist.",
"The first of these conditions already follows from the assumptions on $ m,j$ but the second implies in particular that $ m0$\\rho _{(0)}$ GmF$ exists.",
"Because this is required for every Cauchy surface, the argument presented in the proof of Lemma \\ref {lem:limit_existence_classical_equivalence} shows that $ F$ must be of the stated form.$ As in the classical case we find that the algebra ${A}_{0,j}$ of the massless limit is generated by field operators $\\mathcal {A}_{0,j}(F)$ with $F\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ ranging over the co-closed test one-forms.",
"Just as in the classical case, discussed in Section REF , this implements the gauge equivalence of the Maxwell theory, using the choice of gauge equivalence of .",
"Hence also in the quantum case, the limit exists only if we implement the gauge beforehand.",
"We now turn to the algebraic relations in ${A}_{0,j}$ .",
"For this we view $[O]_{0,j}$ as an equivalence class of a family of limits $\\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) ([O]m,j)$ in the algebras $${\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$$, indexed by the Cauchy surface $$ and the family $ { m,j }m0$ and we set in particular $$\\mathcal {A}$ 0,j(F) [(0,F,0,...)]0,j$.",
"Exploiting the algebraic structure of the algebras $${\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$$ we then find in a natural way\\footnote {This means that the relations below hold for the corresponding limits \\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}}{\\hbox{$\\scriptscriptstyle \\circ $}}$ m,j-1 ) ([O]m,j)$ for each Cauchy surface and for each family of classical solutions $ { m,j }m0$.$ that $\\mathcal {A}_{0,j}(\\alpha F + \\beta F^{\\prime }) &= \\alpha \\mathcal {A}_{0,j}(F) + \\beta \\mathcal {A}_{0,j}(F^{\\prime }) \\\\\\mathcal {A}_{0,j}(F)^* &= \\mathcal {A}_{0,j}(\\mathchoice{{\\m@th \\displaystyle F}nullfont\\hspace{0.0pt}\\hspace{2.0pt} \\overline{\\hspace{-1.66656pt}\\hspace{0.0pt}\\box \\hspace{0.0pt}\\hspace{-0.55542pt}}\\hspace{0.0pt}}{}{}{}$ nullfont $\\m@th \\textstyle F$ nullfont $\\m@th \\scriptstyle F$ nullfont $\\m@th \\scriptscriptstyle F$ ) for all $F \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $\\alpha , \\beta \\in \\mathbb {C}$ , corresponding to the linearity and the hermitian field property.",
"For the canonical commutation relations we note that for all $F,F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ , $G_mF^{\\prime }=E_mF^{\\prime }$ and hence $\\big [ \\mathcal {A}_{0,j}(F) ,\\mathcal {A}_{0,j}(F^{\\prime }) \\big ]&= \\lim \\limits _{m \\rightarrow 0} \\big [ \\mathcal {A}_{m,j}(F) ,\\mathcal {A}_{m,j}(F^{\\prime }) \\big ] \\\\&=\\mathrm {i}\\cdot \\lim \\limits _{m \\rightarrow 0} {F}{F^{\\prime }}\\cdot \\mathbb {1} \\\\&=\\mathrm {i}\\cdot \\lim \\limits _{m \\rightarrow 0} \\langle F,E_F^{\\prime }\\rangle _{\\mathcal {M}}\\\\&= \\mathrm {i}\\, {F}{F^{\\prime }}\\cdot \\mathbb {1}\\,.$ For co-closed test one-forms $F \\in \\Omega ^1_{0,\\delta }$ , the fundamental solutions $E^\\pm _0$ of the massless Klein-Gordon operator are actually also fundamental solutions to Maxwell's equation, i. e. it holds $E_0^\\pm \\delta d F = E_0^\\pm (\\delta d + d \\delta ) F = F$ , so we find that the fields in the zero mass limit are subject to the correct canonical commutation relations.",
"Indeed, using $\\rho _{(\\delta )}E_0 F^{\\prime } = i^* \\delta E_0 F^{\\prime } = i^* E_0 \\delta F^{\\prime } = 0$ and the analogous expression for $F$ , we may rewrite commutator in terms of initial data as ${F}{F^{\\prime }}&=\\langle F, E_0 F^{\\prime } \\rangle _\\mathcal {M}= -\\langle E_0 F , F^{\\prime } \\rangle _\\mathcal {M}\\\\&= \\langle \\rho _{(0)}E_0 F , \\rho _{(d)}E_0 F^{\\prime } \\rangle _\\Sigma - \\langle \\rho _{(d)}E_0 F , \\rho _{(0)}E_0 F^{\\prime } \\rangle _\\Sigma $ in analogy to Equation (REF ).",
"Note that ${F}{F^{\\prime }}$ for $F,F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ is in general degenerate, hence the quantum field theory associated with $\\mathcal {A}_{0,j}$ will in general fail to be local in the sense of Definition REF .",
"However, this is perfectly in line with the free vector potential as presented in .",
"It remains to verify whether $\\mathcal {A}_{0,j}$ solves Maxwell's equation, i. e. if $\\mathcal {A}_{0,j}(\\delta d F) = \\langle j , F\\rangle _\\mathcal {M}$ holds for all $F\\in \\Omega ^1_0(\\mathcal {M})$ .",
"Because $\\delta d F$ is co-closed, the limit $\\mathcal {A}_{0,j}(\\delta dF)$ is well defined.",
"For any Cauchy surface and any family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ we have $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(dF)) =[(m,j,dF$\\mathcal {M}$ ,mdF,0,0, ...)]CCR =[(dm,j,F$\\mathcal {M}$ ,mdF,0,0, ...)]CCR =j,F$\\mathcal {M}$ 1 + [(0,mdF,0,0, ...)]CCR, which is independent of $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"This essentially means that it suffices to consider the source free case, because the second term in Equation (REF ) is $\\Lambda _m\\big (\\mathcal {A}_{m,0}(\\delta dF)\\big )$ .",
"Because $G_m\\delta dF=E_m\\delta dF$ converges to $E_0\\delta dF$ we have $\\lim _{m\\rightarrow 0}\\kappa _m\\delta dF=\\big (\\rho _{(0)}E_0\\delta dF,\\rho _{(d)}E_0\\delta dF\\big )=\\big (\\rho _{(0)}E_0\\delta dF,0\\big ),$ where we have used that $E_0\\delta dF=-E_0d\\delta F$ is closed and hence $\\rho _{(d)}E_0\\delta dF=\\rho _{(n)}dE_0\\delta dF=0$ .",
"To recover Maxwell's equation, we need to verify that the second term in Equation (REF ) vanishes in the limit $m\\rightarrow 0$ for any Cauchy surface.",
"However, this fails in general.",
"Indeed, if $B\\in \\Omega ^1(\\mathcal {M})$ is the solution of the wave equation $\\Box B=0$ with initial data $\\rho _{(0)}B=\\rho _{(d)}B=\\rho _{(n)}B=0$ and $\\rho _{(\\delta )}B\\in \\Omega ^0_0(\\mathcal {M})$ not constant, then $B=E_0F$ for some compactly supported $F\\in \\Omega ^1_0(\\mathcal {M})$ (cf.",
"the proof of Lemma REF ).",
"However, $E_0\\delta dF=-E_0d\\delta F=-d\\delta E_0F=-d\\delta B$ does not vanish, because $\\delta B\\in \\Omega ^0(\\mathcal {M})$ is a function which is not constant.",
"In particular, because $d$ commutes with pull-backs, $\\rho _{(0)}E_0\\delta dF=-d_{\\Sigma }\\rho _{(\\delta )}B\\lnot \\equiv 0$ because $\\rho _{(\\delta )}B$ is not a constant function.",
"Conversely, following the proof of Theorem REF and Lemma REF we see that the limit only vanishes for all Cauchy surfaces if $E_0\\delta dF=0$ , which means that $F\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})+\\Omega ^1_{0,d}(\\mathcal {M})$ .",
"We have encountered a similar situation in the investigation of the classical theory in Section REF (cf.",
"Equation (REF )).",
"There we could get rid of similar remaining terms by restricting the initial data of the field configuration (i. e. of the state of the system) such that the Lorenz constraint is well behaved in the limit.",
"In the quantum scenario, our definition of the massless limit already requires $\\delta j=0$ , but the remaining constraint equation has not been imposed.",
"Indeed, in our present setting, which focuses on observables, the Lorenz constraint does not appear directly at all.",
"Nevertheless, we may impose the desired dynamics in a consistent way by dividing out a corresponding ideal.",
"Note in particular that the limit algebra is not simple, because the skew-symmetric form in Equation (REF ) is degenerate: $\\langle F,E_0\\delta dF^{\\prime }\\rangle _{\\mathcal {M}}=0$ when $\\delta F=0$ .",
"It follows that the operators $\\mathcal {A}_{0,j}(\\delta dF)-\\langle j,F\\rangle _{\\mathcal {M}}\\mathbb {1}$ commute with all other operators in the algebra ${A}_{0,0}$ and they therefore generate a two-sided ideal.",
"In the source free case this ideal is generated by the operators $\\mathcal {A}_{0,j}(\\delta dF)$ , which correspond to $[(0,\\kappa _m\\delta dF,0,\\ldots )]_\\sim ^{\\text{CCR}}$ with $\\kappa _m\\delta dF=(\\rho _{(0)}E_0\\delta dF,0)$ .",
"It is interesting to note that $A_F E_0 \\delta dF$ is a space-like compact solution to the source free Maxwell equation, $\\delta d A_F = -\\delta d E_0 d \\delta F = 0$ , and that it is of the form $A_F=d\\chi $ with the space-like compact function $\\chi -E_0\\delta F$ .",
"Solutions of the form $A_F$ can also be characterized in terms of their initial data, $\\big (\\rho _{(0)}A_F, \\rho _{(d)}A_F\\big ) = \\big (-d_{(\\Sigma )} \\rho _{(0)}\\chi , 0\\big ) \\,.$ Under the correspondence $F\\mapsto E_0F$ of observables (with $\\delta F=0$ ) and space-like compact solutions to Maxwell's equation, the observables $\\delta dF$ therefore generate a subspace that looks like a kind of pure gauge solutions (see for example or ).",
"However, the kind of “gauge equivalence” on the level of the observables, rather than the fields, does not seem to come out of the limiting procedure naturally.",
"It seems plausible that one can recover the correct dynamics by including states in the investigation and formulating conditions on their limiting behaviour, which essentially require that the remaining constraint equations is well behaved in the limit.",
"It is unclear if our limiting procedure can also be improved to directly recover the dynamics without considering states.",
"One idea is to consider the homeomorphisms that propagate the algebras of initial data ${\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ from one Cauchy surface to another.",
"If one can formulate a condition that ensures that these homeomorphisms remain well behaved in the limit, then the resulting limits should have a well behaved time evolution.",
"It would be of interest to develop these ideas and to compare the results with the massless limit of Stueckelberg's theory, which preserves the gauge invariance at all masses at the cost of introducing a coupling to an additional scalar field and all the associated additional complications .",
"We leave the investigation of these worthwhile questions to the future."
],
[
"Conclusion and Outlook",
"We have studied the classical and quantum Proca field in curved spacetimes, using a general setting including external sources and without restrictive assumptions on the spacetime topology.",
"We have shown that the quantum theory is locally covariant in the sense of , where the injectivity of the morphisms is related to the non-degeneracy of the symplectic form.",
"We have shown that the theory depends continuously on the mass $m>0$ , in a way which we have defined.",
"Using specific BU-algebra homeomorphism we mapped families of smeared Proca fields at different masses, initially elements in different BU-algebras, into the BU-algebra of initial data.",
"The topology of the latter algebra then determines a notion of continuity for the family of operators.",
"For $m>0$ we showed that this notion of continuity is independent of the choice of Cauchy surface and of the classical inhomogeneous solutions $\\varphi _{m,j}$ appearing in the homeomorphisms.",
"This result relied crucially on the use of energy estimates.",
"Note that a $C^*$ -Weyl algebra approach is ill-suited for the investigation of the zero mass limit, as one of us has argued in .",
"For the quantum theory we defined the zero mass limit by requiring a continuous family of observables to converge on every Cauchy surface and for every continuous family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ of inhomogeneous classical solutions.",
"(For the classical theory we considered a somewhat simplified setting.)",
"Investigating the zero mass limit we found in both cases that the limit exists and the theory is generated by the class of observables described by co-closed test one-forms.",
"This effectively implements a gauge invariance on the (distributional) solutions to Proca's equation by exact (distributional) one-forms.",
"This is of interest, because in general curved spacetimes the spacetime topology allows different possible choices of gauge invariance (using e. g. closed forms instead).",
"Our limiting procedure naturally leads to the same gauge invariance that was advocated in , using the independent argument that it can account for phenomena such as the Aharonov-Bohm effect and Gauss' law.",
"In the zero mass limit we also find that the quantum fields fulfill the basic properties of linearity, the hermitian field property and the correct CCR, all in line with the massless vector potential of electrodynamics.",
"However, we do not automatically recover the expected Maxwell dynamics.",
"In the classical case, this is caused by a potential divergence in the constraint equations on the initial data of field configurations.",
"This may be avoided by requiring the external source to be conserved, $\\delta j=0$ , and by requiring that the initial data of the configuration also satisfy the constraint equations of Maxwell's theory as given e. g. by Pfenning .",
"In the quantum case we did not clarify if Maxwell's equation can be obtained in the zero mass limit, e. g. by imposing additional conditions on the limits of observables or on states, or by requiring the homeomorphisms that propagate initial data between different Cauchy surfaces to remain well defined in the massless limit.",
"The further development of these ideas might require a detailed investigation of Hadamard states, which is also if interest in its own right.",
"So far these states seem to have been considered only in a restricted class of spacetimes .",
"Furthermore, it would be interesting to make a detailed comparison of our massless limit and the massless limit of Stueckelberg's theory as presented e. g. in .",
"We leave the investigation of these worthwhile questions to the future.",
"Acknowledgements We would like to thank the University of Leipzig, where this research was carried out, and MS would like to thank Prof. Stefan Hollands for helpful comments and discussions.",
"Large parts of this work are adapted from the MSc thesis of MS."
],
[
"Additional Lemmas",
"Let $\\mathfrak {X}$ be a complex vector bundle over a smooth differential manifold $\\mathcal {N}$ .",
"As in Section REF we may define the complete BU-algebra $\\overline{\\mathcal {BU}(\\Gamma _0(\\mathfrak {X}))}$ over $\\Gamma _0(\\mathfrak {X})$ as the direct sum $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )} = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big ) \\,,$ using the outer tensor product of vector bundles (see ).",
"We endow this algebra with the inductive limit topology of the subspaces $\\mathcal {BU}_N = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^N \\Gamma _0(\\mathfrak {X}^{\\boxtimes n}) \\,.$ Note that $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is the completion of the BU-algebra $\\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0(\\mathfrak {X})^{\\otimes n}$ .",
"Lemma 5.1 The complete Borchers-Uhlmann algebra $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is barrelled.",
"Proof: 5.1 The spaces $\\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big )$ of compactly supported sections of a complex vector bundle are LF-spaces, as they are defined as the inductive limit of the Frechét spaces of sections with support in some compact $K_l$ where $\\left\\lbrace K_l \\right\\rbrace _l$ is a fundamental sequence of compact $K_l \\subset \\mathcal {N}$ (see ).",
"Since LF-spaces are barrelled and the direct sum of barrelled spaces is again barrelled , we find for any $N \\in \\mathbb {N}$ that $\\mathcal {BU}_N$ is barrelled.",
"Additionally, the inductive limit of barrelled spaces is barrelled , hence the complete BU-algebra over smooth compactly supported sections $\\Gamma _0(\\mathfrak {X})$ over a complex vector bundle $\\mathfrak {X}$ is barrelled.",
"We will use barrelled spaces in order to apply the following result: Lemma 5.2 Let $X$ be a barrelled locally convex space, let $\\eta :[c,d]\\rightarrow X$ be a continuous map on a closed interval and let $L_m:X\\rightarrow Y$ be a family of continuous linear maps into a locally convex space $Y$ indexed by $m\\in [a,b]$ .",
"If the map $m\\mapsto L_m$ is weakly continuous, i. e. if $m\\mapsto L_mx$ is continuous on $[a,b]$ for each $x\\in X$ , then the map $(m,m^{\\prime })\\mapsto L_m\\eta (m^{\\prime })$ is continuous on $[a,b]\\times [c,d]$ .",
"Proof: 5.2 The weak continuity of $m\\mapsto L_m$ implies that for each $x\\in X$ the image of $m\\mapsto L_mx$ is compact.",
"The family of maps $L_m$ is therefore pointwise bounded.",
"Because $X$ is barrelled we may apply the uniform boundedness principle to find that the maps $L_m$ are equicontinuous.",
"For any $(m_0,m^{\\prime }_0)\\in [a,b]\\times [c,d]$ we set $x \\eta (m^{\\prime }_0)$ and we pick an arbitrary convex open neighbourhood $y+V$ of $y L_{m_0}x$ , where $V$ is an open neighbourhood of 0.",
"By equicontinuity there is an open neighbourhood $U\\subset X$ of 0 such that $L_m(U)\\subset \\frac{1}{2}V$ for all $m\\in [a,b]$ .",
"As $\\eta $ is continuous there is an open neighbourhood $W^{\\prime }\\subset [c,d]$ of $m^{\\prime }_0$ such that $\\eta (W^{\\prime })\\subset x+U$ .",
"Similarly there is an open neighbourhood $W\\subset [a,b]$ of $m_0$ such that $L_mx-y\\in \\frac{1}{2}V$ for all $m\\in W$ .",
"It follows that for all $(m,m^{\\prime })\\in W\\times W^{\\prime }$ $L_m\\eta (m^{\\prime })-y = L_m(\\eta (m^{\\prime })-x) + (L_mx-y)\\in \\frac{1}{2}V+\\frac{1}{2}V\\subset V$ which proves the desired continuity.",
"For our next lemma we will call an element of $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ symmetric if and only if it is totally symmetric in each degree.",
"Lemma 5.3 (Symmetrization of fields) Let $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ denote the linear subspace of the Borchers-Uhlmann algebra of initial data $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ consisting of symmetric elements.",
"Then there is a unique continuous linear surjective projection $S:\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\rightarrow \\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ whose kernel is $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ as defined in Section REF .",
"Proof: 5.3 For each $N\\ge 1$ and each permutation $\\sigma $ of the set $\\lbrace 1,\\ldots ,N\\rbrace $ we introduce the permutation operator $P^{(N)}_{\\sigma }:\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ defined by $\\left(P^{(N)}_{\\sigma }f\\right)\\big (p_1,\\ldots ,p_N\\big ) f\\big (p_{\\sigma (1)},\\ldots ,p_{\\sigma (N)}\\big ) \\,,$ where we view elements of $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ as sections in $\\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N}\\big )$ .",
"The symmetric tensor product $\\big (\\mathcal {D}_{0}(\\Sigma )\\big )^{\\otimes _S N}$ is then the range space of the projection $P^{(N)} \\frac{1}{N!",
"}\\sum \\limits _{\\sigma }P^{(N)}_{\\sigma }.$ Note that each $P^{(N)}_{\\sigma }$ is continuous, because the topology of $\\Gamma _0\\big ( (T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N} \\big ) $ is invariant under the swapping of variables.",
"It follows that $P^{(N)}$ is a continuous surjection.",
"We will first argue that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"For this we note that each $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is of the form $f = \\sum _{i=1}^{k} h_i \\cdot \\big ( -\\mathrm {i}\\mathcal {G}_m(\\psi _i, \\psi ^{\\prime }_i) , 0 , \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i , 0 , 0 ,\\dots \\big ) \\cdot \\tilde{h}_i$ for some $k \\in \\mathbb {N}$ , $h_i, \\tilde{h}_i \\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ and $\\psi _i, \\psi ^{\\prime }_i \\in \\mathcal {D}_{0}(\\Sigma )$ , where we have used the shorthand notation ${\\psi _i}{ \\psi ^{\\prime }_i} = \\langle \\pi _i , \\varphi ^{\\prime }_i \\rangle _\\Sigma - \\langle \\varphi _i , \\pi ^{\\prime }_i \\rangle _\\Sigma $ for $\\psi _i = (\\varphi _i , \\pi _i)$ .",
"If $f\\ne 0$ then its highest degree part is of some degree $N\\ge 2$ and we can write it explicitly, using the above representation, as $f^{(N)} = \\sum _{i=1}^{k} h_i^{(N_i)}\\, \\big ( \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i \\big )\\, \\tilde{h}_i^{(N-2-N_i)} \\,,$ where $h_i^{(N_i)}$ is the highest degree part of $h_i$ and $\\tilde{h}_i^{(N-2-N_i)}$ is either the highest degree part of $\\tilde{h}_i$ or 0.",
"It follows by inspection that $P^{(N)}f^{(N)}=0$ .",
"Now, if $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is non-zero and symmetric and if $f^{(N)}$ is its highest degree part, then $f^{(N)}=P^{(N)}f^{(N)}=0$ , contradicting that $f^{(N)}$ is the highest degree part.",
"It follows that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"We now construct for each degree $N\\ge 2$ two continuous linear maps $\\alpha ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\,,\\\\\\beta ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\,,$ (for $N=2$ we use $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes (N-2)}=\\mathbb {C}$ ) such that $f = P^{(N)}f + \\alpha ^{(N)}f + \\beta ^{(N)}f\\,.$ We start with the observation that $f = P^{(N)}f - \\frac{1}{N!}",
"\\sum \\limits _\\sigma (P^{(N)}_\\sigma - 1) f\\,.$ Every permutation $\\sigma $ can be written as a composition $\\sigma = \\tau _1 \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ 2 l$, where each $ i$ is a transposition of neighbouring indices.",
"We then find $ P(N)= P(N)1P(N)2P(N)l$ and, using a telescoping series,\\begin{equation}\\big (P^{(N)}_\\sigma - 1\\big ) f = \\sum _{i=1}^l \\big (P^{(N)}_{\\tau _i} - 1\\big )\\,P^{(N)}_{\\tau _{i+1}}\\cdots P^{(N)}_{\\tau _{l}}\\, f_m^{(N+2)} \\,.\\end{equation}This is now a sum over terms where the left-most operator $ P(N)i- 1$ yields a commutator.",
"Using the CCR we may reduce this commutator to a term of lower degree, i.\\,e.\\begin{equation}\\big (P^{(N)}_{\\tau _i} - 1\\big ) f^{\\prime } = \\tilde{f}^{\\prime }+g \\,,\\quad \\tilde{f}^{\\prime }\\in \\Gamma _0(\\mathcal {D}_{0}(\\Sigma ))^{\\otimes (N-2)}\\,,\\quad g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\end{equation}for any $ f'0($\\mathcal {D}_{0}(\\Sigma )$ )N$, where $ f'$ depends continuously on $ f'$ and hence so does $ g$.",
"Repeating this procedure for each term in Equation (\\ref {eqn:commuted-elements}) and each term in the sum in Equation (\\ref {eqn:symmetrization_of_field2}) yields a well-defined expression of the form\\begin{equation}f = P^{(N)}f + \\sum _j\\tilde{f}_j+\\sum _jg_j\\,,\\end{equation}where $ j$ runs over some index set, $ fj$ is homogeneous of degree $ N-2$ and $ gj${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$$.",
"Because $ fj$ and $ gj$ depend continuously on $ f$, it suffices to define $ (N)f jfj$ and $ (N)f jgj$.",
"We refer to \\cite [Lemma B.5]{Schambach2016} for more details.$ In Equation (REF ) we may now proceed to symmetrise the term $\\alpha ^{(N)}f$ of degree $N-2$ .",
"Note that elements of degree 0 or 1 are automatically symmetric.",
"By induction we can then show that $f = \\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f +\\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }\\beta ^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,.$ (Here the maps $\\alpha $ are to be omitted when $j=0$ .)",
"We now define $S$ as $S=\\bigoplus _{N=0}^{\\infty }S_N$ in terms of the continuous linear maps $S_N:& \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right) \\,,\\\\&f\\mapsto \\sum \\limits _{j=0}^{N/2}P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,,$ for all $N\\ge 0$ .",
"Note that $S$ is continuous and because the $\\alpha ^{(N)}$ and $\\beta ^{(N)}$ vanish on symmetric elements, $S$ acts as the identity on $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ .",
"It follows from Equation (REF ) that every element $f\\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ can be decomposed into $f=f^{\\prime }+g$ , where $f^{\\prime }$ is symmetric and $g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ .",
"This decomposition is unique, since $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ , and we must have $f^{\\prime }=Sf$ .",
"This entails in particular that $\\mathcal {BU}\\left(\\mathcal {D}_{0}(\\Sigma )\\right)=\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\oplus {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR},$ that $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ and that $S$ is the unique projection with the given range and kernel."
],
[
"Proof of the energy estimate (",
"In this appendix we prove the energy estimate (REF ), which we now restate.",
"Theorem 6.1 Let $P$ be a normally hyperbolic operator on a real vector bundle $V$ over a globally hyperbolic spacetime $M$ and let $\\Sigma \\subset M$ be a smooth, space-like Cauchy surface.",
"For all compact sets $K\\subset \\Sigma $ and $L\\subset \\mathbb {R}$ there is a $C>0$ such that $\\int _{D(K)}\\Vert v^{(r)}\\Vert ^2\\le C\\int _K \\left(*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{{\\left.\\hspace{0.0pt}n^{\\alpha }\\nabla _{\\alpha }v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\right)+C\\int _{D(K)}*{f^{(r)}}^2,$ where $D(k)$ is the domain of dependence and $v^{(r)}$ is a solution to $(P+r)v^{(r)}=f^{(r)}$ .",
"Proof: 6.1 We may identify $M=\\mathbb {R}\\times S$ and $g=-Ndt^2+h_t$ , where $t\\in \\mathbb {R}$ , $N>0$ , $\\Sigma _t \\lbrace t\\rbrace \\times S$ is a smooth spacelike Cauchy surface with metric $h_t$ and $\\Sigma =\\Sigma _0$ .",
"We set $\\xi _{\\alpha } -N\\nabla _{\\alpha }t$ , so that $\\xi ^{\\alpha }$ is a future pointing time-like vector field and $n^{\\alpha } N^{-\\frac{1}{2}}\\xi ^{\\alpha }$ is its normalisation.",
"Without loss of generality we may assume that the auxiliary norm $*{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}$ on $TM$ is given by $2n_{\\alpha }n_{\\beta }+g_{\\alpha \\beta }$ .",
"For the purposes of this proof we choose the connection $\\nabla $ on $V$ to be the one which is compatible with the auxiliary metric on $V$ .",
"Any different choice of connection in (REF ) can easily be accommodated for by adjusting $C$ at the end of the proof.",
"Note that for suitable smooth bundle homomorphisms $A$ and $B$ it holds $P=g^{\\alpha \\beta }\\nabla _{\\alpha }\\nabla _{\\beta }+A^{\\alpha }\\nabla _{\\alpha }+B$ .",
"Let us fix $r$ for now and drop the superscripts on $v$ and $f$ .",
"We define the quantities $T_{\\alpha \\beta }&\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v-\\frac{1}{2}g_{\\alpha \\beta }\\left(*{\\nabla v}^2+*{v}^2\\right)\\,,\\\\P_{\\alpha }&\\xi ^{\\beta }T_{\\alpha \\beta }\\,,\\\\\\epsilon &n^{\\alpha }P_{\\alpha }=\\sqrt{N}n^{\\alpha }n^{\\beta }T_{\\alpha \\beta }\\\\&\\phantom{:}=\\frac{1}{2}\\sqrt{N}\\left((2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v+*{v}^2\\right)\\,,$ where $\\cdot $ refers to the hermitian inner product on $V$ .",
"Note that $\\epsilon \\ge 0$ .",
"We may now choose a $T>0$ such that $D(K)\\subset (-\\infty ,T)\\times S$ and a compact $K^{\\prime }\\subset \\Sigma $ which contains $K$ in its interior.",
"Then we may choose an auxiliary Cauchy surface $\\Sigma ^{\\prime }$ of $(-\\infty ,T)\\times S$ such that $D(K)$ lies to the past of $\\Sigma ^{\\prime }$ , but $\\Sigma ^{\\prime }$ contains $\\Sigma \\setminus K^{\\prime }$ .",
"Furthermore, we may choose a $C\\ge 1$ such that the following inequalities hold on $[0,T]\\times S$ : $N^{\\pm \\frac{1}{2}}\\le C \\,,\\quad \\pm (\\nabla ^{\\alpha }\\xi ^{\\beta }+\\nabla ^{\\beta }\\xi ^{\\alpha })\\le C\\sqrt{N}(2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\,,\\\\\\left|\\nabla _{\\alpha }\\xi ^{\\alpha }\\right|\\le C\\sqrt{N}\\,,\\quad *{R}\\le C\\,,\\quad *{A}\\le C \\quad \\textrm {and}\\quad *{B}\\le C \\,,$ where $R$ is the curvature of $\\nabla $ on $V$ .",
"In addition we may assume that $|r+1|\\le C$ for all $r\\in L$ and that $h_t\\le Ch_{t^{\\prime }}$ on $K^{\\prime }$ for all $t,t^{\\prime }\\in [0,T]$ and similarly for the hermitian metric in $V$ .",
"It will be convenient to introduce $L_t \\Sigma _t\\cap J^-(\\Sigma ^{\\prime })$ for $t\\in [0,T]$ and the “energy” $\\epsilon (t) \\int _{L_t}\\epsilon \\,.$ We now want to estimate the quantity $E(t) \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\epsilon $ for $t\\in [0,T]$ .",
"We note first of all that $\\frac{d}{dt}E(t)\\le \\lim _{\\tau \\rightarrow 0^+}\\tau ^{-1}\\int _{[t,t+\\tau ]\\times L_t}\\epsilon \\le C\\int _{L_t}\\epsilon \\,,$ where the constant $C$ is needed to estimate the factor $\\sqrt{N}$ which arises due to a change of volume form.",
"Furthermore, using Stokes' Theorem: $\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha }=\\epsilon (t)-\\epsilon (0)+\\int _{\\Sigma ^{\\prime }\\cap ([0,t]\\times S)}\\nu ^{\\alpha }P_{\\alpha }\\,,$ where $\\nu ^{\\alpha }$ is the forward unit normal to $\\Sigma ^{\\prime }$ .",
"One may show that the bilinear form $\\nu ^{\\alpha }n^{\\beta }+n^{\\alpha }\\nu ^{\\beta }-g^{\\alpha \\beta }n^{\\gamma }\\nu _{\\gamma }$ is positive definite and $n^{\\gamma }\\nu _{\\gamma }<0$ .",
"This entails that $\\nu ^{\\alpha }P_{\\alpha }\\ge 0$ and hence $\\epsilon (t)-\\epsilon (0)\\le \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha } \\,.$ Furthermore, we may estimate $\\left|\\nabla ^{\\alpha }P_{\\alpha }\\right|\\le \\left|T_{\\alpha \\beta }\\nabla ^{\\alpha }\\xi ^{\\beta } \\right|+\\left|\\xi ^{\\beta }\\nabla ^{\\alpha }T_{\\alpha \\beta }\\right|\\,,$ where $\\nabla ^{\\alpha }T_{\\alpha \\beta }=v\\cdot R_{\\alpha \\beta }\\cdot \\nabla ^{\\alpha }v-v\\cdot B\\cdot \\nabla _{\\beta }v-(r+1)v\\cdot \\nabla _{\\beta }v+f\\cdot \\nabla _{\\beta }v \\,.$ For the term involving $f$ we can use the further estimate $\\left|\\xi ^{\\beta }f\\cdot \\nabla _{\\beta }v\\right|\\le C *{f}\\cdot *{\\nabla v}\\le \\frac{1}{2} C \\left( *{f}^2+*{\\nabla v}^2 \\right) \\,.$ Using our choice of $C$ we can then estimate all the terms in $\\nabla ^{\\alpha }P_{\\alpha }$ to find $\\epsilon (t)\\le \\epsilon (0)+\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}8 C^2\\epsilon +\\frac{1}{2} C *{f}^2$ and consequently $\\frac{d}{dt}E(t)\\le C\\epsilon (t)\\le 8C^3E(t)+C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2.$ Therefore, $\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)\\le C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2$ .",
"With $E(0)=0$ this yields $e^{-8C^2T}E(T)=\\int _0^T\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)dt\\le \\left(C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2\\right)T$ and hence $E(T)\\le C^{\\prime }\\big (\\epsilon (0)+\\int _{D(K^{\\prime })}*{f}^2\\big )$ for a suitable $C^{\\prime }>0$ independent of $r$ .",
"Note that $E(T)\\ge \\int _{D(K)}*{v}^2$ and that $\\epsilon (0)\\le C^{\\prime }\\int _K\\big (*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{n^{\\alpha }\\nabla _{\\alpha }{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\big )$ when we choose $C^{\\prime }$ large enough.",
"Finally, we may shrink $K^{\\prime }$ to $K$ without adjusting the constants $C$ or $C^{\\prime }$ which leads to the desired estimate.",
"[heading=bibintoc,title=References]"
],
[
"The zero mass limit",
"For the main results of this article we will investigate the zero mass limit of the Proca field a in curved spacetime in both the classical and the quantum case.",
"In Section REF we will formulate the key notion of continuity of the field theory with respect to the mass and establish its basic properties.",
"We then define the massless limit in a general, state independent setup first for the classical Proca field in Section REF and then for the quantum Proca field in Section REF .",
"At given points, we compare our results with the theory of the (quantum) vector potential of electromagnetism in curved spacetimes as studied in , ."
],
[
"Continuity in the mass",
"When defining a notion of continuity of the field theory with respect to the mass, the basic problem is that at different masses the smeared fields $\\mathcal {A}_{m,j}(F)$ are elements of different algebras ${A}_{m,j}$ .",
"Indeed, when constructing ${A}_{m,j}$ as a quotient of the BU-algebra, the ideals that implement the dynamics and the commutation relations both depend on the mass.",
"We therefore need to find a way of comparing the Proca fields at different masses with each other.",
"One could try to solve this using the $C^*$ -Weyl algebra to describe the quantum Proca field and the notion of a continuous field of $C^*$ -algebras depending on the mass parameter (cf. ).",
"This would work very nicely, if the theories were described by a weakly continuous family of (non-degenerate) symplectic forms on a fixed linear space (cf.",
", which generalises ).",
"However, as it turns out, this approach is ill-suited for the problem at hand.",
"Indeed, one would like linear combinations of Weyl operators $W_{m,j}(F_i)=\\mathrm {e}^{\\mathrm {i}\\mathcal {A}_{m,j}(F_i)}$ with fixed test-forms $F_i\\in \\Omega _0^1(\\mathcal {M})$ to depend continuously on the mass, but for $j=0$ the norm of an operator like $W_{m,0}\\big ((\\delta d +m_0^2)F\\big )-1$ , with a fixed $F$ and $m_0$ , can be seen to be discontinuous at $m=m_0$ , where the operator vanishes.",
"A different attempt, which we have hinted at in Section REF , is to use the semi-norms $q_{m, j, \\alpha }\\big ( [f]_{m,j} \\big ) = \\inf \\big \\lbrace p_\\alpha (g) : g \\in [f]_{m,j} \\big \\rbrace $ to define a notion of continuity of the theory with respect to the mass $m$ .",
"We could call a family of operators $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ with $O_m \\in {A}_{m,j}$ continuous if and only if the map $m \\mapsto q_{m, j, \\alpha }\\big (O_m\\big )$ is continuous for all $\\alpha $ with respect to the standard topology in $\\mathbb {R}$ .",
"While this definition seems appropriate at first sight, it is non-trivial to show the desirable property that for a fixed $F\\in \\Omega _0^1(\\mathcal {M})$ the smeared field operators $\\mathcal {A}_{m,j}(F)$ vary continuously with $m$ .",
"Even for $j=0$ and considering only the one-particle level, we were unable to prove this.",
"In this paper we therefore opt for the following solution, which makes use of the Borchers-Uhlmann algebra of initial data.",
"For simplicity we first consider the case $j=0$ and a family of operators $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ with $O_m \\in {A}_{m,0}$ .",
"Since we have found for every mass $m>0$ that ${A}_{m,0}$ is homeomorphic to ${\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ , we can map the family $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ to a family of operators in the single algebra ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , which already carries a topology and hence a notion of continuity.",
"When $j\\ne 0$ we combine this idea with the fact that ${A}_{m,j}$ is homeomorphic to ${A}_{m,0}$ .",
"In this way we arrive at the following notion of continuity.",
"Definition 3.1 (Continuity with respect to the mass) Let $j \\in \\Omega ^1(\\mathcal {M})$ be fixed and let $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ be a family of operators with $O_m\\in {A}_{m,j}$ .",
"We call $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ continuous if and only if the map $\\mathbb {R}_+ &\\rightarrow {{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}} \\,,\\\\m &\\mapsto \\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) (Om) is continuous, where $\\Lambda _{m}$ and $\\Psi _{\\varphi _{m,j}}$ are as defined in Section REF and REF and $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m>0}$ is a family of classical solutions to the inhomogeneous Proca equation $(\\delta d + m^2) \\varphi _{m,j} = j$ which depends continuously on $m$ (i. e. $m\\mapsto \\varphi _{m,j}\\in \\Omega ^1(\\mathcal {M})$ is continuous).",
"Equivalently, identifying $O_m = \\big [\\tilde{O}_m\\big ]_{m,j}$ for some $\\tilde{O}_m \\subset {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ , the family $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ is continuous if and only if the map $\\mathbb {R}_+ &\\rightarrow {{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}} \\,,\\\\m &\\mapsto \\big [ \\big ( K_m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) (Om) ]CCR is continuous, with $K_m$ and $\\Gamma _{\\varphi _{m,j}}$ as defined in Section REF and REF .",
"We now aim to establish some desirable properties of this notion of continuity, most importantly that it is independent of the choice of Cauchy surface $\\Sigma $ and of the choice of the continuous family $\\varphi _{m,j}$ of classical solutions.",
"Our arguments will make essential use of the following result for normally hyperbolic operators: Theorem 3.2 Let $P$ be a normally hyperbolic operator on a real vector bundle $V$ over a globally hyperbolic spacetime $M$ .",
"Let $u_0,u_1\\in \\Gamma (V|_{\\Sigma })$ be initial data on a Cauchy surface $\\Sigma $ and $f\\in \\Gamma _0(V)$ .",
"For $r\\in \\mathbb {R}$ , let $u^{(r)}$ be the unique solution to $(P+r)u^{(r)}=f$ with initial data $u_0,u_1$ on $\\Sigma $ .",
"Then $r\\mapsto u^{(r)}$ is a continuous map from $\\mathbb {R}$ to $\\Gamma (V)$ .",
"Proof: 3.1 It suffices to prove continuity at $r=0$ , after shifting $P$ by a constant.",
"We may write $P=\\nabla ^{\\alpha }\\nabla _{\\alpha }+B$ , where $B$ is a bundle endomorphism .",
"Here, $\\nabla _{\\alpha }$ is a connection on $V$ , which may be extended with the Levi-Civita connection to tensor product bundles of $V$ , $TM$ and their dual bundles.",
"We write for $k=0,1,2,\\ldots $ $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k} \\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}(u^{(r)}-u^{(0)})$ and we note that $(P+r)(u^{(r)}-u^{(0)})=-ru^{(0)}$ and hence $(P+r)v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k} = -r\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}u^{(0)}-(B^{(k)}v^{(k,r)})_{\\alpha _1\\cdots \\alpha _k}+\\sum _{l=0}^{k-1}(C^{(k,l)}v^{(l,r)})_{\\alpha _1\\cdots \\alpha _k} \\,,$ where $B^{(k)}$ and $C^{(k,l)}$ are bundle homomorphisms which involve $B$ and the curvature of $\\nabla $ .",
"It follows that $v^{(k,r)}$ solves an inhomogeneous normally hyperbolic equation with the operator $P+B^{(k)}+r$ and an inhomogeneous term determined by $u^{(0)}$ and $v^{(l,r)}$ with $l<k$ .",
"We now first prove by induction over $k$ that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ as $r\\rightarrow 0$ .",
"For $k=0$ this claim is trivial, because $v^{(0,r)}=u^{(r)}-u^{(0)}$ has vanishing initial data for all $r$ .",
"Now suppose that the claim is true for all $0\\le l\\le k-1$ and consider $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k}$ .",
"Using the unit normal vector field $n$ to $\\Sigma $ we may express $v^{(k,r)}_{\\alpha _1\\cdots \\alpha _k}$ as a sum of terms in which all indices are either projected onto the conormal direction or onto the space-like directions cotangent to $\\Sigma $ .",
"If one of the indices is projected onto the space-like directions, then we may commute the derivatives in Equation (REF ) to bring the space-like index to the left.",
"The commutator terms involve the curvature, which is independent of $r$ , and at most $k-2$ derivatives.",
"Hence its initial data vanish as $r\\rightarrow 0$ by the induction hypothesis.",
"Similarly, if the first index is space-like, then the initial data of the term vanish as $r\\rightarrow 0$ by the induction hypothesis, since convergence in $\\Gamma (V|_{\\Sigma })$ entails convergence of all spacelike derivatives.",
"Finally we consider the term where all indices are projected onto the conormal direction.",
"For this term we may use Equation (REF ) to eliminate two normal derivatives in favour of spacelike derivatives and lower order terms.",
"Again the initial data of this term vanish in the limit $r\\rightarrow 0$ by the induction hypothesis.",
"Adding all components together proves that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ as $r\\rightarrow 0$ .",
"At this point our proof uses an energy estimate.",
"To formulate it, we endow the vector bundles $V$ and $TM$ with auxiliary smooth Riemannian metrics, and we denote the corresponding pointwise norms by $\\Vert {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} \\Vert $ .",
"For every compact $K\\subset \\Sigma $ and $L\\subset \\mathbb {R}$ there is a $C>0$ such that for all $r\\in L$ $\\int _{D(K)}*{v^{(r)}}^2\\le C\\int _K \\left(*{ {\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+ *{n^{\\alpha }\\nabla _{\\alpha } {\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\right)+C\\int _{D(K)} *{f^{(r)}}^2 \\,,$ where $D(k)$ is the domain of dependence and $v^{(r)}$ is a solution toAn explicit proof of this estimate is in Appendix .",
"Cf.",
"for an energy estimate of a quite similar form, where the independence of $C$ on $r$ can be established by retracing the steps in the proof.",
"$(P+r)v^{(r)}=f^{(r)}$ .",
"We now apply this result to $T^*M^{\\otimes k}\\otimes V$ instead of $V$ and prove by induction that each $v^{(k,r)}$ converges to 0 in the $L^2$ -sense on every compact set $\\widetilde{K}\\subset M$ .",
"Indeed, $\\widetilde{K}\\subset D(K)$ for some compact $K\\subset \\Sigma $ , so it suffices to apply the above energy estimate to $v^{(k,r)}$ and show that the right-hand side converges to 0.",
"Note that the initial data of $v^{(k,r)}$ converge to 0 in $\\Gamma (V|_{\\Sigma })$ , and hence also in the $L^2$ -norm on every compact $K$ .",
"It remains to consider the source term of Equation (REF ), $-r\\nabla _{\\alpha _1}\\cdots \\nabla _{\\alpha _k}u^{(0)}+\\sum _{l=0}^{k-1}(C^{(k,l)}v^{(l,r)})_{\\alpha _1\\cdots \\alpha _k} \\,.$ Because $u^{(0)}$ is independent of $r$ we see immediately that the first term converges to 0 as $r\\rightarrow 0$ .",
"For $k=0$ the summation vanishes, so the energy estimate proves the desired convergence of $v^{(0,r)}$ .",
"For $k>0$ we use a proof by induction.",
"Assuming that $v^{(l,r)}\\rightarrow 0$ in the $L^2$ -sense as $r\\rightarrow 0$ for all $0\\le l\\le k-1$ , the energy estimate then proves the claim also for $v^{(k,r)}$ .",
"Finally, since $v^{(0,r)}$ and all its derivatives converge to 0 in an $L^2$ sense on every compact set, they also converge in $\\Gamma (V)$ by the Sobolev Embedding Theorem ( ).",
"For us, the following consequence is most relevant: Corollary 3.3 For fixed $F\\in \\Omega ^1_0(\\mathcal {M})$ , the advanced and retarded solutions $E^{\\pm }_mF$ depend continuously on $m\\in \\mathbb {R}$ .",
"Consequently, $E_mF$ is continuous in $m\\in \\mathbb {R}$ and $G_m^\\pm F$ and $G_mF$ are continuous in $m>0$ .",
"Proof: 3.2 We apply Theorem REF to $\\Box +m^2$ with $r=m^2$ .",
"Choosing e. g. $u_0,u_1=0$ and $\\Sigma $ to the past/future of the support of $f$ , we see that $E^{\\pm }_mF$ depend continuously on $m\\in \\mathbb {R}$ , and hence so does $E_mF$ .",
"The continuity of $G^{\\pm }_mF$ and $G_mF$ follows from the formula $G_m^\\pm = (m^{-2}{d\\delta } +1) E_m^\\pm $ as long as $m\\ne 0$ .",
"Let us now return to the continuity of families of observables and verify that it behaves well in the simplest examples.",
"Lemma 3.4 For a fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ and $j\\in \\Omega ^1(\\mathcal {M})$ the family of operators $\\left\\lbrace \\mathcal {A}_{m,j}(F)\\right\\rbrace _{m>0}$ is continuous.",
"Proof: 3.3 We see from the definitions of the maps involved in Definition REF that $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1)($\\mathcal {A}$ m,j(F)) = [( m,j , F $\\mathcal {M}$ , mF,0,0,...)]CCR, where $[{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}]_\\sim ^\\text{CCR}$ is continuous and does not depend on the mass.",
"Because $\\varphi _{m,j}$ depends continuously on $m>0$ , so does $\\langle \\varphi _{m,j} , F \\rangle _{\\mathcal {M}}$ .",
"Furthermore, $G_mF$ is continuous in $m>0$ by Corollary REF and the operators $\\rho _{({\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}})}$ are continuous and independent of $m$ , therefore, the initial data $\\kappa _m F=(\\rho _{(0)}G_m F , \\rho _{(d)}G_m F)$ also depend continuously on $m>0$ .",
"Combining these continuous maps proves the lemma.",
"We have found the desirable property that the quantum fields vary continuously with respect to the mass.",
"Note that this result is in fact independent of the choice of the Cauchy surface $\\Sigma $ , since $\\kappa _m(F)$ is continuous in $m$ for every Cauchy surface.",
"Indeed, we will now show quite generally that the notion of continuity in Definition REF is independent of the choice of the Cauchy surface $\\Sigma $ and of the family of classical solutions $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _m$ .",
"Theorem 3.5 The notion of continuity in Definition REF is independent of the choice of the Cauchy surface $\\Sigma $ and of the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m>0}$ of classical solutions to the inhomogeneous Proca equation.",
"Proof: 3.4 In this proof we will make repeated use of a joint continuity lemma, which we state and prove as Lemma REF in Appendix .",
"This lemma makes use of barrelled locally convex spaces, and we prove in Lemma REF that the complete BU-algebra is such a space.",
"Let $\\left\\lbrace O_m\\right\\rbrace _{m>0}$ be a family of operators with $O_m\\in {A}_{m,j}$ .",
"We first verify the independence of the choice of Cauchy surface.",
"For this we choose two Cauchy surfaces $\\Sigma $ , $\\Sigma ^{\\prime }$ and we consider the family of operators $O^{\\prime }_m \\Psi _{\\varphi _{m,j}}^{-1} (O_m)\\in {A}_{m,0}$ .",
"It then suffices to prove that the continuity of $\\Lambda ^{(\\Sigma )}_m(O^{\\prime }_m)$ implies the continuity of $\\Lambda ^{(\\Sigma ^{\\prime })}_m(O^{\\prime }_m)$ , where we have made the dependence on the Cauchy surfaces explicit.",
"Let us first consider the space of initial data on the Cauchy surface for the wave equation on one-forms, $\\widetilde{\\mathcal {D}}_0(\\Sigma ) \\Omega ^1_0(\\Sigma )\\oplus \\Omega ^1_0(\\Sigma )\\oplus \\Omega ^0_0(\\Sigma )\\oplus \\Omega ^0_0(\\Sigma )$ and its analogue $\\widetilde{\\mathcal {D}}_0(\\Sigma ^{\\prime })$ .",
"For each $m$ we may define a continuous linear map $L_m:\\widetilde{\\mathcal {D}}_0(\\Sigma )\\rightarrow \\widetilde{\\mathcal {D}}_0(\\Sigma ^{\\prime })$ , which propagates the initial data under the wave operator $\\square +m^2$ .",
"By Theorem REF , $L_m$ is weakly continuous.",
"Fixed initial data $\\psi = (\\varphi , \\pi ) \\in \\mathcal {D}_{0}(\\Sigma )$ can be extended to initial data $\\Psi _m\\in \\widetilde{\\mathcal {D}}_0(\\Sigma )$ , using the constraint equations of Theorem REF with $m>0$ and $j=0$ .",
"Note that $\\Psi _m$ depends continuously on the mass $m$ .",
"Because $\\widetilde{\\mathcal {D}}_0(\\Sigma )$ is a barrelled space (cf.",
"the proof of Lemma REF ) we may apply Lemma REF and conclude that $L_m\\Psi _{m^{\\prime }}$ is jointly continuous in $(m,m^{\\prime })$ on $\\mathbb {R}_+\\times \\mathbb {R}_+$ .",
"In particular, $m\\mapsto L_m\\Psi _m$ depends continuously on $m>0$ .",
"Consequently, the map $\\tau _m : \\mathcal {D}_{0}(\\Sigma )\\rightarrow \\mathcal {D}_0(\\Sigma ^{\\prime })$ , which propagates initial data for the Proca field of mass $m$ , is also weakly continuous in $m>0$ .",
"We now extend this result as follows.",
"For $1\\le n\\le N$ we consider the continuous linear map $T^{N,n}_m 1^{\\otimes n-1}\\otimes \\tau _m\\otimes 1^{\\otimes N-n}$ on $\\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes n}\\boxtimes (T^*\\Sigma ^{\\prime }\\oplus T^*\\Sigma ^{\\prime })^{\\boxtimes N-n}\\big )$ , which may be defined using Schwartz' Kernels Theorem.",
"One may extend the proof of Theorem REF and Corollary REF to show that $T^{N,n}_m$ is also weakly continuous in $m>0$ .",
"We then define the map $T^N_m T^{N,1}\\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ TN,2 TN, N$ which is again weakly continuous in $ m>0$, by a repeated application of the joint continuity Lemma \\ref {lem:jointcontinuity}, using the fact that each of the spaces $ 0(($T^*\\Sigma $$T^*\\Sigma $ )n($T^*\\Sigma $ '$T^*\\Sigma $ ')N-n)$ is barrelled.",
"Let us now consider the lift of $ m$ to a continuous linear map $ Tm : $\\mathcal {BU}$ ( $\\mathcal {D}_{0}(\\Sigma )$ )$\\mathcal {BU}$ ( D0('))$ between complete BU-algebras (using sections of the bundle $$T^*\\Sigma $$T^*\\Sigma $$ and its analogue on $ '$).",
"Its action on $ 0(($T^*\\Sigma $$T^*\\Sigma $ )n)$ is simply given by $ TNm$, which shows that $ Tm$ is also weakly continuous in $ m>0$.",
"We note that each $ Tm$ is a homeomorphism and that it maps the ideal $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$$ onto $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}$$.",
"This means that it also maps the closed ideal $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$$ onto $${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}$$ and it descends to a homeomorphism $ Tm$ between the quotient algebras $${\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$$ and $${\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big ( \\mathcal {D}_0(\\Sigma ^{\\prime })\\big )}}{\\; \\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma ^{\\prime }}}}}}$$.",
"The weak continuity of $ Tm$ in $ m>0$ implies the weak continuity of $ Tm$ in $ m>0$.$ The complete algebra $\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}$ is barrelled, as shown in Lemma REF , and hence so is the quotient ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ .",
"Furthermore, because the ideal ${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is a closed subspace of $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ (cf.",
"Section REF ), the quotient ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$ is a dense subspace of ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\;\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ .",
"On this subspace, $\\widetilde{T}_m$ restricts to $\\Lambda ^{(\\Sigma ^{\\prime })}_{m} \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ ( ()m )-1$.",
"Identifying\\begin{equation}\\Lambda ^{(\\Sigma ^{\\prime })}_m(O^{\\prime }_m) = \\widetilde{T}_m \\Lambda ^{(\\Sigma )}_m(O^{\\prime }_m)\\end{equation}we may therefore use the assumed continuity of $ ()m(O'm)$ in $ m>0$ and the known weak continuity of $ Tm$ together with Lemma \\ref {lem:jointcontinuity} to find that $ m(')m(O'm)$ is continuous in $ m>0$.",
"This proves the independence of the choice of $$.$ We now turn to the independence of the choice of classical solutions.",
"Let $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _m$ and $\\left\\lbrace \\varphi ^{\\prime }_{m,j} \\right\\rbrace _m$ specify continuous families of classical solutions to the inhomogeneous Proca equation and fix a Cauchy surface $\\Sigma $ .",
"We denote the initial data of $\\varphi _{m,j}$ and $\\varphi ^{\\prime }_{m,j}$ by $(\\phi _m,\\pi _m)$ and $(\\phi _m^{\\prime },\\pi _m^{\\prime })$ , respectively.",
"For each $m>0$ we now define an algebra homeomorphism $L_m$ on the BU-algebra $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ by setting stipulating that $L_m$ preserves the unit and acts on homogeneous elements of degree 1 as $L_m\\big (0,(\\alpha ,\\beta ),0,0,\\ldots \\big ) \\big (\\mathcal {G}^{(\\Sigma )}\\big ((\\phi _m-\\phi ^{\\prime }_m,\\pi _m-\\pi ^{\\prime }_m),(\\alpha ,\\beta )\\big ),(\\alpha ,\\beta ),0,0,\\ldots \\big ) \\,.$ We can extend each $L_m$ in a unique way to a homeomorphism of the completed BU-algebra $\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}$ , using Schwartz' Kernels Theorem.",
"We denote the extended operator by the same symbol $L_m$ .",
"The action of $L_m$ on a homogeneous element $\\psi ^{(N)}$ of degree $N$ , i. e. on a section $\\psi ^{(N)}\\in \\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N}\\big )$ , can be written out explicitly as a sum of terms of degrees $\\le N$ .",
"Because $\\phi _m,\\pi _m,\\phi _m^{\\prime }$ and $\\pi _m^{\\prime }$ depend continuously on $m>0$ , so does the section $(\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m)$ and also the sections $\\big ((\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m)\\big )^{\\boxtimes n}$ for eachThis may be shown by induction over $n\\ge 1$ , e. g. using the joint continuity Lemma REF and noting that the linear map $\\gamma \\mapsto ((\\phi _m-\\phi ^{\\prime }_m)\\oplus (\\pi _m-\\pi ^{\\prime }_m))\\boxtimes \\gamma $ is weakly continuous in $m>0$ for any section $\\gamma $ of any vector bundle.",
"$n\\ge 1$ .",
"It follows that the components of $L_m\\psi ^{(N)}$ also depend continuously on $m>0$ .",
"Thus we see that $L_m$ is weakly continuous in $m>0$ .",
"Note that $L_m$ preserves the ideal ${\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }$ (just as in the proof of Theorem REF ), and hence it also preserves the closed ideal $\\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}$ .",
"The $L_m$ therefore descend to homeomorphisms $\\tilde{L}_m$ of the quotient algebra ${{\\scalebox {1.2}{{\\overline{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}}{\\ \\overline{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}}$ , and the weak continuity of $L_m$ in $m>0$ implies the weak continuity of $\\tilde{L}_m$ in $m>0$ .",
"We note that $\\tilde{L}_m$ preserves the dense subalgebra ${{\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^{\\mathrm {\\,CCR},\\Sigma }}}}}$ and one may verify directly from Theorem REF and the definitions of the relevant maps that $\\tilde{L}_m$ acts on this subalgebra as $\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ 'm,j-1 'm,j m -1.",
"If $\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 (Om)$ depends continuously on $ m>0$, then so does\\begin{equation}\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}\\end{equation}{\\hbox{$\\scriptscriptstyle \\circ $}}$ 'm,j-1 (Om) = Lm m m,j-1 (Om) by the joint continuity Lemma REF ."
],
[
"The classical case",
"For fixed initial data ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ on a fixed Cauchy surface $\\Sigma $ there is a family of solutions $A_{m,j}$ to the Proca equation of mass $m>0$ with source term $j\\in \\Omega ^1_0(\\mathcal {M})$ .",
"We have seen in Theorem REF that these solutions take the form $\\langle A_{m,j} , F \\rangle _\\mathcal {M}= \\sum \\limits _\\pm \\langle j , G_m^\\mp F \\rangle _{J^\\pm (\\Sigma )} +\\langle {A_{(0)}}, \\rho _{(d)}G_m F \\rangle _\\Sigma - \\langle {A_{(d)}}, \\rho _{(0)}G_m F \\rangle _\\Sigma $ for any fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ .",
"We may think of $F$ as the mathematical representation of an experimental setup which measures the field configuration $A$ through the pairing $\\langle A,F\\rangle _{\\mathcal {M}}$ and we wish to investigate for which $F$ , if any, we can take the limit $m\\rightarrow 0$ in Equation (REF ) above for all choices of $\\Sigma $ and all initial data ${A_{(0)}},{A_{(d)}}$ .",
"Lemma 3.6 For fixed $F \\in \\Omega ^1_0(\\mathcal {M})$ , the limit $m\\rightarrow 0$ of the right-hand side of Equation (REF ) exists for all smooth space-like Cauchy surfaces $\\Sigma $ and all initial data ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ , if and only if $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"Proof: 3.5 Suppose that for a given $F \\in \\Omega ^1_0(\\mathcal {M})$ the right-hand side of Equation (REF ) converges as $m\\rightarrow 0$ for all smooth space-like Cauchy surfaces $\\Sigma $ and all ${A_{(0)}},{A_{(d)}}\\in \\Omega ^1(\\Sigma )$ .",
"Because we can vary the initial data arbitrarily and independently, all three terms in Equation (REF ) must converge separately.",
"In particular, $\\lim _{m\\rightarrow 0}\\rho _{(0)}G_mF$ must exist in a distributional sense.",
"Recall that $G_mF=m^{-2}E_md\\delta F+E_mF$ , where the second term is in $\\Omega ^1(\\mathcal {M})$ and depends continuously on $m\\in \\mathbb {R}$ by Corollary REF .",
"It then follows from the same corollary and from the continuity and linearity of $\\rho _{(0)}$ that $\\rho _{(0)}E_0d\\delta F &= \\lim _{m\\rightarrow 0} \\rho _{(0)}E_md\\delta F\\\\&= \\lim _{m\\rightarrow 0} m^2 \\rho _{(0)}\\left(G_mF-E_mF\\right)\\\\&= \\lim _{m\\rightarrow 0} m^2 \\cdot \\left(\\lim _{m\\rightarrow 0}\\rho _{(0)}G_mF - \\rho _{(0)}E_0F\\right)=0,$ where we used the existence of the limit of $\\rho _{(0)}G_mF$ .",
"Because this holds on every Cauchy surface, the one-form $E_0d\\delta F$ must annihilate every space-like vector at every point.",
"Because all tangent vectors are linear combinations of space-like vectors we conclude that $E_0d\\delta F=0$ and hence also $E_0\\delta dF=E_0(\\delta d+d\\delta )F=0$ .",
"We may then define $F^{\\prime } E_0^+\\delta dF=E_0^-\\delta dF$ and $F^{\\prime \\prime } E_0^+d\\delta F=E_0^-d\\delta F$ and note that these have compact supports.",
"Furthermore, since $\\delta $ and $d$ intertwine with $E_0^+$ on forms, $\\delta F^{\\prime }=0=dF^{\\prime \\prime }$ and $F^{\\prime } + F ^{\\prime \\prime } = E_0^+ (d \\delta + \\delta d) F = F \\,.$ Combining this formula with $G_m^\\pm =E_m^\\pm (m^{-2}d\\delta +1)$ we find $G_m^\\pm F &=E_m^\\pm F^{\\prime } + m^{-2}E_m^\\pm (d\\delta +\\delta d+m^2)F^{\\prime \\prime }\\\\&=E_m^\\pm F^{\\prime } + m^{-2}F^{\\prime \\prime }.$ Substituting this in the first term of Equation (REF ) we see that $\\sum \\limits _\\pm \\langle j , G_m^\\mp F \\rangle _{J^\\pm (\\Sigma )}= \\sum \\limits _\\pm \\langle j , E_m^\\pm F^{\\prime } \\rangle _{J^\\pm (\\Sigma )} + m^{-2} \\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}$ must converge as $m\\rightarrow 0$ .",
"The terms in the first sum converge as $m\\rightarrow 0$ by Corollary REF , and hence the last term must also converge.",
"This clearly implies $\\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}=0$ , showing that $F$ must have the stated form.",
"Conversely, when $F=F^{\\prime }+F^{\\prime \\prime }$ with $\\delta F^{\\prime }=0=dF^{\\prime \\prime }$ and $\\langle j , F^{\\prime \\prime } \\rangle _{\\mathcal {M}}=0$ , then it follows from Equation (REF ) that $G_mF=E_mF^{\\prime }$ , which has a limit as $m\\rightarrow 0$ .",
"Together with Equation (REF ) and the continuity of $\\rho _{(d)}$ and $\\rho _{(0)}$ it follows that the right-side of Equation (REF ) converges as $m\\rightarrow 0$ .",
"Note that $F^{\\prime }$ and $F^{\\prime \\prime }$ are uniquely determined by $F=F^{\\prime }+F^{\\prime \\prime }$ and $\\delta F^{\\prime }=0=dF$ , because $\\Omega ^1_{0,\\delta }(\\mathcal {M}){\\hspace{1.00006pt}\\cap \\hspace{1.00006pt}}\\Omega ^1_{0,d}(\\mathcal {M})=\\lbrace 0\\rbrace $ .",
"Indeed, if $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ satisfies $d\\tilde{F}=\\delta \\tilde{F}=0$ , then also $\\square \\tilde{F}=0$ and hence $\\tilde{F}=0$ by .",
"For a fixed $m>0$ and $j\\in \\Omega ^1(\\mathcal {M})$ there are $F\\in \\Omega ^1_0(\\mathcal {M})$ which define trivial observables in the sense that $\\langle A_{m,j},F\\rangle _{\\mathcal {M}}=0$ for all field configurations (i. e. for all initial data).",
"The following lemma characterizes them: Lemma 3.7 For fixed $m>0$ and $j\\in \\Omega ^1(\\mathcal {M})$ , $F\\in \\Omega ^1_0(\\mathcal {M})$ defines a trivial observable if and only if $F=(\\delta d+m^2) \\tilde{F}$ for some $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ with $\\langle j,\\tilde{F}\\rangle _{\\mathcal {M}}=0$ .",
"Proof: 3.6 Arguing as in the proof of Lemma REF we see that $F$ defines a trivial observable if and only if $G_mF=0$ and $\\langle j,G_m^+F\\rangle _{\\mathcal {M}}=0$ .",
"The first condition is equivalent to $F=(\\delta d+m^2)\\tilde{F}$ for some $\\tilde{F}\\in \\Omega ^1_0(\\mathcal {M})$ by Lemma REF .",
"The second condition then means that $\\langle j,\\tilde{F}\\rangle _{\\mathcal {M}}=0$ .",
"For any fixed $m$ and $j$ one would normally divide out these trivial observables, because they are redundant.",
"For our purposes, however, this is rather awkward, because the space of trivial observables depends on $m$ and $j$ .",
"However, we can remove some of the redundancy in the following way: Theorem 3.8 (Existence of the zero mass limit) Fix $j\\in \\Omega ^1(\\mathcal {M})$ .",
"For $F\\in \\Omega ^1_0(\\mathcal {M})$ , Equation (REF ) admits a massless limit for all initial data on all Cauchy surfaces if and only if there is a $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ such that $F-F^{\\prime }$ is a trivial observable for all $m>0$ .",
"Proof: 3.7 It follows from Lemma REF that $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j, F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"From Equation (REF ) we see that for all $m>0$ it holds $G_mF^{\\prime \\prime }=0$ and $\\langle j,G_m^+F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=m^{-2}\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ , so $F^{\\prime \\prime }=F-F^{\\prime }$ defines a trivial observable for all $m>0$ by Lemma REF and its proof.",
"In other words, for the massless limit it sufficesIt is unclear if there is any remaining redundancy.",
"to consider all co-closed forms $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ .",
"The meaning of this can be quite easily understood under the duality $\\langle {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} , {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}} \\rangle _\\mathcal {M}$ .",
"One finds that ${\\scalebox {1.2}{{\\mathcal {D}^1(\\mathcal {M})}{d\\mathcal {D}^{0}(\\mathcal {M})}}}$ is dual to $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ (see ).",
"Here, $\\mathcal {D}^1(\\mathcal {M})$ denotes the set of distributional one-forms (in a physical sense, these are classical vector potentials), so restricting to co-closed test one-forms is equivalent to implementing the gauge equivalence $A \\rightarrow A + d\\chi $ , for $A \\in \\mathcal {D}^1(\\mathcal {M})$ and $ \\chi \\in \\mathcal {D}^0(\\mathcal {M})$ in the theory.",
"This dual relation is easily checked for $A^{\\prime } = A + d\\chi $ and $F \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ $\\langle A^{\\prime }, F \\rangle _{\\mathcal {M}}&= \\langle A, F \\rangle _{\\mathcal {M}} + \\langle d\\chi , F \\rangle _{\\mathcal {M}} \\\\&= \\langle A, F \\rangle _{\\mathcal {M}} + \\langle \\chi , \\delta F \\rangle _{\\mathcal {M}} = \\langle A, F \\rangle _{\\mathcal {M}} \\,.$ This is a nice result, because it elucidates the gauge equivalence in the Maxwell theory.",
"Note that it is a priori unclear how to implement the gauge equivalence in Maxwell's theory on curved spacetimes due to the non-trivial topology.",
"Maxwell's equation $\\delta d A = 0$ suggests that two solutions that differ by a closed one-form give rise to the same configuration, but one can argue that only exact one-forms should be treated as pure gauge solutions, because the Aharonov-Bohm effect does distinguish between configurations that differ by a form that is closed but not exact .",
"It is gratifying to see that we arrive at a gauge equivalence given by the class of exact forms, simply by keeping the set of linear observables as large as possible in the limit, i. e. $\\Omega ^1_{0,\\delta }(\\mathcal {M})$ .",
"Hence, we have already captured one important feature of the Maxwell theory in the massless limit of the Proca theory!",
"It remains to check whether also the dynamics are well behaved in the massless limit."
],
[
"Dynamics and the zero mass limit",
"In the massless limit one may hope to find a vector potential $A_{0,j}$ satisfying Maxwell's equations $\\delta dA_{0,j}=j$ at least in a distributional sense, i. e. $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}=\\langle j , \\delta d F \\rangle _\\mathcal {M}$ for every test one-form $F\\in \\Omega ^1_0(\\mathcal {M})$ .",
"Note that $\\delta dF$ is co-closed, so by Theorem REF we may substitute $\\tilde{F}=\\delta dF$ in the limit $\\langle A_{0,j} , \\tilde{F} \\rangle _\\mathcal {M}&\\lim \\limits _{m \\rightarrow 0} \\langle A_{m,j}, \\tilde{F} \\rangle _\\mathcal {M}\\\\&= \\lim \\limits _{m \\rightarrow 0}\\Big ( \\sum \\limits _\\pm \\langle j , G_m^\\mp \\tilde{F} \\rangle _{J^{\\pm }(\\Sigma )} +\\langle {A_{(0)}}, \\rho _{(d)}G_m \\tilde{F} \\rangle _\\Sigma - \\langle {A_{(d)}}, \\rho _{(0)}G_m \\tilde{F} \\rangle _\\Sigma \\Big ) \\,$ for any given initial data ${A_{(0)}},{A_{(d)}}$ on any Cauchy surface $\\Sigma $ .",
"However, using $\\lim _{m\\rightarrow 0}G_m^{\\pm }\\delta dF&=\\lim _{m\\rightarrow 0}E_m^{\\pm }\\delta dF= E_0^{\\pm }\\delta dF\\\\&=F - E_0^{\\pm }d\\delta F$ we only find $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}&=\\sum \\limits _\\pm \\langle j , F-E_0^{\\mp }d \\delta F\\rangle _{J^\\pm (\\Sigma )} - \\langle {A_{(0)}}, \\rho _{(d)}E_0 d\\delta F \\rangle _\\Sigma + \\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma \\\\&=\\langle j , F\\rangle _{\\mathcal {M}}- \\sum \\limits _\\pm \\langle j , E_0^{\\mp }d \\delta F\\rangle _{J^\\pm (\\Sigma )}+ \\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma ,\\,$ where we used the fact that $\\rho _{(d)}E_m d\\delta F = - *_{(\\Sigma )}i^* * d E_m d \\delta F = 0$ since $d$ and $E_m$ commute.",
"The second term in Equation (REF ) will not vanish in general (e. g. when $dF=0$ but $\\langle j,F\\rangle _{\\mathcal {M}}\\ne 0$ ).",
"Ergo, the fields $A_{0,j}$ defined as the zero mass limit of the Proca field $A_{m,j}$ will not fulfill Maxwell's equation in a distributional sense.",
"While this might seem surprising at first, it is quite easy to understand when we recall how we have found solutions to Proca's equation, using the massive wave equation (REF ) combined with constraint equations on the initial data to ensure that the Lorenz constraint () is fulfilled.",
"Similarly, one solves Maxwell's equation by specifying a solution to the massless wave equation $(\\delta d + d \\delta )A_{0,j} = j$ and restricting the initial data such that the Lorenz constraint $\\delta A_{0,j} = 0$ is fulfilled.",
"The problem in the massless limit lies with the constraints.",
"Recall from Theorem REF that, in order to implement the Lorenz constraint, we have restricted the initial data by ${A_{(\\delta )}}= m^{-2}\\rho _{(\\delta )}j \\,, \\quad \\textrm {and} \\quad {A_{(n)}}= m^{-2}\\left( \\rho _{(n)}j + \\delta _{(\\Sigma )} {A_{(d)}}\\right) \\,.$ It is obvious that, in general, the resulting ${A_{(\\delta )}}$ and ${A_{(n)}}$ diverge in the zero mass limit, so there is no corresponding solution to Maxwell's equations with the same initial data.",
"In order to keep the dynamics in the zero mass limit, we need to make sure that the constraints are well behaved in the limit.",
"Since we do not want the external source or the initial data to be dependent of the mass, we have to require that ${A_{(\\delta )}}$ and ${A_{(n)}}$ vanish, i. e. we need to specifyThe first equation follows from $\\rho _{(\\delta )}j=0$ on all Cauchy surfaces.",
"$\\delta j &= 0 \\,, \\quad \\textrm {and} \\\\\\delta _{(\\Sigma )} {A_{(d)}}&= -\\rho _{(n)}j \\,.$ This corresponds exactly to the constraints on the initial data for the Maxwell equation which implement the Lorenz gauge (cf.",
"Pfenning ).",
"With these constraints, we can now look at the remaining term of $\\langle A_{0,j} , \\delta d F\\rangle _\\mathcal {M}$ in Equation (REF ).",
"We do this separately for the two summands.",
"Using that $d$ commutes with pullbacks and inserting the constraints on the initial data, we find $\\langle {A_{(d)}}, \\rho _{(0)}E_0 d\\delta F \\rangle _\\Sigma &= \\langle {A_{(d)}}, d_{(\\Sigma )} \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma \\\\&= \\langle \\delta _{(\\Sigma )}{A_{(d)}}, \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma \\\\&= -\\langle \\rho _{(n)}j , \\rho _{(0)}E_0 \\delta F \\rangle _\\Sigma $ For the first summand $\\sum _\\pm \\langle j , E_0^\\mp d\\delta F \\rangle _{\\Sigma ^\\pm }$ we use the partial integration in Equation (REF ) in the proof of Theorem REF and find, using $m=0$ and the constraint $\\delta j = 0$ as specified above, $\\sum _\\pm \\langle j , E_0^\\mp d\\delta F \\rangle _{J^\\pm (\\Sigma )}&= \\sum _\\pm \\langle d \\delta j , E_0^\\mp F \\rangle _{J^\\pm (\\Sigma )} +\\langle \\rho _{(\\delta )}E_0F,\\rho _{(n)}j\\rangle _\\Sigma - \\langle \\rho _{(\\delta )}j,\\rho _{(n)}E_0F\\rangle _\\Sigma \\\\&= \\langle \\rho _{(0)}E_0\\delta F,\\rho _{(n)}j\\rangle _\\Sigma \\,.$ Using the symmetry of the inner product $\\langle {\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}},{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}\\rangle _{\\mathcal {M}}$ we find that the remaining terms of Equation (REF ) cancel when restricting the initial data such that they are well defined in the zero mass limit.",
"We therefore obtain the correct dynamics in that case: $\\langle A_{0,j} , \\delta d F \\rangle _\\mathcal {M}&= \\langle j , F \\rangle _\\mathcal {M}- \\lim \\limits _{m \\rightarrow 0}\\Big (\\sum \\limits _\\pm \\langle j , E_m^\\mp d\\delta F \\rangle _{J^\\pm (\\Sigma )}- \\langle {A_{(d)}}, \\rho _{(0)}E_m d\\delta F \\rangle _\\Sigma \\Big ) \\\\&= \\langle j , F \\rangle _\\mathcal {M}\\,.$ In combination with Theorem REF we have thus shown Theorem 3.9 (The zero mass limit of the Proca field) Let $F\\in \\Omega ^1_0(\\mathcal {M})$ be a test one-form and $j \\in \\Omega ^1(\\mathcal {M})$ an external current.",
"Let $A_{m,j}$ be the solution to Proca's equation specified by initial data ${A_{(0)}}, {A_{(d)}}\\in \\Omega ^1_0(\\Sigma )$ via Theorem REF .",
"Defining the zero mass limit $\\langle A_{0,j} , F \\rangle _\\mathcal {M}= \\lim _{m \\rightarrow 0} \\langle A_{m,j}, F \\rangle _\\mathcal {M}$ of the Proca field, the following holds: The limit exists if and only if $F$ is equivalent to an observable $F^{\\prime }$ (for all $m>0$ ) with $\\delta F^{\\prime } = 0$ , effectively implementing the gauge equivalence of the Maxwell theory.",
"The field $A_{0,j}$ is a Maxwell field, that is, it solves Maxwell's equation, if and only if the current is conserved, $\\delta j = 0$ , and $\\rho _{(n)}j = - \\delta _{(\\Sigma )} {A_{(d)}}$ , implementing the Lorenz gauge.",
"Note that the conservation of the external current $\\delta j = 0$ is not required to solve Proca's equation, but it is necessary to solve Maxwell's equations ($\\delta dA=j$ entails $\\delta j=0$ ).",
"It is therefore not surprising that this condition is also necessary to recover the dynamics in the zero mass limit.",
"In analogy to the quantum theory, we may think of the field configuration $A$ as a state, whereas $F$ is an observable.",
"We then see from the theorem that the limits of observables give rise to the gauge equivalence of the classical vector potential, but additional conditions on the limits of states and external currents are needed in order to recover Maxwell's equation."
],
[
"The quantum case",
"In the quantum case we define the observables in the zero mass limit as follows: Definition 3.10 (Zero mass limit theory) For any fixed $j \\in \\Omega ^1(\\mathcal {M})$ and $O\\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ we say that $[O]_{m,j}\\in {A}_{m,j}$ has a zero mass limit if and only if $\\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) ([O]m,j) exists for all Cauchy surfaces $\\Sigma $ and all families $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ of classical solutions to the inhomogeneous equation $(\\delta d + m^2) \\varphi _{m,j} = j$ which depend continuously on $m$ .",
"Here, $\\Lambda _{m}$ and $\\Psi _{\\varphi _{m,j}}$ are as defined in Section REF and REF .",
"We call the zero mass limit trivial if and only if the above limit vanishes for all Cauchy surfaces $\\Sigma $ and all families $\\lbrace \\varphi _{m,j}\\rbrace _{m\\ge 0}$ .",
"If the zero mass limit exists, we denote its equivalence class modulo trivial observables by $[O]_{0,j}$ .",
"Note that we included $m=0$ in the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"This is done for the following reason.",
"Even when $j=0$ we may choose a non-trivial family $\\left\\lbrace \\varphi _{m,0} \\right\\rbrace _{m\\ge 0}$ and due to the isomorphism $\\Psi _{\\varphi _{m,0}}^{-1}$ we are then considering quantum fluctuations around the classical solutions $\\varphi _{m,0}$ .",
"If the quantum field is to converge, it seems reasonable to require that the classical background field $\\varphi _{m,0}$ also converges.",
"For general sources this implies that $\\varphi _{0,j}$ satisfies Maxwell's equations and hence the current must be conserved, $\\delta j=0$ .",
"We can think of the zero mass limit of an operator $O$ as a family of operators in the algebras ${{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , indexed by $\\Sigma $ and by the family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"Using the properties of the topological algebras ${{\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}}$ , it is not hard to see that the operators $O\\in {\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ which have a zero mass limit form a $^*$ -subalgebra of ${\\mathcal {BU}\\big (\\Omega ^1_0(\\mathcal {M})\\big )}$ in which the operators with a trivial zero mass limit form an ideal.",
"We are interested in the quotient algebra which we denote by ${A}_{0,j}$ and which is generated by $\\mathbb {1}$ and by homogeneous degree-one elements, which we denote by $\\mathcal {A}_{0,j}(F)$ .",
"These are the massless field operators and we can think of them as the massless limits of the field operators $\\mathcal {A}_{m,j}(F)$ .",
"Our next theorem focuses on these field operators.",
"As our main result we determine for which $F\\in \\Omega ^1_0(\\mathcal {M})$ the limit $\\mathcal {A}_{0,j}(F)$ exists.",
"Theorem 3.11 (Existence of the zero mass limit) For given $j\\in \\Omega ^1_{\\delta }(\\mathcal {M})$ , $\\mathcal {A}_{m,j}(F)$ has a zero mass limit $\\mathcal {A}_{0,j}(F)$ if and only if $F\\in \\Omega ^1_0(\\mathcal {M})$ is of the form $F=F^{\\prime }+F^{\\prime \\prime }$ with $F^{\\prime }\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $F^{\\prime \\prime }\\in \\Omega ^1_{0,d}(\\mathcal {M})$ such that $\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ .",
"The zero mass limit is trivial when $F^{\\prime }=0$ .",
"Proof: 3.8 Note that $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F)) =[(m,j,F$\\mathcal {M}$ ,mF,0,0,...)]CCR .",
"Just as in the last paragraph of the proof of Lemma REF we see that all $F$ of the stated form have a limit $\\lim _{m\\rightarrow 0}G_mF=\\lim _{m\\rightarrow 0}E_mF$ and hence the limit of the initial data $\\lim _{m\\rightarrow 0}\\kappa _mF$ exists on every Cauchy surface.",
"By assumption on the $\\varphi _{m,j}$ , $\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}}$ also has a limit as $m\\rightarrow 0$ .",
"Because $[{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}]_\\sim ^\\text{CCR}$ is continuous and independent of $m$ we see that $\\lim _{m\\rightarrow 0}\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F))$ exists for all $ F$ of the stated form.$ When $F^{\\prime }=0$ , then $F=F^{\\prime \\prime }$ and $G_mF=0$ (cf.",
"the proof of Theorem REF ) and hence $\\kappa _mF$ on every Cauchy surface.",
"Furthermore, $\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}}=m^{-2}\\langle j,F^{\\prime \\prime }\\rangle _{\\mathcal {M}}=0$ by Theorem REF and Equation (REF ).",
"Thus the zero mass limit is trivial.",
"Assume that $\\lim _{m\\rightarrow 0}\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(F))$ exists.",
"This means that for each Cauchy surface $$ there is a family of elements $ gm${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$$ such that $ m0 (m,j,F$\\mathcal {M}$ ,mF,0,...)+gm$ exists in $$\\mathcal {BU}$ ($\\mathcal {D}_{0}(\\Sigma )$ )$.",
"Using the projection $ S$ of Lemma \\ref {lem:symmetrization-of-fields}, we have{\\begin{@align}{1}{-1}S\\big (\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )&=\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )\\\\&=S\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )\\,,\\end{@align}}because $ (m,j,F$\\mathcal {M}$ ,mF,0,...)$ is homogeneous of degree 1 and hence symmetric.",
"The continuity of $ S$ then implies that{\\begin{@align}{1}{-1}S\\big (\\lim _{m\\rightarrow 0} \\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )&=\\lim _{m\\rightarrow 0} S\\big (\\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big )+g_m\\big )\\\\&=\\lim _{m\\rightarrow 0} \\big (\\langle \\varphi _{m,j},F\\rangle _{\\mathcal {M}},\\kappa _mF,0,0, \\ldots \\big ),\\end{@align}}exists.",
"This implies that both $ m0m,j,F$\\mathcal {M}$$ and $ m0mF$ exist.",
"The first of these conditions already follows from the assumptions on $ m,j$ but the second implies in particular that $ m0$\\rho _{(0)}$ GmF$ exists.",
"Because this is required for every Cauchy surface, the argument presented in the proof of Lemma \\ref {lem:limit_existence_classical_equivalence} shows that $ F$ must be of the stated form.$ As in the classical case we find that the algebra ${A}_{0,j}$ of the massless limit is generated by field operators $\\mathcal {A}_{0,j}(F)$ with $F\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ ranging over the co-closed test one-forms.",
"Just as in the classical case, discussed in Section REF , this implements the gauge equivalence of the Maxwell theory, using the choice of gauge equivalence of .",
"Hence also in the quantum case, the limit exists only if we implement the gauge beforehand.",
"We now turn to the algebraic relations in ${A}_{0,j}$ .",
"For this we view $[O]_{0,j}$ as an equivalence class of a family of limits $\\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ m,j-1 ) ([O]m,j)$ in the algebras $${\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$$, indexed by the Cauchy surface $$ and the family $ { m,j }m0$ and we set in particular $$\\mathcal {A}$ 0,j(F) [(0,F,0,...)]0,j$.",
"Exploiting the algebraic structure of the algebras $${\\scalebox {1.2}{{\\mathcal {BU}\\big ( \\mathcal {D}_{0}(\\Sigma )\\big ) }{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$$ we then find in a natural way\\footnote {This means that the relations below hold for the corresponding limits \\lim _{m\\rightarrow 0}\\big ( \\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}}{\\hbox{$\\scriptscriptstyle \\circ $}}$ m,j-1 ) ([O]m,j)$ for each Cauchy surface and for each family of classical solutions $ { m,j }m0$.$ that $\\mathcal {A}_{0,j}(\\alpha F + \\beta F^{\\prime }) &= \\alpha \\mathcal {A}_{0,j}(F) + \\beta \\mathcal {A}_{0,j}(F^{\\prime }) \\\\\\mathcal {A}_{0,j}(F)^* &= \\mathcal {A}_{0,j}(\\mathchoice{{\\m@th \\displaystyle F}nullfont\\hspace{0.0pt}\\hspace{2.0pt} \\overline{\\hspace{-1.66656pt}\\hspace{0.0pt}\\box \\hspace{0.0pt}\\hspace{-0.55542pt}}\\hspace{0.0pt}}{}{}{}$ nullfont $\\m@th \\textstyle F$ nullfont $\\m@th \\scriptstyle F$ nullfont $\\m@th \\scriptscriptstyle F$ ) for all $F \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ and $\\alpha , \\beta \\in \\mathbb {C}$ , corresponding to the linearity and the hermitian field property.",
"For the canonical commutation relations we note that for all $F,F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ , $G_mF^{\\prime }=E_mF^{\\prime }$ and hence $\\big [ \\mathcal {A}_{0,j}(F) ,\\mathcal {A}_{0,j}(F^{\\prime }) \\big ]&= \\lim \\limits _{m \\rightarrow 0} \\big [ \\mathcal {A}_{m,j}(F) ,\\mathcal {A}_{m,j}(F^{\\prime }) \\big ] \\\\&=\\mathrm {i}\\cdot \\lim \\limits _{m \\rightarrow 0} {F}{F^{\\prime }}\\cdot \\mathbb {1} \\\\&=\\mathrm {i}\\cdot \\lim \\limits _{m \\rightarrow 0} \\langle F,E_F^{\\prime }\\rangle _{\\mathcal {M}}\\\\&= \\mathrm {i}\\, {F}{F^{\\prime }}\\cdot \\mathbb {1}\\,.$ For co-closed test one-forms $F \\in \\Omega ^1_{0,\\delta }$ , the fundamental solutions $E^\\pm _0$ of the massless Klein-Gordon operator are actually also fundamental solutions to Maxwell's equation, i. e. it holds $E_0^\\pm \\delta d F = E_0^\\pm (\\delta d + d \\delta ) F = F$ , so we find that the fields in the zero mass limit are subject to the correct canonical commutation relations.",
"Indeed, using $\\rho _{(\\delta )}E_0 F^{\\prime } = i^* \\delta E_0 F^{\\prime } = i^* E_0 \\delta F^{\\prime } = 0$ and the analogous expression for $F$ , we may rewrite commutator in terms of initial data as ${F}{F^{\\prime }}&=\\langle F, E_0 F^{\\prime } \\rangle _\\mathcal {M}= -\\langle E_0 F , F^{\\prime } \\rangle _\\mathcal {M}\\\\&= \\langle \\rho _{(0)}E_0 F , \\rho _{(d)}E_0 F^{\\prime } \\rangle _\\Sigma - \\langle \\rho _{(d)}E_0 F , \\rho _{(0)}E_0 F^{\\prime } \\rangle _\\Sigma $ in analogy to Equation (REF ).",
"Note that ${F}{F^{\\prime }}$ for $F,F^{\\prime } \\in \\Omega ^1_{0,\\delta }(\\mathcal {M})$ is in general degenerate, hence the quantum field theory associated with $\\mathcal {A}_{0,j}$ will in general fail to be local in the sense of Definition REF .",
"However, this is perfectly in line with the free vector potential as presented in .",
"It remains to verify whether $\\mathcal {A}_{0,j}$ solves Maxwell's equation, i. e. if $\\mathcal {A}_{0,j}(\\delta d F) = \\langle j , F\\rangle _\\mathcal {M}$ holds for all $F\\in \\Omega ^1_0(\\mathcal {M})$ .",
"Because $\\delta d F$ is co-closed, the limit $\\mathcal {A}_{0,j}(\\delta dF)$ is well defined.",
"For any Cauchy surface and any family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ we have $\\big (\\Lambda _m \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ -1m,j)($\\mathcal {A}$ m,j(dF)) =[(m,j,dF$\\mathcal {M}$ ,mdF,0,0, ...)]CCR =[(dm,j,F$\\mathcal {M}$ ,mdF,0,0, ...)]CCR =j,F$\\mathcal {M}$ 1 + [(0,mdF,0,0, ...)]CCR, which is independent of $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ .",
"This essentially means that it suffices to consider the source free case, because the second term in Equation (REF ) is $\\Lambda _m\\big (\\mathcal {A}_{m,0}(\\delta dF)\\big )$ .",
"Because $G_m\\delta dF=E_m\\delta dF$ converges to $E_0\\delta dF$ we have $\\lim _{m\\rightarrow 0}\\kappa _m\\delta dF=\\big (\\rho _{(0)}E_0\\delta dF,\\rho _{(d)}E_0\\delta dF\\big )=\\big (\\rho _{(0)}E_0\\delta dF,0\\big ),$ where we have used that $E_0\\delta dF=-E_0d\\delta F$ is closed and hence $\\rho _{(d)}E_0\\delta dF=\\rho _{(n)}dE_0\\delta dF=0$ .",
"To recover Maxwell's equation, we need to verify that the second term in Equation (REF ) vanishes in the limit $m\\rightarrow 0$ for any Cauchy surface.",
"However, this fails in general.",
"Indeed, if $B\\in \\Omega ^1(\\mathcal {M})$ is the solution of the wave equation $\\Box B=0$ with initial data $\\rho _{(0)}B=\\rho _{(d)}B=\\rho _{(n)}B=0$ and $\\rho _{(\\delta )}B\\in \\Omega ^0_0(\\mathcal {M})$ not constant, then $B=E_0F$ for some compactly supported $F\\in \\Omega ^1_0(\\mathcal {M})$ (cf.",
"the proof of Lemma REF ).",
"However, $E_0\\delta dF=-E_0d\\delta F=-d\\delta E_0F=-d\\delta B$ does not vanish, because $\\delta B\\in \\Omega ^0(\\mathcal {M})$ is a function which is not constant.",
"In particular, because $d$ commutes with pull-backs, $\\rho _{(0)}E_0\\delta dF=-d_{\\Sigma }\\rho _{(\\delta )}B\\lnot \\equiv 0$ because $\\rho _{(\\delta )}B$ is not a constant function.",
"Conversely, following the proof of Theorem REF and Lemma REF we see that the limit only vanishes for all Cauchy surfaces if $E_0\\delta dF=0$ , which means that $F\\in \\Omega ^1_{0,\\delta }(\\mathcal {M})+\\Omega ^1_{0,d}(\\mathcal {M})$ .",
"We have encountered a similar situation in the investigation of the classical theory in Section REF (cf.",
"Equation (REF )).",
"There we could get rid of similar remaining terms by restricting the initial data of the field configuration (i. e. of the state of the system) such that the Lorenz constraint is well behaved in the limit.",
"In the quantum scenario, our definition of the massless limit already requires $\\delta j=0$ , but the remaining constraint equation has not been imposed.",
"Indeed, in our present setting, which focuses on observables, the Lorenz constraint does not appear directly at all.",
"Nevertheless, we may impose the desired dynamics in a consistent way by dividing out a corresponding ideal.",
"Note in particular that the limit algebra is not simple, because the skew-symmetric form in Equation (REF ) is degenerate: $\\langle F,E_0\\delta dF^{\\prime }\\rangle _{\\mathcal {M}}=0$ when $\\delta F=0$ .",
"It follows that the operators $\\mathcal {A}_{0,j}(\\delta dF)-\\langle j,F\\rangle _{\\mathcal {M}}\\mathbb {1}$ commute with all other operators in the algebra ${A}_{0,0}$ and they therefore generate a two-sided ideal.",
"In the source free case this ideal is generated by the operators $\\mathcal {A}_{0,j}(\\delta dF)$ , which correspond to $[(0,\\kappa _m\\delta dF,0,\\ldots )]_\\sim ^{\\text{CCR}}$ with $\\kappa _m\\delta dF=(\\rho _{(0)}E_0\\delta dF,0)$ .",
"It is interesting to note that $A_F E_0 \\delta dF$ is a space-like compact solution to the source free Maxwell equation, $\\delta d A_F = -\\delta d E_0 d \\delta F = 0$ , and that it is of the form $A_F=d\\chi $ with the space-like compact function $\\chi -E_0\\delta F$ .",
"Solutions of the form $A_F$ can also be characterized in terms of their initial data, $\\big (\\rho _{(0)}A_F, \\rho _{(d)}A_F\\big ) = \\big (-d_{(\\Sigma )} \\rho _{(0)}\\chi , 0\\big ) \\,.$ Under the correspondence $F\\mapsto E_0F$ of observables (with $\\delta F=0$ ) and space-like compact solutions to Maxwell's equation, the observables $\\delta dF$ therefore generate a subspace that looks like a kind of pure gauge solutions (see for example or ).",
"However, the kind of “gauge equivalence” on the level of the observables, rather than the fields, does not seem to come out of the limiting procedure naturally.",
"It seems plausible that one can recover the correct dynamics by including states in the investigation and formulating conditions on their limiting behaviour, which essentially require that the remaining constraint equations is well behaved in the limit.",
"It is unclear if our limiting procedure can also be improved to directly recover the dynamics without considering states.",
"One idea is to consider the homeomorphisms that propagate the algebras of initial data ${\\scalebox {1.2}{{\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )}{{\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}}}}$ from one Cauchy surface to another.",
"If one can formulate a condition that ensures that these homeomorphisms remain well behaved in the limit, then the resulting limits should have a well behaved time evolution.",
"It would be of interest to develop these ideas and to compare the results with the massless limit of Stueckelberg's theory, which preserves the gauge invariance at all masses at the cost of introducing a coupling to an additional scalar field and all the associated additional complications .",
"We leave the investigation of these worthwhile questions to the future."
],
[
"Conclusion and Outlook",
"We have studied the classical and quantum Proca field in curved spacetimes, using a general setting including external sources and without restrictive assumptions on the spacetime topology.",
"We have shown that the quantum theory is locally covariant in the sense of , where the injectivity of the morphisms is related to the non-degeneracy of the symplectic form.",
"We have shown that the theory depends continuously on the mass $m>0$ , in a way which we have defined.",
"Using specific BU-algebra homeomorphism we mapped families of smeared Proca fields at different masses, initially elements in different BU-algebras, into the BU-algebra of initial data.",
"The topology of the latter algebra then determines a notion of continuity for the family of operators.",
"For $m>0$ we showed that this notion of continuity is independent of the choice of Cauchy surface and of the classical inhomogeneous solutions $\\varphi _{m,j}$ appearing in the homeomorphisms.",
"This result relied crucially on the use of energy estimates.",
"Note that a $C^*$ -Weyl algebra approach is ill-suited for the investigation of the zero mass limit, as one of us has argued in .",
"For the quantum theory we defined the zero mass limit by requiring a continuous family of observables to converge on every Cauchy surface and for every continuous family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ of inhomogeneous classical solutions.",
"(For the classical theory we considered a somewhat simplified setting.)",
"Investigating the zero mass limit we found in both cases that the limit exists and the theory is generated by the class of observables described by co-closed test one-forms.",
"This effectively implements a gauge invariance on the (distributional) solutions to Proca's equation by exact (distributional) one-forms.",
"This is of interest, because in general curved spacetimes the spacetime topology allows different possible choices of gauge invariance (using e. g. closed forms instead).",
"Our limiting procedure naturally leads to the same gauge invariance that was advocated in , using the independent argument that it can account for phenomena such as the Aharonov-Bohm effect and Gauss' law.",
"In the zero mass limit we also find that the quantum fields fulfill the basic properties of linearity, the hermitian field property and the correct CCR, all in line with the massless vector potential of electrodynamics.",
"However, we do not automatically recover the expected Maxwell dynamics.",
"In the classical case, this is caused by a potential divergence in the constraint equations on the initial data of field configurations.",
"This may be avoided by requiring the external source to be conserved, $\\delta j=0$ , and by requiring that the initial data of the configuration also satisfy the constraint equations of Maxwell's theory as given e. g. by Pfenning .",
"In the quantum case we did not clarify if Maxwell's equation can be obtained in the zero mass limit, e. g. by imposing additional conditions on the limits of observables or on states, or by requiring the homeomorphisms that propagate initial data between different Cauchy surfaces to remain well defined in the massless limit.",
"The further development of these ideas might require a detailed investigation of Hadamard states, which is also if interest in its own right.",
"So far these states seem to have been considered only in a restricted class of spacetimes .",
"Furthermore, it would be interesting to make a detailed comparison of our massless limit and the massless limit of Stueckelberg's theory as presented e. g. in .",
"We leave the investigation of these worthwhile questions to the future.",
"Acknowledgements We would like to thank the University of Leipzig, where this research was carried out, and MS would like to thank Prof. Stefan Hollands for helpful comments and discussions.",
"Large parts of this work are adapted from the MSc thesis of MS."
],
[
"Additional Lemmas",
"Let $\\mathfrak {X}$ be a complex vector bundle over a smooth differential manifold $\\mathcal {N}$ .",
"As in Section REF we may define the complete BU-algebra $\\overline{\\mathcal {BU}(\\Gamma _0(\\mathfrak {X}))}$ over $\\Gamma _0(\\mathfrak {X})$ as the direct sum $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )} = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big ) \\,,$ using the outer tensor product of vector bundles (see ).",
"We endow this algebra with the inductive limit topology of the subspaces $\\mathcal {BU}_N = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^N \\Gamma _0(\\mathfrak {X}^{\\boxtimes n}) \\,.$ Note that $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is the completion of the BU-algebra $\\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0(\\mathfrak {X})^{\\otimes n}$ .",
"Lemma 5.1 The complete Borchers-Uhlmann algebra $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is barrelled.",
"Proof: 5.1 The spaces $\\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big )$ of compactly supported sections of a complex vector bundle are LF-spaces, as they are defined as the inductive limit of the Frechét spaces of sections with support in some compact $K_l$ where $\\left\\lbrace K_l \\right\\rbrace _l$ is a fundamental sequence of compact $K_l \\subset \\mathcal {N}$ (see ).",
"Since LF-spaces are barrelled and the direct sum of barrelled spaces is again barrelled , we find for any $N \\in \\mathbb {N}$ that $\\mathcal {BU}_N$ is barrelled.",
"Additionally, the inductive limit of barrelled spaces is barrelled , hence the complete BU-algebra over smooth compactly supported sections $\\Gamma _0(\\mathfrak {X})$ over a complex vector bundle $\\mathfrak {X}$ is barrelled.",
"We will use barrelled spaces in order to apply the following result: Lemma 5.2 Let $X$ be a barrelled locally convex space, let $\\eta :[c,d]\\rightarrow X$ be a continuous map on a closed interval and let $L_m:X\\rightarrow Y$ be a family of continuous linear maps into a locally convex space $Y$ indexed by $m\\in [a,b]$ .",
"If the map $m\\mapsto L_m$ is weakly continuous, i. e. if $m\\mapsto L_mx$ is continuous on $[a,b]$ for each $x\\in X$ , then the map $(m,m^{\\prime })\\mapsto L_m\\eta (m^{\\prime })$ is continuous on $[a,b]\\times [c,d]$ .",
"Proof: 5.2 The weak continuity of $m\\mapsto L_m$ implies that for each $x\\in X$ the image of $m\\mapsto L_mx$ is compact.",
"The family of maps $L_m$ is therefore pointwise bounded.",
"Because $X$ is barrelled we may apply the uniform boundedness principle to find that the maps $L_m$ are equicontinuous.",
"For any $(m_0,m^{\\prime }_0)\\in [a,b]\\times [c,d]$ we set $x \\eta (m^{\\prime }_0)$ and we pick an arbitrary convex open neighbourhood $y+V$ of $y L_{m_0}x$ , where $V$ is an open neighbourhood of 0.",
"By equicontinuity there is an open neighbourhood $U\\subset X$ of 0 such that $L_m(U)\\subset \\frac{1}{2}V$ for all $m\\in [a,b]$ .",
"As $\\eta $ is continuous there is an open neighbourhood $W^{\\prime }\\subset [c,d]$ of $m^{\\prime }_0$ such that $\\eta (W^{\\prime })\\subset x+U$ .",
"Similarly there is an open neighbourhood $W\\subset [a,b]$ of $m_0$ such that $L_mx-y\\in \\frac{1}{2}V$ for all $m\\in W$ .",
"It follows that for all $(m,m^{\\prime })\\in W\\times W^{\\prime }$ $L_m\\eta (m^{\\prime })-y = L_m(\\eta (m^{\\prime })-x) + (L_mx-y)\\in \\frac{1}{2}V+\\frac{1}{2}V\\subset V$ which proves the desired continuity.",
"For our next lemma we will call an element of $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ symmetric if and only if it is totally symmetric in each degree.",
"Lemma 5.3 (Symmetrization of fields) Let $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ denote the linear subspace of the Borchers-Uhlmann algebra of initial data $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ consisting of symmetric elements.",
"Then there is a unique continuous linear surjective projection $S:\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\rightarrow \\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ whose kernel is $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ as defined in Section REF .",
"Proof: 5.3 For each $N\\ge 1$ and each permutation $\\sigma $ of the set $\\lbrace 1,\\ldots ,N\\rbrace $ we introduce the permutation operator $P^{(N)}_{\\sigma }:\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ defined by $\\left(P^{(N)}_{\\sigma }f\\right)\\big (p_1,\\ldots ,p_N\\big ) f\\big (p_{\\sigma (1)},\\ldots ,p_{\\sigma (N)}\\big ) \\,,$ where we view elements of $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ as sections in $\\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N}\\big )$ .",
"The symmetric tensor product $\\big (\\mathcal {D}_{0}(\\Sigma )\\big )^{\\otimes _S N}$ is then the range space of the projection $P^{(N)} \\frac{1}{N!",
"}\\sum \\limits _{\\sigma }P^{(N)}_{\\sigma }.$ Note that each $P^{(N)}_{\\sigma }$ is continuous, because the topology of $\\Gamma _0\\big ( (T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N} \\big ) $ is invariant under the swapping of variables.",
"It follows that $P^{(N)}$ is a continuous surjection.",
"We will first argue that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"For this we note that each $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is of the form $f = \\sum _{i=1}^{k} h_i \\cdot \\big ( -\\mathrm {i}\\mathcal {G}_m(\\psi _i, \\psi ^{\\prime }_i) , 0 , \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i , 0 , 0 ,\\dots \\big ) \\cdot \\tilde{h}_i$ for some $k \\in \\mathbb {N}$ , $h_i, \\tilde{h}_i \\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ and $\\psi _i, \\psi ^{\\prime }_i \\in \\mathcal {D}_{0}(\\Sigma )$ , where we have used the shorthand notation ${\\psi _i}{ \\psi ^{\\prime }_i} = \\langle \\pi _i , \\varphi ^{\\prime }_i \\rangle _\\Sigma - \\langle \\varphi _i , \\pi ^{\\prime }_i \\rangle _\\Sigma $ for $\\psi _i = (\\varphi _i , \\pi _i)$ .",
"If $f\\ne 0$ then its highest degree part is of some degree $N\\ge 2$ and we can write it explicitly, using the above representation, as $f^{(N)} = \\sum _{i=1}^{k} h_i^{(N_i)}\\, \\big ( \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i \\big )\\, \\tilde{h}_i^{(N-2-N_i)} \\,,$ where $h_i^{(N_i)}$ is the highest degree part of $h_i$ and $\\tilde{h}_i^{(N-2-N_i)}$ is either the highest degree part of $\\tilde{h}_i$ or 0.",
"It follows by inspection that $P^{(N)}f^{(N)}=0$ .",
"Now, if $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is non-zero and symmetric and if $f^{(N)}$ is its highest degree part, then $f^{(N)}=P^{(N)}f^{(N)}=0$ , contradicting that $f^{(N)}$ is the highest degree part.",
"It follows that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"We now construct for each degree $N\\ge 2$ two continuous linear maps $\\alpha ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\,,\\\\\\beta ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\,,$ (for $N=2$ we use $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes (N-2)}=\\mathbb {C}$ ) such that $f = P^{(N)}f + \\alpha ^{(N)}f + \\beta ^{(N)}f\\,.$ We start with the observation that $f = P^{(N)}f - \\frac{1}{N!}",
"\\sum \\limits _\\sigma (P^{(N)}_\\sigma - 1) f\\,.$ Every permutation $\\sigma $ can be written as a composition $\\sigma = \\tau _1 \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ 2 l$, where each $ i$ is a transposition of neighbouring indices.",
"We then find $ P(N)= P(N)1P(N)2P(N)l$ and, using a telescoping series,\\begin{equation}\\big (P^{(N)}_\\sigma - 1\\big ) f = \\sum _{i=1}^l \\big (P^{(N)}_{\\tau _i} - 1\\big )\\,P^{(N)}_{\\tau _{i+1}}\\cdots P^{(N)}_{\\tau _{l}}\\, f_m^{(N+2)} \\,.\\end{equation}This is now a sum over terms where the left-most operator $ P(N)i- 1$ yields a commutator.",
"Using the CCR we may reduce this commutator to a term of lower degree, i.\\,e.\\begin{equation}\\big (P^{(N)}_{\\tau _i} - 1\\big ) f^{\\prime } = \\tilde{f}^{\\prime }+g \\,,\\quad \\tilde{f}^{\\prime }\\in \\Gamma _0(\\mathcal {D}_{0}(\\Sigma ))^{\\otimes (N-2)}\\,,\\quad g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\end{equation}for any $ f'0($\\mathcal {D}_{0}(\\Sigma )$ )N$, where $ f'$ depends continuously on $ f'$ and hence so does $ g$.",
"Repeating this procedure for each term in Equation (\\ref {eqn:commuted-elements}) and each term in the sum in Equation (\\ref {eqn:symmetrization_of_field2}) yields a well-defined expression of the form\\begin{equation}f = P^{(N)}f + \\sum _j\\tilde{f}_j+\\sum _jg_j\\,,\\end{equation}where $ j$ runs over some index set, $ fj$ is homogeneous of degree $ N-2$ and $ gj${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$$.",
"Because $ fj$ and $ gj$ depend continuously on $ f$, it suffices to define $ (N)f jfj$ and $ (N)f jgj$.",
"We refer to \\cite [Lemma B.5]{Schambach2016} for more details.$ In Equation (REF ) we may now proceed to symmetrise the term $\\alpha ^{(N)}f$ of degree $N-2$ .",
"Note that elements of degree 0 or 1 are automatically symmetric.",
"By induction we can then show that $f = \\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f +\\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }\\beta ^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,.$ (Here the maps $\\alpha $ are to be omitted when $j=0$ .)",
"We now define $S$ as $S=\\bigoplus _{N=0}^{\\infty }S_N$ in terms of the continuous linear maps $S_N:& \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right) \\,,\\\\&f\\mapsto \\sum \\limits _{j=0}^{N/2}P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,,$ for all $N\\ge 0$ .",
"Note that $S$ is continuous and because the $\\alpha ^{(N)}$ and $\\beta ^{(N)}$ vanish on symmetric elements, $S$ acts as the identity on $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ .",
"It follows from Equation (REF ) that every element $f\\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ can be decomposed into $f=f^{\\prime }+g$ , where $f^{\\prime }$ is symmetric and $g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ .",
"This decomposition is unique, since $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ , and we must have $f^{\\prime }=Sf$ .",
"This entails in particular that $\\mathcal {BU}\\left(\\mathcal {D}_{0}(\\Sigma )\\right)=\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\oplus {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR},$ that $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ and that $S$ is the unique projection with the given range and kernel."
],
[
"Proof of the energy estimate (",
"In this appendix we prove the energy estimate (REF ), which we now restate.",
"Theorem 6.1 Let $P$ be a normally hyperbolic operator on a real vector bundle $V$ over a globally hyperbolic spacetime $M$ and let $\\Sigma \\subset M$ be a smooth, space-like Cauchy surface.",
"For all compact sets $K\\subset \\Sigma $ and $L\\subset \\mathbb {R}$ there is a $C>0$ such that $\\int _{D(K)}\\Vert v^{(r)}\\Vert ^2\\le C\\int _K \\left(*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{{\\left.\\hspace{0.0pt}n^{\\alpha }\\nabla _{\\alpha }v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\right)+C\\int _{D(K)}*{f^{(r)}}^2,$ where $D(k)$ is the domain of dependence and $v^{(r)}$ is a solution to $(P+r)v^{(r)}=f^{(r)}$ .",
"Proof: 6.1 We may identify $M=\\mathbb {R}\\times S$ and $g=-Ndt^2+h_t$ , where $t\\in \\mathbb {R}$ , $N>0$ , $\\Sigma _t \\lbrace t\\rbrace \\times S$ is a smooth spacelike Cauchy surface with metric $h_t$ and $\\Sigma =\\Sigma _0$ .",
"We set $\\xi _{\\alpha } -N\\nabla _{\\alpha }t$ , so that $\\xi ^{\\alpha }$ is a future pointing time-like vector field and $n^{\\alpha } N^{-\\frac{1}{2}}\\xi ^{\\alpha }$ is its normalisation.",
"Without loss of generality we may assume that the auxiliary norm $*{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}$ on $TM$ is given by $2n_{\\alpha }n_{\\beta }+g_{\\alpha \\beta }$ .",
"For the purposes of this proof we choose the connection $\\nabla $ on $V$ to be the one which is compatible with the auxiliary metric on $V$ .",
"Any different choice of connection in (REF ) can easily be accommodated for by adjusting $C$ at the end of the proof.",
"Note that for suitable smooth bundle homomorphisms $A$ and $B$ it holds $P=g^{\\alpha \\beta }\\nabla _{\\alpha }\\nabla _{\\beta }+A^{\\alpha }\\nabla _{\\alpha }+B$ .",
"Let us fix $r$ for now and drop the superscripts on $v$ and $f$ .",
"We define the quantities $T_{\\alpha \\beta }&\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v-\\frac{1}{2}g_{\\alpha \\beta }\\left(*{\\nabla v}^2+*{v}^2\\right)\\,,\\\\P_{\\alpha }&\\xi ^{\\beta }T_{\\alpha \\beta }\\,,\\\\\\epsilon &n^{\\alpha }P_{\\alpha }=\\sqrt{N}n^{\\alpha }n^{\\beta }T_{\\alpha \\beta }\\\\&\\phantom{:}=\\frac{1}{2}\\sqrt{N}\\left((2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v+*{v}^2\\right)\\,,$ where $\\cdot $ refers to the hermitian inner product on $V$ .",
"Note that $\\epsilon \\ge 0$ .",
"We may now choose a $T>0$ such that $D(K)\\subset (-\\infty ,T)\\times S$ and a compact $K^{\\prime }\\subset \\Sigma $ which contains $K$ in its interior.",
"Then we may choose an auxiliary Cauchy surface $\\Sigma ^{\\prime }$ of $(-\\infty ,T)\\times S$ such that $D(K)$ lies to the past of $\\Sigma ^{\\prime }$ , but $\\Sigma ^{\\prime }$ contains $\\Sigma \\setminus K^{\\prime }$ .",
"Furthermore, we may choose a $C\\ge 1$ such that the following inequalities hold on $[0,T]\\times S$ : $N^{\\pm \\frac{1}{2}}\\le C \\,,\\quad \\pm (\\nabla ^{\\alpha }\\xi ^{\\beta }+\\nabla ^{\\beta }\\xi ^{\\alpha })\\le C\\sqrt{N}(2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\,,\\\\\\left|\\nabla _{\\alpha }\\xi ^{\\alpha }\\right|\\le C\\sqrt{N}\\,,\\quad *{R}\\le C\\,,\\quad *{A}\\le C \\quad \\textrm {and}\\quad *{B}\\le C \\,,$ where $R$ is the curvature of $\\nabla $ on $V$ .",
"In addition we may assume that $|r+1|\\le C$ for all $r\\in L$ and that $h_t\\le Ch_{t^{\\prime }}$ on $K^{\\prime }$ for all $t,t^{\\prime }\\in [0,T]$ and similarly for the hermitian metric in $V$ .",
"It will be convenient to introduce $L_t \\Sigma _t\\cap J^-(\\Sigma ^{\\prime })$ for $t\\in [0,T]$ and the “energy” $\\epsilon (t) \\int _{L_t}\\epsilon \\,.$ We now want to estimate the quantity $E(t) \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\epsilon $ for $t\\in [0,T]$ .",
"We note first of all that $\\frac{d}{dt}E(t)\\le \\lim _{\\tau \\rightarrow 0^+}\\tau ^{-1}\\int _{[t,t+\\tau ]\\times L_t}\\epsilon \\le C\\int _{L_t}\\epsilon \\,,$ where the constant $C$ is needed to estimate the factor $\\sqrt{N}$ which arises due to a change of volume form.",
"Furthermore, using Stokes' Theorem: $\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha }=\\epsilon (t)-\\epsilon (0)+\\int _{\\Sigma ^{\\prime }\\cap ([0,t]\\times S)}\\nu ^{\\alpha }P_{\\alpha }\\,,$ where $\\nu ^{\\alpha }$ is the forward unit normal to $\\Sigma ^{\\prime }$ .",
"One may show that the bilinear form $\\nu ^{\\alpha }n^{\\beta }+n^{\\alpha }\\nu ^{\\beta }-g^{\\alpha \\beta }n^{\\gamma }\\nu _{\\gamma }$ is positive definite and $n^{\\gamma }\\nu _{\\gamma }<0$ .",
"This entails that $\\nu ^{\\alpha }P_{\\alpha }\\ge 0$ and hence $\\epsilon (t)-\\epsilon (0)\\le \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha } \\,.$ Furthermore, we may estimate $\\left|\\nabla ^{\\alpha }P_{\\alpha }\\right|\\le \\left|T_{\\alpha \\beta }\\nabla ^{\\alpha }\\xi ^{\\beta } \\right|+\\left|\\xi ^{\\beta }\\nabla ^{\\alpha }T_{\\alpha \\beta }\\right|\\,,$ where $\\nabla ^{\\alpha }T_{\\alpha \\beta }=v\\cdot R_{\\alpha \\beta }\\cdot \\nabla ^{\\alpha }v-v\\cdot B\\cdot \\nabla _{\\beta }v-(r+1)v\\cdot \\nabla _{\\beta }v+f\\cdot \\nabla _{\\beta }v \\,.$ For the term involving $f$ we can use the further estimate $\\left|\\xi ^{\\beta }f\\cdot \\nabla _{\\beta }v\\right|\\le C *{f}\\cdot *{\\nabla v}\\le \\frac{1}{2} C \\left( *{f}^2+*{\\nabla v}^2 \\right) \\,.$ Using our choice of $C$ we can then estimate all the terms in $\\nabla ^{\\alpha }P_{\\alpha }$ to find $\\epsilon (t)\\le \\epsilon (0)+\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}8 C^2\\epsilon +\\frac{1}{2} C *{f}^2$ and consequently $\\frac{d}{dt}E(t)\\le C\\epsilon (t)\\le 8C^3E(t)+C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2.$ Therefore, $\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)\\le C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2$ .",
"With $E(0)=0$ this yields $e^{-8C^2T}E(T)=\\int _0^T\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)dt\\le \\left(C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2\\right)T$ and hence $E(T)\\le C^{\\prime }\\big (\\epsilon (0)+\\int _{D(K^{\\prime })}*{f}^2\\big )$ for a suitable $C^{\\prime }>0$ independent of $r$ .",
"Note that $E(T)\\ge \\int _{D(K)}*{v}^2$ and that $\\epsilon (0)\\le C^{\\prime }\\int _K\\big (*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{n^{\\alpha }\\nabla _{\\alpha }{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\big )$ when we choose $C^{\\prime }$ large enough.",
"Finally, we may shrink $K^{\\prime }$ to $K$ without adjusting the constants $C$ or $C^{\\prime }$ which leads to the desired estimate.",
"[heading=bibintoc,title=References]"
],
[
"Conclusion and Outlook",
"We have studied the classical and quantum Proca field in curved spacetimes, using a general setting including external sources and without restrictive assumptions on the spacetime topology.",
"We have shown that the quantum theory is locally covariant in the sense of , where the injectivity of the morphisms is related to the non-degeneracy of the symplectic form.",
"We have shown that the theory depends continuously on the mass $m>0$ , in a way which we have defined.",
"Using specific BU-algebra homeomorphism we mapped families of smeared Proca fields at different masses, initially elements in different BU-algebras, into the BU-algebra of initial data.",
"The topology of the latter algebra then determines a notion of continuity for the family of operators.",
"For $m>0$ we showed that this notion of continuity is independent of the choice of Cauchy surface and of the classical inhomogeneous solutions $\\varphi _{m,j}$ appearing in the homeomorphisms.",
"This result relied crucially on the use of energy estimates.",
"Note that a $C^*$ -Weyl algebra approach is ill-suited for the investigation of the zero mass limit, as one of us has argued in .",
"For the quantum theory we defined the zero mass limit by requiring a continuous family of observables to converge on every Cauchy surface and for every continuous family $\\left\\lbrace \\varphi _{m,j} \\right\\rbrace _{m\\ge 0}$ of inhomogeneous classical solutions.",
"(For the classical theory we considered a somewhat simplified setting.)",
"Investigating the zero mass limit we found in both cases that the limit exists and the theory is generated by the class of observables described by co-closed test one-forms.",
"This effectively implements a gauge invariance on the (distributional) solutions to Proca's equation by exact (distributional) one-forms.",
"This is of interest, because in general curved spacetimes the spacetime topology allows different possible choices of gauge invariance (using e. g. closed forms instead).",
"Our limiting procedure naturally leads to the same gauge invariance that was advocated in , using the independent argument that it can account for phenomena such as the Aharonov-Bohm effect and Gauss' law.",
"In the zero mass limit we also find that the quantum fields fulfill the basic properties of linearity, the hermitian field property and the correct CCR, all in line with the massless vector potential of electrodynamics.",
"However, we do not automatically recover the expected Maxwell dynamics.",
"In the classical case, this is caused by a potential divergence in the constraint equations on the initial data of field configurations.",
"This may be avoided by requiring the external source to be conserved, $\\delta j=0$ , and by requiring that the initial data of the configuration also satisfy the constraint equations of Maxwell's theory as given e. g. by Pfenning .",
"In the quantum case we did not clarify if Maxwell's equation can be obtained in the zero mass limit, e. g. by imposing additional conditions on the limits of observables or on states, or by requiring the homeomorphisms that propagate initial data between different Cauchy surfaces to remain well defined in the massless limit.",
"The further development of these ideas might require a detailed investigation of Hadamard states, which is also if interest in its own right.",
"So far these states seem to have been considered only in a restricted class of spacetimes .",
"Furthermore, it would be interesting to make a detailed comparison of our massless limit and the massless limit of Stueckelberg's theory as presented e. g. in .",
"We leave the investigation of these worthwhile questions to the future.",
"Acknowledgements We would like to thank the University of Leipzig, where this research was carried out, and MS would like to thank Prof. Stefan Hollands for helpful comments and discussions.",
"Large parts of this work are adapted from the MSc thesis of MS."
],
[
"Additional Lemmas",
"Let $\\mathfrak {X}$ be a complex vector bundle over a smooth differential manifold $\\mathcal {N}$ .",
"As in Section REF we may define the complete BU-algebra $\\overline{\\mathcal {BU}(\\Gamma _0(\\mathfrak {X}))}$ over $\\Gamma _0(\\mathfrak {X})$ as the direct sum $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )} = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big ) \\,,$ using the outer tensor product of vector bundles (see ).",
"We endow this algebra with the inductive limit topology of the subspaces $\\mathcal {BU}_N = \\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^N \\Gamma _0(\\mathfrak {X}^{\\boxtimes n}) \\,.$ Note that $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is the completion of the BU-algebra $\\mathbb {C}\\oplus \\bigoplus \\limits _{n= 1}^\\infty \\Gamma _0(\\mathfrak {X})^{\\otimes n}$ .",
"Lemma 5.1 The complete Borchers-Uhlmann algebra $\\overline{\\mathcal {BU}\\big (\\Gamma _0(\\mathfrak {X})\\big )}$ is barrelled.",
"Proof: 5.1 The spaces $\\Gamma _0\\big (\\mathfrak {X}^{\\boxtimes n}\\big )$ of compactly supported sections of a complex vector bundle are LF-spaces, as they are defined as the inductive limit of the Frechét spaces of sections with support in some compact $K_l$ where $\\left\\lbrace K_l \\right\\rbrace _l$ is a fundamental sequence of compact $K_l \\subset \\mathcal {N}$ (see ).",
"Since LF-spaces are barrelled and the direct sum of barrelled spaces is again barrelled , we find for any $N \\in \\mathbb {N}$ that $\\mathcal {BU}_N$ is barrelled.",
"Additionally, the inductive limit of barrelled spaces is barrelled , hence the complete BU-algebra over smooth compactly supported sections $\\Gamma _0(\\mathfrak {X})$ over a complex vector bundle $\\mathfrak {X}$ is barrelled.",
"We will use barrelled spaces in order to apply the following result: Lemma 5.2 Let $X$ be a barrelled locally convex space, let $\\eta :[c,d]\\rightarrow X$ be a continuous map on a closed interval and let $L_m:X\\rightarrow Y$ be a family of continuous linear maps into a locally convex space $Y$ indexed by $m\\in [a,b]$ .",
"If the map $m\\mapsto L_m$ is weakly continuous, i. e. if $m\\mapsto L_mx$ is continuous on $[a,b]$ for each $x\\in X$ , then the map $(m,m^{\\prime })\\mapsto L_m\\eta (m^{\\prime })$ is continuous on $[a,b]\\times [c,d]$ .",
"Proof: 5.2 The weak continuity of $m\\mapsto L_m$ implies that for each $x\\in X$ the image of $m\\mapsto L_mx$ is compact.",
"The family of maps $L_m$ is therefore pointwise bounded.",
"Because $X$ is barrelled we may apply the uniform boundedness principle to find that the maps $L_m$ are equicontinuous.",
"For any $(m_0,m^{\\prime }_0)\\in [a,b]\\times [c,d]$ we set $x \\eta (m^{\\prime }_0)$ and we pick an arbitrary convex open neighbourhood $y+V$ of $y L_{m_0}x$ , where $V$ is an open neighbourhood of 0.",
"By equicontinuity there is an open neighbourhood $U\\subset X$ of 0 such that $L_m(U)\\subset \\frac{1}{2}V$ for all $m\\in [a,b]$ .",
"As $\\eta $ is continuous there is an open neighbourhood $W^{\\prime }\\subset [c,d]$ of $m^{\\prime }_0$ such that $\\eta (W^{\\prime })\\subset x+U$ .",
"Similarly there is an open neighbourhood $W\\subset [a,b]$ of $m_0$ such that $L_mx-y\\in \\frac{1}{2}V$ for all $m\\in W$ .",
"It follows that for all $(m,m^{\\prime })\\in W\\times W^{\\prime }$ $L_m\\eta (m^{\\prime })-y = L_m(\\eta (m^{\\prime })-x) + (L_mx-y)\\in \\frac{1}{2}V+\\frac{1}{2}V\\subset V$ which proves the desired continuity.",
"For our next lemma we will call an element of $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ symmetric if and only if it is totally symmetric in each degree.",
"Lemma 5.3 (Symmetrization of fields) Let $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ denote the linear subspace of the Borchers-Uhlmann algebra of initial data $\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ consisting of symmetric elements.",
"Then there is a unique continuous linear surjective projection $S:\\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\rightarrow \\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ whose kernel is $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ as defined in Section REF .",
"Proof: 5.3 For each $N\\ge 1$ and each permutation $\\sigma $ of the set $\\lbrace 1,\\ldots ,N\\rbrace $ we introduce the permutation operator $P^{(N)}_{\\sigma }:\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ defined by $\\left(P^{(N)}_{\\sigma }f\\right)\\big (p_1,\\ldots ,p_N\\big ) f\\big (p_{\\sigma (1)},\\ldots ,p_{\\sigma (N)}\\big ) \\,,$ where we view elements of $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}$ as sections in $\\Gamma _0\\big ((T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N}\\big )$ .",
"The symmetric tensor product $\\big (\\mathcal {D}_{0}(\\Sigma )\\big )^{\\otimes _S N}$ is then the range space of the projection $P^{(N)} \\frac{1}{N!",
"}\\sum \\limits _{\\sigma }P^{(N)}_{\\sigma }.$ Note that each $P^{(N)}_{\\sigma }$ is continuous, because the topology of $\\Gamma _0\\big ( (T^*\\Sigma \\oplus T^*\\Sigma )^{\\boxtimes N} \\big ) $ is invariant under the swapping of variables.",
"It follows that $P^{(N)}$ is a continuous surjection.",
"We will first argue that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"For this we note that each $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is of the form $f = \\sum _{i=1}^{k} h_i \\cdot \\big ( -\\mathrm {i}\\mathcal {G}_m(\\psi _i, \\psi ^{\\prime }_i) , 0 , \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i , 0 , 0 ,\\dots \\big ) \\cdot \\tilde{h}_i$ for some $k \\in \\mathbb {N}$ , $h_i, \\tilde{h}_i \\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ and $\\psi _i, \\psi ^{\\prime }_i \\in \\mathcal {D}_{0}(\\Sigma )$ , where we have used the shorthand notation ${\\psi _i}{ \\psi ^{\\prime }_i} = \\langle \\pi _i , \\varphi ^{\\prime }_i \\rangle _\\Sigma - \\langle \\varphi _i , \\pi ^{\\prime }_i \\rangle _\\Sigma $ for $\\psi _i = (\\varphi _i , \\pi _i)$ .",
"If $f\\ne 0$ then its highest degree part is of some degree $N\\ge 2$ and we can write it explicitly, using the above representation, as $f^{(N)} = \\sum _{i=1}^{k} h_i^{(N_i)}\\, \\big ( \\psi _i \\otimes \\psi ^{\\prime }_i - \\psi ^{\\prime }_i \\otimes \\psi _i \\big )\\, \\tilde{h}_i^{(N-2-N_i)} \\,,$ where $h_i^{(N_i)}$ is the highest degree part of $h_i$ and $\\tilde{h}_i^{(N-2-N_i)}$ is either the highest degree part of $\\tilde{h}_i$ or 0.",
"It follows by inspection that $P^{(N)}f^{(N)}=0$ .",
"Now, if $f\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ is non-zero and symmetric and if $f^{(N)}$ is its highest degree part, then $f^{(N)}=P^{(N)}f^{(N)}=0$ , contradicting that $f^{(N)}$ is the highest degree part.",
"It follows that $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ .",
"We now construct for each degree $N\\ge 2$ two continuous linear maps $\\alpha ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\,,\\\\\\beta ^{(N)}:&\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\,,$ (for $N=2$ we use $\\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes (N-2)}=\\mathbb {C}$ ) such that $f = P^{(N)}f + \\alpha ^{(N)}f + \\beta ^{(N)}f\\,.$ We start with the observation that $f = P^{(N)}f - \\frac{1}{N!}",
"\\sum \\limits _\\sigma (P^{(N)}_\\sigma - 1) f\\,.$ Every permutation $\\sigma $ can be written as a composition $\\sigma = \\tau _1 \\mathbin {\\mathchoice{\\hbox{$\\scriptstyle \\circ $}}{}{}{}}{\\hbox{$\\scriptstyle \\circ $}}$ 2 l$, where each $ i$ is a transposition of neighbouring indices.",
"We then find $ P(N)= P(N)1P(N)2P(N)l$ and, using a telescoping series,\\begin{equation}\\big (P^{(N)}_\\sigma - 1\\big ) f = \\sum _{i=1}^l \\big (P^{(N)}_{\\tau _i} - 1\\big )\\,P^{(N)}_{\\tau _{i+1}}\\cdots P^{(N)}_{\\tau _{l}}\\, f_m^{(N+2)} \\,.\\end{equation}This is now a sum over terms where the left-most operator $ P(N)i- 1$ yields a commutator.",
"Using the CCR we may reduce this commutator to a term of lower degree, i.\\,e.\\begin{equation}\\big (P^{(N)}_{\\tau _i} - 1\\big ) f^{\\prime } = \\tilde{f}^{\\prime }+g \\,,\\quad \\tilde{f}^{\\prime }\\in \\Gamma _0(\\mathcal {D}_{0}(\\Sigma ))^{\\otimes (N-2)}\\,,\\quad g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}\\end{equation}for any $ f'0($\\mathcal {D}_{0}(\\Sigma )$ )N$, where $ f'$ depends continuously on $ f'$ and hence so does $ g$.",
"Repeating this procedure for each term in Equation (\\ref {eqn:commuted-elements}) and each term in the sum in Equation (\\ref {eqn:symmetrization_of_field2}) yields a well-defined expression of the form\\begin{equation}f = P^{(N)}f + \\sum _j\\tilde{f}_j+\\sum _jg_j\\,,\\end{equation}where $ j$ runs over some index set, $ fj$ is homogeneous of degree $ N-2$ and $ gj${\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$$.",
"Because $ fj$ and $ gj$ depend continuously on $ f$, it suffices to define $ (N)f jfj$ and $ (N)f jgj$.",
"We refer to \\cite [Lemma B.5]{Schambach2016} for more details.$ In Equation (REF ) we may now proceed to symmetrise the term $\\alpha ^{(N)}f$ of degree $N-2$ .",
"Note that elements of degree 0 or 1 are automatically symmetric.",
"By induction we can then show that $f = \\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f +\\sum \\limits _{j=0}^{\\lfloor N/2\\rfloor }\\beta ^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,.$ (Here the maps $\\alpha $ are to be omitted when $j=0$ .)",
"We now define $S$ as $S=\\bigoplus _{N=0}^{\\infty }S_N$ in terms of the continuous linear maps $S_N:& \\big (\\Gamma _0(T^*\\Sigma \\oplus T^*\\Sigma )\\big )^{\\otimes N}\\rightarrow \\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right) \\,,\\\\&f\\mapsto \\sum \\limits _{j=0}^{N/2}P^{(N-2j)}\\alpha ^{(N+2-2j)}\\cdots \\alpha ^{(N)}f \\,,$ for all $N\\ge 0$ .",
"Note that $S$ is continuous and because the $\\alpha ^{(N)}$ and $\\beta ^{(N)}$ vanish on symmetric elements, $S$ acts as the identity on $\\mathcal {BU}_S\\left(\\mathcal {D}_{0}(\\Sigma )\\right)$ .",
"It follows from Equation (REF ) that every element $f\\in \\mathcal {BU}\\big (\\mathcal {D}_{0}(\\Sigma )\\big )$ can be decomposed into $f=f^{\\prime }+g$ , where $f^{\\prime }$ is symmetric and $g\\in {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ .",
"This decomposition is unique, since $\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\cap {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}=\\lbrace 0\\rbrace $ , and we must have $f^{\\prime }=Sf$ .",
"This entails in particular that $\\mathcal {BU}\\left(\\mathcal {D}_{0}(\\Sigma )\\right)=\\mathcal {BU}_S\\big (\\mathcal {D}_{0}(\\Sigma )\\big )\\oplus {\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR},$ that $\\mathrm {ker}{\\left(S\\right)}={\\mathcal {I}}_{\\sim }^\\mathrm {\\,CCR}$ and that $S$ is the unique projection with the given range and kernel."
],
[
"Proof of the energy estimate (",
"In this appendix we prove the energy estimate (REF ), which we now restate.",
"Theorem 6.1 Let $P$ be a normally hyperbolic operator on a real vector bundle $V$ over a globally hyperbolic spacetime $M$ and let $\\Sigma \\subset M$ be a smooth, space-like Cauchy surface.",
"For all compact sets $K\\subset \\Sigma $ and $L\\subset \\mathbb {R}$ there is a $C>0$ such that $\\int _{D(K)}\\Vert v^{(r)}\\Vert ^2\\le C\\int _K \\left(*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{{\\left.\\hspace{0.0pt}n^{\\alpha }\\nabla _{\\alpha }v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\right)+C\\int _{D(K)}*{f^{(r)}}^2,$ where $D(k)$ is the domain of dependence and $v^{(r)}$ is a solution to $(P+r)v^{(r)}=f^{(r)}$ .",
"Proof: 6.1 We may identify $M=\\mathbb {R}\\times S$ and $g=-Ndt^2+h_t$ , where $t\\in \\mathbb {R}$ , $N>0$ , $\\Sigma _t \\lbrace t\\rbrace \\times S$ is a smooth spacelike Cauchy surface with metric $h_t$ and $\\Sigma =\\Sigma _0$ .",
"We set $\\xi _{\\alpha } -N\\nabla _{\\alpha }t$ , so that $\\xi ^{\\alpha }$ is a future pointing time-like vector field and $n^{\\alpha } N^{-\\frac{1}{2}}\\xi ^{\\alpha }$ is its normalisation.",
"Without loss of generality we may assume that the auxiliary norm $*{\\hspace{0.20004pt}\\cdot \\hspace{0.20004pt}}$ on $TM$ is given by $2n_{\\alpha }n_{\\beta }+g_{\\alpha \\beta }$ .",
"For the purposes of this proof we choose the connection $\\nabla $ on $V$ to be the one which is compatible with the auxiliary metric on $V$ .",
"Any different choice of connection in (REF ) can easily be accommodated for by adjusting $C$ at the end of the proof.",
"Note that for suitable smooth bundle homomorphisms $A$ and $B$ it holds $P=g^{\\alpha \\beta }\\nabla _{\\alpha }\\nabla _{\\beta }+A^{\\alpha }\\nabla _{\\alpha }+B$ .",
"Let us fix $r$ for now and drop the superscripts on $v$ and $f$ .",
"We define the quantities $T_{\\alpha \\beta }&\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v-\\frac{1}{2}g_{\\alpha \\beta }\\left(*{\\nabla v}^2+*{v}^2\\right)\\,,\\\\P_{\\alpha }&\\xi ^{\\beta }T_{\\alpha \\beta }\\,,\\\\\\epsilon &n^{\\alpha }P_{\\alpha }=\\sqrt{N}n^{\\alpha }n^{\\beta }T_{\\alpha \\beta }\\\\&\\phantom{:}=\\frac{1}{2}\\sqrt{N}\\left((2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\nabla _{\\alpha }v\\cdot \\nabla _{\\beta }v+*{v}^2\\right)\\,,$ where $\\cdot $ refers to the hermitian inner product on $V$ .",
"Note that $\\epsilon \\ge 0$ .",
"We may now choose a $T>0$ such that $D(K)\\subset (-\\infty ,T)\\times S$ and a compact $K^{\\prime }\\subset \\Sigma $ which contains $K$ in its interior.",
"Then we may choose an auxiliary Cauchy surface $\\Sigma ^{\\prime }$ of $(-\\infty ,T)\\times S$ such that $D(K)$ lies to the past of $\\Sigma ^{\\prime }$ , but $\\Sigma ^{\\prime }$ contains $\\Sigma \\setminus K^{\\prime }$ .",
"Furthermore, we may choose a $C\\ge 1$ such that the following inequalities hold on $[0,T]\\times S$ : $N^{\\pm \\frac{1}{2}}\\le C \\,,\\quad \\pm (\\nabla ^{\\alpha }\\xi ^{\\beta }+\\nabla ^{\\beta }\\xi ^{\\alpha })\\le C\\sqrt{N}(2n^{\\alpha }n^{\\beta }+g^{\\alpha \\beta })\\,,\\\\\\left|\\nabla _{\\alpha }\\xi ^{\\alpha }\\right|\\le C\\sqrt{N}\\,,\\quad *{R}\\le C\\,,\\quad *{A}\\le C \\quad \\textrm {and}\\quad *{B}\\le C \\,,$ where $R$ is the curvature of $\\nabla $ on $V$ .",
"In addition we may assume that $|r+1|\\le C$ for all $r\\in L$ and that $h_t\\le Ch_{t^{\\prime }}$ on $K^{\\prime }$ for all $t,t^{\\prime }\\in [0,T]$ and similarly for the hermitian metric in $V$ .",
"It will be convenient to introduce $L_t \\Sigma _t\\cap J^-(\\Sigma ^{\\prime })$ for $t\\in [0,T]$ and the “energy” $\\epsilon (t) \\int _{L_t}\\epsilon \\,.$ We now want to estimate the quantity $E(t) \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\epsilon $ for $t\\in [0,T]$ .",
"We note first of all that $\\frac{d}{dt}E(t)\\le \\lim _{\\tau \\rightarrow 0^+}\\tau ^{-1}\\int _{[t,t+\\tau ]\\times L_t}\\epsilon \\le C\\int _{L_t}\\epsilon \\,,$ where the constant $C$ is needed to estimate the factor $\\sqrt{N}$ which arises due to a change of volume form.",
"Furthermore, using Stokes' Theorem: $\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha }=\\epsilon (t)-\\epsilon (0)+\\int _{\\Sigma ^{\\prime }\\cap ([0,t]\\times S)}\\nu ^{\\alpha }P_{\\alpha }\\,,$ where $\\nu ^{\\alpha }$ is the forward unit normal to $\\Sigma ^{\\prime }$ .",
"One may show that the bilinear form $\\nu ^{\\alpha }n^{\\beta }+n^{\\alpha }\\nu ^{\\beta }-g^{\\alpha \\beta }n^{\\gamma }\\nu _{\\gamma }$ is positive definite and $n^{\\gamma }\\nu _{\\gamma }<0$ .",
"This entails that $\\nu ^{\\alpha }P_{\\alpha }\\ge 0$ and hence $\\epsilon (t)-\\epsilon (0)\\le \\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}\\nabla ^{\\alpha }P_{\\alpha } \\,.$ Furthermore, we may estimate $\\left|\\nabla ^{\\alpha }P_{\\alpha }\\right|\\le \\left|T_{\\alpha \\beta }\\nabla ^{\\alpha }\\xi ^{\\beta } \\right|+\\left|\\xi ^{\\beta }\\nabla ^{\\alpha }T_{\\alpha \\beta }\\right|\\,,$ where $\\nabla ^{\\alpha }T_{\\alpha \\beta }=v\\cdot R_{\\alpha \\beta }\\cdot \\nabla ^{\\alpha }v-v\\cdot B\\cdot \\nabla _{\\beta }v-(r+1)v\\cdot \\nabla _{\\beta }v+f\\cdot \\nabla _{\\beta }v \\,.$ For the term involving $f$ we can use the further estimate $\\left|\\xi ^{\\beta }f\\cdot \\nabla _{\\beta }v\\right|\\le C *{f}\\cdot *{\\nabla v}\\le \\frac{1}{2} C \\left( *{f}^2+*{\\nabla v}^2 \\right) \\,.$ Using our choice of $C$ we can then estimate all the terms in $\\nabla ^{\\alpha }P_{\\alpha }$ to find $\\epsilon (t)\\le \\epsilon (0)+\\int _{([0,t]\\times S)\\cap J^-(\\Sigma ^{\\prime })}8 C^2\\epsilon +\\frac{1}{2} C *{f}^2$ and consequently $\\frac{d}{dt}E(t)\\le C\\epsilon (t)\\le 8C^3E(t)+C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2.$ Therefore, $\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)\\le C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2$ .",
"With $E(0)=0$ this yields $e^{-8C^2T}E(T)=\\int _0^T\\frac{d}{dt}\\mathrm {e}^{-8C^2t}E(t)dt\\le \\left(C\\epsilon (0)+\\frac{1}{2}C^2\\int _{D(K^{\\prime })}*{f}^2\\right)T$ and hence $E(T)\\le C^{\\prime }\\big (\\epsilon (0)+\\int _{D(K^{\\prime })}*{f}^2\\big )$ for a suitable $C^{\\prime }>0$ independent of $r$ .",
"Note that $E(T)\\ge \\int _{D(K)}*{v}^2$ and that $\\epsilon (0)\\le C^{\\prime }\\int _K\\big (*{{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2+*{n^{\\alpha }\\nabla _{\\alpha }{\\left.\\hspace{0.0pt}v^{(r)} \\vphantom{\\big |} \\right|_{\\Sigma } }}^2\\big )$ when we choose $C^{\\prime }$ large enough.",
"Finally, we may shrink $K^{\\prime }$ to $K$ without adjusting the constants $C$ or $C^{\\prime }$ which leads to the desired estimate.",
"[heading=bibintoc,title=References]"
]
] | 1709.01911 |
[
[
"A security proof of continuous-variable QKD using three coherent states"
],
[
"Abstract We introduce a ternary quantum key distribution (QKD) protocol and asymptotic security proof based on three coherent states and homodyne detection.",
"Previous work had considered the binary case of two coherent states and here we nontrivially extend this to three.",
"Our motivation is to leverage the practical benefits of both discrete and continuous (Gaussian) encoding schemes creating a best-of-both-worlds approach; namely, the postprocessing of discrete encodings and the hardware benefits of continuous ones.",
"We present a thorough and detailed security proof in the limit of infinite signal states which allows us to lower bound the secret key rate.",
"We calculate this is in the context of collective eavesdropping attacks and reverse reconciliation postprocessing.",
"Finally, we compare the ternary coherent state protocol to other well-known QKD schemes (and fundamental repeaterless limits) in terms of secret key rates and loss."
],
[
"Introduction",
"Quantum key distribution (QKD) [1], [2], in principle, provides the most secure form of quantum safe cybersecurity, i.e., protection against a quantum computing attack.",
"As opposed to post quantum cryptography [3], which is based on computationally secure mathematics, QKD exploits the laws of quantum physics to achieve, at least in theory, unbreakable codes.",
"Since QKD was first suggested in 1984, many advances have taken place; from theoretical to proof-of-principle experiments to field tests and even the forming of companies.",
"Even though this seems like the end of the story there are still many advances being made in all of these areas.",
"To this point, in this paper, we look at creating a best-of-both worlds approach to QKD by combining the beneficial practical aspects of the two main implementations of QKD: those using discrete variables (DVs) [1] and those using continuous variables (CVs) [4], [5].",
"To be more specific, we would like to use the simpler encoding and decoding methods from DV QKD but at the same time leverage the simpler and more affordable room temperature hardware components of CV QKD.",
"Recently, the ultimate (optimal) limit for a lossy bosonic channel was discovered and is given by the PLOB bound [6].",
"An interpretation of this result is that no QKD protocol can go beyond this bound without a quantum repeater.",
"In terms of key rate as a function of channel loss (cf.",
"for instance with Fig.",
"6 of [6]) this corresponds to a CV QKD Gaussian protocol with reverse reconciliation using a quantum memory at Alice's side and heterodyne at Bob's side [7].",
"In terms of implementations, below this optimal bound lies the single photon BB84 protocol [8].",
"Both of these two protocols are in terms of the ideal case, i.e., perfect sources and perfect detectors.",
"However, when one considers the realistic version of these two (in the case of DV QKD this corresponds to the decoy state scheme [9], [10]), both become remarkably similar in terms of key rates as a function of loss; except for a slight advantage in key rates for CVs in the low-loss regime and a slight distance advantage in DVs for the high-loss regime.",
"In this realistic scenario, both the DV and the CV QKD schemes sit below the PLOB bound.",
"Ideally we would like to either: (1) find a (realistic) protocol above these two protocols or (2) have a protocol similar to these protocols in terms of key rates but one that leverages the practical benefits of both schemes.",
"With that in mind, we consider a protocol first introduced in 2009 by Zhao et al.",
"[11] that uses binary-phase shift-keying (BPSK) of coherent states, $\\mathop {|\\alpha \\rangle }\\nolimits $ and $\\mathop {|-\\alpha \\rangle }\\nolimits $ , along with homodyne detection.",
"Unfortunately, as one can see, the performance of this protocol is below that of the realistic BB84 with decoy states and the realistic Gaussian modulation CV scheme.",
"In this paper, we consider a ternary-phase shift-keying (TPSK) of coherent states, $\\mathop {|\\alpha \\rangle }\\nolimits _i$ where $i = 0,1,2$ , with homodyne detection.",
"Here each of the three coherent states are phase shifted in phase space by $120^{\\circ }$ , cf.",
"Fig.",
"REF .",
"One may ask the question, what is the motivation of going from two coherent states to three coherent states?",
"Or perhaps why not go to more coherent states straightaway?",
"In terms of the second question, this is easily answered by considering the Zhao et al.",
"paper [11] and our results here.",
"The extension to three states is challenging enough, while the extension to more than three states is a very hard problem if one wants a strong security proof like the one we have given here.",
"In terms of the first question, there are two possible ways to answer this.",
"One way is that we know that at some stage as one increases the number of coherent states there must be a point where it becomes a close approximation to the full Gaussian distribution.",
"So there may be a point where one may not need the entire (continuum) Gaussian distribution.",
"Another way is to consider the affect that decoy state BB84 QKD has on ideal single photon BB84 and draw inspiration from there.",
"Specifically, by increasing the number of pulses from the ideal case of one to say three pulses gives a boost to both the key rate and distance [9], [10].",
"So perhaps we can consider increasing the number of discrete coherent states from two to three (and potentially higher) as a decoy-state-like extension of the BPSK-modulated CV QKD protocol.",
"Figure: Phase space configurations of the ternary coherent state QKD protocol.",
"Note that each subsequent coherent state is 120 ∘ 120^{\\circ } from the other one.",
"Alice's role is to continually and randomly choose from these three options and then send them to Bob who performs homodyne detection on the incoming states by randomly alternating between the QQ and PP quadratures.",
"As is standard, the quantum channel is assumed to be monitored by the eavesdropper, Eve.In this paper, we introduce and rigorously prove the asymptotic security of a new ternary QKD protocol based on three coherent states and homodyne detection.",
"For completeness, we mention that other discrete encodings for CVs have also been considered [12], [13], [14], [15].",
"In the papers by Leverrier and Grangier [12], [13], [14], they considered two different coherent state encodings (i.e., two and four) but in order to analyze the security they `padded out' the states with decoy-like states that effectively resembled a Gaussian distribution from Eve's point of view.",
"This is instigated in order to leverage previous Gaussian encoding security proofs.",
"Finally, in [15] a multi letter phase-shift keying scheme was introduced, where an $N$ number of coherent states can be used.",
"However, the security proof only considered a lossy bosonic channel (i.e., no excess noise).",
"In contrast, we consider a bosonic channel with arbitrary noise.",
"Our results here allow for the significant reduction, compared to Gaussian modulation protocols, in classical post-processing, random-number generation, and classical-communication overheads.",
"Furthermore, by keeping the benefits of CV hardware, our approach has the practical benefits of doing away with single-photon detectors that characterize DV QKD systems.",
"Such detectors are only able to reach their promise of low-noise and high-efficiency only with the addition of cumbersome cryogenics."
],
[
"Outline",
"This paper is structured as follows.",
"We begin by giving more background on the relationship between discrete and Gaussian encodings.",
"This is followed by a description of the steps of our ternary coherent state protocol.",
"Our main result is presented next and consists of a simulation of a TPSK modulated lossy bosonic channel.",
"We end with our conclusion.",
"In what follows, $f^{\\prime }$ denotes ${\\mathrm {d}f\\over \\mathrm {d}z}$ and similarly for higher derivatives.",
"Sometimes we explicitly mention a function's variable (typically $f=f(z)$ ).",
"The symbol $\\overset{\\mathrm {df}}{=}$ stands for `defined'.",
"The von Neumann entropy of a density matrix $\\varpi _A$ is $H(A)_\\varpi \\equiv H(\\varpi _A)\\overset{\\mathrm {df}}{=}-\\mathop {{\\mathrm {Tr}}_{}}[\\varpi _A\\log {\\varpi _A}]$ [16], [17] and it becomes Shannon entropy for classical probability distributions (denoted by $X,Y$ in this paper).",
"We will intensively study the properties of $H(X)$ where $X=\\vec{x}=(x_1,x_2,1-x_1-x_2)$ and so a special name will be reserved for it – the ternary Shannon entropy: $h_3(\\vec{x})\\overset{\\mathrm {df}}{=}-x_1\\log {x_1}-x_2\\log {x_2}-(1-x_1-x_2)\\log {[1-x_1-x_2]}.$ The base of the logarithms is irrelevant but will be set to two throughout the paper.",
"The classical-quantum conditional entropy (entropy conditioned on a classical variable) reads $H(A|Y)=\\sum _{y}p(y)H(\\varpi _A^y)$ .",
"For a classical variable $A=X$ the entropy becomes the standard Shannon conditional entropy $H(X|Y)=-\\sum _{y}p(y)\\sum _xp(x|y)\\log {p(x|y)}$ .",
"Other entropic quantities used in the paper include the classical mutual information $I(X:Y)\\overset{\\mathrm {df}}{=}H(X)+H(Y)-H(XY)=H(Y)-H(Y|X)$ .",
"We will also use the quantum version of the mutual information where one of the registers is quantum and express it as $I(Y:E)=H(E|X) + I(X:E) - H(E|Y)$ .",
"When we say a function $f$ is increasing we mean non-decreasing ($f(x)\\le f(y)$ whenever $x\\le y$ ).",
"Similarly, a decreasing function means a non-increasing function.",
"We will use the convention of [11] for the quadrature operators.",
"They are given by $Q=1/\\sqrt{2}(a+a^\\dagger ),P=1/\\sqrt{2}(a-a^\\dagger )$ and so $\\langle (\\Delta Q)^2\\rangle _\\alpha =\\langle (\\Delta P)^2\\rangle _\\alpha =1/2$ ($\\hbar =1$ ) and $\\langle Q\\rangle _\\alpha =1/\\sqrt{2}(\\alpha +\\bar{\\alpha })$ where $\\alpha =r\\exp {[i\\sigma ]}$ .",
"In our case we have $(\\alpha _x)_{x=0,1,2}$ and $\\sigma _0=0,\\sigma _1=2\\pi /3$ and $\\sigma _2=4\\pi /3$ and $r_i=r$ is a free parameter chosen by the legitimate participants to maximize the secret key rate.",
"A lossy bosonic channel is a Gaussian channel parametrized by has two quantitites.",
"One of them is the transmittance $0\\le \\eta \\le 1$ and the other one the number of thermal photons representing the Gaussian excess noise.",
"For the sake of comparison, we use the definition of excess noise from [11]: $\\delta ={\\langle (\\Delta Q)^2\\rangle _{\\varrho _B}\\over \\langle (\\Delta Q)^2\\rangle _{\\mathop {|0\\rangle }\\nolimits }}-1$ given by Bob's measurement of $\\varrho _B$ .",
"For the simulation scenario we also assume $\\langle (\\Delta Q)^2\\rangle _{\\varrho _B}=\\langle (\\Delta P)^2\\rangle _{\\varrho _B}$ .",
"A quantity called a “mixedness parameter” $\\varepsilon _x\\ge 0$ is upper bounded by Bob's second moments according to (65) of [11] and it is the main estimate of the state in Eve's possession.",
"In our simulation scenario we may setThe variables $\\alpha ,\\delta ,\\varepsilon ,\\gamma $ used in this section should not be confused with those from Sec.",
"., $\\varepsilon \\equiv \\varepsilon _x$ ."
],
[
"Discrete versus Gaussian encoding",
"The most studied QKD schemes are discrete-variable (DV) QKD [1], [2] and continuous-variable (CV) QKD [4] based on a Gaussian encoding.",
"The DV QKD security analysis is very mature but the secret key rates are limited given the discrete nature of the encoding.",
"Higher-dimensional DV QKD scheme have been analyzed [18] but yet to have graduated from the experimental point of view.",
"Gaussian CV QKD offers much generous secret key rates together with a relatively simple experimental realization in terms of the state preparation and detection.",
"But it has also its disadvantages.",
"For instance, the classical postprocessing such as error-correction is computationally demanding and currently not very efficient.",
"The aspiration of CV QKD based on a distribution of discrete signal states holds a promise of combining the best of both worlds.",
"Unlike a Gaussian encoding where the best adversary's strategy is known, the same is not true if the number of signal states is discrete.",
"In fact, to the authors' knowledge, there exists only one paper dealing with the security of such a scheme without assuming nearly anything about the adversary's powers [11].",
"The security proof (and thus the corresponding secret key rate lower bound) is derived by assuming a collective attack and in the asymptotic scenario of an infinite code length.",
"The collective attacks are not the most general eavesdropping scheme.",
"However, it is widely believed that similarly to DV QKD or Gaussian CV QKD, a more general attack strategy does not bring any advantage.",
"For the second point, an asymptotic analysis is not a realistic assumption but it is historically the first step after which a finite-key length analysis typically follows.",
"The number of signal (coherent) states prepared by a sender in [11] is two and the receiver is allowed to measure only the first and second moments of whatever gets through the (unknown) quantum channel.",
"Through a tour-de-force calculation, the authors essentially construct a statistical model of the adversary's quantum states compatible with the legitimate recipient's measurement and maximize the amount of information the adversary can in principle get, following a two-way public discussion.",
"In this way, a secret key rate lower bound is derived.",
"The analysis is achieved by splitting the secret key rate for a reverse reconciliation protocol into three entropic quantities and upper/lower bounding them from the quantities available from the recipient's measurement.",
"In this paper, we follow the same strategy but instead of two signals the communicating parties exchange three coherent signals.",
"This may seems like a small iteration but the opposite is true.",
"We get not only substantially better secret key rate lower bounds but also show the limitation of the approach.",
"The latter point is worth elaborating on.",
"The proof presented in [11] crucially relies on the monotonicity and concavity of the binary Shannon entropy as a function of the absolute value of the overlap of two pure states (not necessarily the signal states).",
"For two signal states, these properties are trivial and they are not proved in [11].",
"The situation dramatically changes for three signal states.",
"Essentially, the result of this paper is the proof that these two crucial properties hold for the ternary Shannon entropy, Eq.",
"(REF ).",
"Only then can the rest of the previous analysis be applied verbatim and that is precisely what we have done.",
"Once these two properties are proven, the rest of the proof follows exactly as in [11] only with a few minor modifications which we will write explicitly.",
"There is a caveat, however.",
"For two signal states, the binary Shannon entropy depends only on the absolute values of the overlap of the signal states.",
"For three states, the ternary entropy depends on three possible overlaps and a certain phase.",
"This wouldn't be a problem if we needed to study the entropy of the density matrix for the signal states only.",
"After all, the participants are those who decide what symmetry (and a probability distribution) the signal states obey and that could greatly simplify the analysis.",
"The problem is that at one point of the previous analysis [11], the purified adversary's state (estimated from Bob's measurement) need not obey any such property and the state must be considered arbitrary.",
"As it is discussed in the first remark of Section REF , in the presence of more than one overlap, the studied function does not even satisfy the (suitably generalized) notion of monotonicity.",
"This is not only surprising but it also affects the applicability of the approach of [11] that we follow here – unlike the case of two signal states, the proof strategy has its limits.",
"Another consequence of our generalization is that unless a generic argument for monotonicity and concavity of the suitable generalized entropy function can be found (taking into account what we have just stated), it is most likely that a completely different approach is needed in order to study discrete CV QKD protocols and their rates for more than three signal states."
],
[
"Description of Ternary Coherent State Protocol",
"Here we outline our ternary (three coherent state) QKD protocol.",
"It goes as follows.",
"Alice prepares one of three possible coherent states $\\mathop {|\\alpha _i\\rangle }\\nolimits $ with probability $p_i=1/3$ , where $i = 0,1,2$ .",
"In Fig.",
"REF , we have a schematic of the phase space depicting how the three coherent states are placed, i.e., sequentially separated by $120^{\\circ }$ .",
"She then sends the randomly selected coherent state to the receiver, Bob, over an insecure quantum channel.",
"It is assumed that this channel could be monitored by Eve.",
"Alice repeats this step many times.",
"Alice's choice for the $i$ th signal (coherent state pulse) is recorded in the variable $x_i$ .",
"Specifically, the labeling goes as: $\\mathop {|\\alpha _0\\rangle }\\nolimits $ is $x_i = 0$ , $\\mathop {|\\alpha _1\\rangle }\\nolimits $ is $x_i = 1$ , and $\\mathop {|\\alpha _2\\rangle }\\nolimits $ is $x_i = 2$ .",
"Bob, upon receiving a sequence of quantum states, randomly performs homodyne detection thereby randomly measuring the quadratures $Q(\\phi )$ for $\\phi =(\\pi /2,-\\pi /6,-5\\pi /6)$ of each of the coherent states.",
"A similar setup was used in [19] but tested on a specific eavesdropping strategy.",
"Bob's measurement results are recorded in the variable $y_i$ .",
"Note that $Q(\\pi /2)\\equiv P$ in Fig.",
"REF .",
"After the transmission, the parties publicly announce the measurement quadratures.",
"One of the quadratures, say $Q(-5pi/6)$ , the measurement data is published which is used to determine the extent of the adversary's maliciousness.",
"These data are subsequently discarded.",
"The remaining data (which we denote as $\\lbrace \\vec{x},\\vec{y}\\rbrace $ ) will be used for the final key generation.",
"For the purpose of reverse reconciliation, Bob sends computes functions $u(\\vec{y})$ and $w(\\vec{y})$ and sends $u(\\vec{y})$ over a public channel to Alice and keeps $w(\\vec{y})$ which is a discrete proto-key (partially correlated with Alice's discrete variable $\\lbrace \\vec{x}$ ).",
"Classical post-processing procedures of error correction and privacy amplification are applied by Alice and Bob in order to extract the final shared secret-key.",
"This final secret bit string is then used as a one-time pad in order to perfectly secure messages."
],
[
"A secret key rate lower bound ",
"In this section, we derive the lower secret key rate for the ternary protocol with respect to a lossy bosonic channel.",
"Mathematically the main results needed for this lower bound (and which are rigorously proven in the Appendix) involve proving that monoticity and concavity both hold for the ternary Shannon entropy, Eq.",
"(REF ).",
"We begin by defining the lower bound of the secret key rate $K$ followed by calculating the individual components of this bound which include Alice and Bob's mutual information and Eve's mutual information.",
"The secret key rate $K$ is lower bounded as $K > I(X:Y) - \\max \\limits _{\\varrho _{ABE}} I(Y:E)$ Eq.",
"(REF ) has its origin in [20] where the one-way private quantum channel capacity was established.",
"The lower bound also differs from [20] in several aspects.",
"(i) The channel is a priori not known and is only partially estimated by the measurements of the legitimate participants.",
"The ambiguity in its identification is an advantage for Eve – the optimization leads to the penalty on the amount of shared secret correlations as if Eve used the best eavesdropping channel compatible with the measurements.",
"This translates into the best channel purification $\\varrho _{ABE}$ held by Eve among all admissible ones in Eq.",
"(REF ), see also Ref. [21].",
"(ii) Our key distribution protocol uses reverse reconciliation where the classical communication (exploited by Eve) is transmitted from Bob to Alice.",
"This results in the appearance of the second term in (REF ) as opposed to [21], [20] dealing with direct reconciliation.",
"(iii) Finally, given the reality of the explicit quantum private code described in Sec.",
", the RHS of (REF ) is a one-shot formula – a natural lower bound to a multi-letter secret key rate formula.",
"A closely related expression for a secret key rate was derived in [22] while focusing solely on the security of QKD."
],
[
"A secret key rate lower bound for a Lossy Bosonic Channel",
"The job here is to maximize the mutual information $I(Y:E)$ in order to find a lower bound on the secret key rate $K$ .",
"In an actual experiment, the classical probability distribution must be measured to be subsequently inserted to the relevant entropic quantities in (REF ).",
"Following [11] we may simulate an actual link by a lossy bosonic channel.",
"This is a realistic model for the atmospheric CV QKD with homodyne measurement.",
"Note that the complementary channel is another lossy bosonic channel and it captures the effect of the environment or an adversary Eve.",
"As is common for QKD, Eve is assumed to control the channel and take an advantage of the generated noise to hide her illicit behavior.",
"As we will see in Section REF , unlike the BPSK case studied in [11] the entropic properties of the investigated density matrix depend not only on the mutual overlaps of the three signal states but also on the overall phase, see the expressions for $d$ in Eq.",
"(REF ) or (REF ).",
"In the simulation scenario for a lossy bosonic channel the phase can be computed as we will show now.",
"We will first consider the zero excess noise case $\\delta ={\\langle (\\Delta Q)^2\\rangle _\\alpha \\over \\langle (\\Delta Q)^2\\rangle _{\\mathop {|0\\rangle }\\nolimits }}-1=0$ (a pure-loss bosonic channel).",
"The estimated quantities become simpler as the recipient's detected states are pure coherent states and similarly for Eve.",
"The parameter $\\varepsilon $ given by (65) in [11] is bounded from above by $U\\equiv U_x=0$ from (65).",
"Hence $\\varepsilon =0$ and (66) together with (C17,C18) of [11] imply $|\\langle \\tilde{\\beta }_i | \\tilde{\\beta }_j \\rangle |=c_u=c_l=\\kappa .$ The RHS is given by $\\kappa \\equiv \\kappa _{ij}=|\\langle \\sqrt{\\eta }\\alpha _i | \\sqrt{\\eta }\\alpha _j \\rangle |$ .",
"Inserting $c_u,c_l$ into (70,71) in [11] we get $d_l=d_u={|\\langle \\alpha _i | \\alpha _j \\rangle |\\over \\kappa }\\overset{\\mathrm {df}}{=}|\\gamma _{ij}|\\equiv |\\gamma |=e^{-{3\\over 2}(1-\\eta )r^2}.$ This quantity is the estimated overlap of the states going to the environment.",
"As expected from the properties of a pure-loss bosonic channel it is the same quantity as $\\kappa $ with $\\eta $ substituted by $1-\\eta $ .",
"We can geometrically interpret the product of inner products in (REF ) (or its special case (REF )) if $\\psi _i$ are coherent states.",
"Then the product $z_{01}z_{12}z_{20}=\\langle \\alpha _0 | \\alpha _1 \\rangle \\langle \\alpha _1 | \\alpha _2 \\rangle \\langle \\alpha _2 | \\alpha _0 \\rangle =e^{-{1\\over 2}(c_{01}^2+c_{12}^2+c_{20}^2)}e^{-i2(A_{01}+A_{12}+A_{20})}$ is written in terms of the sides $c_{ij}$ and area $A_{012}\\overset{\\mathrm {df}}{=}A_{01}+A_{12}+A_{20}$ of the triangle formed by the corresponding three points in phase space.",
"This is the interpretation provided by Lemma REF .",
"We illustrate it on the symmetric case $c_{01}=c_{20}=c_{12}\\equiv c$ of an equilateral triangle for $\\delta =0$ , whose side squared is equal to $c^2=3r^2(1-\\eta )$ found in (REF ).",
"From the new triangle side we deduce, with the help of elementary geometry (essentially Heron's formula), the corresponding area: $A_{012}={1\\over 4}\\big (4c_{01}^2c_{12}^2-(c_{01}^2+c_{12}^2-c_{20}^2)^2\\big )^{1/2}.$ and consequently the phase: $\\vartheta =2A_{012}=r^2{3\\sqrt{3}\\over 2}(1-\\eta )$ .",
"How do we apply it to the $\\delta >0$ case?",
"Here, the situation is slightly different.",
"The effect of a lossy bosonic channel is not only shrinking of the phase space triangle but also increasing the states' variances – environment (Eve) and Bob do not receive a mixture of three pure states but rather of three mixed Gaussian states.",
"Following the general procedure outlined in [11], where only the first and second moments are measured, the overlaps of Eve's state figuring in our simulation scenario are bounded by (70) and (71) in [11].",
"In that case, neither $|\\gamma |$ nor $\\kappa $ are overlaps of the corresponding pure coherent states.",
"More precisely, since Bob measures only the first two moments, the authors of [11] introduced fiducial coherent states $\\mathop {|\\overline{\\beta }_i\\rangle }\\nolimits $ on Bob's side compatible with the measurement of the first moment.",
"Then $\\kappa =|\\langle \\overline{\\beta }_i | \\overline{\\beta }_j \\rangle |$ and as before $\\kappa \\equiv \\kappa _{ij}=|\\langle \\sqrt{\\eta }\\alpha _i | \\sqrt{\\eta }\\alpha _j \\rangle |$ for the case of a lossy bosonic channelAn insight provided by Saikat Guha..",
"This provides the same interpretation for $|\\gamma |$ (Eve's parameters estimated from Bob's measurement) and the phase is then determined according to Lemma REF .",
"The main object of study is a lower bound on the secret key rate, Eq.",
"(REF ).",
"Here we break down the lower bound for the simulated lossy bosonic channel.",
"The central role is played by the ternary Shannon entropy, Eq.",
"(REF ), where $x_k=t_k+1/3$ and $t_k$ is given by (REF )."
],
[
"Eve's and Alice's Mutual Information, $I(X:E)$ ",
"Closely following [11], to get a secret key lower bound, the first quantity to estimate is $I(X:E)<I(X:QE)=h_3(\\vec{x}(Z))$ for $x_k$ restricted to $p_k=1/3$ and $\\langle \\Psi _{EQ}^i | \\Psi _{EQ}^j \\rangle =Z_{ij}={Z}\\exp [{i\\tilde{\\tau }_{ij}}],\\,Z>0$ .",
"As explained in the remark on p. REF , the restriction to $|Z_{ij}|=Z$ is a necessary step for the proof strategy following [11] to go through.",
"Then, from (REF ), we get the explicit form of $x_k$ : $x_1 & ={1\\over 3}\\Big (1+2Z\\cos {\\vartheta \\over 3}\\Big ), \\\\x_{2,3} & ={1\\over 3}\\Big (1-Z\\big (\\cos {\\vartheta \\over 3}\\mp \\sqrt{3}\\sin {\\vartheta \\over 3}\\big )\\Big ).$ Denoting $f\\equiv f_{ij}=F(\\varrho _E^i,\\varrho _E^j)$ to be the fidelity of $\\varrho ^{i(j)}_E=\\mathop {{\\mathrm {Tr}}_{Q}}[\\Psi ^{i(j)}_{EQ}]$ we get $h_3(\\vec{x}(Z,\\vartheta ))\\le h_3(\\vec{x}(f,\\vartheta ))\\le h_3\\big (\\vec{x}((1-\\tilde{\\varepsilon }_0)^{1/2}(1-\\tilde{\\varepsilon }_1)^{1/2}|\\gamma |,\\vartheta )\\big )$ where $0\\le \\tilde{\\varepsilon }_i\\le \\varepsilon $ .",
"The second inequality follows from the proof of monotonicity, Theorem REF , as a special case $p_k=1/3$ .",
"When restricted to the simulation scenario of a lossy bosonic channel, the parameter $\\vartheta $ is a phase whose value we determine with the help of Lemma REF .",
"Before doing so, recall that for $\\delta =0$ the lossy bosonic channel merely “shrinks” the triangle representing the mixture of three coherent states in phase space and the shrinking factor is $1-\\eta $ for Eve's system (see (REF )).",
"Consequently, $\\varrho _E^i$ are pure and Eq.",
"(REF ) can be interpreted as the modulus of their overlap.",
"The next expression used for the secret key estimation is the conditional entropy $H(E|X)$ .",
"It is upper bounded by [11] ${1\\over 3}\\sum _{x}{(1+V_x)\\log {[1+V_x]}-V_x\\log {V_x}},$ where $V_x=\\big (\\langle (\\Delta Q)^2\\rangle _{\\varrho _B}\\langle (\\Delta P)^2\\rangle _{\\varrho _B}\\big )^{1/2}-1/2$ .",
"In the case of a lossy bosonic channel we find $V_x=\\delta /2$ .",
"The third expression needed to be evaluated from the secret key lower bound is $H(E|Y)$ in (62) from [11].",
"In order to do so we have to generalize the conditional probability distribution related to the action of a lossy bosonic channel.",
"We cannot simply take the derived expressions in [11] since for three and more signal states the states cannot all be aligned with a real line in phase space.",
"Instead, we introduce $p(y|x)={1\\over \\pi (1+\\delta )}\\exp {\\bigg [-{|y-\\sqrt{\\eta }\\alpha _x|^2\\over \\delta +1}\\bigg ]}={1\\over \\pi (1+\\delta )}\\exp {\\bigg [-{|y|^2+\\eta r^2-2|y|r\\sqrt{\\eta }\\cos {[\\phi -\\sigma _x]}\\over \\delta +1}\\bigg ]},$ where $y=|y|\\exp {[i\\phi ]}$ and $\\alpha _x=r\\exp {[i\\sigma _x]}$ .",
"For three signal states we take the values of $\\sigma _{0,1,2}$ introduced in Section .",
"To simulate the channel we further use $p(x|y)={1\\over 3}p(y|x)/p(y)$ together with $p(y)=\\sum _{x=0,1,2}p(y|x)p(x)={1\\over 3}{1\\over \\pi (1+\\delta )}\\sum _{x=0,1,2}\\exp {\\bigg [-{|y-\\sqrt{\\eta }\\alpha _x|^2\\over \\delta +1}\\bigg ]}.$ Hence, for example, $p(0|y)={\\exp {\\Big [-{|y-\\sqrt{\\eta }\\alpha _0|^2\\over \\delta +1}\\Big ]}\\over \\sum _{x=0,1,2}\\limits \\exp {\\Big [-{|y-\\sqrt{\\eta }\\alpha _x|^2\\over \\delta +1}\\Big ]}}.$ A straightforward generalization of the derivation of Eqs.",
"(56) and (57) in [11] allows us to lower bound $H(E|Y)$ .",
"The final component is the classical mutual information $I(X:Y)=H(X)-H(X|Y)$ calculated with the help of $p(x|y)$ and $p(y)$ defined above.",
"Now we have all the ingredients we need to find the actual secret key rate lower bound.",
"It is expression (72) given in [11], adapted to the TPSK encoding.",
"It can be written as $K & > \\underbrace{\\log {3}-\\int _{0}^\\infty \\mathrm {\\,d} |y||y|\\int _0^{2\\pi }\\mathrm {\\,d} \\phi p(y)\\sum _{x=0,1,2}p(x|y)\\log {[p(x|y)]}}_{I(X:Y)}\\nonumber \\\\&-\\underbrace{\\big ((1+\\delta /2)\\log {[1+\\delta /2]}-\\delta /2\\log {[\\delta /2]}\\big )}_{H(E|X)}-\\max _{0\\le \\tilde{\\varepsilon }\\le \\varepsilon }\\Bigg [\\underbrace{h_3\\big (\\vec{x}((1-\\tilde{\\varepsilon })|\\gamma |,\\vartheta )\\big )}_{H(X:E)}\\\\&\\left.\\begin{array}{@{}l}\\nonumber {\\displaystyle - \\int _{0}^\\infty \\mathrm {\\,d} |y||y|\\int _0^{2\\pi }\\mathrm {\\,d} \\phi p(y)h_3(\\vec{x}(|\\gamma |,\\vartheta ,p(0|y),p(1|y)))}\\\\{\\displaystyle +\\sum _{x=0,1}\\Bigg [\\bigg ({\\tilde{\\varepsilon }\\over 3}{1+|\\gamma |\\over 1-|\\gamma |}\\bigg )^{1/2}\\Bigg (\\int _{0}^\\infty \\mathrm {\\,d} |y||y|\\int _0^{2\\pi }\\mathrm {\\,d} \\phi p(y){h_3^2\\big (\\vec{x}(|\\gamma |,\\vartheta ,p(0|y),p(1|y))\\big )\\over p(x|y)}\\Bigg )^{1/2}\\Bigg ]}\\\\{\\displaystyle +{\\tilde{\\varepsilon }\\over 1-|\\gamma |}h_3\\big (\\vec{x}(|\\gamma |,\\vartheta ,{1/3},1/3)\\big )}\\end{array}\\!\\!\\right\\rbrace &\\!\\!\\!-H(E|Y).$ For ease of sight we identified the origin of the summands by the expressions in the braces.",
"Figure: Secret key rates as functions of loss 1-η1-\\eta for several values of the channel excess noise δ=(0,0.0004,0.001,0.005,0.01)\\delta =(0,0.0004,0.001,0.005,0.01) (the pink dots).",
"The black curve is the ultimate achievable bound without an energy constraint for δ=0\\delta =0.",
"The orange curve is an achievable bound for δ=0\\delta =0 taking into account the input energy constraint .",
"All curves are functions of the channel loss.The main technical result of this paper – the proofs of monotonicity and concavity of the ternary Shannon entropy – participate in the derivation of $H(E|Y)$ .",
"The reasoning is nearly a verbatim copy of Section IV.C and the Appendices A and C of [11] implying the conditional entropy to be a lower bound on the secret key rate $K$ .",
"In Fig.",
"REF we present the main result of our analysis (applied to a simulated lossy bosonic channel).",
"We plot the secret key lower bound, Eq.",
"(REF ), for several values of the excess noise parameter.",
"Compared to [11], we find better lower bounds as expected from the use of three signals states but also much better threshold values where the rate is zero.",
"It therefore supports the idea that to approach the high rates given by a continuous Gaussian encoding, one would need only a reasonably small number of signal states.",
"This cannot, strictly speaking, be correct for the vicinity of $\\eta =1$ .",
"It is known that the ultimate upper bound for the two-way secret key rate at the presence of zero excess noise is equal to $K=-\\log {[1-\\eta ]}$ [6], a quantity diverging for $\\eta \\rightarrow 1$ .",
"Clearly, for any finite number of discrete signal states $d$ , the maximal secret key rate for $\\eta =1$ is $\\log {d}$ like in our case $d=3$ .",
"Ref.",
"[6] also provided an achievable bound (actually a lower bound based on [7]) by taking into account the input energy constraint.",
"This is depicted in Fig.",
"REF as the orange dotted curve for $\\delta =0$ .",
"The `stairs' on this curve are the consequence of a different optimal energy (input state overlap leading to a different input energy constraint) shown in Fig.",
"REF .",
"An important fact to realize is that even though we have only proved monotonicity and concavity of $h_3$ for $0\\le \\vartheta \\le \\pi /2\\Leftrightarrow \\varepsilon \\le 0$ (for $\\varepsilon $ given by ()), it does not affect the secret key rate lower bound.",
"The optimal input energy falls inside the region $\\varepsilon \\le 0$ .",
"The situation is also depicted in Fig.",
"REF .",
"Figure: The red dots depict the optimizing overlaps rr for δ=0\\delta =0.",
"The black curve is a boundary ε=0\\varepsilon =0 of () (ϑ=π/2\\vartheta =\\pi /2) given by r 2 33 2η=π/2r^2{3\\sqrt{3}\\over 2}\\eta =\\pi /2 (see below Eq.",
"()) below which the proofs of monotonicity and concavity exist (ε≤0\\varepsilon \\le 0)."
],
[
"Differences in an actual QKD experiment",
"The real-world scenario introduces further complications.",
"The channel may not be lossy bosonic (it may not even be described by a stationary process for the duration of the experiment but we will avoid this type of complications).",
"For a stationary channel and in the asymptotic scenario the participants collect enough statistics to reconstruct the channel to estimate the conditional probability distributions $p(y|x)$ and $p(y)$ arbitrarily well.",
"The same applies to the BPSK analysis from [11] but as we already alluded to, there is more degrees of freedom in the ternary case.",
"There are in total three overlaps in the form of three real parameters for a general triple of coherent pure states and in addition there is a phase.",
"In the simulation scenario of a lossy bosonic channel the overlaps if chosen symmetrically by Alice (our assumption) and the phase can be subsequently calculated as done in the previous sectionNote that similarly to [11] we not only calculate the entropy of the input density matrix but also of other, say intermediate, density matrices in order to lower bound the secret key rate.",
"Even there the three real parameters coincide (they can't be interpreted as overlaps, though, see below Eq.",
"(REF )) and the phase can be calculated for a lossy bosonic channel.",
"But in for an actual experiment we can only assume the symmetry of a density matrices directly prepared by Alice.",
"The states where Eve can in principle intervene has no a priori symmetry which translates into their entropy to be dependent on three plus one free parameters.",
"As it turns out (see the discussion in Sec.",
"REF ), the key property of monotonicity of the ternary Shannon entropy does not hold in general and the strategy to lower bound the secret key rate from [11] must be abandoned.",
"How do we overcome this problem here?",
"If the parameters measured by Bob indicate that the incoming states are not symmetrically distributed, the participants assume the closest symmetric distribution that gives Eve the biggest advantage.",
"One could be tempted to take the smallest of the three overlaps and create a symmetric distribution based on it.",
"However, as the example in [23] shows, the entropy of such a density matrix does not necessarily becomes smaller thus indicating more distinguishable quantum states.",
"So a better strategy to introduce a single overlap is called for and it will necessarily reduce the secret key rate.",
"But only this is the situation for which we can follow the proof in [11] once the monotonicity and concavity of the ternary Shannon entropy is proven.",
"The worst case scenario happens if Bob detects only two states, that is, if the channel is so disruptive that it managed to merge two signal states to one quantum state.",
"In that case the secret key rate would be zero and it would probably be better to switch to BPSK.",
"How do we recover the other free parameter, namely the angle?",
"Similarly to the lossy bosonic case, a triple of fiducial coherent states $(\\mathop {|\\overline{\\beta }_i\\rangle }\\nolimits )_{i=0,1,2}$ with the same absolute value of the overlap is introduced.",
"We assume that the triple properly bounds the entropies as described in the previous paragraph, so that the advantage is given to Eve resulting in the key rate reduction.",
"Then we followed the procedure of phase calculation described below Eq.",
"(REF ) following Lemma REF .",
"This is the right phase for the fiducial triple of pure coherent states."
],
[
"Conclusion",
"In conclusion, we introduced and rigorously proved the asymptotic security of a new ternary QKD protocol based on three coherent states and homodyne detection.",
"The motivation for introducing such a protocol is to extract a best-of-both-world's approach to QKD in terms of the encoding and decoding of discrete variable schemes along with the practical hardware of continuous variable schemes.",
"There is, however, the downside that the security proof is very challenging compared to the results for Gaussian modulated continuous-variable QKD protocols.",
"We overcame this challenge by mathematically proving that two crucial properties, monotonicity and concavity, hold for the ternary Shannon entropy.",
"This allowed us to evaluated a lower bound to the secret key rate in the collective attack scenario.",
"Other interesting avenues of research could include considering a four-state extension (if possible, or perhaps using a different method), determining what number of signal states are enough to tend close to the full Gaussian distribution and also a thorough finite-key analysis.",
"This is a lively area of research for many classes of bosonic channels where the lossy bosonic channel is an important subclass [6], [24].",
"A measurement-device-independent (MDI)-QKD [25], [26], [27] version of our scheme presented here would also be interesting as a way of ruling out side channel attacks."
],
[
"Acknowledgement",
"We would like to thank Saikat Guha for helpful discussions.",
"The authors acknowledge support from the U.S. Office of Naval Research (ONR).",
"This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0083.",
"The authors thank Saikat Guha for valuable comments and discussions."
],
[
"Properties of ternary density matrix",
"In this section, we give the calculations needed to prove the main results.",
"To begin with, let $\\varpi =p_0| \\psi _0\\rangle \\!\\langle \\psi _0 |+p_1| \\psi _1\\rangle \\!\\langle \\psi _1 |+p_2| \\psi _2\\rangle \\!\\langle \\psi _2 |$ be a rank-three density operator where $p_0+p_1+p_2=1$ .",
"The state $\\varpi $ takes on a different meaning depending on where it is used.",
"It can be an input density matrix a sender prepares in a lab in which case $p_k=1/3$ and $\\psi _k$ are the signal (coherent) states with a chosen symmetry.",
"Or, it can be Eve's conditioned state based on Bob's measurement.",
"In that case, $p_k$ are arbitrary conditional probabilities $p_k(x|y)$ and $\\psi _k$ are pure states with no obvious symmetry properties [11].",
"Following the Cayley-Hamilton theorem, one finds the coefficients of the characteristic polynomial $\\det {[\\varpi -x\\mathop {{\\mathrm {id}}}\\nolimits ]}=f(x)=ax^3+bx^2+cx+d=0,$ where $a & =1, \\\\b & =-\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ]=-1,\\\\c & ={1\\over 2}((\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ])^2-\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2])={1\\over 2}(1-\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2]),\\\\d & =-{1\\over 6}((\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ])^3-{3}\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ]\\,\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2]+2\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^3])=-{1\\over 6}(1-3\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2]+2\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^3]).$ The last two coefficient become $c & ={1\\over 2}(1-p_0^2-p_1^2-p_2^2-2p_0p_1|z_{01}|^2-2p_1p_2|z_{12}|^2-2p_0p_2|z_{02}|^2),\\\\d & =\\frac{1}{6} \\Big (-1+3 \\left( p_0^2+ p_1^2+p_2^2+2 {p_0} {p_1} |z_{01}|^2+2 {p_0} {p_2} |z_{02}|^2+2 {p_1} {p_2} |z_{12}|^2\\right)\\nonumber \\\\&\\quad -2 \\big ( p_0^3+p_1^3+p_2^3+3 (p_0^2 p_1+p_0p_1^2) |z_{01}|^2+3 (p_0^2 {p_2}+{p_0}p_2^2)|z_{02}|^2+3(p_1^2{p_2}+{p_1}p_2^2)|z_{12}|^2\\nonumber \\\\&\\quad +3{p_0}{p_1}p_2(z_{01}z_{12}z_{20}+c.c)\\big )\\Big ).$ Note that $\\varpi $ in all its roles in the security proof if always a sum of rank-one operators.",
"Hence the trace quantities in Eqs.",
"(REF ) are easy to find.",
"An additional check was performed by calculating the quartic term ${1\\over 24}\\big ((\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ])^4-6(\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ])^2\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2]+3(\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^2])^2 + 8\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ]\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^3]-6\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ^4]\\big )$ and was found to be zero as it should be.",
"We set the overlaps to be $\\langle \\psi _i | \\psi _j \\rangle =z_{ij}=|z|\\exp [{i\\tau _{ij}}]$ and get $c & ={1\\over 2}\\big (1-p_0^2-p_1^2-p_2^2-|z|^2(2p_0p_1+2p_1p_2+2p_0p_2)\\big ),\\\\d & =\\frac{1}{6} \\Big (-1+3 \\left( p_0^2+ p_1^2+p_2^2+2|z|^2( {p_0} {p_1} + {p_0} {p_2} + {p_1} {p_2})\\right)\\nonumber \\\\&\\quad -2 \\big ( p_0^3+p_1^3+p_2^3+3 (p_0^2 p_1+p_0p_1^2) |z|^2+3 (p_0^2 {p_2}+{p_0}p_2^2)|z|^2+3(p_1^2{p_2}+{p_1}p_2^2)|z|^2\\nonumber \\\\&\\quad +6|z|^3{p_0}{p_1}p_2\\cos {\\vartheta } \\big )\\Big ),$ where $\\vartheta =\\tau _{01}+\\tau _{12}+\\tau _{20}$ .",
"The absolute value $|z|$ and the angle $0\\le \\vartheta \\le \\pi $ are not independent and we will revisit the relation below Eq.",
"(REF ) (see also Lemma REF ).",
"Remark It may seem that by setting $|z_{ij}|=|z|,\\forall i,j$ we limit ourselves to a special case of $\\varpi $ .",
"This is indeed true.",
"Quite surprisingly, however, it is the most general case for which one of the studied properties (monotonicity) actually holds.",
"It turns out that the multivariable function studied in this paper, the ternary Shannon entropy (Eq.",
"(REF )), is not monotone decreasing unless $|z_{ij}|=|z|,\\forall i,j$ in which case it reduces to the standard single-variable problem.",
"What does it mean for a multivariable function to be monotone increasing/decreasing?",
"This question is closely related to the existence of sets that cannot be totally ordered (totality means that either $x\\le y$ or $y\\ge x$ holds).",
"An example is $\\mathbb {R}^n$ for $n>1$ which is only a partially ordered set.",
"To this end, one defines the componentwise order [28] of two $n$ -tuples $(x_1,\\dots ,x_n)\\le (y_1,\\dots ,y_n)$ iff $x_i\\le y_i,\\forall i$ .",
"A monotone increasing or decreasing function $f:\\mathbb {R}^n\\mapsto \\mathbb {R}^m$ then satisfies $f(x_1,\\dots ,x_n)\\le f(x_1,\\dots ,x_n)$ and $f(x_1,\\dots ,x_n)\\ge f(x_1,\\dots ,x_n)$ , respectively.",
"The lack of this property (namely not decreasing) means that the strategy outlined in [11] we follow here is simply not applicable.",
"Coefficients, Eqs.",
"(REF ), are used to get the eigenvalues of $\\varpi $ .",
"Following [29] (or Wikipedia for a quick summary) we form $\\Delta _0 & = b^2 - 3 a c=1 - 3 c,\\\\\\Delta _1 & = 2 b^3 - 9 a b c + 27 a^2 d=-2 + 9 c + 27 d$ and define $p & =-{\\Delta _0\\over 3}=\\alpha +\\beta z^2, \\\\q & ={\\Delta _1\\over 27}=\\gamma +\\delta z^2+\\varepsilon z^3.$ They are the coefficients of a reduced cubic $t^3+pt+q$ the general cubic polynomial $f(x)$ can be converted to.",
"The coefficients of $p,q$ from Eqs.",
"REF are given by $\\alpha & = {1\\over 6}(1-3 p_0^2-3p_1^2-3p_2^2)\\le 0,\\\\\\beta & = -(p_0p_1+p_0p_2+p_1p_2)\\le 0,\\\\\\gamma & = {1\\over 27}(-2 + 9 p_0^2 - 9 p_0^3 + 9 p_1^2 - 9 p_1^3 + 9 p_2^2 - 9 p_2^3)\\nonumber \\\\& = {1\\over 27}(3p_1-1)(3p_2-1)(3p_1+3p_2-2)\\lessgtr 0,\\\\\\delta & = {1\\over 27} \\big (18 (p_0 p_1+p_0 p_2+ p_1p_2)- 27 (p_0^2 p_1+p_0 p_1^2+p_0^2 p_2+p_1^2 p_2+p_0p_2^2+p_1p_2^2)\\big )\\le 0 , \\\\\\varepsilon & = -2p_0p_1p_2\\cos {\\vartheta }\\lessgtr 0,$ where we also summarized some basic properties based on $0\\le p_i\\le 1,\\sum _ip_i=1$ .",
"Then, the three roots (the eigenvalues of $\\varpi $ ) are $x_k=t_k-b/(3a)=t_k+1/3$ where $t_k=2\\sqrt{-{p\\over 3}}\\cos {\\bigg ({1\\over 3}\\arccos {\\bigg ({3\\over 2}{q\\over p}\\sqrt{-{3\\over p}}\\bigg )}-{2k\\pi \\over 3}\\bigg )}.$ It is known [29] that $t_0+t_1+t_2 &= 0, \\\\t_0\\ge t_1&\\ge t_2$ hold.",
"Hence $x_0+x_1+x_2=1$ as we expect from $\\mathop {{\\mathrm {Tr}}_{}}[\\varpi ]=1$ but $x_2\\ge 0$ is not satisfied for all $|z|$ and $\\vartheta $ .",
"For example, if $\\psi _1=e^{i\\varphi _1}\\psi _0,\\psi _2=e^{i\\varphi _2}\\psi _0$ then $|z|=1$ and $\\vartheta =\\tau _{01}+\\tau _{12}+\\tau _{20}=0$ .",
"In general, it turns out that $x_2\\ge 0$ is equivalent to $q\\le {1\\over 27}+{p\\over 3}$ which provides a bound on $\\vartheta $ given $|z|$ .",
"Indeed, for $|z|=1$ the only possibility is $\\vartheta =0$ .",
"Something much stronger can be said about the phases if $\\psi _i$ are actual coherent states (either the signal states or the fiducial states we mentioned in the main text).",
"Lemma 1 The phase $\\mathrm {Arg[\\langle \\alpha _i | \\alpha _j \\rangle ]}$ of an inner product of two coherent states $\\mathop {|\\alpha _i\\rangle }\\nolimits $ and $\\mathop {|\\alpha _j\\rangle }\\nolimits $ is a function of $|\\langle \\alpha _i | \\alpha _j \\rangle |$ .",
"Using elementary trigonometry we write $\\langle \\alpha _i | \\alpha _j \\rangle =e^{-{1\\over 2}(r_i^2+r_j^2-2r_ir_j\\cos {[\\sigma _j-\\sigma _i]})}e^{-i2r_ir_j\\sin {[\\sigma _j-\\sigma _i]}}=e^{-{1\\over 2}c_{ij}^2}e^{-i2A_{ij}},$ where $c_{ij}$ a side of triangle opposite to the angle $\\tau _{ji}=\\sigma _j-\\sigma _i$ between the sides $r_i$ and $r_j$ and $A_{ij}$ is the triangle area (it is oriented since $A_{ij}=-A_{ji}$ ).",
"But knowing $r_i,r_j,c_{ij}$ , we can easily calculate the area of the triangle and hence the phase $\\mathrm {Arg{\\,[\\langle \\alpha _i | \\alpha _j \\rangle ]}}=-2A_{ij}=2A_{ji}$ .",
"Hence the phase is much more constrained if $\\psi _i$ are coherent states.",
"This trivial statement (we could also use the relation between $\\sin {}$ and $\\cos {}$ to get the phase) has interesting consequences we exploited in Eq.",
"(REF )."
],
[
"Monotonocity of the ternary Shannon entropy",
"The following result will be a useful tool in the course of our analysis.",
"Theorem 2 (Descartes' rule of signs [30], [31]) Let $p(x)=\\sum _{m=0}^{s}a_{n-m}x^{n-m}$ be a real polynomial of order $n$ where $s\\le n$ and $a_{n-m}\\ne 0$ .",
"Then the number of positive real zeros (including multiplicities) is equal to $V-2k$ where $k\\ge 0$ and $V$ is the number of sign variations of $a_{n-m}$ starting from $a_n$ .",
"Lemma 3 Let $\\varepsilon \\le 0$ .",
"Then $q(z)$ in (REF ) is monotone-decreasing and concave in $z\\in (0,1)$ for all $p_k$ .",
"It has a single positive root $z^\\#\\in (0,1)$ iff $\\gamma >0$ in which case $q(z)\\ge 0$ for $z\\in (0,z^\\#)$ .",
"The monotonicity of $q$ follows from $q^{\\prime }=2\\delta z+3\\varepsilon z^2,$ since $\\delta ,\\varepsilon \\le 0$ .",
"Because of Theorem REF (or just by inspection), there is no positive root of $q(z)$ for $\\gamma <0$ again following from $\\delta ,\\varepsilon \\le 0$ .",
"There is one positive root for $\\gamma >0$ and it has to lie in the interval $(0,1]$ since $q(0)=\\gamma >0$ and $q(1)=\\gamma +\\delta +\\varepsilon \\le \\gamma +\\delta =-{2\\over 27}+2p_0p_1p_2\\le 0$ valid for all $p_k$ .",
"Remark Even more straightforward is to show $p<0$ in $z\\in (0,1)$ (follows from Eqs.",
"(REF ), (REF ) and () by considering $(p_0+p_1+p_2)^2=1$ ).",
"The equality $p=0$ is achieved for $z=0$ and $p_0=p_1=p_2=1/3$ but in order to have future expressions well-defined we will consider the open interval $z\\in (0,1)$ throughout this work.",
"Similarly, we find $p^{\\prime }\\le 0$ .",
"It is useful to know the generic behavior of the central piece of the cubic solutions, Eq.",
"(REF ).",
"That is uncovered in the following lemma.",
"Lemma 4 Let $\\varepsilon \\le 0$ and $g(z)={3\\over 2}{q\\over p}\\sqrt{-{3\\over p}}.$ Then $|g(z)|\\le 1$ , $g(z)\\propto -q(z)$ and $g^{\\prime }\\lessgtr 0$ for $z\\in (0,1)$ .",
"The bound $|g(z)|\\le 1$ follows from the cubic equation discriminant ${q^2\\over 4}+{p^3\\over 27}\\le 0,$ where the inequality is always true for the case of three real roots of a cubic equation [29].",
"This, on the other hand, must be true since $\\varpi $ is a density matrix.",
"Eq.",
"(REF ) can be both positive and negative with its sign always opposite to that of $q(z)$ .",
"This is because ${1\\over p}\\sqrt{-{3\\over p}}<0$ for $z\\in (0,1)$ following from Lemma REF .",
"A related useful fact is that for $\\gamma <0$ we get $g(z)>0$ for $z\\in (0,1)$ .",
"Finally, by writing $g^{\\prime }={3\\sqrt{3}\\over 4}{2pq^{\\prime }-3p^{\\prime }q\\over p^2\\sqrt{-p}}$ and noticing that the denominator is nonnegative we only need to study the behavior of $\\nu _1(z)=2pq^{\\prime }-3p^{\\prime }q$ .",
"First, we find a zero root due to $\\nu _1(z)=-2z\\left(-2\\alpha \\delta +3\\beta \\gamma -3z\\alpha \\varepsilon +z^2\\beta \\delta \\right)$ .",
"The quadratic equation $-2\\alpha \\delta +3\\beta \\gamma -3z\\alpha \\varepsilon +z^2\\beta \\delta =0$ yields two other real roots and, in general, they both may lie in the interval $(0,1)$ .",
"Only when $\\gamma \\ge 0$ , one of the roots is negative.",
"Lemma 5 Let $\\tau (z,n)=\\sqrt{-p}\\cos {h\\over n}$ and $n\\in \\mathbb {Z}_{>1}$ such that $p,p^{\\prime }<0$ , $0\\le h\\le \\pi $ in $z\\in (0,1)$ and $h^{\\prime }>0$ in $\\euI \\subset (0,1)$ .",
"Then ${\\mathrm {\\,d} \\tau (z,n)\\over \\mathrm {\\,d} z}>{\\mathrm {\\,d} \\tau (z,2)\\over \\mathrm {\\,d} z}$ in $\\euI $ .",
"We find ${\\mathrm {\\,d} \\tau (z,n)\\over \\mathrm {\\,d} z}=\\frac{2ph^{\\prime }\\sin {h\\over n}-np^{\\prime }\\cos {h\\over n}}{2n\\sqrt{-p}}.$ The denominator is positive for $z\\in (0,1)$ but there are two competing expressions in the numerator.",
"The first summand is negative, the second one is nonnegative and so the overall sign may be hard to infer.",
"The inequality follows by observing that the nonnegative summand in the numerator of (REF ) remains constant as $n$ increases while the negative one is divided by $n$ and so its overall contribution diminishes.",
"Finally, $\\sin {h\\over n}$ and $\\cos {h\\over n}$ do not change their sign with a growing $n\\ge 2$ and, conveniently, $\\sin {h\\over n}>\\sin {h\\over n+1}$ holds together with $\\cos {h\\over n}<\\cos {h\\over n+1}$ for $n\\ge 2$ as illustrated in Fig.",
"REF .",
"Figure: Properties of some trigonometric functions.Proposition 6 The function $t_0$ is monotone-increasing in $z\\in (0,1)$ for $\\varepsilon \\le 0$ .",
"From (REF ) we get $t^{\\prime }_0={2\\over 3}\\frac{\\sqrt{-p(z)} g^{\\prime }(z) \\sin {\\left(\\frac{1}{3} \\arccos {(g(z))}\\right)}}{ \\sqrt{1-g(z)^2}}-\\frac{p^{\\prime }(z) \\cos {\\left(\\frac{1}{3} \\arccos {(g(z))}\\right)}}{\\sqrt{-p(z)}}.$ Both denominators are non-negative (check zero).",
"Since $|g(z)|\\le 1$ and $\\operatornamewithlimits{ran}{[\\arccos ]}=[0,\\pi ]$ , both trigonometric functions are nonnegative.",
"So the second summand (without the minus) is always negative due to $p^{\\prime }\\le 0$ .",
"The overall expression is thus positive whenever $g^{\\prime }\\ge 0$ .",
"But from Lemma REF we know that $g^{\\prime }<0$ can occur as well so let us assume that for the rest of the proof.",
"For $h\\overset{\\mathrm {df}}{=}\\arccos {g}$ we first observe $h^{\\prime }=-{g^{\\prime }\\over \\sqrt{1-g^2}}$ and so $h^{\\prime }>0$ .",
"We now summon Lemma REF and for it to be useful we show ${\\mathrm {\\,d} \\tau (z,2)\\over \\mathrm {\\,d} z}\\ge 0$ .",
"The case $n=2$ is special since we can use the half-angle formula $\\cos {h\\over 2}=\\sqrt{1+\\cos {h}\\over 2}$ (valid for $-\\pi \\le h\\le \\pi $ ) to be inserted into $\\tau (z,2)=\\sqrt{-p}\\cos {h\\over 2}$ and we get ${\\mathrm {\\,d} \\tau (z,2)\\over \\mathrm {\\,d} z}={ph^{\\prime }\\sin {h}-p^{\\prime }(1+\\cos {h})\\over 2\\sqrt{-2p(1+\\cos {h})}}={-pg^{\\prime }-p^{\\prime }(1+g)\\over 2\\sqrt{-2p(1+g)}}=-{\\left(p(1+g)\\right)^{\\prime }\\over 2\\sqrt{-2p(1+g)}}\\ge 0.$ The inequality $(p(1+g))^{\\prime }\\le 0$ follows from $p(1+g)$ being monotone-decreasing since both $p$ and $pg\\approx q\\sqrt{-3/p}$ are monotone-decreasing (from Lemma REF we know that for $\\varepsilon \\le 0$ the function $q\\lessgtr 0$ is monotone-decreasing and $\\sqrt{-3/p}$ as well and because $\\min _{p_k,z}\\limits {[\\sqrt{-3/p}]}=3$ then $\\sqrt{-3/p}$ merely “stretches” $q$ ).",
"A sum of decreasing functions is decreasing which concludes the proof since according to Lemma REF we have $t_0^{\\prime }{\\sqrt{3}\\over 2}\\equiv {\\mathrm {\\,d} \\tau (z,3)\\over \\mathrm {\\,d} z}>{\\mathrm {\\,d} \\tau (z,2)\\over \\mathrm {\\,d} z}\\ge 0.$ Proposition 7 The function $t_2$ is monotone-decreasing in $z\\in (0,1)$ for $\\varepsilon \\le 0$ .",
"Considering $t_k$ in (REF ) as functions of $p$ and $q$ , it is known [29] that $t_2(p,q)=-t_0(p,-q)$ .",
"The mapping $q\\mapsto -q$ changes the sign of $g$ (and so of $g^{\\prime }$ as well, see (REF ) and (REF )).",
"The proof of Proposition REF then goes through in the same way as for $t_0(p,-q)$ since the trigonometric functions in (REF ) remain nonnegative for $-g$ and the proof “covers” both cases $g^{\\prime }\\ge 0$ and $g^{\\prime }<0$ .",
"Hence, $t^{\\prime }_0(p,-q)>0$ and we conclude that $t^{\\prime }_2<0$ .",
"Corollary 8 The function $t_0+t_1$ is monotone-increasing in $z\\in (0,1)$ for $\\varepsilon \\le 0$ .",
"From Eq.",
"(REF ) we find $t_0+t_1=-t_2$ and because $t_2$ is monotone-decreasing, its negative is monotone-increasing.",
"Remark Notice that we do not claim anything about the monotonocity of $t_1$ and indeed it in general does not hold.",
"Definition 1 ([32]) A function is called Schur-concave iff it is concave and symmetric.",
"Definition 2 Let $\\vec{u}$ be an $\\ell $ -tuple for a non-increasingly ordered sequence $u_0\\ge \\hdots \\ge u_{\\ell -1}$ denoted as $u_k^\\downarrow $ where $u_k\\ge 0$ .",
"We say that $\\vec{u}$ is majorized by $\\vec{v}$ (written as $\\vec{u}\\prec \\vec{v}$ ) iff $\\sum _{k=0}^{m-1} u_k^\\downarrow &\\le \\sum _{k=0}^{m-1} v_k^\\downarrow , \\\\\\sum _{k=0}^{\\ell -1} u_k & = \\sum _{k=0}^{\\ell -1} v_k$ is satisfied for $0\\le m\\le \\ell -1$ .",
"Theorem 9 ([32], Karamata [33]) If a function $f(u)$ is concave then $ f(\\vec{u})\\overset{\\mathrm {df}}{=}\\sum _k f(u_k)$ is Schur-concave and $\\vec{u}\\prec \\vec{v}\\Rightarrow f(\\vec{u})\\ge f(\\vec{v}).$ Remark The function $f(u)=-u\\log {u}$ is concave in $u\\in (0,1)$ and therefore the Shannon entropy $S(\\vec{u})\\overset{\\mathrm {df}}{=}-\\sum _{k=0}^Ku_k\\log {u_k}$ is a Schur-concave function.",
"For $K=2$ , we obtain the ternary Shannon entropy $h_3(\\vec{u})$ , Eq.",
"(REF ).",
"Theorem 10 The von Neumann entropy $H(\\varpi )$ of density matrix (REF ) is a monotone-decreasing function of the overlap $z$ as introduced in (REF ), for all $p_k$ and for all $0\\le \\vartheta \\le \\pi /2$ corresponding to $\\varepsilon \\le 0$ in ().",
"The eigenvalues of $\\varpi $ are $x_0\\ge x_1\\ge x_2$ and satisfy $\\sum _kx_k=1$ .",
"From the discussion below Eq.",
"(REF ) we know when $x_2\\ge 0$ holds and so $(x_k)_{k=0,1,2}$ is a probability distribution.",
"We will identify $\\vec{u}=\\vec{x}(z_1)$ and $\\vec{v}=\\vec{x}(z_2)$ where $0< z_1\\le z_2<1$ .",
"Then, Proposition REF and Corollary REF imply (REF ).",
"Since $\\sum _{k=0}^2u_k=\\sum _{k=0}^2v_k=1$ holds (so () is satisfied) we may write $\\vec{u}\\prec \\vec{v}$ .",
"Following Theorem REF we obtain $h_3(\\vec{u})\\ge h_3(\\vec{v})$ and $H(\\varpi )\\equiv h_3$ concludes the proof."
],
[
"Concavity of the ternary Shannon entropy",
"We now turn our attention to the concavity proof.",
"We start by a proving the convexity of $t_0$ in (REF ) in $z$ .",
"To that end, we set $\\tau (z,n)=\\sqrt{-p}\\cos {h\\over n}$ as in Lemma REF and study the properties of its second derivative ${\\mathrm {\\,d}^2 \\tau (z,n)\\over \\mathrm {\\,d} z^2}&={1\\over 4(-p)^{3/2}}\\bigg (\\cos {h\\over n}\\,{-4p^2h^{\\prime 2}+n^2(2pp^{\\prime \\prime }-p^{\\prime 2})\\over n^2}+\\sin {h\\over n}\\,{-4p(ph^{\\prime })^{\\prime }\\over n}\\bigg ).$ To show ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}\\ge 0$ we have to separately investigate several different cases.",
"We will need a couple of auxiliary results.",
"The proof of concavity will be presented for $\\varepsilon \\le 0$ .",
"The next lemma is the only exception where $\\varepsilon $ is arbitrary.",
"Lemma 11 We find ${-4p^2h^{\\prime 2}+n^2(2pp^{\\prime \\prime }-p^{\\prime 2})\\over n^2}\\ge 0$ for $p$ in (REF ), $h=\\arccos {g}$ for $g$ given by (REF ) and $n=3$ .",
"Using (REF ) in (REF ) the inequality becomes $&27(3qp^{\\prime }-2pq^{\\prime })^2-9(4p^3+27q^2)(2pp^{\\prime \\prime }-p^{\\prime 2})=(\\alpha +\\beta z^2)\\nu _4(z)\\le 0,$ where $\\nu _4(z)=\\beta (4 \\alpha ^3 +27\\gamma ^2)+z^2(8 \\alpha ^2 \\beta ^2+12\\alpha \\delta ^2+18\\beta \\gamma \\delta )+36z^3\\alpha \\delta \\varepsilon +z^4\\big (\\alpha (4 \\beta ^3+27 \\varepsilon ^2)+3\\beta \\delta ^2\\big ).$ Since $\\alpha +\\beta z^2\\equiv p\\le 0$ we have to show that $\\nu _4(z)$ is nonnegative for $z\\in (0,1)$ .",
"Theorem REF reveals a lot of information about $\\nu _4$ through its coefficients.",
"Depending on the sign of $\\varepsilon $ we find from (REF ) Table: NO_CAPTION No matter what the signs of $8 \\alpha ^2 \\beta ^2+12\\alpha \\delta ^2+18\\beta \\gamma \\delta $ and $\\varepsilon $ are there are always two sign changes.",
"Hence $\\nu _4(z)$ has two or none positive roots (for $\\varepsilon =0$ the second row from the top is missing but still there can be two or none positive roots).",
"Let's first assume $\\varepsilon =-2p_0p_1p_2$ (its minimal value given by $\\vartheta =0$ ).",
"In this case we observe $\\nu _4(1)=4\\alpha ^3\\beta +8\\alpha ^2\\beta ^2+\\alpha \\big (4\\beta ^3+3 (2\\delta +3 \\varepsilon )^2\\big )+3\\beta (3\\gamma +\\delta )^2=0.$ Since $\\nu _4^{\\prime }(1)=0$ as well and $\\nu _4^{\\prime \\prime }(1)=4\\left(4\\alpha ^2\\beta ^2+3\\alpha (4 \\beta ^3+2 \\delta ^2+18\\delta \\varepsilon +27 \\varepsilon ^2)+9\\beta \\delta (\\gamma +\\delta )\\right)\\ge 0$ the point $z=1$ is a proper local minimum.",
"The expression $\\nu ^{\\prime }_4$ is a cubic polynomial.",
"Hence it has three roots: one of them is always zero and the greatest one always equals one (the one we found previously).",
"The third root can be both positive or negative and whatever its position is we want to make sure that $\\nu _4\\ge 0$ in the interval $(0,1)$ .",
"Recall that according Theorem REF there must be another positive root of $\\nu _4$ .",
"At first sight it seems impossible because if the third root of $\\nu _4^{\\prime }$ is negative then the segment of $\\nu _4$ in $(0,1)$ must be decreasing ($\\nu _4(0)=\\beta (4 \\alpha ^3 +27\\gamma ^2)\\ge 0$ ).",
"Even if the third root of $\\nu _4^{\\prime }$ lies in $(0,1)$ it can be either a positive local maximum or a stationary point.",
"This is because we showed that $z=1$ is a local minimum, $\\nu _4^{\\prime }$ has only three roots and again because of $\\nu _4(0)\\ge 0$ .",
"So where is the remaining positive root?",
"The only possibility is that $z=1$ is a double root.",
"Indeed, by calculating the discriminant [29] of $\\nu _4$ we find it to be equal to zero.",
"This means that at least two roots coincide.",
"Hence $\\nu _4\\ge 0$ holds for $\\varepsilon =-2p_0p_1p_2$ .",
"For $0<\\vartheta \\le \\pi /2$ the coefficients of the monomials of order 3 and 4 in (REF ) clearly increase and hence no new root can appear in the interval $(0,1)$ .",
"For $\\pi /2<\\vartheta \\le \\pi $ , the monomial order 4 coefficient decreases but $\\varepsilon ^2\\propto \\cos ^2{\\vartheta }$ is a symmetric function and we have seen that $\\nu _4(z)$ had no positive root even when $\\varepsilon <0$ .",
"But now $\\varepsilon >0$ and so again there is no positive root which concludes the proof.",
"Remark The claim holds for any $n\\ge 3$ but we do not make use of it.",
"Lemma 12 The function $(pg^{\\prime })^{\\prime }(z)$ has a single positive root $z^*$ whenever $\\gamma (z)\\ge 0$ and $(pg^{\\prime })^{\\prime }(z)\\le 0$ for $z\\in (0,z^*)$ .",
"We calculate $(pg^{\\prime })^{\\prime }={3\\sqrt{3}\\over 8}{q(9p^{\\prime 2}-6pp^{\\prime \\prime })+4p(-2p^{\\prime }q^{\\prime }+pq^{\\prime \\prime })\\over \\sqrt{-p}p^2}.$ The position of the positive roots is unaffected by the numerical prefactors or by the denominator.",
"Hence, we rewrite only the numerator $\\nu _2(z)=q(9p^{\\prime 2}-6pp^{\\prime \\prime })+4 p (-2p^{\\prime }q^{\\prime }+pq^{\\prime \\prime })$ in terms of Eqs.",
"(REF ) and (): $\\nu _2(z)=-12\\alpha \\beta \\varepsilon z^3+\\left(24 \\beta ^2 \\gamma -28 \\alpha \\beta \\delta \\right)z^2+24\\alpha ^2\\varepsilon z+8\\alpha ^2\\delta -12\\alpha \\beta \\gamma .$ Given $\\alpha ,\\beta ,\\delta ,\\varepsilon \\le 0$ and $\\gamma >0$ there is only one sign change if $8\\alpha ^2\\delta -12\\alpha \\beta \\gamma \\le 0$ .",
"This is indeed satisfied for $\\gamma >0$ .",
"The observation $\\lim _{z\\rightarrow +\\infty }{[(pg^{\\prime })^{\\prime }]}=+\\infty $ concludes the proof.",
"Remark In fact, we can refine the previous lemma by calculating $\\lim _{z\\rightarrow 0}{[(pg^{\\prime })^{\\prime }]}=-\\frac{3}{2} \\sqrt{3} \\left(-\\frac{1}{\\alpha }\\right)^{3/2} (2 \\alpha \\delta -3 \\beta \\gamma )\\le 0$ and expressing $\\nu _2(z)$ with the help of (REF ) and $\\sum _{i}p_i=1$ as $\\nu _2(1)={8\\over 9}\\big (p_0^4+2 p_0^3 (-1+p_1)+(-1+p_1)^2 p_1^2+p_0p_1 (-1-p_1+2 p_1^2)+p_0^2 (1-p_1+3 p_1^2)\\big )\\ge 0.$ Therefore, $z^*\\in (0,1)$ .",
"The minimum on the RHS is achieved for $p_0=p_1=p_2=1/3$ .",
"Lemma 13 The function $(ph^{\\prime })^{\\prime }(z)$ has a single positive root for $z\\in (0,1)$ whenever $\\gamma \\ge 0$ and for all $\\varepsilon \\le 0$ .",
"We write $(ph^{\\prime })^{\\prime }=-\\frac{gpg^{\\prime }+(1-g^2)(pg^{\\prime })^{\\prime }}{(1-g^2)^{3/2}}$ and after inserting Eqs.",
"(REF ), (REF ) and $g^{\\prime \\prime }(z)=\\frac{3\\sqrt{3}\\left(4p(p q^{\\prime \\prime }-3p^{\\prime }q^{\\prime })+3q(5p^{\\prime 2}-2pp^{\\prime \\prime })\\right)}{8(-p)^{-3/2}p^5}$ we get an expression whose numerator reads $\\nu _5=-81q^3 p^{\\prime \\prime }+54q^2\\left(p^{\\prime }q^{\\prime }+pq^{\\prime \\prime }\\right)+q\\left(-12p^3p^{\\prime \\prime }+18p^2p^{\\prime 2}-54pq^{\\prime 2}\\right)+8p^4q^{\\prime \\prime }-16p^3p^{\\prime }q^{\\prime }$ and whose denominator is negative in $(0,1)$ .",
"By inserting Eqs.",
"(REF ) we get a daunting polynomial of degree seven: $\\nu _5&=z^7 \\left(-24\\alpha \\beta ^3\\varepsilon -162\\alpha \\varepsilon ^3-54\\beta \\delta ^2\\varepsilon \\right)+z^6\\left(-56\\alpha \\beta ^3\\delta -378\\alpha \\delta \\varepsilon ^2+48\\beta ^4\\gamma +324\\beta \\gamma \\varepsilon ^2-54\\beta \\delta ^3\\right)\\nonumber \\\\&\\quad +z^5 \\left(324 \\beta \\gamma \\delta \\varepsilon -324 \\alpha \\delta ^2 \\varepsilon \\right)+z^4 \\left(-96 \\alpha ^2 \\beta ^2 \\delta +72 \\alpha \\beta ^3 \\gamma +162 \\alpha \\gamma \\varepsilon ^2-108 \\alpha \\delta ^3-54 \\beta \\gamma \\delta ^2\\right)\\nonumber \\\\&\\quad +z^3 \\left(72 \\alpha ^3 \\beta \\varepsilon +216 \\alpha \\gamma \\delta \\varepsilon +162 \\beta \\gamma ^2 \\varepsilon \\right)+z^2 \\left(-24 \\alpha ^3 \\beta \\delta -162 \\beta \\gamma ^2 \\delta \\right)+z\\left(48 \\alpha ^4 \\varepsilon +324 \\alpha \\gamma ^2 \\varepsilon \\right)\\nonumber \\\\&\\quad +16\\alpha ^4\\delta -24\\alpha ^3\\beta \\gamma +108\\alpha \\gamma ^2\\delta -162\\beta \\gamma ^3.$ Let us first assume $\\varepsilon =-2p_0p_1p_2$ which is the minimal value given by $\\vartheta =0$ .",
"Then, there is an inflection point at $z=1$ : $\\nu _5^{\\prime }(1)=\\nu _5^{\\prime \\prime }(1)=0$ .",
"This indicates a triple root (corroborated by the zero discriminant indicating multiple roots) and so $\\nu _5=f_4(z-1)^3,$ where $f_4=\\sum _{i=0}^4a_iz^i$ .",
"By comparing the coefficients with (REF ) we deduce the coefficient $a_i$ and get $f_4&= -6z^4 \\varepsilon \\left(4 \\alpha \\beta ^3+27 \\alpha \\varepsilon ^2+9 \\beta \\delta ^2\\right)\\nonumber \\\\&\\quad +2z^3 \\left(-4 \\alpha \\beta ^3 (7 \\delta +9 \\varepsilon )-27 \\alpha \\varepsilon ^2 (7 \\delta +9 \\varepsilon )+24 \\beta ^4 \\gamma -27\\beta (-6\\gamma \\varepsilon ^2+\\delta ^3+3 \\delta ^2 \\varepsilon )\\right)\\nonumber \\\\&\\quad -6 z^2 (4 \\alpha ^3+27 \\gamma ^2) (\\alpha (4 \\delta +6 \\varepsilon )-\\beta (6 \\gamma +\\delta ))\\nonumber \\\\&\\quad -6 z (4 \\alpha ^3+27 \\gamma ^2) (2 \\alpha (\\delta +\\varepsilon )-3 \\beta \\gamma )\\nonumber \\\\&\\quad -2(4\\alpha ^3+27\\gamma ^2)(2\\alpha \\delta -3\\beta \\gamma ).$ With the help of the following table Table: NO_CAPTION Theorem REF reveals that there is only one positive root.",
"Note that the degree three coefficient of (REF ) seems too complicated to analytically deduce its sign but our ignorance does not affect the number of sign variations.",
"Now we show that by for any $\\varepsilon \\le 0$ the single root shifts and the inflection disappears.",
"We inspect the coefficients of (REF ) where $\\varepsilon $ appears.",
"Considering $\\gamma \\ge 0$ , the ones accompanying the monomials $z,z^3$ and $z^5$ satisfy $48 \\alpha ^4 \\varepsilon +324 \\alpha \\gamma ^2 \\varepsilon & \\le 0, \\\\72 \\alpha ^3 \\beta \\varepsilon +216 \\alpha \\gamma \\delta \\varepsilon +162 \\beta \\gamma ^2 \\varepsilon & \\le 0,\\\\324 \\beta \\gamma \\delta \\varepsilon -324 \\alpha \\delta ^2 \\varepsilon & \\le 0.$ Hence an increase of $\\varepsilon $ from its minimal values to any $\\varepsilon \\le 0$ will not add a new root in $(0,1)$ .",
"Similarly for the $z^7$ coefficient $\\varepsilon (-24\\alpha \\beta ^3-162\\alpha \\varepsilon ^2-54\\beta \\delta ^2)$ which, due to $-24\\alpha \\beta ^3-162\\alpha \\varepsilon ^2-54\\beta \\delta ^2\\ge 0$ (valid only for the minimal $\\varepsilon $ ), is an increasing function of $\\varepsilon \\le 0$ .",
"This is because $\\alpha \\le 0$ and so (REF ) is a decreasing function of $\\varepsilon \\le 0$ ((REF ) can become negative).",
"Even if (REF ) does not change the sign, the $z^7$ coefficient will always be greater than the one with the minimal $\\varepsilon $ because the overall multiplication by $\\varepsilon \\le 0$ swaps the sign (and so the order).",
"Finally, the $z^4$ and $z^6$ coefficients contain negative factors accompanying $\\varepsilon ^2$ (recall $\\alpha ,\\beta ,\\delta \\le 0$ and $\\gamma \\ge 0$ by assumption).",
"Hence, as $\\varepsilon ^2$ decreases, it effectively increases the coefficients of $z^4$ and $z^6$ .",
"We can conclude that no new root for $z\\in (0,1)$ appears for $\\varepsilon \\le 0$ .",
"Proposition 14 The following relations hold: [row sep=large, column sep=normal] q0 [r,Leftrightarrow,\"Lemma REF \" inner sep=1ex] g0 [r,Rightarrow,\"(i)\"][d,Rightarrow,\"(ii)\"] (pg')'0 (ph')'0 (i) The sought after implication can be reformulated in the language of Lemma REF and REF as $z^\\#\\le z^*$ since $q\\ge 0$ in $(0,z^\\#)$ and $(pg^{\\prime })^{\\prime }\\le 0$ in $(0,z^*)$ .",
"We proceed by setting $q=0$ and, conveniently, the numerator of (REF ) simplifies to $\\nu _2(z)\\big |_{q=0}\\propto -2p^{\\prime }q^{\\prime }+pq^{\\prime \\prime }= 6 \\beta \\varepsilon z^3 +6 \\beta \\delta z^2-6 \\alpha \\varepsilon z-2\\alpha \\delta .$ We ignored the factor $4p$ as it does not affect the position of the roots for $q=0$ .",
"In principle we just need to compare the position of the roots for the polynomials $q$ and $ \\nu _2(z)|_{q=0}$ .",
"However, they are both cubic polynomials and the roots' form is too complicated to determine their relation.",
"It follows from Lemma REF , Lemma REF and the previous remark that the polynomials intersect at a single point in the interval $(z^\\#,z^*)\\subset (0,1)$ and, in addition, the position of the intersection point above or below the $x$ axis informs us about the relation of the two roots.",
"It would not be very helpful to set $q=\\nu _2(z)|_{q=0}$ and solve for $z$ , though.",
"It again leads to a cubic equation and we face a similar problem as before.",
"The trick we will use is the following transformation: $q(z)\\mapsto \\tilde{q}(z)=-6\\beta q=-6\\beta \\gamma -6\\beta \\delta z^2-6\\beta \\varepsilon z^3.$ The new function $\\tilde{q}(z)$ has the same properties as $q(z)$ uncovered in Lemma REF (the minus sign reverses the negative sign of $\\beta $ ).",
"By setting $\\tilde{q}=\\nu _2(z)|_{q=0}$ we obtain another cubic equation $\\tilde{\\mu }(z)=2\\mu (z)=2\\big (6\\beta \\varepsilon z^3 +6\\beta \\delta z^2-3\\alpha \\varepsilon z-\\alpha \\delta +3\\beta \\gamma \\big )=0.$ Its (single) root in $(0,1)$ reveals where $\\tilde{q}$ and $\\nu _2(z)|_{q=0}$ intersect but that also means that by comparing the roots' position of $\\mu $ (or $\\tilde{\\mu }$ ) with $\\nu _2(z)|_{q=0}$ in the interval $(0,1)$ we learn whether $\\tilde{q}$ and $\\nu _2(z)|_{q=0}$ intersected above or below the $x$ axis.",
"So by setting $\\mu (z)=\\nu _2(z)|_{q=0}$ we crucially get a linear equation whose solution reads $z_\\ell =\\frac{-\\alpha \\delta -3\\beta \\gamma }{3\\alpha \\varepsilon }.$ By inserting it back to $\\nu _2(z)|_{q=0}$ we get $\\nu _2(z_\\ell )|_{q=0}=\\frac{2\\beta \\left(\\alpha ^3(27\\gamma \\varepsilon ^2+2 \\delta ^3)+9 \\alpha ^2\\beta \\gamma \\delta ^2-27\\beta ^3\\gamma ^3\\right)}{9\\alpha ^3\\varepsilon ^2}.$ It remains to show $\\nu _2(z_\\ell )|_{q=0}\\le 0$ in order to prove $z^\\#\\le z^*$ .",
"Since $\\alpha ,\\beta \\le 0$ it suffices to show that $\\nu _3(z)=\\alpha ^3 \\left(27\\gamma \\varepsilon ^2+2 \\delta ^3\\right)+9 \\alpha ^2\\beta \\gamma \\delta ^2-27\\beta ^3\\gamma ^3\\le 0$ .",
"Using (REF ) and $\\sum _ip_i=1$ we find $\\nu _3(z)\\overset{\\mathrm {df}}{=}{1\\over 27}f_1f_2$ where $f_1 & = \\big (p_0+2 p_0^3+3 p_0^2 (-1+p_1)-3 p_0 p_1^2+p_1 (-1+3 p_1-2 p_1^2)\\big )^2, \\\\f_2 & = 4 p_0^6+12 p_0^5 (-1+p_1)+p_0^4 (13-27 p_1+24 p_1^2)+p_0^3 (-6+20 p_1-42 p_1^2+28 p_1^3)\\nonumber \\\\& \\quad + p_0^2 (1-5 p_1+24 p_1^2-42 p_1^3+24 p_1^4)+p_1^2 (1-3 p_1+2 p_1^2)^2+p_0 p_1^2 (-5+20 p_1-27 p_1^2+12 p_1^3).$ Since $f_1\\ge 0$ , we have to show $f_2\\le 0$ .",
"We reduced the problem to a task analytically solvable by Mathematica.",
"Indeed, we find $\\max {[f_2]}=0$ subject to $\\gamma \\ge 0$ and $0\\le z_\\ell \\le 1$ .",
"The first inequality is a necessary condition for the initial assumption $q\\ge 0$ in $z\\in (0,z^\\#)$ via Lemma REF .",
"Achieving the maximum implies $\\nu _3=0$ which in turn implies $ \\nu _2(z_\\ell )|_{q=0}=0$ (from (REF )) and so $\\mu (z_\\ell )=0$ (see above (REF )).",
"This finally leads to $\\nu _2(z_\\ell )|_{q=0}=\\tilde{q}(z_\\ell )=0=q(z_\\ell )$ and so $z_\\ell ^\\#=z_\\ell ^*$ which concludes the proof.",
"(ii) Assuming $\\gamma \\ge 0$ as a necessary condition to the current case of interest $q\\ge 0$ $(g\\le 0)$ for $z\\in (0,z^\\#)$ (see Lemma REF and REF ) we find $\\nu _5(0)=2(4\\alpha ^3+27\\gamma ^2)(2\\alpha \\delta -3\\beta \\gamma )\\le 0$ (the different sign in the bottom of the table on page REF is due to $f_4$ being multiplied by $(z-1)^3$ ) and so $(ph^{\\prime })^{\\prime }(0)\\ge 0$ .",
"This is because $(ph^{\\prime })^{\\prime }\\propto -\\nu _5$ .",
"We also notice that $(ph^{\\prime })^{\\prime }\\ge 0$ for $g=0$ .",
"This follows from Eq.",
"(REF ) implying that in this case $(ph^{\\prime })^{\\prime }=-(pg^{\\prime })^{\\prime }$ .",
"But from item (i) of the current lemma we know that $(pg^{\\prime })^{\\prime }\\le 0$ for $g\\le 0$ .",
"Inevitably, the only positive root of $(ph^{\\prime })^{\\prime }$ occurs for $g\\ge 0$ , that is, as long as $g\\le 0$ we get $(ph^{\\prime })^{\\prime }\\ge 0$ as we wanted to show.",
"Remark Note that $(ph^{\\prime })^{\\prime }(0)\\ge 0$ does not contradict $g^{\\prime }(0)=0$ we found in Lemma REF .",
"This could be hastily concluded by looking at Eq.",
"(REF ).",
"But it is true only if $g^{\\prime }=0$ and $\\sqrt{1-g^2}\\ne 0$ .",
"In many cases it is found, however, that for $z=0$ one gets $g^{\\prime }=\\sqrt{1-g^2}=0$ and $\\lim _{z\\rightarrow 0^+}h^{\\prime }\\ne 0$ .",
"Lemma 15 Let $h=\\arccos {g}$ and $(ph^{\\prime })^{\\prime }\\le 0$ .",
"Then $\\cos {h\\over 2}\\,{-4p^2h^{\\prime 2}+3^2(2pp^{\\prime \\prime }-p^{\\prime 2})\\over 4\\times 3^2(-p)^{3/2}}+\\sin {h\\over 2}\\,{-p(ph^{\\prime })^{\\prime }\\over 3(-p)^{3/2}}\\ge 0$ whenever $g\\ge 0$ and for all $\\varepsilon \\le 0$ .",
"Remark The expression resembles part of Eq.",
"(REF ).",
"However, notice $n=2$ in the trigonometric functions and $n=3$ elsewhere.",
"Given $\\tau (z,n)=\\sqrt{-p}\\cos {\\arccos {g}\\over n}$ , a straightforward calculation reveals $&{\\mathrm {\\,d}^2 \\tau (z,n)\\over \\mathrm {\\,d} z^2} \\nonumber \\\\&=\\cos {\\arccos {g}\\over n}\\,{4p^2g^{\\prime 2}+n^2(-1+g^2)(2pp^{\\prime \\prime }-p^{\\prime 2})\\over 4n^2(1-g^2)\\sqrt{-p}p}+\\sin {\\arccos {g}\\over n}\\,{-pgg^{\\prime 2}+(-1+g^2)(pg^{\\prime })^{\\prime }\\over n\\sqrt{1-g^2}(1-g^2)\\sqrt{-p}}.$ We set $n=2$ in the trigonometric functions and $n=3$ elsewhere at which point (REF ) becomes the studied expression (Eq.",
"(REF )) by virtue of (REF ).",
"We multiply both summands by $p\\sqrt{-p}(1-g^2)\\le 0$ and use $\\cos {x\\over 2}=\\sqrt{1+\\cos {x}\\over 2}$ ($-\\pi \\le x\\le \\pi $ ) and $\\sin {x\\over 2}=\\sqrt{1-\\cos {x}\\over 2}$ ($0\\le x\\le 2\\pi $ ).",
"We got the reverse inequality to prove $\\kappa ={4\\over 9}p^2g^{\\prime 2}(1-2g)+(-1+g^2)\\Big ((1+g)(2pp^{\\prime \\prime }-p^{\\prime 2})+{4\\over 3}p(pg^{\\prime })^{\\prime }\\Big )\\le 0.$ For this purpose, we use Eq.",
"(REF ) and deduce $\\kappa ={1\\over p^2}\\bigg ({-}\\nu _4+f_3{1\\over 2}\\sqrt{3\\over -p}\\bigg ),$ where $\\nu _4$ is given by (REF ) and $f_3(q)=-27 q^2q^{\\prime \\prime }+q(-6 pp^{\\prime 2}+18 q^{\\prime 2})-4p^2(pq^{\\prime \\prime }-2p^{\\prime }q^{\\prime }).$ Since in Lemma REF we proved $\\nu _4>0$ we only have to show $f_3\\le 0$ for $\\kappa \\le 0$ to hold.",
"The inequality $f_3\\le 0$ does not hold in general, however.",
"That is not a problem as long as we show that it holds for $(ph^{\\prime })^{\\prime }\\le 0$ .",
"First we assume $\\gamma \\ge 0$ .",
"By contrapositive of Proposition REF (ii) we know $(ph^{\\prime })^{\\prime }\\le 0\\Rightarrow g\\ge 0\\Leftrightarrow q\\le 0.$ Hence we need to show $q\\le 0\\Rightarrow f_3\\le 0$ .",
"For $q=0$ the function $f_3$ becomes $-4p^2(pq^{\\prime \\prime }-2p^{\\prime }q^{\\prime })$ which is proportional to $\\nu _2(z_\\ell )|_{q=0}$ (see (REF )).",
"Its relation to $q$ was studied in Proposition REF and we found $\\nu _2(z_\\ell )|_{q=0}=q(z_\\ell )=0$ for $z_\\ell $ given by (REF ).",
"Therefore, $f_3(0)$ =0.",
"Then, as demanded in (REF ), for any $q<0$ we get $f_3(q)<0$ since Eq.",
"(REF ) is an increasing function of $q$ .",
"This follows from $q^{\\prime \\prime }=6\\varepsilon z\\le 0$ (valid for $\\varepsilon \\le 0$ ) and $-6pp^{\\prime 2}+18q^{\\prime 2}\\ge 0$ by looking at Eqs.",
"(REF ) and (REF ).",
"For $\\gamma <0$ we know from Lemma REF that $g>0$ $(q<0)$ always holds independently on the sign of $(ph^{\\prime })^{\\prime }$ .",
"Therefore $f_3<0$ and the proof goes as outlined above.",
"Proposition 16 The function $t_0$ is convex in $z\\in (0,1)$ .",
"The function $t_0$ is proportional to $\\tau (z,3)=\\sqrt{-p}\\cos {\\arccos {g}\\over 3}$ and so we will focus on proving ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}\\ge 0$ given by (REF ) for $n=3$ .",
"In Lemma REF we presented a proof of nonnegativity of a fraction multiplying $\\cos {\\arccos {g}\\over 3}$ .",
"Both $\\cos {\\arccos {g}\\over 3}\\ge 0$ and $\\sin {\\arccos {g}\\over 3}\\ge 0$ for $|g(z)|\\le 1$ and so ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}\\ge 0$ holds whenever $(ph^{\\prime })^{\\prime }\\ge 0$ .",
"For the rest of the proof assume $(ph^{\\prime })^{\\prime }<0$ .",
"We will construct a lower bound on ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}$ and show it to be nonnegative.",
"To this end, we summon the inequalities $\\sin {h\\over n}>\\sin {h\\over n+1}$ and $\\cos {h\\over n}<\\cos {h\\over n+1}$ (valid for $n\\ge 2$ and visible in Fig.",
"REF for $n=2,3$ ) and substitute $\\sin {h\\over 3}$ and $\\cos {h\\over 3}$ by $\\sin {h\\over 2}$ and $\\cos {h\\over 2}$ , respectively.",
"This is a lower bound on ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}$ and the quantity was proved to be nonnegative in Lemma REF .",
"This concludes the proof.",
"Remark The fact that $g\\lnot <0$ for $(ph^{\\prime })^{\\prime }\\le 0$ is crucial.",
"First of all, it is not clear how to prove the validity of ${\\mathrm {\\,d}^2 \\tau (z,3)\\over \\mathrm {\\,d} z^2}\\ge 0$ given by (REF ) for $(ph^{\\prime })^{\\prime }\\le 0$ .",
"But even the only manageable lower bound, Eq.",
"(REF ), is in some cases not good enough (i.e., non-negative) for $g<0$ and $(ph^{\\prime })^{\\prime }\\le 0$ .",
"Corollary 17 The function $t_2$ is concave in $z\\in (0,1)$ .",
"Similarly to Proposition REF we will make use of $t_2(p,q)=-t_0(p,-q)$ .",
"The mapping $q\\mapsto -q$ changes the sign of $g,g^{\\prime }$ and $h^{\\prime }$ (see Eqs.",
"(REF ), (REF ) and (REF )).",
"Looking at Eq.",
"(REF ), we notice that for $n\\ge 2$ the sign of the trigonometric functions remains unaffected (see Fig.",
"REF for $n=2,3$ ).",
"Similarly for the expression from Lemma REF coming from (REF ).",
"The sign change of $q$ also swaps the sign of $(ph^{\\prime })^{\\prime }$ in (REF ).",
"This is because $(ph^{\\prime })^{\\prime }=p^{\\prime }h^{\\prime }+ph^{\\prime \\prime }$ together with $h^{\\prime \\prime }=-{gg^{\\prime 2}+(1-g^2)g^{\\prime \\prime }\\over (1-g^2)^{3/2}}$ taking into account that $g^{\\prime \\prime }$ changes the sign upon $q\\mapsto -q$ .",
"But both cases ($(ph^{\\prime })^{\\prime }\\ge 0$ and $(ph^{\\prime })^{\\prime }<0$ ) have been separately investigated in Proposition REF .",
"So we conclude $t^{\\prime \\prime }_0(p,-q)\\ge 0$ and so $t^{\\prime \\prime }_2(p,q)\\le 0$ .",
"Lemma 18 Let $(u_i)_{i=0}^2$ be a probability distribution function and $h_3$ the ternary Shannon entropy defined in (REF ).",
"Then $h_3$ is concave in $(0,1)\\times (0,1)\\subset \\mathbb {R}^2$ and for a fixed $u_2\\in (0,1)$ the function $h_3$ is monotone increasing (decreasing) for $u_1<(1-u_2)/2$ ($u_1>(1-u_2)/2$ ).",
"The Hessian matrix $\\mathsf {H}(h_3(\\vec{u}))=\\begin{bmatrix}-\\frac{1}{1-u_1-u_2}-\\frac{1}{u_1} & -\\frac{1}{1-u_1-u_2} \\\\-\\frac{1}{1-u_1-u_2} & -\\frac{1}{1-u_1-u_2}-\\frac{1}{u_2} \\\\\\end{bmatrix}$ is negative definite since $\\mathop {{\\mathrm {Tr}}_{}}{\\mathsf {H}}<0$ and $\\det {\\mathsf {H}}=1/(u_0u_1)+1/(u_0u_2)+1/(u_1u_2)>0$ .",
"The concavity of $h_3$ follows from the positivity of the characteristic polynomial throughout the interval $(u_0,u_1)\\in (0,1)\\times (0,1)$ .",
"We now fix the value of $u_2$ and the equation $\\partial {h_3}/\\partial {u_1}=0$ is satisfied for $u_1=(1-u_2)/2$ .",
"Thanks to the previously proved concavity, it is a local maximum for every $u_2\\in (0,1)$ and thus $(1-u_2)/2$ defines a one-parameter family of local maxima for $h_3$ .",
"Remark Due to the symmetry between $u_2$ and $u_1$ in (REF ) we may fix $u_1$ and get a family of local maxima given by $-2u_2+1$ .",
"The global maximum of $h_3$ at $(u_2,u_1)=(1/3,1/3)$ lies in the intersection of $(1-u_2)/2$ and $-2u_2+1$ .",
"The situation is illustrated in Fig.",
"REF .",
"Figure: The ternary Shannon entropy Eq.",
"() is shown.",
"The blue line depicts u 1 =(1-u 2 )/2u_1=(1-u_2)/2 while the green one is the plot of u 1 =-2u 2 +1u_1=-2u_2+1.Theorem 19 The von Neumann entropy $H(\\varpi )$ of density matrix (REF ) is a concave function of the overlap $z$ as introduced in (REF ), for all $p_k$ and for all $0\\le \\vartheta \\le \\pi /2$ corresponding to $\\varepsilon \\le 0$ in ().",
"The von Neumann entropy $H(\\varpi )$ is given by $ h_3(\\vec{x}(z))$ in (REF ) where $x_k=t_k+1/3$ come from Eq.",
"(REF ).",
"Taking into account $\\sum _{i=0}^2x_i=1$ we get $\\vec{x}:\\mathbb {R}\\mapsto \\mathbb {R}^2$ and the investigated expression ${\\mathrm {d^2}h_3\\over \\mathrm {d}z^2}$ will be written using the following notation: We define $\\mathsf {x^{\\prime }} & =\\begin{bmatrix}x_1^{\\prime } \\\\x_2^{\\prime } \\\\\\end{bmatrix} \\\\\\mathsf {x^{\\prime \\prime }} & =\\begin{bmatrix}x_1^{\\prime \\prime } \\\\x_2^{\\prime \\prime } \\\\\\end{bmatrix}$ and $\\mathsf {h_3^{\\prime }}=\\begin{bmatrix}\\frac{\\partial h_3}{\\partial x_1} \\\\\\frac{\\partial h_3}{\\partial x_2} \\\\\\end{bmatrix}.$ Using the chain rule, the second derivative can be succinctly expressed as ${\\mathrm {d^2}h_3\\over \\mathrm {d}z^2}= (\\mathsf {x^{\\prime }})^\\top \\mathsf {H}(h_3(\\vec{x}))\\,\\mathsf {x^{\\prime }}+(\\mathsf {h_3^{\\prime }})^\\top \\mathsf {x^{\\prime \\prime }},$ where $\\top $ denotes transposition and the dot (matrix) product is implied.",
"Hessian Eq.",
"(REF ) is negative definite according to Lemma REF and so the first summand is negative for any $\\mathsf {x^{\\prime }}$ .",
"In order for the second summand to be nonpositive as well, one possibility is when either the functions $x_1$ and $x_2$ are concave and the two components of $h_3$ nondecreasing or $x_1,x_2$ convex and $h_3$ entry-wise nonincreasing.",
"We proved $x_2^{\\prime \\prime }\\le 0$ in Corollary REF but said nothing about the concavity of $x_1$ .",
"As a matter of fact, it is incomparably more difficult to prove $x_1^{\\prime \\prime }\\le 0$ in spite of the overwhelming numerical evidence.",
"The same numerics suggests that there is a whole class of input probabilities $p_k$ for which $x_1^{\\prime \\prime }=0$ .",
"So no `simple' bounds like those leading to Proposition REF exist.",
"But there is a third possibility of how to make the second summand in (REF ) negative and it is the combination of the two previous cases.",
"We know that $x_0^{\\prime \\prime }\\ge 0$ from Proposition REF and $x_2^{\\prime \\prime }\\le 0$ from Corollary REF .",
"The second summand (REF ) will be negative if we take $x_0$ instead of $x_1$ in Eqs.",
"(REF ) and (REF ) and show $\\frac{\\partial h_3}{\\partial x_0}\\le 0$ and $\\frac{\\partial h_3}{\\partial x_2}\\ge 0$ .",
"Note that the Hessian remains negative definite: $\\mathsf {H}(h_3(\\vec{x}))=\\begin{bmatrix}-\\frac{1}{1-x_0-x_2}-\\frac{1}{x_0} & -\\frac{1}{1-x_0-x_2} \\\\-\\frac{1}{1-x_0-x_2} & -\\frac{1}{1-x_0-x_2}-\\frac{1}{x_2} \\\\\\end{bmatrix}.$ Hence, in spite of the components of $\\mathsf {x^{\\prime }}$ to have different signs (see Propositions REF and REF ), the summand is negative.",
"Also note that we are proving the properties of the same ternary entropy (REF ) since it is equivalent to $h_3(\\vec{x}(z))=-x_0\\log {x_0}-x_2\\log {x_2}-(1-x_0-x_2)\\log {[1-x_0-x_2]}.$ Lemma REF informs us that $\\frac{\\partial h_3}{\\partial x_2}\\ge 0$ and $x_2^{\\prime \\prime }\\le 0$ together with $\\frac{\\partial h_3}{\\partial x_0}\\le 0$ and $x_0^{\\prime \\prime }\\ge 0$ is satisfied in the domain's subset delimited by the blue line ($x_0\\ge (1-x_2)/2$ ) and the green line ($x_0\\le -2x_2+1$ ) depicted in Fig.",
"REF if instead of $u_2,u_1$ we have $x_2,x_0$ (resulting in the same figure).",
"But it turns out that this is precisely the range of $\\vec{x}$ represented by $x_0,x_2$ .",
"To this end, consider the basic property of the cubic roots [29] $t_0\\ge t_1\\ge t_2$ that becomes $x_0\\ge x_1\\ge x_2\\ge 0$ for the eigenvalues of $\\varpi $ .",
"First, using $x_0\\ge x_1$ we write $x_0 & \\ge {x_0+x_1\\over 2}\\nonumber \\\\& = {x_0+x_1+x_2-x_2\\over 2}\\nonumber \\\\& = {1-x_2\\over 2},$ where in the second row we used the normalization condition $\\sum _ix_i=1$ .",
"The last equality leads to one of the desired bounds.",
"For the second bound we start with $x_1\\ge x_2$ to write $x_0 & \\le x_0 -x_2+x_1\\nonumber \\\\& = -2x_2+x_0+x_1+x_2\\nonumber \\\\& = -2x_2+1,$ where the last line provides the other inequality we were looking for.",
"Hence $(\\mathsf {h_3^{\\prime }})^\\top \\mathsf {x^{\\prime \\prime }}\\le 0$ resulting in ${\\mathrm {d^2}h_3\\over \\mathrm {d}z^2}\\le 0$ ."
]
] | 1709.01758 |
[
[
"Automatic Document Image Binarization using Bayesian Optimization"
],
[
"Abstract Document image binarization is often a challenging task due to various forms of degradation.",
"Although there exist several binarization techniques in literature, the binarized image is typically sensitive to control parameter settings of the employed technique.",
"This paper presents an automatic document image binarization algorithm to segment the text from heavily degraded document images.",
"The proposed technique uses a two band-pass filtering approach for background noise removal, and Bayesian optimization for automatic hyperparameter selection for optimal results.",
"The effectiveness of the proposed binarization technique is empirically demonstrated on the Document Image Binarization Competition (DIBCO) and the Handwritten Document Image Binarization Competition (H-DIBCO) datasets."
],
[
"Introduction",
"Document image binarization aims to segment the foreground text in a document from the noisy background during the preprocessing stage of document analysis.",
"Document images commonly suffer from various degradations over time, rendering document image binarization a daunting task.",
"Typically, a document image can be heavily degraded due to ink bleed-through, faded ink, wrinkles, stains, missing data, contrast variation, warping effect, and noise due to lighting variation during document scanning.",
"Though document image binarization has been extensively studied, thresholding of heavily degraded document images remains a largely unexplored problem due to difficulties in modelling different types of document degradations.",
"The Document Image Binarization Competition (DIBCO), and the Handwritten Document Image Binarization Competition (H-DIBCO), held from 2009 to present aim to address this problem by introducing challenging benchmarking datasets to evaluate the recent advancement in document image binarization.",
"However, competition results so forth indicate a scope for improvement in the binarized image quality.",
"The performance of binarization techniques significantly depends on the associated control parameter values [1], i.e., the hyperparameters.",
"Despite the significance of optimal hyperparameter selection for document image binarization, automatic binarization has still not been sufficiently explored.",
"This paper presents an automatic document image binarization technique that uses two band-pass filters for background noise removal, and Bayesian optimization [2] for automatic thresholding and hyperparameter selection.",
"The band-pass filtering method uses a high frequency band-pass filter to separate the fine detailed text from the background, and subsequently a low frequency band-pass filter as a mask to remove noise.",
"The parameters of the two band-pass filtering algorithm include a threshold for removing noise, the mask size for blurring the text, and the window size to be set dynamically depending upon the degree of degradation.",
"Bayesian optimization is used to automatically infer the optimal values of these parameters.",
"The proposed method is simple, robust and fully automated for handling heavily degraded document images.",
"This makes it suitable for use by, e.g., librarians and historians for quick and easy binarization of ancient texts.",
"Since optimum parameter values are selected on-the-fly using Bayesian optimization, the average binarization performance is improved.",
"This is due to each image being assigned its respective ideal combination of hyperparameter values instead of using a global sub-optimal parameter setting for all images.",
"To the best of authors' knowledge, this is the first work in the community that uses Bayesian optimization on binarization algorithm for selecting multiple hyperparameters dynamically for a given input image.",
"Figure: The proposed automatic document image binarization framework."
],
[
"Document binarization methods",
"Numerous document image binarization techniques have been proposed in literature, and are well-documented as part of the DIBCO reports [3], [4], [5], [6], [7], [8], [9].",
"Under normal imaging conditions, a simple global thresholding approach, such as Otsu's method [10], suffices for binarization.",
"However, global thresholding is not commonly used for severely degraded document images with varying intensities.",
"Instead, an adaptive thresholding approach that estimates a local threshold for each pixel in a document is popular [11], [12], [13].",
"In general, the local threshold is estimated using the mean and standard deviation of the pixels in a document image within a local neighborhood window.",
"However, the prime disadvantage of using adaptive thresholding is that the binarization performance depends upon the control parameters such as the window size, that cannot be accurately determined without any prior knowledge of the text strokes.",
"Moreover, methods such as Niblack's thresholding [13] commonly introduce additional noise, while Sauvola's thresholding [12] is highly sensitive to contrast variations.",
"Other popular methods in literature include [14], [15], [16], [1], [17], [18], [19].",
"The binarization algorithms of the winners of competitions are sophisticated, and achieve high performance partially due to thresholding, but primarily by modelling the text strokes and the background to enable accurate pixel classification.",
"Lu et al.",
"[14] modelled the background using iterative polynomial smoothing, and a local threshold was selected based on detected text stroke edges.",
"Lelore and Bouchara [18] proposed a technique where a coarse threshold is used to partition pixels into ink, background, and unknown groups.",
"The method is based on a double threshold edge detection approach that is capable of detecting fine details along with being robust to noise.",
"However, these methods combine a variety of image related information and domain specific knowledge, and are often complex [19].",
"For example, methods proposed in [15], [17] made use of document specific domain knowledge.",
"Gatos et al.",
"[15] estimated the document background based on the binary image generated using Sauvola's thresholding [12].",
"Su et al.",
"[17] used image contrast evaluated based on the local maximum and minimum to find the text stroke edges.",
"Although there exist several document thresholding methods, automatic selection of optimal hyperparameters for document image binarization has received little attention.",
"Gatos et al.",
"[15] proposed a parameter-free binarization approach that depends on detailed modelling of the document image background.",
"Dawoud [20] proposed a method based on cross-section sequence that combines results at multiple threshold levels into a single binarization result.",
"Badekas and Papamarkos [21] introduced an approach that performs binarization over a range of parameter values, and estimates the final parameter values by iteratively narrowing the parameter range.",
"Howe [1] proposed an interesting method that optimizes a global energy function based on the Laplacian image, and automatically estimates the best parameter settings.",
"The method dynamically sets the regularization coefficient and Canny thresholds for each image by using a stability criterion on the resultant binarized image.",
"Mesquita et al.",
"[22] investigated a racing procedure based on a statistical approach, named I/F-Race, to fine-tune parameters for document image binarization.",
"Cheriet et al.",
"[23] proposed a learning framework for automatic parameter optimization of the binarization methods, where optimal parameters are learned using support vector regression.",
"However, the limitation of this work is the dependence on ground truth for parameter learning.",
"This work uses Bayesian optimization [2], [24] to efficiently infer the optimal values of the control parameters.",
"Bayesian optimization is a general approach for hyperparameter tuning that has shown excellent results across applications and disciplines [25], [2]."
],
[
"Proposed Binarization Technique",
"The overall pipeline of the proposed automatic document image binarization technique is presented in Fig.",
"REF using an example image from H-DIBCO 2016 dataset.",
"The binarization algorithm is discussed in detail as follows."
],
[
"Document Image Binarization",
"Given a degraded document image, adaptive thresholding using median filters is first performed that separates the foreground text from the background.",
"The output is a grayscale image with reduced background noise and distortion.",
"The grayscale image is then passed through two band-pass filters separately for further background noise removal.",
"A high frequency band-pass filter is used to separate the fine detailed text from the background, and a low frequency band-pass filter is used for masking that image in order to remove great parts of the noise.",
"Finally, the noise reduced grayscale image is converted into a binary image using Kittler's minimum error thresholding algorithm [26].",
"Figure REF illustrates the overall binarization algorithm for a given degraded document image.",
"The image output generated at each filtering step is presented for better understanding of the background noise removal algorithm.",
"It can be observed from Fig.",
"REF that the input document image is heavily degraded with stains, wrinkles and contrast variation.",
"After performing adaptive median filtering, the image becomes less noisy, and is enhanced further using the two band-pass filtering approach.",
"The final binarized image represents the document image with foreground text preserved and noisy background removed.",
"However, the performance of the binarization algorithm depends upon six hyperparameter values that include two control parameters required by the adaptive median filter, namely the local threshold and the local window size; and four control parameters required by the band-pass filters, namely the mask size for blurring the text, a local threshold and two window size values for high frequency and low frequency band-pass filters.",
"The value of these hyperparameters must be chosen such that quality metrics corresponding to the binarized image (e.g., F-measure, Peak Signal-To-Noise Ratio (PSNR), etc.",
"[3]) are maximized (or error is minimized).",
"This corresponds to an optimization problem that must be solved to arrive at the best combination of hyperparameters.",
"For example, the following optimization problem finds optimal values of $d$ hyperparameters ${\\bf x} = (x_1, x_2, ..., x_d)$ with respect to maximizing the F-measure [3], $\\begin{aligned}& \\underset{\\bf x}{\\text{maximize}}& & \\mathrm {f_{measure}}(\\bf x) \\\\& \\text{subject to}& & min\\_x_1 \\le x_1 \\le max\\_x_1, \\; min\\_x_2 \\le x_1 \\le max\\_x_2, \\\\&&& \\cdots , \\; min\\_x_d \\le x_d \\le max\\_x_d,\\end{aligned}$ where $min\\_x_i$ and $max\\_x_i$ span the search space for parameter $x_i$ .",
"No-reference image quality metrics [27] can be optimized in place of metrics such as F-measure in applications where ground truth reference images are not available.",
"Techniques like grid search, cross-validation, evolutionary optimization, etc.",
"can be used to find optimal values of the hyperparameters in the $d$ -dimensional search space.",
"For large values of $d$ , such approaches tend to be slow.",
"Bayesian optimization [2] efficiently solves such hyperparameter optimization problems as it aims to minimize the number of evaluations of the objective function (e.g., F-measure in this case) required to infer the hyperparameters, and is used in this work in the interest of computational efficiency."
],
[
"Bayesian Optimization",
"Bayesian optimization is a model based approach involving learning the relationship between the hyperparameters and the objective function [24].",
"A model is constructed using a training set known as an initial design that covers the hyperparameter space in a space-filling manner.",
"Statistical designs such as Latin hypercubes and factorial designs [28] can be used for this purpose.",
"The initial design may also be random.",
"The goal is to gather initial information about the parameter space in absence of any prior knowledge.",
"After the model is trained using the initial design, an iterative sampling approach follows.",
"The sampling algorithm uses the information offered by the model to intelligently select additional samples in regions of the hyperparameter space that are likely to lead to the optimal solution (i.e., good binarization performance as quantified by metrics such as F-measure).",
"The model is then refined by extending the training set with the selected samples.",
"Bayesian optimization involves using a Bayesian model such as Gaussian process models [29], and the sampling scheme exploits the mean and variance of prediction offered by the model to select additional samples iteratively.",
"This work considers Gaussian process models in the context of Bayesian optimization.",
"A detailed explanation of Gaussian processes can be found in [29]."
],
[
"Gaussian Processes",
"A Gaussian process (GP) is the multivariate Gaussian distribution generalized to an infinite-dimensional stochastic process where any finite combination of dimensions are jointly-Gaussian [29].",
"A Gaussian process $f$ is completely described by its mean $m$ and covariance $k$ functions, $f({\\bf x}) \\sim \\mathcal {GP}(m({\\bf x}), k({\\bf x, x^{\\prime }}))$ .",
"The mean function incorporates prior domain-specific knowledge, if available.",
"The mean function $m({\\bf x})=0$ is a popular choice without loss of generality in absence of any prior knowledge about the problem at hand.",
"The covariance function $k$ incorporates variation of the process from the mean function and essentially controls the expressive capability of the model.",
"Numerous covariance functions exist in literature including squared exponential (SE) function [25] and the Matérn kernels [29].",
"The SE kernel is a good general-purpose kernal and is used for experiments in this paper.",
"The SE kernel is described as, $k({\\bf x}_i, {\\bf x}_j) = exp\\Big ( -\\frac{1}{2\\theta ^2} \\Vert {\\bf x}_i - {\\bf x}_j\\Vert ^2 \\Big )$ , with $\\theta $ being a hyperparameter that controls the width of the kernel.",
"Let ${\\mathcal {D}}=(X,{\\bf y})$ be a $n$ -point training set.",
"Let $\\bf K$ be the kernel matrix holding the pairwise covariances between points in $X$ , ${\\bf K} =\\begin{bmatrix}k({\\bf x}_1, {\\bf x}_1) & \\dots & k({\\bf x}_1, {\\bf x}_n) \\\\\\vdots & \\ddots & \\vdots \\\\k({\\bf x}_n, {\\bf x}_1) & \\dots & k({\\bf x}_n, {\\bf x}_n)\\end{bmatrix}.$ Let $y_{n+1} = \\hat{y}({\\bf x}_{n+1})$ be the predicted value of a query sample ${\\bf x}_{n+1}$ using the GP model $\\hat{y}$ .",
"Since ${\\bf y}$ and $y_{n+1}$ are jointly-Gaussian by definition of Gaussian processes, $\\begin{bmatrix}{\\bf y} \\\\y_{n+1}\\end{bmatrix}\\sim \\mathcal {N}\\Big ( 0,\\begin{bmatrix}{\\bf K} & {\\bf k}\\\\{\\bf k}^\\intercal & k({\\bf x}_{n+1}, {\\bf x}_{n+1})\\end{bmatrix}\\Big ),$ with ${\\bf k} = [k({\\bf x}_{n+1}, {\\bf x}_1,), \\dots k({\\bf x}_{n+1}, {\\bf x}_n,)]$ .",
"The posterior distribution is calculated as $P(y_{n+1}|{\\mathcal {T}},{\\bf x}_{n+1}) = \\mathcal {N}(\\mu _n({\\bf x}_{n+1}), \\sigma ^2_n({\\bf x}_{n+1}))$ , where, $\\mu _n({\\bf x}_{n+1}) &= {\\bf k}^\\intercal {\\bf K}^{-1} {\\bf y},\\\\\\sigma ^2_n({\\bf x}_{n+1}) &= k({\\bf x}_{n+1}, {\\bf x}_{n+1}) - {\\bf k}^\\intercal {\\bf K}^{-1}{\\bf k}.$ The mean and variance of prediction is used by the sampling process to solve the optimization problem, and described as follows.",
"Table: Evaluation results of popular binarization methods on DIBCO datasets.",
"The * ^\\ast marks the cases where existing binarization methods outperform the proposed approach."
],
[
"Sampling Algorithms",
"A sampling scheme must make a trade-off between exploration of the hyperparameter space, and exploitation of sub-spaces with a high likelihood of containing the optima.",
"The variance estimates provided by the GP offer insight on unexplored regions, while the mean predictions point towards estimates of the behavior of the objective function in a region of interest.",
"Therefore, the model can be an effective tool to select a set of samples from a large number of candidates (e.g., generated randomly), that either enrich the model itself (by sampling unexplored regions), or drive the search towards the optima (by exploiting regions with optimal predicted objective function values).",
"Popular sampling schemes in literature include the expected improvement criterion, the probability of improvement criterion and upper/lower confidence bounds [2] (UCB/LCB).",
"Sampling algorithms are also known as acquisition functions in the context of Bayesian optimization [2].",
"The upper and lower confidence bounds offer a good mix of exploration and exploitation.",
"The probability of improvement favors exploitation much more than exploration, while expected improvement lies in between the two.",
"This work uses the UCB criterion for its balanced sampling characteristics in absence of any problem-specific knowledge.",
"Let $\\mu ({\\bf x})$ and $\\sigma ({\\bf x})$ be the posterior mean and variance of prediction provided by the Gaussian process (GP) model.",
"The value of the UCB criterion corresponding to a set of hyperparameters $\\bf x$ is defined as, $\\alpha _{UCB} ({\\bf x}) = \\mu (\\bf x) + \\beta \\sigma ({\\bf x})$ .",
"This essentially corresponds to exploring $\\beta $ intervals of standard deviation around the posterior mean provided by the GP model.",
"The value of $\\beta $ can be set to achieve optimal regret according to well-defined guidelines [35].",
"A detailed discussion of the sampling approaches is out of scope of this work, and the reader is referred to [2], [25] for a deeper treatment of sampling algorithms, and Bayesian optimization in general.",
"The following section demonstrates the proposed approach on benchmark datasets and compares it to existing approaches."
],
[
"Experimental setup",
"The proposed binarization method has been tested on the images from the DIBCO dataset [3], [5], [7] that consists of machine-printed and handwritten images with associated ground truth available for validation and testing, and the H-DIBCO [4], [6], [8], [9] dataset that consists of handwritten document test images.",
"The performance of the proposed method is compared with the state-of-the-art binarization methods such as [10], [12], [13], [11], [17], [1].",
"Six hyperparameters of the binarization algorithm are automatically selected using Bayesian optimization.",
"These include a local threshold $\\tau _1$ and local window size $ws$ for adaptive median filtering; and local threshold $\\tau _2$ , mask size for blurring the text $ms$ , window size $ws_h$ and $ws_l$ for high frequency and low frequency band-pass filters respectively.",
"The corresponding optimization problem is formulated as, $\\begin{aligned}& \\underset{\\bf x}{\\text{maximize}}& & \\mathrm {f_{measure}}(\\bf x) \\\\& \\text{subject to}& & 0.05 \\le \\tau _1 \\le 0.2, \\; 35 \\le ws \\le 95, \\; 0.05 \\le \\tau _2 \\le 0.5, \\\\&&& 0 \\le ms \\le 10, \\; 200 \\le ws_h \\le 400, \\; 50 \\le ws_l \\le 150,\\end{aligned}$ where ${\\bf x} = (\\tau _1, ws, \\tau _2, ms, ws_h, ws_l)$ .",
"This work uses the Bayesian optimization framework available as part of MATLAB (R2017a).",
"The parameter $\\beta $ of UCB criterion was set to 2.",
"Figure: Document image binarization results obtained on sample test images from the H-DIBCO 2016 dataset."
],
[
"Experimental results",
"The evaluation measures are adapted from the DIBCO reports [3], [4], [5], [6], [7], [8], [9], and include F-measure, Peak Signal-to-Noise Ratio (PSNR), Distance Reciprocal Distortion metric (DRD), Negative Rate Metric (NRM) and Misclassification Penalty Metric (MPM).",
"The binarized image quality is better with high F-measure and PSNR values, and low DRD, MPM and NRM values.",
"For details on the evaluation measures, the reader is referred to [3], [9].",
"Table: Evaluation results on the H-DIBCO 2016 dataset and comparison with top ranked methods from the competition.Table: Comparison results of average F-measure (%), PSNR and DRD values obtained using different binarization methods.Figure: The evolution of F-measure during the Bayesian optimization process.",
"The estimated objective refers to the value of F-measure predicted by the GP model trained as part of the optimization process.",
"The model accurately tracks the value of the objective function.",
"The objective values are negative since the implementation followed the convention of minimizing the objective function rather than maximizing.",
"Therefore, the objective function here is -1*F-Measure-1 * F-Measure.",
"Figure best viewed in color.The experimental results are presented in Table REF .",
"The proposed method is compared to several popular binarization methods on competition datasets from 2009 to 2014.",
"Table REF illustrates the evaluation results on the most recent H-DIBCO 2016 dataset, and a comparison is drawn with the top five ranked methods from the competition, and the state-of-the-art methods.",
"Figure REF highlights the document binarization results for sample test images from the H-DIBCO 2016 dataset and compares with the results obtained from the algorithm of the competition winner.",
"Finally, Table REF presents the average F-measure, PSNR and DRD values across different dataset combinations.",
"In Tables REF -REF , $\\uparrow $ implies that a higher value of F-measure and PSNR is desirable, while $\\downarrow $ implies a lower value of DRD, NRM and MPM is desirable.",
"The $^\\ast $ indicates a case where the result of an existing method is better than the proposed method.",
"It is observed from Table REF and Figure REF that the proposed method achieves higher scores with respect to F-measure, PSNR and DRD, as compared to other methods.",
"However, on closely inspecting Table REF , it can be seen that there are instances where existing methods outperform the proposed method by a close margin (marked as $^\\ast $ ).",
"Nevertheless, with reference to all datasets used in the experiments, the proposed method is found to be most consistent and stable with high F-measure and PSNR, and low DRD, NRM and MPM scores.",
"Table REF empirically evaluates the performance of the proposed method with respect to all 86 images from DIBCO 2009-2016.",
"On an average, the proposed method achieves $90.99\\%$ F-measure and $19.00$ PSNR for all test images under the experimental settings.",
"For DIBCO 2009-2013, the top ranked method [17] from the competition achieves $89.15\\%$ F-measure, and the proposed method outperforms it by achieving $90.21\\%$ accuracy.",
"The top ranked method [1] in DIBCO 2011-2014 competition obtains $91.88\\%$ accuracy, which is marginally higher (by $0.72\\%$ ) than the accuracy achieved using the proposed method ($91.16\\%$ ).",
"The proposed method produces least visual distortions (DRD) in comparison to other methods.",
"Figure REF conveys the accuracy of the GP model trained as part of the Bayesian optimization process.",
"The estimated values of the F-measure (the green curve) are in line with the observed values (obtained by computing F-measure values of the selected samples, represented by the blue curve).",
"This validates the accuracy of the GP model and subsequently, the correctness of the Bayesian optimization process.",
"In general, the Bayesian optimization-based approach used herein can aid in automating state-of-the-art binarization methods."
],
[
"Conclusions",
"A novel binarization technique is presented in this paper that efficiently segments the foreground text from heavily degraded document images.",
"The proposed technique is simple, robust and fully automated using Bayesian optimization for on-the-fly hyperparameter selection.",
"The experimental results on challenging DIBCO and H-DIBCO datasets demonstrate the effectiveness of the proposed method.",
"On an average, the accuracy of the proposed method for all test images is found to be $90.99\\%$ (F-measure).",
"As future work, the ideas presented herein will be scaled to perform preprocessing of images in word spotting algorithms, and hybridization of the proposed technique with existing state-of-the-art binarization methods will be explored."
],
[
"Acknowledgment",
"This work was supported by the Swedish strategic research programme eSSENCE, the Riksbankens Jubileumsfond (Dnr NHS14-2068:1), and the Göran Gustafsson foundation."
]
] | 1709.01782 |
[
[
"Thresholds for hanger slackening and cable shortening in the Melan\n equation for suspension bridges"
],
[
"Abstract The Melan equation for suspension bridges is derived by assuming small displacements of the deck and inextensible hangers.",
"We determine the thresholds for the validity of the Melan equation when the hangers slacken, thereby violating the inextensibility assumption.",
"To this end, we preliminarily study the possible shortening of the cables: it turns out that there is a striking difference between even and odd vibrating modes since the former never shorten the cables.",
"These problems are studied both on beams and plates."
],
[
"Introduction",
"In 1888, the Austrian engineer Josef Melan [6] introduced the so-called deflection theory and applied it to derive the differential equation governing a suspension bridge, modeled as a combination of a string (the sustaining cable) and a beam (the deck), see Figure REF .",
"The beam and the string are connected through hangers.",
"Since the spacing between hangers is usually small relative to the span, the set of the hangers is considered as a continuous membrane connecting the cable and the deck.",
"Figure: Beam (red) sustained by a cable (black) through parallel hangers.Let us quickly outline how the Melan equation is derived; we follow here [15].",
"We denote by $L$ the length of the beam at rest (the distance between towers) and $x\\in (0,L)$ the position on the beam; $p=p(x)$ the live load and $-q<0$ the dead load per unit length applied to the beam; $g=g(x)$ the displacement of the cable due to the dead load $-q$ ; $L_c$ the length of the cable subject to the dead load $-q$ ; $A$ the cross-sectional area of the cable and $E_c$ its modulus of elasticity; $H$ the horizontal tension in the cable, when subject to the dead load $-q$ only; $EI$ the flexural rigidity of the beam; $w=w(x)$ the displacement of the beam due to the live load $p$ ; $h=h(w)$ the additional tension in the cable produced by the live load $p$ .",
"When the system is only subject to the action of dead loads, the cable is in position $g(x)$ while the unloaded beam is in the horizontal position $w\\equiv 0$ , see Figure REF .",
"The cable is adjusted in such a way that it carries its own weight, the weight of the hangers and the weight of the deck (beam) without producing a bending moment in the beam, so that all additional deformations of the cable and the beam due to live loads are small.",
"The cable is considered as a perfectly flexible string subject to vertical dead and live loads.",
"The string is subject to a downwards vertical constant dead load $-q$ and the horizontal component $H>0$ of the tension remains constant.",
"If the mass of the cable is neglected, then the dead load is distributed per horizontal unit.",
"The resulting equation simply reads $Hg^{\\prime \\prime }(x)=q$ (see [15]) so that the cable takes the shape of a parabola with a $\\cup $ -shaped graph.",
"If the endpoints of the string (top of the towers) are at the same level $\\gamma >0$ (as in suspension bridges, see again Figure REF ), then the solution $g$ and the length $L_c$ of the cable are given by: $g(x)\\!=\\!\\gamma \\!+\\!\\frac{q}{2H}x(x-L)\\, ,\\quad g^{\\prime }(x)=\\frac{q}{H}\\left(x-\\frac{L}{2}\\right)\\, ,\\quad g^{\\prime \\prime }(x)=\\frac{q}{H}\\, ,\\quad \\forall x\\in (0,L),$ $ L_c\\!=\\!\\int \\limits _0^{L}\\!\\sqrt{1\\!+\\!g^{\\prime }(x)^2}\\, dx.$ The elastic deformation of the hangers is usually neglected, so that the function $w$ describes both the displacements of the beam and of the cable from its equilibrium position $g$ .",
"This classical assumption is justified by precise studies on linearized models, see e.g.",
"[5].",
"When the live load $p$ is added, a certain amount $p_1$ of $p$ is carried by the cable whereas the remaining part $p-p_1$ is carried by the bending stiffness of the beam.",
"In this case, it is well-known [6], [15] that the equation for the displacement $w$ of the beam is $EI\\, w^{\\prime \\prime \\prime \\prime }(x)=p(x)-p_1(x)\\qquad \\forall x\\in (0,L)\\, .$ At the same time, the horizontal tension of the cable is increased to $H+h(w)$ and the deflection $w$ is added to the displacement $g$ .",
"Hence, according to (REF ), the equation which takes into account these conditions reads $\\big (H+h(w)\\big )\\big (g^{\\prime \\prime }(x)+w^{\\prime \\prime }(x)\\big )=q-p_1(x)\\qquad \\forall x\\in (0,L)\\, .$ Then, by combining (REF )-(REF )-(REF ), we obtain $EI\\, w^{\\prime \\prime \\prime \\prime }(x)-\\big (H+h(w)\\big )\\, w^{\\prime \\prime }(x)-\\frac{q}{H}\\, h(w)=p(x)\\qquad \\forall x\\in (0,L)\\, ,$ which is known in literature as the Melan equation [6].",
"The beam representing the bridge is hinged at its endpoints, which means that the boundary conditions to be associated to (REF ) are $w(0)=w(L)=w^{\\prime \\prime }(0)=w^{\\prime \\prime }(L)=0\\, .$ Theoretical results on the Melan equation (REF ) are quite demanding [3], [4] and this is the reason why it has attracted the attention of numerical analysts [9], [10], [11], [16].",
"In this paper we analyze and quantify the two main nonlinear (and challenging) behaviors of (REF ).",
"The first one is the additional tension of the cable, $h(w)$ which is a nonlocal term and is proportional to the length increment of the cable.",
"Depending on the deflection of the beam, the cable may vary its shape and tension, and such phenomenon is studied in Section where we compute the exact thresholds of shortening, depending on the deflection $w$ .",
"In Theorem REF we show that there is a striking difference between the even and odd vibrating modes of the beam.",
"The second source of nonlinearity is the possible slackening of the hangers which, however, is not considered in (REF ) due to the assumption of inextensibility of the hangers.",
"Indeed, $w$ in (REF ) aims to represent both the deflections of the beam and of the cable, implying that the cable reaches the new position $g+w$ .",
"But since the hangers do not resist to compression, they may slacken so that the cable and the beam move independently and $w$ will no longer represent the displacement of the cable from its original position.",
"This phenomenon is analyzed in detail in Section where we suggest an improved version of (REF ) which also takes into account the slackening of the hangers, see (REF ).",
"In Section we extend this study to a partially hinged rectangular plate aiming to model the deck of a bridge and thereby having two opposite edges completely free: we view these free edges as beams sustained by cables and governed by the Melan equation.",
"The results are complemented with some enlightening figures."
],
[
"Thresholds for cable shortening in a beam model",
"A given displacement of the deck $w\\in C^1([0, L], \\mathbb {R})$ generates an additional tension $h(w)$ in the cable that is proportional to the increment of length of the cable $\\Gamma (w)$ , that is, $ h(w) = \\frac{E_cA}{L_{c}}\\Gamma (w)\\ \\mbox{ where }\\ \\Gamma (w) = \\int \\limits _{0}^{L}\\Big [\\sqrt{1 + \\big (w^{\\prime }(x)+g^{\\prime }(x)\\big )^2}-\\sqrt{1 + g^{\\prime }(x)^2}\\Big ] \\ dx\\, .$ Definition 2.1 We say that a displacement $w$ shortens the cable if $\\Gamma (w)<0$ .",
"There are at least three rude ways to approximate $h(w)$ , by replacing $\\Gamma (w)$ with $-\\tfrac{q}{H}\\!\\int _0^L\\!\\!w(x)dx,\\quad -\\tfrac{q}{H}\\!\\int _0^L\\!\\!w(x)dx+\\int _0^L\\!\\tfrac{w^{\\prime }(x)^2}{2}dx,\\quad -\\tfrac{q}{H}\\!\\!_0^L\\!\\tfrac{w(x)}{\\left[1+\\tfrac{q^2}{H^2}\\left(x-\\tfrac{L}{2}\\right)^2\\right]^{3/2}}dx.$ These approximations are obtained through an erroneous argument.",
"While introducing (REF ), Biot-von Kármán [15] warn the reader by writing whereas the deflection of the beam may be considered small, the deflection of the string, i.e., the deviation of its shape from a straight line, has to be considered as of finite magnitude.",
"However, they later decide to neglect $g^{\\prime }(x)^2$ in comparison with unity.",
"A similar mistake with a different result is repeated by Timoshenko [13], [14].",
"These approximations may lead to an average error of about $5\\%$ for $h(w)$ .",
"Around 1950 the civil and structural German engineer Franz Dischinger emphasized the dramatic consequences of bad approximations on the structures and $5\\%$ turns out to be a too large error.",
"Moreover, since related numerical procedures are very unstable, see [3], [9], [10], [11], also from a mathematical point of view one should analyze the term $h(w)$ with extreme care.",
"Since the displacement of the deck $w$ , created by a live load $p$ , is the solution of the Melan equation (REF ), we study here which loads yield a shortening of the cable.",
"In particular, we analyze the fundamental modes of vibration of the beam so that we consider the following class of live loads: $ p_n(x) =\\rho \\left(\\frac{n \\pi }{L} \\right)^{2} \\left\\lbrace \\left(\\frac{n \\pi }{L} \\right)^{2} EI + H + h\\left(\\rho \\sin \\left(\\tfrac{n \\pi x}{L} \\right)\\right)\\right\\rbrace \\sin \\left(\\frac{n \\pi x}{L} \\right) - \\frac{q}{H}h \\left(\\rho \\sin \\left(\\tfrac{n \\pi x}{L} \\right)\\right)\\quad \\forall n\\in \\mathbb {N},$ for varying values of $\\rho \\in \\mathbb {R}$ .",
"The load $p_n$ consists of a negative constant part $-\\frac{q}{H}h\\big (\\rho \\sin (\\frac{n \\pi x}{L})\\big )$ and a part that is proportional to the fundamental vibrating modes of the beam $\\sin \\left(\\frac{n \\pi x}{L} \\right)$ , which are the eigenfunctions of the following eigenvalue problem: $v^{\\prime \\prime \\prime \\prime }(x)=\\lambda v(x)\\quad (0<x<L)\\, ,\\qquad v(0)=v(L)=v^{\\prime \\prime }(0)=v^{\\prime \\prime }(L)=0\\, .$ The reason of this choice for $p_n$ is that, after some computations, one sees that the resulting displacement $w_n$ (solution of (REF )) is proportional to a vibrating mode: $ w_{n}(x) = \\rho \\sin \\left(\\frac{n \\pi x}{L} \\right)\\quad \\forall x \\in [0,L].$ Whence, $|\\rho |$ measures the amplitude of oscillation of the vibrating mode $w_{n}$ .",
"For every $n \\in \\mathbb {N}$ , we put $\\Gamma _{n}(\\rho ):=\\Gamma (w_n)$ and from (REF ) we infer that $ \\Gamma _{n}(\\rho ) = \\int \\limits _{0}^{L} \\sqrt{1 + \\left[\\dfrac{q}{H}\\left(x - \\dfrac{L}{2} \\right)+\\dfrac{n\\pi }{L}\\rho \\cos \\left(\\frac{n \\pi x}{L} \\right) \\right] ^2} \\ dx - L_{c}\\quad \\forall \\rho \\in \\mathbb {R}.$ In the next result we emphasize a striking difference between even and odd modes.",
"Theorem 2.1 Assume that $\\tfrac{q}{H}<\\tfrac{2}{5}$ .",
"$\\bullet $ If $n \\ge 1$ is even, then $\\Gamma _{n}(\\rho )\\ge 0$ for all $\\rho $ ; therefore, an even vibrating mode cannot shorten the cable.",
"$\\bullet $ If $n \\ge 1$ is odd, then there exists a (unique) critical value $\\rho _{n}^{*}>0$ such that $\\Gamma _{n}(\\rho _{n}^{*})=0$ and $\\Gamma _{n}(\\rho )<0$ for all $\\rho \\in (0,\\rho _{n}^{*})$ ; therefore, odd vibrating modes shorten the cable when their amplitude of oscillation $\\rho $ is within this interval.",
"Theorem REF is proved in Section .",
"The assumption $q/H<2/5$ in Theorem REF is verified in the vast majority of real suspension bridges.",
"For instance, for the numerical data employed in [16], it happens that $q/H=1.739 \\times 10^{-3} \\, [m^{-1}]$ .",
"Moreover, as reported in [8], the sag-span ratio in a suspension bridge always lies in the range $(\\tfrac{1}{12},\\tfrac{1}{8})$ .",
"In view of (REF ), this means that $\\dfrac{L}{12} < g(0) - g \\left( \\dfrac{L}{2} \\right) < \\dfrac{L}{8}\\quad \\mbox{or, equivalently,}\\quad \\dfrac{2}{3L} < \\dfrac{q}{H} < \\dfrac{1}{L}.$ Therefore, the assumption $\\tfrac{q}{H}<\\tfrac{2}{5}$ is valid for any suspension bridges with a span of at least $2.5\\, [m]$ !",
"In any case, numerical results seem to show that the assumption $\\tfrac{q}{H}<\\tfrac{2}{5}$ is not necessary for the validity of Theorem REF .",
"Related to $\\rho _n^*$ , as characterized by Theorem REF , we introduce the quantity $ \\xi _{n}^*=\\rho _{n}^*\\left(\\frac{n \\pi }{L} \\right)^{2} \\left\\lbrace \\left(\\frac{n \\pi }{L} \\right)^{2} EI + H + \\frac{E_cA}{L_{c}} \\Gamma \\big (\\rho _n^*\\sin \\left(\\tfrac{n \\pi x}{L}\\right)\\big )\\right\\rbrace \\quad \\forall n \\in \\mathbb {N},$ which is the amplitude of oscillation of the live load $p_n$ in (REF ) that generates the critical oscillation $w_{n}^*(x)=\\rho _n^*\\sin (\\tfrac{n \\pi x}{L})$ .",
"Throughout this paper, as far as numerical data are needed, we use the parameters taken from [16]: $L = 460\\, [m],\\quad EI = 57 \\times 10^{6}\\, [kN \\cdot m],\\quad E_cA=36\\times 10^{6} \\, [kN],\\quad \\frac{q}{H} = 1.739 \\times 10^{-3} \\, [m^{-1}].$ Table REF shows the critical values of $\\rho _{n}^{*}$ and $\\xi _{n}^{*}$ (according to Theorem REF and (REF )), as functions of some odd values of $n \\in \\mathbb {N}$ .",
"Table: NO_CAPTIONAs stated in Theorem REF , even modes never shorten the cable.",
"This does not mean that odd modes are “worse” or more prone to elongate the cable.",
"On the contrary, thinking of a periodic-in-time oscillation proportional to a vibrating mode (REF ), that is, $\\rho (t)\\sin \\left(\\frac{n \\pi x}{L} \\right)\\quad \\forall x \\in [0,L],\\quad \\forall t>0\\, ,$ with $\\rho (t)$ varying between $\\pm \\overline{\\rho }$ , we reach the opposite conclusion.",
"To see this, in Figure REF we plot the graphs of $\\Gamma _{2}$ and $\\Gamma _{3}$ and we see that $\\max \\lbrace \\Gamma _3(\\overline{\\rho }),\\Gamma _3(-\\overline{\\rho })\\rbrace >\\max \\lbrace \\Gamma _2(\\overline{\\rho }),\\Gamma _2(-\\overline{\\rho })\\rbrace =\\Gamma _2(\\overline{\\rho })\\, .$ Therefore, even if the cable shortens when $\\rho (t)\\in (0,\\rho _3^*)$ for the third mode, the cable itself elongates more than for the second mode when $\\rho (t)<0$ .",
"We come back to this issue in Section .",
"Figure: Increment of cable length in the second (left) and third (right) vibrating modes."
],
[
"Thresholds for hangers slackening in a beam model",
"In this section we estimate the thresholds that provoke the slackening of some hangers.",
"Since the hangers resist to extension but not to compression, if the deck goes too high above its equilibrium position, then the hangers may no longer be considered as rigid inextensible bars.",
"In particular, they will not push upwards the cable in such a way that it loses convexity: the general principles governing the deformation of a finite-length cable under the action of a downwards vertical load (see [15]) indicate that the cable remains convex.",
"This means that if $g+w$ is not convex, then it does not describe the position of the cable anymore.",
"In order to explain how the Melan equation (REF ) should be modified in case of hanger slackening we briefly recall the concept of convexification which can be formalized in several equivalent ways, see [1] for full details.",
"Let $I\\subset \\mathbb {R}$ be a compact interval.",
"The convexification $f^{**}$ of a continuous function $f:I\\rightarrow \\mathbb {R}$ is: $\\bullet $ the pointwise supremum of all the affine functions everywhere less than $f$ ; $\\bullet $ the pointwise supremum of all the convex functions everywhere less than $f$ ; $\\bullet $ the largest convex function everywhere less than or equal to $f$ ; $\\bullet $ the convex function whose epigraph is the closed convex hull of the epigraph of $f$ ; $\\bullet $ the second Fenchel conjugate of $f$ , that is, $f^{**}(x)=\\sup _{y\\in \\mathbb {R}} \\lbrace xy - f^{*}(y) \\rbrace \\ \\ \\forall x\\in I\\, ,\\quad \\mbox{ where }\\quad f^{*}(y)=\\max _{x\\in I}\\lbrace yx-f(x)\\rbrace \\ \\ \\forall y\\in \\mathbb {R}\\, .$ This notion enables us to give the following: Definition 3.1 We say that a displacement $w$ slackens the hangers in some (nonempty) interval $(a,b)\\subset [0,L]$ if the graph of $z:=g+w$ lies strictly above that of its convexification $z^{**}$ in $(a,b)$ .",
"Then, the slackening region $\\mathcal {S}\\subset [0,L]$ is the union of all the slackening intervals, that is, $\\mathcal {S}=\\lbrace x \\in (0,L) \\ | \\ z(x) > z^{**}(x) \\rbrace \\, .$ In the slackening region, not only the Melan equation (REF ) is incorrect but also (REF ) fails since the whole amount of live load is carried by the beam: one has $p_1(x)=0$ for all $x\\in \\mathcal {S}$ .",
"Therefore (REF ) should be replaced with the more reliable equation $EI\\, w^{\\prime \\prime \\prime \\prime }(x)+\\Big (\\chi _\\mathcal {S}(w) - 1\\Big )\\bigg (\\big (H+h(w)\\big )\\, w^{\\prime \\prime }(x)+\\frac{q}{H}\\, h(w)\\bigg )=p(x)\\qquad \\forall x\\in (0,L)$ where $\\chi _\\mathcal {S}(w)$ is the characteristic function (that depends on $w$ ) of the slackening region $\\mathcal {S}$ , see Definition REF .",
"We summarize these results in the following statement.",
"Proposition 3.1 In absence of slackening ($\\mathcal {S}=\\emptyset $ ) the two equations (REF ) and (REF ) coincide; in this case, the solution $w$ represents the displacement of the beam whereas $z$ in (REF ) represents the position of the cable.",
"In presence of slackening ($\\mathcal {S}\\ne \\emptyset $ ) the correct equation is (REF ) and the position of the cable is described by $z^{**}$ .",
"The term $(\\chi _\\mathcal {S}(w)-1)$ adds a further nonlinearity to the Melan equation (REF ).",
"As far as we are aware, there is no general theory to tackle equations such as (REF ).",
"It would therefore be interesting to study its features in detail.",
"Although the exact slackening region is difficult to determine, it is clear that the non-convexity intervals of $z$ in (REF ) represent proper subsets of these regions.",
"Therefore, we have Proposition 3.2 Let $w$ be the solution of (REF ) and let $z$ be as in (REF ).",
"If $\\mathcal {S}\\ne \\emptyset $ , then $\\lbrace x\\in (0,L);\\, z^{\\prime \\prime }(x)\\le 0\\rbrace \\subsetneqq \\mathcal {S}\\, .$ We now apply Proposition REF to the case of the loads $p_n$ in (REF ).",
"Proposition 3.3 Let $p_n$ and $w_n$ be as in (REF ) and (REF ).",
"Let $ C_{n}^{*} := \\frac{q}{H} \\left(\\frac{L}{n \\pi } \\right)^{2}\\quad \\forall n \\in \\mathbb {N}\\, .$ Slackening occurs if and only if $\\rho >C_1^*\\mbox{ when }n=1\\, ,\\qquad |\\rho |>C_n^*\\mbox{ when }n\\ge 2\\, ;$ in this case, the position of the cable is described by $z_n^{**}$ (with $z_n=g+w_n$ ).",
"The proof of Proposition REF is fairly simple.",
"The slackening region of $w_n$ is nonempty if and only if there exists $x\\in (0,L)$ such that $z_n^{\\prime \\prime }(x)<0$ , where $z_{n}(x) =g(x)+w_n(x)= \\gamma + \\frac{q}{2H}x(x-L) + \\rho \\sin \\left(\\frac{n \\pi x}{L} \\right)\\quad \\forall x \\in [0,L].$ This property translates into $\\exists x\\in (0,L)\\quad \\mbox{such that}\\quad \\rho \\sin \\left(\\frac{n \\pi x}{L} \\right)>C_n^*\\, ,$ which is equivalent to (REF ).",
"Figure: Slackening of the third vibrating mode when ρ 3 <-C 3 * \\rho _{3} < -C_{3}^{*} (left) and when ρ 3 >C 3 * \\rho _{3} > C_{3}^{*} (right).Since it is by far nontrivial to determine explicitly the convexification of $z_n$ and the slackening region $\\mathcal {S}_{n}$ , we follow a numerical-geometrical approach, that is, we plot the closed convex hull of the epigraph of $z_n$ .",
"We take again the numerical values (REF ).",
"In order to illustrate the procedure, consider the function $z_{3}$ (with $\\gamma = 0$ , since we are only interested in the shape of the curve), whose slackening threshold is $C_{3}^{*} \\approx 4.1426$ .",
"By putting amplitudes of $\\rho _{3} = \\pm 10$ , we obtained the graphs of $z_{3}$ in Figure REF where the slackening intervals have been highlighted over the horizontal axis, and the closed convex hull of the epigraph of $z_{3}$ has been shaded.",
"Similarly, by putting amplitudes of $\\rho _{5} = \\pm 5$ , we obtained the plots displayed in Figure REF for the graphs of $z_{5}$ (for which $C_{5}^{*} \\approx 1$ .4913): Figure: Slackening of the fifth vibrating mode when ρ 5 <-C 5 * \\rho _{5} < -C_{5}^{*} (left) and when ρ 5 >C 5 * \\rho _{5} > C_{5}^{*} (right).It is worthwhile noticing that the hangers slackening in even modes occurs asymmetrically with respect to the center of the beam but, at the same time, symmetrically with respect to the value of $\\rho _{n}$ .",
"To clarify this point, in Figure REF we display the graphs of $z_{2}$ (where $C_{2}^{*}\\approx 9.3208$ ) when $\\rho _{2} = -20$ , and of $z_{4}$ (where $C_{4}^{*}\\approx 2.3301$ ) when $\\rho _{4} = 8$ .",
"The remaining figures when $\\rho _{2} > C_{2}^{*}$ or $\\rho _{4} < -C_{4}^{*}$ may be obtained by simply reflecting the curves with respect to the center of the beam.",
"Figure: Slackening of the second vibrating mode when ρ 2 <-C 2 * \\rho _{2} <- C_{2}^{*} (left), and of the fourth vibrating mode when ρ 4 >C 4 * \\rho _{4} > C_{4}^{*}(right).The numerical values of $C_n^*$ for $n\\le 10$ are reported in Table REF where we used the parameters as in (REF ).",
"One last issue must be addressed.",
"In some of the pictures in Figures REF , REF and REF we observe that the endpoints of the deck $x=0$ and $x=L$ actually belong to the slackening region $\\mathcal {S}_{n}$ .",
"This is clearly a physically impossible situation since the hangers are not expected to slacken at the endpoints of the beam.",
"Geometrically, one expects instead that the tangent lines to the curve at the endpoints of the beam lie strictly below the graph of $z_{n}$ in $(0,L)$ , that is: $ z_{n}(x) > \\max \\lbrace z_{n}^{\\prime }(0)x, \\ z_{n}^{\\prime }(L)(x-L) \\rbrace \\ \\ \\forall x \\in (0,L), \\ \\ \\forall n \\in \\mathbb {N}.$ Clearly, condition (REF ) is not satisfied for large values of $|\\rho _{n}|$ , but it remains valid even when $|\\rho _{n}|$ is slightly larger than the slackening (and convexity) threshold (REF ).",
"For the first ten vibrating modes, we numerically computed the threshold $\\rho _{n}^{**}$ that ensures condition (REF ), when $|\\rho _n|\\le \\rho _{n}^{**}$ (if $n$ is even) and $\\rho _{n} \\le \\rho _{n}^{**}$ (if $n$ is odd), with the parameters as in (REF ).",
"We obtained the second line in Table REF .",
"Table: NO_CAPTION"
],
[
"Behavior of cables and hangers in a plate model",
"The deck of a real bridge cannot be described by a simple (one-dimensional) beam since it fails to display torsional oscillations.",
"In this section we take advantage of the results so far obtained in order to analyze the vibrating modes of a rectangular plate $\\Omega =(0,\\pi )\\times (-\\ell ,\\ell )$ ($2\\ell >0$ is the width of the plate and $2\\ell \\ll \\pi $ ); for simplicity, we take here $L=\\pi $ .",
"Specifically, we consider a partially hinged plate whose elastic energy is given by the Kirchhoff-Love functional, see [7], [12] for discussions on the boundary conditions and updated derivation of the corresponding Euler-Lagrange equation.",
"From [2] we know that the vibrating modes of the plate $\\Omega $ are obtained by solving the following eigenvalue problem $\\left\\lbrace \\begin{array}{ll}\\Delta ^2 u = \\lambda u\\quad & \\text{for }(x,y)\\in \\Omega \\\\u=u_{xx}=0\\quad & \\text{for }(x,y)\\in \\lbrace 0,\\pi \\rbrace \\times (-\\ell , \\ell )\\\\u_{yy}+\\sigma u_{xx}=u_{yyy}+(2-\\sigma )u_{xxy}=0\\quad & \\text{for }(x,y)\\in (0,\\pi )\\times \\lbrace -\\ell ,\\ell \\rbrace \\, ,\\end{array}\\right.",
"$ where $\\sigma \\in \\left(0 , \\frac{1}{2} \\right)$ is the Poisson ratio.",
"The boundary conditions for $x=0$ and $x=\\pi $ show that the short edges of the plate are hinged, while the conditions for $y=\\pm \\ell $ show that the plate is free on the long edges.",
"Problem (REF ) is the two-dimensional counterpart of (REF ).",
"From [2] we also know that the eigenvalues of (REF ) may be ordered in an increasing sequence of strictly positive numbers diverging to $+\\infty $ .",
"Correspondingly, the eigenfunctions are identified by two indices $m,k\\in \\mathbb {N}_+$ and they have one of the following forms: $W_{m,k}(x,y)=\\varphi _{m,k}(y)\\sin (mx) \\,\\quad \\text{with corresponding eigenvalue } \\nu _{m,k}\\, ,$ $\\overline{W}_{m,k}(x,y)=\\psi _{m,k}(y)\\sin (mx)\\,\\quad \\text{with corresponding eigenvalue } \\mu _{m,k}\\, .$ The $\\varphi _{m,k}$ are odd while the $\\psi _{m,k}$ are even and this is why the $W_{m,k}$ are called torsional eigenfunctions while the $\\overline{W}_{m,k}$ are called longitudinal eigenfunctions.",
"The main difference between these two classes is precisely that $\\overline{W}_{m,k}(x,\\ell )=\\overline{W}_{m,k}(x,-\\ell )$ so that the free edges $y=\\pm \\ell $ are in the same position for longitudinal vibrations, while $W_{m,k}(x,\\ell )=-W_{m,k}(x,-\\ell )$ so that the free edges are in opposite positions for torsional vibrations.",
"We first deal with the slightly more complicated case of torsional vibrating modes.",
"Then the eigenvalues $\\nu _{m,k}$ are the (ordered) solutions $\\lambda > m^4$ of the following equation: $\\sqrt{\\lambda ^{1/2} - m^2}[\\lambda ^{1/2} + (1-\\sigma )m^2]^2 \\tanh (\\ell \\sqrt{\\lambda ^{1/2} + m^2})= \\sqrt{\\lambda ^{1/2} + m^2}[\\lambda ^{1/2} - (1-\\sigma )m^2]^2 \\tanh (\\ell \\sqrt{\\lambda ^{1/2} - m^2})\\, ,$ while the function $\\varphi _{m,k}$ may be taken as $\\varphi _{m,k}(y)=\\big [\\nu _{m,k}^{1/2}-\\!",
"(1\\!-\\!\\sigma )m^2\\big ]\\tfrac{\\sinh \\Big (y\\sqrt{\\nu _{m,k}^{1/2}+m^2}\\Big )}{\\sinh \\Big (\\ell \\sqrt{\\nu _{m,k}^{1/2}+m^2}\\Big )}+\\big [\\nu _{m,k}^{1/2}+\\!",
"(1\\!-\\!\\sigma )m^2\\big ]\\tfrac{\\sin \\Big (y\\sqrt{\\nu _{m,k}^{1/2}-m^2}\\Big )}{\\sin \\Big (\\ell \\sqrt{\\nu _{m,k}^{1/2}-m^2}\\Big )}\\, ,$ see [2].",
"In particular, $\\varphi _{m,k}(\\ell ) = 2 \\sqrt{\\nu _{m,k}}=-\\varphi _{m,k}(-\\ell )$ .",
"We view both the free edges of the plate $y=\\pm \\ell $ as beams connected to a cable and governed by the modified Melan equation (REF ).",
"Then we take the following function as a solution of (REF ): $w_{m,k}(x):=\\alpha W_{m,k}(x,\\ell )= \\alpha \\varphi _{m,k}(\\ell )\\sin (mx)=2\\alpha \\sqrt{\\nu _{m,k}}\\sin (mx)\\quad \\forall x\\in [0,\\pi ],$ for $m,k\\in \\mathbb {N}$ and $\\alpha \\in \\mathbb {R}$ , a function that belongs to the family of eigenfunctions of (REF ), see (REF ), assuming that $L=\\pi $ .",
"As already mentioned, together with $w_{m,k}$ in (REF ), for torsional modes one needs to consider also its companion $-w_{m,k}$ .",
"For longitudinal modes, one has to replace $w_{m,k}$ in (REF ) with $\\overline{w}_{m,k}(x):=\\alpha \\overline{W}_{m,k}(x,\\ell )=\\alpha \\psi _{m,k}(\\ell )\\sin (mx)\\quad \\forall x\\in [0,\\pi ],$ where $\\psi _{m,k}(\\ell )$ depends on the longitudinal eigenvalue $\\mu _{m,k}$ of (REF ); see [2] for the precise characterization of $\\mu _{m,k}$ .",
"For longitudinal modes, the behavior is the same on the two opposite edges.",
"The above discussion, combined with Theorem REF , yields the following statement.",
"Proposition 4.1 Assume that $\\tfrac{q}{H}<\\tfrac{2}{5}$ .",
"$\\bullet $ If $m\\ge 1$ is even, then the vibrating mode (either torsional or longitudinal) cannot shorten the cable.",
"$\\bullet $ If $m\\ge 1$ is odd and the mode is longitudinal, then there exists a (unique) critical value $\\alpha ^*=\\alpha _{m,k}^*>0$ such that for $\\alpha \\in (0,\\alpha ^*)$ both the cables are shortened while for other values of $\\alpha $ no cable is shortened.",
"$\\bullet $ If $m\\ge 1$ is odd and the mode is torsional, then there exists a (unique) critical value $\\alpha ^*=\\alpha _{m,k}^*>0$ such that for $0<|\\alpha |<\\alpha ^*$ one and only one cable is shortened, while for other values of $\\alpha $ no cable is shortened.",
"Following the guideline of Section , one may then determine the exact critical values $\\alpha _{m,k}^*$ (for odd $m$ ).",
"It suffices to consider the critical values $\\rho _n^*$ from Theorem REF and to take $\\alpha _{m,k}^*=\\frac{\\rho _m^*}{\\varphi _{m,k}(\\ell )}\\quad \\mbox{or}\\quad \\alpha _{m,k}^*=\\frac{\\rho _m^*}{\\psi _{m,k}(\\ell )}\\, ,$ depending on whether the vibration is torsional or longitudinal.",
"Regarding slackening and the loss of convexity, the above discussion, combined with Propositions REF and REF , yields the following statement.",
"Proposition 4.2 Let $w$ be the solution of (REF ) and assume that one of the free edges of $\\Omega $ is in position $w$ .",
"Let $z$ be as in (REF ).",
"If $\\mathcal {S}\\ne \\emptyset $ , then $\\lbrace x\\in (0,L);\\, z^{\\prime \\prime }(x)\\le 0\\rbrace \\subsetneqq \\mathcal {S}\\, .$ In particular, if $w_{m,k}$ in (REF ) (resp.",
"$\\overline{w}_{m,k}$ in (REF )) is the position of one of the free edges of $\\Omega $ , then slackening of the hangers on that edge occurs if and only if $\\alpha _1>\\frac{q}{H\\varphi _{1,k}(\\ell )}\\mbox{ when }m=1\\, ,&\\ &|\\alpha _m|>\\frac{q}{Hm^2\\varphi _{m,k}(\\ell )}\\mbox{ when }m\\ge 2\\\\\\Big (\\mbox{resp.",
"}\\alpha _1>\\frac{q}{H\\psi _{1,k}(\\ell )}\\mbox{ when }m=1\\, ,&\\ &|\\alpha _m|>\\frac{q}{Hm^2\\psi _{m,k}(\\ell )}\\mbox{ when }m\\ge 2\\Big )\\, ;$ in this case, the position of the cable is described by $z_{m,k}^{**}$ (with $z_{m,k}=g+w_{m,k}$ , resp.",
"$z_{m,k}=g+\\overline{w}_{m,k}$ ).",
"The final step consists in considering the evolution equation modeling the vibrations of the partially hinged rectangular plate $\\Omega $ .",
"According to [2], this leads to the following fourth-order wave-type equation: $ \\left\\lbrace \\begin{array}{ll}u_{tt}+\\Delta ^2 u = 0\\quad & \\text{for }(x,y,t)\\in \\Omega \\times \\mathbb {R}_+\\\\u=u_{xx}=0\\quad & \\text{for }(x,y,t)\\in \\lbrace 0,\\pi \\rbrace \\times (-\\ell ,\\ell )\\times \\mathbb {R}_+\\\\u_{yy}+\\sigma u_{xx}=u_{yyy}+(2-\\sigma )u_{xxy}=0\\quad & \\text{for }(x,y,t)\\in (0,\\pi )\\times \\lbrace -\\ell ,\\ell \\rbrace \\times \\mathbb {R}_+\\, .\\end{array}\\right.$ We wish to analyze here the evolution of the cable shortening and of the hanger slackening for the torsional vibrating modes $W_{m,k}$ of (REF ); as for the stationary case, the behavior of the longitudinal modes $\\overline{W}_{m,k}$ is simpler.",
"Therefore, we associate to (REF ) the following initial conditions $u(x,y,0) =B W_{m,k}(x,y),\\quad u_{t}(x,y,0)=0\\qquad \\forall (x,y) \\in \\Omega ,$ for some $B\\in \\mathbb {R}$ .",
"The problem (REF )-(REF ) may be solved by separating variables and the solution is $ u_{m,k}(x,y,t) = B \\cos \\big (\\sqrt{\\nu _{m,k}}\\, t\\big )W_{m,k}(x,y)\\qquad \\forall (x,y,t)\\in \\Omega \\times \\mathbb {R}_+\\, .$ Again, we view both the free edges of $\\Omega $ as beams connected to a cable and governed by the modified Melan equation (REF ).",
"Therefore, we consider the restriction to the free edge $y=\\ell $ (the case $y=-\\ell $ being similar) of the function $u_{m,k}$ in (REF ): $ v_{m,k}(x,t):=u_{m,k}(x,\\ell ,t)= B \\cos \\big (\\sqrt{\\nu _{m,k}}\\, t\\big )\\varphi _{m,k}(\\ell )\\sin (mx)\\qquad \\forall (x,t)\\in (0,\\pi )\\times \\mathbb {R}_+\\, ,$ see (REF ).",
"Similarly, for the longitudinal modes, we consider the function $ \\overline{v}_{m,k}(x,t):= B \\cos \\big (\\sqrt{\\mu _{m,k}}\\, t\\big )\\psi _{m,k}(\\ell )\\sin (mx)\\qquad \\forall (x,t)\\in (0,\\pi )\\times \\mathbb {R}_+\\, ,$ see (REF ).",
"We are interested in determining the conditions under which the cables shorten their length (in odd vibrating modes) or when the hangers slacken.",
"Unlike the preceding situations, such conditions will now be observed over a space-time region, because the coefficients representing the amplitude of the expressions (REF ) and (REF ) are periodic functions in time.",
"Concerning the shortening of the cables, we introduce some notations.",
"Let $\\alpha _{m,k}^*>0$ be as in Proposition REF .",
"If $m\\ge 1$ is odd and the mode is longitudinal, put $I_S=\\lbrace t\\ge 0;\\, 0<B\\cos (\\sqrt{\\mu _{m,k}}\\, t)<\\alpha _{m,k}^*\\rbrace \\, ,\\quad I_N=\\mathbb {R}_+\\setminus I_S\\, .$ If $m\\ge 1$ is odd and the mode is torsional, put $I^S=\\lbrace t\\ge 0;\\, 0<|B\\cos (\\sqrt{\\nu _{m,k}}\\, t)|<\\alpha _{m,k}^*\\rbrace \\, ,\\quad I^N=\\mathbb {R}_+\\setminus I^{S}\\, .$ Note that all these sets are nonempty, although $I^N$ may have null measure: this happens if $|B|\\le \\alpha _{m,k}^*$ .",
"Then, from Proposition REF we deduce the following statement.",
"Figure: For B∈[-0.0001,0.0001]B \\in [-0.0001,0.0001], values of t∈I S t\\in I^S (shaded) provoking cable shortening in the third torsional mode.Proposition 4.3 Assume that $\\tfrac{q}{H}<\\tfrac{2}{5}$ .",
"$\\bullet $ If $m\\ge 1$ is even, then the vibrating mode (either torsional or longitudinal) does not shorten the cables for any $t>0$ .",
"$\\bullet $ If $m\\ge 1$ is odd and the mode is longitudinal, then both the cables are shortened if $t\\in I_S$ whereas no cable is shortened if $t\\in I_N$ .",
"$\\bullet $ If $m\\ge 1$ is odd and the mode is torsional, then one and only one cable is shortened when $t\\in I^S$ whereas no cable is shortened if $t\\in I^N$ .",
"Once more we emphasize the striking difference between odd and even modes.",
"Proposition REF is illustrated in Figure REF by shading the sub-regions of the rectangle $(B,t)\\in [-0.0001 , 0.0001]\\times [0,0.05]$ in which $t\\in I^S$ for the third torsional mode.",
"It turns out that for $0<|B|\\lessapprox 0.000032$ , for almost every $t>0$ one (and only one) cable is shortened, whereas for larger values of $|B|$ the white regions (no shortening) have positive measure.",
"Also the slackening of the hangers (and the loss of convexity) in all the vibrating modes is now observed in a space-time region which periodically-in-time reproduces itself.",
"In order to discuss together the longitudinal and torsional cases, we use the same notation to denote the function to be convexified: $ z_{m,k}(x,t)=g(x)+v_{m,k}(x,t)\\ \\Big (\\mbox{resp.",
"}z_{m,k}(x,t)=g(x)+\\overline{v}_{m,k}(x,t)\\Big )\\qquad \\forall (x,t)\\in (0,\\pi )\\times \\mathbb {R}_+\\, ,$ where $v_{m,k}$ and $\\overline{v}_{m,k}$ are as in (REF ) and (REF ).",
"Concerning the non-convexity regions, for a given $B\\in \\mathbb {R}$ they are characterized by the points $(x,t)\\in [0,\\pi ] \\times [0,\\infty )$ that satisfy the inequality: $\\dfrac{\\partial ^2 z_{m,k}}{\\partial x^2}(x,t) \\le 0$ or, equivalently, by the points $(x,t) \\in [0,\\pi ] \\times [0,\\infty )$ in which: $ B m^2 \\varphi _{m,k}(\\ell ) \\cos \\left( \\sqrt{\\nu _{m,k}} \\ t \\right)\\sin \\left(m x \\right) \\ge \\frac{q}{H}\\quad \\Big (\\mbox{resp.",
"}B m^2 \\psi _{m,k}(\\ell ) \\cos \\left( \\sqrt{\\nu _{m,k}} \\ t \\right)\\sin \\left(m x \\right) \\ge \\frac{q}{H}\\Big )\\, .$ Notice that inequality (REF ) defines a region of $\\mathbb {R}^2$ of positive measure only when $|B|>C_{m,k}^{*}$ , where the convexity threshold is now given by: $C_{m,k}^{*} = \\frac{q}{Hm^2\\varphi _{m,k}(\\ell )} \\mbox{ for the torsional modes}, \\ \\ C_{m,k}^{*} = \\frac{q}{Hm^2\\psi _{m,k}(\\ell )} \\mbox{ for the longitudinal modes},$ for every integers $m,k\\ge 1$ .",
"Precisely, given $B \\in \\mathbb {R}$ and integers $m$ and $k$ , let us put: $\\alpha _{m,k}(t) = B \\cos \\left(\\sqrt{\\nu _{m,k}} \\ t \\right)\\quad \\Big (\\mbox{resp.",
"}\\alpha _{m,k}(t) = B \\cos \\left(\\sqrt{\\mu _{m,k}} \\ t \\right)\\Big )\\quad \\forall t \\ge 0.$ Then, as a consequence of Proposition REF , we obtain the following statement.",
"Proposition 4.4 Let $u$ be the solution of (REF ) and assume that one of the free edges of $\\Omega $ is in position $v_{m,k}$ as in (REF ) or $\\overline{v}_{m,k}$ as in (REF ).",
"Let $z_{m,k}$ be as in (REF ), depending on the vibrating mode considered.",
"If $|B|>C_{m,k}^{*}$ , then $\\mathcal {S}\\ne \\emptyset $ .",
"Furthermore, whenever $|\\alpha _{m,k}(t)| > C_{m,k}^{*}$ we have: $\\left\\lbrace x \\in (0,\\pi ) \\ | \\ \\dfrac{\\partial ^2 z_{m,k}}{\\partial x^2}(x,t) \\le 0 \\right\\rbrace \\subsetneqq \\mathcal {S}\\, .$ More precisely, if $|B|>C_{m,k}^{*}$ and if $v_{m,k}$ in (REF ) (resp.",
"$\\overline{v}_{m,k}$ in (REF )) is the position of one of the free edges of $\\Omega $ , then slackening of the hangers on that edge occurs for all $t>0$ such that: $\\alpha _{1,k}(t)>\\frac{q}{H\\varphi _{1,k}(\\ell )}\\mbox{ when }m=1\\, ,&\\ &|\\alpha _{m,k}(t)|>\\frac{q}{Hm^2\\varphi _{m,k}(\\ell )}\\mbox{ when }m\\ge 2\\\\\\Big (\\mbox{resp.",
"}\\alpha _{1,k}(t)>\\frac{q}{H\\psi _{1,k}(\\ell )}\\mbox{ when }m=1\\, ,&\\ &|\\alpha _{m,k}(t)|>\\frac{q}{Hm^2\\psi _{m,k}(\\ell )}\\mbox{ when }m\\ge 2\\Big )\\, ;$ in this case, the position of the cable is described by $z_{m,k}^{**}$ , with $z_{m,k}=g+v_{m,k}$ as in (REF ).",
"Proposition REF defines the slackening regions in the $(x,t)$ -plane.",
"Since these are difficult to determine explicitly, we focus our attention on the non-convexity regions.",
"As a first example, we take the second torsional mode, whose convexity threshold is $C_{2,1}^{*} \\approx 1.52 \\times 10^{-4}$ .",
"In this case, setting $B = \\pm 2 \\times 10^{-4}$ and considering the rectangle $(x,t) \\in [0,\\pi ] \\times [0,0.035]$ , we obtained Figure REF .",
"Figure: Non-convexity region of the second torsional mode for a time-varying amplitude.Similar plots are obtained for the function $z_{3,1}$ , whose convexity threshold is $C_{3,1}^{*} \\approx 4.5 \\times 10^{-5}$ .",
"By taking $B = \\pm 1 \\times 10^{-4}$ , we get the following sub-region of the space-time rectangle $(x,t)\\in [0,\\pi ] \\times [0,0.025]$ defined by Proposition REF : Figure: Non-convexity region of the third torsional mode for a time-varying amplitude.All these plots may also be read by assuming that the right and left pictures represent simultaneously the non-convexity intervals for each cable, as far as torsional vibrations are involved: for any given $t>0$ one should cross horizontally the two pictures in order to find which part of the interval $(0,\\pi )$ of the two cables would be non-convex.",
"In fact, the non-convexity regions are proper subsets of the slackening regions, see Proposition REF .",
"Hence, the slackening regions are slightly wider in the $x$ -direction than the “ellipses” in the above plots.",
"This fact is illustrated in Figure REF where we compare the non-convexity and slackening regions in the third torsional mode: Figure: Slackening region for the third torsional mode"
],
[
"Proof of Theorem ",
"The first step is a technical lemma which involves hyper-geometric integrals: Lemma 5.1 For odd $n \\in \\mathbb {N}$ and $0<\\mu <\\frac{2}{5}$ we have $ G_{n}:= \\int \\limits _{0}^{\\pi / 2} \\dfrac{t \\sin (nt)}{\\sqrt{1 + \\left(\\mu t \\right)^2}} \\, dt\\ \\left\\lbrace \\begin{array}{ll}>0 & \\mbox{if }n \\equiv 1 \\ (\\text{mod} \\ 4)\\\\<0 & \\mbox{if }n \\equiv 3 \\ (\\text{mod} \\ 4).\\end{array}\\right.$ For $t \\in \\mathbb {R}$ such that $| t | < \\dfrac{1}{\\mu }$ , the following power expansion is valid: $\\dfrac{1}{\\sqrt{1 + \\left(\\mu t \\right)^2}} = \\sum _{k=0}^{\\infty } \\binom{-1 / 2}{k} (\\mu t)^{2k}.$ Therefore, since $\\mu < \\frac{2}{5}$ , for all $t \\in \\left[0, \\dfrac{\\pi }{2} \\right]$ we can write: $ G_{n} = \\sum _{k=0}^{\\infty } \\binom{-1 / 2}{k} I_{n,k} \\ \\mu ^{2k}\\quad \\forall n \\in \\mathbb {N},\\quad \\mbox{where}\\quad I_{n,k} = \\int \\limits _{0}^{\\pi / 2} t^{2k + 1} \\sin (nt)\\, dt\\quad \\forall n,k \\in \\mathbb {N}.$ Since we are considering odd values of $n \\in \\mathbb {N}$ , after integrating by parts twice $I_{n,k}$ in (REF ) we obtain: $I_{n,k} = -\\dfrac{2k(2k+1)}{n^2} I_{n,k-1} + \\delta (n) \\dfrac{2k+1}{n^2} \\left( \\dfrac{\\pi }{2} \\right)^{2k}\\quad \\forall k \\ge 1,$ with $I_{n,0} = \\dfrac{\\delta (n)}{n^2}$ , for every odd $n \\in \\mathbb {N}$ , and: $ \\delta (n) = \\left\\lbrace \\begin{aligned}& 1 \\ \\ \\mbox{if } n \\equiv 1 \\ (\\text{mod} \\ 4) \\\\- & 1 \\ \\ \\mbox{if } n \\equiv 3 \\ (\\text{mod} \\ 4).",
"\\\\\\end{aligned}\\right.$ An inductive argument over $k \\ge 1$ allows then to deduce $ I_{n,k} = \\delta (n) \\sum _{j=0}^{k} \\dfrac{(-1)^{k + j}}{n^{2(k+1-j)}} \\dfrac{(2k+1)!}{(2j)!}",
"\\left( \\dfrac{\\pi }{2} \\right)^{2j}\\quad \\forall k \\ge 1.$ Our first claim is that $I_{n,k} > 0$ when $n \\equiv 1 \\ (\\text{mod} \\ 4)$ , and that $I_{n,k} < 0$ when $n \\equiv 3 \\ (\\text{mod} \\ 4)$ , for all $k \\in \\mathbb {N}$ .",
"But, according to (REF ) and the form of expression (REF ), it suffices to show that: $ J_{n,k} := (-1)^{k} \\sum _{j=0}^{k} \\dfrac{(-1)^{j}}{(2j)!}",
"\\left( \\dfrac{n \\pi }{2} \\right)^{2j} > 0\\quad \\forall n \\in \\mathbb {N} \\ \\text{odd}, \\ k \\ge 1.$ In order to prove (REF ), we distinguish two cases.",
"$\\bullet $ Case (A): $k > \\dfrac{\\sqrt{1 + (n \\pi )^2} - 7}{4}$ .",
"Since $n \\in \\mathbb {N}$ is odd, we know that $ 0 = \\cos \\left( \\dfrac{n \\pi }{2} \\right) = \\sum _{j=0}^{k} \\dfrac{(-1)^{j}}{(2j)!",
"}\\left( \\dfrac{n \\pi }{2} \\right)^{2j} + \\sum _{j=k+1}^{\\infty } \\dfrac{(-1)^{j}}{(2j)!}",
"\\left( \\dfrac{n \\pi }{2} \\right)^{2j}.$ We put $a_{j} = \\dfrac{1}{(2j)!}",
"\\left( \\dfrac{n \\pi }{2} \\right)^{2j}$ and observe that, for every $j \\ge 1$ , $\\dfrac{a_{j}}{a_{j-1}} = \\dfrac{(n\\pi )^2}{8j(2j-1)} < 1 \\ \\ \\Longleftrightarrow \\ \\ j > \\dfrac{\\sqrt{1 + (n \\pi )^2} + 1}{4}.$ Hence, the Leibniz criterion can be applied to the tail series $\\sum _{j=k+1}^\\infty (-1)^ja_j$ if $j>\\tfrac{\\sqrt{1+(n\\pi )^2}+1}{4}$ .",
"But since the first ratio to be considered is $a_{k+2}/a_{k+1}$ , the Leibniz criterion may be applied whenever $k + 2 > \\dfrac{\\sqrt{1 + (n \\pi )^2} + 1}{4} \\ \\ \\Longleftrightarrow \\ \\ k > \\dfrac{\\sqrt{1 + (n \\pi )^2} - 7}{4},$ which is precisely the case considered.",
"Therefore, the tail series $\\sum _{j=k+1}^{\\infty } (-1)^{j} a_{j}$ has the same sign as $(-1)^{k + 1}$ .",
"In view of (REF ), the finite sum $\\sum _{j=0}^{k} (-1)^{j} a_{j}$ has the sign of $(-1)^{k}$ , that is, the opposite sign of the tail series.",
"In turn, $J_{n,k} > 0$ in this case, for all odd values of $n \\in \\mathbb {N}$ .",
"$\\bullet $ Case (B): $k < \\frac{\\sqrt{1 + (n \\pi )^2} - 7}{4}$ .",
"We distinguish here further between odd and even values of $k$ .",
"For even $k \\in \\mathbb {N}$ , we may write $J_{n,k}= 1 + \\sum _{i=1}^{k/2}(a_{2i}-a_{2i-1})$ and, since $2i\\le k$ , all the terms in the sum are positive in view of the assumption of case B.",
"Therefore, $J_{n,k}>0$ for even $k$ .",
"For odd $k\\in \\mathbb {N}$ , we may write $J_{n,k}=\\sum _{i=0}^{\\frac{k-1}{2}}(a_{2i+1}-a_{2i})$ and, since $2i+1\\le k$ , all the terms in the sum are positive in view of the assumption of case B.",
"Therefore, $J_{n,k}>0$ also for odd $k$ .",
"Inequality (REF ) is so proved for all $n$ and $k$ .",
"Let us now fix an integer $n \\equiv 1 \\ (\\text{mod} \\ 4)$ (the case when $n \\equiv 3 \\ (\\text{mod} \\ 4)$ follows a completely analogous procedure).",
"As a consequence of (REF ), we obtain the upper bound: $ I_{n,k} = - \\dfrac{2k(2k+1)}{n^2} \\dfrac{(2k-1)!",
"}{n^{2k}} J_{n,k-1} + \\dfrac{2k+1}{n^2} \\left( \\dfrac{\\pi }{2} \\right)^{2k} <\\dfrac{2k+1}{n^2} \\left( \\dfrac{\\pi }{2} \\right)^{2k}\\quad \\forall k \\ge 1.$ Back to (REF ), we may write: $ G_{n} = \\dfrac{1}{n^2} + \\sum _{k=1}^{\\infty } \\binom{-1 / 2}{k} \\left[\\sum _{j=0}^{k} \\dfrac{(-1)^{k + j}}{n^{2(k+1-j)}}\\dfrac{(2k+1)!}{(2j)!}",
"\\left( \\dfrac{\\pi }{2} \\right)^{2j} \\right] \\mu ^{2k}.$ We observe that the binomial coefficient $\\binom{-1 / 2}{k}$ is negative when $k$ is odd and positive otherwise.",
"Furthermore, if we put $b_{k} := \\left| \\binom{-1 / 2}{k} \\right|\\quad \\forall k \\ge 1,$ then one has that $ \\dfrac{b_{k+1}}{b_{k}} = \\dfrac{2k+1}{2k+2} < 1\\quad \\forall k \\ge 1,$ so that $b_{k} \\le b_{1} = 1/2$ for all $k \\ge 1$ .",
"Since in (REF ) all the terms in the sum over $j \\in \\lbrace 0,\\ldots , k \\rbrace $ are strictly positive as a consequence of (REF ), by exploiting (REF ) and (REF ) we obtain $G_{n} > \\dfrac{1}{n^2} - \\sum _{k=1 \\atop k\\ \\textrm {odd}}^{\\infty } \\dfrac{1}{2} \\dfrac{2k+1}{n^2} \\left( \\dfrac{\\mu \\pi }{2} \\right)^{2k} =\\dfrac{1}{n^2} \\left[ 1 - \\dfrac{1}{2} \\sum _{p=0}^{\\infty } (4p+3) \\left( \\dfrac{\\mu \\pi }{2} \\right)^{4p+2} \\right]\\, .$ For every $x \\in (-1,1)$ , the geometric series can be differentiated term by term, that is, $\\dfrac{d}{dx} \\left( \\sum _{p=0}^{\\infty } x^{4p+3} \\right) = \\sum _{p=0}^{\\infty } (4p + 3) x^{4p+2} =\\dfrac{d}{dx} \\left( \\dfrac{x^3}{1 - x^4} \\right) = \\dfrac{x^6 + 3x^2}{(1 - x^4)^2}\\qquad \\forall x \\in (-1,1).$ Hence, we finally infer that $ G_{n} > \\dfrac{1}{n^2} \\left[ 1 - \\dfrac{\\left( \\dfrac{\\mu \\pi }{2} \\right)^{6} +3 \\left( \\dfrac{\\mu \\pi }{2} \\right)^{2}}{2 \\left[ 1 - \\left( \\dfrac{\\mu \\pi }{2} \\right)^{4} \\right]^2} \\right]\\, .$ Some computations show that the right-hand side of (REF ) is strictly positive (at least) when $\\dfrac{\\mu \\pi }{2} < 0.65$ , so in particular, when $\\mu < 0.4$ .",
"This concludes the proof.",
"For the sake of illustration, in Table REF we give the numerical approximation of $G_{n}$ , for odd values of $n \\in \\mathbb {N}$ up to $n=19$ , when $\\mu =1.739 \\times 10^{-3}$ (as in (REF )): Table: NO_CAPTIONIn fact, for every $\\mu \\ge 0$ we know that $G_{n} \\longrightarrow 0$ as $n \\longrightarrow \\infty $ , as a direct consequence of the Riemann-Lebesgue Theorem.",
"This is quite visible also in Table REF .",
"Our second technical result gives a qualitative property of the graph of $\\Gamma _n(\\rho )$ .",
"Lemma 5.2 For all integer $n\\ge 1$ , the map $\\rho \\mapsto \\Gamma _n(\\rho )$ is strictly convex.",
"It suffices to analyze the case when $L = \\pi $ , and so: $\\Gamma _{n}(\\rho ) = \\int \\limits _{0}^{\\pi } \\sqrt{1 + \\left[\\dfrac{q}{H}\\left(x - \\dfrac{\\pi }{2} \\right) +n \\rho \\cos (nx) \\right] ^2} \\ dx - L_{c}\\quad \\forall \\rho \\in \\mathbb {R}, \\ \\ \\forall n \\in \\mathbb {N}.$ After differentiating under the integral sign we obtain the following: $ \\Gamma _{n}^{\\prime }(\\rho ) = \\int \\limits _{0}^{\\pi } \\dfrac{n \\cos (nx) \\left[\\dfrac{q}{H}\\left(x - \\dfrac{\\pi }{2} \\right) +n \\rho \\cos (nx) \\right]}{\\sqrt{1 + \\left[\\dfrac{q}{H}\\left(x - \\dfrac{\\pi }{2} \\right) +n \\rho \\cos (nx) \\right]^2}} \\ dx,$ $\\Gamma _{n}^{\\prime \\prime }(\\rho )=\\int \\limits _{0}^{\\pi }\\dfrac{[n\\cos (nx)]^2}{\\left[1+\\left(\\dfrac{q}{H}\\left(x-\\dfrac{\\pi }{2}\\right)+n\\rho \\cos (nx)\\right)^2\\right]^{3/2}}\\ dx,$ for $\\rho \\in \\mathbb {R}$ and $n \\in \\mathbb {N}$ .",
"Therefore, $\\Gamma _{n}^{\\prime \\prime }(\\rho ) > 0$ , for every $n \\ge 1$ and $\\rho \\in \\mathbb {R}$ , so that $\\Gamma _{n}$ is a strictly convex function all over $\\mathbb {R}$ .",
"In view of (REF ), we see that $ \\Gamma _{n}^{\\prime }(0) = \\dfrac{nq}{H} \\int \\limits _{0}^{\\pi } \\dfrac{\\left(x - \\dfrac{\\pi }{2} \\right)\\cos (nx)}{\\sqrt{1 + \\left( \\dfrac{q}{H} \\right)^2 \\left(x - \\dfrac{\\pi }{2} \\right)^2}} \\ dx.$ If $n$ is even, then the integrand in (REF ) is skew-symmetric with respect to $x=\\pi /2$ and hence $\\Gamma _{n}^{\\prime }(0)=0\\qquad \\mbox{for even }n\\, .$ If $n$ is odd, then we make the substitution $t = x - \\dfrac{\\pi }{2}$ and we note that $\\cos \\left( nt + \\dfrac{n\\pi }{2} \\right) =\\left\\lbrace \\begin{aligned}- &\\sin (nt), \\ \\mbox{if } n \\equiv 1 \\ (\\text{mod} \\ 4) \\\\&\\sin (nt), \\ \\mbox{if } n \\equiv 3 \\ (\\text{mod} \\ 4), \\\\\\end{aligned}\\right.$ for all $n \\ge 1$ and $t \\in \\left[ -\\dfrac{\\pi }{2}, \\dfrac{\\pi }{2} \\right]$ .",
"Therefore, after setting $\\mu =\\dfrac{q}{H}$ , we see that $\\Gamma _{n}^{\\prime }(0)=-2\\mu n \\delta (n) G_n$ if $n$ is odd.",
"From (REF ) and Lemma REF we then infer that $\\Gamma _{n}^{\\prime }(0)<0\\qquad \\mbox{for odd }n\\, .$ Since $\\Gamma _n(0)=0$ for all $n$ , Theorem REF follows by combining Lemma REF with (REF ) and (REF ).",
"Acknowledgments.",
"The first author is partially supported by the PRIN project Partial differential equations and related analytic-geometric inequalities and by GNAMPA-INdAM."
]
] | 1709.01792 |
[
[
"Ergodicity of skew products over linearly recurrent IETs"
],
[
"Abstract We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure."
],
[
"1",
"sectionchapter equationchapter subsubsection .",
"-0.5em -0.5em Ergodicity of skew products over linearly recurrent IETs Jon Chaika Donald Robertson We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure."
],
[
"Introduction",
"Let $T$ be an ergodic measure-preserving transformation on a probability space $(X,{B},\\mu )$ .",
"Given a measurable function $f : X \\rightarrow \\mathbb {R}$ one can consider the skew product $T_f$ on $X \\times \\mathbb {R}$ defined by $T_f(x,t) = (Tx,t + f(x))$ for all $x \\in X$ and all $t \\in \\mathbb {R}$ .",
"It is immediate that $T_f$ preserves $\\mu \\otimes \\nu $ where $\\nu $ is Lebesgue measure on $\\mathbb {R}$ .",
"Atkinson proved that $T_f$ is recurrent with respect to $\\mu \\otimes \\nu $ if and only if $f$ has zero mean and Schmidt proved that $T_f$ is conservative if and only if $f$ has zero mean.",
"It is therefore natural to ask whether $T_f$ is ergodic with respect to $\\mu \\otimes \\nu $ when $f$ has zero mean.",
"In this paper we are interested in the situation where $T$ is an interval exchange transformation.",
"We remind the reader that an interval exchange transformation is specified by a permutation $\\pi $ of $\\lbrace 1,\\dots ,b\\rbrace $ for some $b \\in \\mathbb {N}$ and by positive lengths $\\lambda _1,\\dots ,\\lambda _b$ that sum to 1.",
"Given such data one defines a map $T : [0,1) \\rightarrow [0,1)$ by $Tx = x - \\sum _{j < i} \\lambda _j + \\sum _{\\pi j < \\pi i} \\lambda _j$ for all $x \\in I_i$ where $I_i = [\\lambda _0 + \\cdots + \\lambda _{i-1},\\lambda _0 + \\cdots + \\lambda _i)$ for each $1 \\le i \\le b$ and $\\lambda _0 = 0$ .",
"All interval exchange transformations preserve Lebesgue measure on $[0,1)$ .",
"A permutation $\\pi $ on $\\lbrace 1,\\dots ,b\\rbrace $ is irreducible if there is not $1 \\le k < b$ such that $\\pi (\\lbrace 1,\\dots ,k\\rbrace ) = \\lbrace 1,\\dots ,k\\rbrace $ .",
"Throughout, we only consider interval exchange transformations defined by permutations that are irreducible.",
"For interval exchange transformations with $b = 2$ (i.e.",
"circle rotations) and varying classes of skewing function $f$ the associated skew product $T_f$ was shown to be ergodic by Oren , Hellekalek and Larcher , Pask and Conze and Piękniewska .",
"For special IETs on more intervals Conze and Frączek proved ergodicity for skew products by certain piecewise linear functions.",
"Negative results also occur, as Frączek and Ulcigrai showed typical non-ergodicity for a family of IETs with skewing functions that depend on the intervals of the IETs (which were considered for their relation to certain billiards).",
"In this paper we prove that, for linearly recurrent interval exchange transformations (an analogue of badly approximable rotations for interval exchange transformations) the skew product $T_f$ is ergodic for almost every step function $f$ with zero mean.",
"(By a step function we mean a linear combination of characteristic function of intervals.)",
"Theorem 1.1 Let $T$ be a linearly recurrent interval exchange transformation.",
"For almost every mean-zero step function $f : [0,1) \\rightarrow \\mathbb {R}$ the skew product $T_f$ is ergodic.",
"The terms “linear recurrent” and “almost every” in Theorem REF require some explanation.",
"We first recall the definition of linear recurrence for interval exchange transformations.",
"Let $T$ be an interval exchange transformation and let $\\beta _i = \\lambda _1 + \\cdots + \\lambda _i$ for all $1 \\le i \\le b-1$ .",
"Put $D = \\lbrace \\beta _1,\\dots , \\beta _{b-1} \\rbrace $ .",
"One says that $T$ satisfies the infinite distinct orbits condition if $D \\cap (T^n)^{-1} D = \\varnothing $ for all $n \\in \\mathbb {N}$ .",
"Keane proved that the infinite distinct orbits condition implies $T$ is minimal in the sense that every point has dense orbit.",
"An interval exchange transformation $T$ satisfying the infinite distinct orbits condition is said to be linearly recurrent if $c_3 = \\inf \\lbrace n \\eta (n) : n \\in \\mathbb {N} \\rbrace > 0$ where $\\eta (n)$ is the length of the smallest interval in the partition $D \\cup \\cdots \\cup (T^n)^{-1} D$ of $[0,1)$ .",
"Linear recurrence implies the following statement: that there are constants $c_1,c_2 > 0$ such that every finite orbit $x,Tx,\\dots ,T^{n-1}x$ is $c_1/n$ dense and $c_2/n$ separated.",
"The condition “almost every” in Theorem REF refers to a particular parameterization of mean-zero step functions we now describe.",
"Every step function $f : [0,1) \\rightarrow \\mathbb {R}$ with $d > 0$ discontinuities can be written in the form $f = y_1 1_{[0,x_1)} + \\cdots + y_{d+1} 1_{[x_1 + \\cdots + x_d, 1)}$ for some $y_1,\\dots ,y_{d+1}$ in $\\mathbb {R}$ with $y_i \\ne y_{i+1}$ for all $1 \\le i \\le d$ and some $x_1,\\dots ,x_{d+1}$ in $(0,1)$ that sum to 1.",
"By the jumps of any such $f$ we mean the values $y_2 - y_1,\\dots ,y_{d+1} - y_d$ .",
"The manifold $\\mathcal {C}_d=\\left\\lbrace (x_1,\\dots ,x_{d+1},y_1,\\dots ,y_{d+1}) \\in (0,1)^{d+1} \\times \\mathbb {R}^{d+1}:\\begin{aligned}x_1 y_1 + \\cdots + x_{d+1} y_{d+1} = 0\\\\x_1 + \\cdots + x_{d+1} = 1\\\\y_i \\ne y_{i+1} \\textrm { for all } 1 \\le i \\le d\\end{aligned}\\right\\rbrace $ parameterizes all mean-zero step functions $f : [0,1) \\rightarrow \\mathbb {R}$ with $d$ discontinuities.",
"We equip $\\mathcal {C}_d$ with the metric $\\mathsf {d}$ induced by the $\\ell ^\\infty $ metric on $\\mathbb {R}^{2d+2}$ .",
"In Theorem REF and throughout the paper, a statement is true for almost every mean-zero step function $f$ if, for every $d \\in \\mathbb {N}$ the points in $\\mathcal {C}_d$ for which the statement is false is a null set for the Lebesgue measure class on $\\mathcal {C}_d$ .",
"Our methods also apply to mean-zero step functions $f : [0,1) \\rightarrow \\mathbb {Z}$ .",
"The necessary modifications are to (i) replace $\\mathbb {R}^{d+1}$ with $\\mathbb {Z}^{d+1}$ in the definition of $\\mathcal {C}_d$ and equip $(0,1)^{d+1} \\times \\mathbb {Z}^{d+1}$ with the natural Lebesgue measure; (ii) redefine nudges (cf.",
"Section REF below) to remain $\\mathbb {Z}$ valued.",
"As well as being of intrinsic interest, the resulting skew products are related with $\\mathbb {Z}$ covers of compact translation surfaces.",
"Given such a cover $p : \\tilde{M} \\rightarrow M$ and a direction $\\theta $ the first return map on any line segment $\\Lambda $ transverse to $\\theta $ is an interval exchange transformation $T$ on $\\Lambda $ .",
"The first return dynamics on $p^{-1}(\\Lambda )$ is equivalent to a skew-product $T_f$ where $f : \\Lambda \\rightarrow \\mathbb {Z}$ is a step function.",
"It follows from that, for a zero Lebesgue measure but full Hausdorff dimension set of $\\theta $ the corresponding interval exchange transformation $T$ is linearly recurrent.",
"Given such a direction $\\theta $ the induced skew product $T_f$ and in turn the flow on $\\tilde{M}$ in the direction $\\theta $ are both recurrent provided $f$ has mean zero (cf.",
").",
"The proof of Theorem REF is outlined in Section and the details are given in the subsequent sections.",
"We mention here the following questions, in which we are very interested.",
"Question 1.3 Is Theorem REF true with the assumption of linear recurrence weakened to unique ergodicity (or maybe even just minimality)?",
"Question 1.4 Let $f = 1_{[0,\\frac{1}{2})} - 1_{[\\frac{1}{2}, 1)}$ .",
"Is $T_f$ ergodic as a $\\mathbb {Z}$ -valued skew product for almost every $T$ ?",
"Question 1.5 Let $f(x)=\\cos (2\\pi x)$ .",
"Is $T_f$ ergodic for almost every interval exchange transformation on at least three intervals?",
"J. Chaika is supported in part by NSF grants DMS-135500 and DMS-1452762, the Sloan foundation and a Warnock chair.",
"D. Robertson is grateful for the support of the NSF via grants DMS-1246989 and DMS-1703597.",
"This project (in fact a more ambitious one) began in 2005 as a joint project between the first named author and Pascal Hubert.",
"We thank Pascal Hubert for many helpful conversations."
],
[
"Proving ergodicity",
"Fix a minimal interval exchange transformation $T$ on $[0,1)$ and a measurable function $f : [0,1) \\rightarrow \\mathbb {R}$ .",
"As with all skew products $T_f$ is said to be recurrent if, for every $B \\in {B}$ with $\\mu (B) > 0$ and every $\\epsilon > 0$ one has $\\mu ( B \\cap (T^n)^{-1} B \\cap \\lbrace x \\in [0,1) : T_f^n (x,0) \\in [0,1) \\times (-\\epsilon ,\\epsilon ) \\rbrace ) > 0$ for some $n \\in \\mathbb {Z} \\setminus \\lbrace 0\\rbrace $ .",
"Atkinson proved that $T_f$ is recurrent if and only if $f$ has zero mean.",
"Since $T$ is minimal it then follows from that if $f$ has zero mean then $T_f$ is conservative.",
"It is therefore reasonable to ask – assuming $f$ has zero mean – whether $T_f$ is ergodic with respect to Lebesgue measure $\\mathsf {m}$ on $[0,1) \\times \\mathbb {R}$ .",
"To answer this question one considers the $\\mathbb {R}$ action $V$ defined on $[0,1) \\times \\mathbb {R}$ by $V^v(x,t) = (x,t+v)$ , which preserves $\\mathsf {m}$ and commutes with $T_f$ .",
"Write ${Z}_f$ for the $\\sigma $ -algebra of $T_f$ invariant Borel subsets of $[0,1) \\times \\mathbb {R}$ .",
"As a consequence of and the measure $\\mathsf {m}$ is ergodic for $T_f$ if and only if the closed subgroup $\\mathsf {Ess}(f) = \\lbrace v \\in \\mathbb {R} : \\mathsf {m}(B \\operatorname{\\triangle }(V^v)^{-1} B) = 0 \\textrm { for all } B \\in {Z}_f \\rbrace $ is all of $\\mathbb {R}$ .",
"The members of $\\mathsf {Ess}(f)$ are the essential values of $f$ .",
"Our main result is therefore a consequence of the following theorem.",
"Theorem 2.1 Let $T$ be a linearly recurrent interval exchange transformation.",
"For almost every mean-zero step function $f : [0,1) \\rightarrow \\mathbb {R}$ each of its jumps is an essential value of $T_f$ .",
"[Proof of Theorem REF assuming Theorem REF ] For almost every mean-zero step function $f$ its jumps generate a dense subgroup of $\\mathbb {R}$ .",
"Therefore $\\mathsf {Ess}(f)$ is dense in $\\mathbb {R}$ .",
"Other works, for example , also prove that the jumps of the step function are essential values.",
"They consider step function skew products over rotations $R$ of the circle, for which one can always find infinitely many times $q_n$ such that (Denjoy-Koksma) $\\left| \\displaystyle \\sum _{i=0}^{q_n-1} f(R^ix) \\right| \\le \\mathsf {Var}(f)$ $\\underset{n \\rightarrow \\infty }{\\lim }d(R^{q_n}x,x)=0$ both hold for all $x$ .",
"One then seeks to show there are pairs of level sets of $\\sum _{i=0}^{q_n-1}f(R^ix)$ of definite measure where the values of $\\sum _{i=0}^{q_n-1}f(R^ix)$ differ by the size of particular jump discontinuities of $f$ .",
"In short, one obtains invariance by looking at sets of definite measure at particular times.",
"We do not suspect that something like the Denjoy-Koksma inequality holds in our context.",
"As a substitute, we show that the size of the jumps of the skewing function are essential values by following a pair of nearby points whose values under the skew differ by the size of a jump discontinuity of $f$ for a defnite proportion of an orbit segment.",
"This approach is outlined below and carried out in Section .",
"Such arguments go back at least to Ratner .",
"In order to prove Theorem REF we study for some $B > 0$ the transformation $S_{f,B}$ induced by $T_f$ on the space $X_B := [0,1) \\times [-B,B]$ .",
"This is defined almost everywhere because $T_f$ is recurrent.",
"Normalized Lebesgue measure $\\mathsf {m}_B$ on $X_B$ is $S_{f,B}$ invariant, so $\\mathsf {m}_B$ almost every point $(x,t)$ is generic for an ergodic $S_{f,B}$ invariant probability measure $\\mu _{f,B,(x,t)}$ on $X_B$ .",
"The following theorem (proved in Appendix ) relates vertical invariance of the measures $\\mu _{f,B,(x,t)}$ with the essential values of $T_f$ .",
"Theorem 2.2 Let $T$ be an ergodic interval exchange transformation.",
"Fix $B > 0$ .",
"Suppose that there is $v \\in \\mathbb {R}$ such that an $\\mathsf {m}_B$ positive measure set of $(x,t)$ in $X_B$ is generic for an $S_{f,B}$ invariant probability measure $\\mu _{f,B,(x,t)}$ that is not singular with respect to $V^v \\mu _{f,B,(x,t)}$ .",
"Then every $T_f$ invariant set is $V^v$ invariant.",
"We now describe how Theorem REF will be used to prove Theorem REF .",
"Fix an interval exchange transformation $T$ .",
"Given a pair $(x,t) \\in [0,1) \\times \\mathbb {R}$ , a mean-zero step function $f : [0,1) \\rightarrow \\mathbb {R}$ and $B > 0$ , say that $(x,t)$ and $f$ are right friends at $B$ if there are constants $\\beta > 0$ , $\\delta > 0$ such that, for every discontinuity $p$ of $f$ there is $K \\subset \\mathbb {N}$ infinite such that all of the following properties hold for all $k \\in \\mathbb {N}$ .",
"For all $0 \\le i < 2^{k}$ the transformation $T^i$ is continuous on $[x,x+\\frac{2 \\delta }{2^k}]$ .",
"The family $\\lbrace T^i [x,x+\\frac{2\\delta }{2^k}] : 0 \\le i < 2^{k} \\rbrace $ of intervals is pairwise disjoint.",
"There is $0 \\le i < 2^{k-1}$ with $p \\in T^i[x,x+\\frac{\\delta }{2^k}]$ .",
"No other discontinuity of $f$ belongs to $\\displaystyle \\bigcup _{i=0}^{2^k-1} T^i [x, x+\\tfrac{2\\delta }{2^k}]$ .",
"$\\displaystyle {\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\ge \\beta \\sum _{n=0}^{2^k - 1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)}$ .",
"Note that only REF refers to $t$ .",
"We say that $(x,t)$ and $f$ are left friends at $B$ if the above is true with all occurrences of $[x,x+\\frac{2\\delta }{2^k}]$ and $[x,x+\\frac{\\delta }{2^k}]$ replaced by $[x-\\frac{2\\delta }{2^k},x]$ and $[x-\\frac{\\delta }{2^k},x]$ respectively.",
"Declare $(x,t)$ and $f$ to be friends at $B$ if they are either left friends at $B$ or right friends at $B$ .",
"Theorem REF is now a consequence of the following two results.",
"Theorem 2.3 Let $T$ be an ergodic interval exchange transformation and let $f$ be a mean-zero step function whose jump discontinuities generate a dense subgroup of $\\mathbb {R}$ .",
"If $\\mathsf {m}_B( \\lbrace (x,t) \\in X_B : f \\textrm { and } (x,t) \\textrm { are friends at } B \\rbrace ) \\rightarrow 1$ as $B \\rightarrow \\infty $ then $T_f$ is ergodic with respect to Lebesgue measure on $[0,1) \\times \\mathbb {R}$ .",
"The proof of Theorem REF is given in Section .",
"It is proved by showing that for all jumps $v$ of $f$ the measure $\\mu _{f,B,(x,t)}$ is not singular with respect to $V^v \\mu _{f,B,(x,t)}$ for a positive measure set of $(x,t)$ .",
"By Theorem REF this implies Theorem REF .",
"The next theorem guarantees that the hypothesis of Theorem REF are satisfied when $T$ is a linearly recurrent interval exchange transformation - together with Theorem REF it concludes the proof of Theorem REF .",
"Theorem 2.5 Fix a linearly recurrent interval exchange transformation $T$ .",
"For almost every mean-zero step function $f$ we have $\\mathsf {m}_B( \\lbrace (x,t) \\in X_B : f \\textrm { and } (x,t) \\textrm { are friends at } B \\rbrace ) \\rightarrow 1$ as $B \\rightarrow \\infty $ .",
"The proof of Theorem REF is based on a density points argument with the following steps.",
"The details are given in Section .",
"(Section REF ) Properties REF and REF always hold either on the left or the right.",
"(Section REF ) For any $x,f$ and $k$ we can perturb $f$ to satisfy REF and REF .",
"(Section REF ) For all $f$ and almost every $(x,t)$ (with $|t|<B$ ) there exists infinitely many $k$ satisfying REF .",
"(Section REF ) Small perturbations in $f$ preserve REF .",
"(Section REF ) The preceding steps imply Theorem REF .",
"We conclude this section with some preparatory remarks that will be used implicitly in the proofs of the above results.",
"Firstly, given $d \\in \\mathbb {N}$ and $D \\in \\mathbb {N}$ write $\\mathcal {C}_{d,D}$ for the set of points in $\\mathcal {C}_d$ with $|y_i| \\le D$ for all $1 \\le i \\le d+1$ .",
"Lemma 2.7 The map $\\begin{aligned}\\mathcal {C}_{d,D} \\times [0,1) \\times \\mathbb {R}&\\rightarrow [0,1) \\times \\mathbb {R}\\\\(f,(x,t))&\\mapsto T_f(x,t)\\end{aligned}$ is measurable, and $(f,(x,t)) \\mapsto \\phi (T_f^n(x,t))$ is measurable for every $n$ in $\\mathbb {Z}$ and every measurable function $\\phi $ on $[0,1) \\times \\mathbb {R}$ .",
"Writing $x_0 = 0$ , the map $[0,1)^{d+1} \\times \\mathbb {R}^{d+1} \\times [0,1) \\times \\mathbb {R}&\\rightarrow [0,1) \\times \\mathbb {R}\\\\(x_1,\\dots ,x_{d+1},y_1,\\dots ,y_{d+1},x,t)&\\mapsto \\left( Tx, t + \\sum _{i=0}^d y_i \\cdot 1_{[x_0 + \\cdots + x_i,x_0 + \\cdots + x_{i+1})}(x) \\right)$ is measurable and (REF ) is simply its restriction to $\\mathcal {C}_{d,D} \\times [0,1) \\times \\mathbb {R}$ .",
"Define $G_3(B) = \\lbrace (f,(x,t)) \\in \\mathcal {C}_d \\times X_B : (x,t) \\textrm { is generic for an } S_{f,B} \\textrm { invariant probability measure on } X_B \\rbrace $ for all $B > 0$ .",
"Measurability of the sets $G_3(B)$ follows from Lemma REF .",
"Lastly, for each $B > 0$ and each $\\epsilon > 0$ note that, by Egoroff's theorem, there is set $G_4(B,\\epsilon ) \\subset X_B$ with $\\mathsf {m}_B$ -measure at least $1- \\epsilon $ on which the sequence $\\frac{1}{N} \\sum _{n=0}^{N-1} \\delta _{S_{f,B}^n (x,t)}$ of measures converges uniformly to $\\mu _{f,B,(x,t)}$ .",
"Write $\\mathsf {coll}(B,b) = [0,1) \\times \\Big ( [-B,-B+b] \\cup [B-b,B] \\Big )$ for any $B > 0$ and any $b > 0$ .",
"Note that $\\mathsf {coll}(B,B) = [0,1) \\times [-B,B]$ .",
"For all $b,\\tau > 0$ we have $\\mathsf {m}_{B+b}(\\lbrace (x,t) \\in X_{B+b} : \\mu _{f,B+b,(x,t)}(\\mathsf {coll}(B+b,2b)) \\ge \\tau \\rbrace ) \\le \\frac{1}{\\tau } \\mathsf {m}_{B+b}(\\mathsf {coll}(B+b,2b)) \\rightarrow 0$ as $B \\rightarrow \\infty $ by Markov's inequality.",
"In various proofs below we will make use of the following lemma, which we do not prove.",
"Lemma 2.10 Let $n \\mapsto a_n$ be a sequence that Cesàro converges to $\\alpha $ .",
"Fix $\\epsilon > 0$ and $0 < \\gamma < 1$ .",
"There is $K \\in \\mathbb {N}$ so large that $\\left| \\alpha - \\frac{1}{N-M} \\sum _{n=M}^{N-1} a_n \\right| < \\epsilon $ whenever $N > K$ and $N - M \\ge \\gamma N$ ."
],
[
"Proof of Theorem ",
"In this section we prove Theorem REF .",
"The argument is a modification of a now standard argument that goes back at least to Ratner .",
"[Proof of Theorem REF ] Fix an ergodic interval exchange transformation $T$ on $[0,1)$ and a mean-zero step function $f : [0,1) \\rightarrow \\mathbb {R}$ with coordinates $(x_1,\\dots ,x_{d+1},y_1,\\dots ,y_{d+1})$ .",
"Assume the jump discontinuities $y_{i+1} - y_i$ of $f$ generate a dense subgroup of $\\mathbb {R}$ .",
"Fix a discontinuity $p$ of $f$ .",
"Let $v$ be the jump at $p$ and put $b = |v|$ .",
"Fix $0 < \\eta < 1/10$ .",
"Choose $B > \\max \\lbrace 2y_1,\\dots ,2y_{d+1} \\rbrace /(1 - \\eta )$ so large that $\\mathsf {m}_{B-b} ( \\lbrace (x,t) \\in X_{B-b} : f \\textup { and } (x,t) \\textup { are friends at } B-b \\rbrace ) \\ge 1 - \\eta \\\\\\mathsf {m}_{B}(\\lbrace (x,t) \\in X_{B} : \\mu _{f,B,(x,t)}(\\mathsf {coll}(B,b)) < \\eta \\rbrace ) \\ge 1 - \\frac{b}{B} - \\eta \\\\\\mathsf {m}_{B+b}(\\lbrace (x,t) \\in X_{B+b} : \\mu _{f,B+b,(x,t)}(\\mathsf {coll}(B+b,2b)) < \\eta \\rbrace ) \\ge 1 - \\frac{2b}{B} - \\eta \\\\\\mathsf {m}_{B+b}(\\lbrace (x,t) \\in X_{B+b} : \\mu _{f,B+b,(x,t)}(\\mathsf {coll}(B+b,b)) < \\eta /2 \\rbrace ) \\ge 1 - \\frac{2b}{B} - \\eta $ all hold.",
"The first inequality follows from the hypothesis (REF ) while the rest follow from (REF ).",
"It follows immediately that the four sets above each have $\\mathsf {m}_{B-b}$ measure at least $1 - \\eta $ .",
"Fix $L$ a compact subset of $X_{B-b}$ with $\\mathsf {m}_{B-b}$ measure at least $1 - \\eta $ on which $(x,t) \\mapsto \\mu _{f,B,(x,t)}$ is continuous.",
"From these choices and $\\eta < \\frac{1}{10}$ we can find an $\\mathsf {m}_{B-b}$ positive measure set of $(x,t) \\in X_{B-b}$ with all the following properties.",
"$(x,t)$ and $f$ are friends at $B-b$ .",
"$(x,t) \\in G_3(B+b) \\cap G_3(B)$ .",
"$(x,t)$ belongs to and is a density point of $L \\cap G_4(B,\\eta ) \\cap \\Big ( [0,1) \\times \\lbrace t\\rbrace \\Big )$ .",
"$\\mu _{f,B+b,(x,t)}(\\mathsf {coll}(B+b,2b)) < \\eta $ .",
"$\\mu _{f,B,(x,t)}(\\mathsf {coll}(B,b)) < \\eta $ .",
"$\\mu _{f,B+b,(x,t)}(X_B) \\ge 1 - \\frac{\\eta }{2}$ .",
"Our goal is to prove that for every $(x,t)$ satisfying REF through REF the measures $\\mu _{f,B,(x,t)}$ and $V^v \\mu _{f,B,(x,t)}$ are not mutually singular.",
"Indeed if this goal is realized then, since we have a positive measure set of $(x,t) \\in X_B$ satisfying REF through REF , by Theorem REF all $T_f$ invariant sets will be $V^v$ invariant.",
"In other words $v$ will be an essential value of $f$ .",
"Since $v$ was an arbitrary jump of $f$ and the jumps of $f$ are assumed to generate a dense subgroup of $\\mathbb {R}$ we will conclude that $T_f$ is ergodic.",
"To realize our goal fix $(x,t)$ satisfying REF through REF .",
"Claim It suffices to prove that $\\int g \\,\\mathrm {d}\\mu _{f,B,(x,t)} \\le \\frac{3}{2 - \\eta } \\int V^{-v} g \\,\\mathrm {d}\\mu _{f,B,(x,t)}$ for all $g \\in \\operatorname{\\operatorname{\\mathsf {C}}_\\mathsf {c}}(X_{B-b})$ .",
"[Proof of claim] If $\\mu _1 = \\mu _{f,B,(x,t)}$ and $\\mu _2 = V^{-v} \\mu _{f,B,(x,t)}$ are mutually singular then we can find via REF disjoint compact sets $H_1 \\subset X_B \\setminus \\mathsf {coll}(B,b)$ and $H_2 \\subset X_B$ with $\\mu _i(H_i) > 1 - 2\\eta $ and $\\mu _i(H_{3-i}) = 0$ for each $i \\in \\lbrace 1,2\\rbrace $ .",
"There also exist open sets $W_2 \\subset X_B$ and $W_1 \\subset X_B \\setminus \\mathsf {coll}(B,b)$ so that $H_i \\subset W_i$ and $\\mu _i(W_{3-i}) < \\eta (1 - 2\\eta )$ .",
"We may choose a continuous, non-negative function $0\\le g\\le 1$ such that $1_{H_1} \\le g \\le 1_{W_1}$ .",
"It is then straightforward that $\\int g \\,\\mathrm {d}\\mu _2\\le \\mu _2(W_1)\\le \\eta (1 - 2\\eta )\\le \\eta \\mu _1(H_1)\\le \\eta \\int g \\,\\mathrm {d}\\mu _1\\le \\frac{3 \\eta }{2 - \\eta } \\int g \\,\\mathrm {d}\\mu _2$ holds.",
"But $\\eta < 1/2$ so it must be the case that all quantities above are zero, which is impossible.",
"To establish (REF ) suppose by REF that $(x,t)$ and $f$ are right friends at $B-b$ .",
"(The proof when they are left friends is similar and omitted.)",
"Let $\\beta > 0$ , $\\delta > 0$ be the attendant constants and let $K \\subset \\mathbb {N}$ be the subset associated with our fixed discontinuity $p$ of $f$ .",
"Let $k_1 < k_2 < \\cdots $ be an enumeration of $K$ .",
"By REF and REF we can find (because $[x+\\frac{\\delta }{2^{k_i}},x+\\frac{2\\delta }{2^{k_i}}]$ is a definite proportion of $[x,x+\\frac{2\\delta }{2^{k_i}}]$ ) for all $i$ large enough, some point $z_i$ in $[x,x + \\frac{2\\delta }{2^{k_i}}]$ with the following properties: that $(z_i,t)$ belongs to $L \\cap G_4(B,\\eta )$ and that our discontinuity $p$ is between $T^{\\ell _i}(x)$ and $T^{\\ell _i}(z_i)$ for some $0 \\le \\ell _i < 2^{k_i - 1}$ .",
"Using REF to rule out other discontinuities of $f$ and REF to rule out a second occurrence of $p$ we have $T_f^n(z_i,t)&=\\big ( T^n(z_i),t + f(z_i) + \\cdots + f(T^{n-1} z_i) \\big )\\\\&=\\big ( T^n(z_i),t + f(x) + \\cdots + f(T^{n-1} x) + f(T^{\\ell _i} z_i) - f(T^{\\ell _i} x) \\big )\\\\&=V^v \\big ( T^n(z_i),t + f(x) + \\cdots + f(T^{n-1} x) \\big )$ whenever $2^{k_i} \\ge n > \\ell _i$ .",
"It follows that $|\\!|T_f^n(x,t) - V^{-v} (T_f^n(z_i,t)) |\\!|_2 = |z_i - x|$ whenever $n > \\ell _i$ .",
"To every time $n$ at which $T_f^n(x,t)$ belongs to $X_{B-b}$ corresponds some iterate $r(n)$ of the $S_{f,B}$ orbit of $(x,t)$ .",
"For all such $n$ we also have $T_f^n(z_i,t)$ in $X_B$ and a corresponding iterate $r_i(n)$ of the $S_{f,B}$ orbit of $(z_i,t)$ .",
"That is $S_{f,B}^{r_i(n)}(z_i,t) = T_f^n(z_i,t)$ .",
"Define $U_i &= \\lbrace r(n) : \\ell _i < n < 2^{k_i} \\textup { and } T_f^n(x,t) \\in X_{B-b} \\rbrace \\\\U_i^{\\prime } &= \\lbrace r_i(n) : \\ell _i < n < 2^{k_i} \\textup { and } T_f^n(x,t) \\in X_{B-b} \\rbrace $ for all large enough $i \\in \\mathbb {N}$ .",
"Note that $|U_i| \\ge \\beta \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B+b,B-b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ because $f$ and $(x,t)$ satisfy REF at $B-b$ via REF .",
"By (REF ) we have $|\\!|S_{f,B}^{r(n)}(x,t) - V^{-v} (S_{f,B}^{r_i(n)}(z_i,t)) |\\!|_2 = |z_i - x|$ whenever $T_f^n(x,t) \\in X_{B-b}$ and $n > \\ell _i$ .",
"Claim We have $|U_i|\\ge \\beta (1-2\\eta ) \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B,B]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ for all large enough $i$ .",
"First note that $\\left| \\frac{1}{N} \\sum _{n=1}^N 1_{\\mathsf {coll}(B,b)}(S_{f,B}^n (x,t)) - \\mu _{f,B,(x,t)}(\\mathsf {coll}(B,b)) \\right| < \\eta $ if $N$ is large enough by REF .",
"Thus $\\sum _{n=1}^N 1_{\\mathsf {coll}(B,b)}(S_{f,B}^n (x,t)) \\le 2 \\eta N$ by REF .",
"In terms of $T_f$ this becomes $\\sum _{n=0}^{2^{k_i} - 1} 1_{[-B,-B+b] \\cup [B-b,B]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)\\le 2\\eta \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B,B]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ whenever $i$ is large enough.",
"Combining with (REF ) gives (REF ).",
"Claim We have $|U_i^{\\prime }|\\ge \\beta (1-2\\eta ) \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B,B]} \\left( t + \\sum _{j=0}^{n-1} f(T^j z_i) \\right)$ for all large enough $i$ .",
"[Proof of claim] Certainly $|U_i| = |U_i^{\\prime }|$ .",
"We have $|U_i|\\ge \\sum _{n=2^{k_i - 1}}^{2^{k_i} -1} 1_{[-B+b,B-b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)\\ge \\beta \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B+b,B-b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ because $\\ell _i < 2^{k_i - 1}$ and because $(x,t)$ satisfies REF at $B-b$ via REF .",
"Arguing as in the previous claim, from REF and REF we also have $\\sum _{n=0}^{2^{k_i} - 1} 1_{[-B-b,-B+b] \\cup [B-b,B+b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)\\le 2\\eta \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B-b,B+b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ for all $i$ is large enough.",
"Combining these two estimates gives $|U_i^{\\prime }| \\ge \\beta (1-2\\eta ) \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B-b,B+b]} \\left( t + \\sum _{j=0}^{n-1} f(T^j x) \\right)$ and we conclude that $|U_i^{\\prime }| \\ge \\beta (1-2\\eta ) \\sum _{n=0}^{2^{k_i} - 1} 1_{[-B,B]} \\left( t + \\sum _{j=0}^{n-1} f(T^j z_i) \\right)$ because $T_f^n(x,t) \\in X_{B+b}$ whenever $T_f^n(z_i,t) \\in X_B$ .",
"Turning to the validity of (REF ) fix $g \\in \\operatorname{\\operatorname{\\mathsf {C}}_\\mathsf {c}}(X_{B-b})$ and $\\epsilon > 0$ .",
"For each time $n$ at which $T_f^n(x,t)$ belongs to $X_B$ let $s(n)$ be the corresponding iterate of $S_{f,B}(x,t)$ .",
"For each time $n$ at which $T_f^n(z_i,t)$ belongs to $X_B$ let $s_i(n)$ be the corresponding iterate of $S_{f,B}(z_i,t)$ .",
"Define $W_i &= \\lbrace s(n) : \\ell < n < 2^{k_i} \\textup { and } T_f^n(x,t) \\in X_B \\rbrace \\\\V_i &= \\lbrace s_i(n) : \\ell _i < n < 2^{k_i} \\textup { and } T_f^n(z_i,t) \\in X_B \\rbrace $ for all $i \\in \\mathbb {N}$ large enough, both of which are intervals of natural numbers.",
"Note that $W_i \\supset U_i$ and that $V_i \\supset U_i^{\\prime }$ .",
"If we choose $i$ large enough then $\\left| \\int g \\,\\mathrm {d}\\Phi _B(x,t) - \\frac{1}{|W_i|} \\sum _{n \\in W_i} g(S_{f,B}^n(x,t)) \\right| < \\epsilon $ using $(x,t) \\in G_3(B)$ from REF and Lemma REF via (REF ) and $|W_i| \\ge |U_i|$ .",
"Since $g$ is supported on $X_{B-b}$ we have $\\frac{1}{|W_i|} \\sum _{n \\in W_i} g(S_{f,B}^n(x,t))=\\frac{1}{|W_i|} \\sum _{n \\in U_i} g(S_{f,B}^n(x,t))$ for all $i$ .",
"Since every member of $U_i^{\\prime }$ corresponds to a unique member of $U_i$ we have $\\left|\\frac{1}{|W_i|} \\sum _{n \\in U_i} g(S_{f,B}^n(x,t))-\\frac{1}{|W_i|} \\sum _{n \\in U_i^{\\prime }} (V^{-v} g) (S_{f,B}^n(z_i,t))\\right|\\le \\epsilon $ from (REF ) and uniform continuity of $g$ if $i$ is large enough.",
"Combining the above three with $V_i \\supset U_i^{\\prime }$ and $g \\ge 0$ gives $\\int g \\,\\mathrm {d}\\Phi _B(x,t) \\le 2\\epsilon + \\frac{1}{|W_i|} \\sum _{n \\in V_i} (V^{-v} g) (S_{f,B}^n(z_i,t))$ for all $i$ large enough.",
"The point $(z_i,t)$ is generic for $\\Phi _B(z_i,t)$ and belongs to $G_4(B,\\eta )$ by REF .",
"By (REF ) and $|V_i| \\ge |U_i^{\\prime }|$ we can apply Lemma REF to get $\\left|\\frac{1}{|V_i|} \\sum _{n \\in V_i} (V^{-v} g) (S_{f,B}^n(z_i,t))-\\int V^{-v} g \\,\\mathrm {d}\\Phi _B(z_i,t)\\right|<\\epsilon $ if $i$ is large enough.",
"Since $(x,t) \\in L$ by REF we also have $\\left| \\int V^{-v} g \\,\\mathrm {d}\\Phi _B(z_i,t) - \\int V^{-v} g \\,\\mathrm {d}\\Phi _B(x,t) \\right| < \\epsilon $ for $i$ large enough.",
"These inequalities together with (REF ) imply $\\int g \\,\\mathrm {d}\\Phi _B(x,t)\\le 2 \\epsilon \\left( 1 + \\frac{|V_i|}{|W_i|} \\right) + \\frac{|V_i|}{|W_i|} \\int V^{-v} g \\,\\mathrm {d}\\Phi _B(x,t)$ for our point $(x,t)$ and our function $g \\in \\operatorname{\\operatorname{\\mathsf {C}}_\\mathsf {c}}(X_{B-b})$ .",
"Finally, for every time $n$ at which $T_f^n(x,t)$ belongs to $X_{B+b}$ let $t(n)$ be the corresponding iterate of the $S_{f,B+b}$ orbit of $(x,t)$ .",
"Put $Y_i = \\lbrace t(n) : \\ell _i < n < 2^{k_i} \\textup { and } T_f^n(x,t) \\in X_{B+b} \\rbrace $ for all $i$ large enough and note that $|V_i| \\le |Y_i|$ .",
"Since $(x,t)$ is generic for $\\Phi _{B+b}(x,t)$ by REF we have $\\lim _{N \\rightarrow \\infty } \\frac{|W_i|}{|Y_i|} = \\mu _{f,B+b,(x,t)}(X_B) \\ge 1 - \\frac{\\eta }{2}$ by REF .",
"Choosing $\\epsilon $ small enough (depending only on $\\eta $ ) and $i$ large enough gives $\\int g \\,\\mathrm {d}\\Phi _B(x,t) \\le \\frac{3}{2} \\frac{1}{1 - \\frac{\\eta }{2}} \\int V^{-v} g \\,\\mathrm {d}\\Phi _B(x,t)$ which is (REF )."
],
[
"Proof of Theorem ",
"Fix throughout this section a linearly recurrent interval exchange transformation $T$ and attendant constants $c_1$ , $c_2$ and $c_3$ as in Section .",
"Fix $\\delta = c_2/4$ ."
],
[
"Conditions ",
"Lemma 4.1 For all $x\\in [0,1)$ and $n\\in \\mathbb {N}$ at least one of the the following two possibilities hold.",
"$\\lbrace T^i[x,x+\\frac{2\\delta }{n}] : 0 \\le i < n \\rbrace $ consists of $n$ disjoint intervals.",
"$\\lbrace T^i[x-\\frac{2\\delta }{n},x] : 0 \\le i < n \\rbrace $ consists of $n$ disjoint intervals.",
"The fact that these are disjoint follows from the next result of Boshernitzan.",
"(Indeed if $T^iA$ are disjoint sets for $p\\le i\\le q$ then $T^jA$ are disjoint sets for $0\\le j\\le q-p$ .)",
"Lemma 4.2 () If $T$ satisfies the Keane condition and the distance between any discontinuities of $T^{n+1}$ is $s$ then for any interval $J$ with measure at most $ s$ there exist integers ${p \\le 0 \\le q}$ (which depend on $J$ ) such that $q-p \\ge n$ $T^i$ acts continuously on $J$ for $p \\le i \\le q$ $T^i(J) \\cap T^j(J)= \\varnothing $ for $p \\le i < j \\le q$ .",
"[Proof of Lemma REF ] Linear recurrence implies the discontinuities of $T^n$ are $\\frac{c_2}{n}$ separated.",
"Writing $\\beta _1,\\dots ,\\beta _r$ for the discontinuities of $T$ , it follows that $T^i[\\beta _j,\\beta _j + \\frac{c_2}{n}) \\cap \\lbrace \\beta _1,\\dots ,\\beta _r\\rbrace = \\varnothing $ and $T^i[\\beta _j - \\frac{c_2}{n},\\beta _j) \\cap \\lbrace \\beta _1,\\dots ,\\beta _r\\rbrace = \\varnothing $ for all $1 \\le i \\le n$ and all $1 \\le j \\le r$ .",
"Now consider $T^i (x-\\frac{c_2}{n},x+\\frac{c_2}{n})$ .",
"If it is not an interval then some discontinuity $\\beta $ of $T$ belongs to $T^j (x-\\frac{c_2}{n},x+\\frac{c_2}{n})$ for some $0 \\le j < i$ .",
"If $\\beta \\in [T^j x - \\frac{c_2}{2n},T^j x)$ then by above we have that $T^i [x,x+\\frac{c_2}{2n}) \\cap \\lbrace \\beta _1,\\dots ,\\beta _r\\rbrace = \\varnothing $ for all $0 \\le i < n$ .",
"Similarly, if $\\beta \\in [T^j x, T^j x + \\frac{c_2}{2n})$ we have the other possibility."
],
[
"Conditions ",
"Fix $f$ in $\\mathcal {C}_{d,D}$ and $x \\in [0,1)$ .",
"Let $(x_1,\\dots ,x_{d+1},y_1,\\dots ,y_{d+1})$ be the coordinates of $f$ as in (REF ).",
"Put $\\xi = \\min \\lbrace x_1,\\dots ,x_{d+1} \\rbrace $ .",
"We verify in this subsection that, by perturbing $f$ , we can assume conditions REF and REF are true.",
"We wish to choose $g \\in \\mathcal {C}_{d}$ close enough to $f$ such that REF and REF hold.",
"We construct $g$ by nudging the locations of the discontinuities of $f$ .",
"Specifically, if we wish to move the location of a discontinuity of $f$ to the left or to the right we adjust the values taken by $f$ on the interval to the right of the discontinuity in such a way that the resulting step function still has zero mean.",
"Explicitly, to nudge $f$ by moving its $i$ th discontinuity from $x_1 + \\cdots + x_i$ to $x_1 + \\cdots + x_i + \\zeta $ we replace $f$ with the step function $\\mathsf {nudge}(f,i,\\zeta )$ having coordinates $\\left(x_1,\\dots ,x_i + \\zeta , x_{i+1} - \\zeta ,\\dots , x_{d+1},y_1,\\dots ,y_i,\\frac{y_{i+1} x_{i+1} - \\zeta y_i}{x_{i+1} - \\zeta },\\dots ,y_{d+1} \\right)$ which makes sense provided $|\\zeta | < \\frac{\\xi }{2}$ .",
"Recall that $\\mathsf {d}$ denotes the $\\ell ^\\infty $ metric on the data $(x_1,\\dots ,y_{d+1})$ .",
"Lemma 4.3 If $|\\zeta | < \\frac{\\xi }{2}$ then $\\mathsf {d}(f,\\mathsf {nudge}(f,i,\\zeta )) < \\max \\lbrace |\\zeta |, 8 |\\zeta | D/3 \\xi \\rbrace $ .",
"The values of the skewing function have changed by $\\left| \\frac{y_{i+1} x_{i+1} - \\zeta y_i}{x_{i+1} - \\zeta } - y_{i+1} \\right|=\\left| \\frac{\\zeta y_{i+1} - \\zeta y_i}{x_{i+1} - \\zeta } \\right|\\le \\frac{|\\zeta | 2 D}{|x_{i+1}- \\zeta |}\\le \\frac{|\\zeta | 8D}{3 x_{i+1}}\\le \\frac{|\\zeta | 8D}{3 \\xi }$ in carrying out the nudge.",
"Proposition 4.5 Fix $f$ in $\\mathcal {C}_{d,D}$ , $1 \\le i \\le d$ and $x \\in [0,1)$ .",
"For every $k$ in $\\mathbb {N}$ with $\\frac{c_1 + \\delta }{2^{k-1}} < \\frac{\\xi }{4}$ there is $g$ in $\\mathcal {C}_d$ with $\\mathsf {d}(f,g)\\le \\frac{2c_1 + 3\\delta }{2^{k}} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace $ such that $\\mathsf {d}(g,h) < \\frac{\\delta }{3 \\cdot 2^k}$ implies $x$ and $h$ satisfy REF through REF on the left or on the right.",
"Fix $k \\in \\mathbb {N}$ satisfying (REF ).",
"Assume that the first possibility in Lemma REF is true for $n = 2^k$ .",
"(The alternative is treated similarly.)",
"There are two cases to consider, according to whether there is a time $0 \\le \\ell < 2^{k-1}$ at which $T^\\ell [x,x+\\frac{\\delta }{2^k}]$ contains the $i$ th discontinuity of $f$ .",
"Note that our assumption on $k$ guarantees that each such interval contains at most one discontinuity of $f$ .",
"Case 1: There is such an $\\ell $ .",
"Nudge the discontinuity in $T^\\ell [x,x+\\frac{\\delta }{2^k}]$ by at most $\\frac{\\delta }{2^k}$ so that it lies in $T^\\ell [x+\\frac{\\delta }{3 \\cdot 2^k},x+\\frac{2\\delta }{3 \\cdot 2^k}]$ .",
"For each $0 \\le j < 2^k$ with $j \\ne \\ell $ and $T^j [x,x+\\frac{3\\delta }{2^k}]$ containing a discontinuity of $f$ we nudge the discontinuity of $f$ by at most $\\frac{3 \\delta }{2^k}$ so that it lies in $T^j[x+\\frac{3\\delta }{2^k},x+\\frac{4 \\delta }{2^k}]$ .",
"For the resulting function $g$ we have $\\mathsf {d}(f,g) \\le \\frac{3\\delta }{2^k} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3 \\xi } \\right\\rbrace $ by Lemma REF .",
"Case 2: There is no such $\\ell $ .",
"For each $0 \\le j < 2^k$ with $T^j [x,x+\\frac{3\\delta }{2^k}]$ containing a discontinuity of $f$ we nudge the discontinuity of $f$ by at most $\\frac{3\\delta }{2^k}$ so that it lies in $T^j[x+\\frac{ 3\\delta }{ 2^k},x+\\frac{4 \\delta }{3 \\cdot 2^k}]$ .",
"By linear recurrence $\\lbrace T^j x : 0 \\le j < 2^{k-1} \\rbrace $ is within a distance of at most $\\frac{c_1}{2^{k-1}}$ from the $i$ th discontinuity of $f$ .",
"We nudge it by at most $\\frac{c_1}{2^{k-1}} + \\frac{\\delta }{2^k}$ to lie within some $T^j[x+\\frac{\\delta }{3 \\cdot 2^k},x+\\frac{2 \\delta }{3 \\cdot 2^k}]$ with $0 \\le j < 2^{k-1}$ .",
"For the resulting function $g$ we have $\\mathsf {d}(f,g)\\le \\frac{3\\delta + 2c_1}{2^{k}} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace $ by Lemma REF .",
"In both cases we have constructed a function $g$ with $\\mathsf {d}(f,g)\\le \\frac{3\\delta + 2c_1}{2^{k}} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace $ and the following properties: there is only one discontinuity of $g$ in $\\cup \\lbrace T^j [x,x+\\frac{\\delta }{2^k}] : 0 \\le i < 2^k \\rbrace $ ; there is $0 \\le j < 2^{k-1}$ such that $T^j [x,x+\\frac{\\delta }{2^k}]$ contains the only discontinuity of $g$ .",
"Moreover, by the construction every $h$ in $\\mathcal {C}_d$ with $\\mathsf {d}(g,h) < \\frac{\\delta }{3 \\cdot 2^k}$ also satisfies the above properties.",
"Thus $x$ and any such $h$ satisfy REF through REF on the right (for this $k$ )."
],
[
"Condition ",
"Using the material of Appendix we prove the following theorem.",
"Theorem 4.9 Fix $D > 0$ and $d \\in \\mathbb {N}$ .",
"There is $\\beta > 0$ such that, for every $f \\in \\mathcal {C}_{d,D}$ and every $B > 0$ and every $t \\in (-B,B)$ , almost every $x \\in [0,1)$ satisfies $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\ge \\beta \\sum _{n=0}^{2^k - 1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ for infinitely many $k$ .",
"The first step is to prove the following quantitative version of Atkinson's theorem.",
"Theorem 4.10 Let $T$ be an aperiodic measure-preserving transformation on a probability space $(X,{B},\\mu )$ and let $f : X \\rightarrow \\mathbb {R}$ in $\\operatorname{L}^1(X,{B},\\mu )$ have zero mean.",
"Further assume that there exists $0 < \\gamma < 1$ and $N_0 \\in \\mathbb {N}$ such that $ \\left| \\sum _{n=0}^{N-1} f(T^n x) \\right| < N^\\gamma $ for all $x \\in X$ and all $N \\ge N_0$ .",
"Then for every $\\epsilon > 0$ there is $N_1 \\in \\mathbb {N}$ such that $\\sum _{n=0}^{N-1} 1_{(-\\epsilon ,\\epsilon )} \\left( \\sum _{i=0}^n f(T^i x) \\right) > N^{1-\\gamma -\\epsilon }$ for almost every $x$ whenever $N \\ge N_1$ .",
"First, we claim that it suffices to prove for every $\\eta > 0$ that there is arbitrarily large $N \\in \\mathbb {N}$ with $\\left| \\left\\lbrace 1 \\le L \\le N: \\sum _{m=0}^N 1_{(-\\epsilon ,\\epsilon )} \\left( \\, \\sum _{n=0}^m f(T^i T^L x) \\right) > N^{1-\\gamma -\\epsilon } \\right\\rbrace \\right|>(1-\\eta ) N$ for all $x$ .",
"Denote the subset of $\\lbrace 1,\\dots ,N\\rbrace $ at left by $H_x$ (with the dependence on $N$ implicit).",
"Let $E_N=\\left\\lbrace x \\in [0,1) : \\sum _{n=0}^{N-1} 1_{(-\\epsilon ,\\epsilon )} \\left( \\sum _{i=0}^n f(T^i x) \\right) > N^{1-\\gamma -\\epsilon } \\right\\rbrace $ and notice that if $j \\in H_x$ then $T^jx \\in E_N$ .",
"Now, if we have (REF ) then $\\int _X \\sum _{i=1}^N 1_{E_N}(T^ix) \\,\\mathrm {d}\\mu \\ge (1-\\eta )N$ and so $\\mu (E_N)\\ge \\frac{(1-\\eta )N}{N}$ .",
"Now we prove (REF ).",
"Fix $\\eta > 0$ .",
"If $N$ is large enough then $f(x) + \\cdots + f(T^{N-1} x)$ belongs to $[-N^\\gamma ,N^\\gamma ]$ for all $1 \\le L \\le N$ and all $x \\in X$ .",
"Fix an interval $J \\subset [-N^\\gamma ,N^\\gamma ]$ of length $\\epsilon $ .",
"Let $0 \\le L_1 < \\cdots < L_s \\le N$ be an enumeration of those $1 \\le L \\le N$ at which $f(x) + \\cdots + f(T^{L-1} x)$ belongs to $J$ .",
"We have $\\epsilon >\\left| \\sum _{n=0}^{L_j - 1} f(T^n x) - \\sum _{n=0}^{L_i - 1} f(T^n x) \\right|=\\left| \\sum _{n=0}^{L_j - L_i - 1} f(T^n T^{L_i} x) \\right|$ for all $0 \\le i < j \\le s$ .",
"Therefore $L_i$ belongs to $H_x$ whenever $s - i \\ge N^{1 - \\gamma - \\epsilon }$ holds.",
"Since we can cover $[-N^\\gamma ,N^\\gamma ]$ by at most $\\lceil 2N^\\gamma \\epsilon ^{-1} \\rceil $ intervals of length $\\epsilon $ , it follows that at most $\\lceil 2N^\\gamma \\epsilon ^{-1} \\rceil N^{1 - \\gamma - \\epsilon }$ of the $1 \\le L \\le N$ do not belong to $H_x$ .",
"For $N$ large enough (and independent of $x$ ) we will therefore have $|H_x| > (1-\\eta ) N$ .",
"Lemma 4.13 Fix $0 < \\alpha < 1$ and suppose $g : \\mathbb {N} \\rightarrow \\mathbb {N}$ is non-decreasing and satisfies $n^\\alpha \\le g(n) \\le n$ for infinitely many $n \\in \\mathbb {N}$ .",
"Then for every $0 < \\beta < 1 - 2^{-\\alpha }$ one has $g(2^k) - g(2^{k-1}) \\ge \\beta g(2^k)$ infinitely often.",
"Fix $0 < \\beta < 1 - 2^{-\\alpha }$ .",
"Suppose the conclusion is false.",
"Then there is $K \\in \\mathbb {N}$ such that $(1-\\beta ) g(2^k) < g(2^{k-1})$ for all $k \\ge K$ .",
"Thus $(1 - \\beta )^l g(2^{k+l}) < g(2^k)$ for all $l \\in \\mathbb {N}$ and all $k \\ge K$ by induction.",
"Write $n_j$ for the increasing sequence of times $n \\ge 2^K$ at which $n^\\alpha \\le g(n) \\le n$ holds.",
"For each $j$ fix $l_j \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ with $2^{K+l_j} \\le n_j < 2^{K+l_j + 1}$ .",
"We have $(1-\\beta )^{-(1 + \\ell _j)} g(2^K) \\ge g(2^{K + l_j + 1}) \\ge g(n_j) \\ge n_j^\\alpha \\ge 2^{(K+l_j)\\alpha }$ for all $j \\in \\mathbb {N}$ .",
"But then $\\Big ( (1-\\beta ) 2^\\alpha \\Big )^{l_j}\\le \\frac{2^K}{(1-\\beta ) 2^{K \\alpha }}$ for all $j \\in \\mathbb {N}$ .",
"Taking $j$ large enough gives the desired contradiction because $(1 - \\beta )2^\\alpha > 1$ and so the left hand side goes to infinity with $j$ while the right hand side is independent of $j$ .",
"[Proof of Theorem REF ] Let $B$ and $t$ be given with $-B<t<B$ .",
"It suffices to show that for almost every $x$ we have that there exists infinitely many $k$ so that $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B-t,B-t]} \\left(\\; \\sum _{i=0}^{n-1} f(T^i x) \\right)\\ge \\beta \\sum _{n=0}^{2^k - 1} 1_{[-B-t,B-t]} \\left(\\; \\sum _{i=0}^{n-1} f(T^i x) \\right).$ By Lemma REF it suffices to show that there exists $\\gamma >0$ and an infinite sequence of $N_j$ so that $\\sum _{i=0}^{N_j} 1_{[-B-t,B-t]}\\left(\\; \\sum _{i=0}^{n-1} f(T^i x) \\right)>(N_j)^\\gamma $ holds.",
"Letting $0<\\epsilon <|B|-|t|$ and invoking Theorem REF (which we may do because Theorem REF shows (REF ) is satisfied) gives this condition.",
"Moreover, since Theorem REF produces $0 < \\gamma < 1$ uniform over all step functions $f$ we have that $\\beta $ is uniform over those functions as well."
],
[
"Condition ",
"Here we prove that if REF holds for a specific pair $(f,(x,t))$ then it also holds if $f$ is perturbed a little.",
"Proposition 4.14 Fix $D > 0$ and $d \\in \\mathbb {N}$ .",
"Let $\\beta > 0$ be as in Theorem REF .",
"Fix $B > C > 0$ and $\\tau ,\\theta > 0$ .",
"Suppose given $f \\in \\mathcal {C}_{d,D}$ and $(x,t) \\in X_{B+C}$ and $K \\subset \\mathbb {N}$ infinite such that REF holds with $B+C$ in place of $B$ for all $k \\in K$ ; $(f,(x,t)) \\in G_3(B+C)$ ; $\\mu _{f,(x,t),B+C} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) < \\dfrac{1 - \\tau }{1+\\theta }$ ; all hold.",
"Then there is $K^{\\prime } \\subset K$ cofinite in $K$ such that $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B+C,B-C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\ge \\beta \\tau \\sum _{n=0}^{2^k-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ for all $k \\in K^{\\prime }$ .",
"Let $f$ and $(x,t)$ be as in the hypothesis.",
"By REF we can fix $\\eta > 0$ such that $(1 + \\theta ) \\mu _{f,(x,t),B+C} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) + \\eta < 1 - \\tau $ holds.",
"By REF we have $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B-C,B+C]} \\left(t+ \\sum _{i=0}^{n-1} f(T^i x) \\right)\\ge \\beta \\sum _{n=0}^{2^k-1} 1_{[-B-C,B+C]} \\left(t+ \\sum _{i=0}^{n-1} f(T^i x) \\right)$ for all $k \\in K$ .",
"Now $(f,(x,t)) \\in G_3(B+C)$ by REF , so we can apply Lemma REF to get $L \\in \\mathbb {N}$ so large that $\\frac{1}{N-M} \\sum _{n=M}^{N-1} 1_{\\mathsf {coll}(B+C,2C)} \\left( S_{f,B+C}^n (x,t) \\right)\\le (1 + \\theta ) \\mu _{f,(x,t),B+C} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) + \\eta $ whenever $N > L$ and $N-M \\ge \\beta N$ .",
"There is $K^{\\prime } \\subset K$ cofinite such that $N := \\sum _{n=0}^{2^k-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ is at least $L$ whenever $k \\in K^{\\prime }$ and, choosing $M := \\sum _{n=0}^{2^{k-1}-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ it follows from (REF ) that $N-M \\ge \\beta N$ .",
"Therefore $&\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B-C,-B+C] \\cup [B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\\\\\le &\\left( (1 + \\theta ) \\mu _{f,(x,t),B+C} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) + \\eta \\right) \\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ for all $k \\in K^{\\prime }$ because times $n$ at which $t + \\displaystyle \\sum _{i=0}^{n-1} f(T^i x)$ belongs to $[-B-C,-B+C] \\cup [B-C,B+C]$ are in bijective correspondence with the visits of the $S_{f,B+C}$ orbit of $(x,t)$ to $\\mathsf {coll}(B+C,2C)$ .",
"We therefore have $&\\sum _{n=2^{k-1}}^{2^k - 1} 1_{[-B+C,B-C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\\\\\ge \\;&\\left( 1 - (1+\\theta ) \\mu _{f,(x,t),B+C} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) - \\eta \\right) \\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)$ for all $k \\in K^{\\prime }$ .",
"Using (REF ) and (REF ) gives (REF ) immediately.",
"Recall that if $f$ in $\\mathcal {C}_d$ has coordinates $(x_1,\\dots ,x_{d+1},y_1,\\dots ,y_{d+1})$ then $\\xi $ denotes $\\min \\lbrace x_1,\\dots ,x_{d+1} \\rbrace $ .",
"Theorem 4.18 Fix $D > 0$ and $d \\in \\mathbb {N}$ and $A > 0$ .",
"Let $\\beta > 0$ be as in Theorem REF Put $C = 6 d D A \\frac{c_1}{c_2} + 2dD + 4dAc_1$ and fix $B > C$ .",
"Fix $\\tau > 0$ .",
"Given $f \\in \\mathcal {C}_{d,D}$ and $(x,t)$ and $K \\subset \\mathbb {N}$ satisfying REF , REF and REF we can find $K^{\\prime } \\subset K$ cofinite such that, whenever $k \\in K^{\\prime }$ and $g \\in \\mathcal {C}_{d,D}$ satisfy $\\mathsf {d}(f,g)\\le \\min \\left\\lbrace \\frac{\\xi }{4(d+1)}, \\frac{A(c_1 + \\delta )}{2^k} \\right\\rbrace $ then $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} g(T^i x) \\right)\\ge \\beta \\tau \\sum _{n=0}^{2^k-1} 1_{[-B,B]} \\left( t + \\sum _{i=0}^{n-1} g(T^i x) \\right)$ holds.",
"Let $f \\in \\mathcal {C}_{d,D}$ and $(x,t)$ and $K \\subset \\mathbb {N}$ infinite satisfy REF , REF and REF .",
"Let $K^{\\prime }$ be as in the conclusion of Proposition REF .",
"Fix $k \\in K^{\\prime }$ .",
"Fix $g$ with coordinates $(\\tilde{x}_1,\\dots ,\\tilde{x}_{d+1},\\tilde{y}_1,\\dots ,\\tilde{y}_{d+1})$ satisfying (REF ).",
"Let $I_j$ be the interval with endpoints $x_1 + \\cdots + x_j$ and $\\tilde{x}_1 + \\cdots + \\tilde{x}_j$ for each $1 \\le j \\le d+1$ .",
"These intervals are disjoint by (REF ).",
"We have $|f(x) - g(x)| \\le 2D$ on each such interval.",
"Off these intervals we have $|f(x) - g(x)| \\le \\mathsf {d}(f,g)$ .",
"Put $J_j = \\lbrace 0 \\le i < 2^k : T^i x \\in I_j \\rbrace $ for each $1 \\le j \\le d+1$ and $J = J_1 \\cup \\cdots \\cup J_{d+1}$ .",
"The complement of the intervals $I_i$ is a collection $I^{\\prime }_1,\\dots ,I^{\\prime }_{d+1}$ of $d+1$ disjoint intervals with respective widths at most $x_i$ .",
"Linear recurrence implies $|J_j| \\le \\lceil 2^k |I_j| /c_2 \\rceil \\le \\lceil 2^k \\mathsf {d}(f,g)/c_2 \\rceil $ for all $j$ and that the orbit $x,\\dots ,T^{2^k-1} x$ is in $I^{\\prime }_i$ at most $\\lceil 2^k x_i /c_2 \\rceil $ times.",
"Now for each $0 < n \\le 2^k$ we estimate that $\\left| \\sum _{i=0}^{n-1} f(T^i x) - \\sum _{i=0}^{n-1} g(T^i x) \\right|\\le \\,&\\sum _{i \\in J} |f(T^i x) - g(T^i x) | + \\sum _{i \\notin J} |f(T^i x) - g(T^i x) |\\\\\\le \\,&d \\left( \\frac{ 2^k \\mathsf {d}(f,g)}{c_2} + 1 \\right) 2D + \\sum _{i=1}^{d+1} \\left( \\frac{2^k x_i}{c_2} + 1 \\right) \\mathsf {d}(f,g)\\\\\\le \\,&2dD \\frac{A(c_1 + \\delta )}{c_2} + 2dD + \\frac{A(c_1 + \\delta )}{c_2} + (d+1) A(c_1 +\\delta )\\\\\\le \\,&4dDA \\frac{c_1}{c_2} + 2dD + 2A \\frac{c_1}{c_2} + 4dA c_1\\\\\\le \\,&6 d D A \\frac{c_1}{c_2} + 2dD + 4dAc_1$ using (REF ) and $c_1 \\ge c_2$ and $\\delta = c_2/4$ .",
"This gives the implications $t+\\sum _{i=0}^{n-1} f(T^i x) \\in [-B+C,B-C]\\Rightarrow \\,&t+\\sum _{i=0}^{n-1} g(T^i x) \\in [-B,B]\\\\t+\\sum _{i=0}^{n-1} g(T^i x) \\in [-B,B]\\Rightarrow \\,&t+\\sum _{i=0}^{n-1} f(T^i x) \\in [-B-C,B+C]$ for all $0 < n \\le 2^k$ .",
"Combining with (REF ) we get $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B,B]} \\left(t+ \\sum _{i=0}^{n-1} g(T^i x) \\right)\\ge &\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B+C,B-C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\\\\\ge &\\beta \\tau \\sum _{n=0}^{2^k-1} 1_{[-B-C,B+C]} \\left( t + \\sum _{i=0}^{n-1} f(T^i x) \\right)\\\\\\ge &\\beta \\tau \\sum _{n=0}^{2^k - 1} 1_{[-B,B]} \\left( \\sum _{i=0}^{n-1} t + g(T^i x) \\right)$ for all $k \\in K^{\\prime }$ as desired."
],
[
"Conditions ",
"Fix throughout this subsection $d \\in \\mathbb {N}$ and $D \\in \\mathbb {N}$ .",
"Let $\\beta $ be as in Theorem REF .",
"Given a mean-zero step function $f$ in $\\mathcal {C}_{d,D}$ put $A = 2 (d+1) \\frac{10 \\delta + 6 c_1}{3 \\delta + 3 c_1} \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace $ where $\\xi = \\min \\lbrace x_1,\\dots ,x_{d+1} \\rbrace $ and write $C = 6 d D A c_1/c_2 + 2dD + 4dAc_1$ as in (REF ).",
"Lemma 4.22 Fix $\\tau > 0$ .",
"For every $\\epsilon > 0$ and every $f \\in \\mathcal {C}_{d,D}$ there is $B > 0$ such that the $\\mathsf {m}_{B+C}$ measure of the set of pairs $(x,t) \\in X_{B+C}$ for which the following statement is true is at least $1 - \\epsilon $ : there are $\\theta > 0$ and $K \\subset \\mathbb {N}$ infinite such that the following conditions all hold.",
"$(f,(x,t)) \\in G_3(B+C)$ .",
"$\\mu _{f,B+C,(x,t)} \\Big ( \\mathsf {coll}(B+C,2C) \\Big ) < \\dfrac{1 - \\tau }{1+\\theta }$ .",
"REF with $B+C$ in place of $B$ for all $k \\in K$ .",
"$\\dfrac{A(c_1 + \\delta )}{2^k} \\le \\dfrac{\\xi }{4(d+1)}$ for all $k \\in K$ .",
"$\\dfrac{c_1 + \\delta }{2^{k-1}} < \\dfrac{\\xi }{4}$ for all $k \\in K$ .",
"Fix $\\tau > 0$ , $\\epsilon > 0$ and $f \\in \\mathcal {C}_{d,D}$ .",
"For every $B > 0$ almost every $(x,t) \\in X_{B+C}$ has the property that it is generic for an $S_{f,B}$ invariant probability measure $\\mu _{f,B+C,(x,t)}$ so REF holds almost surely for all $B$ .",
"For REF note that $\\mathsf {m}_{B+C}\\left( \\left\\lbrace (x,t) \\in X_{B+C} : \\mu _{f,B+C,(x,t)}(\\mathsf {coll}(B+C,2C)) \\ge \\frac{1 - \\tau }{1 + \\theta } \\right\\rbrace \\right)\\le \\frac{1 + \\theta }{1-\\tau } \\mathsf {m}_{B+C}(\\mathsf {coll}(B+C,2C))$ by Markov's inequality and that the right-hand side is less than $\\epsilon $ for $B$ large enough independent of $(x,t)$ .",
"Theorem REF implies REF is true for almost all $(x,t) \\in X_{B+C}$ .",
"By removing finitely many points from $K$ we get REF and REF .",
"[Proof of Theorem REF ] Fix $0 < \\tau < 1$ .",
"For each $(x,t) \\in X_B$ and every $q \\in \\mathbb {N}$ we can consider the set $E_B(q,(x,t)) = \\bigcap _{k \\ge q} \\lbrace e \\in \\mathcal {C}_{d,D} : \\textup {one of } \\ref {fr:3},\\ref {fr:4},\\ref {fr:5} \\textup { at } \\beta \\tau \\textup {, fails for } (x,t) \\textup { and } e \\rbrace $ of skewing functions $e$ which fail to be friends with $(x,t)$ because there are no $k$ larger than $q$ for which all of REF , REF , REF at $\\beta \\tau $ hold.",
"Fix $f \\in \\mathcal {C}_{d,D}$ .",
"Claim If $f$ and $(x,t)$ satisfy REF through REF for some $\\theta > 0$ and some $K \\subset \\mathbb {N}$ infinite then $f$ is not a density point of $E_B(q,(x,t))$ .",
"[Proof of claim] Fix $f$ and $(x,t)$ such that REF through REF are satisfied for some $\\theta > 0$ and some $K \\subset \\mathbb {N}$ infinite.",
"By REF , REF and REF we can apply Theorem REF , by which there is $K^{\\prime } \\subset K$ cofinite with the following property: if $h \\in \\mathcal {C}_{d,D}$ satisfies $\\mathsf {d}(f,h) \\le \\min \\left\\lbrace \\frac{\\xi }{4(d+1)}, \\frac{A(c_1 + \\delta )}{2^k} \\right\\rbrace $ for some $k \\in K^{\\prime }$ then $\\sum _{n=2^{k-1}}^{2^k-1} 1_{[-B,B]} \\left(t + \\sum _{i=0}^{n-1} h(T^i x) \\right)\\ge \\beta \\tau \\sum _{n=0}^{2^k - 1} 1_{[-B,B]} \\left(t + \\sum _{i=0}^{n-1} h(T^i x) \\right)$ holds.",
"Fix $k \\in K^{\\prime }$ .",
"By REF we can apply Proposition REF to get $g \\in \\mathcal {C}_d$ with the properties therein.",
"That is, for any $h$ in $\\mathcal {C}_d$ with $\\mathsf {d}(g,h) < \\frac{\\delta }{3 \\cdot 2^k}$ we have that $h$ and $(x,t)$ satisfy REF through REF on either the left or the right for our current value of $k$ .",
"Now $\\mathsf {d}(f,h)&\\le \\frac{2c_1 + 3 \\delta }{2^k} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace + \\frac{\\delta }{3 \\cdot 2^k}\\\\&\\le \\frac{A(c_1 + \\delta )}{2^k}\\\\&\\le \\min \\left\\lbrace \\frac{\\xi }{4(d+1)}, \\frac{A(c_1 + \\delta )}{2^k} \\right\\rbrace $ for any such $h$ upon using (REF ), (REF ) and REF .",
"By the previous paragraph, this implies (REF ) holds.",
"So, for our current value of $k \\in K^{\\prime }$ the pair $(x,t)$ and the function $h$ satisfy REF through REF either on the left or the right with $\\tau \\beta $ in place of $\\beta $ .",
"To summarize, for each $k \\in K^{\\prime }$ every $h$ in the ball (with respect to the $\\mathsf {d}$ metric) centered at $g$ of radius $r(k) = \\frac{\\delta }{3 \\cdot 2^k}$ satisfies REF through REF with $\\tau \\beta $ in place of $\\beta $ either on the left or on the right.",
"This ball is entirely contained within the ball centered at $f$ of radius $R(k) = r(k) + \\frac{2c_1 + 3 \\delta }{2^k} (d+1) \\max \\left\\lbrace 1, \\frac{8D}{3\\xi } \\right\\rbrace $ and $\\inf \\lbrace R(k) / r(k) : k \\in K^{\\prime } \\rbrace > 0$ so taking $k > q$ we conclude that $f$ is not a density point for the set $E_B(q,(x,t))$ .",
"Combining the claim with Lemma REF gives, for almost every $f \\in \\mathcal {C}_{d,D}$ , that $\\mathsf {m}_B(\\lbrace (x,t) \\in X_B : f \\textup { not a density point of } E_B(q,(x,t)) \\rbrace ) \\rightarrow 1$ as $B \\rightarrow \\infty $ .",
"Moreover, the convergence is uniform in $q$ because (as in Lemma REF ) we need only take $B$ large enough that $\\mathsf {m}_{B+C}(\\mathsf {coll}(B+C,2C))$ is small.",
"Fix now any probability $\\eta $ on $\\mathcal {C}_{d,D}$ that is equivalent to Lebesgue measure in every atlas.",
"For each $B$ and every $(x,t) \\in X_B$ the sequence $q \\mapsto E_B(q,(x,t))$ of sets is increasing, so $\\eta \\Big ( \\bigcup _{q \\in \\mathbb {N}} E_B(q,(x,t)) \\Big )\\le \\limsup _{q \\rightarrow \\infty } \\eta ( \\lbrace \\textup {Density points of } E_B(q,(x,t)) \\rbrace )$ holds.",
"But $\\mathsf {m}_B(\\lbrace (x,t) \\in X_B : \\eta ( \\lbrace \\textup {Density points of } E_B(q,(x,t)) \\rbrace ) > \\epsilon \\rbrace ) \\rightarrow 0$ as $B \\rightarrow \\infty $ so we conclude the set $\\mathsf {m}_B \\left( \\left\\lbrace (x,t) \\in X_B : \\eta \\left( \\bigcap _{B=1}^\\infty \\bigcup _{q \\ge 1} E_B(q,(x,t)) \\right) = 0 \\right\\rbrace \\right)\\rightarrow 0$ as $B \\rightarrow \\infty $ .",
"By Fubini's theorem, for almost every $f$ we have (REF )."
],
[
"An ergodic decomposition",
"In this appendix we prove Theorem REF .",
"We will do this using an ergodic decomposition result for quasi-invariant measures due to Schmidt that we reproduce here for convenience.",
"For measures $\\mu ,\\nu $ on a measure space $(X,{B})$ recall that $\\mu $ and $\\nu $ are equivalent written $\\mu \\sim \\nu $ when each is absolutely continuous with respect to the other.",
"Also $\\mu ,\\nu $ are mutually singular written $\\mu \\perp \\nu $ if there is a measurable set $A \\subset X$ with $\\mu (A^\\mathsf {c}) = 0 = \\nu (A)$ .",
"Theorem E.1 () Let $(X,{B})$ be a measurable space and let $F : X \\rightarrow X$ be a $\\mathcal {B}$ measurable map.",
"Fix an $F$ quasi-invariant probability measure $\\mu $ on $(X,{B})$ .",
"There exists a standard Borel space $(Y,{Y})$ , a surjective, measurable map $\\psi : X \\rightarrow Y$ , and a family $y \\mapsto q_y$ of Borel probability measures on $(X,{B})$ with the following properties.",
"The map $y \\mapsto q_y(A)$ is Borel for every $A \\in {B}$ .",
"Against all members of ${B}$ one has $\\mu = \\displaystyle \\int q_y \\,\\mathrm {d}\\rho (y)$ where $\\rho = \\psi \\mu $ .",
"All of the measures $q_y$ are $F$ quasi-invariant and ergodic.",
"$q_y(\\psi ^{-1}(y))=1$ for every $y\\in Y$ .",
"Let ${Z}$ be the $\\sigma $ -algebra of $F$ invariant sets.",
"Let ${C} = \\lbrace \\psi ^{-1}(B) : B \\in {Y}\\rbrace $ .",
"Then ${Z}$ and ${Z}$ are $\\mu $ equivalent.",
"If there is another collection $(Y^{\\prime },{Y}^{\\prime },\\psi ^{\\prime },q^{\\prime })$ satisfying 1. through 4. then there exists a measurable map $\\Theta : Y \\rightarrow Y^{\\prime }$ that is an isomorphism between $(Y,{Y},\\rho )$ and $(Y^{\\prime },{Y}^{\\prime },\\rho ^{\\prime })$ such that $q^{\\prime }_{\\Theta (y)}\\sim q_y$ for $\\rho $ almost every $y \\in Y$ .",
"The following corollary is virtually the same as .",
"We reformulate it slightly for our purposes.",
"Corollary E.2 (cf. )",
"Let $(X,{B})$ be a measurable space and let $F : X \\rightarrow X$ be a ${B}$ measurable map.",
"Fix an $F$ invariant $\\sigma $ -finite measure $\\nu $ on $(X,{B})$ .",
"There exists a Borel space $(Y,{Y})$ , a surjective Borel measurable map $\\psi : X \\rightarrow Y$ , and a family $y \\mapsto p_y$ of $\\sigma $ -finite Borel measures on $(X,{B})$ with the following properties.",
"The map $y \\rightarrow p_y(A)$ is measurable for every $A \\in {B}$ .",
"Against all members of ${B}$ one has $\\nu = \\displaystyle \\int p_y d\\rho (y)$ where $\\rho =\\psi \\nu $ .",
"All of the measures $p_y$ are $F$ invariant and ergodic.",
"All of the measures $p_y$ satisfy $p_y(X \\setminus \\psi ^{-1}(y)) = 0$ .",
"Let ${Z}$ be the $\\sigma $ -algebra of $F$ invariant sets.",
"Let ${C} = \\lbrace \\psi ^{-1}(B) : B \\in {Y} \\rbrace $ .",
"Then ${Z}$ and ${C}$ are $\\nu $ equivalent.",
"If there is another such collection $(Y^{\\prime },{Y}^{\\prime },\\psi ^{\\prime },p^{\\prime })$ satisfying REF through REF then there is a measurable map $\\Theta : Y \\rightarrow Y^{\\prime }$ that is an isomorphism between $(Y,{Y},\\rho )$ and $(Y^{\\prime },{Y}^{\\prime },\\rho ^{\\prime })$ such that $p^{\\prime }_y \\sim p_{\\Theta (y)}$ for $\\rho $ almost every $y$ .",
"Let $\\mu $ be a probability measure that is equivalent to $\\nu $ .",
"Write $\\mu = f \\nu $ where $f$ is a positive, measurable function.",
"Theorem REF applied to $(X,{B},F,\\mu )$ gives $(Y,{Y},\\psi ,q)$ satisfying REF through REF of Theorem REF .",
"Put $p_y = \\frac{1}{f} q_y$ .",
"It is straightforward to verify that $p$ is a disintegration of $\\nu $ and therefore satisfies REF and REF .",
"It inherits the other properties from $q_y$ .",
"Fix an interval exchange transformation $T$ and $f : [0,1) \\rightarrow \\mathbb {R}$ a mean-zero step function.",
"Take $X = [0,1) \\times \\mathbb {R}$ and let ${B}$ be the Borel $\\sigma $ -algebra on $X$ .",
"Apply Corollary REF with $F = T_f$ and $\\nu = \\mathsf {m}$ to get $(Y,{Y})$ , the map $\\psi $ and the family $y \\mapsto p_y$ with the stated properties.",
"Write $\\rho = \\psi \\mathsf {m}$ .",
"Let ${Z}$ be the $\\sigma $ -algebra of $T_f$ invariant sets and let ${C} = \\lbrace \\psi ^{-1}(B) : B \\in {Y} \\rbrace $ .",
"For $b \\in \\mathbb {R}$ define $V^b : X \\rightarrow X$ by $V^b(x,t) = (x,t+b)$ .",
"In preparation for the proof of Theorem REF we verify the following lemmas.",
"Lemma E.3 For every $b \\in \\mathbb {R}$ and $\\mathsf {m}$ almost every $(x,t)$ the measures $V^b p_{\\psi (x,t)}$ and $p_{\\psi (x,t+b)}$ are equivalent.",
"Define $p^{\\prime }_y = V^b p_y$ and $\\psi ^{\\prime }(x,t) = \\psi (V^{-b}(x,t)) = \\psi (x,t-b)$ .",
"We claim that $(Y,{Y},\\psi ^{\\prime },p^{\\prime })$ satisfies REF through REF of Theorem REF .",
"This is easily verified: we only check REF here by observing that $\\psi ^{\\prime } \\mathsf {m}= \\psi \\mathsf {m}$ and calculating $\\iint 1_B \\,\\mathrm {d}p^{\\prime }_y \\,\\mathrm {d}\\rho (y)=\\iint 1_{V^{-b}B} \\,\\mathrm {d}p_y \\,\\mathrm {d}\\rho (y)=\\int 1_{V^{-b} B} \\,\\mathrm {d}\\mathsf {m}=\\int 1_B \\,\\mathrm {d}\\mathsf {m}$ for all $B$ in ${B}$ .",
"We get from REF an automorphism $\\Theta : Y \\rightarrow Y$ such that $p^{\\prime }_{\\Theta (y)} \\sim p_y$ for $\\rho $ almost every $y$ .",
"Thus $p^{\\prime }_{\\Theta (\\psi (x,t))} \\sim p_{\\psi (x,t)}$ for $\\mathsf {m}$ almost every $(x,t)$ .",
"So for $\\mathsf {m}$ almost every $(x,t)$ we have that the intersection $\\psi ^{-1}(\\psi (x,t)) \\cap (\\psi ^{\\prime })^{-1}(\\Theta (\\psi (x,t)))$ is co-null for both measures.",
"It follows that for $\\mathsf {m}$ almost every $(x,t)$ we have both $p_{\\psi (z)} = p_{\\psi (x,t)}$ and $p^{\\prime }_{\\psi ^{\\prime }(z)} = p^{\\prime }_{\\Theta (\\psi (x,t))}$ for $p_{\\psi (x,t)}$ almost every $z$ .",
"In conclusion $p^{\\prime }_{\\psi ^{\\prime }(z)} = p_{\\psi (z)}$ for $\\mathsf {m}$ almost every $z \\in [0,1) \\times \\mathbb {R}$ .",
"In other words $V^b \\psi _{\\psi (x,t+b)} \\sim p_{\\psi (x,t)}$ for $\\mathsf {m}$ almost every $(x,t)$ .",
"Extending $T$ to $[0,1) \\times \\mathbb {R}$ by $T(x,t) = (Tx,t)$ we have the following lemma as well, whose proof is almost identical to that of the previous lemma.",
"Lemma E.4 For $\\mathsf {m}$ almost every $(x,t)$ the measures $T p_{\\psi (x,t)}$ and $p_{\\psi (Tx,t)}$ are equivalent.",
"We also need the following lemmas.",
"Lemma E.5 Fix $b \\in \\mathbb {R}$ .",
"If $p_y$ and $V^b p_y$ are not mutually singular for a set of positive $\\rho $ measure then $p_y$ and $V^b p_y$ are not mutually singular for $\\rho $ almost every $y$ .",
"Put $H = \\lbrace (x,t) \\in X : p_{\\psi (x,t)} \\lnot \\perp V^b p_{\\psi (x,t)} \\rbrace $ .",
"We have $\\mathsf {m}(H) > 0$ by REF and our hypothesis.",
"It follows from Lemma REF that $H$ is $T$ invariant.",
"Since $T$ is ergodic on $[0,1)$ the set $\\lbrace x \\in [0,1) : (x,t) \\in H \\rbrace $ has either null of full Lebesgue measure for every $t \\in \\mathbb {R}$ .",
"So our hypothesis gives a positive measure set of $t \\in \\mathbb {R}$ such that $\\lbrace x \\in [0,1) : (x,t) \\in H \\rbrace $ has full Lebesgue measure.",
"But Lemma REF also implies that $H$ is $V^a$ invariant for all $a \\in \\mathbb {R}$ .",
"So $H$ must be a co-null set.",
"Lemma E.6 If $\\eta _y$ is the $\\rho $ almost-surely defined normalized restriction of $p_y$ to $X_B$ then $y \\mapsto \\eta _y$ is an ergodic decomposition of $\\mathsf {m}_B$ for the transformation $S_{f,B}$ .",
"Since almost every $p_y$ is ergodic for $T_f$ the restriction $\\eta _y$ is almost surely ergodic for the induced transformation $S_{f,B}$ .",
"That $y \\mapsto p_y$ is a disintegration of $\\mathsf {m}_B$ follows from REF .",
"Thus $y \\mapsto \\eta _y$ is an ergodic decomposition of $\\mathsf {m}_B$ .",
"We are now ready to prove Theorem REF .",
"[Proof of Theorem REF ] Fix $b \\in \\mathbb {R}$ such that the set $\\lbrace (x,t) \\in X_B : \\mu _{f,B,(x,t)} \\lnot \\perp V^b \\mu _{f,B,(x,t)} \\rbrace $ has positive $\\mathsf {m}_B$ measure.",
"Uniqueness in the ergodic decomposition implies $\\lbrace (x,t) \\in X_B : p_{\\psi (x,t)}|X_B \\lnot \\perp V^b \\big ( p_{\\psi (x,t)}|X_B \\big ) \\rbrace $ has positive $\\mathsf {m}_B$ measure.",
"If $p_{\\psi (x,t)}|X_B \\lnot \\perp V^b \\big ( p_{\\psi (x,t)}|X_B \\big )$ then we get from the Lebesgue decomposition theorem a measure $\\lambda $ on $[0,1) \\times \\mathbb {R}$ which is absolutely continuous with respect to both measures.",
"So we have $\\lambda \\ll p_{\\psi (x,t)}|X_B \\ll p_{\\psi (x,t)}$ and $\\lambda \\ll V^b (p_{\\psi (x,t)}|X_B) \\ll V^b p_{\\psi (x,t)}$ whence $p_{\\psi (x,t)}$ and $V^b p_{\\psi (x,t)}$ are not mutually singular.",
"Since (REF ) has positive measure Lemma REF implies $p_{\\psi (x,t)}\\lnot \\perp V^b p_{\\psi (x,t)}$ for almost every $(x,t)$ .",
"Since ergodic measures are either mutually singular or equivalent we have $p_{\\psi (x,t)} \\sim V^bp_{\\psi (x,t)}$ for almost every $(x,t)$ .",
"Our goal is to prove that $b$ is an essential value of $f$ .",
"Fix $Z \\in {Z}_f$ a $T_f$ invariant set.",
"We wish to prove that $\\mathsf {m}(Z \\triangle (V^b)^{-1} Z) = 0$ .",
"By REF we may assume there exists $S \\subset Y$ with $Z = \\psi ^{-1}(S)$ .",
"For such $Z$ we have $p_y(Z) \\in \\lbrace 0,1\\rbrace $ for $\\rho $ almost every $y$ .",
"But then (REF ) implies $p_y(Z) = p_y(V^{-b} Z)$ for $\\rho $ almost every $y$ .",
"Finally $\\mathsf {m}( Z \\triangle V^{-b} Z )=\\int p_y (Z \\triangle V^{-b} Z) \\,\\mathrm {d}\\rho (y)=0$ as desired."
],
[
"Quantitative unique ergodicity",
"It follows from work of Boshernitzan that every linearly recurrent interval exchange transformation is uniquely ergodic.",
"In this section we prove the following quantitative version of Boshernitzan's result.",
"Throughout this section we use $b$ to denote the number of intervals of an interval exchange transformation.",
"Theorem F.1 Let $T$ be a linearly recurrent interval exchange transformation.",
"There is $0<\\gamma <1$ so that for any mean-zero step function $f$ we have $\\left| \\sum _{n=0}^{N-1} f(T^n x) \\right| \\le N^\\gamma $ for all large enough $N$ .",
"In fact, this result follows from Section 4 of .",
"Most of this section constitutes a self-contained proof of Theorem REF , which we give for completeness.",
"Our interest in Theorem REF is in deducing from it that Property REF holds for every mean-zero step function $f : [0,1) \\rightarrow \\mathbb {R}$ and almost every $x$ .",
"We begin with some notation for the induction scheme we will use throughout the proof of Theorem REF .",
"Fix a linearly recurrent interval exchange transformation $T$ with $b-1$ discontinuities.",
"Write $I_0$ for $[0,1)$ and let $I_{0,1},\\dots ,I_{0,b}$ be the intervals of continuity of $T$ .",
"Define inductively $I_n = I_{n-1,1}$ and $I_{n,1},\\dots ,I_{n,b}$ as the intervals of the induced transformation $T|I_n$ on $I_n$ .",
"(See Figure REF for a schematic.)",
"Since we require defined linearly recurrent interval exchange transformations to satisfy the Keane condition, the induced transformation $T|I_n$ is also an exchange of $b$ intervals.",
"Given $\\ell > k \\ge 0$ define $r_{k,\\ell }(j) = \\min \\lbrace n \\in \\mathbb {N} : (T|I_k)^n I_{\\ell ,j} \\subset I_\\ell \\rbrace $ for all $1 \\le j \\le b$ .",
"This is the first time the $T|I_k$ orbit of $I_{\\ell ,j}$ returns to $I_\\ell $ .",
"Write $r_{k,\\ell }(x) = \\min \\lbrace n \\in \\mathbb {N} : (T|I_k)^n x \\in I_\\ell \\rbrace $ for all $x \\in I_k$ .",
"Note that $r_{k,\\ell }(j) = r_{k,\\ell }(x)$ for all $x \\in I_{\\ell ,j}$ .",
"Define also for each $k \\in \\mathbb {N}$ a matrix $B_k$ with entries $B_k(i,j) = \\sum _{n=0}^{r_{k,k+1}(j)-1} 1_{I_{k,i}} \\Big ( (T|I_k)^n I_{k+1,j} \\Big )$ for all $1 \\le i,j \\le b$ that count the number of visits of the $T|I_k$ orbit of $I_{k+1,j}$ to $I_{k,i}$ before the orbit visits $I_{k+1}$ .",
"Therefore $B_k(1,j) = 1$ for all $1 \\le j \\le b$ .",
"Note also that $|\\!|B_k e_j |\\!|_1 = B_k(1,j) + \\cdots + B_k(b,j) = r_{k,k+1}(j)$ for all $1 \\le j \\le b$ where $e_1,\\dots ,e_b$ is the standard basis of $\\mathbb {R}^b$ .",
"Figure: The interval I k I_k and some of its subintervals.The entries of the matrix $B_{k,r} := B_k B_{k+1} \\cdots B_{k+r}$ are $B_{k,r}(i,j) = \\sum _{n=0}^{r_{k,k+r+1}(j) - 1} 1_{I_{k,i}} \\Big ( (T|I_k)^n I_{k+r+1,j} \\Big )$ and they count the number of visits of $I_{k+r+1,j}$ to $I_{k,i}$ under $T|I_k$ before it returns to $I_{k+r+1}$ .",
"Therefore $|\\!|B_{k,r} e_j |\\!|_1 = B_{k,r}(1,j) + \\cdots + B_{k,r}(b,j) = r_{k,k+r+1}(j)$ for all $1 \\le j \\le b$ .",
"Our proof of Theorem REF relies on the following facts.",
"Fact F.3 There is a constant $D_1 > 0$ such that $\\frac{1}{D_1} < \\frac{r_{k,l}(i)}{r_{k,l}(j)} < D_1$ for all $k > l\\ge 0$ and all $1 \\le i,j \\le b$ .",
"Fix $x \\in I_l$ .",
"First note that $\\min \\lbrace r_{m,k}(z) : z \\in I_m \\rbrace r_{k,l}(x) \\le r_{m,l}(x) \\le \\max \\lbrace r_{m,k}(z) : z \\in I_m \\rbrace r_{k,l}(x)$ for all $l > k > m$ because each step in the $T|I_k$ orbit of $x$ involves a return of some point in $I_k$ to $I_k$ under $T|I_m$ .",
"Now $c_2 \\le r_{0,l}(x) |I_l| \\le c_1$ for all $l$ by linear recurrence so $\\frac{c_2/|I_l|}{c_1/|I_k|} \\le \\frac{r_{0,l}(x)}{\\max \\lbrace r_{0,k}(z) : z \\in I_0 \\rbrace } \\le \\frac{r_{0,l}(x)}{\\min \\lbrace r_{0,k}(z) : z \\in I_0 \\rbrace } \\le \\frac{c_1/|I_l|}{c_2/|I_k|}$ for all $k$ and $l$ .",
"Taking $m = 0$ in (REF ) and combining with (REF ) shows that $D_1 = \\dfrac{c_1^2}{c_2^2}$ works.",
"Fact F.7 There is a constant $D_2 > 0$ such that $\\frac{1}{D_2} < \\frac{|I_{k,j}|}{|I_{k,i}|} < D_2$ for all $1 \\le i,j \\le b$ and all $k \\in \\mathbb {N}$ .",
"Linear recurrence implies that the discontinuities of $T^{r_{0,k}(j)}$ are $c_1/r_{0,k}(j)$ dense and $c_2/r_{0,k}(j)$ separated.",
"Since $T|I_k$ is continuous on the interior of $I_{k,j}$ and has discontinuities at its endpoints we must have $\\frac{c_2}{\\max \\lbrace r_{0,k}(j) : 1 \\le j \\le b \\rbrace } \\le |I_{k,j}| \\le \\frac{c_1}{\\min \\lbrace r_{0,k}(j) : 1 \\le j \\le b \\rbrace }$ so Fact REF implies $D_2$ can be chosen to be $c_1 D_1/c_2$ .",
"Fact F.8 There are constants $\\rho _1, \\rho _2 > 1$ such that $\\rho _1 |I_{k,j}| \\le |I_k| \\le \\rho _2 |I_{k,j}|$ for all $k \\in \\mathbb {N}$ and all $1 \\le j \\le b$ .",
"In particular $\\rho _1 |I_{k+1}| \\le |I_k| \\le \\rho _2 |I_{k+1}|$ for all $k \\in \\mathbb {N}$ .",
"We have $|I_k|=|I_{k,1}| + |I_{k,2}| + \\cdots + |I_{k,b}|\\le |I_{k,j}| + (b-1) D_2 |I_{k,j}|=(1 + (b-1)D_2) |I_{k,j}|$ and $|I_k|=|I_{k,1}| + |I_{k,2}| + \\cdots + |I_{k,b}|\\ge |I_{k,j}| + \\frac{b-1}{D_2} |I_{k,j}|=\\left( 1 + \\frac{b-1}{D_2} \\right) |I_{k,j}|$ by Fact REF .",
"Applying this when $j=1$ we can take $\\rho _1 = 1 + (b-1)/D_2$ and $\\rho _2 = 1 + (b-1)D_2$ .",
"Fact F.10 There is a constant $D_3 > 0$ such that $|\\!|B_k |\\!|_{1} \\le D_3$ for all $k$ .",
"We have $B_k(i,j)\\le r_{k,k+1}(j)\\le \\frac{r_{0,k+1}(j)}{\\min \\lbrace r_{0,k}(l) : 1 \\le l \\le b \\rbrace }\\le \\frac{c_1}{c_2} \\frac{|I_{k}|}{|I_{k+1}|}\\le \\frac{c_1}{c_2} \\rho _2$ by taking $m = 0$ and $l = k+1$ in (REF ) and then using (REF ) and Fact REF .",
"It then follows that $|\\!|B_k |\\!|_{1} \\le b c_1 \\rho _2 / c_2$ for all $k$ .",
"Fact F.11 There is $r \\in \\mathbb {N}$ such that $B_k B_{k+1} \\cdots B_{k+r}$ is positive for all $k \\in \\mathbb {N}$ .",
"We must produce $r \\in \\mathbb {N}$ such that, for every $i,j,k$ the $T|I_k$ orbit of $I_{k+r+1,j}$ visits $I_{k,i}$ before returning to $I_{k+r+1}$ .",
"It is enough to find $r$ such that, for every $i,j,k$ the $T$ orbit of any point $x \\in I_{k+r+1,j}$ visits $I_{k,i}$ before time $r_{0,k+r+1}(j)$ .",
"By linear recurrence this is the case if $r_{0,k+r+1}(j) |I_{k,i}| \\ge c_1$ .",
"Using Fact REF repeatedly and then (REF ) we have $r_{0,k+r+1}(j) |I_{k,i}|\\ge r_{0,k+r+1}(j) \\frac{|I_k|}{\\rho _2}\\ge r_{0,k+r+1}(j) |I_{k+r+1}| \\frac{\\rho _1^{r+1}}{\\rho _2}\\ge c_2 \\frac{\\rho _1^{r+1}}{\\rho _2}$ and, independent of $i,j,k$ , this will be at least $c_1$ if $r$ is large enough.",
"Fact F.12 There are constants $D_4 \\in \\mathbb {R}$ and $\\gamma < 1$ such that $\\Theta (B_{k,r} e_j, B_{k,r} e_\\ell ) < D_4 \\gamma ^r$ for all $j,\\ell ,k,r$ where $\\Theta $ denotes the angle between two vectors in $\\mathbb {R}^b$ and $e_1,\\dots ,e_b$ is the standard basis of $\\mathbb {R}^b$ .",
"Because positive matrices of a fixed size act as definite contractions in the Hilbert projective metric, Fact REF implies that there exists $\\rho <1$ so that $\\Theta (B_{k,k+r}v,B_{k,k+r}w)<\\rho \\Theta (v,w)$ for any $v,r \\in \\mathbb {R}_+^b$ .",
"Iterating we have $\\Theta (B_{k,k+nr}v,B_{k,k+nr}w)<\\rho ^n$ .",
"Letting $\\gamma =\\rho ^{\\frac{1}{r}}$ and choosing $D_4=\\rho ^{-1}$ we obtain the fact.",
"We use these facts to prove the following lemmas.",
"Lemma F.13 There exists $0<\\zeta _2$ and $E_2$ such that $\\left| \\sum _{n=0}^{N-1} 1_{I_k}(T^nx)- 1_{I_k}(T^n y) \\right|\\le E_2\\max \\lbrace 1,(|I_k|N)^{\\zeta _2}\\rbrace $ for all $x,y \\in [0,1)$ and all $k,N \\in \\mathbb {N}$ .",
"Let $r=\\frac{1}{2} c\\log (N|I_k|)$ .",
"We first consider $x \\in I_{k+r}$ .",
"Let $M$ be the maximal number so that $M < N$ and $T^Mx=(T|I_{k+r})^ax \\in I_{k+r}$ .",
"Put $\\upsilon = \\sum _{i=0}^M 1_{I_k} (T^ix) = |C_{i_1}(B_k \\cdots B_{k+r})|+ \\cdots +|C_{i_{a-1}}(B_k \\cdots B_{k+r})|$ where $i_j$ is defined by $(T|I_{k+r})^jx\\in I_{k+r,i_j}$ .",
"By Fact REF $\\left| \\frac{C_j(B_k \\cdots B_{k+r})}{|C_j(B_k \\cdots B_{k+r})|} - \\frac{\\upsilon }{M} \\right|$ is exponentially small (in $\\log (N|I_k|)$ ) for each $j$ .",
"Now $\\upsilon \\le \\sum _{i=0}^{N-1} 1_{I_k}(T^ix)\\le |C_{\\max }(B_k \\cdots B_{k+r})| + \\upsilon $ and by our choice of $r$ we have that $|C_{\\max }(B_k \\dots B_{k+r})|/M$ is exponentially small in $\\log (N|I_k|)$ so $\\left| \\frac{1}{N}\\sum _{i=0}^{N-1} 1_{I_k}(T^ix)-\\frac{C_j(B_k \\cdots B_{k+r})}{|C_j(B_k \\cdots B_{k+r})|} \\right|$ is exponentially small, establishing the lemma for $x \\in I_{k+r}$ .",
"A general $x \\in [0,1)$ gives another error of at most $|C_{\\max }(B_k \\cdots B_{k+r})|$ .",
"The proof of the next lemma is similar and omitted.",
"Lemma F.14 There is $0 < \\zeta _3 < 1$ and $E_3 > 0$ such that $\\left| \\sum _{n=0}^{N-1} 1_{I_{k,j}} (T^n x) - 1_{I_{k,j}} (T^n y) \\right| < E_3 \\max \\lbrace 1, ( |I_{k,j}| N )^{\\zeta _3} \\rbrace $ for all $k,j,N,x,y$ .",
"Corollary F.15 There is $0 < \\zeta _3 < 1$ and $E_3 > 0$ such that $\\left| \\sum _{n=0}^{N-1} 1_{T^s I_{k,j}} (T^n x) - 1_{T^s I_{k,j}} (T^n y) \\right| < E_3 \\max \\lbrace 1, ( |I_{k,j}| N )^{\\zeta _3} \\rbrace $ for all $k,j,N,x,y,s$ .",
"This is immediate from Lemma REF because $x$ and $y$ therein can be any point.",
"With these lemmas we can prove Theorem REF .",
"[Proof of Theorem REF ] Fix a mean-zero step function (REF ).",
"Set $a_0 = 0$ and $a_{d+1} = 1$ .",
"Put $a_i = x_1 + \\cdots + x_i$ for all $1 \\le i \\le d$ .",
"For each $k \\in \\mathbb {N}$ the partition $\\mathcal {P}_k = \\lbrace T^n I_{k,j} : 0 \\le n < r_{0,k}(j), 1 \\le j \\le b \\rbrace $ of $[0,1)$ is $|I_k|$ dense.",
"Let $\\zeta _3$ be as in Corollary REF and fix $\\frac{1}{2} + \\frac{\\zeta _3}{2} < \\gamma < 1$ .",
"Fix $N \\in \\mathbb {N}$ .",
"For each $1 \\le i \\le d+1$ choose $k$ minimal with $|I_k| \\sqrt{N} < \\sqrt{x_i}$ .",
"Therefore $\\sqrt{x_i} \\le \\rho _2 |I_k| \\sqrt{N}$ by Fact REF .",
"Write the interval $[a_{i-1},a_i)$ as a union of at most $x_i/|I_k|$ intervals from $\\mathcal {P}_k$ together with an interval of length at most $|I_k|$ at each end.",
"Fix $x,y \\in [0,1)$ .",
"By estimating the hits of $x$ and $y$ to the end intervals using linear recurrence, and by comparing hits of $x$ and $y$ to the other intervals using Corollary REF , we obtain $\\left| \\sum _{n=0}^{N-1} 1_{[a_{i-1},a_i)} (T^n x) - \\sum _{n=0}^{N-1} 1_{[a_{i-1},a_i)} (T^n y) \\right|&\\le \\frac{4}{c_2} \\sqrt{N} \\sqrt{x_i} + \\frac{x_i}{|I_k|} E_3 \\max \\lbrace 1, (|I_k| N)^{\\zeta _3} \\rbrace \\\\&\\le \\frac{4 \\sqrt{N}}{c_2}+E_3 \\rho _2 \\sqrt{N}\\max \\left\\lbrace 1, N^{\\frac{\\zeta _3}{2}} \\right\\rbrace $ for all $1 \\le i \\le d+1$ using $\\sqrt{x_i} < 1$ .",
"Combined with (REF ) gives (REF ) by our choice of $\\gamma $ ."
]
] | 1709.01575 |
[
[
"Convolutional Neural Network Based Metal Artifact Reduction in X-ray\n Computed Tomography"
],
[
"Abstract In the presence of metal implants, metal artifacts are introduced to x-ray CT images.",
"Although a large number of metal artifact reduction (MAR) methods have been proposed in the past decades, MAR is still one of the major problems in clinical x-ray CT.",
"In this work, we develop a convolutional neural network (CNN) based open MAR framework, which fuses the information from the original and corrected images to suppress artifacts.",
"The proposed approach consists two phases.",
"In the CNN training phase, we build a database consisting of metal-free, metal-inserted and pre-corrected CT images, and image patches are extracted and used for CNN training.",
"In the MAR phase, the uncorrected and pre-corrected images are used as the input of the trained CNN to generate a CNN image with reduced artifacts.",
"To further reduce the remaining artifacts, water equivalent tissues in a CNN image are set to a uniform value to yield a CNN prior, whose forward projections are used to replace the metal-affected projections, followed by the FBP reconstruction.",
"The effectiveness of the proposed method is validated on both simulated and real data.",
"Experimental results demonstrate the superior MAR capability of the proposed method to its competitors in terms of artifact suppression and preservation of anatomical structures in the vicinity of metal implants."
],
[
"Introduction",
"Patients are usually implanted with metals, such as dental fillings, hip prostheses, coiling, etc.",
"These highly attenuated metallic implants lead to severe beam hardening, photon starvation, scatter, and so on.",
"This brings strong star-shape or streak artifacts to the reconstructed CT images [1].",
"Although a large number of metal artifact reduction (MAR) methods have been proposed during the past four decades, there is still no standard solution [2], [3], [4].",
"Currently, how to reduce metal artifacts remains a challenging problem in the x-ray CT imaging field.",
"Metal artifact reduction algorithms can be generally classified into three groups: physical effects correction, interpolation in projection domain and iterative reconstruction.",
"A direct way to reduce artifacts is to correct physical effects, e.g., beam hardening [5], [6], [7] and photon starvation [8].",
"However, in the presence of high-atom number metals, errors are so strong that the aforementioned corrections cannot achieve satisfactory results.",
"Hence, the metal-affected projections are assumed as missing and replaced with surrogates [9], [10], [11].",
"Linear interpolation (LI) is a widely used MAR method, where the missing data is approximated by the linear interpolation of its neighboring unaffected projections for each projection view.",
"The LI usually introduces new artifacts and distorts structures near large metals [12].",
"By comparison, by employing a priori information, the forward projection of a prior image is usually a more accurate surrogate for the missing data [13], [14], [15], [16], [17].",
"The normalized MAR (NMAR) is a state-of-the-art prior image based MAR method, which applies a thresholding based tissue classification on the uncorrected image or the LI corrected image to remove most of the artifacts and produce a prior image [14].",
"In some cases, artifacts are so strong that some image pixels are classified into wrong tissue types, leading to inaccurate prior images and unsatisfactory results.",
"The last group of methods iteratively reconstruct images from the unaffected projections [18], [19], [20], [21] or weighted/corrected projections [22].",
"With proper regularizations, the artifacts are suppressed in the reconstructed results.",
"However, due to the high complexity of various metal materials, sizes, positions, and so on, it is hard to achieve satisfactory results for all cases using a single MAR strategy.",
"Therefore, several researchers combined two or three types of MAR techniques as hybrid methods [23] [24], fusing the merits of various MAR techniques.",
"Hence, the hybrid strategy has a great potential to obtain more robust and outstanding performance by appropriately compromising a variety of MAR approaches.",
"Recently, deep learning has achieved great successes in the image processing and pattern recognition field.",
"For example, the convolutional neural network (CNN) has been applied to medical imaging for low dose CT reconstruction and artifacts reduction [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].",
"In particular, the concept of deep learning was introduced to metal artifact reduction for the first time in 2017 [29], [35], [31], [36], [37], [38], [39].",
"Park et al.",
"employed a U-Net to correct metal-induced beam hardening in the projection domain [38] and image domain [39], respectively.",
"Their simulation studies showed promising results over hip prostheses of titanium.",
"However, the beam hardening correction based MAR methods have limited capability for artifact reduction in the presence of high-Z metal.",
"Gjesteby et al.",
"developed a few deep learning based MAR methods that refine the performance of the state-of-the-art MAR method, NMAR, with deep learning in the projection domain [29] and image domain [31], [35], respectively.",
"The CNN was used to help overcome residual errors from the NMAR.",
"While their experiments demonstrated that CNN can improve the NMAR effectively, remaining artifacts are still considerable.",
"Simultaneously, based on the CNN, we proposed a general open framework for MAR [36], [37].",
"This paper is a comprehensive extension of our previous work [36].",
"We adopt the CNN as an information fusion tool to produce a reduced-artifact image from some other methods corrected images.",
"Specifically, before the MAR, we build a MAR database to generate training data for the CNN.",
"For each clinical metal-free patient image, we simulate the metal artifacts and then obtain the corresponding corrected images by several representative MAR methods.",
"Without loss of generality, we apply a beam hardening correction (BHC) method and the linear interpolation (LI) method in this study.",
"Then, we train a CNN for MAR.",
"The uncorrected, BHC and LI corrected images are stacked as a three-channel image, which is the input data of CNN and the corresponding metal-free image is the target, and a metal artifact reduction CNN is trained.",
"In the MAR phase, the pre-corrected images are obtained using the BHC and LI methods, and these two images and the uncorrected image are put into the trained CNN to obtain the corrected CNN image.",
"To further reduce the remaining artifacts, we incorporate the strategy of prior image based methods.",
"Specifically, a tissue processing step is introduced to generate a prior from the CNN image, and the forward projection of the prior image is used to replace metal-affected projections.",
"The advantages of the proposed method are threefold.",
"First, we combine the corrected results from various MAR methods as the training data.",
"In the end-to-end CNN training, the information from different correction methods is captured and the merits of these methods are fused, leading to a higher quality image.",
"Second, the proposed method is an open framework, and all the MAR methods can be incorporated into this framework.",
"Third, this method is data driven.",
"It has a great potential to improve the CNN capability if we continue increasing the training data with more MAR methods.",
"The source codes of our proposed method are openhttps://github.com/yanbozhang007/CNN-MAR.git.",
"The rest of the paper is organized as follows.",
"Section II describes the creation of metal artifact database and the training of a convolutional neural network.",
"Section III develops the CNN based MAR method.",
"Section IV describes the experimental settings.",
"Section V gives the experimental results and analyzes properties of the proposed method.",
"Finally, Section VI discusses some relevant issues and concludes the paper."
],
[
"Training of the Convolutional Neural Network ",
"There are two phases to train a convolutional neural network for MAR.",
"First, we generate metal-free, metal-inserted and MAR corrected CT images to create a database.",
"Then, a CNN is constructed and the training data is collected from the established database and used to train the CNN."
],
[
"Establishing a Metal Artifact Database",
"At first, we need to create a CT image database for CNN training.",
"In this database, for each case, metal-free, metal-inserted, and MAR methods processed images are included.",
"In this subsection, we describe how to generate metal-free and metal-inserted CT images, where beam hardening and Poisson noise are simulated.",
"To ensure that the trained CNN works for real cases, instead of using phantoms, we simulate the metal artifacts based on clinical CT images.",
"To begin with, a number of DICOM format clinical CT images are collected from online resources and “the 2016 Low-dose CT Grand Challenge” training dataset [40].",
"In the presence of metal implants, we manually segment metals and store them as small binary images, which represent typical metallic shapes in real cases.",
"Several representative metal-free CT images are selected as benchmark images.",
"For a given benchmark image, its pixel values are converted from CT values to linear attenuation coefficients and denoted as $\\mathbf {x}$ .",
"To simulate polychromatic projection, we need to know the material components in each pixel.",
"Hence, a soft threshold-based weighting method [41] is applied to segment the image $\\mathbf {x}$ into bone and water-equivalent tissue components, denoted as $\\mathbf {x}^b$ and $\\mathbf {x}^w$ , respectively.",
"Pixels with values below a certain threshold $T_1$ are viewed as water equivalent, while pixels above a higher threshold $T_2$ are assumed to be bone.",
"The pixels with values between $T_1$ and $T_2$ are assumed to be a mixture of water and bone.",
"Thus, a weighting function for bone is introduced as $\\omega (x_i)=\\left\\lbrace \\begin{array}{ll}0, & x_i \\le T_1\\\\1, & x_i \\ge T_2\\\\\\frac{x_i - T_1}{T_2 - T_1}, & T_1 < x_i <T_2\\end{array}\\right.",
",$ where $x_i$ is the $i^{th}$ pixel value of $\\mathbf {x}$ .",
"Hence, $\\mathbf {x}^b$ and $\\mathbf {x}^w$ are expressed as $x^b_i = \\omega (x_i)x_i,$ $x^w_i = (1 - \\omega (x_i))x_i.$ Fig.",
"REF gives an example of the image segmentation.",
"Figure: Example of tissue segmentation.",
"(a) The benchmark image, (b) water-equivalent tissue and (c) bone.For an x-ray path $L_j$ , the linear integral of water and bone images are $d^w_j$ and $d^b_j$ , respectively.",
"We have $d^k_j = \\int _{L_j} \\mathbf {x}^k dl ,$ where the superscript “$k$ \" indicates “$w$ ” or “$b$ ”.",
"To simulate polychromatic projections, we need to obtain linear attenuation maps of water and bone at various energies.",
"For each material, the linear attenuation coefficient at the pixel is the product of the known energy-dependent mass attenuation coefficient and the unknown energy-independent density [42].",
"We have $x^k_i(E) = m^k(E) \\rho ^k_i ,$ where $\\rho ^k_i$ is the density of “$k$ ” material at the $i^{th}$ pixel, and $m^w(E)$ and $m^b(E)$ are respectively mass attenuation coefficients at energy $E$ of water and bone.",
"For a given polychromatic x-ray imaging system, let us assume that the equivalent monochromatic energy is $E_0$ .",
"Then, $x_i^b$ and $x_i^w$ can be written as $x^k_i = x^k_i(E_0) = m^k(E_0) \\rho ^k_i.$ Combining Eqs.",
"(REF ) and (REF ), the unknown density $\\rho ^k_i$ can be eliminated.",
"Hence, the energy dependent linear attenuation coefficient for each material is obtained as the following, $x^k_i(E) = \\frac{x^k_i m^k(E)}{m^k(E_0)} .$ For the given x-ray path $L_j$ , the ideal projection measurement $\\bar{y}_j$ recorded by the $j^{th}$ detector bin is $\\begin{array}{ll}\\bar{y}_j& = \\int I(E) \\text{exp} \\left( - \\int _{L_j} (x^w_i(E) + x^b_i(E)) dl \\right) dE\\\\& = \\int I(E) \\text{exp} \\left( - \\int _{L_j} (\\frac{x^w_i m^w(E)}{m^w(E_0)} + \\frac{x^b_i m^b(E)}{m^b(E_0)}) dl \\right) dE \\\\& = \\int I(E) \\text{exp} \\left( - \\frac{m^w(E) d^w_j}{m^w(E_0)} - \\frac{m^b(E) d^b_j}{m^b(E_0)} \\right) dE\\end{array},$ where $I(E)$ is the known energy dependence of both the incident x-ray source spectrum and the detector sensitivity.",
"Because the linear projection $d^w_j$ and $d^b_j$ have been computed in advance, computing the polychromatic projection using Eq.",
"(REF ) is very efficient.",
"Approximately, the measured data follow the Poisson distribution: $y_j \\sim \\text{Poisson} \\lbrace \\bar{y}_j + r_j \\rbrace ,$ where $r_j$ is the mean number of background events and read-out noise variance, which is assumed as a known nonnegative constant [42], [43].",
"Thus, the noisy polychromatic projection $\\mathbf {p}$ for reconstruction can be expressed as: $p_j = -\\text{ln} \\frac{y_i}{\\int I(E) dE},$ The metal-free image is reconstructed using filtered backprojection (FBP), and the image is assumed as reference and denoted as $\\mathbf {x}^{ref}$ .",
"To simulate metal artifacts, one or more binary metal shapes are placed into proper anatomical positions, generating a metal-only image $\\mathbf {x}^m $ .",
"We specify the metal material, and assign metal pixels with linear attenuation coefficient of this material at energy $E_0$ , and set the rest pixels to be zero.",
"Because metals are inserted into patients, pixel values in $\\mathbf {x}^b$ and $\\mathbf {x}^w$ are set to be zero if the corresponding pixels in $\\mathbf {x}^m$ are nonzero.",
"Then, the $d^w_j$ and $d^b_j$ are updated, and the corresponding metal projection is computed using Eq.",
"(REF ).",
"Similar to Eq.",
"(REF ), the ideal projection measurement is $\\scalebox {0.95}{\\bar{y}_j ^* = \\int I(E) \\text{exp} \\left( - \\frac{m^w(E) d^w_j}{m^w(E_0)} - \\frac{m^b(E) d^b_j}{m^b(E_0)} - \\frac{m^m(E) d^m_j}{m^m(E_0)} \\right) dE , }$ where $m^m (E)$ is the mass attenuation coefficient of the metal at energy $E$ .",
"Following the same operations in Eqs.",
"(REF ) and (REF ), the noisy polychromatic projection $\\mathbf {p}^*$ is obtained, and then the image $\\mathbf {x}^{art}$ containing artifacts is reconstructed.",
"Fig.",
"REF shows four samples in the database.",
"The top four rows in Fig.",
"REF are benchmark images, metal-only images, metal-free images and metal-inserted images, respectively.",
"Figure: Representative samples in the database.",
"Each column corresponds to one case.",
"The top four rows are benchmark images, metal-only images, metal-free and metal-inserted images, respectively.",
"The last two rows are images after metal artifact reduction using the BHC and LI, respectively."
],
[
"Simple Metal Artifact Reduction",
"We apply two simple metal artifact reduction methods, the linear interpolation (LI) and beam hardening correction (BHC) [44], to alleviate artifacts.",
"These methods are fast and easy to implement, and there are no manually selected parameters.",
"Moreover, they suppress metal artifacts with different schemes, which have a great potential to provide complementary information for the CNN.",
"In the LI method, the metal-affected projections are identified and replaced with the linear interpolation of their unaffected neighboring projections in each projection view.",
"The LI corrected image is denoted as $\\mathbf {x}^{LI} $ .",
"The BHC approach [44] adopts a first-order model of beam hardening error to compensate for the metal-affected projections.",
"The length of metal $\\lbrace l^m_j \\rbrace $ along each x-ray path is computed by forward projecting the binary metal-only image.",
"The difference $\\lbrace p^m_j \\rbrace $ between the original and LI projections is assumed as the contribution of metal.",
"The correction curve between $\\lbrace l^m_j \\rbrace $ and $\\lbrace p^m_j \\rbrace $ is fitted to the correlation using a least squares cubic spline fit.",
"Finally, the correction curve is subtracted from the original projection to yield the corrected data.",
"The image obtained using BHC is denoted as $\\mathbf {x}^{BHC}$ .",
"The two bottom rows in Fig.",
"REF are four samples of BHC and LI corrected images, where metals are not inserted back into the LI images.",
"For each sample in the database, the original uncorrected image, BHC image and LI image are combined as a three-channel image.",
"The samples in the database are randomly divided into two groups for CNN training and validation.",
"Small image patches of $s \\times t \\times 3$ are extracted from three-channel images, and these patches are assumed as the input data of CNN.",
"Correspondingly, image patches of $s \\times t$ are also obtained from the same positions of the metal-free images, and these patches are assumed as the target of CNN during training.",
"The $r^{th}$ training sample pair is denoted as $\\mathbf {u}_r \\in \\mathbb {R}^{s \\times t \\times 3 } $ and $\\mathbf {v}_r \\in \\mathbb {R}^{s \\times t } $ , $r=1, \\dots , R $ , where $R$ is the number of training samples.",
"The CNN training is to find a function $H: \\mathbb {R}^{s \\times t \\times 3 } \\rightarrow \\mathbb {R}^{s \\times t }$ that minimizes the following cost function [30]: $H = \\text{arg} \\min _H \\frac{1}{R} \\sum _{r = 1} ^R \\Vert H(\\mathbf {u}_r) - \\mathbf {v}_r \\Vert _F ^2 ,$ where $ \\Vert \\cdot \\Vert _F $ is the Frobenius norm.",
"Fig.",
"REF depicts the workflow of our CNN, which is comprised of an input layer, an output layer and $ L=5 $ convolutional layers.",
"The ReLU, a nonlinear activation function defined as $\\text{ReLU} (x) = \\text{max} (0, x)$ , is performed after each of the first $L-1$ convolutional layers.",
"In each layer, the output after convolution and ReLU can be formulated as: $C_l (\\mathbf {u}) = \\text{ReLU} (\\mathbf {W}_l * C_{l-1} (\\mathbf {u}) + \\mathbf {b}_l), l = 1, \\dots , L-1 ,$ where $*$ means convolution, $\\mathbf {W}_l$ and $\\mathbf {b}_l$ denote weights and biases in the $l^{th}$ layer, respectively.",
"We define $C_0 (\\mathbf {u}) = \\mathbf {u}$ .",
"$\\mathbf {W}_l$ can be assumed as an $n_l$ convolution kernel with a fixed size of $c_l \\times c_l$ .",
"$C_l (\\mathbf {u})$ generates new feature maps based on the $(l-1) ^ {th}$ layer’s output.",
"For the last layer, feature maps are used to generate an image that is close to the target.",
"Then, we have: $C_L (\\mathbf {u}) = \\mathbf {W}_L * C_{L-1} (\\mathbf {u}) + \\mathbf {b}_L .$ After the construction of the CNN, the parameter set $\\Theta = \\lbrace \\mathbf {W}_1, \\cdots , \\mathbf {W}_L, \\mathbf {b}_1, \\cdots , \\mathbf {b}_L \\rbrace $ is updated during the training.",
"The estimation of the parameters can be obtained by minimizing the following loss function: $Loss(\\mathbf {U}, \\mathbf {V}, \\Theta ) = \\frac{1}{R} \\sum _{r = 1} ^R \\Vert C_L(\\mathbf {u}_r) - \\mathbf {v}_r \\Vert _F ^2 ,$ where $\\mathbf {U} = \\lbrace \\mathbf {u}_1, \\cdots , \\mathbf {u}_R \\rbrace $ and $\\mathbf {V} = \\lbrace \\mathbf {v}_1, \\cdots , \\mathbf {v}_R \\rbrace $ are the input and target datasets, respectively.",
"Figure: Architecture of the convolutional neural network for metal artifact reduction."
],
[
"CNN-MAR METHOD",
"Because the proposed MAR approach is based on the CNN, it is referred to as CNN-MAR method.",
"It consists of five steps: (1) metal trace segmentation; (2) artifact reduction with the LI and BHC; (3) artifact reduction with the trained CNN; (4) generation of a CNN prior image using tissue processing; (5) replacement of metal-affected projections with the forward projection of CNN prior, followed by the FBP reconstruction.",
"The workflow of CNN-MAR is summarized in Fig.",
"REF .",
"Steps 1 and 5 are the same as our previous work [24], and step 2 has been described in the above subsection.",
"Hence, we only provide the details for the key steps 3 and 4 as follows.",
"Figure: Flowchart of the proposed CNN-MAR method."
],
[
"CNN Processing",
"After the BHC and LI corrections, the original uncorrected image $\\mathbf {x}^{art}$ , BHC image $\\mathbf {x}^{BHC}$ and LI image $\\mathbf {x}^{LI}$ are combined as a three-channel image $\\mathbf {x}^{input}$ .",
"Hence, the image after CNN processing is $\\mathbf {x}^{CNN} = C_L(\\mathbf {x}^{input}) ,$ where the parameters in $C_L$ have been obtained in advance in the CNN training phase.",
"Fig.",
"REF shows an example of the CNN inputs and processed CNN image.",
"All the three input images contain obvious artifacts, as indicated by the arrows 1-3.",
"Although the LI alleviates the artifacts indicated by the arrow 1, it introduces new artifacts indicated by the arrow 4.",
"In the CNN image, the artifacts are remarkably suppressed .",
"Figure: Illustration of the CNN image and CNN prior."
],
[
"Tissue Processing",
"Although the metal artifacts are significantly reduced after the CNN processing, the remaining artifacts are still considerable.",
"We generate a prior image from the CNN image by the proposed tissue processing approach.",
"Because the water equivalent tissues have similar attenuations and are accounted for a dominate proportion in a patient, we assign these pixels with a uniform value to remove most of the artifacts and obtain a CNN prior image.",
"By the k-means clustering on the CNN image, two thresholds are automatically determined and the CNN image is segmented into bone, water and air.",
"To avoid wrong clustering in the case of only a few bone pixels, the bone-water threshold is not less than 350 HU.",
"Additionally, to preserve low-attenuated bones, larger regions are segmented with half of the bone-water threshold, and those regions overlapped with the previously obtained bony regions are also assumed as bone and preserved.",
"Then, we obtain a binary image $\\mathbf {B}$ for water regions with the target pixels setting to be one and the rest setting to be zero.",
"Because it may cause discontinuities at boundary and produce fake edges/structures to directly set all water regions with a constant value [13], [24], we introduce an $N=5$ pixel transition between water and other tissues.",
"Based on the binary image $\\mathbf {B}$ , we introduce a distance image $\\mathbf {D}$ , where the pixel value is set to be the distance between this pixel and its nearest zero pixel if the distance is not greater than $N$ , and is set to be $N$ if it is greater than $N$ .",
"Hence, in the image $\\mathbf {D}$ , most of the water pixels are with the value $N$ , and there is an $N$ pixel transition region, while the other tissues are still zeros.",
"We compute the weighted average of water pixel values: $\\bar{x} ^{CNN, w} = \\frac{\\sum _i {D_i x^{CNN}_i}}{\\sum _i {D_i } } ,$ Thus, the prior image is obtained: $x^{prior}_i = \\frac{D_i}{N} \\bar{x} ^{CNN, w} + (1 - \\frac{D_i}{N}) x^{CNN}_i .$ Finally, to avoid the potential discontinuities at boundaries of metals, the metal pixels are replaced with their nearest pixel values.",
"Fig.",
"REF (f) shows an example of the CNN prior image after the tissue processing.",
"It is clear that the regions of water equivalent tissue are flat and the artifacts are removed.",
"Simultaneously, the bony structures are persevered very well.",
"The CNN prior is beneficial for the projection interpolation.",
"As shown in Fig.",
"REF , the LI is a poor estimation of the missing projections.",
"With the help of forward projection of the CNN prior, the surrogate sinogram is extremely close to the ideal one.",
"Figure: Comparison of sinogram completion.",
"An ROI is enlarged and displayed with a narrower window."
],
[
"Creating a Metal Artifact Database",
"74 metal-free CT images and 15 metal shapes are collected.",
"Various metal implants are simulated, such as dental fillings, spine fixation screws, hip prostheses, coiling, wires, etc.",
"The metal materials include titanium, iron, copper and gold.",
"We carefully adjust the sizes, angles, positions and inserted metal materials so that the simulations are close to clinical cases.",
"In this work, a database is created with 100 cases.",
"To segment water and bone from a benchmark image, thresholds $T_1$ and $T_2$ are set to linear attenuation coefficients corresponding to 100 HU and 1500 HU, respectively.",
"Mass attenuation coefficients of water, bone and metals are obtained for the XCOM database [45].",
"To simulate metal-free and metal-inserted data, an equi-angular fan-beam geometry is assumed.",
"A 120 kVp x-ray source is simulated and each detector bin is expected to receive $2 \\times 10^7$ photons in the case of blank scan [46].",
"There are 984 projection views over a rotation and 920 detector bins in a row.",
"The distance between the x-ray source and the rotation center is 59.5 cm.",
"The metal-free and metal-inserted images are reconstructed by the FBP from simulated sinograms and each image consists of $512 \\times 512$ pixels."
],
[
"CNN Training",
"In Fig.REF , the convolutional kernel is $3 \\times 3$ in each layer.",
"Therefore, the convolutional weights are $3 \\times 3 \\times 3 \\times 32$ in the first layer, $3 \\times 3 \\times 32 \\times 32$ in the second to the fourth layers and $3 \\times 3 \\times 32 \\times 1$ in the last layer.",
"We set the padding to 1 in each layer so that the size of the output image is the same as the input.",
"To train the CNN, images are selected from the database to generate the training data.",
"10,000 patch samples with the size of $64 \\times 64$ are extracted from the selected images.",
"Because the spatial distribution of metal artifacts in an image is not uniform, we design a specific strategy to select training patches.",
"A major proportion of the total training data are those patches with strongest artifacts in each corrected image, and the rest patches are randomly selected.",
"The trained neural networks are very similar with different proportions between 50% to 80%.",
"The obtained training data are randomly divided into two groups.",
"80% of the data is used for training and the rest is for validation during the CNN training.",
"The CNN is implemented in Matlab with the MatConvNet toolbox [47], [48].",
"A GeForce GTX 970 GPU is used for acceleration.",
"The training code runs about 25.5 hours and stops after 2000 iterations."
],
[
"Numerical Simulation",
"Three typical metal artifacts cases are selected from the database to evaluate the usefulness of the proposed method.",
"They are: case 1, two hip prostheses; case 2, two fixation screws and a round metal inserted in bone; case 3, several dental fillings.",
"These cases are not used in the CNN training.",
"The proposed method is compared to the BHC, LI and a famous prior image based method NMAR [14].",
"In the NMAR, a prior image is generated from an original image in the case of smaller metal objects of medium density and from an LI image in the case of strong artifacts.",
"For a comprehensive comparison, we generate prior images from both of the original and LI images for the NMAR, which are referred as to NMAR1 and NMAR2 in this paper, respectively.",
"For a quantitative evaluation, we use the metal-free images as references to compute the root mean square error (RMSE) and the structural similarity (SSIM) index [49]."
],
[
"Real Data",
"The effectiveness of the proposed method is also validated over a clinical data.",
"A patient with a surgical clip is scanned on a Siemens SOMATOM Sensation 16 CT scanner with 120 kVp and 496 mAs using the helical scanning geometry [50].",
"The measurement was acquired with 1160 projection views over a rotation and 672 detector bins in a row.",
"The FOV is 25 cm in radius and distance from the x-ray source to the rotation center is 57 cm."
],
[
"Numerical Simulation",
"Fig.",
"REF shows the reference, uncorrected and corrected images of the bilateral hip prostheses case.",
"The corresponding prior images for the NMAR1, NMAR2 and CNN-MAR are given in Fig.",
"REF .",
"A severe dark strip presents between two hip prostheses in the original image as indicated by the arrow “1”.",
"Although the BHC alleviates these artifacts to some extent, the remaining artifacts are still remarkable.",
"The NMAR1 corrected image also contains strong dark strip in the same location, which is due to its poor prior image.",
"The NMAR method adopts a simple thresholding to segment air, water equivalent tissue, and bone after the image is smoothed with a Gaussian filter [14].",
"Then, air and water regions are set to -1000 HU and 0 HU, respectively.",
"Because of the severe artifacts in the original image, several regions are segmented as wrong tissue types.",
"The NMAR1 prior presents false structures as indicated by the arrows “1” and “2” in Fig.",
"REF (a).",
"The false structural information is propagated to the NMAR1 corrected image.",
"The LI corrected image has moderate artifacts compared to the aforementioned methods.",
"However, the bony structures near the metals, as highlighted in the magnified ROI, are blurred and distorted.",
"This is due to the significant information loss near a large metal.",
"As a result, the NMAR2 prior does not suffer from the wrong segmentation but an inaccurate bony structure as indicated by the arrow “3” in Fig.",
"REF (b).",
"Hence, the NMAR2 corrected images reduce artifacts well and introduce wrong bony structures.",
"By comparison, the CNN image captures tissue structures faithfully from the original, BHC and LI images, and avoids most of the artifacts.",
"Due to the excellent image quality of the CNN image, a good CNN prior is generated, followed by a CNN-MAR image with superior image quality.",
"It is clearly seen from Fig.",
"REF (h) that the artifacts are almost removed completely and the tissue features in the vicinity of metals are faithfully preserved.",
"Fig.",
"REF presents the case 2, where two fixation screws and a metal are inserted in the shoulder blade.",
"The metal artifacts in the original image are moderate, and the BHC is able to remove the bright artifacts (arrow “2”) around the metals and recovered some bony structures.",
"On the contrary, the LI introduces many new artifacts, and most of the bony structures near the metals are lost as indicated by the arrow “1”.",
"Both the NMAR1 and NMAR2 are not able to obtain satisfactory results because it can hardly get a good prior image from the original or LI corrected images.",
"The CNN image restores most of the bony features near the metals, and no new artifacts are introduced.",
"Consequently, the CNN-MAR corrected image is very close to the reference.",
"Fig.",
"REF shows the dental images with multiple dental fillings.",
"The original, BHC, LI and NMAR1 images suffer from severe artifacts, and the NMAR2 has less artifacts.",
"Although none of Figs.",
"REF (b)-REF (d) has a good image quality, the CNN demonstrates an outstanding capacity to preserve the tissue features and avoid most of the strong artifacts simultaneously.",
"Consistent with the previous cases, the CNN-MAR achieves the best image quality.",
"Figure: Case 1: bilateral hip prostheses.",
"(a) is the reference image, (b) is the original uncorrected image, and (c)-(h) are the corrected results by the BHC, LI, NMAR1, NMAR2, CNN and CNN-MAR, respectively.",
"The ROI highlighted by the small square is magnified.",
"The display window is [-400 400] HU.Figure: The prior images for NMAR1, NMAR2 and CNN-MAR in Fig.",
".Figure: Same as Fig.",
"but for case 2: two fixation screws and a metal inserted in the shoulder blade.",
"The display window is [-360 310] HU.Figure: Same as Fig.",
"but for case 3: four dental fillings.",
"The display window is [-1000 1400] HU.Table I lists the RMSEs of the original and corrected images with respect to the reference images, where the metallic pixels are excluded.",
"Because the noise also contributes to the RMSE, the artifact induced error is slightly smaller than the values listed in the table.",
"The BHC, LI and NMAR1 have overall large error.",
"In comparison, the NMAR2 achieves a higher accuracy.",
"The CNN images have comparable accuracy to the NMAR2, and the CNN-MAR achieves the smallest RMSEs for all these three cases.",
"Because the SSIM measures the structural similarity between two images, it is good to evaluate the strength of artifacts [49].",
"The SSIM index lies between 0 and 1, and a higher value means better image quality.",
"Table II lists the SSIM of each image in the numerical simulation study.",
"The BHC has comparable SSIM indices to those of the uncorrected images.",
"The other five MAR methods increase the SSIM significantly.",
"For the LI, NMAR1, NMAR2 and CNN, their ranks are case-dependent.",
"Generally speaking, the NMAR2 and CNN have better image quality.",
"By comparison, the CNN-MAR has the highest SSIM for the three cases, implying its superior and robust artifact reduction capability.",
"Table: RMSE of each image in the numerical simulation study.",
"(Unit: HU).Table: SSIM of each image in the numerical simulation study."
],
[
"Clinical Application",
"Fig.",
"REF shows a patient's head CT image with a surgical clip.",
"The patient is a 59 year-old female with diffused subarachnoid hemorrhage in the basal cisterns and sylvian fissures.",
"The CT angiography demonstrates a left middle cerebral artery aneurysm.",
"She is taken to the operation room and the aneurysm is clipped.",
"She has numerous head CT scans after the surgery for assessment of increased intracranial pressure to rule out rebleeding and hydrocephalus [50].",
"The original, BHC and LI images contain too strong artifacts to provide bleeding information in her brain.",
"The NMAR1 and NMAR2 are able to better alleviate artifacts.",
"However, there still exists obvious artifacts in the images as indicated by the arrow “1”, and bony structures indicated by the arrow “2” are distorted.",
"In comparison, the CNN-MAR achieves the best image quality.",
"As highlighted in the rectangular region, there is only one tiny dark streak, and the bright hemorrhage can be observed clearly.",
"The CNN-MAR demonstrates a superior metal artifact reduction and the potential for diagnostic tasks after the clipping surgery.",
"Figure: The head CT image with a surgical clip.",
"(a) is the original uncorrected image, and (b)-(f) are the corrected results by the BHC, LI, NMAR1, NMAR2 and CNN-MAR, respectively.",
"The display window is [-100 200] HU."
],
[
"Effectiveness of the Tissue Processing",
"To study the effectiveness of the tissue processing, we ignore the tissue processing step and directly assume the CNN images as the prior images.",
"The corresponding corrected images are shown in Fig.",
"REF .",
"Compared to the CNN images in Figs.",
"REF , REF and REF , some artifacts can be alleviated by the forward projection.",
"Nevertheless, most of the streaks that are tangent to the metals are preserved as indicated by the arrows in Fig.",
"REF .",
"By comparison, the tissue processing keeps the major structures and removes the low-contrast features and remaining artifacts.",
"Although the features in the regions of water equivalent tissues are lost after the tissue processing, because the metal-affected projections account for a very small proportion in the sinogram, the missing information is able to be partially recovered from the rest of the unaffected projections.",
"In addition, in the projection replacement step, a projection transition is applied to compensate for the difference between the prior sinogram and the measurements at the boundary of the metal traces [24], which is also beneficial to the information recovery.",
"However, in the presence of large metals, a low-contrast feature in the vicinity of metal may suffer from missing or distortion.",
"Figure: Results obtained by directly adopting a CNN image as the prior image without the tissue processing step.",
"(a)-(c) corresponds to the cases 1-3, respectively."
],
[
"Selection of Input Images (MAR Methods)",
"In this work, the original uncorrected, BHC and LI images are adopted as the input of CNN.",
"We also compare the results with various input images (MAR methods).",
"Here, we apply the original uncorrected, BHC, LI, NMAR1 and NMAR2 images as a five-channel input image, and adopt the original and LI images as a two-channel input image.",
"In addition, the NMAR2 images is employed as a one-channel input image.",
"Fig.",
"REF shows the results of dental fillings case.",
"When NMAR2 is selected as the single input image, the CNN processing is equivalent to the NMAR-CNN method proposed by Gjesteby et al.",
"[31], [35].",
"Because the NMAR2 image has less artifacts, the CNN image and CNN-MAR image have better image quality.",
"Regarding the multi-channel input, it can be seen from the top three rows that the performance of artifacts reduction is improved by introducing more input images.",
"Particularly, compared to the cases of two-channel input images, three-channel input images remarkably improve the image quality.",
"Therefore, introducing the NMAR1 and NMAR2 only brings limited benefits.",
"This effect depends on if the newly introduced input images contain new useful information.",
"As the aforementioned, the BHC and LI belong to different MAR strategies, which provide complementary information.",
"Without the BHC, some artifacts are wrongly classified as tissue structures and preserved, as illustrated in the third row of Fig.",
"REF .",
"On the contrary, the NMAR1 and NMAR2 are obtained based on prior images from the original and LI images, respectively.",
"Hence, they provide limited new information.",
"In summary, as an open MAR framework, the performance of CNN-MAR can be further improved in the near future by incorporating various types of MAR algorithms.",
"Figure: CNN and CNN-MAR results based on different channels of input images.",
"Five-channel: original, BHC, LI, NMAR1 and NMAR2 images.",
"Three-channel (default): original, BHC and LI images.",
"Two-channel: original and LI images.",
"One-channel: NMAR2 image."
],
[
"Architecture of the CNN",
"To study the performance of CNN with respect to different architectures, we adjust the CNN parameters and calculate the average RMSE and SSIM over ten simulated metal artifact cases including the aforementioned three cases.",
"Fig.",
"REF shows the values of RMSE and SSIM using the networks with different number of convolutional layers, number of filters per layer, and the size of each filter.",
"In each subplot, there is only one parameter to be tuned and other parameters are kept as the default ones.",
"The number of neurons in the network increases with the increase of these three parameters, obtaining slightly smaller RMSE and greater SSIM indices.",
"However, because the computational cost rises considerably by using greater parameters, we employ a medium size CNN in this work.",
"Figure: Average RMSE and SSIM of CNN images and CNN-MAR images with respect to various CNN architecture parameters: (a) number of convolutional layers, (b) number of filters/features in each layer and (c) the size of each filter.",
"The default CNN has 5 convolutional layers, 32 filters per layer, and each filter is with the size of 3 ×\\times 3."
],
[
"Training Data",
"We compare the network trained with different numbers of patches.",
"Fig.",
"REF presents the average RMSE and SSIM values over ten cases of our results using the network trained with 100, 500, 2000 and 10000 patches.",
"It is clear that the RMSE decreases and the SSIM increases dramatically by applying more training data.",
"This suggests that the performance of the proposed method strongly depends on the size of training data.",
"Figure: Average RMSE and SSIM values using the CNN trained with different data size.We also compare selection strategies for the training data.",
"The convergence curves of CNN training are presented in Fig.",
"REF (a), and the obtained network after 2000 training epochs is used in this work.",
"It can be observed that the energy of the objective function decreases steadily with the increasing training epoch.",
"In Fig.",
"REF (a), the training and validation data are selected from a subset of the same dataset, which consists of all types of inserted metals.",
"It is clear that the trained CNN works well on the validation data.",
"In Fig.",
"REF (b), the training data and the validation data are selected from the same subset.",
"While the training data is from all types of metals except the multiple dental fillings, and the validation data is from the multiple dental fillings cases.",
"The two separated curves demonstrate an unsatisfactory performance of the obtained CNN on the validation data caused by the difference of the artifact patterns in the two data sets.",
"Hence, it is crucial to include a wider variety of metal artifacts cases as the training data.",
"Figure: The convergence curves of CNN training in terms of energy of loss function versus training epochs.",
"(a) Training data and validation data are selected from the same dataset.",
"(b) Training data and validation data are from different cases in the dataset."
],
[
"Training Epochs",
"The proposed method is tested with different training epochs.",
"Fig.",
"REF compares average RMSE and SSIM values over ten cases of the CNN and CNN-MAR images obtained with the network after 100, 200, 1000 and 2000 training epochs.",
"Obviously, by increasing the training epochs, the RMSE of CNN images decreases steadily and the SSIM increases constantly.",
"After the tissue processing, the image quality of CNN-MAR images is remarkably improved.",
"Likewise, the RMSE and SSIM of CNN-MAR images with respective to training epochs follows the same trend to those of CNN images.",
"Figure: Average RMSE and SSIM values using the CNN trained after different epochs."
],
[
"Discussion And Conclusion",
"From the aforementioned experimental results, it can be seen that the CNN and tissue processing are two mutual beneficial steps.",
"For the CNN step, its strength is to fuse useful information from different sources to avoid strong artifacts.",
"Its drawback is that the CNN can hardly remove all artifacts and mild artifacts typically remain.",
"As to the tissue processing, similar to other prior image based MAR methods, it can remove moderate artifacts and generate a satisfactory prior image.",
"However, in the presence of severe artifacts, the prior image usually suffers from misclassification of tissues.",
"By incorporating the CNN and tissue processing, the CNN training can stop with fewer epochs, and the obtained CNN prior is not affected by tissue misclassification.",
"Their strengths are complementary.",
"The key factors to ensure outstanding performance of the CNN-MAR are twofold: selection of the appropriate MAR methods and preparation of the training data.",
"The former factor provides sufficient information for the CNN to distinguish tissue structures from the artifacts.",
"The later ensures the generality of the trained CNN by involving as many varieties of metal artifacts cases as possible.",
"The forward projection of metal identifies which project data is affected.",
"For data correction/estimation based metal artifact reduction methods, including the proposed method, the performance of artifact reduction may be compromised in the case of inaccurate metal segmentation [51].",
"Fortunately, a few advanced metal segmentation schemes have been reported [50], [52], which can be directly applied to the proposed method.",
"Moreover, the deep learning strategy has been widely used for image segmentation [53].",
"Study of applying the neural network for the robust metal segmentation is planned for our future work.",
"Although the proposed CNN-MAR in this paper works on 2D image slices, it can be directly extended to 3D volumetric images.",
"Along the new dimension, due to different spatial distribution patterns of tissue structures and artifacts, the 3D version may achieve superior performance.",
"Meanwhile, 3D data will require more training time.",
"In conclusion, we have proposed a convolutional neural network based metal artifact reduction (CNN-MAR) framework.",
"It is an open artifact reduction framework that is able to distinguish tissue structures from artifacts and fuse the meaningful information to yield a CNN image.",
"By applying the designed tissue processing technique, a good prior is generated to further suppress artifacts.",
"Both numerical simulations and clinical application have demonstrated that the CNN-MAR can significantly reduce metal artifacts and restore fine structures near the metals to a large extent.",
"In the future, we will increase the training data and involve more MAR methods in the CNN-MAR framework to improve its capability.",
"From a broader aspect, the proposed framework has a great potential for other artifacts reduction problems in the biomedical imaging and industrial applications."
],
[
"Acknowledgment",
"The authors are grateful to Dr. Ying Chu at Shenzhen University for deep discussions and constructive suggestions.",
"The authors also appreciate Mr. Robert D. MacDougall's proofreading.",
"Part of the benchmark images in our database are obtained from “the AAPM 2016 Low-dose CT Grand Challenge”."
]
] | 1709.01581 |
[
[
"Probabilistic Rule Realization and Selection"
],
[
"Abstract Abstraction and realization are bilateral processes that are key in deriving intelligence and creativity.",
"In many domains, the two processes are approached through rules: high-level principles that reveal invariances within similar yet diverse examples.",
"Under a probabilistic setting for discrete input spaces, we focus on the rule realization problem which generates input sample distributions that follow the given rules.",
"More ambitiously, we go beyond a mechanical realization that takes whatever is given, but instead ask for proactively selecting reasonable rules to realize.",
"This goal is demanding in practice, since the initial rule set may not always be consistent and thus intelligent compromises are needed.",
"We formulate both rule realization and selection as two strongly connected components within a single and symmetric bi-convex problem, and derive an efficient algorithm that works at large scale.",
"Taking music compositional rules as the main example throughout the paper, we demonstrate our model's efficiency in not only music realization (composition) but also music interpretation and understanding (analysis)."
],
[
"=1 [pages=1-last]rulerealizeselectcr.pdf [pages=1-last]supplementarycr.pdf"
]
] | 1709.01674 |
[
[
"Existence and stabilization results for a singular parabolic equation\n involving the fractional Laplacian"
],
[
"Abstract In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \\begin{equation*} \\quad (P_{t}^s) \\left\\{ \\begin{split} \\quad u_t + (-\\Delta)^s u &= u^{-q} + f(x,u), \\;u >0\\; \\text{in}\\; (0,T) \\times \\Omega, u &= 0 \\; \\mbox{in}\\; (0,T) \\times (\\mb R^n \\setminus\\Omega), \\quad \\quad \\quad \\quad u(0,x)&=u_0(x) \\; \\mbox{in} \\; {\\mb R^n}, \\end{split} \\quad \\right.",
"\\end{equation*} where $\\Omega$ is a bounded domain in $\\mb{R}^n$ with smooth boundary $\\partial \\Omega$, $n> 2s, \\;s \\in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \\in L^\\infty(\\Omega)\\cap X_0(\\Omega)$ and $T>0$.",
"We suppose that the map $(x,y)\\in \\Omega \\times \\mb R^+ \\mapsto f(x,y)$ is a bounded below Carath\\'eodary function, locally Lipschitz with respect to second variable and uniformly for $x \\in \\Omega$ it satisfies \\begin{equation}\\label{cond_on_f} { \\limsup_{y \\to +\\infty} \\frac{f(x,y)}{y}<\\lambda_1^s(\\Omega)}, \\end{equation} where $\\la_1^s(\\Omega)$ is the first eigenvalue of $(-\\Delta)^s$ in $\\Omega$ with homogeneous Dirichlet boundary condition in $\\mathbb{R}^n \\setminus \\Omega$.",
"We prove the existence and uniqueness of weak solution to $(P_t^s)$ on assuming $u_0$ satisfies an appropriate cone condition.",
"We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results.",
"We also show additional regularity on the solution of $(P_t^s)$ when we regularize our initial function $u_0$."
],
[
"Introduction",
"In this paper, we study the existence and uniqueness of weak solution for the following fractional parabolic equation with singular nonlinearity $\\quad (P_{t}^s) \\left\\lbrace \\begin{split}\\quad u_t + (-\\Delta )^s u &= u^{-q} + f(x,u), \\;u >0\\; \\text{in}\\;\\Lambda _T, \\\\u &= 0 \\; \\mbox{in}\\; \\Gamma _T,\\\\u(0,x)&=u_0(x) \\; \\mbox{in} \\; {\\mathbb {R}^n},\\end{split}\\quad \\right.$ where $\\Lambda _T = (0,T) \\times \\Omega $ , $\\Gamma _T= (0,T) \\times (\\mathbb {R}^n \\setminus \\Omega )$ , $\\Omega $ is a bounded domain in $\\mathbb {R}^n$ with smooth boundary $\\partial \\Omega $ (atleast $C^2$ ), $n>2s,\\; s \\in (0,1),\\; q>0, \\; q(2s-1)<(2s+1)$ and $T>0$ .",
"The map $(x,y)\\in \\Omega \\times \\mathbb {R}^n \\mapsto f(x,y)$ is assumed to be a bounded below Carathéodary function, locally Lipschitz with respect to second variable and uniformly for $x \\in \\Omega $ it satisfies $ { \\limsup _{y \\rightarrow +\\infty } \\frac{f(x,y)}{y}<\\lambda _1^s(\\Omega )}, $ where $\\lambda _1^s(\\Omega )$ is the first eigenvalue of $(-\\Delta )^s$ in $\\Omega $ with (homogeneous) Dirichlet boundary condition in $\\mathbb {R}^n \\setminus \\Omega $ .",
"The fractional Laplace operator $(-\\Delta )^s$ is defined as $ (-\\Delta )^s u(x) = 2C^s_n\\mathrm {P.V.",
"}\\int _{\\mathbb {R}^n} \\frac{u(x)-u(y)}{\\vert x-y\\vert ^{n+2s}}\\,\\mathrm {d}y$ where $\\mathrm {P.V.",
"}$ denotes the Cauchy principal value and $C^s_n=\\pi ^{-\\frac{n}{2}}2^{2s-1}s\\frac{\\Gamma (\\frac{n+2s}{2})}{\\Gamma (1-s)}$ , $\\Gamma $ being the Gamma function.",
"In this article, we will be concerned with the nonlocal problem $(P^s_t)$ that involves the fractional Laplacian.",
"A large variety of diffusive problems in Physics are satisfactorily described by the classical Heat equation.",
"However, anomalous diffusion that follow non-Brownian scaling is nowadays intensively studied with wide range of applications in physics, finance, biology and many others.",
"The governing equations of such mathematical models involve the fractional Laplacian.",
"For a detailed survey on this we refer to [25], [26] and references therein.",
"It is natural to study the local and global existence and stabilization results for such problems.",
"Singular parabolic problems in the local case has been studied by authors in [14], [11], [5].",
"The inspiring point for us was the work of M. Badra et al.",
"[6], where the existence and stabilization results for parabolic problems where the principal part of the equation is the $p$ -Laplacian operator, has been studied when $0<q < 2+ \\frac{1}{p-1}$ .",
"In [9] Bougherara and Giacomoni authors proved the existence of unique mild solution to the problem for all $q>0$ when $u_0 \\in (C_0(\\overline{\\Omega }))^+$ .",
"In the present work we extend the results obtained in [6] to the non-local case.",
"However, there is a substantial difference between local and nonlocal operators.",
"This difference is reflected in the way of construction of sub and super solutions of stationary problems associated to $(P^s_t)$ as well as the validity of the weak comparison principle.",
"Nonethless, we will show that the semi-discretization in time method used in [6] can still be efficient in this case.",
"Coming to the non-local case, singular elliptic equations involving fractional laplacian has been studied by Barios et al.",
"in [8] and Giacomoni et al.",
"in [16].",
"More specifically, existence and multiplicity results for the equation $(-\\Delta )^su = \\lambda u^{-q}+ u^p \\; \\text{in}\\; \\Omega ,\\; u=0 \\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega $ has been shown for $0<q\\le 1$ and $0<p< 2^*_s-1$ where $2^*_s= \\displaystyle \\frac{2n}{n-2s}$ in [8] and $p=2^*_s-1$ in [19].",
"Whereas the case $q>0$ and $p= 2^*_s-1$ has been studied in [16].",
"Concerning the parabolic problems involving the fractional laplacian, we cite [25], [26], [3], [13] and references therein.",
"Caffarelli and Figalli studied the regularity of solutions to fractional parabolic obstacle problem in [10].",
"In [17], authors studied the Hölder estimates for singular problems of the type $(-\\Delta )^s u^m +u_t=0$ where $\\frac{n-2s}{n+2s}<m<1.$ In [18], the summability of solutions with respect to the summability of the data is studied.",
"In [1], authors studied the influence of Hardy potential on the existence and nonexistence of positive solutions for fractional heat equation.",
"To the best of our knowledge there are no works on parabolic equations with fractional laplacian and singular nonlinearity.",
"In this work, we first define the positive cone motivated from the work of [2] and obtain the existence of solutions in this cone for the elliptic problem $(S)$ in section 2 associated to the semi-discretization of $(P^s_t).$ Using this, we proved the existence and uniqueness of solution and its regularity for the parabolic problem (see $(G^s_t)$ in section 2 with bounded source term $h(x,t)$ and principal diffusion operator $(-\\Delta )^s - u^{-q}$ in section 4).",
"Finally using the new uniqueness results for the stationary problem proved in section 5, we prove the existence and uniqueness of solutions to the problem $(P^s_t)$ in section 6.",
"Thanks to nonlinear accretive operators theory, we also find that these solutions are more regular when the regularity assumption is refined on the initial condition.",
"We end our paper by showing that the solution to $(P^s_t)$ converges to the unique solution of its stationary problem as $t \\rightarrow \\infty $ in section 7.",
"In this aim, we extend existence and regularity results about the stationary problem proved in [2]."
],
[
"Functional Setting and Main results",
"We denote the usual fractional Sobolev space by $H^s(\\Omega )$ endowed with the Gagliardo norm $\\Vert u\\Vert _{H^s(\\Omega )}= \\Vert u\\Vert _{L^2(\\Omega )}+ \\left(\\int _\\Omega \\int _\\Omega \\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}~dxdy\\right)^{\\frac{1}{2}}.",
"$ Then we consider the following space $X(\\Omega )= \\left\\lbrace u|\\;u:\\mathbb {R}^n \\rightarrow \\mathbb {R} \\;\\text{is measurable},u|_{\\Omega } \\in L^2(\\Omega )\\;\\text{and}\\; \\frac{\\left(u(x)- u(y)\\right)}{ |x-y|^{\\frac{n+2s}{2}}}\\in L^2(Q)\\right\\rbrace ,$ where $Q=\\mathbb {R}^{2n}\\setminus (\\mathcal {C}\\Omega \\times \\mathcal {C}\\Omega )$ and $\\mathcal {C}\\Omega := \\mathbb {R}^n\\setminus \\Omega $ .",
"The space $X(\\Omega )$ is endowed with the norm defined as $\\Vert u\\Vert _{X(\\Omega )} = \\Vert u\\Vert _{L^2(\\Omega )} +\\left( \\int _{Q}\\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}dxdy\\right)^{\\frac{1}{2}}.$ Now we define the space $ X_0(\\Omega ) = \\lbrace u\\in X(\\Omega ) : u = 0 \\;\\text{a.e.",
"in}\\; \\mathbb {R}^n\\setminus \\Omega \\rbrace $ equipped with the norm $\\Vert u\\Vert _{X_0(\\Omega )}=\\left( C_n^s\\int _{Q}\\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}dxdy\\right)^{\\frac{1}{2}}$ where $C_n^s$ is defined in section 1 and it is well known that $X_0(\\Omega )$ forms a Hilbert space with this norm (see [21]).",
"From the embedding results, we know that $X_0(\\Omega )$ is continuously and compactly embedded in $L^r(\\Omega )$ when $1\\le r < 2^*_s$ and the embedding is continuous but not compact if $r= 2^*_s$ .",
"For each $\\alpha \\ge 0$ , we set $ C_{\\alpha } = \\sup \\left\\lbrace \\int _{\\Omega } |u|^\\alpha dx : \\Vert u\\Vert _{X_0(\\Omega )} = 1\\right\\rbrace .$ Then $C_0= |\\Omega |=$ Lebesgue measure of $\\Omega $ and $\\int _{\\Omega } |u|^\\alpha dx \\le C_\\alpha \\Vert u\\Vert ^{\\alpha }$ , for all $ u \\in X_0(\\Omega )$ .",
"Let us consider a more general problem $(G_t^s)\\left\\lbrace \\begin{split}\\quad u_t + (-\\Delta )^s u &= u^{-q} + h(t,x) ,\\;u>0\\; \\text{in}\\;\\Lambda _T, \\\\u &= 0 \\; \\mbox{in}\\; \\Gamma _T,\\\\u(0,x)&=u_0(x) \\; \\mbox{in} \\; {\\mathbb {R}^n},\\end{split}\\right.$ where $T>0$ , $s\\in (0,1)$ , $h \\in L^\\infty (\\Lambda _T)$ , $q>0$ , $q(2s-1)<(2s+1)$ and $u_0 \\in L^\\infty (\\Omega ) \\cap X_0(\\Omega )$ .",
"In order to define weak solution for the problem $(G_t^s)$ , we need to introduce the following space $\\mathcal {A}(\\Lambda _T):= \\lbrace u : \\; u \\in L^\\infty (\\Lambda _T), \\; u_t \\in L^2(\\Lambda _T), \\; u \\in L^\\infty (0,T;X_0(\\Omega ))\\rbrace .",
"$ We have the following result as a direct consequence of Aubin-Lions-Simon Lemma (see [24]).",
"Lemma 2.1 Suppose $u \\in L^\\infty (0,T;X_0(\\Omega ))$ and $u_t \\in L^2(\\Lambda _T)$ .",
"Then $u \\in C([0,T]; L^2(\\Omega ))$ and the embedding is compact.",
"We now define the notion of weak solution for the problem $(G_t^s)$ .",
"Definition 2.2 We say $u \\in \\mathcal {A}(\\Lambda _T)$ is a weak solution of $(G_t^s)$ if for any compact subset $K \\subset \\Lambda _T$ , $ess\\inf _K u >0$ , for every $\\phi \\in \\mathcal {A}(\\Lambda _T)$ , $\\int _{\\Lambda _T}\\frac{\\partial u}{\\partial t}\\phi ~dxdt + C_n^s\\int _0^T\\int _{Q}\\frac{(u(x)-u(y))(\\phi (x)-\\phi (y))}{|x-y|^{n+2s}}dydxdt=\\int _{\\Lambda _T}( u^{-q}+ h(t,x))\\phi dxdt,$ $u(0,x)= u_0(x)$ a.e.",
"in $\\Omega $ .",
"We remark that because of Lemma REF , we get $\\mathcal {A}(\\Lambda _T) \\subset C([0.T]; L^2(\\Omega ))$ which means that the third point of the above definition makes sense.",
"Now, we define a conical shell $\\mathcal {C}$ as the set of functions $v \\in L^\\infty (\\Omega )$ such that there exist constants $k_1,k_2>0$ such that $\\left\\lbrace \\begin{split}&k_1\\delta ^s(x)\\le v \\le k_2\\delta ^s(x) & \\text{ if } q<1,\\\\&k_1 \\delta ^s(x) \\left(\\ln \\left( {\\frac{r}{\\delta ^s(x)}}\\right)\\right)^{\\frac{1}{2}} \\le v \\le k_2 \\delta ^s(x) \\left(\\ln \\left( {\\frac{r}{\\delta ^s(x)}}\\right)\\right)^{\\frac{1}{2}} & \\text{ if } q=1,\\\\&k_1\\delta ^{\\frac{2s}{q+1}}(x) \\le v \\le k_2 \\delta ^{\\frac{2s}{q+1}}(x) & \\text{ if } q>1,\\end{split}\\right.$ where $\\delta (x) := \\text{dist}(x,\\partial \\Omega )$ for $x \\in \\Omega $ and $r>\\text{diam}(\\Omega )$ .",
"We set $C_0(\\overline{\\Omega }) := \\left\\lbrace u \\in C(\\overline{\\Omega }): \\; u =0 \\;\\text{on} \\; \\partial \\Omega \\right\\rbrace .$ We begin by considering the stationary problem $(S)$ : $\\quad (S) \\left\\lbrace \\begin{split}\\quad u + \\lambda \\left((-\\Delta )^s u - u^{-q}\\right) &= g, \\;\\; \\; u>0\\; \\text{in}\\;\\Omega , \\\\\\quad \\quad \\quad \\quad u &= 0 \\; \\text{in}\\; \\mathbb {R}^n \\setminus \\Omega ,\\\\\\end{split}\\quad \\right.$ where $g \\in L^\\infty (\\Omega )$ and $\\lambda >0$ is a real parameter.",
"The notion of weak solution is defined as follows.",
"Definition 2.3 We say $ u \\in X_0(\\Omega )$ is a weak solution of $(S)$ if for any compact subset $K \\subset \\Omega $ , $ess\\inf _K u >0$ , for every $\\phi \\in X_0(\\Omega )$ , $\\int _\\Omega u\\phi ~dx + \\lambda \\left( C_n^s \\int _Q\\frac{(u(x)-u(y))(\\phi (x)-\\phi (y))}{|x-y|^{n+2s}}~dxdy-\\int _{\\Omega } u^{-q}\\phi ~dx \\right)= \\int _{\\Omega }g\\phi ~dx.",
"$ We prove the following theorem considering the problem $(S)$ .",
"Theorem 2.4 If $g \\in L^\\infty (\\Omega )$ , $q>0$ and ${q}(2s-1)<(2s+1)$ , then for any $\\lambda >0$ , problem $(S)$ has a unique weak solution $u_\\lambda \\in X_0(\\Omega )\\cap {\\mathcal {C} \\cap C^\\alpha (\\mathbb {R}^n)}$ where $\\alpha = s$ if $q<1$ , $\\alpha =s-\\epsilon $ if $q=1$ , for any $\\epsilon >0$ small enough and $\\alpha = \\displaystyle \\frac{2s}{q+1}$ if $q>1$ .",
"In the case $q(2s-1)\\ge (2s+1)$ , we get less regularity on solution of $(S)$ .",
"So we will have a weaker notion of solution in this case for which we define the set $\\Theta := \\lbrace \\phi :\\;\\phi : \\mathbb {R}^n \\rightarrow \\mathbb {R} \\; \\text{measurable and}\\; (-\\Delta )^s\\phi \\in L^\\infty (\\Omega ),\\; \\phi \\equiv 0 \\; \\text{on}\\; \\mathbb {R}^n \\setminus \\Omega ^\\prime ,\\; \\Omega ^\\prime \\Subset \\Omega \\rbrace .$ Theorem 2.5 Let $g \\in L^\\infty (\\Omega )$ , $q>1$ and $q(2s-1) \\ge (2s+1)$ then for any $\\lambda >0$ , there exists a $u_\\lambda \\in L^1(\\mathbb {R}^n)$ satisfying $u\\equiv 0$ in $\\mathbb {R}^n \\setminus \\Omega $ , $\\inf \\limits _K u_\\lambda >0$ for every $K \\Subset \\Omega $ and $\\int _\\Omega u_\\lambda \\phi ~dx + \\lambda \\left( C_n^s \\int _Q\\frac{(u_\\lambda (x)-u_\\lambda (y))(\\phi (x)-\\phi (y))}{|x-y|^{n+2s}}~dxdy-\\int _{\\Omega } u_\\lambda ^{-q}\\phi ~dx \\right)= \\int _{\\Omega }g\\phi ~dx $ for any $\\phi \\in \\Theta $ .",
"Moreover $u^{\\beta }_\\lambda \\in X_0(\\Omega )$ where $\\beta > \\max \\left\\lbrace 1, \\left(1-\\frac{1}{2s} \\right)\\left( \\frac{q+1}{2}\\right) \\right\\rbrace $ but $u_\\lambda \\notin X_0(\\Omega )$ .",
"Definition 2.6 We say that ${u(t)} \\in \\mathcal {C}$ uniformly for each $t \\in [0,T]$ when there exist $\\psi _1, \\psi _2 \\in \\mathcal {C}$ such that $\\psi _1(x) \\le u(t,x) \\le \\psi _2(x)$ a.e.",
"$(t,x)\\in [0,T] \\times \\Omega $ .",
"We prove the following existence and uniqueness result for the problem $(G^s_t)$ using semi-discretization in time with implicit Euler method, Theorem REF , energy estimates and the weak comparison principle.",
"Theorem 2.7 If $h(t,x) \\in L^{\\infty }(\\Lambda _T)$ , $u_0\\in X_0(\\Omega )\\cap \\mathcal {C}$ , $q>0$ and $q(2s-1)<(2s+1)$ , then there exists a unique weak solution $u \\in C([0,T]; X_0(\\Omega ))$ for the problem $(G^s_t)$ such that ${u(t)} \\in \\mathcal {C}$ uniformly for each $t \\in [0,T]$ .",
"Also, $u$ satisfies $\\begin{split}\\int _0^{t} \\int _{\\Omega } \\left( \\frac{\\partial u}{\\partial t}\\right)^2 dxd\\tau & + \\frac{1}{2} \\Vert u(t,x)\\Vert _{X_0(\\Omega )}^2- \\frac{1}{1-q}\\int _{\\Omega } u^{1-q}(t,x)dx \\\\&\\quad = \\int _0^{t} \\int _{\\Omega } h(\\tau , x)\\frac{\\partial u}{\\partial t} dxd\\tau + \\frac{1}{2} \\Vert u_0(x)\\Vert _{X_0(\\Omega )}^2- \\frac{1}{1-q}\\int _{\\Omega } u_0^{1-q}(x)dx\\end{split}$ for any $t \\in [0,T]$ .",
"The solution obtained in above theorem can be shown to be more regular under some extra assumptions as can be seen in the next result.",
"Proposition 2.8 Under the hypothesis of Theorem REF , if $u_0 \\in \\overline{\\mathcal {D}(L)}^{L^{\\infty }(\\Omega )}$ , where ${{\\mathcal {D}(L)}} := \\lbrace v \\in \\mathcal {C} \\cap X_0(\\Omega ): \\; L(v):= (-\\Delta )^sv -v^{-q} \\in L^{\\infty }(\\Omega )\\rbrace $ then the solution to $(G^s_t)$ obtained in Theorem REF belongs to $C([0,T];C_0(\\overline{\\Omega }))$ .",
"Also $u$ satisfies: If $v$ is another solution to $(G^s_t)$ with initial condition $v_0 \\in \\overline{\\mathcal {D}(L)}^{L^\\infty (\\Omega )}$ and nonhomogenous term $b \\in L^\\infty (\\Lambda _T)$ , then for any $t \\in [0,T]$ , $\\Vert u(t,\\cdot )- v(t,\\cdot )\\Vert _{L^\\infty (\\Omega )} \\le \\Vert u_0-v_0\\Vert _{L^\\infty (\\Omega )}+ \\int _0^t \\Vert h(\\tau ,\\cdot )- b(\\tau ,\\cdot ) \\Vert _{L^\\infty (\\Omega )}d\\tau .$ If $u_0 \\in \\mathcal {D}(L)$ and $h \\in W^{1,1}(0,T;L^\\infty (\\Omega ))$ , then $u \\in W^{1,\\infty }(0,T; L^\\infty (\\Omega ))$ , $(-\\Delta )^su + u^{-q} \\in L^\\infty (\\Lambda _T)$ and the following holds true for any $t \\in [0,T]$ , $\\left\\Vert \\frac{du(t,\\cdot )}{dt}\\right\\Vert _{L^\\infty (\\Omega )} \\le \\Vert (-\\Delta )^su_0 + u^{-q}_0+h(0,\\cdot ) \\Vert _{L^\\infty (\\Omega )}+ \\int _0^T \\left\\Vert \\frac{dh(\\tau ,\\cdot )}{dt}\\right\\Vert _{L^\\infty (\\Omega )}d\\tau .",
"$ In order to establish Theorem REF , we need the following result.",
"Theorem 2.9 Suppose $q>0$ , $q(2s-1)<(2s+1)$ and $f : \\Omega \\times \\mathbb {R}^+ \\rightarrow \\mathbb {R}$ be bounded below Carathéodary function satisfying (REF ).",
"Assume that $f$ is locally Lipschitz with respect to second variable uniformly in $\\Omega $ and $\\frac{f(x,y)}{y}$ is decreasing in $\\mathbb {R}^+$ for a.e.",
"$x \\in \\Omega $ .",
"Then the following problem $(Q^s)$ has a unique solution $\\hat{u} \\in {X_0(\\Omega )\\cap {\\mathcal {C}}} \\cap C^\\alpha (\\mathbb {R}^n)$ where $\\alpha = s$ if $q<1$ , $\\alpha =s-\\epsilon $ if $q=1$ , for any $\\epsilon >0$ small enough and $\\alpha = \\displaystyle \\frac{2s}{q+1}$ if $q>1$ .",
": $(Q^s)\\left\\lbrace \\begin{split}(-\\Delta )^s \\hat{u} -{\\hat{u}}^{-q} &= f(x, \\hat{u})\\; \\text{in} \\; \\Omega ,\\\\\\hat{u} &= 0 \\; \\text{in} \\; \\mathbb {R}^n \\setminus \\Omega .\\end{split}\\right.$ Coming back to our original problem $(P^s_t)$ , we have the following theorem : Theorem 2.10 Assume $q>0$ , $q(2s-1)<(2s+1)$ and $f(t,x)$ to be a bounded below Carathéodory function, locally Lipschitz with respect to second variable uniformly in $x \\in \\Omega $ and satisfies (REF ).",
"If $u_0 \\in X_0(\\Omega )\\cap \\mathcal {C}$ , then for any $T>0$ , there exists a unique weak solution $u$ to $(P^s_t)$ such that ${u(t)} \\in \\mathcal {C}$ uniformly for $t \\in [0,T]$ and $u \\in C([0,T]; X_0(\\Omega ))$ .",
"Moreover for any $t \\in [0,T]$ , $\\begin{split}&\\int _0^{t} \\int _{\\Omega } \\left( \\frac{\\partial u}{\\partial t}\\right)^2 dxd\\tau + \\frac{1}{2} \\Vert u(t,x)\\Vert _{X_0(\\Omega )}^2- \\frac{1}{1-q}\\int _{\\Omega } u^{1-q}(t,x)dx \\\\&\\quad = \\int _\\Omega F(x,u(t)) dx + \\frac{1}{2} \\Vert u_0(x)\\Vert _{X_0(\\Omega )}^2- \\frac{1}{1-q}\\int _{\\Omega } u_0^{1-q}(x)dx- \\int _\\Omega F(x,u_0)dx,\\end{split}$ where $F(x,z):= \\int _0^z f(x,z)dz$ .",
"Using Proposition REF , on a similar note we have the following proposition regarding the solution of problem $(P_t^s)$ .",
"Proposition 2.11 Assume that the hypothesis of Theorem REF are true.",
"If $u_0 \\in \\overline{\\mathcal {D}(L)}^{L^\\infty (\\Omega )}$ , then the solution of $(P_t^s)$ belongs to $C([0,T];C_0(\\overline{\\Omega }))$ .",
"Let $\\alpha \\ge 0$ denotes the Lipschitz constant of $f(\\cdot ,x)$ in $[\\underline{u},\\overline{u}]$ , where $\\underline{u}$ and $\\overline{u}$ denotes the sub and super solution respectively of $(Q^s)$ , then the following holds: If $v$ is another weak solution to $(P_t^s)$ with initial condition $v_0 \\in \\overline{\\mathcal {D}(L)}^{L^\\infty (\\Omega )} $ , then $\\Vert u(t,\\cdot )-v(t,\\cdot )\\Vert _{L^\\infty (\\Omega )} \\le \\exp (\\alpha t) \\Vert u_0-v_0\\Vert _{L^\\infty (\\Omega )},\\; 0\\le t\\le T.$ If $u_0 \\in \\mathcal {D}(L)$ , then $u \\in W^{1,\\infty }([0,T];L^\\infty (\\Omega ))$ and $(-\\Delta )^s u + u^{-q} \\in L^\\infty (\\Lambda _T)$ .",
"Also the following holds: $\\left\\Vert \\frac{du(t,\\cdot )}{dt}\\right\\Vert _{L^\\infty (\\Omega )} \\le \\exp (\\alpha t)\\Vert (-\\Delta )^su_0 + u^{-q}_0+f(x,u_0) \\Vert _{L^\\infty (\\Omega )}.$ Finally, we can show the following asymptotic behavior of solutions of $(P_t^s)$ .",
"Theorem 2.12 Under the hypothesis of Theorem REF and the assumption that $y \\mapsto \\frac{f(x,y)}{y}$ is decreasing in $(0,\\infty )$ a.e.",
"$x \\in \\Omega $ , the solutions to $(P_t^s)$ is defined in $(0,\\infty )\\times \\Omega $ and it satisfies $u(t) \\rightarrow \\hat{u} \\text{ in } L^\\infty (\\Omega ) \\text{ as } t \\rightarrow \\infty ,$ where $\\hat{u}$ is defined in Theorem REF .",
"Remark 2.13 We can conclude the results for the problem $(P_t^s)$ in a similar manner when $q>-1$ and $q(2s-1)<(2s+1)$ holds."
],
[
"Existence of solution to $(S)$",
"Basically we prove Theorem REF in this section.",
"Before proving this, we give a Lemma that will be recalled in our work several times as the weak comparison principle.",
"Lemma 3.1 Assume $\\lambda >0$ and $u,v \\in X_0(\\Omega )$ are weak solutions of $&{A_\\lambda }u= g_1 \\; \\text{in}\\; \\Omega ,\\\\&{A_\\lambda }v = g_2 \\; \\text{in}\\; \\Omega $ with $g_1,g_2 \\in L^2(\\Omega )$ such that $g_1 \\le \\; g_2$ , where ${A_\\lambda } :X_0(\\Omega )\\cap \\mathcal {C} \\rightarrow (X_0(\\Omega ))^*$ (dual space of $X_0(\\Omega )$ ) is defined as ${A_\\lambda }(u):= u+\\lambda ((-\\Delta )^su -u^{-q})$ , with $\\lambda >0$ fixed.",
"Then $u\\le v$ a.e.",
"in $\\Omega $ .",
"Moreover, for $g \\in L^\\infty (\\Omega )$ the problem $A_\\lambda u = g \\; \\text{in} \\;\\; \\Omega , \\; u =0 \\; \\text{in}\\;\\; \\mathbb {R}^n\\setminus \\Omega $ has a unique solution in $X_0(\\Omega )$ .",
"Proof.",
"Let $w=(u-v)$ , then $w=w^+-w^-$ where $w^+=\\max \\lbrace w,0\\rbrace $ and $w^-=\\max \\lbrace -w,0\\rbrace $ .",
"Let $\\Omega ^+ := \\lbrace x\\in \\Omega : u(x)> v(x)\\rbrace $ and $\\Omega ^- := \\Omega \\setminus \\Omega ^+$ , then $\\Omega = \\Omega ^+\\cup \\Omega ^-$ .",
"Multiplying (REF ) and () by $w^+$ , integrating over $\\mathbb {R}^n$ on both sides and subtracting, we get $&\\int _{\\Omega ^+} (u-v)^2dx+ \\lambda \\left( C_n^s\\int _Q\\frac{((u-v)(x)-(u-v)(y))(w^+(x)- w^+(y))}{|x-y|^{n+2s}}dxdy\\right.\\\\&\\quad \\quad - \\left.\\int _{\\Omega ^+}\\left(\\frac{1}{v^q}-\\frac{1}{u^q}\\right)(u-v)dx\\right)= \\int _{\\Omega ^+} (g_1-g_2)w^+ dx.$ Since for $(x,y)\\in \\Omega \\times \\mathcal {C}\\Omega $ , $((u-v)(x)-(u-v)(y))(w^+(x)-w^+(y))= (u-v)(x)w^+(x)\\ge 0$ and for $(x,y)\\in \\Omega ^+ \\times \\Omega ^-$ , $((u-v)(x)-(u-v)(y))w^+(x)\\ge 0$ we get $\\begin{split}\\int _{\\Omega ^+} (u-v)^2dx &+ \\lambda \\left(C_n^s \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{((u-v)(x)-(u-v)(y))^2}{|x-y|^{n+2s}}dxdy\\right.\\\\&\\quad \\left.- \\int _{\\Omega ^+}\\left(\\frac{1}{v^q}-\\frac{1}{u^q}\\right)(u-v)dx\\right)\\le \\int _{\\Omega ^+} (g_1-g_2)w^+ dx.\\end{split}$ We can also prove that $A_\\lambda $ is a strictly monotone operator (for definition refer [7]).",
"So left hand side of (REF ) is positive whereas $\\int _{\\Omega ^+}(g_1-g_2)w^+ dx \\le 0$ .",
"Therefore we arrive at a contradiction which implies $u\\le v$ a.e.",
"in $\\Omega $ .",
"Then the uniqueness of (REF ) follows directly.",
"$\\square $ Proof of Theorem REF : For $\\epsilon >0$ , we consider the following approximated problem corresponding to $(S)$ as $(S_\\epsilon )\\left\\lbrace \\begin{split}&u+ \\lambda \\left( (-\\Delta )^su-(u+\\epsilon )^{-q}\\right)=g,\\;\\; \\; u>0 \\; \\text{in} \\; \\Omega ,\\\\& u=0 \\; \\text{in}\\; \\mathbb {R}^n \\setminus \\Omega .\\end{split}\\right.$ Let $X_0^+(\\Omega )= \\lbrace u\\in X_0(\\Omega ): \\; u\\ge 0\\rbrace $ .",
"The energy functional associated to $(S_\\epsilon ): X_0^+(\\Omega ) \\rightarrow \\mathbb {R}$ is $E_\\lambda (u) = \\frac{1}{2} \\int _\\Omega u^2 ~dx + \\frac{\\lambda }{2}\\Vert u\\Vert _{X_0(\\Omega )}^2 -\\frac{\\lambda }{1-q}\\int _{\\Omega }({u}+\\epsilon )^{1-q}~dx- \\int _{\\Omega }gu ~dx$ which can shown to be weakly lower semicontinuous, coercive and strictly convex in $X_0^+(\\Omega )$ .",
"Since $X_0(\\Omega )$ is reflexive and $X_0^+(\\Omega )$ being a closed convex subset of $X_0(\\Omega )$ , $E_\\lambda $ has a unique global minimizer $u_{\\lambda ,\\epsilon } \\in X_0^+(\\Omega )$ i.e.",
"$u_{\\lambda ,\\epsilon } \\ge 0$ a.e.",
"in $\\Omega $ .",
"Let $\\phi _{1,s}$ denotes the normalized first eigenfunction associated with first eigenvalue $\\lambda _{1,s}$ of $(-\\Delta )^s$ with Dirichlet boundary condition in $\\mathbb {R}^n \\setminus \\Omega $ i.e.",
"$(-\\Delta )^s\\phi _{1,s}= \\lambda _{1,s}\\phi _{1,s} \\; \\text{in} \\; \\Omega , \\; \\; \\phi _{1,s}=0 \\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega ,$ where $0< \\phi _{1,s} \\in X_0(\\Omega )\\cap L^{\\infty }(\\Omega )$ is normalized by $\\Vert \\phi _{1,s}\\Vert _{L^2(\\Omega )}=1$ , refer [[22], Proposition 9, p. 8].",
"Also there exists $l>0$ such that $l\\delta ^s(x) \\le \\phi _{1,s}(x)$ for a.e.",
"$x \\in \\Omega $ (see [20]).",
"Since $g \\in L^\\infty (\\Omega )$ , if we choose $m>0$ (depending on $\\lambda ,q$ and $g$ ) small enough such that (in the weak sense) $m \\Vert \\phi _{1,s}\\Vert _\\infty + \\lambda \\lambda _{1,s}m\\Vert \\phi _{1,s}\\Vert _{\\infty }- \\frac{\\lambda }{m^q \\Vert \\phi _{1,s}+\\epsilon \\Vert ^q_\\infty }<g,$ then $m\\phi _{1,s}$ forms a strict subsolution of $(S_\\epsilon )$ (independent of $\\epsilon $ ) i.e.",
"$\\left\\lbrace \\begin{split}&m\\phi _{1,s}+\\lambda \\left( (-\\Delta )^s(m\\phi _{1,s})- \\frac{1}{(m\\phi _{1,s}+\\epsilon )^q}\\right)<g \\; \\text{in}\\; \\Omega ,\\\\&m\\phi _{1,s}=0 \\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega .\\end{split}\\right.$ We define $w_{\\epsilon }:= (m\\phi _{1,s}-u_{\\lambda ,\\epsilon })^+$ with the assumption that $\\text{supp}(w_{\\epsilon })$ has non zero measure and for $t>0$ , $\\zeta (t):= E_\\lambda (u_{\\lambda ,\\epsilon } +tw_{\\epsilon })$ , then $\\zeta ^\\prime (t)&= \\int _{\\Omega }(u_{\\lambda ,\\epsilon } +tw_{\\epsilon })w_{\\epsilon } + \\lambda C_n^s\\int _Q\\frac{( (u_{\\lambda ,\\epsilon } +t w_{\\epsilon })(x)-(u_{\\lambda ,\\epsilon } +tw_{\\epsilon } )(y))(w_{\\epsilon }(x)-w_{\\epsilon }(y))}{|x-y|^{n+2s}}~dxdy \\\\&\\quad -\\lambda \\int _{\\Omega }\\frac{w_{\\epsilon }}{(u_{\\lambda ,\\epsilon }+tw_{\\epsilon }+\\epsilon )^q}-\\int _{\\Omega }gw_{\\epsilon }$ in $(0,1]$ .",
"Since $u_{\\lambda ,\\epsilon }$ is the minimizer of $E_\\lambda $ , $\\lim \\limits _{t \\rightarrow 0^+}{\\zeta ^\\prime }(t)\\ge 0$ .",
"Moreover, convexity of $E_\\lambda $ assures that the map $t \\mapsto \\zeta ^\\prime (t)$ is non decreasing.",
"This implies $0\\le \\zeta ^\\prime (0^+) \\le \\zeta ^\\prime (1)$ .",
"Let us recall the following inequality for any $\\psi $ being a convex Lipschitz function: $(-\\Delta )^s\\psi (u) \\le \\psi ^\\prime (u)(-\\Delta )^su.$ Therefore using this with $\\psi (x)=\\max \\left\\lbrace x,0\\right\\rbrace $ and (REF ), we get $\\zeta ^\\prime (1)\\le \\langle E_\\lambda ^\\prime (m\\phi _{1,s}), w_\\epsilon \\rangle <0$ which is a contradiction.",
"Hence $\\text{supp}(w_\\epsilon )$ must have measure zero which implies $m\\phi _{1,s}\\le u_{\\lambda ,\\epsilon }.$ Using (REF ), we can show that $E_\\lambda $ is Gâteaux differentiable in $u_{\\lambda ,\\epsilon }$ and as a result $u_{\\lambda ,\\epsilon }$ satisfies in the sense of distributions $u_{\\lambda ,\\epsilon } +\\lambda (-\\Delta )^s u_{\\lambda ,\\epsilon }= \\lambda u_{\\lambda ,\\epsilon }^{-q} +g \\; \\text{in}\\; \\Omega .$ Using Proposition $2.9$ of [23], we get $u_{\\lambda ,\\epsilon } \\in C^{1,\\alpha }(\\mathbb {R}^n)$ for any $\\alpha < 2\\sigma -1$ where $2\\sigma >1$ .",
"Also since $g \\in L^\\infty (\\Omega )$ , using Proposition $1.1$ (p. 277) of [20] we get $u_{\\lambda ,\\epsilon } \\in C^s(\\mathbb {R}^n)$ .",
"Now we claim that $u_{\\lambda ,\\epsilon }$ is monotone increasing as $\\epsilon \\downarrow 0^+$ .",
"Let $0<\\epsilon _1<\\epsilon _2$ , then we show that $u_{\\lambda ,\\epsilon _1}> u_{\\lambda ,\\epsilon _2}$ in $\\Omega $ .",
"If possible, let $x_0\\in \\Omega $ be such that $x_0 := \\text{arg} \\;\\min \\limits _{\\overline{\\Omega }} (u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})$ and $u_{\\lambda ,\\epsilon _1}(x_0)\\le u_{\\lambda ,\\epsilon _2}(x_0)$ .",
"Then $(u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})+ \\lambda (-\\Delta )^s(u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})= \\lambda \\left( \\frac{1}{(u_{\\lambda ,\\epsilon _1}+\\epsilon _1)^q}- \\frac{1}{(u_{\\lambda ,\\epsilon _2}+\\epsilon _2)^q}\\right)$ which implies $&(u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})(x_0)+ \\lambda C_n^s \\int _{\\mathbb {R}^n}\\frac{(u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})(x_0)-(u_{\\lambda ,\\epsilon _1}-u_{\\lambda ,\\epsilon _2})(y)}{|x_0-y|^{n+2s}}~dy \\\\&= \\lambda \\left( \\frac{1}{(u_{\\lambda ,\\epsilon _1}(x_0)+\\epsilon _1)^q}- \\frac{1}{(u_{\\lambda ,\\epsilon _2}(x_0)+\\epsilon _2)^q}\\right).$ But we can see that (REF ) is negative whereas () is positive which gives a contradiction.",
"Therefore $x_0 \\in \\partial \\Omega $ and $u_{\\lambda ,\\epsilon _1}> u_{\\lambda ,\\epsilon _2}$ in $\\Omega $ .",
"Thus we get that $u_\\lambda := \\lim \\limits _{\\epsilon \\downarrow 0^+} u_{\\lambda ,\\epsilon } \\ge m \\phi _{1,s}$ .",
"Let $w \\in X_0^+(\\Omega )$ solves the problem $(-\\Delta )^s w = w^{-q} \\; \\text{in}\\; \\Omega ,\\;\\;w =0\\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega .$ Then from the proof of Theorem $1.1$ of [2], we know that $w$ satisfies $&k_1\\phi _{1,s} \\ln ^{\\frac{1}{2}}\\left(\\frac{2}{\\phi _{1,s}}\\right) \\le w \\le k_2 \\phi _{1,s} \\ln ^{\\frac{1}{2}}\\left(\\frac{2}{\\phi _{1,s}}\\right),\\; \\text{if}\\; q=1\\\\&k_1 \\phi _{1,s}^{\\frac{2}{q+1}}\\le w \\le k_2 \\phi _{1,s}^{\\frac{2}{q+1}},\\; \\text{if}\\; q>1 $ where $k_1,k_2>0$ are appropriate constants.",
"Let $\\overline{u}:= M_1w \\in \\mathcal {C} \\cap C_0(\\overline{\\Omega })$ for $M_1>0$ .",
"Then we can choose $M_1>>1$ (independent of $\\epsilon $ ) large enough such that $\\overline{u}+\\lambda \\left( (-\\Delta )^s\\overline{u} - \\frac{1}{(\\overline{u} +\\epsilon )^q}\\right)&= M_1w + \\lambda \\left(\\frac{M_1}{w^q}- \\frac{1}{(M_1w+\\epsilon )^q} \\right)\\\\&\\ge M_1w + \\lambda \\left(\\frac{1}{(M_1w)^q}- \\frac{1}{(M_1w+\\epsilon )^q} \\right)>g \\; \\text{in}\\; \\Omega .$ Using Lemma REF , we get $u_{\\lambda ,\\epsilon } \\le \\overline{u}$ which implies $u_\\lambda \\le \\overline{u} = M_1w$ .",
"Now since $m \\phi _{1,s}\\le u_\\lambda \\le M_1w$ and both $w,\\phi _{1,s} = 0$ in $\\mathbb {R}^n \\setminus \\Omega $ , we get $u_\\lambda = 0$ in $\\mathbb {R}^n \\setminus \\Omega $ .",
"Also $u_\\lambda $ solves $(S)$ in the sense of distributions.",
"Let $\\underline{u}:= M_2w\\in \\mathcal {C} \\cap C_0(\\overline{\\Omega })$ then $M_2>0$ can be chosen small enough so that $M_2^{q+1}\\left( 1+ \\frac{w^{q+1}}{\\lambda }\\right) &\\le 1+ \\frac{g(M_2w)^q}{\\lambda } \\;\\text{in}\\; \\Omega \\\\\\text{i.e.",
"}\\; \\underline{u}+\\lambda (-\\Delta )^s\\underline{u} &< \\frac{\\lambda }{\\underline{u}^q}+g \\;\\text{in}\\; \\Omega .$ This implies that $\\underline{u}$ forms a subsolution to $(S)$ .",
"We claim that $\\underline{u}\\le u_\\lambda $ in $\\Omega $ .",
"If possible, let $x_0\\in \\Omega $ be such that $x_0 := \\text{arg} \\;\\min \\limits _{\\overline{\\Omega }} (u_\\lambda -\\underline{u})$ and $u_\\lambda (x_0)\\le \\underline{u}(x_0)$ .",
"Then using the fact that $u_\\lambda $ is a solution of $(S)$ in the sense of distributions and $\\underline{u}$ is a subsolution of $(S)$ , we get $\\begin{split}(u_\\lambda -\\underline{u})(x_0) &+ \\lambda \\int _{\\Omega }\\frac{(u_\\lambda -\\underline{u})(x_0)-(u_\\lambda -\\underline{u})(y)}{|x_0-y|^{n+2s}}dy\\\\&\\ge (u_\\lambda -\\underline{u})(x_0) + \\lambda (-\\Delta )^s(u_\\lambda -\\underline{u})(x_0) \\ge \\lambda \\left( \\frac{1}{u_\\lambda ^q(x_0)}- \\frac{1}{\\underline{u}^q(x_0)}\\right).\\end{split}$ This gives a contradiction, since left hand side of (REF ) is negative whereas right hand side of (REF ) is positive.",
"Therefore we obtain $\\underline{u} \\le u_\\lambda \\le \\overline{u} $ which implies that $u_\\lambda \\in \\mathcal {C}$ , using (REF ) and ().",
"We now show that $u_\\lambda \\in X_0(\\Omega )$ and is a weak solution to $(S)$ .",
"Since $q(2s-1)<(2s+1)$ , using the behavior of $u_\\lambda $ with respect to $\\delta $ function we get that $\\displaystyle \\int _{\\Omega }u_\\lambda ^{1-q}~dx < +\\infty $ .",
"Also $\\displaystyle \\int _{\\Omega }u_\\lambda ^{-q}\\phi ~dx < +\\infty $ for any $\\phi \\in X_0(\\Omega )$ from Hardy's inequality.",
"Therefore using $\\overline{C_c^\\infty }^{\\Vert \\cdot \\Vert _{X_0(\\Omega )}}= X_0(\\Omega )$ and Lebsegue dominated convergence theorem, we get that for any $\\phi \\in X_0(\\Omega )$ $\\int _{\\Omega } u_\\lambda \\phi + \\lambda C_n^s \\int _Q\\frac{(u_\\lambda (x)- u_\\lambda (y))(\\phi (x)-\\phi (y))}{|x-y|^{n+2s}}~dxdy - \\int _{\\Omega }\\left( \\frac{\\lambda }{u^q_\\lambda }+ g\\right)\\phi ~ dx=0.$ That is $u_\\lambda \\in X_0(\\Omega ) \\cap \\mathcal {C}$ is a weak solution to $(S)$ .",
"By Lemma REF , uniqueness of $u_\\lambda $ follows.",
"Following the proof of Theorem $1.2$ in [2], we get that $u \\in C^\\alpha (\\mathbb {R}^n)$ where $\\alpha = s$ if $q<1$ , $\\alpha =s-\\epsilon $ if $q=1$ , for any $\\epsilon >0$ small enough and $\\alpha = \\displaystyle \\frac{2s}{q+1}$ if $q>1$ .",
"This completes the proof.",
"$\\square $ To prove the next result, we follow Lemma $3.6$ and Theorem $3.7$ of [8].",
"Proof of Theorem REF : Consider the following approximated problem $(P_{\\lambda ,k})\\left\\lbrace \\begin{split}u_k + \\lambda \\left((-\\Delta )^su -\\frac{1}{\\left(u+\\frac{1}{k}\\right)^q} \\right) &= g \\; \\text{in}\\; \\Omega ,\\\\u_k &=0 \\; \\text{in}\\; \\mathbb {R}^n \\setminus \\Omega .\\end{split}\\right.$ By minimization argument we know that the solution $u_k$ to the problem $(P_{\\lambda ,k})$ belongs to $X_0(\\Omega )$ .",
"By weak comparison principle we get $u_k \\le u_{k+1}$ for all $k$ .",
"From the proof of Theorem REF we know that $ m\\phi _{1,s}$ and $\\overline{u} =M_1w$ forms subsolution and supersolution of $(P_{\\lambda ,k})$ respectively independent of $k$ , where $w$ solves (REF ) and $m$ is a sufficiently small whereas $M_1$ is a sufficiently large positive constant.",
"Therefore $0 \\le m\\phi _{1,s} \\le u_k \\le u_{k+1} \\le \\overline{u},\\; \\text{for all}\\; k.$ Since $g \\in L^\\infty (\\Omega )$ so Proposition $1.1$ of [20] gives that $u_k \\in L^\\infty (\\Omega )\\cap C^s(\\mathbb {R}^n)$ for all $k$ .",
"Therefore if $\\tilde{\\Omega }\\Subset \\Omega $ then there exists a constant $c_{\\tilde{\\Omega }}>0$ such that $u_k \\ge c_{\\tilde{\\Omega }} >0 \\; \\text{in}\\; \\tilde{\\Omega }.$ Let $u_\\lambda := \\lim \\limits _{k \\rightarrow \\infty }u_k$ .",
"Then $u_\\lambda $ solves $(S)$ in the sense of distributions.",
"From the proof of Theorem REF we also know that for sufficiently small $M_2>0$ , $\\underline{u} = M_2w$ satisfies $\\underline{u}+ \\lambda ((-\\Delta )^s \\underline{u}- \\underline{u}^{-q}) < g\\; \\text{in}\\; \\Omega .",
"$ Then following the arguments in proof of Theorem REF (refer (REF )) we can show that $\\underline{u} \\le u_\\lambda \\le \\overline{u}$ which implies that $u_\\lambda \\sim d^{\\frac{2s}{q+1}}(x)$ .",
"Now for $ b> 1$ and $\\beta \\ge 1$ , consider the function $\\phi _\\beta : [0,+\\infty ) \\rightarrow [0,+\\infty )$ defined as $\\phi _\\beta (r)= \\left\\lbrace \\begin{split}r^\\beta , \\; \\text{if}\\; 0\\le r<b,\\\\\\beta b^{\\beta -1}r-(\\beta -1)b^\\beta , \\; \\text{if}\\; r\\ge b >1.\\end{split}\\right.$ Then $\\phi _\\beta $ is a lipschitz function with lipschitz constant $\\beta b^{\\beta -1}$ .",
"We have $q>1$ .",
"So let $\\beta > \\max \\left\\lbrace 1, \\left(1-\\frac{1}{2s} \\right)\\left( \\frac{q+1}{2}\\right) \\right\\rbrace \\ge 1.$ Then if $(2\\beta - 1-q)<0$ then from $u_\\lambda \\sim d^{\\frac{2s}{q+1}}(x)$ and (REF ) we get $\\int _{\\Omega } \\frac{\\phi _\\beta ^\\prime (u_\\lambda ) \\phi _\\beta (u_\\lambda )}{u_\\lambda ^q}~dx < +\\infty .$ Since $\\phi ^\\prime _\\beta (u)\\phi _\\beta (u)\\le \\beta u^{2\\beta -1}$ so using (REF ), $u_k\\uparrow u_\\lambda $ as $k \\rightarrow \\infty $ and monotone convergence theorem we get that $\\int _{\\Omega } \\frac{\\phi _\\beta ^\\prime (u_k) \\phi _\\beta (u_k)}{u_k^q}~dx < +\\infty \\; (\\text{independent of } k).$ Also (REF ) holds true when $(2\\beta -1-q)\\ge 0$ which follows from the uniform bound of $\\lbrace u_k\\rbrace $ in $L^\\infty (\\Omega )$ .",
"Since it holds $(-\\Delta )^s \\phi _\\beta (u_k) \\le \\phi _\\beta ^\\prime (u_k) (-\\Delta )^s u_k,$ therefore using (REF ) we get $\\begin{split}\\int _{\\mathbb {R}^n} \\phi _\\beta (u_k) (-\\Delta )^s\\phi _\\beta (u_k) & \\le \\frac{1}{\\lambda }\\int _{\\Omega } (g-u_k)\\phi _\\beta ^\\prime (u_k) \\phi _\\beta (u_k)~dx + \\int _{\\Omega } \\frac{\\phi _\\beta ^\\prime (u_k) \\phi _\\beta (u_k)}{u_k^q}~dx\\\\& \\le \\beta \\left( \\frac{\\Vert g\\Vert _\\infty \\Vert \\overline{u}\\Vert _{L^{2\\beta -1}(\\Omega )}}{\\lambda }+ C \\right),\\end{split}$ where $C>0$ is a constant independent of $k$ .",
"Passing on the limit as $b \\rightarrow \\infty $ we get $\\lbrace u_k^\\beta \\rbrace $ is uniformly bounded in $X_0(\\Omega )$ .",
"By weak lower semicontinuity of norms we have $ \\Vert u_\\lambda ^{\\beta }\\Vert \\le \\liminf _{k \\rightarrow \\infty } \\Vert u_k^{\\beta }\\Vert < +\\infty $ which implies $u_\\lambda ^{\\beta } \\in X_0(\\Omega )$ .",
"Thus $u_\\lambda ^{\\beta } \\in L^{2^*_s}(\\Omega )$ and since $\\beta 2^*_s >1$ we get $u_\\lambda \\in L^1(\\Omega )$ .",
"Now let $\\psi \\in \\Theta $ such that $\\text{supp}(\\psi )= \\tilde{\\Omega }\\Subset \\Omega $ then by Lebesgue dominated convergence theorem we get $\\lim _{k \\rightarrow \\infty } \\int _{\\mathbb {R}^n} u_k (-\\Delta )^s \\psi ~dx = \\int _{\\mathbb {R}^n} u_\\lambda (-\\Delta )^s \\psi ~dx < +\\infty .",
"$ Using (REF ) we get $0 \\le \\left| \\left( \\frac{g-u_k}{\\lambda } + \\frac{1}{\\left(u_k + \\frac{1}{k}\\right)^q} \\right)\\psi \\right| \\le \\left( \\frac{|g| + |\\overline{u}|}{\\lambda }+ \\frac{1}{c_{\\tilde{\\Omega }}^q} \\right)|\\psi | \\in L^1(\\Omega ).$ Therefore using Lebesgue dominated convergence theorem again we obtain $\\int _{\\mathbb {R}^n}u_\\lambda (-\\Delta )^s \\psi = \\lim _{k \\rightarrow \\infty } \\int _\\Omega \\left(\\frac{g-u_k}{\\lambda } + \\frac{1}{\\left(u_k + \\frac{1}{k}\\right)^q}\\right)\\psi ~dx= \\int _\\Omega \\left(\\frac{g-u_\\lambda }{\\lambda } + \\frac{1}{u_\\lambda ^q}\\right)\\psi ~dx.$ We claim that $u_\\lambda \\notin X_0(\\Omega )$ .",
"On contrary if $u_\\lambda \\in X_0(\\Omega )$ then using Lemma $3.1$ of [16] and monotone convergence theorem, we can easily show that (REF ) holds for any $\\psi \\in X_0(\\Omega )$ .",
"Therefore $u_\\lambda \\in X_0(\\Omega )$ solves $(S)$ in the weak sense and we get $\\frac{1}{u^q_\\lambda }= \\frac{1}{\\lambda }(u_\\lambda -g)+ (-\\Delta )^su_\\lambda \\in (X_0(\\Omega ))^*.",
"$ Using (REF ) this implies that $\\int _\\Omega \\overline{u}^{1-q}~dx \\le \\int _\\Omega u^{1-q}_\\lambda ~dx <+\\infty $ which contradicts the definition of $\\overline{u}$ .",
"$\\square $ Now following the proof of Lemma $6.1$ of [19], we can show that (REF ) holds for any $\\psi \\in X_0(\\Omega )$ ."
],
[
"Existence of solution to $(G^s_t)$ and its regularity",
"We prove Theorem REF and Proposition REF in this section.",
"We use the method of semi-discretization in time along with implicit Euler method to prove Theorem REF .",
"Theorem 4.1 If $h(t,x) \\in L^{\\infty }(\\Lambda _T)$ , $u_0\\in X_0(\\Omega )\\cap \\mathcal {C}$ , $q>0$ and $q(2s-1)<(2s+1)$ , then there exists a unique weak solution $u \\in \\mathcal {A}_{\\Lambda _T}\\cap \\mathcal {C}$ of the problem $(G^s_t)$ .",
"Proof.",
"Let $\\Delta _t = \\frac{T}{n}$ and for $0\\le k \\le n$ , define $t_k:= k\\Delta _t $ .",
"Also define $h_k(x):= \\frac{1}{\\Delta _t}\\int _{t_{k-1}}^{t_k} h(\\tau ,x)d\\tau \\; \\text{for}\\; x \\in \\Omega .$ Since $h \\in L^\\infty (\\Lambda _T)$ , we get $h_k \\in L^\\infty (\\Omega )$ and $\\Vert h_k\\Vert _{\\infty } \\le \\Vert h\\Vert _{L^\\infty (\\Lambda _T)}$ .",
"Then we define $h_{\\Delta _t}(t,x):= h^k(x), \\; \\text{when}\\; t \\in [t_{k-1}, t_k),\\; 1\\le k \\le n$ and get that $h_{\\Delta _t} \\in L^\\infty (\\Lambda _T)$ .",
"For $1<p<+\\infty $ , $\\begin{split}\\Vert h_{\\Delta _t}\\Vert _{L^p(\\Lambda _T)}\\le (|\\Omega |T)^{\\frac{1}{p}}\\Vert h\\Vert _{L^\\infty (\\Lambda _T)}\\end{split}$ and $h_{\\Delta _t} \\rightarrow h$ in $L^p(\\Lambda _T)$ as $\\Delta _t \\rightarrow 0$ .",
"Taking $\\lambda = \\Delta _t$ and $g= \\Delta _t h_k + u^{k-1} \\in L^\\infty (\\Omega )$ in $(S)$ , using Theorem REF we define the sequence $\\lbrace u^k\\rbrace \\subset X_0(\\Omega )\\cap \\mathcal {C}$ as solutions to problem $\\left\\lbrace \\begin{split}\\frac{u^k-u^{k-1}}{\\Delta _t} +(-\\Delta )^su^k - \\frac{1}{(u^k)^q}&=h_k \\; \\text{in}\\; \\Omega ,\\\\u^k&=0\\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega ,\\end{split}\\right.$ where $u^0 = u_0 \\in X_0(\\Omega )\\cap \\mathcal {C}$ .",
"Now, for $1\\le k\\le n$ , we define $\\forall t \\in [t_{k-1}, t_k),\\left\\lbrace \\begin{split}u_{\\Delta _t}(t,x)&:= u^k(x)\\\\\\tilde{u}_{\\Delta _t}(t,x) &:= \\frac{(u^k(x)-u^{k-1}(x))}{\\Delta _t}(t-t_{k-1})+u^{k-1}(x).\\end{split}\\right.$ Then $u_{\\Delta _t}$ and $\\tilde{u}_{\\Delta _t}$ satisfies $\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t} +(-\\Delta )^su_{\\Delta _t}- \\frac{1}{u_{\\Delta _t}^q}= h_{\\Delta _t} \\in L^{\\infty }(\\Lambda _T).$ At first, we establish some a priori estimates for $u_{\\Delta _t}$ and $\\tilde{u}_{\\Delta _t}$ independent of $\\Delta _t$ .",
"Multiplying (REF ) by $\\Delta _t u^k$ , integrating over $\\mathbb {R}^n$ and summing from $k=1$ to $n^\\prime \\le n$ , using Young's inequality and (REF ) we get for a constant $C>0$ $\\begin{split}&\\sum _{k=1}^{n^\\prime } \\int _\\Omega (u^k-u^{k-1})u^k dx + \\Delta _t\\sum _{k=1}^{n^\\prime }\\left( \\Vert u^k\\Vert _{X_0(\\Omega )}^2 -\\int _\\Omega (u^k)^{1-q}dx \\right)= \\Delta _t \\sum _{k=1}^{n^\\prime }\\int _{\\Omega }h^k u^k dx\\\\&\\le \\Delta _t \\sum _{k=1}^{n^\\prime }\\int _\\Omega \\frac{|h^k|^2}{2}dx + \\Delta _t \\sum _{k=1}^{n^\\prime } \\int _\\Omega \\frac{|u^k|^2}{2}dx \\le \\frac{T}{2}|\\Omega |\\Vert h\\Vert _{L^\\infty (\\Lambda _T)}^2 + \\frac{C\\Delta _t}{2} \\sum _{k=1}^{n^\\prime }\\Vert u^k\\Vert _{X_0(\\Omega )}^2.\\end{split}$ As inequality $(2.7)$ of Theorem $0.9$ in [6], we can estimate the first term of (REF ) as $\\sum _{k=1}^{n^\\prime } \\int _\\Omega (u^k-u^{k-1})u^k dx = \\frac{1}{2} \\sum _{k=1}^{n^\\prime }\\int _\\Omega |u^k-u^{k-1}|^2 dx +\\frac{1}{2} \\int _\\Omega |u^{n^\\prime }|^2 dx - \\frac{1}{2} \\int _\\Omega |u_0|^2dx.$ Let $v$ and $w$ solves (REF ) and define $\\underline{u}= m w \\; \\text{and}\\; \\overline{u}=M w$ where $m >0$ is small enough and $M>0$ is large enough chosen in such a way that $\\left\\lbrace \\begin{split}(-\\Delta )^s\\underline{u}-\\frac{1}{\\underline{u}^q} &\\le -\\Vert h\\Vert _{L^\\infty (\\Lambda _T)} \\; \\text{in}\\; \\Omega ,\\\\(-\\Delta )^s\\overline{u}-\\frac{1}{\\overline{u}^q} &{\\ge \\Vert h\\Vert _{L^\\infty (\\Lambda _T)}} \\; \\text{in}\\; \\Omega .\\end{split}\\right.$ Since $u_0 \\in \\mathcal {C}$ , we can always choose $\\underline{u}$ and $\\overline{u}$ such that it satisfies the above inequalities and $\\underline{u} \\le u_0 \\le \\overline{u} $ .",
"Applying Lemma REF iteratively we get $\\underline{u} \\le u^k \\le \\overline{u}$ for all $k$ .",
"This implies for a.e.",
"$(t,x) \\in [0,T]\\times \\Omega $ , $\\underline{u}(x) \\le u_{\\Delta _t}(t,x), \\tilde{u}_{\\Delta _t}(t,x)\\le \\overline{u}(x)$ i.e.",
"$u_{\\Delta _t},\\tilde{u}_{\\Delta _t} \\in \\mathcal {C}$ uniformly.",
"Now since $q(2s-1)<(2s+1)$ we can estimate the singular term in (REF ) as $\\Delta _t \\sum _{k=1}^{n^\\prime }\\int _\\Omega (u^k)^{1-q}dx \\le \\;\\left\\lbrace \\begin{split}T \\int _\\Omega \\overline{u}^{1-q}dx < +\\infty \\; \\text{if} \\; q \\le 1, \\\\T \\int _\\Omega \\underline{u}^{1-q}dx < +\\infty \\; \\text{if} \\; q > 1.\\end{split}\\right.$ Since $u^k \\in L^\\infty (\\Omega )$ for all $k$ , by the definition of $u_{\\Delta _t}$ and $\\tilde{u}_{\\Delta _t}$ we easily get that $u_{\\Delta _t},\\tilde{u}_{\\Delta _t} \\; \\text{is bounded in}\\; L^\\infty ([0,T], L^\\infty (\\Omega )).$ We see that for $t \\in [t_{k-1},t_k)$ $\\Vert \\tilde{u}_{\\Delta _t}(t,\\cdot )\\Vert _{X_0(\\Omega )}= \\left\\Vert \\frac{(t-t_{k-1})}{\\Delta _t}u^k+ \\frac{(\\Delta _t- t+t_{k-1})}{\\Delta _t}u^{k-1}\\right\\Vert _{X_0(\\Omega )} \\le \\Vert u^k\\Vert _{X_0(\\Omega )}+ \\Vert u^{k-1}\\Vert _{X_0(\\Omega )}.$ Integrating both sides of (REF ) over $(t_{k-1},t_k)$ and using (REF ), (REF ) we get that $u_{\\Delta _t},\\tilde{u}_{\\Delta _t} \\; \\text{is bounded in}\\; L^2([0,T], X_0(\\Omega )).$ We now try to obtain a second energy estimate.",
"Multiplying (REF ) by $u^k-u^{k-1}$ , integrating over $\\mathbb {R}^n$ and summing from $k=1$ to $n^\\prime \\le n$ , using Young's inequality and (REF ) we get $\\begin{split}&\\Delta _t\\sum _{k=1}^{n^\\prime } \\int _\\Omega \\left(\\frac{u^k-u^{k-1}}{\\Delta _t}\\right)^2 dx + {\\sum _{k=1}^{n^\\prime } \\int _{\\mathbb {R}^n} ((-\\Delta )^su^k(x))(u^k-u^{k-1})(x)dx} -\\sum _{k=1}^{n^\\prime } \\int _\\Omega \\frac{(u^k-u^{k-1})}{(u^k)^{q}}dx \\\\&= \\Delta _t\\sum _{k=1}^{n^\\prime }\\int _{\\Omega }\\frac{h^k (u^k-u^{k-1})}{\\Delta _t}~ dx \\le \\frac{\\Delta _t}{2} \\sum _{k=1}^{n^\\prime }\\left(\\int _\\Omega {|h^k|^2}dx + \\int _\\Omega \\left(\\frac{u^k-u^{k-1}}{\\Delta _t}\\right)^2dx\\right)\\\\\\end{split}$ which implies $\\begin{split}&\\frac{\\Delta _t}{2}\\sum _{k=1}^{n^\\prime } \\int _\\Omega \\left(\\frac{u^k-u^{k-1}}{\\Delta _t}\\right)^2 dx + \\sum _{k=1}^{n^\\prime } \\int _{\\mathbb {R}^n}((-\\Delta )^su^k(x))(u^k-u^{k-1})(x)dx\\\\& \\quad \\quad -\\sum _{k=1}^{n^\\prime } \\int _\\Omega \\frac{(u^k-u^{k-1})}{(u^k)^{q}}dx \\le \\frac{|\\Omega |T}{2}\\Vert h\\Vert _{L^{\\infty }(\\Lambda _T)}^2.\\end{split}$ By convexity of the term $\\frac{-1}{1-q}\\int _\\Omega u^{1-q}dx $ , we have $\\frac{1}{1-q} \\int _\\Omega \\left( (u^{k-1})^{1-q}-(u^k)^{1-q} \\right)dx \\le -\\int _\\Omega \\frac{u^k-u^{k-1}}{(u^k)^q}~dx.$ Also ${\\frac{1}{2} \\left(\\Vert u^k\\Vert _{X_0(\\Omega )}^2-\\Vert u^{k-1}\\Vert _{X_0(\\Omega )}^2\\right) \\le \\int _{\\mathbb {R}^n} ((-\\Delta )^su^k(x))(u^k-u^{k-1})(x)dx}.$ Therefore (REF ) gives $\\begin{split}\\frac{\\Delta _t}{2}\\sum _{k=1}^{n^\\prime }& \\int _\\Omega \\left(\\frac{u^k-u^{k-1}}{\\Delta _t}\\right)^2 dx + \\frac{1}{2} \\left(\\Vert u^{n^\\prime }\\Vert _{X_0(\\Omega )}^2-\\Vert u_0\\Vert _{X_0(\\Omega )}^2\\right)\\\\&\\quad + \\frac{1}{1-q} \\int _\\Omega \\left( (u_0)^{1-q}-(u^{n^\\prime })^{1-q} \\right)dx \\le \\frac{|\\Omega |T}{2}\\Vert h\\Vert _{L^{\\infty }(\\Lambda _T)}^2.\\end{split}$ Integrating over $(t_{k-1},t_k)$ on both sides of (REF ) and using (REF ), we get $\\frac{\\Delta _t}{2} \\int _{\\Lambda _T}\\left|\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}\\right|^2~dxdt < +\\infty $ which implies $\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t} \\; \\text{is bounded in}\\; L^2(\\Lambda _T)\\; \\text{uniformly in}\\; \\Delta _t.$ Using definition of $u_{\\Delta _t}$ and $\\tilde{u}_{\\Delta _t}$ , we have that ${u_{\\Delta _t} \\; \\text{and}\\; \\tilde{u}_{\\Delta _t} \\; \\text{are bounded in}\\; L^\\infty ([0,T]; X_0(\\Omega ))\\; \\text{uniformly in}\\; \\Delta _t.",
"}$ Moreover, there exists a constant $C>0$ (independent of $\\Delta _t$ ) such that $\\Vert u_{\\Delta _t}-\\tilde{u}_{\\Delta _t}\\Vert _{L^\\infty ([0,T];L^2(\\Omega ))} \\le \\max _{1\\le k\\le n}\\Vert u^k - u^{k-1}\\Vert _{L^2(\\Omega )} \\le C(\\Delta _t)^{\\frac{1}{2}}.$ Using (REF ) and (REF ), we get $u_{\\Delta _t} \\; \\text{and}\\; \\tilde{u}_{\\Delta _t} \\; \\text{are bounded in}\\; L^\\infty ([0,T]; X_0(\\Omega )\\cap L^\\infty (\\Omega ))\\; \\text{uniformly in}\\; \\Delta _t.$ Using $\\text{weak}^*$ and weak compactness results, we say that as $\\Delta _t \\rightarrow 0^+$ (i.e.",
"$n \\rightarrow \\infty $ ), upto a subsequence $\\begin{split}\\tilde{u}_{\\Delta _t} {\\text{*}} u ,\\;\\;u_{\\Delta _t} {\\text{*}} v\\; \\text{in}\\; L^\\infty ([0,T]; X_0(\\Omega )\\cap L^\\infty (\\Omega ))\\;\\;\\text{and}\\;\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t} \\rightharpoonup \\frac{\\partial u}{\\partial t} \\; \\text{in}\\; L^2(\\Lambda _T)\\end{split}$ where $u,v \\in L^\\infty ([0,T]; X_0(\\Omega )\\cap L^\\infty (\\Omega ))$ such that $\\frac{\\partial u}{\\partial t} \\in L^2(\\Lambda _T)$ .",
"From (REF ), we confer that $u \\equiv v$ .",
"Also from (REF ), we get that $\\underline{u} \\le u \\le \\overline{u}$ .",
"Thus, $u \\in \\mathcal {A}(\\Lambda _T)\\cap \\mathcal {C}$ .",
"Now we will prove that $u$ is a weak solution to $(G^s_t)$ .",
"First we see that for a.e.",
"$ x \\in \\Omega $ , $\\tilde{u}_{\\Delta _t}(\\cdot ,x) \\in C([0,T])$ .",
"By (REF ), we get that $\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}$ is bounded in $L^2(\\Lambda _T)$ uniformly in $\\Delta _t$ .",
"Also, $\\lbrace \\tilde{u}_{\\Delta _t}\\rbrace $ is a bounded family in $X_0(\\Omega )$ and the embedding of $X_0(\\Omega )$ into $L^2(\\Omega )$ is compact.",
"If we define $W:= \\left\\lbrace u \\in C([0,T];X_0(\\Omega )): \\; \\frac{\\partial u}{\\partial t}\\in L^2(\\Lambda _T)\\right\\rbrace ,$ then by Aubin-Lions-Simon Lemma, the embedding $W$ into $C([0,T];L^2(\\Omega ))$ is compact.",
"Therefore, we get that $\\lbrace \\tilde{u}_{\\Delta _t}\\rbrace $ is compact in $C([0,T];L^2(\\Omega ))$ .",
"Using $\\underline{u} \\le \\tilde{u}_{\\Delta _t}\\le \\overline{u}$ again, we get that $\\lbrace \\tilde{u}_{\\Delta _t}\\rbrace $ is compact in $C([0,T];L^p(\\Omega ))$ , $1<p<\\infty $ and therefore as $\\Delta _t \\rightarrow 0^+$ , upto a subsequence $\\tilde{u}_{\\Delta _t} \\rightarrow u \\; \\text{in}\\; C([0,T];{L^2(\\Omega )}).$ This along with (REF ) gives that as $\\Delta _t \\rightarrow 0^+$ $u_{\\Delta _t} \\rightarrow u \\; \\text{in}\\; L^\\infty ([0,T]; {L^2(\\Omega )}).$ Using $(u_{\\Delta _t}-u)$ as the test function in (REF ), we get $\\int _0^T \\int _{\\mathbb {R}^n}\\left(\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}+(-\\Delta )^s u_{\\Delta _t} - u_{\\Delta _t}^{-q}\\right)(u_{\\Delta _t}-u)dxdt= \\int _{\\Lambda _T}h_{\\Delta _t}(u_{\\Delta _t}-u)dxdt.$ Also using (REF ), we know that $\\int _{\\Lambda _T}\\frac{\\partial u}{\\partial t}(\\tilde{u}_{\\Delta _t}-u)dxdt \\rightarrow 0$ as $\\Delta _t \\rightarrow 0^+$ .",
"Hence $\\begin{split}&\\int _{\\Lambda _T}\\left(\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}-\\frac{\\partial u}{\\partial t}\\right)(\\tilde{u}_{\\Delta _t}-u)dxdt-\\int _{\\Lambda _T} u_{\\Delta _t}^{-q}(u_{\\Delta _t}-u)dxdt\\\\&\\quad {+\\int _0^T \\langle (-\\Delta )^s u_{\\Delta _t},(u_{\\Delta _t}-u)\\rangle dt= \\int _{\\Lambda _T}h_{\\Delta _t}(u_{\\Delta _t}-u)dxdt+ o_{\\Delta _t}(1).",
"}\\end{split}$ By (REF ), we have $ u_{\\Delta _t}^{-q} \\le \\underline{u}^{-q}$ .",
"Also since $\\underline{u}\\le u \\le \\overline{u}$ , we apply Lebesgue Dominated convergence theorem with (REF ) to get $\\int _0^T \\int _\\Omega u_{\\Delta _t}^{-q}(u_{\\Delta _t}-u)dxdt \\le \\int _0^T\\int _\\Omega \\underline{u}^{-q}(u_{\\Delta _t}-u)dxdt = o_{\\Delta _t}(1).$ Similarly using (REF ) and (REF ) along with Lebesgue theorem, we get $\\int _{\\Lambda _T}h_{\\Delta _t}(u_{\\Delta _t}-u)dxdt= o_{\\Delta _t}(1).$ Using integration by parts and the fact that $\\tilde{u}_{\\Delta _t}(0,x)= u(0,x)= u_0$ in a.e.",
"$\\Omega $ , we get $2\\int _{\\Lambda _T}\\left(\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}-\\frac{\\partial u}{\\partial t}\\right)(\\tilde{u}_{\\Delta _t}-u)dxdt= \\int _\\Omega (\\tilde{u}_{\\Delta _t}-u)^2(T)dt.",
"$ Therefore, (REF ) implies $\\frac{1}{2}\\int _\\Omega (\\tilde{u}_{\\Delta _t}-u)^2(T)dt {+ \\int _0^T\\langle (-\\Delta )^s u_{\\Delta _t}- (-\\Delta )^s u,u_{\\Delta _t}-u\\rangle dt}= o_{\\Delta _t}(1)$ where we used the fact that $\\int _0^T \\langle (-\\Delta )^su, u_{\\Delta _t}-u\\rangle dt = o_{\\Delta _t}(1)$ which follows from (REF ).",
"Since $u \\lnot \\equiv 0$ identically in $\\Lambda _T$ , using (REF ) we get $\\int _0^T \\Vert (u_{\\Delta _t}-u)(t,\\cdot )\\Vert _{X_0(\\Omega )}^2dt= o_{\\Delta _t}(1).$ Let $(X_0(\\Omega ))^*$ denotes the dual of $X_0(\\Omega )$ .",
"Then the above equations suggest that as $\\Delta _t \\rightarrow 0$ $(-\\Delta )^su_{\\Delta _t} \\rightarrow (-\\Delta )^su \\; \\text{in}\\; L^2([0,T];(X_0(\\Omega ))^*).$ From (REF ), for any $\\phi \\in X_0(\\Omega )$ , using Hardy's inequality and $q(2s-1)<(2s+1)$ we have $\\int _\\Omega | \\phi (u_{\\Delta _t})^{-q} |dx \\le \\int _{\\Omega }|\\phi ||\\underline{u}^{-q}|dx \\le \\left(\\int _\\Omega \\frac{1}{\\delta ^{2s(q-1)/(q+1)}(x)}dx \\right)^{\\frac{1}{2}}\\left(\\int _\\Omega \\frac{\\phi ^2}{\\delta ^{2s}(x)}dx \\right)^{\\frac{1}{2}}<+\\infty .$ Therefore using Lebesgue Dominated convergence theorem we get $\\frac{1}{(u_{\\Delta _t})^q} \\rightarrow \\frac{1}{u^q} \\; \\text{in}\\; L^\\infty ([0,T];(X_0(\\Omega ))^*) \\; \\text{as}\\; \\Delta _t \\rightarrow 0^+.$ Finally, we get $u \\in \\mathcal {A}(\\Lambda _T)$ and for any $\\phi \\in \\mathcal {A}(\\Lambda _T)$ passing on the limit $\\Delta _t \\rightarrow 0^+$ in $\\begin{split}&\\int _{\\Lambda _T}\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t}\\phi ~dxdt+ \\int _0^T \\int _{\\mathbb {R}^n} (-\\Delta )^su_{\\Delta _t}\\phi ~dxdt-\\int _{\\Lambda _T} \\frac{1}{u_{\\Delta _t}^{q}}\\phi ~dxdt= \\int _{\\Lambda _T}h_{\\Delta _t}\\phi ~dxdt,\\end{split}$ using (REF ), (REF ), (REF ) and (REF ), we get $\\begin{split}&\\int _{\\Lambda _T}\\frac{\\partial u}{\\partial t}\\phi ~dxdt+ \\int _0^T \\int _{\\mathbb {R}^n}(-\\Delta )^su\\phi ~dxdt-\\int _{\\Lambda _T} \\frac{1}{{u}^{q}}\\phi ~dxdt= \\int _{\\Lambda _T}h\\phi ~dxdt.\\end{split}$ That is, $u$ is a weak solution to $(G_t^s)$ .",
"Now we show the uniqueness of $u$ as solution of $(G^s_t)$ such that $u(t,\\cdot ) \\in \\mathcal {C}$ , for all $t \\in [0,T]$ .",
"On contrary, let $v$ such that $v(t,\\cdot ) \\in \\mathcal {C}$ , for all $t \\in [0,T]$ distinct from $u$ be another weak solution to $(G^s_t)$ .",
"Then for any $t \\in [0,T]$ , we have $\\begin{split}\\int _\\Omega \\frac{\\partial (u-v)}{\\partial t}(u-v)(t,x)~dx &+ \\int _{\\mathbb {R}^n}((-\\Delta )^s(u-v))(u-v)(t,x)~dx\\\\& \\quad - \\int _\\Omega \\left( \\frac{1}{u^{q}}- \\frac{1}{v^q}\\right)(u-v)dx=0\\end{split}$ which implies $\\begin{split}\\frac{\\partial }{\\partial t}\\left( \\int _\\Omega \\frac{1}{2} (u-v)^2(t,x)~dx\\right)= -\\Vert (u-v)(t,\\cdot )\\Vert _{X_0(\\Omega )}^2 + \\int _\\Omega \\left( \\frac{1}{u^{q}}- \\frac{1}{v^q}\\right)(u-v)(t,x)dx\\le 0.\\end{split}$ So we see that the function $E:[0,T]\\rightarrow \\mathbb {R}$ defined as $E(t):= \\frac{1}{2} \\int _\\Omega (u-v)^2(t,x)~dx$ is a decreasing function.",
"Then since $u,v$ are distinct, we get $0 < E(t) \\le E(0) =0$ which implies $E(t)=0$ , for all $t\\in [0,T]$ .",
"Hence $u \\equiv v$ .$\\square $ Theorem 4.2 The unique weak solution $u$ of $(G^s_t)$ (as obtained in Theorem REF ) belongs to $ C([0,T]; X_0(\\Omega ))$ and ${u(t)} \\in \\mathcal {C}$ uniformly for each $t \\in [0,T]$ .",
"Also, $u$ satisfies (REF ).",
"Proof.",
"We first show that $u \\in C([0,T]; X_0(\\Omega ))$ and then establish (REF ) in order to complete the proof of this theorem.",
"From (REF ), we already have $u \\in C([0,T];L^2(\\Omega ))$ which implies that the map $\\tilde{u}:[0,T]\\rightarrow X_0(\\Omega )$ defined as $[\\tilde{u}(t)](x):= u(t,x)$ is weakly continuous.",
"Also (REF ) gives $u \\in L^\\infty ([0,T];X_0(\\Omega ))$ , which implies $\\tilde{u}(t) \\in X_0(\\Omega )$ and $\\Vert \\tilde{u}(t)\\Vert _{X_0(\\Omega )}\\le \\lim \\inf \\limits _{t\\rightarrow t_0} \\Vert \\tilde{u}(t)\\Vert _{X_0(\\Omega )}$ for all $t_0 \\in [0,T]$ .",
"Multiplying (REF ) by $u^k-u^{k-1}$ , integrating over $\\mathbb {R}^n$ on both sides and summing from $k=n^{\\prime \\prime }$ to $n^{\\prime }$ ($n^\\prime $ has been considered in (REF )) and using (REF ) we get $\\begin{split}\\frac{\\Delta _t}{2}\\sum _{k=n^{\\prime \\prime }}^{n^\\prime }& \\int _\\Omega \\left(\\frac{u^k-u^{k-1}}{\\Delta _t}\\right)^2 dx + \\frac{1}{2} \\left(\\Vert u^{n^\\prime }\\Vert _{X_0(\\Omega )}^2-\\Vert u^{n^{\\prime \\prime } -1}\\Vert _{X_0(\\Omega )}^2\\right)\\\\&\\quad + \\frac{1}{1-q} \\int _\\Omega \\left( \\left(u^{n^{\\prime \\prime }-1}\\right)^{1-q}-\\left(u^{n^\\prime }\\right)^{1-q} \\right)dx \\le \\sum _{k=n^{\\prime \\prime }}^{n^\\prime } \\int _\\Omega h_{\\Delta _t}(u^k-u^{k-1})dx.\\end{split}$ For any $t_1 \\in [t_0,T]$ , we take $n^{\\prime \\prime }$ and $n^{\\prime }$ such that $n^{\\prime \\prime }\\Delta _t \\rightarrow t_1$ and $n^\\prime \\Delta _t\\rightarrow t_0$ as $\\Delta _t\\rightarrow 0^+$ .",
"Using (REF ), (REF ), (REF ) and (REF ), from above inequality we get $\\begin{split}& \\int _{t_0}^{t_1} \\int _\\Omega \\left( \\frac{\\partial u}{\\partial t}\\right)^2 dxdt+ \\frac{1}{2}\\Vert u(t_1,\\cdot )\\Vert _{X_0(\\Omega )}^2-\\frac{1}{1-q}\\int _\\Omega u(t_1)^{1-q}dx\\\\&\\quad \\le \\int _{t_0}^{t_1} \\int _\\Omega h\\frac{\\partial u}{\\partial t}~ dxdt+ \\frac{1}{2}\\Vert u(t_0,\\cdot )\\Vert _{X_0(\\Omega )}^2-\\frac{1}{1-q}\\int _\\Omega u(t_0)^{1-q}dx.\\end{split}$ Since $u \\in L^\\infty ([0,T];L^p(\\Omega ))$ for $1<p<\\infty $ , passing on the limit $t_1 \\rightarrow t_0^+$ , we get $\\lim \\sup \\limits _{t_1\\rightarrow t_0^+}\\Vert u(t_1,\\cdot )\\Vert _{X_0(\\Omega )}\\le \\Vert u(t_0,\\cdot )\\Vert _{X_0(\\Omega )}.$ Therefore $\\lim \\limits _{t_\\rightarrow t_0^+}\\Vert u(t,\\cdot )\\Vert _{X_0(\\Omega )}= \\Vert u(t_0,\\cdot )\\Vert _{X_0(\\Omega )}$ which implies that $u$ is right continuous on $[0,T]$ .",
"Let us now prove the left continuity and assume $t_1>t_0$ .",
"Let $0< r \\le t_1-t_0$ .",
"Define $\\sigma _r(z):= \\frac{u(z+r)-u(r)}{r}.$ Since $u$ is a weak solution to $(G^s_t)$ , taking $\\sigma _r(u)$ as the test function in $(G^s_t)$ , integrating over $(t_0,t_1)\\times \\mathbb {R}^n$ and using (REF ) we get $\\begin{split}&\\int _{t_0}^{t_1}\\int _\\Omega \\frac{\\partial u}{\\partial t}\\sigma _r(u)~dxdt+ \\frac{1}{2r}\\int _{t_0}^{t_1}\\int _{\\mathbb {R}^n}((-\\Delta )^su(t+r,x)- (-\\Delta )^s u(t,x))dxdt\\\\&\\quad - \\frac{1}{r(1-q)}\\int _{t_0}^{t_1}\\int _\\Omega (u^{1-q}(t+r,x)-u^{1-q}(t,x))dxdt\\ge \\int _{t_0}^{t_1}\\int _\\Omega \\sigma _r(u)dxdt.\\end{split}$ Then it is an easy task to get $\\begin{split}&\\int _{t_0}^{t_1}\\int _\\Omega \\frac{\\partial u}{\\partial t}\\sigma _r(u)~dxdt+ \\frac{1}{2r}\\left(\\int _{t_1}^{t_1+r}\\int _{\\mathbb {R}^n}(-\\Delta )^su(t,x)dxdt- \\int _{t_0}^{t_0+r}\\int _{\\mathbb {R}^n}(-\\Delta )^s u(t,x)dxdt\\right)\\\\&\\quad - \\frac{1}{r(1-q)}\\left(\\int _{t_1}^{t_1+r}\\int _\\Omega u^{1-q}(t,x)dxdt-\\int _{t_0}^{t_0+r}\\int _\\Omega u^{1-q}(t,x)dxdt\\right)\\ge \\int _{t_0}^{t_1}\\int _\\Omega \\sigma _r(u)dxdt{.",
"}\\end{split}$ Since $u$ is right continuous in $X_0(\\Omega )$ , using Lebesgue Dominated Convergence theorem we get the following as $r \\rightarrow 0^+$ : $\\begin{split}\\frac{1}{r} \\int _{t_1}^{t_1+r}\\int _{\\mathbb {R}^n}(-\\Delta )^su(t,x)dxdt &\\rightarrow \\int _{\\mathbb {R}^n}(-\\Delta )^su(t_1,x)dx,\\\\\\frac{1}{r} \\int _{t_0}^{t_0+r}\\int _{\\mathbb {R}^n}(-\\Delta )^su(t,x)dxdt &\\rightarrow \\int _{\\mathbb {R}^n}(-\\Delta )^su(t_0,x)dx,\\\\\\frac{1}{r} \\int _{t_1}^{t_1+r}\\int _\\Omega u^{1-q}(t,x)dxdt & \\rightarrow \\int _\\Omega u^{1-q}(t_1,x)dxdt,\\\\\\frac{1}{r} \\int _{t_0}^{t_0+r}\\int _\\Omega u^{1-q}(t,x)dxdt & \\rightarrow \\int _\\Omega u^{1-q}(t_0,x)dxdt.\\end{split}$ Using these estimates in (REF ), as $r \\rightarrow 0^+$ we get $\\begin{split}& \\int _{t_0}^{t_1} \\int _\\Omega \\left( \\frac{\\partial u}{\\partial t}\\right)^2 dxdt+ \\frac{1}{2}\\Vert u(t_1,\\cdot )\\Vert _{X_0(\\Omega )}^2-\\frac{1}{1-q}\\int _\\Omega u(t_1)^{1-q}dx\\\\&\\quad \\ge \\int _{t_0}^{t_1} \\int _\\Omega h\\frac{\\partial u}{\\partial t}~ dxdt+ \\frac{1}{2}\\Vert u(t_0,\\cdot )\\Vert _{X_0(\\Omega )}^2-\\frac{1}{1-q}\\int _\\Omega u(t_0)^{1-q}dx.\\end{split}$ The inequality (REF ) along with (REF ) gives the equality.",
"Since the map $t \\mapsto \\int _\\Omega u^{1-q}(t,x)dt$ is continuous, $u \\in C([0,T]; X_0(\\Omega ))$ .",
"Also, (REF ) is obtained by taking $t_1=t \\in [0,T]$ and $t_0=0$ .$\\square $ Proof of Theorem REF : The proof follows from Theorem REF and Theorem REF .$\\square $ Next, we present the proof of Proposition REF and end this section.",
"Through this Proposition, the solution obtained above for $(G^s_t)$ can be proved to belong in $C([0,T]; C_0(\\overline{\\Omega }))$ if the initial function $u_0 \\in \\overline{\\mathcal {D}(L)}^{L^\\infty }$ .",
"in Section 2.",
"Proof of Proposition REF : Let $u_0 \\in \\overline{\\mathcal {D}(L)}^{L^\\infty }$ .",
"Let $\\lambda >0$ and $f_1,f_2 \\in L^\\infty (\\Omega )$ .",
"Let $u,v \\in X_0(\\Omega )\\cap \\mathcal {C}\\cap C_0(\\overline{\\Omega })$ be the unique solution to $\\left\\lbrace \\begin{split}u+\\lambda L(u) &= f_1 \\; \\text{in}\\; \\Omega ,\\\\v+\\lambda L(v) &= f_2 \\; \\text{in}\\; \\Omega ,\\end{split}\\right.$ as obtained using Theorem REF .",
"Then obviously, $u,v \\in \\mathcal {D}(L)$ .",
"We define $w:= (u-v-\\Vert f_1-f_2\\Vert _{\\infty })^+$ and taking $w$ as test function, from (REF ) we get $\\int _\\Omega w^2dx +\\lambda \\int _\\Omega (L(u)-L(v))w~dx \\le 0.$ It is easy to compute that $\\displaystyle \\int _\\Omega (L(u)-L(v))w~dx \\ge 0$ .",
"So if $\\text{supp}(w)$ has nonzero measure, then $\\int _\\Omega w^2dx +\\lambda \\int _\\Omega (L(u)-L(v))w~dx > 0$ which contradicts (REF ).",
"Therefore $(u-v)\\le \\Vert f_1 -f_2\\Vert _\\infty $ and if we reverse the roles of $u$ and $v$ then we get $\\Vert u-v\\Vert _\\infty \\le \\Vert f_1 -f_2\\Vert _\\infty $ .",
"This proves that $L$ is m-accretive in $L^\\infty (\\Omega )$ .",
"Let $\\tilde{w} \\in \\mathcal {D}(L)$ and $a,b \\in L^\\infty (\\Lambda _T)$ .",
"Then further proof of Proposition REF can be obtained using Chapter 4, Theorem $4.2$ and Theorem $4.4$ of [7] or following proof of Proposition $0.1$ of [6].$\\square $"
],
[
"Existence of unique solution to $(Q^s)$",
"We give the proof of Theorem REF in this section.",
"Before doing that, we prove a weak comparison principle which is needed to prove Theorem REF .",
"We recall the following discrete Picone identity which will be required to prove the weak comparison principle.",
"Lemma 5.1 (Lemma $6.2$ , [4]) Let $p\\in (1,+\\infty )$ .",
"For $u,v : \\Omega \\subset \\mathbb {R}^n \\rightarrow \\mathbb {R}$ such that $u\\ge 0$ , $v> 0$ , we have $M(u,v)\\ge 0\\; \\text{in}\\; \\mathbb {R}^n \\times \\mathbb {R}^n,$ where $\\displaystyle M(u,v)= |u(x)-u(y)|^p-|v(x)-v(y)|^{p-2}(v(x)-v(y))\\left(\\frac{u(x)^p}{v(x)^{p-1}}- \\frac{u(y)^p}{v(y)^{p-1}} \\right).$ The equality holds in $\\Omega $ if and only if $u=kv$ a.e.",
"in $\\Omega $ , for some constant $k$ .",
"Theorem 5.2 Let $g:\\Omega \\times \\mathbb {R}^+\\rightarrow \\mathbb {R}$ be a Carathéodary function bounded below such that the map $y \\mapsto \\frac{g(x,y)}{y}$ is decreasing in $\\mathbb {R}^+$ for a.e.",
"$x \\in \\Omega $ .",
"Let $u,v \\in L^\\infty (\\Omega )\\cap X_0(\\Omega )$ be such that $u,v>0$ in $\\Omega $ , $\\int _{\\Omega }u^{1-q}~dx <+\\infty , \\;\\int _{\\Omega }v^{1-q}~dx < +\\infty $ and satisfies $\\begin{split}(-\\Delta )^s u \\le \\frac{1}{u^q}+ g(x,u)\\; \\text{and}\\;(-\\Delta )^s v \\ge \\frac{1}{v^q}+ g(x,v) \\; \\text{weakly in }\\; (X_0(\\Omega ))^*.\\end{split}$ Moreover, if there exists $0<w\\in L^\\infty (\\Omega )$ such that $c_1 w \\le u,v\\le c_2 w$ , for $c_1,c_2>0$ constants and $\\int _\\Omega |g(x,c_1w)|w~dx < +\\infty ,\\; \\; \\int _\\Omega |g(x,c_2w)|w~dx < +\\infty ,$ then $u\\le v$ in $\\Omega $ .",
"Proof.",
"For $k>0$ , let us define $u_k := u+\\frac{1}{k}$ and $v_k := v+\\frac{1}{k}$ .",
"Also let $\\phi _k := \\frac{u_k^2-v_k^2}{u_k}\\; \\; \\; \\text{and}\\;\\; \\; \\psi _k := \\frac{v_k^2-u_k^2}{v_k}.$ Since $u,v\\in L^\\infty (\\Omega )$ , obviously $u_k,v_k \\in L^\\infty (\\Omega )$ and thus $u_k,v_k \\in L^2(\\Omega )$ .",
"We assumed $u,v \\in X_0(\\Omega )$ , this implies $u,v \\in H^s(\\Omega )$ .",
"Since $\\Vert u_k\\Vert _{H^s(\\Omega )}= \\Vert u\\Vert _{H^s(\\Omega )}$ and $\\Vert v_k\\Vert _{H^s(\\Omega )}= \\Vert v\\Vert _{H^s(\\Omega )}$ we conclude that $u_k,v_k \\in H^s(\\Omega )$ .",
"Let $\\eta _k := \\frac{v_k^2}{u_k} \\; \\text{and}\\; \\xi _k:= \\frac{u_k^2}{v_k} $ then we claim that $\\eta _k, \\xi _k \\in H^s(\\Omega )$ .",
"Consider $\\begin{split}|\\eta _k(x)-\\eta _k(y)| &= \\left| \\frac{v_k^2(x)- v_k^2(y)}{u_k(x)}- \\frac{v_k^2(y)(u_k(x)-u_k(y))}{u_k(x)u_k(y)}\\right|\\\\&\\le k|v_k(x)-v_k(y)||v_k(x)+v_k(y)|+ \\Vert v_k\\Vert _{L^{\\infty }(\\Omega )}^2 \\frac{|u_k(x)-u_k(y)|}{u_k(x)u_k(y)}\\\\& \\le 2k\\Vert v_k\\Vert _{L^{\\infty }(\\Omega )}|v_k(x)-v_k(y)| + k^2 \\Vert v_k\\Vert _{L^{\\infty }(\\Omega )}^2 |u_k(x)-u_k(y)|\\\\& \\le C(k,\\Vert v_k\\Vert _{L^{\\infty }(\\Omega )})(|v_k(x)-v_k(y)| +|u_k(x)-u_k(y)|),\\end{split}$ where $ C(k,\\Vert v_k\\Vert _{L^{\\infty }(\\Omega )})>0$ is a constant.",
"Since $u_k,v_k \\in H^s(\\Omega )$ , $\\eta _k \\in H^s(\\Omega )$ .",
"Similarly $\\xi _k\\in H^s(\\Omega )$ .",
"Clearly, this implies that $\\phi _k,\\psi _k \\in H^s(\\Omega )$ .",
"We note that $\\phi _k, \\psi _k$ can also be written as $\\phi _k= \\frac{(u-v)(u_k+v_k)}{u_k}\\; \\text{and}\\; \\psi _k= \\frac{(v-u)(v_k+u_k)}{v_k}$ which implies that $\\phi _k,\\psi _k = 0$ in $\\mathbb {R}^n \\setminus \\Omega $ i.e.",
"$\\phi _k,\\psi _k \\in X_0(\\Omega )$ since $\\frac{u_k+v_k}{u_k}$ and $\\frac{u_k+v_k}{v_k}$ in $L^\\infty (\\Omega )$ .",
"We set $\\Omega ^+=\\lbrace x\\in \\Omega : u(x)>v(x)\\rbrace $ and $\\Omega ^-=\\lbrace x\\in \\Omega : u(x)\\le v(x)\\rbrace $ .",
"Then $\\phi _k \\ge 0$ and $\\psi _k \\le 0$ in $\\Omega ^+$ .",
"Let $\\tilde{\\phi }_k = \\chi _{\\Omega ^+}\\phi _k$ and $\\tilde{\\psi }_k = \\chi _{\\Omega ^+}\\psi _k$ .",
"Since $\\phi _k(x)\\le \\phi _k(x)-\\phi _k(y)$ for $(x,y)\\in \\Omega ^+\\times \\Omega ^-$ , we get $&\\int _Q \\frac{|\\tilde{\\phi }_k(x)- \\tilde{\\phi }_k(y)|^2}{|x-y|^{n+2s}}~dxdy\\\\&= \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{| \\phi _k(x)- \\phi _k(y)|^2}{|x-y|^{n+2s}}~dxdy+ 2\\int _{\\Omega ^+}\\int _{\\Omega ^-} \\frac{| \\phi _k(x)|^2}{|x-y|^{n+2s}}~dxdy+ 2 \\int _{\\Omega ^+}\\int _{\\mathcal {C} \\Omega } \\frac{| \\phi _k(x)|^2}{|x-y|^{n+2s}}~dxdy \\\\&\\le \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{| \\phi _k(x)- \\phi _k(y)|^2}{|x-y|^{n+2s}}~dxdy+2\\int _{\\Omega ^+}\\int _{\\Omega ^-} \\frac{| \\phi _k(x)- \\phi _k(y)|^2}{|x-y|^{n+2s}}~dxdy\\\\&\\quad + 2 \\int _{\\Omega }\\int _{\\mathcal {C} \\Omega } \\frac{| \\phi _k(x)|^2}{|x-y|^{n+2s}}~dxdy = \\Vert \\phi _k\\Vert _{X_0(\\Omega )}^2<+\\infty .$ This implies $\\tilde{\\phi }_k \\in X_0(\\Omega )$ since by definition $\\tilde{\\phi }_k =0 $ in $\\mathbb {R}^n\\setminus \\Omega $ .",
"Similarly, $\\tilde{\\psi }_k \\in X_0(\\Omega )$ .",
"Using $\\tilde{\\phi }_k$ and $\\tilde{\\psi }_k$ as test functions in (REF ), we get $\\begin{split}\\int _{\\mathbb {R}^n}((-\\Delta )^s u)\\tilde{\\phi }_k~dx &\\le \\int _{\\Omega ^+}\\left(\\frac{1}{u^q}+ g(x,u)\\right)\\phi _k~dx,\\\\\\int _{\\mathbb {R}^n}((-\\Delta )^s v)\\tilde{\\psi }_k~dx &\\le \\int _{\\Omega ^+}\\left(\\frac{1}{v^q}+ g(x,v)\\right)\\psi _k~dx.\\end{split}$ Consider $\\begin{split}&\\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{(u(x)-u(y))(\\phi _k(x)-\\phi _k(y))}{|x-y|^{n+2s}}~dxdy+ \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{(v(x)-v(y))(\\psi _k(x)-\\psi _k(y))}{|x-y|^{n+2s}}~dxdy\\\\& = \\int _{\\Omega ^+}\\int _{\\Omega ^+}\\frac{(u_k(x)-u_k(y))^2 +(v_k(x)-v_k(y))^2}{|x-y|^{n+2s}}~dxdy\\\\&\\quad \\quad - \\int _{\\Omega ^+}\\int _{\\Omega ^+}\\frac{\\left((v_k(x)-v_k(y))\\left( \\frac{u_k^2(x)}{v_k(x)}-\\frac{u_k^2(y)}{v_k(y)}\\right)+(u_k(x)-u_k(y))\\left( \\frac{v_k^2(x)}{u_k(x)}-\\frac{v_k^2(y)}{u_k(y)}\\right) \\right)}{|x-y|^{n+2s}}~dxdy\\\\& = \\int _{\\Omega ^+}\\int _{\\Omega ^+}\\frac{M(u_k,v_k)+ M(v_k,u_k)}{|x-y|^{n+2s}}~dxdy\\ge 0,\\end{split}$ using Lemma REF with $p=2$ .",
"We have $\\int _{\\Omega ^+}\\left(\\frac{ \\phi _k}{u^q}+ \\frac{ \\psi _k}{v^q}\\right)~dx \\le 0.$ Using this, we get $\\begin{split}&\\int _{\\Omega ^+}\\left( \\frac{1}{u^q}+g(x,u)\\right)\\phi _k~dx+ \\int _{\\Omega ^+}\\left( \\frac{1}{v^q}+g(x,v)\\right)\\psi _k~dx\\\\& \\le \\int _{\\Omega ^+}(g(x,u)\\phi _k+ g(x,v)\\psi _k)~dx= \\int _{\\Omega ^+}\\left(\\frac{g(x,u)}{u}\\left(\\frac{u}{u_k}\\right)- \\frac{g(x,v)}{v}\\left(\\frac{v}{v_k}\\right)\\right)(u_k^2-v_k^2)~dx.\\end{split}$ Since $\\frac{u}{u_k}\\rightarrow 1$ and $\\frac{v}{v_k}\\rightarrow 1$ a.e.",
"in $\\Omega $ as $k\\rightarrow +\\infty $ , using (REF ) and Lebesgue Dominated convergence theorem with (REF ) we get $\\lim \\limits _{k \\rightarrow +\\infty }{\\int _{\\Omega ^+}}(g(x,u)\\phi _k+g(x,v)\\psi _k)~dx=0$ .",
"Therefore (REF ) implies that $\\lim _{k\\rightarrow +\\infty }\\int _{\\Omega ^+ }\\left( \\frac{1}{u^q}+g(x,u)\\right)\\phi _k~dx+ \\int _{\\Omega ^+}\\left( \\frac{1}{v^q}+g(x,v)\\right)\\psi _k~dx \\le 0.$ From (REF ), we have that $\\int _{\\Omega ^+}(((-\\Delta )^s u)\\phi _k+((-\\Delta )^s v)\\psi _k)~dx \\le \\int _{\\Omega ^+}\\left(\\left(\\frac{1}{u^q}+ g(x,u)\\right)\\phi _k+\\left(\\frac{1}{v^q}+ g(x,v)\\right)\\psi _k\\right)~dx,$ We claim that $\\begin{split}&\\int _Q \\frac{(u(x)-u(y))(\\tilde{\\phi }_k(x)-\\tilde{\\phi }_k(y))}{|x-y|^{n+2s}}~dxdy+ \\int _Q \\frac{(v(x)-v(y))(\\tilde{\\psi }_k(x)-\\tilde{\\psi }_k(y))}{|x-y|^{n+2s}}~dxdy\\\\&\\ge \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{(u(x)-u(y))( \\phi _k(x)- \\phi _k(y))+ (v(x)-v(y))(\\psi _k(x)-\\psi _k(y))}{|x-y|^{n+2s}}~dxdy\\end{split}$ To prove this we consider $&\\int _Q \\frac{(u(x)-u(y))(\\tilde{\\phi }_k(x)-\\tilde{\\phi }_k(y))}{|x-y|^{n+2s}}~dxdy+ \\int _Q \\frac{(v(x)-v(y))(\\tilde{\\psi }_k(x)-\\tilde{\\psi }_k(y))}{|x-y|^{n+2s}}~dxdy\\\\&= \\int _{\\Omega ^+}\\int _{\\Omega ^+} \\frac{(u(x)-u(y))( \\phi _k(x)- \\phi _k(y)) + (v(x)-v(y))(\\psi _k(x)-\\psi _k(y))}{|x-y|^{n+2s}}~dxdy\\\\& \\quad +2 \\int _{\\Omega ^+}\\int _{\\Omega ^-}\\frac{(u(x)-u(y))\\phi _k(x)+ (v(x)-v(y))\\psi _k(x)}{|x-y|^{n+2s}}~dxdy \\\\&\\quad \\quad + 2 \\int _{\\Omega ^+}\\int _{\\mathcal {C} \\Omega } \\frac{(u(x)-u(y))\\phi _k(x)+ (v(x)-v(y))\\psi _k(x)}{|x-y|^{n+2s}}~dxdy.$ Since $\\phi _ku_k+\\psi _kv_k =0$ by definition and $\\phi _k+\\psi _k \\le 0$ in $\\Omega ^+$ and $\\Omega ^-$ both, we get $&\\int _{\\Omega ^+}\\int _{\\Omega ^-}\\frac{(u(x)-u(y))\\phi _k(x)+ (v(x)-v(y))\\psi _k(x)}{|x-y|^{n+2s}}~dxdy\\\\&= \\int _{\\Omega ^+}\\int _{\\Omega ^-}\\frac{(u_k(x)-u_k(y))\\phi _k(x)+ (v_k(x)-v_k(y))\\psi _k(x)}{|x-y|^{n+2s}}~dxdy\\\\& = - \\int _{\\Omega ^+}\\int _{\\Omega ^-}\\frac{u_k(y)\\phi _k(x)+ v_k(y)\\psi _k(x)}{|x-y|^{n+2s}}~dxdy \\ge - \\int _{\\Omega ^+}\\int _{\\Omega ^-}\\frac{v_k(y)(\\phi _k(x)+\\psi _k(x))}{|x-y|^{n+2s}}~dxdy\\ge 0.$ Similarly $&\\int _{\\Omega ^+}\\int _{\\mathcal {C} \\Omega } \\frac{(u(x)-u(y))\\phi _k(x)+ (v(x)-v(y))\\psi _k(x)}{|x-y|^{n+2s}}~dxdy\\\\& \\quad \\quad =\\int _{\\Omega ^+}\\int _{\\mathcal {C} \\Omega } \\frac{-(\\phi _k(x)+\\psi _k(x))}{k|x-y|^{n+2s}}~dxdy \\ge 0.$ This establishes our claim.",
"Therefore using (REF ), (REF ), (REF ), (REF ) and Fatou's Lemma , we get $\\begin{split}0 & \\le \\int _{\\Omega ^+}\\int _{\\Omega ^+}\\frac{M(u,v)+ M(v,u)}{|x-y|^{n+2s}}~dxdy\\le \\lim _{k \\rightarrow +\\infty }\\left(\\int _{\\mathbb {R}^n}((-\\Delta )^su)\\tilde{\\phi }_k~dx + \\int _{\\mathbb {R}^n}((-\\Delta )^sv)\\tilde{\\psi }_k~dx\\right)\\\\&\\le \\lim _{k \\rightarrow +\\infty }\\int _{\\Omega ^+}\\left(\\left(\\frac{1}{u^q}+ g(x,u)\\right)\\phi _k+\\left(\\frac{1}{v^q}+ g(x,v)\\right)\\psi _k\\right)~dx \\le 0.\\end{split}$ This implies that $\\int _{\\Omega ^+}\\int _{\\Omega ^+}\\frac{M(u,v)+M(v,u)}{|x-y|^{n+2s}}~dxdy =0.$ Therefore $M(u,v)=0= M(v,u)$ a.e.",
"in $\\Omega ^+$ .",
"So using Lemma REF , we have $u = kv$ a.e.",
"in $\\Omega ^+$ , for some constant $k>0$ .",
"By definition of $\\Omega ^+$ , we have $k>1$ .",
"Consider $\\begin{split}&\\int _{\\Omega ^+}(((-\\Delta )^su)u-((-\\Delta )^s kv)kv)~dx = \\int _{\\Omega ^+}((-\\Delta )^su- (-\\Delta )^s kv)kv)~dx\\\\& = \\int _{\\Omega ^+} ( (-\\Delta )^s(u-kv))kv~dx =2C_n^s\\int _{\\Omega ^+}\\left( P.V.\\int _{\\mathbb {R}^n}\\frac{(u-kv)(x)- (u-kv)(y)}{|x-y|^{n+2s}}dy\\right)kv(x)dx\\\\& = 2C_n^s \\int _{\\Omega ^+}P.V.\\int _{\\Omega ^-}\\frac{(kv-u)(y)}{|x-y|^{n+2s}}kv(x)dxdy \\ge 2C_n^s k^2 \\int _{\\Omega ^+}P.V.\\int _{\\Omega ^-}\\frac{(v-u)(y)}{|x-y|^{n+2s}}v(x)dxdy\\ge 0.\\end{split}$ From (REF ) and (REF ) we get $\\begin{split}\\int _{\\Omega ^+}((-\\Delta )^su)u~dx &\\le \\int _{\\Omega ^+} \\left(\\frac{g(x,kv)}{kv}(kv)^2 + k^{1-q}v^{1-q}\\right)~dx\\; \\text{and}\\;\\\\k^2 \\int _{\\Omega ^+}((-\\Delta )^sv)v~dx & \\ge \\int _{\\Omega ^+} \\left(\\frac{g(x,v)}{v}(kv)^2 + k^2 v^{1-q}\\right)dx\\end{split}$ which implies that $k\\le 1$ by (REF ).",
"This gives a contradiction which implies $u\\le v$ in $\\Omega $ .",
"$\\square $ Proof of Theorem REF : Under the hypothesis on $f$ , we let $l,\\mu >0$ be such that $-l \\le f(x,y)\\le \\mu y+l$ .",
"Let $\\mu $ be such that $0< \\mu < \\lambda _{1,s}(\\Omega )$ .",
"Suppose $w$ is a solution of (REF ).",
"For $\\eta >0$ , we define $\\underline{u}= \\eta w.$ Since $w \\in \\mathcal {C} \\cap C_0(\\overline{\\Omega })$ (see (REF )-()), we can choose $\\eta >0$ small enough such that $\\begin{split}(-\\Delta )^s\\underline{u} -\\frac{1}{\\underline{u}^q} &\\le -l \\le f(x,\\underline{u}) \\; \\text{in}\\; \\Omega ,\\; \\; \\underline{u}=0 \\; \\text{in}\\; \\mathbb {R}^n \\setminus \\Omega .\\end{split}$ Let $0<M,M^\\prime $ and $\\overline{u}= Mw+M^\\prime \\phi _{1,s}$ Let $\\epsilon >0$ and define $\\Omega _\\epsilon := \\lbrace x \\in \\Omega :\\; \\text{dist}(x,\\partial \\Omega )<\\epsilon \\rbrace $ .",
"Since we know that $w=0$ in $\\mathbb {R}^n \\setminus \\Omega $ , we can choose $\\epsilon >0$ small enough such that $0\\le w\\le c$ in $\\Omega _\\epsilon $ where $c>0$ is such that $\\left(M-\\frac{1}{M^q}\\right)\\frac{1}{c^q} \\ge \\mu Mc+l$ which is possible for $c>0$ sufficiently small.",
"Therefore in $\\Omega _\\epsilon $ we get $\\begin{split}(-\\Delta )^s\\overline{u}- \\frac{1}{\\overline{u}^q} &= \\left(M-\\frac{1}{M^q}\\right)\\frac{1}{w^q} + M^\\prime \\lambda _{1,s}\\phi _{1,s}\\ge \\left(M-\\frac{1}{M^q}\\right)\\frac{1}{c^q}+ M^\\prime \\mu \\phi _{1,s} \\\\& \\ge \\mu Mc+l + M^\\prime \\mu \\phi _{1,s} \\ge \\mu Mw +l + M^\\prime \\mu \\phi _{1,s}= \\mu \\overline{u}+l.\\end{split}$ Now consider the set $\\Omega \\setminus \\Omega _\\epsilon = \\lbrace x \\in \\Omega :\\; d(x,\\partial \\Omega )\\ge \\epsilon \\rbrace $ .",
"Then there exists a constant $c_1>0$ (depending on $\\epsilon $ ) such that $0< c_1\\le \\phi _{1,s}$ in $\\Omega \\setminus \\Omega _\\epsilon $ .",
"Since $\\mu < \\lambda _{1,s}$ and $M$ is fixed now, we choose $M^\\prime \\ge \\frac{\\mu M\\Vert w\\Vert _\\infty +l}{c_1(\\lambda _{1,s}-\\mu )}.$ Then in $\\Omega \\setminus \\Omega _\\epsilon $ we get $\\begin{split}(-\\Delta )^s\\overline{u}- \\frac{1}{\\overline{u}^q} = \\left(M-\\frac{1}{M^q}\\right)\\frac{1}{w^q} + M^\\prime \\lambda _{1,s}\\phi _{1,s}\\ge M^\\prime \\lambda _{1,s}\\phi _{1,s} \\ge \\mu Mw +l + M^\\prime \\mu \\phi _{1,s}= \\mu \\overline{u}+l.\\end{split}$ Therefore (REF ) and (REF ) implies that $\\overline{u}$ satisfies $\\begin{split}(-\\Delta )^s\\overline{u} -\\frac{1}{\\overline{u}^q} & \\ge \\mu \\overline{u}+l \\ge f(x,\\overline{u}) \\; \\text{in}\\; \\Omega ,\\; \\; \\overline{u}=0 \\; \\text{in}\\; \\mathbb {R}^n \\setminus \\Omega .\\end{split}$ By construction, $\\underline{u}, \\overline{u}\\in \\mathcal {C}$ .",
"Since $f$ uniformly locally lipschitz with respect to second variable, we can find appropriate constant $K_0>0$ such that the map $t \\mapsto K_0 t + f(x,t)$ is non-decreasing in $[0,\\Vert \\overline{u}\\Vert _{X_0(\\Omega )}]$ , for a.e.",
"$x \\in \\Omega $ .",
"We define an iterative scheme to obtain a sequence $\\lbrace u_k\\rbrace \\subset X_0(\\Omega )\\cap \\mathcal {C}\\cap C_0(\\overline{\\Omega })$ (using Theorem REF ) as solution to the problem $\\left\\lbrace \\begin{split}(-\\Delta )^s u_k -\\frac{1}{u_k^q}+K_0 u_k = f(x,u_{k-1})+K_0 u_{k-1}\\; \\text{in}\\; \\Omega ,\\;\\;u_k =0, \\; \\text{in}\\; \\mathbb {R}^n\\setminus \\Omega ,\\end{split}\\right.$ where $u_0:= \\underline{u}$ .",
"This scheme is well defined because by the choice of $K_0$ and using weak comparison principle (Lemma REF ), we get that ${\\underline{u} \\le } u_k \\le \\overline{u},$ for all $k$ .",
"This implies for each $k$ , right hand side of (REF ) is in $L^\\infty (\\Lambda _T)$ and hence Theorem REF is applicable for (REF ).",
"Again using Lemma REF and monotonicity of the map $t \\mapsto K_0 t + f(x,t)$ , we have that the sequence $\\lbrace u_k\\rbrace $ is a monotone increasing sequence.",
"From (REF ) we have $(\\Delta )^su_k = g_k \\in L^\\infty (\\Omega ^\\prime )$ , where $g_k := u_k^{-q}- K_0 u_k + f(x,u_{k-1})+K_0 u_{k-1} \\le \\underline{u}^{-q}-K_0 \\underline{u}+ f(x,\\overline{u})+ K_0 \\overline{u}$ and $\\Omega ^\\prime $ is a compact subset of $\\Omega $ .",
"Following proof of Theorem $1.2$ of [2], we get that $u_k \\in C^{s-\\epsilon }(\\mathbb {R}^n)$ for each $\\epsilon >0$ small enough when $q=1$ and $u_k\\in C^{\\frac{2s}{q+1}}(\\mathbb {R}^n)$ when $q>1$ .",
"Also since (REF ) holds, we get that $\\lbrace u_k\\rbrace $ is a uniformly bounded sequence in $C_0(\\overline{\\Omega })\\cap \\mathcal {C}$ .",
"Therefore by Arzela Ascoli theorem we know that there exist $\\tilde{u} \\in C_0(\\overline{\\Omega })\\cap \\mathcal {C}$ such that $u_k \\uparrow \\tilde{u}$ in $C_0(\\overline{\\Omega })\\cap \\mathcal {C}$ as $k \\rightarrow \\infty $ .",
"Therefore it must be Cauchy in $C_0(\\overline{\\Omega })\\cap \\mathcal {C}$ and this alongwith (REF ) gives that $\\lbrace u_k\\rbrace $ is Cauchy in $X_0(\\Omega )$ which converges to $\\tilde{u}$ in $X_0(\\Omega )$ .",
"Now passing on to the limits as $k\\rightarrow \\infty $ and using Lebesgue Dominated convergence theorem (since $u_k\\le \\overline{u}$ , for all $k$ ) in (REF ), we obtain $\\tilde{u}$ to be solution to $(Q^s)$ .",
"Lastly, uniqueness of $\\tilde{u}$ follows from Theorem REF .$\\square $"
],
[
"Existence of solution to $(P^s_t)$ and its regularity",
"We devote this section to study the problem $(P^s_t)$ which is our concern for this article.",
"Precisely, we will prove Theorem REF and Proposition REF .",
"Proof of Theorem REF : We will closely make use of arguments in the proof of Theorem REF while proving this theorem.",
"Since $T>0$ , we define $\\Delta _t:= \\frac{T}{n}$ , where $n \\in \\mathbb {N}^*$ .",
"Taking $u^0= u_0$ , we obtain a sequence $\\lbrace u^k\\rbrace \\subset \\mathcal {C} \\cap X_0(\\Omega ) \\subset L^\\infty (\\Omega )$ as solutions to following iterative scheme $\\begin{split}u^k- \\Delta _t \\left((-\\Delta )^s u^k + \\frac{1}{(u^k)^q} \\right) = \\Delta _t f(x,u^{k-1})+u^{k-1}\\; \\text{in}\\; \\Omega .\\end{split}$ Since $u^0 \\in \\mathcal {C}\\cap X_0(\\Omega )$ and $\\Delta _t f(x,u^{k-1})+ u^{k-1} \\in L^\\infty (\\Lambda _T)$ for each $k$ , we can apply Theorem REF to obtain the sequence $\\lbrace u^k\\rbrace \\subset \\mathcal {C} \\cap X_0(\\Omega ) \\subset L^\\infty (\\Omega )$ .",
"In (REF ) and (REF ), we can choose $\\eta ,M, M^\\prime >0$ appropriately such that $\\underline{u} \\le u_0 \\le \\overline{u}$ (since $u_0 \\in \\mathcal {C}$ ).",
"Using $-l \\le f(x,y)\\le \\mu y +l $ and applying Lemma REF iteratively, we can get $\\underline{u} \\le u^k\\le \\overline{u}$ , for all $k$ .",
"We remark that it is clear from definition in (REF ) that $\\underline{u}$ and $\\overline{u}$ are independent of $\\Delta _t$ .",
"Let $u_{\\Delta _t}$ and $\\tilde{u}_{\\Delta _t}$ be as defined in (REF ) alongwith the assumption that $u_{\\Delta _t}(t)= u_0$ , when $t<0$ .",
"Then it is easy to see that (REF ) is satisfied with $h_{\\Delta _t}(t,x):= f(x,u_{\\Delta _t}(t-\\Delta _t,x))$ , for $t \\in [0,T]$ and $x \\in \\Omega $ .",
"Using (REF ), we have $ \\underline{u} \\le u_{\\Delta _t} \\le \\overline{u}$ .",
"Therefore, $h_{\\Delta _t}(t,x) \\le \\mu u_{\\Delta _t}(t-\\Delta _t,x)+l \\in L^\\infty (\\Lambda _T)$ independent of $\\Delta _t$ .",
"Hence we can use similar techniques as in proof of Theorem REF to get $\\begin{split}&u_{\\Delta _t}, \\tilde{u}_{\\Delta _t} \\in L^\\infty ([0,T]; X_0(\\Omega )\\cap \\mathcal {C}), \\; u_{\\Delta _t}, \\tilde{u}_{\\Delta _t} \\in L^\\infty (\\Lambda _T),\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t} \\in L^2(\\Lambda _T),\\\\&\\;\\Vert u_{\\Delta _t} - \\tilde{u}_{\\Delta _t}\\Vert _{L^2(\\Omega )} \\le C(\\Delta _t)^{\\frac{1}{2}}\\; \\text{and}\\; \\frac{1}{(u_{\\Delta _t})^q} \\in L^\\infty ([0,T]; (X_0(\\Omega ))^*)\\end{split}$ uniformly in $\\Delta _t$ .",
"So we can use the Banach Alaoglu theorem and (REF ) to get $u \\in L^\\infty ([0,T];X_0(\\Omega ))$ and $u \\in L^\\infty (\\Lambda _T)$ such that, upto a subsequence, $\\begin{split}u_{\\Delta _t}, \\tilde{u}_{\\Delta _t} {\\text{*}} L^\\infty ([0,T]; X_0(\\Omega ))\\; \\text{and in}\\; L^\\infty (\\Lambda _T),\\;\\frac{\\partial \\tilde{u}_{\\Delta _t}}{\\partial t} \\rightharpoonup \\frac{\\partial u}{\\partial t} \\; \\text{in}\\; L^2(\\Lambda _T)\\end{split}$ as $\\Delta _t \\rightarrow 0^+$ .",
"Also similar to proof of Theorem REF , we get $u_{\\Delta _t}, \\; \\tilde{u}_{\\Delta _t} \\rightarrow u \\; \\text{in}\\; L^\\infty ([0,T];L^2(\\Omega ))\\; \\text{and}\\; u \\in C([0,T];L^2(\\Omega )).$ In addition, if $M>0$ denotes the Lipschitz constant for $f$ then for $t \\in [0,T]$ $\\begin{split}\\Vert h_{\\Delta _t}(t,\\cdot )- f(\\cdot ,u(t,\\cdot ))\\Vert _{L^2(\\Omega )}&= \\Vert f(\\cdot , u_{\\Delta _t}(t-\\Delta _t,\\cdot ))- f(\\cdot ,u(t,\\cdot ))\\Vert _{L^2(\\Omega )}\\\\&\\le M\\Vert u_{\\Delta _t}(t-\\Delta _t,\\cdot )- u(t,\\cdot )\\Vert _{L^2(\\Omega )}.\\end{split}$ From (REF ) and (REF ), we deduce that $h_{\\Delta _t}(t,x) \\rightarrow f(x,u(x))$ in $L^\\infty ([0,T]; L^2(\\Omega ))$ .",
"Finally, following exactly the last part of the proof of Theorem REF , we can show that $u \\in \\mathcal {A}(\\Lambda _T)$ and $u$ is a weak solution to $(P_t^s)$ .",
"It remains to prove the uniqueness.",
"For that, let $v\\in \\mathcal {A}(\\Lambda _T)$ be another weak solution to $(P^s_t)$ .",
"For fix $t_0 \\in [0,T]$ we have $\\begin{split}&\\int _0^{t_0}\\int _{\\Omega } \\frac{\\partial (u-v)}{\\partial t}(u-v)~dxdt+ \\int _0^{t_0}\\int _{\\mathbb {R}^n}((-\\Delta )^s(u-v))(u-v)~dxdt\\\\& \\quad - \\int _0^{t_0}\\int _{\\Omega } \\left( \\frac{1}{u^q}-\\frac{1}{v^q}\\right)(u-v)~dxdt= \\int _0^{t_0}\\int _{\\Omega }(f(x,u(x)- f(x,v(x))))(u-v)~dxdt.\\end{split}$ From (REF ), $u(0,x)=v(0,x)=u_0(x)$ in $\\Omega $ and $f$ being locally Lipschitz uniformly in $\\Omega $ , we get $\\begin{split}&\\frac{1}{2}\\Vert (u-v)(t_0)\\Vert _{L^2(\\Omega )}+ \\int _0^{t_0}\\int _{\\mathbb {R}^n}((-\\Delta )^s(u-v))(u-v)~dxdt- \\int _0^{t_0}\\int _{\\Omega }\\left( \\frac{1}{u^q}-\\frac{1}{v^q}\\right)(u-v)~dxdt\\\\&\\le M \\int _0^{t_0}\\int _{\\Omega }|u-v|^2~dxdt,\\end{split}$ where $M$ is Lipschitz constant for $f$ .",
"From Lemma REF , we know that the operator $A$ is strictly monotone which gives $\\begin{split}0 & < \\int _{0}^{t_0}\\int _{\\Omega }|(u-v)|^2~dxdt+ \\int _0^{t_0}\\int _{\\mathbb {R}^n}((-\\Delta )^s(u-v))(u-v)~dxdt\\\\& \\quad \\quad -\\int _0^{t_0}\\int _{\\Omega } \\left( \\frac{1}{u^q}-\\frac{1}{v^q}\\right)(u-v)~dxdt.\\end{split}$ Using this with (REF ), we get $\\frac{1}{2}\\Vert (u-v)(t_0)\\Vert _{L^2(\\Omega )}\\le M_0\\int _0^{t_0}\\int _{\\Omega }|u-v|^2~dxdt, $ where $M_0>0$ is a constant.",
"By Gronwall's inequality, we get $\\Vert (u-v)(t_0,\\cdot )\\Vert _{L^2(\\Omega )}\\le \\Vert (u-v)(0,\\cdot )\\Vert _{L^2(\\Omega )}\\exp (M_0t_0)$ .",
"Since $u(0,\\cdot )=v(0,\\cdot )$ and this holds for all $t_0\\in [0,T]$ , we get $u\\equiv v$ .",
"This completes the proof.$\\square $ Now we give the proof of Proposition REF .",
"Proof of Proposition REF : Using Proposition REF above and following the proof of Proposition $0.2$ of [6], the result can be similarly obtained.",
"$\\square $"
],
[
"Asymptotic Behavior",
"In this section, we present the proof of Theorem REF .",
"Proof of Theorem REF : Let $\\underline{u}, \\overline{u} \\in \\mathcal {C} \\cap X_0(\\Omega )\\cap C_0(\\overline{\\Omega })$ , be the sub and supersolution respectively to $\\left\\lbrace \\begin{split}(-\\Delta )^s u - \\frac{1}{u^q}&= f(x,u)\\; \\text{in}\\; \\Omega ,\\\\u &=0 \\; \\text{in} \\; \\mathbb {R}^n \\setminus \\Omega ,\\end{split}\\right.$ where $\\underline{u}, \\overline{u}$ is defined in (REF ).",
"We can choose $\\eta >0$ small enough and $M>0$ large enough so that $\\underline{u}\\le u_0 \\le \\overline{u}$ which is possible because we took $u_0 \\in \\mathcal {C} \\cap X_0(\\Omega )$ .",
"Let $u$ be solution of $(P_t^s)$ and $v_1$ and $ v_2$ be unique solutions to $(P^s_t)$ with initial datum $\\underline{u}$ and $\\overline{u}$ .",
"The existence of $v_1$ and $v_2$ are justified through Theorem REF .",
"We claim that $\\underline{u}, \\overline{u}\\in \\overline{\\mathcal {D}(L)}^{L^\\infty (\\Omega )}$ .",
"Let $g,h \\in (X_0(\\Omega ))^{*}$ be functions such that $L(\\underline{u}) = g$ and $L(\\overline{u}) = h$ .",
"Using (REF ), we have $g\\le 0$ and $h \\ge 0$ .",
"Now, let $\\lbrace g_k\\rbrace = \\max \\lbrace g, -k\\rbrace $ , $\\lbrace h_k\\rbrace = \\min \\lbrace h,k\\rbrace $ and $\\lbrace u_k\\rbrace ,\\lbrace w_k\\rbrace $ be two sequences in $\\mathcal {D}(L)$ defined by $L(u_k)= g_k$ , $L(w_k)=h_k$ .",
"Since $L$ is a monotone operator, as Lemma REF we can show a similar kind of weak comparison principle concerning $L$ .",
"Using that, we can get $\\lbrace u_k\\rbrace $ is non increasing while $\\lbrace w_k\\rbrace $ is non decreasing.",
"By definition of $g_k, h_k$ , we can show that $g_k \\rightarrow g$ and $h_k \\rightarrow h$ in $(X_0(\\Omega ))^*$ as $k \\rightarrow \\infty $ .",
"This implies $u_k \\rightarrow \\underline{u}$ and $w_k \\rightarrow \\overline{u}$ in $X_0(\\Omega )$ as $k \\rightarrow \\infty $ .",
"Therefore, upto a subsequence, $u_k \\rightarrow \\underline{u}$ and $w_k \\rightarrow \\overline{u}$ pointwise a.e.",
"in $\\Omega $ as $k \\rightarrow \\infty $ .",
"Using Dini's theorem, we get $u_k \\rightarrow \\underline{u}$ and $w_k \\rightarrow \\overline{u}$ in $L^\\infty (\\Omega )$ as $k \\rightarrow \\infty $ .",
"This proves our claim.",
"Now we can use Theorem REF and Proposition REF to obtain $v_1,v_2 \\in C([0,T];C_0(\\overline{\\Omega }))$ .",
"Taking $\\underline{u}^0= \\underline{u}$ (respectively $\\overline{u}^0= \\overline{u}$ ), we consider the sequence $\\lbrace \\underline{u}^k\\rbrace $ (respectively $\\lbrace \\overline{u}^k\\rbrace $ ) which is non decreasing(respectively non increasing) as solutions to the iterative scheme given by (REF ), for $0<\\Delta _t<1/M$ where $M$ denotes the Lipschitz constant of $f$ on $[\\underline{u}, \\overline{u}]$ .",
"If the sequence $\\lbrace u^k\\rbrace $ denotes the one that is obtained in (REF ), then by the choice of $\\Delta _t$ we can show that $\\underline{u}^k \\le u^k \\le \\overline{u}^k.$ Let $u$ denotes the weak solution of $(P^s_t)$ as obtained in proof of Theorem REF .",
"We can follow the proof of Theorem REF and use (REF ) to obtain $v_1(t)\\le u(t)\\le v_2(t).$ Consider the maps $t\\mapsto v_1(t,x)$ and $t \\mapsto v_2(t,x)$ which are non decreasing and non increasing respectively.",
"Let $v_1(t)\\rightarrow \\tilde{v}_1$ and $v_2(t) \\rightarrow \\tilde{v}_2$ as $t \\rightarrow \\infty $ .",
"Now let $S(t)$ denotes the semigroup on $L^\\infty (\\Omega )$ generated by the given evolution equation $u_t + \\lambda L(u)=f(x,u)$ .",
"Then we know $\\tilde{v}_1 = \\lim _{t^\\prime \\rightarrow +\\infty }S(t^\\prime +t)(\\underline{u})= S(t)\\lim _{t^\\prime \\rightarrow +\\infty }S(t^\\prime )(\\underline{u})= S(t)\\lim _{t^\\prime \\rightarrow +\\infty }v_1(t^\\prime )= S(t)\\tilde{v}_1 $ and analogously, we obtain $\\tilde{v}_2= S(t)\\tilde{v}_1.",
"$ Then $\\tilde{v}_1$ and $\\tilde{v}_2$ are stationary solutions to $(P_t^s)$ i.e.",
"solves $(Q^s)$ .",
"But by uniqueness of solution to $(Q^s)$ as shown in Theorem REF , we get $\\tilde{v}_1 = \\tilde{v}_2= \\hat{u} \\in C(\\overline{\\Omega })$ .",
"Therefore, by Dini's theorem we get $v_1(t) \\rightarrow \\hat{u} \\; \\text{and}\\; v_2(t) \\rightarrow \\hat{u} \\; \\text{in} \\; L^\\infty (\\Omega )\\; \\text{as}\\; t \\rightarrow \\infty .$ Using (REF ), we conclude that $u(t) \\rightarrow \\hat{u}$ in $L^\\infty (\\Omega )$ as $t \\rightarrow \\infty $ .$\\square $ 0.5 Acknowledgements: The authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project \"Singular phenomena in reaction diffusion equations and in conservation laws\"."
]
] | 1709.01906 |
[
[
"A Comparative Study of 2D Numerical Methods with GPU Computing"
],
[
"Abstract Graphics Processing Unit (GPU) computing is becoming an alternate computing platform for numerical simulations.",
"However, it is not clear which numerical scheme will provide the highest computational efficiency for different types of problems.",
"To this end, numerical accuracies and computational work of several numerical methods are compared using a GPU computing implementation.",
"The Correction Procedure via Reconstruction (CPR), Discontinuous Galerkin (DG), Nodal Discontinuous Galerkin (NDG), Spectral Difference (SD), and Finite Volume (FV) methods are investigated using various reconstruction orders.",
"Both smooth and discontinuous cases are considered for two-dimensional simulations.",
"For discontinuous problems, MUSCL schemes are employed with FV, while CPR, DG, NDG, and SD use slope limiting.",
"The computation time to reach a set error criteria and total time to complete solutions are compared across the methods.",
"It is shown that while FV methods can produce solutions with low computational times, they produce larger errors than high-order methods for smooth problems at the same order of accuracy.",
"For discontinuous problems, the methods show good agreement with one another in terms of solution profiles, and the total computational times between FV, CPR, and SD are comparable."
],
[
"Introduction",
" Numerical simulation of fluids typically requires high resolution and large computational power.",
"In industrial settings, high resolution is usually obtained through the computational domain, and not the computational method itself.",
"This is because low-order methods such as finite volume (FV) are employed in simulations.",
"In this paper, a low-order method implies either 1st or 2nd order spatial reconstruction, while a high-order method indicates a solution reconstruction of 3rd order and higher [1].",
"This differs from compressible methods, where a low-order method is 1st order accurate, and high-order is 2nd or 3rd order accurate.",
"While it is possible for FV methods to achieve higher-order spatial reconstruction, the computational cost becomes high in terms of memory access, especially for unstructured grids [2].",
"The solution reconstruction requires information from neighboring elements, and as the order of accuracy is increased, the number of elements required for communication also increases.",
"In contrast, high-order methods only require information at element neighbors, regardless of the order of accuracy.",
"This compact nature is appealing to parallel processing, especially Graphics Processing Unit (GPU) computing.",
"While most practical computations are completed in three-dimensions, two-dimensional problems are still of interest.",
"They are even more appealing towards GPUs, whose low memory storage makes computing on high-resolution three-dimensional problems a issue.",
"In addition, it is not clear how different numerical methods compare with one-another under GPU implementation, even for two-dimensional problems.",
"Various researchers have explored GPU Compute Unified Device Architecture (CUDA) with different numerical methods.",
"Implementation of the FV method for GPUs has been investigated by Castro et al.",
"[3], where the governing equations were the shallow water equations, and Obenschain [4] for unstructured meshes.",
"The parallelism of FV per element is limited, as solutions are reconstructed along element edges before the volume integration step.",
"In contrast, high-order methods have multiple solution states within each element, stored at solution points, which increases parallelism per element.",
"The most developed high-order methods to date include Discontinuous Galerkin (DG), Nodal Discontinuous Galerkin (NDG), Correction Procedure via Reconstruction (CPR), and Spectral Difference (SD).",
"Discontinuous Galerkin (DG) [5], [6], [7], [8], [9], [10] was the first high-order method introduced to hyperbolic equations.",
"There are mutliple approaches to the DG method, depending on how the integration points are chosen.",
"Using Gauss-Legendre points for DG implementation demands computations of surface and volume integrals at each step.",
"This allows for improved accuracy at a cost of increased computational work per step.",
"A more efficient implementation of DG was completed by Hesthaven and Warburton [11], which moved the integration points to element edges (NDG).",
"For an in depth discussion of the implementation of NDG to GPUs, the reader is directed to the paper by Klöckner et.",
"al.",
"[12].",
"The CPR method was developed to improve efficiency of other high-order methods [13], [14], [15], [16], which includes the DG method.",
"The CPR approach allows the equations to be solved in differential form, removing the added surface and volume integration computations present in DG.",
"While this increases the computing speed, the method is not as accurate as the DG approach [1].",
"CPRs application to GPUs was completed by Hoffmann and Zimmerman [17], [18], where significant speed-ups are observed.",
"The SD method is a finite difference-like formulation [19], [20], [21], which uses two sets of points, solution and flux points, where the flux derivative is computed across the flux points to update the solution states.",
"The SD methods application to GPUs was completed by Zimmerman [22] for a three-dimensional system.",
"The aforementioned references layout efficient algorithms and implementation techniques for the numerical methods discussed, and compare the speeds from GPU to Central Processing Unit (CPU) implementations, where significant speed-up results are shown.",
"There has been a comparative study done by Yu et al.",
"[23] on high-order methods using a CPU platform.",
"However, there has been no performance assessment of different methods using a GPU platform.",
"Furthermore, there has been no performance comparison between high-order methods to FV methods on GPUs.",
"Thus, the intent of this paper is to perform a fair comparison in two-dimensions of numerical methods and determine the relative performance between them in terms of total computing speed and accuracy with GPUs.",
"The developed approach for two-dimensions should be extended to three-dimensions in subsequent work in order to account for the known shift in computational cost from two to three-dimensions.",
"To this end, the FV, CPR, DG, NDG, and SD methods are all implemented using GPU CUDA, in similar manners from the references discussed above.",
"Each method is compared at the same order of accuracy and same number of degrees of freedom, with the maximum allowable time-step for a given mesh.",
"The comparison is for the two-dimensional Euler system, for both smooth and discontinuous problems.",
"For discontinuous problems, a shock capturing approach is required.",
"For the FV method, the MUSCL scheme [24], [25], [26] is implemented, while the high-order methods use a slope limiter [7] to limit the order of the solution only at discontinuities.",
"The present study investigates only quadrilateral elements, where the total number of degrees of freedom are held constant between the methods.",
"In addition, each method takes a maximum allowable time-step for stability.",
"This plays an important factor when considering the work to reach a specified final time, since high-order methods are time-step restricted, and this restriction increases with the order of accuracy of the scheme [27], [28].",
"The paper is organized in the following manner.",
"In section 2, each numerical method implemented is discussed briefly.",
"Section 3 outlines the implementation with GPU programming.",
"The results are discussed in section 4, where error analysis and computational time information are discussed in detail.",
"Finally, section 5 draws conclusions from the study."
],
[
"Numerical Methods",
"The hyperbolic conservation law is given by, $\\frac{\\partial {\\mathbf {q}}}{\\partial t} + \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) = 0,$ where ${\\mathbf {q}}$ is the state vector and $\\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}})$ is the divergence of the inviscid flux vector, which takes the following form, $\\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) = \\frac{\\partial {\\mathbf {f}}({\\mathbf {q}})}{\\partial x} + \\frac{\\partial {\\mathbf {g}}({\\mathbf {q}})}{\\partial y}.$ For the two-dimensional Euler equations, $\\mathbf {q}$ is a vector of the conserved variables, $\\mathbf {q} &= \\begin{bmatrix} \\rho \\\\ \\rho u \\\\ \\rho v \\\\ e \\end{bmatrix},$ and ${\\mathbf {f}}({\\mathbf {q}})$ and ${\\mathbf {g}}({\\mathbf {q}})$ are flux vectors, $\\mathbf {f}(\\mathbf {q}) = \\begin{bmatrix} \\rho u \\\\ p + \\rho u^2 \\\\ \\rho u v \\\\ u(e + p) \\end{bmatrix}, \\quad \\quad \\mathbf {g}(\\mathbf {q}) = \\begin{bmatrix} \\rho v \\\\ \\rho u v \\\\ \\ p + \\rho v^2 \\\\ v(e + p) \\end{bmatrix}.$ In Eqns.",
"(REF ) and (REF ), $\\rho $ is the density, $u$ is the x-direction velocity, $v$ is the y-direction velocity, $e$ is the total energy per unit volume, and $p$ is the pressure.",
"To close the system, the ideal gas equation of state is used, $p = (\\gamma -1)(e - \\frac{1}{2} \\rho (u^2 + v^2)).$ The computational domain is discretized with non-overlapping elements, each with volume $V_m$ .",
"Additionally, each element must be transformed into a standard element [29].",
"Within each element, a set of solution points are defined, which stores the solution states."
],
[
"FV Formulation",
"In the FV approach, the solution per element takes on an averaged value.",
"The governing equations are integrated over the elements volume, $V_m$ , $\\int _{V_m} \\left[ \\frac{\\partial {\\mathbf {q}}}{\\partial t} + \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) \\right] dV = 0.$ The solution average, denoted by ${\\mathbf {\\bar{q}}}_m$ is then defined as ${\\mathbf {\\bar{q}}}_m = \\frac{1}{V_m}\\int _{V_m} \\mathbf {\\bar{q} } dV.$ The semi-discretization can then be written in the following well known form for two-dimensional quadrilateral elements, $\\frac{\\partial {\\mathbf {\\bar{q}}}_{i,j}}{\\partial t} + \\frac{1}{\\Delta x} \\left[ \\mathbf {f}_{i+1/2,j} - \\mathbf {f}_{i-1/2,j} \\right]+ \\frac{1}{\\Delta y} \\left[ \\mathbf {g}_{i,j+1/2} - \\mathbf {g}_{i,j-1/2} \\right] = 0.$ In the above formulation, $i$ is the index in the x-direction, while $j$ is the index in the y-direction.",
"To obtain the flux at an interface (say $\\mathbf {f}_{i-1/2,j}$ , which is the left interface of the element) left and right solutions need to be reconstructed at the elements edge first.",
"Once left and right solutions are found at each interface, a Riemann problem is solved to determine the flux value at the interface.",
"The averaged solution is then updated via a time-marching scheme."
],
[
"CPR Formulation",
"Here, the CPR method is described.",
"For a full derivation, see [14].",
"The formulation of the CPR method requires the definition of an arbitrary weighting function ${w}$ .",
"By multiplying the weighting function to Eqn.",
"(1) and integrating over the domain, Eqn.",
"(REF ) becomes $\\int _{V_m} \\left[ \\frac{\\partial {\\mathbf {q}}}{\\partial t} + \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) \\right] {w} dV = 0.$ By applying the Gauss divergence theorem, Eqn.",
"(REF ) is expanded to be $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} \\vec{\\mathbf {F}}({\\mathbf {q}}) \\cdot {\\mathbf {n}} dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) dV = 0.$ Let ${\\mathbf {q}}_m$ approximate the solution ${\\mathbf {q}}$ within the element $V_m$ .",
"Furthermore, the solution is assumed to belong to the space of polynomials of degree $k$ or less (${\\mathbf {q}}_m \\in P^k$ ).",
"Thus, Eqn.",
"(REF ) must satisfy the following, $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}_m}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} \\vec{\\mathbf {F}}({\\mathbf {q}}_m) \\cdot {\\mathbf {n}} dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV = 0.$ There is no requirement enforced on element edges at this point.",
"The normal flux is replaced with a common Riemann flux to enforce element coupling, $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}_m}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} \\vec{\\mathbf {F}}^n_{com}({\\mathbf {q}}_m,{\\mathbf {q}}_{m+})dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV = 0.$ In Eqn.",
"(REF ), ${\\mathbf {q}}_{m+}$ is the solution outside of element $m$ .",
"Next, integration by parts is applied again to the last term in Eqn.",
"(REF ) to yield $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}_m}{\\partial t}{w} dV + \\int _{V_m} {w} \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV+ \\int _{\\partial V_m} {w} \\left[ {\\mathbf {F}}^n_{com} - {\\mathbf {F}}^n({\\mathbf {q}_m}) \\right] dS= 0.$ In the CPR formulation, the last term in Eqn.",
"(REF ) is viewed as a penalty term, which can be lifted to a volume integral by introducing a correction polynomial $\\mathbf {\\delta }_m \\in P^k$ , $\\int _{V_m} w \\mathbf {\\delta }_m dV = \\int _{\\partial V_m} {w} \\left[ {\\mathbf {F}}^n_{com} - {\\mathbf {F}}^n({\\mathbf {q}_m}) \\right] dS.$ The volume integral formulation of Eqn.",
"(REF ) is obtained, $\\int _{V_m} \\left[ \\frac{\\partial {\\mathbf {q}}_m}{\\partial t} + \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) + {\\mathbf {\\delta }}_m \\right] {w} dV= 0.$ If the conservation law is non-linear, then $\\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m)$ does not generally fall into $P^k$ .",
"To resolve the non-linear situation, the term $\\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m)$ is projected into $P^k$ .",
"Then, eliminating the weight and volume integral gives the differential formulation, $\\frac{\\partial {\\mathbf {q}}_m}{\\partial t} + \\Pi \\left[ \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) \\right] + {\\mathbf {\\delta }}_m = 0.$ The weighted residual formulation is reduced to a differential one.",
"Each element must store the solution states at a set of points, called solution points.",
"For the CPR method, within an element $V_m$ , a set of Legendre-Lobatto solution points are defined, as shown in Fig.",
"REF (b).",
"At each solution point $j$ , Eqn.",
"(REF ) must be true, $\\frac{\\partial {\\mathbf {q}}_{m,j}}{\\partial t} + \\Pi _j \\left[ \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_{m}) \\right] + {\\mathbf {\\delta }}_{m,j} = 0.$ Now the calculation of both $\\Pi _j \\left[ \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_{m}) \\right]$ and the correction polynomial, ${\\mathbf {\\delta }}_{m,j}$ , must be completed.",
"The inviscid flux divergence follows a chain rule approach (see [29] for analytical flux derivative).",
"For ${\\mathbf {\\delta }}_{m}$ formulations, see [15].",
"In this work, the correction polynomial is computed using Radau polynomials, which casts $\\mathbf {\\delta }$ into the DG framework [13], improving accuracy.",
"The definition of Legendre-Lobatto solution points brings a sense of efficiency into the method.",
"The solution points occupy edges of elements, thus no interpolation of information to element edges is required, and element coupling becomes straightforward."
],
[
"DG Formulation",
"The DG methods formulation is more straightforward than the CPR method, and more information can be found in Ref.",
"[7].",
"Again, a weighting function $w$ multiplies the conservation law, Eqn.",
"(REF ), and is integrated over the domain, $\\int _{V_m} \\left[ \\frac{\\partial {\\mathbf {q}}}{\\partial t} + \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}) \\right] {w} dV = 0.$ Like the CPR method, integration by parts is performed, and ${\\mathbf {q}}_m$ , which belongs to the space $P^k$ , is allowed to approximate the solution on element $V_m$ , $\\int _{V_m} \\frac{\\partial {\\mathbf {q}_m}}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} \\vec{\\mathbf {F}}({\\mathbf {q}_m}) \\cdot {\\mathbf {n}} dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}_m}) dV = 0.$ The solution and flux polynomials are approximated within each element $i$ over $n$ Gauss-Legendre points as, ${\\mathbf {q}_m} = \\sum _{j=1}^{n} {\\mathbf {q}}_{m,j} \\phi _j, \\\\ \\quad \\vec{\\mathbf {F}}({\\mathbf {q}_{m,j}}) = \\sum _{j=1}^{n} \\vec{\\mathbf {F}}_{m,j} \\phi _j,$ where $\\phi _j$ are the basis functions.",
"If the basis and weighting functions are equal, then the procedure is Galerkin.",
"The surface integral term in Eqn.",
"(REF ) couples elements together and the common flux is again calculated via a Riemann solver.",
"Since the solution points are Gauss-Legendre for the DG method, there is more computational work per time step when compared to the CPR method, since solutions must be interpolated to edges before element coupling.",
"Figure REF (a) shows a typical $P^2$ DG element, where sets of flux points are defined along the edges to communicate solutions.",
"In addition to the interpolation step, volume and surface integral calculations further increase the computational cost of the method.",
"Figure: SD element"
],
[
"NDG Formulation",
"The Nodal DG formulation closely follows the CPR formulation discussed previously.",
"Equation (REF ) is multiplied by a weighting function and integrated to yield the weak form, $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}_m}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} \\vec{\\mathbf {F}}({\\mathbf {q}}_m) \\cdot {\\mathbf {n}} dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV = 0.$ A Riemann flux is used to apply element coupling, and replaces $\\vec{\\mathbf {F}}({\\mathbf {q}}_m) \\cdot {\\mathbf {n}}$ with a common Riemann flux ${\\mathbf {F}}^n_{com}$ , which uses the current and neighboring element information, $\\int _{V_m} \\frac{\\partial {\\mathbf {q}}_m}{\\partial t}{w} dV + \\int _{\\partial V_m} {w} {\\mathbf {F}}^n_{com} dS - \\int _{V_m} \\vec{\\nabla }{w}\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV = 0.$ For the DG approach, a basis set $w_j$ is chosen for the solution space, where $j$ is the index of each solution point.",
"Equation (REF ) is written as the following strong DG form, $\\frac{\\partial }{\\partial t}\\int _{V_m} w_i {\\mathbf {q}}_{m,j} w_j dV - \\int _{\\partial V_m} {w_i} \\left[ \\vec{\\mathbf {F}} \\cdot \\mathbf {n} - {\\mathbf {F}}^n_{com} \\right] dS +\\int _{V_m} {w_i} \\vec{\\nabla }\\cdot \\vec{\\mathbf {F}}({\\mathbf {q}}_m) dV = 0.$ A mass, stiffness, differentiation, and face mass matrices can be formulated, as completed in [12], $M_{i,j} &= \\int _{V_m} w_i w_j dV, \\\\S_{i,j} &= \\int _{V_m} w_i \\nabla w_j dV, \\\\D_{i,j} &= \\left( M_{i,j} \\right)^{-1} S_{i,j}, \\\\M^A_{i,j} &= \\int _{\\partial V_m} w_i w_j dS.$ These matrices are used in Eqn.",
"(REF ) to obtain the following formulation, $\\frac{\\partial {\\mathbf {q}_{m,j}}}{\\partial t} + D \\left[\\vec{\\mathbf {F}}({\\mathbf {q}}_m) \\right] - L\\left[ \\vec{\\mathbf {F}} \\cdot \\mathbf {n} - {\\mathbf {F}}^n_{com} \\right]_A = 0.$ The matrix $L$ , or lifting matrix, acts on the facial degrees of freedom on face $A_m$ .",
"It combines the mathematical aspects of applying the mass matrix on the face, lifting the facial integral to volume integral, and finally applying the inverse mass matrix.",
"Much like CPR, this method also uses Gauss-Lobatto quadrature as the solution points (see Fig.",
"REF (b)), simplifying the element communication step."
],
[
"SD Formulation",
"The SD scheme employs a finite-difference like approach on the conservation laws.",
"The solution is assumed to be in the space $P^k$ , while the flux is assumed to be in the space $P^{k+1}$ .",
"A set of solution points and flux points are defined within each element.",
"Figure REF (c) illustrates the point locations in a SD $P^2$ element.",
"Note how an extra flux point is required per direction for the flux polynomial.",
"The solution states are stored at the solution points while the flux points compute the flux terms.",
"Let $h(\\xi )$ define the degree $k$ Lagrange polynomial at the solution points and $l(\\xi )$ be the degree $(k+1)$ polynomial at the flux points.",
"The coordinates $(x,y)$ are transformed into standard coordinates $(\\xi , \\eta )$ .",
"The solution is reconstructed as tensor products of two one-dimensional polynomials, ${\\mathbf {q}} = \\sum _{j=1}^{k+1} \\sum _{i=1}^{k+1} { \\mathbf {q}}_{i,j} h_i(\\xi ) h_j(\\eta ).$ In a similar manner, the reconstructed flux polynomials are formulated as ${\\mathbf {f}} = \\sum _{j=1}^{k+1} \\sum _{i=0}^{k+1} { \\mathbf {f}}_{i+1/2,j} l_{i+1/2}(\\xi ) h_j(\\eta ), \\\\{\\mathbf {g}} = \\sum _{j=0}^{k+1} \\sum _{i=1}^{k+1} { \\mathbf {g}}_{i,j+1/2} h_i(\\xi ) l_{j+1/2}(\\eta ).$ In this formulation, $i$ and $j$ indicate the points in $x$ and $y$ directions respectfully.",
"The flux polynomials are only continuous within each element.",
"To resolve the discontinuous interface, a Riemann solver is applied at flux points on the interfaces to provide element coupling.",
"Once the fluxes at the interface are augmented to a common value, the flux derivatives are evaluated as $\\frac{\\partial {\\mathbf {f}}}{\\partial \\xi } = \\sum _{r=0}^{k+1} {\\mathbf {f}}_{r+1/2,j} l^{\\prime }_{r+1/2}(\\xi _i), \\\\\\frac{\\partial {\\mathbf {g}}}{\\partial \\eta } = \\sum _{r=0}^{k+1} {\\mathbf {f}}_{i,r+1/2} l^{\\prime }_{r+1/2}(\\eta _j).$ The term $l^{\\prime }(\\xi _i)$ is the derivative of the flux points lagrange polynomial evaluated at the solution point locations $\\xi _i$ ."
],
[
"Shock Capturing",
"To resolve solution discontinuities, the low-order and high-order methods follow two approaches.",
"For the FV method, the second and third order MUSCL schemes are implemented, which is applied during the reconstruction of the solution at element interfaces.",
"The slopes of the reconstructed solutions are limited with the minmod limiter [30].",
"For second order reconstruction, the second order MUSCL scheme is applied, while the third order MUSCL scheme is selected for third order reconstruction.",
"For the high-order methods, the same technique is applied for all schemes, which uses a minmod limiter (similar to FV) to find troubled elements and apply slope limiting.",
"The updated solution is interpolated (if need be) to element edges.",
"Once interpolation is completed, the minmod limiter is applied to reconstruct a second solution based on cell averaged values to the edge.",
"If the difference in these two values is greater than a certain threshold (numerical experiments indicate $> 1.0 \\times 10^{-3}$ gives good solutions) then the cell is marked for limiting, where the new solution is, ${\\mathbf {q}}_{m,j} = \\bar{\\mathbf {q}}_{m} + (x_{m,j} - x {\\rm 0}) {\\rm minmod}\\left(\\frac{\\bar{\\mathbf {q}}_{m+1} - \\bar{\\mathbf {q}}_{m}}{h}, \\frac{\\bar{\\mathbf {q}}_{m} - \\bar{\\mathbf {q}}_{m-1}}{h} \\right).$ In Eqn.",
"(REF ), ${\\mathbf {q}}_{m,j}$ is the solution in element $m$ and solution point $j$ , $x_{m,j}$ is the location of the solution point, $x {\\rm 0}$ is the element midpoint, $h$ is the element size, and $\\bar{\\mathbf {q}}_m$ is the averaged solution in an element.",
"This scheme results in a second order reconstruction which can be applied to any of the high-order methods discussed in this paper."
],
[
"GPU CUDA Overview and Implementation",
"Before discussing the implementation of methods into GPU CUDA, a brief overview of GPU computing is presented, to give the reader a basic understanding of some conventions and algorithms on the GPU.",
"For a more complete discussion, refer to the NVIDIA CUDA programming guide [31]."
],
[
"CUDA Overview",
"Graphics computing is aimed toward image rendering, a largely parallel task.",
"GPUs are built around streaming multiprocessors to complete tasks, which execute hundreds of independent threads.",
"The multiprocessors launch blocks, containing threads, running in parallel.",
"Threads within a block are allowed to share information through the GPU's shared memory (each GPU has a limit on the amount of shared memory available).",
"This architecture is coined as single-instruction-multiple-thread (SIMT) architecture.",
"The blocks are executed through a grid, where no communication is allowed between the threads, and there is no guarantee of which block will finish first.",
"Only after every blocks work is completed can the grid be viewed, and data can be analyzed or seen by other threads if the appropriate memory was written into a GPU's global memory.",
"Global memory can be seen by all threads in all blocks on the GPU, and every thread can write to this memory.",
"However, the cost to write to this memory location can be high (hundreds of clock cycles).",
"So writes into this memory should be completed only when necessary.",
"Global memory can be bound to texture memory to hasten read access.",
"In this implementation, all global memory is also allotted space in the texture memory.",
"Finally, shared memory is used when threads in a block need to communicate information to one another.",
"Typical usage of this occurs during for loops, where one thread needs the information of other threads to perform computations, such as derivatives.",
"There are a few rules to follow when writing CUDA code to help optimize computing speed.",
"The usage of shared memory should be minimized and reused when possible.",
"The storage locations of memory should compliment the SIMT architecture.",
"Threads should be synchronized rarely and in optimal locations.",
"Each thread should write to global memory only once.",
"Some are quite obvious, such as the recycling of shared memory and location of barriers.",
"For storage order of memory, consider the following case: Let's use the Euler system and assume a memory storage where at a single point, memory position 0 is conservation of mass, memory position 1 and 2 are conservation of momentum in x and y, and memory position 3 is conservation of energy.",
"Now, let thread 0 read memory position 0, thread 1 read memory position 1, and so on.",
"One can observe that evaluating components of the field is completed by evaluating different expressions, which means different code, inefficient for SIMT architecture.",
"A better solution is to let thread 0 access memory position 0 of the point, and thread 1 access the memory position 0 of another point, which allows the same expression to be computed by the threads.",
"The final item, one global write, is also self explanatory, since each access to global memory is expensive.",
"It is noted, however, that in some cases this cannot be followed, and allowing multiple writes to global is cheaper than splitting the algorithm.",
"Some conventions are now listed to simplify the algorithms presented, and assist the reader.",
"Threads and blocks are allowed to be multi-dimensional, and have the indexes $t_x$ , $t_y$ , $t_z$ , $b_x$ , and $b_y$ (threads can have three indexes while blocks can have up to two).",
"Memory locations are presented in the following manner: Assume some code variable $u$ , which can be in any of the following memory locations based on the superscript.",
"If the variable has no superscript, it resides in the local memory to the thread.",
"The other three locations are denoted by superscripts $g$ , $t$ , and $s$ to represent global, texture, and shared memory space respectfully.",
"In addition, any memory reads or writes with indexes will be denoted in the following manner: If stored memory needs to be read (say $c$ is the pointer or array which holds the information), and the indexes depend on $i, j,$ and $k$ , then let $v = c[(i,j,k)]$ .",
"Meaning, $v$ now reads information in $c$ at an index location which depends on $i, j$ , and $k$ (not a three dimensional array or pointer)."
],
[
"CUDA Implementation",
"Now that the basic idea is presented, the GPU implementation of each method is presented.",
"Each methods entire implementation will not be discussed, only the residual update and shock capturing algorithm.",
"The remaining functions were implemented according to the algorithms found in Ref.",
"[32].",
"An overview of each method's steps are outlined in Fig.",
"REF .",
"Figure: Overview of methods.",
"Rectangles indicate non-local operations, while other shapes indicate local operations.For each method, the local and non-local operations are shown.",
"A local operation means all information to complete the operation is contained within the element, while non-local means communication must occur between elements.",
"Note that the number of operations listed does nor correlate with the number of functions required.",
"Some operations, local and non-local, can be combined into one function to reduce memory loading and multiple sweeps through the domain."
],
[
"FV CUDA",
"The FV method can be separated into two seperate kernels to update the residual.",
"As shown in Fig.",
"REF , one kernel reconstructs the solution and provides element coupling (both non-local operations) whose output feeds into the flux differentiation kernel.",
"[1]$\\triangleright $ 1 [t]FV_Reconstruct Faces in element $t_x$ = threadIdx.x Current global face $k = \\text{blockIdx}.x * \\text{blockDim}.x + i$ $k < \\text{n}_{e}$ Gather information from neighbors $q_{e_1}[(0...n_v)] = q^t[id_{e_1}(t_x,k)]$ $q_{e_2}[(0...n_v)] = q^t[id_{e_2}(t_x,k)]$ ... Reconstruct left and right solutions $q_L[(0...n_v)] = f(q_{e_1}, q_{e_2} ...)$ $q_R[(0...n_v)] = f(q_{e_1}, q_{e_2} ...)$ Compute the interface flux $\\text{InterfaceFlux}(q_L, q_R, f_n)$ Store interface flux into global memory $f^g[k + (0...n_v)*n_f] = f_n[(0...n_v)]$ The FV_Reconstruct algorithm is outlined in Algorithm 1, which reconstructs the left and right solutions at faces and computes the Riemann flux at the face.",
"Our implementation uses strictly texture memory and registers, with one write to global memory to finish the algorithm.",
"The threads are defined as faces, $t_x$ , in the domain.",
"In all algorithms, multiple elements (or faces) are calculated in one block, increasing the parallelism of the algorithm.",
"The variables $n_v$ and $n_e$ denote the number of state variables and number of elements respectfully.",
"Depending on the degree of the reconstruction polynomial, an appropriate amount of information from neighbors is loaded (the index $e_1$ denotes element 1).",
"Once the data is loaded, the appropriate reconstruction formula is applied and the flux at the interface is computed and stored.",
"To compute the flux derivative, multiple elements are computed per block, and each thread reads the appropriate flux information from texture memory to compute the flux derivative in the element (computed from FV_Reconstruct).",
"The FV method, while simplistic, demands memory transfers from neighboring elements in the domain, which is the major bottleneck in the method."
],
[
"High-Order Methods CUDA",
"For GPU implementation of each high-order method, a general algorithm is presented to compute the flux derivative.",
"The CPR and NDG methods will be lumped together, as implementation of the two is quite similar.",
"The variables $n_{sp1d}$ , $n_{fp1d}$ , $n_{sp}$ , $n_{fp}$ , and $n_{ep}$ denote the number of solution points in one dimension, number of flux points in one dimension, total number of solution points in an element, total number of flux points in an element, and total number of edge points respectfully."
],
[
"DG CUDA",
"As shown in Fig.",
"REF , the decomposition of the DG residual update requires three kernels.",
"The three algorithms (corresponding to three kernels) interpolate the information from solution points to flux points on element edges, couple the elements via a Riemann flux, and compute the volume and surface integrals using information stored at both solution and flux points.",
"[H]DG_Interpolation Point on element face $t_x$ = threadIdx.x Current face in block $t_y$ = threadIdx.y The global face $f = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $f < \\text{n}_{f}$ Gather index information on face $m = id^t_f(t_x,f)$ $l=0$ to $\\text{n}_{sp1d}$ Read solution point information and operate $id = id^t_{sp}[(m,l)]$ $q_l[(0...n_v)] = q_l[(0...n_v)] + cint^t[l] * q^t[id]$ $q^g_l[(t_x,f,0...n_v)] = q_l[(0...n_v)]$ [H]DG_Couple Point on element face $t_x$ = threadIdx.x Current face in block $t_y$ = threadIdx.y The global face $f = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $f < \\text{n}_{f}$ Read data (normals, cell indexes) ... Read in left and right solution) $q_{L}[(0...n_v)] = q^t[(id_{e_1}(t_x,f),0...n_v)]$ $q_{R}[(0...n_v)] = q^t[(id_{e_2}(t_x,f),0...n_v)]$ Compute the interface flux $\\text{InterfaceFlux}(q_L, q_R, f_n)$ Store interface flux into global memory $f^g_n[(t_x,f,0...n_v)] = f_n[(0...n_v)]$ [H]DG_Flux Solution points, current element in block, global block $t_x$ = threadIdx.x $t_y$ = threadIdx.y $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $k < \\text{n}_{e}$ Read state at solution points $q[(0...n_v)] = q^t[(t_x,k,0...n_v)]$ Compute flux into shared memory $f^s[t_x,t_y,(0...nv)] = F(q[(0...n_v)])$ $g^s[t_x,t_y,(0...nv)] = G(q[(0...n_v)])$ Threads need to wait for shared memory to fill syncthreads() Compute volume integral using shared memory $l=0$ to $\\text{n}_{sp}$ Stiffness matrix coefficients $(S_x, S_y) = (S^t_x, S^t_y)[(t_x,l)]$ $Vol[(0...n_v)] = Vol[(0...n_v)] + V([S_x, S_y, f^s, f^y])$ Surface integral next $l=0$ to $\\text{n}_{fp}$ Read integration term $I = I^t[(t_x,l)]$ Read in flux at interface points and compute surface integral $f_n[(0...n_v)] = f^t_n[(l,k,0...n_v)]$ $Sur[(0...n_v)] = Sur[(0...n_v)] - f_n[(0...n_v)]*I$ Assemble flux derivative and store $Res^g[(t_x,k,0...n_v)] = M^{-1}*(Vol[(0...n_v)] + Sur[(0...n_v)])$ The DG_Interpolation kernel runs threads along each point in all the faces in the domain.",
"At each face, the solution point information is read from texture memory, which serves as an index to read the required state at the solution points.",
"The solution states and interpolation coefficients ($cint$ ) are read from textured memory to perform the indicated operation, which is stored in global memory for future access.",
"Note that $cint$ is used in other algorithms to indicate interpolation coefficients, but the coefficients are not the same between the algorithms.",
"To couple the elements, an even simpler kernel (DG_Couple), only demands the left and right information at interfaces, obtained in Algorithm 2.",
"Boundary conditions are imposed on $q_R$ if necessary.",
"In Algorithm 4, the threads run on solution points within elements, and multiple elements are packed within a thread block.",
"A sufficient amount of shared memory is allocated for storage of the flux, and threads are halted while the memory is loaded.",
"Shared memory in this case offers high computational efficiency, since the volume integration loop requires information at other solution points in the element.",
"The surface integral is computed in a similar manner, without the use of shared memory.",
"The flux derivative is assembled and stored in the GPUs global memory."
],
[
"SD GPU",
"Like DG, SD also decomposes nicely into three seperate kernels as shown in Fig.",
"REF : Interpolations, coupling, and flux computation.",
"Two major differences in implementation are the following: SD has no volume integration and each element has interior flux points (not just on the edges).",
"This aspect makes the interpolation more expensive in terms of operations and storage, but the final flux evaluation cheaper.",
"[H]SD_Interpolation Flux points in one direction $t_x$ = threadIdx.x Current element in block $t_y$ = threadIdx.y The global element $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $f < \\text{n}_{e}$ Read states into shared memory $j<\\text{n}_{sp}$ $q^s[(t_x,t_y,0...n_v)] = q^t[t_x,k,0...n_v)]$ syncthreads() Build polynomial at flux points (x-direction) $l=0$ to $\\text{n}_{sp1d}$ $q_x[(0...n_v)] = q_x[(0...n_v)] + cint^t[(l,t_x)] * q^s[(l,t_x,t_y,0...n_v)]$ Compute flux terms $f_x[(0...nv)] = F(q_x[(0...n_v)])$ Store states and flux at flux points $q_{x,y}^g[(t_x,k,0...n_v)] = q_x[(0...n_v)]$ $f_{x,y}^g[(t_x,k,0...n_v)] = f_x[(0...n_v)]$ Repeat for y-direction ...",
"The interpolation must be completed in each coordinate direction separately, and both the solution states and flux terms must be stored in global memory for future use.",
"The SD_Interpolation kernel, outlined in Algorithm 5, sets each thread as a flux point in an element, and takes time to load the solution states into shared memory.",
"Note that for SD, $n_{sp} < n_{fp}$ , regardless of order of accuracy.",
"This shared memory will be used for both interpolation in the x and y-directions.",
"Once storage for x-coordinates are completed, the y-direction terms are computed.",
"For the coupling of elements, the reader is referred back to Algorithm 3.",
"Each element now has left and right solutions available in global memory access, and the DG algorithm can be used to store the interface flux.",
"The only difference is the location of that stored flux.",
"Rather than $f_n^g$ , it is simply stored in $f_{x,y}^g$ , overwriting the original memory from the kernel SD_Interpolation.",
"[H]SD_Flux Flux points in one direction $(n_{sp1d}*n_{fp1d})$ $t_x$ = threadIdx.x Current element in the block and global element $t_y$ = threadIdx.y $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ Solution point and flux point indexes $isp = mod(t_x,n_{sp1d})$ $ifp = t_x/n_{sp1d}$ $k < \\text{n}_{e}$ Read fluxes into shared memory $id_x = id_x^t(isp, ifp)$ $id_y = id_y^t(isp, ifp)$ $f_x^s[(id_x,t_y,0...n_v)] = f^t_{x,y}[(id_x,k,0...n_v)]$ $f_y^s[(id_y,t_y,0...n_v)] = f^t_{x,y}[(id_y,k,0...n_v)]$ syncthreads() Now only run on solution points $ifp < n_{sp1d}$ Flux differentiation on solution points $l=0$ to $\\text{n}_{fp1d}$ Derivative coefficients $c_x = c_x[(isp, l)]$ $c_y = c_y[(ifp, l)]$ Flux derivative per direction $dF_x[(0...n_v)] = dF_x[(0...n_v)] + c_x*f_x^s[(id_x,l,t_y,0...n_v)]$ $dF_y[(0...n_v)] = dF_y[(0...n_v)] + c_y*f_y^s[(id_y,l,t_y,0...n_v)]$ $Res^g[(t_x,k,0...n_v)] = dF_x[(0...n_v)] + dF_y[(0...n_v)]$ The kernel SD_Flux (Algorithm 6) gathers all the flux terms and computes the derivative at the solution points.",
"The threads are allowed to run across solution and flux points in one direction.",
"This allows the flux to be loaded into shared memory in the two coordinate directions $(x,y)$ .",
"The extra flux point thread is stopped, and the algorithm continues to run only on solution points.",
"Implementations of the method were performed without shared memory, and threads only operated across solution points reading the flux values from textured memory.",
"The presented algorithm was found to be around $1\\%$ faster.",
"The derivative of the flux is completed across the flux points and stored at the solution points (these coefficients are all in $c_x$ and $c_y$ ).",
"The final results are written to global memory for time-stepping."
],
[
"CPR / NDG CUDA",
"Both CPR and NDG methods have the unique property that solution and flux point coincide with one another, which enables the entire algorithm to be written in one GPU kernel, as outlined in Fig.",
"REF .",
"The major differences between the two methods are illustrated within the algorithm presented.",
"For CPR, the solution states are loaded into memory, the for loop computes the solution derivatives, and the projections are computed and stored.",
"NDG requires the flux values to be stored and the flux derivatives computed within the for loop.",
"[H]CPR_Flux / (NDG_Flux) Part 1 Max of flux points or solution points $t_x$ = threadIdx.x Current element in the block and global element $t_y$ = threadIdx.y $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ Solution points in x and y directions $i_x = mod(t_x,n_{sp1d})$ $i_y = t_y/n_{sp1d}$ $k < \\text{n}_{e}$ Operate on solution points first $t_x < n_{sp}$ Only CPR - Load solution into shared memory from texture $q^s[(t_x,t_y,0...n_v)] = q^t[(t_x,k,0...n_v)]$ Only NDG - Load solution from texture and store flux values $f^s[(t_x,t_y,0...n_v)] = f(q^t[(t_x,k,0...n_v)])$ syncthreads() $l=0$ to $n_{sp1d}$ Only CPR $c_x = c_x^t[(l,i_x)]$ $c_y = c_y^t[(l,i_y)]$ $dq_x[(0...n_v)] = dq_x[(0...n_v)] + c_x*q^s([i_x,l,t_y,0...n_v)]$ $dq_y[(0...n_v)] = dq_y[(0...n_v)] + c_y*q^s([i_y,l,t_y,0...n_v)]$ Only NDG $d_x = d_x^t[(l,i_x)]$ $d_y = d_y^t[(l,i_y)]$ $df_x[(0...n_v)] = df_x[(0...n_v)] + d_x*f^s([i_x,l,t_y,0...n_v)]$ $df_y[(0...n_v)] = df_y[(0...n_v)] + d_y*f^s([i_y,l,t_y,0...n_v)]$ Only CPR - Compute projections $Proj[(0...n_v)] = P(dq_x, dq_y)$ syncthreads() ... bkbreak The algorithm sets threads to operate over flux points or solution points, whichever is larger (the algorithm can then switch to operating on the other set within the kernel).",
"First, operations are completed over solution points, reading in the solution states and storing the states (flux values for NDG) into shared memory.",
"The shared memory is used in computing derivatives of the states in CPR, or the flux for NDG.",
"The chain rule is used for the flux derivative in CPR.",
"[H]CPR_Flux / (NDG_Flux) Part 2 bkbreak ... $t_x < n_{fp}$ Couple elements (see DG_Couple) ... Store normal flux difference into shared memory ($f^s$ for NDG) $q^s[(t_x,t_y,0...n_v)] = f_x[(0...n_v)]*n_x + f_y[(0...n_v)]*n_y - f^n[(0...n_v)]$ synctheads() $t_x < n_{sp}$ Get number of updates per solution point $n_{upd} = n^t_{upd}[t_x]$ Correct the normal flux (Lift the flux for NDG) $l=0$ to $n_{upd}$ Locations for correction (lifting) $id = id^t[(t_x,l,k])$ Only CPR $Corr[(0...n_v)] = Corr[(0...n_v)] - c^t[id] * q^s[(t_x,t_y,0...n_v)]$ Only NDG $Lift[(0...n_v)] = Lift[(0...n_v)] - L^t[id] * f^s[(t_x,t_y,0...n_v)]$ Only CPR $Res^g[(t_x,k,0...n_v)] = Proj[(0...n_v)] + Corr[(0...n_v)]$ Only NDG $Res^g[(t_x,k,0...n_v)] = df_x[(0...n_v)] + df_y[(0...n_v)] + Lift[(0...n_v)]$ The second part of the algorithm switches the threads to operate on flux points.",
"The coupling of the elements is straightforward, as information is already on element interfaces.",
"The normal flux difference on the flux points is stored into the same shared memory space from Algorithm 7.",
"The threads are switched a final time to operate on solution points, where, based on the method, the normal flux difference stored in shared memory is corrected or lifted and used to update the residual."
],
[
"Shock Capturing",
"The shock capturing algorithm for FV differs significantly from the other methods.",
"In Algorithm 1, the left and right states at element interfaces is computed.",
"Immediately following this step, the solutions can be limited using an appropriate slope limiting routine following second or third order MUSCL reconstruction.",
"Unlike the FV method, the approach used in high-order methods requires extra sweeps through the computational domain.",
"The slope limiting requires solution averages at elements, hence the high-order methods must first build the averaged solution within each element before any slope limiting can be applied.",
"[H]Average The current solution state $t_x$ = threadIdx.x The current element in the block $t_y$ = threadIdx.y Current global element $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $k < n_e$ $l=0$ to $n_{sp}$ Build average $q_m[t_x] = c_m^t[l] * q^t[(t_x,l,k)]$ $q_m^g[(t_x,k)] = q_m[t_x]$ The Average kernel (Algorithm 9) builds the solution averages using information from texture memory and stores the result in global memory space.",
"These averages are used in Limit kernels (Algorithms 10 and 11) for limiting.",
"[H]Limit Part 1 Solution points $t_x$ = threadIdx.x The current element in the block $t_y$ = threadIdx.y Current global element $k = \\text{blockIdx}.x * \\text{blockDim}.y + t_y$ $k < n_e$ Run through points on edges $j < n_{ep}$ Interpolate to edge if necessary $q_l = $ ... Load index locations of neighboring elements $(i_1,i_2) = $ ... Construct the minmod at the edge $q_e = q_m^t[k] + minmod(q_m^t[k]-q_l,q_m^t[i_2]-q_m^t[k],q_m^t[k]-q_m^t[i_1])$ Check if solution is large $|q_l - q_e| > \\epsilon $ $mark=1$ Store the mark into shared memory at edge $tmp_s[(t_x,t_y)] = mark$ syncthreads() ... bkbreak The Limit Part 1 kernel starts by switching the threads to run on the edge points of elements.",
"At each edge point, the solution is read from memory (CPR or NDG) or is interpolated from solution point information (DG or SD).",
"Then the minmod limiter is applied at the edge to detect if the element has a discontinuity, and if there is one, the [H]Limit Part 2 bkbreak ... $mark = 0$ Each edge point runs through the others edge points $l=0$ to $n_{ep}$ $mark = mark + tmp_s[(l,t_y)]$ $mark_s[t_y] = mark$ syncthreads() $j < n_{sp}$ $mark_s[t_y] > 0$ Apply slope limiting now ... Every edge point needs to see the markings of the others, which is completed using a summation.",
"This way, an element with at least one troubled point will give each edge point a value of one.",
"The marking is sent into shared memory so the information can be communicated when the threads switch to operate across the solution points.",
"At each solution point, in shared block $t_y$ , the marking is read from the shared space, and slope limiting is applied if this marking is greater than zero."
],
[
"Results",
"Two test cases are presented for both a smooth and discontinuous problem.",
"For all methods, a three state Runge-Kutta [33] time stepping scheme was applied and the interface fluxes were evaluated using the Rusanov [34] Riemann solver.",
"The time step for each method was computed using the $CFL$ condition as $\\Delta t \\le \\frac{CFL \\Delta x}{|u| + c}.$ The $CFL$ number for high-order methods is known to be quite restrictive in comparison to FV.",
"To ensure a fair comparison with FV, the following convention is used: At the end of a simulation, the error is recorded.",
"A new simulation is completed at a value of $0.5*CFL$ of the previous.",
"Again, the error is recorded.",
"If the percent error between these two errors is less than $0.1 \\%$ , the $CFL$ is termed the maximum $CFL$ .",
"Errors were completed by comparing the averaged solution with the averaged exact solution.",
"For $P^2$ FV, the error was computed by reconstructing the solution along element faces, and then using a quadrature rule to compute an averaged solution [1].",
"Finally, since FV has one solution state per element, while high-order methods have mutliple, the total number of degrees of freedom $NDoF$ between the methods was held constant.",
"Only quadrilateral elements are considered in this work.",
"For one element and a $P^2$ reconstruction for a high-order method, $NDoF = 9$ .",
"To match this, the FV method must have 9 elements.",
"A single Tesla K20c GPU card was used for all simulations.",
"The code was compiled under compute architecture 3.5 using the CUDA toolkit version 6.0.",
"In addition, the -O3 compiler optimization was used as well as the CUDA 64-bit libraries.",
"Double precision is used for all computations.",
"The computational time was nondimensionalized by taubench [35] using the following taubench condition: ./Taubench -n 250000 -s 10.",
"On the GPU workstation used in our simulations, taubench gave a value of 8.274.",
"This produces what is known as a work unit, as suggested by the 1st International Workshop on High-Order Methods [1] when comparing timings from numerical methods.",
"A work unit is a nondimensionalized unit computed by dividing the computational time it takes to complete a simulation by the taubench result."
],
[
"Smooth Problem",
"A vortex propagation case is used as the smooth problem in this paper.",
"The flow of the vortex is characterized in Ref.",
"[36].",
"A mean flow is specified $(\\rho , u, v, p) = (1, 1, 0, 1)$ with fluctuation in the velocity, temperature ($T$ ), and entropy ($S$ ), $(\\delta u, \\delta v) &= \\frac{\\epsilon }{2\\pi } e^{0.5(1-r^2)} \\left( -y, x\\right), \\\\\\delta T &= -\\frac{(\\gamma -1)\\epsilon ^2}{8\\gamma \\pi ^2}e^{1-r^2}, \\\\\\delta S &= 0.$ Here, $r^2 = x^2 + y^2$ and the vortex has strength $\\epsilon = 5$ .",
"An exact solution exists and can be found using $x_e = x - ut$ and $y_e = y - vt$ , where $t$ is the final time.",
"The solution evolves until time $t=1$ and the $L_2$ error norm of $\\rho $ is computed.",
"The domain is taken as $[-5, 5] \\times [-5, 5]$ and periodic conditions are imposed on the boundaries.",
"Discretizations from $20 \\times 20$ to $100 \\times 100$ quadrilateral elements are used in the simulations.",
"Table 1 shows the maximum $CFL$ chosen for the runs, which allowed a less than $0.1\\%$ error change when the $CFL$ was decreased by $1/2$ .",
"Figure: L 2 L_2 density errors using (a) P 1 P^1, (b) P 2 P^2, and (c) P 3 P^3 reconstructions versus work unitTable: Maximum CFLCFL - Smooth problemFigure: Total work unit to finish simulations for (a) P 1 P^1, (b) P 2 P^2, and (c) P 3 P^3The solution errors versus work units for $P^1$ , $P^2$ , and $P^3$ reconstructions are shown in Fig.",
"REF .",
"The overall trend of increasing accuracy with increasing work unit is observed for all methods.",
"It is observed that the high-order methods obtain smaller error thresholds than the FV method for a given work unit (exception for $P^2$ NDG, which has larger errors associated).",
"For $P^1$ reconstructions shown in Fig.",
"REF (a), DG clearly outperforms other methods, but as the order is increased, Fig.",
"REF (b) shows CPR and SD both achieve comparable errors with DG for a given work unit.",
"In the case of CPR, the schemes compact nature is attributed to this, where the operations to compute the flux derivative are contained in one GPU kernel.",
"For SD, Table REF showis that the SD method can take larger time-steps than the other high-order methods, as the CFL is not as restrictive.",
"Similar plots are observed in the comparative study done by Yu et al.",
"[23], where the error is compared with work units for several high-order methods on CPUs.",
"They also observe that CPR has a considerably lower error given a work unit for $P^2$ reconstruction than other methods, while NDG is significantly higher.",
"In the results presented here, SD is comparable to CPR because the maximum allowable time-step for a given temporal error is used.",
"This is different than the approach in Ref.",
"[23], where a constant time-step is implemented.",
"A fourth order reconstruction is also completed, and shown in Fig.",
"REF (c).",
"It illustrates that the SD and CPR methods both obtain the lowest errors for a given work unit for a $P^3$ reconstruction.",
"Figure REF shows the work unit needed to complete a simulation on a given mesh for $P^1$ , $P^2$ , and $P^3$ reconstructions.",
"The obvious trend of the work unit increasing for finer meshes is observed for both orders of accuracy.",
"To complete a full simulation, the FV and CPR methods are the fastest on coarse meshes (Fig.",
"REF (a) and (b)).",
"Small computational domains do not take advantage of the GPU architecture with the optimizations and different memory types discussed in this paper.",
"As the domain is refined and the order is, the high-order schemes can produce solutions faster than FV.",
"The data illustrates that on fine meshes with high-order reconstruction, the high-order CPR, NDG, and SD methods run faster than the FV method as the degrees of freedom are increased.",
"Further increasing the order to $P^3$ , Fig.",
"REF (c), shows the CPR and NDG converge to the same work unit for a given simulation.",
"The SD method, however, is able to complete solutions faster than any other high-order method for $P^3$ reconstruction.",
"The solution errors are recorded for the high-order methods and are shown in Tables REF and REF .",
"For $P^1$ reconstructions, the DG method produces the lowest $L_2$ errors and all methods slopes decay at a rate equivalent to the order of accuracy.",
"A similar trend is observed for $P^2$ errors.",
"Note that the NDG errors are significantly higher, which is due to aliasing issues with the method.",
"Table: High-order error values for smooth problem (P 1 P^1 reconstruction)Table: High-order error values for smooth problem (P 2 P^2 reconstruction)"
],
[
"Discontinuous Problem",
"The next case is a radially expanding shock tube from Toro [37].",
"A domain of size $[-1,1] \\times [-1,1]$ initializes density and pressure $(\\rho , p)$ of $1.0$ inside a radius of $0.4$ .",
"Outside the radius, $\\rho = 0.125$ and $p = 0.1$ .",
"There is no velocity component at the initial time.",
"Rather than using solution errors to check if the CFL is small enough, the residual error is used.",
"Table: Maximum CFL - Discontinuous problemThe solution is ran until a final time of $t=0.25$ , where the density is compared along the centerline, $y=0$ .",
"For the reference solution, the data was taken from the text Riemann Solvers and Numerical Methods for Fluid Dynamics [37] (digitized for use here).",
"As illustrated in Fig.",
"REF (b), all methods have good agreement with the reference solution.",
"Figure: Discontinuous test case results (a) Density contours (b) P 1 P^1 solution comparisonFigure: Computational work per iteration for (a) P 1 P^1 and (b) P 2 P^2 reconstructionFig.",
"REF illustrates the work unit needed per iteration for each method for both $P^1$ and $P^2$ reconstructions.",
"As the computational domain is increased, the benefit of using some high-order methods becomes apparent.",
"For the $P^2$ reconstruction in Fig.",
"REF (b) with 1440k degrees of freedom, CPR is $27\\%$ faster per iteration than FV, while DG is $14\\%$ slower than FV.",
"Figure: Total work for P 1 P^1 reconstruction (a) 160,000 degrees of freedom and (b) 640,000 degrees of freedomFigure: Total work for P 2 P^2 reconstruction (a) 360,000 degrees of freedom and (b) 1,440,000 degrees of freedomThe total computational work is shown in Fig.",
"REF and REF .",
"For $P^1$ reconstructions, the CPR method is the fastest of the high-order methods, while SD and DG take the most time.",
"Once the reconstruction is increased to $P^2$ , a similar trend is observed, however the total work for SD and CPR is nearly identical.",
"This is due to the SD schemes ability to take larger time steps than the CPR approach.",
"However, the time step restrictions on high-order methods are harsher than FV, which enables FV to arrive at the final solution time $25\\%$ faster than the CPR/SD schemes for a $P^2$ reconstruction with 1440k degrees of freedom.",
"This is also the case because of the extra sweeps the high-order methods need to take for discontinuous problems, one for evaluating the average and another to apply the limiting procedure."
],
[
"Conclusions",
"The presented work compares multiple numerical methods implemented on GPUs using CUDA computing.",
"The algorithms for each method were presented, and used to test smooth and discontinuous problems.",
"The maximum allowable time step from the CFL condition was used for each problem, with the number of degrees of freedom held constant across the methods.",
"For the smooth problem, the high-order methods obtained an error threshold at a lower work unit than the FV method.",
"Additionally, the CPR, NDG, and SD methods are capable of arriving at solutions faster than the FV approach as the computational domain is increased.",
"For discontinuous problems, the FV method does produce solutions $25\\%$ faster than the fastest high-order methods, but solution profiles between the methods are similar.",
"The computational work per step shows the CPR and NDG methods are most efficient, but time-step restrictions cause slower solution generation when compared to FV.",
"This two-dimensional approach may provide a foundation for comparisons with three-dimensional methods.",
"The extension is necessary because bottlenecks shift when going from two-dimensional to three-dimensional problems.",
"Additionally, different grids and unstructured mesh cases should be considered."
],
[
"Acknowledgements",
"This research has been supported by a NIAC (NASA Innovative Advanced Concepts) Phase 2 study entitled “An Innovative Solution to NASA's Asteroid Impact Threat Mitigation Grand Challenge and Its Flight Validation Mission Design.\"",
"Additional support has been given by the Vance Coffman Chair Fund.",
"Approved for unlimited release: LA-UR-17-27957."
]
] | 1709.01619 |
[
[
"Synthetic Medical Images from Dual Generative Adversarial Networks"
],
[
"Abstract Currently there is strong interest in data-driven approaches to medical image classification.",
"However, medical imaging data is scarce, expensive, and fraught with legal concerns regarding patient privacy.",
"Typical consent forms only allow for patient data to be used in medical journals or education, meaning the majority of medical data is inaccessible for general public research.",
"We propose a novel, two-stage pipeline for generating synthetic medical images from a pair of generative adversarial networks, tested in practice on retinal fundi images.",
"We develop a hierarchical generation process to divide the complex image generation task into two parts: geometry and photorealism.",
"We hope researchers will use our pipeline to bring private medical data into the public domain, sparking growth in imaging tasks that have previously relied on the hand-tuning of models.",
"We have begun this initiative through the development of SynthMed, an online repository for synthetic medical images."
],
[
"Introduction",
"Computer-aided medical diagnosis is widely used by medical professionals to assist in the interpretation of medical images [1].",
"Recently, deep learning algorithms have shown the potential to perform at higher accuracy than professionals in certain medical image understanding tasks, such as segmentation and classification [2].",
"Along with accuracy, deep learning improves the efficiency of data analysis tremendously, due to its automated and computational nature.",
"Since most medical data is produced in large volumes, and is often 3-dimensional (MRIs, CTs, etc.",
"), it can be cumbersome and inefficient to annotate manually.",
"There is strong interest in computer-aided medical diagnosis systems that rely on deep learning techniques [3].",
"However, due to proprietary and privacy reasons limiting data access [4], the development and advancement of these systems cannot be accelerated by public contributions.",
"It is difficult for medical professionals to make most medical images public without patient consent [5].",
"In addition, the publicly available datasets often lack size and expert annotations, rendering them useless for the training of data-hungry neural networks.",
"The design of these systems is therefore done exclusively by researchers that have access to private data, limiting the growth and potential of this field of research.",
"In the last 10 years, many breakthroughs in deep learning attribute success to extensive public datasets such as ImageNet.",
"The annual ImageNet competition decreased image recognition error rates from 28.2% to 6.7% [6] in the span of 4 years from 2010 to 2014.",
"ImageNet required the work of almost 50,000 people to evaluate, sort, and annotate one billion candidate images [6].",
"This showcases that access to large and accurate datasets is extremely important for building accurate models.",
"However, current research in the field of medical imaging has relied on hand-tuning models rather than addressing the underlying problem with data.",
"We believe that a public dataset for medicine can spark exponential growth in imaging tasks.",
"We propose a novel pipeline for generating synthetic medical images, allowing for the production of a public and extensive dataset, free from privacy concerns.",
"We put this into practice through SynthMed, a public repository for these images."
],
[
"Related Works",
"Researchers across a variety of disciplines have taken private data to the public domain using synthetic data.",
"For example, the U.S. Census collects personally identifiable information (PII) such as occupation, education, income, and geographical data for the US population.",
"Due to the natural specificity of the data, even if sources are de-identified and obfuscated, there is considerable risk of deanonymization [7].",
"This valuable data, which holds many potentially useful statistical correlations, is publicly unavailable because of privacy issues.",
"Reiter, a researcher at Duke University, solved this privacy problem by generating synthetic business census data [8].",
"In 2011, they released the Synthetic Longitudinal Business Database [9], the first publicly available record-level database on business establishments.",
"As seen in Reiter’s research, previous uses of synthetic data to bring private data to the public domain have been done solely with scalar quantities.",
"With the growing power of data-driven computer vision techniques, this paper explores the idea of synthetic data for images.",
"Recent developments of neural networks, specifically the generative adversarial network (GAN) [12], promise the possibility for more realistic image generation.",
"However, images produced by a GAN may often still contain artifacts and noise, due to instabilities in locating a saddle point in the energy landscape.",
"We address this issue by creating a novel image generation pipeline using a pair of GANs to promote increased stability."
],
[
"Data",
"We trained the GAN in Stage-I with retinal vessel segmentations from the DRIVE database [10].",
"DRIVE contains forty pairs of retinal fundi images and vessel segmentation masks manually labeled by two experts.",
"The GAN in Stage-II was trained with segmentation masks, derived from a segmentation network, and corresponding photorealistic images from MESSIDOR [11].",
"We also used DRIVE to train a single GAN model to compare our results.",
"One u-net was trained on DRIVE, and the other u-net was trained on 50 pairs of GAN-produced images.",
"Figure: Example vessel tree segmentation mask and retina fundus image from DRIVE."
],
[
"General Pipeline",
"To generate a high quality synthetic dataset, we propose the use of two GANs, breaking down the generation problem into two parts: Stage-I GAN: Produce segmentation masks that represent the variable geometries of the dataset.",
"Stage-II GAN: Translate the masks produced in Stage-I to photorealistic images.",
"We illustrate the process with retinal fundi images.",
"Figure: Flowchart of proposed pipeline."
],
[
"Generative Adversarial Network",
"The Generative Adversarial Network (GAN), as proposed by Goodfellow et al.",
"[12] in June 2014, involves the competition between two models: the discriminator D and the generator G. D is a binary classifier that classifies the data produced by G as either part of the training set (realistic) or not (unrealistic).",
"G minimizes its loss function by producing data that D will classify as real, as modeled by: $minmax(D,G)=E_{x-p_{data}(x)}[logD(x)] + E_{x-p_{z}}[log(1-D(G(z)))]$ The discriminator is a standard convolutional neural network (CNN) that takes an input image and returns a scalar that represents how real the input image is.",
"There are two convolutional layers that identify 5x5 pixel features and, as with most CNNs, there are fully connected layers at the end.",
"The generator is initialized with a random noise vector while D is trained with a small set of ground truth data.",
"The generator is a deeper neural network, having more convolutional layers and nonlinearities.",
"The noise vector is upsampled and the weights of G are learned through backpropagation, eventually producing data that is classified as real by the discriminator.",
"Further, a key feature of GANs is the ability to produce a larger amount of images than the original dataset.",
"We utilized this novel architecture to create a pipeline that is able to uphold patient privacy and generate a wider variety of realistic images."
],
[
"Stage-I GAN",
"The purpose of the Stage-I GAN is to generate varied segmentation masks.",
"It is based on the deep convolutional generative adversarial network (DCGAN) architecture [13], and built on the TensorFlow platform.",
"This network has demonstrated competitive results [13] while simultaneously improving training stability in comparison to the standard GAN.",
"The distinctive feature of the DCGAN, compared to other generative models, is that it is fully convolutional, meaning convolutional layers were used instead of pooling layers.",
"Pooling layers reduce the spatial size of the representation, and although they improve computational efficiency, they also result in the loss of important features found in medical images.",
"The generator is initialized with a noise vector, which is fed through multiple strided convolutions to generate a synthetic image.",
"We used the cross-entropy loss function to train the discriminator in the Stage-I GAN: $ l_{D}=\\frac{1}{m}\\sum _{i=1}^{m}[log(D(G(z^{i}))) + log(1-D(x^{i}))]$ D is the discriminator, G is the generator, m refers to mini-batch size, z is the corresponding input noise vector, x is the image, and i is the index of the image.",
"The generator’s loss is described by: $l_{G}=\\frac{1}{m}\\sum _{i=1}^{m}log(1-D(x^{i}))$ As a result of these two connected loss functions, the generator and discriminator are constantly competing with each other to minimize their respective loss functions.",
"We trained this model on an NVIDIA Tesla K80 GPU."
],
[
"Stage-II GAN",
"The purpose of Stage-II GAN is to translate segmentation masks to corresponding photorealistic images.",
"Stage-II GAN is also built on the TensorFlow platform.",
"Our model is based on an image-to-image translation network proposed by Isola et al.",
"[14] in November 2016; specifically, a vessel-to-retina implementation built by Costa et al.",
"[15].",
"This network is a special form of GAN known as a conditional generative adversarial network (CGAN).",
"It aims to condition the two networks D and G to a vector y and input image X that represents the mapping between the segmentation mask and photorealistic image.",
"Similar to the regular GAN, the CGAN can be modeled by this function (with the additional input parameter y): $min_{G}max_{D}V(D,G) = $ $ \\mathbb {}{E}_{p_{data}}[logD(x,y)]+ \\mathbb {}{E}_{z-p_{z}}[log(1-D(G(z,y))),y]$ The Stage-II GAN is trained with corresponding pairs of real fundi and segmentations masks in order to find a mapping between the two classes of images.",
"Given a segmentation mask, the model will translate the given geometry to a photorealistic medical image."
],
[
"U-net",
"To evaluate the reliability of our synthetic data, we used it to train a u-net segmentation network which creates a segmentation mask given a photorealistic medical image.",
"The u-net architecture, specifically formulated for biomedical images [16], is derived from an autoencoder architecture that relies on unsupervised learning for dimensionality reduction.",
"The u-net is especially useful for biomedical applications since it does not contain fully connected layers, imposing no restriction on input image size and allowing a significantly higher number of feature channels than a regular CNN.",
"Receptive fields after convolution are also concatenated with receptive fields in the decoding process.",
"This allows the network to use original features along with ones after the up-convolution.",
"Segmentation is an important task in machine learning used to partition an image into relevant parts.",
"It is also especially useful in medicine to outline malignant bodies and abnormalities such as tumors.",
"When examining retinal images, doctors commonly search for microaneurysms in the blood vessels for the diagnosis of diabetic retinopathy.",
"A u-net segmentation network trained with synthetic data, as shown with our results below, can easily automate and improve accuracy for this process.",
"This is just one of the many applications for the data produced by our pipeline in computer-aided medical diagnosis."
],
[
"Evaluation Metrics",
"Our pipeline produced synthetic segmentation masks along with corresponding fundi images.",
"We used this data to train a u-net segmentation network.",
"We evaluated the u-net on test images from the DRIVE database and compared them with the ground truth to calculate an F1 score.",
"We also calculated the variance between the synthetic and real datasets through a Kullback–Leibler (KL) divergence score.",
"When considering GANs, we must analyze the adversarial divergence to calculate the statistical correlation between the generated and original data.",
"The KL divergence score has been the standard to measure this for generative models, calculated by: $KL(P,Q)=\\sum _{i}^{ } P_{i}(ln\\frac{P_{i}}{Q_{i}})$ We also used the universal F1-score, calculated by taking the harmonic mean of precision and recall.",
"This score can display the similarity between two images, which we use to compare the segmentations produced from our synthetically trained u-net and DRIVE-trained u-net to ground truth segmentations."
],
[
"Quantitative Results",
"We received an F1 accuracy rating of 0.8877 for our synthetically trained u-net and an F1 accuracy of 0.8988 for our DRIVE-trained u-net.",
"The negligible difference between the two scores displays the quality of our produced training data.",
"To test for variance, we obtained a KL-divergence score that shows the difference between the distributions of two datasets.",
"The synthetic data score of 4.759 is from comparing the synthetic and real datasets, while the real-data score of 4.212 x 10-4 was measured by comparing two random subsets of the real data.",
"This low score is expected as the two subsets of images are from the same dataset.",
"The synthetic-data score is higher than the real-data score, showing that our synthetic data does not simply copy the original distribution.",
"Figure: Pixel-intensity distribution of real and synthetic datasets."
],
[
"Pipeline Validation",
"To confirm the flexibility of our pipeline, we tested it on a second dataset.",
"Using the BU-BIL database [17] of 35 rat smooth muscle cell images and segmentations as the training data for our pipeline, we were able to produce a synthetic version of the data.",
"We chose this database due to its intense variation.",
"The subject in each image is varied in both shape and position, making it difficult for the GAN to learn which features are relevant.",
"However, through our dual GAN pipeline’s hierarchical generation process, we were able to successfully produce realistic smooth muscle cell images as well as corresponding segmentations.",
"As described by our pipeline, we first generated segmentation masks of the smooth muscle cells using Stage-I GAN.",
"We then transferred the segmentations to Stage-II GAN where they were translated into photorealistic smooth muscle cells.",
"Figure: Graphic displaying examples from BUBIL and acorresponding synthesized dataset.It is important to note this was done on an extremely small dataset of 35 images for both Stage-I and Stage-II to test the limits of our pipeline.",
"The results show that Stage-II was able to learn the correspondence between the segmentation mask and the photorealistic image, but a greater variety of data would be helpful to develop the natural background found in the original images."
],
[
"Discussion",
"Due to the extreme variation of medical imaging data (various illuminations, noise, patterns, etc.",
"), a single GAN is unable to produce a convincing image (see Figure 8).",
"The GAN is unable to determine complex structures, as seen with the poorly defined vessel tree structure and dark spots.",
"It is only able to identify simple features such as general color, shape, and lighting.",
"Figure: Example of a retina image from a single GAN pipeline.This lack of detail is unacceptable for medical image generation, as medical images have many intricacies that must be accurately represented for the data to be usable.",
"Our dual GAN architecture improves the quality of synthetic images by breaking down the challenging task of generating medical images into to hierarchical process.",
"Stacking GANs has been shown to be effective in refining the images produced by a GAN, as seen with Zhang et al [18].",
"This also allows the unstable nature of GANs to be controlled by providing each GAN with a relatively elementary task.",
"Stage-I GAN focuses only on a much lower dimensional problem: generating unique segmentation geometries, while ignoring photorealism.",
"This allows Stage-II GAN to only generate the colors, lighting, and textures of the medical image from the given geometry.",
"Because the geometry is generated in a lower dimensional image by a separate GAN, an unrealistic vessel geometry causes a larger loss compared to a single GAN that produces unrealistic geometries in its high dimensional fundi images.",
"This system allows both GANs in our pipeline to perform at a high level and reach convergence faster, creating images with more realistic geometries and textures than an ordinary single GAN system.",
"In addition, the nature of our pipeline produces a wider variety of images than the original dataset.",
"This is because our pipeline generates images that are between the data that formed the distribution.",
"As shown by Figure 5 and Figure 6, our synthetic dataset keeps the general statistical distribution of the real dataset while producing original images.",
"Our pipeline can produce larger quantities of images for effective use in data-driven machine learning tasks, while avoiding legal concerns regarding patient privacy."
],
[
"Conclusion",
"We have proposed a pipeline that is able to generate medical images for a segmentation task end-to-end, using a pair of generative adversarial networks.",
"Our method decomposes the image generation process into two parts: Stage-I GAN which focuses on creating varied geometries of the segmentation mask and Stage-II GAN which transforms the geometry into a photorealistic image.",
"Given a dataset of real images, it can produce larger amounts of synthetic data that is not an image of any real patient, meaning that data produced by our pipeline can be distributed in the public domain.",
"This is a significant step towards the creation of a public and synthetic medical image dataset, analogous to ImageNet.",
"To further this purpose, we have created an online synthetic medical imaging database known as SynthMed.",
"We plan to populate this database with synthetic data from private research.",
"We hope that future researchers will apply similar synthetic data techniques to provide public access to their private data for the further advancement and development of computer-aided medical diagnosis."
],
[
"Future Work",
"We believe that our pipeline of dual generative adversarial networks can also be applied to fields outside of medical imaging.",
"Specifically, scene generation has been a challenging topic in Computer Vision, due to the complexity and variance of the images.",
"Our two-stage pipeline may be used to simplify the problem where simple features of the scene can be generated using Stage-I GAN and details can be learned through Stage-II.",
"Researchers have shown in the past that single GANs are able to translate manually done photo segmentations to realistic scenes, as seen with Isola et al.",
"in facade generation [14].",
"We hope to optimize Stage-I in future works by exploring other representations of the segmentation masks.",
"For example, instead of the GAN in Stage-I producing an image, generative models trained with other representations such as bezier curves, 2D point clouds, or skeletons may be used to reduce dimensionality in the generation process.",
"This would reduce computational time as well as the chance of artifacts.",
"Using our pipeline for different datasets may require the tuning of hyperparameters for increased effectiveness, as is the case with most neural networks.",
"In addition, the development of deeper and more advanced architectures could be implemented to replace certain networks in our pipeline.",
"Our pipeline relies on a set of accurate data with high variance.",
"For our pipeline to be executed on a variety of medical images, we must have access to private research data.",
"Access to these private collections of images to generate synthetic data is the key to opening up public collaboration for more advanced automated medical image interpretation.",
"Figure: Examples of Synthetic Retinal Data"
],
[
"References",
"[1] Doi, K. (2007).",
"Computer-Aided Diagnosis in Medical Imaging: Historical Review, Current Status and Future Potential.",
"Computerized Medical Imaging and Graphics : The Official Journal of the Computerized Medical Imaging Society, 31(4-5), 198–211.",
"http://doi.org/10.1016/j.compmedimag.2007.02.002 [2] Baris Kayalibay, Grady Jensen, and Patrick van der Smagt.",
"Cnn-based segmentation of medical imaging data.",
"CoRR, abs/1701.03056, 2017.",
"[3] Ishida, T, and S Katsuragawa.",
"“[Overview of computer-Aided diagnosis].” Nihon Igaku Hoshasen Gakkai zasshi.",
"Nippon acta radiologica., U.S. National Library of Medicine, July 2002, www.ncbi.nlm.nih.gov/pubmed/12187835 [4] Secretary, HHS Office of the, and Office for Civil Rights (OCR).",
"“Your Rights Under HIPAA.” HHS.gov, US Department of Health and Human Services, 1 Feb. 2017, www.hhs.gov/hipaa/for-individuals/guidance-materials-for-consumers/index.html.",
"[5] Christopher Cunniff, Janice L.B.",
"Bryne Louanne M. Hudgins, John B. Moeschler, Ann Haskins Olney, Richard M. Pauli, Lauri H. Seaver, Cathy A. Stevens, Christopher Figone.",
"Informed consent for medical photographs, Dysmorphology Subcommittee of the Clinical Practice Committee, American College of Medical Genetics, https://www.acmg.net/staticcontent/staticpages/informed_consent.pdf [6] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and Fei-Fei Li.",
"Imagenet large scale visual recognition challenge.",
"CoRR, abs/1409.0575, 2014.",
"[7] Aditi Ramachandran, Lisa Singh, Edward Porter, Frank Nagle.",
"Exploring Re-identification Risks in Public Domains, Georgetown University, Harvard University, https://www.census.gov/srd/CDAR/rrs2012-13_Exploring_Re-ident_Risks.pdf [8] Jarmin, R. and Louis, T. (2014).",
"[ebook] Washington: U.S. Census Bureau, Center for Economic Studies, https://www2.census.gov/ces/wp/2014/CES-WP-14-10.pdf [9] Satkartar K. Kinney, Jerome P. Reiter, Arnold P. Reznek, Javier Miranda, Ron S. Jarmin, and John M. Abowd.",
"Towards Unrestricted Public Use Business Microdata: The Synthetic Longitudinal Business Database.",
"International Statistical Review, 79(3):362–384, December 2011.",
"[10] J.J. Staal, M.D.",
"Abramoff, M. Niemeijer, M.A.",
"Viergever, and B. van Ginneken.",
"Ridge based vessel segmentation in color images of the retina.",
"IEEE Transactions on Medical Imaging, 23(4):501–509, 2004.",
"[11] Decencière et al..",
"Feedback on a publicly distributed database: the Messidor database.",
"Image Analysis & Stereology, v. 33, n. 3, p. 231-234, aug. 2014.",
"ISSN 1854-5165.",
"[12] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.",
"Generative Adversarial Networks.",
"ArXiv e-prints, June 2014.",
"[13] Alec Radford, Luke Metz, and Soumith Chintala.",
"Unsupervised representation learning with deep convolutional generative adversarial networks.",
"CoRR, abs/1511.06434, 2015.",
"[14] P. Isola, J.-Y.",
"Zhu, T. Zhou, and A.",
"A. Efros.",
"Image-to-Image Translation with Conditional Adversarial Networks.",
"ArXiv e-prints, November 2016 [15] P. Costa, A. Galdran, M. Ines Meyer, M. D. Abramoff, M. Niemeijer, A. M. Mendonca, and A. Campilho.",
"Towards Adversarial Retinal Image Synthesis.",
"ArXiv e-prints, January 2017 [16] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.",
"U-net: Convolutional networks for biomedical image segmentation.",
"CoRR, abs/1505.04597, 2015.",
"[17] D. Gurari, D. Theriault, M. Sameki, B. Isenberg, T. A. Pham, A. Purwada, P. Solski, M. Walker, C. Zhang, J. Y. Wong, and M. Betke.",
"\"How to Collect Segmentations for Biomedical Images?",
"A Benchmark Evaluating the Performance of Experts, Crowdsourced Non-Experts, and Algorithms.\"",
"Winter conference on Applications in Computer Vision (WACV), 8 pp, 2015.",
"[In Press].",
"[18] P. Costa, A. Galdran, M. Ines Meyer, M. Niemeijer, M. D. Abramoff, A. M. Mendonca, and A. Campilho.",
"\"End-to-end Adversarial Retinal Image Synthesis.\""
]
] | 1709.01872 |
[
[
"A Comparison of Audio Signal Preprocessing Methods for Deep Neural\n Networks on Music Tagging"
],
[
"Abstract In this paper, we empirically investigate the effect of audio preprocessing on music tagging with deep neural networks.",
"We perform comprehensive experiments involving audio preprocessing using different time-frequency representations, logarithmic magnitude compression, frequency weighting, and scaling.",
"We show that many commonly used input preprocessing techniques are redundant except magnitude compression."
],
[
"Introduction",
"Many music information retrieval researches using deep learning usually focus on optimising the hyperparameters which specify the network structure.",
"Meanwhile, the audio preprocessing stage is often heuristically decided and not subject to optimisation.",
"Although neural networks are known to be universal function approximators [1], training efficiency and performance may vary significantly with different training methods as well as generic techniques including preprocessing the input data [2].",
"In other words, a neural network can represent any function but it does not mean it can effectively learn any function.",
"Therefore, both empirical decisions and domain knowledge are crucial since choosing between various preprocessing methods can be seen as a non-differentiable choice function, therefore it cannot be optimised using gradient-based learning methods.",
"For example, mel-spectrograms have been preferred over short-time Fourier transform in many tasks [3] because it was considered to have enough information despite of its smaller size, i.e., efficient yet effective.",
"When a time-frequency representation $\\textbf {X}$ is given, one of the most common preprocessing approaches is to apply logarithmic compression, i.e., $\\log (\\textbf {X}+\\alpha )$ where $\\alpha $ can be arbitrary constants such as very small number (e.g.",
"$10^{-7}$ ) or 1.",
"However, the performances of these methods are usually not strictly compared.",
"In this paper, we focus on audio preprocessing strategies for deep convolutional neural networks for music tagging.",
"By providing empirical results with various preprocessing strategies, we aim to demystify the effect of audio preprocessing methods on the performances.",
"This will help researchers design deep learning systems for music research."
],
[
"Experiments and Discussions",
" A representative network structure needs to be defined to compare the effects of audio preprocessing.",
"A ConvNet (convolutional neural networks) with 2D kernels and 2D convolution axes was chosen.",
"This showed a good performance with efficient training in a prior benchmark [4], where the model we selected was denoted k2c2, indicating 2D kernels and convolution axes.",
"As illustrated in Figure REF , homogeneous 2D (3$\\times $ 3) convolutional kernels are used in every convolutional layer.",
"The input has a single channel, 96 mel bins, and 1,360 temporal frames, denoted as (1, 96, 1360).",
"The figures in Table REF denote the number of channels (N), kernel height and kernel width for convolutional layers and subsampling height, subsampling width for max-pooling layers.",
"Here, the height and width corresponds to the frequency- and time-axes respectively.",
"Exponential linear unit (ELU) is used as an activation function in all convolutional layers [5].",
"For the training of music tagger, we used the Million Song Dataset (MSD) [6] with preview audio clips.",
"The training data are 30-second stereo mp3 files with a sampling rate of 22,050Hz and 64 kbps constant bit-rate encoding.",
"For efficient training in our experiments, we downmix and downsample the signals to 12 kHz after decoding and trim the audio duration to 29-second to ensure equal-sized input signals.",
"The short-time Fourier transform and melspectrogram are computed using a hop size of 256 samples (21.3 ms) with a 512-point discrete Fourier transform aggregated to yield 96 mel bins per frame.",
"The preprocessing is performed using Librosa [7] and Kapre [8].",
"Total 224,242 tracks are used and split into train/validation/test sets, 201,672/12,633/28,537 tracks respectively.The network implementation and split setting are online: https://github.com/keunwoochoi/transfer_learning_music and https://github.com/keunwoochoi/MSD_split_for_tagging During training, the binary cross-entropy function is used as a loss function.",
"For the acceleration of stochastic gradient descent, we use adaptive optimisation based on Adam [9].",
"The experiment is implemented in Python with Keras [10] and Theano [11] as deep learning frameworks.",
"In the all experiments, area under curve - of receiver operating characteristic (AUC) is used as a metric.",
"Although it can be lower than 0.5 in theory, AUC practically ranges in [0.5, 1.0] as random and perfect predictions show 0.5 and 1.0 of AUC respectively.",
"The reported AUC scores are all measured on the test set.",
"Figure: Network structure of the 5-layer ConvNet.",
"NN refers to the number of feature maps (which is set to 32 for all layers in this paper) while 𝐖\\textbf {W} refers to the weights matrix of the fully-connected output layer.",
")Table: Details of the ConvNet architecture shown in Figure .",
"2-dimensional convolutional layer is specified by (channel, (kernel lengths in frequency, time)).",
"Pooling layer is specified by (pooling length in frequency and time)"
],
[
"Variance by different initialisation",
"In deep learning, using K-fold cross-validation is not a standard practice for two reasons.",
"First, with large enough data and a good split of train, validation and test sets, the model can be trained with small variance.",
"Second, the cost of hyperparameter search is very high and it makes repeating experiments too expensive in practice.",
"For these reasons, we do not cross-validate the ConvNet in this study.",
"Instead, we present the results of repeated experiments with fixed network and training hyperparameters, such as training example sequences and batch size.",
"This experiment therefore measures the variance of the model introduced by different weight initialisations of the convolutional layers.",
"For this, a normal distribution is used following He et al.",
"[12], which has been shown to yield a stable training procedure.",
"The results are summarised in Figure REF .",
"This shows the AUC scores of 15 repeated experiments on the left as well as their standard deviation on the right.",
"Small standard deviation indicates that we can obtain a reliable, precise score by repeating the same experiments for a sufficient number of times.",
"The two largest differences observed between the average AUC score and that of experiment 4 and 8 (AUC differences of 0.0028 and 0.0026 respectively) indicate that we may obtain up to $\\sim 0.005$ AUC difference among experiment instances.",
"Based on this, we can assume that an AUC difference of $< 0.005$ is non-significant in this paper."
],
[
"Time-frequency representations",
"STFT and melspectrogram have been the most popular input representations for music classification [3].",
"Although sample-based deep learning methods have been introduced, 2-dimensional represenations would be still useful in the near future for efficient training.",
"Melspectrograms provide an efficient and perceptually relevant representation compared to STFT [13] and have been shown to perform well in various tasks [14], [15], [16], [17], [18].",
"However, an STFT is closer to the original signal and neural networks may be able to learn a representation that is more optimal to the given task than mel-spectrograms.",
"This requires large amounts of training data however, as reported in [18] where using melspectrograms outperformed STFT with a smaller dataset.",
"Figure: Performances of predictions with melspectrogram and STFT with varying training data sizes.",
"The numbers above bars indicate the absolute performance differences between melspectrograms and STFTs.Figure REF shows the AUC scores obtained using $\\log $ (STFT) vs. $\\log $ (melspectrogram) while varying the size of the utilised training data.",
"Although there are small differences on AUC scores up to 0.007, neither of them outperforms the other, especially when enough data is provided.",
"This rebuts a previous result in [18] because melspectrograms did not have a clear advantage here even with a small training data size.",
"This may be due to the difference in frequency resolution of the representations used and summarised as follows.",
"STFT in [18]: $6000/129$$=$$46.5$ Hz (256-point FFT with 12 kHz sampling rate) STFT in our work: $6000/257$$=$$23.3$ Hz (512-point FFT with 12 kHz sampling rate) Melspectrogram in [18] and our work: $35.9$ Hz for frequency $<1$ kHz (96 mel-bins and by [19] and [7]) In [18], the frequency resolution of the STFT was lower than that of the melspectrogram to enable comparing them with similar number of frequency bins.",
"On the contrary, STFT of higher frequency resolution is used in our experiment.",
"Nevertheless, it is found to be only as good as melspectrogram in terms of performance, i.e., the network does not take advantage of finer input.",
"This means overall that, in practice, melspectrogram may be preferred since its smaller size leads to reduced computation in training and prediction.",
"The figure also illustrates how much the data size affects the performance.",
"Exponentially increasing data size merely results in a linear AUC improvement.",
"AUC starts to converge at 64% and 100%."
],
[
"Analysis of scaling effects and frequency-axis weights",
"In this section, we discuss the effects of magnitude manipulation.",
"Preliminary experiments suggested that there might be two independent aspects to investigate; i) frequency-axis weights and ii) magnitude scaling of each item in the training set.",
"Examples of frequency-axis weights are illustrated in Figure REF , where different weighting schemes are plotted.",
"Our experiment is designed to isolate these two effects.",
"We tested two input representations log-melspectrogram vs. melspectrogram, with three frequency weighting schemes per-frequency, A-weighting and bypass, as well as two scaling methods $\\times 10$ (on) and $\\times 1$ (off), yielding 2$\\times $ 3$\\times $ 2$=$ 12 configurations in total.",
"We summarise the mechanism of each block as follows.",
"First, there are three frequency weights schemes.",
"[leftmargin=*] per-frequency stdd: Often called spectral whitening.",
"Compute means and standard deviations across time, i.e., per-frequency, and standardise each frequency band using these values.",
"The average frequency response becomes flat (equalised).",
"This method has been used in the literature for tagging [18], singing voice detection [20] and transcription [21].",
"A-weighted: Apply the international standard IEC 61672:2003 A-weighting curve, which approximates human perception of loudness as a function of frequency.",
"Bypass: Do not apply any processing, i.e., $f:\\mathbf {X} \\rightarrow \\mathbf {X}$ Figure: Average frequency magnitude of randomly selected 200 excerpts with three frequency-axis normalisation.",
"A per-sample (excerpt) standardisation follows to remove the effect of different overall average magnitude.There are two other blocks which are mutually independent and also independent to frequency weights schemes.",
"[leftmargin=*] per-sample stdd: Excerpt-wise normalisation with its overall mean and standard deviation, i.e., using statistics across time and frequency of each spectrogram.",
"$\\times $ 10 scaler: Multiply the input spectrogram by 10, i.e., $f:\\mathbf {X} \\rightarrow 10$$\\mathbf {X}$ ."
],
[
"Frequency weighting",
"This process is related to the loudness, i.e., human perception of sound energy [13], which is a function of frequency.",
"The sensitivity of the human auditory system drops substantially below a few hundred HzSee equal-loudness contours e.g.",
"in ISO 226:2003., hence music signals typically exhibit higher energy in the lower range to compensate for the attenuation.",
"This is illustrated in Figure REF , where uncompensated average energy measurements corresponding to the Bypass curve (shown in green) yield a peak at low frequencies.",
"This imbalance affects neural network activations in the first layer which may influence performance.",
"To assess this effect, we tested three frequency weighting approaches.",
"Their typical profiles are shown in Figure REF .",
"In all three strategies, excerpt-wise standardisation is used to alleviate scaling effects (see Section REF ).",
"Our test results show that networks using the three strategies all achieve similar AUC scores.",
"The performance differences within four groups, {1, 1s, 2, 2s} in Figure REF are small and none of them are governing the others.",
"The curves in Figure REF show the average input magnitudes over frequency.",
"These offsets change along frequency, but the change does not seem large enough to corrupt the local patterns due to the locality of ConvNets, and therefore the network is learning useful representations without significant performance differences within each group.",
"Figure: histograms of the magnitude of melspectrogram time-frequency bins with (left) and without (right) logarithmic compression.",
"The number of bins are 100 and both are normalised, i.e., ∑ i=1 100 0.01×y i =1\\sum _{i=1}^{100} 0.01 \\times y_i=1.",
"Log compression significantly affects the histogram, making the distribution Gaussian (left), otherwise extremely skewed (right).",
"This is after standardisation and based on randomly selected 100 tracks from the training set."
],
[
"Analysis of scaling effects",
"We may assume a performance increase if we scale the overall magnitudes for a number of reasons.",
"During training using gradient descent, the gradient of error with respect to weights $\\frac{\\partial E}{\\partial W}$ is proportional to $\\frac{\\partial }{\\partial W} f(W^\\top X)$ where $f$ is the activation function.",
"This means that the learning rate of a layer is proportional to the magnitude of input $X$ .",
"In particular, the first layer usually receives the weakest error backpropagation, hence scaling of the input may affect the overall performance.",
"We tested the effect of this with the results shown in Fig.",
"REF .",
"To this end, consider comparing the same-coloured bars of {1 vs. 1s} and {2 vs. 2s}.",
"Here, the scaling factor is set to 10 for illustration, however many possible values $<$ 100 were tested and showed similar results.",
"In summary, this hypothesis is rebutted – scaling did not affect the performance.",
"The analysis of trained weights revealed that different magnitudes of the input only affects the bias of the first convolutional layer.",
"Training with scaling set to $\\times 10$ results in 3.4 times larger mean absolute value of the biases in the first layer.",
"This is due to batch normalization [22] which compensates for the different magnitudes by normalizing the activations of convolutional layers."
],
[
"Log-compression of magnitudes",
"Lastly, we discuss how logarithmic compression of magnitudes, i.e.",
"decibel scaling, affects performance.",
"This is considered standard preprocessing in music information retrieval.",
"The procedure is motivated by the human perception of loudness [13] which has logarithmic relationship with the physical energy of sound.",
"Although learning a logarithmic function is a trivial task for neural networks, it can be difficult to implicitly learn an optimal nonlinear compression when it is embedded in a complicated task.",
"A nonlinear compression was also shown to affect the performance in visual image recognition using neural networks [23].",
"Figure REF compares the histograms of the magnitudes of time-frequency bins after zero-mean unit-variance standardisation.",
"On the left, a logarithmically compressed melspectrogram shows an approximately Gaussian distribution without any extreme values.",
"Meanwhile, the bins of linear melspectrogram on the right is extremely condensed in a very small range while they range in wider region overall.",
"This means the network should be trained with higher numerical precision to the input, hence more vulnerable to noise.",
"As a result, decibel-scaled melspectrograms always outperform the linear versions as shown in Fig REF , where the same-coloured bars should be compared across within {1 vs. 2} and {1s vs. 2s}.",
"Colours indicate normalization schemes while {1 vs. 1s} and {2 vs. 2s} compare effect, both of which are explained in Section REFDecibel-scaled STFT also outperformed linear STFT in our unreported experiments..",
"Compared to the performance differences while controlling the training set size (the pink bar charts on the right of Figure REF ) the additional work introduced by not using decibel scaling can be roughly estimated by comparing these scores to those networks when the training data size is limited (in pink).",
"While this also depends on other configurations of the task, seemingly twice the data is required to compensate for the disadvantage of not using a decibel scaled representation."
],
[
"Conclusion",
"In this paper, we have shown that some of the input preprocessing methods can affect the performance.",
"We quantify this in terms of the size of the training data required to achieve similar performances.",
"Among several preprocessing techniques tested in this study, only logarithmic scaling of the magnitude resulted in significant improvement.",
"In other words, the network was resilent to most modifications of the input data except logarithmic compression of magnitudes in various time-frequency representations.",
"Althpugh we focused on the music tagging task, our results provide general knowledge applicable in many similar machine-listening problems, e.g., music genre classification or the prediction of environmental sound descriptors."
]
] | 1709.01922 |
[
[
"Soft Proposal Networks for Weakly Supervised Object Localization"
],
[
"Abstract Weakly supervised object localization remains challenging, where only image labels instead of bounding boxes are available during training.",
"Object proposal is an effective component in localization, but often computationally expensive and incapable of joint optimization with some of the remaining modules.",
"In this paper, to the best of our knowledge, we for the first time integrate weakly supervised object proposal into convolutional neural networks (CNNs) in an end-to-end learning manner.",
"We design a network component, Soft Proposal (SP), to be plugged into any standard convolutional architecture to introduce the nearly cost-free object proposal, orders of magnitude faster than state-of-the-art methods.",
"In the SP-augmented CNNs, referred to as Soft Proposal Networks (SPNs), iteratively evolved object proposals are generated based on the deep feature maps then projected back, and further jointly optimized with network parameters, with image-level supervision only.",
"Through the unified learning process, SPNs learn better object-centric filters, discover more discriminative visual evidence, and suppress background interference, significantly boosting both weakly supervised object localization and classification performance.",
"We report the best results on popular benchmarks, including PASCAL VOC, MS COCO, and ImageNet."
],
[
"-1mm -1mm 1]Yi Zhu 1]Yanzhao Zhou 1]Qixiang Ye 2]Qiang Qiu 1]Jianbin Jiao$^\\dag $ [1]University of Chinese Academy of Sciences [2]Duke University {zhuyi215, zhouyanzhao215}@mails.ucas.ac.cn, {qxye, jiaojb}@ucas.ac.cn, [email protected] section/abstract section/introduction section/related section/method section/experiment section/conclusions"
]
] | 1709.01829 |
[
[
"Chain conditions on \\'etale groupoid algebras with applications to\n Leavitt path algebras and inverse semigroup algebras"
],
[
"Abstract The author has previously associated to each commutative ring with unit $R$ and \\'etale groupoid $\\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\\mathscr G$.",
"In this paper we characterize when $R\\mathscr G$ is Noetherian and when it is Artinian.",
"As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okni\\'nski of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras."
],
[
"Introduction",
"Groupoid $C^*$ -algebras have played a crucial role in operator algebra theory since the seminal work of Renault [30]; see also [29], [18].",
"Recently, the author introduced [34], [33] a ring theoretic analogue for the class of ample groupoids [30], [29] over any base commutative ring with unit.",
"Ample groupoids are étale groupoids with a locally compact, Hausdorff and totally disconnected unit space where we recall that a topological groupoid is étale if its structure maps are local homeomorphisms.",
"Note that these algebras were independently introduced over the field of complex numbers slightly later in [15].",
"Algebras of ample groupoids, dubbed Steinberg algebras by Clark and Sims [16], include many important classes of rings including group algebras, commutative algebras over a field generated by idempotents, crossed products of the previous two sorts of rings, inverse semigroup algebras and Leavitt path algebras [2].",
"Over the past few years, there has been a plethora of papers on these algebras, particularly in connection with Leavitt path algebras, cf.",
"[15], [11], [14], [6], [16], [21], [26], [27], [28], [35], [37], [25], [9], [10], [19], [7], [12].",
"In this paper we investigate the ascending and descending chain conditions on ample groupoid algebras.",
"A classical result of Connell [17] says that a group ring $RG$ is Artinian if and only if $G$ is finite and $R$ is Artinian.",
"This was (essentially) extended to semigroups by Zelmanov [39].",
"The finite dimensional Leavitt path algebras over a field were characterized by Abrams, Aranda Pino and Siles Molina [4].",
"The problem of characterizing Noetherian group rings is more complicated.",
"The largest class of groups known to have Noetherian group algebras over a Noetherian coefficient ring is the class of polycyclic-by-finite groups.",
"Thus the best we can hope to achieve for groupoid algebras is to classify the Noetherian property up to the case of group rings, which we achieve.",
"Special cases of our results include the characterization of Noetherian Leavitt path algebras over a field [5] (which we now extend to any base ring) and Okniński's characterization of Noetherian inverse semigroup algebras [23].",
"Groupoid algebras admit an involution and hence are isomorphic to their opposite algebras.",
"Thus there is no difference for them between the left and right Noetherian or Artinian properties and so we omit the adjectives “left\" and “right\" in what follows.",
"Our main result is the following theorem (see below for undefined terminology).",
"Theorem 1 Let $G$ be an ample groupoid and $R$ a commutative ring with unit.",
"The groupoid algebra $RG$ is Noetherian if and only if $G$ has finitely many objects and $RG$ is Noetherian for each isotropy group $G$ of $G$ .",
"The groupoid algebra $RG$ is Artinian if and only if $G$ is finite and $R$ is Artinian.",
"The groupoid algebra $RG$ is semisimple if and only if $G$ is finite and $R$ is a finite direct product of fields whose characteristics do not divide the order of any isotropy subgroup of $G$ .",
"Moreover, in any of these cases $RG$ is a finite direct product of matrix algebras over group algebras $RG$ of isotropy groups $G$ of $G$ .",
"Theorem REF is in a sense a negative result since it indicates that étale groupoid algebras satisfy chain conditions only under very stringent hypotheses.",
"Nonetheless, we recover the known results for Leavitt path algebras and inverse semigroup algebras as special cases."
],
[
"Étale groupoids and their algebras",
"In this paper, following the Bourbaki convention, a topological space will be called compact if it is Hausdorff and satisfies the property that every open cover has a finite subcover."
],
[
"Étale groupoids",
"A topological groupoid $G$ is a groupoid (i.e., a small category each of whose morphisms is an isomorphism) whose object (or unit) space $G^{(0)}$ and arrow space $G^{(1)}$ are topological spaces and whose domain map $\\mathop {d}$ , range map $\\mathop {r}$ , multiplication map, inversion map and unit map $u\\colon G^{(0)}\\rightarrow G^{(1)}$ are all continuous.",
"Since $u$ is a homeomorphism with its image, we often identify elements of $G^{(0)}$ with the corresponding identity arrows and view $G^{(0)}$ as a subspace of $G^{(1)}$ with the subspace topology.",
"A topological groupoid $G$ is étale if $\\mathop {d}$ is a local homeomorphism.",
"This implies that $\\mathop {r}$ and the multiplication map are local homeomorphisms and that $G^{(0)}$ is open in $G^{(1)}$ [31].",
"Note that the fibers of $\\mathop {d}$ and $\\mathop {r}$ are discrete in the induced topology.",
"An étale groupoid is said to be ample[29] if $G^{(0)}$ is Hausdorff and has a basis of compact open sets.",
"In this case $G^{(1)}$ is locally Hausdorff but need not be Hausdorff.",
"Note that any discrete groupoid is ample.",
"A discrete group is the same thing as an ample gropoid with a single object.",
"If $x\\in G^{(0)}$ , then the isotropy group $G_x$ at $x$ is the group of all arrows $g\\colon x\\rightarrow x$ .",
"The orbit $\\mathcal {O}_x$ of $x\\in \\mathcal {G}^{(0)}$ is the set of objects $y$ such that there exists an arrow $g\\colon x\\rightarrow y$ .",
"The isotropy groups of elements in the same orbit are isomorphic."
],
[
"Groupoid algebras",
"Let $G$ be an ample groupoid and $R$ a commutative ring with unit.",
"Define $RG$ to be the $R$ -submodule of $R^{G^{(1)}}$ spanned by the characteristic functions $\\chi _U$ with $U\\subseteq G^{(1)}$ compact open.",
"If $G^{(1)}$ is Hausdorff, then $RG$ consists precisely of the compactly supported, locally constant functions $G^{(1)}\\rightarrow R$ .",
"See [34], [33], [15] for details.",
"The convolution product on $RG$ , defined by $f_1\\ast f_2(g)=\\sum _{\\mathop {d}(h)=\\mathop {d}(g)} f_1(gh^{-1})f_2(h), $ turns $RG$ into an $R$ -algebra.",
"Note that if $U,V\\subseteq G^{(0)}$ are compact open, then $\\chi _U\\ast \\chi _V=\\chi _{U\\cap V}$ .",
"The ring $RG$ is unital if and only if $G^{(0)}$ is compact.",
"If $G$ is discrete, then the elements of $RG$ are just the finitely supported functions $G^{(1)}\\rightarrow R$ and we can identify $RG$ with the category algebra of $G$ (in the sense of Mitchell [22], [38]).",
"That is we can identify the underlying $R$ -module with the free $R$ -module $RG^{(1)}$ on $G^{(1)}$ and equip it with the unique product extending the product on basis elements given by $g\\cdot h = {\\left\\lbrace \\begin{array}{ll} gh, & \\text{if}\\ \\mathop {d}(g)=\\mathop {r}(h)\\\\ 0, & \\text{else.}\\end{array}\\right.",
"}$"
],
[
"Proof of Theorem ",
"We shall first prove Theorem REF under the hypothesis that $G^{(0)}$ is finite.",
"Then we shall show that the Noetherian (and hence Artinian) condition implies that $G^{(0)}$ is finite.",
"Proposition 2 Suppose that $G$ is an ample groupoid with $G^{(0)}$ finite.",
"Let $\\mathcal {O}_1,\\ldots , \\mathcal {O}_k$ be the orbits of $G$ , let $G_i$ be the isotropy group of $\\mathcal {O}_i$ (well defined up to isomorphism) and let $n_i=|\\mathcal {O}_i|$ , for $i=1,\\ldots , k$ .",
"Then $RG\\cong \\prod _{i=1}^k M_{n_i}(RG_i)$ for any commutative ring with unit $R$ .",
"In particular, $RG$ is Noetherian if and only if each $RG_i$ is Noetherian, for $i=1,\\ldots , k$ , $RG$ is Artinian if and only if $R$ is Artinian and $G$ is finite and $RG$ is semisimple if and only if $G$ is finite and $R$ is a finite direct product of fields whose characteristics do not divide the order of any $G_i$ .",
"Since $\\mathop {d}$ is a local homeomorphism and $G^{(0)}$ is discrete, it follows that $G^{(1)}$ is discrete, i.e., $G$ is a discrete groupoid with finitely many objects.",
"Thus $RG$ is the usual category algebra of this groupoid [22], [38] and the isomorphism is then folklore (cf.",
"[36] where it is proved for finite groupoids but only finiteness of the set of objects is used in the proof).",
"Namely, one fixes a basepoint $x_i\\in \\mathcal {O}_i$ (and assumes $G_i=G_{x_i}$ ) and chooses, for each $y\\in \\mathcal {O}_i$ , an arrow $g_y\\colon x_i\\rightarrow y$ .",
"The isomorphism sends an arrow $g\\colon y\\rightarrow z$ in $\\mathcal {O}_i$ to $g_z^{-1}gg_yE_{zy}\\in M_{n_i}(RG_i)$ where we index the rows and columns of matrices in the factor $M_{n_i}(RG_i)$ by $\\mathcal {O}_i$ .",
"Since a finite product of rings is Noetherian (respectively, Artinian) if and only if each factor is and a matrix algebra is Noetherian (respectively, Artinian) if and only if the base of the matrix algebra is, we deduce that $RG$ is Noetherian (respectively, Artinian) if and only if each $RG_i$ is, for $i=1,\\ldots k$ .",
"To complete the proof it suffices to apply a result of Connell that states that a group algebra $RG$ is Artinian if and only if $R$ is Artinian and $G$ is finite [17], to apply Maschke's theorem and to observe that a groupoid is finite if and only if it has finitely many objects and each isotropy group is finite (in which case, it has cardinality $\\sum _{i=1}^k n_i^2|G_i|$ using the above notation).",
"Let us prove a topological lemma, which is essentially a well-known result about Boolean algebras.",
"Lemma 3 Let $X$ be a Hausdorff space with a basis of compact open sets.",
"Then $X$ satisfies the ascending chain condition on compact open subsets if and only if $X$ is finite.",
"Clearly, if $X$ is finite, then it satisfies the ascending chain condition on compact open subsets.",
"Assume now that $X$ satisfies the ascending chain condition on compact open subsets.",
"First observe that $X$ is compact.",
"Indeed, $X$ must have a maximal compact open subset $K$ .",
"If $K\\ne X$ and $x\\in X\\setminus K$ , then there is a compact open neighborhood $U$ of $x$ and $K\\cup U\\supsetneq K$ is a strictly larger compact open subset.",
"Thus $X=K$ .",
"It follows from compactness of $X$ that the compact open subsets are closed under complement and hence $X$ also enjoys descending chain condition on compact open subsets.",
"Hence each point $x\\in X$ is contained in a minimal compact open subset $K_x$ .",
"Suppose that $y\\in K_x\\setminus \\lbrace x\\rbrace $ .",
"Then since $X$ is Hausdorff with a basis of compact open subsets, there is a compact open subset $V$ with $x\\in V\\subseteq K_x$ and $y\\notin V$ .",
"This contradicts the minimality of $K_x$ and we deduce that $K_x=\\lbrace x\\rbrace $ .",
"Thus $X$ is discrete and compact, whence finite.",
"Recall that if $A$ is an ring and $e,f\\in A$ are idempotents, then $Ae\\subseteq Af$ if and only if $ef=e$ .",
"Proposition 4 Let $G$ be an ample groupoid and suppose that $RG$ is Noetherian for some commutative ring with unit $R$ .",
"Then $G^{(0)}$ is finite.",
"Let $U,V\\subseteq G^{(0)}$ be compact and open.",
"Then $\\chi _U,\\chi _V\\in RG$ are idempotents and $RG\\chi _U\\subseteq RG\\chi _V$ if and only if $\\chi _{U\\cap V}=\\chi _U\\ast \\chi _V=\\chi _U$ , that is, if and only if $U\\subseteq V$ .",
"It follows that if $RG$ is Noetherian, then $G^{(0)}$ satisfies the ascending chain condition on compact open subsets and hence is finite by Lemma REF .",
"Theorem REF is now an immediate application of Proposition REF and Proposition REF ."
],
[
"Applications",
"In this section, we use Theorem REF to characterize the Noetherian and Artinian properties for Leavitt path algebras and for inverse semigroup algebras.",
"The results for Leavitt path algebras are due to Abrams, Aranda Pino and Siles Molina [5], [4], in the special case of coefficients in a field; for inverse semigroup algebras the Noetherian result is due to Okniński [23] and the Artinian result is a special case of Zelmanov's results [39]."
],
[
"Leavitt path algebras",
"Let $E=(E^{(0)},E^{(1)})$ be a (directed) graph (or quiver) with vertex set $E^{(0)}$ and edge set $E^{(1)}$ .",
"We use $\\mathop {s}(e)$ for the source of an edge $e$ and $\\mathop {r}(e)$ for the range, or target, of an edge.",
"A vertex $v$ is called a sink if $\\mathop {s}^{-1}(v)=\\emptyset $ and it is called an infinite emitter if $|\\mathop {s}^{-1}(v)|=\\infty $ .",
"The length of a finite (directed) path $\\alpha $ is denoted $|\\alpha |$ .",
"The Leavitt path algebra [3], [8], [1], [2] $L_R(E)$ of $E$ with coefficients in the unital commutative ring $R$ is the $R$ -algebra generated by a set $\\lbrace v\\in E^{(0)}\\rbrace $ of pairwise orthogonal idempotents and a set of variables $\\lbrace e,e^*\\mid e\\in E^{(1)}\\rbrace $ satisfying the relations: $\\mathop {s}(e)e=e=e\\mathop {r}(e)$ for all $e\\in E^{(1)}$ ; $\\mathop {r}(e)e^*= e^*=e^*\\mathop {s}(e)$ for all $e\\in E^{(1)}$ ; $e^*e^{\\prime }=\\delta _{e,e^{\\prime }}\\mathop {r}(e)$ for all $e,e^{\\prime }\\in E^{(1)}$ ; $v=\\sum _{e\\in \\mathop {s}^{-1}(v)} ee^*$ whenever $v$ is not a sink and not an infinite emitter.",
"It is well known that $L_R(E)=RG_E$ for the graph groupoid $G_E$ defined as follows.",
"Let $\\partial E$ consist of all one-sided infinite paths in $E$ as well as all finite paths $\\alpha $ ending in a vertex $v$ that is either a sink or an infinite emitter.",
"If $\\alpha $ is a finite path in $E$ (possibly empty), put $Z(\\alpha )=\\lbrace \\alpha \\beta \\in \\partial E\\rbrace $ .",
"Note that $Z(\\alpha )$ is never empty.",
"Then a basic open neighborhood of $\\partial E$ is of the form $Z(\\alpha )\\setminus (Z(\\alpha e_1)\\cup \\cdots \\cup Z(\\alpha e_n))$ with $e_i\\in E^{(1)}$ , for $i=1,\\ldots n$ (and possibly $n=0$ ).",
"These neighborhoods are compact open.",
"The graph groupoid $G_E$ is the given by: $G_E^{(0)}= \\partial E$ ; $G_E^{(1)} =\\lbrace (\\alpha \\gamma ,|\\alpha |-|\\beta |,\\beta \\gamma )\\in \\partial E\\times \\mathbb {Z}\\times \\partial E\\rbrace \\mid |\\alpha |,|\\beta |<\\infty \\rbrace $ .",
"One has $\\mathop {d}(\\eta ,k,\\gamma )=\\gamma $ , $\\mathop {r}(\\eta ,k,\\gamma )=\\eta $ and $(\\eta ,k,\\gamma )(\\gamma ,m,\\xi ) = (\\eta , k+m,\\xi )$ .",
"The inverse of $(\\eta ,k,\\gamma )$ is $(\\gamma ,-k,\\eta )$ .",
"A basis of compact open subsets for the topology on $G_E^{(1)}$ can be described as follows.",
"Let $\\alpha ,\\beta $ be finite paths ending at the same vertex and let $U\\subseteq Z(\\alpha )$ , $V\\subseteq Z(\\beta )$ be compact open with $\\alpha \\gamma \\in U$ if and only if $\\beta \\gamma \\in V$ .",
"Then the set $(U,\\alpha ,\\beta ,V)=\\lbrace \\alpha \\gamma ,|\\alpha |-|\\beta |,\\beta \\gamma )\\mid \\alpha \\gamma \\in U,\\beta \\gamma \\in V\\rbrace $ is a basic compact open set of $G_E^{(1)}$ .",
"Of special importance are the compact open sets $Z(\\alpha ,\\beta ) = (Z(\\alpha ),\\alpha ,\\beta ,Z(\\beta ))= \\lbrace (\\alpha \\gamma ,|\\alpha |-|\\beta |,\\beta \\gamma )\\in G^{(1)}\\rbrace $ where $\\alpha ,\\beta $ are finite paths ending at the same vertex.",
"There is an isomorphism $L_R(E)\\rightarrow RG_E$ that sends $v\\in E^{(0)}$ to the characteristic function of $Z(\\varepsilon _v,\\varepsilon _v)$ where $\\varepsilon _v$ is the empty path at $v$ and, for $e\\in E^{(1)}$ , it sends $e$ to the characteristic function of $Z(e,\\varepsilon _{\\mathop {r}(e)})$ and $e^*$ to the characteristic function of $Z(\\varepsilon _{\\mathop {r}(e)},e)$ , cf.",
"[15], [37], [13], [26].",
"In [4] the finite dimensional Leavitt algebras over a field were characterized.",
"The Noetherian Leavitt path algebras over a field were determined in [5].",
"We extend these results to arbitrary base rings using groupoid methods (the original proofs were purely algebraic).",
"By a cycle in a directed graph $E$ , we mean a simple, directed, closed circuit.",
"A cycle is said to have an exit if some vertex on the cycle has out-degree at least two.",
"The graph $E$ is said to satisfy condition (NE) if no cycle in $E$ has an exit.",
"The following is presumably well known but I do not know a reference.",
"Proposition 5 Let $E$ be a graph.",
"Then $G_E^{(0)}$ is finite if and only if $E$ is finite and satisfies condition (NE).",
"Suppose first that $G_E^{(0)}=\\partial E$ is finite.",
"Then $E^{(0)}$ must be finite as the cylinder sets $Z(\\varepsilon _v)$ with $v\\in E^{(0)}$ are pairwise disjoint and non-empty.",
"As the cylinder sets $Z(e)$ with $e\\in E^{(1)}$ are pairwise disjoint and non-empty, we deduce that $E^{(1)}$ is finite and hence $E$ is finite.",
"Suppose now that some cycle contains an exit.",
"Then there is a vertex $v$ of out-degree at least two such that there is a cycle $\\alpha $ starting at $v$ .",
"Let $e$ be an edge emitted by $v$ not belonging to $\\alpha $ .",
"Then the cylinder sets $Z(\\alpha ^ne)$ with $n\\ge 0$ are non-empty and pairwise disjoint, again contradicting that $\\partial E$ is finite.",
"Thus $E$ satisfies condition (NE).",
"Conversely, suppose that $E$ is finite and no cycle has an exit.",
"Then any path of length greater than $|E^{(0)}|$ must enter a cycle and it can never leave that cycle.",
"Thus $\\partial E$ consists of those paths of length at most $|E^{(0)}|$ ending at a sink and those infinite paths of the form $\\alpha \\beta \\beta \\cdots $ with $\\alpha $ a path of length at most $|E^{(0)}|$ and $\\beta $ a cycle.",
"As a finite graph has only finitely many cycles (as cycles do not repeat vertices), we conclude that $G_E^{(0)}=\\partial E$ is finite.",
"The next proposition, characterizing isotropy in graph groupoids, should also be considered folklore.",
"Proposition 6 Let $E$ be a graph and $\\gamma \\in \\partial E$ .",
"Then the isotropy group $G_\\gamma $ is trivial unless $\\gamma =\\rho \\zeta \\zeta \\cdots $ where $\\rho $ is a path and $\\zeta $ is a cycle, in which case $G_\\gamma \\cong \\mathbb {Z}$ .",
"An isotropy group element is of the form $g=(\\gamma ,k,\\gamma )$ where $\\gamma =\\alpha \\xi =\\beta \\xi $ with $k=|\\alpha |-|\\beta |$ .",
"Moreover, $g$ is a unit unless $k\\ne 0$ .",
"If $k\\ne 0$ , replacing $g$ by its inverse we may assume that $|\\alpha |>|\\beta |$ .",
"Then $\\alpha = \\beta \\eta $ and $\\xi = \\eta \\xi =\\eta \\eta \\cdots $ .",
"We thus deduce that $\\gamma =\\rho \\zeta \\zeta \\cdots $ with $\\rho $ a path and $\\zeta $ a cycle.",
"Moreover, $G_\\gamma = \\lbrace \\gamma \\rbrace \\times H\\times \\lbrace \\gamma \\rbrace \\cong H\\cong \\mathbb {Z}$ with $H$ a non-trivial subgroup of $\\mathbb {Z}$ .",
"As $R\\mathbb {Z}\\cong R[x,x^{-1}]$ , the Laurent polynomial ring in one-variable over $R$ , we immediately obtain from Proposition REF and Proposition REF , Hilbert's basis theorem and Theorem REF the following result, generalizing [4] and [5].",
"Theorem 7 Let $E$ be a directed graph (i.e., quiver), $R$ a commutative ring with unit and $L_R(E)$ the Leavitt path algebra of $E$ over $R$ .",
"$L_R(E)$ is Noetherian if and only if $R$ is Noetherian, $E$ is finite and no cycle in $E$ has an exit, i.e., $E$ satisfies condition (NE).",
"Moreover, in this case $L_R(E)$ is a finite direct product of matrix algebras over $R$ and $R[x,x^{-1}]$ .",
"$L_R(E)$ is Artinian if and only if $R$ is Artinian and $E$ is finite acyclic, in which case it is a finite direct product of matrix algebras over $R$ .",
"$L_R(E)$ is semisimple if and only if $R$ is a finite direct product of fields and $E$ is finite acyclic."
],
[
"Inverse semigroups",
"An inverse semigroup is a semigroup $S$ such that, for each $s\\in S$ , there exists a unique $s^*$ in $S$ with $ss^*s=s$ , $s^*ss^*=s^*$ ; see [20] for an introduction to inverse semigroup theory.",
"Connections between inverse semigroups, étale groupoids and operator algebras can be found in [30], [29], [18].",
"In particular, each ample groupoid algebra is a quotient of an inverse semigroup algebra [34].",
"It is shown in [34] that if $R$ is a commutative ring with unit and $S$ is an inverse semigroup, then $RS\\cong RG(S)$ where $G(S)$ is Paterson's universal groupoid of $S$ [29], a certain ample groupoid associated to $S$ [29], [18], [34].",
"The full description of this groupoid is a bit complicated but we describe some of its salient features.",
"The set $E(S)$ of idempotents of $S$ is a commutative subsemigroup [20].",
"The unit space of $G(S)$ is the set of homomorphisms from $E(S)$ to $\\lbrace 0,1\\rbrace $ (with the latter viewed as a semigroup under multiplication).",
"Thus $G(S)^{(0)}$ is finite if and only if $E(S)$ is finite as the homomorphisms from an idempotent, commutative semigroup (i.e., a meet semilattice) to $\\lbrace 0,1\\rbrace $ separate points.",
"Moreover, when $E(S)$ is finite $G(S)^{(1)}$ is in bijection with $S$ , the isotropy groups are precisely the maximal subgroups of $S$ and, in fact, $G(S)$ reduces to the so-called underlying groupoid of the inverse semigroup considered in [20] or [32]; see [34] for details.",
"Recall that a maximal subgroup of $S$ is a unit group of a monoid $eSe$ with $e\\in E(S)$ .",
"Thus Theorem REF recovers Okniński's theorem [23] (see also [24]) and Zelmanov's theorem [39] (restricted to inverse semigroups).",
"Theorem 8 Let $S$ be an inverse semigroup and $R$ a commutative ring with unit.",
"$RS$ is Noetherian if and only if $S$ has finitely many idempotents and $RG$ is Noetherian for each maximal subgroup $G$ of $S$ .",
"$RS$ is Artinian if and only if $R$ is Artinian and $S$ is finite.",
"$RS$ is semisimple if and only if $S$ is finite and $R$ is a finite direct product of fields whose characteristics divide the order of no maximal subgroup of $S$ .",
"In any of these cases, $RS$ is isomorphic to a finite direct product of matrix algebras over group algebras of maximal subgroups."
]
] | 1709.01582 |
[
[
"Phase I results with the Large Angle Beamstrahlung Monitor (LABM) with\n SuperKEKB beams"
],
[
"Abstract We report on the SuperKEKB Phase I operations of the Large Angle Beamstrhalung Monitor (LABM).",
"The detector is described and its performance characterized using the synchrotron radiation backgrounds from the last Beam Line magnets.",
"The backgrounds are also used to determine the expected position of the Interaction Point (IP), and the expected background rates during Phase II."
],
[
"Introduction",
"The Large Angle Beamstrahlung Monitor (LABM) is a direct monitor of the beam parameters at the Interaction Point (IP).",
"Built by a collaboration of five institutions headed by WSU, the LABM consists of four narrow angle telescopes which collect light from the IP, divide it into two polarizations and four wavelengths, and count photons by means of 32 total photomultipliers (PMT).",
"The main background separation between beamstrahlung signal and synchrotron radiation backgrounds exploits the very forward angular pattern of radiation from a magnet, whereas the radiation from the very short beam-beam region is much broader in angle [1].",
"The four telescopes account for observation of the emitted light for both beams both above (azimuthal position of 90 degrees, or up) and below (270 degrees, or down) the Beam Pipe (BP).",
"The redundancy improves the measurement of a number of systematics, including possible misalignments of the beam trajectory with the detector axis.",
"The detector works by building observables from the ratio of polarizations, and in particular it determines directly at the IP the relative height of the two beams and their relative vertical offset.",
"This is described graphically through the use of beamstrahlung diagrams, Fig.",
"REF .",
"Direct IP observations are rare, and the direct determination of the beam heights at the IP particularly crucial given the luminosity limitations encountered at KEKB.",
"Figure: Graphic representation of how large beamstrahlung variablesrelate to beam-beam collisions.",
"The beamsare represented as ellipses in the transverse planeat the moment of collision.Solid arrow corresponds to solid beamand dashed arrow to dashed beam.",
"The arrows represent the observedpolarized rates U x ,U y U_x, U_y and produce a different pattern for a differentbeam-beam mismatch.The detector was first installed in Summer 2015, then installed a second time during Fall 2015, and after some upgrades, was operational in February 2016, a few days before beams started circulating.",
"We have taken data continuously during Phase I.",
"The paper describes how we were able to successfully see light coming from the IP, shows that the detector works as specified, and describes the small modifications done to the LABM to overcome the problems seen with two of the four telescopes.",
"The observed background rates are used to perform side-to-side (positron-to-electron) cross checks and to determine the expected background rates during Phase II."
],
[
"Detector description.",
"To accurately study light emitted from the IP, light needs to be extracted from the Beam Pipe, transported to a relatively low radiation area, and its spectral and polarization components counted separately.",
"Each telescope has four mirrors at 45 degrees, at locations between 451 and 477cm from the IP, which reflect incoming light into a Fused Silica Vacuum Window located opposite in Beam Pipe azimuth, see Table REF .",
"Once outside the Beam Pipe, light is transported over a length of 8-10 meters through four Optical Channels, consisting of several dark Aluminum pipes and 45 degrees mirrors, to a sheltered area where light is analyzed inside two Optics Boxes, one for the electron ( or “Oho” side of the detector) and one for the positron beam ( or “Nikko” side of the detector).",
"Each Optics Box has two optical tables, servicing both viewports on that side.",
"Inside the Optics Boxes, light is separated into components and counted.",
"Table: Optical and geometrical parameters of the LABM.",
"The φ\\phi acceptanceis the same for all detectors and equals 1/(40π)1/(40\\pi ), or 0.0080.The first component along the line are the vacuum mirrors, made out of Beryllium and having sizes of 2$\\times 2.8$ mm$^2$ , which corresponds to a square angular acceptance once the mirror inclination is taken into account.",
"The mirrors are passively cooled, with the copper base carrying heat out to copper fingers which are themselves air cooled by small fans.",
"The mirrors are small to minimize RF heating, and their inner edge is 38 mm from the nominal Beam Line.",
"The small mirrors represent the first of two collimators in a simple two-collimator optics to limit the telescope angula acceptance and reject collinear backgrounds.",
"Given that the detector is sensitive in the $350<\\lambda <650$ nm range, diffraction effects are of the order 0.18-0.33 mrad, which are small compared to the about 8 mrad angles of observation for the signal REF .",
"On exiting the BP, light enters the Optical Channel.",
"This is a system of aluminum pipes and 45 degrees mirrors (“Elbows”) to transport light to the Optics Boxes, located in a radiation sheltered area.",
"All mirrors are Surface First UV Enhanced Aluminum Elliptical Mirrors from Edmunds Optics for 45 degrees reflection.",
"The first mirror in each OC is one inch minor axis, the second is 2 inches minor axis, and all subsequent mirrors in every OC are 4 inches minor axis.",
"The first two in each line, called primary and secondary, are remotely controlled for pointing purposes, through stepper motors and their controllers, themselves activated by our Slow Control Software.",
"The aluminum pipes likewise are 1.5 inches inner diameter for the connector between BP and primary mirror, 2 inches for the connector between primary and secondary mirrors, and 4 inches for all subsequent pipes.",
"All OC have black paint in the first two pipes and two to four 1 cm optical baffles at points along the OC to reduce reflections.",
"Each OC is of different length with different turns, as the OC have to dodge numerous pieces of equipment.",
"Over a length of about 10 meters, light is transported through a 1.5 meters concrete floor to the Optics Boxes.",
"Entering the Optics Box the light beam (with a typical divergence of 2.2 mrad) encounters a broad band Wollaston prism from Lambrecht Corporation which splits the beam into two beams, with local transverse polarization $x-$ and $y-$ .",
"The two polarized beams are separated by 20 degrees.",
"Each beam is then guided by two mirrors onto a ruled grating, a type of grating which transmits maximal intensity to the first order peak, from Thorlabs.",
"The rainbow exiting the grating is guided onto 4 PMTs, each observing approximately 75 nm bandwidth, where photons are counted.",
"For Phase I only, gratings with a narrower angular spread were used, to have a “blind” PMT, not illuminated by the incoming beam, for the purpose of measuring Cherenkov backgrounds in the PMT window during data taking.",
"These backgrounds were found to be vanishingly small.",
"Fig.",
"REF shows an Optics Table, which is half an Optics Box schematics.",
"The PMTs are Hamamatsu R6095, and they are mounted on a remotely controllable conveyor belt.",
"They can be switched onto different viewports, and in particular PMTs observing the up telescope can be rotated to observe the down telescope for the purpose of measuring out some of the systematic errors in the determination of beamtrahlung asymmetries which provide information about beam-beam parameters.",
"Numerous pictures of our installation can be found at the website [2].",
"Figure: Schematics of one half of an Optics Box, which views one viewport.The light beam enters the Box, and is split into two polarized beams bya Wollaston prism.",
"Each beam is then spread by a grating and a fourcomponent spectrum is measured by quartets of photomultipliers."
],
[
"Electronics and DAQ.",
"The PMTs signals are amplified, discriminated by a -30mV threshold, and sent through cables to our counting room (which also hosts the electronics of other beam monitors), located respectively 25 and 80 meters (in cable length) away from the two counting boxes.",
"There a CAEN scaler V830 counts data at (currently) 1Hz rate, although our DAQ permits data taking at frequencies far exceeding relevant seismic frequencies to study seismic effects on the beam alignment.",
"A PC donated by KEK collects and writes all data, which include the 32 PMT and various other beam monitors around the ring, obtained through the EPICS network.",
"The PC is accessible through ftp or other Internet protocols, and we are therefore fully independent in analyzing our data as well as monitor correlations.",
"For the analysis presented here, the only SuperKEKB information used were the instantaneous beam currents."
],
[
"Calibration and alignment.",
"At time of assembly, we place a diode in front of the vacuum window and assemble the Optical Channels starting from the beam Pipe.",
"As the Channel is assembled, we can observe visually the reflected image of the diode by removing the mirror and back side of the next elbow.",
"This trick allows us to look inside the OC after a new pipe and elbow are assembled.",
"If the image is not centered, the elbow under study is re-oriented until the image is centered.",
"The process is repeated down the Channel.",
"The last elbow entering the Optics Box can not be aligned this way.",
"However, the last elbow is located about 8 cm in front of the Optics Box entrance, and the previous leg of the Channel is 550 to 570 cm depending on the OC.",
"No significant beam misalignments can occur with such a geometry.",
"PMT calibration by using the conveyor belt and light from a diode was first done successfully in Summer 2015.",
"The calibration is done by illuminating the OB with a fixed light source placed just in front of the vacuum window, while rotating the 16 PMTs on, say, the positron side through the 16 viewports of that side.",
"192 measurements are obtained for 32 degrees of freedom, allowing the determination of the relative efficiency of the 16 PMTs observing the positron side.",
"The procedure was also done on the electron side.",
"In the future, swapping some PMTs from one side to the other side should allow a relative efficiency determination for all 32 PMTs."
],
[
"Phase I data.",
"The goals of Phase I data taking were as follows: first, generally commission the device, second, prove that we could point the telescopes (with an acceptance of approximately 1mrad$^2$ ) to the IP, and third, cross check that the two sides were seeing rates consistent with one another.",
"As only one beam at a time was present in the Beam Pipe, all analysis was done with synchrotron radiation from the last magnets in the Beam line.",
"Tables REF and REF provide the geometrical parameters of detector and last magnets."
],
[
"Bending magnets",
"During Phase I, we measured the synchrotron radiation (SR) emitted when the beams are bent while traveling inside the bending magnets (BM) near the IP.",
"It is convenient to discuss the properties of such SR sweeps before looking at the angular scans.",
"The BMs are located on each side of the IP, and their position respect to the LABM vacuum mirrors is shown in Fig.REF .",
"Since the SR is emitted while the beam progresses through the BM, the image produced is a sweep.",
"The length of the sweep depends on the length L and radius of curvature $\\rho $ of the BM.",
"With the help of Fig.REF , the angle $\\theta _{s}$ subtended by the sweep, as seen from the LABM, is calculated as: $ \\theta _{s} \\approx \\frac{L^{2}}{2\\rho (D+d)}$ where D is the distance from the edge of the bending magnet to the IP, and d the horizontal distance from the vacuum mirror to the IP.",
"Figure: Schematic to find the length of the sweep from the geometry of the interaction region.The angle subtended by the SR sweep is tabulated, for each of the 4 LABM channels, in Table REF .",
"We have a sweep of 1.5 mrad for the Nikko side, and a sweep of 0.8 mrad for the Oho side.",
"We define the spacial resolution of the LABM as the diameter of the spot from which the LABM accepts radiation.",
"For phase I, the angle subtended by this diameter was 1.4 mrad.",
"Therefore, our angular resolution is smaller than the Nikko sweep and larger than the Oho sweep.",
"Therefore, we expect that we will be able to measure the Nikko sweep as an image of length 1.4+1.5+1.4=4.3 mrad, and the Oho sweep as an image of length 1.4 mrad.",
"Scanning the solid angle inside the beam pipe, we can find the angular position of the SR sweep, and then calculate the angular position of the IP.",
"At the exit of the bending magnet, the SR collected by the LABM is emitted at an angle $\\theta $ from the axis of the beam pipe, as shown in Fig.REF .",
"Figure: Schematic to find the angle α\\alpha that separates the sweep from the IP.",
"The IP will be located at an angle α\\alpha below the position of the SR sweep.The angular separation of the IP from the SR sweep can be calculated as $ \\alpha = \\arcsin \\Bigg ( \\frac{D}{d} \\frac{\\sin \\theta }{\\sqrt{1+\\frac{h^{2}}{d^{2}}}} \\Bigg )$ where h is height of the vacuum mirror from the beam pipe axis.",
"To find the position of the IP from the image of the scan, we need to move by an angle $\\alpha $ in direction -Y. Analogously, for an ”up” channel, we would need to move by an angle $\\alpha $ in direction +Y.",
"All the relevant quantities described in this section are shown in Table REF , with the magnets lengths and radii of curvature shown in Table REF .",
"Table: The relevant positions and angles of LABM and BM discussed in this section.",
"dd is distance to the IP.Table: Last magnet parameters before LABM in each Beam Line.",
"ρ\\rho is thecurvature radius and d LABM ,θ LABM d_{LABM},\\theta _{LABM} are the distances and angles of themagnet end to the LABM vacuum mirrors."
],
[
"In the following, all data are pedestal subtracted rates.",
"As stated in Section 2, the telescopes have a double collimator optics.",
"The first collimator is the small vacuum mirror, and the second is a sheet metal collimator placed directly before the 19 mm square Wollaston prism.",
"We started taking data with a 19 mm collimator (no collimator at all, with the acceptance being limited by the size of the Wollaston prism).",
"As we understood the images, we moved to 15 mm collimators, then 12mm, then 8mm.",
"7mm square collimators were installed after Phase I, but 2mm collimators will be the final configuration.",
"Fig.",
"REF shows at a glance the rate of all the Oho PMTs over one day, with 60 mA beams before and after an access during which the collimators were changed from 12mm to 8mm.",
"One can see that the detector is reasonably dark, quiet, and that the measured rates follow from the solid angle reduction by the collimator.",
"We have very often taken data during widely varying beam conditions.",
"The condition for significant data taking is that the rates seen be proportional to the beam current.",
"This is certainly the case for low beam currents, Fig.",
"REF .",
"After our main tests performed at currents of order 100 mA, we parked the detectors at the predicted IP angle.",
"As the beam currents increased, saturation effects started to become apparent, Fig.",
"REF .",
"The curve fits best with a 6.0MHz saturation rate which is in rough agreement with a (predicted) saturation rate of 7.5MHz.",
"In any case each PMT response curve was fitted with its own dead time curve, and we extract the true rates according to $ R_{true}=\\frac{R_{obs}R_{max}}{R_{max}-R_{obs}},$ where $R_{max}$ is the measured saturation rate.",
"Using the specific rate (Hz/mA) allowed us to take long scans to study details of the light reaching our detector.",
"With the detector stable and operating properly the crucial question is if we can deduce the location of the Interaction Point (IP), based on the observed light pattern.",
"If the location can be determined, then direct observation is possible.",
"It is noted that eventually, at nominal conditions, the IP will be by far the brightest spot in our images.",
"However the beamstrahlung signal scales roughly like $I_+I_-^2$ for the electron beam, and also like $1/\\sigma _x^2$ , where $\\sigma _x$ is the transverse horizontal width, common to both beams.",
"If collision data are to be taken, the backgrounds will have to be substracted.",
"Besides finding the IP based on the observed pattern, we measure the expected background rates for that point.",
"The detector response was measured in November 2015 with a long series of runs using a white light source and a point-like white reflector placed just above (or below) the vacuum windows.",
"The measurements focused on the relative calibration of the PMTs, which are known to vary by 30%, whereas each piece in the optical lines had been measured on the bench at WSU with accuracies well below 1%.",
"The source spectrum is constant, and the source is assumed to be unpolarized to high precision.",
"However the source, which consists of a lamp external to the Optical Channel and a small, white diffuser placed directly above or below the vacuum window, needs to be re-positioned from the Up to the Down detector when calibrating the detector, so that different rates can be observed.",
"Subsequently we take data while rotating the PMTs by means of the conveyor belts in the Optics Boxes, in 12 different configurations, both the PMTs relative efficiencies and the relative efficiencies of the optical efficiencies of each channel can be calculated (this corresponds to a fit with 112 data and 24 free parameters).",
"These corrections are convoluted in the analysis presented below.",
"Figure: Oho PMT Rates versus time for March 10, 2016.The last fill with the 12mm collimator and first fill with the 8mm collimator are included in the time window.",
"We notice that the rate, for the same current of the radiating beam, drop to about 1/3 of the initial value.",
"Time (hour) on the horizontal axis.",
"Rate (Hz) on the vertical axis.Figure: Left: Rate-current correlation for one of the LABM PMTs.",
"Rate (solid line) and current (marker) superimposed on the left.",
"Time (hour) on the horizontal axis.",
"Rate (Hz) on the vertical axis.",
"Right: Rate versus current.",
"Current (A) on the horizontal axis.",
"Rate (Hz) on the vertical axis.",
"We notice that the signal from the PMTs is linearly correlated to the current of the radiating beam.With increasing currents we observed a saturation effect due to the discriminators dead time, Fig.",
"REF .",
"With the current electronics, we will run with smaller collimators and light attenuators during Phase II.",
"Figure: Rate-current correlation for one of the LABM PMTs, takenon a high current day.",
"The saturation is due to discriminator dead time."
],
[
"Properties of LABM angular scans.",
"Angular scans were performed to try and identify the IP.",
"By orienting either the primary or the secondary mirror using remotely controlled motors, we were able to accept light coming from different angles inside the Beam Pipe.",
"Since the primary and secondary mirrors are very close (typically 10 to 20 cm out of optical lines exceeding 15 meters from IP to the Optics Box), we have tested numerous times that we could obtain identical images by scanning with only one of the two mirrors, while keeping the other fixed.",
"The angle to step motor conversion factors are all published in our Belle II Twiki page [3], as well as which coordinate and which sign for each motor.",
"We have chosen reference frames where the $+y$ axis always points up and the $+x$ axis is always pointing towards the center of the ring.",
"As a result, two of our angular frames are right-handed and two are left-handed.",
"The scans are done on a 26X41 grid of points, typically spanning 20X20 mrad$^2$ .",
"Each point and each current are taken five times and averaged before moving the telescope, and the specific rates are plotted.",
"Initially, scans were taken with 19 mm collimators (the Wollaston prisms themselves).",
"As we found the luminous areas, we restricted the angular range of the scans and decreased the size of the collimators.",
"Numerous images are always present in our scans, due to reflections inside the Beam Pipes.",
"While the HER pipe is brushed aluminum, highly reflecting at low angles and for all (visible) wavelengths considered, the LER pipe is coated with sputtered titanium nitride(TiN), whose optical characteristics are not well known.",
"We contacted the authors of Ref.",
"[4], and with their help we were able to calculate the low angle reflectivity of TiN, presented in Fig.",
"REF for our lowest and highest wavelengths.",
"We can not automatically assume that the brightest image be the directly observed beam.",
"Reflections can come from light emitted at lower angles than the actual angle of direct observation.",
"Also, reflections at, for example, 45 degrees in azimuth will not produce any significant $x-y$ polarization since the reflection polarizes light at 135 degrees, with equal components along the vertical and horizontal axes.",
"Figure: TiN glancing reflectivity for both s- and p- componentsas a function of glancing angle.",
"First plot: λ=650\\lambda =650 nm.",
"Second plot:λ=350\\lambda =350 nm.The method used to identify the IP in our scans goes as follows: complete a scan.",
"Look for the main sweeps or spots to identify the incoming beam, radiating from the last bending dipole, whose location and strength is known (see Table REF ).",
"The last spot in the sweep (or more precisely, the outermost point of the sweep in the horizontal plane) is where the beam exits the dipole.",
"given the known geometry of the IR, the IP will be a known number of milliradians above or below the last spot.",
"reflected radiation is not necessarily redder and more polarized than the direct image, because reflections can happen near the 45 degrees area of the Beam Pipe, and because reflections are typically from radiation emitted at a lower angle than the direct observation.",
"We check that the polarization and spectral pattern, Our detector has a triangle shaped angular acceptance, with base approximately 1 mrad with current collimators.",
"Both sweeps are fully visible from our vacuum mirrors.",
"The LER sweep will be approximately 3.3 mrad in angular width, the HER will be 1.4 mrad.",
"As saturation effects decrease the quality of a picture, images were taken with PMTs that presented the least saturation."
],
[
"Nikko Down",
"The Nikko side (observing the LER, or positrons) faces a hard last bend (Table REF ).",
"The angular scan in the red x- and y-PMTs are shown in Fig.",
"REF .",
"Figure: Nikko Down angular scan with “red” PMT.",
"First plot: x-polarization.Second plot: y-polarization.",
"The IP is marked as ared square, the structures marked I-IV are discussed in the text.",
"Thewaist of the Beam Pipe at the IP is shown as a red ellipse.The single sweep on the left (reflection I) has characteristics consistent with being the direct image.",
"It has the right angular size, after detector effects are taken into account, as the beam progresses through the sweep, the angle $\\theta $ to the vacuum mirror decreases.",
"We observe a decrease of 60% total intensity from left to right.",
"as mentioned before, the beam sweep, projected to the front of the magnet, is 3cm long.",
"This means that the azimuthal angle to the vacuum mirror changes from about -53 degrees to about -90 degrees.",
"Based on the azimuth, we expect a substantial polarization change across the sweep.",
"This is best done with the “blue” PMT, since the onset of the short magnet regime is faster with higher frequencies.",
"In the four main pixels of the sweep the polarization $R_x/R_y$ goes from 9.3 to 19.4 to 29 to 53.",
"We mark the location of the IP with a red square (Fig.",
"REF ).",
"The rest of the images (Fig.",
"REF ) are reflections off the outer wall of the Beam Pipe.",
"The red ellipses correspond to the waist of the Beam Pipe at the IP.",
"We calculate which angles allow for reflection into our lower telescope window, starting from the nominal beam located at $(-x,0)$ as it sweeps through the last dipole.",
"We find that the equation relating the reflection angles $\\psi $ on the Beam Pipe reads ($r$ is the inner radius of the Beam Pipe) (see Fig.",
"REF ) $\\frac{x}{\\sin {\\phi }}=\\frac{r}{\\cos {(3\\phi )}},\\;\\; \\psi =\\pi /4-2\\phi .$ Figure: Transverse geometry of synchrotron radiation reflections in theBeam Pipe.",
"The position of the radiating point along the sweepis defined by XX, and the angles φ\\phi and ψ\\psi are defined.This equation can be tranformed into a cubic equation in $\\cos ^2{\\phi }$ .",
"One or three solutions are possible.",
"By varying $x$ to evaluate the properties of the reflection sweeps, we find that for $x=0$ (the last point on the sweep) the one solution is $\\psi =\\pi /2$ , indicating a reflection off the top or bottom of the Pipe.",
"These reflections can only be seen very faintly above and below the main sweep.",
"They are faint because the radiation angle is triple the angle of the main sweep, and they are also slightly to the left of the main sweep because the reflection from the early part of the sweep will be slightly towards the right.",
"With increasing $x$ , two solutions become available.",
"The value of $\\psi $ becomes $\\pi /4, 5\\pi /4$ for $x=1$ .",
"Therefore reflections of sweeps that cover an interval of $x$ will have a slope.",
"We can clearly see two such reflections in Fig.",
"REF , Reflections III and IV.",
"Reflection V can not be reconciled with any expected reflection from the last magnet.",
"We believe it is a triple reflection from the second to last magnet.",
"We calculate that such a reflection would show up in $\\theta _y$ approximately 1.5 mrad above the true sweep.",
"The observed sweep is 1.0 mrad above the true sweep."
],
[
"Oho Up",
"The Oho data, observing the HER, or electron, beam, are similar for both the Up and Down detectors.",
"The last bend in this line is a soft bend, so that the sweep is only 30% wider than our angular resolution.",
"The beam in both telescopes will present itself as a small, very bright spot (because all the light from the bend is concentrated in two scan points).",
"The straighter, more reflective Beam Pipe creates more diffuse reflections.",
"The Oho Up data including the IP are shown in Fig.",
"REF .",
"Here, too, broad reflections are seen.",
"The one spot consistent with the sweep is clear.",
"Oho Up showed widespread PMT saturation, so that this image was taken with the “UV” PMT, which only accepted less than 1% of the total light.",
"To obtain cross comparisons with Nikko Down, we used data from 20 mA beams, which had only one PMT (“green”) showing saturation.",
"Oho Up is the only telescope where the $(x,y)$ mapping is inverted, and it shows it by being the only one with a strong $y-$ polarization at the Box.",
"Figure: Oho Up angular scan with “UV” PMT.",
"The IP is marked as ared square."
],
[
" Nikko Up and Oho Down.",
"In the case of Nikko Up, we were unable to obtain images consistent with an angular scan.",
"This was due to a mechanical problem with all the Primary mirrors, which were built from scratch out of metal at Wayne State to make sure they would withstand the high radiation environment.",
"In a commercial remotely controlled mirror, the mirror support is made of somewhat flexible plastics which yields and can transform the linear motion of the stepper motors into the spherical motion of the rotating mirror.",
"In our case, the first version of the mechanism involved one square brass rod sliding into a slightly larger rod, and this transmission was sensitive to slight misalignments between stepper motors and mirrors, to the point where one of the four primary mirrors (the Nikko Up) would move irregularly.",
"The current version, where the transmission is made with sliding forks operating the mirror support knobs, is extremely robust.",
"Nikko Up is also the only one of the four telescopes where the remotely controlled secondary mirror is 100 cm from the primary (every other secondary is within 30 cm).",
"With the primary stuck in an uncertain direction, the secondary mirror here is too far from the vacuum mirror to be usable for the angular scan.",
"Having no back up scanning, we obtained images with bands which were consistent with one of the two mirror controls not actually moving.",
"Oho Down produced good images, but with rates which were 30 times lower than Oho Up.",
"Diode calibration data, which had been used to calibrate the PMTs, were re-analyzed to discover that diode data, too, were weaker by a factor of 30 with respect to Oho Up.",
"This discovery shows that the attenuation takes place in the Optical Channel, and not inside the Beam Pipe where it could not be fixed.",
"To solve this problem, we have established a new protocol for alignment were the diode intensity is measured at every step of the Optical Channel assembly, so that intensity losses are immediately detected and remedied.",
"The detector as presented here had only final diode intensity measurements."
],
[
"Large angle synchrotron radiation.",
"In order to discuss quantitatively the obtained images, it is important to consider some properties of the large angle SR, which we review in this section.",
"Ref.",
"[1] discusses the properties of large angle synchrotron radiation.",
"It considers the classical formulae from Ref.",
"[5], which only offer the intensity as a function of the elevation angle $\\psi $ above the sweep.",
"In Gaussian units, the approximation $\\gamma \\theta \\gg 1$ and the fractional Bessel functions in the limit of large argument, one electron will radiate according to $ \\frac{d^{2}U}{d\\omega d\\psi } \\sim \\frac{e^2}{\\pi c} \\bigg ( \\frac{\\omega \\rho \\psi }{c} \\bigg ) exp\\bigg ( \\frac{-2\\omega \\rho \\psi ^{3}}{3c} \\bigg )$ At large angle, these formulae are unpolarized.",
"At the location where the LABM vacuum mirrors are located, which is directly above or below the beam axis, the elevation angle and the common spherical angle $\\theta $ coincide, and so the latter symbol is used.",
"Following Ref.",
"[1], the magnet length $L$ is introduced, the total radiated energy $U=PL/c$ , and the differentials are such that $d\\psi =4\\pi d\\theta $ .",
"Making the large-angle substitutions $ (1 + \\gamma ^{2} \\theta ^{2}) \\sim u^{2} \\,\\,\\,;\\,\\,\\, \\omega \\sim \\frac{3ct}{\\rho \\theta ^{3}}$ the final double-differential distribution in $(t,u)$ is obtained, $ \\frac{d^{2}U}{du dt} \\sim \\frac{54 \\pi U \\rho }{L} \\bigg ( \\frac{t}{u^5} \\bigg ) exp(-2t)$ The short magnet approximation considers angles much larger than $1/\\gamma $ , Ref. [6].",
"In the large angle approximation, the radiation is polarized according to an azimuthal pattern.",
"Using the same notation, the general form for short magnet radiation is $ \\frac{d^{3}U_{x}}{du dt d\\phi } \\sim \\frac{48 U \\rho }{\\pi L \\gamma } \\frac{cos^{2}(2\\phi )}{u^{4} t^{2}}$ and $ \\frac{d^{3}U_{y}}{du dt d\\phi } \\sim \\frac{48 U \\rho }{\\pi L \\gamma } \\frac{sin^{2}(2\\phi )}{u^{4} t^{2}}$ where the angle $\\phi $ is taken from the direction of the force applied to the electron.",
"The full equations also have exponential factors that suppress radiation at angles larger than the few milliradians considered here.",
"The transition angle between the two regimes is at around $20/\\gamma $ [1].",
"In Table REF , we see that the angles at which the last dipoles are observed are in the “short magnet” region, at 33 and 44 times $1/\\gamma $ respectively.",
"We do not have a precise caluclation, but use the formulae above to extract useful comparisons.",
"From the different angles of observation, we expect a redder, more polarized spectrum on the positron side, and a less polarized, bluer spectrum on the electron side."
],
[
"Cross checks.",
"We use Eq.",
"REF , integrated over the visible spectrum and the solid angle subtended by the primary mirrors, and divided over the number of pixels in a sweep, to calculate SR rates $(Hz/mA)$ on each side and compare them against the true rates calculated according to Eq. (3).",
"Only the rates in pixels identified as belonging to a sweep are used.",
"To compare the experimental rates with the calculations, we use the LABM efficiency calculations from Ryan Gillard's Thesis [7], which finds that the overall photon detection efficiency varies from 2 to 7% depending on wavelength, averaging about 4%.",
"We had to re-run the Monte Carlo because we made the operative decision to use (in Phase I alone) wider ruled gratings than previously planned, and in the process we sacrificed one PMT for background measurements.",
"The overall detector efficiency does not change, but the visible photons are currently detected by three out of four PMTs in each array.",
"In Phase II, having found that backgrounds localized in the Optics Box are negligible, we will return to the nominal gratings.",
"The Up/Down comparisons would be very useful to extract information about the relative angle of radiation with respect to the detector but, as stated above, one detector on each side had problems.",
"Our results are summarized in Table REF and show good agreement (to about 20%, consistent with the error in determining the exact number of pixels in a sweep) for the total rates on both sides.",
"Also the total rates are very similar on the positron and electron side, both as measured and as calculated.",
"This is consistent with both telescopes observing the true beam, and with them having similar efficiency.",
"The ratio green/blue $R_{23}$ is also given (the ratio of true rates of the second and third PMT), and also the polarization ratio, which is the $R_x/R_y$ ratio for a given color.",
"We find that the positron telescope does see highly polarized light consistent with Eqs.",
"(7-8) and that the ratio increases rapidly with increasing frequency.",
"However, the electron telescope sees a bluer light with little polarization, consistent with a smaller angle of observation but not consistent with the azimuthal pattern expected in large anlge radiation.",
"While this is qualitatively correct, we do not have a model to explain in detail the large polarization difference between the two sides.",
"It is noted that in any case, backgrounds will be measured directly during Phase II, during one beam runs, so that a detailed model of large angle SR radiation is not needed.",
"Table: Comparison of rates and calculated intensities.",
"The quantities are described in the text."
],
[
"Signal expectations.",
"In this Section we compare the rates at the calculated IP positions against future beamstrahlung signals.",
"Onyl the normalized rate of the pixel closest to the calculated IP is considered.",
"Beam parameters are listed in Table REF for June 2016.",
"We have assumed 1A currents for each beam.",
"The total beamstrahlung yield, the fractional energy acceptance of the LABM, and the calculated photoelectron rates (summed over all PMTs) are also reported.",
"We have assumed 1A currents for each beam.",
"The total beamstrahlung yield, the fractional energy acceptance of the LABM, and the calculated photoelectron rates (summed over all PMTs) are also reported.",
"Table: Beam parameters and LABM yields for Phase I and Phase II.LABM ϵ\\epsilon means LABM energy acceptance, the fraction of total radiatedenergy intercepted by the vacuum mirror.",
"LABM rates refer to the totalphotoelectron rates(summed over all phototubes), while LABM S/B andcompare the currentlymeasured background rates to the expected signal rates.",
"HER, LER I representPhase I parameters, HER, LER II Phase II parameters.We note the S/B greater than 10 for Phase II on the LER side and of order 1 on the HER side.",
"Regrettably the HER side has a significant amounts of reflection right where the IP is.",
"Further, we are extrapolating the present rates being extrapolated to Phase II.",
"However, there are currently no SR masks installed in SuperKEKB.",
"It is expected that the masks will strongly reduce the noise for rays coming from an angle consistent with the IP.",
"But even without masks, the LABM would be able to perform a variety of studies with background subtraction techniques.",
"It is also noted that the Phase I measurements do not include the expected Touschek halo accompanying the beam, which may contribute extra backgrounds when crossing the final quadrupoles off-axis."
],
[
"Conclusions.",
"The LABM has shown that two of the telescopes work close to specifications.",
"The detector is dark and has worked continuously for 4 months.",
"The problems encountered with the other two telescopes have been overcome by substituting some mechanical parts and by changing the alignment protocol.",
"The main results are: the direct measurement of expected radiation patterns through angular scans.",
"This shows first that the whole system works with some redundancy (scans were performed by moving either the primary or secondary mirrors), and that the modeling of radiation inside the Beam Pipe is of sufficient quality for both the positron and the electron side.",
"from the patterns, the geometric determination of the IP, which is crucial for the LABM.",
"from the determination of the IP, the direct measurement of the expected Phase II backgrounds, although Phase II beams will also have a smaller, spectrally different contribution from the Touschek halo, absent in Phase I, which may contribute some extra background from radiation in the Final Focus quadrupoles.",
"Under the assumption that the Touschek halo is not dominant, the backgrounds are sufficiently low that LABM operations will be possible.",
"They will be further reduced by a planned change in collimators."
],
[
"Acknowledgment",
"We thank Y. Funakoshi for useful discussions.",
"This project has received support from the US-Japan Science and Technology Cooperation Program, and grants Fronteras de la Ciencia Conacyt FOINS-296, Ciencia Basica Conacyt CB-180023, PROFAPI UAS 2013-144, and University of Tabuk grant 1436/052/2."
]
] | 1709.01608 |
[
[
"Non-equilibrium Green's function study of magneto-conductance features\n and oscillations in clean and disordered nanowires"
],
[
"Abstract We explore various aspects of magneto-conductance oscillations in semiconductor nanowires, developing quantum transport models based on the non-equilibrium Green's function formalism.",
"In the clean case, Aharonov-Bohm (AB - h/e) oscillations are found to be dominant, contingent upon the surface confinement of electrons in the nanowire.",
"We also numerically study disordered nanowires of finite length, bridging a gap in the existing literature.",
"By varying the nanowire length and disorder strength, we identify the transition where Al'tshuler-Aronov-Spivak (AAS - h/2e) oscillations start dominating, noting the effects of considering an open system.",
"Moreover, we demonstrate how the relative magnitudes of the scattering length and the device dimensions govern the relative dominance of these harmonics with energy, revealing that the AAS oscillations emerge and start dominating from the center of the band, much higher in energy than the conduction band-edge.",
"We also show the ways of suppressing the oscillatory components (AB and AAS) to observe the non-oscillatory weak localization corrections, noting the interplay of scattering, incoherence/dephasing, the geometry of electronic distribution, and orientation of magnetic field.",
"This is followed by a study of surface roughness which shows contrasting effects depending on its strength and type, ranging from magnetic depopulation to strong AAS oscillations.",
"Subsequently, we show that dephasing causes a progressive degradation of the higher harmonics, explaining the re-emergence of the AB component even in long and disordered nanowires.",
"Lastly, we show that our model qualitatively reproduces the experimental magneto-conductance spectrum in [Holloway et al, PRB 91, 045422 (2015)] reasonably well while demonstrating the necessity of spatial-correlations in the disorder potential, and dephasing."
],
[
"Introduction",
"Quantum transport in nanowires is a widely studied topic with numerous applications, many of which are yet to be explored.",
"Nanowires fabricated from narrow band-gap semiconductors harbor surface confined states due to the pinning of Fermi-level over the conduction band edge [1], [2], [3].",
"This results in the formation of a cylindrical two dimensional electron gas (2DEG) on the surface[4], [5], [3].",
"Cylindrical surface confinement may also be brought about by core-shell heterostructured nanowires [6].",
"Such a surface distribution over the nanowire forms a multiply connected domain, topologically equivalent to a ring.",
"Transport in closed multiply connected structures, such as rings or cylinders can result in interference between paths having completed different number of loops.",
"Such paths pick up different phase factors in a magnetic field, which leads to oscillations in the conductance with periods in multiples of $h/e$ , the Aharonov Bohm (AB) period, depending on the paths involved [7], [8].",
"Experiments performed on core shell nanowires [9] and nanowires with surface confined electronic distributions [10], [11], [3] have revealed AB oscillations.",
"Appreciable and sustained oscillations over a large range of applied fields are however contingent upon the presence of a superficial conduction layer.",
"Hence the magneto-conductance traces can be used to probe the presence of such surface confined states.",
"Such experiments and theoretical studies have also been performed on topological insulator nanowires [12], [13], [14], [15].",
"Moreover, superconductor-semiconductor hybrid nanowire devices are used to study Majorana fermions where a study of magneto-conductance and disorder is critical [16].",
"Further, the conductance of disordered nanowires have revealed the Al'tshuler-Aronov-Spivak (AAS) oscillations[17] with period $h/2e$ .",
"Given various theoretical and experimental endeavors on magneto-conductance of nanowires [18], [19], [20], [21], there is still a lack of a comprehensive theoretical study of various aspects of magneto-transport in both clean and disordered nanowires.",
"In this paper, we try to bridge this gap and extend previous studies by employing non-equilibrium Green's function (NEGF) simulations in clean and disordered nanowires.",
"This paper is organized as follows.",
"We begin by providing a brief review of the origin of oscillations in clean and disordered individual nanowires with axial magnetic field.",
"This is followed by a study of clean nanowires in the clean and ballistic limit, where we highlight the role of surface confinement in producing the AB oscillations.",
"However, in reality, semiconductor nanowires are plagued by various sources of disorder such as short-range unscreened potential impurities and surface roughness, as well as dephasing/decoherence.",
"Accordingly, by considering a range of nanowire lengths and disorder strengths, we show a transition point where the AAS oscillations start dominating over the AB oscillations and thus bridge the gap between the clean and the disordered limits.",
"Further, it is found that the oscillatory behavior depends on the position of the Fermi level, dividing the band into distinct regions each with a dominant type of oscillation, depending on the dispersion of the surface states and the constituent material.",
"This analysis is performed by comparing the scattering length with the device dimension, which provides a guideline to predict the oscillatory behavior.",
"Next, we highlight the conditions for observing the non-oscillatory weak localization corrections.",
"We then study the effects of surface roughness and dephasing on the transmission spectrum.",
"We find that while surface roughness may again lead to the dominance of AAS oscillations, dephasing systematically degrades the oscillations, starting with the higher harmonics which can explain the experimentally observed AB oscillations in disordered nanowires.",
"Lastly, we investigate the previously studied effects in nanowires with a parabolic transverse potential leading to a weaker surface confinement of electrons.",
"Our model produces a reasonably well qualitative reproduction of the experimental magneto-conductance spectrum in Ref.",
"numbers[10], highlighting the interplay of spatially-correlated disorder scattering and dephasing to reproduce the experimentally observed features."
],
[
"Surface conduction in a nanowire",
"In this section we give a brief theoretical overview of the physics behind AB and AAS oscillations and their effects on the conductance of a nanowire, using simple steady state equations.",
"We consider a nanowire subjected to an axial magnetic field, in which the electronic motion may be broken down into two parts: the axial motion along the length of the nanowire, and the motion along the circumferential direction.",
"While traversing the nanowire axially, electrons moving along the circumferential direction interfere, resulting in a number of effects which shall be explored in this work.",
"For the simple case of electrons confined only on the surface of the nanowire, thereby behaving as an infinitely long cylindrical system of radius $R$ , the Hamiltonian is given by $H = \\frac{\\hbar ^2}{2m}\\frac{d^2}{dz^2}+\\frac{1}{2m}\\left(\\frac{-i\\hbar }{R}\\frac{\\partial }{\\partial \\phi }-\\frac{e\\Phi }{2\\pi R}\\right)^2$ .",
"The eigen-energies $E(l)$ satisfy E(l) = 2kz22m*+22m*R2( l-0)2, where $k_z$ represents the wave number along the $\\hat{z}-$ direction, $m^{*}$ is the effective mass, $\\Phi =\\pi R^2B$ is the applied axial magnetic flux , with an axial magnetic field ($B$ ) along z-axis, $\\Phi _0 = \\frac{h}{e}$ -the magnetic flux quantum, $e$ is the electronic charge and $h$ is the Planck's constant.",
"While traversing the nanowire axially, electrons move on the surface along series of rings, and interfere.",
"Quantum corrections to conductivity due to the interference between two time-reversed trajectories have been explored previously [22], [21].",
"The conductance can be evaluated from the phase factors arising from the interference terms, which may be calculated using path integrals by summing over the factors corresponding to paths having given winding numbers K(',t',,t)=(t')|(t)=nCndeS(,A)/ =exp ( i e('-)/h )(n ei2ne/hKn ( ',t',, t )), where $\\mathcal {K}$ and $\\mathcal {K}_n$ are the full propagator, and propagators (without magnetic field, as the magnetic part has already been extracted out) restricted to paths having a winding number $n$ ($C_n$ ) respectively, and $S(\\theta ,\\mathbf {A})$ is the action.",
"From the Landauer formalism, the conductance may be written as $\\mathcal {G}(E)=\\frac{2e^2}{h}\\vert \\mathcal {K}(E)\\vert ^2$ .",
"Now all such paths given by the terms inside the bracket in () interfere.",
"For example, paths with winding numbers $n$ and $-n$ combine to result in an oscillation of amplitude $ \\alpha _n \\mathrm {cos}(4\\pi n e/h)$ having a period $h/2ne$ .",
"The dynamics of such a situation can be captured in the expression for the propagator.",
"The conductivity correction can be given by, $\\Delta \\sigma (H)/\\sigma _0 = \\beta \\left(\\sum _n \\alpha _n \\mathrm {cos}(2\\pi n e/h) \\right) $ , where $\\alpha _n$ and $\\beta $ are constants [23].",
"This can also be seen in a different way in steady state.",
"In the absence of a flux through the nanowire, the angular part of the solutions $(\\Psi (r_0,\\theta ))$ form sinusoids on the surface for a given angular momentum quantum number $l$ , along with an axially propagating component.",
"Assuming in general that a state is a linear combination of mutually orthonormal angular momentum eigenstates, the norm of the angular part is given by () = l=-l=rlexp (il).",
"In the presence of a magnetic field, the rotating states pick up a phase, dependent on the angular momentum eigenstate considered.",
"Assuming that a wave-packet incident from one of the leads has had sufficient time to acquire steady state in its angular part, at any angle $\\phi $ along the circumference, the wave-function is written as a sum of components having different winding numbers(n).",
"However such a description can lead to multi-valued wave functions on the application of magnetic field (including the Peierl's phase factor) for non-integral values of $\\Phi /\\Phi _0$ , as the phase acquired over a loop is not necessarily $2\\pi $ .",
"This technicality can however be avoided by switching into a description of the steady state solution in terms of waves with given winding numbers using the Poisson summation formula, drawing from the discussion by Berry [24].",
"This describes the wave at a point as a sum of waves arriving at that point after traversing different loop numbers: () = n=-Wn(), Wn() = - r()exp (i)exp (i2n) d , where $W_n$ is a wave function with winding number n, and $\\gamma $ interpolates $l$ to non-integral values.",
"On assigning the Peierls' phase factor accordingly, () = n=-Wn()exp( i(2n+)0), which avoids the problem above since $\\Psi (\\phi +2\\pi ) = \\sum _{n=-\\infty }^{\\infty }W_n(\\phi +2\\pi )\\mathrm {exp}\\left( i2\\pi (n+1 + \\phi )\\frac{\\Phi }{\\Phi _0}\\right) \\\\= \\sum _{n=-\\infty }^{\\infty }W_{n+1}(\\phi )\\mathrm {exp}\\left( i2\\pi (n+1 + \\phi )\\frac{\\Phi }{\\Phi _0}\\right) =\\Psi (\\phi ).$ Note that after a revolution, the weight of $W_n$ is transferred to $W_{n+1}$ , as $W_n(\\phi +2\\pi )=W_{n+1}(\\phi )$ .",
"Now, in (), the terms in the summation interfere due to the presence of the phase factors, giving rise to the oscillatory behavior with respect to the magnetic flux $\\Phi $ .",
"Note that, for the disordered case, the phase picked by by $W_n(\\phi )$ is now $\\mathrm {exp}\\left( i2\\pi (n+1 + \\phi )\\frac{\\Phi }{\\Phi _0}\\right)\\mathrm {exp}(ig(n,\\phi ))$ .",
"Here the first exponential term is the magnetic phase factor, and the second exponential term is the disorder potential induced phase factor.",
"Here $g(n,\\phi )$ is the classical action for the corresponding path without the magnetic field.",
"Also, $g(n,\\phi )$ is real for our case of real disorder potential.",
"Now $\\vert \\Psi (\\phi ) \\vert ^2$ has oscillatory components corresponding to the terms (for the general disordered case), Wx() Wy() cos( f(x+y)) cos( 2(x-y)0), where $f(x+y)=g^*(x,\\phi )-g(y,\\phi )$ is the path dependent random phase, which evaluates to zero when $y=-x$ , and is dependent on the disorder configuration along the path traversed by the states labeled by $x,y$ .",
"The path dependent phases are phase factors picked up by rotating waves in the absence of a magnetic field, and in general depend on energy and disorder.",
"However for paths $x$ and $y$ , satisfying $y=-x$ , the phases cancel, as mentioned earlier.",
"Such paths contribute to oscillations with period $h/me$ in flux with even $m$ .",
"Higher harmonics are present too, but with much smaller magnitude corresponding to higher winding numbers or longer paths traversed.",
"The AB oscillations can also be connected to the eigenvalue spectrum of a ring, which is a series of parabolas as noted in (), with their minima on integral values of $\\Phi /\\Phi _0$ .",
"The conductance changes each time a parabola crosses the Fermi level, while the flux is increased.",
"The origin of higher harmonics however cannot be explained by this simple picture.",
"From (), it is seen that the first oscillatory component has a period $\\frac{h}{e}$ .",
"Note that this component arises from paths with path difference equal to the circumference of the nanowire.",
"This can happen when $(x,y)=\\lbrace (1,0)$ $(2,1)\\ldots \\rbrace $ in ().",
"Further, one may also introduce a shift of $\\pi $ to the final (=$\\theta ^{\\prime }$ in ()) angular position of both the interfering paths relative to the initial (=$\\theta $ in ()) angular position, such that one may effectively have a winding number $(x,y)=\\lbrace (1/2,-1/2)$ .",
"Thermal averaging in a suitable energy interval at non zero temperatures and/or ensemble averaging in the presence of multiple parallely connected rings in a cylinder can dampen the $\\frac{h}{e}$ oscillations.",
"The next significant mode observed is the $\\frac{h}{2e}$ periodic oscillation.",
"The primary contributor to this harmonic corresponds to $(x,y)=(1,-1)/(-1,1)$ , which are independent of any random phase, and are consequently robust against disorder (elastic scattering) and other fluctuations.",
"Clearly, the paths involved in this harmonic enclose twice the amount of flux as the paths involved in the $h/e$ harmonic.",
"As seen from (), all oscillations with period $h/ne$ , where $n$ is even (formed by $x=n$ and $y=-n$ ) are independent of random phases, making them theoretically resistant to disorder and thermal averaging, within the scope of this analysis.",
"Ideally, in thin rings, oscillations with period $h/e$ dominate, with higher harmonics appearing with smaller amplitudes.",
"However in disordered rings with finite widths, $h/2e$ periodic oscillations become survive and a crossover occurs where $h/2e$ becomes the dominant component [19].",
"Such behaviour is also expected in cylindrical conductors, either due to impurity based disorder or surface roughness and radial randomness along the cylinder axis.",
"We now proceed to a numerical analysis of the aforesaid features."
],
[
"TIGHT BINDING NEGF MODEL",
"In this section we introduce the system and the non equilibrium Green's function (NEGF) formalism of quantum transport being used in this work.",
"We consider a nanowire with a cubic lattice as shown in Fig.",
"REF .",
"To capture the essential physics, we use a tight binding Hamiltonian with a single basis orbital for each lattice-site.",
"$\\mathcal {H} = \\sum _{\\langle i, j \\rangle }\\left( t\\hat{c}^\\dagger _{i}\\hat{c}^{\\phantom{\\ast }}_{j} + (\\epsilon + U_{i})\\hat{c}^\\dagger _{i}\\hat{c}^{\\phantom{\\ast }}_{i} \\right),$ where $\\epsilon $ is the on-site energy, $t$ is the hopping parameter, $U_i$ is the random potential at each site, and the sum is over nearest neighbours $i,j$ (good approximation in the maximally localized Wannier basis).",
"The operators $\\hat{c}^{\\dagger }_{i}\\hspace{2.84544pt}(\\hat{c}^{\\phantom{\\ast }}_{i})$ represent the creation (annihilation) operators for electrons at site $i$ .",
"Figure: Device schematics: (a) The nanowire device and the leads, along with an axial magnetic field.",
"(b) Cubic lattice structure of the nanowire with a surface confining potential.",
"The region marked in red has lower electronic energy than the region marked in blue.",
"This results in a surface confined electronic distribution (in the red region).For obtaining a cross-section of our choice, in our case a disc, suitable potential has been added to simulate the band offset of the required geometry.",
"A very high band offset works as well as hard wall boundary conditions implemented by brute force shaping of the Hamiltonian.",
"Note that cylindrical symmetry has been assumed throughout the study.",
"In the presence of an external vector potential A, the hopping parameters between sites $m$ and $n$ acquire a Peierl's phase[25], $t_{mn} \\rightarrow t_{mn}e^{i2\\pi \\frac{e}{\\hbar }\\int _{\\mathbf {r_n}}^{{\\mathbf {r_m}}}d\\mathbf {r}\\cdot \\mathbf {A(r,}t)}=t_{mn}e^{2\\pi \\frac{\\Phi }{\\Phi _0}}.$ Transport calculations are performed using the NEGF/Keldysh formalism[26] as discussed in ref.",
"[27].",
"We begin with the one particle retarded and lesser Green's function GR(x,t;x,t')=-i(t-t'){c(x,t),c(x',t')}, G<(x,t;x,t')=ic(x,t),c(x',t'), where $\\mathbf {x},\\mathbf {x}^{\\prime }$ represent the initial and the final states respectively, on the standard time contour.",
"The advanced Green's function is given by $G^A=(G^R)^\\dagger $ .",
"In the non-equilibrium steady state, it admits an energy domain representation after Fourier transforming.",
"In the energy domain real space matrix representation, we have, GR(E) = [(E+i)I-H0-U-(E)]-1, where $\\eta $ is an infinitesimal quantity, $E$ is the Fermi energy (controlled by gate voltage), $I$ is an identity matrix of the same size as the system Hamiltonian $H_0$ , $U$ is the on-site Anderson disorder potential representing unscreened short-range impurity potentials.",
"In this work $U$ serves the following purposes: a) It adds a band offset to form a cylinder/disc for the nanowire geometry, b) it adds a confining potential to confine electrons on the surface, and c) in disordered nanowires, it adds an on-site scattering potential.",
"The Keldysh formalism is capable of including various scattering mechanisms such as electron-electron, electron-phonon (which can be used for dephasing), disorder averaged quantities etc, through suitable self-energy operators.",
"The net self-energy $\\Sigma (E)$ accounts for the leads, and may additionally dress the electrons with appropriate interactions.",
"The implementation of the lead self-energy is detailed in Appendix .",
"Since we are concerned only with the physics of the system in the limit of very small applied bias and not directly on the device performance, for transmission calculations we assume that the bands are linearly dropping in the axial direction with vanishing slope.",
"Transverse potential is introduced without any self-consistent calculations.",
"The current for the non-interacting case is given by [28], J = ehdE(fL(E)-fR(E)) Tr[L(E)G(E)RG(E)], where $\\Gamma _{L/R}$ represent the tunnel coupling to the leads, which are related to the broadening induced by the contact self-energies ($\\Gamma _{L/R} = i(\\Sigma _{L/R}-\\Sigma _{L/R}^{\\dagger })$ ).",
"The quantity $\\mathbf {Tr} [\\Gamma _L(E)G(E)\\Gamma _RG^{\\dagger }(E)]$ is the transmission at the energy $E$ , denoted by $T(E)$ .",
"Now, the conductance through a level of degeneracy $g$ ($g=2$ for a spin-degenerate level) in the limiting case of vanishing applied voltage and temperature is given by $\\mathcal {G} = g\\mathcal {G}_0\\int dE T(E)(f_L(E)-f_R(E))\\vert _{{eV,T=0}}=g\\mathcal {G}_0T(\\mu )$ , with $\\mu $ being the equilibrium Fermi level and $\\mathcal {G}_0=e^2/h$ .",
"In the subsequent sections, the lattice constant is denoted by $a$ .",
"Further, all energies and potentials are specified as a scale-invariant quantity, in terms of the hopping parameter $t$ ."
],
[
"NON-DISORDERED NANOWIRES",
"The presence of an axial magnetic field couples orbital angular momentum states with the field, which causes levels with adjacent angular momentum quantum numbers to shift by one flux quantum.",
"The resulting spectrum is quasi-parabolic and quasi-periodic in nature for small applied fields and can be used to study the sub-band structure.",
"The transmission spectrum is influenced by the electronic distribution, which in turn depends upon the radial confining potential[10].",
"Lower surface confinement is found to impart a lower degree of periodicity to the transmission spectrum.",
"This enables a better identification of the sub-bands.",
"Note that throughout the study, the nanowire is assumed to be in the phase coherent regime $(L\\le l_{\\phi })$ unless mentioned otherwise, with $l_{\\phi }$ being the phase coherence length.",
"Our clean nanowires have a diameter $11a$ .",
"Typical experiments have reported phase coherence lengths of the order of a few hundred nanometers[2], [10], which is longer than the nanowire dimensions considered here."
],
[
"No surface confinement",
"In this case, the transverse potential is zero within the boundaries of the nanowire, i.e.",
"$V(r)=0$ .",
"The plot of the variations of the transmission from its mean (w.r.t.",
"flux) as a function of energy and applied field and its fast fourier transform (FFT) spectrum are depicted in Fig.",
"REF (a) and (b).",
"Note that in the quantity $\\delta T(E,\\Phi )=T(E,\\Phi )-\\langle T(E,\\Phi ) \\rangle _\\Phi $ , the average over flux has been performed over the range shown in panel (a).",
"This holds for all the figures in this work.",
"The transmission resembles the Fock-Darwin spectrum in quantum dots[29], with the transmission increasing in steps at the onset of each level.",
"At higher magnetic fields they however tend to align with the Landau levels[30], [29].",
"Subsequent minima move higher in energy as they represent states with a higher orbital angular momentum, which are confined closer to the surface.",
"The period of the pseudo-oscillations are however greater than one flux quantum ($\\Phi _0=h/e$ ).",
"This happens because the electronic distribution is not confined to the surface.",
"Consequently, electronic paths enclose a smaller flux than the flux through the entire nanowire cross-section.",
"Also, there is no common flux-ratio ($\\Phi /\\Phi _0$ ) that may be defined for all states, as different paths can enclose different fluxes.",
"This explains the breadth of the peak.",
"Figure: The unconfined case: (a) Variation of the transmission, from the mean mean value for each energy, δT(E,Φ)=T(E,Φ)-〈T(E,Φ)〉 Φ \\delta T(E,\\Phi )=T(E,\\Phi )-\\langle T(E,\\Phi ) \\rangle _\\Phi .",
"No AB oscillations are observed.",
"At small fluxes, it resembles the Fock Darwin spectrum, which quickly aligns with the Landau levels.",
"(b) FFT spectrum of transmission for the same device, which again shows the lack of AB oscillations.",
"Instead we have a broader peak, with frequencies smaller than that of the AB oscillations ((Φ/Φ 0 ) -1 =1(\\Phi /\\Phi _0)^{-1}=1)."
],
[
"With surface confinement",
"Now, we add a parabolic transverse potential, to study the effect of surface confinement on the conductance harmonics.",
"The potential, $V(r)$ , has the following form, $U(r)=-eV(r) = -V_0\\left(\\frac{r}{R}\\right)^p, $ where $R$ is the cylinder radius, $U(r)$ is the electronic energy, and the parameters $V_0$ and $p$ are adjusted to ensure strong surface confinement.",
"In Fig.",
"REF , we have plotted $\\delta T(E,\\Phi )=T(E,\\Phi )-\\langle T(E,\\Phi ) \\rangle _\\Phi $ for a nanwire with a transverse potential described by (REF ) with $V_0=0.362t$ and $p=2$ , showing the variation of the transmission from its mean value (w.r.t.",
"flux).",
"As seen from Fig.",
"REF (a) and (b), as the field varies, more elaborate and sustained diamonds are observed in comparison to the case with no confining potential.",
"Energy levels do not align with the Landau levels as quickly as in the non-confined case, highlighting the effect of surface confinement.",
"The presence of a confining potential invariably confines all states near the surface, which diminishes the difference between the higher and the lower angular momentum states, as the effective radius of the distribution is forced to be the same for all angular momentum states as in Eq. .",
"Therefore, the spectrum more or less lies in the same band of energy.",
"Further, proper oscillations are observed, with a strong peak at a frequency corresponding to period $\\frac{h}{e}$ .",
"Figure: The surface confined case with V 0 =0.358tV_0=0.358t and p=2p=2: (a) Variation of the transmission, from the mean value (w.r.t flux) for each energy, with a parabolic radial potential.",
"Diamond shaped structures with a period of h/eh/e, synonymous with AB oscillations, can be observed at lower energies.",
"At higher values of flux (roughly over 10Φ 0 10\\Phi _0), they start moving up, aligning with Landau levels.",
"(b) FFT spectrum of the transmission, showing a peak at (Φ/Φ 0 ) -1 =1(\\Phi /\\Phi _0)^{-1}=1, i.e.",
"the AB peak, at low energies.",
"At higher energies, it gives way to lower frequency components.It is observed from Fig.",
"REF (b) that at higher energies we get smaller frequency (larger period) oscillations, similar to the unconfined case.",
"This is because at higher energies there exist states which lie away from the surface, i.e., in the bulk.",
"It is a result of using a parabolic confining potential which has a minima at the center of the nanowire, with the electronic energy being highest at the center.",
"These states, with high energy, enclose a smaller flux and hence oscillate with larger periods.",
"Figure: Transmission traces/cuts at two different energies(chosen for illustration), with (a) a parabolic surface confining potential, and (b) no surface confining potential, corresponding to Figs.",
"and .",
"The orange curves are taken at E-E c =1.9tE-E_c=1.9t, and the blue curves are taken at E-E c =1.2tE-E_c=1.2t, where E c E_c is the lower band-edge.",
"Much better and sustained oscillations, with the AB period, are observed in the surface confined case shown in (a).",
"The sharp steps are a consequence of using zero temperature.",
"In the unconfined case shown in (b), sudden drops in conductance in the unconfined case arise due to the passing of energy bands over the Fermi energy, as they align with the Landau levels.",
"Note that the boundary where this depopulation occurs for the unconfined case, follows the same shape as seen in Fig.",
", where the transmission drops to zero.Note that the cylindrical symmetry of the nanowire under the influence of the gate potential is crucial for observing good AB oscillations.",
"In its absence, a gate potential which depends upon the azimuth angle would suppress paths traversing the section with higher electron energy, and therefore suppress the oscillations.",
"However, for standard oxide thicknesses, such effects are expected to be minor for thin nanowires [10].",
"In Fig.",
"REF we show the transmission at two energy values (chosen just for illustration) for both the confined and unconfined case respectively.",
"In the unconfined case, we initially observe irregular variations with a period larger than the AB period.",
"This is followed by sudden drops in the transmission, as all the levels align with the Landau levels at high flux, and move up, over the Fermi energy, depopulating those states.",
"Higher surface confinement pushes the point where alignment begins to higher values of flux."
],
[
"DISORDERED NANOWIRES",
"In the weak disorder regime, low temperature conductivity is largely dominated by elastic scattering via impurity potentials.",
"When the system is of comparable size, or smaller than the phase coherence length, transmission is affected by interference between paths.",
"This results in oscillatory behaviour in the weak-localization corrections in multiply connected systems, in addition to aperiodic, noise-like universal conductance fluctuations (UCF)[31].",
"Unlike clean nanowires, in long and weakly disordered nanowires, AB (h/e) component no longer dominates over the AAS (h/2e) component [22], [3], which has been revealed in experiments too[3][20].",
"As mentioned earlier, AB (h/e) oscillations have a random non-magnetic phase, whereas in AAS (h/2e) oscillations, the time-reversed paths show a cancellation of this non-magnetic phase.",
"This occurs due to the paths contributing to AB oscillations facing different environments due to the presence of different disorder configurations.",
"This does not occur in AAS oscillations as both the time reversed paths traverse the same classical path, accumulating the same phase which gets canceled, as seen in () and the discussion following it.",
"The phase change introduced by such rigid elastic scatterers is definite, unlike the phase randomization considered in Sec.",
"REF , as might be expected from electron-electron interactions or exchange with a bath as seen in phonon scattering.",
"Simply stated, here we are concerned with the quantum diffusive regime $(l_\\phi >l_e)$ , whereas in Sec.",
"REF , we move towards the classical diffusive regime $(l_\\phi <l_e)$ .",
"In this study, we have neglected effects of spin orbit interaction (SOI), as for nanowires with small diameters (as has been considered in our study) the spin-relaxation length $l_{\\mathrm {SO}}$ has been found to be much larger than the nanowire length/phase coherence length in experiments conducted in InAs and Ga$_\\mathrm {x}$ In$_{1-\\mathrm {x}}$ As/InP nanowire[32], [33].",
"This is manifested as the appearance of WL corrections instead of weak anti-localization (WAL).",
"Further, Rashba SOI would split the degenerate bands and electrons in each spin-polarized band behave independently of the other band [34].",
"The quantum corrections to the conductivity $(\\delta \\sigma )$ (from the Kubo formula), in diagrammatic terms, are given by the sum of the diffusons (ladder diagrams), and Cooperons (maximally crossed diagrams), which add to the Drude part.",
"For isotropic scatterers in closed systems, the contributions of the ladder diagrams to the conductivity vanish.",
"For a cylindrical electronic distribution penetrated by an axial magnetic field ($\\mathbf {B}=\\nabla \\times \\mathbf {A}$ ), $\\delta \\sigma $ can be obtained by solving the Cooperon equation[19], [-i-D(-2ieA)2+1]C(r,r')=1(r-r'), which gives us the return probability.",
"For a cylinder of radius $R$ , the DC correction is given by[19], = -e2 [ln(lle) +2n=1K0(n2Rl)cos(2n20)], where $l_\\phi =\\sqrt{D\\tau _\\phi }$ is the phase coherence length, $l_e=\\sqrt{D\\tau }$ is the elastic scattering mean free path, and $\\mathrm {K}_0(z=2n\\pi R/l_\\phi )$ is the Macdonald function, representing the magnitude of the harmonic with period $h/(2ne)$ .",
"While analytical theory exists for closed systems in the clean, and the extreme case of complete disorder averaging, real experiments involve finite nanowires connected to metallic leads.",
"We numerically bridge this gap by studying this transition region with finite disordered nanowires.",
"We look at the quantum diffusion regime, where the device dimensions are larger than the mean free path, but smaller than the phase coherence length.",
"The magnitude of the AAS component in the transmission can be used to probe the degree of disorder.",
"However, the application of disorder of large magnitude puts the nanowire into the Anderson localized regime, where the conductance drops exponentially.",
"In a weakly disordered mesoscopic ring (one specific sample with a specific lead configuration), both AB and AAS oscillations are expected.",
"On taking an ensemble average over multiple rings, this phase causes the AB oscillation to die down.",
"However, a weakly disordered nanowire is in itself an ensemble of multiple weakly disordered rings with longitudinal/axial nanowire slices.",
"This suggests a significant AAS component in sufficiently long disordered nanowires.",
"On the other hand, too short a nanowire would permit electrons to leave before they are able to traverse the required path lengths along the circumference to generate the AAS oscillation.",
"This should allow us to identify a transition point where AAS magnitude becomes larger than AB magnitude.",
"Considering both the effect of disorder strength as well as the role of nanowire length in dictating the magnitudes of the oscillation harmonics, we now present conductance traces and corresponding spectra for both cases.",
"We consider long disordered nanowires with a strong/sharp surface confining transverse potential which confines the electrons to one layer on the surface.",
"This permits us to explore the essential physics and simulate long nanowires.",
"We consider nanowires with diameter $10a$ .",
"Figure: T(E,Φ)-〈T(E,Φ)〉 Φ T(E,\\Phi )-\\langle T(E,\\Phi )\\rangle _{\\Phi } for W=1.5tW=1.5t and lengths of the disordered section equal to (a)100a100a, (b)125a125a, (c)150a150a and (d)175a175a, in nanowires with strongly surface-confined electrons.",
"For comparison, the strongly confined clean case is shown in Fig. (a).",
"With increasing length, AB oscillations ((Φ/Φ 0 ) -1 =1(\\Phi /\\Phi _0)^{-1}=1) weaken while AAS ((Φ/Φ 0 ) -1 =2(\\Phi /\\Phi _0)^{-1}=2) oscillations become significant.",
"This leads to a transition point where AAS oscillations start dominating AB.",
"The FFT spectrums shown in Fig.",
"highlight the corresponding harmonic contents.Figure: FFT spectrum of the variation in transmission (Fig. )",
"from the mean for each energy, for W=1.5tW=1.5t and lengths of the disordered section equal to (a)100a100a, (b)125a125a, (c)150a150a and (d)175a175a, in nanowires with strongly surface-confined electrons.",
"With increasing length, AB oscillations ((Φ/Φ 0 ) -1 =1(\\Phi /\\Phi _0)^{-1}=1) weaken while AAS ((Φ/Φ 0 ) -1 =2(\\Phi /\\Phi _0)^{-1}=2) oscillations become significant.",
"This leads to a transition point where AAS oscillations start dominating AB.",
"This is seen more clearly in panels (e) 100a100a and (f) 175a175a, where the limiting cases for the AB and the AAS FFT are shown with the corresponding energy-averaged AAS magnitude shown in green (See Fig. ).",
"Here (a) and (e) show the limit where the AB oscillations dominate, while (d) and (f) show the limit where the strength of the AAS oscillations become significant and comparable to the AB oscillations.Figure: FFT of the variation in transmission from the mean for each energy, averaged over energy, for W=1.5tW=1.5t, corresponding to the systems considered in Fig. .",
"Note that for this disorder value, the nanowires are in the quantum diffusive/weakly localized regime for all energies.",
"One can observe the transition point (shown by the dashed line) beyond which the AAS starts dominating over the AB oscillations.",
"The increase in the magnitude of the AAS component is an effect of increasing phase coherence length with increasing nanowire length, as we are considering phase coherent transport here.",
"Further, the residual AB component in long nanowires can be attributed to the leads, which are clean extensions of the device, and are subject to the same magnetic field.Coherent scattering cannot be captured by a self-energy as it is in general complex.",
"The imaginary part of the self-energy will destroy the coherence [35], [36].",
"Since the disorder is static, it is introduced using a random on-site uncorrelated disorder potential into the Hamiltonian, instead of using a self-energy [37].",
"The disorder potential is uniformly distributed in $[-W/2, W/2]$ , where $W=\\mathrm {constant}\\times t$ , and satisfies $\\langle V(r)\\rangle =0$ and $\\langle V(r)V(r^{\\prime })\\rangle =(W^2/12)\\delta (r-r^{\\prime })$ .",
"Details regarding the implementation of the self-energy are explained in Appendix .",
"For sufficiently long disordered nanowires in the quantum diffusive (phase coherent elastic scattering) regime (range of energies where the scattering length is smaller than the nanowire dimensions), AAS (h/2e) component survives, with a significant decrease in the AB (h/e) component due to `ensemble averaging'/phase-cancellation over the constituent rings/slices.",
"In Fig.",
"REF we show $T(E,\\Phi )-\\langle T(E,\\Phi )\\rangle _{\\Phi }$ for lengths ranging from $100a-175a$ .",
"Fig.",
"REF presents the corresponding FFT spectrums.",
"Beginning with this section, we consider a nanowire with strongly surface-confined electrons for ease of simulation, using a much stronger transverse confining potential than the parabolic transverse potential used in Fig.",
"REF .",
"This explains the sharpness of the harmonics.",
"In Fig.",
"REF (a) with $L=100a$ the effects of disorder are not significant.",
"However, in Fig.",
"REF (d) with $L=175a$ disorder significantly alters the transmission spectrum, resulting in components with twice the AB frequency.",
"Correspondingly, from Fig.",
"REF with $L=100a$ , it is observed that AB oscillations still dominate, even in the coherent scattering regime.",
"A small degradation in AB amplitude is observed for $L=125a$ .",
"However, for $L=175a$ , it is observed that in the quantum diffusive and weakly localized regime, the amplitude of AAS oscillations just exceed that of the AB oscillations, signifying a transition point.",
"The AB oscillation amplitude decreases as a longer nanowire ensures better averaging.",
"This allows us to find a critical length, beyond which the AAS oscillations dominate.",
"This is highlighted in Fig.",
"REF .",
"Note that we observe a plateau forming for the AB component.",
"This may be explained by the presence of the lead self energy in the device Green's function.",
"The leads form a clean extension of the devive, with the same geometry, and is subject to the same magnetic field.",
"Hence, the lead self-energy introduces a spatially in-homogeneous, flux dependent, non local quantity at the sites connected to the leads, unlike the analytical calculations performed for a closed system [22], [19].",
"This results in a non-vanishing contribution of the diffusons to the conductivity.",
"Also, we observe a rise in the AAS component.",
"From (REF ), the magnitude of AAS$\\sim \\mathrm {K}_0\\left(2\\times 2\\pi R/l_\\phi \\right)$ , which is a decreasing function of $\\left(2\\times 2\\pi R/l_\\phi \\right)$ .",
"Since we are considering phase coherent transport, we are inadvertently increasing the phase coherence length with increasing device length.",
"This can roughly explain the rise, but one must be careful as it assumes complete averaging.",
"From a different perspective, the diffusion probabilities are not restricted to the device alone.",
"Since we are considering phase coherent transport in the device, the winding of the electronic paths are directly affected by the leads, as electrons may escape into the leads along their paths.",
"Lastly, the non-monotonic variations in the plot are a consequence of the stochastic nature of the problem in the presence of uniform random disorder.",
"Figure: T(E,Φ)-〈T(E,Φ)〉 Φ T(E,\\Phi )-\\langle T(E,\\Phi )\\rangle _{\\Phi } for L=100aL=100a and a range of disorder strengths (a)W=0.0tW=0.0t, (b)W=1.0tW=1.0t, (c)W=1.6tW=1.6t and (d)W=1.9tW=1.9t.",
"One can observe the transition from the AB dominated regime to the AAS dominated regime on increasing WW.",
"This is inferred by counting the repeating features with increasing flux at any given energy, which changes from being one in panel (a) to two in panel (d).",
"Also, in (c) the AAS harmonic starts dominating from the center of the band (E=6t)(E = 6t) while the AB harmonic is stronger near the band-edges (only the upper band-edge at E=10tE=10t is shown here).",
"This is explained by the variation of the scattering length over the band as shown in Fig.",
"and the following corresponding discussion.",
"Further, the AAS oscillations bear the same phase relation at all energies.Figure: The dominant harmonic contribution as a function of the disorder strength and device length (diameter =10a=10a), as obtained from our numerical results.",
"The boundary between the regions has been smoothly interpolated.",
"The domain yielding a dominant AAS contribution is expected to increase on using a nanowire with a larger diameter due to a reduction in the scattering length.In Fig.",
"REF we observe the seamless transition from the AB dominated regime to the AAS dominated regime on increasing the disorder strength.",
"Further, by gradually introducing disorder into the system, the results progressively start to resemble experimentally observed features [10], [20].",
"Note that while the AB oscillations at different energies are uncorrelated, the AAS oscillations are correlated [38].",
"This may be observed by observing the zero field phase of the harmonics.",
"In Fig.",
"REF we show the numerically obtained dominant harmonic as a function of the length of the disordered section and the disorder strength, summarizing their roles.",
"According to Fig.",
"REF , on increasing the length of the nanowire in Fig.",
"REF the characteristics of disorder should start appearing at smaller values of the disorder strength.",
"Further, we find that a rather strong disorder potential is required for a dominant AAS contribution.",
"However, the domain in $W$ and $L$ yielding a dominant AAS contribution is expected to increase on using nanowires with a larger diameter.",
"The scattering length for a sub-band is given by, le=v(E)21a2W212DOS(E), where $v(E)$ is the group velocity and $\\mathrm {DOS}(E)$ is the density of states of the considered sub-band at energy $E$ .",
"The net scattering legnth, after considering all the sub-bands, is given by a harmonic sum of the scattering lengths of each sub-band [39] (See Appendix for details).",
"Since the scattering length is inversely proportional to the density of states which in turn is proportional to the nanowire diameter, thicker nanowires will have a smaller scattering length.",
"This is elaborated in the discussion following Fig.REF .",
"Note that while this analysis suggests that a larger nanowire shall support a significant AAS contribution, in reality, dephasing limits the system size over which phase coherence is retained.",
"Consequently, the device size over which interference effects may be sustained is limited.",
"The effects of dephasing are explored in detail in Sec.",
"REF .",
"Also, we observe that the FFT spectrum is dominated by the oscillation harmonics, with the UCF hardly contributing.",
"A strongly surface confined distribution is not expected to contribute to UCF, as closed loops on the nanowire surface which do not traverse the circumference do not have a net flux through them ($\\oint \\mathbf {A}\\cdot d\\mathbf {l}=0$ ).",
"Such features may be obtained from bulk transport with an axial magnetic field, or from both the bulk and the surface contributions in the case of non-axial magnetic field.",
"In Fig REF , the results for a disordered nanowire with a magnetic field perpendicular to the axis is shown.",
"In this case, magneto-conductance oscillations are dominated by loops confined on the surface, but not covering the circumference, showing only the UCF component.",
"Such results have been experimentally realized[40].",
"Figure: Magnetic field perpendicular to axis (B ⊥ B_{\\perp }) : (a) Transmission spectrum, and (b) its FFT, for a nanowire of length 125a125a with W=1.5tW=1.5t and a strongly surface confined distribution.",
"Note that Φ=B ⊥ πR 2 \\Phi =B_{\\perp }\\pi R^2.",
"We observe aperiodic UCF, which vary much more slowly and randomly than the periodic AB oscillations.Back to the case of axial field, in the same transmission spectrum, two separate regimes can be observed.",
"Near the center of the band, AB oscillations dominate, whereas on moving away from the enter, AAS oscillations dominates.",
"This can be explained by finding the scattering length as a function of energy, as shown in Fig.",
"REF (detailed in the Appendix ).",
"Here, by the scattering length, we are referring to the mean distance between elastic scattering events.",
"This is different from the transport mean free path, associated with back-scattering (describing the transmission in the diffusive limit[41]), differing by an energy dependent relationship.",
"Figure: Scattering length (in units of the lattice constant(aa)) as a function of energy in the band, for (a)W=1.0tW=1.0t, and (b)W=0.15tW=0.15t (disorder potential ∈[-W/2,W/2]\\in [-W/2,W/2]).",
"The band edges are located at 2t2t and 10t10t.",
"We see that transport is more diffusive in the middle of the band than near the band edges.",
"In order to achieve a transition from the diffusive to the ballistic regime within the band, we must consider a much weaker scattering potential.",
"In (b), we have marked the length L=150aL=150a, which is the length of the nanowire used in Fig. .",
"Note that the dips in the scattering length correspond to the Van Hove singularities, where the associatedhigh density of states enhances scattering.Figure: FFT spectrum of the transmission: (a) For a nanowire of length 150a150a, with disorder potential given by W=0.15tW=0.15t.",
"A significant AAS component is observed in the diffusive regime near the center of the band (E=6tE=6t), which subsides as we move higher in energy to the ballistic regime near the band-edge (E=10tE=10t).",
"Only in the range of energies where the scattering length is smaller than the nanowire length (marked in Fig.",
"), which is roughly E∈[3t,9t]E\\in [3t,9t], we observe a significant AAS component.",
"Outside this range (E>9tE>9t in this figure), the AAS component is much smaller than the AB component.",
"(b) For the same nanowire as in (a), but with the on-site energies (E=5tE=5t and E=7tE=7t) alternating on neighboring sites.",
"Apart from the bandgap at the center of the band, the AAS oscillations dominate in two separate regions around E=7tE=7t and E=9tE=9t.From Fig.",
"REF , we see that the scattering length is smallest near the band center and increases as we move away from the center.",
"This implies that the band center experiences a more diffusive environment than the band edge.",
"This is a consequence of the effective two-dimensional nature of the problem due to strong surface confinement, leading to a growing mobility edge appearing from the band center.",
"A shorter scattering length near the band-center leads to a dominant AAS contribution in the center of the band and not near the band-edges, where the AB component strengthens.",
"This is clearly seen clearly in Fig.",
"REF .",
"Since experiments typically probe the conduction band-edge (lower edge), observing the AAS harmonic becomes harder.",
"On considering a weaker scattering potential, we can observe the diffusive to ballistic transition in the transmission spectrum, as seen in Fig.",
"REF (a).",
"Further, the strength of the AAS harmonic is found to be related to the the relative magnitude of the nanowire length and the scattering length.",
"Also, since (REF ) implies that the scattering length decreases with increasing density of states (detailed in Appendix ), which in turn increases with increasing nanowire diameter; a nanowire with a larger diameter is expected to show stronger AAS oscillations.",
"In Fig.",
"REF (b), where we consider the same nanowire as in Fig.",
"REF (a), but with alternating on-site energies on neighboring sites, we observe a different behavior as a result of the change in the density of states.",
"The AAS oscillations are significant in two regions, around $E=7t$ and $E=9t$ , with its strength displaying a non-monotonic behavior unlike Fig.",
"REF (a).",
"This highlights the effect of dispersion of surface states in governing the strength of the harmonics with energy.",
"These observations may be used as a guideline to extract the elastic scattering length from experiments in suitable situations, by varying the gate voltage and studying the oscillatory components.",
"Assuming ideal surface confinement, this structure of the transport regime seen across the band is a function of the surface dispersion, which enters the scattering length via its dependence on the density of states and the group velocity.",
"Hence a different lattice would have a different structure.",
"Note that, throughout the work, we do not look at the disorder induced band-tail extensions as the leads are infinitely long and clean extensions of the device, which do not support such states."
],
[
"In a nanowire with a surface confined distribution, the oscillation harmonics introduce transmission variations of magnitude $\\approx 1$ .",
"Non-oscillatory weak localization (WL) corrections, which give rise to a decreasing resistance with increasing magnetic field, introduce much smaller variations.",
"In order to observe them separately without being subdued by the large oscillatory AB component and UCF, we have to reduce the oscillations.",
"This may be achieved by doing the following: Using long disordered nanowires to suppress the AB component.",
"Along with point (1) averaging the transmission spectrum over a range of energies, as shown in Fig.",
"REF to kill random fluctuations (UCF).",
"This also helps to kill the AB harmonic as the oscillations at differennt energies are uncorrelated.",
"Such averaging effects maybe present in experiments due to non-zero temperatures, or finite applied bias.",
"Finite energy averaging has been studied earlier in similar contexts [42].",
"Note that the origin of AAS oscillations is same as the conventional WL and in the low field regime only the latter manifests itself.",
"Using an electronic distribution which naturally results in diminished oscillatory components.",
"This may be brought about by weaker surface confinement, as mentioned in Sec.",
"REF .",
"Using a perpendicular magnetic field, which would suppress all harmonics.",
"In Fig.",
"REF we show the non-oscillatory weak localization corrections after suppressing the oscillatory ones.",
"At few values of energy, we still observe an initially decreasing transmission, which may be attributed to a large UCF component.",
"With energy averaging, the transmission rises at all energies, for small values of flux.",
"Further, as mentioned above the origin of AAS oscillation are same as the pure weak-localization corrections.",
"Therefore the for small flux values, the transmission traces in Fig.",
"REF (d) as well as Fig.",
"REF (a) rise as well.",
"Figure: δT(E,Φ)=T(E,Φ)-〈T(E,Φ)〉 Φ \\delta T(E,\\Phi )=T(E,\\Phi )-\\langle T(E,\\Phi ) \\rangle _\\Phi : (a) Parallel field : Nanowire of length 175a175a, W=1.5tW=1.5t, and strong surface confinement.",
"We have introduced a finite temperature by averaging over an energy interval=0.31t =0.31t.",
"(b) Perpendicular field : Nanowire of length 125a125a, W=1.5tW=1.5t, and strong surface confinement, corresponding to the system in Fig.",
", with no energy averaging.",
"As expected, we observe that the transmission initially rises with increasing flux, as the phase relationships between the disorder induced localized states are destroyed.",
"Note that the magnitude of the weak localization correction is ≈0.1-0.3\\approx 0.1-0.3, which can get easily subdued by the oscillatory components.",
"Also, on extending panel (a) to higher flux values, we recover oscillatory behavior which eventually start dominating.",
"But on extending (b) we get the same transmission values as shown in Fig.",
"as it's the same system."
],
[
"SURFACE ROUGHNESS",
"We now proceed to detail the effects of surface roughness/random corrugations.",
"We model the surface roughness by a radius $R(\\phi ,z)$ , which varies randomly as a function of the azimuth angle and the axial distance.",
"This induces random variations in the hopping parameters and the flux across each cross-sectional disc along the nanowire axis.",
"The variations in the radius are described by an uncorrelated noise, with $R(\\phi ,z)-R_0\\in [-\\Delta _R/2,\\Delta _R/2]$ , such that $\\langle R(\\phi ,z)\\rangle =R_0$ ($R_0$ is the radius without surface roughness), and $\\langle R(\\phi ,z)R(\\phi ^{\\prime },z^{\\prime })\\rangle =(\\Delta _\\mathrm {R}^2/12)\\delta (\\phi -\\phi ^{\\prime })\\delta (z-z^{\\prime })$ .",
"Note that we do not consider Anderson disorder while studying surface roughness, resulting in a diffusive regime due to random hoppings instead of random on-site potentials.",
"Now, we adopt a generalized Harrison [43] (power-law) scaling of the hopping parameters with respect to the corresponding bond distances, which gives us, tx,x'=tx,x'(a2+(r(x)-r(x'))2a), where $\\mathbf {x}\\equiv \\left(\\phi ,z\\right)$ , and $a$ is the bond distance in the absence of surface roughness.",
"$\\zeta <0$ represents the sensitivity of the hopping parameter to variations in the bond distance.",
"We introduce two cases cases which we discuss subsequently: First, $\\zeta =-2$ along with minor realistic surface roughness (specified in terms for radial variations as the corresponding areal variations are negligible) in Fig.",
"REF ; second $\\zeta =0$ along with severe roughness (specified in terms of transverse areal variations) in Fig.",
"REF .",
"In Fig.",
"REF , we implement a realistic surface roughness using $\\Delta _\\mathrm {R} = 0.4\\mathrm {R}_0$ and $\\zeta =-2$ , which induce minor variations in the cross-sectional area with standard deviation $\\sigma _\\mathrm {A}=-0.04\\mathrm {A}_0$ , and variations in the hopping given by $\\sigma _\\mathrm {\\tilde{t}}= 0.23\\mathrm {t}$ .",
"Figure: Surface roughness without on-site Anderson disorder: (a)δT(E,Φ)\\delta T(E,\\Phi ), and (b) its FFT, of a clean nanowire of length L=100aL=100a, with surface roughness described by ζ=-2\\zeta =-2 and Δ R =0.4R 0 \\Delta _\\mathrm {R} = 0.4\\mathrm {R}_0.",
"We observe a degradation of the AB component, accompanied by the dominance of the AAS component in general introduced primarily by the hopping disorder (except at E=7.1tE=7.1t, which shows up as an odd AB peak in the FFT spectrum, at that energy).",
"The AAS contribution is clearly visible in Fig. (a).",
"Note that, we do not observe magnetic depopulation, as seen in Fig .",
"Further, as explained in the text, the skewed hopping disorder reduces the bandwidth of transmission.When the variation in the flux experienced by different planes is of the order of the flux quantum $(\\Phi _0=h/e)$ , it should destroy the phase relationships and consequently the flux periodic oscillations.",
"However, for all practical values of $\\Delta _{\\mathrm {R}}$ , the corresponding variation in the cross-sectional surface area and hence the flux penetrating each cross-sectional disc is too small to induce significant flux variations.",
"Therefore, the effects of surface roughness are dominated by random hopping parameters.",
"In this case (shown in Fig.REF ), we observe a degradation of the AB component, accompanied by a significant contribution from the AAS component.",
"Moreover, while on-site Anderson disorder and surface roughness both lead to the emergence of AAS oscillations, only the latter decreases the bandwidth (difference of upper and lower band-edges) of transmission which decreases on the mean value of the hopping parameters.",
"The reduction in the bandwidth is a consequence of the power-law dependence of the hopping parameters on the bond distances (see (REF )) which reduces its mean value thereby skewing its distribution.",
"Figure: Surface roughness without on-site Anderson disorder: (a) Variation of the transmission from the mean value for each energy δT(E,Φ)\\delta T(E,\\Phi ), and (b) its FFT, without SR, in a nanowire with strong surface confinement for comparison.",
"The 'discretized' diamonds are due to the presence of a finite number of states.",
"In (c), we see δT(E,Φ)\\delta T(E,\\Phi ), and (d) its FFT, of a clean nanowire of length L=100aL=100a, but with surface roughness, described by ζ=0\\zeta =0 and Δ A =0.4A 0 \\Delta _\\mathrm {A} = 0.4\\mathrm {A}_0.",
"Note that, in panels (b) and (d), the energy axis goes into the page.",
"Here E=6t\\mathrm {E}=6t and E=10t\\mathrm {E}=10t are the band-center and the upper band-edge respectively.",
"A degradation of the sharp features is observed in the transmission spectra as a result of SR, which is manifested as a smoother decay of the corresponding FFT spectrum.",
"The strong low frequency peaks ((Φ/Φ 0 ) -1 <1(\\Phi /\\Phi _0)^{-1}<1) correspond to the slow drop in transmission due to magnetic depopulation.",
"The magnitudes of the harmonics have been found to be same.",
"Further, at higher values of flux, the AB oscillation diamonds have degraded.Now, in Fig.",
"REF (c) and (d), we implement strong roughness with the cross-sectional area $\\mathrm {A}(\\mathrm {z})\\in [-\\Delta _A/2,\\Delta _A/2]$ (where $\\Delta _A=0.4\\mathrm {A}_0$ ) with a uniform distribution.",
"It doesn't suppress the harmonics, but rather causes a large drop in the transmission induced by magnetic depopulation.",
"The flux penetrating each cross-sectional disc along the nanowire axis is given by, $\\tilde{\\Phi }(\\mathrm {z}) = \\Phi (\\mathrm {A}(\\mathrm {z})/\\mathrm {A}_0)$ , where $\\Phi $ is the flux in the absence of surface roughness, and $\\mathrm {A}(\\mathrm {z})$ is the cross-sectional area.",
"Note that, the same variation in cross-sectional area induces larger variations in the flux, at higher values of flux as $\\Delta _\\Phi =B\\Delta _A=\\langle \\Phi \\rangle \\Delta _A/\\langle A\\rangle $ .",
"The effects of this can be observed in Fig.",
"REF (c) from the stronger degradation of the AB component at higher values of flux, than at lower values.",
"Also note that we consider perfect surface confinement.",
"However, for large values of $\\Delta _A$ as the electronic distribution is subjected to a large variation in the magnetic flux, the system may effectively be considered as possessing a radially smeared/spread-out electronic distribution (weaker surface confinement).",
"From the discussion following Fig.",
"REF , the magnetic depopulation is justified as the energy bands rise quickly with the flux on decreasing the surface confinement.",
"In conclusion, surface roughness can have varying and contrasting effects with the observed behavior being dependent on the material and the sample under consideration.. For considerable and realistic values of $\\zeta $ , the effect of surface roughness is dominated by the random hopping parameters, which results in the dominance of the AAS component.",
"For values of $\\zeta $ close to zero, we observe a magnetic depopulation.",
"Further, the degradation of the AB component occurs only at higher values of flux while the AAS component never dominates.",
"Also, large variations in the nanowire cross-sectional area may produce flux variations of the order of the flux quantum when the nanowire is subjected to strong magnetic fields, destroying the oscillatory magneto-conductance features."
],
[
"DEPHASING",
"Magneto-conductance oscillations arise from the phase picked up over closed loops.",
"For a particular harmonic to survive, the phase coherence length should be greater than the corresponding constituent path lengths.",
"In (REF ), the amplitude of the harmonics is given by the Macdonald function $(\\mathrm {K}_0(z=(n2\\pi R/l_\\phi )))$ , which exponentially decays $(\\sim e^{-z}/\\sqrt{z})$ to zero for $z\\rightarrow \\infty $ .",
"This suggests that when the $l_\\phi <2\\pi n R$ , then the amplitude of the corresponding oscillatory component with period $h/(2ne)$ exponentially vanishes.",
"To clearly see its effect, phase relaxation (rather randomization) has been included, which should be able to diminish/eliminate the oscillations.",
"The dephasing is implemented in the self-consistent Born approximation (or the non crossing approximation) by a phenomenological dephasing model [36]$^{,}$ [44]$^{,}$ [45], to emulate electron-electron and electron-phonon interactions.",
"Beginning with the current in lead $x$ , Ix=2edE2[x<(E)G>(E)-x>(E)G<(E)], the kinetic equation for the lesser Green's function, with the lead and elastic interaction self-energies given by $\\Sigma _{C}$ and $\\Sigma _{S}$ , respectively, G<(E)=GR(E)C<(E)GA(E)+GR(E)S<(E)GA(E), reduces the current to, I=Icoh+Iincoh, Icoh/incoh=2edE2[x<(E)(GR(E)C/S>(E)GA(E)) -x>(E)(GR(E)C/S<(E)GA(E))].",
"Now, the general dephasing self-energy is given by, [RS(E)]= D[GR(E)], [S</>(E)]= D[G</>(E)], GR(E)=[(E+i)I-H-RC(E)-RS(E)], where $\\tilde{D}$ is an operator, whose form depends on the dephasing scheme and strength.",
"Now (REF ), (REF ), (REF ) and (REF ) may be solved numerically to get the current.",
"However, further reduction gives us the following relations for the transmissions, I=dE2(Tcoh(E)+Tincoh(E)), Tcoh(E)=Tr[L GR(E)R GA(E)], Tincoh(E)=Tr[L GR(E)K(E) GA(E)], where $K(E)$ is obtained self-consistently by, K(E)=D[GR(E)(R(E)+K(E))GA(E)].",
"For the momentum relaxing scheme, the scattering self energy is local in its action and is therefore diagonal in its real space matrix representation.",
"Therefore, $\\tilde{D}[L]_{i,j}=D[L]_{i,j}\\delta _{i,j}$ , for a matrix $[L]_{i,j}$ , with $D$ being the dephasing strength.",
"This reduces (REF ) and (REF ) to, [S(E)]i,j<=kM(j,k)[GRC< GA]k,ki,j, [M]i,j = DI-D|[GR(E)]i,j|2i,j, K=k[M]j,k[GR2 GA]k,k.",
"The scattering strength is given by $D\\sim 1/\\tau $ , which is a measure of the correlation of the dephasing scattering potential $U(r)$ , $D(E,r,r^{\\prime })=D\\delta (r-r^{\\prime }) \\propto \\mathinner {\\langle {U(r)|U(r^{\\prime })}\\rangle }$ .",
"In semiconductor nanowires at low temperatures, low energy dephasing scattering is dominated by elastic electron-electron interactions[46] and electron-phonon interactions [47], [46].",
"Also, acoustic phonon scattering is nearly elastic and randomizes the momentum of the electronic distribution.",
"In that case, $D=\\zeta ^2k_BT/(\\rho ^2)$ , where $\\zeta $ is the deformation potential, $\\rho $ is the density, and $v$ is the longitudinal sound velocity.",
"This permits us to consider these processes together in the phenomenological dephasing model, once the total scattering rate is accounted for $(1/\\tau = 1/\\tau _{e-e}+1/\\tau _{e-ph})$ .",
"We have not used Büttiker probes, as they are phenomenological and appropriate for inelastic scattering, such as longitudinal electron-phonon (e-ph) interactions, which have been neglected in this study.",
"Figure: Dephasing - Clean case : FFT spectrum of the variation in transmission from the mean value for each energy δT(E,Φ)\\delta T(E,\\Phi ), (a)without dephasing and (b)with momentum relaxing dephasing in a clean nanowire of length L=50a\\mathrm {L}=50a with strong surface confinement.",
"The energy axis goes into the page.",
"The difference between (b) and (a) can be observed by comparing the decreased magnitudes of the Fourier components in (b).Figure: Dephasing - Disordered case (W=0.75tW=0.75t) : (a) Variation of the transmission from the mean value for each energy δT(E,Φ)\\delta T(E,\\Phi ), and (b) its FFT, without dephasing, in a disordered nanowire of length L=50a\\mathrm {L}=50a with strong surface confinement.",
"In (c), we see δT(E,Φ)\\delta T(E,\\Phi ), and (d) its FFT, of the same nanowire, but with momentum relaxing dephasing (D=0.4 2 \\mathrm {D}=0.4^2).",
"The energy axis goes into the page.",
"A degradation in the amplitudes of the harmonics is observed in the FFT spectrum (from (b) to (d)).",
"Further, the degradation for the AB peak is smaller than the higher harmonics, increasing its dominance.",
"Consequently, the distinct diamond shaped structures representing the AB oscillations become the dominant feature in the transmission spectrum.",
"Here E=6t\\mathrm {E}=6t and E=10t\\mathrm {E}=10t are the band-center and the upper band-edge respectively.Now, as the magnitude of dephasing is gradually increased, the phase coherence length should decrease, and fall behind the required length to sustain each winding number.",
"This effect should be observable in the FFT spectrum as a systematic degradation of the oscillatory part, with the highest harmonics vanishing one by one on increasing the dephasing potential.",
"In the case of clean nanowire, it is seen from Fig.",
"REF that the magnitude of the AB oscillations die down.",
"Further, in the case of disordered wires, we observe a degradation in all the harmonics, with the higher harmonics degrading much faster than the AB harmonic.",
"This leaves us with a relatively dominant AB contribution.",
"This can be noticed by comparing Figs.",
"REF (c) and (d).",
"Also, at the edges of the steps in the transmission, we encounter Van Hove singularities in the density of states, which increase the scattering rate.",
"This results in smoothed out steps.",
"We have until now, considered a dephasing rate which is constant with respect to energy.",
"In reality, scattering rates depend on energy[48], being typically of the form $\\tau (E)\\propto E^r$ .",
"When an energy dependent scattering rate is taken into account in the local dephasing model, the degradation of the harmonics becomes energy dependent.",
"Note that the effect of dephasing is very different compared to the effect of surface roughness.",
"While dephasing kills the higher harmonics one by one, creating a relative dominance of the AB component, surface roughness may lead to the relative dominance of the AAS component, similar to the case with local potential disorder considered in Sec REF .",
"This is evident from Figs.",
"REF and REF .",
"This observation may serve as a guideline to pinpoint the source of features observed in experiments."
],
[
"DISORDERED AND INCOHERENT NANOWIRES WITH WEAK SURFACE CONFINEMENT",
"Having explored disorder scattering, as well as dephasing in nanowires with a strongly surface confined electronic distribution, it remains to be seen how a weaker surface confinement, like the case shown in Fig.",
"REF , affects the results in the presence of disorder.",
"To this end, we study a disordered nanowire of length $25a$ with a parabolic surface confining potential described by (REF ) with $V_0=0.4098t$ and $p=2$ , as shown in Fig.",
"REF , to qualitatively investigate the underlying physics.",
"Figure: δT(E,Φ)=T(E,Φ)-〈T(E,Φ)〉 Φ \\delta T(E,\\Phi )=T(E,\\Phi )-\\langle T(E,\\Phi ) \\rangle _\\Phi , for each energy for a parabolic transverse potential/weak surface confinement: (a) In the absence of any disorder potential (W=0)(W=0).",
"(b) In the presence of a spatially uncorrelated disorder with W=1tW=1t.",
"The angular momentum sub-bands are observed.",
"(c) In the presence spatially uncorrelated disorder with W=1tW=1t as well as dephasing characterized by D=0.4 2 D=0.4^2, resulting in smoother variations.",
"(d) In the presence of a spatially Gaussian correlated disorder with standard deviation σ=2a\\sigma =2a and W=1tW=1t.",
"(e) Same as (d), but with dephasing characterized by D=0.4 2 D=0.4^2.",
"Note that, in all the panels, the conduction band edge (not shown) lies at ≈-0.1t\\approx -0.1t.On comparing Fig.",
"REF with Fig.",
"3 in Ref.",
"numbers[10], which shows the magneto-conductance spectrum in a InAs nanowire, one makes three observations: First, the angular momentum sub-band structure is clearly observed from both our results and the experimental data in Ref. numbers[10].",
"Note that we show a larger range of energy and consequently, many sub-bands are visible.",
"Second, as compared to Fig.",
"REF (b), the experimental data in Ref.",
"numbers[10] displays much smoother fluctuations in the magneto-conductance spectrum along with fluctuating behavior within each transmission diamond.",
"Further, the fluctuations are no longer limited to the transmission step edges, but it extends into the transmission plateaus too.",
"This may arise from two sources namely, correlated disorder and dephasing.",
"Now, as seen from (REF ), the scattering rate for an uncorrelated disorder potential is proportional to the density of states (detailed in Appendix ).",
"Accordingly, the scattering rate is peaked and much larger at the Van-Hove singularities than elsewhere.",
"Hence, the effect of disorder is largely limited to band-edges.",
"However, if a disorder potential with a long range real-space correlation ($\\langle V(\\mathbf {r})V(\\mathbf {r}^{\\prime })\\rangle \\ne \\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })$ ) is introduced, then there would be contributions from propagators at different wavevectors weighted by the corresponding momentum-space correlation function ($\\langle V(\\mathbf {q})V(\\mathbf {q}^{\\prime })\\rangle \\ne \\delta (\\mathbf {q}+\\mathbf {q}^{\\prime })$ ) in the self-energy given by (), which would then enter the scattering time via ().",
"As a result, even for wavevectors not located at the Van-Hove singularities, the scattering rate picks up contributions from the nearby Van-Hove singularities.",
"Therefore, a disorder potential with long-range real-space correlation introduces larger fluctuations within the transmission plateaus compared to uncorrelated disorder.",
"This is evident from comparing the spatially uncorrelated disorder in Fig.",
"REF (b) with Fig.",
"REF (d) where the disorder potential at each point is still $\\in [-\\frac{W}{2},\\frac{W}{2}]$ where $W=1t$ , but along with a Gaussian spatial correlation with standard deviation $\\sigma =2a$ , i.e., $\\langle V(\\mathbf {r})V(\\mathbf {r}^{\\prime })\\rangle =a^2\\frac{W^2}{12}\\mathrm {exp}\\left(\\frac{|\\mathbf {r}-\\mathbf {r}^{\\prime }|^2}{2\\times (2a)^2}\\right)$ .",
"For disorder potentials with the same strength, the Gaussian correlated disorder yields more fluctuations, in particular, within the transmission plateaus.",
"Note that while this analysis is valid for weak disorder, the conclusion holds even for strong disorder where, at each order of perturbation in $V$ , the disorder with long-range real-space correlation yields more fluctuations.",
"Additionally, dephasing can introduce smoother fluctuations, especially within the transmission plateaus.",
"This suggests an interplay of correlated disorder and dephasing in the experiment, both of which must be included in an accurate description of the experiment [10].",
"Third, oscillatory features, including even the AB oscillations, seem smeared out in the experiment.",
"This may be attributed to a significant disorder potential.",
"In fact, Fig.",
"REF (e), with a Gaussian spatially-correlated disorder with standard-deviation $2a$ shows the best qualitative agreement with the experiment.",
"AAS oscillations are not observed in the simulations as our nanowires are shorter than the ones considered in Figs.",
"REF and REF even though a weaker surface confinement renders the observation of AAS oscillations more difficult.",
"In spite of this, our short nanowires qualitatively reproduce the experimentally observed features reasonably well which too do not have the AAS oscillations.",
"This may partly be attributed to the nanowire not being sufficiently long for the transverse potential present, or the presence of significant dephasing."
],
[
"Conclusion",
"We have employed the NEGF formalism to systematically analyze magneto-conductance oscillations in nanowires in the presence of an axial magnetic field, demonstrating the effects of disorder, roughness and dephasing.",
"In the ballistic limit AB oscillations are dominant contingent upon the surface confinement of the electronic distribution in the nanowire.",
"By studying disordered nanowires of different lengths, we show the parameter space which leads to a significant AAS harmonic.",
"We also demonstrated that the relative magnitudes of the scattering length and the device dimensions dictate the ballistic and quantum diffusive regimes within the energy bands, thereby determining the relative dominance of the AB and the AAS oscillations with energy.",
"We find that the AAS oscillations begin dominating from the center of the band, while typical experiments probe the low-energy physics near the conduction band-edge (bottom of the band).",
"This should effectively increase the required disorder strength and/or nanowire length required to see a significant AAS contribution.",
"We then showed the ways of suppressing the oscillatory WL corrections to reveal the non-oscillatory WL correction.",
"Lastly the effects of surface roughness and dephasing on the magnitude as well as the components of the oscillations were studied, revealing a key difference in their effects on the harmonics.",
"While surface roughness may have contrasting effects of dominant AAS oscillations or magnetic depopulation depending on the sensitivity of the hopping parameters to the roughness, dephasing systematically degrades harmonics, beginning with the higher ones.",
"These additional factors can explain the unexpected suppression of the AAS content and the consequent relative dominance of AB oscillations even in disordered nanowires [10].",
"Finally, we considered nanowires with a parabolic transverse potential, demonstrating the necessity of spatially-correlated disorder potential and dephasing to yield qualitative agreement with magneto-conductance experiments in [Holloway et al, PRB 91, 045422 (2015)].",
"In conclusion, our comprehensive results capture the physics and satisfactorily provide qualitative agreement with experimental features while motivating further experimental research.",
"Acknowledgments: The authors AL and BM acknowledge support from IIT Bombay SEED grant and ISRO-RESPOND grant.",
"KG and JB acknowledge support from the Natural Sciences and Engineering Research Council of Canada.",
"AL acknowledges useful discussions with Arnab Manna.",
"Materials supporting the claims shall be made available on reasonable requests."
],
[
" Contact self energies",
"For clean nanowires, to reduce computational complexity, only 1 layer (cross section) is taken as the device.",
"Rest of it is accounted for by the contacts [49].",
"To simulate a long nanowire subject to an axial magnetic field, while having a single layer as the device, leads need to have the same geometry and the same magnetic field as the device.",
"We find the surface Green's function iteratively to calculate the contact self-energies.",
"One may keep the leads free from magnetic field.",
"But, in order to keep the magnetic field divergence-free, it would necessitate either the presence of additional transverse field components to compensate for the axial field gradient, or the use of a gradual ramping (and consequently a much longer device, increasing computational expense) to approximately ensure zero divergence.",
"The surface Green's function $(g_s)$ is given by, gs-(E+i-Hlead)+gs-1=0, lead=gs , where $\\beta $ is the coupling between the transverse layers in the device and $\\tau $ is the coupling matrix between the device and the lead.",
"In our case, $\\tau =\\beta $ .",
"For a disordered nanowire, the self-energy is calculated in the same way as the ballistic case.",
"However, in this case, the device has a finite length/number of planes to model a disordered nanowire of the corresponding length connected to ideal leads.",
"Note that we have kept the same axial magnetic field in the leads as it cannot be abruptly terminated $(\\nabla \\cdot \\vec{B}=0)$ ."
],
[
"Density of states and group velocity of rolled 2D square lattice (cylindrical)",
"Assuming spin degeneracy, a band's contribution to the density of states (DOS) is given by [50], $\\mathrm {DOS}(\\mathrm {E}) = \\frac{2}{l}\\sum _i\\int dk \\delta (k-k_i) \\Bigr |\\frac{\\partial \\epsilon }{\\partial k_{\\parallel }} \\Bigr |^{-1},$ where $l=2\\pi /a$ is the length of the first Brillouin zone, $\\epsilon (\\mathbf {k})$ is the dispersion relation, and $k_is$ satisfy $E=\\epsilon (k_i)$ .",
"We follow a procedure, similar to the one given in Ref. [50].",
"However, for our problem, we cannot use a low energy approximation.",
"We need the distribution over the entire band.",
"$\\epsilon (\\mathbf {k}) = \\alpha +2\\beta \\mathrm {cos}(k_{\\parallel }a)+2\\beta \\mathrm {cos}(k_{\\perp }a) \\quad (\\mathbf {k}=\\mathbf {k_{\\parallel }}+\\mathbf {k_{\\perp }}).$ Defining a circumferential vector $\\vec{R} = N\\vec{a_1}$ , where $\\vec{a_1}$ , is the reciprocal lattice basis vector along the circumference, we get, $\\Delta k_{\\perp }=\\Bigr |\\mathbf {k}\\cdot \\frac{\\vec{R}}{R}\\Bigr |.$ The total DOS is obtained by summing up the sub-band contributions $\\mathrm {DOS}(E,n)$ .",
"DOS(E) = n=0N-1DOS(E,n) =n=0N-111(2)2-( E--2cos(2nN))2.",
"Also the group velocity, for each band is given by | v(E,n)| = |1k(k,n)|=(2a)sin2(ka) =a( (2)2-[ E--2cos(2nN ) ]2 ) (1/2)."
],
[
"Scattering length",
"Here we derive the scattering length using disorder averaging [51].",
"A random on-site potential (uniformly drawn from $\\left[-\\frac{W}{2},\\frac{W}{2}\\right]$ ) with the following properties is considered.",
"$\\langle V(\\mathbf {r})\\rangle =0 \\qquad \\langle V(\\mathbf {r})V(\\mathbf {r}^{\\prime })\\rangle =a^2\\frac{W^2}{12}\\delta (\\mathbf {r}-\\mathbf {r}^{\\prime }).$ where the $\\langle \\ldots \\rangle $ stands for disorder average.",
"From the theory of disorder averaging, the first order term in the self energy, $\\Sigma ^{(1)}=\\int \\frac{d\\mathbf {q_1}}{(2\\pi )^2}\\langle V(\\mathbf {q_1})\\rangle $ vanishes (using (REF )).",
"The second order term is, (2)(E, k) = dq1(2)2dq2(2)2V(q1)G0(k+q2,E)V(q2) =a2W2d DOS()G0(E,) For weak disorder, we can use the first Born approximation, in which all but the second order term of the full diagrammatic perturbative expansion are discarded.",
"The imaginary part of the self energy, obtained by using the Sokhotski-Plemelj formula, gives the scattering rate, 1sc(E)=-2Im(E)=2a2W212DOS(E), le(E) = v(E)sc(E) = v(E)21a2W212DOS(E).",
"Using Eqs.",
"() and (), the scattering length at energy E for the $n^{th}$ sub-band is, le(E,n)a= 122W2M(E,n)DOS(E,n), M(E,n)= ( (2)2-[ E--2cos(2nN ) ]2)12, DOS(E,n)= 1(2)2-( E--2cos(2nN))2.",
"The scattering rates are then summed up over all sub-bands [39] using (), le(E)-1 = 1Nn=1Nle(E,n)-1."
],
[
"Parameters",
"All energies are specified in units of the hopping parameter $t=\\hbar ^2/(2ma^2)=0.61$ eV, where $m=9.1\\times 10^{-31}$ kg is the electronic mass, and $a=0.79$ nm is the lattice constant.",
"This scaling renders the actual value of $t$ seemingly irrelevant, instead manifesting its importance through the length of the nanowire and the realization of actual disorder strength relative to $t$ .",
"These are available from experimental data.",
"Our clean nanowires have a diameter equal to $11a$ , and the disordered nanowires have diameter equal to $10a$ .",
"Disorder potentials, surface roughness parameters and dephasing strengths have been specified in the corresponding figure captions."
]
] | 1709.01623 |
[
[
"Hybrid Finite Element - Spectral Method for the Fractional Laplacian:\n Approximation Theory and Efficient Solver"
],
[
"Abstract A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions.",
"The scheme is based on reformulating the original problem posed over $\\Omega$ on the extruded domain $\\mathcal{C}=\\Omega\\times[0,\\infty)$ following Caffarelli and Silvestre (2007).",
"The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme.",
"Finite elements are used in the direction parallel to the problem domain $\\Omega$, and an appropriate spectral method is used in the extruded direction.",
"The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $\\Omega$.",
"We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues.",
"We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem.",
"The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient.",
"Numerical examples in two and three dimensions show that the approach is quasi-optimal in terms of complexity."
],
[
"Introduction",
"Over the last few years, non-local and fractional order models models have seen a surge in interest in a wide variety of application areas such as anomalous diffusion, material science, image processing, finance and electromagnetic fluids .",
"Compared with local, integer order equations, the linear algebraic systems arising from fractional order models are generally dense, and can be difficult to solve efficiently.",
"In the present work, we explore how structural sparsity can be leveraged to solve a fractional order Poisson problem in quasi-optimal complexity.",
"Let $\\Omega \\in C^{2}$ or a convex polyhedron in $\\mathbb {R}^{d}$ .",
"One of the many possible ways of defining a fractional order Laplacian on $\\Omega $ uses the spectral information of the integer order operator.",
"Let $0<\\lambda _{0}\\le \\lambda _{1}\\le \\dots $ and $\\phi _{0},\\phi _{1},\\dots $ be the eigenvalues and eigenfunctions of the regular Laplacian $\\left\\lbrace \\begin{array}{rlrl}-\\Delta \\phi _{m}\\left(\\vec{x}\\right)&=\\lambda _{m}\\phi _{m}\\left(\\vec{x}\\right), &&\\vec{x}\\in \\Omega , \\\\\\phi _{m}\\left(\\vec{x}\\right) &=0, && \\vec{x}\\in \\partial \\Omega ,\\end{array}\\right.",
"$ normalised so that $\\left|\\!\\left|\\phi _{m}\\right|\\!\\right|_{L^{2}}=1$ .",
"The eigenfunctions $\\left\\lbrace \\phi _{m}\\right\\rbrace _{m=0}^{\\infty }$ form a complete orthonormal basis of $L^{2}\\left(\\Omega \\right)$ .",
"This means that any function $u\\in L^{2}\\left(\\Omega \\right)$ can be expanded as $u=\\sum _{m=0}^{\\infty }u_{m}\\phi _{m} \\qquad \\text{with } u_{m}= \\left(u,\\phi _{m}\\right)_{L^{2}}.",
"$ In particular, $\\left(-\\Delta \\right) u\\left(\\vec{x}\\right)&= \\sum _{m=0}^{\\infty }u_{m}\\lambda _{m}\\phi _{m}\\left(\\vec{x}\\right),$ while the spectral fractional Laplacian of order $s\\in (0,1)$ is given by $\\left(-\\Delta \\right)^{s}u\\left(\\vec{x}\\right)&= \\sum _{m=0}^{\\infty } u_{m}\\lambda _{m}^{s} \\phi _{m}\\left(\\vec{x}\\right).$ As $s\\rightarrow 0$ , the identity is recovered, whereas the usual, integer order Laplacian is recovered as $s\\rightarrow 1$ .",
"We are interested in solving the fractional order Poisson problem $\\left\\lbrace \\begin{array}{rlrl}\\left(-\\Delta \\right)^{s}u\\left(\\vec{x}\\right) &= f\\left(\\vec{x}\\right), && \\vec{x}\\in \\Omega ,\\\\u(\\vec{x})&=0, && \\vec{x}\\in \\partial \\Omega \\end{array}\\right.",
"$ with given right-hand side $f$ .",
"The spectral definition is not the only possibility to define a fractional order Laplacian on $\\Omega $ ; other choices include the so-called integral fractional Laplacian, defined as $\\left(-\\Delta \\right)_{I}^{s} u\\left(\\vec{x}\\right) = C(d,s) \\operatorname{p.v.}",
"\\int _{\\mathbb {R}^{d}} \\; d\\vec{y} ~ \\frac{u(\\vec{x})-u(\\vec{y})}{\\left|\\vec{x}-\\vec{y}\\right|^{d+2s}}$ where $C(d,s) = \\frac{2^{2s}s\\Gamma \\left(s+\\frac{d}{2}\\right)}{\\pi ^{d/2}\\Gamma \\left(1-s\\right)}$ is a normalisation constant and $\\operatorname{p.v.",
"}$ denotes the Cauchy principal value of the integral .",
"If $\\Omega =\\mathbb {R}^{d}$ , the two definitions coincide, but they are different for bounded domains .",
"In previous work , , we demonstrated that adaptive finite elements and multigrid methods can be used to solve fractional equations based on the integral definition in quasi-optimal complexity.",
"Existing solution methods for the fractional Poisson problem involving the spectral definition of the fractional Laplacian generally follow one of two different paths: exploit the Dunford-Taylor integral representation of the fractional power of the discretized integral order Laplacian , ; or, interpret the fractional operator as a Dirichlet-to-Neumann map of a singular elliptic problem embedded in $d+1$ space dimensions - the so-called extruded problem approach , .",
"used finite elements to discretize the extruded problem, along with a problem specific multigrid solver , while is based on a $hp$ -FEM discretization.",
"In this work, we also approximate the extruded problem.",
"However, while we use finite elements in the direction parallel to the problem domain, we introduce a spectral method in the extruded direction.",
"By a careful choice of expansion functions in the spectral method, we recover a quasi-optimal method.",
"This work is structured as follows: In sec:notation, we introduce the necessary notation as well as the extruded problem associated with the spectral fractional Laplacian.",
"We briefly discuss the eigenfunctions of the extruded problem in sec:eigenf-extend-probl, which are then used to discretize the extruded problem in sec:probl-discr.",
"In sec:error-bound, we derive an a priori error bound which is explicit in the mesh size $h$ on $\\Omega $ and the spectral order on $[0,\\infty )$ .",
"The method requires suitable approximation of the true eigenvalues of the standard Laplacian over $\\Omega $ .",
"We describe in sec:choice-appr-eigenv how the approximations can be obtained in an efficient manner.",
"In sec:solution-linear-system, we give details on the solution of the resulting linear systems using a multigrid solver.",
"Finally, in sec:numerical-examples, numerical results are presented that confirm quasi-optimal complexity of our algorithm."
],
[
"Notation",
"Let $\\Omega $ be a subdomain of $\\mathbb {R}^{d}$ as above, then we define the Sobolev space $H^{s}\\left(\\Omega \\right)$ to be $H^{s}\\left(\\Omega \\right)&:=\\left\\lbrace u\\in L^{2}\\left(\\Omega \\right) \\mid \\left|\\!\\left|u\\right|\\!\\right|_{H^{s}\\left(\\Omega \\right)} < \\infty \\right\\rbrace ,$ equipped with the norm $\\left|\\!\\left|u\\right|\\!\\right|_{H^{s}\\left(\\Omega \\right)}^{2}&= \\left|\\!\\left|u\\right|\\!\\right|_{L^{2}\\left(\\Omega \\right)}^{2} + \\int _{\\Omega }\\; d\\vec{x} \\int _{\\Omega }\\; d\\vec{y} \\frac{\\left(u(\\vec{x})-u(\\vec{y})\\right)^{2}}{\\left|\\vec{x}-\\vec{y}\\right|^{d+2s}}.$ The space $\\widetilde{H}^{s}\\left(\\Omega \\right)$ is defined as $\\widetilde{H}^{s}\\left(\\Omega \\right)&=\\left\\lbrace u\\in L^{2}\\left(\\Omega \\right) \\mid \\left|u\\right|_{\\widetilde{H}^{s}}<\\infty \\right\\rbrace ,$ where the norm is given by $\\left|u\\right|_{\\widetilde{H}^{s}}^{2}&= \\sum _{m=0}^{\\infty }u_{m}^{2}\\lambda _{m}^{s},$ where $u_{m}$ are defined in (REF ).",
"For $s>1/2$ , $\\widetilde{H}^{s}\\left(\\Omega \\right)$ coincides with the space $H_{0}^{s}\\left(\\Omega \\right)$ defined to be the closure of $C_{0}^{\\infty }\\left(\\Omega \\right)$ with respect to the $H^{s}\\left(\\Omega \\right)$ -norm, whilst for $s<1/2$ , $\\widetilde{H}^{s}\\left(\\Omega \\right)$ is identical to $H^{s}\\left(\\Omega \\right)$ .",
"In the critical case $s=1/2$ , $\\widetilde{H}^{s}\\left(\\Omega \\right)\\subset H^{s}_{0}\\left(\\Omega \\right)$ , and the inclusion is strict.",
"(See for example .)",
"The spaces $\\widetilde{H}^{s}\\left(\\Omega \\right)$ are a useful vehicle to describe the properties of the spectral fractional Laplacians: For instance, suppose $f\\in \\widetilde{H}^{r}\\left(\\Omega \\right)$ , $r\\ge -s$ , and $f=\\sum _{m=0}^{\\infty }f_{m}\\phi _{m}\\left(\\vec{x}\\right)$ with $f_{k}=\\left(f,\\phi _{m}\\right)_{L^{2}}$ then the solution $u$ to the fractional Poisson problem $(\\ref {eq:fracPoisson})$ of order $s$ with right-hand side $f$ is given by $u&=\\sum _{m=0}^{\\infty } u_{m}\\phi _{m}(\\vec{x}), \\qquad u_{m}=f_{m}\\lambda _{m}^{-s}, $ and hence $u\\in \\widetilde{H}^{r+2s}\\left(\\Omega \\right)$ .",
"A more detailed regularity theory for spectral Poisson problems can be found in the work of .",
"We also define the weighted norms on a generic domain $\\mathcal {D}$ for a non-negative weight function $\\omega $ by $\\left|\\!\\left|u\\right|\\!\\right|_{L^{2}_{\\omega }}^{2} &= \\int _{\\mathcal {D}} \\omega \\left|u\\right|^{2}, &\\left|u\\right|_{H^{1}_{\\omega }}^{2} &= \\int _{\\mathcal {D}} \\omega \\left|\\nabla u\\right|^{2}, \\\\\\left|\\!\\left|u\\right|\\!\\right|_{H^{1}_{\\omega }}^{2} &= \\left|\\!\\left|u\\right|\\!\\right|_{L^{2}_{\\omega }}^{2}+\\left|u\\right|_{H^{1}_{\\omega }}^{2},$ along with the associated weighted spaces $L^{2}_{\\omega }\\left(\\mathcal {D}\\right) &= \\left\\lbrace u \\text{ measurable } \\mid \\left|\\!\\left|u\\right|\\!\\right|_{L^{2}_{\\omega }}<\\infty \\right\\rbrace , &H^{1}_{\\omega }\\left(\\mathcal {D}\\right)&= \\left\\lbrace u\\in L^{2}_{\\omega }\\left(\\mathcal {D}\\right) \\mid \\left|\\!\\left|u\\right|\\!\\right|_{H^{1}_{\\omega }}<\\infty \\right\\rbrace .$ Building on the work of , showed that the fractional Poisson problem $(\\ref {eq:fracPoisson})$ can be recast as a problem over the extruded domain $\\mathcal {C}=\\Omega \\times [0,\\infty )$ : $\\left\\lbrace \\begin{array}{rlrl}-\\nabla \\cdot y^{\\alpha } \\nabla U\\left(\\vec{x},y\\right) &= 0, && \\left(\\vec{x},y\\right)\\in \\mathcal {C}, \\\\U\\left(\\vec{x},y\\right) &= 0, && \\left(\\vec{x},y\\right)\\in \\partial _{L}\\mathcal {C} := \\partial \\Omega \\times [0,\\infty ), \\\\\\frac{\\partial U}{\\partial \\nu ^{\\alpha }}\\left(\\vec{x}\\right) &= d_{s}f\\left(\\vec{x}\\right), && \\vec{x}\\in \\Omega ,\\end{array} \\right.",
"$ where $\\alpha = 1-2s$ , $d_{s} = 2^{1-2s}\\frac{\\Gamma \\left(1-s\\right)}{\\Gamma \\left(s\\right)}$ , and $\\frac{\\partial U}{\\partial \\nu ^{\\alpha }}\\left(\\vec{x}\\right)&= -\\lim _{y\\rightarrow 0^{+}}y^{\\alpha }\\frac{\\partial U}{\\partial y}\\left(\\vec{x},y\\right),$ with the solution to $(\\ref {eq:fracPoisson})$ recovered by taking the trace of $U$ on $\\Omega $ , i.e.",
"$u=\\operatorname{tr}_{\\Omega }U$ .",
"We define the solution space $\\mathcal {H}^{1}_{\\alpha }\\left(\\mathcal {C}\\right)$ on the semi-infinite cylinder $\\mathcal {C}$ as $\\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right)&=\\left\\lbrace V\\in H^{1}_{y^{\\alpha }}\\left(\\mathcal {C}\\right) \\mid V=0 \\text{ on } \\partial _{L}\\mathcal {C}\\right\\rbrace ,$ with norm $\\left|\\!\\left|V\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}= \\left|V\\right|_{H^{1}_{y^{\\alpha }}}$ .",
"The weak formulation of the extruded problem $(\\ref {eq:extendedProblem})$ consists of seeking $U\\in \\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right)$ such that: $\\int _{{\\cal C}}y^{\\alpha }\\nabla U\\cdot \\nabla V &= d_{s}\\left\\langle f,\\operatorname{tr}_{\\Omega }V\\right\\rangle \\quad \\forall V \\in \\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right).",
"$ Using a trace inequality , the Lax-Milgram Lemma shows that the extruded problem is well-posed."
],
[
"Eigenfunctions of the Extruded Problem",
"We seek a solution of the extruded problem using classical separation of variables: $U\\left(\\vec{x},y\\right)=\\Phi \\left(\\vec{x}\\right) \\Psi \\left(y\\right)$ .",
"Then $\\frac{-\\Delta _{\\vec{x}}\\Phi }{\\Phi }=\\frac{\\partial _{y}y^{\\alpha }\\partial _{y}\\Psi }{y^{\\alpha }\\Psi }=A,$ where $A$ is a constant that is independent of $\\vec{x}$ and $y$ .",
"The boundary condition on the lateral face of the cylinder $\\mathcal {C}$ , shows that $\\Phi =\\phi _{m}$ and $A=\\lambda _{m}$ for $m\\in \\mathbb {N}$ thanks to $(\\ref {eq:Eig})$ .",
"The associated function $\\Psi $ in the extruded direction must therefore satisfy $\\partial _{y}y^{\\alpha }\\partial _{y}\\Psi =\\lambda _{m}y^{\\alpha }\\Psi , $ or, equivalently, $\\partial _{y}^{2}\\Psi + \\frac{\\alpha }{y}\\partial _{y}\\Psi - \\lambda _{m}\\Psi =0.$ Choosing the normalisation $\\Psi \\left(0\\right)=1$ gives $\\Psi \\left(y\\right)=\\psi _{m}\\left(y\\right)&:= c_{s}\\left(\\lambda _{m}^{1/2}y\\right)^{s} K_{s}\\left(\\lambda _{m}^{1/2}y\\right), $ where $c_{s}= 2^{1-s} / \\Gamma \\left(s\\right)$ .",
"Moreover $\\frac{\\partial \\psi _{m}}{\\partial \\nu ^{\\alpha }}= d_{s}\\lambda _{m}^{s},$ so that $\\int _{0}^{\\infty }y^{\\alpha } \\psi _{m}\\psi _{n}&={\\left\\lbrace \\begin{array}{ll}d_{s}\\frac{\\lambda _{m}^{s}-\\lambda _{n}^{s}}{\\lambda _{m}-\\lambda _{n}} & \\text{if } m\\ne n,\\\\sd_{s}\\lambda _{m}^{s-1} & \\text{if } m=n,\\end{array}\\right.}",
"\\multicolumn{2}{l}{\\text{and}}\\\\\\int _{0}^{\\infty }y^{\\alpha } \\psi _{m}^{\\prime }\\psi _{n}^{\\prime }&={\\left\\lbrace \\begin{array}{ll}d_{s}\\frac{\\lambda _{m}\\lambda _{n}^{s}-\\lambda _{n}\\lambda _{m}^{s}}{\\lambda _{m}-\\lambda _{n}} & \\text{if } m\\ne n,\\\\(1-s)d_{s}\\lambda _{m}^{s} & \\text{if } m=n.\\end{array}\\right.",
"}$ The solution to the extruded problem $(\\ref {eq:extendedProblem})$ is then given by $U\\left(\\vec{x},y\\right)&=\\sum _{m=0}^{\\infty } u_{m}\\phi _{m}(\\vec{x})\\psi _{m}\\left(y\\right) \\quad \\text{where } u_{m}=\\lambda _{m}^{-s}f_{m}, $ whilst $u\\left(\\vec{x}\\right)=\\sum _{m=0}^{\\infty }u_{m}\\phi _{m}\\left(\\vec{x}\\right)$ as in (REF ).",
"The separable solution (REF ) forms the basis for our choice of discretization of the extruded problem to be described in the next section.",
"The chief advantage of this approach is that the extruded problem involves only integer order derivatives but come as the price of having to deal with a degenerate weight $y^{\\alpha }$ ."
],
[
"Discretization of the Extruded Problem",
"We propose to approximate the variational problem (REF ) using a Galerkin scheme with the subspace consisting of standard low order nodal finite elements of order $k$ in the $\\vec{x}$ -variable and a spectral method in the $y$ -direction.",
"To this end, we let $\\mathcal {T}_{h}$ be a shape regular, globally quasi-uniform triangulation of $\\Omega $ , and let $V_{h}=\\left\\lbrace u_{h}\\in H^{1}_{0}\\left(\\Omega \\right) \\mid {\\left.\\hspace{0.0pt}u_{h} \\vphantom{\\big |} \\right|_{K} }\\in \\mathbb {P}_{k}\\left(K\\right) ~\\forall K\\in \\mathcal {T}_{h}\\right\\rbrace .$ Ideally, we would like to use $y$ -basis functions given by (REF ).",
"Unfortunately, this would require knowledge of the true eigenvalues of the integer order Laplacian over $\\Omega $ .",
"Instead, for a given spectral expansion order $M$ , we use an approximation $\\widetilde{\\lambda }_{m} \\approx \\lambda _{m}$ in place of the true eigenvalues in (REF ): $\\widetilde{\\psi }_{m}\\left(y\\right)&:= c_{s}\\left(\\widetilde{\\lambda }_{m}^{1/2}y\\right)^{s} K_{s}\\left(\\widetilde{\\lambda }_{m}^{1/2}y\\right).$ The Galerkin subspace for the extruded problem is then taken to be $\\mathcal {V}_{h,M}=\\left\\lbrace U_{h,M}=\\sum _{m=0}^{\\widetilde{M}-1}u_{h,m}\\left(\\vec{x}\\right)\\widetilde{\\psi }_{m}\\left(y\\right) \\mid u_{h,m}\\in V_{h} \\right\\rbrace \\subset \\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right).$ The selection of the approximate eigenvalues is discussed in sec:choice-appr-eigenv.",
"In particular, if two or more of the approximate eigenvalues are “close” then we retain only a single eigenvalue, thereby reducing the dimension of $\\mathcal {V}_{h,M}$ to $\\mathcal {N}:=\\dim \\mathcal {V}_{h,M}=n\\times \\widetilde{M}$ , where $n:=\\dim V_{h}$ and $\\widetilde{M}\\le M$ is the number of distinct approximate eigenvalues.",
"We return to this point in sec:size-reduct-appr.",
"In the analysis, it will be useful to consider the semi-discrete space $\\mathcal {V}_{M}=\\left\\lbrace U_{M}=\\sum _{m=0}^{M-1}u_{m}\\left(\\vec{x}\\right)\\widetilde{\\psi }_{m}\\left(y\\right) \\mid u_{m}\\in H^{1}_{0}\\left(\\Omega \\right) \\right\\rbrace \\subset \\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right).$ The Galerkin approximation consists of seeking $U_{h,M} \\in \\mathcal {V}_{h,M}$ such that $\\int _{{\\cal C}}y^{\\alpha }\\nabla U_{h,M}\\cdot \\nabla V &= d_{s}\\left\\langle f,\\operatorname{tr}_{\\Omega }V\\right\\rangle \\quad \\forall V \\in \\mathcal {V}_{h,M}, $ with the approximation of the fractional Poisson problem given by $u_{h,M}&:=\\operatorname{tr}_{\\Omega }U_{h,M}.$ We wish to obtain an estimate for the error $u-u_{h,M}$ in this approximation.",
"The trace inequality implies that $\\left|\\!\\left|u-u_{h,M}\\right|\\!\\right|_{\\widetilde{H}^{s}} &\\le C \\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }},$ where the constant is independent of $k$ , $M$ and $h$ .",
"Hence, in order to bound $u-u_{h,M}$ , it suffices to bound the term on the right-hand side - the discretization error of the extruded problem (REF )."
],
[
"A Priori Error Estimate",
"We first consider the error in the approximation given by the semi-discrete Galerkin scheme on the space $\\mathcal {V}_{M}$ .",
"The following result shows how the error depends on $M$ and on how well the approximate eigenvalues $\\left\\lbrace \\widetilde{\\lambda }_{m}\\right\\rbrace $ match the true eigenvalues $\\left\\lbrace \\lambda _{m}\\right\\rbrace $ .",
"Lemma 1 Let $M\\in \\mathbb {N}$ and $U\\in \\mathcal {H}^{1}_{\\alpha }\\left(\\mathcal {C}\\right)$ be the solution of the extruded problem.",
"Then $\\inf _{V_{M}\\in \\mathcal {V}_{M}}\\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&= d_{s}\\sum _{m=0}^{\\infty }\\beta _{m}u_{m}^{2}\\lambda _{m}^{s},\\multicolumn{2}{l}{\\text{where}}\\\\\\beta _{m}&={\\left\\lbrace \\begin{array}{ll}g\\left(s, \\widetilde{\\lambda }_{m} / \\lambda _{m}\\right) & m=0,\\dots ,M-1, \\\\1 & m\\ge M,\\end{array}\\right.",
"}\\multicolumn{2}{l}{\\text{and}}\\\\g\\left(s, \\rho \\right)&=1-\\frac{1}{(1-s)\\rho ^{s} + s \\rho ^{s-1}}.$ Without loss of generality, we may write $U=\\sum _{m=0}^{\\infty } u_{m}\\phi _{m}(\\vec{x})\\psi _{m}\\left(y\\right)$ , and consider $V_{M}=\\sum _{m=0}^{M-1} \\alpha _{m}u_{m}\\phi _{m}(\\vec{x})\\widetilde{\\psi }_{m}\\left(y\\right)$ , where $\\alpha _{m}\\in \\mathbb {R}$ will be chosen below.",
"Direct computation gives $\\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}_{\\alpha }^{1}}^{2}&= \\sum _{m=0}^{M-1}\\sum _{n=0}^{M-1} u_{m}u_{n} \\left\\langle \\phi _{m}\\left(\\psi _{m}-\\alpha _{m}\\widetilde{\\psi }_{m}\\right),\\phi _{n}\\left(\\psi _{n}-\\alpha _{n}\\widetilde{\\psi }_{n}\\right)\\right\\rangle _{\\mathcal {H}_{\\alpha }^{1}} \\\\&\\quad + 2\\sum _{m=0}^{M-1}\\sum _{n=M}^{\\infty } u_{m}u_{n} \\left\\langle \\phi _{m}\\left(\\psi _{m}-\\alpha _{m}\\widetilde{\\psi }_{m}\\right),\\phi _{n}\\psi _{n}\\right\\rangle _{\\mathcal {H}_{\\alpha }^{1}} \\\\& \\quad + \\sum _{m=M}^{\\infty }\\sum _{n=M}^{\\infty } u_{m}u_{n} \\left\\langle \\phi _{m}\\psi _{m},\\phi _{n}\\psi _{n}\\right\\rangle _{\\mathcal {H}_{\\alpha }^{1}}.$ To deal with the first term, we observe that for arbitrary smooth functions $h_{1}$ and $h_{2}$ there holds $\\left\\langle \\phi _{m}\\left(\\vec{x}\\right) h_{1}\\left(y\\right),\\phi _{n}\\left(\\vec{x}\\right)h_{2}\\left(y\\right)\\right\\rangle _{\\mathcal {H}_{\\alpha }^{1}}&= \\int _{\\mathcal {C}}y^{\\alpha } \\nabla \\left[\\phi _{m}\\left(\\vec{x}\\right) h_{1}\\left(y\\right)\\right]\\cdot \\nabla \\left[\\phi _{n}\\left(\\vec{x}\\right) h_{2}\\left(y\\right)\\right] \\\\&= \\int _{\\Omega } \\phi _{m} \\phi _{n} \\int _{0}^{\\infty } y^{\\alpha } h_{1}^{\\prime \\prime } h_{2}^{\\prime } +\\int _{\\Omega } \\nabla _{\\vec{x}}\\phi _{m}\\cdot \\nabla _{\\vec{x}} \\phi _{n} \\int _{0}^{\\infty } y^{\\alpha } h_{1} h_{2} \\\\&= \\delta _{nm}\\left(h_{1},h_{2}\\right)_{m}$ where the inner product in the final equality is defined to be $\\left(h_{1},h_{2}\\right)_{m} &= \\int _{0}^{\\infty }y^{\\alpha } h_{1}^{\\prime } h_{2}^{\\prime } + \\lambda _{m} \\int _{0}^{\\infty }y^{\\alpha } h_{1}h_{2},$ with the induced norm denoted by $\\left|\\!\\left|\\cdot \\right|\\!\\right|_{m}=\\sqrt{\\left(\\cdot ,\\cdot \\right)_{m}}$ .",
"In particular, from eq:massSpec,eq:stiffnesSpec we obtain $\\left|\\!\\left|\\psi _{m}\\right|\\!\\right|_{m}^{2} = d_{s}\\lambda _{m}^{s}$ .",
"Therefore $\\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}_{\\alpha }^{1}}^{2}&= \\sum _{m=0}^{M-1}u_{m}^{2}\\left|\\!\\left|\\psi _{m}-\\alpha _{m}\\widetilde{\\psi }_{m}\\right|\\!\\right|_{m}^{2} + \\sum _{m=M}^{\\infty }u_{m}^{2} \\left|\\!\\left|\\psi _{m}\\right|\\!\\right|_{m}^{2}.$ The coefficients $\\left\\lbrace \\alpha _{m}\\right\\rbrace $ are chosen to minimise the right-hand side.",
"A simple computation reveals that the optimal choice is $\\alpha _{m}= \\cos ^{2} \\theta _{m}$ , where $\\cos \\theta _{m}&= \\frac{\\left(\\psi _{m},\\widetilde{\\psi }_{m}\\right)_{m}}{\\left|\\!\\left|\\psi _{m}\\right|\\!\\right|_{m} \\left|\\!\\left|\\widetilde{\\psi }_{m}\\right|\\!\\right|_{m}}= \\sqrt{1-g\\left(s, \\widetilde{\\lambda }_{m} / \\lambda _{m}\\right)},$ so that $\\left|\\!\\left|\\psi _{m}-\\alpha _{m}\\widetilde{\\psi }_{m}\\right|\\!\\right|_{m}^{2}&= \\left|\\!\\left|\\psi _{m}\\right|\\!\\right|_{m}^{2}\\sin ^{2}\\theta _{m} = d_{s}\\lambda _{m}^{s}\\sin ^{2}\\theta _{m}= d_{s}\\lambda _{m}^{s}g\\left(s, \\widetilde{\\lambda }_{m} / \\lambda _{m}\\right)$ and the result follows as claimed.",
"Observe that if the approximate eigenvalue coincides with the true eigenvalue, $\\widetilde{\\lambda }_{m}=\\lambda _{m}$ , then $g\\left(s,\\widetilde{\\lambda }_{m} / \\lambda _{m}\\right)=0$ as one would expect.",
"By continuity, if the approximate eigenvalue is sufficiently close to the true eigenvalue, then $g\\left(s,\\widetilde{\\lambda }_{m} / \\lambda _{m}\\right)$ will be small, meaning that $\\mathcal {V}_{M}$ will be a good approximation to $\\mathcal {H}_{\\alpha }^{1}\\left(\\mathcal {C}\\right)$ .",
"The next result gives an error bound for the fully discrete scheme: Theorem 2 Let $f\\in \\widetilde{H}^{r}\\left(\\Omega \\right)$ , for $r\\ge -s$ , and choose $M$ sufficiently large such that $\\lambda _{M}^{-(r+s)/2}\\sim h^{\\min \\lbrace k,r+s\\rbrace }$ .",
"Assume that for $0\\le m\\le M-1$ it holds that $g\\left(s, \\widetilde{\\lambda }_{m} / \\lambda _{m}\\right)\\le \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace } $ and that $\\left(\\frac{\\widetilde{\\lambda }_{m}}{\\lambda _{m}}\\right)^{s}, \\left(\\frac{\\lambda _{m}}{\\widetilde{\\lambda }_{m}}\\right)^{1-s} &\\le c_{\\sigma }^{2} $ with a positive constant $c_{\\sigma }$ that is independent of $h$ .",
"Moreover, assume that there exist positive constants $C_{0}$ , $C_{1}$ independent of $h$ such that the following two inequalities hold for any $\\vec{\\gamma }\\in \\mathbb {R}^{M}$ : $\\sum _{m,n=0}^{M-1}\\gamma _{m}\\gamma _{n} \\int _{\\Omega } \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) &\\le C_{0} \\log (\\lambda _{M}) \\sum _{m=0}^{M-1}\\gamma _{m}^{2}\\left|\\!\\left|\\phi _{m}-\\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2}, \\\\\\sum _{m,n=0}^{M-1}\\gamma _{m}\\gamma _{n} \\int _{\\Omega } \\nabla \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right) \\cdot \\nabla \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) &\\le C_{1} \\log (\\lambda _{M}) \\sum _{m=0}^{M-1}\\gamma _{m}^{2}\\left|\\!\\left|\\nabla \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\right|\\!\\right|_{L^{2}}^{2}, $ where $\\pi _{h}$ is the Scott-Zhang interpolant .",
"Then, the solution $U_{h,M}$ to the discretized extruded problem (REF ) satisfies $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }} & \\le C \\left|f\\right|_{\\widetilde{H}^{r}}h^{\\min \\lbrace k,r+s\\rbrace }\\sqrt{\\left|\\log h\\right|},$ where $C$ is independent of $h$ .",
"By Céa's Lemma, the discretization error is bounded by $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}&\\le C\\inf _{V_{h,M}\\in \\mathcal {V}_{h,M}} \\left|\\!\\left|U-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}.$ By analogy with the proof of lem:semianalyticBound, we choose $V_{h,M}\\in \\mathcal {V}_{h,M}$ to be $V_{h,M}&=\\sum _{m=0}^{M-1}\\alpha _{m}u_{m}\\left(\\pi _{h}\\phi _{m}\\right)\\left(\\vec{x}\\right)\\widetilde{\\psi }_{m}\\left(y\\right),$ where $\\alpha _{m}=\\cos \\theta _{m}$ and $\\pi _{h}$ is the Scott-Zhang interpolant .",
"The triangle inequality gives $\\left|\\!\\left|U-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }} &\\le \\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }} + \\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}.$ The first term is easily estimated thanks to lem:semianalyticBound,eq:eigenvalues: $\\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&= d_{s}\\sum _{m=0}^{M-1}u_{m}^{2}\\lambda _{m}^{s}g\\left(s,\\widetilde{\\lambda }_{m} / \\lambda _{m}\\right) + d_{s}\\sum _{m=M}^{\\infty }u_{m}^{2}\\lambda _{m}^{s} \\nonumber \\\\&\\le d_{s}h^{2\\min \\lbrace k,r+s\\rbrace }\\sum _{m=0}^{M-1}u_{m}^{2}\\lambda _{m}^{r+2s} + d_{s} \\lambda _{M}^{-(r+s)} \\sum _{m=M}^{\\infty } u_{m}^{2} \\lambda _{m}^{r+2s} \\nonumber \\\\&\\le d_{s}h^{2\\min \\lbrace k,r+s\\rbrace } \\left|u\\right|_{\\widetilde{H}^{r+2s}}^{2}, $ where we recall $M$ is chosen large enough such that $\\lambda _{M}^{-(r+s)/2}\\sim h^{\\min \\lbrace k,r+s\\rbrace }$ .",
"Turning to the second term, elementary manipulation gives $&\\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}\\\\=& \\sum _{m=0}^{M-1}\\sum _{n=0}^{M-1} \\alpha _{m}\\alpha _{n}u_{m}u_{n}\\int _{\\mathcal {C}}y^{\\alpha } \\nabla \\left[\\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\widetilde{\\psi }_{m}\\right]\\cdot \\nabla \\left[\\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right)\\widetilde{\\psi }_{n}\\right]\\\\=& \\sum _{m=0}^{M-1}\\sum _{n=0}^{M-1} \\alpha _{m}\\alpha _{n}u_{m}u_{n} \\left\\lbrace \\int _{\\Omega }\\nabla \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\cdot \\nabla \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) \\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}\\widetilde{\\psi }_{n} \\right.\\\\&\\qquad \\left.",
"+ \\int _{\\Omega }\\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right) \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) \\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{\\prime }\\widetilde{\\psi }_{n}^{\\prime }\\right\\rbrace \\\\\\le & \\sum _{m=0}^{M-1}\\sum _{n=0}^{M-1} \\alpha _{m}\\alpha _{n}u_{m}u_{n} \\left\\lbrace \\int _{\\Omega }\\nabla \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\cdot \\nabla \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) \\sqrt{\\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{2}}\\sqrt{\\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{n}^{2}} \\right.\\\\&\\qquad \\left.",
"+ \\int _{\\Omega }\\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right) \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) \\sqrt{\\int _{0}^{\\infty }y^{\\alpha }\\left(\\widetilde{\\psi }_{m}^{\\prime }\\right)^{2}}\\sqrt{\\int _{0}^{\\infty }y^{\\alpha }\\left(\\widetilde{\\psi }_{n}^{\\prime }\\right)^{2}}\\right\\rbrace \\\\\\le & \\log (\\lambda _{M}) \\sum _{m=0}^{M-1} \\alpha _{m}^{2}u_{m}^{2} \\left\\lbrace C_{1}\\left|\\!\\left|\\nabla \\phi _{m}-\\nabla \\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2} \\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{2}+ C_{0}\\left|\\!\\left|\\phi _{m}-\\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2} \\int _{0}^{\\infty }y^{\\alpha }\\left(\\widetilde{\\psi }_{m}^{\\prime }\\right)^{2}\\right\\rbrace \\\\\\le & \\max \\lbrace C_{0},C_{1}\\rbrace \\log (\\lambda _{M}) \\sum _{m=0}^{M-1} u_{m}^{2} \\left\\lbrace \\left|\\!\\left|\\nabla \\phi _{m}-\\nabla \\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2} \\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{2}+ \\left|\\!\\left|\\phi _{m}-\\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2} \\int _{0}^{\\infty }y^{\\alpha }\\left(\\widetilde{\\psi }_{m}^{\\prime }\\right)^{2}\\right\\rbrace ,$ where we used (REF ), (), and that $\\alpha _{m}^{2}\\le 1$ .",
"Standard properties of the Scott-Zhang interpolant give $\\left|\\!\\left|\\nabla \\phi _{m}-\\nabla \\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}&\\le Ch^{k}\\left|\\phi _{m}\\right|_{H^{k+1}}\\le Ch^{k}\\lambda _{m}^{(k+1)/2},\\\\\\left|\\!\\left|\\phi _{m}-\\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}&\\le Ch^{k+1}\\left|\\phi _{m}\\right|_{H^{k+1}}\\le Ch^{k+1}\\lambda _{m}^{(k+1)/2},$ while, from eq:massSpec,eq:stiffnesSpec, $\\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{2} &= d_{s}s\\widetilde{\\lambda }_{m}^{s-1}, &\\text{and} &&\\int _{0}^{\\infty }y^{\\alpha }\\left(\\widetilde{\\psi }_{m}^{\\prime }\\right)^{2} &= d_{s}(1-s)\\widetilde{\\lambda }_{m}^{s}.$ Hence, $\\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&\\le C \\log (\\lambda _{M}) h^{2k} \\sum _{m=0}^{M-1} u_{m}^{2}\\lambda _{m}^{k+1}\\widetilde{\\lambda }_{m}^{s-1} + C \\log (M) h^{2k+2}\\sum _{m=0}^{M-1}u_{m}^{2}\\lambda _{m}^{k+1}\\widetilde{\\lambda }_{m}^{s}\\\\&\\le C \\left|u\\right|_{\\widetilde{H}^{r+2s}}^{2} \\left|\\log h\\right| \\left[h^{2k} \\max _{m=0,\\dots ,M-1}\\lambda _{m}^{k-(r+s)}\\left(\\frac{\\lambda _{m}}{\\widetilde{\\lambda }_{m}}\\right)^{1-s} \\right.",
"\\\\&\\qquad \\left.+ h^{2k+2}\\max _{m=0,\\dots ,M-1}\\lambda _{m}^{k+1-(r+s)}\\left(\\frac{\\widetilde{\\lambda }_{m}}{\\lambda _{m}}\\right)^{s}\\right],$ where we used the fact that $\\log (\\lambda _{M}) \\sim \\left|\\log h\\right|$ .",
"Thanks to assumption (REF ), we obtain $\\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&\\le C \\left|u\\right|_{\\widetilde{H}^{r+2s}}^{2} \\left|\\log h\\right|{\\left\\lbrace \\begin{array}{ll}\\lambda _{M-1}^{-(r+s)}\\left(h^{2k}\\lambda _{M-1}^{k}+h^{2k+2}\\lambda _{M-1}^{k+1}\\right) & \\text{if } 0\\le r+s\\le k, \\\\h^{2k}\\left(1+h^{2}\\lambda _{M-1}^{k+1-(r+s)}\\right) & \\text{if } k\\le r+s \\le k+1, \\\\h^{2k}& \\text{if } r+s\\ge k+1,\\end{array}\\right.",
"}$ Recalling that $M$ is chosen such that $\\lambda _{M}^{-(r+s)/2}\\sim h^{\\min \\lbrace k,r+s\\rbrace }$ , we obtain $\\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}&\\le C\\left|u\\right|_{\\widetilde{H}^{r+2s}}h^{\\min \\lbrace k,r+s\\rbrace }\\sqrt{\\left|\\log h\\right|}.",
"$ Finally, by combining eq:spectralBound,eq:feBound, we deduce that $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}&\\le C\\left|\\!\\left|U-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}\\\\&\\le C\\left(\\left|\\!\\left|U-V_{M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}+\\left|\\!\\left|V_{M}-V_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}\\right) \\\\&\\le C\\left|u\\right|_{\\widetilde{H}^{r+2s}} h^{\\min \\lbrace k,r+s\\rbrace }\\sqrt{\\left|\\log h\\right|}\\\\&= C\\left|f\\right|_{\\widetilde{H}^{r}} h^{\\min \\lbrace k,r+s\\rbrace }\\sqrt{\\left|\\log h\\right|},$ since $\\left|u\\right|_{\\widetilde{H}^{r+2s}}=\\left|f\\right|_{\\widetilde{H}^{r}}$ .",
"thm:errorBound contains two types of assumption.",
"Assumptions (REF ) and (REF ) concern the approximation of the exact eigenvalues $\\left\\lbrace \\lambda _{m}\\right\\rbrace $ by $\\left\\lbrace \\widetilde{\\lambda }_{m}\\right\\rbrace $ , which will discussed in the next section.",
"On the other hand, assumptions (REF ) and () concern the orthogonality of the finite element approximation error of the eigenfunctions.",
"In lieu of the absence of a proof of the validity of (REF ) and () in general, we provide a justification in the cases where $\\Omega $ is either an interval on the real line, or the unit disc in the plane.",
"Example 1: Suppose $\\Omega $ is the unit interval, $\\mathcal {T}_{h}$ is a uniform mesh with nodes $x_{j}=jh$ , $j=0,\\dots ,n$ , and $\\left\\lbrace \\Phi _{j}\\right\\rbrace $ are the piecewise linear Lagrange basis functions.",
"The Scott-Zhang interpolant of the eigenfunction $\\phi _{m}\\left(x\\right)=\\frac{1}{\\sqrt{\\pi }}\\sin \\left(m \\pi x\\right)$ of the integer order Laplacian is given by $\\pi _{h}\\phi _{m}= \\vec{c}_{m}^{h}\\cdot \\vec{\\Phi }$ , where $\\vec{\\Phi }$ is the vector of finite element basis functions, and $\\vec{c}_{m}^{h}=\\left\\lbrace \\phi _{m}\\left(x_{j}\\right)\\right\\rbrace _{j=0}^{n}=\\left\\lbrace \\frac{1}{\\sqrt{\\pi }}\\sin \\left(m\\pi x_{j}\\right)\\right\\rbrace _{j=0}^{n}$ is the finite element coefficient vector.",
"Moreover, the $L^{2}$ -projection of $\\phi _{m}$ onto the space of piecewise linear functions $V_{h}$ is given by $\\vec{c}_{m}^{L^{2}}\\cdot \\vec{\\Phi }$ , with coefficient vector $\\vec{c}_{m}^{L^{2}}=\\left\\lbrace \\int _{0}^{1}\\phi _{m}\\left(x\\right)\\Phi _{j}\\left(x\\right)\\right\\rbrace _{j=0}^{n}$ .",
"Hence, the left-hand side of (REF ) can be written as $\\int _{\\Omega } \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right)&= \\delta _{mn} - \\vec{c}_{m}^{h}\\cdot \\vec{c}_{n}^{L^{2}} - \\vec{c}_{m}^{L^{2}}\\cdot \\vec{c}_{n}^{h} + \\vec{c}_{m}^{h}\\cdot M_{FE}\\vec{c}_{n}^{h},$ where $M_{FE}$ is the mass matrix, and we used the orthogonality of the eigenfunctions $\\phi _{m}$ and $\\phi _{n}$ .",
"Now $\\int _{0}^{1}\\phi _{m}\\left(x\\right)\\Phi _{j}\\left(x\\right) &= \\frac{1}{\\sqrt{\\pi }} \\sin \\left(m\\pi x_{j}\\right) \\frac{2-2\\cos \\left(m\\pi h\\right)}{\\pi ^{2}hm^{2}},$ and it follows that $\\vec{c}_{m}^{h}$ and $\\vec{c}_{m}^{L^{2}}$ are collinear.",
"Moreover, we recognise that $\\vec{c}_{m}^{h}$ are in fact the orthogonal eigenvectors of the tridiagonal mass matrix.",
"Therefore we have shown that $\\int _{\\Omega } \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right) &= \\delta _{mn}\\left|\\!\\left|\\phi _{m}-\\pi _{h}\\phi _{m}\\right|\\!\\right|_{L^{2}}^{2},$ and (REF ) holds (without the factor $\\log (\\lambda _{M})$ ).",
"A similar argument applies for ().",
"Example 2: $\\Omega $ is the unit disc.",
"In this case, we verify numerically that (REF ) and () hold.",
"In fig:sCSI, we plot $\\rho \\left(D_{0}^{-1}E_{0}\\right)$ and $\\rho \\left(D_{1}^{-1}E_{1}\\right)$ versus $M$ , where $E_{0,mn} &= \\int _{\\Omega } \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right), \\\\E_{1,mn} &= \\int _{\\Omega } \\nabla \\left(\\phi _{m}-\\pi _{h}\\phi _{m}\\right)\\cdot \\nabla \\left(\\phi _{n}-\\pi _{h}\\phi _{n}\\right),$ for $0\\le m,n \\le M-1$ and $D_{0}$ and $D_{1}$ are the diagonals of $E_{0}$ and $E_{1}$ respectively.",
"In order for assumptions (REF ) and () to be satisfied, it suffices to show that $\\rho \\left(D_{k}^{-1}E_{k}\\right)&\\le C_{k}\\log (\\lambda _{M}), \\quad k=1,2.",
"$ In fig:sCSI we present the numerical values of the quantities appearing in (REF ) for a globally quasi-uniform mesh with about 4,000 vertices for $M\\in \\left\\lbrace 1,\\dots ,100\\right\\rbrace $ , which suggests that (REF )–() are valid for this case.",
"Figure: Numerical verification that () and () hold in the case of the unit disc."
],
[
"Choice of Approximate Eigenvalues $\\widetilde{\\lambda }_{m}\\approx \\lambda _{m}$",
"How can we find approximations $\\widetilde{\\lambda }_{m}$ for the eigenvalues $\\lambda _{m}$ of the standard Laplacian that satisfy conditions (REF ) and (REF ) in thm:errorBound while, ideally, keeping the number of distinct approximate eigenvalues $\\widetilde{M}$ as small as possible?",
"The following technical lemma will be useful: Lemma 3 Let $s\\in (0,1)$ , $0\\le \\varepsilon \\le \\min \\left\\lbrace \\frac{e}{2}\\frac{\\min \\lbrace s,1-s\\rbrace }{\\max \\lbrace s,1-s\\rbrace },1\\right\\rbrace $ and $\\kappa _{s}=\\sqrt{\\frac{2}{e}\\frac{1}{s(1-s)}}$ .",
"If $\\left|\\log \\rho \\right| \\le \\kappa _{s}\\sqrt{\\varepsilon }\\quad \\text{ then }\\quad g\\left(s,\\rho \\right)&\\le \\varepsilon \\text{ and } \\max \\lbrace \\rho ^{s},\\rho ^{s-1}\\rbrace \\le e.$ Set $\\gamma =\\log \\rho $ and assume $\\left|\\gamma \\right| \\le \\kappa _{s}\\sqrt{\\varepsilon }\\le \\frac{1}{\\max \\lbrace s,1-s\\rbrace }$ .",
"Now, by Taylor's Theorem, $(1-s)\\exp {s\\gamma }+s\\exp {(s-1)\\gamma }&= (1-s) \\left[1+s\\gamma + \\frac{1}{2}s^{2}\\gamma ^{2}\\exp {s\\xi }\\right] \\\\&\\quad + s\\left[1+(s-1)\\gamma + \\frac{1}{2}(s-1)^{2}\\gamma ^{2}\\exp {(s-1)\\xi }\\right]$ for some $\\xi $ between 0 and $\\gamma $ and therefore $(1-s)\\exp {s\\gamma }+s\\exp {(s-1)\\gamma }&\\le 1 + \\frac{s(1-s)}{2}\\gamma ^{2}\\exp {\\max \\lbrace s,1-s\\rbrace \\left|\\gamma \\right|}\\\\&\\le 1 + \\frac{s(1-s)}{2} \\frac{2}{e} \\frac{1}{s(1-s)}\\varepsilon e \\\\&= 1+ \\varepsilon \\\\&\\le \\frac{1}{1-\\varepsilon }.$ Hence $g\\left(s,\\rho \\right)=1-\\frac{1}{(1-s)\\exp {s\\gamma }+s\\exp {(s-1)\\gamma }}&\\le \\varepsilon ,$ and $\\max \\lbrace \\rho ^{s},\\rho ^{s-1}\\rbrace &\\le \\exp {\\kappa _{s} \\max \\lbrace s,1-s\\rbrace \\sqrt{\\varepsilon }}\\le e.$ The lemma shows that, in order to satisfy both (REF ) and (REF ), it suffices that the ratio $\\widetilde{\\lambda }_{m} / \\lambda _{m}$ satisfies $\\left|\\log \\widetilde{\\lambda }_{m} / \\lambda _{m}\\right|\\le \\kappa _{s} \\lambda _{m}^{(r+s)/2}h^{\\min \\lbrace k,r+s\\rbrace }$ ."
],
[
"Approximation of Upper Part of the Spectrum - Weyl Asymptotics",
"If $m\\left(\\lambda \\right)$ designates the number of eigenvalues that are smaller than $\\lambda \\ge 0$ , then Weyl's conjecture reads $m\\left(\\lambda \\right) &= \\left(2\\pi \\right)^{-d}\\omega _{d}\\left|\\Omega \\right|\\lambda ^{d/2} - \\frac{1}{4}\\left(2\\pi \\right)^{1-d}\\omega _{d-1}\\left|\\partial \\Omega \\right|\\lambda ^{(d-1)/2}+o\\left(\\lambda ^{(d-1)/2} \\right), $ where $\\omega _{d}=\\frac{\\pi ^{d/2}}{\\Gamma \\left(1+d/2\\right)}$ is the volume of the unit ball in $\\mathbb {R}^{d}$ .",
"For more details on the exact conditions under which Weyl's law has been shown to be valid, see e.g.",
", .",
"Neglecting all lower order terms on the right-hand side of (REF ) motivates the eigenvalue approximation $\\widetilde{\\lambda }_{m}^{\\text{Weyl}}:= C_{d}\\left(\\frac{m}{\\left|\\Omega \\right|}\\right)^{2/d}$ with $C_{d}= 4\\pi \\Gamma \\left(1+d/2\\right)^{2/d}$ .",
"Taking $\\lambda =\\lambda _{m}$ in (REF ), one obtains that $\\widetilde{\\lambda }_{m}^{\\text{Weyl}}$ satisfies $\\widetilde{\\lambda }_{m}^{\\text{Weyl}} &= \\lambda _{m}\\left[1-C\\lambda _{m}^{-1/2}+o\\left(\\lambda _{m}^{-1/2}\\right)\\right].$ Combining (REF ) with lem:bound shows (REF ) is satisfied for sufficiently large $\\lambda _{m}$ and $g(s,\\widetilde{\\lambda }_{m}^{\\text{Weyl}} / \\lambda _{m})\\le \\frac{C}{\\lambda _{m}}.$ Therefore, the Weyl approximation $\\widetilde{\\lambda }_{m}^{\\text{Weyl}}$ satisfies (REF ), provided that $\\lambda _{m}^{-1}=\\mathcal {O}\\left(\\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace }\\right)$ , which will be the case for all $m\\ge m_{0}$ , where $m_{0} =\\mathcal {O}\\left(h^{-d\\min \\lbrace k,r+s\\rbrace /(1+r+s)}\\right)=\\mathcal {O}\\left(n^{\\min \\lbrace k,r+s\\rbrace /(1+r+s)}\\right).$ We expect Weyl's conjecture to provide a good estimate for the eigenvalues in the upper part of the spectrum where $m_{0}\\le m\\le M$ .",
"We illustrate the approximation of the spectrum using Weyl's conjecture in the case $\\Omega =B(0,1)\\subset \\mathbb {R}^{2}$ (for which the exact eigenvalues $\\lambda _{m}$ are known).",
"fig:bounds shows the quantities on each side of inequality (REF ) for the choice $\\widetilde{\\lambda }_{m}=\\widetilde{\\lambda }_{m}^{\\text{Weyl}}$ , where $h$ corresponds to a quasi-uniform triangulation of the unit disc using about one million nodes.",
"We observe that: the inequality (REF ) holds for the Weyl approximation in all but for the first few eigenvalues; and that $g\\left(s,\\widetilde{\\lambda }_{m}^{\\text{Weyl}} / \\lambda _{m}\\right)$ asymptotically behaves like $\\lambda _{m}^{-1}$ with only a mild variation with $s$ .",
"The quantity appearing on the left-hand side of the inequality (REF ) depends on the fractional order $s$ , and decreases as $m$ increases.",
"The right-hand side, however, depends on the fractional order, the mesh size $h$ , the order $k$ of the finite element space, and increases as $m$ increases.",
"Here, we plot the right-hand side of the inequality for $s+r\\in \\lbrace 0.75,1.25\\rbrace $ and $k=1$ .",
"We observe that as the mesh is refined, the number of eigenvalues approximated using the Weyl conjecture which fail to satisfy inequality (REF ) grows.",
"Figure: Condition () requires gs,λ ˜ m Weyl /λ m ≤λ m r+s h 2min{k,r+s} g\\left(s, \\widetilde{\\lambda }_{m}^{\\text{Weyl}} / \\lambda _{m}\\right)\\le \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace }.We display gs,λ ˜ m Weyl /λ m g\\left(s, \\widetilde{\\lambda }_{m}^{\\text{Weyl}} / \\lambda _{m}\\right) and λ m r+s h 2min{k,r+s} \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace } for r+s∈{0.75,1.25}r+s\\in \\lbrace 0.75, 1.25\\rbrace and k=1k=1.Here, hh corresponds to a triangulation of the unit disc with about one million nodes.It can be observed that Weyl's conjecture gives a good approximation of the eigenvalues for the upper part of the spectrum."
],
[
"Finite Element Approximation of Lower Part of the Spectrum",
"The numerical example in the previous section shows that an alternative approach to Weyl's conjecture is required to approximate the smaller eigenvalues $\\lambda _{m}$ , $m=0,\\dots ,m_{0}$ .",
"We propose to use the finite element method to approximate the lower part of the spectrum.",
"The solution of the linear system which arises in the fully discrete Galerkin problem entails the assembly of the mass matrix and the stiffness matrix for the Laplacian on the domain $\\Omega $ using finite elements which can also be used to compute approximate eigenvalues of the Laplacian.",
"As a matter of fact, since we are using a multigrid solver, coarser discretizations of the same problem are also readily available, meaning that we can compute eigenpairs $\\left(\\widetilde{\\lambda }_{m,H}^{FE}, \\vec{\\Phi }_{m,H}\\right)$ on the coarser grids: $S_{FE,H}\\vec{\\Phi }_{m,H} &=\\widetilde{\\lambda }_{m,H}^{FE} M_{FE,H}\\vec{\\Phi }_{m,H}$ where $S_{FE,H}$ and $M_{FE,H}$ are stiffness and mass matrix for a mesh size $H\\ge h$ .",
"It is known that the approximate eigenvalues obtained using a finite element discretization satisfy (see e.g.",
"or ) $\\lambda _{m}\\le \\widetilde{\\lambda }_{m,H}^{\\text{FE}}\\le \\lambda _{m} + C_{m} H^{2k}\\lambda _{m}^{k+1}=\\lambda _{m}\\left(1 + C_{m} H^{2k}\\lambda _{m}^{k}\\right), $ where $C_{m}$ may grow as $m\\rightarrow \\infty $ .",
"In particular, if $C_{m} H^{2k}\\lambda _{m}^{k}$ is sufficiently small, then $\\log \\left(1+C_{m}H^{2k}\\lambda _{m}^{k}\\right)\\approx C_{m}H^{2k}\\lambda _{m}^{k}$ is small, and, according to lem:bound, condition (REF ) is satisfied and $g\\left(s, \\widetilde{\\lambda }_{m,H}^{FE} / \\lambda _{m}\\right)\\le C H^{4k}\\lambda _{m}^{2k}.$ This means that (REF ) will be satisfied by choosing $H$ small enough that $H^{4k}\\lambda _{m}^{2k}=\\mathcal {O}\\left(\\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace }\\right)$ for $0\\le m\\le m_{0}$ , or, equally well $H\\le C{\\left\\lbrace \\begin{array}{ll}h^{\\frac{\\min \\lbrace k,r+s\\rbrace }{1+r+s}\\frac{1+2k}{2k}} & \\text{if } 0\\le r+s\\le 2k, \\\\h^{1/2} & \\text{if } r+s\\ge 2k.\\end{array}\\right.",
"}$ We illustrate these estimates by considering the case of the unit disc.",
"In fig:boundsFE, we show $g\\left(s, \\widetilde{\\lambda }_{m,H}^{FE} / \\lambda _{m}\\right)$ for $m=0,\\dots ,19$ obtained using several mesh sizes $H\\ge h$ and finite elements of order $k=1$ .",
"We also plot the quantity appearing on the right-hand side of inequality (REF ).",
"It can be seen that even very coarse discretizations lead to approximations that satisfy (REF ).",
"Moreover, halving $H$ decreases $g\\left(s,\\widetilde{\\lambda }_{m,H}^{FE}\\right)$ by a factor of 16, as suggested by (REF ).",
"It can also be seen that $g\\left(s, \\widetilde{\\lambda }_{m,H}^{FE} / \\lambda _{m}\\right)$ grows in proportion to $\\lambda _{m}^{2}$ , as suggested by (REF ).",
"The results confirm the expectation that the finite element approximation of the eigenvalues in the lower part of the spectrum provide a good choice for $\\widetilde{\\lambda }_{m}$ .",
"Figure: Condition () requires gs,λ ˜ m,H FE /λ m ≤λ m r+s h 2min{k,r+s} g\\left(s, \\widetilde{\\lambda }_{m,H}^{FE} / \\lambda _{m}\\right)\\le \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace }.We display gs,λ ˜ m,H FE /λ m g\\left(s, \\widetilde{\\lambda }_{m,H}^{FE} / \\lambda _{m}\\right) for several choices of coarsened mesh sizes HH against λ m r+s h 2min{k,r+s} \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace } for r+s∈{0.75,1.25}r+s\\in \\lbrace 0.75, 1.25\\rbrace and k=1k=1.Here, hh corresponds to a triangulation of the unit disc with about one million nodes.It can be observed that the finite element approximation of the eigenvalues provides a good choice for the approximation of the lower part of the spectrum."
],
[
"Size Reduction of the Approximation Space",
"Suppose we have a candidate sequence of approximate eigenvalues $\\widetilde{\\lambda }_{m}$ , $m=0,\\dots ,M$ .",
"These might coincide with the exact eigenvalues, if they are known, or could be obtained by a combination of finite element and Weyl approximations as described earlier.",
"In general, both the finite element approximations $\\widetilde{\\lambda }_{m}^{\\text{FE}}$ and the approximations $\\widetilde{\\lambda }_{m}^{\\text{Weyl}}$ from Weyl's law will be distinct.",
"This implies that the number of approximate eigenvalues is $\\widetilde{M}=M$ , and therefore the dimension of the approximation space $\\mathcal {V}_{h,M}$ would be $\\mathcal {N}=nM$ .",
"However, it is unnecessary for the approximate eigenvalues to be in one to one correspondence with the true eigenvalues.",
"For instance, if two true eigenvalues are close together, then a single approximate eigenvalue should suffice for both.",
"This effectively reduces the number of approximate eigenvalues to $\\widetilde{M}\\le M$ .",
"Accordingly, we propose to minimise the number of distinct eigenvalues $\\left\\lbrace \\widetilde{\\lambda }_{m}\\right\\rbrace _{m=0}^{\\widetilde{M}-1}$ whilst still satisfying the bounds of eq:eigenvalues,eq:eigenvalues2.",
"Employing lem:bound, we select a new set of approximations $\\widehat{\\lambda }_{m}$ by choosing $\\widehat{\\lambda }_{0}=\\widetilde{\\lambda }_{0}$ , and for $m\\ge 1$ , $\\widehat{\\lambda }_{m}&={\\left\\lbrace \\begin{array}{ll}\\widehat{\\lambda }_{m-1} & \\text{if } \\left|\\log \\frac{\\widehat{\\lambda }_{m-1}}{\\widetilde{\\lambda }_{m}}\\right| \\le \\kappa _{s} \\min \\left\\lbrace \\left(\\widetilde{\\lambda }_{m}^{\\text{Weyl}}\\right)^{(r+s)/2}h^{\\min \\lbrace k,r+s\\rbrace },\\sqrt{\\frac{e}{2}\\frac{\\min \\lbrace s,1-s\\rbrace }{\\max \\lbrace s,1-s\\rbrace }},1\\right\\rbrace \\\\\\widetilde{\\lambda }_{m} &\\text{otherwise}\\end{array}\\right.}.",
"$ Here, we have used the fact that the Weyl approximations $\\widetilde{\\lambda }_{m}^{\\text{Weyl}}$ bound the exact eigenvalues from below.",
"We will see in the numerical examples in sec:numerical-examples (e.g.",
"fig:spacedimensiondisc) that this procedure results in $\\widetilde{M} \\ll M$ .",
"To illustrate the method, we again consider the case where the domain is chosen to be the unit disc.",
"In fig:boundsHybrid, we display $g\\left(s, \\widehat{\\lambda }_{m} / \\lambda _{m}\\right)$ , where $\\widehat{\\lambda }_{m}$ is obtained by collapsing eigenvalue approximations obtained through finite element approximation and Weyl's law as described above.",
"We observe that (REF ) remains valid.",
"Figure: Condition () requires gs,λ ^ m /λ m ≤λ m r+s h 2min{k,r+s} g\\left(s, \\widehat{\\lambda }_{m} / \\lambda _{m}\\right)\\le \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace }.We display gs,λ ^ m /λ m g\\left(s, \\widehat{\\lambda }_{m} / \\lambda _{m}\\right) and λ m r+s h 2min{k,r+s} \\lambda _{m}^{r+s}h^{2\\min \\lbrace k,r+s\\rbrace } for r+s∈{0.75,1.25}r+s\\in \\lbrace 0.75, 1.25\\rbrace and k=1k=1.Here, hh corresponds to a triangulation of the unit disc with about one million nodes.We observe that () is still satisfied."
],
[
"Solution of the Linear Algebraic System",
"Let $\\left\\lbrace \\Phi _{i}\\right\\rbrace _{i=1}^{n}$ denote the nodal basis functions of the finite element solution space $V_{h}$ , then the solution of the discretized fractional Poisson problem can be written as $u_{h,M}\\left(\\vec{x}\\right)=\\sum _{i=1}^{n}d_{i}\\Phi _{i}\\left(\\vec{x}\\right)$ .",
"Here, for ease of notation, we assume that the eigenvalue approximations $\\widetilde{\\lambda }_{m}$ , $m=0,\\dots ,\\widetilde{M}-1$ , are all distinct.",
"Obviously, this can easily be achieved by relabelling the reduced set of eigenvalues resulting from the procedure described by (REF ).",
"The solution of the extruded problem (REF ) can be written in the form $U_{h,M}\\left(\\vec{x},y\\right)=\\sum _{m=0}^{\\widetilde{M}-1}\\sum _{i=1}^{n}c_{i,m}\\Phi _{i}\\left(\\vec{x}\\right)\\widetilde{\\psi }_{m}\\left(y\\right)\\in \\mathcal {V}_{h,M}$ with the coefficients $\\left(c_{i,m}\\right)=\\vec{U}_{h,M}$ obtained by solving the linear system $\\left(M_{FE}\\otimes S_{\\sigma }+ S_{FE}\\otimes M_{\\sigma }\\right)\\vec{U}_{h,M}=\\vec{F}_{h,M},$ where $M_{FE} &= \\left(\\int _{\\Omega }\\Phi _{i}\\Phi _{j}\\right), &S_{FE} &= \\left(\\int _{\\Omega }\\nabla \\Phi _{i}\\nabla \\Phi _{j}\\right), \\\\M_{\\sigma } &= \\left(\\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}\\widetilde{\\psi }_{n}\\right), &S_{\\sigma } &= \\left(\\int _{0}^{\\infty }y^{\\alpha }\\widetilde{\\psi }_{m}^{\\prime }\\widetilde{\\psi }_{n}^{\\prime }\\right), \\\\\\vec{F}_{h,M} &= \\vec{f}_{h} \\otimes \\vec{1}_{\\widetilde{M}},&\\vec{f}_{h}&= \\left(d_{s} \\left\\langle f_{h},\\Phi _{i}\\right\\rangle \\right).$ Here, $\\vec{1}_{\\widetilde{M}}$ is the vector of ones of length $\\widetilde{M}$ .",
"The approximation to the solution of the fractional Poisson problem is then obtained by taking the trace of $U_{h,M}$ on $\\Omega $ : $u_{h,M}&=\\operatorname{tr}_{\\Omega }U_{h,M}= \\sum _{i=1}^{n}\\left(\\sum _{m=0}^{\\widetilde{M}-1}c_{i,m}\\right)\\Phi _{i}\\left(\\vec{x}\\right),$ where we recall the normalisation $\\widetilde{\\psi }_{m}\\left(0\\right)=1$ .",
"In matrix form, the trace operator is given by $I\\otimes \\vec{1}_{\\widetilde{M}}^{T}\\in \\mathbb {R}^{n\\times \\mathcal {N}}$ , so that $\\vec{u}_{h,M}=\\left[I\\otimes \\vec{1}_{\\widetilde{M}}^{T}\\right]\\vec{U}_{h,M}$ .",
"eq:massSpec,eq:stiffnesSpec show that both the spectral mass and stiffness matrices are symmetric and dense.",
"In order to compute the solution of (REF ) efficiently, we consider the Cholesky factorisation of $M_{\\sigma }$ : $M_{\\sigma }=LL^{T}$ where $L$ is lower triangular; and the eigenvalue decomposition of $L^{-1}S_{\\sigma }L^{-T}$ $S_{\\sigma }&=LP\\Lambda P^{T}L^{T}$ where $\\Lambda $ is diagonal and $P$ is orthogonal.",
"Each factorisation can be computed in $\\mathcal {O}\\left(\\widetilde{M}^{3}\\right)$ operations.",
"These factorisations allow the matrix appearing in (REF ) to be factorised as $M_{FE}\\otimes S_{\\sigma }+ S_{FE}\\otimes M_{\\sigma }&= \\left[ I\\otimes \\left(LP\\right)\\right] \\left[(M_{FE}\\otimes \\Lambda + S_{FE}\\otimes I\\right]\\left[I\\otimes \\left(LP\\right)^{T}\\right] .$ with the inverse given by $\\left(M_{FE}\\otimes S_{\\sigma }+ S_{FE}\\otimes M_{\\sigma }\\right)^{-1}&= \\left[I\\otimes \\left(L^{-T}P\\right)\\right] \\left[M_{FE}\\otimes \\Lambda + S_{FE}\\otimes I\\right]^{-1} \\left[I\\otimes \\left(P^{T}L^{-1}\\right)\\right].$ Using this form of the inverse to write down an explicit expression for the solution $\\vec{U}_{h,M}$ of (REF ) and then inserting into (REF ), taking account of the right-hand side and applying the discrete trace operator gives $\\vec{u}_{h,M}=&\\left[I\\otimes \\left(P^{T}L^{-1}\\vec{1}_{\\widetilde{M}}\\right)^{T}\\right] \\left[M_{FE}\\otimes \\Lambda + S_{FE}\\otimes I\\right]^{-1} \\left[\\vec{f}_{h}\\otimes \\left(P^{T}L^{-1}\\vec{1}_{\\widetilde{M}}\\right)\\right] \\nonumber \\\\=& \\sum _{m=0}^{\\widetilde{M}-1} w_{m}^{2} \\left[M_{FE}\\Lambda _{mm} + S_{FE}\\right]^{-1} \\vec{f}_{h}.",
"$ Here, $\\vec{w}$ denotes the weight vector $\\vec{w}=P^{T}L^{-1}\\vec{1}_{\\widetilde{M}}\\in \\mathbb {R}^{\\widetilde{M}}$ , which can be computed in $\\mathcal {O}\\left(\\widetilde{M}^{2}\\right)$ operations and stored for reuse.",
"It remains to compute the action of the inverse $\\left[M_{FE}\\otimes \\Lambda + S_{FE}\\otimes I\\right]^{-1}$ .",
"This is accomplished using a conjugate gradient solver with standard geometric multigrid preconditioner for the solution of the systems $M_{FE}\\Lambda _{mm}+S_{FE}$ , $m=0,\\dots ,\\widetilde{M}-1$ , meaning that each system can be solved in $\\mathcal {O}\\left(n\\right)$ operations.",
"In summary, the setup of the multigrid solver, the prefactorisation of the matrices and the precomputation of $\\vec{w}$ will cost $\\mathcal {O}\\left(n+\\widetilde{M}^{3}\\right)$ operations, and each solve will cost $\\mathcal {O}\\left(n\\widetilde{M}\\right)=\\mathcal {O}\\left(\\mathcal {N}\\right)$ operations.",
"The parallelisation of the solution procedure can take advantage of the fact that each of the solves in (REF ) is independent."
],
[
"Piecewise Linear Finite Element Approximation on the Unit Disk",
"Consider the problem $\\left\\lbrace \\begin{array}{rlrl}\\left(-\\Delta \\right)^{s}u&=f && \\text{in } \\Omega =B(0,1)\\subset \\mathbb {R}^{2}\\\\u&=0 && \\text{on }\\partial \\Omega ,\\end{array}\\right.$ where $f=\\left(1-\\left|\\vec{x}\\right|^{2}\\right)^{r-1/2}\\in \\widetilde{H}^{r-\\varepsilon }\\left(\\Omega \\right)$ , for all $\\varepsilon >0$ .",
"We approximate the solution for $s\\in \\lbrace 0.25,0.75\\rbrace $ and $r\\in \\lbrace 0.5, 2\\rbrace $ using piecewise linear finite elements (i.e.",
"$k=1$ ).",
"The true eigenvalues and eigenfunctions of the Laplacian $\\left\\lbrace \\begin{array}{rlrl}-\\Delta \\phi _{k,\\ell }&=\\lambda _{k,\\ell }\\phi _{k,\\ell } && \\text{in } \\Omega \\\\u&=0 && \\text{on }\\partial \\Omega ,\\end{array}\\right.$ are given by $\\phi _{k,0}&=\\frac{1}{\\sqrt{\\pi }J_{1}\\left(\\alpha _{0,k}\\right)}J_{0}\\left(\\alpha _{0,k}r\\right), && k\\ge 1, \\\\\\phi _{k,\\ell }&=\\frac{\\sqrt{2}}{\\sqrt{\\pi }J_{\\ell +1}\\left(\\alpha _{\\ell ,k}\\right)} J_{\\ell }\\left(\\alpha _{\\ell ,k}r\\right) \\cos \\left(\\ell \\theta \\right), && k\\ge 1, \\ell \\ge 1,\\\\\\phi _{k,-\\ell }&=\\frac{\\sqrt{2}}{\\sqrt{\\pi }J_{\\ell +1}\\left(\\alpha _{\\ell ,k}\\right)} J_{\\ell }\\left(\\alpha _{\\ell ,k}r\\right) \\sin \\left(\\ell \\theta \\right),&& k\\ge 1, \\ell \\ge 1, \\\\\\lambda _{k,\\ell }&=\\lambda _{k,-\\ell }=\\alpha _{\\ell ,k}^{2},$ where $J_{\\ell }$ are the Bessel functions of the first kind and $\\alpha _{\\ell ,k}$ are the zeros of $J_{\\ell }$ .",
"Although the true eigenvalues are known for this case, we do not use this information in the definition of the solution space $\\mathcal {V}_{h,M}$ .",
"Instead, we use the approximations obtained via finite elements and Weyl's law detailed above.",
"In order to assess the overall accuracy, we evaluate the error in the approximation using the expression: $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&= \\left|\\!\\left|U\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2} - 2\\left\\langle U,U_{h,M}\\right\\rangle _{\\mathcal {H}^{1}_{\\alpha }} + \\left|\\!\\left|U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2} \\\\&= \\left|\\!\\left|U\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2} - d_{s}\\left\\langle f,u_{h,M}\\right\\rangle .$ Then, expanding the data $f$ as a Bessel series $f&= \\sum _{k,\\ell } f_{k,\\ell }\\phi _{k,\\ell }, \\quad \\text{where} \\quad f_{k,\\ell }=\\left(f,\\phi _{k,\\ell }\\right)_{L^{2}} = \\delta _{\\ell ,0}2^{r+1/2}\\sqrt{\\pi }\\Gamma \\left(r+1/2\\right) \\frac{J_{r+1/2}\\left(\\alpha _{0,k}\\right)}{\\alpha _{0,k}^{r+1/2}J_{1}\\left(\\alpha _{0,k}\\right)},$ we obtain $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}&= d_{s}\\sum _{k,\\ell } f_{k,\\ell }^{2}\\lambda _{k,\\ell }^{-s} - d_{s}\\left\\langle f,u_{h,M}\\right\\rangle \\\\&= d_{s} \\left\\lbrace 2^{2r+1}\\pi \\Gamma \\left(r+1/2\\right)^{2} \\sum _{k=1}^{\\infty } \\left(\\frac{J_{r+1/2}\\left(\\alpha _{0,k}\\right)}{\\alpha _{0,k}^{s+r+1/2}J_{1}\\left(\\alpha _{0,k}\\right)}\\right)^{2} - \\left\\langle f,u_{h,M}\\right\\rangle \\right\\rbrace .$ In practice, we truncate the summation but keep sufficiently many terms that the error from the truncation is negligible in comparison with the error in the Galerkin scheme.",
"In fig:errorhdisc, we plot the $\\mathcal {H}^{1}_{\\alpha }$ -error with respect to the mesh size $h$ .",
"It is observed that the error decays as predicted by thm:errorBound.",
"In fig:errorNdisc, we again show the $\\mathcal {H}^{1}_{\\alpha }$ -error, this time with respect to the total number of degrees of freedom $\\mathcal {N}$ .",
"Letting $n=\\dim V_{h}$ , we have $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}&\\le C\\left|f\\right|_{\\widetilde{H}^{r}}h^{\\min \\lbrace k,r+s\\rbrace }\\sqrt{\\left|\\log h\\right|}\\le C\\left|f\\right|_{\\widetilde{H}^{r}}n^{-\\min \\lbrace k,r+s\\rbrace /d}\\sqrt{\\log n}.$ Suppose that the number of distinct eigenvalue approximations behaves like $\\widetilde{M}=\\mathcal {O}\\left(\\log ^{p} n\\right)$ for some $p\\ge 0$ .",
"Then the total number of degrees of freedom is $\\mathcal {N}=n\\widetilde{M}=\\mathcal {O}\\left(n \\log ^{p} n\\right)$ .",
"That is to say, the total number of degrees of freedom scales like the number of degrees of freedom in the usual, integer order case, apart from the logarithmic factor.",
"In this case, we would obtain quasi-optimal $\\mathcal {H}^{1}_{\\alpha }$ -error convergence: $\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }} \\le C \\left|f\\right|_{\\widetilde{H}^{r}}\\mathcal {N}^{-\\min \\lbrace k,r+s\\rbrace /d} \\log ^{q}\\mathcal {N} $ for some $q\\ge 0$ up to a logarithmic factor.",
"It is observed in fig:errorNdisc that this behaviour is observed in practice.",
"In fact, $\\widetilde{M}=\\mathcal {O}\\left(\\log ^{p} n\\right)$ for some exponent $p\\ge 1$ , as can be seen from fig:spacedimensiondisc, and the method displays quasi-optimal complexity as observed in fig:errorNdisc.",
"In order to assess the efficiency of the solver for the linear algebraic system, in fig:iterationsdisc we show the average number of iterations of multigrid preconditioned conjugate gradient necessary to solve the systems $M_{FE}+\\Lambda _{mm}S_{FE}$ , $m=0,\\dots ,\\widetilde{M}-1$ .",
"Observe that roughly 10 iterations are required for convergence independently of problem size, regularity of the data or fractional order.",
"Finally, we display timing results for setup and solution in fig:timingsdisc.",
"It can be seen that both the setup time for the solver (which includes the approximation of eigenvalues) and solution of the resulting linear system of equations scale as $\\mathcal {O}\\left(n\\right)$ , where $n$ is the number of degrees of freedom in the finite element discretization.",
"Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error for the fractional Poisson problem with right-hand side f=1-x → 2 r-1/2 f=\\left(1-\\left|\\vec{x}\\right|^{2}\\right)^{r-1/2} on the unit disc with piecewise linear finite elements (k=1k=1).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.The error decay of h min{k,r+s} h^{\\min \\lbrace k,r+s\\rbrace } predicted by thm:errorBound is observed.Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error with respect to the total number of degrees of freedom 𝒩\\mathcal {N} on the unit disc with piecewise linear finite elements (k=1k=1).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.Quasi-optimal convergence is obtained.",
"(Compare with the optimal order given in ().",
")Figure: Number of distinct eigenvalues M ˜\\widetilde{M}.s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.The number of distinct eigenvalue approximations M ˜\\widetilde{M} grows like Clog p nC\\log ^{p} n for some p≥1p\\ge 1.Figure: Average number of multigrid preconditioned conjugate gradient iterations.s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.We observe that 10 iterations are sufficient for convergence, independent of problem size, right-hand side regularity and fractional order.Figure: Timings of setup and solution.s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.It can be seen that both setup of the solver, which includes the approximation of eigenvalues, and solution of the resulting linear system of equations scale roughly as 𝒪n\\mathcal {O}\\left(n\\right), where nn is the number of degrees of freedom of the finite element discretization."
],
[
"Piecewise Quadratic Finite Element Approximation on the Unit Square",
"Consider now the approximation of the fractional Poisson problem on the unit square: $\\left\\lbrace \\begin{array}{rlrl}\\left(-\\Delta \\right)^{s}u&=f && \\text{in } \\Omega =[0,1]^{2}\\\\u&=0 && \\text{on }\\partial \\Omega ,\\end{array}\\right.$ where $f\\left(\\vec{x}\\right)=\\left[x_{1}x_{2}(1-x_{1})(1-x_{2})\\right]^{r-1/2}$ .",
"This time we use piecewise quadratic finite elements in order to demonstrate the flexibility of the approach.",
"The exact eigenvalues and eigenfunctions are known and can be used to compute the $\\mathcal {H}^{1}_{\\alpha }$ -error using the expression $&\\left|\\!\\left|U-U_{h,M}\\right|\\!\\right|_{\\mathcal {H}^{1}_{\\alpha }}^{2}\\\\=& d_{s}\\left\\lbrace 4\\pi ^{2} \\Gamma \\left(r+1/2\\right)^{4} \\right.\\\\&\\qquad \\times \\sum _{p,q=0}^{\\infty } \\frac{1}{\\pi ^{4r+2s}} \\frac{1}{\\left(2p+1\\right)^{2r} \\left(2q+1\\right)^{2r}\\left[(2p+1)^{2}+(2q+1)^{2}\\right]^{s}} J_{r}\\left(\\pi (p+1/2)\\right)^{2} J_{r}\\left(\\pi (q+1/2)\\right)^{2} \\\\&\\qquad \\left.- \\left\\langle f,u_{h,M}\\right\\rangle \\right\\rbrace .$ As before, we wish to assess the convergence rate of our procedure when the eigenvalues are approximated using Weyl's law and finite element approximations for the definition of the solution space.",
"In fig:errorhsquare,fig:errorNsquare, we show the $\\mathcal {H}^{1}_{\\alpha }$ -error versus $h$ and $\\mathcal {N}$ respectively.",
"It can be seen that the error bound of thm:errorBound is satisfied, and that quasi-optimal convergence with respect to $\\mathcal {N}$ is again obtained.",
"Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error for the fractional Poisson problem with right-hand side f=x 1 x 2 (1-x 1 )(1-x 2 ) r-1/2 f=\\left[x_{1}x_{2}(1-x_{1})(1-x_{2})\\right]^{r-1/2} on the unit square with piecewise quadratic finite elements (k=2k=2).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.The error decay of h min{k,r+s} h^{\\min \\lbrace k,r+s\\rbrace } predicted by thm:errorBound is observed.Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error with respect to the total number of degrees of freedom 𝒩\\mathcal {N} on the unit square with piecewise quadratic finite elements (k=2k=2).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.Quasi-optimal convergence is obtained.",
"(Compare with the optimal order given in ().)"
],
[
"Piecewise Linear Finite Element Approximation on the Unit Cube",
"Finally, consider a fractional Poisson problem in three dimensions on the unit cube: $\\left\\lbrace \\begin{array}{rlrl}\\left(-\\Delta \\right)^{s}u&=f && \\text{in } \\Omega =[0,1]^{3}\\\\u&=0 && \\text{on }\\partial \\Omega ,\\end{array}\\right.$ We use piecewise linear finite element approximation and compute the true $\\mathcal {H}^{1}_{\\alpha }$ -error in similar fashion as before.",
"Here, $f\\left(\\vec{x}\\right)=\\left[x_{1}x_{2}x_{3}(1-x_{1})(1-x_{2})(1-x_{3})\\right]^{r-1/2}$ .",
"In fig:errorhcube,fig:errorNcube, we plot the $\\mathcal {H}^{1}_{\\alpha }$ -error versus $h$ and $\\mathcal {N}$ respectively.",
"It can be seen that the error bound of thm:errorBound is satisfied, and that quasi-optimal convergence with respect to $\\mathcal {N}$ is again observed.",
"Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error for the fractional Poisson problem with right-hand side f=x 1 x 2 x 3 (1-x 1 )(1-x 2 )(1-x 3 ) r-1/2 f=\\left[x_{1}x_{2}x_{3}(1-x_{1})(1-x_{2})(1-x_{3})\\right]^{r-1/2} on the unit cube with piecewise linear finite elements (k=1k=1).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.The error decay of h min{k,r+s} h^{\\min \\lbrace k,r+s\\rbrace } predicted by thm:errorBound is observed.Figure: ℋ α 1 \\mathcal {H}^{1}_{\\alpha }-error with respect to the total number of degrees of freedom 𝒩\\mathcal {N} on the unit cube with piecewise linear finite elements (k=1k=1).s=0.25s=0.25 on the left, s=0.75s=0.75 on the right.Quasi-optimal convergence is obtained.",
"(Compare with the optimal order given in ().)"
],
[
"Conclusion",
"A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions.",
"The scheme is based on reformulating the original problem posed over $\\Omega $ on the extruded domain $\\mathcal {C}=\\Omega \\times [0,\\infty )$ following .",
"The resulting degenerate elliptic integer order PDE is approximated using a hybrid FEM-spectral scheme.",
"Finite elements are used on $\\Omega $ , whilst an appropriate spectral method is used in the extruded direction.",
"The spectral part of the scheme requires suitable approximations of the true eigenvalues of the usual Laplacian over $\\Omega $ .",
"We derive an a priori error estimate which takes account of the error arising from the approximation of the true eigenvalues, and present a strategy for choosing suitable approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem.",
"The resulting system of linear algebraic equations is decomposed into blocks which are solved using standard iterative solvers such as multigrid and conjugate gradient.",
"Numerical examples in two and three dimensions show that the approach is quasi-optimal in terms of complexity."
]
] | 1709.01639 |
[
[
"Torsion subgroups of quasi-abelianized braid groups"
],
[
"Abstract This article extends the works of Gon\\c{c}alves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group.",
"We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient.",
"In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity.",
"We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group."
],
[
"Introduction",
"Let $V$ be a finite dimensional complex vector space and $W \\subset \\mathrm {GL}(V)$ be an irreducible complex reflection group.",
"We define ${A}$ to be the set of reflection hyperplanes for $W$ and $V^\\textrm {reg}=V \\setminus \\cup _{H \\in {A}} H$ .",
"Let $x_0 \\in V^\\textrm {reg}$ .",
"We consider $P=\\pi _1(V^\\textrm {reg},x_0)$ the pure braid group of $W$ and $B=\\pi _1(V^\\textrm {reg}/W,\\overline{x}_0)$ its braid group where $\\overline{x}_0$ is the image of $x_0$ in $V^\\textrm {reg}/W$ .",
"We have the following defining exact sequence (see [2] for more on this subject) ${1 [r]& P ^-{}[r] & B ^-{}[r] & W[r] & 1}$ giving rise to the abelian extension of $W$ ${0 [r]& P^\\textrm {ab} ^-{i}[r] & B/[P,P] ^-{p}[r] & W[r] & 1}$ This last extension whose order has been studied in [1] will play a key role in the sequel.",
"Let us say that $B/[P,P]$ is the relative abelianization of the braid group of $W$.",
"Let us also remind that as a $W$ -module $P^\\textrm {ab}$ is the permutation module $\\mathbb {Z}{A}$ (see [7]).",
"We get a very concrete criterion for a subgroup of $W$ to embed in $B/[P,P]$ in terms of the stabilizers of the hyperplanes of $W$ (Theorem REF ); for the case of the symmetric group $\\mathfrak {S}_n$ (and more generally for a wide class of complex reflection groups including the real ones) and the standard braid group $B_n$ , we are able to describe all finite subgroups of $B/[P,P]$ (see Corollary REF and Proposition REF ).",
"In particular we show that every odd-order subgroup of $\\mathfrak {S}_n$ can be embedded in $B_n/[P_n,P_n]$ and that every odd-order group can be embedded in $B_\\infty /[P_\\infty ,P_\\infty ]$ where $B_\\infty $ (resp.",
"$P_\\infty $ ) denotes the infinite (pure) braid group.",
"We are also able to give a complete list of the irreducible reflection groups such that $B/[P,P]$ is a Bieberbach group (Corollary REF ).",
"Let us start with the notations we will use in this article.",
"The group $W$ acts on ${A}$ : if $H$ is the hyperplane of the reflection $s \\in W$ , then $wH$ is the hyperplane of the reflection $wsw^{-1}$ .",
"Moreover $wH=H$ if and only if $wsw^{-1}=s$ .",
"For $H \\in {A}$ , we denote by $N_H=\\lbrace w \\in W, wH=H\\rbrace = \\lbrace w \\in W, sw=ws\\rbrace $ the stabilizer of $H$ which is also the centralizer of any reflection of $W$ with hyperplane $H$ .",
"For a subgroup $G$ of $W$ and $H \\in {A}$ , the stabilizer of $H$ under the action of $G$ is $N_H\\cap G$ .",
"We denote it by $N_{H,G}$ .",
"Since $W$ is generated by reflections, we then deduce $w H=H$ for every $H \\in {A}$ if and only $w \\in Z(W)$ where $Z(W)$ is the centre of $W$ .",
"In particular, the group $\\overline{W}=W/Z(W)$ acts faithfully on ${A}$ .",
"All along this article, we will consider subgroups $G$ of $W$ and subgroups $\\overline{G}$ of $\\overline{W}$ .",
"For a subgroup $G$ of $W$ , we denote by $\\overline{G}$ its direct image in $\\overline{W}$ .",
"Finally since $W$ is finite, we may assume that $\\langle \\cdot , \\cdot \\rangle $ is an hermitian product invariant under $W$ .",
"For $H \\in {A}$ , we set $C_H=\\lbrace w \\in W,\\ \\forall \\,x \\in H^\\perp , \\ w(x)=x\\rbrace $ the parabolic subgroup of $W$ associated to $H^\\perp $ .",
"Let us end this introduction with a cohomological lemma that we will need later.",
"0 Lemma 1 Let $G$ be a finite group, $X$ a set on which $G$ acts and $\\mathbb {Z}X$ the corresponding permutation module.",
"Then $H^1(G,\\mathbb {Z}X)$ is trivial.",
"Proof.",
"Decomposing $X$ into $G$ -orbits $X=\\sqcup _{A \\in X/G} A$ , we get $H^1(G,\\mathbb {Z}X) = \\bigoplus _{A \\in X/G} H^1(G,\\mathbb {Z}A)$ .",
"Choosing for each $A \\in X/G$ a representative $a$ and using Shapiro's isomorphism ([3]) we obtain that $H^1(G,\\mathbb {Z}X) = \\bigoplus _{A \\in X/G} H^1(G_a,\\mathbb {Z})$ where $G_a$ is the stabilizer of $a$ .",
"But $G_a$ acts trivially on $\\mathbb {Z}$ and then $H^1(G_a,\\mathbb {Z})=\\mathrm {Hom}_{\\textrm {gr.",
"}}(G_a,\\mathbb {Z})=\\lbrace 0\\rbrace $ since $G_a \\subset G$ is finite."
],
[
"A criterion for finite subgroup",
"In this subsection, we give a criterion for a subgroup of $W$ to lift in $B/[P,P]$ .",
"The criterion (Theorem REF ) has many consequences such that the description of the finite subgroups of $B_n/[P_n,P_n]$ and $B_\\infty /[B_\\infty ,B_\\infty ]$ which are explored in the following subsections.",
"0 Theorem 2 Let $G$ be a subgroup of $W$ then the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(G) ^-{p}[r] & G[r] & 1}$ splits if and only if for every $H \\in {A}$ we have $N_H \\cap G \\subset C_H$ .",
"When these conditions are fulfilled, $G$ identifies to a finite subgroup of $B/[P,P]$ and $\\overline{G}$ is isomorphic to $G$ .",
"Moreover $p^{-1}(G)$ is a crystallographic group.",
"Let $\\widetilde{G}$ be a torsion subgroup of $B/[P,P]$ then $\\widetilde{G}$ is finite and is isomorphic to its image in $W$ and the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(p(\\widetilde{G})) ^-{p}[r] & p(\\widetilde{G})[r] & 1}$ splits.",
"Moreover every subgroup of $B/[P,P]$ isomorphic through $p$ to $p(\\widetilde{G})$ is conjugated to $\\widetilde{G}$ by an element of $P^\\textrm {ab}$ and the intersection of the normalizer of $\\widetilde{G}$ in $B/[P,P]$ with $P^\\textrm {ab}$ is isomorphic to $\\mathbb {Z}{A}/G$ .",
"Proof.",
"Let us consider the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(G) ^-{p}[r] & G[r] & 1}$ as an element in $H^2(G,P^\\textrm {ab})$ .",
"This is the image of the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & B/[P,P] ^-{p}[r] & W[r] & 1}$ through the restriction map $\\textrm {Res}\\colon H^{2}(W,P^\\textrm {ab}) \\rightarrow H^2(G, P^\\textrm {ab})$ .",
"Following the proof of Proposition 5 of [1], we get the following commutative diagram $\\displaystyle {H^2(W,P^\\textrm {ab}) [r] ^{\\textrm {Res}}[d] & \\left[\\bigoplus _{H \\in {A}} H^2(N_H,\\mathbb {Z})\\right]^W [r]^{\\textrm {Res}}[d]& \\left[\\bigoplus _{H \\in {A}} \\mathrm {Hom}_{\\textrm {gr}}(N_H,\\mathbb {C}^\\times ) \\right]^W ^{\\textrm {Res}}[d]\\\\H^2(G,P^\\textrm {ab}) [r] & \\left[\\bigoplus _{H \\in {A}} H^2(N_H\\cap G,\\mathbb {Z})\\right]^G [r]&\\left[\\bigoplus _{H \\in {A}} \\mathrm {Hom}_{\\textrm {gr}}(N_H \\cap G,\\mathbb {C}^\\times ) \\right]^G }$ Hence the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(G) ^-{p}[r] & G[r] & 1}$ can be identified with the family of group homomorphisms $r_H: N_H \\cap G \\rightarrow \\mathbb {C}^\\times $ given by the restriction to the line $H^\\perp $ whose kernel is by definition $C_H$ .",
"In this case, $G \\cap ZW=\\lbrace 1\\rbrace $ since for every non trivial element $w$ of the center of $W$ and every hyperplane $H \\in {A}$ , we have $r_H(w) \\ne 1$ .",
"So $\\overline{G}$ and $G$ are isomorphic.",
"Finally, $p^{-1}(G)$ is a crystallographic group since $G$ acts faithfully on ${A}$ since $G \\cap ZW=\\lbrace 1\\rbrace $ .",
"Let us now consider $\\widetilde{G}$ a torsion subgroup of $B/[P,P]$ .",
"For $x \\in \\widetilde{G}$ , let us first show that the order of $x$ is the same that the order of $p(x)\\in W$ .",
"Let $m$ be the order of $x$ and assume that $p(x)^n=1$ for some $n$ dividing $m$ and $n \\ne m$ .",
"Then $x^n \\in P$ but $x^n \\notin [P,P]$ .",
"But $P^\\textrm {ab}$ is a torsion-free group, so it is impossible.",
"The map $p:\\widetilde{G} \\rightarrow W$ is then injective and $\\widetilde{G}$ is finite and $p: \\widetilde{G} \\rightarrow p(\\widetilde{G})$ is an isomorphism.",
"The $P^\\textrm {ab}$ -conjugacy classes of subgroups of $B/[P,P]$ isomorphic to $p(\\widetilde{G})$ through $p$ are in bijection with $H^{1}(p(\\widetilde{G}),P^\\textrm {ab})$ (see [3]).",
"But $H^1(p(\\widetilde{G}),P^\\textrm {ab})$ is trivial (Lemma ) and we get the result.",
"Finally, let $x \\in P^\\textrm {ab}$ such that $x\\widetilde{G}x^{-1}=\\widetilde{G}$ .",
"Then for every $g \\in \\widetilde{G}$ , we have $xgx^{-1}g^{-1} \\in \\widetilde{G} \\cap P^\\textrm {ab}$ .",
"But $\\widetilde{G}$ is finite and $P^\\textrm {ab}$ is torsion-free, hence $xgx^{-1}g^{-1}=1$ .",
"We then deduce $gxg^{-1}=x$ for all $g \\in \\widetilde{G}$ and so $\\displaystyle x= \\sum _{{C}\\in {A}/p(\\widetilde{G})} \\alpha _{C}\\left(\\sum _{H \\in {C}} c_H\\right)\\,.$ Since $w \\in W$ has a finite order lifting in $B/[P,P]$ if and only if the extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(\\langle w \\rangle ) ^-{p}[r] & \\langle w \\rangle [r] & 1}$ is split, we get the following corollary.",
"0 Corollary 3 An element $w \\in W$ has a lifting in $B/[P,P]$ of finite order if and only if $\\langle w \\rangle \\cap N_H \\subset C_H$ for every $H \\in {A}$ .",
"We can then easily generalize Theorem 2.5 of [6].",
"0 Corollary 4 Let $w \\in ZW \\setminus \\lbrace 1\\rbrace $ ; the order of every lifting of $w$ in $B/[P,P]$ is infinite.",
"Let $w \\in W$ such that $w$ has two eigenvalues one of those is 1 then the order of every lifting of $w$ in $B/[P,P]$ is infinite.",
"In particular, if $w \\in W$ has order 2 or is a reflection then the order of every lifting of $w$ in $B/[P,P]$ is infinite.",
"Proof.",
"Let $w \\in ZW \\setminus \\lbrace 1\\rbrace $ then $w \\in N_H$ for every $H \\in {A}$ and $w \\notin C_H$ .",
"So Theorem REF gives the result.",
"This result follows also directly from the fact that, when $W$ is irreducible $p^{-1}(ZW)$ is a free abelian group of rank $|{A}|$ generated by $P^\\textrm {ab}$ and the class $\\underline{\\mathbf {z}}$ of the path $t \\mapsto \\exp (2i\\pi t/|Z(W)|)x_0$ (see [6]).",
"Let us now consider $w \\in W$ with $w$ having two eigenvalues 1 and $\\lambda \\ne 1$ .",
"Steinberg's theorem implies that there exists a reflection in $W$ such that its hyperplane $H$ contains the eigenspace of $w$ associated to 1.",
"Then $H^\\perp $ is contained in $\\ker (w-\\lambda \\mathrm {id})=\\ker (w -\\mathrm {id})^\\perp $ .",
"Thus the restriction of $w$ on $H^\\perp $ is $\\lambda \\mathrm {id}\\ne \\mathrm {id}$ .",
"If $w \\in W$ has order 2 then either $w = -\\mathrm {id}\\in ZW \\setminus \\lbrace 1\\rbrace $ or $w$ has two eigenvalues 1 and $-1$ .",
"If $w \\in W$ is a reflection, its eigenvalues are 1 and $\\lambda $ for some $\\lambda \\ne 1$ .",
"Remark 5 In fact, Theorem 2.5 of [6] and its proof can easily be extended to the case of $w \\in W$ such that $w$ has two eigenvalues one of those is 1.",
"So the previous corollary was already known but we give here a different proof.",
"0 Lemma 6 If $w \\in W$ is such that a power of $w$ has only infinite lifting in $B/[P,P]$ then every lifting of $w$ in $B/[P,P]$ has infinite order.",
"This is in particular the case when $w$ is of even order or if a power of $w$ is a non trivial element of $ZW$ .",
"Proof.",
"If $x$ is a lifting of $w$ then $x^n$ is a lifting of $w^n$ .",
"We may interpret Theorem REF as a kind of local property in the following sense: a subgroup $G$ of $W$ identifies to a subgroup of $B/[P,P]$ if and only if every element of $G$ has a finite order lifting in $B/[P,P]$ .",
"More precisely, we get the following corollary.",
"0 Corollary 7 Let $G$ be a subgroup of $W$ .",
"The extension ${1 [r]& P^\\textrm {ab} ^-{i}[r] & p^{-1}(G) ^-{p}[r] & G[r] & 1}$ is split if and only if every element of $G$ has a finite order lifting in $B/[P,P]$ .",
"Proof.",
"Both conditions are equivalent to the following: for every $H \\in {A}$ and every $w \\in G \\cap N_H$ , $w \\in C_H$ ."
],
[
"Examples and Applications",
"We start this subsection by describing the elements of the groups in the infinite series that can be lifted in their restricted braid group.",
"We then study the groups $W$ such that $B/[P,P]$ is a Bieberbach group and finally we give a list of groups $W$ (containing the Coxeter groups) such that every odd-order element of $W$ has a finite order lifting in $B/[P,P]$ .",
"For an integer $n$ , we denote by $\\mathbb {U}_n$ the set on $n^\\textrm {th}$ root of unity in $\\mathbb {C}$ .",
"Let us consider $(e_1,\\ldots , e_r)$ the canonical basis of $\\mathbb {C}^r$ .",
"For $w \\in G(de,e,r)$ , we denote by $\\sigma _w \\in \\mathfrak {S}$ the image of $w$ through the natural homomorphism $G(de,e,r) \\rightarrow \\mathfrak {S}_r$ .",
"Thus, there exists a family $(a_i)_{1 \\le i \\le r} \\in {\\mathbb {U}_{de}}^r$ such that $w(e_i)=a_i e_{\\sigma _{w}(i)}$ for all $i \\in \\lbrace 1,\\ldots , r\\rbrace $ .",
"Let us write $\\sigma _w=\\sigma _1\\cdots \\sigma _s$ as a product of disjoint cycles.",
"For such a cycle $\\sigma =(i_1,\\ldots , i_\\alpha )$ , we denote by $p_{w,\\sigma }=a_{i_1}\\cdots a_{i_\\alpha } \\in \\mathbb {U}_{de}$ .",
"0 Corollary 8 Finite order lifting for the infinite series.",
"Let $w \\in G(de,e,r)$ .",
"If $w$ has a finite order lifting in $B/[P,P]$ then either $\\sigma _w \\ne \\mathrm {id}$ and for every cycle $\\sigma $ appearing in $\\sigma _w$ , we have $p_{w,\\sigma }=1$ , or $\\sigma _w =\\mathrm {id}$ and for every $i \\ne j \\in \\lbrace 1,\\ldots ,r\\rbrace $ the order of $a_i{a_j}^{-1}$ is a multiple of the order of $a_i$ .",
"If $w$ is of odd order and for every cycle $\\sigma $ appearing in $\\sigma _w$ , we have $p_{w,\\sigma }=1$ then $w$ has a finite order lifting in $B/[P,P]$ .",
"When $d \\ge 2$ , then $w$ has a finite order lifting in $B/[P,P]$ if and only if $w$ is of odd order and for every cycle $\\sigma $ appearing in $\\sigma _w$ , we have $p_{w,\\sigma }=1$ .",
"When $d=1$ , $w$ has a finite order lifting in $B/[P,P]$ if and only if $w$ is of odd order and for $w$ for every cycle $\\sigma $ of $\\sigma _w$ , we have $p_{w,\\sigma }=1$ or $\\sigma _w =\\mathrm {id}$ and for every $i \\ne j \\in \\lbrace 1,\\ldots ,r\\rbrace $ the order of $a_i{a_j}^{-1}$ is a multiple of the order of $a_i$ .",
"Proof.",
"The result is clear for $r=1$ .",
"Let us assume that $r \\ge 2$ .",
"Let us consider $w \\in W$ with a finite order lifting in $B/[P,P]$ and $\\sigma $ a cycle of $\\sigma _w$ of length greater than 1.",
"Set $i \\ne j$ in the support of $\\sigma $ and $\\zeta \\in \\mathbb {U}_{de}$ and consider the hyperplane $H_{i,j,\\zeta }=\\lbrace (z_1,\\ldots ,z_r) \\in \\mathbb {C}^r, \\ z_i=\\zeta z_j\\rbrace .$ For $\\ell \\in \\mathbb {Z}$ , if $w^\\ell \\in N_{H_{i,j,\\zeta }}$ then $\\lbrace {\\sigma _w}^\\ell (i),{\\sigma _w}^\\ell (j)\\rbrace =\\lbrace i,j\\rbrace $ .",
"But $\\sigma _w$ is of odd order (Corollary REF ), hence ${\\sigma _w}^\\ell (i)=i$ and ${\\sigma _w}^\\ell (j)=j$ .",
"We then deduce that $\\ell $ is a multiple of the length $\\alpha $ of $\\sigma $ : we can write $\\ell = \\alpha m$ .",
"Then the restriction of $w^\\ell $ to the space $\\textrm {Span}(e_{i_1},\\ldots , e_{i_\\alpha })$ (where $\\sigma =(i_1,\\ldots , i_\\alpha )$ ) is ${p_{w,\\sigma }}^m \\mathrm {id}$ .",
"But from Corollary REF , we get that $w^\\ell \\in C_{H_{i,j,\\zeta }}$ , hence ${p_{w,\\sigma }}^m=1$ .",
"Since we can choose $m=1$ , we get the result.",
"If $i$ is a fixed point of $\\sigma _w$ .",
"We have $w(e_i)=a_ie_i$ .",
"We want to show that $a_i=1$ when there exists a non trivial orbit $\\sigma $ of $\\sigma _w$ , we consider $j$ in the support of $\\sigma $ .",
"If $w^\\ell $ in $N_{H_{i,j,1}}$ then $\\textrm {Span}(e_i,e_j)$ is stable by $w^\\ell $ .",
"Hence $\\ell $ is a multiple of the length of $\\sigma $ .",
"Hence $w^\\ell (e_j)=e_j$ by the preceeding point.",
"But $w^\\ell (e_j- e_i)=e_j- e_i$ since $w^\\ell \\in C_{H_{i,j,1}}$ (Corollary REF ).",
"Hence $w^\\ell (e_i)=e_i$ .",
"Let us now consider the case where $\\sigma _w=\\mathrm {id}$ then we can write $w(e_i)=a_ie_i$ for all $i$ .",
"For $i\\ne j$ , we have $w^\\ell \\in N_{H_{i,j,1}}$ if and only if ${a_i}^\\ell ={a_j}^\\ell $ .",
"If $w^\\ell \\in N_{H_{i,j,1}}$ then $w^\\ell (e_i-e_j)=(e_i -e_j)$ (since $w^\\ell \\in C_{H_{i,j,\\ell }}$ ).",
"Hence ${a_i}^\\ell ={a_j}^\\ell $ implies ${a_i}^\\ell ={a_j}^\\ell =1$ .",
"So for every $i \\ne j$ the order of $a_i{a_j}^{-1}$ is a multiple of the order of $a_i$ and $a_j$ .",
"Conversely, assume that $p_{w,\\sigma }=1$ for every cycle $\\sigma $ of $\\sigma _w$ and $w$ is of odd order and let us apply the criterion of Theorem REF to show that $w$ has a finite order lifting in $B/[P,P]$ .",
"Let $1 \\le i \\le r$ and $H_i=\\lbrace (z_1,\\ldots , z_r) \\in \\mathbb {C}^r, \\ z_i= 0\\rbrace $ .",
"If $w^\\ell \\in N_{H_i}$ then $w^\\ell (e_i) \\in \\mathbb {C}^\\times e_i$ .",
"Hence $\\ell $ is a multiple of the length of the cycle $y=(i,i_1,\\ldots , i_\\alpha )$ of $\\sigma _w$ whose support contains $i$ .",
"So that the restriction of $w^\\ell $ to $\\textrm {Span}(e_i,e_{i_1},\\ldots , e_{i_\\alpha })$ is the identity since $p_{w,\\sigma }=1$ .",
"So $w^\\ell \\in C_{H_i}$ .",
"For $i \\ne j \\in \\lbrace 1,\\ldots , r\\rbrace $ and $\\zeta \\in \\mathbb {U}_{de}$ , if $w^\\ell \\in N_{H_{i,j,\\zeta }}$ then $\\lbrace {\\sigma _w}^\\ell (i),{\\sigma _w}^\\ell (j)\\rbrace = \\lbrace i,j\\rbrace $ .",
"If $i$ and $j$ belongs to different orbits $\\sigma $ and $\\sigma ^{\\prime }$ under $\\sigma _w$ then necessarily ${\\sigma _w}^\\ell (i)=i$ and ${\\sigma _w}^\\ell (j)=j$ .",
"Hence $\\ell $ is a multiple of the lengths of $\\sigma $ and $\\sigma ^{\\prime }$ .",
"But $p_{w,\\sigma }=p_{w,\\sigma ^{\\prime }}=1$ so that $w^\\ell (e_i)=e_i$ and $w^\\ell (e_j)=e_j$ and $w^\\ell \\in C_{H_{i,j,\\zeta }}$ .",
"If $i$ and $j$ are in the same orbit $\\sigma $ under $\\sigma _w$ then we also have ${\\sigma _w}^\\ell (i)=i$ and ${\\sigma _w}^\\ell (j)=j$ because if ${\\sigma _w}^\\ell (i)=j$ and ${\\sigma _w}^\\ell (j)=i$ then $\\sigma _w$ would be of even order and $w$ too, which is not.",
"Hence $\\ell $ is a multiple of the length of $\\sigma $ and since $p_{w,\\sigma }=1$ , we get that $w^\\ell \\in C_{i,j,\\zeta }$ .",
"Let us now consider the case $d\\ge 2$ .",
"From the first two parts, the only thing to show is that if $w$ verifies $\\sigma _w = \\mathrm {id}$ and $w$ has finite order lifting in $B/[P,P]$ then $w=\\mathrm {id}$ .",
"Let $i \\in \\lbrace 1,\\ldots ,r\\rbrace $ .",
"We have $w(e_i)=a_i e_i$ .",
"Hence $w \\in N_{H_i}$ where $H_i=\\lbrace (z_1,\\ldots , z_r) \\in \\mathbb {C}^r, \\ z_i= 0\\rbrace $ .",
"We then deduce that $w \\in C_{H_i}$ that is $a_i=1$ .",
"Let us consider the case $d=1$ .",
"From the first two parts, the only thing to show is that if $\\sigma _w=\\mathrm {id}$ and for $i\\ne j \\in \\lbrace 1,\\ldots ,r\\rbrace $ , the order of $a_i{a_j}^{-1}$ is a multiple of the order of $a_i$ then $w$ has a finite order lifting in $B/[P,P]$ .",
"For $i \\ne j \\in \\lbrace 1,\\ldots , r\\rbrace $ and $\\zeta \\in \\mathbb {U}_{de}$ , if $w^\\ell \\in N_{H_{i,j,\\zeta }}$ then $w^\\ell $ stabilizes three lines in the plane $\\textrm {Span}(e_i,e_j)$ ; namely $\\mathbb {C}e_i,\\mathbb {C}e_j$ and the line ${H_{i,j,\\zeta }}^\\perp $ .",
"So ${a_i}^\\ell ={a_j}^\\ell $ and then ${a_i{a_j}^{-1}}^\\ell =1$ .",
"Hence ${a_i}^\\ell = {a_j}^\\ell = 1$ and then $w^\\ell \\in C_{H_{i,j,\\zeta }}$ .",
"Corollary REF gives then the result.",
"Example 9 Let $j = -1/2+ i \\sqrt{3}/2$ .",
"The preceding corollary ensures us that $w=\\mathrm {diag\\,}(j,j^2,1\\ldots , 1) \\in G(3,3,r)$ verifies $\\sigma _w = \\mathrm {id}$ and has a finite order lifting in $B/[P,P]$ .",
"In particular, for $r=2$ , $\\mathrm {diag\\,}(j,j^2)$ gives an example of an element of $G(3,3,2)$ of order 3 that has a finite order lifting in $B/[P,P]$ but such that no element of order 3 of $\\mathfrak {S}_2$ (since there are no such element) lifts in $B/[P,P]$ .",
"More generally, if $de = 2^m q$ with $q$ odd, let $\\zeta $ of $q^\\textrm {th}$ root of unity.",
"Then $w=\\mathrm {diag\\,}(\\zeta ,\\zeta ^{-1},1,\\ldots , 1) \\in G(de,de,r)$ has a finite order lifting in $B/[P,P]$ .",
"Remark 10 Let $G$ be a subgroup of $G(de,e,r)$ where $d \\ge 2$ which lifts in $B/[P,P]$ .",
"Then Corollary REF shows that $G \\cap D=\\lbrace 1\\rbrace $ where $D$ is the subgroup of diagonal matrices of $G(de,e,r)$ .",
"In particular $G$ is isomorphic to its image in $\\mathfrak {S}_r$ .",
"Moreover the criterion of Corollary REF shows also that every odd-order subgroup of $\\mathfrak {S}_r$ lifts into $B/[P,P]$ .",
"So that the isomorphism classes of finite subgroups of $B/[P,P]$ are the same that the isomorphism classes of odd-order subgroups of $\\mathfrak {S}_r$ .",
"The preceding example shows that this is not the case when $d=1$ .",
"In [6], Marin shows that $B/[P,P]$ is a Bieberbach group for $G_4$ and $G_6$ .",
"We are able to give the list of the irreducible group such that $B/[P,P]$ is a Bieberbach group (Corollary REF ).",
"But we start with two small groups that we can study by hands.",
"0 Corollary 11 For $W=G_5$ , the group $B/[P,P]$ is a Bieberbach group of holonomy group $\\mathfrak {A}_4$ and dimension 8.",
"For $W=G_7$ , the group $B/[P,P]$ is a Bierberbach group of holonomy group $\\mathfrak {A}_4$ and dimension 14.",
"Proof.",
"In $G_5$ , there are elements of order $1,2,3,4,6,12$ .",
"Corollary REF ensures us that we have only to study the case of the elements of order 3.",
"There are 26 of them shared in 8 conjugacy classes.",
"There are two elements in the center of $G_5$ .",
"There are 16 reflections.",
"Both cases are treated by previous results.",
"It remains to consider two conjugacy classes which are products of an element of order 3 of $ZG_5$ with a reflection and are not reflections: they have two eigenvalues which are $j$ and $j^2$ (the third non trivial roots of unity).",
"Such an element is contained in the stabilizer of a hyperplane but not in the corresponding $C_H$ .",
"So Theorem REF allows us to conclude.",
"The situation is exactly the same for $G_7$ since the elements of $G_7$ have order $1,2,3,4,6,12$ and every element of order 3 of $G_7$ is contained in the $G_5$ subgroup of $G_7$ .",
"0 Corollary 12 Bieberbach groups.",
"For $W=G(de,e,r)$ , then $B/[P,P]$ is a Bieberbach group if and only if $r=1$ or $r=2$ and $d \\ge 2$ or $r=2$ , $d=1$ and $e=2^m$ for some $m$ .",
"Among the exceptionnal groups, the group $B/[P,P]$ is a Bieberbach group for $W=G_4, G_5, G_6, G_7, G_{10}, G_{11}, G_{14}, G_{15}, G_{18}, G_{19}, G_{25}, G_{26}.$ Proof.",
"Assume that for $G(de,e,r)$ , $B/[P,P]$ is a Bieberbach group.",
"Corollary REF ensures us that $(1,2,3)\\in \\mathfrak {S}_r$ has a finite order lifting in $B/[P,P]$ .",
"Hence $r \\le 2$ .",
"If $r=2$ and $d=1$ , then Example REF shows that $e$ has to be a power of 2.",
"Conversely, if $r=1$ , then $B=B/[P,P]=\\mathbb {Z}$ .",
"If $r=2$ and $d \\ge 2$ then a diagonal matrix $D \\in G(de,e,2)$ has a finite order lifting only if $D=\\mathrm {id}$ (since for every $i\\in \\lbrace 1,2\\rbrace $ we have $a_i=p_{D,\\lbrace i\\rbrace }=1$ ).",
"And a non diagonal matrix is of even order so it does not have a finite order lifting in $B/[P,P]$ .",
"For $r=2,d=1,e=2^m$ , let $D=\\mathrm {diag\\,}(\\zeta ,\\zeta ^{-1})$ be a diagonal matrix of $G(e,e,2)$ with $\\zeta \\in \\mathbb {U}_e$ .",
"If $\\zeta \\ne 1$ then the order of $\\zeta ^2$ is half the order of $\\zeta $ (since $e$ is a power of 2) and by Corollary REF , $D$ does not have a finite order lifting in $B/[P,P]$ .",
"A non diagonal matrix is of even order so it does not have a finite order lifting in $B/[P,P]$ .",
"For the exceptionnal groups, we use the package [5] of [8] to compute stabilizers of hyperplanes and parabolic subgroups associated to the orthogonal line.",
"We now give a corollary generalizing Corollary 2.7 of [6] and Theorem 16 of [4].",
"0 Corollary 13 Real reflection groups.",
"Let $W$ be a real reflection group.",
"Then every odd-order element has a finite order lifting in $B/[P,P]$ .",
"More generally, if $W$ is a reflection group for which the roots of unity inside its field of definition all have for order of power of 2 (The irreducible groups verifying this property are the following $G_8,G_9,G_{12},G_{13},G_{22}, G_{23}, G_{24}, G_{28},G_{29},G_{30},G_{31},G_{35},G_{36},G_{37}$ for the exceptional ones, and $G(de,e,r)$ with $d$ and $e$ powers of 2 and $r \\ge 2$ , $\\mathfrak {S}_r$ , $G(d,1,1)$ with $d$ power of 2 and $G(e,e,2)$ with $e \\ge 3$ for the infinite series), then every odd-order element has a finite order lifting in $B/[P,P]$ .",
"For those groups, let $G \\subset W$ then $p^{-1}(G)$ the inverse image of $G$ in $B/[P,P]$ is a Bieberbach group if and only if $G$ is a 2-subgroup of $W$ .",
"Again, for those groups, let $G \\subset W$ with $|G|$ odd then $B/[P,P]$ has a finite group isomorphic to $G$ and $p^{-1}(G)$ is a crystallographic group.",
"Proof.",
"Let $w$ be an odd-order element of $W$ and $H \\in {A}$ .",
"If $w \\in N_H$ then $w$ stabilizes $H^\\perp $ and the restriction of $w$ to $H^\\perp $ is still of odd order.",
"In particular the eigenvalue of $w$ along $H^\\perp $ is of odd order.",
"But by hypothesis, this eigenvalue is a root of unity belonging to the field of definition of $W$ .",
"Its order is then a power of 2.",
"Hence $w$ acts trivially on $H^\\perp $ which means that $w \\in C_H$ .",
"If $G$ is a 2-subgroup of $W$ then every element in $p^{-1}(G)$ has infinite order thanks to Corollary REF .",
"Let $P_0=p^{-1}(Z(W)) \\overset{{\\rm {\\textrm { \\tiny gr.",
"}}}}{\\simeq } \\mathbb {Z}^{|{A}|}$ (see [6]), the extension ${0 [r]& P_0 ^-{}[r] & p^{-1}(G) ^-{}[r] & \\overline{G}[r] & 1}$ gives the result (let us remind that $\\overline{G}$ is the direct image of $G$ in $W/ZW$ ).",
"With Theorem REF , we get the converse.",
"If $G$ is not a 2-group, then it contains an element of odd order.",
"The first part ensures us that this element may be lifted in $p^{-1}(G)$ as a finite order element.",
"Let us now consider $G \\subset W$ whose order is odd.",
"Then, for every $H \\in {A}$ and every $w \\in G\\cap N_H$ , we have $w \\in C_H$ by the above argument since the order of $w$ is odd.",
"Hence Theorem REF gives the result.",
"Remark 14 The irreducible complex reflection groups $W$ such that every element of odd order has a finite order lifting in $B/[P,P]$ are the same as the list of Corollary REF that is to say $G_8, G_9, G_{12}, G_{13}, G_{22}, G_{23}, G_{24}, G_{28}$ , $G_{29}, G_{30}, G_{31}, G_{35}, G_{36}, G_{37}$ for the exceptional ones, and $G(d,1,1)$ with $d$ a power of 2, $G(de,e,r)$ with $d$ and $e$ powers of 2 and $r \\ge 2$ , $\\mathfrak {S}_r$ and $G(e,e,2)$ with $e \\ge 3$ .",
"The preceding corollary shows that all these groups verify the property.",
"For the exceptional ones, a [8] computation using [5] gives that there are no more exceptional group such that every element of odd order has a finite order lifting in $B/[P,P]$ .",
"Let us now study the infinite series and consider $G(de,e,r)$ such that every element of odd order has a finite order lifting in $B/[P,P]$ .",
"If $r=1$ , since $B/[P,P]=\\mathbb {Z}$ then $G(d,1,1)$ should have only one odd order element, so that $d$ is a power of 2.",
"Let us assume that $r \\ge 2$ and $d \\ge 2$ .",
"For every $\\zeta \\in \\mathbb {U}_{de} \\setminus \\lbrace 1\\rbrace $ , the element $\\mathrm {diag\\,}(\\zeta ,\\zeta ^{-1},1,\\ldots , 1)$ does not have a finite order lifting in $B/[P,P]$ (see Corollary REF ).",
"It implies that $de$ has to be a power of 2.",
"Let us consider the case where $r \\ge 3$ and $d=1$ .",
"For $\\zeta \\in \\mathbb {U}_{de} \\setminus \\lbrace 1\\rbrace $ , the element $\\mathrm {diag\\,}(\\zeta ,\\zeta ,\\zeta ^{-2},1,\\ldots ,1)$ does not have a finite order lifting in $B/[P,P]$ (see Corollary REF ).",
"It implies that $de$ has to be a power of 2.",
"When $r=2$ et $d=1$ , we get $G(e,e,2)$ which verify the property.",
"Remark 15 Following Cayley's argument, every odd-order group may be identified to a subgroup of $\\mathfrak {S}_n$ for $n$ large enough and thus to a subgroup of $B_n/[P_n,P_n]$ thanks to Corollary REF .",
"In the course of writing, Gonçalves, Guaschi and Ocampo have communicated to us that they independently got this same result."
],
[
"Infinite Braid Group",
"Let $B_\\infty $ the infinite braid group.",
"This is the direct limit of the family $(B_n)_{n \\in \\mathbb {N}}$ where $B_n$ is the standard braid group on $n$ strands and the map $B_n \\rightarrow B_{n+1}$ consists to add one strand on the right.",
"The pure braid group $P_\\infty $ is the direct limit of the family $(P_n)_{n \\in \\mathbb {N}}$ where $P_n$ is the pure braid group on $n$ strands.",
"Since $[P_{n-1},P_{n-1}]=[P_n,P_n] \\cap B_n$ , $B_{n-1}/[P_{n-1},P_{n-1}]$ may be identified to a subgroup of $B_n/[P_n,P_n]$ and then also to a subgroup of the direct limit of the family $(B_n/[P_n,P_n])_{n \\in \\mathbb {N}}$ which is nothing else than $B_{\\infty }/[P_\\infty ,P_\\infty ]$ .",
"Moreover, with the same kind of arguments $P_{\\infty }/[P_\\infty ,P_\\infty ]$ is the direct limit of the groups $P_n/[P_n,P_n]$ and is a free abelian group on the set of 2-sets of the infinite set $\\mathbb {N}^*$ .",
"Finally, the group $\\mathfrak {S}_\\infty $ may be defined as the direct limit of the family $(\\mathfrak {S}_n)_{n\\in \\mathbb {N}}$ where an element of $\\mathfrak {S}_{n-1}$ is seen as an element of $\\mathfrak {S}_n$ fixing $n$ .",
"But $\\mathfrak {S}_\\infty $ may also be viewed as the group of permutation of $\\mathbb {N}^*$ whose support is finite.",
"0 Proposition 16 For every odd-order group $G$ , there exists a subgroup of $B_{\\infty }/[P_\\infty ,P_\\infty ]$ isomorphic to $G$ and $B_{\\infty }/[P_\\infty ,P_\\infty ]$ contains no even order element.",
"Proof.",
"The group $B_{\\infty }/[P_\\infty ,P_\\infty ]$ is the direct limit of the family $(B_n/[P_n,P_n])_{n \\in \\mathbb {N}}$ .",
"Moreover Proposition 2.4 of [6] ensures us that the maps $B_{n-1}/[P_{n-1},P_{n-1}] \\rightarrow B_n/[P_n,P_n]$ are injective for all positive $n$ .",
"Then the map $B_n/[P_n,P_n] \\rightarrow B_\\infty /[P_\\infty ,P_\\infty ]$ is injective for all $n$ .",
"Let us now consider $G$ a group of odd order.",
"Following Cayley's argument, we let $G$ acts on itself by left translation, so that $G$ identifies to an odd-order group of $\\mathfrak {S}_{|G|}$ the symmetric group on $|G|$ letters.",
"But Corollary REF ensures us that $G$ identifies to a subgroup of $B_{|G|}/[P_{|G|},P_{|G|}]$ which itself is a subgroup of $B_\\infty /[P_\\infty ,P_\\infty ]$ .",
"Let $x$ be an even order element of $B_{\\infty }/[P_\\infty ,P_\\infty ]$ , then it belongs to $B_n/[P_n,P_n]$ for some $n$ .",
"But $B_n/[P_n,P_n]$ has no even order element.",
"We can be even more precise.",
"0 Proposition 17 Let $\\mathfrak {S}_\\infty $ be the permutation group of the infinite set $\\mathbb {N}^*=\\lbrace 1,2,\\ldots \\rbrace $ with finite support.",
"We have the following extension ${1 [r]& {P_\\infty }^\\textrm {ab} ^-{}[r] & B_\\infty /[P_\\infty ,P_\\infty ] ^-{p}[r] & \\mathfrak {S}_\\infty [r] & 1}$ and ${P_\\infty }^\\textrm {ab}$ identifies to the free abelian group over the set ${P}_2$ of 2-subsets of $\\mathbb {N}^*$ .",
"A finite subgroup of $B_\\infty /[P_\\infty ,P_\\infty ]$ is of odd order and maps isomorphically to $\\mathfrak {S}_\\infty $ through $p$ .",
"For every odd-order subgroup $G$ of $\\mathfrak {S}_\\infty $ , there exists a group homomorphism $s:G \\rightarrow B_\\infty /[P_\\infty ,P_\\infty ]$ such that $ps=\\mathrm {id}_G$ .",
"Moreover two such group homomorphisms are conjugated by an element of ${P_\\infty }^\\textrm {ab}$ and the intersection of the normalizer of a finite subgroup $\\widetilde{G}$ of $B_{\\infty }/[P_\\infty ,P_\\infty ]$ with ${P_\\infty }^\\textrm {ab}$ is isomorphic to $\\mathbb {Z}{P}_2/p(\\widetilde{G})$ .",
"Proof.",
"A finite subgroup of $B_\\infty /[P_\\infty ,P_\\infty ]$ is contained in some $B_n/[P_n,P_n]$ and hence of odd order (Corollary REF ).",
"Moreover since $P_\\infty /[P_\\infty ,P_\\infty ]$ is a free abelian group, the surjective map $p$ preserves the order of the finite order elements (we have already seen this in the proof of theorem REF ).",
"So a finite subgroup of $B_\\infty /[P_\\infty ,P_\\infty ]$ maps isomorphically onto a subgroup of $\\mathfrak {S}_\\infty $ through $p$ .",
"Let $G$ be a finite subgroup of $\\mathfrak {S}_\\infty $ .",
"Then it is a subgroup of $\\mathfrak {S}_n$ for some $n$ and hence embeds in $B_n/[P_n,P_n]$ and so in $B_\\infty /[P_\\infty ,P_\\infty ]$ .",
"The second part follows from the fact that $H^1(G,P_\\infty /[P_\\infty ,P_\\infty ])=\\lbrace 0\\rbrace $ since $P_\\infty /[P_\\infty ,P_\\infty ]$ is a permutation module (Lemma ).",
"Finally, let $x \\in {P_\\infty }^\\textrm {ab}$ .",
"Then, as in the proof of Theorem REF , $x$ verifies $x\\widetilde{G}x^{-1}=\\widetilde{G}$ if and only if $gxg^{-1}=x$ for every $g \\in \\widetilde{G}$ ."
],
[
"Some More Examples",
"We have seen that every odd-order subgroup of $\\mathfrak {S}_n$ can be lifted in $B_n/[P_n,P_n]$ .",
"This has been applied to show that every odd-order group $G$ can be embedded in $B_\\infty /[P_\\infty ,P_\\infty ]$ .",
"Here we describe some odd-order subgroups of finite symmetric groups verifying a stronger property: they do not meet any stabilizer of a hyperplane.",
"The study of such subgroups is quite natural because they are the subgroups of $W$ that acts freely on ${A}$ .",
"Moreover, when $G$ is a subgroup of $W$ such that for every $H \\in {A}$ , $G \\cap N_H = \\lbrace 1\\rbrace $ then the criterion of Theorem REF is clearly verified for $G$ .",
"But we have in fact a stronger property: $H^2(G,\\mathbb {Z}{A})$ is trivial.",
"In order to prove this, it suffices to consider the orbits of ${A}$ under $G$ and to apply Shapiro's isomorphism."
],
[
"The case of the symmetric group",
"To study the case of the symmetric group, let us introduce the following notation.",
"For an integer $n$ , we denote by $F_n$ the set $F_n=\\lbrace \\sigma \\in \\mathfrak {S}_n, \\quad \\sigma \\textrm { is of cycle type } k^{[n/k]} \\textrm { or } 1^{[1]}k^{[(n-1)/k]}\\textrm { with } k \\textrm { odd}\\rbrace \\,.$ 0 Proposition 18 Let $G$ be a subgroup of $\\mathfrak {S}_n$ .",
"For $i \\ne j$ , we denote by $C(i,j)$ the centralizer of the transposition $(i,j)$ : this is also the stabilizer $N_{H_{i,j}}$ .",
"Then, $G \\cap C(i,j)= \\lbrace \\mathrm {id}\\rbrace $ for all $i\\ne j$ if and only if $G \\subset F_n$ .",
"Proof.",
"We have $C_{i,j}=\\mathfrak {S}_{n-2}\\times \\langle (i,j) \\rangle $ .",
"In particular, $w \\in C_{i,j}$ if and only if $\\lbrace w(i),w(j)\\rbrace =\\lbrace i,j\\rbrace $ .",
"Let now $G$ such that $G\\cap C(i,j)=\\lbrace \\mathrm {id}\\rbrace $ for all $i \\ne j$ .",
"If $G$ has an element $w$ of order $n=2k$ then $w^k$ is a non trivial product of transpositions and then belongs to $C(i,j)$ for some $(i,j)$ .",
"So every element of $G$ is of odd order and every cycle of an element of $G$ is of odd length (to see this, we could also have applied the fact the $G$ lifts into $B/[P,P]$ thanks to Theorem REF ).",
"Moreover every non trivial element $w$ of $G$ has at most one fixed point: if there are two or more, then let $\\lbrace i,j\\rbrace $ be two such fixed points, then $w \\in C(i,j)$ .",
"We then deduce that the length of every non trivial cycle of $w \\in G$ is the same.",
"Indeed if $c_1$ and $c_2$ are two non trivial cycles of length $k_1 < k_2$ then $w^{k_1} \\ne 1$ and has more than one fixed point.",
"This gives the wanted cycle decomposition.",
"Conversely, assume that $G \\subset F_n$ .",
"For $g \\in F_n$ , every power of $g$ is still in $F_n$ since every cycle of $g$ will decomposes in cycles of length a divisor of an odd number.",
"So for every $i\\ne j$ , the set $\\lbrace i,j\\rbrace $ is not stable by any non trivial element of $G$ .",
"Example 19 Group of odd order.",
"Let $G$ be a group of odd order.",
"When $G$ acts on itself by left multiplication, then the cycle decomposition of an element $g \\in G \\subset \\mathfrak {S}_{|G|}$ is a product of $[G:\\langle g \\rangle ]$ cycles of length the order of $g$ which is odd.",
"And so $G \\subset F_{|G|}$ which a bit more precise that what was needed in Proposition REF .",
"Let us know study a bit Frobenius groups who will provide some more examples.",
"Let $G$ be a Frobenius group with kernel $N$ and $H$ a Frobenius complement.",
"The group $G$ is a disjoint union $G= N \\sqcup \\bigsqcup _{g \\in G/H} (gHg^{-1}\\setminus {1})\\,.$ 0 Lemma 20 The group $G$ acts on $G/H$ with the following properties.",
"((ii)) ($i$ ) An element $n \\in N \\setminus \\lbrace 1\\rbrace $ has no fixed point on $G/H$ .",
"($ii $ ) For $g \\in G$ , an element of $gHg^{-1}\\setminus {1}$ has only one fixed point on $G/H$ namely $gH$ .",
"In particular, $G$ acts faithfully on $G/H$ .",
"Proof.",
"((ii)) ($i$ ) If $ngH=gH$ then $g^{-1}ng \\in H$ and $n \\in gHg^{-1}$ which is absurd thanks to the set partition of $G$ .",
"($ii $ ) If $ghg^{-1}g^{\\prime }H=g^{\\prime }H$ (with $h \\in H \\setminus \\lbrace 1\\rbrace $ ) then $g^{\\prime -1}ghg^{-1}g^{\\prime } \\in H$ and $h \\in H \\cap g^{-1}g^{\\prime }Hg^{\\prime -1}g$ .",
"Hence, the set partition of $G$ ensures us that $g^{-1}g^{\\prime } \\in H$ .",
"0 Corollary 21 For every $g \\in G$ , the cycle decomposition of $g \\in G$ seen as a permutation of $G/H$ is the following ((ii)) ($i$ ) if $g \\in N$ , then $g$ is a product of $[G:H]/k$ cycles of length $k$ where $k$ is the order of $g$ .",
"($ii $ ) if $g \\in g^{\\prime }Hg^{\\prime -1} \\setminus \\lbrace 1\\rbrace $ then $g$ is a product of $([G:H]-1)/k$ cycles of length $k$ where $k$ is the order of $g$ .",
"Proof.",
"It is trivial when $g =1$ .",
"Let us consider $g \\ne 1$ .",
"((ii)) ($i$ ) For $g^{\\prime }H \\in G/H$ and $1 \\le \\ell <k$ , we have $g^\\ell g^{\\prime }H \\ne g^{\\prime }H$ since $g^\\ell \\in N \\setminus \\lbrace 1\\rbrace $ .",
"Hence the orbit of $g^{\\prime }H$ under $\\langle g \\rangle $ has $k$ elements.",
"($ii $ ) Let $1 \\le \\ell < k$ and $g^{\\prime \\prime }H \\in G/H$ with $g^{\\prime \\prime }H \\ne g^{\\prime }H$ .",
"We have $g^\\ell g^{\\prime \\prime }H\\ne g^{\\prime \\prime }H$ since $g^\\ell \\in g^{\\prime }Hg^{\\prime -1} \\setminus \\lbrace 1\\rbrace $ and $g^{\\prime \\prime }H \\ne g^{\\prime }H$ .",
"Hence the orbit of $g^{\\prime \\prime }H$ under $\\langle g \\rangle $ has $k$ elements.",
"0 Corollary 22 Let $K$ be a subgroup of a Frobenius group of odd order, then $K$ is contained in $F_{(G:H)}$ .",
"This last corollary associated to Corollary REF generalizes Corollary 3.10 and 3.11 of [6] and Theorem 7 of [4]."
],
[
"The infinite series",
"Let us introduce the following subset of $G(de,e,r)$ $F(de,e,r)=\\lbrace w \\in G(de,e,r), \\textrm {the cycle type of } \\sigma _w \\textrm { is } [k]^{r/k} \\textrm { with } k \\textrm { odd and }p_{w,\\sigma }=1 \\textrm { for every } \\sigma \\textrm { cycle of } \\sigma _w\\rbrace $ where $\\sigma _w$ is permutation associated to $w$ .",
"0 Proposition 23 Let us consider $G$ a subgroup of $G(de,e,r)$ with $d \\ge 2$ and ${A}$ be the set of hyperplanes of $G(de,e,r)$ .",
"Then $G \\cap N_H= \\lbrace 1\\rbrace $ for every $H \\in {A}$ if and only if $G \\subset F(de,e,r)$ .",
"Proof.",
"Assume that $G \\cap N_H= \\lbrace 1\\rbrace $ for every $H \\in {A}$ then from Theorem REF we deduce that every element of $G$ is of odd order and so for every $w \\in G$ and every $\\sigma $ cycle of $\\sigma _w$ is of odd order and from Corollary REF that $p_{w,\\sigma }=1$ for every $\\sigma $ cycle of $\\sigma _w$ and every $w \\in G$ .",
"Moreover consider $w \\in G$ such that $\\sigma _w$ has at least two cycles of different lengths.",
"Suppose that these lengths are $\\ell _1 < \\ell _2$ .",
"Then $w^{\\ell _1} \\in G \\setminus \\lbrace 1\\rbrace $ and stabilizes every hyperplane of the form $\\lbrace z_i=0\\rbrace $ where $i$ belongs to the orbit of length $\\ell _1$ .",
"Conversely, assume that $G \\subset F(de,e,r)$ .",
"If $w \\in G$ stabilizes the hyperplane $\\lbrace z_i=0\\rbrace $ then $\\sigma _w(i)=i$ and hence $\\sigma _w=\\mathrm {id}$ since every cycle of $\\sigma _w$ has the same length.",
"Since $p_{w,\\sigma }=1$ for every $\\sigma $ cycle of $\\sigma _w=1$ , we get that $w=1$ .",
"If $w \\in G$ stabilizes $H_{i,j,\\zeta }$ for $i \\ne j$ and $\\zeta \\in \\mathbb {U}_{de}$ then $\\lbrace \\sigma _w(i),\\sigma _w(j)\\rbrace =\\lbrace i,j\\rbrace $ .",
"Since every cycle of $\\sigma _w$ is of odd length then $\\sigma _w(i)=i$ and $\\sigma _w(j)=j$ .",
"Hence $w$ stabilizes $\\lbrace z_i=0\\rbrace $ so $w=1$ ."
]
] | 1709.01853 |
[
[
"On the Operator Jensen-Mercer Inequality"
],
[
"Abstract Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity.",
"This paper is two folded.",
"First, we present a Mercer-type inequality for operators without assuming convexity nor operator convexity.",
"Yet, this form refines the known inequalities in the literature.",
"Second, we present a log-convex version for operators.",
"We then use these results to refine some inequalities related to quasi-arithmetic means of Mercer's type for operators."
],
[
"To proceed in details we fix some notations and terminologies.",
"We assume that ${H}$ and ${K}$ are Hilbert spaces, $\\mathbb {B}\\left( {H} \\right)$ and $\\mathbb {B}\\left( {K} \\right)$ are ${{C}^{*}}$ -algebras of all bounded operators on the appropriate Hilbert spaces.",
"An operator $A$ is called self-adjoint if $A=A^*$ .",
"A self-adjoint operator is called positive if $\\left\\langle Ax,x \\right\\rangle \\ge 0$ for all $x\\in {H}$ , and we then write $A\\ge 0$ .",
"We denote by $A>0$ if it is a positive invertible operator.",
"For self-adjoint operators $A,B\\in \\mathbb {B}\\left( {H} \\right)$ , we say $A\\le B$ if $B-A\\ge 0$ .",
"The spectrum of $A\\in \\mathbb {B}\\left( {H} \\right)$ is denoted by $\\sigma \\left( A \\right)$ .",
"A linear map $\\Phi :\\mathbb {B}\\left( {H} \\right)\\rightarrow \\mathbb {B}\\left( {K} \\right)$ is said to be positive if $\\Phi \\left( A \\right)\\ge 0$ whenever $A\\ge 0$ and $\\Phi $ is called unital if $\\Phi \\left( {{\\mathbf {1}}_{{H}}} \\right)={{\\mathbf {1}}_{{K}}}$ (we use the symbol $\\mathbf {1}$ for the identity operator).",
"If the function $f:I\\subseteq \\mathbb {R}\\rightarrow \\mathbb {R}$ is convex, then the so-called Jensen inequality $f\\left( \\sum \\limits _{i=1}^{n}{{{w}_{i}}{{x}_{i}}} \\right)\\le \\sum \\limits _{i=1}^{n}{{{w}_{i}}f\\left( {{x}_{i}} \\right)},$ holds for some positive scalars ${{w}_{1}},\\ldots ,{{w}_{n}}$ with $\\sum \\nolimits _{i=1}^{n}{{{w}_{i}}}=1$ and ${{x}_{i}}\\in I$ .",
"Our motivation for this paper arose from a paper by Mercer [7], which is connected with a remarkable variant of the inequality (REF ).",
"His result says: Theorem 1.1 [7] If $f$ is a convex function on $\\left[ m,M \\right]$ , then $f\\left( M+m-\\sum \\limits _{i=1}^{n}{{{w}_{i}}{{x}_{i}}} \\right)\\le f\\left( M \\right)+f\\left( m \\right)-\\sum \\limits _{i=1}^{n}{{{w}_{i}}f\\left( {{x}_{i}} \\right)},$ for all ${{x}_{i}}\\in \\left[ m,M \\right]$ and all ${{w}_{i}}\\in \\left[ 0,1 \\right]$ $\\left( i=1,\\ldots ,n \\right)$ with $\\sum \\nolimits _{i=1}^{n}{{{w}_{i}}}=1$ .",
"There are many versions, variants and generalizations for the inequality (REF ); see for example [1], [2], [9].",
"A similar arguing with application of functional calculus gives Jensen-Mercer's operator inequality without operator convexity assumptions.",
"More precisely, the following theorem is proved in [5]: Theorem 1.2 [5] Let ${{A}_{1}},\\ldots ,{{A}_{n}}\\in \\mathbb {B}\\left( {H} \\right)$ be self-adjoint operators with spectrum in $\\left[ m,M \\right]$ and let ${{\\Phi }_{1}},\\ldots ,{{\\Phi }_{n}}:\\mathbb {B}\\left( {H} \\right)\\rightarrow \\mathbb {B}\\left( {K} \\right)$ be positive linear maps with $\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{\\mathbf {1}}_{{H}}} \\right)}={{\\mathbf {1}}_{{K}}}$ .",
"If $f:\\left[ m,M \\right]\\subseteq \\mathbb {R}\\rightarrow \\mathbb {R}$ is convex function, then $f\\left( \\left( M+m \\right){{1}_{K}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)\\le \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{1}_{K}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)}.$ Moreover, in the same paper the following series of inequalities was proved $& f\\left( \\left( M+m \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right) \\nonumber \\\\& \\le \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}+\\frac{\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-M{{\\mathbf {1}}_{{K}}}}{M-m}f\\left( m \\right)+\\frac{m{{\\mathbf {1}}_{{K}}}-\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}}{M-m}f\\left( M \\right) \\\\& \\le \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)}\\nonumber .$ Analogues of this result for other kinds of functions have been established; see [3], [4], [6].",
"An outline of the paper is as follows: In Theorem REF , inspired by the idea of Mićić et al.",
"[8], we shall generalize the Jensen-Mercer's operator inequality to the context of twice differentiable functions.",
"Theorem REF deals with the case, log-convex functions.",
"In such a way, we obtain improvements of some recent results, known from the literature.",
"Special cases of the main theorems are studied to recover other inequalities of Mercer's type."
],
[
"In the following theorem we give an extension of Jensen-Mercer's operator inequality given in Theorem REF .",
"Theorem 2.1 Let ${{A}_{1}},\\ldots ,{{A}_{n}}\\in \\mathbb {B}\\left( {H} \\right)$ be self-adjoint operators with spectrum in $\\left[ m,M \\right]$ and let ${{\\Phi }_{1}},\\ldots ,{{\\Phi }_{n}}:\\mathbb {B}\\left( {H} \\right)\\rightarrow \\mathbb {B}\\left( {K} \\right)$ be positive linear maps with $\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{\\mathbf {1}}_{{H}}} \\right)}={{\\mathbf {1}}_{{K}}}$ .",
"If $f:\\left[ m,M \\right]\\subseteq \\mathbb {R}\\rightarrow \\mathbb {R}$ is continuous twice differentiable function such that $\\alpha \\le f^{\\prime \\prime }\\le \\beta $ with $\\alpha ,\\beta \\in \\mathbb {R}$ , then $& \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)} \\nonumber \\\\&\\quad -\\beta \\left\\lbrace \\left( M+m \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-Mm{{\\mathbf {1}}_{{K}}}-\\frac{1}{2}\\left( {{\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)}^{2}}+\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( A_{i}^{2} \\right)} \\right) \\right\\rbrace \\nonumber \\\\& \\le f\\left( \\left( M+m \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right) \\\\& \\le \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)} \\nonumber \\\\&\\quad -\\alpha \\left\\lbrace \\left( M+m \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-Mm{{\\mathbf {1}}_{{K}}}-\\frac{1}{2}\\left( {{\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)}^{2}}+\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( A_{i}^{2} \\right)} \\right) \\right\\rbrace .",
"\\\\\\nonumber $ The idea of the proof is the following: It is well-known that for any convex function $f$ and $m\\le t\\le M$ we have $f\\left( t \\right)=f\\left( \\frac{M-t}{M-m}m+\\frac{t-m}{M-m}M \\right)\\le L\\left( t \\right),$ where $L\\left( t \\right)\\equiv \\frac{M-t}{M-m}f\\left( m \\right)+\\frac{t-m}{M-m}f\\left( M \\right).$ According to the assumption, the function ${{g}_{\\alpha }}\\left( t \\right)\\equiv f\\left( t \\right)-\\frac{\\alpha }{2}{{t}^{2}}$ $\\left( m\\le t\\le M \\right)$ is convex.",
"On account of (REF ), we have $f\\left( t \\right)\\le L\\left( t \\right)-\\frac{\\alpha }{2}\\left\\lbrace \\left( M+m \\right)t-Mm-{{t}^{2}} \\right\\rbrace .$ Since $m\\le M+m-t\\le M$ , we can replace $t$ with $M+m-t$ , which gives us $f\\left( M+m-t \\right)\\le {{L}_{0}}\\left( t \\right)-\\frac{\\alpha }{2}\\left\\lbrace \\left( M+m \\right)t-Mm-{{t}^{2}} \\right\\rbrace ,$ where ${{L}_{0}}\\left( t \\right)\\equiv L\\left( M+m-t \\right)=f\\left( M \\right)+f\\left( m \\right)-L\\left( t \\right).$ Using functional calculus for the operator $m{{\\mathbf {1}}_{{K}}}\\le \\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}\\le M{{\\mathbf {1}}_{{K}}}$ , we infer that $\\begin{aligned}f\\left( \\left( M+m \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)&\\le {{L}_{0}}\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right) \\\\&\\quad -\\frac{\\alpha }{2}\\left\\lbrace \\left( M+m \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-Mm{{\\mathbf {1}}_{{K}}}-{{\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)}^{2}} \\right\\rbrace .\\end{aligned}$ On the other hand, by applying functional calculus for the operator $m{{\\mathbf {1}}_{{H}}}\\le {{A}_{i}}\\le M{{\\mathbf {1}}_{{H}}}$ in (REF ), we get $f\\left( {{A}_{i}} \\right)\\le L\\left( {{A}_{i}} \\right)-\\frac{\\alpha }{2}\\left\\lbrace \\left( M+m \\right){{A}_{i}}-Mm{{\\mathbf {1}}_{{H}}}-A_{i}^{2} \\right\\rbrace .$ Applying positive linear maps ${{\\Phi }_{i}}$ and summing, we have $\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)}\\le L\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)-\\frac{\\alpha }{2}\\left\\lbrace \\left( M+m \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-Mm{{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( A_{i}^{2} \\right)} \\right\\rbrace .$ Combining the two inequalities (REF ) and (REF ), we get ().",
"The inequality (REF ) follows similarly by taking into account that $L\\left( t \\right)-\\frac{\\beta }{2}\\left\\lbrace \\left( M+m \\right)t-Mm-{{t}^{2}} \\right\\rbrace \\le f\\left( t \\right),\\qquad \\text{ }m\\le t\\le M.$ The details are left to the reader.",
"Hence, we have the conclusion.",
"This expression has the advantage of using twice differentiable functions instead of convex functions used in Theorem REF .",
"Here we give an example to clarify the situation in Theorem REF .",
"Example 2.1 Taking $f\\left( t \\right)=\\sin t\\text{ }\\left( 0\\le t\\le 2\\pi \\right)$ , $A=\\left( \\begin{matrix}\\frac{\\pi }{4} & 0 \\\\0 & \\frac{\\pi }{2} \\\\\\end{matrix} \\right)$ and $\\Phi \\left( A \\right)=\\frac{1}{2}Tr\\left[ A \\right]$ .",
"After simple computations (by putting $m=\\frac{\\pi }{4}$ and $M=\\frac{\\pi }{2}$ ), we get $0.9238\\approx f\\left( \\left( M+m \\right)-\\Phi \\left( A \\right) \\right)\\nless f\\left( M \\right)+f\\left( m \\right)-\\Phi \\left( f\\left( A \\right) \\right)\\approx 0.8535.$ This example shows that (REF ) may fail without the convexity assumption.",
"On the other hand, $\\begin{aligned}0.9238&\\approx f\\left( \\left( M+m \\right)-\\Phi \\left( A \\right) \\right) \\\\& \\lneqq f\\left( M \\right)+f\\left( m \\right)-\\Phi \\left( f\\left( A \\right) \\right)-\\alpha \\left\\lbrace \\left( M+m \\right)\\Phi \\left( A \\right)-Mm-\\frac{1}{2}\\left\\lbrace \\Phi {{\\left( A \\right)}^{2}}+\\Phi \\left( {{A}^{2}} \\right) \\right\\rbrace \\right\\rbrace \\approx 0.9306, \\\\\\end{aligned}$ i.e., our approach can fill this gap.",
"Importantly, under convexity assumption, a strong result related to Theorem REF hold: Remark 2.1 It is instructive to observe that $\\left( M+m \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}-Mm{{\\mathbf {1}}_{{K}}}-\\frac{1}{2}\\left( {{\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)}^{2}}+\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( A_{i}^{2} \\right)} \\right)\\ge 0.$ One the other hand, if $f$ is convex then $\\alpha \\ge 0$ .",
"This shows that () can provide a much stronger bound than (REF ).",
"(Of course, the inequality (REF ) is also sharper than (REF ).)"
],
[
"In the sequel, we will briefly review some known properties related to log-convex functions.",
"A positive function defined on an interval (or, more generally, on a convex subset of some vector space) is called $log $ -convex if $\\log f\\left( x \\right)$ is a convex function of $x$ .",
"We observe that such functions satisfy the elementary inequality $f\\left( \\left( 1-v \\right)a+vb \\right)\\le {{\\left[ f\\left( a \\right) \\right]}^{1-v}}{{\\left[ f\\left( b \\right) \\right]}^{v}},\\qquad \\text{ }0\\le v\\le 1$ for any $a,b\\in I$ .",
"$f$ is called $log $ -concave if the inequality above works in the reversed way (that is, when $\\frac{1}{f}$ is $\\log $ -convex).",
"Because of the arithmetic-geometric mean inequality, we also have $f\\left( \\left( 1-v \\right)a+vb \\right)\\le {{\\left[ f\\left( a \\right) \\right]}^{1-v}}{{\\left[ f\\left( b \\right) \\right]}^{v}}\\le \\left( 1-v \\right)f\\left( a \\right)+vf\\left( b \\right),$ which says that any log-convex function is a convex function.",
"This is of interest to us because (REF ) can be written as $f\\left( t \\right)\\le {{\\left[ f\\left( m \\right) \\right]}^{\\frac{M-t}{M-m}}}{{\\left[ f\\left( M \\right) \\right]}^{\\frac{t-m}{M-m}}}\\le L\\left( t \\right),\\qquad \\text{ }m\\le t\\le M$ where $L\\left( t \\right)$ is as in (REF ).",
"With the inequality (REF ), we can present the following result, which can be regarded as an extension of Theorem REF to log-convex functions.",
"The proof is left to the reader as an exercise.",
"Theorem 2.2 Let all the assumptions of Theorem REF hold except that $f:\\left[ m,M \\right]\\rightarrow \\left( 0,\\infty \\right)$ is log-convex.",
"Then $\\begin{aligned}f\\left( \\left( M+m \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)} \\right)&\\le {{\\left[ f\\left( m \\right) \\right]}^{\\frac{\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)-m{{\\mathbf {1}}_{{K}}}}}{M-m}}}{{\\left[ f\\left( M \\right) \\right]}^{\\frac{M{{\\mathbf {1}}_{{K}}}-\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( {{A}_{i}} \\right)}}{M-m}}} \\\\& \\le \\left( f\\left( M \\right)+f\\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( f\\left( {{A}_{i}} \\right) \\right)}.\\end{aligned}$"
],
[
"Throughout this section we will assume that $\\mathbf {A}=\\left( {{A}_{1}},\\ldots ,{{A}_{n}} \\right)$ , where ${{A}_{i}}\\in \\mathbb {B}\\left( {H} \\right)$ are positive invertible operators with $\\sigma \\left( {{A}_{i}} \\right)\\subseteq \\left[ m,M \\right]$ for some scalars $0<m<M$ .",
"$\\mathbf {\\Phi } =\\left( {{\\Phi }_{1}},\\ldots ,{{\\Phi }_{n}} \\right)$ , where ${{\\Phi }_{i}}:\\mathbb {B}\\left( {H} \\right)\\rightarrow \\mathbb {B}\\left( {K} \\right)$ are positive linear maps.",
"Here $C\\left( \\left[ m,M \\right] \\right)$ is the set of all real valued continuous functions on an interval $\\left[ m,M \\right]$ .",
"A function $f\\in C\\left( \\left[ m,M \\right] \\right)$ is called operator increasing if $f$ is operator monotone, i.e., $A\\le B$ implies $f\\left( A \\right)\\le f\\left( B \\right)$ , for all self-adjoint operators $A$ and $B$ on a Hilbert space ${H}$ with $\\sigma \\left( A \\right),\\sigma \\left( B \\right)\\subseteq \\left[ m,M \\right]$ .",
"A function $f\\in C\\left( \\left[ m,M \\right] \\right)$ is said to be operator decreasing if $-f$ is operator monotone.",
"In [5] the following expression is defined, which the authors calls the operator quasi-arithmetic mean of Mercer's type: ${{\\widetilde{M}}_{\\varphi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right)\\equiv {{\\varphi }^{-1}}\\left( \\left( \\varphi \\left( M \\right)+\\varphi \\left( m \\right) \\right){{\\mathbf {1}}_{{K}}}-\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi \\left( {{A}_{i}} \\right) \\right)} \\right).$ Theorem 3.1 Let $\\varphi ,\\psi \\in C\\left( \\left[ m,M \\right] \\right)$ be two stricly monotonic functions.",
"(i) If either $\\psi o{{\\varphi }^{-1}}$ is convex and ${{\\psi }^{-1}}$ is operator increasing, or $\\psi o{{\\varphi }^{-1}}$ is concave and ${{\\psi }^{-1}}$ is operator decreasing, then ${{\\widetilde{M}}_{\\varphi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right)\\le {{\\widetilde{M}}_{\\psi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right).$ (ii) If either $\\psi o{{\\varphi }^{-1}}$ is concave and ${{\\psi }^{-1}}$ is operator increasing, or $\\psi o{{\\varphi }^{-1}}$ is convex and ${{\\psi }^{-1}}$ is operator decreasing, then the inequality in (REF ) is reversed.",
"These interesting inequalities were firstly discovered by Matković et al.",
"[5].",
"By virtue of Theorem REF , we have the following result: Theorem 3.2 Let $\\varphi ,\\psi \\in C\\left( \\left[ m,M \\right] \\right)$ be two strictly monotonic functions and $\\psi o{{\\varphi }^{-1}}$ is twice differentiable function.",
"(i) If $\\alpha \\le {{\\left( \\psi o{{\\varphi }^{-1}} \\right)}^{^{\\prime \\prime }}}$ with $\\alpha \\in \\mathbb {R}$ and ${{\\psi }^{-1}}$ is operator monotone, then ${{\\widetilde{M}}_{\\varphi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right)\\le {{\\psi }^{-1}}\\left\\lbrace \\psi \\left( {{\\widetilde{M}}_{\\psi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right) \\right)-\\alpha \\diamond \\left( m,M,\\varphi ,\\mathbf {A},\\mathbf {\\Phi } \\right) \\right\\rbrace ,$ where $\\begin{aligned}\\diamond \\left( m,M,\\varphi ,\\mathbf {A},\\mathbf {\\Phi } \\right)&\\equiv \\left( \\varphi \\left( M \\right)+\\varphi \\left( m \\right) \\right)\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi \\left( {{A}_{i}} \\right) \\right)}-\\varphi \\left( M \\right)\\varphi \\left( m \\right){{\\mathbf {1}}_{{K}}} \\\\&\\quad -\\frac{1}{2}\\left( {{\\left( \\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi \\left( {{A}_{i}} \\right) \\right)} \\right)}^{2}}+\\sum \\limits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi {{\\left( {{A}_{i}} \\right)}^{2}} \\right)} \\right).\\end{aligned}$ (ii) If ${{\\left( \\psi o{{\\varphi }^{-1}} \\right)}^{^{\\prime \\prime }}}\\le \\beta $ with $\\beta \\in \\mathbb {R}$ and ${{\\psi }^{-1}}$ is operator monotone, then the reverse inequality is valid in (REF ) with $\\beta $ instead of $\\alpha $ .",
"Make the substitution $f=\\psi o{{\\varphi }^{-1}}$ in () and replace ${{A}_{i}}$ , $m$ and $M$ with $\\varphi \\left( {{A}_{i}} \\right)$ , $\\varphi \\left( m \\right)$ and $\\varphi \\left( M \\right)$ respectively, we get $\\psi \\left( {{\\widetilde{M}}_{\\varphi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right) \\right)\\le \\psi \\left( {{\\widetilde{M}}_{\\psi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right) \\right)-\\alpha \\diamond \\left( m,M,\\varphi ,\\mathbf {A},\\mathbf {\\Phi } \\right).$ Since ${{\\psi }^{-1}}$ is operator monotone, we can get the conclusion.",
"The other case follows in a similar manner from (REF ).",
"In the same spirit, we infer from Theorem REF the following result: Theorem 3.3 Let $\\varphi ,\\psi \\in C\\left( \\left[ m,M \\right] \\right)$ be two strictly monotonic functions.",
"If $\\psi o{{\\varphi }^{-1}}$ is log-convex function and ${{\\psi }^{-1}}$ is operator increasing, then ${{\\widetilde{M}}_{\\varphi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right)\\le {{\\psi }^{-1}}\\left\\lbrace {{\\left[ \\psi \\left( m \\right) \\right]}^{\\frac{\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi \\left( {{A}_{i}} \\right) \\right)}-\\varphi \\left( m \\right){{\\mathbf {1}}_{{K}}}}{\\varphi \\left( M \\right)-\\varphi \\left( m \\right)}}}{{\\left[ \\psi \\left( M \\right) \\right]}^{\\frac{\\varphi \\left( M \\right){{\\mathbf {1}}_{{K}}}-\\sum \\nolimits _{i=1}^{n}{{{\\Phi }_{i}}\\left( \\varphi \\left( {{A}_{i}} \\right) \\right)}}{\\varphi \\left( M \\right)-\\varphi \\left( m \\right)}}} \\right\\rbrace \\le {{\\widetilde{M}}_{\\psi }}\\left( \\mathbf {A},\\mathbf {\\Phi } \\right).$ Remark 3.1 By choosing adequate functions $\\varphi $ and $\\psi $ , and appropriate substitutions, we can obtain some improvement concerning operator power mean of Mercer's type.",
"We leave the details of this idea to the interested reader, as it is just an application of our main results.",
"In the end of the article, we show the example such that there is no relationship between Theorems REF and REF .",
"Here, we restrict ourselves to the power function $f\\left( t \\right)={{t}^{p}}$ with $p<0$ .",
"Example 3.1 It is sufficient to compare (REF ) and the first inequality of (REF ).",
"We take $m=1, M=3$ .",
"Setting $ g(t) = \\frac{M-t}{M-m} m^p +\\frac{t-m}{M-m} M^p -\\frac{p(p-1)M^{p-2}}{2} \\left\\lbrace (M+m) t - M m -t^2\\right\\rbrace -\\left( m^{\\frac{M-t}{M-m}} M^{\\frac{t-m}{M-m}}\\right)^p.$ Little calculation shows $g(2) \\approx -0.0052909$ when $p=-0.2$ , while $g(2) \\approx 0.0522794$ when $p=-1$ .",
"We thus conclude that there is no ordering the R.H.S.",
"of (REF ) and the first inequality of (REF ).",
"Acknowledgements.",
"The authors are grateful to Dr. Trung Hoa Dinh for fruitful discussion and revising the manuscript.",
"$^\\dagger $ Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran.",
"E-mail address: [email protected] $^\\ddagger $ Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan.",
"E-mail address: [email protected]"
]
] | 1709.01808 |
[
[
"Convolutional Gaussian Processes"
],
[
"Abstract We present a practical way of introducing convolutional structure into Gaussian processes, making them more suited to high-dimensional inputs like images.",
"The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel.",
"This allows us to gain the generalisation benefit of a convolutional kernel, together with fast but accurate posterior inference.",
"We investigate several variations of the convolutional kernel, and apply it to MNIST and CIFAR-10, which have both been known to be challenging for Gaussian processes.",
"We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance.",
"We hope that this illustration of the usefulness of a marginal likelihood will help automate discovering architectures in larger models."
],
[
"Inter-domain approximations",
"Implementation of convolutional kernels requires only straightforward modifications of existing code that implements an inducing variable GP approximation.",
"All inducing point methods like FITC [1], variations on (Power-) EP [2] or variational free energy [3], [4] rely on the covariance of inducing variables and between observations and inducing variables: $\\mathbf {K}_{\\mathbf {uu}}&= \\left[\\right] \\,, \\\\\\mathbf {K}_{\\mathbf {fu}}&= \\left[\\right] \\,.$ In normal inducing point approximations, $$ and $$ are simply evaluations of the latent GP of interest: $[]_n = f_n &= f(_n) \\,, && []_m = u_m = f(_m) \\,.$ The resulting covariances are simply evaluations of the kernel of the GP prior $k(, ^{\\prime })$ .",
"Inter-domain approximations [5] simply result in a different expression for elements of $\\mathbf {K}_{\\mathbf {uu}}$ and $\\mathbf {K}_{\\mathbf {fu}}$ , and so only require a modification to the evaluation of these matrices – the computation of the rest of the learning objective remains unchanged.",
"As shown in the main text, our proposed inference method is an inter-domain method, and therefore can be implemented with the same small modification.",
"The advantage of this is that all changes to the inference can be encapsulated in the kernel."
],
[
"Exploiting convolutions",
"A large bottleneck for the implementation is summation of kernel evaluations over numerous patches.",
"A general implementation could simply extract the patches from the image, compute the kernel, and sum: $\\left[\\mathbf {K}_{\\mathbf {fu}}\\right]_{nm} = k_{fu}(_n, _m) = \\sum _p k_g(_n^{[p]}, _m) \\,.$ This can be implemented as evaluating a large $PN\\times M$ kernel matrix, reshaping to $P\\times N\\times M$ , and summing over the first dimension.",
"For general kernels, this is required.",
"However, if the kernel is stationary, i.e.",
"$k_g(, ^{\\prime }) = k_g({- ^{\\prime }})$ , the first step is computing the matrix of pairwise distances between all patches and inducing points.",
"For general inputs this still doesn't help, but in this case neighbouring inputs overlap strongly, since they're all patches from the same image.",
"By expanding the euclidean distance as $(^{[p]}_n - _m)^2 = {^{[p]}_n}^{[p]}_n - 2{^{[p]}_n}_m + _m_m$ we see that the inner product of $_m$ along all patches of $$ is a convolution operation (figure REF ).",
"Additionally, the inner product of the patches with themselves is also a convolution of the squared image with a window of ones.",
"This allows the euclidean distance to be computed in $\\left(\\log E\\right)$ rather than $\\left(E\\right)$ .",
"An additional speed benefit comes from being able to leverage the highly optimised code for convolutions on GPUs that was developed in light of the popularity of convnets.",
"Figure: Pictorial representation of convolution required in stationary kernels.",
"A single dot product with a patch is highlighted."
],
[
"The convolutional GP can not learn any function, but is nonparametric",
"Here we show that there are sets of functions that a convolutional GP places no probability mass on.",
"As a consequence, these functions will also never have density in the posterior, and can not be learned, regardless of the amount of observed data.",
"This observation is particularly interesting, as the kernel still defines a nonparametric model.",
"This places convolutional kernels in an interesting middle-ground, not seen in common examples of kernels.",
"This constraint helps the convolutional kernel to generalise well, at the expense of possibly leaving some signal unexplained.",
"However, as discussed in the main text, it is possible to get the best of both worlds by using a sum of a convolutional kernel and a kernel which spreads its probability mass more widely, and letting the marginal likelihood determine their relative weighting."
],
[
"Background: nonparametric models and consistency",
"Nonparametric methods are usually justified by a desire to build a method that is universally consistent, by which is meant that an optimal solution is found in the limit of infinite data.",
"For example, in regression the unknown function may be any continuous function, and we would like our solution to be able to come arbitrarily close.",
"Achieving this would require, at least, being able to represent any continuous function arbitrarily closely (the universal approximation property), and for Bayesian models, a prior that places probability mass over this entire space [6].",
"Neural networks have been shown to be able to approximate any function arbitrary closely in the limit of having an infinite number of basis functions (i.e.",
"hidden units) [7], [8].",
"Kernel methods such as Gaussian processes, kernel regression or SVMs all implicitly use basis functions, possibly infinitely many, through their kernel.",
"We can find a representation of the basis functions they implicitly use through Mercer's theorem [9], [10], [11], which represents a kernel in terms of its eigenvalues and orthogonal eigenfunctions: $k(, ^{\\prime }) = \\sum _{i=1}^\\infty \\lambda _i \\phi _i() \\phi _i^*(^{\\prime }) \\,.$ The span of the functions $\\left\\lbrace \\sqrt{\\lambda _i}\\phi _i\\right\\rbrace _i$ determines exactly the functions that a kernel method can capture.",
"For SVMs and kernel regression, the estimated function lies in the RKHS spanned by these bases, while the sample paths of a Gaussian process can be constructed as the infinite sum of eigenfunctions weighted by Gaussian random variables [10]: $f(\\cdot ) = \\sum _{i=1}^\\infty w_i \\phi _i(\\cdot ) \\,, && w_i \\sim {0, \\lambda _i} \\,.$"
],
[
"Degenerate kernels",
"We call any kernel with only a finite number of non-zero eigenvalues degenerate (following [11]).",
"Degenerate kernels can not lead to models that are universal approximators, as any function that contains a component of an eigenfunction with a zero eigenvalue, will be outside the RKHS and outside the set of functions the GP can generate.",
"Furthermore, such a function can have an arbitrarily large deviation from functions that can be represented, simply by adding a larger component of the eigenfunction with zero eigenvalue.",
"From this, we can easily see that an infinite number of non-zero eigenvalues is necessary for universal approximation.",
"We can alternatively see this by considering that a degenerate kernel with $B$ non-zero eigenvalues can be expressed as a finite basis function model.",
"In these models, we can fully specify any function with knowledge of $B$ function values.",
"The function value at a $B+1$ th input is therefore also fully constrained.",
"We can construct a function outside the RKHS or prior simply by adding a perturbation to the constrained function which does not pass through the $B+1$ th output.",
"Common degenerate kernels arise from considering parametric models, like linear or polynomial models."
],
[
"Universal kernels",
"[12] introduced the concept of a universal kernel, which has an RKHS which is dense in the space of all continuous functions, i.e.",
"for every continuous function, there is a function in the RKHS with an arbitrarily small maximum deviation.",
"Gaussian processes based on universal kernels have sample paths which are arbitrarily close to any continuous functionThis follows from [13], which shows that Gaussian processes assign non-zero probability to functions that are close to functions in the RKHS of their kernel..",
"The universal consistency arguments for SVM classification and kernel regression by [14] and [15] rely on using universal kernels.",
"[16] further characterise the properties required for universal kernels.",
"Most common non-degenerate kernels, like the squared exponential, are also universal [16]."
],
[
"Convolutional kernels: nonparametric but not universal",
"Here, we show that convolutional kernels fall between degenerate and universal kernels in terms of their representational capacity.",
"We first show that we can construct a collection of inputs which fully constrains the function value at a different input, as was discussed for degenerate kernels.",
"We then follow on to show that unlike degenerate kernels, we can still arbitrarily specify the function at an infinite number of distinct points, showing that the kernel can not implicitly be using a finite number of basis functions.",
"Claim Weighted covariance kernels are not universal, and Gaussian processes based on them do not place probability on (or near) all continuous functions.",
"Consider $W\\times H$ sized images with $w\\times h$ sized patches.",
"If $w < W$ and $h < H$ , we will have $P > 1$ patches in each image.",
"There are $WH$ images with a single pixel switched on, and $wh$ distinct patches $\\left\\lbrace _{i}\\right\\rbrace _{i=1}^{wh}$ .",
"We can organise the evaluations of $g(\\cdot )$ for each of the $wh$ distinct patches in the vector $\\in ^{wh}$ , where $\\left[\\right]_i = g(_i)$ .",
"If we consider $N \\le WH$ image inputs with a single pixel switched on, we can obtain the function values $\\in ^N$ through the linear transformation $= WQ\\,.$ $W \\in ^{N\\times P}$ has the patch weights as rows, and the matrix $Q \\in ^{P\\times wh}$ contains a 1 at $Q_{ni}$ when the $n$ th image contains the patch $i$ , and zero elsewhere.",
"The matrix $WQ$ has size $N\\times wh$ .",
"This implies that for $N = wh + 1$ , one of the function values in $$ will be fully determined by the responses of the previous images.",
"As a consequence, the kernel matrix for these inputs has to have some zero eigenvalues because the matrix $\\mathbb {E}\\left[ \\mathbf {ff}\\right] &= \\mathbf {K}_{\\mathbf {ff}}= \\mathbb {E}\\left[WG WG\\right] = WQK_GQW$ has rank at most $wh$ , which shows that all functions with evaluations $$ with a component in the null space of $\\mathbf {K}_{\\mathbf {ff}}$ have no density under the prior.",
"The construction of a singular kernel matrix $\\mathbf {K}_{\\mathbf {ff}}$ also implies that the kernel is not strictly positive definite, and therefore not universal [17].",
"Claim Convolutional kernels are nonparametric, in that they can not be represented as a finite basis function model.",
"If the kernel only had a finite number of non-zero eigenvalues and the model could be expressed as a finite basis function model, all functions would admit the representation: $f() = ()\\,.$ where $: ^D \\rightarrow ^I$ .",
"The corresponding kernel matrix would have at most $\\operatorname{rank}I$ .",
"We choose an $N$ distinct images by with a distinct patch in the top left corner of the image of size $w\\times h$ , all other pixels being zero.",
"Because patches overlap, we get $wh$ distinct patch responses which influence $f()$ as well as the influence from all the zero patches: $f() = \\sum _i g(^{[i]}) w_i + w_0g({\\bf 0})$ We collect the patch responses in $G\\in ^{N\\times wh}$ , with the weights $\\in ^{wh}$ , with the image evaluations becoming $= G$ .",
"We obtain the covariance of $$ : $\\left[\\mathbf {K}_{\\mathbf {ff}}\\right]_{nn^{\\prime }} = {g}{\\sum _{i=1}^{wh}\\sum _{j=1}^{wh} g\\left(_n^{[i]}\\right) g\\left(_{n^{\\prime }}^{[j]}\\right) w_iw_j} = \\sum _{i=1}^{wh}\\sum _{j=1}^{wh} k_g\\left(_n^{[i]}, _{n^{\\prime }}^{[j]}\\right) w_iw_j \\,.$ This covariance matrix can be obtained by reducing down the $Nwh\\times Nwh$ covariance matrix between all patches.",
"If we choose a universal kernel for $k_g(\\cdot , \\cdot )$ , this matrix will always be positive definite.",
"The reduced matrix is also positive definite since: $\\mathbf {K}_{\\mathbf {ff}}= \\sum _{nn^{\\prime }} a_na_{n^{\\prime }} \\sum _{i=1}^{wh}\\sum _{j=1}^{wh} \\left[\\mathbf {K}_{\\mathbf {gg}}\\right]_{nin^{\\prime }j} w_iw_j = \\sum _{nin^{\\prime }j} \\left[\\mathbf {K}_{\\mathbf {gg}}\\right]_{nin^{\\prime }j} a_nw_i a_{n^{\\prime }}w_j > 0$ This contradicts the model being parametric, which would allow the rank of $\\mathbf {K}_{\\mathbf {ff}}$ to be at most $I$ ."
],
[
"Remarks",
"The existence of non-degenerate kernels which are not universal may not come as a surprise to theoreticians, particularly due to the effort required for proving universality.",
"For example, [16] place strong requirements on the form of the implicit features of the kernel, which are likely not satisfied by convolutional kernels.",
"Despite the prescience of theory, convolutional kernels provide an interesting and practically useful example of such kernels."
],
[
"Variational bound for separate representation of latent GPs",
"In the main text (sections 3.3 & 3.4) we saw two examples of models with additive structure that required separate representation of their inducing outputs.",
"The weighted convolution + RBF experiment required this due to the inducing inputs lying in separate spaces, while the multi-channel convolutional kernel required this due to separate inter-domain inducing outputs being required to find the distribution over the GP output.",
"In both cases, we can construct the GP output of interest from components in the same space as the inducing variables: $f_{sum}() &= f_{rbf}() + \\sum _p w_p g(^{[p]}) && f_{multi}() = \\sum _{p=1}^P\\sum _{c=1}^C w_{pc} g_c\\left(^{[pc]}\\right) \\\\f_{rbf}(\\cdot ) &\\sim \\left(0, k_{rbf}(\\cdot , \\cdot )\\right) && g_c(\\cdot ) \\sim \\left(0, k_g(\\cdot , \\cdot )\\right) \\\\g(\\cdot ) &\\sim \\left(0, k_g(\\cdot , \\cdot )\\right) &&$ Here we show in detail how inference is done in these models, and how no expensive operations are performed on matrices larger than $M\\times M$ ."
],
[
"Defining the inducing variables",
"We first choose our inducing variables.",
"For the summed GP, we choose $M$ evaluations of $f_{rbf}(\\cdot )$ and $g(\\cdot )$ each (giving $2M$ inducing variables), while for the multi-channel GP we choose $M$ evaluations of all $C$ colour channel outputs $g_c(\\cdot )$ (giving $MC$ inducing variables).",
"We can construct the sparse approximate posterior by conditioning the prior on these variables.",
"The form of the posterior is exactly the same as usual: $f(\\cdot )|\\sim \\left(\\mathbf {k}_{}(\\cdot )\\mathbf {K}_{\\mathbf {uu}}, k(\\cdot , \\cdot ) - \\mathbf {k}_{}(\\cdot )\\mathbf {K}_{\\mathbf {uu}}\\mathbf {k}_{}(\\cdot )\\right) \\,.$ As with usual inter-domain approximations, the task is to find the correct covariances, again by finding the covariances from equations (10) and (11)."
],
[
"Summed kernels",
"For $f_{sum}(\\cdot )$ , the cross-covariance $k_{fu}(, )$ will be the regular kernel evaluation for inducing points on $f_{rbf}(\\cdot )$ , and the appropriate cross-covariance for the inducing patches.",
"We order the inducing point covariances above the inducing patch covariances in the matrix $\\mathbf {K}_{\\mathbf {fu}}$ .",
"Additionally, $\\mathbf {K}_{\\mathbf {uu}}$ will be a block-diagonal $2M\\times 2M$ matrix, with the inducing point $M\\times M$ matrix for $f_{rbf}(\\cdot )$ in the top left, and the inducing patch covariances in the bottom right.",
"No cross terms between $f_{rbf}(\\cdot )$ and $g(\\cdot )$ appear, as they are independent in the prior.",
"$k(, _{img}) &= {f_{rbf}, g}{\\left(f_{rbf}() + \\sum _p w_p g(^{[p]})\\right)f_{rbf}(_{img})} = k_{rbf}(, _{img}) \\\\k(, _{patch}) &= {f_{rbf}, g}{\\left(f_{rbf}() + \\sum _p w_p g(^{[p]})\\right)g(_{patch})} = k_{g}(, _{patch}) \\\\k(_{img}, _{patch}) &= {f_{rbf}, g}{f_{rbf}(_{img})g(_{patch})} = 0$"
],
[
"Multi-channel kernels",
"For $f_{multi}(\\cdot )$ the situation is similar, with the difference that we only have $M$ inducing inputs, but $MC$ inducing outputs.",
"If we order the inducing variables by colour, we get an $N\\times MC$ $\\mathbf {K}_{\\mathbf {fu}}$ matrix (as in equation (21)), and a block-diagonal $\\mathbf {K}_{\\mathbf {uu}}$ , as: ${\\lbrace g_c\\rbrace _{c=1}^C}{g_{c}()g_{c^{\\prime }}(^{\\prime })} = k_g(, ^{\\prime }) \\delta _{cc^{\\prime }} \\,.$ This process is slightly different compared to usual inducing variable approximations, and the even the case for summed kernels, as the number of inducing variables is larger than the number of inducing inputs.",
"As a curiosity, and not necessarily a practical method of implementation, we would like to point out that we could view this process as having multi-output inducing variables.",
"The function $g: ^{wh}\\rightarrow ^C$ could collect all $g_c(\\cdot )$ s, as one $^C$ variable."
],
[
"Inference with block-diagonal $\\mathbf {K}_{\\mathbf {uu}}$ matrices",
"In the previous section, we saw how to find the conditional process.",
"Here we show that the marginal likelihood bound $\\textsc {ELBO} = \\sum _i {q(f(_i)}{\\log p(y_i\\,|\\,f(\\mathbf {x}_i))} - \\textsc {KL}[q(\\mathbf {u})||p(\\mathbf {u})]$ can be computed without operations on matrices larger than $M\\times M$ , despite using more than $M$ inducing variables, regardless of the mean-field assumptions between inducing variables."
],
[
"Approximate posterior marginals",
"The bound requires computation of the marginals of the approximate posterior $q(f(_i))$ .",
"This requires marginalising the conditional approximate posterior over $$ (the same procedure as in [18], [4]).",
"This is where equations (6-8) come from.",
"We simply substitute in the $\\mathbf {K}_{\\mathbf {fu}}$ and $\\mathbf {K}_{\\mathbf {uu}}$ matrices from the previous section, and simplify using the block-diagonal structure in $\\mathbf {K}_{\\mathbf {uu}}$ .",
"We refer to each group of inducing variables ($C$ in total) that are correlated in a block of $\\mathbf {K}_{\\mathbf {uu}}$ as $_c$ , and the corresponding covariance matrices $\\mathbf {K}_{_c_{c^{\\prime }}}$ and $_{_c}()$ .",
"We similarly split the variational parameters $$ and $\\mathbf {S}$ into blocks of the same size $_c$ and $\\mathbf {S}_{cc^{\\prime }}$ .",
"$\\mu _i &= _{}()\\mathbf {K}_{\\mathbf {uu}}= \\sum _c _{_c}()\\mathbf {K}_{_c_{c}}_c \\\\\\sigma ^2_i &= k(\\mathbf {x}_i, \\mathbf {x}_i)+ _{}()\\mathbf {K}_{\\mathbf {uu}}^{-1}(\\mathbf {S}-\\mathbf {K}_{\\mathbf {uu}})\\mathbf {K}_{\\mathbf {uu}}^{-1}_() \\nonumber \\\\&= k(_i,_i) + \\sum _{cc^{\\prime }}_{_c}()\\mathbf {K}_{_c_{c^{\\prime }}}\\mathbf {S}_{cc^{\\prime }}\\mathbf {K}_{_c_{c^{\\prime }}}_{_c}() + \\sum _{c}_{_c}()\\mathbf {K}_{_c_{c}}_{_c}()$ In both cases outlined above, $\\mathbf {K}_{_c_{c^{\\prime }}}$ is $M\\times M$ .",
"If a mean-field approximation is chosen $\\mathbf {S}_{cc^{\\prime }} = 0$ when $c \\ne c^{\\prime }$ .",
"This does not impact the number or size of any inverses, only requiring less parameters and avoiding a summation over $c^{\\prime }$ ."
],
[
"KL divergence",
"The second term in the bound requires the KL divergence between the prior and posterior distribution over the inducing variables, which we can again simplify using knowledge of the block-diagonal structure.",
"$&{q()}{p()} = \\frac{1}{2}\\left(\\left(\\mathbf {K}_{\\mathbf {uu}}\\mathbf {S}\\right) + \\mathbf {K}_{\\mathbf {uu}}- MC + \\log \\frac{{\\mathbf {K}_{\\mathbf {uu}}}}{{\\mathbf {S}}}\\right) \\\\& \\qquad = \\frac{1}{2}\\left(\\sum _c\\left(\\mathbf {K}_{_c_{c}}\\mathbf {S}_{cc}\\right) + \\sum _c_c\\mathbf {K}_{_c_{c}}_c - MC + \\sum _c \\log {\\mathbf {K}_{_c_{c}}} - \\log {\\mathbf {S}}\\right)$ Now, the determinant of $\\mathbf {S}$ , which may be of size $MC\\times MC$ remains.",
"Luckily, we are free to choose the parameterisation of this matrix.",
"We parameterise this matrix as $\\mathbf {S}= LL$ , which makes $\\log {\\mathbf {S}} = 2\\sum \\log L$ ."
],
[
"Summary",
"Here we showed that when the prior covariance of the inducing outputs is block-diagonal, the inference requires only requires expensive matrix operations on each of the blocks separately, regardless of the posterior correlations taken into account.",
"This allows efficient inference for the summed and multi-channel convolutional kernels considered here."
],
[
"Inter-domain inducing variables for general invariances",
"We finally briefly show that the inter-domain trick used for convolutional kernels can also be applied to kernels that give rise to Gaussian processes with arbitrary invariances.",
"Invariant kernels have been discussed before, notably by [19] and [20], [21], [22].",
"[23], [24] also provides an accessible discussion.",
"Here, we review the connection between kernels resulting in invariant functions and a summation structure which allows our inter-domain trick to be applied."
],
[
"Specifying invariances in kernels",
"An invariance can be formalised by placing equality constraints on $f(\\cdot )$ under transformations of the input.",
"Consider a collection of transformations from the input space to itself $g_i:\\mathcal {X} \\rightarrow \\mathcal {X}$ .",
"Making $f(\\cdot )$ invariant to these transformations specifies that $f() = f(g_i()) && \\forall \\in \\mathcal {X} && \\forall i \\,.$ [19] and [20] discuss that this requirement is equivalent to invariance under every composition of transformations as well.",
"For example, if $g_1(\\cdot )$ and $g_2(\\cdot )$ are translations upwards and to the right respectively, we must also have invariance to a translation up and to the right $f() = f(g_1(g_2()))$ .",
"The set of compositions of all transformations forms a group $G$ .",
"[20] show that in order for samples $f(\\cdot )$ to be invariant to all compositions of transformations, the kernel must be argumentwise invariant: $k(, ^{\\prime }) = k(g(), g^{\\prime }(^{\\prime })) && \\forall ,^{\\prime }\\in \\mathcal {X} && g,g^{\\prime }\\in G \\,.$ The elements of the group $g\\in G$ are all compositions of the transformations $g_i$ , defined above."
],
[
"Constructing invariant kernels",
"The requirement stated above does not directly help with constructing invariant models.",
"Three main methods have been proposed, which are neatly discussed by [23].",
"For our purposes, we are mainly interested in the “summation over orbit” method, as this gives a structure almost identical to the convolutional kernel.",
"[19] and [20] show that an argumentwise invariant kernel can be constructed by summing some base kernel over the orbits of $$ and $^{\\prime }$ .",
"The orbit of a point $$ with respect to a group $G$ is defined as the set of all points obtained from applying each element of $G$ to $$ : $\\mathcal {O}_G() = \\left\\lbrace g()\\mid g\\in G\\right\\rbrace $ .",
"The resulting kernel becomes: $k_{invariant}(, ^{\\prime }) = \\sum _{\\tilde{}\\in \\mathcal {O}_G()} \\sum _{\\tilde{}^{\\prime }\\in \\mathcal {O}_G(^{\\prime })} k_{base}(\\tilde{}, \\tilde{}^{\\prime }) \\,.$ The relation between invariances and the addition structure is further investigated by [25].",
"Kernels constructed in this way have the same computational issues as convolutional kernels: evaluating the kernel for a single pair of points requires $P^2$ base kernel evaluations, where $P$ is the size of the orbit.",
"For example, we could make a fully translation invariant kernel by considering translations by 1 pixel upwards, downwards and to the left and right, while clipping and zero-padding edges.",
"For images of size $W\\times H$ the orbit would consist of all $W-1\\times H-1$ translated images.",
"For MNIST this would give $P^2 = (27\\times 27)^2 = 5.3\\cdot 10^5$ , which is again impractical."
],
[
"Inter-domain inducing variables for invariant kernels",
"The invariant kernel above can also be obtained by considering a model that sums a base function $f_{base}() \\sim \\left(0, k_{base}(\\cdot , \\cdot )\\right)$ over the orbit of $$ : $f() = \\sum _{\\tilde{}\\in \\mathcal {O}_G()} f_{base}(\\tilde{}) \\,.$ In this construction, the base function $f_{base}(\\cdot )$ takes the place of the patch response function $g(\\cdot )$ from the convolutional kernel, allowing us to use the same inter-domain trick.",
"Instead of using normal inducing inputs, we place the inducing inputs in $f_{base}(\\cdot )$ instead.",
"We then obtain the covariances: $k(, ) &= \\sum _{\\tilde{}\\in \\mathcal {O}_G()} k_{base}(\\tilde{}, ) \\,, \\\\k(, ^{\\prime }) &= k_{base}(, ^{\\prime })\\,.$ Just like with the convolutional kernel, this reduces the cost of evaluating the required kernels significantly."
],
[
"Summary",
"The structure of kernels resulting in GPs that are invariant to specified transformations is almost identical to that of convolutional kernels, allowing the same inter-domain trick to be used to speed up inference.",
"We present the derivation here, but leave empirical demonstration and evaluation to future work.",
"0pt"
]
] | 1709.01894 |
[
[
"Throughput Optimal Decentralized Scheduling of Multi-Hop Networks with\n End-to-End Deadline Constraints: II Wireless Networks with Interference"
],
[
"Abstract Consider a multihop wireless network serving multiple flows in which wireless link interference constraints are described by a link interference graph.",
"For such a network, we design routing-scheduling policies that maximize the end-to-end timely throughput of the network.",
"Timely throughput of a flow $f$ is defined as the average rate at which packets of flow $f$ reach their destination node $d_f$ within their deadline.",
"Our policy has several surprising characteristics.",
"Firstly, we show that the optimal routing-scheduling decision for an individual packet that is present at a wireless node $i\\in V$ is solely a function of its location, and \"age\".",
"Thus, a wireless node $i$ does not require the knowledge of the \"global\" network state in order to maximize the timely throughput.",
"We notice that in comparison, under the backpressure routing policy, a node $i$ requires only the knowledge of its neighbours queue lengths in order to guarantee maximal stability, and hence is decentralized.",
"The key difference arises due to the fact that in our set-up the packets loose their utility once their \"age\" has crossed their deadline, thus making the task of optimizing timely throughput much more challenging than that of ensuring network stability.",
"Of course, due to this key difference, the decision process involved in maximizing the timely throughput is also much more complex than that involved in ensuring network-wide queue stabilization.",
"In view of this, our results are somewhat surprising."
],
[
"Introduction",
"For multi-hop networks serving real-time applications, data packets typically have a deadline and it is important to ensure that a maximum fraction of the packets reach their destination within the deadline.",
"Currently the performance metric of throughput optimality, i..e, the property of ensuring queue stability for maximal set of arrival vectors, is widely popular in designing network control policies.",
"Backpressure policy is known to be throughput optimal under very general conditions [1], [2].",
"Similarly the Q-CSMA scheme, which combines backpressure routing along with the CSMA algorithm to find the maximum weight matching, is known to be throughput optimal for scheduling traffic in wireless networks under link interference constraints [3], [4].",
"However, the goal of a throughput maximizing policy does not amount to ensuring that the packets meet stringent end-to-end deadlines.",
"Thus, for example, the backpressure policy, or its wireless version Q-CSMA, is known to have poor performance with respect to average delays [5], [6], [7].",
"Thus, we consider the problem of scheduling packets for multi-hop wireless interference network in which data packets have stringent deadlines that has to be met.",
"However, designing an optimal policy for maximizing the timely throughput is much more complicated than ensuring the throughout optimality since the timely throughput attained by a policy is “highly sensitive\" to its routing-scheduling decisons.",
"This is the case because the utility earned from a single packet “drops to 0\" in a discontinuous fashion as soon as its age has crossed its deadline, and hence even a minute fluctuation in the network bandwidth or a small deviation from the optimal decision affects the timely thoughput significantly.",
"For a network control policy, it is highly desirable that it is decentralized, meaning that the wireless nodes do not need to know the “global\" state of the system denoted $X(t)$ .",
"The backpressure policy is decentralized, in the sense that a node needs to know the queue lengths of only its neighbouring nodes, i.e, the nodes that are connected to it via an outgoing link.",
"However, as depicted in Fig.",
"REF , it seems unlikely that the timely throughput maximization problem has a decentralized solution.",
"In this paper we derive a new class of policies that not only maximize the timely throughput, but are also highly decentralized in the sense that a wireless ndoe only needs to know the age of the packets present with it in order to make scheduling decisions.",
"This eliminates the need to share any information amongst the nodes.",
"In a companion paper [8], we developed a theoretical framework and proposed decentralized policies that maximize the timely-throughput in a multi-hop stochastic network.",
"This paper extends the ideas therein to the set-up of wireless networks in which the links suffer from wireless interference.",
"Figure: Making optimal scheduling decisions for meeting deadline constraints is a challenging problem that requires the knowledge of the network state.",
"Consider the decision process involved in routing packets at a wireless node ii.",
"The link pairs (1,i),(i,j)(1,i),(i,j) and (2,i),(i,k)(2,i),(i,k) interefere with each other.",
"Thus, either the pair (1,i),(i,k)(1,i),(i,k) or (2,i),(i,j)(2,i),(i,j) can be activated simultaneously.",
"Since Flow 2's traffic faces downstream congestion, it might be “optimal\" to exclusively focus on scheduling Flow 1's packets.",
"This is true because a packet sent on link (i,k)(i,k) may have to wait for long period due to traffic congestion, and hence will not be able to make it to its destination node within deadline.",
"Thus the links (1,i)(1,i) and (i,j)(i,j) should be given priority.",
"However, since (1,i)(1,i) and (i,j)(i,j) interfere, one further has to make a choice between activating one link amongst these two.",
"It may be the case that the packets that can be scheduled on (i,j)(i,j) are “nearing their deadline\", and hence link (i,j)(i,j) should be prioritized over (1,i)(1,i).",
"Alternatively, it may be the case that these packets have little chances of making it to the destination, because their “deadline has almost crossed\", and (i,j)(i,j) has a low channel reliability.",
"In this case, link (1,i)(1,i) must be activated.",
"We thus notice that the decisions have to be made on the basis of the state of the network.",
"Thus, a knowledge of the complete network state is required at each time tt in order to maximize the timely throughput.",
"The presence of wireless interference makes the problem further difficult since a choice of an independent set (whose number is exponential in |E||E|) of links has to be made."
],
[
"Contributions and Past Works",
"We consider the problem of designing efficient scheduling policy for multihop wireless networks in which the data packets have a deadline associated with their deliveries.",
"As described in Fig.",
"REF , the timely throughput is “highly sensitive\" to the routing-scheduling decisons because the utility earned from a single packet drops to 0 in a discontinuous fashion as soon as its age crosses its deadline.",
"Thus, we cannot use the fluid model, which is commonly used in combination with Lyapunov techniques in order to establish throughput optimality.",
"We have to resort to directly solving the stochastic network model rather than its fluid approximation.",
"The policy provided by us is decentralized, and its complexity scales linearly with the network size, thus addressing the crucial problem of meeting end-to-end packet deadlines, that is typically encountered in multi-hop scheduling in wireless networks.",
"We pose the problem of finding the timely throughput maximizing policy as an Markov Decision Process (MDP) in Section .",
"However, the MDP formulation is intractible since the resulting policy is centralized.",
"We then devise a novel approach to decentralized stochastic control under constraints in Section by replacing the hard versions of the i) link interference, and ii) network-wide bandwidth availability constraints, by their “softer versions\" which involve time averages of the corresponding constraint violations.",
"This relaxation is in inspired by Whittle's relaxation for the Restless Multiarmed Bandit problem.",
"This relaxation of the constraints to their soft versions leads us to a constrained MDP (CMDP).",
"We first deal exclusively with the optimal scheduling problem under link-level average constraints, and hence ignore the wireless interference completely.",
"Since the primal CMDP is intractible we then consider the dual version of the CMDP, and show that it is much simpler to solve.",
"This reduction in complexity results from the realization that the Lagrangian with the multiplier set equal to $\\lambda =\\lbrace \\lambda _{\\ell }\\rbrace $ can be viewed as the sum of the rewards earned by the individual data packets, where the reward of a single packet is equal to its timely throughput minus the price it pays for using network bandwidth.",
"Since the optimization of reward of an individual packet can be performed independently of other packets present in the network, this means that the evaluation of dual function $D(\\lambda )$ , and consequently its optimization ($\\max _{\\lambda \\ge 0}D(\\lambda )$ ) can be performed in a decentralized manner.",
"We then use the strong duality property of linear programs to deduce a highly decentralized policy denoted $\\pi ^\\star $ for the primal CMDP that can be computed in a distributed way.",
"We then come back to original problem of scheduling under wireless interference constraints.",
"We introduce a new variant of the commonly used CSMA protocol in order to allocate the total bandwidth amongst various independent sets in a decentralized way.",
"Then, we consider a scaling of the network, under which the total available bandwidth, and the traffic arrival rates are scaled by a parameter $N$ .",
"The total bandwidth is then divided into $N$ “orthogonal\" channels, and CSMA protocol is used on each of them independently.",
"The policy $\\pi ^\\star $ , that was optimal under link-level average bandwidth constraints is then modified to yield a policy $\\tilde{\\pi }$ for the network with interference.",
"We then use show that as $N\\rightarrow \\infty $ , the timely throughput of $\\tilde{\\pi }$ approaches within $O(\\sqrt{N})$ of the optimal policy, and hence $\\tilde{\\pi }$ is asymptotically optimal.",
"The proof relies on the structure of the policy $\\tilde{\\pi }$ , combined with a “bandwidth smoothing\" that is achieved by using independent CSMA counters on each of the orthogonal subchannels.",
"We highlight some key differences between the manner in which we utilize the CSMA protocol, from the commonly used CSMA.",
"CSMA has been commonly used to provide decentralized channel access.",
"The CSMA “aggression parameter\" can be modulated, as in [9] based on the network queue lengths in order to ensure the throughput optimality property.",
"Since the focus of these works has been on throughput optimality, by letting the aggression rate $r_\\ell $ of the CSMA counters continually adapt to the mismatch between the traffic arrival intensity at link $\\ell $ and the time-average bandwidth provided to the link $\\ell $ (which is an increasing function of $r_\\ell $ ), the Q-CSMA scheme achieves its throughput optimality.",
"However, since in our case, the contribution of a single packet towards the timely throughput heavily depends on the actual value of the channel capacities on each link $\\ell $ belonging to its source-destination path, we need to ensure that the bandwidth fluctuations of each link $\\ell $ in the network are “small\" over the entire duration that a packet remains in the network.",
"Thus, we divide the total available bandwidth into $N$ sub-channels, each of unit bandwidth, and assume that each link $\\ell $ utilizes independent CSMA counters on each of these $N$ orthogonal sub-channels in order to attain channel access.",
"Due to “central limit theorem type\" bandwiwdth smoothing, and due to the property of $\\tilde{\\pi }$ , we are able to ensure that the network timely throughput is not affected much by having CSMA based channel access mechanism, as opposed to a centralized controller.",
"This optimality result can be viewed as a large deviations control of timely throughput of wireless networks.",
"The CSMA aggression parameter $r$ is adjusted based on the feedback received on the link prices, and we utilize a gradient decent based scheme in order to converge to a locally optimal solution.",
"Such a deficiency arises because the optimization problem is non-convex.",
"The problem of scheduling traffic in order to satisfy timely throughput constraints of multiple clients being served by a single-hop wireless network is fairly well-understood by now [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"For this single-hop network, simple to implement greedy policies are shown to be optimal.",
"However, the problem of extending the approach to a multi-hop network has been open for quite some time, though several heuristics have been proposed [20], or a fraction of the maximum achievable timely-throughput has been attained [21]."
],
[
"System Description",
"Network Description : The wireless network is described by a graph $G=(V,\\cal {E})$ , where $V$ is the set of wireless nodes, and $\\cal {E}$ is the set of directed edges of the form $(i,j),~i,j\\in V$ .",
"Associated with each edge $\\ell =(i,j)$ is a rate $C_\\ell $ denoting that node $i$ can attempt transmission of $C_\\ell $ packets to node $j$ in a single time slot.",
"A packet transmission can fail due to to either the unreliability of the wireless channel, or the interference caused by other concurrent wireless transmissions.",
"Assuming that no interfering links are transmitting, a transmission on a link $\\ell $ in a time slot $t$ succeeds with a probability $p_\\ell $ .",
"The quantity $p_\\ell $ is called the channel reliabilty of link $\\ell $ .",
"Throughout we assume that the random outcomes of packet transmissions are independent across time and links.",
"Interference Model : We model wireless interference constraints via an “edge interference graph\".",
"It is an undirected graph in which each vertex corresponds to a wireless link from the set $\\mathcal {E}$ .",
"There is an undirected edge between two nodes $\\ell _1,\\ell _2$ if and only if the links $\\ell _1$ and $\\ell _2$ interfere, i.e., if $\\ell _1,\\ell _2$ are activated simultaneously, then the packet transmissions occurring on both the links fail.",
"Let $\\cal {I}$ be the set comprising of the maximal independent sets of the link interference graph.",
"We denote the maximal independent sets by $IS_1,IS_2,\\ldots $ .",
"Each maximal independent set $IS \\in \\cal {I}$ consists of links $\\ell \\in \\mathcal {E}$ that do not interfere with each other, because they are not connected by an edge in the edge interference graph.",
"Moreover any link $\\ell $ that does not belong to $IS$ definitely interferes with at least one link in $IS$ because $IS$ is maximally independent.",
"Therefore, at any time $t$ , a scheduling policy $\\pi $ is restricted to choosing an $IS\\in \\mathcal {I}$ and activating the corresponding links.",
"Multiple Flows: The network is shared by $F$ flows, where each flow $f$ has a source node $s_f$ and a destination node $d_f$ .",
"We suppose that time is slotted, and the network evolves at tim slots $t=1,2,\\ldots $ .",
"The time-duration of a single time-slot is equal to the time taken to attempt transmission of $C_\\ell $ packets on link $\\ell \\in \\mathcal {E}$ .",
"The packets for flow $f$ arrive at their source $s_f$ in an i.i.d.",
"fashion across time.",
"We assume that the packet arrival process is uniformly bounded across flows and time.",
"Since a source-destination pair $(s_f,d_f)$ need not be a link, packets will typically need to traverse multihop paths before reaching their destination."
],
[
"Multi-hop Timely Throughput ",
"We define the timely throughput metric for a multihop wireless network described in the previous section.",
"Let $\\tau \\ge 1$ be an integer which represents the deadline for packet deliver, i.e., it is the maximum allowable end-to-end delay that a packet may incur.",
"Multihop Timely Throughput: We define the timely throughput $\\bar{D}_f $ of a flow $f$ as the average number of packets per unit time of flow $f$ that reach the destination node $d_f$ within $\\tau $ time-slots from the time they are generated at the source node $s_f$Throughout the paper, we assume that the averages corresponding to stochastic processes of interest converge almost surely.",
"This is not restrictive since we are optimizing over finite state Markov processes, which always admit a stationary policy that is optimal.",
"Since a time-homogenous finite-state Markov chain is necessarily positive recurrent, they admit almost sure limits.",
"Thus, we can replace $\\limsup $ by $\\lim $ etc., $\\bar{D}_f := \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\mathbb {E}\\left\\lbrace \\sum _{t=1}^{T}D_f(t)\\right\\rbrace ,$ where $D_f(t)$ is the number of flow $f$ packets delivered at time $t$ to the destination node $d_f$ within $\\tau $ time-slots from the time they were generated at $s_f$ .",
"The expectation is under the probability measure induced by the scheduling policy $\\pi $ under use, the packet arrival processes and the random states of wireless links.",
"We will denote the average value of $D_f(t)$ by $\\bar{D}_f$ , so that $\\bar{D}_f$ is the timely throughput of flow $f$ .",
"Similarly, time-averages corresponding to a stochastic process $X(t)$ will be denoted by $\\bar{X}$ .",
"Network State: The state of an individual packet at time $t$ is described by, the node $i\\in V$ at which it is present, and, the age (time elapsed since generation) of the packet.We note that the age of a packet can be easily deduced by time-stamping it at the time it was generated at its source node.",
"The state of the network at time $t$ , denoted $X(t)$ , is described by specifying the state of each packet that is present in the network, for each flow $f$ .",
"We note that for conventional networks that are designed to be throughput or delay optimal do not need to keep track of the ages of packets.",
"However, in our case the age of a packet has to be taken into account because a packet delivered after its age has crossed $\\tau $ does not contribute to timely throughput.",
"Policy: A policy $\\pi $ maps $X(t)$ , the system state at time $t$ , to an action $U(t)$ , which is uniquely described by the following i) the independent set $IS$ that has to be activated at time $t$ , ii) the packets that are to be transmitted on each link $\\ell \\in IS$ .",
"Thus, a policy $\\pi $ has to make both routing and scheduling decisions simultaneously.",
"The MDP for Timely Throughput Maximization: The problem of maximizing a weighted sum of timely throughputs can be posed as the following MDP, $\\max _\\pi \\sum _f \\beta _f \\bar{D}_f,$ where $\\pi $ is a policy, and the weights $\\beta _f\\ge 0$ allow the network operator to prioritize the various packets on the basis of their relative importance of their timely-throughputs.",
"Complexity of Solving the MDP (REF ): The size of the state-space corresponding to MDP (REF ) is $\\left(|V|\\tau \\right)^B$ , where $B$ is a bound on the number of packets that can be present in the network at any given time.",
"Thus it is computationally infeasible to solve (REF ).",
"Secondly, the resulting optimal policy will prescribe the optimal action $U(t)$ as a function of the network state $X(t)$ , and hence calls for a centralized controller.",
"This again is quite impractical."
],
[
"Orthogonal Channels",
"Our results will be asymptotic in nature, i.e., the resulting policy will be shown to be nearly optimal as the network capacity is scaled to $\\infty $ .",
"Therefore, we will slightly enhance the system model.",
"We will assume that the total bandwidth available to the wireless network is equal to $K$ units, divided into $K$ orthogonal sub-channels, so that each orthogonal channel has access to a unit amount of bandwidth.",
"At each time $t$ , a scheduling policy $\\pi $ can activate an independent set of links on each of the $K$ orthogonal sub-channels.",
"Denote by $I_m(t)\\ge 0$ the amount of bandwidth provided to independent set $IS_m$ at time $t$ , with $\\sum _{m}I_m(t)=K$ .",
"We suppose that the data rate attainable per unit bandwidth on link $\\ell $ is $C_\\ell $ packets/slot.",
"Hence the number of packet transmissions that can be scheduled on a link $\\ell $ at time $t$ is given by $C_\\ell \\sum _{m:\\ell \\in IS_m}I_m(t)$ .",
"In the below, we let $I^\\pi (t)$ denote the vector with entries $I^\\pi _m(t)$ , while $\\bar{I}^\\pi $ contains their time-averages.",
"Let $\\pi $ be any history dependent policy that decides the following at each time $t$ : i ) $I^\\pi _m(t)$ , the i) number of sub-channels allocated to independent set $IS_m$ , and ii) the packets scheduled on each link $\\ell \\in \\mathcal {E}$ .",
"The timely throughput maximization problem can now be re-stated as, $&\\max _{\\pi } \\sum _{f}\\beta _f \\bar{D}_f, \\\\&\\mbox{ s.t. }",
"\\sum _m I^\\pi _m(t) \\le K, ~~\\forall t=1,2,\\ldots .$ under the Markov Decision Process Model."
],
[
"Relaxing the constraints",
"Problem (REF )-() imposes a “hard\" constraint () on the bandwidth consumption of a feasible policy $\\pi $ in every time-slot $t=1,2, \\ldots $ .",
"We will now consider a simpler version of the bandwidth constraint, by replacing the constraint $\\sum _m I^\\pi _m(t) \\le K, ~~\\forall t=1,2,\\ldots $ by a “soft\" version that requires only that the time-average bandwidth consumption by $\\pi $ is less than $K$ units.",
"We pose this relaxation of the timely throughput maximization problem (REF ) as, $&\\max _{\\pi } \\sum _{f}\\beta _f \\bar{D}_f\\mbox{, such that } \\\\&\\qquad \\sum _m\\bar{I}^\\pi _m \\le K , $ where $\\bar{I}_m$ denotes the time-average of the bandwidth “consumed\" by the independent set $IS_m$ .",
"The relaxed constraint now allows a policy $\\pi $ to utilize more than $K$ units of bandwidth at any time-slot $t$ .",
"We note that that the relaxed constraint () $\\sum _m\\bar{I}^\\pi _m \\le K$ is still a constraint of a “global spatial nature\" This is true because the choice of $I_m(t)$ affects the amount of bandwidth available to each link $\\ell \\in IS_m$ , and hence must necessarily depend on the state of the packets present at each link $\\ell \\in IS_m$ ..",
"Thus, a An optimal $\\pi $ has to achieve a “global spatial coordination\" amongst the various links $\\ell \\in \\mathcal {E}$ , and thus the problem (REF )-() is challenging.",
"Hence we will now further relax the constraints, and this will lead us to a somewhat weaker form of the hard interference constraints under which any concurrent transmissions on two interfering links fail.",
"We observe that under a policy $\\pi $ , the transmission rate obtained by a link $\\ell $ at time $t$ is given by $C^\\pi _\\ell (t) := C_\\ell \\sum _{m:\\ell \\in IS_m}I_m(t),$ while the time-average transmission rate that it receives is given by $\\bar{C}_{\\ell }^\\pi := C_{\\ell } \\left(\\sum _{m:\\ell \\in \\mathcal {E}_m} \\bar{I}_m \\right).$ The relation between $C^{\\pi }_\\ell (t)$ and $I(t)$ can be viewed as interference constraint.",
"Thus, the “Relaxed Problem\" in which the hard interference and the hard bandwidth availability constraints have been relaxed to their average versions is as follows, $&\\max _{\\pi } \\sum _{f}\\beta _f \\bar{D}_f\\mbox{, such that } \\\\&~ \\bar{C}_{\\ell }^\\pi \\le C_{\\ell } \\left(\\sum _{m:\\ell \\in IS_m} \\bar{I}_m \\right),\\\\&~\\sum _{m}\\bar{I}_m \\le K.$ We interpret the constraints ()-() as follows.",
"Once a feasible vector $\\bar{I}=\\lbrace \\bar{I}_m\\rbrace $ satisfying $\\sum _m\\bar{I}_m\\le K$ has been fixed, the average link bandwidths $\\bar{C}^{\\pi }_\\ell $ get fixed according to ().",
"The constraint () does not impose any constraint on instantaneous link bandwidths $C^{\\pi }_\\ell (t)$ .",
"They also do not impose any interference constraints, i.e., the edge interference graph model described in Section is no longer valid.",
"Hence two links $\\ell ,\\hat{\\ell }$ that are connected in edge interference graph, are allowed to carry out concurrent packet transmissions without packet dropsEquivalently we can assume that each link $\\ell $ has access to unlimited bandwidth, and that the channel it uses is orthogonal to the channels used by other links.. Notice that though we have rid ourselves of the hard interference constraints (REF ), we have retained the constraints i) the average bandwidth available to a link $\\ell $ depends on the average bandwidth allocation vector $\\bar{I}$ , and, ii) the total average bandwidth available to the wireless network, i.e., the constraint $ \\sum _m\\bar{I}^\\pi _m \\le K$ .",
"We have only relaxed the constraints (REF ) imposed by the “wireless interference system\" since they are overly restrictive.",
"This relaxation, in a certain sense, is equivalent to “weaking\" the wireless interference constraints, i.e., under the relaxed constraints an admissible scheduling policy $\\pi $ does not need to assign an independent set $IS$ on each of the $K$ sub-channels.",
"However, as will be shown in this paper, solving this relaxed problem does yield a near-optimal solution to the original problem when the network traffic is scaled to $\\infty $ , and hence the efforts to solve the relaxed problem will not go in vain.",
"Connections with Whittle's Relaxation for the MABP The relaxation that we have introduced above is in the sprit of Whittle's relaxation [22] for Restless Multi-Armed bandit problem (MABP) [23] which can be posed as an MDP.",
"In the MABP set-up, a controller has to play one “arm\" at each time-slot $t=1,2,\\ldots $ , and he obtains a reward which is a function of the state of the arm that he currently played.",
"The various arms of the bandit are essentially controlled Markov processes.",
"Whittle's relaxation for the MABP is to replace the hard constraint that only a single arm be played at each time, by a softer constraint that requires that the player plays a single arm only on an average.",
"Since in the relaxed problem, the “decision processes\" for the arms are decoupled, the relaxed problem is much easier to solve, and the complexity reduces drastically as compared to the MABP.",
"The Whittle's index policy then activates the arm that attains the “highest gain in reward\" from activating it.",
"See [23] for details.",
"In our set-up, the packets that are to be routed over the wireless network are the analogues of bandit arms in the MABP.",
"The player is the policy $\\pi $ that has to be designed by the network operator, the constraints are the wireless interference constraints.",
"The wireless interference constraints are obviously much more complex than the constraints imposed in the MABP in that only a single independent set/arm has to be played in a time-slot.",
"However, as we will see, the idea of Whittle's relaxation does turn out to be useful even in this complex setup.",
"In view of the above discussion, our approach to solving the original problem (REF )-() will be as follows.",
"We will first solve the much simpler relaxed problem (REF )-() since it is tractable and admits a neat decentralized solution.",
"Denote the solution to (REF )-() by $\\pi ^\\star $ .",
"Once $\\pi ^\\star $ has been obtained, we will modify it appropriately, and combine it with CSMA protocol.",
"This will yield us a policy $\\tilde{\\pi }$ that is feasible for the original problem (REF ).",
"Then, we will show that $\\tilde{\\pi }$ is asymptotically optimal in the limit the traffic arrival rates, and available bandwidth $K$ are scaled to $\\infty $ .",
"Remark 1 We note that the relaxed version of the link capacity constraints ()-() still induces a weaker version of the interference constraint.",
"In later sections we will show that an optimized version of the CSMA protocol resolves the bandwidth allocation problem in a decentralized fashion.",
"Moreover, since the amount of bandwidth it assigns to each link $\\ell $ is nearly a constant, i.e., its stochastic fluctuations are small, we would be interested in developing a scheduling policy under the assumption that the bandwidth for each link $\\ell $ has been fixed.",
"The reason why we derive the scheduling policy under fixed average link bandwidths is that we will require the solution of this problem in order to optimize the CSMA protocol.",
"We begin by addressing the relaxed problem (REF )-() for the case when the link-level bandwidths have been fixed at $C^{av}$ ."
],
[
"Scheduling under Link-Level Average Bandwidth Constraints",
"In this section we will be concerned with maximizing the total timely throughput under the constraint that the average bandwidth provided to each link $\\ell $ is less than or equal to $C^{av}_{\\ell }$ .",
"Equivalently, the constraints ()-() in the problem (REF )-() will be replaced by the constraints $\\bar{C}_{\\ell }^\\pi \\le C^{av}_{\\ell }, \\forall \\ell \\in \\mathcal {E},$ where $C^{av}_{\\ell }$ is the bound on average bandwidth of link $\\ell $ .",
"We also let $C^{av}=\\lbrace C^{av}_\\ell \\rbrace _{\\ell \\in E}$ .",
"Thus, in this Section we will solve the following CMDP, $f\\left(C^{av}\\right):&=\\max _{\\pi } \\sum _{f}\\beta _f \\bar{D}_f\\mbox{, such that } \\\\&~ \\bar{C}_{\\ell }^\\pi \\le C^{av}_{\\ell },\\forall \\ell \\in E.$ We will obtain a computationally simple and decentralized solution.",
"However, as discussed in Section , a naive approach to solve the above CMDP in its primal form using the linear programming approach is impractical owing to the curse of dimensionality, and the requirement of a centralized controller.",
"In order to develop a decentralized and computationally feasible iterative solution, we consider the dual problem associated with the primal CMDP (REF )-().",
"For a scheduling policy $\\pi $ , the Lagrangian corresponding to (REF )-() is given by, $&\\mathcal {L}(\\pi ,\\mu ) \\\\&= \\sum _{f} \\beta _f \\bar{D}_f - \\sum _\\ell \\mu _\\ell \\left(\\bar{C}^\\pi _\\ell - C^{av}_{\\ell }\\right)\\\\&= \\sum _{f} \\beta _f \\bar{D}_f - \\sum _\\ell \\mu _\\ell \\bar{C}^\\pi _\\ell + \\sum _{\\ell } \\mu _\\ell C^{av}_{\\ell },$ where $\\mu _\\ell \\ge 0$ is the multiplier associated with average link capacity constraint $\\bar{C}^\\pi _\\ell \\le C^{av}_{\\ell }$ , and $\\mu =\\lbrace \\mu _\\ell \\rbrace _{\\ell \\in E}$ is the vector containing these multipliers.",
"We note that in the above, the policy $\\pi $ under consideration is the primal variable.",
"The dual function $D(\\mu )$ is then given by, $D(\\mu )& = \\max _{\\pi } \\mathcal {L}(\\pi ,\\mu ) \\\\&= \\left(\\max _{\\pi } \\sum _{f} \\beta _f \\bar{D}_f - \\sum _\\ell \\mu _\\ell \\bar{C}^\\pi _\\ell \\right)+\\sum _{\\ell }\\mu _\\ell C^{av}_{\\ell },$ where we note that only the term within the braces $\\left(\\cdot \\right)$ depends on the policy $\\pi $ ."
],
[
"Decentralized computation of Dual Function $D(\\mu )$",
"In order to evaluate the dual function at value $\\mu $ , the following problem needs to be solved, $\\max _{\\pi } \\sum _{f} \\beta _f \\bar{D}_f - \\sum _\\ell \\mu _\\ell \\bar{C}^\\pi _\\ell .$ Next, we make the following important observation.",
"The cost $\\sum _{\\ell } \\mu _\\ell \\bar{C}^\\pi _\\ell $ as well as the reward $\\beta _f \\bar{D}_f$ is the sum of the individual costs incurred by packets, i.e., $\\sum _{\\ell }\\mu _\\ell \\bar{C}^\\pi _\\ell &=\\sum _{\\ell } \\mu _\\ell \\sum _f \\bar{C}^\\pi _{\\ell ,f}\\\\&=\\sum _f \\left( \\sum _{\\ell } \\mu _{\\ell }\\bar{C}^\\pi _{\\ell ,f} \\right)\\\\& = \\sum _f \\sum _{\\sigma _f} \\left( \\sum _{\\ell } \\mu _{\\ell }\\bar{C}^\\pi _{\\ell ,\\sigma _f} \\right),$ where $\\bar{C}^\\pi _{\\ell ,f}$ is the average bandwidth consumption by packets belonging to flow $f$ on link $\\ell $ , the index $\\sigma _f$ labels packets of flow $f$ , and $\\bar{C}^\\pi _{\\ell ,\\sigma _f}$ denotes the average amount of link $\\ell $ bandwidth consumed by packet $\\sigma _f$ .",
"A similar decomposition holds for the timely-throughput reward too.",
"This decomposition property yields us the following algorithm to compute the dual function $D(\\mu )$ ."
],
[
"Highly Decentralized Packet Level Policy",
"In this section, we will fix the value of dual variable at $\\mu $ , and focus exclusively on maximizing the following “cumulative reward\" earned by a policy $\\pi $ in a decentralized way, $\\sum _f \\sum _{\\sigma _f} \\left( \\beta _f \\bar{D}_{\\sigma _f} - \\sum _{\\ell } \\mu _{\\ell }\\bar{C}^\\pi _{\\ell ,\\sigma _f} \\right),$ where $\\bar{D}_{\\sigma _f}$ denotes the probability that packet $\\sigma _f$ is delivered to its destination node within its deadline.",
"Maximization of the cumulative reward will yield us the value of dual function $D(\\mu )$ .",
"Maximizing (REF ) using Dynamic Programming : The state $X(t)$ of the system at time $t$ is mentioned by describing the flow $f$ and age for each packet present at each node $i\\in V$ .",
"We can then solve for the optimal $\\pi $ that maximizes the cumulative reward (REF ) using Dynamic Programming i.e., $R + V(x) = \\max _{u} \\left( R_{inst}(x,u) + \\mathbb {E}_{u}\\left\\lbrace V(y) \\right\\rbrace \\right),$ where $R$ is the optimal average reward, $V(x)$ is the transient reward function associated with the system beginning in state $x$ , and $R_{inst}(x,u)$ is the one-step reward earned when the system state is $x$ , and control $u$ is applied.",
"The instantaneous reward $R_{inst}$ includes the reward earned due to timely delivery of packets, and the cost paid due to using the link bandwidth, i.e., $\\mu _{\\ell }$ amount of price is incurred upon using unit amount of link $\\ell $ bandwidth.",
"Solving the Dynamic programming equation (REF ), and implementing the resulting policy leads to several technical difficulties: The number of variables involved in solving (REF ) is equal to the size of the state space.",
"If we assume that the total number of packets in the network is bounded by $B$ , the state space size is exponential in $B$ (one has to mention the location and age of each packet present in the network).",
"The optimal policy calls for a centralized controller in order to be implemented, i.e., the control input at time $t$ , $U(t)=\\pi ^\\star (X(t))$ is a function of the system state $X(t)$ .",
"Thus, the nodes need to share their information with all the other nodes in every time-slot.",
"Our key result is that the cumulative reward can be maximized by maximizing the cumulative rewards earned by each individual packets.",
"We re-collect the decomposition principle (REF ), which says that the instantaneous reward asociated with the cumulative reward function (REF ) is the sum of rewards of individual packets, i.e., $\\sum _f \\sum _{\\sigma _f} \\left(\\sum _t \\beta _f D_{\\sigma _f}(t) + \\sum _{\\ell } \\mu _\\ell C_{\\ell ,\\sigma _f}(t) \\right),$ where $D_{\\sigma _f}(t)=1$ only if the packet $\\sigma _f$ is delivered at time $t$ to its destination node $d_f$ , and is 0 otherwise, while $C_{\\ell ,\\sigma _f}(t)$ is the amount of bandwidth utilized by the packet $\\sigma _f$ at time $t$ on link $\\ell $ .",
"Since the total cost decomposes into the cost incurred by individual packets (REF ), and since the reward of an individual packet can be optimized independently of other packets, it then follows that the cumulative reward (REF ) can be optimized by implementing a “packet-by-packet optimal policy\".",
"Thus, we introduce the following MDP which is concerned with optimizing the trajectory of a single packet from its source to destination.",
"Single Packet Optimal Transportation Problem : Consider the following dynamic optimization problem.",
"At time $t=0$ , a single packet is generated at its source node $s_f$ .",
"Thereafter its evolution is jointly decided by the scheduling action applied at each of the node it traverses, and the prevailing channel state.",
"Thus, if its transmission is attempted on a link $\\ell $ at any time $t$ , then the transmission succeeds with a probability $p_\\ell $ which is the reliability of link $\\ell $ .",
"Moreover, the link $\\ell $ that is utilized for transmision charges a price of $\\mu _{\\ell }$ from the packet.",
"After a sequence of transmissions occurring at different nodes $i\\in V$ , if the packet manages to reach the destination $d_f$ before time $\\tau $ , then it earns a reward of $\\beta _f$ units.",
"The problem is to design a scheduling policy so as to maximize the net reward earned while transporting a unit packet from source to destination.",
"In order to do so, we realize that the state of the packet at time $t$ is described by the node $i$ at which it is present, and its age, i.e.",
"the time that has elapsed since it was generated at time $t=0$ at the source node.",
"Solving the following DP equations yields the solution to the Single Packet Optimal Transportation Problem, $&V(i,s) = \\max _{\\ell =(i,j)}\\left(\\max _{u} \\left\\lbrace \\mu _{\\ell } u + P(\\ell ,u) V(j,s+1) \\right.\\right.\\\\&\\left.\\left.~~+ (1-P(\\ell ,u))V(i,s+1) \\right\\rbrace \\right),$ where $u\\in \\lbrace 0,1\\rbrace $ represents the amount of bandwidh utilized for transmission, i.e., $u=0$ for not transmitting, and $u=1$ for transmitting.",
"The joint action comprising the decisions $(\\ell ,u)$ ranges over all the choices of a transmission link $\\ell $ , or not transmitting the packet at all.",
"We will denote the optimal policy thus obtained by solving the above DP as $\\pi ^{\\star }_{f}(\\mu )$ , with the subscript $f$ denoting that the solution depends upon the flow $f$ that the individual packet $\\sigma _f$ belongs to.",
"The size of the state space involved in solving the Single Packet Transportation Problem is equal to the number of nodes in the network $|V|$ times the deadline threshold $\\tau $ , i.e., $|V| \\tau $ .",
"Moreover, the optimal decision for a packet $\\sigma _f$ at any time $t$ depends only on its state, i.e., its age and location.",
"Thus, it can be implemented in a distributed fashion, i.e., the node $i$ at which the packet is present simply looks up the optimal action to be taken, and implements it.",
"It does not need to know the state of packets present at other nodes, or even the states of other packets present at the node $i$ .",
"Notice that this was not the case in implmenting the solution to (REF ).",
"Lemma 1 The policy $\\pi ^{\\star }(\\mu )$ that maximizes the Lagrangian $\\mathcal {L}(\\pi ,\\mu )$ , or equivalently satisfies $\\mathcal {L}(\\pi ,\\mu ) = D(\\mu )$ implements the solution of the Single Packet Optimal Transportation Problem for each packet $\\sigma _f$ of each flow $f$ .",
"Thus, we have $\\pi ^{\\star }(\\mu )= \\otimes _{f} \\pi ^{\\star }_{f}(\\mu )$ ."
],
[
"Obtaining the optimal prices $\\mu ^\\star $",
"In the previous section, we derived an algorithm that computes the value of dual function $D(\\mu )$ , and $\\pi ^{\\star }(\\mu )$ , i.e, the policy that maximizes the Lagrangian $\\mathcal {L}(\\cdot ,\\mu )$ .",
"However, in order to solve the dual MDP corresponding to the CMDP (REF )-(), we need to solve the following dual problem, $\\min _{\\mu \\ge 0} D(\\mu ).$ We will use sub-gradient descent method in order to converge to optimal link-prices $\\mu ^\\star $ .",
"In the below, $k$ denotes the iteration index, and $\\bar{C}^{\\pi ^{\\star }(\\mu )}_{\\ell }$ denotes the average bandwidth consumption on link $\\ell $ under the application of policy $\\pi ^{\\star }(\\mu )$ that can be calculated by solving the DP equations (REF ) for each flow $f$ .",
"In order to implement sub-gradient descent algorithm, each link $\\ell $ needs to iterate on its price $\\mu _{\\ell }\\ell (t)$ as follows $\\mu _{\\ell }(t+1) =\\Pi \\left[ \\mu _{\\ell }(t) + \\alpha (t) \\left(\\bar{C}^{\\pi ^{\\star }(\\mu (t))}_{\\ell }-C^{av}_{\\ell }\\right) \\right], t=1,2,\\ldots ,$ we have used the same index $t$ to label the time-slots, and the sub-gradient descent iterations.where $\\Pi \\left[\\cdot \\right]$ projects the iterates onto a suitable compact set.",
"Since the dual problem (REF ) is convex, we have, Figure: A two-layered iterative algorithm that solves the Policy Optimization Problem.",
"Notice that the link-price tuner requires the value of average bandwidth congestion on each link ℓ\\ell that results under the application of policy π(μ)\\pi (\\mu ).Lemma 2 The price iterations (REF ) converge to the price vector $\\mu ^\\star $ that solves the dual problem (REF ).",
"We note that in order to carry out the price iterations we need to compute the quantities $\\bar{C}^{\\pi ^{\\star }(\\mu (t))}_{\\ell }$ .",
"This task is computationally expensive, and moreover, the assumption that there is a central entity that has knowledge of the network characteristics, is an unrealistic one.",
"We provide a distributed scheme in the theorem below.",
"We summarize the results obtained in this section by concluding with the following Theorem.",
"Theorem 1 (Scheduling under Average Link Bandwidth constraints) Consider the problem of optimal scheduling for packets having end-to-end deadline constraints under link-level average constraints $\\bar{C}_{\\ell }^\\pi \\le C^{av}_{\\ell }, \\forall \\ell \\in \\mathcal {E}$ , i.e., the problem (REF )-().",
"The optimal policy is given by $\\pi ^{\\star }=\\otimes _{f}\\pi _f^{\\star }(\\mu ^{\\star })$ , where $\\mu ^\\star $ is the solution to the dual problem (REF ).",
"It implements the policy $\\pi _f^{\\star }(\\mu ^{\\star })$ that is the solution to the single packet transportation problem with link prices set to $\\mu ^\\star $ , for each packet belonging to flow $f$ .",
"Hence in order to make decisions regarding a packet present at a node $i\\in V$ , the node $i$ only needs to know the age of the packet.",
"The vector $\\mu ^\\star $ of optimal prices can be obtained by performing the gradient descent iterations (REF ).In between two successive updates of the price $\\mu (t)$ , the DP iterations (REF ) can be performed with price set to $\\mu (t)$ in order to evaluate the quantity $\\bar{C}^{\\pi ^{\\star }(\\mu (t))}_{\\ell }$ .",
"This involves a link $\\ell $ to obtain the value of value function evaluated at its outgoing links, i.e., if $\\ell =(i,j)$ then all nodes $j:(j,k)\\in \\mathcal {E}$ need to share $V(j,\\cdot )$ .",
"Hence, the price updates and value iterations can be performed in a distributed way.",
"In practice, the iterations need to be performed using the data that is available during the network operation.",
"Thus, in Section REF we briefly discuss a stochastic approximation based scheme which is an “online learning\" algorithm that guarantees convergence to the optimal policy.",
"Remark 2 It must be noted that in this section we have addressed only a sub-problem concerning the relaxed version (REF )-() of the timely throughput maximization problem.",
"Thus, the following must be noted, Since the policy $\\pi ^{\\star }(\\mu ^{\\star })$ is designed to satisfy link-level bandwidth constraints only on an average, its instantaneous bandwidth consumption might exceed $C^{av}_\\ell $ , i.e., $C^{\\pi ^{\\star }(\\mu ^{\\star })}_\\ell (t)>C^{av}_\\ell $ is a possibility.",
"As mentioned in Section , we have not considered the problem of channel access yet.",
"Thus, so far we have explicitly assumed that interfering links have been provided orthogonal channels amounting to unlimited bandwidth via some mechanism.",
"That is to say, at each time $t$ , the set of available orthogonal sub-channels is allocated amongst the various network links $\\ell \\in E$ in such a manner that any two interfering links $\\ell ,\\hat{\\ell }\\in \\mathcal {E}$ are allocated sub-channels that are orthogonal.",
"Furthermore there is no bandwidth constraint on these orthogonal channels.",
"The problem of designing a decentralized channel access mechanism is highly non-trivial, and will be addressed in Section .",
"We have not addressed the constraint that the cumulative bandwidth consumed by the network at any time $t$ should be less than or equal to $K$ units.",
"In Appendix REF we apply the gradient descent algorithm in order to further optimize over the vector of bandwidth allocation $\\bar{I}$ , and hence solve the problem (REF )-().",
"The scheme discussed therein is impractical because it assumes that there is a centralized controller, and the scheme involves tuning the bandwidth allocated to each independent set in the set $\\mathcal {I}$ .",
"The number of independent sets grows exponentially with the number of links $|\\mathcal {E}|$ , the scheme cannot be justifed for practical purrposes.",
"This brings us to the CSMA protocol."
],
[
"CSMA for Decentralized Channel Access",
"We briefly discuss the CSMA protocol that we will utilize in order to obtain decentralized channel access.",
"The CSMA scheme used by us is significntly different from the commonly used CSMA, and the differences will be pointed out at the end of this section."
],
[
"Randomized Channel Access",
"We begin with a brief discussion of the CSMA protocol.",
"We will now slightly augment the discrete time-slot model introduced earlier, in order to accomodate the CSMA scheme to be implemented in conjuntion with a scheduling policy.",
"Thus, we will now assume that a small portion of each time-slot is devoted to making channel access decisions.",
"We will call this dedicated time duration within each time-slot as a “minislot\".",
"Thus, a time-slot is divided into a mini-slot and a data-slot, with the former reserved for making channel access decisions, and the latter for packet transmissions.",
"At the beginning of each minislot, each link $\\ell \\in \\mathcal {E}$ waits for a random amount of time duration that is exponentially distributed with mean value of $1\\slash r_\\ell $ .",
"The quantity $r_\\ell $ is called the aggression parameter of link $\\ell $ .",
"We will denote this random wait-time as counter.",
"During a minislot, each link $\\ell \\in \\mathcal {E}$ continually senses the carrier in order to detect packet transmissions from any of its conflicting links.",
"At the expiry of its counter, if the link $\\ell $ finds that none of its conflicting linksa link $\\hat{\\ell }$ such that $(\\ell ,\\hat{\\ell })$ is an edge in the edge interference graph.",
"is transmitting, then it makes the decision to attempt a packet transmission in the current data-slot.",
"Since the support of an exponential random variable is the entire real line, we will truncate the wait counters to some large enough threshold value so that the probability that a counter value exceeds this threshold is vanishingly small.",
"Thus, it is assumed that the duration of a minislot is much longer than the average value of waiting time, $1/r_{\\ell }$ .",
"Under the above assumptions, the following fact is easily verified.",
"Lemma 3 Under the above described randomized channel access scheme, the probability that a link $\\ell \\in \\mathcal {E}$ gets channel access to transmit a packet in a data-slot $t$ is given by, $p(\\ell ;r) = \\frac{r_\\ell }{\\sum _{\\hat{\\ell }\\in N(\\ell )}r_{\\hat{\\ell }}},$ where the vector $r:=\\left(r_1,r_2,\\ldots ,r_{|\\mathcal {E}|}\\right)$ , and $N(\\ell )$ is the set of links $\\hat{\\ell }$ that interfere with the link $\\ell $ .",
"Equivalently, the bandwidth available to link $\\ell $ under the CSMA-$r$ protocol is equal to $\\frac{r_\\ell }{\\sum _{\\hat{\\ell }\\in N(\\ell )}r_{\\hat{\\ell }}}$ .",
"We will denote the above randomized channel access mechanism as CSMA-$r$ .",
"The CSMA algorithm is decentralized because each link carries out sensing and channel access independently of other links in the network, and hence it does not require a centralized co-ordinator to ensure that average bandwidth constraints are satisfied.",
"We will use $p(r):=\\lbrace p(\\ell ;r)\\rbrace _{\\ell \\in \\mathcal {E}}$ to denote the vector consisting of average link bandwidths under the CSMA-$r$ protocol.",
"Remark 3 We note that the CSMA model considered by us is significantly different from the existing commonly used CSMA scheme as in [24], [25].",
"Under the commonly used CSMA, the set of links active at any time $t$ , is described by a Markov process, with its state-space equal to $\\mathcal {I}$ , i.e., the set of independent sets of the link interference graph.",
"However in our set-up, the set of active links, i.e., $IS(t)$ , is i.i.d.",
"across each time-slot $t=1,2,\\ldots $ .",
"Such a construction is required because we need to guarantee that the temporal bandwidth fluctuations are minimal, which is necessary in order to ensure optimality of scheduling policy with respect to timely-throughput metric.",
"More concretely, we cannot allow for large amount of fluctuations in link-bandwidths.",
"Thus, for example, the contribution of a single packet to the timely throughput is an intricate function of the bandwidth availability across various links $\\ell \\in \\mathcal {E}$ over a time horizon of $\\tau $ time-slots, which is the time that the packet spends in the multi-hop network.",
"This is in contrast with the network queue stability problems, where temporal fluctuations in bandwidth availability do not affect the throughput as long as the average link-bandwidth remains the same [1].",
"The primary reason why such a control on bandwidth fluctuation is required, is because the “current utility\" of a packet depends on its “age\", i.e., the time it has spent in the network."
],
[
"Capacity Scaling and Asymptotically Optimal Policy",
"We will develop a decentralized scheduling policy and show that is asymptotically optimal if the cumulative networkwide-available bandwidth is scaled to $\\infty $ .",
"In order to develop the policy, we will combine the solution of the relaxed problem with the CSMA protocoland then analyze its timely throughput in this limiting regime.",
"We begin by formally defining the network scaling that we employ.",
"Recall the definition of $f(C^{av})$ as in (REF ) $f(C^{av}) := \\max _{\\pi : \\bar{C}^\\pi \\le C^{av}} \\sum _f \\beta _f \\bar{D}_f.$ Now, assume that the packet arrivals for each flow $f$ are random and follow the Bernoulli distribution with parameters $\\left(1,A_f\\right)$ .",
"We now introduce a performance metric that is similar to $f(C^{av})$ .",
"Let us assume that at the beginning of each time-slot $t=1,2,\\ldots $ , each link $\\ell \\in \\mathcal {E}$ is now available with a probability $p(\\ell ;r)$ , which is the activation probability of link $\\ell $ under the CSMA-$r$ protocol (REF ).",
"The stochastic availability of the links is used to model the random activations of links by the CSMA scheme.",
"Now define $\\tilde{f}_{1}(p(r)) := \\max _{\\pi : C^\\pi (t)\\le C^{CSMA_r}_{1}(t)} \\sum _f \\beta _f \\bar{D}_f,$ where $C^{CSMA_r}_{1}(t)=\\lbrace C^{CSMA_r}_{1,\\ell }(t)\\rbrace _{\\ell \\in \\mathcal {E}}$ is the vector of bandwidths allocated at time $t$ under the CSMA-$r$ protocol applied to a unit bandwidth.",
"The subscript 1 in the above stands for the fact that only a single independent set is to be activated at each time-slot $t$ .",
"Let us now consider a sequence of wireless networks.",
"For the $N$ -th network in the sequence, we have that the packet arrivals for each flow $f$ are distributed according to Bernoulli $\\left(N,A_f\\right)$ .",
"Also, the network has access to $N$ units of bandwidth, and hence can now activate $N$ independent sets simultaneously in any time-slot $t$ .",
"The channel access mechanism in the $N$ -th network is as follows.",
"The network uses $N$ independent CSMA counters for channel access on $N$ orthogonal channels.",
"Each link $\\ell $ , at the beginning of each mini-slot, generates $N$ i.i.d.",
"backoff counters which are exponential with mean $1\\slash r_\\ell $ , one for each orthogonal channel.",
"Then, it uses a single counter on the corresponding channel in order to apply the CSMA scheme on it.",
"Hence, the number of orthogonal channels available to link $\\ell $ at each time $t$ is distributed according to Bernoulli $(N,p(\\ell ;r))$ .",
"Define $\\tilde{f}_{N}(p(r)) := \\max _{\\pi : C^\\pi (t)\\le C_{N}^{CSMA_r}(t)} \\sum _f \\beta _f \\bar{D}_f,$ where the sub-script $N$ in $C_N^{CSMA_r}(t):=\\lbrace C_{N,\\ell }^{CSMA_r}(t)\\rbrace $ denotes that the network has $N$ orthogonal channels available to it, and superscript $r$ denotes that independent CSMA counters with aggression parameter $r$ are used on each of them separately.",
"We note that the quantity $\\tilde{f}_N(p(r))$ is less than or equal to $f(Np(r))$ because of the following observation.",
"The set of policies that qualify while evaluating $f(Np(r))$ have no constraint on instantaneous bandwidths during individual time-slots $t=1,2,\\ldots $ .",
"However during the computation of $\\tilde{f}_{N}(p(r))$ , the set of allowable policies can utilize only $C^{CSMA_{r}}_{N}(t)$ amount of links at time $t$ .",
"While since under the CSMA scheme, the infinite horizon average bandwidth consumption is equal to $Np(r)$ , the set of policies that are feasible during the evaluation of $\\tilde{f}_N(p(r))$ are automatically feasible for the evaluation of $f(Np(r))$ .",
"Lemma 4 We have, $\\tilde{f}_N(p(r)) \\le f(Np(r)).$ Remark 4 It is this hard, per time restriction on the available bandwidth that makes the optimal scheduling problem for CSMA network much more challenging, since a scheduler now has to prioritize amongst the packets based on the global state of the network, thus requiring a centralized controller.",
"However, as will be shown now, it is possible to overcome this limitation if the optimal prices $\\mu ^\\star $ are utilized appropriately while making scheduling decisions.",
"Next, we show that the relative difference between $f(Np(r))$ and $\\tilde{f}_N(p(r))$ asymptotically vanishes as the network capacity is scaled to $\\infty $ , i.e., $\\frac{f(Np(r)) - \\tilde{f}_{N}(p(r))}{f(Np(r)) }\\rightarrow 0$ as $N\\rightarrow \\infty $ .",
"Hence, asymptotically nothing is lost due to restraining the link capacities to those made available by the CSMA algorithm.",
"Our proof relies on constructing a decentralized scheduling algorithm for the CSMA network, denoted $\\tilde{\\pi }$ , for which the timely-throughput is within $O\\left(\\sqrt{N}\\right)$ of $f(Np(r))$ .",
"We now describe our scheduling algorithm $\\tilde{\\pi }$ .",
"Let $\\pi ^{\\star }$ denote the policy that is optimal for the scheduling problem under the link-level average bandwidth constraints given by $p(r)$ , i.e., $\\pi ^\\star $ solves the problem (REF )-() with $C^{av}$ set equal to $p(r)$ .",
"$\\pi ^\\star $ can be obtained as in Theorem REF .",
"Construction of $\\tilde{\\pi }$: It follows from Theorem REF that the policy $\\pi ^{\\star }$ makes packet-based decisions at each node $i\\in V$ .",
"Since $\\pi ^{\\star }$ does not make decisions based on instantaneous bandwidth availability $C^{CSMA_r}_{N}(t)$ , it is not a feasible policy for scheduling under the CSMA protocol applied to $N$ orthogonal channels.",
"Thus, one cannot utilize $\\pi ^{\\star }$ in order to schedule packets for the $N$ -th scaled network.",
"Now, if at some timeslot $t$ it occurs that according to $\\pi ^{\\star }$ a node $i$ has to utilize more than $C^{CSMA_r}_{N,\\ell }(t)$ amount of bandwidth on a link $\\ell $ , then the node $i$ simply chooses a maximal subset of the packets meant for transmission on link $\\ell $ subject to total bandwidth utilization less than $C^{CSMA_r}_{N,\\ell }(t)$ .",
"The selection of the set of packets meant for transmission on link $\\ell $ can be made according to some rule that has been fixed apriori before the network operation begins at time $t=0$ .",
"The policy $\\tilde{\\pi }$ is essentially $\\pi ^\\star $ truncated according to $C^{CSMA_r}_{N}(t)$ .",
"Theorem 2 Consider the sequence of “scaled CSMA-$r$ networks\" as defined above operating under the policy $\\tilde{\\pi }$ .",
"We then have that $\\frac{f(Np(r)) - \\tilde{f}_{N}(p(r))}{f(Np(r)) } = O\\left(\\frac{1}{\\sqrt{N}}\\right),$ where $\\tilde{f}_{N}(p(r))$ is the maximum timely-throughput attainable by the $N$ -th CSMA network in the sequence.",
"In the below, we drop the reference to the scale $N$ , and the CSMA aggression vector $r$ , e.g.",
"$C_N^{CSMA_r}(t)$ becomes $C^{CSMA}(t)$ .",
"The following arguments are based on analysis of the evolutions of policies on an appropriately constructed probability space.",
"Let us denote by $r_0$ the (average) reward earned by policy $\\pi ^{\\star }$ under the average bandwidth constraint on link $\\ell $ equal to $NC(r)$ .",
"Firstly note that the reward collected by the policy $\\tilde{\\pi }$ (denoted by $r_1$ ) does not increase if it were to, instead of dropping a packet because of violation of instantaneous capacity $C^{CSMA}_\\ell (t)$ , schedule it as dictated by $\\pi ^{\\star }$ , but no reward is given to it if this packet is delivered to its destination node (denoted by $r_2$ ).",
"However $r_2$ is more than the reward if now a penalty of $\\beta _f$ units per packet was imposed for scheduling a packet via utilizing “capacity in excess of $C^{CSMA}(t)$ \" at some link $\\ell \\in \\mathcal {E}$ , but it were given a reward in case this packet reaches the destination node (denoted by $r_3$ ).",
"$r_3$ is certainly more than the reward which $\\pi ^{\\star }$ earns if it is penalized an amount equal to the sum of the excess bandwidths (in excess of $C^{CSMA}(t)$ ) that its links utilize (denoted by $r_4$ ) multiplied by $\\beta _f$ , since any individual packet may be scheduled multiple times by utilizing excess bandwidth.",
"Thus, the difference $r_0-r_4$ is less than the sum of the excess bandwidths utilized by the links operating under the policy $\\pi ^{\\star }$ , scaled by $\\max _f \\beta _f$ 's.",
"Next, we will derive a bound on the excess capacity utilization.",
"Consider the system operation under the policy $\\pi ^\\star $ .",
"Based on the above arguments, we thus have that, (let all $\\beta _f\\equiv 1$ ), $r_0-r_4 \\le \\lim _{T\\rightarrow \\infty } \\frac{1}{T}\\mathbb {E}\\sum _{t=1}^{T}\\sum _{\\ell \\in \\mathcal {E}} \\left(C^{\\pi ^\\star }_\\ell (t)- C^{CSMA}_\\ell (t)\\right)^+$ Note that $&\\left(C^{\\pi ^\\star }_\\ell (t)- C^{CSMA}_\\ell (t)\\right)^+\\\\&= \\left( \\left(C^{\\pi ^\\star }_\\ell (t)- C_\\ell (r)\\right) + \\left(C_\\ell (r) -C^{CSMA}_\\ell (t)\\right)\\right)^+ \\\\&\\le \\left(C^{\\pi ^\\star }_\\ell (t)- C_\\ell (r)\\right)^{+} + \\left(C_\\ell (r) -C^{CSMA}_\\ell (t)\\right)^{+}.$ We also note that $C^{\\pi ^\\star }_\\ell (t) = \\sum _{f}\\sum _{\\tau }C^{\\pi ^\\star }_{f,\\ell ,\\tau }(t),$ where $C^{\\pi ^\\star }_{f,\\ell ,\\tau }(t)$ denotes the bandwidth utilization at time $t$ on link $\\ell $ by packets of flow $f$ that have an age of $\\tau $ time-slots.",
"Similarly, $C_\\ell (r) = \\sum _{f}\\sum _{\\tau }C_{f,\\ell ,\\tau },$ where $C_{f,\\ell ,\\tau }$ denotes the average bandwidth utilization on link $\\ell $ by packets of flow $f$ that have an age of $\\tau $ time-slots.",
"Using (REF ), (REF ) and (REF ) we upperbound the term $\\left(C^{\\pi ^\\star }_\\ell (t)- C^{CSMA}_\\ell (t)\\right)^+$ in the r.h.s.",
"of (REF ) as, $\\left(C^{\\pi ^\\star }_\\ell (t)- C^{CSMA}_\\ell (t)\\right)^+ &\\le \\left(\\sum _{f,\\tau }\\left(C_{f,\\ell ,\\tau }(t)- C_{f,\\ell ,\\tau }\\right)^+\\right) \\\\&+ \\left(C^{CSMA}_\\ell (t)- C_{\\ell }(r)\\right)^+$ Combining (REF ) with the above, we obtain that $& r_0-r_4 \\\\&\\le \\lim _{t\\rightarrow \\infty } \\frac{1}{T}\\sum _{t=1}^{T}\\sum _{f,\\tau } MAD\\left(C_{f,\\ell ,\\tau }(t)\\right) + MAD\\left(C^{CSMA}_\\ell (t)\\right)\\\\&= O(\\sqrt{N}) + O(\\sqrt{N})$ where $N$ is the scaling parameter for packet arrivals.",
"Thus we have that, $r_o-r_4 \\le O\\left(\\sqrt{N}\\right).$ Since the quantity $f(Np(r))$ scales linearly with $N$ , this completes the proof.",
"Next, we show that if the parameter $r$ of the CSMA schem is chosen appropriately so as to optimize the bandwidths allocated to various independent sets in $\\mathcal {I}$ “optimally\", then the policy $\\tilde{\\pi }$ is also asymptotically optimal for the original problem.",
"Theorem 3 Let $OPT$ denote the value of the relaxed problem (REF )-() that was obtained by relaxing the original timely-throughput maximization problem (REF )-().",
"There exists a value of the aggression parameter $r^\\star $ such that for the CSMA $r^{\\star }$ network operating under the policy $\\tilde{\\pi }$ , we have that the timely throughput $\\left(\\sum _f \\bar{D}_f\\right)_{\\tilde{\\pi }}$ is greater than or equal to $OPT-O(\\sqrt{N})$ , i.e., $\\frac{OPT - \\left(\\sum _f \\bar{D}_f\\right)_{\\tilde{\\pi }}}{OPT} = O\\left(\\frac{1}{\\sqrt{N}}\\right),$ and hence asymptotically CSMA $r^{\\star }$ utilized in combination with $\\tilde{\\pi }$ is asymptoticaly optimal for the timely throughput maximization problem (REF )-().",
"Let us denote by the set $\\mathcal {S}$ , the set of vectors that describe the instantaneous average bandwidths available to each independent set.",
"Thus, $\\mathcal {S} &= \\left\\lbrace \\bar{I}_m(t): \\bar{I}_m(t)\\mbox{ is average bandwidth available to}\\right.\\\\&\\qquad \\left.",
"IS_m \\mbox{ at time } t \\right\\rbrace .$ The closure of the set $\\mathcal {S}$ , i.e.",
"$\\bar{\\mathcal {S}}$ then coincides with the set $\\lbrace \\bar{I}:\\bar{I}_m\\ge 0,\\sum _m\\bar{I}_m = K \\rbrace $ .",
"In particular, $\\bar{I}^\\star $ , the time-average bandwidth that is optimal for the relaxed problem (REF )-(), also lies in the set $\\mathcal {S}$ .",
"Let $tp_1$ be the timely-throughput of the policy that attains the maximum while evaluating $f\\left( C_\\ell ( \\sum _{m:\\ell \\in IS_m} \\bar{I}^\\star _m )\\right)$ .",
"Also, let $tp_2$ the timely throughput of the scheduling policy that is optimal when applied in conjunction with the CSMA-$r^{\\star }$ , where the parameter $r^{\\star }$ is chosen so that the expected bandwidths allocated at each time-slot are given by $\\bar{I}^\\star $ .",
"Such an $r^{\\star }$ exists because any allocation in the set $S$ can be obtained through an appropriate choice of $r$ .",
"It then follows from Theorem REF that the difference between $tp_1$ and $tp_2$ is $O(\\sqrt{N})$ , and hence asymptotically, as $N\\rightarrow \\infty $ , the optimal throughput achievable under the CSMA protocol is the same as the solution of the relaxed problem (REF )-().",
"Remark 5 Utilizing multiple independent copies of CSMA protocol allows us to smoothen the bandwidth fluctuations for a time duration equal to the deadline $\\tau $ , which is the time taken by a packet to reach its deadline.",
"This helps us in ensuring that a single packet that is generated during the time-slot $s$ , views the link-capacities $\\lbrace C_{\\ell }(t)\\rbrace _{\\ell \\in \\mathcal {E}},t\\in [s,s+\\tau ]$ as nearly equal to their average values $\\lbrace \\bar{C}_{\\ell }\\rbrace $ for its entire lifetime in the network.",
"However, since the number of packets generated by the network also scales linearly in $N$ , hence it is not trivial, in the light of utilizing a complicated policy such as $\\tilde{\\pi }$ , to ensure that a packet receives its “right share of bandwidths\" over its entire lifetime, one that ensures that the timely throughput is not affected."
],
[
"Obtaining $r^{\\star }$",
"Though Theorem REF ensures the existence of an $r^{\\star }$ such that the combination of CSMA $r^{\\star }$ and $\\tilde{\\pi }$ can be used to attain the network timely throughput capacity in a decentralized fashion, it does not discuss how to obtain $r^{\\star }$ .",
"Since obtaining the performance of $\\tilde{\\pi }$ as a function of CSMA parameter $r$ , and the scaling parameter $N$ is a difficult problem, we will instead optimize the timely throughput under the average bandwidth constraint (REF )-().",
"Consider the following problem, dubbed the CSMA Optimization Problem.",
"Define $F(r) := \\sup \\left\\lbrace f(C^{av}) : \\mbox{ s.t. }",
"C^{av}_\\ell = \\frac{r_\\ell }{\\sum _{\\hat{\\ell }\\in N(\\ell ) } r_{\\hat{\\ell }}},\\forall \\ell \\in E\\right\\rbrace ,$ CSMA optimization problem is $\\max _{r\\ge 0} F(r).$ We now turn our attention towards obtaining its solution $r^\\star $ in a distributed manner.",
"Next, we compute the gradient of the function $F(\\cdot )$ with respect to the CSMA aggression parameter $r$ which can be used in the gradient-descent method for optimizing the function $F(\\cdot )$ .",
"In the below, we let $R_\\ell = r_\\ell + \\sum _{\\hat{\\ell }\\in N(\\ell )}r_{\\hat{\\ell }},$ be the “cumulative aggression\" associated with link $\\ell $ and its neighbouring links.",
"The following results are easily derived.",
"Lemma 5 $p(\\ell ;r) = \\frac{r_\\ell }{R_\\ell },$ so that for a link $\\hat{\\ell }$ that interferes with link $\\ell $ , i.e., $\\hat{\\ell }\\in N(\\ell )$ , we have that, $\\frac{\\partial p(\\ell ;r)}{\\partial r_{\\hat{\\ell }}} = -\\frac{r_\\ell }{R_\\ell ^2},$ while, $\\frac{\\partial p(\\ell ;r)}{\\partial r_{\\ell }} = \\frac{R_\\ell - r_\\ell }{R_\\ell ^2}.$ Hence it follows from Lemma REF (see Appendix) that, $\\frac{\\partial F}{\\partial r_{\\ell }} = \\sum _{\\hat{\\ell }\\in N(\\ell )} \\frac{-\\mu ^{\\star }_{\\hat{\\ell }}(C(r)) }{R^2_{\\hat{\\ell }}} + \\frac{\\left(R_{\\ell }-r_{\\ell }\\right)\\mu ^{\\star }_\\ell (C(r))}{R^2_\\ell }$ After having derived the explicit expressions for the gradients, we are now in a position to apply the gradient-descent scheme in order to optimize the CSMA-$r$ protocol, Theorem 4 Consider the CSMA optimization problem (REF ), with the function $F(\\cdot )$ defined as in (REF ).",
"Denote by $\\mu ^\\star (C(r)):=\\lbrace \\mu ^{\\star }_{\\ell }(C(r))\\rbrace _{\\ell \\in \\mathcal {E}}$ the value of link prices that solve the dual problem (REF ) with average link bandwidth constraints set equal to $\\frac{r_\\ell }{\\sum _{\\hat{\\ell }\\in N(\\ell )}r_{\\hat{\\ell }}}$ (see (REF )).",
"In the below, variable $k$ denotes the iteration index associated with “$r$ updates\".",
"Consider the following iterative algorithm in which each link $\\ell \\in \\mathcal {E}$ tunes its parameter $r_\\ell $ according to (in the below, $\\mu ^k_\\ell $ is to be read as $\\mu ^\\star _\\ell (C(r^k))$ , similarly for $R^k_\\ell $ etc.",
"), $&r^{k+1}_\\ell = \\Gamma \\left(r^k_\\ell + \\gamma ^k \\left(\\sum _{\\hat{\\ell }\\in N(\\ell )} \\frac{-\\mu ^k_{\\hat{\\ell }} }{(R^k_{\\hat{\\ell }})^2} + \\frac{\\left(R^k_{\\ell }-r^k_{\\ell }\\right)\\mu ^k_\\ell }{(R^k_\\ell )^2}\\right) \\right), \\\\&\\forall \\ell \\in E, k = 1,2,\\ldots ..$ The above iterations converge to a locally optimal value of $r$ for the problem (REF ).",
"The complexity of this algorithm is $O(|\\mathcal {E}|)$ .",
"Remark 6 Let us break down the various components involved in performing the iterations (REF ).",
"An update of the $r^k_{\\ell }$ involves access to the values of $\\mu ^\\star _\\ell (C(r^k))$ and the quantity $R^k_{\\ell }$ .",
"Since $R^k_{\\ell }=\\sum _{\\hat{\\ell }\\in N(\\ell )}r^k_{\\hat{\\ell }}$ , the quantity $R^k_{\\ell }$ is easily available if we allow the links to share their values $r^k_{\\ell }$ with their neighbours.",
"Now, the quantity $\\mu ^\\star _\\ell (C(r^k))$ can be computed by performing gradient-descent and value iterations as in Theorem (REF ) by setting the link bandwidths at $p(r)$ (or $C(r)$ ).",
"This involves the nodes to share the values of value function with their neighbours.",
"The overall scheme thus requires information sharing amongst neighbouring nodes only.",
"Section REF discusses the problem of searching the optimal policy using online learning methods that use data available during the operation of network."
],
[
"Simulation Results",
"We now carry out simulations to test the performance of the policy that was shown to be asymptotically optimal in Theorem REF .",
"However note that the CSMA modulator of Theorem REF that solves the CSMA optimization problem converges to an $r^\\star $ that is only locally optimal.",
"However, simulation results show that the resulting policy is quite good in practice.",
"We will refer to the policy simply as the “optimal policy\", with the understanding that it is using an $r^\\star $ that may not be optimal, and also that even if the $r^\\star $ were to be globally optimal, the policy of Theorem REF is asymptotically optimal in the limit the network scale $N\\rightarrow \\infty $ ."
],
[
"Policy Description",
"We compare the performance of the optimal policy with a version of the Q-CSMA policy [4], [9], [3] that has been adapted to be relevant to the problem of maximizing the timely throughput.",
"We denote this policy as Q-CSMA with EDF-Shortest Path, which is described below.",
"Q-CSMA with EDF-Shortest Path: The Q-CSMA algorithm [4] has been shown to throughput optimal for wireless networks in which interference is modeled using edge intereference graph.",
"The Q-CSMA algorithm uses a decentralized channel access mechanism in which the CSMA aggression parameter $r$ is tuned in accordance with the current queue lengths $Q(t)$ .",
"We describe a discretized version of the Q-CSMA algorithm that was discussed in [4].",
"A single time-slot is divided into a control mini-slot and a data slot.",
"It is during the control mini-slot, that decisions regarding channel access are made.",
"A single control mini-slot is divided into $W$ sub-slots.",
"During the control mini-slot for time-slot $t$ , each link $\\ell $ generates a number $w_\\ell (t)$ uniformly at random from the set $\\lbrace 1,2,\\ldots ,W\\rbrace $ .",
"The quantity $W$ is called the window-size.",
"The link $\\ell $ then declares an “intent\" during the sub-slot $w_\\ell (t)$ in case none of its neighbouring links have declared an intent by the sub-slot $w_{\\ell }(t)$ .",
"At the end of the control mini-slot, if the link $\\ell $ does not hear intent from any of its neighbours, and if none of its neighboring links were transmitting during data slot $t$ , then the link $\\ell $ transmits during the data-slot for time $t$ with a probability equal to $e^{Q_{\\ell }(t)}/1+Q_{\\ell }(t)$ , where $Q_{\\ell }(t)=\\sum _f Q_{f,\\ell }(t)$ is the cumulative queue length at link $\\ell $ .",
"Also, if multiple neighboring links declare intent in the same sub-slot, then none of them transmits data during the corresponding data-slot.",
"Note that the Q-CSMA provides only channel access decisions, but not the routing or packet scheduling decisions that prioritize based on the age of packets.",
"Thus, we will combine the Q-CSMA with EDF discipline which will enable it to make routing decisions, and also the earliest deadline first (EDF) policy which will allow it to prioritize the scheduling of packets that are “closer\" to their deadline.",
"For such a policy, the bandwidth allocated across the network links during a data-slot $t$ are decided by the Q-CSMA algorithm described above.",
"At the end of the control-slot, each link $\\ell $ arranges the packets with it in increasing order of their time until deadline.",
"Then, the link $\\ell $ schedules them on the shortest path route, subject to the instantaneous link bandwidth of link $\\ell $ that has been provided to it by the Q-CSMA.",
"In case there are multiple shortest paths that connect the link $\\ell $ to the destination node for a flow $f$ , then the link $\\ell $ chooses from amongst them uniformly at random while scheduling packets for flow $f$ .",
"Figure: A multihop wireless network shared by two flows.",
"The source-destination pairs are (1,4)(1,4) and (4,1)(4,1).",
"Channel reliabilities for the links are provided within braces, eg.",
"reliability of link (2,3)(2,3) is equal to .7.7.",
"It is assumed that any links that share a node interfere with each other.",
"Figure: A multihop wireless network shared by two flows with the source-destination given by (1,4)(1,4) and (2,4)(2,4).",
"Channel reliabilities for the links are provided within braces, and is equal to .5.5 for all of the network links.",
"It is assumed that any two links that share a node interfere with each other.",
"The corresponding link interference graph is shown in Fig.",
".Figure: Link interference graph for the network of Fig.",
"."
],
[
"Network Set-Up",
"We simulate the policies for the networks shown in Fig.",
"REF and Fig.",
"REF .",
"We assume that any two links which share a node will be affected by wireless interference, and hence will be connected by an edge in the link interference graph.",
"Thus, two links $\\ell _1 = (i_1,j_1)$ and $\\ell _2=(i_2,j_2)$ interefere if either of the following conditions is satisfied $i_1=i_2,i_1=j_2,j_1=i_1,j_1=j_2$ .",
"Throughout, we assume that all links have a transmission capacity of 1 pkt/time-slot.",
"For the Q-CSMA with EDF-SP policy, we set the window length of the control mini-slot to be equal to 10 sub-slots.",
"We assume that for the unscaled network, the arrivals for each flow $f$ at each time $t$ are distributed according to Bernoulli $(1,.8)$ ."
],
[
"Results",
"We fix the relative end-to-end deadline for the flows to be equal to 10 time-slots, and vary the network scale $N$ of Theorem REF .",
"The resulting timely throughputs are plotted in Fig.",
"REF and Fig.",
"REF .",
"We observe that the normalized timely throughputs (timely throughput/$N$ ) converge to the asymptotic ($N\\rightarrow \\infty $ ) timely throughputs quite quickly.",
"Even with the scale $N=4$ , the normalized timely throughput has equilibriated to the asymptotic timely throughput.",
"Secondly, we observe that the performance obtained by using the $r^\\star $ that was derived in Theorem REF is near-optimal.",
"Since the cumulative mean arrival rate for the network shown in Fig.",
"REF is equal to $1.6$ units, its maximum achievable normalized timely-throughput is less than or equal to $1.6$ pkts/time-slot.",
"As seen in Fig.",
"REF , the timely throughput of the optimal scheme is quite close to this upper bound.",
"We then fix the scale of the networks at $N=4$ , and vary the end-to-end relative deadlines for the flows.",
"The results are plotted in Fig.",
"REF and Fig.",
"REF .",
"We observe that the performance of the optimal policy is much superior to that of the Q-CSMA with EDF-SP.",
"This is primarily because it utilizes the link prices $\\lambda _{\\ell }$ in order to make decisions.",
"Since this automatically allows the packets to be prioritized according to the probability that they will be able to reach their destination within the deadline.",
"Figure: A plot of the normalized timely throughput for the network of Fig.",
"as the scale of the network is varied.",
"We notice that the scaled throughput equilibriates quite “fast\" at a scaling of 4, hence hinting that the sub-optimality bounds of O(1 N)O(\\frac{1}{\\sqrt{N}}) derived in Theorem might be pessimistic, and might be further improved upon.Figure: A plot of the normalized timely throughput for the network of Fig.",
"as the scale of the network is varied.",
"Similar to the observation made in the plot of Fig.",
", we notice that the scaled throughput equilibriates quite “fast\" at a scaling of 4.",
"Thus, the sub-optimality bounds of O(1 N)O(\\frac{1}{\\sqrt{N}}) derived in Theorem might be possibly further improved upon.Figure: A plot of the timely throughput of the network shown in Fig.",
"as the relative deadline of the flows is varied.",
"Network scale, i.e., the parameter NN of Theorem and Theorem is set to 4.",
"We observe that the Q-CSMA based EDF-SP policy performs poorly with respect to bandwidth allocation amongst the packets since it does not use optimally the “information\" regarding the relative deadlines and ages of the packets.",
"In contrast, the optimal policy uses the link prices λ ℓ \\lambda _{\\ell } in order to make routing-scheduling decisions, and hence attains a much higher timely throughput.",
"The link prices enable us to prioritize the packets in proportion to the probability of their successful delivery to the destination wihin their deadline.Figure: A plot of the timely throughput of the network shown in Fig.",
"as the relative deadline of the flows is varied while keeping the network scale (the parameter NN of Theorem and Theorem ) fixed at N=4N=4.",
"Similar to the observation made in plot of Fig.",
", we observe that the policy of Theorem outperforms the Q-CSMA based EDF-SP policy by a huge margin."
],
[
"Centralized Optimization of Bandwidth Allocation $\\bar{I}$",
"We will now provide a centralized algorithm that performs optimization over the bandwidth allocation vector $\\bar{I}$ that solves (REF )-().",
"Throughout this section we will assume that there is a centralized controller that knows the wireless network $G$ , and its link-interference graph.",
"Thus it knows the various independent sets $IS$ in the set $\\mathcal {I}$ .",
"This asumption is relaxed in Section , where CSMA protocol is utilized for decentralized channel access.",
"The algorithms discussed in this section are based on the commonly used optimization technique of gradient-descent-method, and the discussion is very straightforward.",
"Define the following, $f(C^{av}) := \\max _{\\pi : \\bar{C}^\\pi \\le C^{av}} \\sum _f \\beta _f \\bar{D}_f,$ where the inequality $\\bar{C}^\\pi \\le C^{av}$ between two vectors is to be taken componentwise.",
"Our concern in this section will be to solve the following optimization problem, $\\max &\\qquad f(C^{av})\\\\\\mbox{ s.t. }",
"C^{av}_\\ell &= C_\\ell \\left(\\sum _{m:\\ell \\in C_m} \\bar{I}_m \\right),\\forall \\ell \\in E,\\\\\\mbox{ and } \\sum _{m}\\bar{I}_m &= K.$ Assumption 1 We will design an iterative algorithm based on gradient descent that will converge to the value of $\\bar{I}^\\star $ , which is the optimal bandwidth allocation vector.",
"Denote by $\\bar{I}^k$ the value of average bandwidth allocation vector at iteration $k$ .",
"The central controller keeps track of $\\bar{I}^k$ .",
"It then updates it according to, $\\bar{I}_m^{k+1} = \\bar{I}^k + \\alpha ^k \\frac{\\partial f }{\\partial \\bar{I}_{m}}, m = 1,2,\\ldots ,M,$ where the quantity $\\frac{\\partial f }{\\partial \\bar{I}_{m}}$ is the rate of change of the maximum achievable timely-throughput under the constraint on average bandwidth consumption on link $\\ell $ set to $C_{\\ell } \\left(\\sum _{m:\\ell \\in IS_m} \\bar{I}_m \\right)$ .",
"The iterates stated above are also projected onto the feasible set $\\lbrace \\bar{I}: \\bar{I}_m\\ge 0~\\forall m,\\sum _{m}\\bar{I}_m = K\\rbrace $ .",
"Next, we derive an explicit expression for the quantities $\\frac{\\partial f }{\\partial \\bar{I}_{m}}$ .",
"Lemma 6 For the problem (REF )-() since the Lagrange multiplier $\\mu _\\ell $ associated with the constraint $\\bar{C}_{\\ell }^\\pi \\le C^{av}_{\\ell }$ can be interpreted as shadow price, we have, $\\frac{\\partial f}{\\partial C^{av}_\\ell } = \\mu ^{\\star }_\\ell (C^{av}), \\forall \\ell \\in \\mathcal {E},$ where $\\mu ^{\\star }(C^{av})$ is the vector that solves the dual problem (REF ) and can be obtained by the gradient descent method as discussed in Section REF .",
"Since the average bandwidth available to a link $\\ell $ is the sum of the bandwidths provided to each independent set that it is part of, we have that $\\frac{\\partial f}{\\partial \\bar{I}_m} = \\sum _{\\ell \\in IS_m} \\mu ^{\\star }_\\ell (C^{av}).$"
],
[
"Online Learning using Multiple Time-Scales Stochastic Approximation",
"We now use the technique of stochastic approximation [26], [27], [28] in order to solve the problem of searching for the optimal policy in case the network parameters are unknown.",
"Recall that the following 3 iterations were performed Value Iterations for solving Dynamic Programming equations (REF ), which yield the policies $\\pi ^{\\star }_f(\\mu )$ .",
"Gradient descent iterations (REF ) for solving the dual problem (REF ).",
"Gradient descent iterations (REF ) for obtaining the locally optimal CSMA aggression parameter $r^{\\star }$ in problem (REF )-(REF ).",
"We will now combine these algorithms using multi timescale stochastic approximation [29], [28] so as to obtain a single online learning algorithm.",
"We note that the constraints () imposed in the definition of $f(C^{av})$ involve average values of link bandwidth consumption and hence the instantaneous bandwidth utilization by the optimal policy corresponding to the evaluation of $f(C^{av})$ can exceed $C^{av}_\\ell $ .Indeed, for the policy that is optimal under link-level average bandwidth constraints, if the instantaneous state of the packets present at time $t$ at a link $\\ell $ is such that their cumulative bandwidth demand is in excess of $C^{av}_{\\ell }$ , then the link $\\ell $ would simply charge them a price of $\\mu ^{\\star }_\\ell $ , and provide them excess capacity.",
"In contrast, the CSMA-$r$ access protocol is oblivious to the state of the packets present at various network links.",
"It allocates bandwidth in an i.i.d.",
"fashion, without any knowledge of the state of the packets present at various links $\\ell \\in \\mathcal {E}$ , and it also imposes a hard constraint on the cumulative bandwidth consumption, i.e.",
"$\\sum _m I_{m}(t) = N$ .",
"Hence, we make the following crucial assumption.",
"Assumption 2 During the “learning\" phase of network operation when we are solving for the optimal CSMA parameter $r^{\\star }$ , we will allow the links to utilize bandwidths in excess of that provided by the CSMA scheme.",
"This can be achieved by assuming that each link $\\ell \\in \\mathcal {E}$ has spare bandwidth available to it during this phase, and moreover this bandwidth is reserved for its usage, i.e., no link interference takes place.",
"In view of the above discussion, we will use different symbols to denote the bandwidths allocated by the CSMA and that actually utilized by the network.",
"The bandwidth allocated by CSMA protocol, i.e., $C^{CSMA}_{\\ell }(t)$ , may not be equal to the actual bandwidth utilized by the routing policy, which is denoted $C^{U(t)}_{\\ell }(t)$ .",
"The superscript $U(t)$ denotes that the value is decides by the actual scheduling decision implemented at time $t$ .",
"We note that typically $C^{U(t)}_{\\ell }(t)$ will necessarily exceed $C^{CSMA}_{\\ell }(t)$ not only because the Q-learning iterations make packet-based decisions, but also due to the fact that price iterations that solve the dual problem rely on bandwidth utilization $C^{U(t)}_{\\ell }(t)$ exceeding the bandwidth availability $C^{CSMA}_{\\ell }(t)$ .",
"We will thus use Assumption REF .",
"Furthermore since we are searching for the value of $r$ that maximizes the timely throughput under average bandwidth constraints, the scaling parameter $N$ is irrelevant, and the algorithm proposed below can be used with $N=1$ .",
"Theorem 5 We propose the following 3-layered iterative algorithm that converges to the optimal CSMA parameter $r^{\\star }$ that solves the problem (REF )-(REF ), and the optimal policy for the relaxed problem $\\pi ^{\\star }(C(r^{\\star }))$ .",
"$&Q^f(i,\\tau ,j) = Q^f(i,\\tau ,j)\\left(1-\\alpha _t\\right)\\\\&+\\alpha _t\\left\\lbrace -\\mu _{(i,j)} + \\mathbb {1}(i=d_f) + \\max _{\\tilde{j}}Q(i^{+},(\\tau +1)\\wedge B,\\tilde{j})\\right\\rbrace ,\\\\\\\\&\\mu _{\\ell }(t+1) = \\Gamma \\left[ \\mu _\\ell (t) + \\beta _t \\left(C^{U(t)}_{\\ell }(t)-C^{CSMA}_{\\ell }(t) \\right) \\right],\\\\\\\\&r_\\ell (t+1) \\\\&= \\Gamma \\left(r_\\ell (t) + \\gamma _t \\left\\lbrace \\mu _{\\ell }(t) \\left(C^{CSMA}_\\ell (t) - C^{CSMA}_\\ell (t)^2\\right) \\right\\rbrace \\right), \\\\\\\\&t = 1,2,\\ldots .$ where $\\sum _t\\alpha _t = \\infty ,\\sum _t\\alpha ^2_t<\\infty ,\\sum _t\\beta _t = \\infty ,\\sum _t\\beta ^2_t<\\infty ,\\sum _t\\gamma _t = \\infty ,\\sum _t\\gamma ^2_t<\\infty $ and also $\\beta _t=o(\\alpha _t),\\gamma _t=o(\\beta _t)$ .",
"The roles of the three layers that comprise the algorithm are described below.",
"Layer 1 [Q Learning]: Learns the optimal scheduling policy to solve the Single Packet Transportation Problem that is paramterzied by the current link prices $\\mu (t)$ .",
"Layer 2 [Online Gradient Descent for Price ]: is provided “target link capacities $\\lbrace C^{CSMA}_\\ell \\rbrace _{\\ell \\in \\mathcal {E}}$ from Layer 3, and adjusts the link-prices $\\lbrace \\mu _\\ell \\rbrace _{\\ell \\in \\mathcal {E}}$ so that the “average traffic intensity\" resulting from the policy produced by Layer 1, i.e., $C^{tr}$ respects the target link capacities $C^{CSMA}$ .",
"Iterations are based on the sub-gradient descent method (REF ).",
"Layer 3 [CSMA Learning] : gets access to the link prices $\\lbrace \\mu _\\ell \\rbrace _{\\ell \\in \\mathcal {E}}$ from Layer 2, and modulates the aggression parameter $r$ of the CSMA-$r$ protocol.",
"It performs the bandwidth optimization by converging to the optimal aggression rate $r^{\\star }$ .",
"Figure REF depicts the 3-layered hierarchial structure of the proposed algorithm.",
"Remark 7 Notice that the iterations asociated with tuning the parameter $r$ can be guaranteed to converge to only a local optima of the function $F(\\cdot )$ .",
"However, gradient descent schemes in general suffer from this drawback unless the function $F(\\cdot )$ is shown to be convex.",
"For our problem, it is not easy to the convexity of the function $F(r)$ in the parameter $r$ .",
"One can use noisy perturbations (see [30], [27]) in order to ensure that the iterations do not get stuck at a critical point that is not local optima.",
"If we utilize methods like simulated annealing, then the $r$ iterations are guaranteed to converge to a global optima.",
"However the convergence speed of simulated annealing procedure is too slow to be of any practical application.",
"Figure: A hierarchial view of the 3 components of our proposed algorithm.",
"We notice that so far we have assumed that the individual components can be computed instantaneously.We realize that due to the 3 layered structure, and complex multi-timescale nature of the above algorithm, the proposed algorithms may not be practical to implement.",
"Thus, we now consider a somewhat related problem that involves scheduling packet transmissions under constraints on link-bandwidths.",
"Next, we introduce this objective."
]
] | 1709.01672 |
[
[
"Total Generalized Variation for Manifold-valued Data"
],
[
"Abstract In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting.",
"We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms.",
"We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup.",
"Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data."
],
[
"Introduction",
"In this work we introduce and explore a generalization of second-order total generalized variation (TGV) regularization for manifold-valued data.",
"The TGV functional has originally been introduced in [29] for the vector space setting as generalization of the total variation (TV) functional [73], which is extensively used for regularization in image processing and beyond.",
"The advantage of TV regularization compared to, e.g., classical $H^1$ regularization approaches, is that jump discontinuities can be much better reconstructed.",
"This can be seen in the function space setting since functions of bounded variation, as opposed to Sobolev functions, can have jump discontinuities.",
"It is also reflected in numerical realizations where TV minimization allows to effectively preserve sharp interfaces.",
"A disadvantage of TV regularization, however, is its tendency to produce piecewise constant results even in non-piecewise-constant regions, which is known as the staircasing effect.",
"Employing a regularization with higher order derivatives, such as second-order TV regularization, overcomes this drawback, but again does not allow for jump discontinuities.",
"As a result, a lot of recent research aims at finding suitable extensions of TV that overcome the staircasing effect, but still allow for jumps [31], [75], [21].",
"While infimal-convolution-type approaches can be seen as the first methods to achieve this, the introduction of the TGV functional (of arbitrary order $k$ ) in 2010 finally provided a complete model for piecewise smooth data with jump discontinuities.",
"This is achieved by an optimal balancing between first and higher order derivatives (up to the order $k$ ), which is carried out as part of the evaluation of the functional.",
"We refer to [29] for more motivation and to [25] for a detailed analysis of TGV in the context of inverse problems.",
"In particular, second-order TGV, which balances between first and second-order derivatives, achieves piecewise linear – as opposed to piecewise constant – image reconstructions while it still allows for jumps.",
"This renders second-order TGV a well suited model for piecewise smooth images and can be seen as the motivation of the use of second-order TGV regularization in a plethora of applications [57], [26], [27], [58], [65].",
"Up to now, TGV regularization was only available for vector space data and applications are hence limited to this situation.",
"In various problems of applied sciences, however, the data do not take values in a linear space but in a nonlinear space such as a smooth manifold.",
"Examples of manifold-valued data are circle and sphere-valued data as appearing in interferometric SAR imaging [62], wind directions [74], orientations of flow fields [3], [79], and color image processing [33], [89], [56], [60].",
"Other examples are data taking values in the special orthogonal group $SO(3)$ expressing vehicle headings, aircraft orientations or camera positions [87], Euclidean motion group-valued data [72] as well as shape-space data [63].",
"Another prominent manifold is the space of positive (definite) matrices endowed with the Fisher-Rao metric [70].",
"This space is the data space in DTI [69].",
"It is a Cartan-Hadamard manifold and as such it has particularly nice differential-geometric properties.",
"DTI allows to quantify the diffusional characteristics of a specimen non-invasively [11], [54] which is helpful in the context of neurodegenerative pathologies such as schizophrenia [44] or autism [5].",
"Because of the natural appearance of these nonlinear data spaces quite a lot of recent work deals with them.",
"Examples are wavelet-type multiscale transforms for manifold-valued data [87], [51], [91], [90], [92] as well as manifold-valued partial differential equations [85], [35], [48].",
"Work on statistics on Riemannian manifolds can be found in [66], [23], [43], [24], [68], [42].",
"Optimization problems for manifold-valued data are for example the topic of [2], [1], of [49] and of [52] with a view towards learning in manifolds.",
"We also mention related work on optimization in Hadamard spaces [10], [9], [7] and on the Potts and Mumford-Shah models for manifold-valued data [94], [81].",
"TV functionals for manifold-valued data have been considered from the analytical side in [46], [47], [45]; in particular, the existence of minimizers of certain TV-type energies has been shown.",
"A convex relaxation based algorithm for TV regularization for $\\mathbb {S}^1$ -valued data was considered in [82], [37].",
"Approaches for TV regularization for manifold-valued data are considered in [61] which proposes a reformulation as multi-label optimization problem and a convex relaxation, in [50] which proposes iteratively reweighted minimization, and in [93] which proposes cyclic and parallel proximal point algorithms.",
"An exact solver for the TV problem for circle-valued signals has been proposed in [80].",
"Furthermore, [15] considers half-quadratic minimization approaches, which may be seen as an extension of [50], and [19] considers an extension of the Douglas-Rachford algorithm for manifold-valued data.",
"Applications of TV regularization to shape spaces may be found in [13], [78].",
"TV regularization with a data term involving an imaging type operator has been considered in [14] where a forward-backward type algorithm is proposed to solve the corresponding inverse problem involving manifold-valued data.",
"As with vector space data, TV regularization for manifold-valued data has a tendency to produce piecewise constant results in regions where the data is smooth.",
"As an alternative which prevents this, second-order TV type functionals for circle-valued data have been considered in [18], [20] and, for general manifolds, in [10].",
"However, similar to the vector space situation, regularization with second-order TV type functionals tends towards producing solutions which do not preserve the edges as desired.",
"To address this drawback, the vector space situation guides us to considering TGV models and models based on infimal convolution to address this issue.",
"The most difficult part in this respect is to define suitable notions in the manifold setup which have both reasonable analytic properties on the one hand and which are algorithmically realizable on the other hand.",
"Concerning infimal-convolution type functionals such as $\\operatorname{TV}$ –$\\operatorname{TV}^2$ infimal convolutions, a first effort towards a generalization to the manifold setting has been made in the recent conference proceeding [17] which has later been extended in an arXiv preprint [16].",
"The present manuscript [28], which was submitted to arXiv at the same time as [16], emerged independently.",
"In this work, we focus on TGV and aim at providing a thorough study of reasonable generalizations of TGV for piecewise smooth, manifold-valued data by first investigating crucially required properties of generalizations of TGV to the manifold setting.",
"Then we propose suitable extensions that fulfill these properties and which are, in addition, computationally feasible.",
"In this respect, it is important to note that due to the cascadic nature of TGV (as opposed to infimal-convolution-type functionals), its generalization to manifolds requires a conceptually new approach.",
"For this reason, both the techniques we propose for a generalization of TGV as well as our algorithmic setting are rather different from the one of [17], which uses a mid-point approach to generalize $\\operatorname{TV}$ –$\\operatorname{TV}^2$ infimal convolutions and gradient descent for numerical minimization."
],
[
"Contributions",
"The contributions of the paper are as follows: (i) we lay the foundations for – and provide concrete realizations of – a suitable variational model for second-order total generalized variation for manifold-valued data, (ii) we provide algorithms for the proposed models, (iii) we show the potential of the proposed algorithms by applying them to synthetic and real data.",
"Concerning (i), we use an axiomatic approach.",
"We first formalize reasonable fundamental properties of vector-valued TGV which should be conserved in the manifold setting.",
"Then we propose two concrete realizations which we show to fulfill the axioms.",
"The one is based on parallel transport and the other is motivated by the Schild's approximation of parallel transport.",
"We obtain well-posedness results for TGV-based denoising of manifold valued data for both variants.",
"Concerning (ii) we provide the details for a numerical realization of variational regularization for general manifolds using either of the two proposed generalizations of TGV.",
"We build on the well-established concept of cyclic proximal point algorithms (using the inexact variant).",
"The challenging part and the main contribution consists in the computation of the proximal mappings of the TGV atoms involved.",
"In particular, for the class of symmetric spaces, we obtain closed form expressions of the related derivatives in terms of Jacobi fields for the Schild variant; for the parallel transport variant, we obtain closed form expressions for the related derivatives for general symmetric spaces up to a derivative of a certain parallel transport term for which we can still analytically compute the derivative for the class of manifolds considered in the paper.",
"Concerning (iii), we provide a detailed numerical investigation of the proposed generalization and its fundamental properties.",
"Furthermore, we provide experiments with real and synthetic data and a comparison to existing methods."
],
[
"Outline of the paper",
"The paper is organized as follows.",
"The topic of Section is the derivation of suitable TGV models for manifold-valued data.",
"We start out with a detailed discussion of fundamental properties expected by a reasonable TGV version for manifold-valued data.",
"To this end, we first reconsider vector-space TGV in a form suitable for our purposes in Subsection REF .",
"Then we derive an axiomatic extension of $\\operatorname{TGV}_\\alpha ^2$ to the manifold setting where we – for a better understanding – first consider the univariate case in Subsection REF .",
"Next we suggest two realizations which we call the Schild variant of TGV and the parallel transport variant of TGV and show that these realizations indeed fulfill all desired properties.",
"Then, in Subsection REF , we extend the axiomatic framework to the bivariate setting and provide bivariate versions of the Schild variant of TGV and the parallel transport variant of TGV, respectively, that we show to fulfill the required axioms.",
"In Section we then provide existence and lower semi-continuity results for the proposed variants of TGV from which the existence of minimizers for the TGV-regularized denoising problems of manifold-valued data follows.",
"The topic of Section is the algorithmic realization of the proposed models.",
"We start out recalling the concept of a cyclic proximal point algorithm (CPPA).",
"Then, in Subsection REF , we consider the implementation of the CPPA for manifold-valued TGV.",
"We identify certain TGV atoms whose proximal mappings are challenging to compute.",
"The necessary derivations needed for their computation are provided for the Schild variant of TGV in Subsection REF , and, for the parallel transport variant of TGV, in Subsection REF .",
"The topic of Section is the numerical evaluation of the proposed schemes.",
"There, we first carry out a detailed numerical evaluation of the proposed model for synthetic data.",
"We test extreme parameter choices and ensure consistency of the results of our numerical method with a reference implementation for vector spaces.",
"Then we consider various application scenarios and compare to existing methods.",
"Finally, we draw conclusions in Section ."
],
[
"Basic Notation and Concepts from Differential Geometry",
"Throughout this work, $\\mathcal {M}\\subset \\mathbb {R}^K$ will always denote a complete manifold with a Riemannian structure (with its canonical metric connection, its Levi-Civita connection).",
"Let us explain these notions more precisely.",
"We consider a manifold $\\mathcal {M}$ with a Riemannian metric which is a smoothly varying inner product $\\langle \\cdot ,\\cdot \\rangle _p$ in the tangent space of each point $p$ .",
"(We note that any paracompact manifold can be equipped with a Riemannian structure.)",
"As usual, we will frequently omit the dependence on $p$ in the notation in the following.",
"According to the Hopf-Rinow theorem, the complete manifold $\\mathcal {M}$ is geodesically complete in the sense that any geodesic can be prolongated arbitrarily.",
"Here a geodesic is a curve $\\gamma $ of zero acceleration, i.e., $\\frac{D}{dt} \\frac{d}{dt} \\gamma = 0,$ where $\\frac{D}{dt}$ denotes the covariant derivative along the curve $\\gamma $ (see below for details on the covariant derivative).",
"Geodesics are invariant under affine reparametrizations and usually identified with their image in $\\mathcal {M}$ .",
"(We parametrize them on $[0,1]$ or, more generally, on $[t_0,t_1]$ with $t_0<t_1$ , depending on the context.)",
"We always denote by $d: \\mathcal {M}\\times \\mathcal {M}\\rightarrow [0,\\infty )$ the distance on $\\mathcal {M}$ which is induced by the Riemannian structure and note that, since $\\mathcal {M}$ is complete, for any $a,b \\in \\mathcal {M}$ there is always a geodesic from $a$ to $b$ whose length equals $d(a,b)$ .",
"Such geodesics are called length minimizing geodesics between $a$ and $b$ .",
"By $T_a\\mathcal {M}$ we denote the tangent space of $\\mathcal {M}$ at $a \\in \\mathcal {M}$ and by $T\\mathcal {M}$ the tangent bundle of $\\mathcal {M}$ .",
"Further, we need the notion of parallel transport.",
"The parallel transport of a vector $v \\in T_a \\mathcal {M}$ with $a \\in \\mathcal {M}$ to a point $b\\in \\mathcal {M}$ along a curve $\\gamma :[0,1] \\rightarrow \\mathcal {M}$ such that $\\gamma (0) = a$ , $\\gamma (1) = b$ is the vector $V_1 = V(1) \\in T_b \\mathcal {M}$ , where $V:[0,1] \\rightarrow T\\mathcal {M}$ is given as the solution of the ODE $\\frac{D}{dt}V(t) = 0$ on $[0,1]$ with the initial condition $V(0)=v,$ where the covariant derivative $\\frac{D}{dt}$ is taken along the curve $\\gamma $ .",
"The covariant derivative $\\frac{D}{dt}$ is induced by the Levi-Civita connection $\\nabla _X Y$ of the Riemannian manifold $\\mathcal {M}$ ; a connection $\\nabla _X Y$ is a $C^\\infty $ -linear mapping w.r.t.",
"the vector field $X$ and a derivation w.r.t.",
"the vector field $Y.$ (Connections are needed since, in a general manifold $\\mathcal {M}$ , there is no a priori canonical way to define the directional derivative in direction $X$ of a vector field $Y$ .)",
"For the covariant derivative $\\frac{D}{dt}Y$ along a curve $\\gamma $ , a direction $X$ along the curve $\\gamma $ is given by $\\frac{d}{dt}\\gamma $ and the vector field $Y$ is given along the curve; hence, if $\\frac{d}{dt}\\gamma \\ne 0,$ $\\frac{D}{dt}Y$ is just given by $\\nabla _{\\frac{d}{dt}\\gamma } Y.$ In a Riemannian manifold $\\mathcal {M},$ there is a (uniquely determined) canonical metric connection, its Levi-Civita connection.",
"The Levi-Civita connection is the only connection which is symmetric (i.e.,$\\nabla _X Y = \\nabla _Y X$ for any vector fields $X,Y$ which commute) and which is compatible with the metric (i.e., in terms of the induced covariant derivative which we need later on, $\\frac{d}{dt} \\langle X,Y \\rangle =$ $\\langle \\frac{D}{dt}X,Y\\rangle $ + $\\langle X, \\frac{D}{dt}Y\\rangle $ for any two vector fields $X,Y$ along a curve $\\gamma ,$ along which the covariant derivatives are taken.)",
"For an account on Riemannian geometry we refer to the books [76], [38] or to [59].",
"For later use, we fix some further notation.",
"To this end, let $a,b \\in \\mathcal {M},$ $v \\in T_a \\mathcal {M},$ and $\\gamma :[0,L] \\rightarrow \\mathcal {M}$ be a curve connecting $a$ and $b$ such that $\\gamma (t_0) = a$ , $\\gamma (t_1) = b$ with $t_0<t_1$ , $t_0,t_1 \\in [0,L].$ We define $\\exp _a(v) = \\psi (1)$ where $\\psi :[0,1]\\rightarrow \\mathcal {M}$ is the unique geodesic such that $\\psi (0) = a$ , $\\frac{d}{dt}\\psi (0) = v$ , $\\exp (v) = \\exp _a(v)$ when it is clear from the context that $v \\in T_a\\mathcal {M}$ , $\\log _a(b) = \\lbrace z \\in T_a\\mathcal {M} \\, |\\, \\exp (z) = b \\text{ and } [0,1] \\ni t \\mapsto \\exp (tz) \\text{ is a length-minimizing geodesic}\\rbrace $ , in case $\\gamma $ is a geodesic and length-minimizing between $a$ and $b$ on $[t_0,t_1]$ , we denote $\\log ^\\gamma _a(b) = (t_1 - t_0)\\frac{d}{dt}\\gamma (t_0),$ i.e., the vector in $\\log _a(b)$ that is parallel to $\\frac{d}{dt}\\gamma (t_0)$ and such that $\\exp _a(\\log ^\\gamma _a(b)) = b$ , $\\operatorname{pt}^\\gamma _{a,b}(v) = z \\in T_b M$ , where $z$ is the vector resulting from the parallel transport of $v$ from $a$ to $b$ along the curve $\\psi $ that reparametrizes $[t_0,t_1]$ to $[0,1]$ in an affine way, i.e., $\\psi (t) = \\gamma (t_0 + t(t_1 - t_0))$ such that $\\psi (0) = a$ , $\\psi (1) = b$ , $\\operatorname{pt}_{a,b}(v) = \\lbrace \\operatorname{pt}^\\psi _{a,b}(v) \\, |\\, \\psi \\text{ is a length-minimizing geodesic connecting } a \\text{ and } b\\rbrace $ , $\\operatorname{pt}_b(v) = \\operatorname{pt}_{a,b}(v)$ , $\\operatorname{pt}^\\gamma _{a,b}(v) = \\operatorname{pt}^\\gamma _b(v)$ when it is clear from the context that $v \\in T_a\\mathcal {M}$ , for $t \\in [0,1]$ , $[a,b]_t = \\lbrace \\psi (t) \\,|\\, \\psi : [0,1] \\rightarrow \\mathcal {M}\\text{ is a length-minimizing geodesic with } \\psi (0) = a, \\, \\psi (1) = b \\rbrace .$ We extend this definition for $t \\in \\mathbb {R}\\setminus [0,1]$ by extending the corresponding geodesic.",
"We also note that throughout the paper we identify sets having only one element with the corresponding element.",
"We further need the concept of geodesic variations for the existence results in Section and the derivation of the algorithms in Section .",
"A variation of a curve $\\gamma :I \\rightarrow \\mathcal {M}$ defined on an interval $I$ is a smooth mapping $V:I \\times J \\rightarrow \\mathcal {M},$ $J$ an interval containing 0, such that $V(s,0) = \\gamma (s)$ for all $s \\in I.$ A variation is called geodesic, if all curves $s \\mapsto V(s,t)$ are geodesics for any $t$ in $J.$"
],
[
"Definition of TGV for manifold-valued data",
"The goal of this section is to define a discrete total generalized variation (TGV) functional of second order for manifold-valued data.",
"To this end, we first state some fundamental properties of the TGV functional in infinite and finite dimensional vector spaces.",
"The definition of a generalization of TGV to the manifold setting will then be driven by the goal of preserving these fundamental properties of vector-space TGV."
],
[
"TGV on vector spaces",
"We first recall the definition of TGV on infinite-dimensional vector spaces via its minimum representation which is, according to results in [30], [88], [25], [26] covering the second- and general order case as well as the scalar and vectorial case, equivalent to the original definition as provided in [29].",
"Then we present a discretization which is slightly different from the one typically used.",
"Definition 2.1 (Minimum representation of TGV in vector spaces) For $\\alpha _0,\\alpha _1 \\in (0,\\infty )$ , $u \\in L^1_{\\operatorname{loc}}(\\Omega )^K$ with $\\Omega \\subset \\mathbb {R}^d$ a bounded Lipschitz domain, we define $ \\operatorname{TGV}_\\alpha ^2(u) = \\min _{w \\in \\operatorname{BD}(\\Omega ,\\mathbb {R}^{d})^K} \\alpha _1 \\Vert \\mathrm {D}u - w\\Vert _{\\mathcal {M}} + \\alpha _0 \\Vert \\mathcal {E}w\\Vert _{\\mathcal {M}} .$ Here , $\\Vert \\cdot \\Vert _{\\mathcal {M}}$ denotes the Radon norm in the space of Radon measures $\\mathcal {M}(\\Omega ,X)^K:= (C_0(\\Omega ,X)^K)^*$ with $X \\in \\lbrace \\mathbb {R}^{d},S^{d\\times d}\\rbrace $ , $S^{d\\times d}$ the space of symmetric matrices and further $\\operatorname{BD}(\\Omega ,\\mathbb {R}^d)^K:= \\lbrace w \\in L^1(\\Omega ,\\mathbb {R}^d)^K \\, |\\, \\mathcal {E}w \\in \\mathcal {M}(\\Omega ,S^{d\\times d})^K\\rbrace $ .",
"The derivatives $\\mathrm {D}u \\in \\mathcal {D}(\\Omega ,\\mathbb {R}^{d})^K$ and $\\mathcal {E}w \\in \\mathcal {D}(\\Omega ,S^{d\\times d })^K$ are defined in the weak sense by $\\langle \\mathrm {D}u ,\\varphi \\rangle &= -\\langle u , \\operatorname{div}\\varphi \\rangle , \\quad \\varphi \\in C^\\infty _c (\\Omega ,\\mathbb {R}^{d})^K, \\\\\\langle \\mathcal {E}w ,\\varphi \\rangle &= -\\langle w , \\operatorname{div}\\varphi \\rangle , \\quad \\varphi \\in C^\\infty _c (\\Omega ,S^{d\\times d})^K ,$ with $\\operatorname{div}\\varphi = ( \\operatorname{div}\\varphi ^1,\\ldots ,\\operatorname{div}\\varphi ^K)\\in C^\\infty _c (\\Omega )^K$ for $\\varphi = (\\varphi ^1,\\ldots ,\\varphi ^K) \\in C^\\infty _c (\\Omega ,\\mathbb {R}^{d})^K$ and, for $\\varphi = (\\varphi ^ 1,\\ldots ,\\varphi ^ K) \\in C^\\infty _c (\\Omega ,S^{d\\times d})^K$ with $\\varphi ^ i = (\\varphi ^ i_1,¸\\ldots ,\\varphi ^ i_d) \\in C^\\infty _c (\\Omega ,S^{d\\times d})$ , we denote $\\operatorname{div}\\varphi = (\\operatorname{div}\\varphi ^1,\\ldots ,\\operatorname{div}\\varphi ^K) \\in C^\\infty _c (\\Omega ,\\mathbb {R}^d)^K$ with $\\operatorname{div}\\varphi ^i = (\\operatorname{div}\\varphi ^i_1,\\ldots ,\\operatorname{div}\\varphi ^i_d)\\in C^\\infty _c (\\Omega ,\\mathbb {R}^d)$ .",
"See [25], [26] for details.",
"Note that $\\operatorname{TGV}_\\alpha ^2(u)$ is finite if and only if $u\\in \\operatorname{BV}(\\Omega )^K$ and, in this case, the minimum is actually obtained [25].",
"Hence the term $\\min $ in the above definition is justified.",
"For a discretization and generalization later on, it is convenient to list some of the main properties of second-order $\\operatorname{TGV}$ in function space (see [25], [26]): (P1) If the minimum in (REF ) is obtained at $w=0$ , then $\\operatorname{TGV}_\\alpha ^2(u) = \\alpha _1 \\operatorname{TV}(u)$ , (P2) If the minimum in (REF ) is obtained at $w=\\mathrm {D}u$ , then $\\operatorname{TGV}_\\alpha ^2(u) = \\alpha _0 \\operatorname{TV}^ 2(u)$ , (P3) $\\operatorname{TGV}_\\alpha ^2(u) = 0$ if and only if $u$ is affine.",
"Here, $\\operatorname{TV}^ 2$ denotes a second-order TV functional which can be defined as $\\operatorname{TV}^ 2(u) = \\Vert \\mathrm {D}^ 2 u\\Vert _{\\mathcal {M}} $ where $\\mathrm {D}^ 2u$ denotes the second-order distributional derivative of $u$ and the above quantities are finite if and only if $\\mathrm {D}^ 2u \\in \\mathcal {M}(\\Omega ,S^ {d\\times d})^K$ .",
"We note that $w=0$ and $w=\\mathrm {D} u $ can trivially be obtained when $u$ is constant and affine, respectively, but $w=0$ is for instance also obtained under some symmetry conditions when $u$ is piecewise constant or when the ratio $\\alpha _0/\\alpha _1$ is large enough [29], [67].",
"Using the minimum representation above, a discretization of $\\operatorname{TGV}_\\alpha ^2$ in vector spaces on two-dimensionals grids can be given as follows.",
"Definition 2.2 (Discrete isotropic and anisotropic TGV in vector spaces) Set $U = \\mathbb {R}^K$ .",
"For $u=(u_{i,j})_{i,j} \\in U^{N\\times M}$ with $u_{i,j} \\in U$ , we define the discrete second-order TGV functional as $ \\operatorname{TGV}_\\alpha ^2(u) = \\min _{w \\in U^{(N-1)\\times M } \\times U^{N\\times (M-1)} }\\alpha _1 \\Vert \\mathrm {D}u - w\\Vert _1 + \\alpha _0 \\Vert \\mathcal {E}w\\Vert _1$ where $\\mathrm {D}u := (\\delta _{x+} u,\\delta _{y+} u) \\in U^{(N-1)\\times M } \\times U^{N\\times (M-1) }$ and $\\mathcal {E}w := \\mathcal {E}(w^1,w^2):= ( \\delta _{x-} w^1,\\delta _{y-} w^2,\\tfrac{\\delta _{y-} w^1 + \\delta _{x-} w^2 }{2}) \\in U^{(N-2)\\times M } \\times U^{N\\times (M-2) } \\times U^{(N-1)\\times (M-1) }$ with $\\delta _{x+}$ , $\\delta _{y+}$ and $\\delta _{x-}$ , $\\delta _{y-}$ being standard forward and backward differences, respectively.",
"Introducing a parameter $p \\in [1,\\infty )$ , the one-norms are given as $ \\Vert w\\Vert _1 = \\sum _{i,j} |(w_{i,j}^1,w_{i,j}^2)|_p \\text{ for } w \\in U^{(N-1)\\times M } \\times U^{N\\times (M-1)} $ and $ \\Vert z\\Vert _1 = \\sum _{i,j} \\left|\\left( \\begin{matrix}z_{i,j}^1 & z_{i,j}^3 \\\\ z_{i,j}^3 & z_{i,j}^2\\end{matrix}\\right) \\right|_p \\text{ for } z \\in U^{(N-2)\\times M } \\times U^{N\\times (M-2) } \\times U^{(N-1)\\times (M-1)}, $ where we extend the signals by zero to have the same size.",
"Here, $|(w^1,w^2)|_p:= \\left( |w^1|^p + |w^2|^p\\right)^{1/p}$ and $\\left|\\left(\\begin{matrix}z^1 & z^3 \\\\ z^3 & z^2\\end{matrix}\\right)\\right|_p:= \\left( |z^1|^p + |z^2|^p + 2|z^3|^p\\right)^{1/p}$ with $|\\cdot | $ the Euclidean norm on $U=\\mathbb {R}^K$ .",
"Note that we incorporate a pointwise $\\ell ^p$ -norm with $p \\in [1,\\infty )$ in the definition of $\\operatorname{TGV}_\\alpha ^2$ .",
"In the function space setting of Definition REF , this corresponds to choosing the appropriate dual norm on $X \\in \\lbrace \\mathbb {R}^d,S^{d\\times d}\\rbrace $ .",
"There, any choice of point-wise norm (also beyond $\\ell ^p$ -norms) yields equivalent functionals and function spaces, with the Euclidean norm being used in the first paper on TGV [29] and the most popular choices being $\\ell ^p$ norms with $p \\in \\lbrace 1,2\\rbrace $ .",
"The focus of this paper is the case $p=1$ since this is more frequently used in the manifold context because of being more amenable to numerical realization.",
"In the theory part we include the case of $p \\in [1,\\infty )$ since it can be done without additional effort.",
"We remark that the above discretization of $\\operatorname{TGV}_\\alpha ^2$ is slightly different from the standard one as provided, e.g., in [29].",
"The purpose of this is to achieve consistency of the zero set of both the continuous and discrete version as follows.",
"Also, note that, notation-wise, we do not distinguish between the continuous and the discrete version of TGV (and of TV and $\\operatorname{TV}^2$ ).",
"In the following, we will always refer to discrete versions.",
"Proposition 2.3 (Zero set of $\\operatorname{TGV}_\\alpha ^2$ in vector spaces) For $u\\in U^{N\\times M}$ we have that $\\operatorname{TGV}_\\alpha ^2(u) = 0$ if and only if $u$ is affine, i.e., there exist $a,b,c \\in U$ such that $u_{i,j} = ai + bj + c $ .",
"We provide only the basic steps: Setting $w = \\mathrm {D}u$ we get that $\\mathcal {E}w = \\mathcal {E}\\mathrm {D}u = \\left(\\delta _{x-}\\delta _{x+} u , \\delta _{y-}\\delta _{y+} u , \\tfrac{ \\delta _{y-}\\delta _{x+} u + \\delta _{x-}\\delta _{y+} u }{2}\\right).$ Now it is easy to see that if $u$ is linear as above, $\\mathcal {E}\\mathrm {D}u = 0$ and hence the choice $w = \\mathrm {D}u$ renders $\\operatorname{TGV}_\\alpha ^2(u) $ to be zero.",
"Conversely, $\\operatorname{TGV}_\\alpha ^2(u) = 0$ implies $w = \\mathrm {D}u$ and hence $\\mathcal {E}\\mathrm {D}u = 0$ .",
"It is easy to see that $\\delta _{x-}\\delta _{x+} u = \\delta _{y-}\\delta _{y+} u = 0$ implies that $u$ is of the form $ u_{i,j} = rij + ai + bj + c $ with $r,a,b,c \\in U$ .",
"But in this case, the last component of $\\mathcal {E}\\mathrm {D}u$ being zero implies that $ r = \\delta _{y-}\\delta _{x+} u = -\\delta _{x-}\\delta _{y+} u = -r $ , hence $r=0$ .",
"The latter proposition shows that, also after discretization, the kernel of $\\operatorname{TGV}_\\alpha ^2$ consists exactly of (discrete) affine functions.",
"In fact, this is one of the fundamental properties of $\\operatorname{TGV}$ (corresponding to (P3)) which should also be transferred to an appropriate generalization of TGV for manifolds.",
"Regarding appropriate counterparts of (P1) and (P2), we introduce the following definition.",
"Definition 2.4 Using the notation of Definition REF , we define $\\mathrm {D}^ 2u = \\mathcal {E}Du$ and $ \\operatorname{TV}(u) = \\Vert \\mathrm {D}u\\Vert _1, \\quad \\text{and} \\quad \\operatorname{TV}^ 2(u) = \\Vert \\mathrm {D}^ 2u\\Vert _1.",
"$ In summary, using the discretized version of $\\operatorname{TV}$ , $\\operatorname{TV}^2$ and $\\operatorname{TGV}_\\alpha ^2$ and the discrete notion of affine functions as in Proposition REF , we can conclude that the properties (P1) to (P3) of TGV in continuous vector spaces also transfer to its discretization."
],
[
"Univariate $\\operatorname{TGV}_\\alpha ^2$ on manifolds",
"When moving from the vector space to the manifold setting, a main difference is that vectors representing derivatives can no longer be made independent of their location, but are attached to a base point on the manifold.",
"In other words, in the Euclidean setting, all tangent spaces at all locations can be identified, which is not possible for manifolds in general.",
"Accordingly, when aiming to extend $\\operatorname{TGV}$ to the manifold setting, important questions are how to introduce the vector fields $(w_{i,j})_{i,j}$ appearing in the minimum representation, how to define $\\mathcal {E}w $ for such elements, and, most importantly, how to define a suitable distance-type function between such tangent vectors sitting in different points.",
"In the following, we describe our main ideas to resolve these questions.",
"In order to allow the reader to understand the underlying ideas more easily, we consider the univariate setting first.",
"Let $u = (u_i)_i$ be a finite sequence of points in a manifold $\\mathcal {M}$ with distance $d:\\mathcal {M}\\times \\mathcal {M}\\rightarrow [0,\\infty )$ .",
"Our first goal is to suitably extend the notion of forward differences $(\\delta _{x+}u)_i = u_{i+1} - u_i$ and introduce auxiliary variables $w_i$ which they can be compared to.",
"To this end, the central idea is to identify tangent vectors $w_i$ in the space $T_{u_i}\\mathcal {M}$ with point-tuples, i.e., $w_i \\simeq [u_i,y_i]$ with $y_i \\in \\mathcal {M}$ .",
"In the vector-space case, this can be done via the correspondence $w_i = y_{i} - u_i$ .",
"For manifolds, the correspondence can be established via the exponential and the logarithmic map.",
"That is, any $w_i \\in T_{u_i}\\mathcal {M}$ can be assigned to a unique point-tuple $[u_i,y_i]$ such that $\\exp _{u_i}(w_i) = y_i$ .",
"Conversely, any point-tuple $[u_i,y_i]$ can be assigned to (generally multiple) tangent vectors $w_i$ such that $w_i \\in \\log _{u_i}(y_i)$ .",
"(Note that the ambiguities in logarithmic map result from non-uniqueness of distance-minimizing geodesics, which is a rather degenerate situation in the sense that it only occurs for a set of points with measure zero [53], [34].)",
"Figure REF visualizes the correspondence between tangent vectors $w$ and point-tuples $[u,y]$ .",
"Figure: A three-point section of a signal uu together with tangent vectors ww represented by the endpoints yy.",
"The red vectors show the tangent vectors at the signal points, the blue dotted lines indicate the geodesics t↦exp u (tw)t \\mapsto \\exp _u(tw) and the yy are the endpoints exp u (w)\\exp _u(w).",
"The black lines indicate geodesic interpolations between the signal points and are for visualization purposes only.Exploiting this correspondence, our approach is to work with a discrete “tangent bundle” that is defined as the set of point-tuples $[u,y]$ with $u,y \\in \\mathcal {M}$ , rather than with the continuous one.",
"We refer to the elements in this discrete “tangent bundle” as tangent tuples.",
"The identification of forward differences with a tangent tuple is then naturally given via $(\\delta _{x+}u)_i = [u_i,u_{i+1}]$ .",
"The discretized vector-space version of second-order TGV (for which there is a one-to-one correspondence of tangent vectors and point tuples) can then in the univariate case be rewritten as $\\operatorname{TGV}_\\alpha ^2(u)& = \\min _{(w_i)_i } \\sum _{i} \\alpha _1 \\big | (\\delta _{x+} u)_i - w_i \\big | + \\alpha _0 \\big | (\\delta _{x-}w)_{i} \\big | \\\\& = \\min _{ (y_i)_i } \\sum _{i} \\alpha _1 D([u_i,u_{i+1}],[u_i,y_i]) + \\alpha _0 D([u_i,y_i] , [u_{i-1},y_{i-1}]),$ where we define $D([x,y],[u,v]) = \\big |[x,y] - [u,v]\\big | $ and $[x,y] - [u,v] = (y-x) - (v-u)$ .",
"Note that, in this context, $(w_i)_i$ , $(y_i)_i$ always denote finite sequences of points with their length being the same as the one of $\\delta _{x+}u$ and, whenever out of bound indices are accessed in such a summation, we assume the signals to be extended such that the evaluation of $D$ yields zero cost.",
"Hence, in order to extend TGV to the manifold-setting, we are left to appropriately measure the distance of two tangent tuples, i.e., generalize the expression $D([x,y],[u,v]) = |[x,y] - [u,v]|$ .",
"In case both tuples have the same base-point on the manifold, a simple and rather natural generalization is to measure their distance by the distance of their endpoints, that is, we set $D([x,y],[x,v]) = d(y,v)$ .",
"In the other case, i.e., for evaluating $D([u_i,y_i] , [u_{i-1},y_{i-1}])$ for $u_i \\ne u_{i-1}$ , there is no such simple or even unique generalization since the two tangent tuples belong to different (tangential) spaces.",
"A quite direct approach to overcome this is to first employ the concept of parallel transport on manifolds to shift both tangent tuples to the same location, and then to measure the distance of their endpoints.",
"One possible generalization for TGV proposed in this paper builds on this idea.",
"This is not the only possible way of measuring the distance of two tangent tuples.",
"Alternatively, one can for instance build on a discrete approximation of parallel transport, i.e., on a variant of the Schild approximation, to measure distance of point tuples at different locations.",
"This second approach is simpler to realize numerically, and it can be build on simpler differential-geometric tools, while the first one is more straight-forward.",
"Both approaches have their motivation and, given this ambiguity in possible generalizations, we take a more systematic approach to the topic.",
"That is, we first formulate fundamental properties that we require from reasonable generalizations of TGV to the manifold setting.",
"Then we translate these properties to requirements on admissible distance-type functions for measuring the distance of two tangent tuples.",
"Having established the later, we propose two possible concrete realizations that fulfill the required properties.",
"Axiomatic extension.",
"Assuming $D:\\mathcal {M}^2 \\times \\mathcal {M}^2 \\rightarrow [0,\\infty )$ to be a distance-type function for tangent tuples, we define a generalization of second-order TGV for univariate manifold-valued data as $ \\text{M-TGV}^2_\\alpha (u)= \\min _{(y_i)_i } \\sum _{i} \\alpha _1 d(u_{i+1},y_i) + \\alpha _0 D( [u_i,y_i],[u_{i-1},y_{i-1}]).$ In this context, we note that, while with TGV for discrete and continuous vector spaces it is known that the minimum in the above expression is attained, this is not clear a-priori for our generalization.",
"In order not to lose focus, we shift the discussion on existence to Section , noting at this point that for all versions of $\\text{M-TGV}^2_\\alpha $ proposed in this work, existence can be shown.",
"Accounting for the fact that the tangent tuples represent vectors in the tangent space, some basic identifications and requirements for the distance-type function $D$ follow naturally.",
"Zero elements in the discrete “tangent bundle” for instance correspond to tangent tuples of the form $[u,u] $ with $u \\in \\mathcal {M}$ and the distance of $[x,x]$ and $[y,y]$ should be zero.",
"Also, the distance of two identical elements should also be zero, i.e., $D([x,y],[x,y]) = 0$ , and, in the vector space case, the distance function should reduce to the difference of the corresponding tangent vectors.",
"Our goal is now to further restrict possible choices of distance-type function in order to obtain a meaningful generalization of TGV.",
"To this aim, we want to ensure that appropriate counterparts of the properties (P1) to (P3) are preserved for the resulting version of $\\text{M-TGV}^2_\\alpha $ also in the manifold setting.",
"In order to make this more precise, we first have to generalize the involved concepts.",
"We start with the notion of “affine”.",
"Given that the geodesics in a manifold play the role of straight lines in vector space, a natural generalization for $u=(u_i)_i $ a finite sequence in $ \\mathcal {M}$ to be affine is to require that all points $(u_i)_i$ are on a single geodesic at equal distance.",
"A difficulty that arises in connection with this definition is that, in general, as opposed to the vector space case, a geodesic connecting two points is not necessarily unique.",
"As a consequence, even though all points might be at the same distance when following a joint geodesic, the distance of each single pair of points in the manifold is not necessarily equal.",
"To account for that, we require in addition that the geodesic connecting all points also realizes the shortest connection between all involved points on the geodesic locally.",
"Definition 2.5 Let $u = (u_i)_{i=0}^ n$ be a finite sequence of points on a manifold $\\mathcal {M}$ .",
"We say that $u$ is geodesic if there exists a geodesic $\\gamma :[0,L] \\rightarrow \\mathcal {M}$ parametrized with unit speed such that $\\gamma (i L/n) = u_i$ for $i=0,\\ldots ,n$ and $d(u_i,u_{i+1}) = L/n$ for $i=0,\\ldots ,n-1$ , i.e., $\\gamma |_{[iL/n,(i+1)L/n]}$ is a geodesic of minimal length connecting $u_i$ and $u_{i+1}$ for all $i$ .",
"A second issue arising from non-uniqueness of geodesics is the fact that, even though every triplet of points in a signal $(u_i)_i$ might be connected by a geodesic, we cannot conclude that all points are connected by a unique geodesic.",
"Consequently, as a reasonable generalization of TGV will typically act local, in particular will be based on three point stencils, we cannot hope to obtain more than a local assertion for signals in the zero set of TGV.",
"To account for that, we introduce the following notion.",
"Definition 2.6 Let $u = (u_i)_{i=0}^ n$ be a finite sequence of points on a manifold $\\mathcal {M}$ .",
"We say that $u$ is locally geodesic if for each $i \\in \\lbrace 2,\\ldots ,n-1\\rbrace $ , the sequence $(u_{j})_{j=i-1}^{i+1}$ is geodesic.",
"As one might expect, in the situation that subsequent points of a signal are sufficiently close such that they admit a unique connecting geodesic we obtain equivalence of the notions of geodesic and locally geodesic signals.",
"In this respect, we again note that, in the general non-local situation, the existence of unique minimizing geodesics is true for any set of points outside a set of measure zero [53], [34].",
"Lemma 2.7 If a sequence of points $u = (u_i)_i $ in $ \\mathcal {M}$ is such that the length minimizing geodesic connecting each pair $u_i$ , $u_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1$ is unique, then $u$ is locally geodesic if and only if it is geodesic.",
"If $u$ is geodesic, it is obviously locally geodesic.",
"Now assume that $u$ is locally geodesic and that the length minimizing geodesic connecting each pair $u_i$ , $u_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1$ is unique.",
"Then there exists a geodesic connecting $ u_0$ , $ u_1$ , $ u_2$ at equal distance, i.e., the distance between two subsequent points equals the length of the geodesic segment.",
"We proceed recursively: Now assume that $ u_{i-2}$ , $ u_{i-1}$ , $ u_{i}$ are connected by a geodesic at equal distance.",
"As $u$ is locally geodesic also $ u_{i-1}$ , $ u_{i}$ , $ u_{i+1}$ are connected by a geodesic.",
"Now by uniqueness, the two geodesics between $u_i$ and $u_{i-1}$ must coincide.",
"Hence all points $u_{i-2}$ till $u_{i+1}$ are on the same geodesic at equal speed.",
"Proceeding iteratively, the result follows.",
"In order to investigate the counterparts of properties (P1) and (P2), we need to define generalizations of the $\\operatorname{TV}$ and $\\operatorname{TV}^2$ functionals to the manifold setting.",
"For $\\operatorname{TV}$ , a natural generalization is to set, for $u=(u_i)_i$ in $\\mathcal {M}$ , $ \\operatorname{TV}(u) = \\sum \\nolimits _i d(u_{i+1},u_i) ,$ see also [93].",
"For $\\operatorname{TV}^2$ , a generalization is not that immediate.",
"In [10], second-order TV was essentially (and up to a constant) defined as $\\operatorname{TV}^2(u) = 2\\sum _i \\inf _{c \\in [u_{i-1},u_{i+1}]_{\\frac{1}{2}} } d(c,u_{i}), $ which, in the vector space setting reduces to $ \\operatorname{TV}^2(u) = 2\\sum \\nolimits _i d(\\tfrac{u_{i-1}+u_{i+1}}{2},u_{i}) = \\sum \\nolimits _i \\big |u_{i-1} -2u_i + u_{i+1} \\big |.$ However, as can be deduced by considering $\\operatorname{TV}^2$ as special case of the different versions of TGV proposed below, this is not the only generalization which fulfills such a property.",
"We will call a function an admissible generalization of $\\operatorname{TV}^2$ whenever, in the vector space setting, it reduces to $\\operatorname{TV}^ 2$ as in Equation (REF ) above.",
"Using these prerequisites, we now formulate our requirements on an appropriate generalization of $\\operatorname{TGV}$ to the manifold setting of the form (REF ).",
"(M-P1) In the vector space setting, $\\text{M-TGV}^2_\\alpha $ reduces to the univariate version of (REF ).",
"(M-P2) If the minimum in (REF ) is attained at $(y_i)_i = (u_i)_i$ , i.e., the tangent tuples $[u_i,y_i]$ all correspond to zero vectors, then $\\text{M-TGV}^2_\\alpha (u) = \\alpha _1 \\operatorname{TV}(u)$ with TV as in (REF ).",
"(M-P3) If the minimum in (REF ) is attained at $(y_i)_i = (u_{i+1})_i$ , i.e., the tangent tuples $[u_i,y_i]$ all correspond to $(\\delta _{x+} u)_i$ , then $\\text{M-TGV}^2_\\alpha (u) = \\alpha _0 \\operatorname{TV}^ 2(u)$ with $\\operatorname{TV}^ 2$ an admissible generalization of $\\operatorname{TV}^ 2$ .",
"(M-P4) $\\text{M-TGV}^2_\\alpha (u) = 0$ if and only if $u$ is locally geodesic according to Definition REF .",
"As the following Proposition shows, the Properties (M-P1) to (M-P3) already follow from basic requirements on the distance function that arise from the intuition that tangent tuples represent tangent vectors.",
"The most restrictive property is (M-P4), which requires an additional assumption on the distance function.",
"Proposition 2.8 Assume that the function $D:\\mathcal {M}^2 \\times \\mathcal {M}^2 \\rightarrow [0,\\infty )$ is such that $D([x,x],[u,u]) = 0$ for any $x,u \\in \\mathcal {M}$ , $D([x,y],[u,v]) = \\big | (y-x) - (v-u)\\big |$ in case $\\mathcal {M}= \\mathbb {R}^K$ .",
"Then, for $\\text{M-TGV}^2_\\alpha $ as in (REF ), the properties (M-P1) to (M-P3) hold.",
"If we further assume that for any geodesic three-point signal $(v_j)_{j=i-1}^{i+1}$ it follows that $D([v_i,v_{i+1}],[v_{i-1},v_i]) = 0$ , then $\\text{M-TGV}^2_\\alpha (u) = 0$ for any locally geodesic signal $u = (u_i)_i$ .",
"Conversely, assume that for any three-point signal $(v_j)_{j=i-1}^{i+1}$ , such that the geodesic connecting each pair $v_i$ , $v_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1 $ is unique, $D([v_i,v_{i+1}],[v_{i-1},v_i]) = 0$ implies that $(v_j)_{j=i-1}^{i+1}$ is geodesic.",
"Then, for any signal $(u_i)_i$ where the geodesic connecting each pair $u_i$ , $u_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1$ is unique, $\\text{M-TGV}^2_\\alpha (u) = 0$ implies that $u$ is geodesic.",
"In the vector space case we get by our assumptions and since $d(x,u) = |x-u|$ , that $\\begin{aligned}\\text{M-TGV}^2_\\alpha (u)= \\min _{(y_i)_i } &\\sum \\nolimits _{i} \\alpha _1 \\big |u_{i+1} - u_i - ( y_i - u_i)\\big | + \\alpha _0 \\big |(y_i - u_i) - (y_{i-1} - u_{i-1})\\big | \\\\=\\min _{(w_i)_i } &\\sum \\nolimits _{i} \\alpha _1 \\big | (\\delta _{x+} u)_i - w_i\\big | + \\alpha _0 \\big |(\\delta _{x-}w)_i \\big |\\end{aligned}$ which coincides with the univariate version of $\\operatorname{TGV}$ as in (REF ), hence (M-P1) holds.",
"Now in case the minimum in (REF ) is achieved for $(y_i)_i = (u_i)_i$ we get that $ \\text{M-TGV}^2_\\alpha (u) = \\sum \\nolimits _i\\alpha _1 d(u_{i+1},u_i) + \\alpha _0 D([u_i,u_i],[u_{i-1},u_{i-1}]) = \\alpha _1 \\operatorname{TV}(u) $ and (M-P2) holds.",
"Similarly, if the minimum in (REF ) is achieved for $(y_i)_i = (u_{i+1})_i$ we get that $ \\text{M-TGV}^2_\\alpha (u) = \\sum \\nolimits _i \\alpha _0 D([u_i,u_{i+1}],[u_{i-1},u_{i}])$ which is an admissible generalization of $\\operatorname{TV}^2$ since, by assumption, $D([u_i,u_{i+1}],[u_{i-1},u_{i}]) = \\big | u_{i+1}-2u_i + u_{i-1}\\big |$ in case $\\mathcal {M}= \\mathbb {R}^K$ .",
"Hence (M-P3) holds.",
"Now in case $(u_i)_i$ is locally geodesic, we can choose $(y_i)_i = (u_{i+1})_i $ to estimate $\\text{M-TGV}^2_\\alpha $ from above with the right-hand side in (REF ) and obtain that $\\text{M-TGV}^2_\\alpha (u) =0 $ .",
"Conversely, in case $\\text{M-TGV}^2_\\alpha (u)=0$ we get that necessarily $(y_i)_i = (u_{i+1})_i $ and $\\text{M-TGV}^2_\\alpha $ reduces to the right hand side in (REF ).",
"By the assumption on $D$ , this implies that $u$ is locally geodesic and by Lemma REF the result follows.",
"Guided by the aim of fulfilling the requirements of Proposition REF , in the following, we propose two possible choices for $D(\\cdot ,\\cdot )$ which result in two different concrete versions of $\\text{M-TGV}^2_\\alpha $ .",
"Both variants follow the general idea of first transporting different tangent tuples to the same location in $\\mathcal {M}$ and then measuring the distance there.",
"The main difference between the two variants will be the way the transport is carried out.",
"The first concept is motivated by the so called Schild's ladder approximation of parallel transport.",
"Its implementation requires only basic differential geometric concepts and is therefore presented first.",
"The second realization, which we call the parallel transport variant, requires more differential geometric concepts [76], [38] and is therefore presented afterwards.",
"However, the Schild's ladder variant can be seen as an approximation of the parallel transport variant as explained below.",
"Realization via Schild's-approximation.",
"Let $[x,y]$ and $[u,v]$ be two tuples in $\\mathcal {M}^ 2$ for which we want to define $D([x,y],[u,v])$ and assume for the moment that distance-minimizing geodesics are unique.",
"Motivated by the Schild's ladder approximation of parallel transport [39], [55], we consider the following construction (see Figure REF ): take $c$ to be the midpoint of the points $v$ and $x$ , i.e., $c = [x,v]_{\\frac{1}{2}};$ set $y^{\\prime } = [u,c]_2$ , i.e., reflect $u$ at $c$ .",
"Then, we claim that the distance-type function $D([x,y],[u,v]) = d(y,y^{\\prime })$ fulfills the requirements of Proposition REF .",
"The above construction may be motivated as follows.",
"To compare the tangent vectors $\\log _x(y)$ and $\\log _u(v)$ , which correspond to the tangent tuples $[x,y]$ and $[u,v]$ , we could move them to the same point using parallel transport and then take the norm induced by the Riemannian metric in the corresponding tangent space.",
"Using the above construction to obtain $y^{\\prime }$ from $x$ and $[u,v]$ , the tangent vector $\\log _x(y^{\\prime })$ can be seen as an approximation of the parallel transport of $\\log _u(v)$ to $x$ (in the limit $x \\rightarrow u$ ) [39], [55].",
"The difference of this vector to $\\log _x(y)$ is then approximated by $d(y,y^{\\prime })$ .",
"We notice that, as can be easily seen, this Schild's ladder approximation of parallel transport, i.e., the construction of $y^{\\prime }$ and $\\log _x(y^{\\prime })$ as above, is exact in the vector space case.",
"Figure: Approximate parallel transport of log u (v)\\log _u(v) to xx via the Schild's ladder construction.Hence we propose to measure the deviation between the tangent tuples $[x,y]$ and $[u,v]$ by transporting $[u,v]$ to $x$ using the construction above to obtain the tangent tuple $[x,y^{\\prime }]$ and then to compare the resulting tangent tuples sitting in the same point $x$ by measuring the distance of their endpoints $y,y^{\\prime }$ .",
"Since in general geodesics are not unique, we have to minimize over all possible constructions as above, i.e., consider all midpoints of all length-minimizing geodesics.",
"This yields the following distance-type function $ \\text{D}_{\\text{S}}([x,y],[u,v]) = \\min _{y^{\\prime } \\in \\mathcal {M}} d(y^{\\prime },y) \\text{ such that } y^{\\prime } \\in [u,c]_2 \\text{ with } c \\in [x,v]_{\\frac{1}{2}}.$ It is immediate that $\\text{D}_{\\text{S}}$ is positive and zero for identical elements.",
"Furthermore if $\\text{D}_{\\text{S}}([x,y],[u,v])= 0$ then $[x,y]$ can be interpreted to be equal to the approximate parallel transport of $[u,v]$ to $x$ .",
"Indeed, in the vector space case, $\\text{D}_{\\text{S}}([x,y],[u,v])= 0$ if and only if $v-u$ is parallel to $y-x$ and has the same length and direction.",
"Plugging in $\\text{D}_{\\text{S}}$ in (REF ) we define $ \\begin{aligned}\\text{S-TGV}^2_\\alpha (u)= \\min _{(y_i)_i } &\\sum \\nolimits _{i} \\alpha _1 d(u_{i+1},y_i) + \\alpha _0 \\text{D}_{\\text{S}}\\big ([u_i,y_{i}],[u_{i-1},y_{i-1}]\\big ).",
"\\\\\\end{aligned}$ Regarding the properties (M-P1) to (M-P4), we then get the following result.",
"Theorem 2.9 The $\\text{S-TGV}^2_\\alpha $ functional as in (REF ) satisfies (M-P1) and (M-P3).",
"If a finite sequence of points $u=(u_i)_i$ is locally geodesic, then $\\text{S-TGV}^2_\\alpha (u) = 0$ .",
"Further, if $\\text{S-TGV}^2_\\alpha (u) = 0$ and the geodesic connecting each pair $u_i$ , $u_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1$ is unique, then $u$ is geodesic.",
"We verify the assumptions of Proposition REF for $D = \\text{D}_{\\text{S}}$ .",
"It can be easily seen that $\\text{D}_{\\text{S}}([x,x],[u,u]) = \\text{D}_{\\text{S}}([x,y],[x,y]) = 0$ for $x,u,y \\in \\mathcal {M}$ .",
"Also, in the case $\\mathcal {M}= \\mathbb {R}^K$ , the approximate parallel transport as in the definition of $\\text{D}_{\\text{S}}$ above coincides with the parallel shift of vectors.",
"Hence $\\text{D}_{\\text{S}}([x,y],[u,v]) = \\text{D}_{\\text{S}}( [0,y-x],[0,v-u]) = \\big |(y-x) - (v-u) \\big | $ and (M-P1) to (M-P3) holds.",
"Now if $v = (v_j)_{j=i-1}^ {i+1}$ is locally geodesic, then $v_i \\in [v_{i},v_{i}]_{\\frac{1}{2}}$ and $v_{i+1} \\in [v_{i-1},v_i]_2$ , hence $\\text{D}_{\\text{S}}([v_i,v_{i+1}],[v_{i-1},v_i]) = 0.$ Conversely, $\\text{D}_{\\text{S}}([v_i,v_{i+1}],[v_{i-1},v_i]) = 0$ implies that there exists $y^{\\prime } \\in [v_{i-1},v_i]_2$ such that $d(y^{\\prime },v_{i+1}) = 0$ .",
"But this implies that $v_{i+1} \\in [v_{i-1},v_i]_2$ and, by the assumption on unique geodesics, it follows that $v$ is geodesic.",
"Remark 2.10 Assuming that, for each $i$ , $u_i$ and $y_i$ are sufficiently close such that they are connected by a unique length minimizing geodesic, we get that $[u_i,[u_i,y_i]_{\\frac{1}{2}}]_2 = y_i$ and hence $ \\text{D}_{\\text{S}}([u_i,u_{i+1}],[u_i,y_i]) = d(u_{i+1},y_i)$ .",
"Consequently, in this situation, an equivalent definition of $\\text{S-TGV}^2_\\alpha $ can be given as $\\begin{aligned}\\text{S-TGV}^2_\\alpha (u)= \\min _{(y_i)_i } &\\sum \\nolimits _{i} \\alpha _1 \\text{D}_{\\text{S}}([u_i,u_{i+1}],[u_i,y_i]) + \\alpha _0 \\text{D}_{\\text{S}}\\big ([u_i,y_{i}],[u_{i-1},y_{i-1}]\\big ) \\\\\\end{aligned}$ This definition regards the mapping $(u_i)_i\\mapsto ([u_i,u_{i+1}])_i$ as discrete gradient operator, mapping from $\\mathcal {M}$ to the discrete tangent space, and exclusively works in the discrete tangent space with $\\text{D}_{\\text{S}}$ the canonical distance-type function.",
"Remark 2.11 We note that an alternative choice for $\\text{D}_{\\text{S}}$ would have been to transport $[u_{i},y_{i}]$ to $u_{i-1}$ rather than $[u_{i-1},y_{i-1}] $ to $u_i$ .",
"However, since the distance $d(u_{i+1},y_i)$ in $\\text{S-TGV}^2_\\alpha $ as in (REF ) can be interpreted as evaluating the difference of the forward difference $(\\delta _{x+}u)_i $ with $[u_i,y_{i}]$ in the center point $u_i$ , it seems natural to evaluate also the backward difference of the signal $([u_i,y_i])_i$ at the center point.",
"Remark 2.12 We also note that, as opposed to the vector space setting, the distance function $\\text{D}_{\\text{S}}$ is in general not symmetric, i.e., $\\text{D}_{\\text{S}}([x,y],[u,v]) \\ne \\text{D}_{\\text{S}}([u,v],[x,y])$ .",
"To obtain symmetry, alternative definitions of $\\text{D}_{\\text{S}}$ could be given as $ \\tilde{\\text{D}_{\\text{S}}}([x,y],[u,v]) = \\text{D}_{\\text{S}}([x,y],[u,v]) + \\text{D}_{\\text{S}}([u,v],[x,y]) $ or $ \\tilde{\\text{D}_{\\text{S}}}([x,y],[u,v]) = \\min _{c_1,c_2} \\, d(c_1,c_2) \\quad \\text{subject to } c_1 \\in [x,v]_{\\frac{1}{2}} \\text{ and } c_2 \\in [u,y]_{\\frac{1}{2}}.$ For the sake of simplicity and in order to obtain the relation with parallel transport, however, we have defined $\\text{D}_{\\text{S}}$ as in (REF ).",
"Realization via parallel transport.",
"The Schild's ladder construction defined above can be seen as a discrete approximation of parallel transport.",
"Alternatively, we can use the identification of point-tuples $[u,v],[x,y]$ with vectors in the tangent space at $u$ and $x,$ respectively, and use the parallel transport directly to transport the respective vectors to a common base point.",
"That is, assuming – for the moment – uniqueness of length-minimizing geodesics, we can identify each $[u,v]$ with $w \\in T_u \\mathcal {M}$ such that $v = \\log _u (w)$ and compare $\\operatorname{pt}_x(w)$ , the vector resulting from the parallel transport of $w$ to $T_x \\mathcal {M}$ , to $\\log _x(y)\\in T_x \\mathcal {M}$ .",
"This yields a distance-type function for two point-tuples $[u,v]$ , $[x,y]$ as $\\text{D}_{\\text{pt}}([x,y],[u,v]) = \\big | \\log _x(y) - \\operatorname{pt}_x(\\log _u(v))\\big |_x$ where $\\big |\\cdot \\big |_x$ denotes the norm in $T_x\\mathcal {M}$ .",
"Comparing this to $\\text{D}_{\\text{S}}$ we note that, besides using a different notion of transport, we now measure the distance with the norm in the tangent space rather than with the distance of endpoints.",
"The reason for doing so is that, in this situation, approximating the norm in the tangent space via the distance of endpoints does not yield further simplification since the the parallel transport forces us to work in the tangent space anyway.",
"Now in general, length minimizing geodesics are not necessarily unique.",
"To deal with this issue, we have defined the $\\log $ mapping and the parallel transport to be set-valued (see Section REF ), i.e., for $u,v \\in \\mathcal {M}$ , not necessarily close, and $w \\in T_v\\mathcal {M}$ we define $\\operatorname{pt}_u(w) \\subset T_u\\mathcal {M}$ to be the set of all vectors in $T_u\\mathcal {M}$ which can be obtained by parallel-transporting $w$ to $T_u\\mathcal {M}$ along a length minimizing geodesic.",
"Note that by isometry of the parallel transport, the length of all such vectors is the same but by holonomy their orientation might be different.",
"In order to adapt the above definition of $\\text{D}_{\\text{pt}}{}$ to this situation, we generalize the distance function $\\text{D}_{\\text{pt}}{}$ for tangent tuples $[x,y]$ and $[u,v]$ as $ \\text{D}_{\\text{pt}}([x,y],[u,v]):= \\min _{\\begin{array}{c}z_1 \\in \\log _x(y) \\\\ z_2 \\in T_x\\mathcal {M}\\end{array} } \\big |z_1 - z_2\\big |_{x} \\text{ such that } z_2 \\in \\operatorname{pt}_x(w) \\text{ with } w \\in \\log _u(v).$ Using this notation, a parallel-transport-based version of $\\text{M-TGV}^2_\\alpha $ , denoted by $\\text{PT-TGV}^2_\\alpha $ , can then be defined for the general situation as $\\text{PT-TGV}^2_\\alpha (u)= \\min _{(y_i)_i } \\sum \\nolimits _{i} \\alpha _1 d(u_{i+1},y_i) + \\alpha _0 \\text{D}_{\\text{pt}}( [u_i,y_i],[u_{i-1},y_{i-1}]) .$ Similarly to the Schild's-version, we then get the following result.",
"Theorem 2.13 The functional $\\text{PT-TGV}^2_\\alpha $ satisfies (M-P1) and (M-P3).",
"If a finite sequence of points $u=(u_i)_i$ is locally geodesic, then $\\text{PT-TGV}^2_\\alpha (u) = 0$ .",
"Further, if $\\text{PT-TGV}^2_\\alpha (u) = 0$ and the geodesic connecting each pair $u_i$ , $u_{i^{\\prime }}$ with $|i-i^{\\prime }|\\le 1$ is unique, then $u$ is geodesic.",
"The proof is similar to the one of Theorem REF below considering the bivariate setting.",
"Remark 2.14 Since the expression $d(u_{i+1},y_i)$ can be seen as approximation of $|\\log _{u_i}(u_{i+1}) - w_i|_{u_i}$ , where $w_i \\in \\log _{u_i}(y_i)$ , an alternative definition of $\\text{PT-TGV}^2_\\alpha $ (assuming uniqueness of geodesics for simplicity) only in terms of tangent vectors is given as $\\widetilde{\\text{PT-TGV}^2_\\alpha }(u)= \\min _{w_i \\in T_{u_i}\\mathcal {M}} \\sum \\nolimits _{i} \\alpha _1 \\big |\\log _{u_i}(u_{i+1}) - w_i \\big |_{u_i} + \\alpha _0 \\big | w_i - \\operatorname{pt}_{u_i}(w_{i-1})\\big |_{u_i} .$ We believe, however, that the originally proposed version is preferable since the fact that the first term in $\\text{PT-TGV}^2_\\alpha $ only involves the standard manifold distance of two points simplifies the numerical realization.",
"That is, with the standard distance we are able to solve certain subproblems (proximal mappings) in the algorithm explicitly whereas the other version would require additional inner loops (see Section for details)."
],
[
"Bivariate $\\operatorname{TGV}_\\alpha ^2$ on manifolds",
"The goal of this section is to extend $\\text{M-TGV}^2_\\alpha $ to the bivariate case.",
"To this aim, we first observe that the bivariate version of $\\operatorname{TGV}_\\alpha ^2$ for vector spaces as in (REF ) can be written as $ \\operatorname{TGV}_\\alpha ^2(u) = \\min _{(w_{i,j}^1)_{i,j},(w_{i,j}^2)_{i,j}} \\sum \\nolimits _{i,j} \\alpha _1 \\bigg ( \\big | (\\delta _{x+}u)_{i,j}- w_{i,j}^ 1\\big |^p + \\big | (\\delta _{y+}u)_{i,j}- w_{i,j}^ 2\\big |^p \\bigg )^{1/p} \\\\+\\alpha _0 \\bigg ( \\big | w^1 _{i-1,j} - w^1_{i,j}\\big |^p + \\big | w^2 _{i,j-1} - w^2_{i,j}\\big |^p + 2^{1-p} \\big | (w^1 _{i,j-1} - w^1_{i,j}) + (w^2 _{i-1,j} - w^2_{i,j})\\big |^p \\bigg ) ^{1/p},$ where we set all norms $|\\cdot |^p$ to be zero whenever out of bound indices are involved.",
"The first four summands in the above definition of $\\operatorname{TGV}_\\alpha ^2$ can all be transferred to the manifold setting using the previously introduced distance-type functions for tangent tuples.",
"The additional difficulty of the bivariate situation arises form the fifth term, which first combines both differences and sums of tangent tuples and then measures the norm.",
"Again possible generalizations are not unique and we employ an axiomatic approach to propose reasonable choices.",
"Axiomatic extension.",
"Denote by $D(\\cdot ,\\cdot )$ one of the two previously introduced distance functions for tangent tuples and assume that $D^{\\text{sym}} : \\mathcal {M}^ 2 \\times \\mathcal {M}^ 2\\times \\mathcal {M}^ 2\\times \\mathcal {M}^ 2 \\rightarrow \\mathbb {R}$ generalizes the fifth term in (REF ), which corresponds to the mixed derivatives.",
"A bivariate version of $\\text{M-TGV}^2_\\alpha $ can then be given as $ \\begin{aligned}\\text{M-TGV}_\\alpha ^ 2(u)= \\min _{y^ 1_{i,j},y^ 2_{i,j}} &\\alpha _1 \\sum \\nolimits _{i,j} \\Big ( d(u_{i+1,j},y^ 1_{i,j})^p + d(u_{i,j+1},y^ 2_{i,j})^p \\Big ) ^{1/p}\\\\+& \\alpha _0 \\sum \\nolimits _{i,j} \\Big ( D\\big ([u_{i,j},y^1_{i,j}],[u_{i-1,j},y^1_{i-1,j}]\\big )^p + D\\big ([u_{i,j},y^2_{i,j}],[u_{i,j-1},y^2_{i,j-1}]\\big )^p \\\\&\\qquad + 2^{1-p} D^{\\text{sym}} ([u_{i,j},y^1_{i,j}],[u_{i,j},y^2_{i,j}],[u_{i,j-1},y^1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}])^p \\Big )^{1/p}.\\end{aligned}$ The basis for this generalization is again the representation of tangent vectors with tangent tuples, only that now for each $u_{i,j}$ we consider two tangent vectors $w^1_{i,j}, w^ 2_{i,j}$ and corresponding points $y^1_{i,j}, y^ 2_{i,j}$ in order to represent horizontal and vertical derivatives, see Figure REF .",
"Remember that in this paper, we focus on the case $p=1,$ for which we derive an algorithmic realization later on.",
"However, since it causes no additional effort in this section, we provide a formulation for all $p \\in [1,\\infty )$ here.",
"Figure: A three-by-three point section of a bivariate signal uu together with tangent vectors ww represented by endpoints yy.",
"The blue, dotted lines indicate the geodesics t↦exp(tw)t \\mapsto \\exp (tw) connecting the signal points uu with the endpoints yy.",
"The black lines indicate a piecewise geodesic interpolation of the signal points and are for visualization purposes only.We now extend the requirements (M-P1) to (M-P4) to the bivariate case.",
"The generalization of TV to bivariate manifold-valued data is quite straightforward [93].",
"For $u = (u_{i,j})_{i,j}$ we define $ \\operatorname{TV}(u) = \\sum \\nolimits _{i,j} \\Big ( d(u_{i+1,j},u_{i,j})^p + d(u_{i,j+1},u_{i,j})^p \\Big )^{1/p} .$ A generalization of second-order $\\operatorname{TV}$ is again less straightforward and we call any functional $\\operatorname{TV}^2$ acting on $u = (u_{i,j})_{i,j}$ an admissible generalization of $\\operatorname{TV}^2$ if it reduces, in the vector space setting, to $\\operatorname{TV}^2 $ as given in Definition REF .",
"Note that our version of $\\operatorname{TV}^2$ differs from the definition of $\\operatorname{TV}^ 2$ as given in [10] since we use a symmetrization of the mixed derivatives.",
"A generalization of affine functions for the bivariate manifold setting is given as follows.",
"Definition 2.15 Let $u = (u_{i,j})_{i,j}$ be a finite sequence of points on a manifold $\\mathcal {M}$ .",
"We say that $u$ is (locally) geodesic if, for each $(i_0,j_0)$ , the univariate signals $ (u_{i,j_0})_{i} $ , $ (u_{i_0,j})_{j}$ and $(u_{i_0+k,j_0-k})_{k}$ are (locally) geodesic.",
"Note that this indeed generalizes the notion of affine for vector space data.",
"While the first two conditions ensure that $u$ is of the form $u_{i,j} = aij + b i + c j + d$ , the last condition ensures that $a = 0$ , i.e., no mixed terms occur.",
"In the vector space case the third condition is equivalent to requiring that $(u_{i_0+k,j_0+k})_{k}$ is (locally) geodesic.",
"In the manifold setting, this is not equivalent in general and hence our definition of geodesic is somewhat anisotropic.",
"The reason for using the anti-diagonal ($(i_0 + k,j_0-k)$ ) and not the diagonal ($(i_0 + k,j_0+k)$ ) direction is that the former arises naturally from the usage of forward-backward differences, as will become clear in the discussion after Proposition REF .",
"An alternative would also be to require both $(u_{i_0+k,j_0+k})_{k}$ and $(u_{i_0+k,j_0-k})_{k}$ to be geodesic.",
"This, however, seems rather restrictive and for general manifolds such signals might even not exist.",
"Hence we do not impose this additional restriction.",
"As a direct consequence of the corresponding result in the univariate setting, we obtain equivalence of the notion of locally geodesic and geodesic if neighboring points are sufficiently close.",
"Lemma 2.16 Let $u = (u_{i,j})_{i,j} $ be a finite sequence in $ \\mathcal {M}$ .",
"If the length minimizing geodesic connecting any two points $u_{i,j}$ and $u_{i^{\\prime },j^{\\prime }}$ with $\\max \\lbrace |i-i^{\\prime }|,|j-j^{\\prime }|\\rbrace \\le 1$ is unique, then $u$ is locally geodesic if and only if it is geodesic.",
"With these prerequisites, we now state our requirements for a reasonable generalization of TGV in the bivariate case.",
"(M-P1') In the vector space setting, $\\text{M-TGV}^2_\\alpha $ reduces to vector-space TGV as in (REF ).",
"(M-P2') If the minimum in (REF ) is attained at $(y^ 1_{i,j})_{i,j} =(y^ 2_{i,j})_{i,j} = (u_{i,j})_{i,j}$ , i.e., the tangent tuples all correspond to zero vectors, then $\\text{M-TGV}^2_\\alpha (u) = \\alpha _1 \\operatorname{TV}(u)$ with TV as in (REF ).",
"(M-P3') If the minimum in (REF ) is attained at $(y^ 1_{i,j})_{i,j} = (u_{i+1,j})_{i,j}$ and $(y^ 2_{i,j})_{i,j} = (u_{i,j+1})_{i,j}$ , i.e., the tangent tuples $[u_{i,j},y^1_{i,j}]$ and $[u_{i,j},y^2_{i,j}]$ all correspond to $(\\delta _{x+} u)_{i,j}$ and $(\\delta _{y+} u)_{i,j}$ , respectively, then $\\text{M-TGV}^2_\\alpha (u) = \\alpha _0 \\operatorname{TV}^ 2(u)$ with $\\operatorname{TV}^ 2$ an admissible generalization of $\\operatorname{TV}^ 2$ .",
"(M-P4') $\\text{M-TGV}^2_\\alpha (u) = 0$ if and only if $u$ is locally geodesic according to Definition REF .",
"Those properties translate to requirements for the involved distance-type functions as follows.",
"Proposition 2.17 Assume that the function $D:\\mathcal {M}^2 \\times \\mathcal {M}^2 \\rightarrow [0,\\infty )$ satisfies the assumptions of Proposition REF and assume that $\\text{D}^{\\text{sym}}$ is such that $\\text{D}^{\\text{sym}}([x,x],[x,x],[u,u],[u,u]) = 0$ for any $x,u \\in \\mathcal {M}$ and, in case $\\mathcal {M}= \\mathbb {R}^K$ , $\\text{D}^{\\text{sym}}([{u_{\\circ ,\\circ }},{y^1_{\\circ ,\\circ }}],[{u_{\\circ ,\\circ }},{y^2_{\\circ ,\\circ }}],[{u_{\\circ ,-}},{y^1_{\\circ ,-}}],[{u_{-,\\circ }},{y^2_{-,\\circ }}]) \\\\ = \\big | {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^1_{\\circ ,-}}- {u_{\\circ ,-}}) + {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^2_{-,\\circ }}- {u_{-,\\circ }})\\big |.$ Then, for $\\text{M-TGV}^2_\\alpha $ as in (REF ), the properties (M-P1') to (M-P3') hold.",
"If further $\\text{D}^{\\text{sym}}([v_{i,j},v_{i+1,j}],[v_{i,j},v_{i,j+1}],[v_{i,j-1},v_{i+1,j-1}],[v_{i-1,j},v_{i-1,j+1}]) = 0,$ for any geodesic three-by-three signal $v= (v_{i,j})_{i,j}$ , then $\\text{M-TGV}^2_\\alpha (u) = 0$ for any locally geodesic $u$ .",
"Conversely, assume that for any three-by-three signal in $v= (v_{i,j})_{i,j}$ where the geodesic connecting each pair $v_{i,j}$ , $v_{i^{\\prime },j^{\\prime }}$ are unique and where $(v_{i,j})_i$ and $(v_{i,j})_j$ are geodesic, it holds that $\\text{D}^{\\text{sym}}([v_{i,j},v_{i+1,j}],[v_{i,j},v_{i,j+1}],[v_{i,j-1},v_{i+1,j-1}],[v_{i-1,j},v_{i-1,j+1}]) = 0,$ implies also $(v_{i+k,j-k})_{k=-1}^1$ being geodesic.",
"Then, for any $u= (u_{i,j})_{i,j}$ such that the geodesic connecting each pair $u_{i,j}$ and $u_{i^{\\prime },j^{\\prime }}$ with $\\max \\lbrace |i-i^{\\prime }|,|j-j^{\\prime }|\\rbrace \\le 2$ is unique, we get that $\\text{M-TGV}^2_\\alpha (u) = 0$ implies $u$ being geodesic.",
"See Section REF in the Appendix.",
"Similar to the univariate case, the most restrictive requirement for $\\text{D}^{\\text{sym}}$ is that, in case of uniqueness of length-minimizing geodesics, for $(u_{i,j})_{i,j}$ being locally geodesic in horizontal and vertical direction, it holds that $\\text{D}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}]) = 0 \\\\\\Longleftrightarrow (u_{i+k,j-k})_k \\text{ is locally geodesic}.$ Let us discuss, again temporarily assuming uniqueness of geodesics, a general strategy to design the function $\\text{D}^{\\text{sym}}$ such that this property holds.",
"We consider the situation around a point ${u_{\\circ ,\\circ }}$ and denote by $w^1_{\\circ ,\\circ },w^2_{\\circ ,\\circ },w^1_{\\circ ,-},w^2_{-,\\circ }$ four tangent vectors corresponding to the four tangent tuples $[{u_{\\circ ,\\circ }},{y^1_{\\circ ,\\circ }}],$ $[{u_{\\circ ,\\circ }},{y^2_{\\circ ,\\circ }}],$ $[{u_{\\circ ,-}},{y^1_{\\circ ,-}}],$ $[{u_{-,\\circ }},{y^2_{-,\\circ }}]$ at which $\\text{D}^{\\text{sym}}$ is evaluated, see Figure REF .",
"The evaluation of $\\text{D}^{\\text{sym}}$ at these points should, in the vector space case, correspond to one of the equivalent formulations $ \\big | w^1_{\\circ ,\\circ }- w^1_{\\circ ,-} + w^2_{\\circ ,\\circ }- w^2_{-,\\circ } \\big | = \\big | \\big ( w^1_{\\circ ,\\circ } + w^2_{\\circ ,\\circ } \\big ) - \\big ( w^1_{\\circ ,-} + w^2_{-,\\circ }\\big ) \\big | = \\big | \\big ( w^1_{\\circ ,\\circ }- w^1_{\\circ ,-}\\big ) -\\big ( w^2_{-,\\circ } - w^2_{\\circ ,\\circ } \\big ) \\big |.",
"$ Disregarding for the moment the fact that the corresponding tangent tuples live on different locations, the difficulty here is how to define algebraic operations, i.e., sum or difference operations, on two tangent tuples.",
"For the sum, there is no direct way of doing so, the difference operation, however, quite naturally transfers to tangent tuples as $ [x,y_1] - [x,y_2] = (y_1 - x) - (y_2 - x) = y_1 - y_2 = [y_2,y_1].",
"$ Hence, in the situation that all tangent tuples live on the same location, we can define $\\text{D}^{\\text{sym}}$ by first evaluating pair-wise differences of the four point tuples and then measuring the distance of the two resulting tuples by $D(\\cdot ,\\cdot )$ .",
"Now in the general situation, we first need to transport tangent tuples to the same location.",
"As described in Section REF , we have means of doing this that build either on parallel transport or its Schild's approximation.",
"Taking this into account, we note that the evaluation of $\\text{D}^{\\text{sym}}$ in the vector space case can be written purely in terms of distances of tangent-vector-differences in various ways, e.g., $ \\big | w^1_{\\circ ,\\circ }- w^1_{\\circ ,-} + w^2_{\\circ ,\\circ }- w^2_{-,\\circ } \\big | = \\big | \\big (w^1_{\\circ ,\\circ }- w^1_{\\circ ,-}\\big ) -\\big (w^2_{-,\\circ }- w^2_{\\circ ,\\circ } \\big ) \\big | = \\big | \\big (w^1_{\\circ ,-} - w^2_{\\circ ,\\circ }\\big ) - \\big ( w^1_{\\circ ,\\circ } - w^2_{-,\\circ } \\big ) \\big |.$ While any order of taking the differences is equivalent in the vector space case, in the manifold setting the order defines how the corresponding tangent tuples need to be transported to the same location.",
"Looking again at Figure REF , the simplest idea seems to be to transport the tangent tuple $[u_{\\circ ,-},y_{\\circ ,-}^1]$ corresponding to $w_{\\circ ,-}^ 1$ and the tangent tuple $[u_{-,\\circ ,-},y_{-,\\circ }^2]$ corresponding to $w_{-,\\circ }^ 2$ to the point ${u_{\\circ ,\\circ }}$ and carry out all operations there.",
"The drawback of this simple solution can be found when looking at the conditions of Proposition REF .",
"There, the main difficulty is to ensure that $\\text{D}^{\\text{sym}}([{u_{\\circ ,\\circ }},{u_{+,\\circ }}],[{u_{\\circ ,\\circ }},{u_{\\circ ,+}}],[{u_{\\circ ,-}},{u_{+,-}}],[{u_{-,\\circ }},{u_{-,+}}]) = 0$ allows to conclude that the signal is geodesic in the anti-diagonal direction.",
"The situation that is relevant for this condition is when both $(u_{i,j_0})_i$ and $(u_{i_0,j})_j$ are locally geodesic for each $(i_0,j_0)$ and the endpoints $y$ coincide with the respective signal points, i.e., $y_{i,j}^1 = u_{i+1,j}$ and $y_{i,j}^2 = u_{i,j+1}$ (see Figure REF , left).",
"In this situation, in order to fulfill the above condition, we need to know something about the transported tangent tuples, e.g., if $[u_{\\circ ,-},y_{\\circ ,-}^1]$ and $[u_{-,\\circ },y_{-,\\circ }^2]$ corresponding to $w_{\\circ ,-}^1$ and $w_{-,\\circ }^2$ are both transported to ${u_{\\circ ,\\circ }}$ , we would need to know for instance that the tangent tuple $[{u_{\\circ ,\\circ }},x]$ resulting from the transport of $[u_{\\circ ,-},y_{\\circ ,-}^1]$ to ${u_{\\circ ,\\circ }}$ points to ${u_{+,\\circ }}$ , i.e., $x = {u_{+,\\circ }}$ .",
"Due to holonomy however, even if the transport is carried out along a geodesic, there is, to the best of our knowledge, no way of obtaining such a result.",
"That is, the vector corresponding to the transported tangent tuple $[{u_{\\circ ,\\circ }},x]$ might be arbitrarily rotated such that $x$ is far away from ${u_{+,\\circ }}$ .",
"On the other hand, a well-known fact for manifolds is that the parallel transport of a vector that is tangential to a geodesic along the geodesic itself again results in a vector that is tangential to the geodesic.",
"As we will see, this implies an equivalent assertion for both our variants for transporting tangent tuples.",
"That is, the transport of a tangent tuple corresponding to a tangential vector of a geodesic along the geodesic itself again results in a tangent tuple that corresponds to a tangential vector of the geodesic.",
"In the above-described particular situation (see again Figure REF , left), this means for example that, since the points ${u_{\\circ ,+}},{u_{\\circ ,\\circ }},{u_{\\circ ,-}}$ are on a geodesic and $w^ 2_{\\circ ,\\circ }\\simeq [u_{\\circ ,\\circ },u_{\\circ ,+}]$ corresponds to a tangent vector that is tangential to this geodesic, we know that the transport of $[u_{\\circ ,\\circ },u_{\\circ ,+}]$ to ${u_{\\circ ,-}}$ will result in the tangent tuple $[{u_{\\circ ,-}},{u_{\\circ ,\\circ }}]$ .",
"More generally, for the situation that both $(u_{i,j_0})_i$ and $(u_{i_0,j})_j$ are locally geodesic for each $(i_0,j_0)$ , it means that we can always transport the tangent tuples corresponding to $w_{\\circ ,\\circ }^1$ and $w_{\\circ ,-}^1$ in horizontal direction and the tangent tuples corresponding to the $w_{\\circ ,\\circ }^ 2 $ and $w_{-,\\circ }^ 2 $ in vertical direction and still know something about the transported tangent tuples.",
"In view of (M-P4'), a natural approach to define $\\text{D}^{\\text{sym}}$ in the general situation is hence to restrict ourselves to these particular transport directions.",
"Of course, we also want to define $\\text{D}^{\\text{sym}}$ in an as-simple-as-possible way, meaning that we want to carry out as few transport operations as possible.",
"A quick case study in Figure REF , shows that we have to transport at least three times and that, accounting for the facts that we need to generalize the expression (REF ) and that we can take differences but not sums of tangent tuples in the same location, there is only one possibility to achieve this.",
"As highlighted in Figure REF on the right for the general situation, our proposed approach is to transport the tangent tuple corresponding to $w^ 2_{\\circ ,\\circ }$ down to ${u_{\\circ ,-}}$ giving $[{u_{\\circ ,-}}, \\tilde{y}_{\\circ ,\\circ }^2]$ , the tangent tuple corresponding to $w^1_{\\circ ,\\circ }$ left to ${u_{-,\\circ }}$ giving $[{u_{-,\\circ }}, \\tilde{y}_{\\circ ,\\circ }^1]$ and take the differences, which results in the tangent tuples $[{y^1_{\\circ ,-}},\\tilde{y}_{\\circ ,\\circ }^2] \\simeq d^2$ and $ [\\tilde{y}_{\\circ ,\\circ }^1,{y^2_{-,\\circ }}] \\simeq d^1$ .",
"The distance of the tangent tuples corresponding to $d^1$ and $d^2$ can then be measured with one of our previously defined distance functions, i.e., by evaluating $D([\\tilde{y}_{\\circ ,\\circ }^1,{y^2_{-,\\circ }}],[{y^1_{\\circ ,-}},\\tilde{y}_{\\circ ,\\circ }^2])$ , which again requires one transport operation.",
"This corresponds to the reformulation of $\\big | w^1_{\\circ ,\\circ }- w^1_{\\circ ,-} + w^2_{\\circ ,\\circ }- w^2_{-,\\circ } \\big | = \\big | \\big ( w^2_{\\circ ,\\circ } -w^1_{\\circ ,-} \\big ) - \\big ( w^2_{-,\\circ } - w^1_{\\circ ,\\circ } \\big ) \\big |.",
"$ In the particular situation of Figure REF , left, due to the usage of the particular transport directions, the two tangent tuples resulting from the transport of the tangent tuples corresponding to $w^ 1_{\\circ ,\\circ }$ and $w^ 2_{\\circ ,\\circ }$ and taking the corresponding difference operations will be $[{u_{\\circ ,\\circ }},{u_{-,+}}]$ and $[{u_{+,-}},{u_{\\circ ,\\circ }}]$ (corresponding to $d^1 $ and $d^2$ ), whose distance will then be measured by $D([{u_{\\circ ,\\circ }},{u_{+,-}}],[{u_{-,+}},{u_{\\circ ,\\circ }}])$ .",
"Using the assumptions as in Proposition REF , $D([{u_{\\circ ,\\circ }},{u_{+,-}}],[{u_{-,+}},{u_{\\circ ,\\circ }}]) = 0$ then allows us to conclude that ${u_{-,+}},{u_{\\circ ,\\circ }},{u_{+,-}}$ are on a distance minimizing geodesic, hence the assumptions for (M-P4') will be fulfilled.",
"Figure: Two three-by-three point sections of a bivarate signal.",
"Left: Signal points (uu), tangent vectors (redww) and endpoints (blueyy) in the particular situation that M-TGV α 2 =0\\text{M-TGV}^2_\\alpha =0.",
"Right: Signal with tangent vectors (darkgreenw ˜\\tilde{w}) corresponding to transported tangent tuples, transported endpoints (bluey ˜\\tilde{y}) and (graydd) corresponding to difference of tangent tuples.The dotted lines indicate geodesics that are determinted by tangent vectors and the black lines show a piecewise geodesic interpolation of the signal points.The above-described strategy is now used to extend both $\\text{S-TGV}^2_\\alpha $ and $\\text{PT-TGV}^2_\\alpha $ to the bivariate setting.",
"As such, the only difference will be how to carry out the transport operations and which of the two functions $\\text{D}_{\\text{S}}$ , $\\text{D}_{\\text{pt}}{}$ is used to measure the distance of tangent tuples.",
"Realization via Schild's-approximation.",
"We now realize the axiomatic setting for bivariate TGV for manifold-valued data of Section REF using the proposed Schild's approximation.",
"In particular, we use the Schild's approximation of parallel transport to obtain an instance of the distance-type function $D^{\\text{sym}}.$ We define the following bivariate version $\\text{S-TGV}^2_\\alpha $ of bivariate TGV for manifold-valued data using the Schild's approximation by $ \\begin{aligned}\\text{S-TGV}_\\alpha ^ 2(u)= \\min _{y^ 1_{i,j},y^ 2_{i,j}} \\alpha _1 \\sum \\nolimits _{i,j} & \\Big ( d(u_{i+1,j},y^ 1_{i,j})^p + d(u_{i,j+1},y^ 2_{i,j})^p \\Big )^{1/p}\\\\+\\alpha _0 \\sum \\nolimits _{i,j} & \\Big ( \\text{D}_{\\text{S}}\\big ([u_{i,j},y^1_{i,j}],[u_{i-1,j},y^1_{i-1,j}]\\big )^p + \\text{D}_{\\text{S}}\\big ([u_{i,j},y^2_{i,j}],[u_{i,j-1},y^2_{i,j-1}]\\big )^p \\\\&+ 2^{1-p}\\text{D}_{\\text{S}}^{\\text{sym}}([u_{i,j},y^1_{i,j}],[u_{i,j},y^2_{i,j}],[u_{i,j-1},y^1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}])^p \\Big )^{1/p}.\\end{aligned}$ Here $\\text{D}_{\\text{S}}$ is given as in Equation (REF ) and $\\text{D}_{\\text{S}}^{\\text{sym}}$ is defined by $\\text{D}_{\\text{S}}^{\\text{sym}}([{u_{\\circ ,\\circ }},{y^1_{\\circ ,\\circ }}],[\\tilde{u}_{\\circ ,\\circ },{y^2_{\\circ ,\\circ }}],[{u_{\\circ ,-}},{y^1_{\\circ ,-}}]&,[{u_{-,\\circ }},{y^2_{-,\\circ }}])= \\\\\\min \\nolimits _{r^ 1,r^2} \\text{D}_{\\text{S}}([r^ 1,{y^2_{-,\\circ }}],[{y^1_{\\circ ,-}},r^ 2]) \\quad & \\text{s.t. }",
"r ^1 \\in [{u_{\\circ ,\\circ }},c^ 1]_2 \\text{ with } c^1 \\in [{u_{-,\\circ }},{y^1_{\\circ ,\\circ }}]_{\\frac{1}{2}} \\\\& \\text{and }r ^2 \\in [\\tilde{u}_{\\circ ,\\circ },c^ 2]_2 \\text{ with } c^2 \\in [{u_{\\circ ,-}},{y^2_{\\circ ,\\circ }}]_{\\frac{1}{2}}.",
"$ We note that, for the sake of a formally correct definition, we have introduced $\\tilde{u}_{\\circ ,\\circ }$ , but in fact we will always choose $\\tilde{u}_{\\circ ,\\circ } = {u_{\\circ ,\\circ }}$ since $\\text{D}_{\\text{S}}^{\\text{sym}}$ effectively depends only on seven variables.",
"Also note that $\\text{D}_{\\text{S}}^{\\text{sym}}$ exactly carries out the construction of $\\text{D}^{\\text{sym}}$ as described in Section REF , using the Schild's approximation as shown in Figure REF for transporting tangent tuples and $\\text{D}_{\\text{S}}$ to measure their difference.",
"In the notation of Figure REF , right, we have that $\\tilde{y}_{\\circ ,\\circ }^1 = r^1$ and $\\tilde{y}_{\\circ ,\\circ }^2 = r^2$ correspond to the transported endpoints, $[u_{-,\\circ },r^1] \\simeq \\tilde{w}_{\\circ ,\\circ }^1 $ and $[u_{\\circ ,-},r^2] \\simeq \\tilde{w}^2_{\\circ ,\\circ } $ to the transported tangent tuples and $[r^ 1,{y^2_{-,\\circ }}] \\simeq d^1 $ and $[{y^1_{\\circ ,-}},r^ 2] \\simeq d^2 $ to the differences of tangent tuples whose distance is measured with $\\text{D}_{\\text{S}}$ .",
"Due to our careful design of $\\text{S-TGV}^2_\\alpha $ we get the following result.",
"Theorem 2.18 The $\\text{S-TGV}^2_\\alpha $ functional as in (REF ) satisfies the properties (M-P1') to (M-P3').",
"If $u = (u_{i,j})_{i,j}$ is locally geodesic, then $\\text{S-TGV}^2_\\alpha (u) = 0$ .",
"If $u$ is such that the geodesic connecting each pair $u_{i,j}$ and $u_{i^{\\prime },j^{\\prime }}$ with $\\max \\lbrace |i-i^{\\prime }|,|j-j^{\\prime }|\\rbrace \\le 2$ is unique and $\\text{S-TGV}^2_\\alpha (u) = 0$ , then $u$ is geodesic.",
"See Section REF in the Appendix.",
"Realization via parallel transport.",
"Using parallel transport in order to realize a symmetrized gradient as described in Section REF yields a parallel transport version $\\text{PT-TGV}^2_\\alpha $ of TGV for bivariate manifold-valued data case as follows.",
"We let $ \\begin{aligned}\\text{PT-TGV}^2_\\alpha (u)= \\min _{y^ 1_{i,j},y^ 2_{i,j}} \\alpha _1 \\sum \\nolimits _{i,j} & \\big ( d(u_{i+1,j},y^ 1_{i,j})^p + d(u_{i,j+1},y^ 2_{i,j})^p \\Big )^{1/p} \\\\+\\alpha _0 \\sum \\nolimits _{i,j} & \\Big ( \\text{D}_{\\text{pt}}{}\\big ([u_{i,j},y^1_{i,j}],[u_{i-1,j},y^1_{i-1,j}]\\big )^p + \\text{D}_{\\text{pt}}{}\\big ([u_{i,j},y^2_{i,j}],[u_{i,j-1},y^2_{i,j-1}]\\big )^p \\\\& + 2^{1-p} \\text{D}_{\\text{pt}}^{\\text{sym}}\\big ([u_{i,j},y^1_{i,j}],[u_{i,j},y^2_{i,j}],[u_{i,j-1},y^1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}]\\big )^p \\Big )^{1/p}.\\end{aligned}$ Here $\\text{D}_{\\text{pt}}{}$ is given as in (REF ) and we define $\\text{D}_{\\text{pt}}^{\\text{sym}}([{u_{\\circ ,\\circ }},{y^1_{\\circ ,\\circ }}],[\\tilde{u}_{\\circ ,\\circ },{y^2_{\\circ ,\\circ }}],[{u_{\\circ ,-}},{y^1_{\\circ ,-}}],[{u_{-,\\circ }},{y^2_{-,\\circ }}]) =\\min _{r^ 1,r^ 2} \\text{D}_{\\text{pt}}{}([r^1,{y^2_{-,\\circ }}],[{y^1_{\\circ ,-}},r^ 2]) & \\\\\\text{s.t. }",
"r^1 \\in \\exp (\\operatorname{pt}_{{u_{-,\\circ }}}(w^1)) \\text{ with } w^ 1 \\in \\log _{{u_{\\circ ,\\circ }}}({y^1_{\\circ ,\\circ }}) & \\\\\\text{and } r^2 \\in \\exp (\\operatorname{pt}_{{u_{\\circ ,-}}}(w^ 2)) \\text{ with } w^ 2 \\in \\log _{\\tilde{u}_{\\circ ,\\circ }}({y^2_{\\circ ,\\circ }}) &.$ Note that, as for $\\text{D}_{\\text{S}}^{\\text{sym}}$ , for the sake of a formal correctness, we have introduced $\\tilde{u}_{\\circ ,\\circ }$ , but in fact we will always choose $\\tilde{u}_{\\circ ,\\circ } = {u_{\\circ ,\\circ }}$ since also $\\text{D}_{\\text{pt}}^{\\text{sym}}$ effectively depends only on seven variables.",
"Also, we note again that $\\text{D}_{\\text{pt}}^{\\text{sym}}$ realizes exactly the construction of $\\text{D}^{\\text{sym}}$ as discussed in Section REF (see also Figure REF ), only that now the parallel transport is used to transport tangent tuples and the distance of two tangent-tuple-differences is again measured with $\\text{D}_{\\text{pt}}{}$ .",
"Due to our careful construction, the following result follows easily.",
"Theorem 2.19 The $\\text{PT-TGV}^2_\\alpha $ functional as in (REF ) satisfies the properties (M-P1') to (M-P3').",
"If $u = (u_{i,j})_{i,j}$ is locally geodesic, then $\\text{PT-TGV}^2_\\alpha (u) = 0$ .",
"If $u$ is such that the geodesic connecting each pair $u_{i,j}$ and $u_{i^{\\prime },j^{\\prime }}$ with $\\max \\lbrace |i-i^{\\prime }|,|j-j^{\\prime }|\\rbrace \\le 2$ is unique and $\\text{PT-TGV}^2_\\alpha (u) = 0$ , then $u$ is geodesic.",
"See Section REF in the Appendix."
],
[
"Existence results for TGV for manifold-valued data",
"This section provides existence results for the minimization problem appearing the definition of $\\text{M-TGV}^2_\\alpha $ as well as for variational $\\text{M-TGV}^2_\\alpha $ -regularized denoising.",
"For the sake of brevity, we only consider the bivariate setting and note that the univariate counterpart follows as special case.",
"We first state the main existence results in a general setting, that builds on lower semi-continuity of the involved distance-type function.",
"The latter is then proven in subsequent lemmata.",
"Theorem 3.1 Assume that $D:\\mathcal {M}^2 \\times \\mathcal {M}^2 \\rightarrow [0,\\infty )$ and $\\text{D}^{\\text{sym}}:\\mathcal {M}^ 2 \\times \\mathcal {M}^ 2 \\times \\mathcal {M}^ 2 \\times M^ 2 \\rightarrow [0,\\infty )$ are distance-type functions that are lower semi-continuous.",
"Then, for any $u = (u_{i,j})_{i,j}$ there exist $(y^1_{i,j})_{i,j}$ , $(y^2_{i,j})_{i,j}$ in $\\mathcal {M}$ that attain the minimum in the definition of $\\text{M-TGV}^2_\\alpha $ as in Equation (REF ).",
"Further, $\\text{M-TGV}^2_\\alpha $ is lower semi-continuous and for any $(f_{i,j})_{i,j} $ in $\\mathcal {M}$ there exists a solution to $ \\min _{u}\\, \\text{M-TGV}^2_\\alpha (u) + \\sum _{i,j} d(u_{i,j},f_{i,j})^2.$ Also, all previously introduced distance-type functions $\\text{D}_{\\text{S}}$ , $\\text{D}_{\\text{S}}^{\\text{sym}}$ , $\\text{D}_{\\text{pt}}{}$ , $\\text{D}_{\\text{pt}}^{\\text{sym}}{}$ are lower semi-continuous and a solution to (REF ) exists if $\\text{M-TGV}^2_\\alpha $ is replaced by $\\text{S-TGV}^2_\\alpha $ or $\\text{PT-TGV}^2_\\alpha $ .",
"In particular, all the models proposed in this paper have a minimizer.",
"The existence results in the general setting follow by standard arguments and the Hopf-Rinow theorem and we only provide a sketch of proof.",
"Regarding existence for the evaluation of $\\text{M-TGV}^2_\\alpha $ , we note that all involved terms are positive and, for fixed $u$ , any infimizing sequence $((y^1)^n,(y^2)^n)_n$ is bounded due to the first two distance functions involved in $\\text{M-TGV}^2_\\alpha $ as in Equation (REF ).",
"Hence it admits a convergent subsequence and from lower semi-continuity of all the involved terms, it follows that the limit is a minimizer for the evaluation of $\\text{M-TGV}^2_\\alpha $ .",
"Now take any sequence $(u^n)_n$ converging to some $u$ for which, without loss of generality, we assume that $\\liminf _n \\text{M-TGV}^2_\\alpha (u^n) = \\lim _n \\text{M-TGV}^2_\\alpha (u^n)$ .",
"We can pick elements $((y^1)^n,(y^2)^n)$ for which the minimum in $\\text{M-TGV}^2_\\alpha (u^n)$ is attained.",
"Again by boundedness of $(u^n)_n$ and the first two distance functions in the definition of $\\text{M-TGV}^2_\\alpha $ , also $((y^1)^n,(y^2)^n)_n$ is bounded and we can extract a subsequence $((y^1)^{n_k},(y^2)^{n_k})_k$ converging to some $(\\hat{y}^ 1,\\hat{y}^ 2)$ .",
"Defining $E(\\cdot )$ such that $\\text{M-TGV}^2_\\alpha (u) = \\inf _{y^1,y^2} E(u,y^1,y^ 2)$ , we obtain from lower semi-continuity of all involved terms that $\\text{M-TGV}^2_\\alpha (u) \\le E(u,\\hat{y}^ 1,\\hat{y}^ 2) \\le \\liminf _k E(u^{n_k},(y^1)^{n_k},(y^2)^{n_k})& = \\liminf _k \\text{M-TGV}^2_\\alpha (u^{n_k})$ and since $\\liminf _k \\text{M-TGV}^2_\\alpha (u^{n_k}) = \\liminf _n \\text{M-TGV}^2_\\alpha (u^n)$ , $\\text{M-TGV}^2_\\alpha $ is lower semi-continuous.",
"Given lower semi-continuity of $\\text{M-TGV}^2_\\alpha $ , existence for (REF ) follows by similar arguments.",
"Regarding the particular realizations $\\text{S-TGV}^2_\\alpha $ and $\\text{PT-TGV}^2_\\alpha $ , it suffices to show lower semi-continuity of the involved distance-type functions, which is done on the sequel.",
"We are now left to show lower semi-continuity of the proposed distance-type functions $\\text{D}_{\\text{S}}$ , $\\text{D}_{\\text{pt}}{}$ , $\\text{D}_{\\text{S}}^{\\text{sym}}$ and $\\text{D}_{\\text{pt}}^{\\text{sym}}$ .",
"To this aim, we exploit that they are all defined by minimizing a distance-type function subject to a constraint.",
"Using this joint structure, we first provide a standard abstract lower semi-continuity result that covers this setting and reduces lower semi-continuity to a closedness condition on the constraint.",
"Proposition 3.2 Take $K,N \\in \\mathbb {N}$ and functions $G:\\mathcal {M}^N\\times \\mathcal {M}^K \\rightarrow [0,\\infty )$ and $C:\\mathcal {M}^N \\rightarrow \\mathcal {P}(\\mathcal {M}^K)$ such that $G$ is lower semi-continuous, $ C(x) \\ne \\emptyset $ for all $x \\in \\mathcal {M}^N$ and for any bounded sequence $(x^n)_n $ with elements in $\\mathcal {M}^N$ and $(y^n)_n $ such that $y^n \\in C(x^n)$ for all $n$ , there exist subsequences $(x^{n_k})_k$ and $(y^{n_k})_k$ converging to some $x \\in \\mathcal {M}^N$ , $y \\in \\mathcal {M}^K$ such that $y \\in C(x)$ .",
"Define $S: \\mathcal {M}^N \\rightarrow [0,\\infty )$ as $ S(x) = \\inf _{y \\in \\mathcal {M}^K} G(x,y) \\text{ such that }y \\in C(x).",
"$ Then, for any $x \\in \\mathcal {M}^N$ there exists $y\\in C(x)$ such that $S(x) = G(x,y)$ .",
"Further, $S$ is lower semi-continuous.",
"See Section REF in the Appendix.",
"In the application of this result to the proposed distance-type functions, the constraint $y \\in C(x)$ will correspond to constraining points to particular positions on distance minimizing geodesics.",
"Consequently, in order to verify the closedness condition, the general result that the set of shortest geodesics connecting two points is closed w.r.t.",
"perturbation of these points will be necessary and is provided as follows.",
"Lemma 3.3 Let $\\mathcal {M}$ be a (geodesically) complete Riemannian manifold.",
"Let $(p^n)_n$ and $(q^n)_n$ be two sequences in $\\mathcal {M}$ converging to $p$ and $q$ , respectively.",
"Let each $\\gamma ^ n :[0,1] \\rightarrow \\mathcal {M}$ be a distance-minimizing geodesic such that $\\gamma ^ n(0) = p^n$ , $\\gamma ^n(1) = q^n$ .",
"Then, there exists a distance-minimizing geodesic $\\gamma :[0,1] \\rightarrow \\mathcal {M}$ such that $\\gamma (0) = p$ , $\\gamma (1) = q$ and a subsequence $(\\gamma ^{n_k})_k$ such that $\\gamma ^{n_k} \\rightarrow \\gamma $ uniformly on $[0,1]$ .",
"Furthermore, extending the geodesics to any interval $[a,b]$ with $[0,1] \\subset [a,b]$ we get that, up to subsequences, $(\\gamma ^{n_k})_{n_k}$ also converges uniformly to $\\gamma $ on $[a,b]$ .",
"See Section REF in the Appendix.",
"We are now in the position of showing closedness of the constraints appearing in the definitions of $\\text{D}_{\\text{S}}$ and $\\text{D}_{\\text{S}}^{\\text{sym}}$ .",
"Lemma 3.4 Let $((x^n,y^n,u^n,v^n))_n$ be a bounded sequence in $\\mathcal {M}^4$ and take $((c^n,\\tilde{y}^n))_n$ in $\\mathcal {M}^2$ such that $ c^n \\in [x^n,v^n]_{\\frac{1}{2}} \\quad \\text{and}\\quad \\tilde{y}^n \\in [u^n,c^n]_2.",
"$ Then there exist subsequences $((x^{n_k},y^{n_k},u^{n_k},v^{n_k}))_k$ and $((c^{n_k},\\tilde{y}^{n_k}))_k$ converging to $(x,y,u,v)$ and $(c,\\tilde{y})$ , respectively, such that $ c \\in [x,v]_{\\frac{1}{2}} \\quad \\text{and}\\quad \\tilde{y} \\in [u,c]_2.",
"$ In particular, $C:\\mathcal {M}^4\\rightarrow \\mathcal {P}(\\mathcal {M}^2)$ defined as $C(x,y,u,v):= \\lbrace (c^{\\prime },y^{\\prime }) \\,|\\, c^{\\prime } \\in [x,v]_{\\frac{1}{2}}, y^{\\prime } \\in [u,c^{\\prime }]_2 \\rbrace $ satisfies the assumption of Proposition REF .",
"Since each $c^n$ is on a length-minimizing geodesic between $x^n$ and $v^n$ , we get that $d(x^n,c^n) \\le d(x^n,v^n)$ and hence $(c^n)_n$ is bounded.",
"Also $d(c^n,\\tilde{y}^n) = d(c^n,u^n)$ and hence $(\\tilde{y}^n)_n$ is bounded.",
"Consequently we can pick subsequences $((x^{n_k},y^{n_k},u^{n_k},v^{n_k}))_k$ and $((c^{n_k},\\tilde{y}^{n_k}))_k$ converging to $(x,y,u,v)$ and $(c,\\tilde{y})$ .",
"Now with $\\gamma _c^{n_k}:[0,1]\\rightarrow \\mathcal {M}$ being a shortest geodesic connecting $x^{n_k}$ and $v^{n_k}$ such that $\\gamma _c^{n_k} (1/2) = c^{n_k}$ we get by Lemma REF that, up to a subsequence, $\\gamma _c^{n_k} \\rightarrow \\gamma _c$ uniformly, where $\\gamma _c$ is a again a shortest geodesic connecting $x$ and $v$ .",
"Consequently, $c = \\gamma (1/2) \\in [x,v]_{\\frac{1}{2}}$ .",
"Now pick $\\gamma _{\\tilde{y}}^{n_k}:[0,1]\\rightarrow \\mathcal {M}$ to be a length minimizing geodesic between $\\gamma _{\\tilde{y}}^{n_k}(0) = u^{n_k}$ and $\\gamma _{\\tilde{y}}^{n_k}(1)= c^{n_k}$ such that $\\gamma _{\\tilde{y}}^{n_k}(2) = \\tilde{y}^{n_k}$ .",
"By Lemma REF we get that, up to subsequences, $\\gamma ^{n_k}_{\\tilde{y}}$ converges uniformly to a geodesic $\\gamma _{\\tilde{y}}$ on $[0,2]$ such that $\\gamma _{\\tilde{y}}$ is length-minimizing between $\\gamma _{\\tilde{y}}(0) = u$ and $\\gamma _{\\tilde{y}}(1) = c$ .",
"By uniform convergence on $[0,2]$ we get that $y = \\lim _{k}y^{n_k} = \\lim _k \\gamma ^{n_k}_{\\tilde{y}}(2) = \\gamma _{\\tilde{y}}(2)$ and the assertion follows.",
"Combining this with the general assertion of Proposition REF , existence and lower semi-continuity results for $\\text{D}_{\\text{S}}$ and $\\text{D}_{\\text{S}}^{\\text{sym}}$ follow as direct consequences.",
"Lemma 3.5 The minimum in the definition of $\\text{D}_{\\text{S}}$ as in Equation (REF ) is attained and $\\text{D}_{\\text{S}}$ is lower semi-continuous.",
"Also, the minimum in the definition of $\\text{D}_{\\text{S}}^{\\text{sym}}$ as in Equation (REF ) is attained and $\\text{D}_{\\text{S}}^{\\text{sym}}$ is lower semi-continuous.",
"See Section REF in the Appendix.",
"Using similar techniques, existence and lower semi-continuity results for the parallel transport variants can be established as follows.",
"Lemma 3.6 The minimum in the definition of $\\text{D}_{\\text{pt}}{}$ as in Equation (REF ) is attained and $\\text{D}_{\\text{pt}}{}$ is lower semi-continuous.",
"Also, the minimum in the definition of $\\text{D}_{\\text{pt}}^{\\text{sym}}$ as in Equation (REF ) is attained and $\\text{D}_{\\text{pt}}^{\\text{sym}}$ is lower semi-continuous.",
"See Section REF in the Appendix."
],
[
"Algorithmic approach to TGV for manifold-valued data",
"In order to algorithmically approach denoising problems using TGV regularization in the manifold setting, we employ the concept of cyclic proximal point algorithms (CPPAs).",
"A reference for cyclic proximal point algorithms in vector spaces is [22].",
"In the context of Hadamard spaces, the concept of CPPAs was developed by [8], where it is used to compute means and medians.",
"In the context of variational regularization methods for nonlinear, manifold-valued data, they were first used in [93].",
"More precisely, the reference [93] deals with TV regularization as well as classical smooth methods for manifold-valued data.",
"The CPPA approach was later used for higher-order TV-type methods in [18] for circle-valued data and in [10] for manifold-valued data."
],
[
"Principle of a CPPA.",
"The idea of a CPPAs is to compose a functional $f: \\mathcal {M} \\rightarrow \\mathbb {R}$ into basic atoms $f_i$ and then to compute the proximal mappings of the $f_i$ in a cyclic, iterative way.",
"More precisely, assume that $f = \\sum \\nolimits _{i=1}^n f_i$ and consider the proximal mappings [64], [41], [6] $\\operatorname{prox}_{\\lambda f_i}: \\mathcal {M} \\rightarrow \\mathcal {M}$ given as $ \\operatorname{prox}_{\\lambda f_i} x = \\operatorname{argmin}_y f_i(y) + \\tfrac{1}{2 \\lambda } d(x,y)^2.$ One cycle of a CPPA then consists of applying each proximal mapping $\\operatorname{prox}_{\\lambda f_i}$ once in a prescribed order, e.g., $\\operatorname{prox}_{\\lambda f_1},$ $\\operatorname{prox}_{\\lambda f_2},$ $\\operatorname{prox}_{\\lambda f_3}, \\ldots ,$ or, generally, $\\operatorname{prox}_{\\lambda f_{\\sigma (1)}},$ $\\operatorname{prox}_{\\lambda f_{\\sigma (2)}},$ $\\operatorname{prox}_{\\lambda f_{\\sigma (3)}},$ $\\ldots ,$ where the symbol $\\sigma $ is employed to denote a permutation.",
"The cyclic nature is reflected in the fact that the output of $\\operatorname{prox}_{\\lambda f_{\\sigma (i)}}$ is used as input for $\\operatorname{prox}_{\\lambda f_{\\sigma (i+1)}}.$ Since the $i$ th update is immediately used for the $(i+1)$ st step, it can be seen as a Gauss-Seidel-type scheme.",
"A CPPA now consists of iterating these cycles, i.e., in the $k$ th cycle, we have $x_{i+1}^{(k)} = \\operatorname{prox}_{\\lambda _k f_{\\sigma (i)}}x_i^{(k)},$ and $x_{1}^{(k+1)}$ is obtained by applying $\\operatorname{prox}_{\\lambda _k f_{\\sigma (n)}}$ to $x_{n}^{(k)}.$ Here, $n$ denotes the number of elements in the cycle.",
"During the iteration, the parameter $\\lambda _k$ of the proximal mappings is successively decreased.",
"In this way, the penalty for deviation from the previous iterate is successively increased.",
"It is chosen in a way such that the sequence $(\\lambda ^k)_k$ is square-summable but not summable.",
"Provided that this condition on the stepsize parameters holds, the cyclic proximal point algorithm can be shown to converge to the optimal solution of the underlying minimization problem, at least in the context of Hadamard manifolds and convex $(f_i)_i$ [9].",
"For general manifolds, the proximal mappings (REF ) are not globally defined, and the minimizers are not unique, at least for general possibly far apart points; cf.",
"[41], [6].",
"This is a general issue in the context of manifolds that are – in a certain sense – a local concept involving objects that are often only locally well defined.",
"In case of ambiguities, we hence consider the above objects as set-valued quantities.",
"Furthermore, we cannot guarantee – and in fact do not expect — the second distance-type functions $D$ in the definition of the $\\text{M-TGV}^2_\\alpha $ functional to be convex.",
"Hence convergence of the CPPA algorithm to a globally optimal solution cannot be ensured.",
"It thus should be seen as an approximative strategy.",
"Nevertheless, as will be seen in the numerical experiments section, we experience a good convergence behavior of the CPPA algorithm in practice.",
"This was also observed in previous works, where the CPPA algorithm was employed to minimize second-order TV-type functionals [18], [10], which are also non-convex.",
"In order to be precise, we further point out that we approximately compute the proximal mappings of the distance-type functions $D$ ; we hence use an inexact proximal point algorithm.",
"We also point out that a parallel proximal point algorithm has been proposed in [93].",
"Here the proximal mappings of the $f_i$ are computed in parallel and then averaged using intrinsic means.",
"In order to reduce the costs for averaging, an approximative variant of the parallel proximal point algorithm has been proposed in [93] as well.",
"Principally, the cyclic proximal point algorithm actually applied in this paper might be replaced by this parallel strategy; the derivations in the following provide all information necessary."
],
[
"Algorithms for manifold-valued TGV",
"We employ a cyclic proximal point algorithm for the minimization of a TGV-regularized variational approach in the manifold context.",
"For simplicity, we consider the univariate setting first; all aspects discussed in the univariate situation are prototypic and basic for the bivariate situation.",
"Starting with the general version of TGV for manifolds as in (REF ), we aim to solve the denoising problem $ \\min _{(u_i)_i,(y_i)_i } \\quad \\tfrac{1}{2} \\sum _{i} d(u_{i},h_i)^2 + \\sum _{i} \\alpha _1 d(u_{i+1},y_i)+ \\sum _{i} \\alpha _0 D( [u_i,y_i],[u_{i-1},y_{i-1}]),$ where $(h_i)_i$ , being a finite sequence in $\\mathcal {M}$ , denotes the given data.",
"We decompose (REF ) into the atoms $ \\begin{split}&g_i(u) := \\tfrac{1}{2} d(u_{i},h_i)^2; \\qquad g^{\\prime }_i(u,y) := \\alpha _1 d(u_{i+1},y_i); \\\\&g^{\\prime \\prime }_i(u,y) := \\alpha _0 D( [u_i,y_i],[u_{i-1},y_{i-1}]).\\end{split} $ Now we use this decomposition into the atoms $g_i,g^{\\prime }_i, g^{\\prime \\prime }_i$ for the decomposition (REF ) for the CPPA.",
"We apply the iteration (REF ) to this decomposition.",
"We remark that the data terms $g_i$ are not coupled w.r.t.",
"the index $i$ .",
"The same applies to the $g^{\\prime }_i$ .",
"This allows the parallel computation of the proximal mappings of the $g_i$ for all $i$ and, separately, of the $g^{\\prime }_i$ for all $i$ .",
"The $g^{\\prime \\prime }_i$ are actually coupled.",
"However, grouping even and odd indices, we see that the $g^{\\prime \\prime }_i,$ $i$ even, are not coupled and that the $g^{\\prime \\prime }_i,$ $i$ odd, are not coupled.",
"So we may compute their proximal mappings in parallel.",
"Together, this results in a cycle of length four per iteration, one step for the $g_i,$ one step for the $g^{\\prime }_i$ and two steps (even, odd) for the $g^{\\prime \\prime }_i.$ In the following, the task is to compute the proximal mappings of the atoms $g_i,g^{\\prime }_i, g^{\\prime \\prime }_i$ of (REF ).",
"To this end, we will from now on always assume that the involved points are locally sufficiently close such that there exist unique length minimizing geodesics connecting them.",
"We note that this restriction is actually not severe: in the general (non-local) situation we have to remove the cut points – which are a set of measure zero – to end up with the corresponding setup.",
"We remark that, as we will see, an explicit computation of the proximal mapping is possible for $g_i$ and $g_i^{\\prime }$ , but not for $g_i^{\\prime \\prime }$ .",
"We believe that it is an important feature of the proposed definition of manifold-TGV via point-tuples that the proximal mappings of the first term $g^{\\prime }_i$ is still explicit and hence only one part of the overall problem does not allow an explicit proximal mapping.",
"Indeed, also existing generalizations of second-order TV to the manifold setting incorporate one part with non-explicit proximal mappings [10].",
"Hence, from the algorithmic viewpoint, the step from second-order TV to our proposed version of TGV does not introduce essential additional computational effort.",
"For the data atoms $g_i,$ the proximal mappings $\\operatorname{prox}_{\\lambda g_i}$ are given by $(\\operatorname{prox}_{\\lambda g_i})_{j}(u) ={\\left\\lbrace \\begin{array}{ll}[u_{i},h_{i}]_t, & \\text{ if } i=j, \\\\u_{j}, & \\text{ else, } \\\\\\end{array}\\right.",
"}\\qquad \\text{ where } t= \\tfrac{\\lambda }{1+\\lambda }.$ They have been derived in [41].",
"We recall that the symbol $[\\cdot ,\\cdot ]_t$ denotes the point reached after time $t$ on the (non unit speed) geodesic starting at the first argument reaching the second argument at time 1.",
"Further, the proximal mappings of the distance terms $g^{\\prime }_i$ have a closed form representation as well (see [93]), which is given by $\\operatorname{prox}_{\\lambda g^{\\prime }_i } (u,y)={\\left\\lbrace \\begin{array}{ll}[u_{i+1},y_i]_t, &\\text{at position } (i+1,1), \\\\[y_i,u_{i+1}]_t, &\\text{at position } (i,2), \\\\u_j , &\\text{at position } (j,1), \\ j \\ne i+1, \\\\y_j &\\text{at position } (j,2), \\ j \\ne i, \\\\\\end{array}\\right.",
"}$ where $t = \\lambda \\alpha _1 /d(u_{i+1},y_i),$ if $\\lambda \\alpha _1 < \\tfrac{1}{2},$ and $t = 1/2$ else.",
"Note that for determining the position, we view $u,y$ as column vectors of a matrix with two columns.",
"The next task is to compute the proximal mappings of the $g^{\\prime \\prime }_i.$ Unfortunately, no closed form for the proximal mappings of the $g^{\\prime \\prime }_i$ is available.",
"Instead, we use a subgradient descent scheme to compute these proximal mappings, i.e., to solve the problem $ \\min _{u,y} \\quad \\tfrac{1}{2} \\sum \\nolimits _{j} d(u_{j},u^{\\prime }_j)^2 + \\tfrac{1}{2} \\sum \\nolimits _{j} d(y_{j},y^{\\prime }_j)^2 +\\lambda \\alpha _0 D( [u_i,y_i],[u_{i-1},y_{i-1}])$ where the $u^{\\prime }_j,y^{\\prime }_j$ are the input of the proximal mapping.",
"A subgradient descent scheme has already been used to compute proximal mappings in [10].",
"Looking at (REF ), the optimal $u,y$ fulfill $u_j = u^{\\prime }_j, y_j = y^{\\prime }_j$ whenever $j \\notin \\lbrace i,i-1\\rbrace $ .",
"Hence we may restrict ourselves to consider the four arguments $u_i,y_i,u_{i-1},y_{i-1}.$ Further the gradient of the mapping $u_{j} \\mapsto \\tfrac{1}{2} \\sum _{j} d(u_{j},u^{\\prime }_j)^2$ is given as $-\\log _{u_{j}}u^{\\prime }_{j}.$ (For background information on (Riemannian) differential geometry, we refer to the books [76], [38].)",
"So we have to compute the (sub)gradient of the mapping $D( [u_i,y_i],[u_{i-1},y_{i-1}])$ as a function of $u_i,y_i,u_{i-1},y_{i-1}$ .",
"To this end, we have to specify the setting to the two versions of $D$ as proposed in Section ; the respective derivations are topics of Section REF and Section REF .",
"We next consider the bivariate denoising problem.",
"We use $\\text{M-TGV}^2_\\alpha $ regularization with the $\\text{M-TGV}^2_\\alpha $ functional given in (REF ) where, as pointed out, we focus on $p=1$ .",
"We obtain the denoising problem $ \\min _{(u_{i,j})_{i,j},(y_{i,j}^1)_{i,j},(y_{i,j}^2)_{i,j}} \\quad &\\tfrac{1}{2} \\sum \\nolimits _{i,j} d(u_{ij},h_{ij})^2 + \\alpha _1 \\sum \\nolimits _{i,j} d(u_{i+1,j},y^ 1_{i,j}) + d(u_{i,j+1},y^ 2_{i,j})\\\\& +\\alpha _0 \\sum \\nolimits _{i,j} D\\big ([u_{i,j},y^1_{i,j}],[u_{i-1,j},y^1_{i-1,j}]\\big ) + D\\big ([u_{i,j},y^2_{i,j}],[u_{i,j-1},y^2_{i,j-1}]\\big ) \\\\& +\\alpha _0 \\sum \\nolimits _{i,j} D^{\\text{sym}} ([u_{i,j},y^1_{i,j}],[u_{i,j},y^2_{i,j}],[u_{i,j-1},y^1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}])$ where the $(h_{ij})_{ij}$ denotes the bivariate data.",
"We decompose the objective functional in (REF ) into the atoms $ & g^{(1)}_{ij}(u,y^1,y^2) := \\tfrac{1}{2} d(u_{i},h_i)^2; \\\\& g^{(2)}_{ij}(u,y^1,y^2) := \\alpha _1 d(u_{i+1,j},y^ 1_{i,j});\\qquad g^{(3)}_i(u,y^1,y^2) := \\alpha _1 d(u_{i,j+1},y^ 1_{i,j}); \\\\& g^{(4)}_{ij}(u,y^1,y^2) := \\alpha _0 D\\big ([u_{i,j},y^1_{i,j}],[u_{i-1,j},y^1_{i-1,j}]\\big ) ; \\\\& g^{(5)}_{ij}(u,y^1,y^2) := \\alpha _0 D\\big ([u_{i,j},y^2_{i,j}],[u_{i,j-1},y^2_{i,j-1}]\\big ) ; \\\\&g^{(6)}_{ij}(u,y^1,y^2) := \\alpha _0 D^{\\text{sym}} ([u_{i,j},y^1_{i,j}],[u_{i,j},y^2_{i,j}],[u_{i,j-1},y^1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}]).$ We use the decomposition w.r.t.",
"the atoms $g^{(k)}_{ij}$ as the decomposition (REF ) for the CPPA, and apply the iteration (REF ) w.r.t.",
"the derived decomposition.",
"The proximal mappings $\\operatorname{prox}_{\\lambda g^{(1)}_{ij}}$ are given by (REF ), and the proximal mappings $\\operatorname{prox}_{\\lambda g^{(2)}_{ij}},\\operatorname{prox}_{\\lambda g^{(3)}_{ij}}$ are given by (REF ).",
"The proximal mappings $\\operatorname{prox}_{\\lambda g^{(4)}_{ij}},\\operatorname{prox}_{\\lambda g^{(5)}_{ij}}$ are computed as in the univariate case explained above.",
"It remains to compute the proximal mappings of the atoms $g^{(6)}_{ij}.$ As before, there is no explicit formula available and we use a subgradient descent for their computation as well.",
"The computation of the corresponding derivatives are topics of Section REF and Section REF ."
],
[
"Riemannian gradients for the Schild version",
"We here derive the derivatives needed to compute the proximal mappings of the mappings $D$ and $D^{\\text{sym}}$ in Section REF when specifying to the Schild's variant.",
"We start out to compute the derivative of the mapping $\\text{D}_{\\text{S}}$ given by $ \\text{D}_{\\text{S}}(u_{i-1},y_{i-1},u_{i},y_{i}) = \\text{D}_{\\text{S}}([u_i,y_i],[u_{i-1},y_{i-1}]) = d([u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2,y_{i}).$ Recall that, for our computations, we assume that geodesics are unique, which is, as pointed out above, the case up to a set of measure zero.",
"We directly see that $\\text{D}_{\\text{S}}$ is symmetric w.r.t.",
"interchanging $y_{i-1},u_{i}$ .",
"Hence we only have to compute the differential with respect to one of these variables $y_{i-1},u_{i}$ .",
"Furthermore, the gradient $\\nabla _{y_{i}} \\text{D}_{\\text{S}}$ w.r.t.",
"the fourth argument $y_{i}$ of $\\text{D}_{\\text{S}}$ for $y_{i}$ with $y_{i}\\ne [u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2$ is just the well-known gradient of the Riemannian distance function which is known to be given by [38], [4] $\\nabla _{y_{i}} \\text{D}_{\\text{S}}=- \\log _{y_{i}}S(u_{i-1},y_{i-1},u_{i}) /{\\big |\\log _{y_{i}}S(u_{i-1},y_{i-1},u_{i}) \\big |}$ where $|\\cdot |$ is the norm in the tangent space associated with the point $y_{i}$ , i.e., the square root of the Riemannian scalar product.",
"Here, $S(u_{i-1},y_{i-1},u_{i}) = [u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2$ denotes the result of applying the Schild's construction to the respective arguments.",
"In order to determine the gradients w.r.t.",
"the other variables we have to apply the adjoint of the respective differentials to the gradient of the distance function w.r.t.",
"the first argument.",
"More precisely, we have to calculate $\\nabla _{u_{i-1}} \\text{D}_{\\text{S}}&=-T_1\\left({\\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i}}/{\\big | \\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i} \\big |}\\right), \\\\\\nabla _{y_{i-1}} \\text{D}_{\\text{S}}&=-T_2\\left({\\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i}}/{\\big | \\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i} \\big |}\\right),$ where $T_1$ is the adjoint of the differential of the mapping $u_{i-1} \\mapsto [u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2$ , and where $T_2$ is the adjoint of the differential of the mapping $y_{i-1} \\mapsto [u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2$ .",
"The differential w.r.t.",
"$u_{i}$ is obtained by symmetry as pointed out above.",
"In the following we derive these adjoint mappings in terms of Jacobi fields.",
"To this aim, we first explain the notion of a Jacobi field and then point out the connection with geodesic variations.",
"In a Riemannian manifold $\\mathcal {M},$ the Riemannian curvature tensor $R$ is given by $R(X,Y)Z = \\nabla _X \\nabla _Y Z - \\nabla _Y \\nabla _X Z - \\nabla _{[X,Y]} Z$ where $X,Y,Z$ are vector fields and $\\nabla $ denotes the Levi-Civita connection, and where $[X,Y]=XY-YX$ denotes the Lie bracket of the vector fields $X,Y.$ A Jacobi field $Y$ along a geodesic $\\gamma $ is the solution of the differential equation $\\tfrac{D}{ds}\\tfrac{D}{ds}Y + R(\\tfrac{d}{ds}\\gamma ,Y)\\tfrac{d}{ds}\\gamma =0.$ The space of Jacobi fields is a $2N$ -dimensional linear space where $N$ denotes the dimension of $\\mathcal {M}.$ The connection to geodesic variations is as follows: the “derivative” vector field $Y(s) = \\tfrac{d}{dt}|_{t=0}V(s,t)$ of a geodesic variation $V$ is a Jacobi field, and conversely, any Jacobi field is obtained from a geodesic variation.",
"Further details may be found in the books [34], [76], [38].",
"We also need the notion of (Riemannian) symmetric spaces for the formulation of the next lemma.",
"A symmetric space is a Riemannian manifold for which its curvature tensor $R$ is invariant under parallel transport, $R(\\operatorname{pt}_{x,y} X, \\operatorname{pt}_{x,y} Y) \\operatorname{pt}_{x,y} Z = \\operatorname{pt}_{x,y} R(X,Y)Z$ where $X,Y,Z$ are tangent vectors at $x$ and $\\operatorname{pt}_{x,y}$ denotes the parallel transport from $x$ to $y$ along a curve $\\gamma $ (here not reflected in the notation).",
"Then, in particular, the covariant derivative of the curvature tensor $R$ along any curve equals zero in a symmetric space.",
"A reference for symmetric spaces is [34].",
"Lemma 4.1 The mapping $T_1$ of Equation (REF ) can be computed using Jacobi fields.",
"In particular, we explicitly have for the class of symmetric spaces and points with $\\text{D}_{\\text{S}}\\ne 0$ that $\\nabla _{u_{i-1}} \\text{D}_{\\text{S}}= \\operatorname{pt}_{S(u_{i-1},y_{i-1},u_{i}),u_{i-1}}\\left({\\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i}}/{\\big | \\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i} \\big |}\\right),$ which means that the gradient of the distance function w.r.t.",
"the second argument is reflected at $[u_{i},y_{i-1}]_{\\tfrac{1}{2}},$ or in other words, parallel transported from the Schild point $[u_{i-1},[u_{i},y_{i-1}]_{\\tfrac{1}{2}} ]_2$ to $u_{i-1}$ and multiplied by $-1.$ The proof is given in Section REF .",
"Regarding the fact that we consider points with $\\text{D}_{\\text{S}}\\ne 0,$ we remark that the points $u_i$ , $u_{i-1}$ , $y_i$ , $y_{i-1}$ such that $\\text{D}_{\\text{S}}(u_i,u_{i-1},y_i,y_{i-1})=0$ which corresponds to $y_i = [u_{i-1},[u_i,y_{i-1}]_{\\frac{1}{2}}]_2$ form a set of measure zero.",
"On this zero set, for instance, the four-tuple consisting of the four zero-tangent vectors sitting in $u_i,u_{i-1},[u_{i-1},[u_i,y_{i-1}]_{\\frac{1}{2}}]_2,y_{i-1}$ belong to the subgradient of $\\text{D}_{\\text{S}}.$ Lemma 4.2 The gradient of the function $\\text{D}_{\\text{S}}$ for points with $\\text{D}_{\\text{S}}\\ne 0$ given by (REF ) w.r.t.",
"the variable $y_{i-1}$ is given by $\\nabla _{y_{i-1}} \\text{D}_{\\text{S}}=-T_4\\left( T_3\\left({\\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i}}/{\\big | \\log _{S(u_{i-1},y_{i-1},u_{i})} y_{i} \\big |}\\right)\\right), \\\\ $ where $T_3$ is the adjoint of the derivative of the mapping $m \\mapsto [u_{i-1},m]_2,$ and where $T_4$ is the adjoint of the derivative of the mapping $y_{i-1} \\mapsto [u_{i},y_{i-1}]_{\\tfrac{1}{2}},$ that is, $T_2$ as in () is given as $T_2 = T_4 \\circ T_3$ .",
"Further $T_3$ and $T_4$ can be computed using Jacobi fields.",
"The proof is given in Section REF .",
"For the class of symmetric spaces, we make the mappings $T_3,T_4$ more explicit.",
"Lemma 4.3 Let $\\mathcal {M}$ be a symmetric space.",
"Let $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ denote the geodesic connecting $\\gamma (0)=u_{i-1}$ and $\\gamma (1)=m=[u_{i},y_{i-1}]_{\\tfrac{1}{2}}.$ We consider an orthonormal basis $v_n$ of eigenvectors of the self-adjoint Jacobi operator $J \\mapsto R(\\tfrac{\\gamma ^{\\prime }(1)}{ |\\gamma ^{\\prime }(1) |} ,J) \\tfrac{\\gamma ^{\\prime }(1)}{ |\\gamma ^{\\prime }(1) |}$ with corresponding eigenvalues $\\lambda _n,$ and $v_1$ tangent to $\\gamma .$ W.r.t.",
"this basis, the mapping $T_3$ is given by $T_3: \\ \\sum \\nolimits _n \\alpha _n \\operatorname{pt}_{[u_{i},y_{i-1}]_{\\frac{1}{2}},S(u_{i-1},y_{i-1},u_{i})} v_n \\mapsto \\sum \\nolimits _n \\alpha _n f(\\lambda _n) v_n,$ where, using the symbol $d$ for the distance between $u_{i-1}$ and $[u_{i},y_{i-1}]_{\\frac{1}{2}},$ $f(\\lambda _n) ={\\left\\lbrace \\begin{array}{ll}2, & \\quad \\text{if} \\ \\lambda _n = 0, \\\\{\\sin (2 \\sqrt{\\lambda _n} d)}/{\\sin (\\sqrt{\\lambda _n} d)}, & \\quad \\text{if} \\ \\lambda _n > 0, \\quad d < \\pi /\\sqrt{\\lambda _n},\\\\{\\sinh (2 \\sqrt{-\\lambda _n} d)}/{\\sinh (\\sqrt{-\\lambda _n} d)}, & \\quad \\text{if} \\ \\lambda _n < 0.\\end{array}\\right.",
"}$ Here, the $\\alpha _n$ are the coefficients of the corresponding basis representation.",
"Further, let $\\xi :[0,1]\\rightarrow \\mathcal {M}$ be the geodesic connecting $y_{i-1}=\\xi (0)$ and $u_{i}=\\xi (1)$ and $w_n$ be an orthonormal basis of eigenvectors of $J \\mapsto R(\\frac{\\xi ^{\\prime }(0)}{ |\\xi ^{\\prime }(0)|} ,J) \\frac{\\xi ^{\\prime }(0)}{ |\\xi ^{\\prime }(0) |}$ with eigenvalues $\\mu _n,$ and $w_1$ tangent to $\\xi .$ Then, w.r.t.",
"this basis, the mapping $T_4$ is given by $T_4: \\ \\sum \\nolimits _n \\beta _n \\operatorname{pt}_{y_{i-1},[u_{i},y_{i-1}]_{\\frac{1}{2}}} w_n \\mapsto \\sum \\nolimits _n \\beta _n g(\\lambda _n) w_n,$ where, using the symbol $d^{\\prime }$ for the distance between $y_{i-1}$ and $u_{i},$ $g(\\lambda _n) ={\\left\\lbrace \\begin{array}{ll}1/2, & \\quad \\text{if} \\ \\lambda _n = 0, \\\\{\\sin (\\frac{1}{2} \\sqrt{\\lambda _n} d^{\\prime })}/{\\sin (\\sqrt{\\lambda _n} d^{\\prime })}, & \\quad \\text{if} \\ \\lambda _n > 0,\\quad d < \\pi /\\sqrt{\\lambda _n},\\\\{\\sinh (\\frac{1}{2} \\sqrt{-\\lambda _n} d^{\\prime })}/{\\sinh (\\sqrt{-\\lambda _n} d^{\\prime })}, & \\quad \\text{if} \\ \\lambda _n < 0.\\end{array}\\right.",
"}$ Here, the $\\beta _n$ are the coefficients of the corresponding basis representation.",
"The proof is given in Section REF .",
"Next, we consider the Riemannian gradients for the Schild version $D^{\\text{sym}}=\\text{D}_{\\text{S}}^{\\text{sym}}$ as given in (REF ) which are needed for the CPPA for the bivariate TGV functional.",
"To this end, we that note that $\\text{D}_{\\text{S}}^{\\text{sym}}$ can be expressed in terms of the function $S$ of (REF ) by $\\text{D}_{\\text{S}}^{\\text{sym}}([u_{i,j},y_{i,j}^ 1],[u_{i,j},y^ 2_{i,j}],[u_{i,j-1},y^ 1_{i,j-1}],[u_{i-1,j},y^2_{i-1,j}])= d( S(y_{i,j-1}^1,r^2,r^1), y_{i-1,j}^2) \\\\\\text{s.t. }",
"r ^1 = S(u_{i,j},y^1_{i,j},u_{i-1,j}), \\, r ^2 = S(u_{i,j},y^2_{i,j},u_{i,j-1}).",
"$ Hence, the mapping $\\text{D}_{\\text{S}}^{\\text{sym}}$ (which depends on seven arguments) is an expression of the Riemannian distance function and three realizations of $S.$ We have seen how to differentiate $S$ in Lemma REF and Lemma REF .",
"We derive the gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ by iterated application of these mappings and the concepts of these lemmata.",
"We point out the symmetry of $\\text{D}_{\\text{S}}^{\\text{sym}}$ with respect to $u_{i-1,j},y^1_{i,j},$ and with respect to $u_{i,j-1}, y^2_{i,j}$ .",
"This reduces the task to actually considering five different arguments.",
"The derivative of $\\text{D}_{\\text{S}}^{\\text{sym}}$ is provided in the following proposition.",
"Its proof is a direct consequence of the previous results and the decomposition of $\\text{D}_{\\text{S}}^{\\text{sym}}$ into decompositions of the function $S$ and the distance function $d$ as above.",
"Proposition 4.4 We consider constellations of points with $\\text{D}_{\\text{S}}^{\\text{sym}}\\ne 0.$ Then, the gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"the variable $y^1_{i,j}$ is given by $\\nabla _{y^1_{i,j}} \\text{D}_{\\text{S}}^{\\text{sym}}=- T_4 \\circ T_3 \\circ \\tilde{T}_4 \\circ \\tilde{T}_3\\left({\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2}/{\\big |\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2 \\big |}\\right), \\\\ $ with the adjoint operators $T_3$ , $T_4$ given as in Lemma REF formed w.r.t.",
"the points $u_{i,j},u_{i-1,j},y^1_{i,j}$ and the adjoint operators $\\tilde{T}_3$ , $\\tilde{T}_4$ given as in Lemma REF formed w.r.t.",
"the points $y_{i,j-1}^1,r^2,r^1.$ The gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"$u_{i-1,j}$ has the same form by symmetry.",
"Similarly, the gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"the variable $y^2_{i,j}$ is given by $\\nabla _{y^2_{i,j}} \\text{D}_{\\text{S}}^{\\text{sym}}=- T_4 \\circ T_3 \\circ \\tilde{T}_4 \\circ \\tilde{T}_3\\left({\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2}/{\\big |\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2 \\big |}\\right), \\\\ $ with the adjoint operators $T_3$ , $T_4$ as in Lemma REF formed w.r.t.",
"the points $u_{i,j},u_{i,j-1},y^2_{i,j}$ and the adjoint operators $\\tilde{T}_3$ , $\\tilde{T}_4$ as in Lemma REF formed w.r.t.",
"the points $y_{i,j-1}^1,r^2,r^1$ .",
"The gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"$u_{i,j-1}$ has the same form by symmetry.",
"The gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"the variable $y_{i,j-1}^ 1$ is given by $\\nabla _{y_{i,j-1}^ 1} \\text{D}_{\\text{S}}^{\\text{sym}}=- T_1\\left({\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2}/{\\big |\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2 \\big |}\\right), \\\\ $ with the adjoint operator $T_1$ given as in Lemma REF formed w.r.t.",
"the points $y_{i,j-1}^1,r^2,r^1$ .",
"The gradient with respect to the variable $y_{i-1,j}^ 2$ is simply given by $\\nabla _{y^2_{i-1,j}} \\text{D}_{\\text{S}}^{\\text{sym}}=- {\\log _{y_{i-1,j}^2} S(y_{i,j-1}^1,r^2,r^1)}/{\\big |\\log _{y_{i-1,j}^2} S(y_{i,j-1}^1,r^2,r^1) \\big |}.",
"\\\\ $ Finally, the gradient of $\\text{D}_{\\text{S}}^{\\text{sym}}$ w.r.t.",
"the variable $u_{i,j}$ is given by $\\nabla _{u_{i,j}} \\text{D}_{\\text{S}}^{\\text{sym}}=-T_1 \\circ T_4 \\circ T_3\\left(\\tfrac{\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2}{\\big |\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2 \\big |}\\right) \\\\ -\\tilde{T}_1\\circ T_4 \\circ T_3\\left(\\tfrac{\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2}{\\big |\\log _{S(y_{i,j-1}^1,r^2,r^1)} y_{i-1,j}^2 \\big |}\\right).$ Here the adjoint operators $T_3$ and $T_4$ are given as in Lemma REF formed w.r.t.",
"the points $y_{i,j-1}^1,r^2,r^1$ , $T_1$ is given as in Lemma REF formed w.r.t.",
"the points $u_{i,j},y_{i,j}^1,u_{i-1,j}$ and $\\tilde{T}_1$ is given as in Lemma REF formed w.r.t.",
"the points $u_{i,j},y_{i,j}^2,u_{i,j-1}$ ."
],
[
"Riemannian gradients for the parallel transport version",
"We here derive the derivatives needed to compute the proximal mappings in Section REF when specifying to the parallel transport variant.",
"We use the short-hand notation $F(u_i,u_{i-1},y_i,y_{i-1}) = \\text{D}_{\\text{pt}}([u_i,y_i],[u_{i-1},y_{i-1}]).$ For the following computations recall that, by our assumption of uniqueness of length-minimizing geodesics, the $\\log $ mapping, initially defined as set-valued, always maps to single points in the tangent space.",
"Lemma 4.5 The function $F$ given by (REF ) is symmetric with respect to interchanging $(u_i,y_i)$ with $(u_{i-1},y_{i-1}).$ In particular, for points with $F \\ne 0,$ the gradient of $F$ w.r.t.",
"the third variable $y_{i},$ is given by $\\nabla _{y_{i}} F(u_i,u_{i-1},y_i,y_{i-1}) =\\nabla _{y_{i-1}} F(u_{i-1}, u_i,y_{i-1}, y_i).$ Further, again for points with $F \\ne 0$ , the gradient of the function $F$ w.r.t.",
"the first component variable $u_{i},$ is given by $\\nabla _{u_{i}} F(u_i,u_{i-1},y_i,y_{i-1}) =\\nabla _{u_{i-1}} F(u_{i-1}, u_i,y_{i-1}, y_i).$ The proof is given in Section REF by specifying $F=F_0$ in Lemma REF .",
"Regarding the fact that we consider points with $F \\ne 0$ in the above statement, we remark that the points $u_i,u_{i-1},y_i,y_{i-1}$ such that $F(u_i,u_{i-1},y_i,y_{i-1})=0$ form a set of measure zero.",
"Only on this zero set, $F$ is not differentiable.",
"Further, in this case, the four-tuple consisting of the four zero-tangent vectors sitting in $u_i,u_{i-1},y_i,y_{i-1}$ belong to the subgradient of $F.$ We note that Lemma REF tells us that, in order to compute the gradient of the second order type difference $F$ , we only need to compute the respective gradient of $F$ w.r.t.",
"$u_{i-1}$ and $y_{i-1}.$ This is done in the following.",
"Lemma 4.6 The gradient of the function $F$ for points with $F \\ne 0$ w.r.t.",
"the variable $y_{i-1}$ is given by $\\nabla _{y_{i-1}} F = T\\left(\\left({\\log _{u_{i-1}} y_{i-1} - \\operatorname{pt}_{u_{i},u_{i-1}} \\log _{u_i} y_i}\\right)/{\\big | \\log _{u_{i-1}} y_{i-1} - \\operatorname{pt}_{u_{i},u_{i-1}} \\log _{u_i} y_i \\big |} \\right), $ where $T = (d_{y_{i-1}} \\log _{u_{i-1}})^\\ast $ is the adjoint of the (Fréchet) derivative (denoted by the symbol $d_{y_{i-1}}$ ) of the $\\log $ mapping w.r.t.",
"the variable $y_{i-1}.$ $T$ can be computed using Jacobi fields.",
"The proof is given in Section REF .",
"In Riemannian symmetric spaces the above mapping $T$ can be made more explicit.",
"Lemma 4.7 Let $\\mathcal {M}$ be a symmetric space.",
"Consider the geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connecting $\\gamma (0)=u_{i-1}$ and $\\gamma (1)=y_{i-1},$ and let $d$ denote the distance between $u_{i-1}$ and $y_{i-1}.$ Let $(v_n)_n$ be an orthonormal basis of eigenvectors of the self-adjoint Jacobi operator $J \\mapsto R(\\frac{\\gamma ^{\\prime }(0)}{ |\\gamma ^{\\prime }(0) |},J)\\frac{\\gamma ^{\\prime }(0)}{ |\\gamma ^{\\prime }(0) |}$ with $v_1$ tangent to $\\gamma ,$ and denote, for each $n$ , the eigenvalue associated with $v_n$ by $\\lambda _n.$ W.r.t.",
"this basis, the operator $T$ of Lemma REF can be represented by $T: \\ \\sum _n \\alpha _n v_n \\mapsto \\sum _n \\alpha _n f(\\lambda _n) \\operatorname{pt}_{u_{i-1},y_{i-1}} v_n,$ where the $\\alpha _n$ are the coefficients of the corresponding basis representation and the function $f,$ depending on the sign of $\\lambda _n$ is given by $f(\\lambda _n) ={\\left\\lbrace \\begin{array}{ll}1, & \\quad \\text{if} \\ \\lambda _n = 0, \\\\{\\sqrt{\\lambda _n} d}/{\\sin (\\sqrt{\\lambda _n} d)}, & \\quad \\text{if} \\ \\lambda _n > 0, \\quad d < \\pi /\\sqrt{\\lambda _n},\\\\{\\sqrt{-\\lambda _n} d}/{\\sinh (\\sqrt{-\\lambda _n} d)}, & \\quad \\text{if} \\ \\lambda _n < 0.\\end{array}\\right.",
"}$ The proof is given in Section REF .",
"Finally, we consider the gradient of $F$ w.r.t.",
"the variable $u_{i-1}.$ Lemma 4.8 The gradient of the function $F$ for points with $F \\ne 0$ w.r.t.",
"the variable $u_{i-1}$ is given by $\\nabla _{u_{i-1}} F_1 = \\sum \\alpha _n v_n, $ where the $(v_n)_n$ form an orthonormal basis of the tangent space at $u_{i-1},$ and the coefficients $\\alpha _n$ are given by $\\alpha _n =\\frac{d}{dt}|_{t=0} \\big | L^n_t - B^n_t \\big |& = \\left\\langle \\tfrac{L(u_{i-1})- B(u_{i-1})}{\\left| L(u_{i-1})- B(u_{i-1}) \\right|} ,\\tfrac{D}{dt}|_{t=0} L_t^n - \\tfrac{D}{dt}|_{t=0} B_t^n \\right\\rangle .$ Here $L_t^n,B_t^n$ denote the vector fields $L,B$ (defined by $L: u_{i-1} \\mapsto \\log _{u_{i-1}}y_{i-1}$ and $B: u_{i-1} \\mapsto \\operatorname{pt}_{u_{i},u_{i-1}} z,$ where $z = \\log _{u_{i}}y_{i},$ ) along the (specific) geodesic $t \\mapsto \\exp _{u_{i-1}}tv_n,$ $t \\in [0,1],$ determined by $v_n,$ i.e., $L^n_t = \\log _{\\exp _{u_{i-1}}tv_n}y_{i-1} \\quad \\text{and}\\quad B^n_t = \\operatorname{pt}_{u_{i},\\exp _{u_{i-1}}tv_n} z,$ with $z = \\log _{u_{i}}y_{i}.$ The proof is given in Section REF .",
"The precise computation of $\\frac{D}{dt}|_{t=0} L_t^n$ in symmetric spaces is topic of Lemma REF .",
"Further, the computation of $\\frac{D}{dt}|_{t=0} B_t^n$ is carried out for the manifolds explicitly considered in this paper: this is done for the sphere in Lemma REF , and for the space of positive matrices in Lemma REF .",
"Lemma REF and its proof may serve as a prototypic guide for deriving similar expressions for other symmetric spaces such as the rotation groups or the Grassmannians for instance.",
"We note that the approach is by no means restricted to the two considered classes of spaces and might serve as a guide for other manifolds; we only did not derive a more explicit representation on the general level of symmetric spaces.",
"We further note that numerical differentiation of the particular term is a second option as well.",
"Lemma 4.9 Assume that the manifold $\\mathcal {M}$ is a symmetric space.",
"Let $(v_n)_n$ be an orthonormal basis of eigenvectors of the self-adjoint Jacobi operator $J \\mapsto R(\\frac{\\gamma ^{\\prime }(0)}{ |\\gamma ^{\\prime }(0) |} ,J) \\frac{\\gamma ^{\\prime }(0)}{ |\\gamma ^{\\prime }(0) |}$ where the (constant speed) geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connects $u_{i-1} = \\gamma (0)$ and $y_{i-1} = \\gamma (1),$ and where $R$ denotes the Riemannian curvature tensor.",
"For each $n$ , we denote by $\\lambda _n$ the eigenvalue associated with $v_n.$ The covariate derivatives $\\frac{D}{dt}|_{t=0} L_t^n,$ of the vector fields $L^n_t = \\log _{\\exp _{u_{i-1}}tv_n}y_{i-1}$ at $t=0$ can be computed jointly for all $n $ using Jacobi fields as follows: $\\frac{D}{dt}|_{t=0} L_t^n ={\\left\\lbrace \\begin{array}{ll}-v_n, &\\text{if} \\quad \\lambda _n = 0, \\\\- d \\sqrt{\\lambda _n} \\ \\frac{ \\cos ( \\sqrt{\\lambda _n} d)}{\\sin ( \\sqrt{\\lambda _n} d)} \\ v_n,& \\text{if} \\quad \\lambda _n > 0, \\quad d < \\pi /\\sqrt{\\lambda _n}, \\\\- d \\sqrt{-\\lambda _n} \\ \\frac{ \\cosh ( \\sqrt{-\\lambda _n} d)}{\\sinh ( \\sqrt{-\\lambda _n} d)} \\ v_n,& \\text{if} \\quad \\lambda _n < 0.\\end{array}\\right.}",
"$ Here $d = d(u_{i-1},y_{i-1})$ denotes the length of the geodesic connecting $u_{i-1},y_{i-1}.$ (If the term $\\sqrt{\\lambda _n} d = 0$ in the denominators of the second line in (REF ), then $u_{i-1} = y_{i-1},$ and the formula is still valid since we are facing a removable singularity then.)",
"The proof is given in Section REF .",
"Lemma 4.10 Consider the unit sphere $S^2$ embedded into euclidean space $\\mathbb {R}^3$ .",
"For $u_i, u_{i-1}$ with $u_i \\ne u_{i-1}$ , the differential $\\frac{D}{dt}|_{t=0} B_t^n$ is given by $\\tfrac{D}{dt}|_{t=0} B_t^n = \\tfrac{D}{dt}|_{t=0} \\operatorname{pt}_{u_{i},\\exp _{u_{i-1}}tv_n} z ={\\left\\lbrace \\begin{array}{ll}\\ 0 \\quad & \\text{ for } v_n \\ \\Vert \\log _{u_{i-1}}u_i, \\\\\\ {\\mathrm {L}}_\\omega \\operatorname{pt}_{u_{i},u_{i-1}} z,\\qquad & \\text{ for } v_n \\perp \\log _{u_{i-1}}u_i, \\ |v_n |=1, \\end{array}\\right.",
"}$ and $v_n$ to the left of $\\log _{u_{i-1}}u_i$ (otherwise multiplied by $-1$ accounting for the change of orientation).",
"Here the skew-symmetric matrix ${\\mathrm {L}}_\\omega =\\begin{pmatrix}0 & \\omega \\\\ -\\omega & 0\\end{pmatrix}$ is taken w.r.t.",
"the basis $\\lbrace \\log _{u_{i-1}}u_i,$ $(\\log _{u_{i-1}}u_i)^\\perp \\rbrace $ of the tangent space $T_{u_{i-1}},$ and $\\omega $ is given by $\\omega =\\tfrac{1}{\\sin d} - \\tfrac{1}{\\tan d},\\quad \\text{where } d = d(u_{i},u_{i-1}).$ For general $v_n,$ $\\tfrac{D}{dt}|_{t=0} B_t^n = \\tfrac{D}{dt}|_{t=0} \\operatorname{pt}_{u_{i},\\exp _{u_{i-1}}tv_n} z =\\big \\langle v_n, w \\big \\rangle \\ {\\mathrm {L}}_\\omega \\operatorname{pt}_{u_{i},u_{i-1}} z,$ where $w$ is the vector determined by $w \\perp \\log _{u_{i-1}}u_i, \\ |w|=1,$ and $w$ is to the left of $\\log _{u_{i-1}}u_i.$ In other words, we have to multiply the second line of (REF ) with the signed length of the projection of $v_n$ to the normalized vector $(\\log _{u_{i-1}}u_i)^\\perp $ .",
"If $u_i = u_{i-1},$ then the differential $\\frac{D}{dt}|_{t=0} B_t^n = 0$ (which is consistent with letting $d \\rightarrow 0$ in the above formulae.)",
"The proof is given in Section REF .",
"Lemma 4.11 Let $\\mathcal {M}$ be the space of symmetric positive definite matrices.",
"Then, the covariate derivative $\\frac{D}{dt}|_{t=0} B_t^n$ (which is a tangent vector sitting in $u_{i-1}$ ) is given by the following sum of matrices $\\tfrac{D}{dt}|_{t=0} B_t^n = (T-\\tfrac{1}{2}S) + (T-\\tfrac{1}{2}S)^\\top ,$ where $(T-\\tfrac{1}{2}S)^\\top $ denotes the transpose of the matrix $T-\\tfrac{1}{2}S.$ The matrix $S$ is determined in terms of elementary matrix operations of the data (by (REF ) in the proof of the statement).",
"The matrix $T$ is determined in terms of elementary matrix operations and the solution of a Sylvester equation (by (REF ) in the proof of the statement with the Sylvester equation given by (REF ) there).",
"The proof is given in Section REF .",
"Summing up, we have computed the derivatives of all building blocks necessary to compute the derivative of $F(u_i,u_{i-1},y_i,y_{i-1}) = \\text{D}_{\\text{pt}}([u_i,y_i],[u_{i-1},y_{i-1}])$ for the non-degenerate case $F \\ne 0.$ Remark 4.12 For the bivariate version of the parallel transport based $\\text{M-TGV}^2_\\alpha $ realization of Section REF , we can use the analogue of the decomposition (REF ) with $\\text{D}_{\\text{S}}, \\text{D}_{\\text{S}}^{\\text{sym}}$ replaced by the corresponding parallel transport versions (REF ) and (REF ).",
"Then we can use this analogue of the decomposition (REF ) and apply the CPPA iteration (REF ) to this decomposition.",
"The proximal mappings $\\operatorname{prox}_{\\lambda g^{(1)}_{ij}}$ are given by (REF ), $\\operatorname{prox}_{\\lambda g^{(2)}_{ij}},\\operatorname{prox}_{\\lambda g^{(3)}_{ij}}$ are given by (REF ) as for the Schild variant above.",
"The proximal mappings $\\operatorname{prox}_{\\lambda g^{(4)}_{ij}},\\operatorname{prox}_{\\lambda g^{(5)}_{ij}}$ are computed as in the univariate case considered in Section REF .",
"In order to compute the proximal mappings of the atoms $g^{(6)}_{ij}$ using a subgradient descent, it is necessary to differentiate the mapping $\\text{D}_{\\text{pt}}^{\\text{sym}}$ of (REF ) (which is the analogue of $\\text{D}_{\\text{S}}^{\\text{sym}}$ for the Schild case) w.r.t.",
"its seven arguments.",
"As in the Schild case, it is possible to decompose the mapping into simpler functions which we have already considered in Section REF .",
"We do not carry out the rather space consuming derivation here."
],
[
"Numerical results",
"This section provides numerical results for synthetic and real signals and images.",
"We first describe the experimental setup.",
"We carry out experiments for $S^1$ , $S^2$ and $\\mathrm {Pos}_3$ (the manifold of symmetric positive definite $3 \\times 3$ matrices equipped with the Fisher-Rao metric) valued data.",
"$S^1$ data is visualized by the phase angle, and color-coded as hue value in the HSV color space when displaying image data.",
"We visualize $S^2$ data either by a scatter plot on the sphere as in Figure REF , or by a color coding as in Figure REF .",
"Data on the $\\mathrm {Pos}_3$ manifold is visualized by the isosurfaces of the corresponding quadratic forms.",
"More precisely, the ellipse visualizing the point $f_p$ at voxel $p$ are the points $x$ fulfilling $(x-p)^\\top f_p (x-p) = c,$ for some $c>0.$ To quantitatively measure the quality of a reconstruction, we use the manifold variant of the signal-to-noise ratio improvement $\\mathrm {\\Delta SNR}= 10 \\log _{10} \\left( \\frac{\\sum _{ij} d(g_{ij}, f_{ij})^2 }{\\sum _{ij} d(g_{ij}, u_{ij})^2}\\right) \\mathrm {\\,dB},$ see [86], [93].",
"Here $f$ is the noisy data, $g$ the ground truth, and $u$ a regularized restoration.",
"A higher $\\mathrm {\\Delta SNR}$ value means better reconstruction quality.",
"For adjusting the model parameters $\\alpha _0,\\alpha _1$ of $\\text{M-TGV}^2_\\alpha $ , it is convenient to parametrize them by $ \\alpha _0 = r \\frac{(1 - s)}{\\min (s, 1-s)}, \\quad \\text{and}\\quad \\alpha _1 = r \\frac{s}{\\min (s, 1-s)},$ so that $r \\in (0, \\infty )$ controls the overall regularization strength and $s \\in (0,1)$ the balance of the two TGV penalties.",
"For $s \\rightarrow 0$ we get $\\alpha _0 \\rightarrow \\infty $ and $\\alpha _1 = r,$ so that $\\operatorname{TGV}$ minimization approximates the minimization of $\\operatorname{TV}$ modulo a linear term which can be added at no cost.",
"For $s \\rightarrow 1$ we have $\\alpha _0 = r$ and $\\alpha _1 \\rightarrow \\infty $ which corresponds to pure second-order TV regularization.",
"One may think of $r$ as the parameter mainly depending on the noise level, and of $s$ as the parameter mainly depending on the geometry of the underlying image.",
"Figure REF illustrates the influence on $r$ and $s$ for a synthetic $S^1$ -valued image corrupted by von Mises noise with concentration parameter $\\kappa = 5.$ There and in most of the following experiments, we observed satisfactory results for the fixed value $s = 0.3.$ Based on these observations, we suggest to use $s = 0.3$ as a starting point, to decrease or increase it if the image is dominated by edges or smooth parts, respectively.",
"We have implemented the presented methods in Matlab 2016b.",
"We use the toolbox MVIRT [10] for the basic manifold operations such as log, exp and parallel transport, and for parts of the visualizationImplementation available at https://github.com/kellertuer/MVIRT.",
"We used 100000 iterations for all experiments with univariate data and 1000 iterations for the image data.",
"The cooling sequence $(\\lambda ^k)_{k \\in \\mathbb {N}}$ used as stepsize in the gradient descent for computing the non-explicit proximal mappings was chosen as $\\lambda ^k = \\lambda ^0 k^{-\\tau }$ with $\\tau = 0.55.$ For the univariate spherical data we observed faster convergence when using a stagewise cooling, i.e., letting the sequence fixed to $\\lambda _0$ for 500 iterations in the first stage, use the moderate cooling $\\tau = 0.35$ in the second stage until iteration 1000 and then the cooling $\\tau = 0.55$ afterwards.",
"Figure: Effect of the model parameters of S-TGV using the parametrization by r>0r >0 and s∈(0,1)s \\in (0,1) according to () for an S 1 S^1-valued image.A higher value of rr results in a stronger smoothing.For small values of s,s, the edges are well-preserved but some staircasing effects appear.",
"For high values of s,s, the linear trends are recovered but the edges are smoothed out.",
"When using an intermediate value such as s=0.3,s =0.3, we observe a combination of positive effects of TV\\operatorname{TV} and TV 2 \\operatorname{TV}^2 regularization: rather sharp edges and good recovery of linear trends."
],
[
"Comparison between Schild variant and parallel transport variant.",
"First we compare the two proposed realizations of manifold TGV: the parallel transport variant and the Schild variant.",
"Figure REF shows the results with both variants for some typical univariate signals for the $S^1,$ the $S^2$ and the $\\mathrm {Pos}_3$ manifold.",
"(The $S^1$ -valued signal was corrupted by von Mises noise with concentration parameter $\\kappa = 10.$ The $S^2$ -valued signal was taken from [10] and corrupted by applying the exponential mapping of Gaussian distributed tangent vectors as in [10] with $\\sigma = 0.1.$ The $\\mathrm {Pos}_3$ -valued signal was corrupted by applying the exponential mapping of Gaussian noise distributed on the tangent vectors with $\\sigma = 0.2.$ The $r$ -parameters were chosen as $r= 1$ for the spherical signals and as $r = 0.2$ for the $\\mathrm {Pos}_3$ signal.",
"In all cases $s = 0.3$ .)",
"The results for the spherical data appear very similar.",
"For the $\\mathrm {Pos}_3$ manifolds, there are slight visible differences at some points, e.g.",
"near the discontinuity of the second to fourth position of the signal and at the last position.",
"In summary, we observe a qualitatively similar result in the sense that the geodesic parts are well reconstructed and that the edges are preserved.",
"In the following, we focus on the Schild variant."
],
[
"Comparison of manifold TGV with vector-space TGV.",
"In order to validate both our generalizations of TGV to the manifold setting as well as numerical feasibility of our optimization algorithms, we carry out a comparison to the vector-space case.",
"This is done for $S^1$ , the unit sphere in $\\mathbb {R}^2$ .",
"By generating a signal with values that are strictly contained in one hemisphere, we can unroll the signal and compare to vector-space TGV-denoising in a way that the same results can be expected.",
"We carried out this comparison for synthetic data without noise and different values of the balancing parameter $s$ .",
"We tested the setting of $s=0.3$ , which comprises a good balance between first and second order terms, as well as the rather extreme settings of $s=0.05$ and $s = 0.95$ .",
"The regularization parameter $r$ was fixed to $r=1$ .",
"In order to approximate the ground-truth solution of second-order TGV denoising in the vector space setting, we implemented the Chambolle-Pock algorithm [32] for this situation.",
"To ensure a close approximation of the ground truth for all parameter settings, we carried out a dedicate stepsize tuning to accelerate convergence of the algorithm, computed $2\\times 10^{5}$ iterations and ensured optimality by measuring the duality gap.",
"The result of this evaluation can be found in Figure REF .",
"It can be observed that the qualitative properties of the numerical solution obtained with the manifold-TGV code are similar to the ones of the approximate ground-truth of the vector space setting, confirming overall feasibility of our model and implementation.",
"For the case $s = 0.05$ , one can see in particular in the right part of the bottom plot (starting with the first plateau after 180 on the x-axis), that the solution is piecewise constant up to a linear part, which is approximately the same for all four plateaus.",
"This is what one would expect for the extreme case $s\\rightarrow 0$ , as in this case, $\\operatorname{TGV}$ minimization mimics TV minimization up to a globally linear term.",
"On the other hand, for the case $s = 0.95$ , one can see that the solution is piecewise linear with no jumps.",
"This can be expected from second-order TV minimization, which coincides with second order $\\operatorname{TGV}$ minimization for $s \\rightarrow 1$ .",
"In particular, the piecewise linear part on the left is approximated well, while the plateaus on the right are not well captured.",
"The case $s=0.3$ provides a good compromise here: The linear parts on the left are still well captured, but the solution still admits jumps on the plateau parts, as can be seen in particular outermost right jump after point.",
"We also note that there are still some differences of the solution obtained with the manifold code and the Chambolle-Pock result interpreted as approximate ground truth of the vector space case, in particular for the cases $s=0.05$ and $s=0.95$ .",
"We believe that this is mostly an issue of the algorithmic realization rather than the model itself caused by the numerical solutions obtained for the more extreme cases $s=0.05$ and $s=0.95$ ."
],
[
"Basic situations.",
"The aim of this experiment is to investigate the performance of the proposed manifold-TGV model for different basic situations and noise levels.",
"The experiment is carried out on a univariate signal on the two-sphere $S^2$ , that comprises jumps in the signal as well as its derivative and is composed of piecewise constant and geodesic signals.",
"The results of the experiment for different noise levels (no noise, intermediate noise, strong noise), a fixed value of $r=1$ and different values of $s$ , namely $s \\in \\lbrace 0.05,0.95,0.3\\rbrace $ , can be found in Figure REF .",
"As can be seen there, the manifold-TGV functional regularizes the data quite well and is able to achieve good results, even in the case with relatively strong noise where it is hard to see any structure in the data.",
"As in the previous experiment, the choice $s=0.05$ promotes piecewise constant solutions, which is naturally best for regions where the signal is piecewise constant, but leads to “staircasing” in smooth regions, as can be particularly seen in the geodesic parts of the strong-noise case.",
"The choice $s=0.95$ approximates geodesics well, but does not allow for jumps and also produces some oscillations in the smooth parts of the case with strong noise.",
"Again the choice $s=0.3$ is a good compromise.",
"Even though it does not reconstruct jumps as well as the TV-like version, it still allows for jumps and reconstructs the geodesic parts rather accurately.",
"Figure: Results for univariate S 2 S^2 data.",
"Top: Ground truth data and reconstruction.",
"Middle: Data with intermediate noise and reconstruction.",
"Bottom: Data with strong noise and reconstruction.",
"From left to right: Data, S-TGV α 2 \\text{S-TGV}^2_\\alpha regularized reconstruction for s=0.05s = 0.05, s=0.95s = 0.95, s=0.3s=0.3, respectively.",
"The color gradation indicates the ordering of the signal."
],
[
"Results for synthetic data",
"Next we investigate the denoising performance of $\\text{M-TGV}^2_\\alpha $ on synthetic images.",
"We compare the results of the proposed algorithm with the results of manifold TV regularization [93], and with the result of second-order manifold TV, denoted by $\\operatorname{TV}^2$ [10].",
"The model parameter of first-order TV is denoted by $\\alpha $ and that of $\\operatorname{TV}^2$ by $\\beta .$ For all algorithms, 1000 iterations were used.",
"First we look at $S^2$ -valued images.",
"As in [10], the noisy data $f$ were created from the original image $g$ by $f_{ij} = \\exp _{g_{ij}} \\eta _{ij}$ where $\\eta _{ij}$ is a tangent vector at $g_{ij}$ and both its components are Gaussian distributed with standard deviation $\\sigma = \\frac{4}{45}\\pi .$ For comparison with first order TV we scanned the parameter range $\\alpha = 0.1, 0.2, \\ldots , 1$ and the special parameters $\\alpha = 0.22$ and $\\alpha = 3.5 \\cdot 10^{-2}$ given in [10] and the corresponding implementation.",
"Similarly, for $\\operatorname{TV}^2$ we scanned the parameter range $\\beta = 1, 5, 10, 15 \\ldots , 30,$ and the special parameters $\\beta = 8.6$ and $\\beta = 29.5$ suggested in [10].",
"As no beneficial effect of combining first and second-order TV was observed in [10], we used pure $\\operatorname{TV}$ and $\\operatorname{TV}^2$ regularization.",
"For the proposed method, we fixed $s=0.3$ and report the best result among the six parameters $r = 0.05, 0.1, 0.15, \\ldots , 0.3.$ The results in Figure REF show that the second-order methods give a significantly smoother result than first order TV and that they do not suffer from staircasing effects.",
"On the flipside, $\\operatorname{TV}^2$ regularization results in an undesired smoothing effect of the edges between the tiles, best seen at the bottom left tile.",
"The proposed TGV regularization preserves these edges which results in an improved reconstruction quality.",
"Figure: Comparison of S-TGV with first and second-order total variationon an S 2 S^2-valued image from .",
"The spherical values are visualized accordingto the scheme (c) which means that the north pole is white, the south pole is blackand the equator gets hue values according to its longitude.Next we look at $\\mathrm {Pos}_3$ -valued images.",
"Such images appear naturally in diffusion tensor imaging (DTI), so we briefly describe the setup.",
"DTI is a medical imaging modality based on diffusion weighted magnetic resonance images (DWI) which for example allows to reconstruct fiber tract orientations [11].",
"A DWI captures the diffusivity of water molecules with respect to a direction $v \\in \\mathbb {R}^3.$ The relation between a diffusion tensor $f(p)$ and the DWIs $D_v(p)$ at the voxel $p$ is given by the Stejskal-Tanner equation $ D_v(p) = A_0 e^{- b \\ v^\\top f(p) v}$ with constants $b,A_0>0.$ A standard noise model for the DWIs is the Rician model which assumes a complex-valued Gaussian noise in the original frequency domain measurements [12].",
"This means that assuming the model (REF ) the actual measurement in direction $v$ at pixel $p$ is given by $D^{\\prime }_v(p) = ((X+ D_v(p))^2 + Y^2)^{1/2},$ with the Gaussian random variables $ X,Y \\sim N(0,\\sigma ^2).$ Building on this model, we impose the noise as follows.",
"From the synthetic tensor image, we compute DWIs according to (REF ) with respect to 21 different directions.",
"Then we impose Rician noise to all derived DWIs, and we estimate the corresponding diffusion tensor image $f$ using a least squares approach on (REF ).",
"Due to the noise, some of the fitted tensors might not be positive definite and thus have no meaningful interpretation.",
"To handle such invalid tensor we exclude them from the data fidelity term.",
"On the other hand, we keep the corresponding pixels in the regularizing term so that we achieve a reasonable inpainting.",
"In Figure REF , we illustrate the effects of the regularization for a synthetic $\\mathrm {Pos}_3$ -valued image.",
"For comparison with (combined) first and second-order TV, we scanned all combinations of the parameters $\\alpha = 0, 0.01, 0.035, 0.1, 0.3125, 0.375,$ and $\\beta = 0.02, 0.05, 0.125, 0.625.$ As before, this comprises the parameters used in [10] and the corresponding implementation for that manifold.",
"For the proposed method, we report the best result among the possible combinations of the parameters $r = 0.06, 0.08,0.09, 0.1, 0.12$ and $s = 0.3, 0.35, 0.4, 0.45.$ In the example we observe that combined $\\operatorname{TV}$ /$\\operatorname{TV}^2$ regularization yields a similar reconstruction quality as pure first order $\\operatorname{TV},$ whereas S-TGV gives a significantly higher reconstruction quality.",
"Figure: Denoising results for a synthetic Pos 3 \\mathrm {Pos}_3-valued image corrupted by Rician noise.Combined TV/TV 2 \\operatorname{TV}/\\operatorname{TV}^2-regularization yields a similar reconstruction quality as first order TV,\\operatorname{TV},whereas S-TGV gives a significantly higher reconstruction quality."
],
[
"Results for real data",
"We illustrate the effects of TGV regularization on real manifold-valued signals and images.",
"First we look at smoothing a time series of wind directions.",
"The natural data space for a signal of orientations is the unit circle $S^1.$ The present data shows wind directions recorded every hour in 2016 by the weather station SAUF1, St. Augustine, FL.",
"; see Figure REFData available at http://www.ndbc.noaa.gov/historical_data.shtml.. We observe that the proposed method smoothes the orientations while respecting the phase jump from $-\\pi $ to $\\pi $ and preserving linear trends in the data.",
"Figure: Top: Hourly wind directionsat weather station SAUF1 (St. Augustine, FL) in the year 2016.Bottom: S-TGV (s=,s = \\protect , r=r = \\protect )smoothes the signal while not over-smoothing jump discontinuities and properly dealing with the phase jump from -π-\\pi to π\\pi and preserving linear trends in the data.Next we look at smoothing of interferometric synthetic aperture radar (InSAR) images.",
"Synthetic aperture radar (SAR) is a radar technique for sensing the earth's surface from measurements taken by aircrafts or satellites.",
"InSAR images consist of the phase difference between two SAR images, recording a region of interest either from two different angles of view or at two different points in time.",
"Important applications of InSAR are the creation of accurate digital elevation models and the detection of terrain changes; cf.",
"[62], [71].",
"As InSAR data consists of phase values that are defined modulo $2\\pi ,$ their natural data space is the unit circle.",
"In Figure REF , we illustrate the effect of S-TGV to an InSAR image taken from [83]Data available at https://earth.esa.int/workshops/ers97/papers/thiel/index-2.html.. We observe a clear smoothing effect while sudden phase changes are preserved.",
"Figure: Left: InSAR image from .Right: Result of S-TGV with r=,r = \\protect , s=.s = \\protect .The image is smoothed and sharp edges are preserved.At last we consider real DTI data which was taken from the Camino project [36]Data available at http://camino.cs.ucl.ac.uk/.",
"In Figure REF , we see an axial slice of a human brain (slice 20 of the Camino data).",
"We also display a zoom to the corpus callosum which connects the right and the left hemisphere.",
"The original tensors were computed from the diffusion weighted images by a least squares method based on the Stejskal-Tanner equation.",
"We observe that the proposed method smoothes the tensors in the corpus callosum while it preserves the sharp edges to the adjacent tissue.",
"Also observe the inpainting of the invalid tensors.",
"Figure: Left: Diffusion tensor image of a human brain (axial slice).The magnified image shows the corpus callosum.",
"Note the missing (invalid) tensors at several voxels.",
"Right: Result of S-TGV regularization with r=,r = \\protect ,s=.s = \\protect .The streamlines are smoothed whereas sharp transitions are preserved.Invalid voxels are reasonably inpainted."
],
[
"Conclusion",
"In this work, we have introduced and explored a notion of second-order total generalized variation (TGV) regularization for manifold-valued data.",
"First, we have derived a variational model for total generalized variation for manifold-valued data.",
"For this purpose, we have used an axiomatic approach.",
"We have first formalized reasonable fundamental properties of vector-valued TGV that should be conserved in the manifold setting.",
"Then we have proposed two realizations which we have shown to fulfill the required axioms.",
"The realization based on parallel transport is rather natural – although not straight forward – from the point of view of differential geometry.",
"The realization based on the Schild's ladder may be seen as an approximation of the parallel transport variant.",
"It requires less differential geometric concepts and it is easier to realize numerically while yielding comparable numerical results as shown in the experimental section.",
"Existence results for $\\text{M-TGV}^2_\\alpha $ -based denoising have been obtained for the proposed variants.",
"Next, we have derived an algorithm for the proposed model.",
"To this end we built on the well-established concept of a cyclic proximal algorithm.",
"As main contribution, we have performed the challenging task to derive all quantities necessary to compute the proximal mappings of the involved atoms.",
"Finally, we have conducted a numerical study of the proposed scheme.",
"The experiments revealed that M-TGV regularization reliably removes noise while it preserves edges and smooth trends.",
"A quantitative comparison on images with groundtruth indicates that TGV regularization improves upon first and second-order TV for manifold-valued data.",
"An interesting topic of future research are improved numerics such as earlier stopping criteria."
],
[
"Acknowledgment",
"Kristian Bredies and Martin Holler acknowledge support by the Austrian Science Fund (FWF) (Grant P 29192).",
"Martin Storath acknowledges support by the German Research Foundation DFG (Grant STO1126/2-1).",
"Andreas Weinmann acknowledges support by the German Research Foundation DFG (Grants WE5886/4-1, WE5886/3-1)."
],
[
"Proofs for Section 2",
"[Proof of Proposition REF ] Given that $D$ satisfies the assumptions of Proposition REF and the particular form of $\\text{D}^{\\text{sym}}$ in the vector-space case, it is immediate that $\\text{M-TGV}^2_\\alpha $ reduces to vector-space TGV, hence (M-P1') holds.",
"Now in case the minimum in (REF ) is attained at $y_{i,j}^1 = y_{i,j}^2 = u_{i,j}$ we get, since $\\text{D}^{\\text{sym}}([u_{i,j},u_{i,j}],[u_{i,j},u_{i,j}],[u_{i,j-1},u_{i,j-1}],[u_{i-1,j},u_{i-1,j}]) = 0 $ for all $i,j$ , that $\\text{M-TGV}^2_\\alpha (u) = \\alpha _1 \\sum _{i,j} \\Big ( d(u_{i+1,j},u_{i,j})^p + d(u_{i,j+1},u_{i,j})^p \\Big )^{1/p} = \\alpha _1 \\operatorname{TV}(u).",
"$ On the other hand, in case the minimum is attained at $y_{i,j}^1 = u_{i+1,j}$ and $y_{i,j}^2 = u_{i,j+1}$ we get, in the vector space situation, that $\\text{M-TGV}^2_\\alpha (u) =\\alpha _0 \\sum _{i,j} \\Big ( | u_{i+1,j}- 2u_{i,j} + u_{i-1,j}|^p+ | u_{i,j+1}- 2u_{i,j} + u_{i,j-1}|^p \\\\+ 2^{1-p} | (u_{i+1,j}- u_{i,j}) - (u_{i+1,j-1} - u_{i,j-1}) + (u_{i,j+1}- u_{i,j}) - (u_{i-1,j+1} - u_{i-1,j}) |^p \\Big )^{1/p}$ which coincides with $\\alpha _0 \\operatorname{TV}^2$ and $\\operatorname{TV}^2$ as in Definition REF .",
"Now suppose that $(u_{i,j})_{i,j}$ is locally geodesic.",
"Choosing $y_{i,j}^1 = u_{i+1,j}$ and $y_{i,j}^2 = u_{i,j+1}$ for all $i,j$ we get, as in the proof of Theorem REF , that $\\text{M-TGV}^2_\\alpha $ is bounded above by $ \\alpha _0 \\sum _{i,j}\\text{D}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}]) ,$ which is zero by assumption.",
"Now conversely, assume that $\\text{M-TGV}^2_\\alpha (u) = 0$ and any two points in $u_{i,j}$ , $u_{i^{\\prime },j^{\\prime }}$ with $\\max \\lbrace |i-i^{\\prime }|,|j-j^{\\prime }|\\rbrace \\le 2$ are connected by unique geodesics.",
"As in the proof of Proposition REF this implies that both $(u_{i,j})_i$ and $(u_{i,j})_j$ are locally geodesic as univariate signals.",
"Also we get that $ 0 = \\sum _{i,j} \\text{D}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}]), $ which implies by assumption, that $(u_{i+k,j-k})_k$ is locally geodesic.",
"Hence, by definition, $(u_{i,j})_{i,j}$ is locally geodesic as bivariate signal.",
"Finally, by Lemma REF , $u$ is geodesic.",
"[Proof of Theorem REF ] It suffices to verify the assumptions of Proposition REF .",
"It is easy to see that, since $\\text{D}_{\\text{S}}([x,x],[u,u])= 0$ , also $\\text{D}_{\\text{S}}^{\\text{sym}}([x,x],[x,x],[u,u],[u,u]) = 0$ .",
"Further, in the vector space setting, we get that $c^1 = \\frac{{u_{-,\\circ }}+ {y^1_{\\circ ,\\circ }}}{2}$ and $c^2 = \\frac{{u_{\\circ ,-}}+ {y^2_{\\circ ,\\circ }}}{2}$ as well as $r^1 = {u_{-,\\circ }}+ {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ and $r^2 = {u_{\\circ ,-}}+ {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ .",
"Consequently, $\\text{D}_{\\text{S}}([r^ 1,{y^2_{-,\\circ }}],[{y^1_{\\circ ,-}},r^ 2]) & = | {y^2_{-,\\circ }}- \\big ( {u_{\\circ ,-}}+ {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}\\big ) -\\big ( {u_{\\circ ,-}}+ {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- {y^1_{\\circ ,-}}\\big ) | \\\\& = \\big | {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^1_{\\circ ,-}}- {u_{\\circ ,-}}) + {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^2_{-,\\circ }}- {u_{-,\\circ }})\\big |$ and, using Proposition REF , (M-P1') to (M-P3') follow.",
"Now suppose that $(u_{i,j})_{i,j}$ is a geodesic three-by-three signal.",
"Hence, with the notation as in the definition of $\\text{D}_{\\text{S}}^{\\text{sym}}$ , $c^1 = u_{i,j} \\in [u_{i-1,j},u_{i+1,j}]_{\\frac{1}{2}}$ and we can choose $r^1 = u_{i,j} \\in [u_{i,j},u_{i,j}]_2$ .",
"Similar, $c^2 = u_{i,j} \\in [u_{i,j-1},u_{i,j+1}]_{\\frac{1}{2}}$ and we can again choose $r^2 = u_{i,j} \\in [u_{i,j},u_{i,j}]_2$ .",
"Consequently, $\\text{D}_{\\text{S}}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}])\\\\ \\le \\text{D}_{\\text{S}}([u_{i,j},u_{i-1,j+1}],[u_{i+1,j-1},u_{i,j}]).$ But, as shown in the proof of Theorem REF , the right-hand-side vanishes since $(u_{i-k,j+k})_k$ is geodesic as univariate signal.",
"Now conversely, assume that $u=(u_{i,j})_{i,j}$ is a three-by-three signal such that both $(u_{i,})_i$ and $(u_{i,j})_j$ are locally geodesic and that $ 0 = \\text{D}_{\\text{S}}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}]).",
"$ By the assumption on uniqueness of geodesics we get, again with the notation as in the definition of $\\text{D}_{\\text{S}}^{\\text{sym}}$ , that $c^1 = u_{i,j}$ and hence $r^1 = u_{i,j}$ and also that $c^2 = u_{i,j}$ and hence $r^2 = u_{i,j}$ .",
"Consequently, $\\text{D}_{\\text{S}}^{\\text{sym}}(\\ldots )$ reduces to $\\text{D}_{\\text{S}}([u_{i,j},u_{i-1,j+1}],[u_{i+1,j-1},u_{i,j}])$ .",
"But the latter being zero implies that $u_{i-1,j+1} \\in [u_{i+1,j-1},u_{i,j}]_2$ and, consequently, $u$ is geodesic.",
"Hence also the remaining assumptions of Proposition REF are satisfied and the assertion follows.",
"The following well-known fact on the parallel transport in manifolds will be required in the proof of Theorem REF .",
"For the sake of completeness, we provide a short proof.",
"Lemma A.1 Let ${u_-}$ , ${u_\\circ }$ and ${u_+}$ be three points in a manifold such that there exists a pairwise distance minimizing geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ such that $\\gamma (0) = {u_-}$ , $\\gamma (1/2) = {u_\\circ }$ , $\\gamma (1) = {u_+}$ .",
"Then $\\log _{u_-}^\\gamma ({u_\\circ }) = \\operatorname{pt}^\\gamma _{u_-}(\\log ^\\gamma _{u_\\circ }({u_+}))$ .",
"For $t \\in [0,1]$ , set $w(t) = (1/2)\\frac{d}{dt}\\gamma (\\frac{t}{2})$ .",
"Then $w$ satisfies $\\frac{D}{dt}w(t) = 0$ on $[0,1]$ , $w(0) = \\log _{u_-}^\\gamma ({u_\\circ })$ .",
"By definition of the parallel transport, $\\operatorname{pt}_{{u_\\circ }}^ \\gamma (\\log _{u_-}^\\gamma ({u_\\circ })) = w(1) = (1/2)\\frac{d}{dt}\\gamma (1/2) = \\log ^\\gamma _{u_\\circ }({u_+}).$ Applying $\\operatorname{pt}^\\gamma _{{u_-}}$ on both sides, the result follows.",
"[Proof of Theorem REF ] It suffices to verify the assumptions of Proposition REF .",
"Since the parallel transport is isometric, reduces to the identity if starting- and endpoint coincide, and since $\\big |\\log _x(x)\\big |_x = 0$ for all $x \\in \\mathcal {M}$ , it is easy to see that $\\text{D}_{\\text{pt}}^{\\text{sym}}([x,x],[x,x],[u,u],[u,u]) = 0$ .",
"In the vector space setting, with the notation as in the definition of $\\text{D}_{\\text{pt}}^{\\text{sym}}$ , we get that $w^ 1 = {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ and $r^ 1 = {u_{-,\\circ }}+{y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ as well as $w^ 2 = {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ and $r^ 2 = {u_{\\circ ,-}}+{y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}$ .",
"Consequently, from the properties of $\\text{D}_{\\text{pt}}{}$ it follows that $\\text{D}_{\\text{pt}}^{\\text{sym}}([{u_{\\circ ,\\circ }},{y^1_{\\circ ,\\circ }}],[{u_{\\circ ,\\circ }},{y^2_{\\circ ,\\circ }}],[{u_{\\circ ,-}},& {y^1_{\\circ ,-}}],[{u_{-,\\circ }},{y^2_{-,\\circ }}]) = \\\\&= \\text{D}_{\\text{pt}}{}([{u_{-,\\circ }}+{y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }},{y^2_{-,\\circ }}],[{y^1_{\\circ ,-}},{u_{\\circ ,-}}+{y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}]) \\\\&= \\big |{y^2_{-,\\circ }}- \\big ( {u_{-,\\circ }}+{y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}) - \\big ( {u_{\\circ ,-}}+{y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- {y^1_{\\circ ,-}}\\big ) \\big | \\\\&= \\big | {y^1_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^1_{\\circ ,-}}- {u_{\\circ ,-}}) + {y^2_{\\circ ,\\circ }}- {u_{\\circ ,\\circ }}- ({y^2_{-,\\circ }}- {u_{-,\\circ }})\\big |$ and from Proposition REF it follows that (M-P1') to (M-P3') holds.",
"Now let $u = (u_{i,j})_{i,j}$ be a three-by-three geodesic signal and choose $y^1_{i,j} = u_{i+1,j}$ and $y^2_{i,j} = u_{i,j+1}$ for each $(i,j)$ .",
"Denoting by $\\gamma ^ 1$ and $\\gamma ^ 2$ distance minimizing geodesics such that $\\gamma ^ 1(0) = u_{i-1,j}$ , $\\gamma ^ 1(1/2) = u_{i,j}$ , $\\gamma ^ 1(1) = u_{i+1,j}$ and $\\gamma ^ 2(0) = u_{i,j-1}$ , $\\gamma ^ 2(1/2) = u_{i,j}$ , $\\gamma ^ 2(1) = u_{i,j-1}$ , respectively, Lemma REF implies that $\\log _{u_{i-1,j}}^ {\\gamma ^ 1}(u_{i,j}) = \\operatorname{pt}_{u_{i-1,j}}(\\log ^ {\\gamma ^ 1}_{u_{i,j}}(u_{i+1,j}))$ as well as $\\log _{u_{i,j-1}}^ {\\gamma ^ 2}(u_{i,j}) = \\operatorname{pt}_{u_{i,j-1}}(\\log ^ {\\gamma ^ 2}_{u_{i,j}}(u_{i,j+1}))$ .",
"Hence we can choose $r^ 1 = r^ 2 = u_{i,j}$ and $\\text{D}_{\\text{pt}}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}])\\\\ \\le \\text{D}_{\\text{pt}}{}([u_{i,j},u_{i-1,j+1}],[u_{i+1,j-1},u_{i,j}]) = 0,$ where the last term is zero since $(u_{i-k,j+k})_k$ is locally geodesic as univariate signal.",
"Now conversely, assume that $u = (u_{i,j})_{i,j}$ is a three-by-three signal such that the geodesics connecting each pair of points are unique and such that $(u_{i,j})_i$ and $(u_{i,j})_j$ are locally geodesic.",
"By Lemma REF and uniqueness of geodesics we get $\\log _{u_{i-1,j}}(u_{i,j}) = \\operatorname{pt}_{u_{i-1,j}}(\\log _{u_{i,j}}(u_{i+1,j}))$ as well as $\\log _{u_{i,j-1}}(u_{i,j}) = \\operatorname{pt}_{u_{i,j-1}}(\\log _{u_{i,j}}(u_{i,j+1}))$ .",
"Again by uniqueness of geodesics, with the notation as in the definition of $\\text{D}_{\\text{pt}}^{\\text{sym}}$ , we conclude that $r^ 1 = r^ 2 = u_{i,j}$ hence, $0 = \\text{D}_{\\text{pt}}^{\\text{sym}}([u_{i,j},u_{i+1,j}],[u_{i,j},u_{i,j+1}],[u_{i,j-1},u_{i+1,j-1}],[u_{i-1,j},u_{i-1,j+1}])\\\\ = \\text{D}_{\\text{pt}}{}([u_{i,j},u_{i-1,j+1}],[u_{i+1,j-1},u_{i,j}]).$ From the properties of $\\text{D}_{\\text{pt}}{}$ it hence follows that the points $u_{i+1,j-1}$ , $u_{i,j}$ , $u_{i-1,j+1}$ are on a joint, length minimizing geodesic at equal distance.",
"Hence $u$ is geodesic.",
"This ensures that all assumptions of Proposition REF are fulfilled and hence the assertion follows."
],
[
"Proofs for Section 3",
"[Proof of Proposition REF ] Take $(y^n)_n$ such that $y^n \\in C(x) $ and $S(x) = \\lim _{n} G(x,y^n)$ .",
"By assumption $(y^n)_n$ admits a subsequence $(y^{n_k})_k$ converging to some $y \\in C(x)$ .",
"By lower semi-continuity of $G$ it follows that $S(x) = G(x,y)$ .",
"For lower semi-continuity of $S$ now take $(x^n)_n$ in $\\mathcal {M}^N$ converging to some $x \\in \\mathcal {M}^N$ for which, without loss of generality, we assume that $\\liminf _n S(x^n) = \\lim _n S(x^n)$ .",
"Pick $y^n \\in C(x^n)$ such that $S(x^n) = G(x^n,y^n)$ .",
"By assumption and uniqueness of limits we can obtain a subsequences $(x^{n_k})_k$ and $(y^{n_k})_k$ converging to $x$ and $y$ , respectively, such that $y \\in C(x)$ .",
"We conclude that $ S(x) \\le G(x,y) \\le \\liminf _k G(x^{n_k},y^{n_k}) = \\liminf _k S(x^{n_k}) = \\liminf _n S(x^n) $ and the assertion follows.",
"[Proof of Lemma REF ] The proof relies on the following fact: for any bounded subset $\\mathcal {N}$ of $\\mathcal {M}$ , there is a constant $D$ such that, if the length of any geodesic $\\psi :[r,s] \\rightarrow \\mathcal {M}$ with $\\psi ([r,s]) \\subset \\mathcal {N}$ is smaller than $D$ , then $\\psi $ is a unique distance minimizing geodesic between $\\psi (r)$ and $\\psi (s),$ and the Jacobi fields have no zero along this geodesic $\\psi ;$ see for instance [34].",
"In addition, there is a constant $L$ which is uniform for all such geodesics such that, with $f:[r,s] \\times [0,1] \\rightarrow \\mathcal {M}$ a geodesic variation of $\\psi $ with $f(\\cdot ,0) = \\psi $ , we have $ d(\\psi (t),f(t,\\tau )) \\le L \\max \\lbrace d(\\psi (r),f(r,\\tau )), d(\\psi (s),f(s,\\tau ))\\rbrace $ for all $\\tau \\in [0,1]$ and all $t \\in [r,s]$ .",
"In order to obtain uniform convergence of $(\\gamma ^n)_n$ , our approach is now to use compactness arguments and subdivide the curves into small segments where these assertions hold.",
"At first we define the bounded set $\\mathcal {N}$ .",
"To this aim, define $B= \\lbrace (p^n,q^n)\\,|\\, n\\in \\mathbb {N}\\rbrace \\cup \\lbrace (p,q)\\rbrace $ , $K = \\sup \\lbrace d(p^{\\prime },q^{\\prime }) \\,|\\, (p^{\\prime },q^{\\prime }) \\in B\\rbrace < \\infty $ and $\\mathcal {N}$ to be the union of all images of shortest geodesics connecting two points $p^{\\prime }$ and $q^{\\prime }$ with $(p^{\\prime },q^{\\prime }) \\in B$ .",
"Then, for any $z \\in \\mathcal {N}$ there is a shortest geodesic $\\psi :[r,s] \\rightarrow \\mathcal {M}$ such that $(\\psi (r),\\psi (s)) \\in B$ and $z \\in \\psi ([r,s])$ , and we get that $d(p,z) \\le d(p,\\psi (r)) + d(\\psi (r),z) \\le d(p,\\psi (r)) + d(\\psi (r),\\psi (s)) \\le \\sup _{n \\in \\mathbb {N}}d(p,p^n) + K$ , hence $\\mathcal {N}$ is bounded.",
"Further, by construction, $\\gamma ^n([0,1]) \\subset \\mathcal {N}$ for any $n$ .",
"With this choice of $\\mathcal {N}$ , we now choose the constant $D$ as above and we choose $l \\in \\mathbb {N}$ large enough such that $l \\ge 2\\frac{d(p,q)}{D}$ and subdivide the interval $[0,1]$ into the $l+1$ points $0,\\frac{1}{l},\\frac{2}{l},\\ldots ,1$ .",
"Then, by compactness, we can find a subsequence $(\\gamma ^{n_k})_k$ such that $\\gamma ^{n_k} (t)$ converges for all $t \\in \\lbrace 0,\\frac{1}{l},\\ldots ,1\\rbrace $ .",
"We set $z^{t}:= \\lim _{k\\rightarrow \\infty } \\gamma ^{n_k}(t)$ .",
"Then $ d(z^{t},z^{t+1/l}) = \\lim _{k \\rightarrow \\infty } d(\\gamma ^{n_k}(t),\\gamma ^{n_k}(t+1/l)) \\le \\frac{1}{l}\\lim _{k \\rightarrow \\infty }d(p^{n_k},q^{n_k}) = \\frac{d(p,q)}{l} < D. $ Hence, for each $l$ , there is a unique shortest geodesic connecting $z^{t}$ and $z^{t+1/l}$ and we define the curve $\\gamma :[0,1] \\rightarrow \\mathcal {M}$ as the concatenation of these geodesics, parametrized proportionally.",
"Then $\\gamma (0) = p$ , $\\gamma (1) = q$ and the length of $\\gamma $ is given as $ \\sum _{t\\in \\lbrace 0,\\frac{1}{l},\\ldots ,1-\\frac{1}{l}\\rbrace } d(z^{t},z^{t+1/l}) \\le \\frac{d(p,q)}{l}l = d(p,q).",
"$ Hence the length of $\\gamma $ is less or equal to the distance of its two endpoints which implies that $\\gamma $ is a shortest geodesic connecting $p$ and $q$ ; see for instance [38].",
"Defining $\\psi ^t = \\gamma |_{[z_{t},z_{t+1/l}]}$ we get that the image of $\\psi ^t$ is in $\\mathcal {N}$ and its length is less than $D$ .",
"Defining the geodesic variation $f^{t,k}:[t,t+1/l] \\times [0,1]$ as $f^{t,k}(\\eta ,\\tau ) = \\big [ [z^{t},\\gamma ^{n_k}(t)]_\\tau ,[z^{t+1/l},\\gamma ^{n_k}(t+\\frac{1}{l})]_\\tau \\big ] _\\theta ,$ with $\\theta $ chosen such that $(1-\\theta ) t + \\theta (t+\\frac{1}{l}) = \\eta $ , we get, for sufficiently large $k$ (such that the brackets are single-valued), that $ d(\\gamma (\\eta ),\\gamma ^{n_k}(\\eta )) = d(\\psi ^{t}(\\eta ),f^{t,k}(\\eta ,1)) \\le L \\max \\lbrace d(z^{t},\\gamma ^{n_k}(t)) , d(z^{t+1/l},\\gamma ^{n_k}(t+\\frac{1}{l})) \\rbrace .$ Hence $\\gamma ^{n_k} \\rightarrow \\gamma $ uniformly on $[0,1]$ .",
"Now consider an arbitrary interval $[a,b]\\supset [0,1]$ .",
"First we uniquely extend the geodesics $(\\gamma ^{n_k})_{k}$ , $\\gamma $ to this interval.",
"Then, $\\tilde{\\mathcal {N}}:=\\lbrace \\gamma ^{n_k}([a,b])\\, |\\, k \\in \\mathbb {N}\\rbrace \\cup \\gamma ([a,b])$ is a bounded set and we can again pick a constant $\\tilde{D}$ such that any geodesic in $\\tilde{\\mathcal {N}}$ with length smaller than $\\tilde{D}$ is a unique length minimizing geodesic between its start- and endpoint.",
"Now since geodesics have constant speed, we note that, for any $a^{\\prime },b^{\\prime }$ such that $[a^{\\prime },b^{\\prime }] \\subset [a,b]$ , the length of $\\gamma ^{n_k}|_{[a^{\\prime },b^{\\prime }]}$ equals $|b^{\\prime }-a^{\\prime }|$ times the length of $\\gamma ^{n_k}|_{[0,1]}$ which in turn is equal to $|b^{\\prime }-a^{\\prime }|d(p^{n_k},q^{n_k})$ .",
"But the latter is uniformly bounded by convergence of $(p^{n_k})_k$ , $(q^{n_k})_k$ , and hence we can pick a uniform $\\epsilon >0$ such that the length of $\\gamma ^{n_k}|_{[a^{\\prime },b^{\\prime }]}$ (and of $\\gamma |_{[a^{\\prime },b^{\\prime }]}$ ) is less than $\\tilde{D}$ whenever $|b^{\\prime }-a^{\\prime }| \\le 2\\epsilon $ .",
"Our approach is now to show that, whenever $\\gamma ^{n_k}$ converges uniformly to $\\gamma $ on an interval $I$ with $[0,1] \\subset I \\subset [a,b]$ , uniform convergence (up to subsequences) still holds if we extend the interval by $\\epsilon $ on both sides (up to the boundary of $[a,b]$ ).",
"By induction, the claimed assertion then follows.",
"We show this extension result by considering the extension from $[0,1]$ to $[0,1+\\epsilon ]$ , the general case follows similarly and by symmetry.",
"For this purpose, we observe that for each $k$ , $\\gamma ^{n_k}|_{[1-\\epsilon ,1+\\epsilon ]}$ is a unique length minimizing geodesic between $\\gamma ^{n_k}(1-\\epsilon )$ and $\\gamma ^{n_k}(1+\\epsilon )$ , both of which, up to subsequences, converge to some limit points $q_{-\\epsilon }$ and $q_{\\epsilon }$ , respectively.",
"Hence, employing the first result of this lemma, we obtain that, again up to subsequences, $\\gamma ^{n_k}|_{[1-\\epsilon ,1+\\epsilon ]}$ converges uniformly to some $\\psi :[1-\\epsilon ,1+\\epsilon ]\\rightarrow \\mathcal {M}$ with $\\psi (1-\\epsilon ) = q_{-\\epsilon }$ , $\\psi (1+\\epsilon ) = q_\\epsilon $ being a length minimizing geodesics between those points.",
"But by uniform convergence of $\\gamma ^{n_k}$ on $[0,1]$ and uniqueness of geodesics, we get that $\\psi |_{[1-\\epsilon ,1]} = \\gamma |_{[1-\\epsilon ,1]}$ .",
"Hence they coincide also on $[1-\\epsilon ,1+\\epsilon ]$ and the assertion follows.",
"[Proof of Lemma REF ] For the assertion on $\\text{D}_{\\text{S}}$ , apply Proposition REF with $C:\\mathcal {M}^4 \\rightarrow \\mathcal {P}(\\mathcal {M}^2)$ , $C(x,y,u,v):= \\lbrace (c^{\\prime },y^{\\prime }) \\,|\\, c^{\\prime } \\in [x,v]_{\\frac{1}{2}}, y^{\\prime } \\in [u,c^{\\prime }]_2 \\rbrace $ and $G((x,y,u,v),(c,\\tilde{y})) = d(\\tilde{y},y)$ .",
"For the assertion on $\\text{D}_{\\text{S}}^{\\text{sym}}$ , apply Proposition REF with $C:\\mathcal {M}^8 \\rightarrow \\mathcal {P}(\\mathcal {M}^4)$ , $C(u^1,v^1,u^2,v^2,x^1,y^1,x^2,y^2):=$ $\\big \\lbrace (\\tilde{c}^1,\\tilde{r}^1,\\tilde{c}^2,\\tilde{r}^2) \\,|$ $(\\tilde{c}^1,\\tilde{r}^1,\\tilde{c}^2,\\tilde{r}^2) \\in [x^2,v^1]_{\\frac{1}{2}} \\times [u^1,\\tilde{c}^1]_2$ $\\times [x^1,v^2]_{\\frac{1}{2}} \\times [u^2,\\tilde{c}^2]_2 \\big \\rbrace $ and $G:\\mathcal {M}^ 8 \\times \\mathcal {M}^ 4 \\rightarrow \\mathbb {R}$ according to $G((u^1,v^1,u^2,v^2,x^1,y^1,x^2,y^2),(c^1,r^1,c^2,r^2)):=\\text{D}_{\\text{S}}([r^1,y^2],[y^1,r^2]).$ Our next goal is to provide the proof of Lemma REF , stating existence and lower semi-continuity results for the parallel-transport-based distance-type functions.",
"Regarding $\\text{D}_{\\text{pt}}{}$ , we note that a proof based on a statement similar to Proposition REF (as done for the Schild's variant) seems possible.",
"However, since this would require a generalization of Proposition REF that involves metrics on the tangent bundle we chose to work out the proof for $\\text{D}_{\\text{pt}}{}$ directly in order to avoid introducing additional technicalities and notation.",
"Lemma A.2 The parallel-transport-based distance-type functional $\\text{D}_{\\text{pt}}$ is lower semi-continuous.",
"In particular, the minimum in (REF ) is attained for any $x,y,u,v \\in \\mathcal {M}.$ We consider sequences $x^n \\rightarrow x,$ $y^n \\rightarrow y, $ $u^n \\rightarrow u,$ and $v^n \\rightarrow v,$ and show that $\\displaystyle \\text{D}_{\\text{pt}}([x,y],[u,v]) \\le \\liminf \\nolimits _n \\text{D}_{\\text{pt}}([x^n,y^n],[u^n,v^n]).$ Here, by (REF ) $\\displaystyle \\text{D}_{\\text{pt}}([x,y],[u,v])= \\inf \\nolimits _{z,w,\\gamma } \\big | z - \\operatorname{pt}^\\gamma _{\\gamma (1),\\gamma (0)} w \\big | = \\inf \\nolimits _{z,w,\\gamma } \\big | \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - w \\big |,$ where $ z \\in \\log _x(y),$ $ w \\in \\log _u(v),$ and $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ is a shortest geodesic connecting $\\gamma (0)=x$ and $\\gamma (1)=u.$ We note that the second equation in (REF ) is true since the parallel transport is an isometry in a Riemannian manifold.",
"We start out choosing subsequences $x^{n_k} \\rightarrow x,$ $y^{n_k} \\rightarrow y,$ $u^{n_k} \\rightarrow u,$ $v^{n_k} \\rightarrow v,$ such that $ \\displaystyle \\lim \\nolimits _{k \\rightarrow \\infty } \\text{D}_{\\text{pt}}([x^{n_k},y^{n_k}],[u^{n_k},v^{n_k}]) = \\liminf \\nolimits _n \\text{D}_{\\text{pt}}([x^n,y^n],[u^n,v^n]).$ Then, for each $k \\in \\mathbb {N},$ we choose tangent vectors $z^{n_k},w^{n_k},$ and shortest geodesics $\\gamma ^{n_k}$ such that $\\displaystyle \\left| \\operatorname{pt}^{\\gamma ^{n_k}}_{\\gamma ^{n_k}(0),\\gamma ^{n_k}(1)} z^{n_k} - w^{n_k} \\right| \\le \\text{D}_{\\text{pt}}([x^{n_k},y^{n_k}],[u^{n_k},v^{n_k}]) + \\tfrac{1}{n_k}.$ Here, $z^{n_k}$ is sitting in the tangent space at the point $x^{n_k},$ $w^{n_k}$ is sitting in the tangent space at the point $u^{n_k},$ and $\\gamma ^{n_k}$ is one of the (potentially non-unique) shortest geodesics connecting the points $\\gamma ^{n_k}(0)= x^{n_k}$ and $\\gamma ^{n_k}(1)= u^{n_k}.$ The parallel transport is understood along $\\gamma ^{n_k}.$ By Lemma REF there is a subsequence $\\gamma ^{n_l}$ of $\\gamma ^{n_k}$ (for convenience, we suppress iterated indices and write $\\gamma ^{n_l}$ instead of $\\gamma ^{n_{k_l}}$ in the following) such that $\\gamma ^{n_l} \\rightarrow \\gamma $ uniformly on $[0,1],$ for some shortest geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connecting $\\gamma (0)=x$ and $\\gamma (1)=u.$ Since $x^{n_l}$ converges to $x,$ the geodesic connecting $x^{n_l}$ and $x$ is unique for sufficiently large $l$ .",
"The same is true for $u^{n_l}$ and $u.$ So we may identify the tangent vectors $z^{n_l}$ sitting at $x^{n_l},$ and the tangent vectors $w^{n_l}$ sitting at $u^{n_l},$ with their parallel transported versions $\\displaystyle \\bar{z}^{n_l} = \\operatorname{pt}_{x_{n_l},x} z_{n_l}, \\qquad \\bar{w}^{n_l} = \\operatorname{pt}_{u_{n_l},u} w_{n_l},$ along the corresponding unique geodesic.",
"Note that the $\\bar{z}^{n_l}$ are sitting in the common point $x,$ and that the $\\bar{w}^{n_l}$ are sitting in the common point $u.$ The sequences $\\bar{z}^{n_l}$ is bounded since parallel transport is an isometry and $ z^{n_l} \\in \\log _{x^{n_l}}(y^{n_l}),$ where both $x^{n_l}$ and $y^{n_l}$ converge.",
"The analogous statements hold for $\\bar{w}^{n_l}$ with the same argument.",
"So, by going to subsequences $\\bar{z}^{n_r}$ of $\\bar{z}^{n_l},$ and $\\bar{w}^{n_r}$ of $\\bar{w}^{n_l},$ we get that $\\displaystyle \\bar{z}^{n_r} \\rightarrow z, \\qquad \\bar{w}^{n_r} \\rightarrow w,$ for a tangent vector $z$ sitting in $x$ and a tangent vector $w$ sitting in $u.$ We have that the limit $ z \\in \\log _x(y),$ and that $ w \\in \\log _u(v),$ since the exponential map from $T\\mathcal {M}\\rightarrow \\mathcal {M}$ is differentiable in a Riemannian manifold which implies $\\exp _x z = \\lim _{r \\rightarrow \\infty } \\exp _{x^{n_r}} z^{n_r} = \\lim _{r \\rightarrow \\infty } y^{n_r} = y, $ and $\\exp _u w = v.$ We are now prepared to estimate $| \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - w |$ which allows us to bound $\\text{D}_{\\text{pt}}([x,y],[u,v])$ from above; see (REF ).",
"Since the parallel transport is an isometry, we have $\\displaystyle \\big | \\operatorname{pt}^ \\gamma _{\\gamma (0),\\gamma (1)} z - w \\big | = \\big | \\operatorname{pt}_{u,u^{n_r}} \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - \\operatorname{pt}_{u,u^{n_r}} w \\big |.$ We further have that $\\displaystyle \\varepsilon _r := \\big | \\operatorname{pt}_{u,u_{n_r}} \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - \\operatorname{pt}^{\\gamma ^{n_r}}_{\\gamma ^{n_r}(0),\\gamma _{n_r}(1)} \\operatorname{pt}_{x,x^{n_r}} z \\big |\\rightarrow 0 \\qquad \\text{ as } \\quad r \\rightarrow \\infty $ which is a consequence of the parallel transport along nearby geodesics being differentiably dependent on the variation of the geodesics.",
"Using (REF ) together with (REF ), we have $ \\big | \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - w \\big |& \\le \\big | \\operatorname{pt}^{\\gamma ^{n_r}}_{\\gamma ^{n_r}(0),\\gamma ^{n_r}(1)} \\operatorname{pt}_{x,x^{n_r}} z - \\operatorname{pt}_{u,u^{n_r}} w \\big | + \\varepsilon _r \\\\& \\le \\big | \\operatorname{pt}^{\\gamma ^{n_r}}_{\\gamma ^{n_r}(0),\\gamma ^{n_r}(1)} \\operatorname{pt}_{x,x^{n_r}} \\bar{z}^{n_r} - \\operatorname{pt}_{u,u^{n_r}} \\bar{w}_{n_r} \\big |+ |\\bar{z}^{n_r} - z | + |\\bar{w}^{n_r} - w |+ \\varepsilon _r.",
"$ The second inequality is a consequence of the triangle inequality together with the fact that parallel transport is an isometry.",
"We take the limit w.r.t.",
"$r$ on the right hand side of (REF ): by (REF ), we have that $\\bar{z}^{n_r} \\rightarrow z,$ and that $\\bar{w}^{n_r} \\rightarrow w $ and by (REF ) that $\\varepsilon _r \\rightarrow 0$ which implies $\\displaystyle \\big | \\operatorname{pt}^\\gamma _{\\gamma (0),\\gamma (1)} z - w \\big |& \\le \\lim _{r \\rightarrow \\infty } \\big | \\operatorname{pt}^{\\gamma ^{n_r}}_{\\gamma ^{n_r}(0),\\gamma ^{n_r}(1)} \\operatorname{pt}_{x,x^{n_r}} \\bar{z}^{n_r} - \\operatorname{pt}_{u,u^{n_r}} \\bar{w} ^{n_r} \\big |\\\\& \\le \\lim _{r \\rightarrow \\infty } \\left( \\text{D}_{\\text{pt}}([x^{n_r},y^{n_r}],[u^{n_r},v^{n_r}]) + \\tfrac{1}{n_r} \\right) = \\liminf _n \\text{D}_{\\text{pt}}([x^n,y^n],[u^n,v^n]).",
"$ The second inequality in (REF ) is a consequence of our choice of $z^{n_k},w^{n_k},$ and $\\gamma ^{n_k}$ in (REF ) and the definition of $\\bar{z}^{n_l},\\bar{w}^{n_l}$ as in (REF ), and the equality in the last line follows by our choice of subsequences in (REF ).",
"Then passing to the infimum according to (REF ) shows (REF ) which shows the first assertion.",
"To see that the infimum in (REF ) is attained for any $x,y,u,v \\in \\mathcal {M}$ , we choose the constant sequences $x^n:= x,$ $y^n := y, $ $u^n := u,$ and $v^n := v,$ and apply the previous result to these sequences.",
"This shows the second assertion and completes the proof.",
"Having shown Lemma REF we can now employ Proposition REF to show existence and lower semi-continuity results for $\\text{D}_{\\text{pt}}^{\\text{sym}}$ .",
"As a preparation we need the following lemma.",
"Lemma A.3 Let $u^n \\rightarrow u,$ $\\tilde{u}^n \\rightarrow \\tilde{u},$ and $y^n \\rightarrow y$ in the complete manifold $\\mathcal {M}.$ We consider a sequence $(r^n)_n$ with $r^n \\in \\exp (\\operatorname{pt}_{\\tilde{u}^n}(w^n)) \\quad \\text{ with } \\quad w^n \\in \\log _{u^n}y^n.",
"$ Then there is a subsequence $(r^{n_k})_k$ which converges and a limit $r = \\lim _{k \\rightarrow \\infty } r^{n_k}$ such that $r$ fulfills $r \\in \\exp (\\operatorname{pt}_{\\tilde{u}}(w)) \\quad \\text{ for some } \\quad w \\in \\log _{u}y.",
"$ The present proof essentially employs the techniques already used in the proof of Lemma REF .",
"For this reason we keep the following arguments rather short.",
"The sequence $\\bar{w}^n := \\operatorname{pt}_{u}(w^n)$ is bounded since parallel transport is an isometry.",
"So, by going to a subsequence $\\bar{w}^{n_l}$ of $\\bar{w}^{n},$ we get that $\\bar{w}^{n_l} \\rightarrow w$ for a tangent vector $w$ sitting in $u.$ We have that the limit $ w \\in \\log _u(y),$ since the exponential map $T\\mathcal {M}\\rightarrow \\mathcal {M}$ is differentiable which implies $\\exp _u w = \\lim _{l \\rightarrow \\infty } \\exp _{u^{n_l}} w^{n_l} = \\lim _{l \\rightarrow \\infty } y^{n_l} = y.$ For each $l$ we choose a distance-minimizing geodesic $\\gamma ^{n_l}$ connecting $\\gamma ^{n_l}(0)$ $= u^{n_l}$ with $\\gamma ^{n_l}(1)$ $= \\tilde{u}^{n_l}.$ Then we use Lemma REF to choose a subsequence of geodesics $\\gamma ^{n_k}$ of $\\gamma ^{n_l}$ (suppressing iterated subindices) which uniformly converges to a geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connecting $\\gamma (0) = u$ with $\\gamma (1)=\\tilde{u}$ .",
"Then, since parallel transport along nearby geodesics is differentiably dependent on the variation of the geodesics and by an argument similar to the one used for the convergence in Equation (REF ), $\\operatorname{pt}_{\\tilde{u}^{n_k} ,\\tilde{u}} \\operatorname{pt}^{\\gamma ^{n_k}}_{u^{n_k},\\tilde{u}^{n_k}} w^{n_k}$ $\\rightarrow \\operatorname{pt}^ \\gamma _{u,\\tilde{u}} w$ as $k \\rightarrow \\infty .$ Then, by the continuity of the exponential map, $r = \\lim _{k \\rightarrow \\infty } r^{n_k}$ $= \\lim _{k \\rightarrow \\infty } \\exp (\\operatorname{pt}^{\\gamma ^{n_k}}_{u^{n_k},\\tilde{u}^{n_k}}(w^{n_k})) = \\exp (\\operatorname{pt}^ \\gamma _{u,\\tilde{u}}(w)),$ i.e., $r^{n_k}$ converges and (REF ) holds true which was the assertion to show.",
"Finally, the proof of Lemma REF follows as consequence.",
"[Proof of Lemma REF ] For $\\text{D}_{\\text{pt}}{}$ , this is the assertion of Lemma REF .",
"For $\\text{D}_{\\text{pt}}^{\\text{sym}}$ , we apply Proposition REF with $C:\\mathcal {M}^8\\rightarrow \\mathcal {P}(\\mathcal {M}^2)$ , $C(u^1,v^1,u^2,v^2,x^1,y^1,x^2,y^2):=$ $\\lbrace \\exp (\\operatorname{pt}_{x^2}w) \\,|\\, w \\in \\log _{u^1}(v^1)\\rbrace $ $\\times \\lbrace \\exp (\\operatorname{pt}_{x^1}w) \\, |\\, w \\in \\log _{u^2}(v^2)\\rbrace ,$ and $G:\\mathcal {M}^ 8 \\times \\mathcal {M}^ 2\\rightarrow \\mathbb {R}$ according to $G((u^1,v^1,u^2,v^2,x^1,$ $y^1,x^2,y^2),(r^1,r^2)):=\\text{D}_{\\text{pt}}{}([r^1,y^2],[y^1,r^2]) .$ The Lemmata REF and REF ensure that $C$ and $G$ satisfy the assumption of Proposition REF ."
],
[
"Proofs for Section 4",
"[Proof of Lemma REF ] Let us consider the Schild's ladder mapping $u_{i-1} \\mapsto S(u_{i-1},y_{i-1},u_{i}) = [u_{i-1},[u_{i},y_{i-1}]_{\\frac{1}{2}}]_2$ as a function of $u_{i-1}$ .",
"Since the points $y_{i-1},u_{i}$ are fixed, so is their midpoint $m=[u_{i},y_{i-1}]_{\\frac{1}{2}}.$ Now $S(u_{i-1},y_{i-1},u_{i})$ is obtained by evaluating the geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connecting $u_{i-1}=\\gamma (0)$ and $m=\\gamma (1)$ at time $2,$ thus considering $\\gamma (2).$ Hence, the differential $T_1$ of $S$ w.r.t.",
"$u_{i-1}$ is related to the Jacobi fields along $\\gamma $ as follows: consider those Jacobi fields $\\mathcal {J}$ along $\\gamma $ for which $J(1)=0$ which means that they belong to geodesic variations leaving $m$ fixed.",
"Then the adjoint of the mapping $J(0) \\mapsto J(2),\\qquad J \\in \\mathcal {J}$ equals $T_1$ .",
"If the manifold is a Riemannian symmetric space, then the mapping $u_{i-1} \\mapsto S(u_{i-1},y_{i-1},u_{i}) = [u_{i-1},[u_{i},y_{i-1}]_{\\frac{1}{2}}]_2$ is a geodesic reflection, see, e.g., [40].",
"We consider an orthonormal basis $(v_n)_n$ of eigenvectors of the self-adjoint Jacobi operator $J \\mapsto R(\\frac{\\gamma ^{\\prime }(1)}{ |\\gamma ^{\\prime }(1) |} ,J) \\frac{\\gamma ^{\\prime }(1)}{|\\gamma ^{\\prime }(1) |}$ with corresponding eigenvalues associated $\\lambda _n,$ and $v_1$ tangent to $\\gamma ^{\\prime }(1).$ Then the mapping $J(0) \\mapsto J(2)$ , $ J \\in \\mathcal {J},$ in (REF ) can be written as $\\displaystyle \\sum \\nolimits _n \\alpha _n f(\\lambda _n) \\operatorname{pt}_{m,u_{i-1}} v_n \\mapsto - \\sum \\nolimits _n \\alpha _n f(\\lambda _n) \\operatorname{pt}_{m,S(u_{i-1},y_{i-1},u_{i})} v_n,$ where the $\\alpha _n$ are the coefficients of the corresponding basis representation and the function $f,$ depending on the sign of $\\lambda _n$ , is given by $f(\\lambda _n) = d,$ if $\\lambda _n = 0,$ by $f(\\lambda _n) = (\\sqrt{\\lambda _n} )^{-1} \\sin (\\sqrt{\\lambda _n} d) ,$ if $\\lambda _n > 0$ and by $f(\\lambda _n) = (\\sqrt{-\\lambda _n} )^{-1} \\sinh (\\sqrt{-\\lambda _n} d) ,$ if $\\lambda _n < 0,$ where $d$ is the distance between $m$ and $u_{i-1}$ (which equals the distance between $m$ and $S(u_{i-1},y_{i-1},u_{i}).$ ) This results from the fact that $f(\\lambda _n) = x(0),$ the evaluation at 0 of the solution $x(t)$ of the scalar initial value problem $x^{\\prime \\prime } = - \\lambda _n d^2 x, \\quad x(1) = 0, x^{\\prime }(1)= -d.$ We observe that (REF ) states that the mapping $J(0) \\mapsto J(2)$ , $ J \\in \\mathcal {J},$ equals the identity multiplied with $-1$ (up to parallel transport).",
"(We note that this can also be concluded from the derivations in [40] near Theorem 1.)",
"Hence the adjoint of the mapping $J(0)\\mapsto J(2)$ coincides with the parallel transport multiplied by $-1$ as in (REF ), in particular, it is an isometry, and the proof is complete.",
"[Proof of Lemma REF ] We observe that the concatenation of the mappings $m \\mapsto [u_{i-1},m]_2$ and $y_{i-1} \\mapsto [u_{i},y_{i-1}]_{\\frac{1}{2}}$ equals the mapping $y_{i-1} \\mapsto [u_{i-1},[u_{i},y_{i-1}]_{\\frac{1}{2}}]_2$ .",
"Hence, we conclude from the discussion leading to () in connection with the chain rule the validity of (REF ).",
"It remains to express $T_3,T_4$ in terms of Jacobi fields.",
"Concerning $T_3$ we consider the geodesic $\\gamma $ connecting $u_{i-1}=\\gamma (0)$ and $m=\\gamma (1).$ $T_3$ is related to the Jacobi fields along $\\gamma $ as follows: consider those Jacobi fields $\\mathcal {J}$ along $\\gamma $ for which $J(0)=0$ which means that they belong to geodesic variations leaving $u_{i-1}$ fixed.",
"Then, the adjoint of the mapping $\\displaystyle J(1) \\mapsto J(2),\\qquad J \\in \\mathcal {J}$ equals $T_3.$ Concerning $T_4,$ we consider the geodesic $\\xi $ connecting $y_{i-1}=\\xi (0)$ and $u_{i}=\\xi (1).$ $T_4$ is related to the Jacobi fields $\\mathcal {J}^{\\prime }$ along $\\xi $ for which $J(1)=0$ by the mapping $\\displaystyle J(0) \\mapsto J(\\tfrac{1}{2}),\\qquad J \\in \\mathcal {J}^{\\prime }.$ [Proof of Lemma REF ] We consider the assertion of Lemma REF for the mapping $T_3.$ By the proof of Lemma REF , we have to determine the adjoint of the mapping given by (REF ) more explicitely.",
"Since the covariant derivative of the curvature tensor equals zero in a symmetric space, the differential equation for the Jacobi fields $J$ in terms of the scalar coefficient $x$ of the vector field $\\operatorname{pt}_{m,[m,u_{i-1}]_t}v_n$ obtained by parallel transport of the eigenvector $v_n$ along $\\gamma $ reads $x^{\\prime \\prime } = - \\lambda _n d^2 x, \\quad x(0) = 0.$ Solving this scalar ODE shows (REF ).",
"Then, $f(\\lambda _n)$ corresponds to the value $x(2)$ of the solution of the scalar ODE of the previous line with additional boundary condition $x^{\\prime }(1)=1$ .",
"Solving this scalar ODE yields (REF ).",
"We have already derived the corresponding statement for $T_4$ in [10].",
"Our next goal is to show Lemma REF .",
"To this end, we introduce the mapping $F_t$ which is a slight extension of $\\text{D}_{\\text{pt}}{}$ for different parameters as follows.",
"For $t \\in [0,1],$ we consider the mapping $F_t: \\mathcal {M}\\times \\mathcal {M}\\times \\mathcal {M}\\times \\mathcal {M}\\rightarrow [0,\\infty ),$ given by $\\displaystyle F_t(u_i,u_{i-1},y_i,y_{i-1}) = \\big | \\operatorname{pt}_{0,t} \\log _{u_i}y_i - \\operatorname{pt}_{1,t} \\ \\log _{u_{i-1}}y_{i-1} \\big |, $ where, for the definition of $F_t$ and in the following Lemma, we use the shorthand notation $\\operatorname{pt}_{s,t}$ to denote the parallel transport from $[u_i,u_{i-1}]_s$ to $[u_i,u_{i-1}]_t$ and the norm on the right hand side is the one introduced by the Riemannian scalar product in the point $[u_i,u_{i-1}]_t$ .",
"Note that $F_0(u_i,u_{i-1},y_i,y_{i-1}) = \\text{D}_{\\text{pt}}([u_i,y_i],[u_{i-1},y_{i-1}])$ , so in order to obtain the derivative of $\\text{D}_{\\text{pt}}{}$ it suffices to differentiate $F_0$ w.r.t.",
"its four arguments.",
"The following lemma shows that we can also consider $F_t$ instead.",
"Lemma A.4 The function $F_t$ given by (REF ) is independent of $t.$ The Riemannian scalar product is invariant under parallel transport, i.e., for any tangent vectors $v,v^{\\prime }$ sitting in the arbitrary point $x,$ and the parallel transport along any curve $\\gamma $ with any points $y,z$ sitting on that curve, we have $ \\langle \\operatorname{pt}_{x,y}v , \\operatorname{pt}_{x,y} v^{\\prime } \\rangle _y = \\langle \\operatorname{pt}_{x,z}v , \\operatorname{pt}_{x,z} v^{\\prime } \\rangle _z.$ Hence, for all $s,t \\in [0,1],$ $\\displaystyle \\big | \\operatorname{pt}_{0,s} \\log _{u_i}y_i - \\operatorname{pt}_{1,s} \\ \\log _{u_{i-1}}y_{i-1} \\big |^2&= \\big | \\operatorname{pt}_{t,s} \\operatorname{pt}_{0,t} \\log _{u_i}y_i - \\operatorname{pt}_{t,s} \\operatorname{pt}_{1,t} \\ \\log _{u_{i-1}}y_{i-1} \\big |^2 \\\\&= \\big | \\operatorname{pt}_{0,t} \\log _{u_i}y_i - \\operatorname{pt}_{1,t} \\ \\log _{u_{i-1}}y_{i-1} \\big |^2.$ Since the first expression in (REF ) equals the square of $F_s,$ and the last expression equals the square of $F_t,$ this shows that $F_s=F_t$ for all $s,t \\in [0,1].$ As pointed out below Lemma REF , Lemma REF is obtained by specifying $F=F_0$ in the following lemma.",
"Lemma A.5 The function $F_t$ given by (REF ) is symmetric with respect to interchanging $(u_i,y_i)$ with $(u_{i-1},y_{i-1})$ i.e., $F_t(u_i,u_{i-1},y_i,y_{i-1}) = F_t(u_{i-1}, u_i,y_{i-1}, y_i).$ In particular, for points with $F_t \\ne 0,$ the gradient of the function $F_t$ w.r.t.",
"the third variable $y_{i},$ is given by $\\nabla _{y_{i}} F_t(u_i,u_{i-1},y_i,$ $y_{i-1}) =\\nabla _{y_{i-1}} F_t(u_{i-1}, u_i,y_{i-1}, y_i),$ where $\\nabla _{y_{i-1}} F_t$ denotes the derivative of $F_t$ w.r.t.",
"the fourth argument.",
"Further, again for points with $F_t \\ne 0$ , the gradient of the function $F_t$ w.r.t.",
"the first component variable $u_{i},$ is given by $\\nabla _{u_{i}} F_t(u_i,u_{i-1},y_i,y_{i-1}) =\\nabla _{u_{i-1}} F_t(u_{i-1}, u_i,y_{i-1}, y_i),$ where $\\nabla _{u_{i-1}} F_t$ denotes the derivative of $F_t$ w.r.t.",
"the second argument.",
"By the definition of $F_t$ in (REF ), we have $F_t(u_i,u_{i-1},y_i,y_{i-1}) = F_{1-t}(u_{i-1}, u_i,y_{i-1}, y_i).$ By Lemma REF , $F_t$ is independent of $t.$ Together, this implies the identity $F_t(u_i,u_{i-1},y_i,y_{i-1}) = F_t(u_{i-1}, u_i,y_{i-1}, y_i)$ which is the first assertion of the lemma.",
"The following two assertions on the gradients are then an immediate consequence of this symmetry property.",
"[Proof of Lemma REF ] We note that $F$ agrees with the function $F_0$ with $F_t$ as in Equation (REF ) by definition and, consequently, also with $F_1$ by Lemma REF .",
"For fixed $u_i$ , $u_{i-1}$ and $y_i$ , we decompose the mapping $y_{i-1} \\mapsto F_1(u_i,u_{i-1},y_i,y_{i-1})$ into the mappings $G,H$ , i.e., $F_1 = H \\circ G,$ where $\\displaystyle G(y_{i-1}) = \\log _{u_{i-1}} y_{i-1}$ locally maps the manifold $\\mathcal {M}$ to the tangent space at $u_{i-1},$ and $H(w) = \\left| w - z\\right|, \\qquad \\text{ where } z = \\operatorname{pt}_{u_{i},u_{i-1}} \\log _{u_i} y_i,$ maps from the tangent space at $u_{i-1}$ to the positive real numbers.",
"The differential of $H$ (as a map defined on the tangent bundle) is given by $d_\\xi H (\\eta ) = \\langle \\frac{\\xi - z }{|\\xi - z |} , \\eta \\rangle , $ for $\\xi \\ne z.$ Here $\\xi $ is a point in the tangent space at $u_{i-1},$ and $\\eta $ is the direction in the tangent space with respect to which the differentiation is performed.",
"Hence, the gradient of $H$ at $\\xi $ equals $\\frac{\\xi - z }{ |\\xi - z |}.$ So the gradient of $H$ at the point $\\xi = \\log _{u_{i-1}} y_{i-1}$ equals $\\nabla H (\\log _{u_{i-1}} y_{i-1})= {\\left(\\log _{u_{i-1}} y_{i-1}-z \\right)}/{\\left| \\log _{u_{i-1}} y_{i-1} - z \\right|}.$ In order to get the gradient of $F_1,$ we have to multiply $\\nabla H (\\log _{u_{i-1}} y_{i-1})$ with the adjoint of the differential of $G.$ The mapping $G$ is related to the Jacobi fields along the geodesic $\\gamma :[0,1]\\rightarrow \\mathcal {M}$ connecting the points $\\gamma (0)= u_{i-1}$ and $\\gamma (1) = y_{i-1}$ as follows.",
"Consider the collection $\\mathcal {J}$ of Jacobi fields $J$ along $\\gamma $ for which $J(0)=0$ , which means that they belong to geodesic variations leaving $u_{i-1}$ fixed.",
"Then the mapping $J(1) \\mapsto J^{\\prime }(0),\\quad J \\in \\mathcal {J}.$ equals the differential of $G.$ [Proof of Lemma REF ] We use the basis $\\lbrace v_n\\rbrace $ to express the operator $T$ of Lemma REF (which is given as the adjoint of the derivative of $G$ defined by (REF ) evaluated at $y_{i-1}$ ).",
"We use the expression (REF ) for the derivative of $G.$ Since the covariant derivative of the curvature tensor equals zero in a symmetric space, the differential equation for the Jacobi fields $J$ in terms of the scalar coefficient $x$ of the vector field $\\operatorname{pt}_{u_{i-1},[u_{i-1},y_{i-1}]_t}v_n$ obtained by parallel transport of the eigenvector $v_n$ along $\\gamma $ reads $x^{\\prime \\prime } = - \\lambda _n d^2 x, \\quad x(0) = 0.$ This shows (REF ).",
"Further, $f(\\lambda _n)$ corresponds to the value $x^{\\prime }(0)$ of the solution of the scalar ODE of the previous line with additional boundary condition $x(1) = 1$ .",
"Solving this scalar ODE yields (REF ).",
"[Proof of Lemma REF ] We again note that $F$ agrees with the function $F_1$ introduced in Equation (REF ).",
"So our task is to determine the gradient of the function $F_1$ for points with $F_1 \\ne 0$ given by (REF ) w.r.t.",
"the variable $u_{i-1}.$ We analyze the structure of $F_1$ as a function of $u_{i-1}.$ To this end, we consider the two vector fields $L: u_{i-1} \\mapsto \\log _{u_{i-1}}y_{i-1}$ and $B: u_{i-1} \\mapsto \\operatorname{pt}_{u_{i},u_{i-1}} z,$ where $z = \\log _{u_{i}}y_{i}$ introduced above.",
"Remember that $\\operatorname{pt}_{u_{i},u_{i-1}}$ denotes the parallel transport from the point $u_{i}$ to the (varying) point $u_{i-1}$ along a shortest geodesic connecting these points.",
"We note that the parallel transport here depends on the varying end point $u_{i-1}$ .",
"Further note that $z = \\log _{u_{i}}y_{i}$ does not depend on $u_{i-1}$ and is therefore fixed.",
"Using this notation we may write (REF ) as $ F_1 (u_i,\\cdot ,y_i,y_{i-1}): u_{i-1} \\mapsto \\big |L(u_{i-1}) - B (u_{i-1}) \\big | .$ In order to differentiate (REF ) w.r.t.",
"$u_{i-1}$ we need some more preparation.",
"Recall that the Levi-Civita connection is metric.",
"Hence, for any two vector fields $V_t,W_t$ along a geodesic $\\gamma ,$ $\\tfrac{d}{dt}\\langle V_t, W_t \\rangle = \\langle \\tfrac{D}{dt} V_t, W_t \\rangle + \\langle V_t, \\tfrac{D}{dt} W_t \\rangle .$ Recall that we use the symbol $\\frac{D}{dt}$ to denote the covariant derivative of the corresponding vector field along the curve $\\gamma .$ Thus, for any two vector fields $V_t,W_t$ with $V_t \\ne W_t,$ we have $\\tfrac{d}{dt} \\big | V_t - W_t \\big | &= \\tfrac{1}{ 2 \\big | V_t - W_t \\big |} \\cdot 2 \\langle V_t- W_t, \\tfrac{D}{dt} V - \\tfrac{D}{dt} W \\rangle \\\\& = \\langle \\tfrac{V_t- W_t}{\\big | V_t - W_t \\big |} , \\tfrac{D}{dt} V - \\tfrac{D}{dt} W \\rangle .$ Since we have chosen the $v_n$ to be an orthogonal basis of the tangent space at $u_{i-1},$ the coordinate representation of the gradient in this basis is given as the directional derivatives w.r.t.",
"the basis vectors.",
"The curves $t \\mapsto \\exp _{u_{i-1}}tv_n$ precisely yield these tangent vectors.",
"This explains (REF ), (REF ), as well as the first identity in (REF ).",
"The second identity in (REF ) is a consequence of (REF ).",
"[Proof of Lemma REF ] We notice that the proof of this lemma uses well-known facts on the connection of Jacobi fields and the exponential map (see for instance the books [76], [38]) which is the reason why we kept it rather short.",
"We consider the Jacobi field $J^n$ associated with the following geodesic variation $f^n(s,t) = \\exp _{c^n(t)} \\left(s \\log _{c^n(t)}y_{i-1}\\right), \\quad \\text{ where }c^n(t) = \\exp _{u_{i-1}}tv_n.",
"$ Then the desired covariant derivative is connected with the Jacobi field $J^n$ associated with the geodesic variation $f^n$ by $\\tfrac{D}{dt}|_{t=0} L_t^n = \\tfrac{D}{ds}|_{s=0} J^n(s), \\text{ where }\\quad J^n(s) = \\tfrac{d}{dt}|_{t=0}f^n(s,t).",
"$ This identity can be seen as follows.",
"By the definition of $f^n,$ we have $L_t^n = \\log _{c(t)}y_{i-1} = \\tfrac{d}{ds}|_{s=0} f^n(s,t).$ Hence, $\\tfrac{D}{dt}|_{t=0} L_t^n &= \\tfrac{D}{dt}|_{t=0} \\tfrac{d}{ds}|_{s=0} f^n(s,t)\\\\& = \\tfrac{D}{ds}|_{s=0} \\tfrac{d}{dt}|_{t=0} f^n(s,t) = \\tfrac{D}{ds}|_{s=0} J^n(s),$ which shows (REF ).",
"We further notice that $J^n(0) = v_n, \\quad J^n(1) = 0.",
"$ The first part of (REF ) is a consequence of $J^n(0) = \\tfrac{d}{dt}|_{t=0} f^n(0,t) = \\tfrac{d}{dt}|_{t=0}c^n(t) = v_n.$ The second equality of (REF ) is a consequence of the mapping $t \\mapsto f^n(1,t)=y_{i-1},$ being constant.",
"We notice that (REF ) determines $J^n$ uniquely which, in turn, yields the desired derivative of $L^n$ via (REF ) as being equal to $(J^n)^{\\prime }(0) := \\frac{D}{ds}|_{s=0} J^n$ .",
"So it only remains to determine this uniquely determined Jacobi field $J^n$ .",
"Since $\\mathcal {M}$ is a symmetric space, and thus the curvature tensor is parallel, and since $v_n$ is an eigenvector of the Jacobi operator, we end up with the problem of determining $x^{\\prime }(0)$ where $x$ is the solution of the scalar boundary value problem $x^{\\prime \\prime }(s) + d^2 \\lambda _n x(s) = 0, \\quad x(0) = 1, \\ x(1) = 0,$ where $d=d(y_{i-1},u_{i-1})= |\\gamma ^{\\prime }(t)|$ for all $t \\in [0,1].$ Here, $\\lambda _n$ is the corresponding eigenvalue of the Jacobi operator.",
"Depending on the sign of $\\lambda _n,$ the solution of (REF ) is given as follows.",
"If $\\lambda _n = 0,$ $x(s)= 1 - s,$ and so $x^{\\prime }(0) = -1.$ If $\\lambda _n > 0,$ the general solution of the ODE is $x(s)=$ $\\alpha \\cos ( d \\sqrt{\\lambda _n} s ) + $ $\\beta \\sin ( d \\sqrt{\\lambda _n} s ) $ with real parameters $\\alpha ,\\beta .$ Then $x(0) = 1$ implies $\\alpha =1$ which, in turn, yields using $0 =x(1) =$ $ \\cos ( d \\sqrt{\\lambda _n} ) + $ $\\beta \\sin ( d \\sqrt{\\lambda _n})$ that $\\beta = - \\cos ( d \\sqrt{\\lambda _n} )/\\sin ( d \\sqrt{\\lambda _n}).$ Hence, $x^{\\prime }(0) = - \\cos ( d \\sqrt{\\lambda _n} )/\\sin ( d \\sqrt{\\lambda _n})$ $\\cdot \\ d \\sqrt{\\lambda _n}.$ If $\\lambda _n < 0,$ an analogous calculation replacing the trigonometric polynomials by their hyperbolic analogues yields that $x^{\\prime }(0) = - \\cosh ( d \\sqrt{-\\lambda _n} )/$ $\\sinh ( d \\sqrt{-\\lambda _n})$ $\\cdot \\ d \\sqrt{-\\lambda _n}.$ This shows (REF ) and completes the proof.",
"[Proof of Lemma REF ] In order to covariantly differentiate the mapping $t \\mapsto B^n_t = \\operatorname{pt}_{u_{i},\\exp _{u_{i-1}}tv_n} z,$ we differentiate the mapping $t \\mapsto P^n_t z:= \\operatorname{pt}_{\\exp _{u_{i-1}}tv_n,u_{i-1}} \\operatorname{pt}_{u_{i},\\exp _{u_{i-1}}tv_n} z$ in the tangent space $T_{u_{i-1}}\\mathcal {M}$ of $u_{i-1}.$ This follows from the relation between parallel transport and the covariant derivative, see for instance [76].",
"If $v_n$ is parallel to $ \\log _{u_{i-1}}u_i$ , then the mapping in (REF ) is constant, and therefore, its derivative is 0 which shows the first statement in (REF ).",
"We show the second part of (REF ).",
"We may assume that $v_n $ is orthogonal to $ \\exp ^{-1}_{u_{i-1}}u_i.$ We have to differentiate the mapping in (REF ), which means calculating $\\lim _{t \\rightarrow 0} \\frac{1}{t}(P^n_t-P^n_0).$ Since the parallel transport is an isometry, the differential of (REF ) is an infinitesimal rotation (up to the identity) applied to $z.$ We start out to calculate $P^n_t-P^n_0$ in the basis of $T_{u_{i-1}}\\mathcal {M}.$ We note that $P^n_t-P^n_0$ corresponds to the holonomy along the (spherical) triangle $\\Delta $ connecting the points $u_{i},$ $\\exp _{u_{i-1}}tv_n$ and $u_{i-1}.$ We observe that the rotation angle $\\alpha _t$ of the rotation $P^n_t-P^n_0$ equals the spherical excess $A_t+B_t+C_t-\\pi $ of the triangle $\\Delta _t,$ where $A_t$ is the angle at $u_{i},$ $C_t$ is the angle at $u_{i-1}$ and $B_t$ is the angle at $\\exp _{u_{i-1}}tv_n$ of the triangle $\\Delta _t.$ Hence, $\\lim _{t \\rightarrow 0} \\frac{1}{t}(P^n_t-P^n_0) =\\begin{pmatrix}0 & \\lim _{t \\rightarrow 0} \\frac{\\alpha _t}{t} \\\\ -\\lim _{t \\rightarrow 0} \\frac{\\alpha _t}{t} & 0\\end{pmatrix}\\operatorname{pt}_{u_i,u_{i-1}}, \\quad \\text{where } \\ \\alpha _t = A_t+B_t+C_t-\\pi .$ Here, the first identity is a consequence of the chain rule combined with $\\alpha _0=0.$ Since $\\frac{\\sin t}{t} = 1+ o(1),$ and since $C_t = \\pi /2$ by the orthogonality of $v_n $ and $\\log _{u_{i-1}}u_i,$ we get $\\omega =\\lim _{t \\rightarrow 0} \\frac{\\alpha _t}{t} =\\lim _{t \\rightarrow 0} \\tfrac{A_t+B_t+C_t-\\pi }{\\sin t}= \\lim _{t \\rightarrow 0} \\left( \\tfrac{A_t}{\\sin t} + \\tfrac{B_t-\\pi /2}{\\sin t} \\right).$ We recall that $d = d(u_{i},u_{i-1})$ and use the following identities for spherical triangles with an angle of $\\pi /2$ (cf.",
"[84]) $A_t = \\arctan \\left(\\tfrac{\\tan t}{\\sin d}\\right), \\quad B_t = \\arctan \\left(\\tfrac{\\tan d}{\\sin t}\\right).$ Using the Taylor expansion of the $\\arctan $ function w.r.t.",
"0 we obtain that $\\lim _{t \\rightarrow 0} \\frac{A_t}{\\sin t} =\\lim _{t \\rightarrow 0} \\frac{\\frac{\\tan t}{\\sin d} + O\\left(\\left(\\frac{\\tan t}{\\sin d} \\right)^3\\right)}{\\sin t} =\\lim _{t \\rightarrow 0} \\frac{1}{\\cos t\\sin d} = \\frac{1}{\\sin d}.$ Further, by L'Hospital's rule, $\\lim _{t \\rightarrow 0} \\frac{B_t-\\pi /2}{\\sin t} &=\\lim _{t \\rightarrow 0} \\frac{\\arctan \\left(\\frac{\\tan d}{\\sin t}\\right)-\\pi /2}{\\sin t}=\\lim _{t \\rightarrow 0} \\frac{-\\frac{\\tan d}{\\sin ^2 t} \\cdot \\cos t }{1+\\left(\\frac{\\tan d}{\\sin t}\\right)^2} \\cdot \\frac{1}{\\cos t}\\\\&= \\lim _{t \\rightarrow 0} \\left(-\\tfrac{1}{\\tan d}\\right) \\cdot \\left( 1- \\frac{1}{1+\\left(\\tfrac{\\tan d}{\\sin t}\\right)^2} \\right) = -\\tfrac{1}{\\tan d}.$ Now, we combine (REF ) with (REF ) and conclude, using (REF ), that $\\omega =\\tfrac{1}{\\sin d} - \\tfrac{1}{\\tan d}.$ Together with (REF ), this shows (REF ).",
"To see (REF ), we notice that the connection is linear w.r.t.",
"the direction of differentiation.",
"Therefore, (REF ) is a consequence of (REF ) and the expression $\\langle v_n, w \\rangle $ equals the coefficient of the corresponding linear combination.",
"If $u_i = u_{i-1},$ the mapping in (REF ) is constant; hence differentiation of this mapping yields zero which implies that the differential $\\frac{D}{dt}|_{t=0} B_t^n = 0.$ This shows the last assertion and completes the proof.",
"[Proof of Lemma REF ] Since the space of positive matrices is a Riemannian symmetric space representable as quotient of matrix Lie groups there are explicit formulae for the objects of Riemannian geometry such as the $\\exp $ mapping and the parallel transport available.",
"The corresponding formulae may be found in, e.g., [77].",
"Our first task is to explicitly express the mapping $B^n_t$ in the space of positive matrices which form a symmetric space.",
"We use the notation $\\gamma _t:[0,1]\\rightarrow \\mathcal {M}$ to denote the geodesic starting in $u_{i-1}$ with direction $v,$ i.e., $\\displaystyle \\gamma _t := \\exp _{u_{i-1}}tv =u_{i-1}^{\\tfrac{1}{2}}\\mathrm {Exp} \\left( u_{i-1}^{-\\tfrac{1}{2}} \\ tv \\ u_{i-1}^{-\\tfrac{1}{2}} \\right)u_{i-1}^{\\tfrac{1}{2}}.$ Here, $\\mathrm {Exp}$ denotes the ordinary matrix exponential.",
"Then, $B^n_t$ may be expressed in the space of positive matrices by $\\displaystyle B^n_t = \\operatorname{pt}_{u_i,\\gamma _t} z =u_{i}^{\\tfrac{1}{2}} \\ \\bar{\\gamma _t}^{\\tfrac{1}{2}}\\ \\bar{z}\\ \\bar{\\gamma _t}^{\\tfrac{1}{2}}\\ u_{i}^{\\tfrac{1}{2}}$ where $\\displaystyle \\bar{\\gamma _t} = u_{i}^{-\\tfrac{1}{2}} \\ \\gamma _t \\ u_{i}^{-\\tfrac{1}{2}},\\quad \\text{and}\\quad \\bar{z} = u_{i}^{-\\tfrac{1}{2}} \\ z \\ u_{i}^{-\\tfrac{1}{2}}.$ (We refer for instance to [77] for the corresponding formulae for the parallel transport.)",
"The covariant derivative in the space of positive matrices may be expressed in terms of the ordinary derivative of a curve in the vector space of matrices plus some “correction terms”(as for instance explained in [77]).",
"In our situation, we have $\\displaystyle \\tfrac{D}{dt}|_{t=0} B_t^n = \\tfrac{d}{dt}|_{t=0} B_t^n- \\tfrac{1}{2} \\left(v u_{i-1}^{-1} \\operatorname{pt}_{u_i,u_{i-1}} z +\\operatorname{pt}_{u_i,u_{i-1}} z u_{i-1}^{-1} v \\right).$ We denote the terms in brackets in (REF ) by $\\displaystyle S + S^\\top , \\qquad S := v u_{i-1}^{-1} \\operatorname{pt}_{u_i,u_{i-1}} z.$ We further have that, by (REF ), $\\displaystyle \\operatorname{pt}_{u_i,u_{i-1}} z = B^n_0 = u_{i}^{\\tfrac{1}{2}} \\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ \\bar{z}\\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ u_{i}^{\\tfrac{1}{2}}, \\quad \\text{ where} \\quad \\bar{u}_{i-1} = u_{i}^{-\\tfrac{1}{2}} \\ u_{i-1} \\ u_{i}^{-\\tfrac{1}{2}},$ and so we may explicitly express $S$ in terms of matrix multiplications by $\\displaystyle S := v u_{i-1}^{-1}u_{i}^{\\tfrac{1}{2}} \\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ \\bar{z}\\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ u_{i}^{\\tfrac{1}{2}}.$ In view of (REF ) and (REF ), we have to compute $\\frac{d}{dt}|_{t=0} B_t^n$ in order to derive an explicit representation of $\\frac{D}{dt}|_{t=0} B_t^n$ in terms of matrices.",
"Differentiating (REF ), we have that $\\displaystyle \\tfrac{d}{dt}|_{t=0} B_t^n =u_{i}^{\\tfrac{1}{2}} \\ \\left(\\tfrac{d}{dt}|_{t=0} \\bar{\\gamma _t}^{\\tfrac{1}{2}}\\right)\\ \\bar{z}\\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ u_{i}^{\\tfrac{1}{2}}+u_{i}^{\\tfrac{1}{2}} \\ \\bar{u}_{i-1}^{\\tfrac{1}{2}}\\ \\bar{z}\\ \\left(\\tfrac{d}{dt}|_{t=0} \\bar{\\gamma _t}^{\\tfrac{1}{2}}\\right)\\ u_{i}^{\\tfrac{1}{2}}.$ Introducing the notation $\\displaystyle X := \\tfrac{d}{dt}|_{t=0} \\bar{\\gamma _t}^{\\tfrac{1}{2}},\\quad T:= u_{i}^{\\tfrac{1}{2}} \\ X \\bar{z}\\ \\bar{u}_{i-1}^{\\tfrac{1}{2}} \\ u_{i}^{\\tfrac{1}{2}},$ the derivative $\\frac{d}{dt}|_{t=0} B_t^n$ in (REF ) may be rewritten as $\\tfrac{d}{dt}|_{t=0} B_t^n = T + T^\\top .$ We express $X$ more explicitly now.",
"To that end, let $f(A)= A^{1/2}$ be the matrix square root function (which is unambiguously defined for positive matrices).",
"We have, by the inverse function theorem, that $\\mathrm {d}f_{A^2}(Z) = Y,$ where $A Y+Y A = Z,$ meaning that, at the point $A$ , the directional derivative of $f$ in direction $Z$ is given by the solution $Y$ of the right-hand Sylvester equation.",
"Hence, in order to get $\\frac{d}{dt}|_{t=0} \\bar{\\gamma _t}^{\\tfrac{1}{2}},$ we have to solve the following Sylvester equation for $X$ , $\\bar{\\gamma _0}^{\\tfrac{1}{2}} X + X \\bar{\\gamma _0}^{\\tfrac{1}{2}} = \\bar{v}\\quad \\text{where} \\quad \\bar{v} := u_{i}^{-\\tfrac{1}{2}} v u_{i}^{-\\tfrac{1}{2}}.$ Summing up, $X = \\tfrac{d}{dt}|_{t=0} \\bar{\\gamma _t}^{\\tfrac{1}{2}}\\quad \\text{ is the solution of } \\quad \\bar{u}_{i-1}^{\\tfrac{1}{2}} X + X \\bar{u}_{i-1}^{\\tfrac{1}{2}} = \\bar{v}.$ Plugging (REF ) and (REF ) together with (REF ) into (REF ) shows that $\\frac{D}{dt}|_{t=0} B_t^n = (S-\\tfrac{1}{2}T) + (S-\\tfrac{1}{2}T)^\\top $ which completes the proof."
]
] | 1709.01616 |
[
[
"Parameterized complexity of machine scheduling: 15 open problems"
],
[
"Abstract Machine scheduling problems are a long-time key domain of algorithms and complexity research.",
"A novel approach to machine scheduling problems are fixed-parameter algorithms.",
"To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory."
],
[
"Introduction",
"Algorithms for machine scheduling problems form one of the core applications of combinatorial optimization.",
"In those problems, we are generally given a finite set $J$ of jobs with certain characteristics, and we must find a schedule for processing the jobs on one or more machines, which also may have their individual specifications.",
"Typical characteristics of a job $j$ are its processing time $p_j\\in \\mathbb {N}$ , its release date $r_j\\in \\mathbb {N}$ , its due date $d_j\\in \\mathbb {N}$ , or its importance reflected by a weight $w_j\\in \\mathbb {N}$ .",
"Jobs may be subject to precedence constraints enforcing some jobs to be completed before other jobs start.",
"Also, jobs may be preempted, or may be required to be processed without preemption.",
"Machine characteristics typically include their speed or whether they are capable of processing a certain type of job.",
"Usually, one is not only searching for a feasible schedule that respects all constraints, but additionally optimizes some objective function.",
"Classical objectives include the minimization of the makespan or the sum of weighted completion times (which is equivalent to minimizing the weighted average completion time).",
"Since the inception of the field in the 1950s, thousands of research papers have been devoted to understanding the complexity of scheduling problems.",
"A significant portion of investigated problems turned out to be $\\text{NP}$ -hard.",
"In consequence, algorithm designers proposed algorithms for these problems that yield approximate solutions in polynomial time.",
"In 1999, [72] listed 10 of the most prominent open problems around polynomialtime approximation algorithms for $\\text{NP}$ -hard scheduling problems at that time.",
"Only recently, a different algorithmic approach has been put forward for solving $\\text{NP}$ -hard scheduling problems: fixedparameter algorithms.",
"The idea in fixedparameter algorithms is to accept exponential running times, which are seemingly inevitable in solving $\\text{NP}$ -hard problems, but to restrict them to certain aspects of the problem, which are captured by parameters [25], [30], [68].",
"More formally, fixedparameter algorithms solve an instance of size $n$ with parameter $k$ in $f(k)\\cdot \\text{poly}(n)$ time for some computable, typically superpolynomial, function $f$ .",
"Thus, fixedparameter algorithms can solve even large instances of $\\text{NP}$ -hard scheduling problems if the parameter takes only small values, the function $f$ grows only moderately, and the polynomial degree in $n$ is small [11].",
"Moreover, problems that are even difficult to approximate can be approximated efficiently and well in real-world instances using fixedparameter approximation algorithms that exploit small parameters of real-world data [13].",
"While fixedparameter algorithms are now a wellinvestigated area of algorithmics, their systematic application to scheduling problems has gained momentum only recently [10], [23], [24], [40], [41], [43], [46], [64].",
"This already led to the transfer of proof techniques from parameterized complexity to the world of scheduling, such as $n$ -fold integer programming [48], color coding, problem kernelization [11], and $\\text{W}$ -hardness [9], [16], [12], [14], [66], [28], [42].",
"It also led to the transfer of techniques from mathematical programming to parameterized complexity, such as convex integer programming [66] or parameterizing by structural properties of the integer feasible polytope of linear programs [44].",
"In the following, we summarize known results and list open problems regarding the parameterized complexity of scheduling problems on a single machine, on parallel identical machines, and in shop scheduling environments.",
"We do not claim these problems to be the most important ones, but expect that their resolution (in one way or the other) will lead to the discovery of further new approaches both in parameterized complexity and scheduling theory, thus stimulating further research with both practical and theoretical significance.",
"In this sense, we hope that this work will be appealing and inspiring both to researchers with a scheduling background, as well as to researchers with a parameterized complexity background.",
"For the latter, we exhibit some fixedparameter tractability results that appeared before the advent of parameterized complexity and thus were not explicitly described as such.",
"We start by giving preliminaries about machine scheduling and parameterized complexity in sec:preliminaries.",
"Then we look at singlemachine problems in sec:singlemachineproblems, and parallel identical machines in sec:parallelmachinescheduling.",
"We finally consider the broad class of shop scheduling problems in sec:shopscheduling."
],
[
"Preliminaries",
"This section defines basic notions of scheduling theory, parameterized complexity theory, and approximation algorithms."
],
[
"Scheduling theory",
"Throughout this work, we use the standard three-field notation of scheduling problems due to [36].",
"This allows us to denote many problems as a triple $\\alpha |\\beta |\\gamma $ , where $\\alpha $ is the machine environment, $\\beta $ are job characteristics and scheduling constraints, and $\\gamma $ is the objective function.",
"We consider the following machine environments $\\alpha $ , which are described in detail in the following.",
"Singlemachine environments are denoted by the symbol 1, parallel identical machines are denoted by $P$ , job shop, open shop, and flow shop environments are denoted by $J$ , $O$ , and $F$ , respectively.",
"The symbol describing the machine environment can be followed by an integer restricting the number of machines (for example, $P2$ indicates two parallel identical machines).",
"If the symbol is followed by $m$ (for example, $Pm$ ), this indicates that the number of machines is an arbitrary constant.",
"If the symbol is neither followed by a number nor $m$ , then the number of machines is assumed to be given as part of the input.",
"Each job $j$ has to be processed for a given amount $p_j\\in \\mathbb {N}$ of time (its processing time) on a single machine.",
"The machine can process only one job at a time."
],
[
"Parallel identical machines (denoted by “P” in the $\\alpha $ -field)",
"We are given a number $m$ of parallel identical machines (that is, of the same speed).",
"Each job $j$ has to be processed by exactly one machine for a given amount $p_j\\in \\mathbb {N}$ of time and each machine can process only one job at a time."
],
[
"Unrelated parallel machines (denoted by “R” in the $\\alpha $ -field)",
"We are given a number $m$ of machines.",
"Each job $j$ has to be processed by exactly one machine $i$ for a given amount $p_{ij}\\in \\mathbb {N}$ of time and each machine can process only one job at a time."
],
[
"Shop scheduling problems",
"In shop scheduling problems, we are given a set $M$ of machines and a set $J$ of $n$ jobs, where each job $j$ consists of $n_j\\in \\mathbb {N}$ operations.",
"In the most general setting, the operations of each job are partially ordered: an operation of a job can only start once its preceding operations are completed.",
"Processing an operation $o_{ij}$ of job $j$ , where $i\\in \\lbrace 1,\\dots ,n_j\\rbrace $ , requires time $p_{ij}$ on a certain machine $\\mu _{ij}$ .",
"Each job can be processed by at most one machine at a time and each machine can process at most one operation at a time.",
"The three most important classes of shop scheduling problems are: Open shop scheduling, denoted by “O” in the $\\alpha $ -field: the processing order of the operations of each job is unrestricted and must be decided by an algorithm.",
"The only constraint is that each job has exactly one operation on each machine.",
"Job shop scheduling, denoted by “J” in the $\\alpha $ -field: the operations of each job are totally ordered, yet each job may have a distinct total order on its operations.",
"Herein, several operations may require processing on the same machine and not every job may have operations on all machines.",
"Flow shop scheduling, denoted by “F” in the $\\alpha $ -field: all jobs have the same set of operations with the same total order."
],
[
"Job characteristics",
"For job characteristics, we add qualifiers to the $\\beta $ -field.",
"In this survey, we consider the following types of job characteristics."
],
[
"Precedence constraints",
"If jobs are restricted by precedence constraints, then we add the qualifier “prec” to the $\\beta $ -field to indicate that jobs are only allowed to start after their predecessors are completed.",
"In the case where the partial order induced by the precedence constraints is the disjoint union of total orders, we write “chains” instead of “prec”."
],
[
"Processing time restrictions",
"If all jobs $j$ have unit processing time, we add “$p_j=1$ ” to the $\\beta $ -field.",
"If all jobs $j$ have an equal processing time $p$ , we add “$p_j=p$ ” to the $\\beta $ -field.",
"In case where processing times are bounded from above by some constant $c$ , we add “$p_j\\le c$ ” to the $\\beta $ -field.",
"If the processing times are restricted to values in some set $P\\subseteq \\mathbb {N}$ , we add “$p_j\\in P$ ” to the $\\beta $ -field."
],
[
"Machine restrictions",
"If each job $j$ can be processed only on a subset $M_j$ of machines, we add “$M_j$ ” to the $\\beta $ -field."
],
[
"Release dates",
"Each job $j$ may have a release date $r_j$ , before which it cannot be started.",
"In this case, we add “$r_j$ ” to the $\\beta $ -field."
],
[
"Preemption",
"Per default, we assume that jobs are not allowed to be preempted.",
"Otherwise, we add “pmtn” to the $\\beta $ -field."
],
[
"Multiprocessor jobs",
"When a job $j$ may occupy size$_j$ machines during execution, we add the qualifier “size$_j$ ” to the $\\beta $ -field; such problems are known as “multiprocessor scheduling”.",
"If the number of required machines is restricted to values in some set $S\\subseteq \\mathbb {N}$ , we add “$\\text{size}_j\\in S$ ” to the $\\beta $ -field."
],
[
"Batch setup times",
"When jobs are partitioned into batches and a sequencedependent setup time $s_{pq}$ is needed when switching from a job of batch $p$ to a job of batch $q$ , then, in accordance with [1], we add “$\\text{ST}_{\\text{sd},b}$ ” to the $\\beta $ -field."
],
[
"Objective functions",
"The objective functions $\\gamma $ that we aim to minimize are the makespan $C_{\\max }=\\max \\lbrace C_j\\mid j\\in J\\rbrace $ , where $C_j$ is the completion time of job $j$ in a schedule, and the weighted sum of completion times, $\\sum _{j\\in J}w_jC_j$ , when each job $j\\in J$ also has a weight $w_j\\in \\mathbb {N}$ .",
"If each job $j\\in J$ has a due date $d_j\\in \\mathbb {N}$ , we also minimize the total weighted tardiness $\\sum _{j\\in J}w_jT_j$ , where $T_j=\\max \\lbrace 0,d_j-C_j\\rbrace ,\\text{ and }$ the weighted number of tardy jobs, $\\sum _{j\\in J}w_jU_j$ , where $U_j=0$ if job $j$ is finished by its due date and $U_j=1$ otherwise.",
"This is equivalent to maximizing throughput—the weighted number of jobs that get finished by their due dates.",
"Generally, we will drop the “$j\\in J$ ” subscript under the sum and we will refer to the unitweight variants of the scheduling problems by simply dropping the $w_j$ from the objective functions."
],
[
"Parameterized complexity",
"An instance of a parameterized problem $\\Pi \\subseteq \\Sigma ^*\\times \\mathbb {N}$ is a pair $(x,k)$ consisting of the input $x$ and the parameter $k$.",
"A parameterized problem $\\Pi $ is fixedparameter tractable (FPT) if there is an algorithm solving any instance of $\\Pi $ with parameter $k$ and size $n$ in $f(k) \\cdot \\text{poly}(n)$ time for some computable function $f$ .",
"Such an algorithm is called a fixedparameter algorithm.",
"Note that fixedparameter tractability for a parameter $k$ is a much stronger property than polynomialtime solvability for constant $k$ : a fixedparameter algorithm running in $O(2^k\\cdot n)$ time takes polynomial time even for $k\\in O(\\log n)$ .",
"In contrast, an algorithm running in time $O(n^k)$ is polynomial for constant $k$ , but is often impractical even for small values of $k$ .",
"The main goal of parameterized complexity is determining which parameterized problems have fixedparameter algorithms.",
"These can efficiently solve $\\text{NP}$ -hard problems in applications where the parameter takes only small values and the degree of the polynomial in $n$ is small.",
"Notably, a problem may be fixedparameter tractable with respect to one parameter, but might be not with respect to another.",
"How to find parameters that are both small in applications and lead to fixedparameter algorithms, is a research branch on its own [29], [50], [69]."
],
[
"Problem kernelization",
"Parameterized complexity enabled a mathematical formalization of polynomialtime data reduction with provable performance guarantees: kernelization.",
"It has been successfully applied to obtain effective polynomialtime data reduction algorithms for many $\\text{NP}$ -hard problems [38], [52] and also led to techniques for proving lower bounds on the effectivity of polynomialtime data reduction [65], [17].",
"A kernelization algorithm for a parameterized problem $\\Pi \\subseteq \\Sigma ^*\\times \\mathbb {N}$ reduces an instance $(x,k)$ to an instance $(x^{\\prime },k^{\\prime })$ in polynomial time such that the size of $x^{\\prime }$ and $k^{\\prime }$ depends only on $k$ and such that $(x,k)\\in \\Pi $ if and only if $(x^{\\prime },k^{\\prime })\\in \\Pi $ .",
"The instance $(x^{\\prime },k^{\\prime })$ is called a problem kernel.",
"Note that it is the parameter that allows us to measure the effectivity of polynomialtime data reduction since an absolute statement like “the data reduction shrinks the input by at least one bit” for an $\\text{NP}$ -hard problem would imply that we can solve $\\text{NP}$ -hard problems in polynomial time."
],
[
"Parameterized intractability",
"To show that a problem is presumably not fixedparameter tractable, there is a parameterized analogue of $\\text{NP}$ hardness.",
"The parameterized analogue of $\\text{P}$${}\\subseteq {}$$\\text{NP}$ is a hierarchy of complexity classes $\\text{FPT}$${}\\subseteq {}$$\\text{W}$ [1]${}\\subseteq {}$$\\text{W}$ [2]${}\\subseteq {}\\dots \\subseteq {}$$\\text{XP}$ , where $\\text{XP}$ is the class of parameterized problems that are solvable in polynomial time for constant parameter values.",
"The working hypothesis is that all inclusions in this hierarchy are proper, paralleling the famous $\\text{P}\\ne \\text{NP}$ conjecture.",
"A parameterized problem $\\Pi $ is (weakly) W[$t$ ]-hard for some $t\\in \\mathbb {N}$ if any problem in W[$t$ ] has a parameterized reduction to $\\Pi $ : a parameterized reduction from a parameterized problem $\\Pi _1$ to a parameterized problem $\\Pi _2$ is an algorithm mapping an instance $(x,k)$ to an instance $(x^{\\prime },k^{\\prime })$ in time $f(k)\\cdot \\text{poly}(|x|)$ such that $k^{\\prime }\\le g(k)$ and $(x,k)\\in \\Pi _1\\iff (x^{\\prime },k^{\\prime })\\in \\Pi _2$ , where $f$ and $g$ are arbitrary computable functions.",
"By definition, no W[$t$ ]-hard problem is fixedparameter tractable unless FPT${}={}$ W[$t$ ].",
"A parameterized problem is strongly $\\text{W}[1]$ -hard if it is $\\text{W}[1]$ -hard even if all numbers in the input are encoded in unary."
],
[
"Approximation algorithms",
"Since we only consider minimization problems, we introduce approximation terminology only for minimization problems.",
"An $\\alpha $ approximation is a solution to an optimization problem with cost at most $\\alpha $ times the cost of an optimal solution.",
"A polynomialtime approximation scheme (PTAS) is an algorithm that computes a $(1+\\varepsilon )$ approximation in polynomial time for any constant $\\varepsilon >0$ .",
"An efficient polynomialtime approximation scheme (EPTAS) is an algorithm that computes a $(1+\\varepsilon )$ approximation in time $f(1/\\varepsilon )\\cdot \\text{poly}(n)$ for some computable function $f$ and $\\varepsilon >0$ .",
"Finally, a fully polynomialtime approximation scheme (FPTAS) is an algorithm that computes a $(1+\\varepsilon )$ approximation in time $\\text{poly}(n,1/\\varepsilon )$ for any $\\varepsilon >0$ .",
"It is known that any problem having an EPTAS is fixedparameter tractable parameterized by the cost of an optimal solution [22].",
"Thus, showing $\\text{W}[1]$ hardness of a problem parameterized by the optimum solution cost shows that it neither has an EPTAS nor an FPTAS unless $\\text{FPT}{}=\\text{W}[1]$ .",
"Moreover, strongly $\\text{NP}$ -hard optimization problems with polynomially bounded objective functions do not have FPTASes unless $\\text{P}=\\text{NP}$ [32].",
"In this section, we consider various single-machine problems: sec:single:twt studies total weighted tardiness as objective function, sec:single:twct the total weighted completion time, sec:single:wtj the weighted number of tardy jobs, and, finally, sec:forbidden considers a variant with forbidden start and end times of jobs."
],
[
"Total weighted tardiness",
"The problem 1||$\\Sigma w_jT_j$ is strongly NP-hard [58], [62], yet has two easier special cases: 1|$p_j{=}p$ |$\\Sigma w_jT_j$ is solvable in polynomial time via a reduction to the classical linear assignment problem, whereas 1||$\\Sigma T_j$ is solvable in pseudopolynomial time [58], [62].",
"To be more exact, [58]'s pseudo polynomialtime algorithm is a fixedparameter algorithm for 1||$\\Sigma T_j$ parameterized by the maximum processing time $p_{\\max }$ that also works for 1||$\\Sigma w_jT_j$ with agreeable processing times and weights, that is, $p_i<p_j$ implies $w_i\\ge w_j$ .",
"Although 1||$\\Sigma w_jT_j$ is strongly $\\text{NP}$ -hard and thus cannot have pseudo polynomialtime algorithms unless $\\text{P}$${}={}$$\\text{NP}$ , it is interesting whether the fixedparameter tractability result for 1||$\\Sigma w_jT_j$ with agreeable processing times and weights holds in general: Open Problem 1 Is $1||\\Sigma w_jT_j$ fixedparameter tractable parameterized by the maximum processing time $p_{\\max }$ ?",
"Also note that, given that 1|$p_j{=}p$ |$\\Sigma w_jT_j$ is solvable in polynomial time, it is interesting to see whether 1||$\\Sigma w_jT_j$ is fixedparameter tractable with respect to the number $\\bar{p}$ of distinct processing times."
],
[
"Total weighted completion time",
"[3] showed that $1|\\text{prec}{}|\\Sigma w_jC_j$ is a special case of Weighted Vertex Cover.",
"Vertex Cover is one of the most wellstudied problems in parameterized complexity theory, in particular in terms of problem kernels; small kernels for Weighted Vertex Cover were established by [27].",
"It is interesting which kernelization algorithms carry over to $1|\\text{prec}{}|\\Sigma w_jC_j$ .",
"However, a problem kernel for $1|\\text{prec}{}|\\Sigma w_jC_j$ whose size is bounded by a function of the optimum is not interesting—the optimum is at least the weighted sum of all processing times and thus already bounds the input size without data reduction.",
"Remarkably, Vertex Cover admits a (randomized) algorithm yielding a problem kernel with size polynomial in the difference between the optimum and a lower bound given by the relaxation of the natural ILP [53].",
"Since $1|\\text{prec}{}|\\Sigma w_jC_j$ is a special case of Weighted Vertex Cover, and thus allows for a likewise natural ILP formulation [3], the following question is interesting: Open Problem 2 Does $1|\\text{prec}{}|\\Sigma w_jC_j$ admit a problem kernel of size polynomial in the difference between the optimum and the lower bound obtained from its natural LP relaxation?",
"We point out that any partial progress on this question would be interesting: be it a randomized kernelization algorithm or even a partial kernel that bounds only the number of jobs and not necessarily their weights or processing times.",
"The question is complicated by the fact that each vertex in the Weighted Vertex Cover instance created by [3], and thus each variable in the corresponding ILP, corresponds to a pair of jobs in the $1|\\text{prec}{}|\\Sigma w_jC_j$ instance, such that known data reduction rules for Weighted Vertex Cover do not allow for a straightforward interpretation in terms of jobs."
],
[
"Throughput or weighted number of tardy jobs",
"In a survey on open questions in maximum throughput scheduling, [75] asked whether there is a polynomialtime algorithm for $1|r_j,p_j{\\le }c|\\Sigma U_j$ for constant $c$ .",
"Similarly, the $\\text{NP}$ hardness of the weighted case for constant $c$ is open.",
"Parameterized complexity can serve as an intermediate step towards resolving these questions.",
"Open Problem 3 Are $1|r_j|\\Sigma U_j$ and $1|r_j|\\Sigma w_jU_j$ $\\text{W}[1]$ -hard parameterized by the maximum processing time $p_{\\max }$ ?",
"Currently, even containment in the parameterized complexity class $\\text{XP}$ is open.",
"The special case $1||\\Sigma U_j$ is polynomialtime solvable, whereas $1||\\Sigma w_jU_j$ is weakly $\\text{NP}$ -hard and solvable in pseudopolynomial time [60], [47].",
"[43] gave fixedparameter algorithms for $1||\\Sigma w_jU_j$ simultaneously parameterized by any two out of the following three parameters: the number of distinct due dates, the number of distinct processing times, and the number of distinct job weights.",
"[28] showed that $1|\\text{prec}{},p_{j}{=}1|\\Sigma U_i$ is $\\text{W}$ [1]-hard parameterized by the number of tardy jobs but fixedparameter tractable with respect to this parameter if the partial order induced by the precedence constraints has constant width.",
"Herein, the width of a partial order is the size of a largest set of mutually incomparable jobs."
],
[
"Forbidden start and end times",
"Machine scheduling problems with forbidden start and end times model the situation when an additional resource, subject to unavailability constraints, is required to start or finish a job.",
"For example, this resource might be operators of chemical experiments, which serves as a major motivation for such problems.",
"For makespan minimization on a single machine, [15] gave an algorithm that runs in $n^{O(\\tau ^2)}$ time for $\\tau $ forbidden start times and $n$ jobs; this was improved by [70] to $n^{O(\\tau )}$ time.",
"For the highmultiplicity encoding of the input—given by binary numbers $n_t$ encoding the number of jobs having the same forbidden start and end times—[31] showed a polynomialtime algorithm if the number $\\tau $ of forbidden times is constant.",
"All of these results leave open the possibility for fixedparameter tractability of the problem parameterized by $\\tau $ .",
"Open Problem 4 Is makespan minimization on a single machine with $\\tau $ forbidden start and end times fixedparameter tractable parameterized by $\\tau $ ?",
"In this section, we consider scheduling on parallel identical machines.",
"sec:parallelfewproctimes considers the makespan objective, sec:prec-sched-fpt the makespan objective with precedence constraints, sec:tardyjobs the weighted number of tardy jobs, sec:jit considers just-in-time scheduling, and, finally, sec:preempt considers variants with allowed preemption of jobs."
],
[
"Makespan",
"[2] showed an EPTAS for $P{}||C_{\\max }$ , which implies that $P{}||C_{\\max }$ is fixedparameter tractable parameterized by the makespan $C_{\\max }$ [22].",
"This was improved by [66] to a fixedparameter algorithm for the maximum processing time $p_{\\max }$ of any job; we generally expect $p_{\\max } \\ll C_{\\max }$ .",
"The running time of both algorithms is doublyexponential in the parameter.",
"An improved algorithm whose running time depends singlyexponentially on $p_{\\max }$ was proposed by [48].",
"Very recently, [49] strengthened this result by giving an algorithm with singleexponential dependence $p_{\\max }$ even for the high-multiplicity encoding, when, for each processing time $p\\le p_{\\max }$ , the number of jobs with processing time $p$ is encoded in binary.",
"[23] generalized the result of [66] by showing that $R{}||C_{\\max }$ is fixedparameter tractable parameterized by $p_{\\max }$ and the rank of the matrix $(p_{ij})$ giving the processing time $p_{ij}$ of job $j$ on machine $i$ .",
"In a breakthrough result, [34] showed that $P{}||C_{\\max }$ is polynomialtime solvable when the number $\\overline{p}$ of distinct processing times is constant, that is, the problem is in $\\text{XP}$ .",
"Their result holds even for the high-multiplicity encoding of the input, where the number $m$ of machines and the number $n_j$ of jobs with processing time $p_j$ are encoded in binary for each $j\\in \\lbrace 1,\\dots ,\\overline{p}\\rbrace $ .",
"A close inspection of their result reveals that it is a fixed-parameter algorithm when the processing times $p_j$ are encoded in unary.",
"Despite all this progress, the following problem remains open.",
"Open Problem 5 Is $P{}||C_{\\max }$ with high-multiplicity encoding fixedparameter tractable parameterized by the number $\\overline{p}$ of distinct processing times, that is, when job processing times $p_j$ and the number of jobs for each processing time are encoded in binary?",
"To find a fixedparameter algorithm for the high-multiplicity encoding of $P{}||C_{\\max }$ parameterized by $\\overline{p}$ , one difficulty is outputting an optimal schedule, whose obvious encoding requires at least $m$ bits (to store how many jobs of each processing time are processed on each machine) and is therefore neither polynomial in the input size nor bounded by a function of $\\overline{p}$ .",
"Such problems might be possibly overcome by using the framework of [18]."
],
[
"Makespan and precedence constraints",
"The parameterized complexity of $P|\\text{prec}{}|C_{\\max }$ and special cases has extensively been studied with respect to the width of the partial order induced by the precedence constraints: the width of a partial order is the size of its largest antichain—a set of pairwise incomparable jobs.",
"The special case $P2|\\text{chains}{}|C_{\\max }$ is weakly $\\text{NP}$ -hard even for partial orders of width three [37], [12].",
"Since this excludes fixedparameter algorithms using the partial order width as parameter already on two machines, it has been tried to use the partial order width and the maximum processing time as parameters simultaneously.",
"[16] showed that even $P|\\text{prec}{},p_j{=}1|C_{\\max }$ is $\\text{W}[2]$ -hard parameterized by the partial order width and by the number of machines.",
"Later, [12] showed that combining partial order width with maximum processing time does not even yield fixedparameter algorithms on two machines.",
"More precisely, they showed that even $P2|\\text{prec}{},p_j{\\in }\\lbrace 1,2\\rbrace |C_{\\max }$ is $\\text{W}[2]$ -hard parameterized by the partial order width; and so is $P3|\\text{prec}{},\\text{size}_j{\\in }\\lbrace 1,2\\rbrace |C_{\\max }$ .",
"Further restricting this problem, one arrives at a longstanding open problem due to [32] of whether $P3|\\text{prec}{},p_j {=} 1|C_{\\max }$ is $\\text{NP}$ -hard or polynomialtime solvable.",
"In fact, the complexity is open for any constant number of machines.",
"Thus, as pointed out by [12], it would be surprising to show $\\text{W}[1]$ hardness of this problem for any parameter, since this would exclude polynomialtime solvability unless $\\text{FPT}{} = \\text{W}[1]$ .",
"Open Problem 6 Is $P3|\\text{prec}{},p_j {=} 1|C_{\\max }$ fixedparameter tractable parameterized by the width of the partial order induced by the precedence constraints?",
"Note that a negative answer would basically answer the open question of [32] on whether the problem is polynomialtime solvable, whereas a positive answer would be in strong contrast to the $\\text{W}[2]$ hardness of the slight generalizations $P|\\text{prec}{},p_j{=}1|C_{\\max }$ , $P2|\\text{prec}{},p_j{\\in }\\lbrace 1,2\\rbrace |C_{\\max }$ , and $P3|\\text{prec}{},\\text{size}_j{\\in }\\lbrace 1,2\\rbrace |C_{\\max }$ considered by [16] and [12]."
],
[
"Throughput or weighted number of tardy jobs",
"[7] showed that $Pm|r_j,p_j{=}p|\\Sigma w_jU_j$ is polynomialtime solvable.",
"According to [75], this is open when the number of machines is not a constant.",
"A first step to resolving this question is solving the following problem.",
"Open Problem 7 Are $P|r_j,p_j{=}p|\\Sigma w_jU_j$ and $P|r_j,p_j{=}p|\\Sigma U_j$ fixedparameter tractable parameterized by the number of machines?",
"A negative answer to this question would also be interesting since it is open whether these problems are even $\\text{NP}$ -hard if the number $m$ of machines is part of the input.",
"Notably, the special case $P|p_j{=}p|\\Sigma w_jU_j$ is polynomialtime solvable via a reduction to the classical linear assignment problem but $P|p_j{=}p,\\text{pmtn}|\\Sigma w_jU_j$ is strongly $\\text{NP}$ -hard [19]."
],
[
"Interval scheduling or just-in-time scheduling",
"An important special case of maximizing the throughput on parallel identical machines is interval scheduling, where each job $j$ has a weight $w_j$ , a fixed start time, and a fixed end time.",
"The goal is to schedule a maximumweight subset of jobs nonpreemptively on $m$ parallel identical machines.",
"As always, each machine can process only one job at a time.",
"Since we can interpret this problem as minimizing the total weight of jobs not meeting their due dates, where each job $j$ has a processing time $p_j$ , a release date $r_j$ , and a due date $d_j$ such that $d_j-r_j=p_j$ , we denote the problem as $P|d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ .",
"This problem is equivalent to maximizing the weighted number of “just-in-time” jobs on parallel identical machines [76].",
"[4] show that this problem is solvable in $O(n^2\\log n)$ time.",
"However, they showed that the variant $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ is $\\text{NP}$ -hard and solvable in $O(n^{m+1})$ time.",
"Open Problem 8 Is $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ fixedparameter tractable parameterized by the number of machines?",
"This problem is a special case of $R|d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ where each job $j$ has a processing time $p_{ij}\\in \\mathbb {\\lbrace }p_j,\\infty \\rbrace $ on machine $i$ .",
"In that sense, any positive answer to cis (in form of a fixedparameter algorithm) can serve as a first step towards a fixedparameter algorithm for $R|d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ also; after all, the $O(n^{m+1})$ -time algorithm of [4] for $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ was generalized to $R|d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ with essentially the same running time [76].",
"We point out that, if we only slightly relax the condition $d_j-r_j=p_j$ to $d_j-r_j\\le \\lambda p_j$ for any constant $\\lambda >0$ , then the problem is weakly $\\text{NP}$ -hard for $m=2$ and strongly $\\text{W}$ [1]-hard parameterized by $m$ [14], even when all jobs are allowed to be processed on all machines and only checking whether one can finish all jobs by their due date.",
"Using a construction due to [40], $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ can be seen to be a special case of the Job Interval Selection problem introduced by [67], where each job has multiple possible execution intervals and we have to select one execution interval for each job in order to process it: we model the machines by pairwise disjoint segments of the real line and each job has an interval in each segment belonging to a machine it can be processed on.",
"Via this relation to Job Interval Selection, one can model $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ as Colorful Independent Set in an interval graph with at most $mn$ intervals colored in $n$ colors and having at most $m$ intervals of each color [11], where the task is to find a maximumweight independent set of intervals with mutually distinct colors.",
"Colorful Independent Set is fixedparameter tractable parameterized by the number of colors in the solution on interval (and even on chordal) graphs [11], [9], which implies that $P|M_j,d_j{-}r_j{=}p_j|\\Sigma w_jU_j$ is fixedparameter tractable parameterized by the number of jobs that can be scheduled in an optimal solution.",
"To answer cis, one could try to show that Colorful Independent Set in interval graphs is fixedparameter tractable parameterized by the maximum number of intervals of any color."
],
[
"Allowed preemption",
"The three problems $P|\\text{pmtn}{},r_j|\\Sigma C_j$ , $P|\\text{pmtn}|\\Sigma U_j$ , and $P|\\text{pmtn}|\\Sigma T_j$ are all NP-hard [26], [59], [57], yet the special cases with equal processing times are polynomialtime solvable [7], [8], [56].",
"Open Problem 9 Are the above problems fixedparameter tractable parameterized by the number $\\overline{p}$ of distinct processing times?",
"[56] survey related problems; as of now, the complexity status of $P2|\\text{pmtn}{},p_j {=} p|\\Sigma w_jT_j$ for constant $p$ is open.",
"Parameterized hardness might serve as an intermediate step towards settling it.",
"Open Problem 10 Is $P2|\\text{pmtn}{},p_j {=} p|\\Sigma w_jT_j$ $\\text{W}$ [1]-hard parameterized by the processing time $p$ of all jobs?",
"We now consider the open shop, job shop, and flow shop scheduling problems.",
"sec:jcmax considers the makespan objective, sec:cmaxapr considers fixedparameter approximation schemes, sec:setup takes into account setup times, and, finally, sec:jsthroughput considers variants of maximizing throughput."
],
[
"Makespan",
"[51] give a computational complexity dichotomy of $J||C_{\\max }$ and $O||C_{\\max }$ into polynomialtime solvable cases and $\\text{NP}$ -hard cases depending on the maximum processing time $p_{\\max }$ of operations, the maximum number $n_{\\max }$ of operations per job, and an upper bound on the makespan $C_{\\max }$ .",
"All problems are $\\text{NP}$ -hard even if all of the listed parameters are simultaneously bounded from above by 4 [51], [78], which fully settles the parameterized complexity of $O||C_{\\max }$ and $J||C_{\\max }$ with respect to these parameters.",
"However, in all hardness reductions of [51], the number of machines is necessarily unbounded: when bounding both the number $m$ of machines and the makespan $C_{\\max }$ , then shop scheduling problems are trivial, since the overall number of operations in the input is at most $m\\cdot C_{\\max }$ .",
"This makes the parameters $p_{\\max }$ and $n_{\\max }$ interesting for fixedparameter algorithms for shop scheduling problems with a fixed number of machines (or with the number of machines as an additional parameter).",
"For example, both $F3||C_{\\max }$ and $O3||C_{\\max }$ are strongly $\\text{NP}$ -hard [35], [33], yet $Om||C_{\\max }$ is fixedparameter tractable parameterized by maximum processing time $p_{\\max }$ : if the maximum machine load is $\\ell _{\\max }\\ge m^2 p_{\\max }$ , then the makespan of the instance is $\\ell _{\\max }$ [74].",
"Otherwise, there are at most $\\ell _{\\max }< m^2 p_{\\max }$ jobs and the instance can be solved using brute force.",
"For $F3||C_{max}$ , the analogous question is open: Open Problem 11 Is $F3||C_{\\max }$ fixedparameter tractable parameterized by the maximum processing time $p_{\\max }$ ?",
"This question is likewise interesting for $F3|\\text{no-wait}|C_{\\max }$ , where “no-wait” means that each operation of a job has to start immediately after completion of the preceding operation of the same job.",
"The $\\text{NP}$ hardness of $F3|\\text{no-wait}|C_{\\max }$ was a longstanding open question [71].",
"Notably, for the threemachine job shop scheduling problem, the above question has a negative answer: even the special case $J3|p_{ij}{=}1|C_{\\max }$ is $\\text{NP}$ -hard [61].",
"Open Problem 12 Is $J3||C_{\\max }$ fixedparameter tractable parameterized by the maximum processing time $p_{\\max }$ and the maximum number $n_{\\max }$ of operations per job?"
],
[
"Makespan approximation",
"Job and open shop scheduling problems are often $\\text{APX}$ -hard [78] but also $\\text{NP}$ -hard for a constant number of machines [35], [33], [61] and many other constant parameters [51].",
"Thus, they are amenable neither to approximation schemes nor fixedparameter algorithms.",
"However, while having a constant number of machines does not help to get polynomialtime algorithms for these problems, it helps tremendously in getting PTASes [73], [39], [45].",
"Remarkably, these are not simply PTASes for a fixed number of machines running in time, say $n^{O(m/\\varepsilon )}$ : In terms of [63], they are actually fixedparameter tractable approximation schemes (FPT-AS) for the parameters $m$ and $\\varepsilon $ , running in time $f(m,\\varepsilon )\\cdot \\text{poly}(n)$ .",
"While not necessarily being a PTAS, an FPT-AS using the number $m$ of machines as parameter and running in $O(2^{m/\\varepsilon }\\cdot n)$ time might be practically more valuable than a PTAS running in $O(n^{1/\\varepsilon })$ time or even an FPTAS running in $O((n+1/\\varepsilon )^3)$ time.",
"[73] compute a $(1+\\varepsilon )$ approximation for $O||C_{\\max }$ in $f(m,\\varepsilon )+O(n \\log n)$ time, [39] computes a $(1+\\varepsilon )$ approximation for $F||C_{\\max }$ in $f(m,\\varepsilon )+O(n^{3.5})$ time.",
"Yet for $J||C_{\\max }$ , [45] need an additional parameter: they compute a $(1+\\varepsilon )$ approximation for $J||C_{\\max }$ in $f(m,n_{\\max },\\varepsilon )+O(n)$ time, where $n_{\\max }$ is the maximum number of operations of a job.",
"Open Problem 13 Is there an algorithm that yields a $(1+\\varepsilon )$ approximation for $J||C_{\\max }$ in $f(m,\\varepsilon )\\cdot \\text{poly}(n)$ time?",
"Giving a negative answer to this question seems challenging: the obvious approach would be proving that $J||C_{\\max }$ is $\\text{W}[1]$ -hard parameterized by $C_{\\max }+m$ .",
"However, as discussed in sec:jcmax, $J||C_{\\max }$ is trivially fixedparameter tractable parameterized by $C_{\\max }+m$ , so that this approach is inapplicable."
],
[
"Makespan with sequencedependent setup times",
"In the following, we consider a variant $O|p_{ij}=1,\\text{ST}_{\\text{sd},b}|C_{\\max }$ of open shop with unit processing times where the jobs are partitioned into $g$ batches and machines require a sequencedependent batch setup time $s_{pq}\\in \\mathbb {N}$ when switching from jobs of batch $p$ to jobs of batch $q$ .",
"The case with unit processing times models applications where large batches of items have to be processed and processing individual items in each batch takes significantly less time than setting up the machine for the batch.",
"Alternatively one can interpret this problem as an open shop problem where machines or experts have to travel between jobs in different locations in order to process them [5].",
"As a generalization of the Traveling Salesperson problem, $O|p_{ij}=1,\\text{ST}_{\\text{sd},b}|C_{\\max }$ is $\\text{NP}$ -hard already for one machine.",
"As shown by [10], the problem is solvable in $O(2^gg^2+mn)$ time if each batch contains more jobs than there are machines.",
"For the case where batches may contain less jobs, they showed that the problem is solvable in $O(n\\log n)$ time if both the number $m$ of machines and the number $g$ of batches are constants.",
"Open Problem 14 Is $O|p_{ij}=1,\\text{ST}_{\\text{sd},b}|C_{\\max }$ fixedparameter tractable parameterized by the number $g$ of batches?",
"We point out that, for this problem, it would be desirable to design a fixedparameter algorithm that works also for the highmultiplicity encoding, which compactly encodes the number of jobs in each batch in binary.",
"The fixedparameter algorithms of [10] also work for the highmultiplicity encoding as long as they only need to compute the minimum makespan."
],
[
"Throughput or weighted number of tardy jobs",
"Since checking for a schedule with makespan $L$ is equivalent to checking whether all jobs can meet a due date of $L$ , all $\\text{NP}$ hardness results from $J||\\Sigma C_j$ and $O||\\Sigma C_j$ in sec:jcmax carry over to $J||\\Sigma U_j$ and $O||\\Sigma U_j$ .",
"In fact, these throughput maximization variants turn out to be even harder: now, even the special cases $J2|p_{ij}{=}1|\\Sigma w_jU_j$ and $J2|p_{ij}{=}1,r_j|\\Sigma U_j$ on two machines are NP-hard [55], [77].",
"The situation is more relaxed for open shop scheduling: $O|p_{ij}{=}1,r_j|\\Sigma U_j$ is $\\text{NP}$ -hard for an unbounded number of machines [54], whereas $O|p_{ij}{=}1|\\Sigma w_jU_j$ is even fixedparameter tractable parameterized by the number $m$ of machines [21].",
"For $Om|p_{ij}{=}1,r_j|\\Sigma w_jU_j$ , we only know polynomialtime solvability for a constant number of machines [6].",
"Open Problem 15 Is $O|p_{ij}{=}1,r_j|\\Sigma w_jU_j$ fixedparameter tractable parameterized by the number of machines?",
"It is known that the special case $O|p_{ij}{=}1,r_j|C_{\\max }$ is polynomialtime solvable and that $O|p_{ij}{=}1,r_j|\\Sigma w_jU_j$ is equivalent to the problem $P|p_{j}{=}m,\\text{pmtn}{}_{\\mathbb {Z}},r_j|\\Sigma w_jU_j$ of scheduling jobs with processing time $m$ on $m$ parallel identical machines with release dates and preemption allowed at integer times [20]."
],
[
"Acknowledgements",
"Matthias Mnich is supported by DFG grant MN 59/4-1 and ERC Starting Grant 306465 (BeyondWorstCase); he expresses his gratitude towards Alexander Grigoriev for enlightening discussions and many helpful comments.",
"René van Bevern is supported by the Russian Foundation for Basic Research, project 16-31-60007 mol_a_dk, and by the Ministry of Science and Education of the Russian Federation under the 5-100 Excellence Program; he thanks Gerhard J. Woeginger for pointing out prob:o3.",
"Both authors thank Martin Koutecký for pointing out that $P||C_{\\max }$ is fixedparameter tractable for the number of distinct processing times when those are encoded in unary and Gerhard J. Woeginger for pointing out that $O3||C_{\\max }$ is fixedparameter tractable when parameterized by the maximum processing time.",
"They also thank the reviewers for their detailed and helpful remarks which lead to an improved presentation."
]
] | 1709.01670 |
[
[
"Upsilon type concordance invariants"
],
[
"Abstract To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $\\Upsilon^C$.",
"For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\\' ath-Stipsicz-Szab\\' o upsilon invariant.",
"Furthermore, to three such regions $C$, $C^+$ and $C^- $ we associate invariants $\\Upsilon_{C^\\pm, C}$ generalising Kim-Livingston secondary invariant.",
"We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots."
],
[
"Introduction",
"In [17] Ozsváth and Szabó, by essentially studying the Floer homology [3] of certain Lagrangian tori in the $g$ -fold symmetric product of a genus $g$ Riemann surface, found a package of three-manifold invariants called Heegaard Floer homology.",
"In [16] they used this circle of ideas to define a related package of knot invariants named knot Floer homology.",
"See [23] for an extensive exposition of this topic.",
"Knot Floer homology has been used to produce knot concordance invariants by many authors [22], [15], [24], [7].",
"The purpose of this note is to show that all these constructions can be seen as particular cases of a more general construction.",
"Our investigation is mainly motivated by the following applications."
],
[
"In [8] Lidman and Moore characterized $L$ -space pretzel knots.",
"They found that a pretzel knot has an $L$ -space surgery if and only if it is a torus knot $T_{2, 2n+1}$ for some $n \\ge 1$ , or a pretzel knot in the form $P(-2, 3, q)$ for some $q \\ge 7$ odd.",
"Motivated by the exploration started by Wang [27], and Livingston [10] one may wonder if $L$ -space pretzel knots of the form $P(-2,3,q)$ are concordant to algebraic knots.",
"Theorem 1.1 None of the $L$ -space pretzel knots $P(-2,3,q)$ , with $q\\ge 7$ odd, is conconcordant to a sum of algebraic knots.",
"Notice that for these knots the obstruction found in [27] vanish."
],
[
"In [11] Friedl, Livingston and Zentner asked whenever a sum of torus knots is concordant to an alternating knot.",
"In [28] Zemke used involutive Floer homology [5] to prove that certain connected sums of torus knots are not concordant to Floer thin knots.",
"Floer thin knots are upsilon-alternating, meaning that $\\Upsilon _K(t) = -\\tau (K) \\cdot (1-|1-t|)$ .",
"A straightforward argument shows that a sum of positive torus knots is upsilon-alternating if and only if it is a connected sum of $(2, 2n+1)$ torus knots and indeed alternating.",
"However, when both positive and negative torus knots are involved this obstruction can vanish.",
"Proposition 1.2 The knot $K=T_{8,5}\\# - T_{6,5} \\# -T_{4,3}$ is upsilon-alternating but not concordant to a Floer thin knot.",
"The connected sum formula (Theorem REF ) employed in the proof of Proposition REF is used in [1] to decide which sums of two torus are concordant to alternating knots."
],
[
"A quick review of knot Floer homology",
"An Alexander filtered, Maslov graded chain complex is a finitely-generated, $\\mathbb {Z}$ -graded, $(\\mathbb {Z}\\oplus \\mathbb {Z})$ -filtered chain complex $C= (\\bigoplus _{\\mathbf {x}\\in B} \\mathbb {Z}_2[U, U^{-1}], \\partial )$ such that $\\partial $ is $\\mathbb {Z}_2[U,U^{-1}]$ -linear and given a basis element $\\mathbf {x}\\in B$ , $\\partial \\mathbf {x}= \\sum _\\mathbf {y}n_{\\mathbf {x}, \\mathbf {y}}U^{m_{\\mathbf {x},\\mathbf {y}}} \\cdot \\mathbf {y}$ for suitable coefficients $ n_{\\mathbf {x}, \\mathbf {y}} \\in \\mathbb {Z}_2$ , and non-negative exponents $m_{\\mathbf {x}, \\mathbf {y}} \\ge 0$ , the multiplication by $U$ drops the homological (Maslov) grading $M$ by two, and the filtration levels (denoted by $A$ and $j$ ) by one.",
"An Alexander filtered, Maslov graded chain complex is said of knot type if in addition $H_*(C, \\partial )= \\mathbb {Z}_2[U, U^{-1}]$ graded so that $\\text{deg}U=-2$ .",
"An Alexander filtered, Maslov graded chain complex can be pictorially described as follows: picture each $\\mathbb {Z}_2$ -generator $U^m \\cdot \\mathbf {x}$ of $C$ on the planar lattice $\\mathbb {Z}\\times \\mathbb {Z}\\subset \\mathbb {R}^2$ in position $\\left(A(\\mathbf {x})-m, -m \\right) \\in \\mathbb {Z}\\times \\mathbb {Z}$ , label each $\\mathbb {Z}_2$ -generator $U^m \\cdot \\mathbf {x}$ of $C$ with its Maslov grading $M(\\mathbf {x})-2m\\in \\mathbb {Z}$ , connect two $\\mathbb {Z}_2$ -generators $U^n \\cdot \\mathbf {x}$ and $U^m \\cdot \\mathbf {y}$ with a directed arrow if in the differential of $U^n \\cdot \\mathbf {x}$ the coefficient of $U^m \\cdot \\mathbf {y}$ is non-zero.",
"In [16] Ozsváth and Szabó show how to associate to a knot $K \\subset S^3$ a knot type complex $CFK^\\infty (K)$ whose filtered chain homotopy type only depends on the isotopy class of $K$ .",
"For a concise introduction to the background material see [6]."
],
[
"Hom's invariance principle",
"Denote by $\\mathcal {CFK}$ the set of knot type complexes up to filtered chain homotopy.",
"Say that two knot type complexes are stably equivalent $C_1 \\sim C_2$ if there exist Alexander filtered, Maslov graded, acyclic chain complexes $A_1$ and $A_2$ such that $C_1 \\oplus A_1 \\simeq C_2 \\oplus A_2$ .",
"The quotient set $\\mathcal {CFK}/_\\sim $ has a natural group structure: the sum is given by tensor product, the class of zero is the one represented by the Floer chain complex of the unknot $CFK^\\infty (U)$ , and the inverse of the class of a complex $C$ is the one represented by its dual complex $\\text{Hom}(C,\\mathbb {Z}_2[U, U^{-1}])$ .",
"Theorem 2.1 (Hom [6]) The map $K \\mapsto CFK^\\infty (K)$ associating to a knot $K \\subset S^3$ its knot Floer complex descends to a group homomorphism $\\mathcal {C} \\rightarrow \\mathcal {CFK}/_\\sim $ .",
"Summarizing, in order to produce a concordance invariant $\\mathcal {C} \\rightarrow \\mathbb {Z}$ one only needs to produce a map $f : \\mathcal {CFK}\\rightarrow \\mathbb {Z}$ such that $f(C_* \\oplus A_*)=f(C_*)$ for every Alexander filtered, Maslov graded, acyclic chain complex $A_*$ ."
],
[
"Upsilon type invariants",
"Inspired by the exposition in [9] we use knot Floer homology to define some more concordance invariants.",
"We start with a definition.",
"Definition 3.1 A region of the plane $C \\subset \\mathbb {R}^2$ is said to be a south-west region if it is non-empty and $(\\overline{x}, \\overline{y}) \\in C \\Rightarrow \\lbrace (x,y) \\ | \\ x\\le \\overline{x}, y \\le \\overline{y}\\rbrace \\subseteq C$ .",
"Let $C$ be a south-west region of the plane.",
"For $t \\in \\mathbb {R}$ let $C_t=\\lbrace (x,y)\\ | \\ (x-t, y-t) \\in C\\rbrace $ denote the translate of $C$ in the direction of $v_t=(t,t)$ .",
"Given a knot type complex $K_*$ consider the map induced on $H_0$ by the inclusion $K_*(C_t) \\hookrightarrow K_*$ , where $K_*(C_t)$ denotes the subcomplex spanned by the generators of $K_*$ lying in $C_t$ .",
"Since $C_t \\subseteq C_{t^{\\prime }}$ for $t \\le t^{\\prime }$ , and $\\bigcup _{t \\in \\mathbb {R}} C_t= \\mathbb {R}^2$ , a cycle representing the generator of $H_0(K_*)=\\mathbb {Z}_2$ will eventually be contained in $K_*(C_t)$ .",
"Thus, for $t$ big enough the inclusion $H_0(K_*(C_t)) \\rightarrow H_0( K_*)$ is a surjective map.",
"Let $\\Upsilon ^C(K_*)$ be the minimum $t\\in \\mathbb {R}$ such that $K_*(C_t) \\hookrightarrow K_*$ induces a surjection on $H_0$ .",
"Here we are using the Maslov grading as homological grading so that $H_{2i}(K_*)=\\mathbb {Z}_2$ and zero otherwise.",
"Lemma 3.2 Suppose that $C$ is a south-west region.",
"If $K_*$ and $K^{\\prime }_*$ are two stably equivalent knot type complexes then $\\Upsilon ^C(K_*)=\\Upsilon ^C(K^{\\prime }_*)$ .",
"The surjectivity of the map induced in homology by the inclusion $K_*(C_t) \\hookrightarrow K_*$ is not infected if we sum an acyclic complex $A$ on the right and a subcomplex of the same acyclic on the left.",
"Corollary 3.3 Suppose that $C \\subset \\mathbb {R}^2$ is a south-west region.",
"Given a knot $K\\subseteq S^3$ set $\\Upsilon ^C(K)= \\Upsilon ^C(CFK^\\infty (K))$ .",
"Then $\\Upsilon ^C(K)$ is a concordance invariant.",
"$\\square $"
],
[
"The classical upsilon invariant",
"Choose the lower half-space $H_t = \\left\\lbrace \\frac{t}{2}\\cdot A+\\left(1-\\frac{t}{2} \\right) \\cdot j \\le 0 \\right\\rbrace $ as south-west region.",
"As $t$ ranges in $[0,2]$ we get a one-parameter family of invariants of knot type complexes $\\Upsilon _t(K_*)=\\Upsilon ^{H_t}(K_*)$ .",
"According to Corollary REF this provides a one-parameter family of knot concordance invariants.",
"More specifically, set $\\Upsilon _K(t)=-2 \\cdot \\Upsilon ^{H_t}(CFK^\\infty (K)) \\ .",
"$ In [9] Livingston proves that the invariant $\\Upsilon _K(t)$ agrees with the upsilon invariant defined by Ozsváth, Stipsicz and Szabó [24]."
],
[
"Regions for Rasmussen's $h_i$ invariants",
"For $s\\ge 0$ choose as south-west region $Q_s=\\lbrace A\\le s , j\\le 0\\rbrace $ .",
"This leads to a one-parameter family of knot concordance invariants $V_K(s)=-2\\Upsilon ^{Q_s}(K)$ .",
"These concordance invariants are equivalent to the one introduced by Rasmussen in [22].",
"We justify the equivalence by proving that the same relation with correction terms pointed out and discussed in [13] holds.",
"Proposition 3.4 Let $K \\subseteq S^3$ be a knot and $q\\ge 2g(K) -1$ be an integer.",
"Denote by $W_q(K)$ the $q$ -framed two-handle attachment along $K$ to $D^4$ , so that $S^3_q(K)= \\partial W_q(K)$ .",
"For any integer $m \\in [-q/2, q/2)$ let $\\mathfrak {s}_m\\in \\text{Spin}^c(S^3_q(K))$ denote the restriction to $S^3_q(K)$ of a $\\text{Spin}^c$ structure $\\mathfrak {t}_m$ on $W_q(K)$ such that $ \\mathinner {\\langle { c_1(\\mathfrak {s}), [\\widehat{F} ] }\\rangle }+q=2m$ , where $\\widehat{F} \\subset W_q(K) $ denotes a capped-off Seifert surface for $K$ .",
"Then $ d(S^3_q(K), \\mathfrak {s}_m)= \\frac{(q-2m)^2-q}{4q}+ V_K(m) \\ ,$ where $d$ denotes the Heegaard Floer correction term introduced in [19].",
"Suppose that $z_1, \\dots , z_k \\in CFK^\\infty (K)$ are the cycles with Maslov grading zero representing the generator of $H_0(CFK^\\infty (K))=\\mathbb {Z}_2$ .",
"If $Q_{s,t}$ denotes the translate of $Q_s=\\lbrace A\\le s , j\\le 0\\rbrace $ in the $(t,t)$ -direction then $Q_{s,t} =\\lbrace \\max (A-s,j)\\le t\\rbrace $ .",
"Thus, $V_K(s)= -2 \\cdot \\min _{i} \\max (A(z_i)-s, j(z_i))$ We now prove that the very same min-max formula can be used to compute the correction terms of the $q$ -framed surgery.",
"Let $q$ and $m$ be as above.",
"Remove a ball from $W_q(K)$ , turn the resulting two-handle cobordism upside-down, and change orientation in order to get a cobordism $X: S^3_q(K) \\rightarrow S^3$ .",
"According to [16], the map induced in homology by the inclusion $C\\lbrace \\max (A-m,j)\\le 0\\rbrace \\hookrightarrow C\\lbrace j \\le 0\\rbrace $ represents the map $F_X: HF^-(S^3_q(K), \\mathfrak {s}_m) \\rightarrow HF^-(S^3)$ induced by $X$ .",
"Since $\\text{gr}(F_X(\\xi ) ) - \\text{gr}(\\xi )= \\frac{c_1(\\mathfrak {s}_m)^2- 2 \\chi (X) - 3 \\sigma (X)}{4} \\ ,$ where $\\xi $ denotes the generator of the tower of $HF^-(S^3_q(K), \\mathfrak {s}_m)$ , we can conlude that $d(S^3_q(K), \\mathfrak {s}_m)= d+\\frac{(q-2m)^2-q}{4q} \\ , $ where $d$ denotes the Maslov grading of the generator of the tower of $H_*(C\\lbrace \\max (A-m,j)\\le 0\\rbrace ) \\simeq HF^-(S^3_q(K), \\mathfrak {s}_m)$ .",
"Since the inclusion $C\\lbrace \\max (A-m,j)\\le 0\\rbrace \\hookrightarrow C\\lbrace j \\le 0\\rbrace $ sends the generator of the tower of $HF^-(S^3_q(K), \\mathfrak {s}_m)$ to a $U^n$ -multiple of the one of $HF^-(S^3)\\simeq H_*(C\\lbrace j \\le 0\\rbrace )$ , if $z_1, \\dots , z_k \\in CFK^\\infty (K)$ denote the Maslov grading zero cycles for the generator of $H_*(CFK^\\infty (K))$ we have that $d= \\max _i M(U^{n_i}\\cdot z_i)= \\max _i M(U^{n_i} \\cdot z_k)- 2n_i= -2 \\min _i n_i$ , where $n_i$ is the minimum $n\\ge 0$ such that $U^n \\cdot z_i \\in C\\lbrace \\max (A-m,j)\\le 0\\rbrace $ .",
"Since $n_i=\\max (A(z_i)-m,j(z_i))$ this proves that $d=-2\\min _i \\max (A(z_i)-m ,j(z_i))=V_K(m)$ , and we are done."
],
[
"Estimates on the slice genus",
"Suppose that $C$ is a south-west region.",
"Associated to $C$ there is a height function $h_C(x)= \\min \\lbrace t\\in \\mathbb {R}\\text{ such that } (x, 0) \\in C_t \\rbrace \\ , $ where $C_t$ denotes as usual the translate of $C$ in the $v_t= (t,t)$ direction.",
"The height function $h_C$ relates the upsilon invariant of the region $C$ to the slice genus.",
"Theorem 3.5 Let $C$ be a south-west region.",
"Given a knot $K \\subset S^3$ the inequality $\\max \\lbrace \\Upsilon ^C(K), \\Upsilon ^C(-K)\\rbrace \\le h_C(g_4(K))$ holds, where $g_4(K)$ denotes the slice genus of $K$ .",
"First of all notice that $h_C(x)$ is a monotone increasing function: since $C$ is a south-west region, $(x-\\delta , 0 ) \\in C_{\\Upsilon ^C(K)}$ for $\\delta >0 $ .",
"Thus, $h_C(x-\\delta )\\le h_C(x)$ .",
"Let $\\nu ^+= \\min _i\\lbrace V_K(i)=0 \\rbrace $ .",
"From the definition of the height function $h_C(x)$ and the fact that $C$ is a south-west region one immediately conclude that $\\lbrace A \\le \\nu ^+ , j \\le 0\\rbrace \\subseteq C_{h_C(\\nu ^+)}$ .",
"The fact that $V_K(\\nu ^+)=0$ ensures that the south-west region $\\lbrace A \\le \\nu ^+ , j \\le 0\\rbrace $ contains a cycle generating $H_0(CFK^\\infty (K))$ and consequently (because of the inclusion) that so does the translate $C_{h_C(\\nu ^+)}$ .",
"This proves that $\\Upsilon ^C(K) \\le h_C(\\nu ^+)$ .",
"On the other hand, according to Rasmussen [22] $\\nu ^+ \\le g_4(K)$ , thus $\\Upsilon ^C(K) \\le h_C(\\nu ^+) \\le h_C(g_4(K))$ .",
"By doing the same argument for $-K$ instead of $K$ we get that $\\Upsilon ^C(-K) \\le h_C(g_4(-K))=h_C(g_4(K))$ , and we are done.",
"Example 3.6 If we choose $C=\\lbrace t/2A+(1-t/2)j\\le 0\\rbrace $ as in the classical upsilon invariant (Section REF ) one has $h_C(x)=t/2 \\cdot x$ .",
"In this case Equation REF leads to the inequality $|\\Upsilon _K(t)|=2 \\cdot \\max \\lbrace \\Upsilon ^C(K),\\Upsilon _K^C(-K)\\rbrace \\le 2 h_C(g_4(K)) =t g_4(K)$ , where the first identity is due to the identity $\\Upsilon _K^C(-K)=-\\Upsilon ^C(K)$ (which is not valid for any $C$ ).",
"Compare this with [24]."
],
[
"Secondary invariants",
"Roughly speaking, upsilon type invariants measure how far one needs to travel north-east in the $(A,j)$ plane in order to see a cycle generating $H_0(CFK^\\infty )$ appear.",
"As suggested by Kim and Livingston in [7], other concordance invariants could be obtained my measuring how far one should go in order to see realized some expected homologies.",
"Suppose that two south-west regions $C^+$ and $C^-$ are given.",
"Given a knot type complex $K_*$ one can consider the maps induced in homology by the inclusions $K_*(C^+_t) \\hookrightarrow K_*$ and $K_*(C^+_t) \\hookrightarrow K_*$ (here we are using again the notation of the beginning of Section ).",
"For $\\gamma _\\pm =\\Upsilon ^{C_\\pm }(K_*)$ one gets surjections $H_0(K_*(C^+_{\\gamma _+})) \\rightarrow H_0(K_*)$ and $H_0(K_*(C^-_{\\gamma _-})) \\rightarrow H_0(K_*)$ .",
"Denote by $\\mathcal {Z}^+$ and $\\mathcal {Z}^-$ the set of cycles in $K_*(C^+_{\\gamma _+})$ and $K_*(C^-_{\\gamma _-})$ respectively projecting on the generator of $H_0(K_*)$ .",
"Suppose now that a third south-west region $C$ has been fixed.",
"Since $H_0(K_*)=\\mathbb {Z}_2$ , for $t \\in \\mathbb {R}$ large enough there will be a 1-chain $\\beta \\in K_1$ realizing a homology between a 0-cycle in $\\mathcal {Z}^+$ and one in $\\mathcal {Z}^-$ .",
"We define $\\Upsilon _{C^\\pm , C}(K_*)$ as the minimum $t \\in \\mathbb {R}$ for which a cycle in $\\mathcal {Z}^+$ represents inside $K_*(C^+_{\\gamma _+})+ K_*(C^-_{\\gamma _-}) + K_*(C_t)$ the same homology class of a cycle in $\\mathcal {Z}^-$ .",
"We set $\\Upsilon _{C^\\pm , C}(K_*)=-\\infty $ in the eventuality that $\\mathcal {Z}^+ \\cap \\mathcal {Z}^- \\ne \\emptyset $ .",
"Lemma 4.1 Suppose that $C^+$ , $C^-$ and $C$ are given south-west regions.",
"If $K_*$ and $K^{\\prime }_*$ are two stably equivalent knot type complexes then $\\Upsilon _{C^\\pm , C}(K_*)=\\Upsilon _{C^\\pm , C}(K^{\\prime }_*)$ .",
"Suppose that $K^{\\prime }_*=K_*\\oplus A$ is obtained from $K_*$ by adding an acyclic complex $A$ .",
"Set $\\gamma _\\pm =\\Upsilon ^{C_\\pm }(K_*)=\\Upsilon ^{C_\\pm }(K^{\\prime }_*)$ , and denote by $\\mathcal {Z}^\\pm (K_*)$ and $\\mathcal {Z}^\\pm (K^{\\prime }_*)$ the set of cycles projecting to the generator through $H_0(K_*(C^\\pm _{\\gamma _\\pm })) \\rightarrow H_0(K_*)$ and $H_0(K^{\\prime }_*(C^\\pm _{\\gamma _\\pm })) \\rightarrow H_0(K^{\\prime }_*)$ respectively.",
"We prove that $\\Upsilon _{C^\\pm , C}(K_*)=\\Upsilon _{C^\\pm , C}(K^{\\prime }_*)$ by proving the two inequalities.",
"Suppose by contradiction that there exists $t< \\Upsilon _{C^\\pm ,C}(K^*)$ for which a cycle $z^+ \\in \\mathcal {Z}^+(K^{\\prime }_*)$ gets identified with a cycle in $z^-\\in \\mathcal {Z}^-(K^{\\prime }_*)$ in $K^{\\prime }_*(C_t) +K^{\\prime }_*(C^+_{\\gamma _+})+ K^{\\prime }_*(C^-_{\\gamma _-})$ .",
"Pick a 1-chain $\\beta ^{\\prime }_t\\in K^{\\prime }_*(C_t)+K^{\\prime }_*(C^+_{\\gamma _+})+ K^{\\prime }_*(C^-_{\\gamma _-})$ such that $z^+- z^-=\\partial \\beta ^{\\prime }_t$ , and write $\\beta ^{\\prime }_t= \\beta _t+ a$ with $\\beta _t \\in K_*(C_t)+K_*(C^+_{\\gamma _+})+ K_*(C^-_{\\gamma _-})$ and $a \\in A$ .",
"Notice that $z^+=z^+_K+a^+$ and $z^-=z^-_K+a^- $ , for some $a^+, a^- \\in A$ , $z^+_K\\in \\mathcal {Z}^+(K_*)$ , and $z^-_K \\in \\mathcal {Z}^-(K_*)$ .",
"By rewriting the relation $z^+- z^-=\\partial \\beta ^{\\prime }_t$ we get that $ (z^+_K -z_K^- -\\partial \\beta _t) + (a^+-a^- - \\partial a)=0$ , from where we can conclude that $z^+_K -z_K^-=\\partial \\beta _t$ .",
"This contradicts the fact that $\\Upsilon _{C^\\pm ,C}(K_*)$ is the minimum $t$ for which such an homology exists, and proves that $\\Upsilon _{C^\\pm ,C}(K_*)\\le \\Upsilon _{C^\\pm ,C}(K^{\\prime }_*)$ .",
"The reverse inequality has a similar proof.",
"Corollary 4.2 For a knot $K \\subset S^3$ set $\\Upsilon _{C^\\pm , C}(K)=\\Upsilon _{C^\\pm , C}(CFK^\\infty (K))$ .",
"Then $\\Upsilon _{C^\\pm , C}(K)$ defines a knot concordance invariant.",
"$\\square $"
],
[
"Breaking points",
"Summarizing, given south-west regions $C^+$ , $C^-$ and $C \\subset \\mathbb {R}^2$ we get a map $\\Upsilon _{C^\\pm , C}: \\mathcal {CFK}/_\\sim \\rightarrow [ -\\infty , + \\infty )$ .",
"In [7] Kim and Livingston produce south-west regions for which the condition $\\mathcal {Z}^+ \\cap \\mathcal {Z}^- = \\emptyset $ is guaranteed.",
"Lemma 4.3 (Kim-Livingston) For $t \\in [0,2]$ let $\\Upsilon _t: \\mathcal {CFK}/_\\sim \\rightarrow \\mathbb {R}$ denotes the stable equivalence invariant associated to the lower half-space $H_t$ of Section REF .",
"Suppose that $K_*$ is a knot type complex such that $\\Upsilon _t(K_*)$ as function of $t \\in [0,2]$ is non smooth at $t=t^*$ .",
"Furthermore, suppose that the derivative of $\\Upsilon _t(K_*)$ at $t=t^*$ has a positive jump, meaning that $ \\Delta \\Upsilon ^{\\prime }_{t}(K_*)= \\lim _{\\epsilon \\rightarrow 0} \\left( \\Upsilon ^{\\prime }_{t+\\epsilon }(K_*) - \\Upsilon ^{\\prime }_{t-\\epsilon }(K_*) \\right)$ is positive at $t=t^*$ .",
"Then for $\\delta >0$ small enough $C^-= H_{t^*-\\delta }$ and $\\ C^+= H_{t^*+\\delta }$ give two south-west regions such that $\\mathcal {Z}^+ \\cap \\mathcal {Z}^- = \\emptyset $ .",
"$\\square $ We say that the upsilon function $\\Upsilon _t(K_*)$ of a knot type complex $K_*$ has a breaking point at $t=t^*$ if for a small perturbation $ \\delta >0$ , $C^-= H_{t^*-\\delta }$ and $C^+= H_{t^*+\\delta }$ are two south-west regions such that $\\mathcal {Z}^+ \\cap \\mathcal {Z}^- = \\emptyset $ .",
"In what follows, the cycles in $\\mathcal {Z}^+$ and $\\mathcal {Z}^-$ are referred to as at the positive and the negative exceptional cycles of the breaking point.",
"Lemma REF says that the singularities of $\\Upsilon _t(K_*)$ (points where $\\Upsilon _t(K_*)$ is non-smooth) at which $\\Delta \\Upsilon ^{\\prime }_{t}(K_*)>0$ are in fact breaking points.",
"In the notation of Proposition REF set $\\Upsilon _{C,t}^{(2)}(K_*) = -2 \\cdot ( \\Upsilon _{H_{t \\pm \\delta }, C}(K_*) - \\Upsilon _t(K_*)) \\ .$ for $\\delta >0$ small enough.",
"This provides a one-parameter family of knot concordance invariants $ \\Upsilon _{C,t}^{(2)}(K)= \\Upsilon _{C,t}^{(2)}(CFK^\\infty (K))$ .",
"Notice that the invariant $\\Upsilon _{K,t}^{(2)}(s)=\\Upsilon _{H_s,t}^{(2)}(K)$ is exactly the secondary upsilon invariant introduced by Kim and Livingston in [7]."
],
[
"Floer thin knots",
"A knot $K \\subset S^3$ is called Floer thin if its knot Floer homology groups $\\widehat{HFK}_{*,*}(K)$ are concentrated on a diagonal, meaning that $\\widehat{HFK}_{i,j}(K)=0$ if $i-j\\ne \\delta $ for a suitable constant $\\delta $ .",
"Examples of Floer thin knots are alternating and quasi-alternating knots [14], [12] (in these cases $\\delta =-\\sigma /2$ , where $\\sigma $ denotes the knot signature).",
"In [21] Petkova shows that for a Floer thin knot the chain homotopy type of $CFK^\\infty (K)$ can be completely reconstructed from its Ozsváth-Szabó tau invariant $\\tau =\\tau (K)$ and its Alexander polynomial $\\Delta _K=a_0+ \\sum _{s>0} a_s (T^s+T^{-s})$ .",
"More precisely we have that: $CFK^\\infty (K)$ has exactly $|a_s|$ generators with $A=s$ and $j=0$ , $CFK^\\infty (K)=(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}]) \\oplus (\\bigoplus _i Q_i \\otimes \\mathbb {Z}_2[U,U^{-1}])$ , where $S_\\tau $ is a staircase complex, and the $Q_i$ 's are square complexes as the one shown in Figure REF .",
"Notice that $A=\\bigoplus _i Q_i \\otimes \\mathbb {Z}_2[U,U^{-1}]$ is acyclic.",
"Consequently, up to acyclics, for a Floer thin knot $K$ we have that $CFK^\\infty (K)= S_{\\tau (K)} \\otimes \\mathbb {Z}_2[U,U^{-1}]$ .",
"Figure: The square complex QQ (left), and the staircase complex S τ S_\\tau (τ≤0\\tau \\le 0 on the center, τ>0\\tau >0 on the right)."
],
[
"Three-parameter upsilon invariants of thin knots",
"We show how to compute some upsilon type invariants in the case of Floer thin knots.",
"Choose as south-west region $ C=\\left\\lbrace \\frac{t}{2}\\cdot A+\\left(1-\\frac{t}{2} \\right) \\cdot j\\le 0 \\right\\rbrace \\cup \\left\\lbrace \\frac{s}{2}\\cdot A+\\left(1-\\frac{s}{2} \\right) \\cdot j \\le \\phantom{\\frac{t}{2}} \\hspace{-8.5359pt} q\\right\\rbrace \\ .$ As the parameters $s,t \\in [0,1]$ and $q \\in \\mathbb {R}$ vary, the concordance invariant $\\Upsilon ^C$ gives rise to a three-parameter family of concordance invariants $\\Upsilon _K(t,s,q)$ collapsing to the classical upsilon invariant when $t=s$ and $q=0$ .",
"Let us compute $\\Upsilon ^C(K)= \\Upsilon ^C(CFK^\\infty (K))$ for a Floer thin knot $K$ .",
"Suppose first that $\\tau = \\tau (K)$ is positive.",
"Since $CFK^\\infty (K)= S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}] \\oplus A$ with $S_\\tau $ staircase shaped as in Figure REF and $A$ acyclic, Hom's principle shows that $\\Upsilon ^C(CFK^\\infty (K))=\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ .",
"Let $C_\\gamma $ denote the translate of $C$ in the $v_\\gamma =(\\gamma , \\gamma )$ direction.",
"The south-west region $C_\\gamma $ contains a generator of $H_0(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}]) = \\mathbb {Z}_2$ as soon as it contains one of the $x_i$ generators of Figure REF .",
"Thus, $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ is the minimum $\\gamma $ such that $\\frac{t}{2} \\cdot A(x_i)+\\left(1-\\frac{t}{2} \\right) \\cdot j(x_i)\\le \\gamma \\ \\ \\text{ or } \\ \\ \\frac{s}{2}\\cdot A(x_i)+\\left(1-\\frac{s}{2} \\right) \\cdot j(x_i)-q \\le \\gamma $ for at least one of the $x_i$ generators, hence $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ is computed by the expression: $ \\min _i \\min \\left\\lbrace \\frac{t}{2} A(x_i)+\\left(1-\\frac{t}{2} \\right) j(x_i), \\frac{s}{2} A(x_i)+\\left(1-\\frac{s}{2} \\right) j(x_i)-q\\right\\rbrace \\ .$ Plugging in $A(x_{i})=\\tau -i$ and $j(x_{i})=i$ we get $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])= \\min _i \\min \\lbrace (1-t)i + \\tau , (1-s)i + \\tau -q \\rbrace $ from where the identity $ \\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])= \\min \\left\\lbrace \\frac{t}{2} \\tau ,(1-s) \\left\\lceil \\frac{\\tau }{2}+\\frac{q}{t-s}\\right\\rceil +\\frac{s\\tau -2q}{2} \\right\\rbrace $ follows for $t\\ne s$ .",
"For $t=s$ one can easily see that $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])=\\frac{t}{2} \\tau - \\max \\lbrace 0, q \\rbrace .$ If $\\tau <0$ then the situation is somehow easier: there is only one 0-cycle generating $H_0(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])= \\mathbb {Z}_2$ , namely $z=\\sum _i x_i$ .",
"Thus, in this case $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ is computed by the following expression: $ \\max _i \\min \\left\\lbrace \\frac{t}{2} A(x_i)+\\left(1-\\frac{t}{2} \\right) j(x_i), \\frac{s}{2} A(x_i)+\\left(1-\\frac{s}{2} \\right) j(x_i)-q\\right\\rbrace \\ .$ By substituting the values of $A(x_{i})$ and $j(x_{i})$ we get $ \\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])= \\min \\left\\lbrace (1-s) \\left\\lfloor \\frac{\\tau }{2}+\\frac{q}{t-s}\\right\\rfloor +\\frac{s\\tau -2q}{2}, \\frac{2-t}{2} \\tau \\right\\rbrace $ if $t\\ne s$ .",
"If $t=s$ we have the identity $\\Upsilon ^C(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])=\\frac{2-t}{2} \\tau - \\min \\lbrace 0, q \\rbrace .$ As an immediate corollary of this discussion we get the following proposition.",
"Proposition 5.1 Suppose that $K \\subset S^3$ is a Floer thin knot.",
"Then, $ \\Upsilon _K(t) = -\\tau (K) \\cdot (1-|t-1|) \\ .",
"$ $\\square $ Notice that the Ozsváth-Stipsicz-Szabó upsilon function of a Floer thin knot $K \\subset S^3$ has only one singularity at $t=1$ where it actually has a breaking point if $\\Delta \\Upsilon ^{\\prime }_{t=1}(K)=2\\tau (K)>0$ .",
"We now compute the Kim-Livingston secondary invariant of these singularities.",
"Proposition 5.2 Suppose that $K \\subset S^3$ is a Floer thin knot.",
"Then $ \\Upsilon ^{(2)}_{K,1}(s) =(1-\\tau (K))\\cdot |1-s| -1 $ if $\\tau (K)>0$ , and $\\Upsilon ^{(2)}_{K,1}(s) =-\\infty $ otherwise.",
"In the notation of Section REF , we would like to compute $\\Upsilon _{C,1}^{(2)}(K_*)$ for $C=\\lbrace s/2A+ (1-s/2)j \\le 0\\rbrace $ and $K_*=S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}]$ .",
"If $\\tau >0$ , the upsilon function $\\Upsilon _t(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ has a breaking point at $t=1$ .",
"The exceptional sets $\\mathcal {Z}^+$ and $\\mathcal {Z}^-$ of this breaking point are easy to identify: there is only one positive and one negative exceptional cycle, namely $z^+=x_0$ and $z^-= x_{\\tau }$ (see Figure again).",
"A quick inspection of the same Figure reveals that a 1-chain realising a homology between $z^+$ and $z^-$ is given by $b=\\sum _i y_i$ .",
"Notice that there is exactly one such chain since $H_1(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])=0$ , and $\\partial $ vanishes on chains with even Maslov grading.",
"Thus, $ \\Upsilon _{C,1}^{(2)}(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])= -2\\left( \\max _i \\left( \\frac{s}{2} A(y_i) +\\left(1-\\frac{s}{2} \\right) j(y_i) \\right) - \\frac{\\tau }{2} \\right) .",
"$ Plugging in $A(y_i)=\\tau -i$ and $j(y_i)=i+1$ the claimed identity can be deduced by algebraic manipulation.",
"If $\\tau \\le 0$ , there is only one 0-cycle generating $H_0(S_\\tau \\otimes \\mathbb {Z}_2[U,U^{-1}])$ , namely $z= \\sum _i x_i$ .",
"Thus $\\mathcal {Z}^+ \\cap \\mathcal {Z}^-= \\lbrace z\\rbrace $ and we conclude that $\\Upsilon ^{(2)}_{K,1}(s)=-\\infty $ ."
],
[
"$L$ -space knots",
"Another interesting class of knots is provided by $L$ -space knots.",
"Recall that a rational homology sphere $Y$ is an $L$ -space if $\\widehat{HF}(Y, \\mathfrak {s})= \\mathbb {Z}_2$ in every $\\text{Spin}^c$ structure.",
"This happens for example in the case of a lens space $Y=L(p,q)$ whence the name.",
"A knot $K \\subset S^3$ is said to be an $L$ -space knot if it has a positive surgery $S^3_p(K)$ that is an $L$ -space.",
"Basic examples of $L$ -space knots are positive torus knots.",
"The homotopy type of the master complex of an $L$ -space knot can be reconstructed from its Alexander polynomial.",
"More precisely suppose that $K \\subset S^3$ is a genus $g$ $L$ -space knot.",
"According to [18] its Alexander polynomial can be written in the form $\\Delta _K(t) = 1-t^{\\alpha _1}+ \\dots -t^{\\alpha _{2k-1}}+t^{\\alpha _{2k}}$ , with $0=\\alpha _0<\\alpha _1< \\dots <\\alpha _{2k}=2g$ .",
"Staring from the sequence $a_i=\\alpha _{i}-\\alpha _{i-1}$ recording the jumps between consecutive exponents of the monomials appearing in the Alexander polynomial, construct a chain complex $S_*(K)= S_*(a_1, \\dots , a_{2k} )$ as follows.",
"Set $S_*(a_1, \\dots , a_{2k} )= \\mathbb {Z}_2\\lbrace x_0, \\dots , x_k, y_0, \\dots , y_{k-1} \\rbrace \\otimes \\mathbb {Z}_2[U,U^{-1}]$ , and consider the differential ${\\left\\lbrace \\begin{array}{ll}\\ \\partial x_i= 0 \\ \\ \\ i=0, \\dots , k \\\\\\ \\partial y_i=x_i+ x_{i+1} \\ \\ \\ i=0, \\dots , k-1\\end{array}\\right.}",
"\\ .$ Define ${\\left\\lbrace \\begin{array}{ll}\\ A(x_i)=n_i \\\\\\ j(x_i)= m_i\\\\\\ M(x_i)=0\\end{array}\\right.}",
"\\ \\ \\ \\ \\text{ and } \\ \\ \\ \\ \\ \\ \\ {\\left\\lbrace \\begin{array}{ll}\\ A(y_i)=n_i \\\\\\ j(y_i)=m_{i+1} \\\\\\ M(y_i)=1\\end{array}\\right.",
"}$ where $ \\ {\\left\\lbrace \\begin{array}{ll}\\ n_i=g-\\sum _{j=0}^{i}a_{2j} \\\\\\ n_0=0\\end{array}\\right.",
"}\\ \\ \\ \\ \\ \\ {\\left\\lbrace \\begin{array}{ll}\\ m_i=\\sum _{j=1}^i a_{2j-1} \\\\\\ m_0=0\\end{array}\\right.}",
"\\ ,$ and coherently extend these gradings to $\\mathbb {Z}_2\\lbrace x_0, \\dots , x_k, y_0, \\dots , y_{k-1} \\rbrace \\otimes \\mathbb {Z}_2[U,U^{-1}]$ so that multiplication by $U$ drops the Maslov grading $M$ by $-2$ , and the Alexander filtration $A$ as well as the algebraic filtration $j$ by $-1$ .",
"In [20] Peters proves that there is a chain homotopy equivalence $CFK^\\infty (K) \\simeq S_*(K)$ ."
],
[
"Kim-Livingston secondary invariant of $L$ -space knots",
"Let us compute the upsilon invariant $\\Upsilon _t(S_*)$ of a staircase complex $S_*= S_*(a_1, \\dots , a_{2k})$ .",
"Since the lower half-space $t/2\\cdot A+\\left(1-t/2 \\right) \\cdot j\\le \\gamma $ contains a cycle generating $H_0(S_*)$ as soon as it contains one of the $x_i$ generators, we have that $ \\Upsilon _t(S_*)= \\min _i \\left\\lbrace \\frac{t}{2}n_i+\\left(1-\\frac{t}{2} \\right) m_i\\right\\rbrace \\ .$ Thus, for an $L$ -space knot $K \\subset S^3$ one has $\\Upsilon _K(t)= -2 \\cdot \\min _i \\left\\lbrace \\left(n_i-m_i \\right)\\frac{t}{2} +m_i\\right\\rbrace =- \\min _i \\lbrace \\alpha _i t + m_i \\rbrace \\,$ as already pointed out by Ozsváth, Stipsicz and Szabó in [24].",
"From Equation REF it is clear where the upsilon function $\\Upsilon _t(S_*)$ of a staircase complex $S_*$ has its breaking points.",
"A parameter $t$ is a singularity for $\\Upsilon _t(S_*)$ if and only if the $\\min $ on the left hand side of (REF ) is realised by more than one index $i$ .",
"These singularities are breaking points since at these points $\\Delta \\Upsilon ^{\\prime }_t>0$ .",
"Notice that there are no other breaking points since at a regular parameter $t$ the half space $t/2\\cdot A+\\left(1-t/2 \\right) \\cdot j \\le \\Upsilon _t(S_*)$ contains (on its boundary line) exactly one $x_i$ generator.",
"Proposition 6.1 Let $S_*= S_*(a_1, \\dots , a_{2k})$ be a staircase complex.",
"Suppose that $t$ is a breaking point of $\\Upsilon _t(S_*)$ , then $ \\Upsilon _{S_*, t}^{(2)}(s)= -2\\left( \\max _{i_-\\le j < i_-} \\left\\lbrace \\frac{s}{2}n_j+\\left(1-\\frac{s}{2} \\right) m_{j+1}\\right\\rbrace - \\Upsilon _t(S_*) \\right) \\ , $ where $i_-$ and $i_+$ denote respectively the minimum and the maximum index realizing the minimum in Equation REF .",
"If $t$ is a breaking point of $\\Upsilon _t(S_*)$ then the half-space $ \\frac{t}{2}\\cdot A+\\left(1-\\frac{t}{2} \\right) \\cdot j \\le \\Upsilon _t(S_*) $ contains (actually on its boundary line) exactly those $x_i$ generators which have index $i \\in \\lbrace 0, \\dots , k\\rbrace $ realizing the minimum in the expression of Equation REF .",
"The exceptional sets $\\mathcal {Z}^+$ and $\\mathcal {Z}^-$ of such a singularity both contain exactly one 0-cycle: $z^+=x_{i_+}$ and $z_-=x_{i_-}$ respectively.",
"Notice that since $H_1(S_*)=0$ , there is only one 1-chain realizing a homology between these cycles, namely $\\beta =\\sum _{j=i_-}^{i_+-1} y_j$ .",
"Thus, in the notation of Equation (REF ) of Section REF , we have that $\\Upsilon _{H_{t \\pm \\delta }, H_s}(S_*) = \\max _{i_-\\le j < i_+} \\left\\lbrace \\frac{s}{2}n_j+\\left(1-\\frac{s}{2} \\right) m_{j+1}\\right\\rbrace $ from where the formula follows."
],
[
"A connected sum formula",
"One of the fundamental properties of the Ozsváth-Stipsicz-Szabó upsilon invariant is its additivity property $ \\Upsilon _t(A_* \\otimes B_*) = \\Upsilon _t(A_*) + \\Upsilon _t (B_*) \\ ,$ turning $\\Upsilon _t$ into a group homomorphism from $\\mathcal {CFK} /_\\sim $ to the group of piecewise linear functions $[0,2] \\rightarrow \\mathbb {R}$ .",
"General upsilon type invariants and their secondary counterparts do not enjoy this property.",
"In this section we prove a connected sum formula for the Kim-Livingston secondary invariant of staircase complexes.",
"Theorem 6.2 Let $A_*=S_*(a_1, \\dots , a_{2n})$ and $B_*=S_*(b_1, \\dots , b_{2m})$ be staircase complexes.",
"Suppose that $\\Upsilon _t(A_*)$ has a breaking point at a point $t=s$ where $\\Upsilon _t(B_*)$ is smooth.",
"Then, $ \\Upsilon ^{(2)}_{A_* \\otimes B_*,s}(s)= \\Upsilon ^{(2)}_{A_*,s}(s) \\ .$ Denote by $x_0, \\dots , x_{n}$ and $z_0, \\dots , z_m$ the Maslov grading zero generators of the staircases of $A_*$ and $B_*$ respectively.",
"Similarly, denote by $y_0, \\dots , y_{n-1}$ and $w_0, \\dots , w_{m-1}$ their Maslov grading one generators.",
"The fact that $\\Upsilon _t( B_*)$ is smooth at $t=s$ guarantees that the half-space $ \\frac{s}{2}\\cdot A+\\left(1-\\frac{s}{2} \\right) \\cdot j \\le \\Upsilon _s(B_*) $ only contains (actually on its boundary line) one 0-cycle $z=z_r$ generating $H_0(B_*)$ .",
"Since $A_*$ is a staircase complex, the set of its exceptional cycles $\\mathcal {Z}^+$ and $\\mathcal {Z}^-$ at $t=s$ both include exactly one 0-cycle.",
"Denote those cycles by $x^+=x_k$ and $x^-=x_h$ respectively.",
"In this notation, the set of exceptional cycles of $A_* \\otimes B_*$ at $t=s$ are given by $\\mathcal {Z}^+=\\lbrace x_k \\otimes z_r\\rbrace $ and $\\mathcal {Z}^-=\\lbrace x_h \\otimes z_r\\rbrace $ .",
"Given a chain $\\xi = \\sum _i \\xi _i$ with $\\xi _1, \\dots , \\xi _n$ homogenous with respect to both the Alexander and the algebraic grading, set $ E_s(\\xi )=\\max _i \\left\\lbrace \\frac{s}{2}\\cdot A(\\xi _i)+\\left(1-\\frac{s}{2} \\right) \\cdot j(\\xi _i) \\right\\rbrace \\ .$ In this notation $E_s(x) \\le \\gamma $ if and only if the chain $\\xi $ is contained in the subcomplex of the lower half-space $s/2\\cdot A+\\left(1-s/2 \\right) \\le \\gamma $ .",
"It is easy to find a 1-chain realizing a homology between $x_k \\otimes z_r$ and $ x_h \\otimes z_r$ : $\\partial \\left( \\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r\\right)=\\sum _{\\ell =a}^{b-1} \\partial y_\\ell \\otimes z_r =\\sum _{\\ell =a}^{b-1} (x_\\ell +x_{\\ell +1} )\\otimes z_r=x_k\\otimes z_r - x_h\\otimes z_r \\ .",
"$ If we prove that between the 1-cycles realizing a homology between $x_k \\otimes z_r$ and $ x_h \\otimes z_r$ this is the one with minimal $E_s$ then we conclude that $\\Upsilon ^{(2)}_{A_* \\otimes B_*,s}(s) &=-2 \\left( E_s \\left(\\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r \\right) - \\Upsilon _t(A_* \\otimes B_*) \\right) \\\\&= -2 \\left( E_s \\left(\\sum _{\\ell =a}^{b-1} y_\\ell \\right) +E_s( z_r) - \\Upsilon _t(A_*) - \\Upsilon _t (B_*) \\right)\\\\&=-2 \\left( E_s \\left(\\sum _{\\ell =a}^{b-1} y_\\ell \\right) - \\Upsilon _t(A_*) \\right) =\\Upsilon ^{(2)}_{A_*,s}(s)$ and we are done.",
"Let us prove that $\\beta =\\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r$ is a cycle minimizing $E_s(\\beta )$ in the class of 1-cycles realizing homologies between $x_k \\otimes z_r$ and $ x_h \\otimes z_r$ .",
"From the fact that for a staircase complex $\\partial (x)=0$ for those $x$ 's with homogenous even Maslov degree, one can conclude that any 1-chain realizing an homology between $x_k \\otimes z_r$ and $ x_h \\otimes z_r$ differs from $\\beta $ by the boundary of an element in $A_1 \\otimes B_1$ .",
"In other words, such a 1-chain should be of the form $ \\partial \\left( \\sum _{i,j} \\epsilon _{ij} y_i \\otimes w_j \\right) + \\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r $ for some coefficients $\\epsilon _{ij}\\in \\mathbb {Z}_2$ .",
"Obviously, we have that $E_s \\left( \\partial \\left( \\sum _{i,j} \\epsilon _{ij} y_i \\otimes w_j \\right) + \\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r \\right) \\ge E_s\\left( \\sum _{\\ell =a}^{b-1} y_\\ell \\otimes z_r \\right)$ provided that none of the generators $y_a\\otimes z_r, y_{a+1}\\otimes z_r, \\dots , y_{b-1}\\otimes z_r $ appears as a component of $ \\partial \\left( \\sum _{i,j} \\epsilon _{ij} y_i \\otimes w_j \\right) =\\sum _{i,j} \\epsilon _{ij} \\partial y_i \\otimes w_j + \\sum _{i,j} \\epsilon _{ij} y_i \\otimes \\partial w_j \\ .$ On the other hand if so happens for some $y_i \\otimes z_r$ , after cancellation the summand $\\sum _{j} \\epsilon _{ij} y_i \\otimes \\partial w_j= \\sum _j \\epsilon _{ij} y_i \\otimes z_{j} + \\epsilon _{ij} y_i \\otimes z_{j+1} $ has a component of the form $y_i \\otimes z_\\mu $ for some $\\mu \\ne k$ .",
"Thus, since $E_s( y_i \\otimes z_\\mu ) =E_s(y_i)+ E_s(z_\\mu )> E_s(y_i)+ \\Upsilon _s(B_*)=E_s(y_i)+ E_s(z_r)=E_s(y_i \\otimes z_r)$ , also in this case the inequality in (REF ) holds, and we are done.",
"According to Feller and Krcatovitch [2] the Ozsváth-Stipsicz-Szabó upsilon function $\\Upsilon _{p,q}(t)$ of the $(p,q)$ torus knot can be computed recursively by means of the formula $ \\Upsilon _{p,q}(t)=\\Upsilon _{p-q, q}(t)+ \\Upsilon _{q+1,q}(t)$ .",
"Thus, $\\Upsilon _K(t)=\\Upsilon _{8,5}(t)-\\Upsilon _{6,5}(t)- \\Upsilon _{4,3}(t)=\\Upsilon _{6,5}(t)+ \\Upsilon _{4,3}(t)+ \\Upsilon _{3,2}(t)-\\Upsilon _{6,5}(t)- \\Upsilon _{4,3}(t)=\\Upsilon _{3,2}(t)$ proving that $K$ is an upsilon-alternating knot.",
"Now suppose by contradiction that there exists a Floer thin knot $J$ such that $T_{6,5}\\# T_{4,3} \\sim T_{8,5} \\# J$ .",
"The upsilon function of the torus knot $T_{6,5}$ has its singularities at $t=2/5,4/5,6/5, 8/5$ while the one of $J$ has its only singularity at $t=1$ .",
"The upsilon function of the torus knot $T_{4,3}$ and $T_{8,5}$ on the other hand both have a singularity at $t=2/3$ .",
"Thus, as consequence of Theorem REF we have that $\\Upsilon _{T_{4,3}, 2/3}^{(2)}\\left(\\frac{2}{3}\\right)=\\Upsilon _{T_{6,5}\\# T_{4,3}, 2/3}^{(2)}\\left(\\frac{2}{3}\\right)=\\Upsilon _{T_{8,5}\\# J, 2/3}^{(2)}\\left(\\frac{2}{3}\\right)=\\Upsilon _{T_{8,5}, 2/3}^{(2)}\\left(\\frac{2}{3}\\right) \\ .$ We claim that $\\Upsilon _{T_{4,3}, 2/3}^{(2)}\\left(2/3\\right)\\ne \\Upsilon _{T_{8,5}, 2/3}^{(2)}\\left( 2/3\\right)$ .",
"In fact, by Proposition REF we have that $\\Upsilon _{T_{4,3}, 2/3}^{(2)}\\left(2/3\\right) =-4/3$ and $\\Upsilon _{T_{8,5}, 2/3}^{(2)}\\left(2/3\\right)=- 20/3$ ."
],
[
"Algebraic Knots",
"Suppose that $Z \\subset 2$ is a planar complex curve given by the equation $f(x,y)=0$ .",
"Recall that a point $p \\in Z$ is said to be regular if the partial derivatives $ \\partial f/\\partial x$ and $\\partial f/\\partial y$ do not both vanish at $p$ .",
"A point that is not regular is said to be singular.",
"In what follows by an isolated plane curve singularity $(Z,p)$ we mean a planar complex curve $Z$ with an isolated singularity at $p \\in Z$ .",
"Without loss of generality we can always suppose $p$ to be the origin of 2.",
"Let $(Z, 0)$ be an isolated plane curve singularity.",
"A small sphere $S^3_\\epsilon (0)$ centred at the origin intersects $Z$ transversally in a link $K= S^3_\\epsilon (0) \\cap Z$ .",
"This is the link of the plane curve singularity and in a neighbourhood of the origin $Z$ looks like a cone over it.",
"If the link $K \\subset S^3$ is actually a knot we say that $(Z,0)$ is cuspidal.",
"Knots arising from this construction are called algebraic knots.",
"Naturally attached to a plane curve singularity $(Z,0)$ there is an arithmetic object capturing informations about the complex geometry of its germ.",
"Given an analytic parametrization $\\varphi (z)$ of $Z$ around 0 consider the pull-back homomorphism $\\varphi ^*: [x,y]] \\rightarrow [z]]$ defined by $g\\mapsto g \\circ \\varphi $ .",
"Set $ S= \\big \\lbrace s \\in \\mathbb {Z}_{\\ge 0} \\ | \\ g(\\varphi (z))= z^sh(z)\\text{ for some } g\\in [x,y]], \\ h \\in [z]] \\text{ with } h(0) \\ne 0 \\big \\rbrace \\ .$ It is esay to see that $S$ is a semigroup, meaning that $0 \\in S$ and if $a, b \\in S$ so is $a+b$ .",
"The semigroup of a cuspidal singularity $(Z,0)$ is related to its knot $K$ via the Alexander polynomial $ \\Delta _K(t)= \\sum _{s \\in S } t^s- t^{s+1}$ Notice that this is a finite sum since the semigroup of a plane curve singularity eventually covers all the positive integers.",
"Any cuspidal plane curve singularity $(Z,0)$ has a parametrization of the form $x=z^a$ , $y=z^{q_1}+ \\dots + z^{q_n}$ for some positive integers $q_1<q_2< \\dots <q_n$ .",
"Such a representation is unique if we further assume $gcd(a, q_1, \\dots , q_i)$ to not divide $q_{i+1}$ and $gcd(a, q_1, \\dots , q_n)=1$ .",
"The sequence $(a; q_1, \\dots , q_n)$ is the Puiseaux characteristic sequence of the cuspidal singularity $(Z,0)$ , and the number $a$ is its Puiseaux exponent.",
"It is a fundemantal fact of the theory of plane curve singularities [25] that starting from the Puiseaux characteristic sequence of a cuspidal singularity one can reconstruct both its semigroup and the topology of its link.",
"Theorem 7.1 Let $(Z,0)$ be a cuspidal plane curve singularity with Puiseaux characteristic sequence $(a;q_1, \\dots , q_n)$ .",
"Set $D_i=gcd(a,q_1, \\dots , q_i)$ , $s_1=q_1$ and $ s_i= \\frac{aq_1+D_1(q_2-q_1)+ \\dots + D_{i-1}(q_i-q_{i-1}) }{D_{i-1}} $ for $i=0, \\dots , n-1$ .",
"Then the link $K$ of $(Z,0)$ is the $(n-1)$ -fold iterated cable of the $(a/D_1,q_1/D_1)$ torus knot with cabling coefficients $(D_{i-1}/D_i, s_{i-1}/D_i)$ , $i=2, \\dots , n$ .",
"Furthermore, the semigroup of $(Z,0)$ is generated by $\\lbrace a, s_1, \\dots , s_n\\rbrace $ .",
"$\\square $ From the viewpoint of Heegaard Floer theory, algebraic knots are interesting since they provide a good source of examples of $L$ -space knots [4].",
"Because of Equation REF , the staircase of an algebraic knot can be recovered from the semigroup of its singularity.",
"More precisely, suppose that $(Z,0)$ is a plane curve singularity giving rise to a genus $g$ algebraic knot $K$ .",
"The semigroup $S$ of $(Z,0)$ determines a colouring of $\\lbrace 0, \\dots , 2g-1\\rbrace $ : color by red the numbers in $S \\cap \\lbrace 0, \\dots , 2g-1\\rbrace $ and by blue the ones in its complement $(\\mathbb {Z}\\setminus S) \\cap \\lbrace 0, \\dots , 2g-1\\rbrace $ .",
"By counting the gaps between blue and red numbers as suggested by Figure REF we get two sequences of numbers $r_1, \\dots r_g$ and $b_1, \\dots , b_g$ .",
"As a consequence of the general recipe discussed at the beginning of Section one can see that $CFK^\\infty (K)\\simeq S_*(r_1,b_1, \\dots , r_g, b_g)$ .",
"Figure: The semigroup of the plane curve singularity x 5 +y 3 =0x^5+y^3=0 is generated by 5 and 3.",
"Its link is the torus knot T 5,3 T_{5,3}.",
"The associated staircase can be computed from the colouring above by counting the gaps between blue and red numbers.",
"In this case r 1 =1,r 2 =1,r 3 =2r_1=1, r_2=1, r_3=2, b 1 =2,b 2 =1,b 3 =1b_1=2, b_2=1, b_3=1, and CFK ∞ (T 5,3 )=S * (red1,blue2,red1,blue1,red2,blue1)CFK^\\infty (T_{5,3})= S_*(\\leavevmode {\\color {red}1},\\leavevmode {\\color {blue}2},\\leavevmode {\\color {red}1},\\leavevmode {\\color {blue}1}, \\leavevmode {\\color {red}2}, \\leavevmode {\\color {blue}1})."
],
[
"$L$ -space pretzel knots",
"We now proceed to the proof of Theorem REF .",
"Suppose that $C$ is a south-west region.",
"For every $x \\in \\mathbb {R}$ we can consider the truncated south west region $C_x= C \\cap \\lbrace A \\le x\\rbrace $ .",
"This leads to a one-parameter family of upsilon type invariants $\\Upsilon ^{C_x}(K_*)$ .",
"Since $C_x \\subseteq C$ we have that $\\Upsilon ^{C_x}(K_*) \\le \\Upsilon ^C(K_*)$ .",
"Furthermore for $x$ large enough $\\Upsilon ^{C_x}(K_*)= \\Upsilon ^C(K_*)$ .",
"Set $\\eta _C(K_*)= \\min \\left\\lbrace x \\ | \\ \\Upsilon ^{C_x}(K_*) = \\Upsilon ^C(K_*) \\right\\rbrace \\ .$ Obviously, this is an invariant of stable equivalence.",
"For a knot $K \\subset S^3$ denote by $\\eta _C(K)= \\eta _C(CFK^\\infty (K))$ the associated knot concordance invariance.",
"Lemma 7.2 Suppose that $(Z,0)$ is a cupsidal plane curve singularity with Puiseaux sequence $(a;q_1, \\dots , q_n)$ .",
"Denote by $K$ the algebraic knot associated to $(Z,0)$ and by $S$ its semigroup.",
"Let $n(S)$ be the maximum among the integers $n \\ge 0$ such that $S \\cap \\mathbb {Z}_{\\le na} = \\lbrace 0, a, 2a, \\dots , na \\rbrace \\ .$ Choose $C= \\lbrace 1/a \\cdot A +(1 - 1/a) \\cdot j \\le 0 \\rbrace $ .",
"Then, $ \\eta _C(K)=\\left(1 - \\frac{1}{a} \\right) \\tau (K)- (a-1) \\ n(S) \\ .",
"$ Let $g$ be the genus of $K$ .",
"Colour by red and blue the numbers in $\\lbrace 0, 1, \\dots , 2g-1 \\rbrace $ as specified by $(R=S \\cap \\lbrace 0, 1, \\dots , 2g-1 \\rbrace , B= (\\mathbb {Z}\\setminus S) \\cap \\lbrace 0, 1, \\dots , 2g-1 \\rbrace )$ and by recording the gaps between red and blue numbers form the sequences $r_1, \\dots , r_g$ and $b_1, \\dots , b_g$ suggested by Figure REF .",
"By the definition of $n(S)$ we have that $r_1=\\dots = r_n=1$ , $b_1= \\dots = b_n= a-1$ , and $1\\le b_i<a-1$ for $i=n+1 , \\dots , g$ .",
"Thus, $CFK^\\infty (K) \\simeq S_*(1, a-1, \\dots , 1, a-1, r_{n+1}, b_{n+1}, \\dots , r_g, b_g)$ where the pair $(1,a-1)$ repeats $n=n(S)$ times.",
"Denote by $x_0, \\dots , x_g$ the Maslov grading zero generators of the staircase for $CFK^\\infty (K)$ .",
"Similarly, denote by $y_1, \\dots , y_g$ its Maslov grading one generators.",
"Consider the half-space $C_\\gamma $ of the $(A,j)$ plane defined by $1/a \\cdot A +(1 - 1/a) \\cdot j \\le \\gamma $ .",
"We claim that for $\\gamma = \\Upsilon ^C(K)=- 2 \\cdot \\Upsilon _{2/a}(K)=\\tau (K)/a$ the only Maslov grading zero generators contained in $C_\\gamma $ are $x_1, \\dots , x_n$ .",
"For this purpose define $E(x_i)=1/a \\cdot A(x_i) +(1 - 1/a) \\cdot j(x_i) \\ .",
"$ Obviously $E(x_i) \\le \\gamma $ if and only if $x_i$ is in $C_\\gamma $ .",
"A quick computation reveals that $E(x_i)=\\gamma $ for $i=1, \\dots , n$ and consequently that $x_1, \\dots , x_n$ are actually contained in the boundary line of $C_\\gamma $ .",
"We claim that $E(x_i) > \\gamma $ for any $i >n$ .",
"For $k \\ge 1$ we have that $E(x_{n+k}) &= E(x_n)- \\frac{1}{a} \\sum _{i=1}^k b_{n+i} + \\left( 1 - \\frac{1}{a} \\right) \\sum _{i=1}^k r_{n+i} \\\\&= \\gamma + \\frac{1}{a} \\left( -\\sum _{i=1}^k b_{n+i} + \\left( a - 1 \\right) \\sum _{i=1}^k r_{n+i} \\right) \\ .$ On the other hand, $ \\left( a - 1 \\right) \\sum _{i=1}^k r_{n+i} \\ge k \\cdot (a-1) > \\sum _{i=1}^k b_{n+i} \\ , $ proving that $E(x_{n+k}) \\ge \\gamma + 1/a \\cdot (\\text{something positive})>\\gamma $ , and we are done.",
"Since the only generators with Maslov grading zero in $C_\\gamma $ are $x_1, \\dots , x_n$ we conclude that $C_\\gamma \\cap \\lbrace A \\le x+ \\gamma \\rbrace $ contains a cycle generating $H_0(CFK^\\infty (K))$ provided $x + \\gamma \\ge \\min \\lbrace A(x_1) , \\dots , A(x_n) \\rbrace =g-n(a-1)$ .",
"Thus, $\\eta _C(K)+\\gamma =g-n(a-1)$ .",
"Plugging in $\\gamma = 1/a \\cdot \\tau (K)$ , $g= \\tau (K)$ , and $n=n(S)$ the claim follows.",
"Using the skein relation at a negative crossing we find that the symmetrized Alexander polynomial $\\Delta _q(t)$ of a $P(-2,3,q)$ pretzel knot ($q \\ge 7$ odd) is given by $\\Delta _q (t)= (t-1+t^{-1})\\Delta _{2,q}(t) + (t^{\\frac{1}{2}}-t^{-\\frac{1}{2}})\\Delta _{2,q+3}(t) $ , where $\\Delta _{2,p}(t)$ denotes the Alexander polynomial of the $(2,p)$ torus link.",
"Since $P(-2,3,q)$ is an $L$ -space knot, this leads to the conclusion that $CFK^\\infty (P(-2,3,q))\\simeq S_*(1,2,1, 1, \\dots , 1, 1,2,1) \\ , $ from where one computes $\\tau (P(-2,3, q))=(q+3)/2$ , and $\\eta _C(P(-2,3,q))= (q-3)/3$ for $C= \\lbrace 1/3 \\cdot A +2/3 \\cdot j \\le 0 \\rbrace $ .",
"Notice that $\\Upsilon _{P(-2,3,q)}(t)=-2 \\cdot \\Upsilon _t(S_*(1,2,1,\\dots ,1,2,1))$ has its only singularities at $t=2/3, 1, 4/3$ .",
"Suppose by contradiction that for some $q \\ge 7$ odd the pretzel knot $P(-2, 3, q)$ is concordant to a sum of algebraic knots $K_1 \\# \\dots \\# K_m$ .",
"For $i=1, \\dots , m$ let $(Z_i,0)$ be a plane curve singularity with knot $K_i$ .",
"Denote by $S_i$ the semigroup of $(Z_i,0)$ and by $a_i$ its Puiseaux exponent.",
"According to Wang [26] the Ozsváth-Stipsicz-Szabó upsilon invariant $\\Upsilon _K(t)$ of an algebraic knot has its first singularity at $t=2/a$ where $a$ denotes its Puiseaux exponent.",
"Since $ \\Upsilon _{P(-2,3,q)}(t)= \\Upsilon _{\\#_i K_i}(t) = \\sum _i \\Upsilon _{K_i}(t)$ , and $\\Delta \\Upsilon ^{\\prime }_{K_i}(t) \\ge 0$ , this leads to the conclusion that either $a_i=3$ , or $a_i=2$ .",
"Notice that, as consequence of Theorem REF , if $K=((T_{p,q})_{p_1, q_1} \\dots )_{p_n, q_n}$ is the knot of a cuspidal plane curve singularity $(Z,0)$ with Puiseaux exponent $a$ then $a=p\\cdot (p_1 \\dots p_n)$ .",
"Since for every $i$ the Puiseaux exponent of $K_i$ is either 3 or 2, we conclude that $K_i$ is either a $(3,p)$ torus knot, or a $(2,k)$ torus knot.",
"An argument along the line of [9] reveals that $\\eta _C(K_1 \\# \\dots \\# K_m)= \\eta _C(K_1) + \\dots +\\eta _C(K_m)$ .",
"A direct computation shows that for the $(2,2k+1)$ torus knot $\\eta _C=2/3 \\cdot k$ .",
"Thus, as consequence of Lemma REF we have that $ \\eta _C(P(-2,3, q))= \\eta _C(K_1 \\# \\dots \\# K_m)= \\frac{2}{3}\\sum _{i} \\tau (K_i)-2 \\sum _j n(S_j)\\ , $ where the second sum is extended only to the $(3,p)$ torus knot summands.",
"Plugging in $\\sum _{i} \\tau (K_i)= \\tau (K_1 \\# \\dots \\# K_m)= \\tau (P(-2,3, q))=(q+3)/2$ and $\\eta _C(P(-2,3,q))= (q-3)/3$ we get that $-2=-2\\sum _j n(S_j)$ and consequently that $K_1 \\# \\dots \\# K_m$ is either of the form $T_{3,4}\\# J$ , or $T_{3,5} \\#J$ where $J$ is a sum of $(2,n)$ torus knots and hence alternating.",
"This leads to a contradiction since a knot of this form has $\\tau = - \\sigma /2$ while for a pretzel knot of the form $P(-2, 3, q)$ we have $(q+3)/2 =\\tau \\ne -\\sigma /2=(q+1)/2$ ."
],
[
"Acknowledgements",
"The author would like to thanks András Stipsicz, András Némethi, and Paolo Aceto.",
"The author was partially supported by the NKFIH grant K112735."
]
] | 1709.01594 |
[
[
"Characterizing spin transport: detection of spin accumulation via\n magnetic stray field"
],
[
"Abstract Spin transport in electric conductors is largely determined by two material parameters - spin diffusion length and spin Hall angle.",
"In metals, these are typically determined indirectly by probing magnetoresistance in magnet/metal heterostructures, assuming knowledge of the interfacial properties.",
"We suggest profiling the charge current induced spin Hall spin accumulation in metals, via detection of the magnetic stray field generated by the associated static magnetization, as a direct means of determining spin transport parameters.",
"We evaluate the spatial profile of the stray field as well as the Oersted field generated by the charge current.",
"We thus demonstrate that such a charge current induced spin accumulation is well within the detection limit of contemporary technology.",
"Measuring the stray fields may enable direct access to spin-related properties of metals paving the way for a better and consistent understanding of spin transport therein."
],
[
"Introduction",
"The field of spintronics investigates the interplay between the spin (magnetic) and charge degrees of freedom in a solid-state system [1], [2].",
"Initial experimental techniques have focused on the electronic or optical detection of the magnetization, where the latter is controlled or initialized via an external magnetic field.",
"It has subsequently been realized that the magnetization direction can also be manipulated via spin-polarized charge currents utilizing the phenomenon of spin-transfer torques (STT) [3], [4], [5].",
"The physics underlying STT may be understood with reference to a simple model in which the magnetization results from the localized d-electrons while the mobile s-electrons mediate transport.",
"Due to an exchange coupling between the s and d electrons, the mobile s-electrons experience a torque exerted by the magnetization.",
"Reciprocally, the magnetization experiences an equal and opposite torque.",
"This technique has successfully been employed for magnetization switching and domain wall motion, and forms the basis for a number of devices such as racetrack [6] and STT-magnetoresistive random access memories [7].",
"While the mechanism for spin-polarization of current relies on the conductor magnetization in the above mentioned devices, pure spin currents have also been generated and detected in non-magnetic materials, with spin-orbit interaction enabling interconversion between charge and spin currents [8], [9], [10].",
"Although there are a number of microscopic mechanisms contributing to this interconversion [11], a simple picture is provided by asymmetric scattering from impurities.",
"An electron experiences, due to spin-orbit interaction, a spin-dependent impurity potential and scattering probability in the transverse direction (see Fig.",
"REF ).",
"Thus, a charge flow leads to a spin current in the transverse direction and vice-versa.",
"This phenomenon has been termed spin Hall effect (SHE) and the conversion efficiency is quantified by the so-called spin Hall angle ($\\theta $ ).",
"Since the spin current cannot escape the material, a spin accumulation builds up close to the conductor edges so that the diffusive backflow compensates the SHE current at the edge.",
"This spin accumulation decays exponentially over a distance, called spin diffusion length ($\\lambda $ ), from the interface and is well described within a diffusive transport theory [12].",
"In heterostructures comprising a magnet (F) and a non-magnetic metal (N)Non-magnetic here shall denote metals that do not show long-range magnetic order such as ferro- or ferrimagnetism., the transport and magnetization electrons may be spatially separated.",
"One mechanism for STT in these systems is via the SHE mediated accumulation of electron spins at the interface, when a charge current is driven in N. In addition to altering or moving the magnetic textures, STT also enables injection of pure spin currents into the magnetic material.",
"This interplay between electronic and magnonic spin currents [14] is exemplified by phenomena like spin pumping [15], [16], electrical spin injection [17], spin Seebeck effect [18], [19], [20], and spin Hall magnetoresistance (SMR) [21], [22], [23], [24].",
"Different methods for spin accumulation detection are necessary in different materials.",
"In semiconductors, direct spatially resolved optical detection has been achieved via Kerr rotation measurements [25], [26] and recently, Stamm et al.",
"reported (non-spatially resolved) detection of the spin accumulation in metal thin films [27].",
"The latter turns out to be challenging in metals due to their small electromagnetic field penetration depths and the resultig Kerr angles of the order of 10e-9rad.",
"Typical techniques employed in metals rely therefore on examining an effect of the spin accumulation and constitute an indirect measurement.",
"For example, the N thickness dependence of SMR in an F$|$ N heterostructure allows inferring the spin Hall angle, but the approach relies on accurate knowledge about the interface and the interplay between the material systems [28], [29].",
"These interfacial properties are not easily determined and vary in a wide range [14].",
"Figure: Schematic illustration of SHE mediated spin separation and accumulation in a metallic strip.",
"The conductor is assumed long with width z 0 z_0 and thickness y 0 y_0.",
"A charge current density j e j_{\\mathrm {e}} flows along the xx-direction.",
"Due to spin Hall effect (SHE), the conduction electrons are scattered in different directions depending on their spin polarization: Up-spins (red; polarized along 𝐲 ^\\mathbf {\\hat{y}}), e. g., are deflected in the -z-z-direction, while down-spins (blue; polarized along 𝐲 ^\\mathbf {\\hat{y}}) are deflected in the +z+z-direction.",
"This results in an accumulation of spin-polarized electrons at the surfaces of the strip.",
"The resulting magnetization close to the edges and the magnetic fields induced by these moments are illustrated, respectively, by colored arrows and black lines.",
"The field lines indicate the net magnetic stray field around the strip, i. e. the sum of the stray and Oersted fields.Here we suggest to detect SHE mediated spin accumulation, and thus characterize spin transport parameters, in a metallic strip by measuring the magnetic `stray' field resulting from the non-equilibrium magnetization associated with the spin-polarized electrons.",
"While the net magnetic moment in the system vanishes, a finite magnetization is generated near the boundaries of the metal.",
"We evaluate the ensuing stray field analytically within a simplified model as well as numerically, and find that the field is well within the detection range of the state-of-the-art sensing techniques such as NV centers[30], [31], [32], magnetic force microscopy[33], [34], scanning SQUID magnetometers[35], [36], or muon spin resonance [37].",
"We further show that the magnetic stray field of spin accumulation may exceed and can be disentangled from the Oersted field arising due to the current flow, that generates the spin accumulation via SHE, using their distinct spatial profiles.",
"The proposed method thus enables a direct access to important spin transport properties - spin diffusion length $\\lambda $ and the spin-Hall angle $\\theta $ in metals - while circumventing the difficulties associated with F$|$ N interfaces.",
"The paper is organized as follows: In Section , we derive the spin accumulation profile in the metallic strip (Fig.",
"REF ) and obtain an analytic expression for the magnetic stray field at large (compared to $\\lambda $ ) distances from the surface.",
"Section discusses the spatial magnetic field distribution evaluated using the approximate analytic expression as well as numerically.",
"In Section we evaluate the Oersted field distribution generated by the charge current in the strip.",
"We discuss the field distribution for a multilayer system in Sec.",
"and demonstrate that the Oersted field can be reduced significantly by allowing a counterflow of current in an adjacent layer.",
"We conclude with discussion of experimental issues and a short summary of our results in Section ."
],
[
"Spin accumulation and magnetic stray field",
"We consider a long metallic strip with width $z_0$ and thickness $y_0$ which supports a charge current density of $j_{\\mathrm {e}}$ driven by an electric field $E_0 \\mathbf {\\hat{x}}$ along its length (Fig.",
"REF ).",
"The general current response in a non-magnetic conductor including SHE reads [12]: $\\left( \\begin{array}{c}\\mathbf {j}_{e} \\\\ \\mathbf {j}_{sx} \\\\ \\mathbf {j}_{sy} \\\\ \\mathbf {j}_{sz}\\end{array} \\right)= \\sigma _N\\left( \\begin{array}{cccc}1 & \\theta \\mathbf {\\hat{x}} \\times & \\theta \\mathbf {\\hat{y}} \\times & \\theta \\mathbf {\\hat{z}} \\times \\\\\\theta \\mathbf {\\hat{x}} \\times & 1 & 0 & 0 \\\\\\theta \\mathbf {\\hat{y}} \\times & 0 & 1 & 0 \\\\\\theta \\mathbf {\\hat{z}} \\times & 0 & 0 & 1\\end{array} \\right)\\left( \\begin{array}{c}\\mathbf {E} \\\\ - \\nabla \\mu _{sx} / {2e} \\\\ - \\nabla \\mu _{sy} / {2e} \\\\ - \\nabla \\mu _{sz} / {2e}\\end{array} \\right),$ where $\\sigma _N$ is the conductivity, $e (>0)$ is the electronic charge, $\\theta $ is the spin Hall angle, $\\mathbf {j}_{e}$ is the charge current density, $\\mathbf {E}$ is the applied electric field, and $\\mathbf {j}_{si}$ is the spin current density polarized in the $i$ -direction ($i=x,y,z$ ).",
"$\\mu _{si}$ is the corresponding spin chemical potential, which obeys the diffusion equation: [38], [39], [40] $\\nabla ^2 {\\mu }_{si} = \\frac{{\\mu }_{si}}{\\lambda ^2},$ with the spin diffusion length $\\lambda $ .",
"The boundary conditions for (REF ) are vanishing spin current flow normal to all interfaces, which in the chosen coordinate system read: $j_{si}^{y} (y = \\pm y_0/2) = 0 \\quad \\text{and}\\quad j_{si}^{z} (z = \\pm z_0/2) = 0.",
"$ Here the superscript denotes the spatial direction of spin current flow while the subscript represents the spin polarization direction.",
"The diffusion equation (REF ) with the boundary conditions [Eq.",
"(REF )] admits the solution [41]: $\\mu _{sy}(\\mathbf {r}) & = - 2 e \\theta \\lambda E_0 \\frac{\\sinh {(z/\\lambda )}}{\\cosh {(z_0/(2\\lambda ))}}, \\\\\\mu _{sz}(\\mathbf {r}) & = 2 e \\theta \\lambda E_0 \\frac{\\sinh {(y/\\lambda )}}{\\cosh {(y_0/(2\\lambda ))}}, $ where $\\mathbf {r}$ is the position vector.",
"As detailed in Appendix , the spin accumulation density is related to the spin chemical potential by [40] $n_{si} = \\frac{\\sigma _N}{2 e^2 D} \\mu _{si},$ where $D=\\lambda ^2/\\tau $ denotes the electron diffusion constant [42], [12] and $\\tau $ is the spin-flip time.",
"The $n_{si}$ are defined as $n_{si}=n_{\\uparrow }-n_{\\downarrow }$ , where the subscript arrows $\\uparrow , \\downarrow $ denote the up- and down-polarized spins for the respective quantization axes.",
"The spin accumulation is thus spatially localized to a region within $\\sim \\lambda $ from the surfaces.",
"While the exact evaluation of the magnetic field arising from this charge-current induced magnetization requires numerics, analytical expressions can be obtained in the limit of $\\lambda \\ll r_p$ , where $\\mathbf {r}_p$ is the position vector of the point at which the magnetic field is measured.",
"We refer to this as the `far-field limit'.",
"Relegating the details to Appendix , the magnetic field distribution $\\mathbf {B}(\\mathbf {r}_p)$ in this limit is evaluated as: $\\mathbf {B}(\\mathbf {r}_p) & = \\frac{\\mu _0\\gamma \\hbar }{8\\pi } \\frac{j_e \\theta \\tau }{e} \\left(\\frac{1}{\\cosh \\left(\\frac{y_0}{2\\lambda }\\right)} - \\frac{1}{\\cosh \\left(\\frac{z_0}{2\\lambda }\\right)}\\right)\\mathbf {F}(y_0,z_0;\\mathbf {r}_p), \\\\\\mathbf {F}(y_0,z_0;\\mathbf {r}_p) & = \\sum _{\\sigma _{1},\\sigma _{2} = \\pm 1} \\frac{2(y_p-\\sigma _1 y_0/2)}{(y_p-\\sigma _1 y_0/2)^2+(z_p- \\sigma _2 z_0/2)^2} \\mathbf {\\hat{y}} + \\frac{2(z_p- \\sigma _2 z_0/2)}{(y_p- \\sigma _1 y_0/2)^2+(z_p- \\sigma _2 z_0/2)^2} \\mathbf {\\hat{z}}, $ where $\\mu _0$ is the permeability of free space and $\\gamma $ is the gyromagnetic ratio in the metal.",
"From the expression above, we note that a high aspect ratio leads to larger stray field.",
"Thus it is desirable to have the metal in the shape of a film."
],
[
"Magnetic stray field: spatial profile ",
"We next compute the spatial distribution of the stray field originating from spin accumulation for the example of a platinum (Pt) conductor.",
"The material parameters employed [28] are spin Hall angle $\\theta =0.08$ , electric conductivity $\\sigma _{\\mathrm {N}}={9.52e6}{A/Vm}$ , spin diffusion length $\\lambda ={4}{nm}$ , spin flip time $\\tau ={60}{ps}$ This value is calculated from Ref.",
"[11], Table I and Ref.",
"11 therein.",
"and $\\gamma \\hbar =\\mu _{\\text{B}}={9.27e-24}{J/T}$ .",
"For the geometric dimensions of the Pt strip we choose $y_0={2}{nm}$ and $z_0={30}{nm}$ and we assume a current density $j_{\\mathrm {e}}={1e10}{A/m^2}$ .",
"Figure: Schematic illustration of the spin accumulation in the platinum strip, shown together with the calculated spin accumulation as a function of yy resp.",
"zz.With these material parameters, we calculate the spin accumulation at the surfaces $|n_{sy}(z=\\pm z_0/2)|={7.8e25}{m^{-3}}$ and $|n_{sz}(y=\\pm y_0/2)|={1.9e25}{m^{-3}}$ .",
"This corresponds to a net spin polarisation of about 0.1 percent present at the interface Here, we have compared the calculated $n_{si}$ to the experimentally determined free electron density in platinum thin films, $n={1.6e28}{m^{-3}}$ (see Ref. fischermean1980)..",
"As evident from Eqs.",
"(REF ) and (), the spin polarisation decays exponentially with decay length $\\lambda $ into the body of the metal, as shown in Fig.",
"REF .",
"Figure: Magnetic stray field profile 𝐁(𝐫 p )\\mathbf {B}(\\mathbf {r}_{\\mathrm {p}}) of spin accumulation in the conducting strip evaluated (a) and (b) numerically as well as (c) analytically using Eq.",
"().",
"The white arrows indicate the magnetic field direction, the color encodes its magnitude, where white regions indicate fields above 200T{200}{T}.",
"The transparent (solid) gray rectangle depicts the cross-section of the metal strip for the numerical (analytical) evaluation.",
"The pink solid line represents the 20T{20}{T} contour line.",
"Panels (b) and (c) show a zoom-in around the top-right edge of the strip to compare the numerical and analytical model.The corresponding spatial distribution of the magnetic stray field calculated numerically (see Appendix) is plotted in Fig.",
"REF a.",
"Here, the gray transparent box indicates the conductor cross-section.",
"The stray field diverges at the edges of the strip, exceeding ${20}{T}$ within a radius of about $d={5}{nm}$ (Fig.",
"REF a).",
"The stray field calculated using Eq.",
"(REF ) matches the numerical solution very well at large distances (Fig.",
"REF c).",
"Near the conducting strip, however, the approximation (REF ) leads to significant errors.",
"In the far-field limit, the stray field decays $\\sim 1/r_{\\mathrm {p}}^3$ ."
],
[
"Oersted field: spatial profile ",
"Relegating the evaluation details to Appendix , we discuss the magnetic field distribution of the Oersted field $\\mathbf {B}_{\\mathrm {oer}}(\\mathbf {r}_{\\mathrm {p}})$ created by the charge current flow in the conductor.",
"Figure REF shows the spatial distribution of the Oersted field around the conductor.",
"It has its maximum of about ${16}{T}$ at the left and right edge of the strip.",
"In the far-field limit, the Oersted field decays proportional to $1/r_{\\mathrm {p}}$ as expected for the far-field.",
"Thus, at large distances the Oersted field dominates the stray field.",
"This is also illustrated in Fig.",
"REF , where the ratio $|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|$ is plotted as a function of the sensor position $\\mathbf {r}_{\\mathrm {p}}$ including white solid line indicating $|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|=1$ .",
"Nevertheless, the spatial dependence of the Oersted field significantly differs from that of the magnetic stray field of spin accumulation.",
"Thus, using a spatially resolved magnetic field sensing technique should allow to disentangle the SHE induced stray field from the Oersted field.",
"Figure: Oersted field 𝐁 oer (𝐫 p )\\mathbf {B}_{\\mathrm {oer}} (\\mathbf {r}_{\\mathrm {p}}) as a function of the sensor position 𝐫 p \\mathbf {r}_{\\mathrm {p}}.",
"Panel (b) depicts a zoom-in of the upper right edge and panel (c) shows the total magnetic field |𝐁 tot |=|𝐁+𝐁 oer ||\\mathbf {B}_{\\mathrm {tot}}|=|\\mathbf {B}+\\mathbf {B_{\\text{oer}}}|.Figure: a.",
"|𝐁|/|𝐁 oer ||\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}| as a function of y p y_p and z p z_p.",
"The white solid line represents the |𝐁|/|𝐁 oer |=1|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|=1 contour line indicating that the spin accumulation induced stray field exceeds the oersted field significantly.",
"Areas, where |𝐁|/|𝐁 oer ||\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}| exceeds 5 are displayed in white.",
"The gray (semi-transparent) rectangle depicts the cross-section of the metal strip.",
"b. Close-up of the edge region of the strip.",
"c. |𝐁||\\mathbf {B}| and |𝐁 oer ||\\mathbf {B_\\mathrm {oer}}| as a function of dd for the sensor position depicted in Fig. .",
"The solid red (black) line corresponds to the full numerical (analytical, i.e.",
"()) computation of |𝐁||\\mathbf {B}|, while the blue line depicts |𝐁 oer ||\\mathbf {B_{\\text{oer}}}|.",
"We find |𝐁|/|𝐁 oer |>1|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|>1 for d≲6nmd\\lesssim {6}{nm}."
],
[
"Trilayer geometry ",
"In order to suppress the contribution of the Oersted field to the total magnetic field, we suggest a trilayer sample geometry where the strip consists of two conducting layers with a thin insulating layer (thickness $d_{\\text{ins}}$ ) in between.",
"We consider the upper layer (thickness $y_0$ ) to have a large spin Hall angle $\\theta $ , while the spin Hall angle of the lower conducting layer (thickness $y^{\\prime }_0$ ) vanishes.",
"In the following we discuss the situation, where current flows through both conducting layers with equal magnitude but opposite signs.",
"In the near field, the trilayer geometry reduces the Oersted field contribution.",
"As we assume the spin Hall angle in the bottom conducting layer to be zero, the stray field of the top layer is not affected by the bottom layer.",
"As a consequence, the ratio $B/B_{\\text{oer}}$ can be increased significantly.",
"Figure: a. Oersted field 𝐁 oer,TL \\mathbf {B_{\\text{oer,TL}}} as a function of y p y_p and z p z_p for the proposed trilayer sample.",
"The semi-tranparent rectangles depict the cross-sections of the two metal strips.",
"b.",
"Total magnetic field 𝐁 tot =𝐁+𝐁 oer,TL \\mathbf {B_{\\text{tot}}}= \\mathbf {B}+\\mathbf {B_{\\text{oer,TL}}} close to the edge of the upper conductive strip.",
"c. Oersted field of the same region for comparison.For a quantitative analysis, we calculate both the stray field and the Oersted field around the trilayer geometry as a function of the sensor position $\\mathbf {r}_{\\mathrm {p}}$ .",
"We here set $y_0=y_0^{\\prime }={2}{nm}$ , $d_{\\text{ins}}={2}{nm}$ and $z_0={30}{nm}$ $z_0$ can be chosen large compared to $y_0$ without significantly decreasing the stray field!",
"and leave the material parameters unchanged.",
"Figure REF shows the calculated Oersted field for this trilayer geometry.",
"Compared to the above discussed single-layer geometry (see Fig.",
"REF ), we observe a significant suppression of the Oersted field.",
"The ratio $B/B_{\\text{oer,TL}}$ , plotted in Fig.",
"REF a, shows maxima around the edges of the top strip where the stray field clearly dominates the Oersted field.",
"In particular, we find that the ratio of stray field and Oersted field, $B/B_{\\text{oer}}$ , is $5.5$ at $d={5}{nm}$ and $3.4$ at $d={10}{nm}$ .",
"Thus the contribution of the spin accumulation to the total magnetic field around the conductor is easily detectable and quantifiable in the presented geometry.",
"Table REF lists the $y$ -components of magnetic field $B_y$ and magnetic field gradient $\\partial B_y/\\partial y$ for a sample-sensor distance of $d={5}{nm}$ and $d={20}{nm}$ (cf.",
"Fig REF ).",
"The $d$ -dependence of stray and Oersted fields is depicted in Fig.",
"REF b.",
"We find that the Oersted farfield around the proposed trilayer sample decays proportional to $1/r_{\\mathrm {p}}^2$ , compared to $1/r_{\\mathrm {p}}$ for the Oersted field of a single conducting layer.",
"Besides, fig.",
"REF b shows the $1/r_{\\mathrm {p}}^3$ -dependence of the stray field.",
"As a consequence, in trilayer geometry, up to $d={100}{nm}$ away from the edges, the magnetic stray field exceeds the Oersted field of the two conducting layers.",
"Figure: a. and c. Field ratio |𝐁|/|𝐁 oer,TL ||\\mathbf {B}|/|\\mathbf {B_{\\text{oer,TL}}}| as a function of y p y_p and z p z_p for the proposed trilayer sample.",
"The white solid line represents the |𝐁|/|𝐁 oer |=1|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|=1 contour line.",
"Areas, where |𝐁|/|𝐁 oer |>10|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|>10 are also shaded in white.",
"The gray (semi-transparent) rectangle depicts the cross-section of the metal strip.",
"b.",
"|𝐁||\\mathbf {B}|, |𝐁 oer,TL ||\\mathbf {B_{\\text{oer,TL}}}| and |𝐁 oer ||\\mathbf {B_{\\text{oer}}}| as a function of dd for the sensor position depicted in Fig. .c.",
"|𝐁||\\mathbf {B}| and |𝐁 oer ||\\mathbf {B_\\mathrm {oer}}| as a function of dd for the sensor position depicted in Fig. .",
"The solid red line corresponds to the full numerical computation of |𝐁||\\mathbf {B}|, while the blue line depicts |𝐁 oer ||\\mathbf {B_{\\text{oer}}}| of the trilayer configuration.",
"We find |𝐁|/|𝐁 oer |>1|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|>1 for d≲50nmd\\lesssim {50}{nm}.Table: yy-components of magnetic field B y B_y and magnetic field gradient ∂B y /∂y\\partial B_y/\\partial y for a sample-sensor distance of d=10nmd={10}{nm} and d=5nmd={5}{nm} (trilayer sample geometry)."
],
[
"Discussion and Summary ",
"We consider magnetic force microscopy (MFM) as a potential candidate for the measurement of the stray field profile [46], [47] and estimate the sensitivity required.",
"The force acting on a MFM tip is $\\mathbf {F}=(\\mathbf {m}\\cdot \\nabla )\\mathbf {B}$ , where $\\mathbf {m}=(0,m,0)$ , $m\\approx {1e-13}{emu}={1e-16}{Am^2}$ is typical magnetic moment of a MFM tip [48].",
"Using $\\partial B_y/\\partial y$ from Tab.",
"REF , we expect a force in $y$ -direction, $|F_y|$ , of ${46}{fN}$ (for $d={10}{nm}$ ) or ${156}{fN}$ ($d={5}{nm}$ ), respectively.",
"The state-of-the-art sensitivity concerning force measurements using MFM is about ${10}{fN}$ at room temperature [49].",
"Besides, Mamin et al.",
"[50] have reported the detection of aN forces with MFM operated at ${100}{mK}$ .",
"Using MFM in frequency-modulated detection mode, force gradient sensitivities down to ${0.14}{N/m}$ have been reported [51].",
"This is well below the expected stray field force gradients $\\partial F_y/\\partial y={7.8}{N/m}$ ($d={20}{nm}$ ) and $\\partial F_y/\\partial y={45}{N/m}$ ($d={5}{nm}$ ).",
"In summary, we have discussed a direct method to detect spin accumulation in a non-magnetic metal strip.",
"The proposed approach is based on the measurement of the magnetic stray field arising from the electron spin accumulation close to the surfaces of the metal strip .",
"To this end, we have derived an analytical expression for the spin accumulation and the corresponding magnetic stray field around a non-magnetic, metallic strip with rectangular cross-section.",
"Based on this, we proposed a sample geometry for a future experiment and calculated the spatial distribution of the magnetic stray field.",
"We showed that the stray field is large enough for detection using the state-of-the-art sensing techniques.",
"Besides, we compared the stray field to the Oersted field around the non-magnetic conductor and found that for the proposed trilayer sample geometry, the Oersted field is dominated by the stray field near the edges of the conducting strip.",
"Such a direct detection of spin accumulation should enable a reliable measurement of spin transport properties, such as spin diffusion length and spin Hall angle, in metals thereby circumventing interfacial complexities."
],
[
"Acknowlegdments",
"We acknowledge funding from DFG via Priority program 1538 Spin-Caloric Transport (Project GO 944/4) and SPP1601 (HU 1896/2).",
"AK is funded by A. v. Humboldt foundation."
],
[
"Relation between spin chemical potential and spin density",
"In order to derive the relation between spin chemical potential and spin density in a non-magnetic conductor, we consider a degenerate non-magnetic gas and obtain relation between spin accumulation and spin density using a two spin channel model.",
"The discussion herein borrows heavily from Ref. fabiansemiconductor2007.",
"For a Fermion gas, we have: $n = \\int g(E) f(E - \\mu ) dE,$ where $n$ denotes the density of electrons, $g(E)$ is the density of states per unit volume, $\\mu $ is the chemical potential, and $f(x) = 1/(\\exp {(x/k_b T)} + 1)$ is the Fermi function.",
"For a two spin model, the above equation becomes: $n_{\\uparrow ,\\downarrow } = \\int g_{\\uparrow ,\\downarrow }(E) f(E - \\mu _{\\uparrow ,\\downarrow }) dE.$ Here, the subscript arrows $\\uparrow , \\downarrow $ denote two opposite spin direction (up and down, resp.",
"left and right).",
"For a non-magnetic conductor, $g_{\\uparrow }(E) = g_{\\downarrow }(E) = g(E)/2$ with $g(E)$ as the total density of states per unit volume.",
"In addition, we define the following notation: $n & = n_{\\uparrow } + n_{\\downarrow }, \\\\s & = n_{\\uparrow } - n_{\\downarrow }, \\\\\\mu & = \\frac{\\mu _{\\uparrow } + \\mu _{\\downarrow }}{2}, \\\\\\mu _{s} & = \\frac{\\mu _{\\uparrow } - \\mu _{\\downarrow }}{2}.$ Further we assume, $\\mu _s \\ll \\mu $ for a linear response theory.",
"Having defined the above notation we proceed to express $s$ in terms of spin chemical potential $\\mu _s$ .",
"$s & = n_{\\uparrow } - n_{\\downarrow }, \\nonumber \\\\& = \\frac{1}{2} \\int g(E) ( f(E - \\mu _{\\uparrow }) - f(E - \\mu _{\\downarrow }) dE.",
"\\\\$ We notice the following relations: $\\mu _{\\uparrow ,\\downarrow } & = \\mu \\pm \\mu _s, \\\\\\therefore f(E - \\mu _{\\uparrow ,\\downarrow }) & = f(E - (\\mu \\pm \\mu _s)), \\\\& = f(E - \\mu ) \\pm \\frac{\\partial f}{\\partial \\mu } \\mu _s, \\\\\\therefore f(E - \\mu _{\\uparrow }) - f(E - \\mu _{\\downarrow }) & = 2 \\frac{\\partial f}{\\partial \\mu } \\mu _s.$ Since $\\partial f/ \\partial \\mu $ for a degenerate gas is approximately $\\delta (E - \\mu )$ [40], we obtain on substitution of the above equation in Eq.",
"(REF ).",
"$s & = \\int g(E) \\delta (E - \\mu ) \\mu _s dE, \\nonumber \\\\& = g(\\mu ) \\mu _s.",
"$ Hence we have a relation between spin density and spin chemical potential via density of states at the chemical potential.",
"It might however be desirable to express the the above relation in terms of commonly used parameters such as conductivity ($\\sigma $ ) and diffusion constant ($D$ ).",
"This is achieved by comparing the diffusion current formulations used in Refs.",
"chentheory2013 and fabiansemiconductor2007.",
"The one dimensional particle diffusion current density in the formulation used in Ref.",
"fabiansemiconductor2007 is given by: $J_{n} = - D \\frac{\\partial n}{\\partial x},$ where $D$ is the diffusion constant of the material, and subscript $n$ reminds us that we are talking about a particle current.",
"Correspondingly we can write net spin “particle” current: $J_{n_\\uparrow } - J_{n_\\downarrow } & = - D \\frac{\\partial (n_\\uparrow - n_\\downarrow )}{\\partial x}, \\nonumber \\\\& = - D g(\\mu ) \\frac{\\partial \\mu _s}{\\partial x}, $ where we used Eq.",
"(REF ) in the last step above.",
"Using formulation used in Ref.",
"chentheory2013, we have $J_{s} = - \\frac{\\sigma }{2 e} \\frac{\\partial \\mu _s}{\\partial x}.$ Please note that the current density above has been expressed in units of charge current density for convenience [38].",
"In order to compare the above expression to Eq.",
"(REF ), we need to divide the above equation by elementary charge ($e$ ) throughout.",
"On comparison of the two particle current, we obtain the following relation: $g(\\mu ) = \\frac{\\sigma }{2 e^2 D}.$ Hence using the above equation in conjunction with Eq.",
"(REF ), we obtain the desired relation between spin imbalance density and spin accumulation: $s = \\frac{\\sigma }{2 e^2 D} \\mu _s.$ Thus, for the $y$ ($z$ )-polarized electrons, we get $n_{sy} = \\frac{\\sigma _N}{2 e^2 D} \\mu _{sy} \\qquad \\text{and}\\qquad n_{sz} = \\frac{\\sigma _N}{2 e^2 D} \\mu _{sz}$"
],
[
"Magnetic field of spin accumulation",
"The magnetic flux density (in the following referred to as magnetic field) originating from a magnetic moment $\\mathbf {m}$ is given by[52] $\\mathbf {B} = \\frac{\\mu _0}{4 \\pi r^{\\prime 3}} \\left[3 (\\mathbf {m} \\cdot \\hat{\\mathbf {r}}^{\\prime }) \\hat{\\mathbf {r}}^{\\prime } - \\mathbf {m} \\right],$ where $\\mathbf {r}^{\\prime }$ is the position vector from the magnetic moment to the point at which the flux density is calculated.",
"To obtain the magnetic stray field arising from the spin accumulation in the conducting strip, we integrate the contribution of the magnetic moments within the volume of the strip.",
"With $n_s(\\mathbf {r})$ being the spin accumulation density at point $\\mathbf {r}=(x,y,z)$ with spin polarization $\\hat{\\mathbf {n}}$ , the orientation-dependent magnetic moment density is $\\gamma \\hbar /2~ \\hat{\\mathbf {n}} n_s(\\mathbf {r})$ , where $\\gamma $ denotes the gyromagnetic ratio of the material, respectively.",
"Note that we treat the magnetic fields generated by the magnetic moments oriented along $\\mathbf {\\hat{z}}$ and $\\mathbf {\\hat{y}}$ initially independently and then calculate the vector sum of the magnetic fields.",
"We obtain for the magnetic field at $\\mathbf {r}_p=(x_p,y_p,z_p)$ , caused by the magnetization $\\gamma \\hbar / 2 \\hat{\\mathbf {n}} n_s(\\mathbf {r})$ present in an infinitesimal volume element $dx\\,dy\\,dz$ around $\\mathbf {r}$ $d\\mathbf {B} = \\frac{\\mu _0}{8 \\pi \\left|\\mathbf {r}_p-\\mathbf {r}\\right|^3} \\left[3 \\frac{\\gamma \\hbar \\hat{\\mathbf {n}} n_s(\\mathbf {r}) \\cdot \\left(\\mathbf {r}_p-\\mathbf {r}\\right)}{ \\left|\\mathbf {r}_p-\\mathbf {r}\\right|^2} \\left(\\mathbf {r}_p-\\mathbf {r}\\right) - \\gamma \\hbar \\hat{\\mathbf {n}} n_s(\\mathbf {r}) \\right]dx\\,dy\\,dz.$ Employing Eqs.",
"(REF ) and (REF ), we calculate the magnetic field distribution outside the conductor originating from the spin accumulation.",
"Due to the translational symmetry of the problem with respect to the $\\mathbf {\\hat{x}}$ -axis, the magnetic field does not depend on $x_{\\text{p}}$ which we choose to be 0.",
"We begin with the integration along the $\\mathbf {\\hat{x}}$ and $\\mathbf {\\hat{y}}$ -direction considering only the accumulation of $\\mathbf {\\hat{y}}$ -polarized spins, i. e. the contribution from $n_{sy}$ .",
"This magnetic field contribution is called $\\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {y}}}$ in the following.",
"Note that the integration can be done easily as $n_{sy}$ does not depend on $x$ and $y$ .",
"We obtain $\\int _{x=-\\infty }^{\\infty } \\int _{y=-y_0/2}^{y_0/2} d\\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {y}}}(\\mathbf {r},\\mathbf {r}_p)= \\frac{\\mu _0\\gamma \\hbar n_{sy}(z)}{8\\pi } \\left[\\left( \\begin{array}{c}0 \\\\\\frac{2\\left(y_p-\\frac{y_0}{2}\\right)}{\\left(y_p-\\frac{y_0}{2}\\right)^2+\\left(z_p-z\\right)^2} \\\\\\frac{2\\left(z_p-z\\right)}{\\left(y_p-\\frac{y_0}{2}\\right)^2+\\left(z_p-z\\right)^2}\\end{array} \\right) -\\left( \\begin{array}{c}0 \\\\\\frac{2\\left(y_p+\\frac{y_0}{2}\\right)}{\\left(y_p+\\frac{y_0}{2}\\right)^2+\\left(z_p-z\\right)^2} \\\\\\frac{2\\left(z_p-z\\right)}{\\left(y_p+\\frac{y_0}{2}\\right)^2+\\left(z_p-z\\right)^2}\\end{array} \\right)\\right] dz.$ An analogous integration along the $\\mathbf {\\hat{x}}$ and $\\mathbf {\\hat{z}}$ -direction for the magnetic field contribution from $n_{sz}$ yields $\\int _{x=-\\infty }^{\\infty } \\int _{z=-z_0/2}^{z_0/2} d\\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {z}}}(\\mathbf {r},\\mathbf {r}_p)= \\frac{\\mu _0\\gamma \\hbar n_{sz}(y)}{8 \\pi } \\left[\\left( \\begin{array}{c}0 \\\\\\frac{2\\left(y_p-y\\right)}{\\left(y_p-y\\right)^2+\\left(z_p-\\frac{z_0}{2}\\right)^2} \\\\\\frac{2\\left(z_p-\\frac{z_0}{2}\\right)}{\\left(y_p-y\\right)^2+\\left(z_p-\\frac{z_0}{2}\\right)^2}\\end{array} \\right) -\\left( \\begin{array}{c}0 \\\\\\frac{2\\left(y_p-y\\right)}{\\left(y_p-y\\right)^2+\\left(z_p+\\frac{z_0}{2}\\right)^2} \\\\\\frac{2\\left(z_p+\\frac{z_0}{2}\\right)}{\\left(y_p-y\\right)^2+\\left(z_p+\\frac{z_0}{2}\\right)^2}\\end{array} \\right)\\right] dy.$ For a full quantitative modelling of the magnetic field distribution in the surrounding of the conductor, we perform the remaining integration over the $y$ - and $z$ -dimensions of Eqs.",
"(REF ) and (REF ) numerically.",
"To this end, we use the spatially dependent spin accumulation density from Eqs.",
"(REF ) – (REF ).",
"Before discussing the numerical results below, we turn to a simplified picture where we consider all spins to be located at the conductor's surface (as indicated in Fig.",
"REF )—a situation with can be treated analytically.",
"This approximation agrees well with the exact solution when the point of interest is located much further away from the conductor as compared to the spin relaxation length ($\\sim $ 1 nm for platinum).",
"In this case, we approximate the spin accumulation density for the $\\mathbf {\\hat{y}}$ -polarized electrons as $n_{sy}(z) \\approx \\tilde{n}_{sy} \\left[\\delta \\left(z-\\frac{z_0}{2}\\right) - \\delta \\left(z+\\frac{z_0}{2}\\right)\\right]$ with $\\delta (x)$ the Dirac delta distribution and $\\tilde{n}_{sy}:=\\int _0^{z_0/2}n_{sy}(z)\\,dz$ .",
"Combining Eqs.",
"(REF ) and (REF ), we get $\\tilde{n}_{sy} = -\\frac{j_e \\theta \\lambda ^2}{D e}\\left(1-\\frac{1}{\\cosh \\left(\\frac{z_0}{2\\lambda }\\right)}\\right)$ Performing the integration for the $z$ -direction in Eq.",
"(REF ) we obtain for the magnetic stray field caused by the spin accumulation $\\tilde{n}_{sy}$ $\\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {y}}}(\\mathbf {r}_p)= \\frac{\\mu _0 \\gamma \\hbar \\tilde{n}_{sy}}{8 \\pi }\\mathbf {F}(y_0,z_0;\\mathbf {r}_p)$ with $\\mathbf {F}(y_0,z_0;\\mathbf {r}_p) =\\left[\\left( \\begin{array}{c}0 \\\\\\frac{2(y_p-y_0/2)}{(y_p-y_0/2)^2+(z_p-z_0/2)^2} \\\\\\frac{2(z_p-z_0/2)}{(y_p-y_0/2)^2+(z_p-z_0/2)^2}\\end{array} \\right) -\\left( \\begin{array}{c}0 \\\\\\frac{2(y_p+y_0/2)}{(y_p+y_0/2)^2+(z_p-z_0/2)^2} \\\\\\frac{2(z_p-z_0/2)}{(y_p+y_0/2)^2+(z_p-z_0/2)^2}\\end{array} \\right)\\right.\\\\\\left.",
"-\\left( \\begin{array}{c}0 \\\\\\frac{2(y_p-y_0/2)}{(y_p-y_0/2)^2+(z_p+z_0/2)^2} \\\\\\frac{2(z_p+z_0/2)}{(y_p-y_0/2)^2+(z_p+z_0/2)^2}\\end{array} \\right) +\\left( \\begin{array}{c}0 \\\\\\frac{2(y_p+y_0/2)}{(y_p+y_0/2)^2+(z_p+z_0/2)^2} \\\\\\frac{2(z_p+z_0/2)}{(y_p+y_0/2)^2+(z_p+z_0/2)^2}\\end{array} \\right)\\right]$ For the magnetic field contribution of the $z$ -polarized electrons, we find correspondingly $\\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {z}}}(\\mathbf {r}_p)=-\\frac{\\mu _0 \\gamma \\hbar \\tilde{n}_{sz}}{8 \\pi }\\mathbf {F}(y_0,z_0;\\mathbf {r}_p)$ with $\\tilde{n}_{sz} = \\frac{j_e \\theta \\lambda ^2}{D e}\\left(1-\\frac{1}{\\cosh \\left(\\frac{y_0}{2\\lambda }\\right)}\\right).$ In total, the magnetic field at point $\\mathbf {r}_p$ arising from the spin polarization in the conducting strip is given by $\\mathbf {B}(\\mathbf {r}_p)= \\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {y}}}(\\mathbf {r}_p) + \\mathbf {B}^{\\hat{\\mathbf {n}}\\parallel \\hat{\\mathbf {z}}}(\\mathbf {r}_p) \\\\= \\frac{\\mu _0\\gamma \\hbar }{8 \\pi } \\frac{j_e \\theta \\tau }{e} \\left(\\frac{1}{\\cosh \\left(\\frac{y_0}{2\\lambda }\\right)} - \\frac{1}{\\cosh \\left(\\frac{z_0}{2\\lambda }\\right)}\\right)\\mathbf {F}(y_0,z_0;\\mathbf {r}_p) $ Obviously, the magnetic stray field is proportional to the spin Hall angle, the spin-flip time and the applied current density through the conductor.",
"Regarding the geometry, a square cross-section of the conductor (i. e. $y_0=z_0$ ) would imply a vanishing stray field as Eq.",
"REF shows.",
"This is a consequence of the symmetry of the problem and holds for the analytical approximation as well as for the full numerical calculation.",
"As we are interested in maximizing the stray field around the conductor, we suggest a very thin ($y_0\\lesssim 3\\,\\text{nm}$ ) metal strip with $z_0\\gg y_0$ for the experimental investigation of the calculated stray field."
],
[
"Oersted field",
"The magnetic field induced by an infinitesimal conductor cross-section $dy\\,dz$ around $\\mathbf {r}$ can be written as we assume the conductor to be aligned along the $x$ -axis.",
"$d\\mathbf {B}_{\\text{oer}}=\\frac{\\mu _0}{2\\pi \\left|\\mathbf {r}_p-\\mathbf {r}\\right|^2} \\mathbf {j}\\times \\left(\\mathbf {r}_p-\\mathbf {r}\\right)dy\\,dz.$ The total Oersted field arising from the (uniform) current density $\\mathbf {j}=j_{\\mathrm {e}}\\hat{\\mathbf {x}}$ in the conducting strip can thus be calculated by integrating $d\\mathbf {B}_{\\text{oer}}$ over the cross-section of the strip.",
"The integral can be solved analytically but the resulting expression is unwieldy and therefore not given here.",
"Figure REF shows the spatial distribution of the Oersted field around the conductor.",
"It has its maximum of about ${16}{T}$ at the left and right edge of the strip.",
"For $r_{\\mathrm {p}}\\gg y_0,z_0$ , the Oersted field decays proportional to $1/r_{\\mathrm {p}}$ as expected for the farfield of a current in a wire.",
"Thus, in the farfield, the Oersted fields dominates the stray field.",
"This is also illustrated in Fig.",
"REF , where the ratio $|\\mathbf {B}|/|\\mathbf {B_{\\text{oer}}}|$ is plotted as a function of the sensor position $\\mathbf {r}_{\\mathrm {p}}$ .",
"Only for small distance from the conducting strip, the stray field exceeds the Oersted field.",
"Nevertheless, the spatial dependence of the Oersted field significantly differs from that of the magnetic stray field of spin accumulation.",
"Thus, using a spatially resolved magnetic field sensing technique would in principle allow to differentiate between stray field and Oersted field."
]
] | 1709.01820 |
[
[
"Impact of Non-potential Coronal Boundary Conditions on Solar Wind\n Prediction"
],
[
"Abstract Predictions of the solar wind at Earth are a central aspect of space weather prediction.",
"The outcome of such a prediction, however, is highly sensitive to the method used for computing the magnetic field in the corona.",
"We analyze the impact of replacing the potential field coronal boundary conditions, as used in operational space weather prediction tools, by non-potential conditions.",
"For this, we compare the predicted solar wind plasma parameters with observations at 1 AU for two six-months intervals, one at solar maximum and one in the descending phase of the current cycle.",
"As a baseline, we compare with the operational Wang-Sheeley-Arge model.",
"We find that for solar maximum, the non-potential coronal model and an adapted solar wind speed formula lead to the best solar wind predictions in a statistical sense.",
"For the descending phase, the potential coronal model performs best.",
"The Wang-Sheeley-Arge model outperforms the others in predicting high speed enhancements and streamer interactions.",
"A better parameter fitting for the adapted wind speed formula is expected to improve the performance of the non-potential model here."
],
[
"Introduction",
"Finding suitable coronal boundary conditions for simulations of the inner heliosphere is a crucial point in solar wind prediction.",
"While a magnetohydrodynamics (MHD) simulation in the coronal domain (like in the MHD-Around-a-Sphere, MAS, model [20], [16], [31], [30]) would provide accurate boundary data, it is, on today's computers, too time-consuming for use in operational space weather forecasting.",
"There exist, however, a number of simplified coronal simulation methods, which use, to different degrees, extrapolation and simplifications.",
"Most notably there are the Potential Field Source Surface (PFSS) method [1], [38], the magnetofrictional (MF) method [46], [40], [47], and the Current Sheet Source Surface (CSSS) method [49], [27], [26].",
"For the outer corona, the Schatten Current Sheet (SCS) method [37] is commonly employed.",
"An empirical wind speed formula is then used to compute the boundary conditions for the heliospheric simulation from the coronal simulation data at the interface between the two domains.",
"Other methods, for example [28], [24], [34], do not simulate the corona (and heliosphere) at all, but rely on purely empirical methods to forecast the solar wind.",
"Recent research by [25] proposes an alternative approach as compromise between full MHD simulations and semi-empirical methods.",
"They compute the structure of the solar winds and its parameters by combining a large number of one-dimensional wind profiles along open magnetic field lines.",
"The form of and the parameters for the empirical wind speed formula are a field of research in their own right, although strongly tied to the coronal simulation in use.",
"Common forms are the Wang-Sheeley (WS) model [41], [42], [3], the Wang-Sheeley-Arge (WSA) model [4] and the Distance from the Coronal Hole Boundary (DCHB) model [30].",
"While the DCHB model has a physics-base explanation of its parameters (which still are varied), the parameters in the WS and WSA formulas are free and adapted for every period considered.",
"[33] have attempted to find optimal parameters for WS, WSA and DCHB by exhaustive search, using a simplified method method in the heliospheric domain.",
"[27], [26] determine the coefficients for their wind speed prediction, which is based on the WS formula, by fitting parameters to a quadratic function using observations that were mapped back to the source surface of their simulation.",
"Over the last years, there have been a number of attempts to compare and validate the different approaches for predicting the solar wind speed.",
"Comparisons of solar wind data of potential and magnetohydrodynamic (MHD) simulations in [32] show that, if time-dependent effects can be neglected, the potential method provides a reasonable approximation to the MHD method, although there still are notable differences.",
"[7] have compared the magnetic structure and the resulting wind speed distribution from potential and non-potential simulations at 21.5 $R_\\odot $ for two solar maximum dates.",
"They have identified considerable differences between the two types of coronal simulations: The non-potential model has more complex magnetic structures, more open flux and, using the WSA wind speed formula, leads to higher predicted wind speeds for the two dates.",
"The present work extends this by considering two time intervals of six months each at solar maximum and the descending phase of the solar cycle, and continuing the simulation to 1 AU for comparison with observational data.",
"[12], [13] present an extensive evaluation of a number of coronal and heliospheric models that are available at the Community Coordinated Modeling Centre.",
"They find that there is not a single candidate that performs best, but that each model has its strengths and weaknesses.",
"In this work we aim to contribute to the efforts of bringing an order into the multitude of solar wind prediction methods by comparing the impact of the PFSS, MF and WSA method for computing the coronal boundary conditions for solar wind simulations.",
"To investigate the difference between potential and non-potential boundary conditions we distinguish here between WSA and PFSS: We perform the PFSS coronal simulation with the same input data and the same method as the MF one (that is, Air Force Data Assimilative Photospheric Flux Transport (ADAPT) synoptic $B_r$ maps), except that we compute the potential field solution in inner corona.",
"The WSA run, in contrast, uses the National Oceanic and Atmospheric Administration / National Weather Service Space Weather Prediction Center (NOAA/NSW SWPC) operational method.",
"This is also based on a PFSS model for the inner corona, but using observational magnetograms from the National Solar Observatory (NSO) Global Oscillation Network Group [9] directly as input.",
"We pick one six-month interval at solar maximum and one in the descending phase of the solar cycle for comparison and do both a statistical and an event-based comparison.",
"The remainder of the paper is structured as follows: In Section we describe the data and the simulation methods that we use in this work.",
"Section details the solar wind models that are used for the different runs.",
"We present our results in Section and conclude and give an outlook on future work in Section ."
],
[
"Data and Methods ",
"The models were set up as follows (see also Table REF ): WSA model driven by daily updated GONG magnetograms; PFSS which used the DuMFriC PFSS model https://github.com/antyeates1983/pfss driven by ADAPT magnetograms; MF model which used the DuMFriC non-potential (NP) MF model driven by ADAPT magnetograms.",
"Table: Description of the solar wind modelsThe time intervals chosen for simulation are May 1 to October 31 in 2014 (solar maximum, CR 2149 to 2156) and 2016 (descending phase, CR 2176 to 2183) respectively.",
"We chose these times for their position in the solar activity cycle and due to data availability.",
"All Enlil and OMNI solar wind data is transformed to the Heliocentric Earth Equatorial (HEEQ) coordinate system for the comparisons."
],
[
"ADAPT Dataset",
"The photospheric boundary conditions for the coronal simulation are derived from Air Force Data Assimilative Photospheric Flux Transport (ADAPT) synoptic $B_r$ maps [5], [10], [11].",
"The ADAPT maps are constructed from GONG magnetograms by evolving them using a photospheric flux transport model which is based on the Worden-Harvey model [45].",
"New data are assimilated into the model once per day (weather permitting) and maps are output at twelve hour cadence.",
"In the current version, the ADAPT data set consists of an ensemble of twelve realizations which account for model parameter uncertainties in the supergranular flow.",
"We picked realization number one for our experiments.",
"This choice is expected to have only a minimal influence on the results (see [43])."
],
[
"Coronal and Inner Heliospheric Simulation ",
"For the Potential Field Source Surface (PFSS) model [1], [38], the magnetic field between 1 $R_\\odot $ and 2.5 $R_\\odot $ , i.e., in the inner corona, is computed by extrapolation from the photospheric magnetic field, assuming that the field is is current free ($\\nabla \\times \\mathbf {B} = 0$ ) and radial at 2.5 $R_\\odot $ .",
"The non-potential coronal model simulates the evolution of the large-scale magnetic field between 1 $R_\\odot $ and 2.5 $R_\\odot $ using the magneto-frictional (MF) method [46], [40], [47].",
"Here, the velocity $\\mathbf {v}$ is approximated by the magneto-frictional form: $\\mathbf {v} = \\nu ^{-1} (\\mathbf {J} \\times \\mathbf {B}/B^2)$ , where $\\mathbf {J} = \\nabla \\times \\mathbf {B}$ and $\\nu $ is a friction coefficient.",
"This enforces the relaxation of the magnetic field towards a nonlinear force-free state where $\\mathbf {J}\\times \\mathbf {B}=0$ .",
"The MF model allows for a gradual build-up and conservation of magnetic energy and electric currents in the corona.",
"The temporal evolution of $\\mathbf {B} = \\nabla \\times \\mathbf {A}$ is driven by photospheric $B_r$ maps from which the update $\\partial \\mathbf {A}/\\partial t = - \\mathbf {E}$ for the vector potential $\\mathbf {A}$ is computed.",
"The method used for the electric field reconstruction is described in [43] and based on work by [2], [8], [14].",
"The coronal simulation uses a grid that is equally spaced in $\\rho $ , $s$ , $\\phi $ , where $\\rho = \\ln \\left(r/R_\\odot \\right)$ and $s=\\cos \\theta $ in terms of spherical coordinates $(r,\\theta \\,\\phi )$ .",
"The resolution is $60 \\times 180 \\times 360$ .",
"For extrapolation of the magnetic field in the outer corona (2.5 $R_\\odot $ to 21.5 $R_\\odot $ ) we use the Schatten Current Sheet (SCS) method [37].",
"Here, we solve for a potential field using the absolute values of the radial magnetic field component at $r = 2.5 R_\\odot $ and the assumption that $\\mathbf {B} \\xrightarrow[r \\rightarrow \\infty ]{} 0$ .",
"Then, the field line direction is restored where $B_r<0$ at 2.5 $R_\\odot $ , producing infinitesimally thin current sheets.",
"The magnetic field at 21.5 $R_\\odot $ and the expansion factor and coronal hole boundary distance as described in Sec.",
"are used to compute the boundary conditions for the solar wind software Enlil [22], [21], which simulates the solar wind in the heliosphere.",
"In this work, we are interested in the wind speed, proton density and magnetic field at Earth.",
"We run Enlil with low resolution, which means 256 cells in $r$ , 30 cells in $\\theta $ and 90 cells in $\\phi $ ."
],
[
"Solar Wind Models ",
"As described in the introduction, most solar wind models consist of two parts.",
"An inner model covering the domain from $R_{\\odot }$ to $21.5R_{\\odot }$ and an outer model covering the domain from $21.5R_{\\odot }$ to some outer boundary such as 1.7AU.",
"The following sections detail the inner models used in this study to determine the boundary conditions for the outer model.",
"The baseline model is the WSA model which is used operationally worldwide.",
"In this paper, we introduce a new approach to modelling the coronal part of the solar wind modelling chain in Section REF , consisting of a potential or non-potential reconstruction of the coronal field using the DuMFriC code driven by ADAPT maps feeding into a alternative empirical formulation for the solar wind speed based on the DCHB model by [30].",
"This provides alternative boundary conditions to those obtained from the WSA model."
],
[
"Wang-Sheeley-Arge Model ",
"The empirical solar wind model used in operational space weather forecasting at the NOAA/NSW SWPC is the Wang-Sheeley-Arge (WSA) model [4].",
"It computes the wind speed $v_r$ from the flux tube expansion factor $f_s = \\left(\\frac{R_\\odot }{R_{s}}\\right)^2 \\left(\\frac{B_r(R_\\odot )}{B_r(R_s)}\\right)$ , with $R_s = 2.5 R_\\odot $ in our case, and $\\theta _b$ (in degrees), which is the minimum distance of a fieldline footpoint from a coronal hole boundary in the photosphere.",
"The WSA formula reads $v_r(f_s, \\theta _b) = v_{slow} + \\frac{v_{fast} - v_{slow}}{(1 + f_s)^{\\alpha }} \\left[ \\beta - \\gamma e^{-(\\theta _b/\\omega )^\\delta }\\right]^{\\iota }.$ There are eight free parameters, including the fast and slow wind speed $v_{fast}$ and $v_{slow}$ .",
"The WSA formula was developed from the Wang-Sheeley (WS) model [41], [42], [3], which only uses $f_s$ as an input parameter and reads $ v_r(f_s) = v_{slow} + \\frac{v_{fast} - v_{slow}}{(f_s)^\\alpha },$ and the Distance from the Coronal Hole Boundary (DCHB) formula [30] which only uses $\\theta _b$ and reads $v_r(\\theta _b) = v_{slow} + (v_{fast} - v_{slow})\\left[1 + \\tanh \\left(\\frac{\\theta _b - \\epsilon }{w}\\right) \\right],$ with $\\epsilon $ the thickness of the slow flow band and $w$ the width over which flow is raised to coronal hole values.",
"Recent research suggests that $\\theta _b$ is more important than $f_s$ for determining the wind speed, and in some cases the presence of $f_s$ actually weakens the predictive power of the WSA formula [33].",
"However, [18] found evidence that the expansion factor might influence the distribution of fast solar wind deep inside coronal holes, and that both factors ($\\theta _b$ and $f_s$ ) might be important to accurately model the solar wind."
],
[
"Modified DCHB Model ",
"All three formulas above were developed for input data using a relatively coarse grid, not resolving much of the fine-scale structure of the magnetic field and small (or thin) coronal holes.",
"The derived solar wind speed and density maps in [7] used Equation REF without modifying the parameters used in the WSA model.",
"Here, for our MF and PFSS simulation, we devised a new formula, tentatively cutting down on free parameters and leaving out the expansion factor term.",
"Our tentative empirical wind speed formula reads $ v_r(\\theta _b) = v_{slow} + (v_{fast} - v_{slow})(\\theta _b \\cdot \\omega )^\\delta ,$ with $\\theta _b$ in radians, $\\omega > 1$ , $\\delta < 1$ , $v_{slow} \\in \\left[100, 400\\right]$ and $v_{fast} \\in \\left[500, 1000\\right]$ .",
"Note that if we measured $\\theta _b$ in degrees we would need $\\omega < 1$ .",
"Because we use the same high resolution input and coronal hole detection method for the PFSS method we use the same wind speed formula.",
"As a first rough parameter fit we use $v_{slow} = 200$ $km s^{-1}$ , $v_{fast} = 700$ $km$ $s^{-1}$ , $\\omega = 7$ and $\\delta = 1/2.5$ .",
"We find overall that the optimal form of the wind speed formula and the optimal parameter fit are very sensitive to the resolution of the photospheric magnetograms or synoptic maps, the resolution of the coronal simulation and the sensitivity of the coronal hole detection algorithm."
],
[
"Results ",
"In order to compare the solar wind predictions at L1 provided by Enlil when it is driven by the different models, we first produce the boundary conditions separately by the MF, PFSS and WSA model as described in the previous sections.",
"For the MF model, we run DuMFriC with a four-month ramp-up time, i.e., we start the simulation in January of the respective year, in order to make sure that it has reached a valid state by May.",
"For every day, we then do a separate Enlil run with the new boundary conditions derived from the current observational data."
],
[
"Validation Metrics for Continuous Variables",
"We first do a comparison in terms of statistical metrics.",
"Figures REF , REF and REF show scatter plots of the wind speed $v_r$ , the magnetic field magnitude $B_{mag}$ and the density $d$ , respectively.",
"For a quantitative comparison we evaluated the Root Mean Square Error $RMSE=\\sqrt{\\frac{\\sum _{1}^{n} (f_i-o_i)^2}{n}}$ ; $bias=\\frac{\\sum _{1}^{n} (f_i-o_i)}{n}$ , and the Pearson correlation coefficient $\\rho \\in [-1, 1]$ that describes the linear correlation between the simulated and the observed quantities.",
"Tables REF , REF and REF summarize the models' performance in these metrics for all variables and both time intervals.",
"Figure: Scatter plots and histograms for the wind speed v r v_r at L1 for 2014 (left) and 2016 (right).",
"Units are km/skm/s.",
"The x axis corresponds to the simulation data, the y axis corresponds to the OMNI data.",
"The top row corresponds to the MF model, the middle row to the PFSS model, and the bottom row to the WSA model.Figure: Scatter plots and histograms for B mag B_{mag} for 2014 (left) and 2016(right).",
"Units are nTnT.",
"The x axis corresponds to the simulation data, the y axis corresponds to the OMNI data.",
"The top row corresponds to the MF model, the middle row to the PFSS model, and the bottom row to the WSA model.Figure: Scatter plots and histograms for density for 2014 (left) and 2016 (right).",
"Units are p/m 3 p/m^{3}.",
"The x axis corresponds to the simulation data, the y axis corresponds to the OMNI data.",
"The top row corresponds to the MF model, the middle row to the PFSS model, and the bottom row to the WSA model.Table: Time-series statistics wind speed.",
"The RMSE is in km/skm/s.Table: Time-series statistics B mag B_{mag}.",
"The RMSE is in nTnT.Table: Time-series statistics density.",
"The RMSE is in p/m 3 p/m^{3}.As we have expected due to the existence of currents in the MF model, this model gives a better correlation with the observed wind speed for solar maximum than the PFSS model, and also than the operational WSA model.",
"Looking at the correlation coefficient for the wind speed and density, the potential 2016 (descending phase) results are better than the 2014 (solar maximum) results.",
"For the magnetic field magnitude, MF performs best for both years.",
"It is interesting to note, however, that this changes when we exclude the time intervals with CMEs from our analysis.",
"In this case, the WSA model performs best in 2016, and the PFSS in 2014 [44].",
"For the density, the MF model again performs better than the potential one in 2014, and not as good in 2016.",
"The RMSE gives a different picture.",
"Here, the two potential models mostly outperform the non-potential one.",
"For the wind speed, the 2014 results have better (lower) values than the 2016 results, while it is the other way around for $B_{mag}$ , and mixed for the density.",
"The reason for the poor performance of the MF model can be observed in the scatter plots: The slope of a best-fit line through the data points, is not one.",
"This systematic error could be corrected by optimizing the solar wind speed formula and its parameters.",
"We will briefly discuss such an optimization in Section .",
"We conclude that the statistics do not give us the one best method here, and none of the methods are particularly good in terms of real correlation and matching the observations on a one-to-one basis.",
"However, we can see a tendency that the non-potential model mostly improves the results at solar maximum."
],
[
"Event-based Validation",
"In addition to the metrics defined in the previous section, event-based validation is crucial in assessing the various models.",
"For the purpose of this study, the focus is on the arrival time of slow-to-fast stream interaction regions (SIRs).",
"The error on the arrival time of SIRs is typically of the order of one day, and so larger than the error on arrival time of CMEs.",
"The arrival of SIRs can trigger a geomagnetic storm response and the fast solar wind is associated with high electron fluences in the radiation belts.",
"One of the drivers of developing the MF model was to get a better characterization of coronal holes, associated with the fast solar wind.",
"The SIR detection algorithm was originally developed in [23] and further refined in [17], [12], [13].",
"We use here, for comparability, the thresholds as given in [12].",
"In Figure REF we show how our wind models perform in terms of predicting SIRs and high speed enhancements (HSEs), i.e., abrupt accelerations in $v_r$ .",
"Figure: SIR detection results for 2014 (top) and 2016 (bottom).",
"Detected SIRs from OMNI observations (top row) are marked in red and and copied down to the other rows (second row is MF, third row is PFSS, bottom row WSA) for easier comparison, as well as the velocity profile (red line in simulation plots).",
"Detected events for the simulations are highlighted in blue.",
"Dashed vertical red (OMNI) and blue (simulations) lines mark detected HSEs.There exists a great number of metrics, or skill scores, for assessing the “goodness” of a binary forecast (cf., for example, [6]).",
"They are usually based on the number $n_{hit}$ of events that very correctly predicted, the number $n_{miss}$ of events that were missed in the prediction and the number $n_{false}$ of false alarms.",
"One such skill score, which we will use here, is the critical success index or threat score (TS, see, e.g., [36] and references therein) $TS=\\frac{n_{hit}}{n_{hit}+n_{miss}+n_{false}}$ .",
"The TS assumes that it is most important to avoid missing an event (i.e., keep $n_{miss}$ low), even on the cost of increasing the number of false alarms.",
"Table: SIR detection performance.Table REF summarizes the SIR prediction results.",
"Overall, the WSA model performs best in predicting the SIRs correctly.",
"It has the highest TS for both time intervals, with PFSS performing equally well in this metric for the descending phase.",
"The difference is that PFSS has one more miss and one more false alarm in the 2016 interval.",
"The MF model has no false alarms for this case, but a lot of misses.",
"For solar maximum, the result is not as clear.",
"The MF model performs slightly better than the PFSS model in terms of TS and false alarms.",
"The WSA model has a better TS and a higher hit rate, but also more false alarms.",
"Looking at the velocity profile in Figure REF , we can see that for some of the missed SIRs in the MF and PFSS model, there actually is a peak around that time, only not as high as required for the SIR detection algorithm.",
"This, as in the scatter plots, points to a systematic error.",
"We conclude that a better parameter optimization for the solar wind speed formula might improve the performance here.",
"In addition, an adaption of the SIR detection thresholds would lead to more hits, but, as mentioned before, we have decided to leave the thresholds as in [12] for better comparability."
],
[
"Conclusion and Future Work ",
"We have compared the performance of three different simulation pipelines for solar wind prediction, using GONG or ADAPT maps as input, non-potential or potential models for the solar corona, and Enlil for extrapolating the results to 1AU.",
"These models were compared with OMNI observations of wind speed, magnetic field strength, and density.",
"We did both a statistical analysis and an evaluation of their performance in predicting SIRs; and we considered two six-months periods, one at solar maximum and one in the descending phase of the solar cycle.",
"We observed a difference in performance of the models due to the phase in the solar cycle.",
"At solar maximum, the non-potential model performed best in the statistical metrics, while the potential and WSA model were better in the descending phase.",
"The better performance of the MF model during solar maximum was expected, as this model, in contrast to the potential ones, includes currents in the corona, which are more frequent at this time.",
"In the event-based validation, the operational WSA model always performed best in terms of hits and threat score, but it also produced the most false alarms.",
"Future work will include testing the effect of re-introducing the expansion factor term (in adapted form) into the empirical wind speed formula for the DuMFriC potential and non-potential simulations.",
"The models then have to be tested with different input data, i.e., from different observatories and different forms of synoptic maps, and extended from simulating the purely ambient wind to including CMEs.",
"The usage of a local method for computing the electric field, avoiding the “halos” as described in [43], would be expected to improve the MF model [48].",
"The next crucial step will be to use automatic optimization methods for parameter fitting.",
"Although doing this was out of scope of this paper, we give here some considerations concerning automatic parameter optimization.",
"It is generally assumed that the major part of the wind speed evolution happens close to the Sun, i.e., between the Sun and 0.3 AU [35], [39], although [19] found that, especially at solar minimum and for intermediate wind speeds, there is still some significant change in the wind speed between 0.385 AU and 1 AU.",
"An optimization using simulation data at 1 AU, however, is very time consuming as it requires running an MHD simulation as e.g.",
"Enlil or an appropriate approximation (see [29] for a comparison of techniques).",
"Therefore, as a first attempt, we could try and compare the speed distributions of the simulation at 0.1 AU and observations at 1 AU for receiving a parameter fit.",
"We expect this to yield better results during solar maximum than during solar minimum.",
"In order to evaluate a suitable method for fitting the parameters there are a number of metrics for the distribution of the solar wind speed values that could be considered.",
"The metrics we propose to consider are the root mean squared error (RMSE), the correlation factor, and the shape of the histogram, i.e.",
"the difference of the mean values, the difference of the variance and the difference of the skewness.",
"Also, a simultaneous (multi-objective) optimization of a subset of these metric and optimization of single objectives should be tested.",
"A multi-objective optimization determines the Pareto front of multiple fitness functions, that is, the points where improving the score for one objective would worsen the score for the other.",
"Here, another question to consider is how to choose the \"best\" point of the resulting set: by minimizing one of the objectives, or by using some kind of trade-off?",
"There exist a number of sophisticated methods alone for the purpose of choosing the the optimal point on the Pareto front.",
"We have seen in Section , though, that all these metrics are only of limited value for parameter optimization for operational space weather forecasting.",
"As noted by other authors [23], [17] before, an optimal correlation factor or RMSE is no guarantee for a good forecast w.r.t.",
"relevant space weather events.",
"Therefore, parameter fitting should ultimately be based on the ability to predict space weather events, such as the arrival time of stream interaction regions and high speed enhancements (see Section REF ).",
"This, however, is computationally very expensive, as it requires the whole simulation pipeline to be run a lot of times.",
"We leave this as an interesting direction for future research which has to combine mathematical, physical and computer science skills."
],
[
"Acknowledgments",
"The work utilizes data produced collaboratively between Air Force Research Laboratory (AFRL) and the National Solar Observatory (NSO).",
"The ADAPT model development is supported by AFRL and AFOSR.",
"The input data utilized by ADAPT are obtained by NSO/NISP (NSO Integrated Synoptic Program).",
"NSO is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under a cooperative agreement with the National Science Foundation (NSF).",
"The OMNI data were obtained from the GSFC/SPDF OMNIWeb interface at http://omniweb.gsfc.nasa.gov.",
"MW and ARY thank STFC and the AFOSR for financial support."
]
] | 1709.01730 |
[
[
"Evaluating Content-centric vs User-centric Ad Affect Recognition"
],
[
"Abstract Despite the fact that advertisements (ads) often include strongly emotional content, very little work has been devoted to affect recognition (AR) from ads.",
"This work explicitly compares content-centric and user-centric ad AR methodologies, and evaluates the impact of enhanced AR on computational advertising via a user study.",
"Specifically, we (1) compile an affective ad dataset capable of evoking coherent emotions across users; (2) explore the efficacy of content-centric convolutional neural network (CNN) features for encoding emotions, and show that CNN features outperform low-level emotion descriptors; (3) examine user-centered ad AR by analyzing Electroencephalogram (EEG) responses acquired from eleven viewers, and find that EEG signals encode emotional information better than content descriptors; (4) investigate the relationship between objective AR and subjective viewer experience while watching an ad-embedded online video stream based on a study involving 12 users.",
"To our knowledge, this is the first work to (a) expressly compare user vs content-centered AR for ads, and (b) study the relationship between modeling of ad emotions and its impact on a real-life advertising application."
],
[
"Introduction",
"The proceedings are the records of a conference.This is a footnote ACM seeks to give these conference by-products a uniform, high-quality appearance.",
"To do this, ACM has some rigid requirements for the format of the proceedings documents: there is a specified format (balanced double columns), a specified set of fonts (Arial or Helvetica and Times Roman) in certain specified sizes, a specified live area, centered on the page, specified size of margins, specified column width and gutter size."
],
[
"The Body of The Paper",
"Typically, the body of a paper is organized into a hierarchical structure, with numbered or unnumbered headings for sections, subsections, sub-subsections, and even smaller sections.",
"The command \\section that precedes this paragraph is part of such a hierarchy.This is a footnote.",
"handles the numbering and placement of these headings for you, when you use the appropriate heading commands around the titles of the headings.",
"If you want a sub-subsection or smaller part to be unnumbered in your output, simply append an asterisk to the command name.",
"Examples of both numbered and unnumbered headings will appear throughout the balance of this sample document.",
"Because the entire article is contained in the document environment, you can indicate the start of a new paragraph with a blank line in your input file; that is why this sentence forms a separate paragraph."
],
[
"Type Changes and ",
"We have already seen several typeface changes in this sample.",
"You can indicate italicized words or phrases in your text with the command \\textit; emboldening with the command \\textbf and typewriter-style (for instance, for computer code) with \\texttt.",
"But remember, you do not have to indicate typestyle changes when such changes are part of the structural elements of your article; for instance, the heading of this subsection will be in a sans serifAnother footnote here.",
"Let's make this a rather long one to see how it looks.",
"typeface, but that is handled by the document class file.",
"Take care with the use ofAnother footnote.",
"the curly braces in typeface changes; they mark the beginning and end of the text that is to be in the different typeface.",
"You can use whatever symbols, accented characters, or non-English characters you need anywhere in your document; you can find a complete list of what is available in the User's Guide ."
],
[
"Math Equations",
"You may want to display math equations in three distinct styles: inline, numbered or non-numbered display.",
"Each of the three are discussed in the next sections."
],
[
"Inline (In-text) Equations",
"A formula that appears in the running text is called an inline or in-text formula.",
"It is produced by the math environment, which can be invoked with the usual \\begin ...\\end construction or with the short form $ ...$.",
"You can use any of the symbols and structures, from $\\alpha $ to $\\omega $ , available in ; this section will simply show a few examples of in-text equations in context.",
"Notice how this equation: $\\lim _{n\\rightarrow \\infty }x=0$ , set here in in-line math style, looks slightly different when set in display style.",
"(See next section)."
],
[
"Display Equations",
"A numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the equation environment.",
"An unnumbered display equation is produced by the displaymath environment.",
"Again, in either environment, you can use any of the symbols and structures available in ; this section will just give a couple of examples of display equations in context.",
"First, consider the equation, shown as an inline equation above: $\\lim _{n\\rightarrow \\infty }x=0$ Notice how it is formatted somewhat differently in the displaymath environment.",
"Now, we'll enter an unnumbered equation: $\\sum _{i=0}^{\\infty } x + 1$ and follow it with another numbered equation: $\\sum _{i=0}^{\\infty }x_i=\\int _{0}^{\\pi +2} f$ just to demonstrate 's able handling of numbering."
],
[
"Citations",
"Citations to articles , , , , conference proceedings or maybe books , listed in the Bibliography section of your article will occur throughout the text of your article.",
"You should use BibTeX to automatically produce this bibliography; you simply need to insert one of several citation commands with a key of the item cited in the proper location in the .tex file .",
"The key is a short reference you invent to uniquely identify each work; in this sample document, the key is the first author's surname and a word from the title.",
"This identifying key is included with each item in the .bib file for your article.",
"The details of the construction of the .bib file are beyond the scope of this sample document, but more information can be found in the Author's Guide, and exhaustive details in the User's Guide by Lamport Lamport:LaTeX.",
"This article shows only the plainest form of the citation command, using \\cite."
],
[
"Tables",
"Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite.",
"To ensure this proper “floating” placement of tables, use the environment table to enclose the table's contents and the table caption.",
"The contents of the table itself must go in the tabular environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules.",
"Again, detailed instructions on tabular material are found in the User's Guide.",
"Immediately following this sentence is the point at which Table REF is included in the input file; compare the placement of the table here with the table in the printed output of this document.",
"Table: Frequency of Special CharactersTo set a wider table, which takes up the whole width of the page's live area, use the environment table* to enclose the table's contents and the table caption.",
"As with a single-column table, this wide table will “float” to a location deemed more desirable.",
"Immediately following this sentence is the point at which Table REF is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.",
"Table: Some Typical CommandsIt is strongly recommended to use the package booktabs and follow its main principles of typography with respect to tables: Never, ever use vertical rules.",
"Never use double rules.",
"It is also a good idea not to overuse horizontal rules."
],
[
"Figures",
"Like tables, figures cannot be split across pages; the best placement for them is typically the top or the bottom of the page nearest their initial cite.",
"To ensure this proper “floating” placement of figures, use the environment figure to enclose the figure and its caption.",
"This sample document contains examples of .eps files to be displayable with .",
"If you work with pdf, use files in the .pdf format.",
"Note that most modern systems will convert .eps to .pdf for you on the fly.",
"More details on each of these are found in the Author's Guide.",
"Figure: A sample black and white graphic.Figure: A sample black and white graphicthat has been resized with the includegraphics command.As was the case with tables, you may want a figure that spans two columns.",
"To do this, and still to ensure proper “floating” placement of tables, use the environment figure* to enclose the figure and its caption.",
"And don't forget to end the environment with figure*, not figure!",
"Figure: A sample black and white graphicthat needs to span two columns of text.Figure: A sample black and white graphic that hasbeen resized with the includegraphics command."
],
[
"Theorem-like Constructs",
"Other common constructs that may occur in your article are the forms for logical constructs like theorems, axioms, corollaries and proofs.",
"ACM uses two types of these constructs: theorem-like and definition-like.",
"Here is a theorem: Let $f$ be continuous on $[a,b]$ .",
"If $G$ is an antiderivative for $f$ on $[a,b]$ , then $\\int ^b_af(t)\\,dt = G(b) - G(a).$ Here is a definition: If $z$ is irrational, then by $e^z$ we mean the unique number that has logarithm $z$ : $\\log e^z = z.$ The pre-defined theorem-like constructs are theorem, conjecture, proposition, lemma and corollary.",
"The pre-defined definition-like constructs are example and definition.",
"You can add your own constructs using the amsthm interface .",
"The styles used in the \\theoremstyle command are acmplain and acmdefinition.",
"Another construct is proof, for example, Suppose on the contrary there exists a real number $L$ such that $\\lim _{x\\rightarrow \\infty } \\frac{f(x)}{g(x)} = L.$ Then $l=\\lim _{x\\rightarrow c} f(x)= \\lim _{x\\rightarrow c}\\left[ g{x} \\cdot \\frac{f(x)}{g(x)} \\right]= \\lim _{x\\rightarrow c} g(x) \\cdot \\lim _{x\\rightarrow c}\\frac{f(x)}{g(x)} = 0\\cdot L = 0,$ which contradicts our assumption that $l\\ne 0$ ."
],
[
"Conclusions",
"This paragraph will end the body of this sample document.",
"Remember that you might still have Acknowledgments or Appendices; brief samples of these follow.",
"There is still the Bibliography to deal with; and we will make a disclaimer about that here: with the exception of the reference to the book, the citations in this paper are to articles which have nothing to do with the present subject and are used as examples only."
],
[
"Headings in Appendices",
"The rules about hierarchical headings discussed above for the body of the article are different in the appendices.",
"In the appendix environment, the command section is used to indicate the start of each Appendix, with alphabetic order designation (i.e., the first is A, the second B, etc.)",
"and a title (if you include one).",
"So, if you need hierarchical structure within an Appendix, start with subsection as the highest level.",
"Here is an outline of the body of this document in Appendix-appropriate form: Generated by bibtex from your .bib file.",
"Run latex, then bibtex, then latex twice (to resolve references) to create the .bbl file.",
"Insert that .bbl file into the .tex source file and comment out the command \\thebibliography."
],
[
"More Help for the Hardy",
"Of course, reading the source code is always useful.",
"The file acmart.pdf contains both the user guide and the commented code.",
"The authors would like to thank Dr. Yuhua Li for providing the matlab code of the BEPS method.",
"The authors would also like to thank the anonymous referees for their valuable comments and helpful suggestions.",
"The work is supported by the GS501100001809National Natural Science Foundation of Chinahttp://dx.doi.org/10.13039/501100001809 under Grant No.",
": GS50110000180961273304 and [http://www.nnsf.cn/youngscientsts]GS501100001809Young Scientsts' Support Program."
]
] | 1709.01684 |
[
[
"Interacting Attention-gated Recurrent Networks for Recommendation"
],
[
"Abstract Capturing the temporal dynamics of user preferences over items is important for recommendation.",
"Existing methods mainly assume that all time steps in user-item interaction history are equally relevant to recommendation, which however does not apply in real-world scenarios where user-item interactions can often happen accidentally.",
"More importantly, they learn user and item dynamics separately, thus failing to capture their joint effects on user-item interactions.",
"To better model user and item dynamics, we present the Interacting Attention-gated Recurrent Network (IARN) which adopts the attention model to measure the relevance of each time step.",
"In particular, we propose a novel attention scheme to learn the attention scores of user and item history in an interacting way, thus to account for the dependencies between user and item dynamics in shaping user-item interactions.",
"By doing so, IARN can selectively memorize different time steps of a user's history when predicting her preferences over different items.",
"Our model can therefore provide meaningful interpretations for recommendation results, which could be further enhanced by auxiliary features.",
"Extensive validation on real-world datasets shows that IARN consistently outperforms state-of-the-art methods."
],
[
"Introduction",
"Recommendation is a fundamental task to enable personalized information filtering, thus to mitigate the information overload problem [34].",
"The goal is to learn user preferences from historical user-item interactions, based on which recommend relevant items.",
"In reality, user preferences often evolve over time, affected by dynamic user inclinations, item perception and popularity.",
"Temporal context therefore has been recognized as an important type of information for modeling the dynamics of user preferences.",
"It has extensive applications, ranging from movie recommendation [3], music recommendation [15], to location recommendation [43].",
"Most existing methods [17], [15], [25], [43], [5] model the temporal dynamics by extending the latent factor model (LFM) [31] with handcrafted features, so as to describe certain temporal patterns of user-item interactions.",
"For example, they either bin user-item interactions into time windows, assuming similar user behavioral patterns in the same window [15], [43], or adopt a time decay function to under-weight the interactions occurring deeper into the past [17], [25].",
"The handcrafted features, though proven to be effective, cannot capture complex temporal patterns in reality [39].",
"More importantly, these methods cannot automatically select important interaction records in user-item interaction history when modeling user preferences.",
"This greatly limits their application in real-world scenarios where user-item interactions can often happen accidentally.",
"Recently, recurrent neural network (RNN) [28] based methods have emerged as a promising approach to model the temporal dynamics of user preferences [10], [14], [39].",
"RNN captures both the latent structures in historical user-item interactions – through hidden units – and their dynamics along the temporal domain.",
"Unlike LFM based methods, these methods are nonparametric, thus can learn inherent dynamics that are more complex and suitable for making recommendations.",
"A specific type of gated RNN, i.e.",
"Long Short-Term Memory (LSTM) [12], is employed by the state-of-the-art recommendation method [39] to model both user and item dynamics.",
"The gating mechanism is adopted to balance the information flow from the current and previous time steps, thus can more effectively preserve historical information over time for recommendation.",
"Figure: Example of user-item interactions determined by dependent user dynamics and item dynamics.",
"The numbers below user and item history are the attention scores inferred by our proposed model, which are used to select relevant time steps in user and item history to accurately predict the user's preference over the item.Nevertheless, LSTM models the gate w.r.t.",
"each hidden unit instead of the whole time step, making it difficult to interpret the importance of each time step for the final recommendation.",
"More importantly, gates for modeling user dynamics and item dynamics so far are learned separately.",
"In real-world scenarios, however, user and item dynamics are dependent on each other, and can jointly affect user-item interactions.",
"Consider the target user Bob in Figure REF , who is interested in both formal clothing (e.g., leather shoes and trousers) and casual clothing (e.g., casual shoes and shorts), as described by his purchasing history.",
"We observe that Bob buys a pair of formal jeans, which were historically bought by users with various interests.",
"The interaction between Bob and the formal jeans is therefore determined by his interest in formal clothing and the inherent property of the formal jeans, namely, the formal style.",
"Such an interest and property could only be learned from historical user-item interactions when no additional auxiliary features are given.",
"Therefore, to accurately capture Bob's preference over the formal jeans, the recommendation model should be able to identify the important time steps of Bob's purchasing history when he bought formal clothing.",
"Similarly, in the history of formal jeans it should be able to identify time steps when they were bought by users who are also interested in formal style clothing, thus to capture the item property relevant to Bob's interest.",
"In this paper, we introduce the Interacting Attention-gated Recurrent Network (IARN) which adopts the attention model to measure the relevance of each time step of user history and item history for recommendation.",
"In particular, we propose a novel attention scheme which allows IARN to learn the relevance – measured by attention scores – of time steps in user and item history in an interacting way, so as to capture the dependencies between user and item dynamics in shaping user-item interactions.",
"As a result, IARN can selectively memorize different time steps of a user's history when predicting her preferences over different items, thereby providing meaningful interpretations for the prediction.",
"For instance, attention scores learned by IARN for the example in Figure REF are shown under the user and item history in the figure (note this example is based on our results on a real-world dataset).",
"IARN could be further enhanced by incorporating auxiliary features of users or items.",
"In this paper we provide methods to integrate IARN with auxiliary features organized in a flat or a hierarchical structure.",
"More specifically, our main contributions include: [noitemsep,nolistsep,leftmargin=*] We extend recurrent networks for modeling user and item dynamics with a novel gating mechanism, which adopts the attention model to measure the relevance of individual time steps of user and item history for recommendation.",
"We design a novel attention scheme which allows the user- and item-side recurrent networks to interact with each other, thus to capture the dependencies between user and item dynamics to improve recommendation accuracy and interpretability.",
"We propose the IARN method implementing the interacting attention-gate as described above, and show how it can be further enhanced by auxiliary features organized in different structures.",
"We conduct extensive experiments to evaluate the proposed IARN method on six real-world datasets, demonstrating that IARN consistently outperforms state-of-the-art methods."
],
[
"Related Work",
"This section provides an overview of state-of-the-art recommendation methods related to our work.",
"We review them from two orthogonal perspectives: (1) the underlying recommendation models; (2) the incorporation of side information for recommendation."
],
[
"Underlying Recommendation Models",
"The recently pervasive recommendation methods can be broadly categorized into two types, namely, the latent factor model based methods and the neural network based ones.",
"Latent Factor Model.",
"Due to the high efficiency, state-of-the-art recommendation methods have been dominated by Latent Factor Model (LFM).",
"It decomposes the high-dimensional user-item rating matrix into low-dimensional user and item latent matrices.",
"A panoply of algorithms have been proposed to date based on LFM, including matrix factorization (MF) [18], Bayesian personalized ranking (BPR) [27], collective matrix factorization (CMF) [32], factorization machine (FM) [26], SVD++ [16], to name a few.",
"Despite of their success, LFM based methods suffer from the following essential limitations.",
"First of all, they merely leverage global statistical information of user-item interaction data, while cannot capture fine-grained regularities in the latent factors [24].",
"Second, LFM based recommendation methods generally learn latent representations of users and items in a linear fashion, which may not be always suitable in real-world scenarios.",
"Besides, most LFM based methods ignore the temporal dynamics of user preferences, assuming that the future user-item interactions are known in advance, which is contradictory with the real-world application.",
"There are a few LFM based methods specifically designed for fusing temporal information, which will be reviewed in section 2.2.",
"Neural Networks.",
"Stemming from the success in related domains (e.g., computer vision, speech recognition, and natural language processing), Neural Network (NN) based methods have recently attracted a considerable amount of interests from the recommendation community.",
"In contrast to LFM based recommendation methods, NN based methods have shown to be highly effective in capturing local item relationships by modeling item co-occurrence in individual users' interaction records.",
"Typical methods are User2Vec [6] and Item2Vec [1], which are inspired by word embedding techniques [20], [21].",
"Furthermore, NN based models can learn nonlinear latent representations through the activation functions (e.g., sigmoid, ReLU [22]).",
"For instance, Suvash et al.",
"propose the AutoRec [30] recommendation method based on autoencoders [11].",
"He et al.",
"propose neural collaborative filtering [9] to learn non-linear interactions between users and items.",
"Recently, the Recurrent Neural Network (RNN) based methods [39], [14], [13], [10] have gained significant enhancement in recommendation thanks to the ability of preserving historical information over time for recommendation.",
"These methods learn time-varying representations of users/items (i.e., hidden-states) in each time step, by taking into account both the present and historical data.",
"The learned states can be used for generating recommendations for the future, therefore being more realistic and attractive for real-world applications.",
"To sum up, NN based methods possess essential advantages and have shown to be more effective to enhance recommendation performance."
],
[
"Incorporating Side Information",
"To better model user preferences thus to further improve recommendation performance, many researchers endeavor to incorporate side information, i.e., information complementing user-item interactions, into recommendation models.",
"Here we focus on the literature with consideration of two types of side information related to our work, i.e., temporal context and auxiliary features.",
"Temporal Context.",
"It has been well recognized that user preferences change over time.",
"This can be due to drifting user inclinations for item, or the constantly changing item perception and popularity when new selection emerges [17], [14].",
"Hence, recommendation methods that capture temporal dynamics of user preferences could provide improved recommendation performance.",
"In the branch of LFM based methods, some take temporal information into consideration based on time windows, assuming user-item interactions in the same window have similar patterns.",
"For instance, Koenigstein et al.",
"[15] and Yuan et al.",
"[43] propose such methods for music and Point-of-Interest recommendation.",
"A disadvantage is that they regard all interactions within the considered time window equally, completely ignoring the relationships of interactions among different windows.",
"In addition, binning user-item interactions aggravates the data sparsity problem.",
"Some other LFM based methods attempt to address these issues by adopting a time decay function to under-weight the instances as they occur deeper into the past.",
"These include TimeSVD++ proposed by Koren [17] and HeteRS proposed by Pham et al.",
"[25].",
"However, these methods could not capture other types of temporal patterns, e.g., certain user-item interactions could be driven by the long-term interest of a user which could not be modeled in a decay manner.",
"In fact, all LFM based methods handle temporal context by creating handcrafted features, thus cannot capture complex temporal patterns in reality.",
"Contrarily, RNN based methods are nonparametric, thus can learn inherent dynamics of user preferences that are more complex.",
"For instance, Hidasi et al.",
"[10] propose a RNN based approach for session-based recommendation.",
"Hosseini et al.",
"[13] introduce a recurrent Poisson factorization framework for recommendation.",
"Among different RNN models, Long Short-Term Memory (LSTM) [12] has gained much popularity in recommendation due to their capability in dealing with the gradient vanishing problem [44].",
"Jing et al.",
"[14] present a LSTM based method to estimate when a user will return to a site and what her future listening behavior will be.",
"Wu et al.",
"[39] propose a LSTM based method, i.e., recurrent recommender network (RRN), to model user and item dynamics.",
"This is the most closely related work to ours.",
"However, one major shortcoming of these gated RNN based methods is that the learned gate lacks interpretability, limiting further improvements of recommendation accuracy.",
"More importantly, these methods model user and item dynamics separately, thus failing to capture their dependencies and their joint effects on user-item interactions.",
"Motivated by the attention scheme in human foveal vision, attention mechanism has been employed by NN based methods to cope with the data noisy problem by identifying relevant parts of the input for the prediction task.",
"It has been applied in a broad spectrum of disciplines, from natural language processing [42] to computer vision [23], [40].",
"However, how to effectively exploit the attention mechanism in recommender systems is still an open research question.",
"To the best of our knowledge, we are the first to propose recurrent network based recommendation method that integrates attention mechanism to automatically learn the relevance of individual time steps for recommendation, so as to enhance both recommendation interpretability and accuracy.",
"More importantly, we design a novel attention scheme that allows user- and item-side recurrent networks to interact with each other, thus to capture the dependencies between user and item dynamics and their joint effects on user-item interactions.",
"Auxiliary Features.",
"Better representations of users and items can also be obtained by incorporating auxiliary features into recommendation, due to the rich semantic information encoded by them.",
"Most existing feature-based recommendation approaches are built upon LFM.",
"These methods are either designed to incorporate features in a flat structure or a hierarchy.",
"For instance, the popular CMF [32] and FM [26] are designed for integrating flat features into recommendation.",
"Recently it has been found that feature hierarchies, i.e., hierarchically organized features, can be more effective in boosting the accuracy as well as the interpretability of recommendation.",
"He et al.",
"[8] devise a visually-aware recommendation model by manually defining the feature hierarchy influence on items.",
"Yang et al.",
"[41] design a recommendation method that automatically learns feature hierarchy influence on user/item by a parameterized regularization traversing from root to leaf features.",
"More recently, Sun et al.",
"[35] introduce a unified recommendation framework that seamlessly incorporates both vertical and horizontal dimensions of feature hierarchies for effective recommendation.",
"In this paper, we show how to incorporate features organized in both flat and hierarchical structures into our model.",
"Note that although in other domains like nature language processing, a few work [42] attempts to integrate hierarchies into RNN model, there is few such kind of approach in recommendation.",
"Hence, we are the first to explore the effect of features organized in different structures together with recurrent networks to learn optimal representations of users and items for improved recommendation interpretability and accuracy."
],
[
"Interacting Attention-gated Recurrent Networks",
"Given the historical user-item interaction data as the input, we aim to learn high-quality hidden representations for both users and items, which are then used for subsequent recommendation.",
"The extracted representations are expected to: 1) capture the temporal dynamics contained in both user and item history with physical interpretability of each time step; 2) learn the dependencies between user and item dynamics in shaping user-item interactions; 3) extract semantically rich information from training data through the incorporation of auxiliary features.",
"To achieve these goals, we propose the Interacting Attention-gated Recurrent Network (IARN), which is composed of three modules: Attention-gated Recurrent Module, Interacting Attention Module and Feature Encoder.",
"They are designed correspondingly to the three aforementioned goals.",
"The overall architecture of IARN is illustrated in Figure REF .",
"It employs two recurrent networks to learn compact and effective hidden representations for the paired user and item, respectively.",
"Each recurrent network is composed of an Attention-gated Recurrent Module, an Interacting Attention Module, and a Feature Encoder.",
"Instead of behaving independently, the two recurrent networks interact with each other to model the dependencies between user and item dynamics.",
"Input and Output Layers.",
"We first describe the input and output layer of IARN, then in the following subsections we will elaborate on the three modules in a top-down fashion to explain step by step how the input and output layers are connected to achieve our goal.",
"Let $\\mathcal {U}$ and $\\mathcal {V}$ be the user and item set, respectively.",
"As input, each user $i\\in \\mathcal {U}$ is described by a sequence $\\mathbf {x}_i$ , which contains the representations (e.g., the embeddings) of all the items rated by her, ordered by the rating time.",
"Similarly, each item $j\\in \\mathcal {V}$ is described by a sequence $\\mathbf {x}_j$ that contains the representations of all users who have rated the item, ordered by the rating time.",
"Through the three modules, IARN learns from $\\mathbf {x}_i$ and $\\mathbf {x}_j$ the hidden representation of user $i$ , denoted by $\\tilde{\\mathbf {u}}_{i}$ , and the hidden representation of item $j$ , denoted by $\\tilde{\\mathbf {v}}_{j}$ .",
"$\\tilde{\\mathbf {u}}_{i}$ and $\\tilde{\\mathbf {v}}_{j}$ are then used to predicted user $i$ 's preference rating $\\tilde{r}_{ij}$ over item $j$ via an inner product operation: $\\vspace{-3.61371pt}\\tilde{r}_{ij} = \\langle \\tilde{\\mathbf {u}}_{i}, \\tilde{\\mathbf {v}}_{j} \\rangle $"
],
[
"Attention-gated Recurrent Module",
"In order to learn high-quality hidden representations $\\tilde{\\mathbf {u}}_{it}$ and $\\tilde{\\mathbf {v}}_{jt}$ , we propose the Attention-gated Recurrent Module to preserve the information of previous time steps with relevance modeled by attention scores, which are obtained by the Interactive Attention Module that will be present in section REF .",
"Specifically, we construct two attention-gated recurrent modules for the paired user and item, respectively.",
"It should be noted that these two modules do not share parameters, since users and items are not expected to share similar hidden representations.",
"This makes our method different from Siamese Network [4], a well-known method for object comparison.",
"Both user- and item-side Attention-gated Recurrent Modules contain two layers, namely, a recurrent layer and a fully-connected layer.",
"The recurrent layer models the temporal dynamics of users and items as hidden-states, while the fully-connected layer transform the hidden-states of users and items in the last time step to the hidden representations for prediction.",
"We first describe the full-connected layer, then introduce in detail the recurrent layer.",
"User- and Item-Side Fully-Connected Layers.",
"Denote the last hidden-states of user- and item-side recurrent layers as $\\mathbf {u}_{iT_i}$ and $\\mathbf {v}_{jT_j}$ , respectively.",
"The hidden representations $\\tilde{\\mathbf {u}}_{i}$ and $\\tilde{\\mathbf {v}}_{j}$ are transformed from these hidden-states by non-linear transformations: $\\vspace{-3.61371pt}{\\left\\lbrace \\begin{array}{ll}\\tilde{\\mathbf {u}}_{i} = g(\\widetilde{\\mathbf {W}}_u \\cdot \\mathbf {u}_{iT_i} + \\tilde{b}_u) \\\\\\tilde{\\mathbf {v}}_{j} = g(\\widetilde{\\mathbf {W}}_v \\cdot \\mathbf {v}_{jT_j} + \\tilde{b}_v)\\end{array}\\right.",
"}$ Herein, $\\widetilde{\\mathbf {W}}_u$ and $\\widetilde{\\mathbf {W}}_v$ are linear transformation parameters of the user- and item-side layers, respectively; $\\tilde{b}_u$ and $\\tilde{b}_v$ are the bias terms; $g$ is the activation function, for which we use the Parametric Rectified Linear Unit (PReLU) [7].",
"PReLU allows the output of the unit to be either positive or negative, thus is more suitable for representing users/items – intuitively, a user could either like or dislike certain types of items (e.g., action movies), and an item could either be of a specific type or not.",
"User-Side Attention-gated Recurrent Layer.",
"Given the user $i$ whose corresponding input sequence is $\\mathbf {x}_i = \\lbrace \\mathbf {x}_{i1}, \\mathbf {x}_{i2}, \\ldots \\rbrace $ .",
"We denoted the attention score at time step $t$ by $a_{it}$ , which is a scalar value between $[0,1]$ inferred by the Interacting Attention Module.",
"The hidden-state of user $i$ at time $t$ is then modeled as $\\vspace{-3.61371pt}\\mathbf {u}_{it} = (1-a_{it}) \\cdot \\mathbf {u}_{i(t-1)} + a_{it} \\cdot \\mathbf {u}_{it}^{\\prime }$ where $\\mathbf {u}_{i(t-1)}$ is the hidden-state in the previous time step and $\\mathbf {u}_{it}^{\\prime }$ is the candidate state value obtained by fully incorporating the input at the current time step: $\\mathbf {u}_{it}^{\\prime } = g(\\mathbf {W}_u \\cdot \\mathbf {u}_{i(t-1)} + \\mathbf {H}_u \\cdot E_u(\\mathbf {x}_{it}) + b_u)$ where $\\mathbf {W}_u$ and $\\mathbf {H}_u$ are respectively the linear transformation parameters for the previous and current time steps; $b_u$ is the bias term; and $E_u(\\cdot )$ is the Feature Encoder that transforms the original user sequence by considering auxiliary features, which will be detailed in section REF .",
"We use ReLU for the activation function $g$ .",
"Equation REF balances the contributions of the input of the current candidate hidden-state and the previous hidden-state with an attention gate described by the attention score $a_{it}$ .",
"Attention gates with high scores will focus more on the current input than previous hidden-states, while recurrent gates with low attention scores will ignore the current input and inherit more information from previous time steps.",
"The attention score therefore quantifies the importance of individual time steps in the final prediction.",
"Item-Side Attention-gated Recurrent Layer.",
"Similarly, for the item-side recurrent layer, we model the the hidden-state as follows ${\\left\\lbrace \\begin{array}{ll}\\mathbf {v}_{jt} = (1-a_{jt}) \\cdot \\mathbf {v}_{j(t-1)} + a_{jt} \\cdot \\mathbf {v}_{jt}^{\\prime } \\\\\\mathbf {v}_{jt}^{\\prime } = g(\\mathbf {W}_v \\cdot \\mathbf {v}_{j(t-1)} + \\mathbf {H}_v \\cdot E_v(\\mathbf {x}_{jt}) + b_v)\\end{array}\\right.",
"}$ where $\\mathbf {x}_{jt}$ is the input of item $j$ at time $t$ ; $\\mathbf {W}_v$ , $\\mathbf {H}_v$ , and $b_v$ are the network parameters; $a_{jt}$ is the attention score that serves as attention gate; and $E_v(\\cdot )$ is the Feature Encoder for transforming item sequences, introduced in section REF ."
],
[
"Interacting Attention Module",
"We propose the Interacting Attention Module for both users and items to measure the saliency and relevance of the input in each time step to rating prediction.",
"The key point in this module is that the inferred attention score should not only consider the current time step in the sequence on its own side, but also take into account the information of the other side so as to model the interacting dependency between the paired user and item.",
"User-Side Interacting Attention Module.",
"To maximize the utility of the input sequence, we model the saliency score based on both the input observation at the current time step and the information from neighboring observations in both directions.",
"This is achieved by using a bi-directional RNN [29], which includes a forward layer and a backward layer, as depicted in Figure REF .",
"The attention score $a_{it}$ at time step $t$ in Equation REF on the user side is modeled as: $a_{it} = \\sigma (\\mathbf {M}_u{^\\top } \\cdot \\tanh (\\mathbf {L}_u \\cdot (\\overset{\\rightarrow }{\\mathbf {u}}_{it}; \\overset{\\leftarrow }{\\mathbf {u}}_{it}; \\overset{\\rightarrow }{\\mathbf {v}}_{jT_j}; \\overset{\\leftarrow }{\\mathbf {v}}_{j1}) + b_u^{\\prime } )))$ Wherein a two-layer network is used to calculate the attention score: $\\mathbf {L}_u$ is a matrix as the parameter of the fusion layer that fuses both directional layers of our bi-directional RNN; $b_u^{\\prime }$ is the bias term of the fusion layer; and $\\mathbf {M}_u$ is the weight vector of the second layer; $\\sigma $ is sigmoid function applied as the activation function to control the attention score to lie between $[0,1]$ ; $(;)$ denotes the concatenation among vectors; $\\overset{\\rightarrow }{\\mathbf {u}}_{it}$ and $\\overset{\\leftarrow }{\\mathbf {u}}_{it}$ perform as the summary of context information around time step $t$ in the user sequence $x_i$ .",
"Specifically, $\\vspace{-3.61371pt}\\begin{split}\\overset{\\rightarrow }{\\mathbf {u}}_{it} = g(\\overrightarrow{\\mathbf {W}}_u\\cdot E_u(\\mathbf {x}_{it}) + \\overrightarrow{\\mathbf {H}}_u\\cdot \\overset{\\rightarrow }{\\mathbf {u}}_{i(t-1)} + \\overset{\\rightarrow }{b}_u)\\\\\\overset{\\leftarrow }{\\mathbf {u}}_{it} = g(\\overleftarrow{\\mathbf {W}}_u\\cdot E_u(\\mathbf {x}_{it}) + \\overleftarrow{\\mathbf {H}}_u\\cdot \\overset{\\leftarrow }{\\mathbf {u}}_{i(t+1)} + \\overset{\\leftarrow }{b}_u)\\end{split}$ Therefore, $\\overset{\\rightarrow }{\\mathbf {u}}_{it}$ summarizes the sequence from the beginning to time $t$ , while $\\overset{\\leftarrow }{\\mathbf {u}}_{it}$ summarizes the sequence from the end to time $t$ .",
"Similarly, $\\overset{\\rightarrow }{\\mathbf {v}}_{jT_j}$ $\\overset{\\leftarrow }{\\mathbf {v}}_{j1}$ in Equation REF are the summary of the paired item sequence $\\mathbf {x}_j$ , whose calculation will be introduced later in detail by Equation REF .",
"They are concatenated together with the summary of the user-side sequence, and used as input of the fusion layer.",
"In this way, the resulting attention score $a_{it}$ is used to characterize the relevance of the current time step $t$ of user sequence $\\mathbf {x}_i$ conditioned on the paired item sequence $\\mathbf {x}_j$ .",
"Item-Side Interactive Attention Module.",
"Similarly, for item-side, we have $a_{jt} = \\sigma (\\mathbf {M}_v{^\\top } \\cdot \\tanh (\\mathbf {L}_v \\cdot (\\overset{\\rightarrow }{\\mathbf {v}}_{jt}; \\overset{\\leftarrow }{\\mathbf {v}}_{jt}; \\overset{\\rightarrow }{\\mathbf {u}}_{iT_i}; \\overset{\\leftarrow }{\\mathbf {u}}_{i1}) + b_v^{\\prime })))$ where $\\mathbf {L}_v, b_v^{\\prime }$ are the parameters of the fusion layer, and $\\mathbf {M}_v$ is the weight vector of the second layer; $\\overset{\\rightarrow }{\\mathbf {v}}_{jt}$ and $\\overset{\\leftarrow }{\\mathbf {v}}_{jt}$ perform as the summary of the context information around time step $t$ in the item sequence $\\mathbf {x}_j$ : $\\begin{split}\\overset{\\rightarrow }{v}_{jt} = g(\\overrightarrow{\\mathbf {W}}_v\\cdot E_v(\\mathbf {x}_{jt}) + \\overrightarrow{\\mathbf {H}}_v\\cdot \\overset{\\rightarrow }{\\mathbf {v}}_{j(t-1)} + \\overset{\\rightarrow }{b}_v)\\\\\\overset{\\leftarrow }{\\mathbf {v}}_{jt} = g(\\overleftarrow{\\mathbf {W}}_v\\cdot E_v(\\mathbf {x}_{jt}) + \\overleftarrow{\\mathbf {H}}_v\\cdot \\overset{\\leftarrow }{v}_{j(t+1)} + \\overset{\\leftarrow }{b}_v)\\end{split}$ The summary of user sequence, i.e., $\\overset{\\rightarrow }{u}_{iT_i}$ $\\overset{\\leftarrow }{u}_{i1}$ , are taken as input for modeling the attention score $a_{jt}$ , so as to condition the learning of $a_{jt}$ on the paired user sequence $\\mathbf {x}_i$ .",
"By modeling the attention of each time step in both the user- and item-side networks, our method can capture the interacting dependency and the joint effects of user and item dynamics on user preferences.",
"It thus enable us to gain “second order” insights such as how user preferences are determined by the dynamics of user inclinations and the change of item perception/popularity together."
],
[
"Feature Encoder",
"We now introduce Feature Encoder, which is used to extract semantically rich information from the input data for learning high-quality hidden-states.",
"Here we focus on Feature Encoder for processing item-side input, as features of items are in generally richer than users (e.g., the datasets we will take for validation in section ).",
"It is however non-trivial to adapt our method for processing the user-side input when auxiliary features of users are given.",
"We consider two structures of feature organizations, namely, flat structure and hierarchy.",
"Formally, let $\\mathcal {F}$ denote the set of features organized in a flat structure or a hierarchy.",
"Each item $j\\in \\mathcal {V}$ is affiliated with a subset of features $\\mathcal {F}(j) = \\lbrace f_j^1, f_j^2, \\ldots , f_j^L\\rbrace $ .",
"The effect of feature $f_j^k$ is modeled as a linear transformation function, denoted by $\\mathbf {M}_j^k$ , that projects the input $\\mathbf {x}_{jt}$ for all $1\\le t\\le T_j$ to a new space determined by the feature (i.e., the column space of $\\mathbf {M}_j^k$ ) $\\mathbf {M}_j^k\\cdot \\mathbf {x}_{jt}$ The problem is how to combine the effects of different features of $\\mathcal {F}(j)$ to project the input for best learning the hidden-states.",
"Considering feature organizations, we design our Feature Encoder as follows.",
"Flat Feature Encoder.",
"In the case when features are organized in a flat structure, we simply add the effects of different features together.",
"Formally, for the input $\\mathbf {x}_{jt}$ , the combined effects of all affiliated features $\\mathcal {F}(j)$ are given by $E_v(\\mathbf {x}_{jt}) = \\sum \\nolimits _{k=1}^L \\mathbf {M}_j^k \\cdot \\mathbf {x}_{jt}$ Hierarchical Feature Encoder.",
"In the case when $\\mathcal {F}(j)$ is a feature hierarchy, let $f_j^1$ be the feature in the leaf layer and $f_j^L$ be the root feature.",
"Intuitively, features in top-layers (close to root in the hierarchy) provide more general description of the item, while those in bottom-layers (close to the leaf layer in the hierarchy) provide more refined description.",
"Inspired by the recursive nature of a hierarchy, we consider the recursive parent-children relationships between features in connected layers from the root to leaf layer.",
"In every two connected-layers, the input will be first projected by the parent feature, then by the child feature.",
"By doing so, they will be first mapped to a more general feature space, and then mapped to a more semantically refined feature space.",
"The effects of all affiliated features in different layers will be combined recursively, such that the input can be sequentially mapped to more refined spaces.",
"Formally, for the input $\\mathbf {x}_{jt}$ , the combined effects of all affiliated features $\\mathcal {F}(j)$ are given by $E_v(\\mathbf {x}_{jt}) = (\\mathbf {M}_j^1 \\cdot (\\mathbf {M}_j^2 \\ldots \\cdot (\\mathbf {M}_j^L \\cdot \\mathbf {x}_{jt})\\ldots )) = \\prod \\nolimits _{k=1}^L \\mathbf {M}_j^k \\cdot \\mathbf {x}_{jt}$"
],
[
"End-to-End Parameter Learning",
"Given the training data $\\mathcal {D}_{train}$ containing $N$ instances in the form of $(i, j, r_{ij}, time stamp)$ , IARN learns the involved parameters by minimizing the mean squared error loss function: $\\mathcal {J} = \\frac{1}{N}\\sum _{r_{ij}\\in \\mathcal {D}_{train}} (\\tilde{r}_{ij} - r_{ij})^2$ Since all the modules and the above loss function are analytically differentiable, IARN can be readily trained in an end-to-end manner.",
"In the learning process, parameters are updated by the back-propagation through time (BPTT) algorithm [38] in the recurrent layers of the Attention-gated Recurrent Module and the Interacting Attention Module, and by normal back-propagation in other parts.",
"We use RMSprop [36] to adaptively update the learning rate, which has proven to be highly effective for training neural networks.",
"To prevent over-fitting, we use dropout [33] to randomly drop hidden units of the network in each iteration during the training process."
],
[
"Comparison with Recurrent Network based Methods",
"Comparison with RNN- and LSTM-backbone.",
"One could argue that our framework can also employ two RNN or LSTM as the backbone for user- and item-side recurrent networks.",
"However, the major downside of RNN- and LSTM-backbone is two-fold.",
"First, RNN- and LSTM-backbone cannot provide interpretable recommendation results either due to the lack of gates (RNN-backbone), or the gates modeled as multi-dimensional vectors (LSTM-backbone).",
"In contrast, gates in IARN are represented by attention scores in scalar values, therefore IARN can provide meaningful interpretations on the relevance of each time step for recommendation.",
"Second, RNN- or LSTM-backbone models user dynamics and item dynamics separately, thus can only learn fixed attention scores for each user and item.",
"The attention scores for a specific user (item) actually indicate the general importance (e.g., the frequency) of each item (user) in purchased history of this user (item), which may not be effective in predicting specific user-item interactions.",
"Unlike them, the novel attention scheme designed for IARN can learn different attention scores for an individual user (item) when interacting different items (users), thus can model the dependencies between user and item dynamics.",
"In addition to the above, when compared with LSTM-backbone, IARN has less parameters, so is less prone to be over-fitting.",
"Moreover, IARN uses the bi-directional recurrent network to model attention gates, which helps to maximize the utility of the input data.",
"Comparison with TAGM.",
"IARN is inspired by the Temporal Attention Gated Model (TAGM) [23] recently proposed for sequence classification.",
"IARN inherits the bi-directional attention gates from TAGM, however, our attention scheme is specifically designed with the purpose of recommendation in mind.",
"The nature of recommendation requires proper modeling user-item interactions, for which we design the Interacting Attention Module for modeling the interacting attention for both users and items.",
"This allows IARN to capture the dependencies between user and item dynamics, making IARN particularly suitable for modeling user-item interactions."
],
[
"Experiments and Results",
"In this section, we conduct experiments to evaluate the performance of IARN on six real-world datasets.",
"We aim to answer the following research questions: (1) How do the interacting attention scheme and feature encoder of IARN contribute to recommendation performance and interpretability?",
"(2) How effective is IARN compared to state-of-the-art recommendation methods in both rating prediction and personalized ranking?",
"They will be addressed by section REF and section REF , respectively."
],
[
"Experimental Setup",
"Datasets.",
"To evaluate the effectiveness of IARN, we utilize six real-world datasets, namely Netflix prize dataset, MovieLens, and four Amazon Web store datasets introduced by McAuley et al.",
"[19], i.e., Electronic, Home, Clothing, Sport.",
"Each data point in these datasets is a tuple – (user id, item id, rating, time stamp).",
"Specifically, the Netflix dataset is a large movie rating dataset scaled from 1 to 5 with a step size of 1, which is collected between November 1999 to December 2005.",
"MovieLens is also a personalized movie rating dataset collected from September 1995 to March 2015 with ratings ranging from 0.5 to 5.0 with a step size of 0.5.",
"Besides, it also contains for each movie the genre information as features in a flat structure.",
"The Amazon Web store datasets are collected from Amazonhttps://www.amazon.com/, which is a large on-line shopping website, including electronics, clothing, etc.",
"The time span is from May 1996 to July 2014.",
"In addition, there is an item category hierarchy associated with each of the four datasets.",
"We sample the datasets such that only users and items with more than 3 ratings are preserved.",
"Table REF summarizes the statistics of all the considered datasets.",
"Table: The statistics of datasets, where #U_av_T (#I_av_T) is the average length of sequences w.r.t.",
"users (items);Comparison Methods.",
"We compare with the following state-of-the-art algorithms, 1) MF [18]: matrix factorization as the basic latent factor model (LFM) aiming at rating prediction; 2) BPR [27]: Bayesian personalized ranking as the basic LFM designed for item ranking; 3) TimeSVD++ [17]: LFM with the incorporation of temporal context; 4) HieVH [35]: LFM integrating feature hierarchies; 5) Item2Vec [1]: the basic neural network (NN) model; 6) NCF [9]: neural collaborative filtering replacing the inner product with non-linear network layers for item ranking; 7) MP2Vec [37]: NN model considering auxiliary features.",
"Note that methods designed for incorporating feature hierarchies can also handle features in a flat structure, by considering all features in the same level; similarly, methods designed for incorporating features in a flat structure can also handle feature hierarchies by flattening them into flat structures, with the loss of certain structural information.",
"To investigate the effect of attention-gates and our novel attention scheme, we also compare the following IARN variants using different recurrent networks as the backbone, a) RNN-backbone: the basic variant using RNN as the backbone of user- and item-side recurrent neural networks; b) LSTM-backbone: the variant using LSTM as the backbone; c) TAGM-backbone: the variant using TAGM as the backbone; d) IARN-Plain: the variant of our proposed attention-gated recurrent networks integrated with the interacting attention scheme; e) IARN: the upgraded version of IARN-Plain by fusing auxiliary features.",
"Note that LSTM-backbone is similar to [39] which also employs LSTM as the backbone; while TAGM-backbone is a completely new method which is adapted from TAGM for recommendation.",
"Given their same goal in modeling temporal dynamics, we compare them together.",
"Evaluation Metrics.",
"We adopt Root Mean Square Error (RMSE) and Area Under the ROC Curve (AUC) to measure the performance of rating prediction and personalized ranking, respectively.",
"The smaller RMSE and the larger AUC, the better the performance.",
"We split all the datasets into training and test data according to the following time stamps: June 1st, 2005 for Netflix dataset; January 1st, 2010 for MovieLens dataset; and January 1st, 2014 for the four Amazon datasets.",
"The data before these time stamps are treated as training data, while the rest are considered as the test data.",
"Parameter Settings.",
"We empirically find out the optimal parameter settings for each comparison method.",
"For all the methods, we set the dimension of the latent factor $d = 25$ on Netflix and MovieLens datasets, and $d = 50$ on the four Amazon datasets.",
"We apply a grid search in $\\lbrace 10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}\\rbrace $ for the learning rate and regularization coefficient.",
"For TimeSVD++, $decay\\_rate = 0.4; bin = 30$ .",
"For HieVH, $\\alpha = 0.01$ .",
"For MP2Vec, $\\alpha = 0.1$ .",
"For all recurrent networks mentioned in this work (RNN-backbone, LSTM-backbone, TAGM-backbone, IARN) as well as NCF, the number of hidden units is set to 64 which is selected as the best configuration from the option set {32, 64, 128} based on a held-out validation set.",
"To avoid potential over-fitting, the dropout value is validated from the option set {0.00, 0.25, 0.50}.",
"Model training is performed using a RMSprop stochastic gradient descent optimization algorithm with mini-batches of 50 pairs of user-item interactions.",
"All the gradients are clipped between -10 and 10 to prevent exploding [2]."
],
[
"Effects of Attention and Feature Encoder",
"Attention.",
"In order to investigate the impact of the proposed attention scheme, we compare the performance (measured by RMSE) of IARN-Plain with different recurrent networks as the backbone, including RNN-backbone, LSTM-backbone, and TAGM-backbone.",
"To understand their capability in modeling temporal dynamics of users and item history in different lengths, we test their performance on different configurations of the datasets by constraining the minimum length of user and item input sequences.",
"A grid search in $\\lbrace 3, 10, 20, 30, 50, 100\\rbrace $ is applied for the minimum length of sequences on all the datasets, excluding Clothing and Sport since there are few users possessing long length sequences in these two datasets.",
"Due to space limitation, we only show the results on four Amazon datasets as depicted by Figure REF , however similar observations as below can be obtained on all the datasets.",
"As the minimum length of input sequences increases, the performance of all methods generally improves, indicating that sufficient temporal context could ensure recurrent network based methods to better model the dynamics of user preferences.",
"The performance of gated recurrent networks, i.e., LSTM-backbone, TAGM-backbone, and IARN-plain, is generally better than the non-gated recurrent network, i.e., RNN-backbone.",
"Such a difference is minor when the minimum sequence length is less than a threshold (e.g., 30), and becomes significant with the further growth of the sequence length.",
"This shows the benefit of gating mechanism in effectively preserving historical information deep into the past for recommendation.",
"The observation further explains the close performance of different methods on Clothing dataset, whose sequences are mostly less than 30 and the average sequence length (i.e., around 6, Table REF ) is significantly smaller than all the other datasets.",
"The overall performance of TAGM-backbone and IARN-Plain, is better than that of LSTM-backbone.",
"LSTM-backbone adopts multi-dimensional gates w.r.t.",
"each hidden unit, which can be more easily over-fitted than the (bi-directional) attention-gates employed by TAGM-backbone and IARN-Plain.",
"With respect to attention-gated recurrent networks, IARN-Plain outperforms TAGM-backbone across all different configurations of minimum sequence length.",
"This is mainly due to the fact that TAGM-backbone learns user and item dynamics separately, i.e., only fixed attention scores for user history and item history are learned (LSTM-backbone suffers from the same issue).",
"Whereas equipped with our novel attention scheme, IARN-Plain can adaptively learn different attention scores for user (item) history when the user (item) interacts with different items (users).",
"Such a comparison clearly shows the advantage of our proposed attention scheme for modeling the dependencies between user and item dynamics.",
"Overall, IARN-Plain achieves the best performance across all different configurations of all the datasets, especially when the sequence length gets larger.",
"On average, the relative improvements w.r.t.",
"the second best method are 2.54% with minimum length $= 50$ and 11.65% with minimum length $= 100$ .",
"This implies the remarkable advantage of IARN-Plain in dealing with long sequences.",
"Feature Encoder.",
"We further examine the effectiveness of auxiliary features which are organized in either a flat or hierarchical structure on all the datasets, excluding Netflix which does not contain any auxiliary features.",
"The results are given by Figure REF .",
"By integrating auxiliary features IARN outperforms IARN-Plain across all the datasets, with 1.19% lift ($p$ -value $<0.01$ ) in RMSE on average.",
"This clearly indicates the benefit of considering feature encoder in our proposed IARN approach.",
"Figure: SportInterpretation by IARN.",
"The attention scores learned by IARN for individual time steps in user and item history can help quantify the relevance of each time steps in user and item history for recommendation.",
"We now qualitatively analyze such attention scores to investigate their effects on providing meaningful interpretations for recommendation.",
"Figure REF shows the attention scores learned by IARN on examples of four datasets.",
"In each of the four examples, we can observe varying attention scores assigned to different time steps in both user and item history.",
"Such attention scores can effectively capture the target user's preference related to the inherent property of the target item, as inferred from the data.",
"For example in MovieLens dataset, IARN learns high attention scores for the time steps in user history when the user was watching movies of genre “Adventure” and “Action”.",
"These time steps are highly indicative of his potential preference over the target item, i.e., “The Lost World: Jurassic Park”.",
"In contrast, low attention scores are assigned to those time steps when he was watching movies of other genres, e.g., “Drama”.",
"IARN thus can selectively memorize most relevant time steps of the user's history in predicting his preference over the target item.",
"Similarly, IARN can also select the most relevant time steps in item history to characterize the inherent genre of the item, i.e., those time steps when it was being watched by users who share the same interest as the target user, i.e., “Adventure” and “Action” movies (e.g., “Aliens”).",
"Similar observations can be noted in the other three examples.",
"For instance in the Sport dataset, IARN can infer the most relevant time steps in the user history when the user bought hiking related item; and in the item history when the item was bought by users who like hiking.",
"Such dependency between the relevance of time steps in user history and in item history is highly useful for discovering the link between the target user and item, and thus provides strong interpretations for the recommendation results.",
"Table: Performance of rating prediction (measured by RMSE) and personalized ranking (measured by AUC) of all comparison methods on the six real-world datasets.",
"The best performance is boldfaced; the runner up is labeled with “*”.",
"The results of HieVH and MP2Vec on Netflix is not available (marked by “–”) due to the lack of feature information in the Netflix dataset."
],
[
"Comparative Results",
"Rating Prediction.",
"The left side of Table REF presents the rating prediction performance on the six real-world datasets.",
"BPR, Item2Vec, NCF, and MP2Vec are excluded since RMSE cannot be applied to these methods.",
"BPR and NCF optimize ranking based objective function.",
"Item2Vec and MP2Vec learn the embeddings of items and then adopt the similarity score between item embeddings to predict recommendations, instead of minimizing the difference between the real ratings and the estimated ratings.",
"Several interesting observations can be obtained.",
"It is unsurprising that MF – as the basic LFM – considering no auxiliary information, performs the worst among all the considered methods.",
"By integrating temporal context into LFM, TimeSVD++ outperforms MF.",
"This confirms that modeling temporal dynamics of user preferences can significantly improve the recommendation performance.",
"HieVH is also a LFM based approach, which takes into account the influence of both vertical and horizontal dimensions of feature hierarchies on recommendation.",
"It outperforms MF, and even slightly exceeds TimeSVD++, confirming the effectiveness of auxiliary features for better recommendation.",
"Our proposed approach – IARN, consistently outperforms the other methods in the comparison pool, with an average performance gain (w.r.t.",
"the second best method) of 8.58% on RMSE.",
"Pair-wised t-test demonstrates that the improvements of IARN on all the datasets are significant ($p-$ value$<0.01$ ).",
"Such big enhancement clearly shows the effectiveness of the integration of interacting attention scores as well as auxiliary features.",
"Ranking Performance.",
"We further evaluate the ranking quality of items recommended by the methods in the comparison pool.",
"Results are shown on the right side of Table REF .",
"A number of meaningful findings can be noted from the table.",
"In terms of the LFM based methods, TimeSVD++ and HieVH outperform MF by taking temporal context and feature hierarchies into account, respectively.",
"This observation further verifies the usefulness of the two types of side information for better recommendations.",
"For NN based method, the fact that the performance of MP2Vec is better than that of Item2Vec and NCF also helps to reach the same conclusion, as MP2Vec considers auxiliary features while Item2Vec and NCF do not.",
"The superior performance of NCF over Item2Vec shows the effectiveness of hidden layers in neural networks for modeling non-linear user-item interactions.",
"In both LFM based methods and NN based methods, those specifically designed for personalized ranking, i.e., BPR and NCF, perform better than methods for rating prediction, i.e., MF and Item2Vec, which strongly confirms the conclusion that methods designed for personalized ranking are more efficient than rating prediction methods for the item ranking problem [27].",
"Our proposed approach IARN generally achieves the best performance on item ranking when compared with the other considered methods.",
"This demonstrates the effectiveness of IARN in modeling user and item dynamics for improving recommendation performance.",
"However, the performance improvements of IARN on ranking prediction is far behind those on rating prediction.",
"The underlying explanation is that the objective function of IARN aims to minimize the squared error between the observed ratings and the estimated ratings, which is just in accordance with the definition of RMSE.",
"IARN is therefore more effective on rating prediction.",
"We leave it as future work the improvement of IARN on item ranking."
],
[
"Conclusions",
"User preferences often evolve over time, thus modeling their temporal dynamics is essential for recommendation.",
"This paper proposes the Interacting Attention-gated Recurrent Network (IARN) to accommodate temporal context for better recommendation.",
"IARN can not only accurately measure the relevance of individual time steps of user and item history for recommendation, but also capture the dependencies between user and item dynamics in shaping user-item interactions.",
"We further show that IARN can be easily integrated with auxiliary features for enhanced recommendation performance.",
"Extensive validation on six real-world datasets demonstrates the superiority of IARN against other state-of-the-art methods.",
"For future work, we intend to further improve the effectiveness of IARN on the item ranking problem."
],
[
"Acknowledgement",
"This work is partially funded by the Social Urban Data Lab (SUDL) of the Amsterdam Institute for Advanced Metropolitan Solutions (AMS).",
"This work is partially supported by the SIMTech-NTU Joint Lab on Complex Systems."
]
] | 1709.01532 |
[
[
"Wireless Networks for Mobile Edge Computing: Spatial Modeling and\n Latency Analysis (Extended version)"
],
[
"Abstract Next-generation wireless networks will provide users ubiquitous low-latency computing services using devices at the network edge, called mobile edge computing (MEC).",
"The key operation of MEC, mobile computation offloading (MCO), is to offload computation intensive tasks from users.",
"Since each edge device comprises an access point (AP) and a computer server (CS), a MEC network can be decomposed as a radio access network (RAN) cascaded with a CS network (CSN).",
"Based on the architecture, we investigate network constrained latency performance, namely communication latency (comm-latency) and computation latency (comp-latency) under the constraints of RAN coverage and CSN stability.",
"To this end, a spatial random network is modeled featuring random node distribution, parallel computing, non-orthogonal multiple access, and random computation-task generation.",
"Given the model and the said network constraints, we derive the scaling laws of comm-latency and comp-latency with respect to network-load parameters (density of mobiles and their task-generation rates) and network-resource parameters (bandwidth, density of APs/CSs, CS computation rate).",
"Essentially, the analysis involves the interplay of theories of stochastic geometry, queueing, and parallel computing.",
"Combining the derived scaling laws quantifies the tradeoffs between the latencies, network coverage and network stability.",
"The results provide useful guidelines for MEC-network provisioning and planning by avoiding either of the cascaded RAN or CSN being a performance bottleneck."
],
[
"Introduction",
"One key mission of 5G systems is to provide users ubiquitous computing services (e.g., multimedia processing, gaming and augmented reality) using servers at the network edge, called mobile edge computing (MEC) [1].",
"Compared with cloud computing, MEC can dramatically reduce latency by avoiding transmissions over the backhaul network, among many other advantages such as security and context awareness [2], [3].",
"Most existing work focuses on designing MEC techniques by merging two disciplines: wireless communications and mobile computing.",
"In this work, we explore a different direction, namely the design of large-scale MEC networks with infinite nodes.",
"To this end, a model of MEC network is constructed featuring spatial random distribution of network nodes, wireless transmissions, parallel computing at servers.",
"Based on the model and under network performance constraints, the latencies for communication and computation are analyzed by applying theories of stochastic geometry, queueing, and parallel computing.",
"The results yield useful guidelines for MEC network provisioning and planning.",
"To realize the vision of Internet-of-Things (IoT) and smart cities, MEC is a key enabler providing ubiquitous and low latency access to computing resources.",
"Edge servers in proximity of users are able to process a large volume of data collected from IoT sensors and provide intelligent real-time solutions for various applications, e.g., health care, smart grid, and autonomous driving.",
"Due to its promising potential and the interdisciplinary nature, many new research issues arise in the area of MEC and are widely studied in different fields (see e.g., [3], [4], [5]).",
"In the area of MEC, one research thrust focuses on designing techniques for enabling low-latency and energy-efficient mobile computation offloading (MCO), which offloads computation intensive tasks from mobiles to the edge servers [6], [7], [8], [9], [10], [11], [12], [13].",
"In [6], considering a CPU with a controllable clock, the optimal policy is derived using stochastic-optimization theory for jointly controlling the MCO decision (offload or not) and clock frequency with the objective of minimum mobile energy consumption.",
"A similar design problem is tackled in [7] using a different approach based on Lyapunov optimization theory.",
"Besides MCO, the battery lives of mobile devices can be further lengthened by energy harvesting [8] or wireless power transfer [9].",
"The optimal policies for MEC control are more complex as they need to account for energy randomness [8] or adapt the operation modes (power transfer or offloading) [9].",
"Designing energy-efficient MEC techniques under computation-deadline constraints implicitly attempts to optimize the latency-and-energy tradeoff.",
"The problem of optimizing this tradeoff via computation-task scheduling is formulated explicitly in [10] and [11] and solved using optimization theory.",
"In addition, other design issues for MEC are also investigated in the literature such as optimal program partitioning for partial offloading [12] and data prefetching based on computation prediction [13].",
"Recent research in MEC focuses on designing more complex MEC systems for multiuser MCO [14], [15], [16], [17], [18], [19], [20].",
"One important issue is the joint radio-and-computation resource allocation for minimizing sum mobile energy consumption under their deadline constraints.",
"The problem is challenging due to the multiplicity of parameters and constraints involved in the problem including multi-user channel states, computation capacities of servers and mobiles, and individual deadline and power constraints.",
"A tractable approach for solving the problem is developed in [14] for a single-cell system comprising one edge server for multiple users.",
"Specifically, a so-called offloading priority function is derived that includes all the parameters and used to show a simple threshold based structure of the optimal policy.",
"The problem of joint resource allocation in multi-cell systems is further complicated by the existence of inter-cell interference.",
"An attempt is made in [15] to tackle this problem using optimization theory.",
"In distributed systems without coordination, mobiles make individual offloading decisions.",
"For such systems, it is proposed in [16] that game theory is applied to improve the performance of distributed joint resource allocation in terms of latency and mobile energy consumption.",
"Cooperation between edge servers (or edge clouds) allows their resource pooling and sharing, which helps overcome their limitations in computation capacity.",
"Algorithms for edge-cloud cooperation are designed in [17] based on game theory that enables or disables cooperation so as to maximize the revenues of edge clouds under the constraint of meeting mobiles' computation demands.",
"Compared with the edge cloud, the central cloud has unlimited computation capacity but its long distance from users can incur long latency for offloading.",
"Nevertheless, cooperation between edge and central clouds is desirable when the formers are overloaded.",
"Given such cooperation, queueing theory is applied in [18] to analyze the latency for computation offloading.",
"On the other hand, cooperation between edge clouds can support mobility by migrating computation tasks between servers.",
"Building on the migration technology, a MEC framework for supporting mobility is proposed in [19] to adapt the placements of offloaded tasks in the cloud infrastructure depending on the mobility of the task owners.",
"Besides offloaded tasks, computing services can be also migrated to adapt to mobility but service migration can place a heavy burden on the backhaul network or result in excessive latency.",
"To address this issue, the framework of service duplication by virtualization is proposed in [20].",
"Prior work considers small-scale MEC systems with several users and servers/clouds, allowing the research to focus on designing complex MCO techniques and protocols.",
"On the other hand, it is also important to study a large-scale MEC network with infinite nodes as illustrated in Fig.",
"REF , which is an area not yet explored.",
"From the practical perspective, such studies can yield guidelines and insights useful for operators' provisioning and planning of MEC networks.",
"Figure: A MEC network where mobiles offload computation tasks to computer servers (CSs) by wireless transmission to access points (APs).Figure: The decomposition view of the MEC network.Figure: The spatial model of a MEC network."
],
[
"Modeling Wireless Networks for Mobile Edge Computing ",
"In the past decade, stochastic geometry has been established as a standard tool for modeling and designing wireless networks, creating an active research area [21].",
"A rich set of spatial point processes such as Poisson point process (PPP) and cluster processes have been used to model node locations in a wide range of wireless networks such as cellular networks [22], heterogeneous networks [23], and cognitive radio networks [24].",
"Based on these network models and applying mathematical tools from stochastic geometry, the effects of most key physical-layer techniques on network performance have been investigated ranging from multi-antenna transmissions [25] to multi-cell cooperation [26].",
"Recent advancements in the area can be found in numerous surveys such as [27].",
"Most existing work in this area shares the same theme of how to cope with interference and hostility of wireless channels (e.g., path loss and fading) so as to ensure high coverage and link reliability for radio access networks (RAN) or distributed device-to-device networks.",
"In contrast, the design of large-scale MEC networks in Fig.",
"REF has different objectives, all of which should jointly address two aspects of network performance, namely wireless communication and edge computing.",
"Modeling a MEC network poses new challenges as its architecture is more complex than a traditional RAN and can be decomposed as a RAN cascaded with a computer-server network (CSN) as illustrated in Fig.",
"REF .",
"The power of modeling MEC networks using stochastic geometry lies in allowing network performance to be described by a function of a relatively small set of network parameters.",
"To be specific, as shown in Fig.",
"REF , the process of mobiles is parametrized by mobile density, the RAN by channel bandwidth and access-point (AP) density, and the CSN by CS density and CS computation capacity.",
"Besides the parameters, the performance of a MEC network is measured by numerous metrics.",
"Like small-scale systems (see e.g., [10] and [11]), the link-level performance of the MEC network is measured by latency, which can be divided into latency for offloading in the RAN, called communication latency (comm-latency) and latency for computing at CSs, called computation latency (comp-latency).",
"At the network level, the coverage of the RAN of a MEC network is typically measured by connectivity probability (also called coverage probability [27]), quantifying the fraction of users having reliable links to APs.",
"A similar metric, called stability probability, can be defined for measuring the stability of the CSN, quantifying the fraction of CSs having finite comp-latency.",
"There exist potentially complex relations between these four metrics that are regulated by the said network parameters.",
"Existing results focusing solely on RAN (see e.g., [27]) are insufficient for quantifying these relations.",
"Instead, it calls for developing a more sophisticated analytical approach integrating theories of stochastic geometry, queueing, and parallel computing.",
"Last, it is worth mentioning that comm-latency and comp-latency have been extensively studied in the literature mostly for point-to-point systems using queueing theory (see e.g., [28], [29]).",
"However, studying such latency in large-scale networks is much more challenging due to the existence of interference between randomly distributed nodes.",
"As a result, there exist only limited results on comm-latency in such networks [30], [31], [32].",
"In [30], the comm-latency given retransmission is derived using stochastic geometry for the extreme cases with either static nodes or nodes having high mobility.",
"The analysis is generalized in [31] for finite mobility.",
"Then the approach for comm-latency as proposed in [30] and [31] is further developed in [32] to integrate stochastic geometry and queueing theory.",
"Compared with these studies, the current work considers a different type of network, namely the MEC network, and explores a different research direction, namely the tradeoff between comm-latency and comp-latency under constraints on the mentioned network-level performance metrics."
],
[
"Contributions",
"This work represents the first attempt on modeling a large-scale MEC network using stochastic geometry.",
"The proposed model has several features admitting tractable analysis of network latency performance.",
"First, the locations of co-located pairs of CS and AP and the mobiles are distributed as two independent homogeneous PPPs.",
"Second, multiple access is enabled by spread spectrum [33], which underpins the technology of code-domain Non-Orthogonal Multiple Access (NOMA) to be deployed in 5G systems for enabling massive access [34].",
"Using the technology, interference is suppressed by a parameter called spreading factor, denoted as $G$ , at the cost of data bandwidth reduction.",
"Third, each mobile randomly generates a computation task in every time slot.",
"Last, each CS computes multiple tasks simultaneously by parallel computing realized via creating a number of virtual machines (VMs), where the so called input/output (I/O) interference in parallel computing is modelled [35].",
"In this work, we propose an approach building on the spatial network model and the joint applications of tools from diversified areas including stochastic geometry, queueing, and parallel computing.",
"Though the network performance analysis relies on well known tools, their applications are far more than straightforward.",
"In fact, new challenges arise from the coupling of communication and edge computing in the MEC network.",
"For example, the simple server model (with memoryless service time) in the traditional queueing theory is now replaced with a more complex MEC server model featuring dynamic virtual machines and their I/O interference.",
"As another example, the random computing-task arrivals are typically modelled as a single stochastic process in conventional computing/queueing systems but the current model has to account for numerous network features ranging from random node distributions to multiple access.",
"The complex network model introduces new technical challenges that call for the development of a systematic framework for studying the MEC network performance and deployment, which forms the theme of this work.",
"The main contributions are summarized below.",
"Modeling a MEC network using stochastic geometry: As mentioned, this work presents a novel model of a large-scale MEC network constructed using stochastic geometry.",
"Given the complexity of the network, the contribution in network modeling lies in proposing a model that is not only sufficiently practical but at the same time allows a tractable approach of analyzing network latency performance, by integrating stochastic geometry, parallel computing, and queuing theory.",
"The results and insights are summarized as follows.",
"Communication latency: The expected comm-latency for an offloaded task, denoted as $\\mathsf {T}_{\\textrm {comm}}$ , is minimized under a constraint on the network connectivity probability.",
"This is transformed into a constrained optimization problem of the spreading factor $G$ .",
"Solving the problem yields the minimum $\\mathsf {T}_{\\textrm {comm}}$ .",
"The result shows that when mobiles are sparse, the full bandwidth should be allocated for data transmission so as to minimize $\\mathsf {T}_{\\textrm {comm}}$ .",
"However, when mobiles are dense, spread spectrum with large $G$ is needed to mitigate interference for satisfying the network-coverage constraint, which increases $\\mathsf {T}_{\\textrm {comm}}$ .",
"As a result, the minimum $\\mathsf {T}_{\\textrm {comm}}$ diminishes inversely proportional to the channel bandwidth and as a power function of the allowed fraction of disconnected users with a negative exponent, but grows sub-linearly with the expected number of mobiles per AP (or CS).",
"In addition, $\\mathsf {T}_{\\textrm {comm}}$ is a monotone increasing function of the task-generation probability per slot that saturates as the probability approaches one.",
"Analysis of RAN offloading throughput: The RAN throughput, which determines the load of the CSN (see Fig.",
"REF ), can be measured by the expected task-arrival rate at a typical AP (or CS).",
"The rate is shown to be a quasi-concave function of the expected number of mobiles per AP, which first increases and then decreases as the ratio grows.",
"In other words, the expected task-arrival rate is low in both sparse and dense networks.",
"The maximum rate is proportional to the bandwidth.",
"Computation latency Analysis: First, to maximize CS computing rates, it is shown that the dynamic number of VMs at each CS should be no more than a derived number to avoid suffering rate loss due to their I/O interference.",
"Then to ensure stable CSN, it is shown that the resultant maximum computing rate should be larger than the task-arrival rate scaled by a factor larger than one, which is determined by the allowed fraction of unstable CSs.",
"Based on the result for parallel computing, tools from stochastic geometry and M/M/m queues are applied to derive bounds on the expected comp-latency for an offloaded task, denoted as $\\mathsf {T}_{\\textrm {comp}}$ .",
"The bounds show that the latency is inversely proportional to the maximum computing rate and linearly proportional to the total task-arrival rate at the typical CS (or AP).",
"Consequently, $\\mathsf {T}_{\\textrm {comp}}$ is a quasi-concave function of the expected number of users per CS (or AP) while $\\mathsf {T}_{\\textrm {comm}}$ is a monotone increasing function.",
"Network provisioning and planning: Combining the above results suggest the following guidelines for network provisioning and planning.",
"Given a mobile density, the AP density should be chosen for maximizing the RAN offloading throughput under the network-coverage constraint.",
"Then sufficient bandwidth should be provisioned to simultaneously achieve the targeted comm-latency for offloading a task.",
"Last, given the mobile and RAN parameters, the CS computation capacities are planned to achieve the targeted comp-latency for a offloaded task as well as enforcing the network-stability constraint.",
"The derived analytical results simplify the calculation in the specific planning process."
],
[
"Modeling MEC Networks",
"In this section, a mathematical model of the MEC network as illustrated in Fig.",
"REF is presented."
],
[
"Network Spatial Model",
" APs (and thus their co-located CSs) are randomly distributed in the horizontal plane and are modelled as a homogeneous PPP $\\Omega = \\lbrace Y\\rbrace $ with density $\\lambda _b$ , where $Y \\in {R}^{2}$ is the coordinate of the corresponding AP.",
"Similarly, mobiles are modelled as another homogeneous PPP $\\Phi = \\lbrace X\\rbrace $ independent of $\\Omega $ and having the density $\\lambda _m$ .",
"Define a MEC-service zone for each AP, as a disk region centered at $Y$ and having a fixed radius $r_0$ , denoted by $\\mathcal {O}(Y, r_0)$ , determined by the maximum uplink transmission power of each mobile (see Fig.",
"REF ).",
"A mobile can access a AP for computing if it is covered by the MEC-service zone of the AP.",
"It is possible that a mobile is within the service ranges of more than one AP.",
"In this case, the mobile randomly selects a single AP to receive the MEC service.",
"As illustrated in Fig.",
"REF , combining the randomly located MEC-service zones, $\\cup _{Y \\in \\Omega } \\mathcal {O}(Y, r_0)$ , forms a coverage process.",
"Covered mobiles are referred to as active ones and others inactive since they remain silent.",
"To achieve close-to-full network coverage, let the fraction of inactive mobiles be no more than a small positive number $\\delta $ .",
"Then the radius of MEC-service zones, $r_0$ , should be set as $r_0 = \\sqrt{\\frac{\\ln \\frac{1}{\\delta }}{\\pi \\lambda _b }}$ [27].",
"Given $r_0$ , the number of mobiles covered by an arbitrary MEC service zone follows a Poisson distribution with mean $\\lambda _m \\pi r_0^2$ .",
"Consider a typical AP located at the origin.",
"Let $X_0$ denote a typical mobile located in the typical MEC service zone $\\mathcal {O}(o, r_0)$ .",
"Without loss of generality, the network performance analysis focuses on the typical mobile."
],
[
"Model of Mobile Task Generation",
" Time is divided into slots having a unit duration.",
"Consider an arbitrary mobile.",
"A computation task is randomly generated in each slot with probability $p$ , referred to as the task-generation rate0.1cmThe random task generation is an abstracted model allowing tractable analysis, and it is widely used in the literature in the same vein.",
"The statistics of task generation can be empirically measured by counting the number of user service requests, which is shown in [36] and [37] to be bursty and periodical.",
"It is interesting to use a more general task generation model, which is outside the scope of current work..",
"The generated tasks are those favorable offloading in terms of energy efficiency such that offloading can save more energy than the local computing.",
"The analysis on the offloading favorable condition will be given in the sequel.",
"Task generations over two different slots are assumed to be independent.",
"The mobile has a unit buffer to store at most a single task for offloading.",
"A newly generated task is sent for offloading when the buffer is empty or otherwise computed locally.",
"This avoids significant queueing delay that is unacceptable in the considered case of latency-sensitive mobile computation.",
"For simplicity, offloading each task is assumed to require transmission of a fixed amount data.",
"The transmission of a single task occupies a single frame lasting $L$ slots.",
"The mobile checks whether the buffer is empty at the end of every $L$ slots and transmits a stored task to a serving AP.",
"Define the task-offloading probability as the probability that the mobile's buffer is occupied, denoted as $p_L$ .",
"Equivalently, $p_L$ gives the probability that at least one task is generated within one frame: $p_L=1-(1-p)^L.$ Thereby, the task-departure process at a mobile follows a Bernoulli process with parameter $p_L$ provided the radio link is reliable (see discussion in the sequel)."
],
[
"Radio Access Model",
"Consider an uplink channel with the fixed bandwidth of $B$ Hz.",
"The channel is shared by all mobiles for transmitting data containing offloaded tasks to their serving APs.",
"The CDMA (or code-domain NOMA) is applied to enable multiple access.",
"For CDMA based on the spread-spectrum technology, each mobile spreads every transmitted symbol by multiplying it with a pseudo-random (PN) sequence of chips (1s and $-1$ s), which is generated at a much higher rate than the symbols and thereby spreads the signal spectrum [33].",
"The multiple access of mobiles is enabled by assigning unique PN sequences to individual users.",
"A receiver then retrieves the signal sent by the desired transmitter by multiplying the multiuser signal with the corresponding PN sequence.",
"The operation suppresses inference and de-spreads the signal spectrum to yield symbols.",
"Let $G$ denote the spreading factor defined as the ratio between the chip rate and symbol rate, which is equivalent to the number of available PN sequences.",
"The cross-correlation of PN sequences is proportional to $\\frac{1}{G}$ and approaches to zero as $G$ increases.",
"As a result, the interference power is reduced by the factor of $G$ [33].0.1cmFor the special case of synchronous multiuser transmissions, orthogonal sequences (e.g., Hadamard sequences) can be used instead of PN sequences to achieve orthogonal access [33].",
"However, the maximum number of simultaneous users is $G$ , making the design unsuitable for massive access.",
"On the other hand, the price for spread spectrum is that the bandwidth available to individual mobiles is reduced by $G$ , namely $\\frac{B}{G}$ .",
"Remark 1 (CDMA vs. OFDMA) While CDMA is expected to enable non-orthogonal access in next-generation systems, orthogonal frequency division multiple access (OFDMA) has been widely deployed in existing system.",
"However, OFDMA limits the number of simultaneous users to be no more than the number of orthogonal sub-channels.",
"Compared with OFDMA, CDMA separates different users by PN sequences.",
"The number of possible PN sequences can be up to $2^G -1$ with $G$ being the spreading factor (sequence length).",
"In theory, an equal number of simultaneous users can be supported by CDMA that can be potentially much larger than that by OFDMA.",
"Allowing non-orthogonality via CDMA provides a graceful tradeoff between the system-performance degradation and the number of simultaneous users, facilitating massive access in 5G.",
"The current analysis of comm-latency can be straightforwardly extended to OFDMA by removing interference between scheduled users.",
"For unscheduled users, comm-latency should include scheduling delay and the corresponding analysis is standard (see e.g., [38]).",
"Uplink channels are characterized by path-loss and small-scale Rayleigh fading.",
"Assuming transmission by a mobile with the fixed power $\\eta $ , the received signal power at the AP is given by $\\eta g_X |Y - X|^{-\\alpha }$ , where $\\alpha $ is the path-loss exponent, the $\\exp (1)$ random variable (RV) $g_X$ represents Rayleigh fading and $|X-Y|$ denotes the Euclidian distance between $X$ and $Y$ .",
"Based on the channel model, the power of interference at the typical AP $Y_0$ , denoted by $I$ , can be derived as follows.",
"Among potential interferers for the typical AP, the fraction of $\\delta $ is outside MEC-service zones.",
"Given random task generation discussed earlier, each interferer transmits with probability $p_L$ .",
"Consequently, the active interferers form a PPP given by $\\tilde{\\Phi }$ with density $(1-\\delta )p_L\\lambda _m$ resulting from thinning $\\Phi $ .",
"It follows that the interference power $I$ can be written as $I = \\frac{1}{G}\\sum _{X \\in \\tilde{\\Phi }} \\eta g_X |X|^{-\\alpha }$ , where the factor $\\frac{1}{G}$ is due to the spread spectrum.",
"Consider an interference-limited radio-access network where channel noise is negligible.",
"The received $\\mathsf {SIR}$ of the typical mobile is thus given as $\\mathsf {SIR}_0 = \\frac{g_{X_0} |X_0|^{-\\alpha }}{\\frac{1}{G}\\sum _{X \\in \\tilde{\\Phi }} \\eta g_X |X|^{-\\alpha }}.$ The condition for successful offloading is that $\\mathsf {SIR}$ exceeds a fixed threshold $\\theta $ depending on the coding rate.",
"Specifically, given $\\theta $ , the spectrum efficiency is $\\log _2(1+\\theta )$ (bits/sec/Hz) [27].",
"It follows that to transmit a task having a size of $\\ell $ bits within a frame, the frame length $L$ should satisfy $L =\\frac{G\\ell }{B \\cdot t_0 \\cdot \\log _2(1+\\theta )}$ (in slots) where $t_0$ is the length of a slot (in sec).",
"Define the minimum time for transmitting a task using the full bandwidth $B$ as $\\mathsf {T}_{\\min } = \\frac{\\ell }{B \\cdot t_0\\cdot \\log _2(1+\\theta )}$ for ease of notation, giving $L = G\\mathsf {T}_{\\min }$ .",
"Assumption 1 (Slow Fading) We assume that channels vary at a much slower time scale than that for mobile computation.",
"To be specific, the mobile locations and channel coefficients $\\lbrace g_X\\rbrace $ remain fixed in the considered time window of computation offloading.",
"Remark 2 (Fast Fading) In the presence of sufficiently high mobility, the channel variation can be faster than edge computation, resulting in fast fading.",
"In this case, a mobile facing an unfavorable channel can rely on retransmission to exploit the channel variation for reliable offloading.",
"Nevertheless, this results in retransmission delay and thereby increases comm-latency.",
"It is straightforward to analyze the extra latency in a large-scale network by applying an existing method (see e.g., [30]).",
"By Assumption REF , mobiles' $\\mathsf {SIR}$ s remain constant and thereby mobiles can be separated into connected and disconnected mobiles.",
"To be specific, a mobile is connected to an AP if the corresponding SIR is above the threshold $\\theta $ or otherwise disconnected.",
"Figure: Markov chain modeling the tasks queueing for computation at the typical CS where Λ\\Lambda is the arrival rate and μ(m)\\mu (m) is the computation rate given mm waiting tasks.We consider both synchronous and asynchronous multiuser transmissions defined in existing wireless standards such as 3GPP LTE.",
"For synchronous transmissions, the frame boundaries of different users are aligned so as to facilitate protocols such as control signaling and channel feedback.",
"Synchronization incurs network overhead for implementing a common clock as well as increases latency.",
"For asynchronous transmissions, the said constraint on frame boundaries is not applied and thus the transmission of each mobile is independent of those of others.",
"The transmissions modes lead to different task-arrival models for CSs.",
"Specifically, given synchronous transmissions, the offloaded tasks arrive at a CS in batches and periodically as illustrated in Fig.",
"REF (a).",
"The number of arrival tasks in each batch is random depending on the number of connected mobiles in the same MEC-service zone.",
"On the other hand, given asynchronous transmissions, the offloaded tasks arrive at an AP at different time instants as illustrated in Fig.",
"REF (b)."
],
[
"Parallel-Computing Model",
"Upon their arrivals at APs, tasks are assumed to be delivered to CSs without any delay and queue at the CS buffer for computation on the first-come-first-served basis.",
"Moreover, each CS is assumed to be provisioned with large storage modelled as a buffer with infinite capacity.",
"At each CS, parallel computing of multiple tasks is implemented by creating virtual machines (VM) on the same physical machine (PM) [35].",
"VMs are created asynchronously such that a VM can be added or removed at any time instant.",
"It is well known in the literature that simultaneous VMs interfere with each other due to their sharing common computation resources in the PM e.g., CPU, memory, buses for I/O.",
"The effect is called I/O interference that reduces the computation speeds of VMs.",
"The model of I/O interference as proposed in [35] is adopted where the expected computation time for a single taskThe latency caused by creating and releasing VMs is not explicitly considered.",
"It is assumed to be part of computation time., denoted by $T_c$ , is a function of the number of VMs, $m$ : $T_c(m) = T_0 (1+d)^{m-1}, $ where $T_0$ is the expected computation time of a task in the case of a single VM ($m=1$ ) and $d$ is the degradation factor due to I/O interference between VMs.",
"One can observe that $T_c$ is a monotone increasing function of $d$ .",
"For tractability, we assume that the computation time for a task is an $\\exp (T_c)$ RV following the common assumption in queueing theory [35]."
],
[
"CS Queuing Model",
"The general approach of analyzing comp-latency relies on the interplay between parallel-computing and queueing theories.",
"In particular, for the case of asynchronous offloading, the task arrival at the typical AP is approximated as a Poisson process for the following reasons.",
"Due to the lack of synchronization between mobiles, the time instants of tasks arrivals are approximately uniform in time.",
"Furthermore, at different time instants, tasks are generated following i.i.d.",
"Bernoulli distributions based on the model in Section REF .",
"It is well known that the superposition of independent arrival process behaves like a Poisson process [39].",
"Assumption 2 For the case of asynchronous offloading, given $N$ connected mobiles and the spreading factor $G$ , the task arrivals at the typical AP are approximated as a Poisson process with the arrival rate of $\\Lambda (N, G)= \\frac{N p_L }{L} = \\frac{N p_L }{G \\mathsf {T}_{\\min }}$ .",
"The Poisson approximation is shown by simulation to be accurate in Appendix III.",
"Given the Poisson arrival process and exponentially distributed computation time, the random number of tasks queueing at the typical CS can be modelled as a continuous-time Markov chain as illustrated in Fig.",
"REF [28].",
"In the Markov chain, $\\Lambda $ denotes the task-arrival rate in Assumption REF and $\\mu (k)$ denotes the CS-computation rate (task/slot) given $k$ tasks in the CS.",
"The CS-computation rate is maximized in the sequel by optimizing the number of VMs based on the queue length.",
"Last, the result-downloading phase is not considered for brevity.",
"First, the corresponding latency analysis is similar to that for the offloading phase.",
"Second, the latency for downloading is negligible compared with those for offloading.",
"The reasons are that computation results typically have small sizes compared with offloaded tasks and furthermore downlink transmission rates are typically much higher than uplink rates."
],
[
"Performance Metrics",
"The network performance is measured by two metrics: comm-latency and comp-latency.",
"The definitions of metrics build on the design constraints for ensuring network connectivity and stability defined as follows.",
"Definition 1 (Network Coverage Constraint) The RAN in Fig.",
"REF is designed to be $\\epsilon $ -connected, namely that the portion of mobiles is no less than $(1-\\epsilon )$ , where $0< \\epsilon \\ll 1$ .",
"The fraction of connected mobiles is equivalent to the success probability, a metric widely used for studying the performance of random wireless networks [27].",
"For the MEC network, the success probability is renamed as connectivity probability and defined for the typical mobile as the following function of the spreading factor $G$ : $\\mathsf {p}_c(G) = \\mathrm {Pr}\\left( \\mathsf {SIR}_0 \\ge \\theta \\right),$ where $\\mathsf {SIR}_0$ is given in (REF ).",
"Then the network coverage constraint can be written as $\\mathsf {p}_c(G) \\ge (1 - \\epsilon )$ .",
"Under the connectivity constraint, most mobiles are connected to APs.",
"Then the comm-latency, denoted as $\\mathsf {T}_{\\textrm {comm}}$ , is defined as the expected duration required for a connected mobile to offload a task to the connected AP successfully.",
"The latency includes both waiting time at the mobile's buffer and and the transmission time.",
"Next, consider the computation load of the typical AP.",
"Since the number of mobiles connected to the AP is a RV, there exists non-zero probability that the AP is overloaded, resulting in infinite queueing delay.",
"In this case, the connected mobiles are referred to as being unstable.",
"To ensure most mobiles are stable, the following constraint is applied on the network design.",
"Definition 2 (Network Stability Constraint) The CSN in Fig.",
"REF is designed to be $\\rho $ -stable, namely that the fraction of stable CSs is no less than $(1-\\rho )$ , where $0< \\rho \\ll 1$ .",
"The fraction $\\rho $ is equivalent to the probability that the typical CS is stable, denoted as $\\mathsf {p}_s$ .",
"Under the stability constraint, most connected mobiles are stable.",
"Then the comp-latency, denoted by $\\mathsf {T}_{\\textrm {comp}}$ , is defined for the typical connected mobile as the expected duration from the instant when an offloaded task arrives at the serving CS until the instant when the computation of the task is completed, which includes both queueing delay and actual computation time.",
"Last, given the above definitions, the network is referred to as being communication-limited (comm-limited) if $\\mathsf {T}_{\\textrm {comm}} \\gg \\mathsf {T}_{\\textrm {comp}}$ and computation-limited (comp-limited) if $\\mathsf {T}_{\\textrm {comm}} \\ll \\mathsf {T}_{\\textrm {comp}}$ ."
],
[
"Communication Latency Analysis",
"In this section, the comm-latency defined in the preceding section is analyzed building on results from the literature of network modeling using stochastic geometry.",
"Then the latency is minimized by optimizing the spreading factor for CDMA, which regulates the tradeoff between the transmission rates of connected mobiles and network-connectivity performance."
],
[
"Feasible Range of Spreading Factor",
" As mentioned, the spreading factor $G$ is a key network parameter regulating the tradeoff between network coverage and comm-latency.",
"To facilitate subsequent analysis, under the network constraint in Definition REF , the feasible range of $G$ is derived as follows.",
"The result is useful for minimizing the comm-latency in the next sub-section.",
"To this end, consider the connectivity probability defined in (REF ).",
"Using a similar approach as the well-known one for deriving network success probability using stochastic geometry (see e.g., [22]), we obtain the following result with the proof omitted for brevity.",
"Lemma 1 (Connectivity Probability) Given the spreading factor $G$ , the connectivity probability of a typical mobile is given as $\\mathsf {p}_c(G) = \\frac{1 - \\exp \\left(- \\xi (G)\\right)}{\\xi (G)}, $ where $\\xi (G)$ is defined as $\\xi (G)=\\frac{2(1-\\delta )\\left( 1 - (1-p)^{G \\mathsf {T}_{\\min }} \\right)\\ln \\delta ^{-1}}{\\alpha } \\mathcal {B}( \\alpha ) \\left( \\frac{\\lambda _m}{\\lambda _b}\\right)\\left( \\frac{\\theta }{G} \\right)^{\\frac{2}{\\alpha }},$ and $\\mathcal {B}( \\alpha ) \\triangleq \\int _{0}^{1}\\kappa ^{\\frac{2}{\\alpha }-1} (1-\\kappa )^{-\\frac{2}{\\alpha }}\\mathrm {d} \\kappa $ denotes the Beta function.",
"Recall that the network coverage constraint in Definition REF requires that $\\mathsf {p}_c(G)\\ge (1-\\epsilon )$ .",
"Note that $G$ is an important system parameter affecting both the transmission rates and the connectivity probability as elaborated in the following remark.",
"Remark 3 (Transmission Rates vs. Connectivity) The spreading factor $G$ of CDMA controls the tradeoff between mobile transmission rates and network connectivity probability.",
"On one hand, increasing $G$ reduces the bandwidth, $\\frac{B}{G}$ , available to each mobile, thereby reducing the transmission rate and increasing comm-latency.",
"As the result, given longer frames with the task-generation rate being fixed, more mobiles are likely to have tasks for offloading at the beginning of each frame, increasing the density of interferers.",
"On the other hand, growing $G$ suppresses interference power by the factor $G$ via spread spectrum.",
"As a result, the connectivity probability grows.",
"Given the two opposite effects, one should expect that in the case of a stringent connectivity constraint, either small or large value for $G$ is preferred but no the moderate ones.",
"Next, the effects of the spreading factor as discussed in Remark REF are quantified by deriving the feasible range of $G$ under the connectivity constraint.",
"Define the Lambert function, $W(x)$ , as the solution for the equation $W(x)e^{W(x)}=x$ .",
"Then using the result in Lemma REF , the coverage constraint $\\mathsf {p}_c(G) \\ge (1 - \\epsilon )$ is equivalent to $\\xi (G) \\le \\mathcal {F}(\\epsilon )$ with the function $\\mathcal {F}(\\epsilon )$ defined as $\\mathcal {F}(\\epsilon )= W\\left( -\\frac{e^{-\\frac{1}{1-\\epsilon }}}{1 - \\epsilon } \\right) + \\frac{1}{1- \\epsilon }.$ Notice that $\\lim _{\\epsilon \\rightarrow 0}\\frac{d}{d\\epsilon }W\\left( -\\frac{e^{-\\frac{1}{1-\\epsilon }}}{1 - \\epsilon } \\right) = 1$ .",
"Moreover, $W\\left( -\\frac{e^{-\\frac{1}{1-\\epsilon }}}{1 - \\epsilon } \\right) = -1$ at $\\epsilon = 0$ .",
"It follows that from these two results that $\\mathcal {F}(\\epsilon )$ can be approximated as $\\mathcal {F}(\\epsilon ) \\approx 2\\epsilon , \\qquad \\epsilon \\ll 1.$ In addition, $\\xi (G)$ is maximized at the point of $G=g_0$ of which the existence and uniqueness are proved in Lemma REF .",
"If $\\xi (g_0)\\le \\mathcal {F}(\\epsilon )$ , it is straightforward that any $G$ satisfies the condition of (REF ).",
"Otherwise, the feasible range of $G$ satisfying the connectivity is provided in Proposition REF .",
"Lemma 2 (Properties of $\\xi (G)$ ) The function $\\xi (G)$ in (REF ) attains its maximum at $G = g_0$ with $g_0=\\frac{\\alpha W\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right) + 2}{{\\alpha \\mathsf {T}_{\\min }} \\ln (1-p)}.$ Moreover, $\\xi (G)$ is monotone increasing in the range $[-\\infty , g_0]$ and monotone decreasing in the range $[g_0, \\infty ]$ .",
"Proof: See Appendix REF .",
"$\\Box $ Proposition 1 (Feasible Range of Spreading Factor) Under the network connectivity constraint, the feasible range of $G$ is $G \\ge 1$ if $\\xi (g_0)\\le \\mathcal {F}(\\epsilon )$ , where $g_0$ is given in (REF ).",
"If $\\xi (g_0) > \\mathcal {F}(\\epsilon )$ , the feasible range of $G$ is $\\mathcal {S}=\\mathcal {S}_1\\bigcup \\mathcal {S}_2$ where $\\mathcal {S}_1=\\lbrace G\\in {Z}^+|1\\le G\\le g_a\\rbrace , \\qquad \\mathcal {S}_2=\\lbrace G\\in {Z}^+| G\\ge g_b \\rbrace ,$ where $g_a$ and $g_b$ are the two roots of the equation $\\xi (G) = \\mathcal {F}(\\epsilon )$ .",
"Based on Lemma REF , the function $\\xi (G)$ is monotone increasing over $\\mathcal {S}_1$ but monotone decreasing over $\\mathcal {S}_2$ .",
"In addition, if $g_a < 1$ , $\\mathcal {S}_1$ is empty and the feasibility range of $G$ reduces to $\\mathcal {S}_2$ ."
],
[
"Communication Latency",
" Recall that the comm-latency of connected mobiles $\\mathsf {T}_{\\textrm {comm}}$ comprises the expected waiting time for offloaded tasks at mobiles, denoted as $\\mathsf {T}_{\\textrm {comm}}^{(a)}$ , and transmission delay, denoted as $\\mathsf {T}_{\\textrm {comm}}^{(b)}$ .",
"Consider the expected waiting time.",
"Recalling that the offloading protocol in Section REF , the first task arrival during $L$ slots is delivered to the offloading buffer and the subsequent tasks are forwarded to the local computation unit.",
"Let $K$ denote the slot index when an offloaded task arrives at the offloading buffer.",
"It follows that the probability distribution of $K$ follows a conditional geometric distribution, i.e., $\\Pr (K=k)=\\frac{p (1-p)^{k-1}}{1-(1-p)^L}$ , where $k = 1, 2, \\cdots , L$ and the normalization term $1-(1-p)^L$ gives the probability that at least one task arrives during a single frame.",
"Thereby, the expected waiting time is given as $\\mathsf {T}_{\\textrm {comm}}^{(a)} = \\sum _{k=1}^L(L-k) \\frac{p(1-p)^{k-1}}{1-(1-p)^L}&= \\frac{L}{1-(1-p)^L}-\\frac{1}{p}.$ Next, consider the transmission time for a single task in a frame that spans $L$ slots.",
"Recall that $L=G\\mathsf {T}_{\\min }$ where $\\mathsf {T}_{\\min }$ is the minimum time for transmitting a task as defined earlier.",
"Combining $\\mathsf {T}_{\\textrm {comm}}^{(b)} = G\\mathsf {T}_{\\min }$ and $\\mathsf {T}_{\\textrm {comm}}^{(a)}$ in (REF ) gives the following result.",
"Lemma 3 (Comm-Latency) Given the spreading factor $G$ , the comm-latency of the typical mobile $\\mathsf {T}_{\\textrm {comm}} $ (in slot) is given as $\\mathsf {T}_{\\textrm {comm}}(G)= G{\\mathsf {T}_{\\min }}+ \\frac{G \\mathsf {T}_{\\min }}{1 - (1-p)^{G \\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $\\mathsf {T}_{\\min }$ is the minimum time for transmitting a task using full bandwidth.",
"Next, consider the minimization of the comm-latency over the spreading factor $G$ .",
"Using (REF ), it is straightforward to show that the comm-latency $\\mathsf {T}_{\\textrm {comm}}(G)$ is a monotone increasing function of $G$ .",
"Therefore, minimizing comm-latency is equivalent to minimizing $G$ .",
"It follows from Proposition REF that the minimum of $G$ , $G^*=\\underset{G\\in {\\mathcal {S}}}{\\min } G$ , is given as $G^* = \\left\\lbrace \\begin{aligned}&g_b, &&\\mathcal {S}_1 = \\emptyset ,\\\\&1, && \\text{otherwise}.\\end{aligned}\\right.$ Substituting $G^*$ into (REF ) gives the minimum comm-latency as shown in the following theorem.",
"Theorem 1 (Minimum Comm-Latency) By optimizing the spreading factor $G$ , the minimum comm-latency (in slot), denoted as $\\mathsf {T}_{\\textrm {comm}}^* $ , is given as follows.",
"If $\\mathcal {S}_1$ in (REF ) is non-empty, $\\mathsf {T}_{\\textrm {comm}}^*= \\mathsf {T}_{\\min }+ \\frac{\\mathsf {T}_{\\min } }{1 - (1-p)^{\\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $\\mathsf {T}_{\\min } = \\frac{\\ell }{B \\cdot t_0 \\cdot \\log _2(1+\\theta )}$ .",
"If $\\mathcal {S}_1$ is empty, $\\mathsf {T}_{\\textrm {comm}}^*= g_b\\mathsf {T}_{\\min }+ \\frac{g_b \\mathsf {T}_{\\min } }{1 - (1-p)^{g_b \\mathsf {T}_{\\min }}} - \\frac{1}{p}, $ where $g_b$ is specified in Proposition REF .",
"Consider the second case in Theorem REF .",
"The comm-latency $\\mathsf {T}_{\\textrm {comm}}^*$ can be approximated in closed-form if $g_b\\mathsf {T}_{\\min }$ is sufficiently large.",
"For this case, $\\left[1 - (1-p)^{g_b \\mathsf {T}_{\\min }}\\right]\\approx 1$ and thus the function $\\xi (G)$ in (REF ) can be approximated as $\\xi (G) \\approx \\frac{2(1-\\delta )\\ln \\delta ^{-1}}{\\alpha } \\mathcal {B}( \\alpha ) \\left( \\frac{\\lambda _m}{\\lambda _b}\\right) \\left(\\frac{\\theta }{G}\\right)^{\\frac{2}{\\alpha }}.$ It follows from Theorem REF and (REF ) that if $\\mathcal {S}_1$ is empty and $g_b\\mathsf {T}_{\\min }$ is large, $\\mathsf {T}_{\\textrm {comm}}^*\\approx 2 g_b\\mathsf {T}_{\\min } - \\frac{1}{p}, $ where $g_b\\approx \\left[ \\frac{(1-\\delta )\\ln \\delta ^{-1}\\mathcal {B}( \\alpha ) }{\\alpha \\epsilon }\\left( \\frac{\\lambda _m}{\\lambda _b}\\right)\\right]^{\\frac{\\alpha }{2}}\\theta .$ Remark 4 (Sparse Network vs.",
"Dense Network) The first and second cases in Theorem REF correspond to sparse and dense networks, respectively, as measured by the mobile-to-AP density ratio $\\lambda _m/\\lambda _b$ .",
"In the first case ($\\mathcal {S}_1\\ne \\emptyset $ ), the network is sufficiently sparse, namely the ratio $\\lambda _m/\\lambda _b$ is sufficiently small, such that the optimal spreading factor $G^*=1$ and the resultant comm-latency is independent of the ratio as shown in the theorem.",
"In other words, for this case, it is optimal to allocate all bandwidth for increasing the transmission rate instead of reducing it for the purpose of suppressing interference to satisfy the network connectivity constraint.",
"In contrast, in the second case ($\\mathcal {S}_1= \\emptyset $ ), the network is relatively dense and it is necessary to apply spread spectrum to reduce interference so as to meet the connectivity requirement, corresponding to $G^* > 1$ .",
"As the result, the minimum comm-latency scales with the density ratio as $\\mathsf {T}_{\\textrm {comm}}^*\\propto \\left(\\frac{\\lambda _m}{\\lambda _b}\\right)^{\\frac{\\alpha }{2}}$ as one can observe from (REF ).",
"Remark 5 (Effects of Network Parameters) Substituting $\\mathsf {T}_{\\min } = \\frac{\\ell }{B \\cdot t_0\\cdot \\log _2(1+\\theta )}$ into (REF ) gives that for a relatively dense network, the comm-latency scales as $\\boxed{\\mathsf {T}_{\\textrm {comm}}^*\\propto \\frac{\\ell }{B}\\left(\\frac{\\lambda _m}{\\epsilon \\lambda _b}\\right)^{\\frac{\\alpha }{2}} - \\frac{1}{p}.",
"}$ The scaling laws shows the effects of network parameters including the task size $\\ell $ , bandwidth $B$ , mobile density $\\lambda _m$ and AP density $\\lambda _b$ , and the task-generation probability per slot $p$ ."
],
[
"Task-Arrival Rates at APs/CSs",
"The offloading throughput of the RAN represents the load of the CSN (see Fig.",
"REF ).",
"The throughput can be measured by the expected task-arrival rate (in number of tasks per slot) at the typical AP (equivalently the typical CS).",
"Its scaling law with the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ , is not straightforward due to several factors.",
"To be specific, the total bandwidth is fixed, the spread factor grows nonlinearly with $\\lambda _m/\\lambda _b$ , and the likelihood of task-generation probability per frame varies with the frame length.",
"To address this issue, the task arrivals at the typical AP are characterized as follows.",
"Consider the case of asynchronous offloading.",
"Based on the model in Section REF , the probability that a mobile generates a task for offloading in each frame is $p_L^*=1 - (1-p)^{L^*}, $ where $L^*$ is the frame length given the optimal spreading factor $G^*$ in (REF ).",
"The expected task-offloading rate (in number of tasks per slot) for the typical mobile, denoted as $\\beta ^*$ , is given as $\\beta ^* = \\frac{p_L^*}{L^*}$ .",
"Since $L^* = G^*\\mathsf {T}_{\\min }$ , $\\beta ^*=\\frac{1 - (1-p)^{G^*\\mathsf {T}_{\\min }}}{G^*\\mathsf {T}_{\\min }}.$ where $\\beta ^*=p_L^* \\cdot G^*\\mathsf {T}_{\\min }$ .",
"Let $\\bar{\\Lambda }^*$ denote the expected task-arrival rate at the typical AP (or CS).",
"Then $\\bar{\\Lambda }^* = \\bar{N}\\beta ^*$ where $\\bar{N}$ is the expected number of mobiles connected to the AP.",
"Since $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ , $\\bar{\\Lambda }^* = (1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}\\beta ^*.$ Remark 6 (Effects of Network Parameters) Using (REF ), (REF ) and (REF ), one can infer that $\\bar{\\Lambda }^* \\propto \\left\\lbrace \\begin{aligned}& p\\frac{\\lambda _m}{\\lambda _b}, && \\frac{\\lambda _m}{\\lambda _b}\\rightarrow 0,\\\\&B\\left(\\frac{\\lambda _m}{\\lambda _b}\\right)^{-\\frac{\\alpha }{2}+1}, && \\frac{\\lambda _m}{\\lambda _b}\\rightarrow \\infty .\\end{aligned}\\right.$ The first case corresponds to a sparse network whose performance is not limited by bandwidth and interference.",
"Then the expected task arrival-rate grows linearly with the task-generation probability per slot, $p$ , and the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ .",
"For the second case, in a dense network that is bandwidth-and-interference limited, the rate grows linearly the bandwidth $B$ , but decreases with $\\lambda _m/\\lambda _b$ .",
"The reason for the decrease is the bandwidth for offloading is reduced so that a larger spreading factor is available for suppressing interference to meet the network-coverage requirement.",
"Consequently, the load for the CSs is lighter for a dense (thus comm-limited) network, reducing comp-latency as shown in the sequel.",
"Consider tasks arrivals for the case of synchronous offloading.",
"Unlike the asynchronous counterpart with arrivals spread over each frame, the tasks from mobiles arrive the typical AP at the beginning of each frame.",
"Thus, it is useful to characterize the expected number of task arrivals per frame, denoted as $\\bar{A}^*$ , which can be written as $\\bar{A}^* = \\bar{N} p_L^*$ .",
"It follows that $\\bar{A}^* =(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b} p_L^*.",
"$ Remark 7 (Effects of Network Parameters) In a dense network ($\\lambda _m/\\lambda _b\\rightarrow \\infty $ ), it can be obtained from (REF ), (REF ), and (REF ) that $p_L^* \\approx 1$ .",
"Then it follows from (REF ) that the expected number of tasks per frame increases linearly with the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ ."
],
[
"Computation Latency Analysis: Asynchronous Offloading",
"This section aims at analyzing the comp-latency of the asynchronous offloading where task arrival and departure are randomly distributed over time.",
"Given the Markov model of Fig.",
"REF , we derive the network stability condition in Definition REF and bounds of the average comp-latency."
],
[
"Optimal Control of VMs",
"On one hand, creating a large number of VMs at the typical CS can slow down its computation rate due to the mentioned I/O interference between VMs.",
"On the other hand, too few VMs can lead to marginal gain from parallel computing.",
"Therefore, the number of VMs should be optimally controlled based on the number of waiting tasks.",
"To this end, let $\\mu (m)$ denote the computation rate given $m$ VMs.",
"Given the computation model in (REF ), it follows from $\\mu (m)=m/T_c(m)$ that: $\\mu (m) = \\frac{m}{T_0}(1+d)^{1-m}.$ By analyzing the derivative of $\\mu (m)$ , one can find that the function is monotone increasing before reaching a global maximum and after that it is monotone decreasing.",
"Thereby, the value of $m$ that maximizes $\\mu (m)$ , denoted as $m_{\\max }$ , can be found with the integer constraint $m_{\\max } = \\textrm {round}\\left(\\frac{1}{\\ln (1+d)}\\right),$ where $\\textrm {round}(x)$ rounds $x$ to the nearest integer.",
"The said properties of the function $\\mu (m)$ and the derived $m_{\\max }$ in (REF ) suggest the following optimal VM-control policy.",
"Proposition 2 (Optimal VM Control) To maximize the computation rate at the typical CS, the optimal VM-control policy is to create $m_{\\max }$ VMs if there are a sufficient number of tasks for computation or otherwise create as many VMs as possible until the buffer is empty.",
"Consequently, the maximum computation rate, denoted as $\\mu ^*(m)$ , given $m$ tasks at the CS (being computed or in the buffer) is $\\mu ^*(m) = \\left\\lbrace \\begin{aligned}&\\frac{m}{T_0}(1+d)^{1-m}, &&1\\le m \\le m_{\\max }, \\\\&\\frac{m_{\\max }}{T_0}(1+d)^{1-m_{\\max }}, &&m > m_{\\max },\\end{aligned}\\right.$ where $m_{\\max }$ is given in (REF ).",
"For ease notation, the maximum computation rate, $\\mu ({m_{\\max }})$ , is re-denoted as $\\mu _{\\max }$ hereafter."
],
[
"Computation Rates under Network Stability Constraint",
" This subsection focuses on analzying the condition for the maximum computation rate of the typical CS to meet the network stability constraint in Definition REF .",
"The analysis combines the results from queueing theory, stochastic geometry and parallel computing.",
"The said constraint requires $\\rho $ -fraction of mobiles, or equivalently $\\rho $ -fraction of CSs, to be stable, namely that comp-latency is finite.",
"According to queuing theory, stabilizing a typical CS requires that the task-arrival rate $\\Lambda $ should be strictly smaller than the maximum departure rate $\\mu ^*(m_{\\max })$ : $\\Lambda <\\mu _{\\max }$ [28].",
"Note that the former is a RV proportional to the random number of mobiles, $N$ , connected to the typical CS while the latter is a constant.",
"Then the stability probability $\\mathsf {p}_s$ is given as $\\mathsf {p}_s =\\Pr [\\Lambda <\\mu _{\\max }]=\\Pr \\left[N <\\frac{\\mu _{\\max }}{\\beta ^*}\\right],$ where $\\beta ^*$ is the task-offloading rate given in (REF ).",
"It follows from the network spatial model that $N$ is a Poisson distributed RV with the mean $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ .",
"Using the distribution and (REF ) and applying Chernoff bound, we can obtain an upper bound on the maximum computation rate required to meet the stability constraint as shown below.",
"Proposition 3 (Computation Rates for $\\rho $ -Stability) For the CSN to be $\\rho $ -stable, a sufficient condition for the maximum computation rate of the typical CS is given as $\\mu _{\\max }\\ge \\bar{\\Lambda }^*\\cdot \\exp \\left(W\\left(\\frac{-\\ln (\\rho )}{\\bar{N} e} - \\frac{1}{e}\\right)+1\\right),$ where $W(\\cdot )$ is the Lambert function, the expected mobiles connected to the typical CS $\\bar{N}=(1-\\delta )(1-\\epsilon )\\frac{\\lambda _m}{\\lambda _b}$ , and $\\bar{\\Lambda }^*$ represents the expected arrival rate given in (REF ).",
"Proof: See Appendix REF .",
"$\\Box $ The above result shows that to satisfy the network-stability constraint, the maximum computation rate of each CS, $\\mu _{\\max }$ , should be larger than the expected task-arrival rate, $\\bar{\\Lambda }^*$ , scaled by a factor larger than one, namely the exponential term in (REF ).",
"Moreover, the factor grows as the stability probability $(1-\\rho )$ increases.",
"Last, it is useful for subsequent analysis to derive the expected arrival rate conditioned on that the typical CS is stable as shown below.",
"Lemma 4 (Expected Task-Arrival Rates for Stable CSs) Given that the typical CS is stable, the expected task-arrival rate is given as $\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]=\\bar{\\Lambda }^* \\left(1-\\frac{\\Pr (N = \\left\\lfloor R\\right\\rfloor )}{1 - \\rho }\\right),$ where $R=\\frac{\\mu _{\\max }}{\\beta ^*}$ measures the maximum number of mobiles the CS can serve, $\\beta ^*$ is the task-offloading rate per mobile in (REF ), $\\bar{N}$ and $\\bar{\\Lambda }^*$ follow those in Proposition REF , and the Poisson distribution function $\\Pr (N = n )=\\frac{\\bar{N}^ne^{-\\bar{N}}}{n!",
"}$ .",
"Proof: See Appendix REF .",
"$\\Box $"
],
[
"Expected Computation Latency",
"In this subsection, the expected comp-latency, $\\mathsf {T}_{\\textrm {comp}}$ , is analyzed using the Markov chain in Fig.",
"REF and applying queueing theory.",
"Exact analysis is intractable due to the fact that the departure rate $\\mu (m)$ in the Markov chain is a non-linear function of state $m$ .",
"This difficulty is overcome by modifying the Markov chain to give two versions corresponding to a M/M/m and a M/M/1 queues, yielding an upper and a lower bounds on $\\mathsf {T}_{\\textrm {comp}}$ , respectively.",
"First, consider upper bounding $\\mathsf {T}_{\\textrm {comp}}$ .",
"To this end, the departure rate $\\mu (m)$ in the Markov chain in Fig.",
"REF with the following lower bound obtained by fixing all exponents as $(1-m_{\\max })$ : $\\mu ^-(m) = {\\left\\lbrace \\begin{array}{ll}\\frac{m}{T_0}(1+d)^{1-m_{\\max }}, ~~~1\\le m \\le m_{\\max }, \\\\\\frac{m_{\\max }}{T_0}(1+d)^{1-m_{\\max }}, ~~~m > m_{\\max }.\\end{array}\\right.",
"}$ As a result, the modified Markov chain is a M/M/$m_{\\max }$ queue.",
"The corresponding waiting time, denoted as $\\mathsf {T}_{\\textrm {comp}}^+$ , upper bounds $\\mathsf {T}_{\\textrm {comp}}$ since it reduces the computation rate.",
"Applying classic results on M/M/m queues (see e.g., [28]), the waiting time, $\\mathsf {T}_{\\textrm {comp}}^+$ , for task arrival rate $\\Lambda $ is $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda ) &= \\frac{m_{\\max }}{\\mu ^-(m_{\\max })} + \\frac{\\tau \\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^{m_{\\max }}}{m_{\\max }!\\mu ^-(m_{\\max })\\left( 1-\\frac{\\Lambda }{m_{\\max }\\mu ^-(m_{\\max })} \\right)^2} , $ where the coefficient $\\tau $ is given as $\\tau = \\left[\\sum _{m=0}^{m_{\\max }-1} \\frac{1}{m!",
"}\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^m + \\sum _{m=m_{\\max }}^{\\infty }\\frac{m_{\\max }^{m_{\\max }-m}}{m_{\\max }!",
"}\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^m\\right]^{-1}.",
"$ Using (REF ), (REF ) and (REF ), the upper bound is given in the following theorem.",
"02 Theorem 2.A (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.",
"The average comp-latency is upper bounded as $\\mathsf {T}_{\\textrm {comp}} \\le \\frac{m_{\\max }}{\\mu _{\\max }} + \\left(\\frac{m_{\\max }}{\\mu _{\\max }}\\right)^2\\cdot \\frac{\\bar{\\Lambda }^*}{(m_{\\max }-1)!\\left( m_{\\max }-1 \\right)^2}\\cdot \\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right),$ where $ R$ follows that in Lemma REF , and $\\bar{\\Lambda }^*$ and $\\mu _{\\max }$ are specified in (REF ) and (REF ), respectively.",
"Proof: See Appendix REF .",
"$\\Box $ Note that the positive factor $\\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right)$ accounts for Poisson distribution of mobiles.",
"Next, a lower bound on $\\mathsf {T}_{\\textrm {comp}}$ is obtained as follows.",
"One can observe from the Markov chain in Fig.",
"REF that for states $m \\le m_{\\max }$ , the departure rates are smaller than the maximum, $\\mu _{\\max }$ .",
"The reason is that for these states, there are not enough tasks for attaining the maximum rate by parallel computing.",
"Then replacing all departure rates in the said Markov chain with the maximum $\\mu _{\\max }$ leads to a lower bound on $\\mathsf {T}_{\\textrm {comp}}$ .",
"The resultant Markov chain corresponds to a M/M/1 queue.",
"Then using the modified Markov chain and the well-known results from M/M/1 queue (see e.g., [28]), the comp-latency for given arrival rate $\\Lambda $ can be lower bounded as $\\mathsf {T}_{\\textrm {comp}}(\\Lambda ) \\ge \\frac{1}{\\mu _{\\max }-\\Lambda }.$ By taking expectation over $\\Lambda $ and applying Jensen's inequality, $\\mathsf {T}_{\\textrm {comp}}= \\mathsf {E}[\\mathsf {T}_{\\textrm {comp}}(\\Lambda )]\\ge \\mathsf {E}\\left[\\left.\\frac{1}{\\mu _{\\max }-\\Lambda }\\right|\\Lambda <\\mu _{\\max }\\right]\\ge \\frac{1}{\\mu _{\\max }-\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]}.$ Using (REF ) and Lemma REF , we obtain the following result.",
"Theorem 2.B (Comp-Latency for Asynchronous Offloading) Consider asynchronous offloading.",
"The average comp-latency is lower bounded as $\\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max } - \\bar{\\Lambda }^*\\cdot \\left(1-\\frac{\\Pr (N = \\lfloor R\\rfloor )}{1- \\rho }\\right)},$ where $ R$ follows that in Lemma REF , and $\\bar{\\Lambda }^*$ and $\\mu _{\\max }$ are specified in (REF ) and (REF ), respectively.",
"Remark 8 (Computation-Resource Provisioning) Consider a MEC network provisioned with sufficient computation resources, $\\mu _{\\max }/\\bar{\\Lambda }^*\\gg 1$ .",
"It follows from Theorem REF $\\nonumber \\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max }}\\left(1 + \\frac{c_1\\bar{\\Lambda }^*}{\\mu _{\\max }}\\right),$ where $c_1$ is a constant.",
"This lower bound has a similar form as the upper bound in Theorem REF .",
"From these results, one can infer that the comp-latency for asynchronous offloading can be approximated written in the following form: $\\boxed{\\mathsf {T}_{\\textrm {comp}}\\approx \\frac{c_2}{\\mu _{\\max }}\\left(1 + \\frac{c_3\\bar{\\Lambda }^*}{\\mu _{\\max }}\\right), \\qquad \\frac{\\mu _{\\max }}{\\bar{\\Lambda }^*}\\gg 1,}$ where $\\lbrace c_2, c_3\\rbrace $ are constants.",
"The result suggests that to contain comp-latency, the provisioning of computation resources for the MEC network must consider two factors.",
"First of all, the maximum computation rate, $\\mu _{\\max }$ , for each CS must be sufficient large.",
"At the same time, the computation rate must scale linearly with the total arrival rate such that the computation resource allocated for a single offloaded task, measured by the ratio $\\mu _{\\max }/\\bar{\\Lambda }^*$ , is sufficiently large."
],
[
"Energy Efficiency",
" Based on the above analytical results so far, the subsection tempts to discuss the energy savings of offloading than local computing.",
"First, the energy consumption of offloading, denoted by $\\mathsf {E}_{\\mathrm {off}}$ , can be derived via multiplying mobile's transmission power $P$ by the offloading duration $G^* \\mathsf {T}_{\\min }$ .",
"To satisfy the minimum average signal strength at the boundary of the MEC service zone, $P$ should scale with the radius ${r_0}$ as $P\\propto r_0^{\\alpha }$ .",
"Recalling $r_0\\propto \\lambda _b^{-\\frac{1}{2}}$ and $\\mathsf {T}_{\\min }\\propto \\ell $ where $\\ell $ is the task size, the resultant energy consumption of $\\mathsf {E}_{\\mathrm {off}}$ is given as $\\mathsf {E}_{\\mathrm {off}}=c_4 \\frac{G^* \\ell }{{\\lambda _b}^{\\frac{\\alpha }{2}}},$ where $c_4$ is a constant depending on the minimum signal strength and $\\mathsf {T}_{\\min }$ .",
"Next, it is well studied in[6] that the optimal $\\mathsf {E}_{\\textrm {loc}}$ is proportional to ${\\ell }^3$ , and inversely proportional to the square of the deadline requirement which could be set as the total latency $\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}}$ without loss of generality.",
"Thus, $\\mathsf {E}_{\\textrm {loc}}$ is given as $\\mathsf {E}_{\\textrm {loc}}=c_5 \\frac{{\\ell }^3}{(\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}})^2},$ where $c_5$ is a constant depending on the chip architecture.",
"where $c_5$ is a constant depending on the chip architecture.",
"As a result, the condition of energy savings, namely $\\mathsf {E}_{\\mathrm {off}}<\\mathsf {E}_{\\textrm {loc}}$ , is given in terms of $\\ell $ and $\\lambda _b$ as $\\ell ^2 \\lambda _b^{\\frac{\\alpha }{2}}>\\frac{c_4}{c_5}{G^*}\\left(\\mathsf {T}_{\\textrm {comm}}+\\mathsf {T}_{\\textrm {comp}}\\right)^2.$ It is observed that the right-side of (REF ) is dominantly affected by the expected number of mobiles $\\frac{\\lambda _m}{\\lambda _b}$ (see (REF ), (REF ), (REF ) and (REF )).",
"In other words, given a mobile density $\\lambda _m$ and the task size $\\ell $ , there exists the minimum density of APs $\\lambda _b$ to satisfy the condition in (REF ).",
"Then under the condition in (REF ), the energy savings due to offloading is given as $\\mathsf {E}_{\\mathrm {loc}}-\\mathsf {E}_{\\mathrm {off}}$ with $\\mathsf {E}_{\\mathrm {off}}$ and $\\mathsf {E}_{\\mathrm {loc}}$ given in (REF ) and (REF ), respectively."
],
[
"MEC Network Provisioning and Planning",
" Combining the results from the preceding analysis on comm-latency and comp-latency yields some guidelines for the provisioning and planning of a MEC network as discussed below.",
"Assume that the network is required to support computing for mobiles with density $\\lambda _m$ with targeted expected comm-latency $\\mathsf {T}_{\\textrm {comm}}$ and comp-latency $\\mathsf {T}_{\\textrm {comp}}$ .",
"The network resources are quantified by the bandwidth $B$ , the density of AP (or CS) $\\lambda _b$ , the maximum computing rate of each CS $\\mu _{\\max }$ .",
"First, consider the planning of the RAN.",
"Combining the above results suggest the following guidelines for network provisioning and planning.",
"As shown in Section REF , under the network-coverage constraint $(1-\\epsilon )$ , the expected task-arrival rate at a AP, representing the RAN offloading throughput, is a quasi-concave function of the expected number of mobiles per AP, $\\lambda _m/\\lambda _b$ , with a global maximum.",
"Therefore, given a mobile density $\\lambda _m$ , the AP density should be chosen for maximizing the RAN offloading throughput.",
"Next, based on results in Theorem REF and (REF ), sufficient large channel bandwidth $B$ should be provisioned to achieve the targeted $\\mathsf {T}_{\\textrm {comm}}$ for given mobile and AP densities, mobile task-generation rates, and task sizes.",
"Next, consider the planning of the CSN.",
"Under the network-stability constraint, the maximum CS computing rate for parallel computing should be planned to be larger than the expected task-arrival rate scaled by a factor larger than one, which is determined by the allowed fraction of unstable CSs (see Proposition REF ).",
"Then, the maximum computing rate should be further planned to achieve the targeted $\\mathsf {T}_{\\textrm {comp}}$ for computing an offloaded task using Theorems REF and REF .",
"In the preceding section, the process of asynchronous task arrival at a CS can be approximated using a Markov chain, allowing tractable analysis of comp-latency using theories of M/M/$m$ and M/M/1 queues.",
"This approach is inapplicable for synchronous offloading and the resultant periodic task arrivals at the CS.",
"Though tractable analysis in general is difficult, it is possible for two special cases defined as follows.",
"Definition 3 (Special Cases: Light-Traffic and Heavy-Traffic) A light-traffic case refers to one that the task-arrival rate is much smaller than the computation rate such that the queue at the CS is always empty as observed by a new arriving task.",
"In contrast, a heavy-traffic case refers to one that the task-arrival rate is close to the computation rate such that there are always at least $m_{\\max }$ tasks in the queue.",
"The comp-latency for these two special cases are analyzed to give insights into the performance of CSN with underloaded CSs and those with overloaded CSs."
],
[
"Expected Computation Latency with Light-Traffic ",
" First, the dynamics of the task queue at the typical CS is modelled as follows.",
"Recall that task arrivals are periodical, occurring at the beginning of every frame.",
"Consider the typical CS.",
"Let $\\mathcal {Q}_t$ and $\\mathcal {A}_t$ denote be the numbers of existing and arriving tasks at the beginning of frame $t$ , respectively, and $\\mathcal {C}_t$ the number of departing tasks during frame $t$ .",
"Then the evolution of $\\mathcal {Q}_t$ can be described mathematically as $\\mathcal {Q}_{t+1} = \\max \\left[ \\mathcal {Q}_t + \\mathcal {A}_{t+1} - \\mathcal {C}_t, 0 \\right].$ The general analysis of comp-latency using (REF ) is difficult.",
"The main difficulty lies in deriving the distribution of $\\mathcal {C}_t$ that depends on the number of VMs that varies continuously in time since the computation time for simultaneous tasks are random and inter-dependent.",
"To overcome the difficulty, consider the case of light-traffic where a number of offloaded tasks arrives at the typical CS to see an empty queue and an idling server.",
"Correspondingly, the evolution equation in (REF ) is modified such that given $\\mathcal {A}_{t+1}\\ne 0 $ , $\\mathcal {Q}_t = \\mathcal {C}_t = 0 $ , yielding $\\mathcal {Q}_{t+1} = \\mathcal {A}_{t+1}$ .",
"Next, given this simple equality, deriving the expected comp-latency reduces to analyzing the latency for computing a random number of $\\mathcal {A}$ tasks at the CS, which arrives at the beginning of an arbitrary frame.",
"Without loss of generality, the tasks are arranged in an ascending order in terms of computation time and referred to as Task $1, 2, \\cdots , \\mathcal {A}$ .",
"Moreover, let $\\mathcal {L}_n$ denote the expected computing time for Task $n$ and hence $\\mathcal {L}_1 \\le \\mathcal {L}_2 \\le \\cdots \\le \\mathcal {L}_{\\mathcal {A}}$ .",
"Then the expected comp-latency $\\mathsf {T}_{\\textrm {comp}}$ can be written in terms of $\\lbrace \\mathcal {L}_n\\rbrace $ as $\\mathsf {T}_{\\textrm {comp}} = \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} {\\mathcal {L}_n} }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right].",
"$ To obtain bounds on $\\mathsf {T}_{\\textrm {comp}}$ in closed form, a useful result is derived as follows.",
"Given $m$ VMs, recall that the computation time of a task follows the exponential distribution with the mean being the inverse of the computation rate $\\mu (m)$ in (REF ).",
"Using the memoryless property of exponential distribution, a useful relation between $\\lbrace \\mathcal {L}_n\\rbrace $ is obtained as $\\mathcal {L}_n =\\mathcal {L}_{n-1}+ \\frac{1}{\\mu (m)}=\\left\\lbrace \\begin{aligned}&\\mathcal {L}_{n-1}+ \\frac{1}{\\mu _{\\max }}, && 1\\le n\\le \\mathcal {A}-m_{\\max }+1,\\\\&\\mathcal {L}_{n-1} +\\frac{1}{\\mu (\\mathcal {A} - n +1)}, && \\text{otherwise},\\end{aligned}\\right.$ with $\\mathcal {L}_0=0$ .",
"Note that $\\mu (1) \\le \\mu (m) \\le \\mu _{\\max }$ for all $m$ .",
"Thus, it follows from (REF ) that $\\frac{n}{\\mu _{\\max }} \\le \\mathcal {L}_n \\le \\frac{n}{\\mu (1)}.",
"$ Substituting (REF ) into (REF ) gives $\\frac{1}{\\mu _{\\max }}\\cdot \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} n }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right]\\le \\mathsf {T}_{\\textrm {comp}} \\le \\frac{1}{\\mu (1)} \\cdot \\mathsf {E}_{\\mathcal {A}}\\left[\\left.\\frac{\\sum _{n=1}^{{\\mathcal {A}}} n }{\\mathcal {A}}\\right|\\mathcal {A}>0 \\right].$ Recalling the number of arriving tasks $\\mathcal {A}$ follows a Poisson RV with mean $\\bar{A}^*$ of (REF ), bounds on the comp-latency is obtained shown in the following theorem.",
"Theorem 3 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with light-traffic.",
"The expected comp-latency can be bounded as $\\nonumber \\frac{1}{2\\mu _{\\max }}\\left(1+\\frac{\\bar{A}^*}{1-e^{-\\bar{A}^*}}\\right)\\le \\mathsf {{T}}_{\\textrm {comp}} \\le \\frac{1}{2\\mu (1)}\\left(1+\\frac{\\bar{A}^*}{1-e^{-\\bar{A}^*}}\\right).$ where $\\bar{A}^*$ is the expected number of arriving tasks to the typical CS per frame given in (REF ).",
"Remark 9 (Comparison with Asynchronous Offloading) From the results in the above theorem, one can infer that for the current case, the comm-latency can be approximated as $\\boxed{\\mathsf {{T}}_{\\textrm {comp}} \\approx \\left\\lbrace \\begin{aligned}&\\frac{1}{\\mu (1)}, && \\bar{A}^* \\rightarrow 0,\\\\&\\frac{\\bar{A}^*}{2\\mu (m_{\\max })},&& \\bar{A}^* \\gg 1.\\end{aligned}\\right.",
"}$ Comparing the expression with the counterpart for asynchronous offloading in (REF ), it is unclear which case leads to longer comp-latency.",
"However, simulation shows that in general, synchronizing offloading tends to incur longer latency by overloading CSs and thereby suffering more from I/O interference in parallel-computing."
],
[
"Expected Computation Latency with Heavy-Traffic",
"This subsection focuses on analyzing the expected comp-latency, $\\mathsf {T}_{\\textrm {comp}}$ , for the case of heavy-traffic as defined in Definition REF .",
"For this case, with the queue being always non-empty, the equation in (REF ) describing the queue evolution reduces to $\\mathcal {Q}_{t+1} = \\mathcal {Q}_t + \\mathcal {A}_t - \\mathcal {C}_t$ .",
"The key step in deriving $\\mathsf {T}_{\\textrm {comp}}$ is to apply the said equation to the analysis of the expected queue length.",
"The technique involves taking expectation of the squares of the two sides of the equation as follows: $\\mathsf {E}\\left[ \\mathcal {Q}^2_{t+1} \\right]= \\mathsf {E}\\left[ \\left(\\mathcal {Q}_t + \\mathcal {A}_t - \\mathcal {C}_t\\right)^2\\right]&=\\mathsf {E}\\left[ \\mathcal {Q}^2_t \\right]+\\mathsf {E}\\left[\\left( \\mathcal {A}_t - \\mathcal {C}_t \\right)^2\\right]+2\\mathsf {E}\\left[{\\mathcal {Q}}_t (\\mathcal {A}_t - \\mathcal {C}_t) \\right].$ Since $\\mathcal {Q}_t$ , $\\mathcal {A}_t$ and $\\mathcal {C}_t$ are independent of each other and $\\mathsf {E}\\left[ \\mathcal {Q}^2_{t+1} \\right]=\\mathsf {E}\\left[ \\mathcal {Q}^2_t\\right]$ given the stable CS, $\\mathsf {E}\\left[ \\mathcal {Q}\\right] = \\frac{\\mathsf {E}\\left[\\mathcal {A}^2\\right] + \\mathsf {E}\\left[\\mathcal {C}^2\\right] - 2\\mathsf {E}\\left[\\mathcal {A}\\right]\\mathsf {E}\\left[\\mathcal {C}\\right]}{2\\left( \\mathsf {E}\\left[\\mathcal {C}\\right] - \\mathsf {E}\\left[\\mathcal {A}\\right] \\right)}, $ where the subscripts $t$ of $\\mathcal {Q}_t$ , $\\mathcal {A}_t$ and $\\mathcal {C}_t$ are omitted to simplify notation.",
"Given the number of connected mobiles, $N$ , the number of arrival tasks $\\mathcal {A}$ follows a Poisson distribution with the first and second moments being $\\mathsf {E}\\left(\\mathcal {A}\\mid N \\right)={N} p_L^*$ and $\\mathsf {E}\\left(\\mathcal {A}^2\\mid N \\right)=N p_L^*+(Np_L^*)^2$ respectively, where the task-offloading probability $p_L^*$ is given in (REF ).",
"Next, under the heavy-traffic assumption, the total computation rate of the CS is $\\mu _{\\max }$ .",
"It follows that the departure process at the typical CS is Poisson distributed where the first and second moments are $\\mathsf {E}[\\mathcal {C}]= \\mu _{\\max }L^*$ and $\\mathsf {E}[\\mathcal {C}^2]= \\mu _{\\max }L^*+[ \\mu _{\\max }L^*]^2$ , respectively.",
"Substituting the results into (REF ) gives $\\mathsf {E}\\left[ \\mathcal {Q}\\mid N \\right] =\\frac{{N} p_L^*}{\\mu _{\\max }L^*-{N} p_L^*}+\\frac{1}{2}\\cdot \\left(\\mu _{\\max }L^*-{N} p_L^*\\right)+\\frac{1}{2}.",
"$ In addition, to satisfy the condition for stabilizing the CS, the arrival rate $\\mathsf {E}\\left[\\mathcal {A}\\right]$ should be strictly smaller than the departure rate $\\mathsf {E}[\\mathcal {C}]$ .",
"This places a constraint on the maximum of $N$ , namely that $N \\le \\left\\lfloor R \\right\\rfloor $ with $ R$ defined in Lemma REF .",
"Under this constraint, applying Little's theorem obtains the expected comp-latency $\\mathsf {T}_{\\textrm {comp}}$ as $\\mathsf {T}_{\\textrm {comp}}& = \\left\\lbrace \\mathsf {E}\\left[\\left.",
"\\frac{\\mathsf {E}\\left[ \\mathcal {Q}\\mid N \\right]}{\\mathsf {E}\\left[ \\mathcal {A}\\mid N \\right]}\\right|N \\le \\left\\lfloor R \\right\\rfloor \\right]-\\frac{1}{2}\\right\\rbrace \\cdot L^*.",
"$ Combining (REF ) and (REF ) yields the main result of this sub-section as shown below.",
"Theorem 4 (Comp-Latency for Synchronous Offloading) Consider the case of synchronous offloading with heavy-traffic.",
"The expected comp-latency is given as $\\mathsf {T}_{\\textrm {comp}}=\\left\\lbrace \\frac{1}{p_L^*}\\mathsf {E}\\left[\\frac{1}{ R-N}\\right] +\\frac{1}{2}\\left( R + \\frac{1}{p_L^*}\\right) \\mathsf {E}\\left[\\frac{1}{N}\\right] - 1\\right\\rbrace L^*,$ where the constant $ R$ and the distribution of $N$ follow those in Lemma REF .",
"Remark 10 (Comparison with Asynchronous Offloading) By applying Jensen's inequality, the comp-latency of (REF ) can be lower bounded as $\\mathsf {T}_{\\textrm {comp}}&\\ge \\frac{1}{\\mu _{\\max }-c_4\\bar{\\Lambda }^*}+\\frac{L^*}{2}\\cdot \\frac{\\mu (m_{\\max })}{c_4\\bar{\\Lambda }^*}+\\frac{1}{c_4\\bar{\\Lambda }^*}- L^*,$ where $c_4$ is a constant.",
"Since for the case of heavy-traffic, the task-arrival rate $c_4\\bar{\\Lambda }^*$ approaches the maximum computation rate $\\mu _{\\max }$ , $\\boxed{\\mathsf {T}_{\\textrm {comp}}\\ge \\frac{1}{\\mu _{\\max }-c_4\\bar{\\Lambda }^*},\\qquad c_4\\bar{\\Lambda }^*\\rightarrow \\mu _{\\max }.",
"}$ The above lower bound has the same form as the asynchronous-offloading counterpart in (REF ).",
"Both diverge as the task-arrival rate approaches the maximum computation rate."
],
[
"Simulation Results",
" In this section, analytical results on comm-latency and comp-latency are evaluated by simulation.",
"The simulation parameters have the following default settings unless specified otherwise.",
"The densities of APs and mobiles are $\\lambda _b = 2 \\times 10^{-2} \\ \\mathrm {m}^{-2}$ and $\\lambda _m = 5 \\times 10^{-2} \\ \\mathrm {m}^{-2}$ , respectively.",
"The $\\mathsf {SIR}$ threshold is set as $\\theta = 1$ dB and the path-loss exponent is $\\alpha = 3$ .",
"For the network coverage parameter $\\delta $ is $\\delta = 10^{-2}$ , corresponding to the radius of MEC service zone being $r_0 = 12 \\mathrm {m}$ .",
"The total bandwidth is $B = 6$ MHz.",
"The data size per task is fixed as $\\ell = 0.5 \\times 10^{6}$ bits.",
"The single-task computation time $T_0$ in the parallel-computation model is set as $T_0 = 0.1$ (sec) and the factor arising from I/O interference is $d = 0.2$ .",
"The task generation probability per slot is $p = 0.2$ .",
"The parameters $\\epsilon $ and $\\rho $ are both set as $0.05$ .",
"Figure: Effect of task generating rate.We present the comparisons between Monte Carlo simulations ($10^4$ realizations) and analytical results in all figures.",
"For each realization, both mobiles and APs are distributed in the plane based on the PPPs.",
"Each mobile randomly generates an offloading task and transmits it to its corresponding AP.",
"Then the AP generates VMs and performs the computation upon the task arrivals.",
"A queue of tasks will appear if the task arrival rate is large than the computation rate (e.g., too many tasks arrived at the AP at the same time).",
"The VM will be released when the corresponding task is computed.",
"Fig.",
"REF compares expected comm-latency and comp-latency for the case of asynchronous offloading.",
"The effects of mobile density $\\lambda _m$ and task generating rate $p$ are investigated and several observations can be made.",
"As shown in Fig.",
"REF (a), the expected comp-latency as a function of the mobile density is observed to exhibit the quasi-concavity described in Remark REF .",
"In contrast, the expected comm-latency is a monotone increasing function following the scaling law in Remark REF .",
"These properties lead to the partitioning of the range of mobile density into three network-operation regimes as indicated in Fig.",
"REF (a).",
"In particular, the middle range corresponds to comp-limited regime while others are comm-limited.",
"Next, consider the effect of task-generation rate at mobiles, specified by the task-generation probability $p$ .",
"Both types of latency are observed to converge to corresponding limits as the rate grows.",
"Their different scaling laws result in the partitioning of the range of task-generation rate into comm-limited and comp-limited regimes.",
"Last, one can observe from both figures that the lower bound on comp-latency as derived in (REF ) is tighter than the upper bond therein.",
"Fig.",
"REF compares expected comm-latency and comp-latency for the case of synchronous offloading.",
"The same observations in the case of synchronous offloading also apply in the current case except that the quasi-concavity of the expected comp-latency with respect to mobile density is not shown in the considered range.",
"Some new observations can be made as follows.",
"Comparing Fig.",
"REF and REF shows that synchronizing offloading results in longer comp-latency.",
"Next, the center of the comp-limited range in Fig.",
"REF (a) corresponds to the case of heavy-traffic studied in Section REF .",
"Consequently, the derived upper bound on expected comp-latency for this case is tight.",
"For other ranges of mobile density, the bounds derived for the case of light traffic are tighter.",
"Last, Fig.",
"REF (b) shows that the expected comp-latency is tightly approximated by bounds derived for the light-traffic case when the task-arrival rate is small ($\\le 0.3$ ) and by that for the heavy-traffic case when the rate is large ($> 0.7$ ), validating the results.",
"Figure: Effect of task generating rate."
],
[
"Conclusion Remarks",
"In this work, we have first studied the network-constrained latency performance of a large-scale MEC network, namely comm-latency and comp-latency under the constraints of RAN connectivity and CSN stability.",
"To study the tradeoffs between these metrics and constraints and model the cascaded architecture of RAN and CSN, the MEC network has been modelled using stochastic geometry featuring diversified aspects of wireless access and computing.",
"Based on the model, the average comm-latency and comp-latency have been analyzed by applying the theories of stochastic geometry, queuing and parallel computing.",
"In particular, their scaling laws have been derived with respect to various network parameters ranging from the densities of mobiles and APs to the computation capabilities of CSs.",
"The results provide useful guidelines for MEC-network provisioning and planning to avoid either the RAN or CSN being a performance bottleneck.",
"The current work can be extended in several directions.",
"In this work, we consider a single type of computation task of which the task size and the average computation time are identical.",
"However, considering different types of tasks makes the latency analysis more challenging but is of practical relevance.",
"Next, studying a large-scale hierarchical fog computing network comprising mobiles, edge cloud and central cloud is aligned with recent advancements in edge computing.",
"Last, considering advanced techniques such as VM migration and cooperative computing to reduce the latency will be another promising direction."
],
[
"Proof of Lemma ",
" The first derivative of $\\xi (G)$ for $G$ given in (REF ) can be derived as: $\\frac{\\partial \\xi (G)}{\\partial G} = -\\frac{\\Delta }{\\alpha }\\left( G^{-\\frac{2 + \\alpha }{\\alpha }}\\left( \\alpha G \\mathsf {T}_{\\min } \\ln (1-p) (1-p)^{G \\mathsf {T}_{\\min }} - 2(1-p)^{G \\mathsf {T}_{\\min }} + 2 \\right) \\right), $ where $\\Delta = \\frac{2(1-\\delta )\\ln \\delta ^{-1}}{\\alpha } \\mathcal {B}( \\alpha ) \\left( \\frac{\\lambda _m}{\\lambda _b}\\right) \\theta ^{\\frac{2}{\\alpha }}$ .",
"The existence of $g_0$ is easily proved because (REF ) is strictly positive and negative when $G \\rightarrow 0$ and $G \\rightarrow \\infty $ respectively.",
"Next, the solution of $g_0$ to solve $\\frac{\\partial \\xi (G)}{\\partial G} = 0$ is $g_0 =\\frac{\\alpha W\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right) + 2}{\\alpha \\mathsf {T}_{\\min } \\ln (1-p)}$ , where $W(x)$ is the Lambert function.",
"Since the value inside the Lambert function is negative, there are two candidates for $g_0$ : one is from the principle branch of Lambert function $W\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right)$ , and the other is the lower branch $W_{-1}\\left( -\\frac{2}{\\alpha }e^{-\\frac{2}{\\alpha }} \\right)$ .",
"The principle branch makes $g_0=0$ , but the lower branch satisfies $g_0>0$ , completing the proof."
],
[
"Proof of Proposition ",
" Applying Chernoff bound on $\\mathsf {p}_s$ in (REF ) makes $\\mathsf {p}_s &\\ge 1- \\exp \\left(- \\frac{\\mu _{\\max }}{\\beta ^*} \\ln \\left(\\frac{ \\mu _{\\max } }{\\bar{N} \\beta ^*}\\right) +\\frac{\\mu _{\\max }}{\\beta ^*}-\\bar{N}\\right)\\ge 1 - \\rho ,$ which is equivalent to $\\mu _{\\max }\\ge \\bar{\\Lambda }^*\\cdot \\exp \\left(W\\left(\\frac{-\\ln (\\rho )}{\\bar{N} e} - \\frac{1}{e}\\right)+1\\right)$ as Proposition REF shows."
],
[
"Proof of Lemma ",
" Noting the expect task arrival of stable CSs is proportional to the average number of connected mobiles.",
"Therefore, we have $\\mathsf {E}[\\Lambda |\\Lambda <\\mu _{\\max }]= \\beta ^{*} \\cdot \\frac{\\sum _{n=1}^{ R } n \\cdot \\frac{\\bar{N}^{n} e^{-\\bar{N}}}{n!",
"}}{1- \\rho } =\\bar{N}\\beta ^* \\left(1-\\frac{\\Pr (N = \\left\\lfloor R\\right\\rfloor )}{1 - \\rho }\\right)$ , where $\\Pr (N = \\left\\lfloor R\\right\\rfloor ) = \\frac{\\bar{N}^{\\left\\lfloor R\\right\\rfloor } e^{-\\bar{N}}}{\\left\\lfloor R\\right\\rfloor !",
"}$ , ending the proof."
],
[
"Proof of Theorem ",
" Substituting $\\tau = 1$ into (REF ) and then applying $\\left(\\frac{\\Lambda }{\\mu ^-(m_{\\max })}\\right)^{m_{\\max }} \\le \\frac{\\Lambda }{\\mu ^-(m_{\\max })}$ as well as $\\left(1-\\frac{\\Lambda }{m_{\\max }\\mu ^-(m_{\\max })} \\right)^2\\le \\left(1-\\frac{1}{m_{\\max }}\\right)^2$ give an upper bound of $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda )$ as $\\mathsf {T}_{\\textrm {comp}}^+(\\Lambda ) \\le \\frac{m_{\\max }}{\\mu ^-(m_{\\max })} + \\frac{\\frac{\\Lambda }{\\mu ^-(m_{\\max })}}{m_{\\max }!\\mu ^-(m_{\\max })\\left( 1-\\frac{1}{m_{\\max }} \\right)^2}.",
"$ The spatial average on (REF ) based on Lemma REF gives the final result in Theorem REF ."
],
[
"Appendix II: Notation Table",
" Table: Summary of NotationIn Fig.",
"REF , the the comp-latency of the Poisson arrival assumption of task-arrival process is compared with that under the Poisson arrival assumption of Assumption REF , showing that the tasks arrival process at AP can be well approximated to the Poisson task arrival assumption.",
"Fig.",
"REF represents the effects of AP density, which is shown that the same observations are made in the case of decreasing mobile density.",
"Figs.",
"REF and REF show the effects of the computation capability of CS, the computation time $T_0$ and the degradation factor $d$ , on the comp-latency for asynchronous and synchronous offloading cases, respectively.",
"Both of the comp-latencies increase exponentially when the edge-computing capability worse, i.e., $T_0$ or $d$ becomes larger, which agrees with the intuition.",
"Finally, we plot the ratio between the comp-latency of asynchronous and synchronous offloading cases in Fig.",
"REF , which are always less than one because the comp-latency of the asynchronous offloading is always smaller than the synchronous counterpart.",
"Specifically, in Fig.",
"REF (a), the ratio first stays constant when mobiles is sparse and then decreases with mobile density grows.",
"As mobiles becomes denser, the number of offloading tasks at the beginning of each frame increases, resulting in severe I/O interference than the asynchronous counterpart of which the task arrivals are distributed over frame.",
"On the other hand, in Fig.",
"REF (b), it is observed that the ratio grows and then converges to a constant when $p$ increases.",
"In other words, the gap of comp-latency between two offloading cases is becoming smaller when the task arrival becomes heaver, aligning with the discussion in Remark REF ."
]
] | 1709.01702 |
[
[
"Constraints on Flavored 2d CFT Partition Functions"
],
[
"Abstract We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e.",
"when the partition functions are \"flavored\".",
"We begin with a new proof of the transformation law for the modular transformation of such partition functions.",
"Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory.",
"We improve previous upper bounds on the state with the greatest \"mass-to-charge\" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory.",
"We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers.",
"Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise."
],
[
"Introduction",
"Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs).",
"It is also a special case of crossing symmetry of CFT correlation functions [1], so aside from its intrinsic interest it is useful as a simpler setting in which to explore many conformal bootstrap ideas and techniques [2], [3], [4].",
"A particularly appealing generalization of the conformal bootstrap equations is to consider correlation functions in the presence of nonlocal operators, since this enlarges the set of CFT data that can be studied.",
"In general, including nonlocal operators is a difficult problem, since their behavior under conformal transformations may be quite complicated.",
"However, one case where the problem remains tractable is when we consider modular invariance in the presence of a chemical potential in 2d CFTs.",
"A chemical potential corresponds to inserting the nonlocal operator $y^{J_0} \\equiv e^{2 \\pi i z J_0}$ into the partition function, $Z( \\tau , z) \\equiv {\\rm tr}\\left( q^{L_0- \\frac{c}{24}} \\bar{q}^{\\bar{L}_0-\\frac{c}{24}} y^{J_0} \\right),$ where $J_0$ is the zero mode of a conserved current and $L_0, \\bar{L}_0$ are Virasoro generators.",
"The resulting partition function is no longer modular invariant, but nevertheless has a well-defined and theory-independent transformation law [5]: $Z(\\frac{a \\tau +b}{c \\tau +d}, \\frac{c z}{c \\tau + d}) = e^{ \\pi i k \\left( \\frac{ c z^2 }{c \\tau +d} - \\frac{ c \\bar{z}^2}{c \\bar{\\tau }+d}\\right)} Z(\\tau , z) .$ This transformation law was used to constrain the spectrum of charges in general 2d CFTs in [6].",
"Proofs of (REF ) so far [5], [7], [8], [6] either are fairly complicated and technical or else apply to special cases such as free boson constructions, and so it is not clear what if any general lessons might be learned from them.We emphasize that we do not assume supersymmetry; additional techniques are available to proof the transformation law for the elliptic genus in the case of supersymmetric theories, see e.g.",
"[9].",
"However, the very simple form of (REF ) suggests it should have an equally simple derivation.",
"Moreover, inserting the nonlocal operator $y^{J_0}$ is equivalent to turning on a background gauge field $A^\\mu $ coupled to the conserved current $J^\\mu $ , which suggests that one might be able to prove (REF ) by studying the CFT's effective action for $A_\\mu $ .",
"We will begin this paper in section 2 by providing such a proof, and its generalization to a non-abelian symmetry current $J^{a \\mu }$ .",
"Starting in section 3, we perform several analyses of the constraints that follow from (REF ) and its non-abelian generalization using linear programming and semi-definite programming methods.",
"Our main results are as follows."
],
[
"Abelian Bounds",
"We begin by reproducing, and improving the results of [6], bounding properties of theories with an abelian current.",
"We place an upper bound on the dimension of the lightest charged state, $ \\begin{split} \\Delta _{*} = \\frac{c}{\\alpha } +\\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.",
"\\end{split} $ This bound is qualitatively similar to than the bounds in [4], [10] of non-charged states.",
"We also improve the bound on the smallest “mass-to-charge” ratio in the theory.",
"These bounds are qualitatively related to the Weak Gravity Conjecture (WGC), though gauge fields in the gravity duals are Chern-Simons fields rather than Maxwell fields.",
"Provocatively, we find numerical evidence for a bound on the mass-to-charge ratio that scales at large $c$ as $\\sqrt{c}$ , consistent with the bulk gravity expectation.",
"This is stronger than the bounds in [6], which scale as $c$ .",
"The improvement again comes from increasing the number of derivatives of the characters used in the analysis.",
"Then, we discuss the bound on the charge gap $Q_*$ .",
"Without any further assumptions, the numerical bound of charge gap is always $Q_*=1$ for all $c$ .",
"We study two examples of $c=2$ and 8 by turning on both a gap in dimension $\\Delta _*$ and in charge $Q_*$ .",
"There are kinks in the $\\Delta _*$ and $Q_*$ plots which can potentially be associated with full CFTs.",
"Lastly, we consider in detail spectra that extremize various gaps.",
"We use the extremal functional method, as well as extra information contained in the charged spectra to study candidate theories at $c=1$ and 8.",
"At $c=8$ we find that the level 1 $E_{8}$ Sugawara theory saturates both the gap in dimension (as was found in [10]) and in charge.",
"Using this, we are able to reproduce the full low lying spectrum, including charge assignments.",
"When the symmetry current $J^a$ is non-abelian, it is more appropriate to consider bounds on the dimensions of different representations in the theory.",
"We will mainly focus for specificity on the case where the gauge group $G$ is $SU(2)$ and the level $k$ is 1, though our methods easily generalize to any algebra and level; the main advantage of $k=1, G=SU(2)$ is that convergence is fastest here, so our numerical results are most precise.",
"We first obtain bounds on the gap to all non-vacuum states in non-abelian theories.",
"As the extended symmetry imposes additional relations on the spectrum, one may have hoped for stronger bounds.",
"The results, however, are similar to those found in the abelian, or even non-flavored case.",
"In particular, at large $c$ we find a bound of the form, $ \\begin{split} \\Delta _{*} = \\frac{c}{\\alpha } + \\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.",
"\\end{split} $ The real power of the modular constraints on the flavored partition function come from the ability to impose constraints independently on different representations.",
"Taking advantage of this, we search for a bound on the gap to non-vacuum states transforming in the trivial representation, with no constraints imposed for other representations.",
"At small $c$ , there are interesting “kinks” at values of $c$ where the bounds on the gap in the neutral sector is minimized.",
"We focus on the case $c=3$ , and use the extremal functional methods to find the low lying degeneracies at this kink.",
"Reassuringly, we find integer degeneracies.",
"This numerical spectral information allows us to guess an exact partition function saturating the bound in the neutral sector.",
"In fact we find multiple partition functions are allowed if we are somewhat liberal in what spins are allowed for states in the theory.The partition functions found are not strictly modular invariant, but invariant under a subgroup generated by $S$ and $T^{n}$ for $n=2$ or $n=4$ .",
"If all states must have integer or half-integer spins, we find a unique partition function, $ \\begin{split} Z(\\tau , \\bar{\\tau }, z, \\bar{z})&= \\frac{1}{4} \\sum _{a,b,a^{\\prime },b^{\\prime }=0}^1 (-1)^{a b^{\\prime } + a^{\\prime } b} \\left|\\theta \\left[^a_b\\right](\\tau ,\\frac{z}{2})\\right|^4 | \\theta [^{a^{\\prime }}_{b^{\\prime }}](\\tau ,0)|^2\\,.",
"\\end{split} $ Allowing quarter-integer spins leads to multiple allowed partition functions that maximize the gap.",
"Perhaps the most interesting aspect of this analysis however is not the specific partition function for this case, but rather that fact that searching for constraints in a representation dependent manner yields structure hidden to a flavor-blind analysis.",
"This means that the extremal functional analysis allows one to “discover” a larger class of partition functions when flavored information is included than when it is not.",
"Moreover, uncovering flavored information can potentially split the degeneracy between theories with the same spectrum and therefore the same partition function, allowing us to address the age-old question of whether one can “taste the shape of a drum.” Finally, we continue to refine our representation dependent analysis.",
"For the case of $SU(2)_{1}$ we prove analytically that the theory either contains all representation, or the partition function splits into a product of the diagonal Sugawara partition function and a neutral, modular invariant partition function.",
"After this work was completed, the paper [11] appeared on arXiv also considering modular bootstrap constraints on theories with conserved currents, though the analysis there did not use the flavored partition function."
],
[
"Partition Function Transformation and Background Gauge Fields",
"In this section, we will present an argument for the transformation law (REF ) based on the effective action obtained upon integrating out the CFT in the presence of a background gauge field $A_\\mu $ .",
"Previous treatments have pointed out that in the present context there are two different notions of the partition function that are natural.",
"One of these is the canonical partition function $Z(\\tau ,z)$ defined by (REF ).",
"Following [5], we will refer to an alternate definition as the “path integral” $Z_{\\rm PI}(\\tau , z)$ : $Z_{\\rm PI}(\\tau , z) \\sim e^{ \\pi k B(\\tau , z) } Z(\\tau , z) , \\qquad B(\\tau , z) \\equiv \\frac{z^2+\\bar{z}^2}{2{\\rm Im}(\\tau )}.$ Under $z^{\\prime } = \\frac{z}{c \\tau +d}, \\tau ^{\\prime } = \\frac{a \\tau +b}{c \\tau +d}$ , the factor $B$ is easily seen to transform as $B(\\tau ^{\\prime },z^{\\prime }) &=& B(\\tau ,z) - i \\left( \\frac{ c z^2}{c\\tau +d} - \\frac{c \\bar{z}^2}{c \\bar{\\tau }+d} \\right).$ The important point about the extra factor $B(\\tau ,z)$ is that its transformation cancels the transformation of $Z(\\tau ,z)$ , leaving $Z_{\\rm PI}$ invariant.",
"The basic idea is that $Z_{\\rm PI}$ should be the result of performing a path integral over the torus, and so should be modular invariant.",
"In free boson constructions, one can explicitly see how this factor is generated by the Legendre transform from the Lagrangian to the Hamiltonian [5].",
"However, we would like to see how this arises in a general CFT, without making any reference to a specific form of a Lagrangian.",
"We will begin with the case of an abelian current, and then consider the generalization to a non-abelian symmetry."
],
[
"Modular transformation and the ground state energy",
"First, let us discuss in more detail how to define the “path integral” function $Z(\\tau ,z)$ , what ambiguities are allowed in this definition, and why they do not affect the transformation law (REF ).",
"In order to be invariant under modular transformations, we will need to define the path integral to be invariant under diffeomorphisms and rigid rescalings $w \\rightarrow \\lambda ^{-1} w$ : $d \\Psi e^{ - S_{\\tau }[\\Psi ]} = d\\Psi ^{\\prime } e^{ - S_{\\tau ^{\\prime }} [\\Psi ^{\\prime }]}.$ Here, $\\Psi $ are all the fields of the CFT.",
"As we review in appendix , these two symmetries are sufficient to imply that the path integral defined as an integral over this measure, $Z_{\\rm PI}(\\tau , z) &\\equiv & \\int d \\Psi e^{- S_{\\tau }[\\Psi ] -\\frac{i }{2\\pi } \\int _\\tau A_{\\bar{w}} J^{\\bar{w}}}, \\qquad A_{\\bar{w}} = -i \\frac{z}{2 {\\rm Im}(\\tau )},$ is invariant under modular transformations: $Z_{\\rm PI}\\left( \\frac{a \\tau +b}{c \\tau +d}, \\frac{c z}{c\\tau +d}\\right) &=& Z_{\\rm PI}(\\tau , z).$ Different choices of regulators will change $\\log Z_{\\rm PI}$ by local terms.",
"However, the local terms allowed by diffeomorphism invariance and scale invariance do not affect the transformation law.",
"For instance, one can shift the effective action by a local term proportional to $\\int _\\tau d^2 x \\sqrt{g} A_\\mu A^\\mu \\sim \\int _\\tau dw d\\bar{w} A_w A_{\\bar{w}} \\sim \\frac{z \\bar{z}}{4 {\\rm Im}(\\tau )}.$ This term arises in the difference between a regulator that preserves the vector current $J^\\mu $ symmetry and one that preserves the axial current $\\epsilon ^{\\mu \\nu }J_\\nu $ .",
"However, it is easily seen to be both Weyl invariant and diffeomorphism invariant, and is invariant under modular transformations.",
"So its coefficient is irrelevant for our purposes, and from now on we will neglect such terms without loss of generality.",
"Now, the next question is how do we relate the “path integral” $Z_{\\rm PI}$ to the partition function $Z$ ?",
"The key point is that turning on a background field $A_\\mu $ not only turns on a chemical potential, but it can also shift the ground state energy, since at fixed $\\beta $ such a shift affects only the overall normalization of the path integral.",
"In the example of the free boson, this energy shift is seen explicitly by doing a Legendre transform, but we can see it in full generality by considering the effective action for $A_\\mu $ .",
"To see the shift, it is sufficient to calculate the ground state energy, so we can take the limit of the torus where $\\tau = i \\frac{\\beta }{2 \\pi } , \\beta \\gg 1 $ .",
"In this limit, the torus becomes a cylinder, and the effective action is conformally related to that in flat space, where it is universal and known in closed form.",
"Including the action for a background metric as well, we can write $\\log Z = \\int d^2 x \\sqrt{-g} \\left( \\frac{c}{48 \\pi } R \\Box ^{-1} R + \\frac{k}{8 \\pi } F^{\\mu \\nu } \\Box ^{-1} F_{\\mu \\nu } \\right).$ Because of the inverse Laplacians, the mapping to the cylinder is a bit subtle.",
"For the metric contribution, it is easiest to work with the Wess-Zumino anomaly action directly, $S_{\\rm WZ} = \\frac{c}{24\\pi } \\int d^2 x\\sqrt{-g} \\left( \\sigma R + (\\partial \\sigma )^2 \\right)$ , and take $\\sigma (w) = w+\\bar{w}$ , which reproduces the standard ground state energy shift $-\\frac{c}{12}$ from the Schwarzian derivative.",
"By contrast, the gauge field term in (REF ) is invariant under Weyl transformations, and its contribution to the ground state energy just comes from evaluating the non-local term on the cylinder.",
"To avoid ambiguities associated with the inverse Laplacian, it is clearest to use the fact that the effective action is the generating functional for the $J^\\mu $ correlators, so we know that we can equivalently write the gauge field part of $\\log Z$ as $W_A[A^\\mu ] = \\int \\frac{d^2 x d^2 x^{\\prime }}{(2\\pi i )^2} A_\\mu (x) A_\\nu (x^{\\prime }) \\langle J^\\mu (x) J^\\nu (x^{\\prime }) \\rangle .$ On the plane, $\\langle J^{\\bar{w}}(w) J^{\\bar{w}}(w^{\\prime })\\rangle = \\frac{k}{(w-w^{\\prime })^2}$ .",
"Mapping to the cylinder and taking $A_{\\bar{w}}$ to be constant, we haveWe performed this integration as follows.",
"First, shift $w \\rightarrow w+w^{\\prime }$ to eliminate $w^{\\prime }$ and immediately do the $d^2 w^{\\prime }$ integral, producing just a factor of the volume $2 \\pi \\beta $ of the torus.",
"Passing to $t,\\theta $ coordinates: $W_A[A_\\mu ] = - \\frac{\\beta k A_{\\bar{w}}^2}{2\\pi } \\int _{-\\beta /2}^{\\beta /2} dt \\int _0^{2\\pi } d\\theta \\frac{1}{(e^{\\frac{t+i \\theta }{2}} -e^{\\frac{-t-i \\theta }{2}})^2} .$ If we do the $\\theta $ integral first, this vanishes, except when $t=0$ where it is divergent; the integral over $\\theta $ is proportional to $\\delta (t)$ .",
"We avoid this subtlety if we do the $t$ integral first, in which case we obtain $W_A[A_\\mu ] =- \\frac{\\beta k A_{\\bar{w}}^2}{2\\pi } \\int _0^{2 \\pi } d \\theta \\frac{\\sinh \\frac{\\beta }{2}}{\\cos \\theta - \\cosh \\frac{\\beta }{2}} = \\beta k A_{\\bar{w}}^2 .$ $W_A[A_\\mu ] \\rightarrow \\frac{k}{(2\\pi i)^2} \\int d^2 w d^2 w^{\\prime } \\frac{A_{\\bar{w}}^2}{\\left( e^{\\frac{w-w^{\\prime }}{2}} - e^{\\frac{w^{\\prime }-w}{2}}\\right)^2} = \\beta k A_{\\bar{w}}^2.$ Combining the above with a symmetric combination from $A_w$ , we put everything together to obtain the ground state energy: $E_0 = -\\lim _{\\beta \\rightarrow \\infty } \\beta ^{-1} \\log Z_{\\rm PI} = - \\frac{c}{12} + \\delta E , \\qquad \\delta E = -k (A_w^2+ A_{\\bar{w}}^2).$ Therefore, the path integral differs from the canonical partition function by an extra factor $e^{-\\beta \\phantom{.}",
"\\delta E}$ , which in turn produces the factor $- \\pi k B(\\tau ,z)$ in (REF ,REF ).",
"So at last we see that this factor is universally the contribution to the partition function from the shift in the ground state energy due to the background gauge field.",
"Summarizing, the canonical partition function $Z(\\tau ,z)$ in (REF ) is defined to have a ground state energy $-\\frac{c}{12}$ .",
"However, any path integral over the torus using a regulator that preserves diffeomorphisms and rescalings will have a ground state energy equal to $-\\frac{c}{12} - k (A_w^2 + A_{\\bar{w}}^2)$ , plus possible terms that do not affect the modular transformation of $Z(\\tau ,z)$ ."
],
[
"Non-abelian current transformation",
"The generalization to the case of a non-abelian is straightforward, and can be made as follows.",
"Unlike in the abelian case, the effective action is not quadratic.",
"However, we can write it formally as the sum over all connected diagrams: $W_A[A^{a \\mu } ] = \\sum _{n=1}^\\infty \\left(\\prod _{i=1}^n \\int \\frac{d^2 w_i}{2\\pi i} A^{a_i}_w \\right) \\langle J^{a_1}(w_1) \\dots J^{a_n}(w_n) \\rangle _{\\rm conn} .$ As before, we want to set $A_w^a, A_{\\bar{w}}^a$ to be constant on the cylinder and integrate over $d^2 w_i$ .",
"For the part quadratic in $A$ , the computation proceeds just as in the abelian case, we simply have an extra index for the different components of $J^a_w$ .",
"The background field couples as $\\frac{-i}{2\\pi } \\int _\\tau A_\\mu ^a J^{\\mu a}, \\qquad A_w^a = -i \\frac{\\bar{z}^a}{2 {\\rm Im}(\\tau )}.$ $A$ 's contribution to the ground state energy is $\\delta E \\cong -k((A_w^a)^2+(A_{\\bar{w}}^a)^2),$ which transforms under modular transformations as $-\\beta \\delta E \\rightarrow -\\beta \\delta E - i \\pi k \\left( \\frac{c (z^a)^2}{c\\tau +d} - \\frac{c (\\bar{z}^a)^2}{c \\bar{\\tau }+d} \\right).$ That leaves the contribution from the higher-point functions.",
"We can always write these in terms of lower-point function by using the recursive formula $J^a(w)J^b(0) \\sim \\frac{k \\delta ^{ab}}{w^2} + \\frac{f^{abc} J^c(0)}{w},$ where $\\sim $ means `up to non-singular terms'.",
"The $\\frac{k \\delta ^{ab}}{w^2}$ piece manifestly generates disconnected diagrams - it produces the two-point function times the $(n-2)$ -point function - so it does not contribute to the effective action for higher-point correlators.",
"But, since we multiply the correlator by $A^a_w$ in the effective action, the $f^{abc}$ term also gives no contribution for constant $A_w^a$ : $A_w^a J^a(w) A_w^b J^b(0) \\sim \\frac{k A_w^2 }{ w^2} + A_w^a A_w^b f^{abc} \\frac{J^c(0)}{w} = \\frac{k A_w^2 }{w^2}$ since $A^a A^b f^{abc} =0$ .",
"Therefore only the two-point functions contribute."
],
[
"Basic setup",
"In all the cases we consider, we will assume the presence of a conserved current $J^a$ in the theory.",
"In general, it is convenient to separate the stress tensor $T$ of the theory into a Sugawara stress tensor piece and a residual piece: $T^{(0)} \\equiv T - T^{\\rm sug},\\qquad T^{\\rm sug} = \\frac{1/2}{k+\\widetilde{h}_G} \\sum _{a=1}^{|G|} : J^a J^a : ,$ where $\\widetilde{h}_G$ is the dual Coxter number, because the modes of $T^{(0)}$ commute with the modes of $J^a$ .",
"Furthermore among themselves they form a Virasoro algebra with central charge $c^{(0)} = c- c^{\\rm sug}, \\qquad c^{\\rm sug} =\\frac{k |G|}{k+\\widetilde{h}_G} .$ Similarly, we can separate the Virasoro generators $L_n = L_n^{(0)}+L_n^{\\rm sug}$ and the weights $h = h^{(0)} + h^{\\rm sug}$ into a part that comes from $T^{(0)}$ and a part that comes from $T^{\\rm sug}$ .",
"For most representations, the distinction between $T$ and $T^{(0)}$ will not make much difference, since the partition function just counts states at each level.",
"However, for the special cases with shortening conditions, some descendants becomes null and do not contribute to the partition function, and this is easier to see using the modes of $T^{(0)}$ .",
"The characters of the Kac-moody algebra $\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})={\\rm Tr}_{V_{\\mathbf {\\mu },k}}q^{L_0^{\\rm sug}-\\frac{c^{\\rm sug}}{24}}e^{2\\pi i \\mathbf {z}\\cdot \\mathbf {H}_0}$ are constructed by acting modes of $J^a$ on some highest weight state which has weight $\\mathbf {\\mu }$ .See e.g.",
"[12] for a standard introduction.",
"Here, ${\\bf H}_0$ is the vector of Cartan generators of the algebra.",
"In the case of an abelian symmetry, ${\\bf H_0} = J_0$ and the characters for a generic primary are simply $ q^{ - \\frac{c^{\\rm sug}-1}{24}}e^{2 \\pi i z Q}/\\eta (\\tau )$ .",
"In the case of a non-abelian symmetry, the characters are more complicated.",
"Some descendants of such a highest weight state may be null so it is non-trivial to write down its form.",
"However, for the purpose of the modular bootstrap, the only property of such characters we use is that the characters transform covariantly $\\mathcal {X}_{\\mathbf {\\mu },k}\\left(-\\frac{1}{\\tau },\\frac{\\mathbf {z}}{\\tau }\\right)=e^{\\frac{i \\pi k \\mathbf {z}^2}{\\tau }}\\sum _{\\mathbf {\\mu ^\\prime }}S_{\\mathbf {\\mu \\mu ^\\prime }}^k\\mathcal {X}_{\\mathbf {\\mu ^\\prime },k}(\\tau ,\\mathbf {z}) ,$ where the matrix $S$ depends on the symmetry group and level $k$ .",
"These characters do not include the modes of $T^{(0)}$ yet.",
"Since the algebra generated by modes of $J^a$ is completely orthogonal to that generated by modes of $T^{(0)}$ , the character generated by the full extended algebra simply factorizes into a Kac-Moody character and a Virasoro character $\\mathcal {X}_{\\mathbf {\\mu },k,h}(\\tau , \\mathbf {z}) = \\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z}) \\mathcal {X}_{h^{(0)}}(\\tau )~.$ Like the simple Virasoro character, the character is different if the primary saturates the unitarity bound: $\\mathcal {X}_{h^{(0)}} (\\tau ) = \\left\\lbrace \\begin{array}{cc} \\frac{q^{h^{(0)}-\\frac{c^{(0)}-1}{24}}}{\\eta (\\tau )} & h^{(0)} >0 \\\\\\frac{(1-q)q^{-\\frac{c^{(0)}-1}{24}}}{\\eta (\\tau )} & h^{(0)}=0 \\end{array} \\right.",
".$ The same goes for the anti-holomorphic part.",
"The full partition function is $Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}}) =\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}}d_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}} \\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}}) \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\bar{\\tau }) ~.$ In the above equation the $\\bar{\\mu }$ means the representation of the anti-holomorphic part.",
"When we are dealing with a non-abelian symmetry, it will be convenient to define a matrix $M_{\\mu , \\bar{\\mu }}$ whose components are the coefficients of the contributions to the partition function from the different representations: $ \\begin{split} M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}&= \\sum _{h, \\bar{h}}d_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }},h,\\bar{h}} \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\bar{\\tau })\\\\Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}})&= \\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}} M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\,.",
"\\end{split} $ Modular transformations on the partition function translates into a specific modular transformation of the matrix $M_{\\mu , \\bar{\\mu }}$ .",
"To see this transformation law, simply separate out the transformation law of $Z$ into its irrep constituents: $ \\begin{split} 0&=Z(-\\frac{1}{\\tau }, -\\frac{1}{\\bar{\\tau }}, \\frac{\\mathbf {z}}{\\tau },\\frac{\\mathbf {\\bar{z}}}{\\bar{\\tau }}) -e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)} Z(\\tau , \\bar{\\tau }, \\mathbf {z},\\mathbf {\\bar{z}}) \\\\&=\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}} M\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}\\left(-\\frac{1}{\\tau },\\frac{\\mathbf {z}}{\\tau }\\right)\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}\\left(-\\frac{1}{\\bar{\\tau }},\\frac{\\bar{\\mathbf {z}}}{\\bar{\\tau }}\\right)-e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)}M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\mathcal {X}_{\\mathbf {\\mu },k}(\\tau ,\\mathbf {z})\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\\\&=\\sum _{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\left(\\sum _{\\mathbf {\\mu }^{\\prime },\\mathbf {\\bar{\\mu }^{\\prime }}}S_{\\mathbf {\\mu ^\\prime \\mu }}^kM\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mathbf {\\mu ^{\\prime }},\\mathbf {\\bar{\\mu }^{\\prime }}}\\bar{S}_{\\mathbf {\\bar{\\mu }^\\prime \\bar{\\mu }}}^k- M(\\tau ,\\bar{\\tau })_{\\mathbf {\\mu },\\mathbf {\\bar{\\mu }}}\\right)e^{i \\pi k \\left(\\frac{\\mathbf {z}^2}{\\tau }-\\frac{\\bar{\\mathbf {z}}^2}{\\bar{\\tau }}\\right)}\\mathcal {X}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\bar{\\mathcal {X}}_{\\mathbf {\\bar{\\mu }},k}(\\bar{\\tau },\\bar{\\mathbf {z}})\\,, \\end{split} $ where we have used the transformation rule, (REF ), and the definition (REF ).",
"Stripping off the characters, the above crossing equation is equivalent to a crossing equation for the matrix $0 = M(\\tau ,\\bar{\\tau })_{\\mu ,\\bar{\\mu }} - S^T_{\\mu ,\\mu ^\\prime }M\\left(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}\\right)_{\\mu ^\\prime ,\\bar{\\mu }^\\prime }\\bar{S}_{\\bar{\\mu }^\\prime ,\\bar{\\mu }}\\,.$ For the constraints on theories with non-abelian currents, equation (REF ) is the form of the constraint that we will use.",
"For each bootstrap question we will input the symmetry group and level $k$ ."
],
[
"Semidefinite Projective Functionals and the Extremal Method",
"To be self-contained, we will briefly review linear and semidefinite programming methods as applied to the modular bootstrap; for more thorough reviews and some examples of applications, see e.g.",
"[4], [10], [13], [14], [15], [16], [17], [18], or [19], [20], [21], [22], [23], [24], [25] for reviews and some of the original papers developing methods in the standard bootstrap that we will adopt directly.",
"The starting point is equation (REF ), which can be written $0 &=& \\sum _{h, \\bar{h}, \\mu ^{\\prime }, \\bar{\\mu }^{\\prime } } d_{\\mu , \\bar{\\mu }, h, \\bar{h} } \\left( F_{\\mu , \\bar{\\mu }, h, \\bar{h}}\\right)_{\\mu ^{\\prime }, \\bar{\\mu }^{\\prime }}(\\tau , \\bar{\\tau }) , \\\\\\left( F_{\\mu , \\bar{\\mu }, h, \\bar{h}}\\right)_{\\mu ^{\\prime }, \\bar{\\mu }^{\\prime }}(\\tau , \\bar{\\tau }) &\\equiv & \\left[ \\delta _{\\mu \\mu ^{\\prime }} \\delta _{\\bar{\\mu } \\bar{\\mu }^{\\prime }} \\mathcal {X}_{h^{(0)}}(\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(\\tau ) - S_{\\mu ^{\\prime } \\mu } S_{\\bar{\\mu }^{\\prime }, \\bar{\\mu }} \\mathcal {X}_{h^{(0)}}(-1/\\tau ) \\mathcal {X}_{\\bar{h}^{(0)}}(-1/\\bar{\\tau }) \\right] .\\nonumber \\\\$ The occupation numbers $d_{\\mu , \\bar{\\mu }, h,\\bar{h}}$ are all non-negative, and include in particular the vacuum $d_{\\rm vac}=1$ .",
"One is generally interested in proving that there exist states in the theory with various properties, for instance that there exists a state with $\\Delta < \\Delta _{\\rm max}$ for some value of $\\Delta _{\\rm max}$ .",
"Let us abstractly call a choice of such properties “$P$ ”.",
"Then, one can prove that there is at least one state in the theory with properties $P$ as long as one can find a linear functional $\\rho $ that maps the characters to real numbers such that it is positive on the vacuum and also positive on all states not satisfying $P$ .",
"In equations, $\\rho ({\\rm vac})= 1 , \\textrm { and } \\rho (F_{\\mu , \\bar{\\mu }, h, \\bar{h}} ) \\ge 0 \\textrm { unless } (\\mu , \\bar{\\mu }, h, \\bar{h}) \\textrm { satisfies } P.$ The normalization $\\rho ({\\rm vac})=1$ is conventional.",
"If such a linear functional $\\rho $ exists, then there must be a state in the theory with the properties $P$ , otherwise $\\rho $ acting on equation (REF ) would imply $0 \\ge 1$ .",
"Usually we will be interested in not just one choice of $P$ but a continuous family of choices $P_s$ parameterized by a continuous variable (or variables) $s$ .",
"Typically, $s$ will be something like the bound $\\Delta _{\\rm max}$ in the example above, so that as one decreases $s$ the set of states with property $P_s$ grows and therefore the set of linear functionals that are positive on all such states shrinks.",
"Critical values $s_*$ of $s$ where the set of such linear functionals vanishes are especially interesting: aside from giving the best possible bounds, at these points one can use the “extremal functional method” [19] to determine all of the occupation numbers $d_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ .",
"The basic idea behind this is that for any $s$ , the space of functions $F_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ spanned by states satisfying $P_s$ is a polytope where $-F_{\\rm vac}$ is inside the polytope for $s<s_*$ and outside the polytope for $s>s_*$ .",
"At exactly $s_*$ , $-F_{\\rm vac}$ passes through one of the faces of the polytope, so there is a unique positive semidefinite linear combination of the states satisfying $P_{s_*}$ that cancels the contribution from the vacuum in (REF ).",
"In practice, we have to work with finite-dimensional projections of the full space of functions $F_{\\mu , \\bar{\\mu }, h, \\bar{h}}$ , but one optimistically expects to converge to a unique solution as the dimensionality of the projected space increases.",
"We will encounter some exceptions that we will discuss as we come to them."
],
[
"Abelian Bounds",
"In this section, we perform a systematic numerical analysis on the bounds on the gap in dimensions and charges, as well as on the smallest charge-to-mass ratio allowed in a theory with a $U(1)$ current."
],
[
"Semi-definite Programming with Continuous Charge $Q$",
"For the abelian case, for simplicity we will not use the full Kac-Moody characters, but rather just the Virasoro characters $\\chi _h(q)$ : $Z(\\tau , z) =\\sum _{h, \\bar{h}, Q, \\bar{Q}} d_{Q, \\bar{Q}, h , \\bar{h}} y^Q \\bar{y}^{\\bar{Q}} \\chi _h(q) \\chi _{\\bar{h}}(\\bar{q}).$ where $q= e^{2\\pi i \\tau }$ , $\\bar{q} = e^{-2\\pi i \\bar{\\tau }}$ , $y =e^{2\\pi i z}$ , $\\bar{y} = e^{-2\\pi i \\bar{z}}$ .",
"We will consider left-right symmetric theories with $c = \\bar{c}$ , and for simplicity we set $\\bar{z} = 0$ .",
"As in [4], we reduce the characters using the $S$ invariant factor $|\\tau |^{\\frac{1}{2}}|\\eta (\\tau )|^2$ .",
"Furthermore we reduce the partition function as $\\hat{Z}(\\tau , z) \\equiv |e^{\\frac{i\\pi z^2}{2\\tau } }|^2|\\tau |^{\\frac{1}{2}}|\\eta (\\tau )|^2 Z(\\tau , z) ,$ so that $\\hat{Z}(\\tau , z)$ is invariant under $ \\left( \\tau \\mapsto -\\frac{1}{\\tau }, z \\mapsto \\frac{z}{\\tau } \\right)$ .",
"The characters are reduced into $\\hat{\\chi }_0(q) = e^{\\frac{i\\pi z^2}{2\\tau } } \\tau ^{\\frac{1}{4}}q^{-\\frac{c-1}{24}}(1-q) ,~~~~\\hat{\\chi }_h(q) y^Q = e^{\\frac{i\\pi z^2}{2\\tau } } \\tau ^{\\frac{1}{4}}q^{-\\frac{c-1}{24}}q^h y^Q .$ We consider linear functionals of the form $\\rho \\equiv \\sum _{m+n+k/2 {\\rm ~odd,~} k {\\rm ~even}}\\left.",
"\\alpha _{m,n,k} \\ \\partial _{t}^m \\, \\partial _{\\bar{t}}^n\\,\\partial _{w}^k\\right|_{t=\\bar{t}=w=0},$ where the change of variable $\\tau = i e^{t}$ and $z =we^{\\frac{t}{2}}$ is made so that $t \\mapsto -t$ and $w^2 \\mapsto -w^2$ under $S$ transformations.In this expression for $\\rho $ , we have not used any $\\bar{z}$ derivatives and so do not use information about the anti-holomorphic charge $\\bar{Q}$ .",
"This is mainly for simplicity and efficiency; it is in principle straightforward, though more computationally intensive, to use $\\bar{Q}$ information as well.",
"Later in this section we will in fact perform one analysis where we keep $\\bar{z}$ derivatives to demonstrate this point.",
"We arrive at a two variable functional $\\rho _l(\\Delta ,Q) &\\equiv \\rho \\left[ \\hat{\\chi }_{\\frac{\\Delta -l}{2}}(\\tau )\\hat{\\bar{\\chi }}_{\\frac{\\Delta +l}{2}}(\\bar{\\tau }) y^Q +\\hat{\\chi }_{\\frac{\\Delta +l}{2}}(\\tau )\\hat{\\bar{\\chi }}_{\\frac{\\Delta -l}{2}}(\\bar{\\tau }) \\right] , \\nonumber \\\\\\rho ({\\rm vac}) &\\equiv \\rho \\left[ \\hat{\\chi }_{0}(\\tau )\\hat{\\bar{\\chi }}_{0}(\\bar{\\tau }) \\right],$ where we assume the spectrum is parity symmetric.",
"Part of the challenge of the abelian analysis is that we do not assume charge quantization, i.e.",
"technically we allow the gauge group to be $\\mathbb {R}$ instead of $U(1)$ , which means that we have to deal with not just one but two continuous parameters, $\\Delta $ and $Q$ .",
"This complicates the application of positive semi-definite approaches, since these are based on constructing positive functionals of the characters and in general the space of such functionals is more complicated for multiple variables than for a single variable.",
"In particular, for a single variable, positive semi-definite functionals can be written without loss of generality as a sum of squares plus a linear term times a sum of squares.",
"For multiple variables, such a parameterization is no longer completely general.",
"One way to deal with this issue is simply to discretize in, say, $Q$ , but we find that such an approach becomes difficult to implement in practice since the discretization needs to become very fine to prevent the numeric search from picking functionals that become negative in between the discretization points.",
"The approach we take is instead to limit the search space to functionals that are still a sum of squares plus a linear term times sums of squares.",
"In the limit of very high order polynomials, one might expect that such functionals can approximate the extremal functionals arbitrarily well.",
"In any case, while such functionals might not give the best possible bounds, they nevertheless produce valid bounds.",
"Even restricting to polynomial functionals, there remains a practical problem of how to implement the search over such functionals using available software for semi-definite programming analyses.",
"In appendix , we discuss how to massage this problem into an appropriate form for use with SDPB."
],
[
"Bound on Dimension of Lightest Charged State",
"With the flavored partition function we can bound the dimension $\\Delta _*$ of lightest charged state in any theory with a U(1).",
"The bound for different $c$ is summerized in Table.",
"REF .",
"We extrapolate the bound values to $n_D \\rightarrow \\infty $ using a linear function of $\\frac{1}{n_D}$ similar to what is done in [10] for $c \\le 100$ where the convergence of the bound values is significant.",
"Then we extrapolate the bounds to $n_D\\rightarrow \\infty $ and $c \\rightarrow \\infty $ by fitting the finite $n_D$ and $c$ results to a linear function of $\\frac{1}{c}$ and $1/n_D$ and extrapolating, as shown in Fig.",
"REF .",
"In this test we take $\\tau = \\frac{i \\beta }{2 \\pi }$ with $\\beta $ real to avoid the complication of spin.",
"It is not understood a priori why the form of this fit works, but empirically it agrees well with the data at large $c$ and $n_D$ .",
"Table: Bounds on the dimension of lightestcharged state assuming the theory has U(1) symmetry, as a function of cc and the number n D n_D of derivatives used in the bootstrap functionals.Figure: Bounds of the dimension oflightest charged state assuming the theory has U(1) symmetry.",
"Theextrapolated gaps at n D →∞n_D \\rightarrow \\infty with the trendline.Similarly to the results of [10], extrapolating in $n_{D}$ and then $c$ provides a parametrically stronger bound than the finite $n_{D}$ analysis.",
"$\\Delta _{*} = \\frac{c}{\\alpha } + \\mathcal {O}(1)\\,, \\ \\ \\ \\alpha > 8\\,.$ This bound (REF ) is similar to the bounds on the non-charged state found in [4], [10], though quantitatively different.",
"The bound in [4] is parametrically weaker, which is not surprising since that analysis did not use the spins of the characters, and did not perform any extrapolation in the number of derivatives.",
"The bound in [10] is more analogous since spins and extrapolations were used; €” the result there is very slightly stronger ($\\alpha \\sim 9$ ) than (REF ) for the charged spectrum."
],
[
"Bound on Charge-to-Mass Ratio",
"In this section, we will present results that there must be a state in the theory with a charge-to-mass ratio $r \\equiv \\frac{Q c}{12 \\Delta } = \\frac{Q}{8 G_N m},$ above some critical value $r_*$ , whose value we will determine numerically.The value of $r^*$ increases as the number $n_D$ of derivatives used increases and the numeric accuracy improves, though we emphasize that even for low number of derivatives the values of $r_*$ are a valid bound proving that a state must exist in the theory with $\\frac{Q}{8 G_N m} > r_*$ .",
"We try to find the linear functional thatWe also impose a dimension cutoff $\\rho (\\Delta ,Q) \\ge 0,~~~ \\Delta >\\frac{100c}{12}$ , because we want the constraint to be a little stronger that not only a state which saturates the ratio bound exists but also the state must have finite dimension.",
"Different dimension cutoffs do not result in significantly different functionals and bounds.",
"It just helps the algorithm to find a functional that only is negative at finite $\\Delta $ .",
"$\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ |Q| \\le Q_{\\Delta } \\equiv \\frac{12 r \\Delta }{c}~.$ We drop the spin index $l$ by only taking functionals of the form $\\rho \\equiv \\sum _{m+k/2 {\\rm ~odd,~} k {\\rm ~even}} (\\partial _t+ \\partial _{\\bar{t}})^m \\partial _w^k ~.$ For the functional to be positive in a bounded region we make a change of variables of the form $Q^2 = \\frac{\\tilde{Q}^2 Q_{\\Delta }^2}{\\tilde{Q}^2 +1 }~.$ $|\\tilde{Q}| \\ge 0$ means $|Q| \\le Q_{\\Delta }$ , inspired by [10].",
"First, we show in Fig.",
"REF the bound on $r_*$ as a function of $c$ .",
"By inspection, one can see that the larger $c$ is, the longer it takes for the bounds to converge.",
"To see how the bound depends on the number $n_D$ of derivatives in more detail, in Fig.",
"REF we focus on a specific value of $c$ , $c=10^5$ and show the resulting bound on $r$ as a function of the number $n_D$ of derivatives allows in the functional $\\rho $ .",
"The best fit as power law suggests that the optimal bound on $r^*$ might be significantly better, i.e.",
"$(r^{*})^{-1} \\ll 1 $ .",
"For comparison, the result in [6] was $\\left(\\frac{8 G_N m}{Q}\\right)_* = (r_*)^{-1} < 4 \\sqrt{\\pi } = 7.1$ .",
"Figure: Bound of mass-to-charge ratio as a function of cc; a trend line ∝c -1/2 \\propto c^{-1/2} is shown for comparison.",
"The extrapolation (“n D =∞n_D=\\infty ” points) and error bars are computed by performing a fit as a function of n D n_D and extrapolating to n D →∞n_D\\rightarrow \\infty as described in the text.In [6], it was also shown that even with a small number $n_D$ of derivatives, one could obtain a bound on $\\frac{\\Delta }{Q}$ at large $c$ that scaled like $\\sim c $ .",
"There is an intriguing possibility however that the true bound scales like $c^{1/2}$ , and that this is obscured because it takes more and more derivatives to reach this optimal bound as $c$ increases.",
"The basic idea for why one might expect a $c^{1/2}$ scaling is that in higher dimensions, the scaling of the WGC limit can easily be read off by demanding that the binding energy from gravity and a Coulomb force cancel each other out.",
"In the AdS$_3$ case, one can think of the binding energy from gravity as $ \\frac{3\\Delta ^2}{c}$ , whereas from a $k=1$ Chern-Simons gauge field exchange it is $Q^2$ ; demanding equality would set $\\frac{\\Delta }{Q} \\approx \\sqrt{c/3}$ , i.e.",
"$\\frac{8 G_N m}{Q} \\approx 6.9 c^{-1/2}$ .",
"One can read off the coefficients by looking at the vacuum conformal block for Virasoro and Kac-Moody algebras in the limit $z\\sim 1$ [26], [27].",
"For comparison, in Fig.",
"REF we have also shown a trend line at $c^{-1/2}$ , which becomes further below our best numeric bound with $n_D=29$ derivatives as $c$ increases.",
"We can try to estimate the optimal bound by taking our result at each $c$ as a function of $n_D$ and extrapolating to $n_D = \\infty $ .",
"The “$n_D=\\infty $ ” points we show in Fig.",
"REF fit our results starting at $n_D \\ge 15$ in order to get the extrapolation.",
"However, there is significant uncertainty in the resulting estimate, as can be gauged by the fact that performing the fit starting at smaller or larger values of $n_D$ gives different answers.",
"In Fig.",
"REF we have shown the bound as a function of $n_D$ , where one can explicitly see that the bound is still changing rapidly as a function of $n_D$ even at the upper range of what we have been able to achieve numerically.",
"In Fig.",
"REF , the error bars we have shown indicate the range over all the different positive values we obtain if we perform the fit over $n_D \\ge 11, n_D \\ge 13, \\dots n_D \\ge 19$ .",
"With better numerical accuracy at large values of $c$ , it should be possible to more firmly establish this scaling behavior.",
"Figure: Upper bound on 8G N m Q\\frac{8 G_N m}{Q} (that is, there existsa state below the bound) as the number n D n_D of derivatives usedin the semidefinite programming analysis increases, for thespecific case c=10 5 c=10^5.",
"The bound value is still changing rapidlyat n D =41n_D = 41."
],
[
"Bound on Lowest Charge",
"Next we will focus on bounds on the lowest charge $Q_*$ of all charged states in the theory.",
"We will first consider the charge $Q$ only, and then see how to do better by including information on dimensions and spins.",
"To determine the upper bound on the gap to the smallest $|Q|$ of all the charged states, we want to find a linear functional satisfying the following conditions: $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge 0 \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge 0 {\\rm ~~~And~~~} |Q| \\ge Q_*$ The resulting bound on $Q_*$ is shown in Fig.",
"REF .",
"The result is somewhat surprisingly always just $Q_* = 1$ .",
"This may be because in some theories with $c\\le 1$ a state of $Q=1$ saturates the bound and theories of larger $c$ can be constructed as a direct product of such theories and other algebras.",
"Figure: We obtain an upper bound, shown here, on the smallest nonzero charge Q * Q_*; the bound isQ * ≤1Q_* \\le 1 for all cc.In any case, we next turn to including information on dimensions by bounding gaps in both $Q$ and $\\Delta $ simultaneously – $Q_*$ as the lowest charge of charged states and $\\Delta _*$ as the lowest dimension of all non-vacuum states.",
"To obtain such a bound, the linear functional $\\rho $ should satisfy $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho (\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge \\Delta _* \\nonumber \\\\\\rho (\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge \\Delta _* {\\rm ~~~And~~~} |Q| \\ge Q_*$ We take the linear functional to have no spin information.",
"Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\Delta _* at c=2c=2.",
"The region near the kink in the left plot is magnified and shown in the right plot.At each individual $c$ , the bound carves out a region in a two-dimensional parameter space.",
"The exclusion curve of an $c=2$ example is shown in Fig.",
"REF .",
"There is a kink at $\\Delta _* \\approx 0.5$ which has a bound $Q_* \\le 1$ .",
"It would be interesting to identify what if any theory lives at this kink.",
"Finally, the simutaneous $\\Delta _*$ and $Q_*$ approach can be even more powerful if we turn on spin.",
"For this case the example we choose is $c=8$ .",
"In [10] the Sugawara theory $E_8$ lattice shows up as a kink of bounds on lowest dimension of scalar primaries.",
"We seek a linear functional $\\rho $ satisfying $\\rho ({\\rm vac}) \\ge 0,& \\nonumber \\\\\\rho _0(\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge \\Delta _* \\nonumber \\\\\\rho _l(\\Delta ,0) \\ge 0,&~~~ \\Delta \\ge |l| \\nonumber \\\\\\rho _0(\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge \\Delta _* {\\rm ~~~And~~~} |Q| \\ge Q_* \\nonumber \\\\\\rho _l(\\Delta ,Q) \\ge 0,&~~~ \\Delta \\ge |l| {\\rm ~~~And~~~} |Q| \\ge Q_*$ for all $l \\in \\mathbb {Z}_{\\ge 0}$ .",
"Figure: Bound on the charge gap Q * Q_* and the scalar dimension gapΔ * \\Delta _* at c=8c=8.The two parameter plot of $\\Delta _*$ and $Q_*$ is shown in Fig.",
"REF .",
"Note that we find that $Q_*$ at $\\Delta _*=0$ is smaller than 1, better than the bound obtained without $Q_*$ informationIt is also possible that by assuming integer spins we throw away the theory that saturates the $Q_{*}=1$ bound.. More interestingly, we see a sharp kink at $\\Delta _{*}=2$ .",
"In the next subsection we will analyze the extremal functional of this kink and see that this kink is the $E_8$ lattice CFT, and we will obtain the dimension and charge spectrum of the low lying states."
],
[
"Extremal Functional Analysis",
"In this subsection, we will use extremal functional analyses to determine the partition function saturating various bounds."
],
[
"Maximal $r_*$ at {{formula:9d4b5483-4fe8-4eda-a64b-878604bb43b5}}",
"Our first application of extremal function methods will be to the bound on the charge-to-mass ratio $r$ .",
"Since our bounds have converged for $c=1$ and we can consider the extremal functional $\\rho $ for this case; by design, $\\rho $ is non-negative on the space of states we allow, and so the states in the theory must be at the places where $\\rho $ vanishes.",
"The functional depends on both $\\Delta $ and $Q$ , so the extremal spectrum contains more data as shown by Fig.",
"REF .",
"Figure: Projection functional ρ\\rho of c=1c=1 as a function ofΔ\\Delta and QQ computed at N=29N=29.",
"The black regions are where ρ≤0\\rho \\le 0; according to our criterion (), ρ\\rho is ≥0\\ge 0 in the allowed region Q≤12r * Δ cQ\\le \\frac{12 r_* \\Delta }{c}, so the extremal spectrum comprises states where ρ=0\\rho =0 in this region.",
"There is a line of small black dots where ρ=0\\rho =0 along the Q=0Q=0 axis that are difficult to see and so we plot ρ\\rho along this line in Fig.",
".The zeros of $\\rho $ occur at points and can be difficult to see in Fig.",
"REF .",
"In Fig.",
"REF , we show $\\rho $ along two particularly relevant lines: the neutral ($Q=0$ ) states, and the states that saturate the mass-to-charge ratio, i.e.",
"$Q= \\frac{12 r_* \\Delta }{c}$ .",
"Figure: The functional restricted at Q=0Q=0 is shown on theleft.",
"The states that saturate the best bound on8G N m Q\\frac{8 G_N m}{Q} is shown on the right."
],
[
"Sequential $\\Delta _*$ and {{formula:b25e4990-736d-44d3-b86f-624d64285424}} Approach - Revisiting the {{formula:b4d2cd11-1472-4199-8de0-93f481d9bb5d}} Lattice",
"In Sec.",
"REF we found a kink in the simultaneous $\\Delta _*$ and $Q_*$ approach with spin information at $c=8$ .",
"In order to show that the kink is indeed the level 1 $E_8$ lattice, we can look for extremal flavored partition functions by using a “two-step” approach where we first solve for the spectrum of dimensions and then solve for the spectrum of charges.",
"The idea is that we can use the extremal functional method on the unflavored partition function, maximizing the gap in dimensions of operators.",
"This step is just the standard extremal functional method and it will just reproduce previous results [10].",
"Then, having fixed the weights of the states in the theory, we can impose a gap in the charge of the states in the theory, allowing only the weights $(h,\\bar{h})$ found previously.",
"For concreteness, we will focus on the case $c=8$ as a representative example.",
"As shown in [10], the gap in dimensions is maximized at $\\Delta _*=2$ by the $E_8$ theory at $k=1$ (this theory can be described as 8 free bosons on an $E_8$ lattice), and the extremal functional method allows one to independently derive the partition function of this theory.",
"We have reproduced the extremal functionals $\\rho _\\ell (\\Delta )$ at each spin $\\ell $ in Fig.",
"REF , which implies the spectrum is ${\\left\\lbrace \\begin{array}{ll}\\Delta = 2,4,6,\\ldots ,& l=0,\\\\\\Delta = l,l+2,l+4,\\ldots ,& l\\ne 0 .\\\\\\end{array}\\right.",
"}$ Figure: Extremal functional of c=8c=8 theory, at n D =35n_D=35.Moving on to the spectrum of charges, we find that the gap $Q_*$ is maximized at $Q_* = \\frac{1}{\\sqrt{2}}$ .",
"The corresponding extremal functionals $\\rho _{\\Delta , \\ell }(Q)$ at each dimension $\\Delta $ and spin $\\ell $ are plotted in Fig.",
"REF .",
"We note that this is not the flavored partition function that one obtains if one turns on a chemical potential for the charge $J = \\partial \\phi $ in the $E_8$ lattice description; that choice corresponds to the spectrum of charges $\\frac{1}{2}\\mathbb {Z}$ rather than $\\frac{1}{\\sqrt{2}} \\mathbb {Z}$ .",
"Instead, if one chooses one of the length-2 vectors $\\vec{\\alpha }$ in the $E_8$ lattice, then $J \\equiv V_{\\vec{\\alpha }} \\equiv \\frac{1}{\\sqrt{2}} \\left( e^{\\vec{\\alpha }\\cdot \\vec{\\phi }} + e^{-\\vec{\\alpha } \\cdot \\vec{\\phi }} \\right)$ has $k=1$ , and the lowest charged states include for instance $V_{2\\vec{\\alpha }}$ , with charge $\\frac{1}{\\sqrt{2}}$ .",
"One can see in Fig.",
"REF that the extremal functional has zeros at around $0, \\pm \\frac{1}{\\sqrt{2}}$ and $\\pm \\frac{2}{\\sqrt{2}}$ for all $\\Delta , \\ell $ , and we expect that this would continue to be true at $ \\frac{n}{\\sqrt{2}}$ for all $n \\in \\mathbb {Z}$ as the number $n_D$ of derivatives used in the analysis approaches infinity.",
"Because these dimensions and charges appear to follow such a simple pattern, we will proceed by assuming this pattern continues.",
"Then, with the allowed weights $\\Delta , \\ell $ and charges $Q$ fixed in the theory, solving the modular bootstrap equation reduces to a linear programming problem, which is much more efficient numerically.Furthermore, since this linear programming analysis fixes the partition function for us to be a particular flavoring of the $E_8$ theory, by uniqueness it will be the correct one, justifying the original Ansatz.",
"We obtain the occupation numbers indicated in Table REF , where we have flavored separately by both holomorphic and anti-holomorphic charges $Q$ and $\\bar{Q}$ .",
"We show the occupation numbers of the conserved $ \\ell = 1$ currents assuming the extremal charge spectrum $Q= \\frac{n}{2}$ (right).",
"In addition, it is straightforward to repeat the analysis assuming $Q = \\frac{n}{\\sqrt{2}}$ (left) for comparison.",
"In both cases, we obtain a total of 248 currents each in the holomorphic and anti-holomorphic part, but distributed differently among different charges in the two cases.",
"Table: Occupation numbers from a linear programming analysis.",
"The left table assumes states at Q∈1 2ℤQ \\in \\frac{1}{\\sqrt{2}} \\mathbb {Z}, whereas the right assumes Q∈1 2ℤQ \\in \\frac{1}{2} \\mathbb {Z}.Figure: Extremal functional of QQ at n D =19n_D=19 when the gap on QQ is maximizedat 1 2\\frac{1}{\\sqrt{2}}."
],
[
"Bounds on gaps in operator dimensions",
"Next we turn to a numeric analysis of gaps in non-abelian theories.",
"In some cases, the results are somewhat stronger or weaker depending on whether or not we allow for states that saturate the unitarity bound $2 k h \\ge Q^2$ , which we will refer to as “extremal states”, and whether or not we impose gaps in all charge sectors.",
"We will present results starting with the strongest assumptions first.",
"In all cases, we will present only the results for $SU(2)$ gauge group at level $k=1$ .",
"We have analyzed larger gauge groups and higher levels and the results are qualitatively similar, though the rate of numeric convergence is worse; some preliminary results for $SU(2)$ with $k=2$ are shown in appendix .",
"To begin, we will set the gap in all representations to be the same and restrict to the partition function at $q=\\bar{q}$ ; the resulting gap value will be the lowest dimension of the primary operators.",
"This “uniform bound” is shown in Fig.",
"REF .",
"We have actually done two slightly different analysis, which are compared to each other on the right in Fig.",
"REF .",
"These analyses differ in whether or not we allow states in the non-trivial representations with $h^{(0)}= \\bar{h}^{(0)}=0$ , which saturate the unitarity bound in both the holomorphic and anti-holomorphic sectors and which we will refer to as “extremal states;” in the analysis where such states are included, the “gap” for each representation is defined as the smallest $\\Delta ^{(0)}$ among the non-extremal states.",
"As one can see, the difference between the results of the two analyses is significant at small $c$ but becomes negligible as $c$ approaches $\\infty $ .",
"Ultimately, this result does not tell us much more than one learns from previous similar analyses without flavored information; all we learn here is that there must be some state in the theory with $\\Delta ^{(0)}$ below some value, which is very similar to the bound on the same quantity from the unflavored modular bootstrap.",
"Figure: Bound on SU(2) k=1k=1 gaps in Δ\\Delta universal for allrepresentations for 1≤c≤1001 \\le c \\le 100.",
"Left: Boundsobtained with different values of n D n_D when extremal states arenot allowed.",
"Dashed lines from blue to red are computed data of3≤n D ≤293 \\le n_D \\le 29.",
"The solid black line is extrapolated fromdata of 11≤n D ≤29 11 \\le n_D \\le 29 using a function linear in1/n D 1/n_D.",
"Right: Bounds extrapolated to n D =∞n_D = \\infty forthe analysis without extremal states compared to the result whenextremal states are allowed.",
"The difference is negligible atlarge cc but significant at small cc.Next, however, we will turn to an analysis that maximizes the bounds separately in different sectors, and this is where we will start to find something qualitatively new compared with what is possible with the unflavored modular bootstrap.",
"In particular, we will maximize the gap in the trivial representation, and not impose any constraint on the gaps in the other representations.",
"In equations, our conditions are $\\rho _{\\lambda ,\\bar{\\lambda }}(\\Delta ^{(0)})\\ge 0 {\\rm ~when~}{\\left\\lbrace \\begin{array}{ll}\\Delta ^{(0)} \\ge \\Delta _{*},&\\lambda =(0){\\rm ~and~} \\bar{\\lambda }=(0),\\\\\\Delta ^{(0)} \\ge 0,&\\lambda {\\rm ~or~} \\bar{\\lambda }\\ne (0),\\\\\\end{array}\\right.",
"}$ in SU(2) $k=1$ , weight $\\lambda $ (or $\\bar{\\lambda }$ ) takes values $(0)$ or $(\\frac{1}{2})$ .",
"The resulting bound on the neutral sector gap is shown as a function of $c$ in Fig.",
"REF .By contrast with the previous subsection, here we find that the bound is exactly the same whether or not we allow extremal ($h^{(0)}=\\bar{h}^{(0)}=0$ ) states in the non-trivial representations.",
"Notably, there is a minimum of about $\\Delta _* =1$ near $c \\equiv c^{(0)}+1 =3$ .",
"We next turn to a more detailed study of this point.",
"Figure: Left: The upper bound on the gap in the dimension ofprimaries, Δ * \\Delta _{*}, in the trivial representationobtained at increasing derivative order of the linear functional(from blue to red, up to n D =29n_D = 29).",
"The black curve is theextrapolated value.",
"Right: Blown-up plot of the-near minimalΔ * \\Delta _{*} region.",
"The minimal isexpected at 2.00≤c≤2.042.00\\le c \\le 2.04,Δ * ≈0.995\\Delta _{*} \\approx 0.995."
],
[
"Spin-Independent Analysis",
"Our $q=\\bar{q}$ analysis in subsection (REF ) found a minimum gap bound at $c=3$ .",
"Using the extremal functional method [19], the dimensions and the degeneracies of states at this point can be extracted, with numerical accuracy being best for the lowest dimension states.",
"The dimensions of states occur at the zeros of the extremal functional, plotted in Fig.",
"REF .",
"Figure: Extremal functional of SU(2) k=1k=1, c=3c=3 spectrum.Furthermore, we find that the occupation numbers of the lowest-energy states of the partition function are uniquely determined to be $M_{(0),(0)}(\\beta ) &\\approx \\chi _{0}(\\beta ) + 28\\chi _{1}(\\beta ) + 76\\chi _{2}(\\beta ) +274 \\chi _{3}(\\beta )+\\ldots \\\\M_{(0),(\\frac{1}{2})}(\\beta ) =M_{(\\frac{1}{2}),(0)}(\\beta ) &\\approx 8\\chi _{0.5}(\\beta ) + 48\\chi _{1.5}(\\beta )+\\ldots \\\\M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\beta ) &\\approx 8\\chi _{0.5}(\\beta ) + 48\\chi _{1.5}(\\beta )+\\ldots $ The subscript on $\\chi _{\\Delta ^{(0)}}$ denotes the non-Sugawara dimension of the state.",
"At this point, the analysis takes $\\tau \\equiv \\frac{i \\beta }{2\\pi }$ to be pure imaginary, so no information on spins is used: $\\chi _{\\Delta ^{(0)} } = q^{-\\frac{c}{12}} \\left\\lbrace \\begin{array}{cc} \\prod _{n=2}^\\infty (1-q^n)^{-2}, & \\Delta ^{(0)}=0 , \\\\q^{\\Delta ^{(0)}} \\prod _{n=1}^\\infty (1-q^n)^{-2}, & \\Delta ^{(0)} >0 \\end{array} \\right.$ We find that a manifestly modular invariant partition function that reproduces this is $Z(\\beta )_{z=\\bar{z}} &=& \\frac{1}{2 \\eta ^6 } \\left( \\left( \\theta _3^2 - \\theta _4^2 \\right) \\theta _2^4(\\frac{z}{2}) + \\left( \\theta _2^2 + \\theta _4^2 \\right) \\theta _3^4(\\frac{z}{2}) + \\left( \\theta _3^2 - \\theta _2^2 \\right) \\theta _4^4(\\frac{z}{2}) \\right) \\\\&=& \\frac{1}{2 \\eta ^6} \\left( \\theta _2^2 \\theta _2^4(\\frac{z}{2}) + \\theta _3^2 \\theta _3^4(\\frac{z}{2}) + \\theta _4^2 \\theta _4^4(\\frac{z}{2}) + \\left( \\theta _3^2 - \\theta _2^2 - \\theta _4^2 \\right) \\theta _1^4(\\frac{z}{2}) \\right) ,$ where $\\theta _i\\equiv \\theta _i(z=0)$ .",
"It is straightforward to check that the occupation numbers are non-negative, so that this partition function is unitary, modular invariant, and extremizes the scalar gap.",
"Therefore (REF ) is the correct partition function by uniqueness."
],
[
"Spin-Dependent Analysis",
"In the previous subsection, we used extremal functional techniques to determine a unique partition function on the subspace $q=\\bar{q}$ when the gap in the scalar sector was maximized for $c=3$ .",
"We can gain much more information about the theory by relaxing the constraint $q=\\bar{q}$ and varying $q,\\bar{q}$ independently; in particular, the analysis becomes sensitive to the spins $h-\\bar{h}$ of the spectrum.",
"We could continue to use semi-definite programming methods, but they converge less quickly for independent $q,\\bar{q}$ than they do for $q=\\bar{q}$ .",
"Instead, we can use the fact that we know the spectrum of dimensions from the $q=\\bar{q}$ analysis, and the fact that spin is quantized.",
"This allows us to fix the allowed values of $h,\\bar{h}$ to a discrete set, turning the problem into a linear programming problem and thereby making the analysis much more efficient.",
"There is a remaining ambiguity, however, which is that we have to make a choice about what spins are allowed.",
"We find that if we allow only integer spins, there is no allowed partition function and in fact we can reduce the bound on the gap somewhat to about 2/3.",
"If instead we allow fractional spins, then we find a few different possibilities depending on what spins we allow.",
"We will begin with the conventional case where we allow integer and half-integer total spins, $h-\\bar{h}$ .",
"The $SU(2)$ Sugawara characters are such that $M_{(0),(0)}$ only has integer spins, $M_{(0),(1/2)}$ and $M_{(1/2),(0)}$ only has quarter spins and $M_{(1/2),(1/2)}$ can have integer and half integer spins.",
"Then to meet the requirement $M_{(0),(0)}$ and $M_{(1/2),(1/2)}$ can only have spins $\\frac{2n}{4}$ and $M_{(0),(1/2)}$ and $M_{(1/2),0}$ can only have spins $\\frac{2n+1}{4}$ .",
"Performing the linear programming analysis for such a spectrum (and continuing to maximize the gap in the neutral sector) leads to the following unique set of weights and occupation numbers $d$ :The reader may notice that the numbers at each dimension in eq.",
"(REF ) do not match the total number of states in the table.",
"The reason is that without knowledge of spin, there are null states that could not be taken into account in (REF ).",
"For instance, at level 2, there are a total of 84 states, as compared with 76 in (REF ), because of the 8 conserved currents at spin 1 that consequently have 8 “null” descendants at $\\Delta = 2$ .",
"At $\\Delta \\ge 3$ , the presence of such null states causes states to get reorganized in increasingly complicated ways and it is easiest to check the number of states is the same by constructing the full partition function.",
"Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThe non-Sugawara dimensions $\\Delta ^{(0)}$ and spins $\\ell ^{(0)}$ are just $\\Delta ^{(0)} = h^{(0)} + \\bar{h}^{(0)}, \\ell ^{(0)} = h^{(0)}- \\bar{h}^{(0)}$ .",
"States are evenly divided between $\\ell ^{(0)} = + | \\ell ^{(0)}| $ and $\\ell ^{(0)} = - | \\ell ^{(0)}|$ at each weight, and the occupation numbers for the $(\\mu , \\bar{\\mu }) = (0, \\frac{1}{2})$ representations are the same as for $(\\frac{1}{2}, 0)$ .",
"To get the full characters one multiplies the non-Sugawara Virasoro characters $\\chi _h(\\tau )$ (i.e.",
"generated by the modes of $T^{(0)} = T- T^{(\\rm sug)}$ ) by the Weyl characters $\\chi _\\lambda ^{(k)}(\\tau ,z)$ , which in this case areSee eg [12], eqs (14.176) and (15.244).",
"$\\chi _\\lambda ^{(1)}(\\tau ,z) = \\frac{1}{\\eta } \\sum _{m \\in \\mathbb {Z}+\\lambda } q^{m^2} y^m .$ After some trial and error, we find that the corresponding flavored partition function is reproduced by $Z(\\tau , \\bar{\\tau }, z, \\bar{z}) = \\frac{1}{4|\\eta |^{6}} \\sum _{a,b,a^{\\prime },b^{\\prime }=0}^1 (-1)^{a b^{\\prime } + a^{\\prime } b} \\left|\\theta \\left[^a_b\\right](\\tau ,\\frac{z}{2})\\right|^4 | \\theta [^{a^{\\prime }}_{b^{\\prime }}](\\tau ,0)|^2 ,$ in Jacobi/Erderlyi notation $\\theta _1 = \\theta [{1 \\atop 1}], \\theta _2 = \\theta [{1 \\atop 0}], \\theta _3 = \\theta [{0 \\atop 0}], \\theta _4 = \\theta [ { 0 \\atop 1}]$ .",
"As this candidate partition function is half integrally modded, it is a little unfamiliar.",
"A natural guess is that it arises as a $\\mathbb {Z}_{2}$ orbifold of a fully modular invariant theory.There is actually a history of extremal theories arising in such a fashion [28], [29], [30].",
"Indeed it is possible to project this onto a fully modular invariant partition function.",
"Taking the unflavored expression for simplicity, $ \\begin{split} Z^{\\rm (inv)}(\\tau , \\bar{\\tau })&=\\frac{1}{2}\\left(Z(\\tau , \\bar{\\tau }, 0,0)+Z(\\tau +1, \\bar{\\tau }+1, 0,0)+Z(-1/(\\tau +1), -1/(\\bar{\\tau }+1), 0,0)\\right) \\end{split} $ Unfortunately, we have not been able to identify a theory corresponding to (REF ).",
"It is straightforward to check by exhausting the possibilities that the central charge and the number of spin-1 conserved currents (11) is not consistent with this partition function being associated with a pure Sugawara theory for some Lie algebra.",
"While other choices for the quantization of spin are less conventional, they are still of some interest.For instance [31].",
"Another possibility we have considered is that the non-Sugawara part of the spin, i.e.",
"$h^{(0)}-\\bar{h}^{(0)}$ , is an integer or a half-integer.",
"Because of the contribution to the weight from the Sugawara part of the stress tensor, in this case the states in the $(\\frac{1}{2}, 0)$ and $(0,\\frac{1}{2})$ representations have quarter-integer spins.",
"Performing the linear programming analysis making this Ansatz for the spins, we find not just a unique solution but in fact a family of solutions given by the following occupation numbers: Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONThese partition functions satisfy crossing for all $x$ .",
"This one-parameter family is shown graphically in Fig.",
"REF , where we perform the linear programming analysis with the degeneracy $d$ of the $(\\mu , \\bar{\\mu })= (\\frac{1}{2} ,0), (h^{(0)}, \\bar{h}^{(0)}) = (\\frac{1}{4}, \\frac{1}{4})$ chosen by hand and look at how several other degeneracies depend on this choice.",
"By inspection of the above table, demanding that all occupation numbers be non-negative integers imposes $x\\in \\lbrace 0,1, \\dots , 4\\rbrace $ .",
"It would be interesting to know if all or any of these partition functions correspond to underlying physical CFTs.",
"Figure: The linear programming analysis finds a range of possible partition functions if we allow the physical spins to take quarter integer values.",
"When we fix one of the degeneracies dd by hand, in this case that of the weight (μ,μ ¯,h (0) ,h ¯ (0) )=(1 2,0,1 4,1 4)(\\mu , \\bar{\\mu }, h^{(0)}, \\bar{h}^{(0)})= (\\frac{1}{2}, 0 , \\frac{1}{4}, \\frac{1}{4}), all other degeneracies become uniquely determined, so that we find a one-parameter family of solutions.",
"The degeneracy on the x-axis here is 4+x4+x in the notation used in the text.The different values of $x$ here correspond to partition functions that have the same spectrum of dimensions $\\Delta = h + \\bar{h}$ , but which can be distinguished by their representation content, i.e.",
"through the “flavored” partition function.They are similar in this respect to multiple different CFTs at $c=24$ that have the same spectrum but different underlying symmetries [32]."
],
[
"Constraints on Representation Content",
"In this final subsection, we will consider the question of what representations are forced to be present in a theory.",
"The gravitational AdS dual of any such constraints would imply that even if certain representations were not present among the perturbative degrees of freedom in some theory, they would have to be present non-perturbatively.",
"The strongest condition one might try to prove is that all theories have all representations present.",
"This would however be too ambitious since there are simple counter-examples, but one might still try to prove restrictive constraints on which representations can be absent.",
"We will only be able to take a very modest step in this direction and prove some simple results for $SU(2)$ .",
"For instance, without referring to numerical methods, we will prove that an $SU(2), k=1$ partition function either has all representations, or else its flavored partition function factorizes into a Sugawara theory partition function times a non-flavored partition function, assuming left-right symmetry.",
"We begin by proving this $k=1$ result.",
"The flavored partition function splits into four representations $M(\\tau , \\bar{\\tau }) = \\left(\\begin{array}{cc}M_{(0),(0)}(\\tau , \\bar{\\tau })& M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })\\\\M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })& M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\tau , \\bar{\\tau })\\end{array}\\right) .$ Modular invariance requires $M(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) = SM(\\tau , \\bar{\\tau })S~,$ with $S$ matrix $S = \\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cc}1 & 1 \\\\1 & -1\\end{array}\\right)~.$ Because there are only two (assuming $(\\frac{1}{2},0)$ and $(0,\\frac{1}{2})$ are symmetric) different nontrivial representations, and the modular transformation manifestly forces at least one to be present, we can delete only one of them.",
"What if we set $M_{(\\frac{1}{2}),(0)}(\\tau , \\bar{\\tau })=M_{(0),(\\frac{1}{2})}(\\tau ,\\bar{\\tau })=0$ ?",
"In this case, the $(1,2)$ entry of matrix equation (REF ) is $\\frac{1}{2}\\left( M_{(0),(0)}(\\tau , \\bar{\\tau })-M_{(\\frac{1}{2}),(\\frac{1}{2})}(\\tau , \\bar{\\tau }) \\right) = 0 .$ The two diagonal representations have to be the same and therefore the “non-Sugawara” $\\tau $ -dependence of the flavored partition function is just an overall flavor-independent prefactor $M_{(0),(0)}(\\tau , \\bar{\\tau })$ that factors out.",
"Similarly, if we set $M_{(\\frac{1}{2}, \\frac{1}{2})}(\\tau ,\\bar{\\tau })=0$ , then the $(2,2)$ entry of (REF ) is $M_{(\\frac{1}{2}),(0)}(\\tau ,\\bar{\\tau }) = \\frac{1}{2} M_{(0),(0)}(\\tau , \\bar{\\tau })$ , so again the non-Sugawara $\\tau $ -dependence factors out completely.",
"In this case, the residual “Sugawara” matrix is just a symmetric holomorphic plus anti-holomorphic matrix, i.e.",
"$\\left( \\begin{array}{cc} 2 & 1 \\\\ 1 & 0 \\end{array} \\right) = \\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & 0 \\end{array} \\right) +\\left( \\begin{array}{cc} 1 & 0 \\\\ 1 & 0 \\end{array} \\right)$ .",
"Beyond $SU(2)$ $k=1$ , similar arguments can be used to somewhat narrow down the possible combinations of representations in any partition function with non-abelian currents.",
"Speficially, we can prove for SU(2) any $k$ , the partition function factorizes into a Sugawara partition function times a flavor-independent partition function if only diagonal representations are allowed.",
"We again begin with the transformation rule $M(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) = SM(\\tau , \\bar{\\tau })S~,$ where now the transformation matrix is $S_{(l) (l^\\prime )} = \\sqrt{\\frac{2}{k+2}} \\sin \\left(\\frac{\\pi }{k+2} (l+1)(l^\\prime +1)\\right) .$ Since we allow only diagonal representations, we can write $M_{(l) (r)}(\\tau , \\bar{\\tau }) &= \\delta _{(l) (r)} f_l , \\\\M_{(l) (r)}(-\\frac{1}{\\tau },-\\frac{1}{\\bar{\\tau }}) &= \\delta _{(l) (r)} g_l~,$ for some arbitrary functions $f_l$ and $g_l$ .",
"The matrix equation can be written as $g_\\alpha S_{\\alpha \\beta }=S_{\\alpha \\beta } f_\\beta .$ For $\\beta = 0$ , $g_\\alpha \\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right) =f_0 \\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right)$ so $g_\\alpha = f_0$ for all $\\alpha $ unless $\\sin \\left(\\frac{\\pi }{k+2} (\\alpha +1)\\right) = 0$ .",
"But the unitary bound $\\alpha <k+1$ does not allow this to happen.",
"So all $g_\\alpha $ should be equal.",
"Therefore, all diagonal representations $M_{(l) (l)}(\\tau , \\bar{\\tau })$ have to be equal, and the $\\tau $ -dependence $f_0(\\tau )$ of $M(\\tau , \\bar{\\tau })$ completely factors out."
],
[
"Discussion and Future Directions",
"One of the main goals of this paper has been to demonstrate how systematic numeric bootstrap techniques can be applied to flavored partition functions.",
"We have considered several specific analyses, but there are many more that could be done.",
"Here we will discuss a few potential future directions.",
"Some of the analyses we have discussed raise questions that could be answered with improved numeric efficiency so that the results could converge to the optimal bound.",
"One such case is the bound on the charge-to-mass ratio, where improved accuracy at large $c$ could more firmly establish the large $c$ scaling of the bound.",
"Another case is the application of our nonabelian extremal methods to larger $k$ and larger symmetry groups.",
"As either of these gets larger, the convergence rate becomes slower and so we have focused on the most efficient case, $SU(2)$ at level $k=1$ , to demonstrate that here the extremal functional method can be used to determine the full partition function of the theory maximizing the gap in the neutral sector.",
"It is interesting that the point maximizing the gap has integer occupation numbers, and it would be interesting to know if this is part of a general pattern or just an exceptional case.",
"Our preliminary analysis of $SU(2)$ at level $k=2$ has not converged well enough to answer this question, but perhaps this would be possible with additional innovations or more computing power.",
"Of course, if it turns out that integer occupation numbers is a generic feature of maximal gap spectra, it would be interesting to understand the underlying reason.",
"As part of this question, one might consider whether the gap should be maximized in just the neutral sector or in several charged sectors.",
"Having integer occupation numbers is a necessary but not sufficient condition for a partition function to have an underlying CFT.",
"Generally, it would be interesting to develop more techniques for determining a CFT once its partition function is known.",
"One way is simply to use the regular bootstrap but restricting all dimensions to those that appear in the partition function.",
"Usually, this is a significant improvement since it reduces the regular bootstrap problem to a linear programming problem; however, for rational theories, the large degeneracy at each level severely mitigates how helpful this additional information is.",
"Another possible approach one could try would be to use the partition function formulated as the four-point function of twist operators, $Z \\propto \\langle \\sigma _2 \\sigma _2 \\sigma _2 \\sigma _2\\rangle $ , to include the partition function together with $\\langle \\phi \\phi \\phi \\phi \\rangle $ and the “mixed” correlator $\\langle \\phi \\phi \\sigma _2 \\sigma _2\\rangle $ for some local operator $\\phi $ .",
"One could also try to make contact at $c=24$ with the Schellekens classification [32] in terms of Neimeier lattices, by rederiving this constraint using only the flavored modular bootstrap.",
"The modular bootstrap alone cannot constrain the number of currents, since they simply contribute a constant to the partition function, but a constant no longer satisfies the correct transformation law after flavoring.",
"Looking farther afield, one of the main motivations for developing a proof of the transformation law (REF ) in terms of background fields was that this might be easier to generalize.",
"There are many theories in 2d with higher spin currents, and one could generalize our derivation to such cases.",
"The correlators of higher spin currents do not have a simple universal generating functional like spin-one currents do, but their correlators are severely constrained by holomorphicity and crossing, and recursion relations are known in many cases.",
"Potentially, one could work out the transformation rule in a case-by-case fashion.",
"More ambitiously, one could try to generalize to $d>2$ .",
"The very interesting recent work [33] on a sort of modular invariance for lens spaces in higher dimension is tantalizing from this point of view.",
"Again, one would face the issue that correlators of currents in $d>2$ are not universal, but one could nevertheless try to obtain a constraint on the partition function in terms of the data in the $\\langle J(x_1) \\dots J(x_n)\\rangle $ correlators.",
"Somewhat more abstractly, one of the appealing features of understanding the flavored partition function better is that, by turning on background fields, we are exploring constraints beyond the class of those that can be seen by inserting local operators.",
"There are many such constraints on CFTs that are invisible in the standard bootstrap; the partition function itself can be though of as one such generalization since mapping to the torus (equivalently, inserting twist operators $\\sigma _2$ ) involves imposing new boundary conditions, and adding background fields is another kind of generalization.",
"It would be very interesting to understand what additional constraints could be obtained by imposing crossing symmetry of correlators in the presence of background fields.",
"Understanding the transformation law (REF ) as a statement about crossing symmetry for the four-point function $\\langle \\sigma _2 \\sigma _2 \\sigma _2 \\sigma _2\\rangle $ would be a useful warm-up case and could potentially give insight into how to think about more general correlators."
],
[
"Acknowledgments",
"We thank Scott Collier, Shamit Kachru, Jared Kaplan, Emanuel Katz, and Per Kraus for useful conversations.",
"ED was supported in part by the National Science Foundation under grant NSF-PHY-1316699 and by the Stanford Institute for Theoretical Physics and in part by the Simons Collaboration Grant 488655 on the Non-Perturbative Bootstrap.",
"ALF and YX were supported in part by the US Department of Energy Office of Science under Award Number DE-SC-0010025, and ALF was supported in part by the Simons Collaboration Grant on the Non-Perturbative Bootstrap."
],
[
"Path Integral Modular Transformation",
"In this appendix, we review how diffeomorphism invariance and rigid rescalings imply the relation $Z_{\\rm PI}\\left( \\frac{a \\tau +b}{c\\tau +d}, \\frac{c z}{c \\tau +d} \\right) = Z_{\\rm PI}(\\tau ,z).$ We begin with invariance of the path integral measure: $d \\Psi e^{ - S_{\\tau _a, \\tau _b}[\\Psi ]} = d\\Psi ^{\\prime } e^{ - S_{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} [\\Psi ^{\\prime }]}.$ Here, $\\Psi $ are all the fields of the CFT, and to keep track of the torus before and after conformal transformations we have introduced $\\tau _a, \\tau _b$ for two of its corners (i.e.",
"the four corners are at $0,\\tau _a, \\tau _b$ , and $\\tau _a+\\tau _b$ ).",
"Under rescalings, the operators ${\\cal O}$ and parameters $\\tau _a, \\tau _b$ transform as ${\\cal O}(w,\\bar{w}) \\rightarrow {\\cal O}^{\\prime }(w,\\bar{w}) = \\lambda ^{-h} \\bar{\\lambda }^{-\\bar{h}}{\\cal O}(\\lambda ^{-1} w, \\bar{\\lambda }^{-1} \\bar{w}), \\qquad (\\tau _a, \\tau _b) \\rightarrow (\\tau _a^{\\prime },\\tau _b^{\\prime })= (\\lambda \\tau _a, \\lambda \\tau _b).$ In particular, for a conserved current $J^\\mu $ , we have $\\int _{\\tau _a, \\tau _b} dw d\\bar{w} J^w (w) = \\int _{\\tau _a, \\tau _b} dw d\\bar{w}\\Big ( \\bar{\\lambda } J^{\\prime w}(\\lambda w)\\Big ) = \\lambda ^{-1} \\int _{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} dw d\\bar{w} J^{\\prime w}(w) .$ and consequently $d \\Psi e^{ - S_{\\tau _a, \\tau _b} [\\Psi ] - \\frac{i}{2\\pi } \\int _{\\tau _a, \\tau _b} dw d\\bar{w} A_w J^w} = d \\Psi ^{\\prime } e^{- S_{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b}[\\Psi ^{\\prime }] - \\frac{i}{2\\pi } \\int _{\\tau ^{\\prime }_a, \\tau ^{\\prime }_b} dw d\\bar{w} \\lambda ^{-1} A_w J^w }.$ Integrating both sides obtains the relation $(A_w, \\tau _a, \\tau _b) \\cong (\\lambda ^{-1} A_w, \\lambda \\tau _a, \\lambda \\tau _b)$ .",
"To obtain the transformation under $U: \\tau \\rightarrow \\frac{\\tau }{\\tau +1}$ , we take $(\\tau _a, \\tau _b) = (\\tau , \\tau +1) \\cong (\\tau ,1),$ where the congruence $\\cong $ follows from a large diffeomorphism cutting the torus along the line from 1 to $\\tau +1$ and sewing it back to the line from $\\tau +1$ to $\\tau +2$ .",
"By inspection of the chemical potential term $\\frac{1}{2\\pi i} \\int _{\\tau ,1} dw d\\bar{w} A_w J^w = 2 \\pi i {\\rm Im}(\\tau ) A_w \\bar{J}_0$ , we read off that $A_w = - i \\frac{\\bar{z}}{2 {\\rm Im}(\\tau )}.$ Finally, we take $\\lambda = (\\tau +1)^{-1}$ , so $(\\tau _a^{\\prime },\\tau _b^{\\prime })=(\\frac{\\tau }{\\tau +1}, 1)$ and $A_w^{\\prime } = (\\tau +1) A_w$ .",
"Therefore, $\\bar{z}^{\\prime } = 2 i {\\rm Im}(\\tau ^{\\prime }) A_w^{\\prime } = \\frac{\\bar{z}}{\\bar{\\tau }+1}.$ The transformation under $T: \\tau \\rightarrow \\tau +1$ is trivial, since $\\tau $ and $\\tau +1$ are related by a large diffeomorphism without any need for a rescaling, so $\\lambda =1$ , and neither $A_w$ nor $z$ transform.",
"All other modular transformations are generated from $T$ and $U$ ."
],
[
"A “Systematic” Treatment to Multivariate Problems",
"The bootstrap of flavored partition function introduces another continuous quantum numbers $Q$ in addition to the scaling dimension of $\\Delta $ .",
"Unlike the unflavored bootstrap where the problem is rigorously converted to a semidefinite programming problem, bootstrap problems with more than one variables do not have a simple and rigorous conversion to semidefinite programming problems.",
"One can choose to discretize the second variable $Q$ and hope that the bound converges at very small $\\delta Q$ .",
"However, the bound obtained in this way is not rigorous.",
"The linear functional can be negative in between discrete $Q$ 's or at large enough $Q$ ."
],
[
"Multivariate Positive Definite Functionals",
"Whether any real positive semidefinite polynomials (PSD) can be written as sum of squares of real polynomials (SOS) is known as the Hilbert's 17th problem.",
"Hilbert himself proves the special case for univariate polynomials is true.",
"But for multivariate polynomials it is later proven that PSD is a sum of squares of real rational functions.",
"We do not like rational functions because we have much less numerical control over them than polynomials.",
"Although we cannot find a clean SOS representation of multivariate PSD, if we only consider the subset of strictly positive polynomials we can still represent them by SOS in the following cases: Workaround 1: multiply by a common denominator $p(x_1,x_2)$ is positive definite polynomial (PD, also denote as $p(x_1,x_2) > 0$ ) then $p_g(x_1,x_2) = (1+x_1^2+x_2^2)^g p(x_1,x_2)$ is a sum of square of polynomial (SOS) for some $g$ .",
"[34] Workaround 2: region is bounded For a compact region $\\mathcal {S}$ defined by $f_i(\\vec{x})\\ge 0$ over set of function $f_i$ , any polynomial strictly positive in $\\mathcal {S}$ can be written as the following form $p = \\sum _I s_I(\\vec{x}) f_{i_1} f_{i_2}\\ldots $ where $s_I(\\vec{x})$ are sum of squares.",
"$I$ denotes some combinations of $f_i$ 's.",
"[35] The hope is that PD can approximate PSD well enough so that in practise we can still resort to SOS.",
"Numerically, solvers like SDPB never give nonnegative polynomials with exact zeros, so in practise we never actually encounter any counterexamples.",
"Another reason to be hopeful is from the proof that PSD can be approximated as closely as desired by SOS [36].",
"There is a possible loophole – the positive region of the polynomial has to be bounded.",
"In unflavored case the region that is frequently used is $\\Delta >\\Delta _{\\star }$ , which is a rare special case of unbounded region.",
"In practise, there is risk of not covering the full space of PD.",
"Although not rigorously, one can hope that by multiplying the $(1+x_1^2+x_2^2)^g$ factors of higher and higher $g$ we lose less and less."
],
[
"Multivariate Problems and SDPB",
"In this subsection we discuss how to rewrite the semidefinite polynomial programming with 2 variables into a form suitable for SDPB [37] solver.",
"SDPB solves univariate “Polynomial Matrix Program” (PMP) question stated as follows: $&\\text{maximize } y_0 + \\sum _n b_n y_n \\nonumber \\\\&\\text{such that } M_j^0(x) + \\sum _n y_n M_j^n(x) \\ge 0 \\nonumber \\\\&\\text{for all } x\\ge 0 \\text{ and } 1\\le j \\le J.$ where $M$ matrices are symmetric matrices of polynomials of $x$ .",
"In SDPB, the PMP question is internally mapped to an SDP question since $ M_j^0(x) + \\sum _n y_n M_j^n(x) \\ge 0$ if and only if $M_j^0(x) + \\sum _n y_n M_j^n(x) =~ {\\rm tr}\\left[ Y_{A} Q_{A}(x) \\right] +x\\, {\\rm tr}\\left[ Y_{B} Q_{B}(x)\\right]$ for some $Y_A, Y_B \\ge 0$ .",
"This equation is also the (2.8) of 1502.02033.",
"We are instead trying to solve the problem for two variable cases.",
"Here for modular bootstrap we are in the special case where the symmetric matrices $M_j$ are one by one, in other words, single polynomials $p_j$ .",
"For simplicity here we only deal with this one dimensional case.",
"Generalization to more dimensions and more variables is very easy.",
"The question is stated as follows: $&\\text{maximize } y_0 + \\sum _n b_n y_n \\nonumber \\\\&\\text{such that } p_j^0(x_1,x_2) + \\sum _n y_n p_j^n(x_1,x_2) \\ge 0 \\nonumber \\\\&\\text{for all } x_1\\ge 0 \\text{ and all $x_2$ and } 1\\le j \\le J.$ Since SDPB only allows one variable to be bounded we cannot add more constraints on the variables.",
"The $x_1>0$ is needed in SDPB because we usually choose the input $\\Delta \\ge \\Delta _*$ .",
"The second variable can be the U(1) charge $Q$ , which is not constrained to be positive number.",
"If one does want to bound the second variable one can make change of variable.",
"We use the symbol $F_j$ to represent the linear functional $F_j(x_1,x_2) \\equiv p_j^0(x_1,x_2) + \\sum _n p_j^n(x_1,x_2) y_n$ Similar to the univariate case, we assume that $F_j\\ge 0$ is equivalent to finding $Y_{A,j},~Y_{B,j} \\ge 0$ , so that $F_j(x_1,x_2) =~ {\\rm tr}\\left[ Y_{A,j} Q_{A}(x_1,x_2) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} Q_{B}(x_1,x_2)\\right]$ Here we introduce “bilinear basis” $\\vec{q}(X)$ so that $Q(X) = \\vec{q}\\vec{q}^T$ spans the space of polynomials of $X$ .",
"An easy example of bilinear basis is $\\vec{q}(x) = \\lbrace 1,x,x^2,\\ldots \\rbrace $ .",
"The bilinear basis of two or more variables can be factored out as a kronecker product of bilinear bases of each single variables $Q_A (x_1,x_2) &= Q_{A1}(x_1) \\otimes Q_2(x_2) \\nonumber \\\\Q_B (x_1,x_2) &= Q_{B1}(x_1) \\otimes Q_2(x_2)$ We define $d_i$ to be the $x_i$ degree of the polynomial $F$ .",
"the dimensions of the matrices $Q$ are $&{\\rm dim} Q_{A1} = \\delta _{A1} = [d_1/2]+1 \\nonumber \\\\&{\\rm dim} Q_{B1} = \\delta _{B1} = [(d_1-1)/2]+1 \\nonumber \\\\&{\\rm dim} Q_{2} = \\delta _{2} = [d_2/2]+1$ After factoring out $Q_A$ and $Q_B$ the function $F_j$ is written as $F_j(x_1,x_2) =~ {\\rm tr}\\left[ Y_{A,j}\\big ( Q_{A1}(x_1) \\otimes Q_2(x_2)\\big ) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} \\big ( Q_{B1}(x_1) \\otimes Q_2(x_2)\\big ) \\right]$ Since a polynomial is fixed if we know its value at $(d+1)$ different points, we can simply evaluate the above equation at $(d_2+1)$ values of $x_2$ in order to reduce the equation to have only one variable $x_1$We have not investigated what choices of the $x_{2,k}$ s are optimal.",
"In practice, we have taken them to be $x_{2,k} = 2^{k-1}$ .",
"$&F_{j,k}(x_1) = F_j(x_1,x_{2,k}) =~ {\\rm tr}\\left[ Y_{A,j}\\big ( Q_{A1}(x_1) \\otimes Q_2(x_{2,k})\\big ) \\right] +x_1 {\\rm tr}\\left[ Y_{B,j} \\big ( Q_{B1}(x_1) \\otimes Q_2(x_{2,k})\\big ) \\right]$ The above $(d_2+1)$ equations are equivalent to (REF ).",
"In the following we omit the $j$ index because the same equation works for all $j$ .",
"Now the form is already in single variable and is very close to the form of (REF ).",
"The only difference is the numerical matrices $Q_2(x_{2,k})$ .",
"Here we can play a trick by shuffling the $(d_2+1)$ equations with linear combination $&\\sum _l \\alpha _{kl} F_l = {\\rm tr}\\left[ Y_{A,j} \\big (Q_{A1}(x_1) \\otimes \\sum _l \\alpha _{kl} Q_2(x_{2,l})\\big ) \\right] + (\\text{B part})$ for some dimension $(d_2+1)$ square matrix $\\alpha _{kl}$ .",
"In fact, the space of symmetric $Q_2$ matrices is only $(d_2+1= 2\\delta _2 -1)$ dimensional space since it spans the space $(d_2+1)$ dimensional polynomials.",
"That means we can always find some $\\alpha _{kl}$ which picks up the the orthornormal basis of the polynomial space.",
"Further we can perform an arbitrary $GL_{\\delta _2}$ transformation on $Q_2$ $Y &\\mapsto G YG^{-1} \\nonumber \\\\\\sum _l \\alpha _{kl} Q_2(x_{2,l}) &\\mapsto G^{-1} \\sum _l \\alpha _{kl}Q_2(x_{2,l}) G$ so that the orthornormal basis maps to the symmetric matrix basis $G^{-1} \\sum _l \\alpha _{kl} Q_2(x_{2,l}) G = E^{r(k) s(k)}$ where $E^{rs} = \\delta _i^r \\delta _j^s + \\delta _j^r \\delta _i^s$ .",
"Then $\\sum _l \\alpha _{kl} F_l = {\\rm tr}\\left[ Y_A Q_{1A}(x_1) \\otimes E^{r(k)s(k)}\\right] + x_1{\\rm tr}\\left[ Y_B Q_{1B}(x_1) \\otimes E^{r(k)s(k)}\\right]$ Compared to (REF ), we can turn double variable programming of polynomial into single variable programming of symmetric polynomial matrices by substitution $M_j^0 &= \\sum _k \\sum _l E^{r(k) s(k)} \\alpha _{kl}P_j^0(x_1,x_{2,l})\\nonumber \\\\M_j^n &= \\sum _k \\sum _l E^{r(k) s(k)} \\alpha _{kl}P_j^n(x_1,x_{2,l})$ Since $Q_2$ span a $(d_2+1 = 2\\delta _2 -1)$ dimension space it means only the diagonal and next-to-diagonal elements will be nonzero.",
"If further we only have even powers of $x_2$ , the matrices will be diagonal.",
"The procedure defined from (REF ) to (REF ) is not the most efficient algorithm to obtain a single-variable matrix basis to input in (REF ).",
"In practice, the algorithm we actually follow is computationally more straightforward, and is as follows.",
"We can rewrite (REF ) as $M_j(x_1)_{tu} =Y_{A,j}^{rs,tu}\\, Q_{A1}(x_1)_{rs} + (\\text{B parts}) ~,$ where $r,s,t,u$ are matrix indices.",
"Given $M_j(x_1)$ , SDPB can optimize this single-variable problem.",
"Rather than obtaining $M_j(x_1)$ by the procedure outlined above, we can instead start directly from equation (REF ), which says $F_{j,k}(x_1) = Y_{A,j}^{rs,tu}\\, Q_{A1}(x_1)_{rs}\\,Q_2(x_{2,k})_{tu} + (\\text{B parts}) ~.$ Combining the two equations $F_{j,k}(x_1) = M_j(x_1)^{tu}\\, Q_2(x_{2,k})_{tu} ~.$ The point is that both $F_{j,k}(x_1)$ and $Q_2(x_{2,k})_{tu} $ are known, so $M$ can be obtained by solving the above equation, after which it can be fed into SDBP.",
"Concretely, first flatten the $(t,u)$ indices $ \\alpha := (t_\\alpha ,u_\\alpha )$ to put $M_{tu}$ into $(\\delta _2+1)(\\delta _2+2)/2$ dimensional array ${\\bf M}_\\alpha $ and $Q_{k,tu}$ into $(d_2+1)\\times (\\delta _2+1)(\\delta _2+2)/2$ array $Q_{k,\\alpha }$ .",
"Then solve the linear equations for ${\\bf M}_\\alpha $ .",
"To be concrete, in this paper we explicitly choose the flatten map ${\\bf M} := \\big (\\text{diag of $M$},~\\text{next-to-diag},~\\cdots \\big )~.$ A specific solution can be found by taking the SVD decomposition of $Q$ , $Q = U \\, W \\, V^T$ where diagonal matrix $W = \\left( \\ \\widetilde{W} \\quad O \\ \\right)$ has the same rank $(d_2+1)$ as $Q$ .",
"Then take $B^{T} = V \\, W^\\prime \\, U^T$ where $W^\\prime = \\left(\\begin{array}{c}\\widetilde{W}^{-1} \\\\ O\\end{array} \\right)$ is the pseudo inverse of $W$ .",
"Contract $B^T$ both sides of (REF ), $B^T {\\bf F} &= V \\, W^\\prime \\, U^T Q {\\bf M}= V \\, W^\\prime \\, U^T U \\, W \\, V^T {\\bf M}= V \\, \\left(\\begin{array}{cc}I_{d_2+1}& O \\\\ O & O\\end{array}\\right) \\, V^T {\\bf M}$ where it's useful to block-decompose the unitary matrix $V$ as $V = \\left(\\begin{array}{cc}V_{11}&V_{12}\\\\ V_{21}&V_{22}\\end{array}\\right)~.$ Continue to simplify the right hand side, $V^TB^T{\\bf F} &= \\left(\\begin{array}{cc}V_{11}&V_{12}\\\\ O & O\\end{array}\\right) {\\bf M} \\nonumber \\\\\\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf F} &= \\left( \\ I_{d_2+1} \\quad (V_{11}^T)^{-1}V_{21}^T \\ \\right){\\bf M} ~.$ From (REF ) we know that there are only $(d_2+1)$ independent elements are unique.",
"Any dependent element can be removed by a $GL_{\\delta _2}$ transformation defined by REF .",
"We choose ${\\bf M}$ such that only the first $(d_2+1)$ elements are zero, and the solution can be written as ${\\bf M} = \\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf F}~.$ Since $F_j(x_1,x_2)$ is the linear functional of input polynomials defined by (REF ).",
"Our single variable input matrices should be substituted in the same way, leading to ${\\bf M}_{j}^n &= \\big ( (V_{11}^T)^{-1} \\ O \\big )V^TB^T{\\bf P}_{j}^n \\nonumber \\\\{\\bf M} &\\equiv {\\bf M}_{j}^0 \\, y_0 + \\sum _n {\\bf M}_{j}^n \\, y_n\\nonumber \\\\\\left( {\\bf P}_{j}^n \\right)_k &\\equiv {\\rm Flatten}\\left( p_j^n(x_1,x_{2,k}) \\right)$ Finally, we inverse (REF ) to put flattened array ${\\bf M}_j^n$ 's back into matrix $M_j^n$ 's."
],
[
"$k=2, SU(2)$ Analysis",
"Here we present some preliminary results on our methods applied to the group $SU(2)$ at level $k=2$ .",
"Our results are qualitatively similar to the $k=1$ case, though with worse numeric accuracy due to the slower convergence.",
"In Fig.",
"REF , we show the bound on the gap $\\Delta _*$ to the lightest neutral state in the theory, which is minimized to be $\\Delta _* \\approx 1.344$ at $c \\approx 2.715$ .",
"Unfortunately, at the point where the bound is minimized, the occupation numbers from our analysis for some of the lowest few states are not particularly close to integers.",
"It is not clear whether this indicates that such a point is not associated with an underlying CFT or if we simply have not converged to sufficient precision.",
"The occupation numbers for the lightest neutral state and charged state are shown as a function of $c$ in Fig.",
"REF .",
"The lightest neutral state is close to $d=74$ , however the lightest charged state, which is even lighter is relatively far from the nearest integer, $d \\approx 7$ .",
"Another possibility is that one ought to maximize the gap in not only the neutral sector but also in one or more charged sectors; it would be interesting to pursue this or other conditions further.",
"Figure: Bound on the gap Δ * \\Delta _* to the lightest neutral state for SU(2)SU(2) at level k=2k=2.",
"The bound is minimized at c≈2.715c \\approx 2.715.Figure: Occupation numbers for the lightest neutral (left) and charged (right) states from the extremal functional analysis with SU(2)SU(2) at k=2k=2 as a function of cc.",
"The optimal bound is at c≈2.715c\\approx 2.715, indicated by a vertical line; horizontal lines are shown at integers.s"
]
] | 1709.01533 |
[
[
"Depression and Self-Harm Risk Assessment in Online Forums"
],
[
"Abstract Users suffering from mental health conditions often turn to online resources for support, including specialized online support communities or general communities such as Twitter and Reddit.",
"In this work, we present a neural framework for supporting and studying users in both types of communities.",
"We propose methods for identifying posts in support communities that may indicate a risk of self-harm, and demonstrate that our approach outperforms strong previously proposed methods for identifying such posts.",
"Self-harm is closely related to depression, which makes identifying depressed users on general forums a crucial related task.",
"We introduce a large-scale general forum dataset (\"RSDD\") consisting of users with self-reported depression diagnoses matched with control users.",
"We show how our method can be applied to effectively identify depressed users from their use of language alone.",
"We demonstrate that our method outperforms strong baselines on this general forum dataset."
],
[
"Introduction",
"Mental health remains a major challenge in public health care.",
"Depression is one of the most common mental disorders and 350 million people are estimated to suffer from depression worldwide [49].",
"In 2014 an estimated 7% of all U.S. adults had experienced at least one major depressive disorder depression2015.",
"Suicide and self-harm are major related concerns in public mental health.",
"Suicide is one of the leading causes of death [7], and each suicide case has major consequences on the physical and emotional well-being of families and on societies in general.",
"Therefore identifying individuals at risk of self-harm and providing support to prevent it remains an important problem [19].",
"Social media is often used by people with mental health problems to express their mental issues and seek support.",
"This makes social media a significant resource for studying language related to depression, suicide, and self-harm, as well as understanding the authors' reasons for making such posts, and identifying individuals at risk of harm [11].",
"Depression and suicide are closely related given that depression is the psychiatric diagnosis most commonly associated with suicide.",
"Research has demonstrated that forums are powerful platforms for self-disclosure and social support seeking around mental health concerns [16], [33].",
"Such support forums are often staffed by moderators who are mental health experts, trained volunteers, or more experienced users whose role is to identify forum posts suggesting that a user is at risk of self-harm and to provide support.",
"Studies have shown that self expression and social support are beneficial in improving the individual's state of the mind [47], [8] and, thus such communities and interventions are important in suicide prevention.",
"However, there are often thousands of user posts published in such support forums daily, making it difficult to manually identify individuals at risk of self-harm.",
"Additionally, users in acute distress need prompt attention, and any delay in responding to these users could have adverse consequences.",
"Therefore, identifying individuals at risk of self-harm in such support forums is an important challenge.",
"Identifying signs of depression in general social media, on the other hand, is also a difficult task that has applications for both better understanding the relationship between mental health and language use and for monitoring a specific user's state (e.g., in the context of monitoring a user's response to clinical care).",
"In this work we propose and evaluate a framework for performing self-harm risk assessment and for identifying depression in online forums.",
"We present a general neural network architecture for combining posts into a representation of a user's activity that is used to classify the user.",
"To address the challenge of depression risk assessment over the general forums, we introduce a large-scale novel Reddit dataset that is substantially larger than the existing data and has a much more realistic number of control users.",
"The dataset contains over 9,000 users with self-reported depression diagnoses matched with over 107,000 control users.",
"We apply our approach to (1) identify the users with depression on a general forum like Reddit, and to (2) estimate the risk of self-harm indicated by posts in a more specific mental-health support forum.",
"Our methods perform significantly better on both datasets than strong existing methods, demonstrating that our approach can be used both to identify depressed users and to estimate the risk of self-harm posed by individual posts."
],
[
"Related Work",
"There is a growing body of related work analyzing mental health-related discourse and language usage in social media to better discover and understand mental health related concerns [41], [17], [14], [11], [35], [46], [12], [1], [36], [4].",
"To investigate NLP methods for identifying depression and PTSD users on Twitter, a shared task [13] at the 2nd Computational Linguistics and Clinical Psychology Workshop (CLPsych 2015) was introduced where the participants evaluated their methods on a dataset of about 1800 Twitter users.",
"Other work has used data from approximately 900 Reddit.com users to support self-reported diagnosis detection [31].",
"Previous work identifying depression and other mental health problems, including the methods participating in CLPsych 2015 (e.g.",
"works by [40] and [39]) heavily rely on utilizing features such as LIWC [38], topic modeling, manual lexicons, or other domain-dependent application-specific features.",
"Aside from the effort required to design effective features, these approaches usually model the problem with respect to the selected features and ignore other indicators and signals that can improve prediction.",
"In contrast, our model only relies on text and is not dependent on any external or domain-specific features.",
"Existing self-reported diagnosis detection datasets contain a limited number of both control users and diagnosed users.",
"In contrast to this, we construct a new dataset with over 9,000 depressed users matched with a realistic number of control users.",
"In addition to general studies addressing mental health, related work has also specifically studied suicide and self-harm through social media [23], [45], [20], [18], [15].",
"Recently, CLPsych 2016 [21] investigated approaches for detecting the self-harm risk of mental health forum posts [34].",
"Most related work in this area uses variations of linear classifiers with some sort of feature engineering; successful methods have employed: a combination of sparse (bag-of-words) and dense (doc2vec) representation of the target forum posts [26], a stack of feature-rich Random Forest and linear Support Vector Machine (SVM) [32], an RBF SVM classifier utilizing similar sets of features [6], and various contextual and psycholinguistic features [9], [10].",
"In contrast to the above works, our model does not use any general or domain specific feature engineering; it learns appropriate representations of documents by considering only their textual content.",
"Our proposed models consist of a shared architecture based on a CNN, a merge layer, model-specific loss functions, and an output layer (as we will describe in §).",
"While our model shares similarities with CNN-based models in prior work [25], [27], [50], it focuses on learning representations of user's posts and combining the post representations into an overall representation of the user's activity.",
"In the case of self-harm risk assessment, we experiment with several loss functions to determine whether considering the ordinal nature of self-harm risk labels (i.e., green, amber, red, and crisis) can improve performance.",
"Evaluation results suggest that the model variant using this loss function is more robust than our other variants."
],
[
"Depression dataset construction.",
"We created a new dataset to support the task of identifying forum users with self-reported depression diagnoses.",
"The Reddit Self-reported Depression Diagnosis (RSDD) dataset was created by annotating users from a publicly-available Reddit datasethttps://files.pushshift.io/reddit/.",
"Users to annotate were selected by identifying all users who made a post between January 2006 and October 2016 matching a high-precision diagnosis pattern.e.g., “I was just diagnosed with depression.” Users with fewer than 100 posts made before their diagnosis post were discarded.",
"Each of the remaining diagnosis posts was then viewed by three layperson annotators to decide whether the user was claiming to have been diagnosed with depression; the most common false positives included hypotheticals (e.g., “if I was diagnosed with depression”), negations (e.g., “it's not like I've been diagnosed with depression”), and quotes (e.g., “my brother announced `I was just diagnosed with depression' ”).",
"Only users with at least two positive annotations were included in the final group of diagnosed users.",
"A pool of potential control users was identified by selecting only those users who had (1) never posted in a subreddit related to mental health, and (2) never used a term related to depression or mental health.",
"These restrictions minimize the likelihood that users with depression are included in the control group.",
"In order to prevent the diagnosed users from being easily identified by the usage of specific keywords that are never used by the control users, we removed all posts by diagnosed users that met either one of the aforementioned conditions (i.e., that was posted in a mental health subreddit or included a depression term).",
"For each diagnosed user and potential control user, we calculated the probability that the user would post in each subreddit (while ignoring diagnosed users' posts made to mental health subreddits).",
"Each diagnosed user was then greedily matched with the 12 control users who had the smallest Hellinger distance between the diagnosed user's and the control user's subreddit post probability distributions, excluding control users with 10% more or fewer posts than the diagnosed user.",
"This matching approach ensures that diagnosed users are matched with control users who are interested in similar subreddits and have similar activity levels, preventing biases based on the subreddits users are involved in or based on how active the users are on Reddit.",
"This yielded a dataset containing 9,210 diagnosed users and 107,274 control users.",
"On average each user in the dataset has 969 posts (median 646).",
"The mean post length is 148 tokens (median 74).",
"The Reddit Self-reported Depression Diagnosis (RSDD) dataset differs from prior work creating self-reported diagnoses datasets in several ways: it is an order of magnitude larger, posts were annotated to confirm that they contained claims of a diagnosis, and a realistic number of control users were matched with each diagnosed user.",
"The lists of terms related to mental health, subreddits related to mental health, high-precision depression diagnosis patterns, and further information are availablehttp://ir.cs.georgetown.edu/data/reddit_depression/.",
"We note that this dataset has some (inevitable) caveats: (i) the method only captures a subpopulation of depressed people (i.e.",
"those with self-reported diagnosis), (ii) Reddit users may not be a representative sample of the population as a whole, and (iii) there is no way to verify whether the users with self-reported diagnoses are truthful."
],
[
"Self-harm assessment.",
"For self-harm risk assessment we use data from mental health forum posts from ReachOut.com, which is a successful Australian support forum for young people.",
"In addition to providing peer-support, ReachOut moderators and trained volunteers monitor and participate in the forum discussions.",
"The NAACL 2016 Computational Linguistics and Clinical Psychology Workshop [21] released a Triage dataset containing 65,024 forum posts from ReachOut, with annotations for 1,227 posts indicating the author's risk of self-harm [34].",
"The annotations consist of one of four labels: green (indicating no action is required from ReachOut's moderators), amber (non-urgent attention is required), red (urgent attention is required), and crisis (a risk that requires immediate attention)."
],
[
"Ethical concerns.",
"Social media data are often sensitive, and even more so when the data are related to mental health.",
"Privacy concerns and the risk to the individuals in the data should always be considered [22], [44], [3].",
"We note that the risks associated with the data used in this work are minimal.",
"This assessment is supported by previous work on the ReachOut dataset [34], on Twitter data [13], and on other Reddit data [31].",
"The RSDD dataset contains only publicly available Reddit posts.",
"Annotators were shown only anonymized posts and agreed to make no attempts to deanonymize or contact them.",
"The RSDD dataset will only be made available to researchers who agree to follow ethical guidelines, which include requirements not to contact or attempt to deanonymize any of the users.",
"Additionally, for the ReachOut forum data that was explicitly related to mental health, the forum's rules require the users to stay anonymous; moderators actively redact any user identifying information.",
"Figure: The general neural network architecture shared among our user and post classification models.",
"Each input (e.g., each of a user's posts) is processed by a convolutional network and merged to create a vector representation of the user's activity.",
"This vector representation is passed through one or more dense layers followed by an output layer that performs classification.",
"The type of input received, merge operation, and output layer vary with the specific model."
],
[
"Methodology",
"We describe a general neural network architecture for performing text classification over multiple input texts.",
"We propose models based on this architecture for performing two tasks in the social media and mental health domains that we call self-harm risk classification and detecting depression.",
"The task of self-harm risk classification is estimating a user's current self-harm risk given the user's post on a mental health support forum and the previous posts in the thread.",
"The task of detecting depressions in users is identifying Reddit users with self-reported depression diagnoses given the users' post histories (excluding posts containing mental health keywords or posted in subreddits related to mental health).",
"While both tasks are focused on predicting a user's mental health status, they differ in both the type of classification performed (i.e., estimating severity on a four point scale vs. boolean classification) and in the amount of data available.",
"Our general architecture is based on a two step process: (1) identifying relevant features in each input text, and (2) combining the features observed in the model's inputs to classify the user.",
"Figure: The convolutional network component of our architecture.",
"A convolutional layer takes a series of terms as input (a) and applies ll filters to a kk-term sliding window to derive feature values for each window or region (b); k=2k=2 and l=3l=3 shown here.",
"A max pooling layer considers non-overlapping region sequences of length nn (b) and keeps the highest feature value for the sequence (c); n=3n=3 shown here."
],
[
"Shared Architecture",
"Our proposed models share a common architecture that takes one or more posts as input, processes the posts using a convolutional layer to identify features present in sliding windows of text, merges the features identified into a vector representation of the user's activity, and uses a series of dense layers to perform classification on the merged vector representation.",
"The type of merging performed and the output layers are properties of the model variant, which we describe in detail in the following section.",
"Convolutional networks have commonly been applied to the task of text classification, such as by Kim:2014.",
"We use categorical cross-entropy as a loss function with both methods, but also experiment with other loss functions when performing severity classification.",
"First, the model takes one or more posts as input and processes each post with a convolutional network containing a convolutional layer and a pooling layer.",
"This process is illustrated with a max pooling layer in Figure REF .",
"The convolutional layer applies filters to a sliding window of $k$ terms (a) and outputs a feature value for each sliding window region and each filter (b).",
"The same filters are applied to each window; each filter can be viewed as a feature detector and the overall process can be conceptualized as looking for windows of terms that contain specific features.",
"The features are not specified a priori through feature engineering, but instead are learned automatically when the model is trained.",
"After identifying the features present in each region (i.e., sliding window), a max pooling layer considers non-overlapping regions of length $n$ and keeps the highest feature value for each region (c).",
"This step eliminates the regions (i.e., sliding windows) that do not contain useful features, which reduces the size of the convolutional network's output.",
"The same convolutional network is applied to each input post, meaning that the model learns to look for the same set of features in each.",
"After each input post has been processed by a convolutional network, the output of each convolutional network is merged to create a representation of the user's activity across all input posts.",
"This representation is processed by one or more dense layers (i.e., fully connected layers) with dropout [42] before being processed by a final output layer to perform classification.",
"The type of output layer is dependent on the model variant.",
"Our shared model architecture is illustrated in Figure REF .",
"The architecture's hyperparameters (e.g., the sliding window size $k$ , the number of convolutional filters used, and type of pooling) also vary among models and are described in §.",
"Both the convolutional and dense layers use ReLU activations [37] in all model variants.",
"Table: The hyperparameters used by each model."
],
[
"Depression detection",
"Our model for depression detection takes a user's posts as input and processes each post with a convolutional network.",
"Each convolutional network performs average pooling to produce its output.",
"These post representations are then merged with a second convolutional layer to create a user representation; we found this approach led to more stable performance than using a second average pooling or max pooling layer.",
"The user representation created by the merge step is then passed to one or more dense layers before being passed to a dense output layer with a softmax activation function to perform classification.",
"The number of dense layers used is a hyperparameter described in §.",
"Categorical cross-entropy is used as the model's loss function."
],
[
"Self-harm risk assessment",
"Our model for self-harm risk classification takes two inputs: the target post being classified and the prior posts (if any) in the target post's thread.",
"The prior posts provide context and are thus useful for estimating the risk of self-harm present in the target post.",
"The two inputs are both processed by a convolutional network as in user-level classification, but in this case the convolutional network's outputs correspond to a representation of the target post and to a representation of the target post's context (i.e., the prior posts in the thread).",
"Given that these two outputs represent different aspects, they are merged by concatenating them together.",
"This merged representation is then passed to one or more dense layers and to an output layer; the type of output layer depends on the loss function used.",
"There are four self-harm risk assessment model variants in total: Categorical Cross Ent.",
"uses an output layer with a softmax activation function, and categorical cross-entropy as its loss function.",
"This mirrors the output layer and loss function used in the user level classification model.",
"MSE uses an output layer with a linear activation function, and mean squared error as its loss function.",
"The model's output is thus a single value; to perform classification, this output value is rounded to the nearest integer in the interval $[0, t - 1]$ , where $t$ is the number of target classes.",
"The final two loss functions perform metric learning rather than performing classification directly.",
"They learn representations of a user's activity and of the four self-harm risk severity labels; classification is performed by comparing the euclidean distance between a representation of a user's activity (produced by the final layer) and each of the four severity label representations.",
"Class Metric: Let $d$ be the size of the output layer and $X$ be the layer's $d$ -dimensional output.",
"Class Metric learns a $d$ -dimensional representation of each class $C_i$ such that $||X - C_i||_2$ is minimized for the correct class $i$ ; this is accomplished with the loss function: $ L_{i,p,n} = \\max \\big (0, ||X_i - C_p||_2 - ||X_i - C_n||_2 + \\alpha \\big ) $ where $C_p$ is the correct (i.e., positive) class for $X_i$ , $C_n$ is a randomly chosen incorrect (i.e., negative) class, and $\\alpha $ is a constant to enforce a minimum margin between classes.",
"Classification is performed by computing the similarity between $X_i$ and each class $C_j$ .",
"Class Metric (Ordinal) extends Class Metric to enforce a margin between ordinal classes as a function of the distance between classes.",
"Given a ranked list of classes such that more similar classes have closer rankings, that is $\\forall i$ $sim(C_i, C_{i \\pm 1}) > sim(C_i, C_{i \\pm 2})$ , we incorporate the class distance into the margin such that more distant incorrect class labels must be further away from the correct class label in the metric space.",
"The loss function becomes $ L_{i,p,n} = \\max \\big (0, \\;\\; ||X_i - C_p||_2 \\;\\; - \\\\||X_i - C_n||_2 + \\alpha |p-n| \\big )$ where $|p-n|$ causes the margin to scale with the distance between classes $p$ and $n$ ."
],
[
"Experiments",
"In this section, we describe the model hyperparameters used and present our results on the depression detection and self-harm risk assessment tasks.",
"To facilitate reproducibility we provide our code and will provide the Reddit depression dataset to researchers who sign a data usage agreementhttp://ir.cs.georgetown.edu/data/reddit_depression/."
],
[
"Experimental setup.",
"The hyperparameters used with our models are shown in Table REF .",
"The severity risk assessment models' hyperparameters were chosen using 10-fold cross validation on the 947 ReachOut training posts, with 15% of each fold used as validation data.",
"The depression identification model's hyperparameters were chosen using the Reddit validation set.",
"The depression identification model's second convolutional layer (i.e., the layer used to merge post representations) used filters of length 15, a stride of length 15, and the same number of filters as the first convolutional layer.",
"All models were trained using stochastic gradient descent with the Adam optimizer [28].",
"The hyperparameters that varied across models are shown in Table REF .",
"The convolution size, number of convolutional filters, pooling type, pooling length, and number of dense layers was similar across all post models.",
"Class balancing was performed with Categorical Cross Ent.",
"by weighting classes inversely proportional to their frequencies, whereas sampling an equal number of instances for each class worked best with the other methods."
],
[
"Addressing limited data.",
"The post classification models' input consists of skip-thought vectors [29]; each vector used is a 7200-dimensional representation of a sentence.",
"Thus, the convolutional windows used for post classification are over sentences rather than over terms.",
"This input representation was chosen to mitigate the effects of the ReachOut dataset's relatively small size.",
"The skip-thought vectors were generated from the the ReachOut forum dataset by sequentially splitting the posts in the training set into sentences, tokenizing them, and training skip-thoughts using Kiros et al.",
"'s implementation with the default parameters.",
"Sentence boundary detection was performed using the Punkt sentence tokenizer [30] available in NLTK [5].",
"These 2400-dimensional forum post skip-thought vectors were concatenated with the 4800-dimensional book corpus skip-thought vectors available from [29].",
"Experiments on the training set indicated that using only the ReachOut skip-thought vectors slightly decreased performance, while using only the book corpus skip-thought vectors substantially decreased performance.",
"As input the post models received the last 20 sentences in each target post and the last 20 sentences in the thread prior to the target post; any prior sentences are ignored."
],
[
"Depression detection.",
"The data used for depression detection was described in §.",
"As baselines we compare our model against the FastText classifier [24] and MNB and SVM classifiers [48] using features from prior work.",
"We tune FastText's hyperparameters on the validation set.",
"Specifically, we consider a maximum n-gram size $\\in [1,2,3,4,5]$ , an embedding size $\\in [50,100, 150]$ , and a learning rate $\\in [0.05, 0.1, 0.25, 0.5]$ as suggested in the documentation.",
"We consider two sets of features for the MNB and SVM classifiers.",
"The first set of features is the post content itself represented as sparse bag of words features (BoW baselines).",
"The second set of features (feature-rich baselines) comprises a large set of features including bag of words features encoded as sparse weighted vectors, external psycholinguistic features captured by LIWChttp://liwc.wpengine.com/ pennebaker2015development, and emotion lexicon features [43].",
"Since our problem is identifying depression among users, psycholinguistic signals and emotional attributes in the text are potentially important features for the task.",
"These features (as described in §) have been also previously used by successful methods in the Twitter self-reported diagnosis detection task [13].",
"Thus, we argue that these are strong baselines for our self-reported diagnosis detection task.",
"We apply count based and TF-IDF based feature weighting for bag of words features.",
"We perform standard preprocessing by removing stopwords and lowercasing the input text.During experimentation, we found TF-IDF sparse feature weighting to be superior than other weighting schemes.",
"Additional features such as LDA topics and $\\chi ^2$ feature selection did not result in any further improvements.",
"The data is split into training, validation, and testing datasets each containing approximately 3,000 diagnosed users and their matched control users.",
"The validation set is used for tuning development and hyperparameter tuning of our models and the baselines.",
"The reported results are on the test set.",
"The depression detection models' input consisted of raw terms encoded as one-hot vectors.",
"We used an input layer to learn 50-dimensional representation of the terms.",
"For each target user, the CNN received up to $n_{post}$ posts containing up to $n_{term}$ terms.",
"In this section we present results for two values of $n_{post}$ .",
"The earliest post approach (CNN-E) takes each user's $n_{post}=400$ earliest posts as input.",
"The random approach (CNN-R) samples $n_{post}=1500$ random posts from each user.",
"We empirically set $n_{term}=100$ with both approaches.",
"We later analyze the model's performance as $n_{post}$ and $n_{term}$ vary in §REF and as the post selection strategy varies in §REF .",
"Table: Performance identifying depressed users on the Reddit test set.The differences between the CNN and baselines are statistically significant (McNemar's test, p<0.05p<0.05).Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 test set.Results for the other methods are from .Differences in performance between the following pairs are statistically significant (McNemar's test, p<0.05p<0.05):Categorical Cross Ent.",
"and Class Metric, MSE and Categorical Cross Ent., MSE and Class Metric (Ordinal), and Class Metric (Ordinal) and Class Metric.Table: Self-harm risk assessment performance on the ReachOut CLPsych '16 training set using 10-fold cross validation.",
"Categorical Cross Ent.",
"performs substantially worse than on the test set, while MSE performs substantially better.",
"Class Metric (Ordinal) continues to perform well.",
"The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.",
"and MSE, Categorical Cross Ent.",
"and Class Metric, Categorical Cross Ent.",
"and Class Metric (Ordinal), MSE and Class Metric, and Class Metric and Class Metric (Ordinal).Results.",
"The results of identifying depressed users for our model and baselines are shown in Table REF .",
"Our proposed model outperforms the baselines by a large margin in terms of recall and F1 on the diagnosed users (increases of 41% and 16%, respectively), but performs worse in terms of precision.",
"As described later in the analysis section, the CNN identifies language associated with negative sentiment across a user's posts.",
"Figure: Sensitivity of the CNN-R model to the parameters n posts n_{posts} (a) and n terms n_{terms} (b) on RSDD's validation set.",
"F1 increases as n posts n_{posts} does (a), but the rate of increase slows as n posts n_{posts} surpasses 1000.The trend for n terms n_{terms} is less clear (b), but the highest F1 is achieved at n terms =100n_{terms}=100.In Figure (a) the parameter n terms n_{terms} was fixed to 100, and in Figure (b) n posts n_{posts} was fixed to 1500.Figure: Empirical cumulative distribution functions (CDF) of the number of posts per user (a) and the post length (b) in the RSDD dataset."
],
[
"Self-harm risk classification.",
"We train our methods to label the ReachOut posts and compare them against the top methods from CLPsych '16.",
"We use the same experimental protocol as was used in CLPsych '16; our methods were trained on the 947 training posts and evaluated on the remaining 280 testing posts.",
"We used 15% of the 947 training posts as validation data.",
"We report results using the same metrics used in CLPsych, which were: the macro-averaged F1 for the $amber$ , $red$ , and $crisis$ labels (non-green posts); the macro-averaged F1 of $ green $ posts vs. $ amber \\cup red \\cup crisis $ (flagged posts); and the macro-averaged F1 of $ green \\cup amber $ vs. $ red \\cup crisis $ (urgent posts).",
"The non-green F1 was used as the official CLPsych metric with the intention of placing emphasis on classification performance for the non-green categories (i.e., those that required some response).",
"The binary flagged meta-class was chosen to measure models' abilities to differentiate between posts that require attention and posts that do not, and the binary urgent meta-class was chosen to measure their abilities to differentiate between posts that require quick responses and posts that do not.",
"In addition to macro-averaged F1, CLPsych also reported the accuracy for each category.",
"We additionally report F1 macro-averaged over all classes.",
"Results.",
"The results on the self-harm risk assessment task for our models and for the current best-performing methods (briefly explained in §) are shown in Table REF .",
"We also report a baseline result which is based on a SVM classifier with bigram features.",
"When measured by non-green F1, the official metric of the CLPsych '16 Triage Task, our proposed models perform up to 19% better than the best existing methods.",
"Similarly, our models perform up to 11% better when measured with an F1 macro-averaged across all categories (i.e., all column) and up to 5% better with measured accuracy across all categories.",
"Categorical Cross Ent.",
"performs best in all of these cases, though the difference between the performance of Categorical Cross Ent.",
"and Class Metric with an ordinal margin is not statistically significant.",
"We also evaluate the performance of our methods on the training set using 10-fold cross validation to better observe performance differences (Table REF ).",
"All model variants perform substantially better on the training set than on the test set.",
"This is partially explained by the fact that the models were tuned on the training set, but the large difference in some cases (e.g., the increase in the highest non-green F1 from 0.50 to 0.87) suggest there may be qualitative differences between the datasets.",
"The best-performing method on the test set, Categorical Cross Ent., performs the worst on the training set; worst-performing method on the test set, MSE, performs the best on the training set.",
"Class Metric (Ordinal) performs well on both the testing and training sets, however, suggesting that it is more robust than the other methods.",
"Furthermore, there is no statistically significant difference between Class Metric (Ordinal) and the best-performing method on either dataset."
],
[
"Posts per user and post length",
"In this section we consider the effects of the maximum number of posts per user (i.e., $n_{post}$ ) and the maximum post length (i.e., $n_{term}$ ) on the Reddit dataset.",
"To do so we train the CNN-R model as described in §REF and report F1 on the validation set.",
"When varying $n_{post}$ we set $n_{term}=100$ , and when varying $n_{term}$ we set $n_{post}=1500$ .",
"As shown in Figure REF , the best performance of the CNN-R model is reached when it considers 100 terms in posts and up to 1750 posts for each user.",
"F1 increases as $n_{post}$ increases, up to the maximum tested value of 1750 (Figure REF ).",
"There is relatively little change in F1 from $n_{post}=1250$ to $n_{post}=1750$ , however, so we use $n_{post}=1500$ in our experiments for efficiency reasons.",
"As shown in Figure REF , approximately 20% of users have more than 1500 posts.",
"The effect of the maximum post length is not consistent (Figure REF ), but performance is maximized at $n_{term}=100$ .",
"As shown in Figure REF , approximately 40% of posts are longer than 100 terms.",
"Table: Models' performance on RSDD's validation set with different post selection strategiesand values of n post n_{post}.",
"CNN-E corresponds to the earliest strategy with n post =400n_{post}=400and CNN-R corresponds to the random strategy with n post =1500n_{post}=1500.Table: Example phrases that strongly contributed to a user's depression classification on the RSDD dataset."
],
[
"Post selection",
"For users with more than the maximum number of posts $n_{post}$ , a post selection strategy dictates which posts are used as input to the model.",
"Table REF shows the effect of the post selection strategy on the Reddit dataset's validation set.",
"Selecting a user's earliest posts performs the worst regardless of $n_{post}$ 's value, though the differences in F1 are smaller when $n_{post}=400$ .",
"Randomly selecting posts for each user performs the best across all metrics when $n_{post}=1500$ , with a large increase in precision over selecting users' earliest posts and a small increase over choosing users' latest posts.",
"Table: Self-harm risk assessment performance on the ReachOut CLPsych '17 test set.All methods perform substantially worse than on the CLPsych '16 test data.The difference in performance between the following method pairs are statistically significant (McNemar's test, p<0.05p<0.05): Categorical Cross Ent.",
"and MSE, and MSE and Class Metric (Ordinal)."
],
[
"Phrases contributing to classification",
"In this section we analyze the language that strongly contributed to the identification of depressed users on the Reddit dataset.",
"Unfortunately, it is impossible to show entire Reddit posts without compromising users' anonymity; we found that even when a post is paraphrased, enough information remains that it can easily be identified using a Web search engine.",
"For example, one Reddit post that strongly contributed to the author's classification as a depressed user contained the mention of a specific type of abuse and several comments vaguely related to this type of abuse.",
"We attempted to paraphrase this post, but found that any paraphrase containing general language related to both the type of abuse and to the user's comments was enough to identify the user.",
"Thus, to protect the anonymity of the users in our dataset, we do not publish posts in any form.",
"Rather than publishing posts, we identify key phrases in posts from users who were correctly identified as being depressed.",
"Phrases from eight self-reported depressed users are shown in Table REF ; to prevent these phrases from being used to identify users, we retain only the top phrase from each user.",
"These phrases were identified by using the model's convolutional filter weights to identify posts in the validation dataset that are strongly contributing to the model's classification decision, and then using the convolutional filter weights to identify the phrase within each post that most strongly contributed to the post's classification (i.e., had the highest feature values).",
"In keeping with the design of our dataset, terms related to depression or diagnoses are not present.",
"Instead, the model identifies phrases that often could be associated with a negative sentiment or outlook.",
"For example, “my whole” could be part of a negative comment referring to the poster's whole life.",
"It should be noted that the model makes classification decisions based on the occurrence of phrases across many posts by the same user.",
"Though one can imagine how the phrases shown here could be used to convey negative sentiment, the presence of a single such phrase is not sufficient to cause the model to classify a user as depressed."
],
[
"CLPsych '17 shared task",
"In this section we report results on the 2017 CLPsych Workshop's self-harm risk classification task.The 2017 test data was released after the initial version of this manuscript had been completed.",
"An official overview paper for CLPysch '17 is not yet available at the time of writing.",
"While CLPsych '17 featured the same self-harm risk classification task as CLPsych '16 (§REF ), new test data was used to conduct the evaluation.",
"This provides an opportunity to further evaluate our model on the task of self-harm risk assessment and to conduct an error analysis.",
"The methods were configured and evaluated in the same manner as described in §REF .The results in this section differ slightly from the methods' results as reported by CLPsych '17.",
"Here the methods were trained on only CLPsych '16 training data to match the experimental setup described earlier, whereas the methods were trained on both the CLPsych '16 training and test data in the official results reported by CLPsych '17.",
"Results are shown in Table REF .",
"All methods perform substantially worse than they performed on the CLPsych '16 test data as measured by non-green, urgent, and overall F1.",
"The trends across methods remain similar, however, with Categorical Cross Ent.",
"performing the best as measured by non-green and overall F1, and with no statistically significant difference between Class Metric (Ordinal) and the best performing method.",
"Notably, the methods' flagged F1 scores do not see a similar decrease on the CLPsych '17 data.",
"This suggests that the decreased performance is being caused by an inability to distinguish between the non-green classes (i.e., amber, red, and crisis).",
"The importance of differentiating between the red and crisis classes increased with the 2017 shared task, because the proportion of crisis labels in the data increased from 0.4% (2016 testing) and 4% (2016 training) to 11% (2017 testing).",
"The methods rarely classify a post as crisis, however, causing an increase in the number of misclassifications on the 2017 testing data.",
"For example, Class Metric (Ordinal) classified only four posts from the 2017 test data as crisis, and it classified no posts from the 2016 test data as crisis.",
"We leave improving the model to better identify crisis posts as future work."
],
[
"Conclusion",
"In this work, we argued for the close connection between social media and mental health, and described a neural network architecture for performing self-harm risk classification and depression detection on social media posts.",
"We described the construction of the Reddit Self-reported Depression Diagnosis (RSDD) datasethttp://ir.cs.georgetown.edu/data/reddit_depression/, containing over 9,000 users with self-reported depression diagnoses matched with over 107,000 similar control users; the dataset is available under a data usage agreement.",
"We applied our classification approach to the task of identifying depressed users on this dataset and found that it substantially outperformed strong existing methods in terms of Recall and F1.",
"While these depression detection results are encouraging, the absolute values of the metrics illustrate that this is a challenging task and worthy of further exploration.",
"We also applied our classification approach to the task of estimating the self-harm risk posed by posts on the ReachOut.com mental health support forum, and found that it substantially outperformed strong previously-proposed methods.",
"Our approach and results are significant from several perspectives: they provide a strong approach to identifying posts indicating a risk of self-harm in social media; they demonstrate a means for large scale public mental health studies surrounding the state of depression; and they demonstrate the possibility of sensitive applications in the context of clinical care, where clinicians could be notified if the activities of their patients suggest they are at risk of self-harm.",
"Furthermore, large-scale datasets such as the one presented in this paper can provide complementary information to existing data on mental health which are generally relatively smaller collections."
]
] | 1709.01848 |
[
[
"Scalar curvature, flat Borromean rings, and the 3-body problem"
],
[
"Abstract This paper explains unexpected links between the 3 topics in the title and frames them in a large canvas."
],
[
"History",
"The basic ideas of geometry go back to Euclid and Archimedes, followed later by Gauss, Minkowski and Weyl.",
"Topology also has ancient roots as evidenced by Alexander the Great cutting the Gordian Knot, and the Popes whose coat of arms became the Borromean rings.",
"Isaac Newton, standing on the shoulders of Galileo and beyond Papal influence, used Greek ellipses to explain planetary motion and left to posterity the 3-body problem.",
"The 3 Borromean rings of polished steel at the entrance to the Newton Institute in CambridgeImage reproduced by kind permission of Peter Robinson, Bradshaw Foundation ©John Robinson/Edition Limitée Paris 2007 provided me with inspiration, in August 2017, at the conference celebrating the 60th birthday of Simon Donaldson.",
"Tantalized by Misha Gromov's lecture on scalar curvature, I decided to combine the two topics and relate them to Newton's planetary orbits.",
"$\\includegraphics [width=6cm]{ini.jpg}$ Visionary ideas are what we mathematicians grapple with, but illustrative special examples help the understanding.",
"They also lead us by analogy to more general and realistic examples.",
"The master whose style I try to emulate is the greatest mathematical physicist since Newton, James Clerk Maxwell.",
"He understood dynamical stability, with its topological underpinnings, more thoroughly than anyone else and I offer this short note as a tribute to him."
],
[
"The Topology and Geometry of Borromean Rings",
"The linking of circles in 3-space is familiar to us all and led to the formation of chains.",
"Gauss, Faraday and Maxwell explained electro-magnetism in terms of such links.",
"The beauty and subtlety of the 3 Borromean rings is that each pair are unlinked but the triple holds together.",
"This mysterious property had, in medieval times, theological (even Trinitarian) overtones which is why it appealed to Popes, and would-be popes, like the Borromeo family.",
"Newton might have heard of this mystery but, as a secret unitarian in Trinity College, he would have kept off such dangerous ground.",
"We all know that geometry and topology are siblings.",
"Geometry is about measurement and topology is about shape.",
"This family unity is embodied in the great theorem of Gauss, which starts locally with the area of spherical triangles, or polygons, and concludes globally with the formula for the Euler member of a closed surface as the average of the Gauss or scalar curvature (divided by $2\\pi $ ).",
"It is natural to ask if there is a similar formula relating the geometry and topology of Borromean rings.",
"For this to make sense the 3 rings must be geometrical, not just topological.",
"Each ring should be planar or spherical, with a boundary and the 3 rings should be placed in 3-space so that their mutual topology is Borromean.",
"Looking at the sculptured steel outside the Newton Institute raises a challenging question, rather like the Rubik Cube or the model sailing boats in wine bottles.",
"How are they made or dismantled?",
"If you are the artist commissioned to make the sculpture, what precise instructions do you give the craftsman in the workshop?",
"For example, suppose you wanted to make the sculpture on the left of the Institute consisting of 3 squares of flat steel, each with a smaller square hole inside it?",
"The craftsman would need exact measurements and their tolerance : how accurate does he have to be?",
"This sounds, and is, standard engineering design.",
"The sculptor, or his mathematical consultant, has to do \"the sums\" and instruct the craftsman, confident that it will give a stable work of art.",
"Having done this once for the first steel sculpture at the Newton Institute, the design team then have to go through a similar procedure with the square replaced by a rhombus or a triangle, for the two sculptures on the right.",
"As your sculptures attract attention the team might be flooded with orders, but each client would have different requirements.",
"Some might want polygons with many sides, perhaps infinitely many, so that polygons become ellipses (with Newtonian approval).",
"Clearly you will need an efficient algorithm to implement the process.",
"Does your firm have mathematical consultants of the necessary quality?",
"If they studied at Trinity or St. John's (the parent Colleges of the Newton Institute) then you might be in luck.",
"As a Trinity man I feel it incumbent on me to try.",
"The next sections provide my mathematical report on these problems."
],
[
"The Iso-Perimetric Inequality",
"In the spirit of Maxwell I will consider a simple model based on the actual steel sculptures at Newton's Institute.",
"My Borromean rings will be modelled by flat regular polygons $C_i $ $(i=1,2,3)$ with $N$ sides ($N\\geqslant 3$ ), having a border of width $\\delta $ .",
"See the figure for $N=4$ and one polygon $C$ .",
"$\\includegraphics [width=10cm]{square.pdf}$ The engineering problem is to take 3 such regular $N$ -gons $C_1,C_2,C_3$ of the same width $\\delta $ and then place them in 3-space (or on a 3-sphere) so that they form a Borromean triple.",
"Clearly the holes in the middle of the $N$ -gons cannot be too small.",
"We look for an inequality which will give the critical size $d$ , so that a Borromean triple of such (flat bordered) $N$ -gons can be constructed provided $\\delta <d$ .",
"Fortunately there is a famous inequality which is precisely what we need.",
"It is the iso-perimetric inequality.",
"Moreover this inequality has been generalized over the centuries and goes by various names.",
"The simplest and most beautiful is due to Minkowski and concerns areas $A$ of plane convex sets $P,Q$ .",
"Minkowski considered the join of $P$ and $Q$ , formed by all segments $\\lambda p+(1-\\lambda )q$ with $p\\in P, q\\in Q$ and $0\\leqslant \\lambda \\leqslant 1$ .",
"He discovered that the function $A(\\lambda P+(1-\\lambda ) Q)$ is a quadratic in $\\lambda $ whose first and last coefficients are $A(P)$ and $A(Q)$ .",
"The middle term is called the Minkowski mixed area $A(P,Q)$ .",
"The key result is that the quadratic (3.1) is indefinite, i.e.",
"$A(P,Q)^2 \\geqslant 4 A(P) A(Q)$ with equality only if $P=Q$ (up to translation and scale).",
"The Isoperimetric Inequality follows by taking $Q$ to be a circle $S$ of (small) radius $\\delta /2$ whose motion around the boundary $\\partial Q$ of $Q$ sweeps out a border of width $\\delta $ .",
"If $P$ is not convex, replacing it by its convex hull only sharpens the inequality.",
"Note that the motion of the rolling ”marble\" requires $\\delta $ to be small relative to sharp bends of $\\partial Q$ .",
"But since the formula is exactly quadratic (not just approximately) we can start with a smooth boundary $\\partial Q$ , fix our $\\delta $ and then let $\\delta \\rightarrow 0$ .",
"Thus (3.2) holds for polygonal as well as smooth boundaries.",
"There are deep relations between the Gauss Theorem on scalar curvature and Minkowski's Theorem on convex bodies.",
"They have both had extensive generalizations, in particular to higher dimensions.",
"They both sit at the crossroads of Geometry, Topology and Dynamics exemplified by the title of this paper.",
"But, as stated in the introduction, my aim in this short paper is to illustrate by well-chosen examples, not to develop general theory.",
"The examples will start with specially symmetric ones, such as regular polygons, but since we are interested in topologically stable features we can deform to much more general configurations.",
"Let us now return to our regular $N$ -gon with vertices on a circle of radius $r$ .",
"The formula for the area $A_N(r)$ is by elementary trigonometry $A_N(r)=\\frac{r^2N}{2} \\sin \\frac{2\\pi }{N}$ Note that as $N\\rightarrow \\infty $ this converges to the formula $\\pi r^2$ which is how Archimedes computed $\\pi $ .",
"As a function of $r$ it is just quadratic, as noted more generally by Minkowski, so that the area of the bordered $N$ -gon is precisely given by the first 2 non-zero terms of the Taylor expansion $A_N(r,\\delta )=\\frac{(2r\\delta -\\delta ^2)N}{2} .\\sin {\\frac{2\\pi }{N}}$ For both Gauss and Minkowski we can follow the formula by continuity as the polygon changes.",
"The difference is that, for Gauss, (3.4) would be an approximate formula only valid for small $\\delta $ relative to the sharp bends on the boundary.",
"For Minkowski by contrast convexity makes (3.4) is exact and $\\delta $ can be made uniformly small irrespective of the sharp bends in the boundary, when the quadratic ”error term\" would lead to delta functions of boundary curvature (angles) for a polygon.",
"Globally, for a surface of constant curvature, Minkowski and Gauss give the same (topological) answer.",
"In this global form curvature can be both positive and negative, but the average has topological content.",
"This principle extends to all generalizations.",
"In the next section I will describe the pure topology of Borromean triples and show that it has hidden depths."
],
[
"The Topology of Borromean Rings",
"When topology became formalized as Algebraic Topology, in the 20th Century, Borromean triples were seen to be related to other triples such as (4.1) the associator, that measures deviation from associativity, as in the octonions, (4.2) the fundamental 3-form of a simple Lie group, starting with the unit quaternions, (4.3) the vertices of a triangle, (4.4) the Massey triple product in homotopy theory.",
"What is the right context of our example?",
"There are many but the essential features are (4.5) 3 continuous closed curves in 3-space with the Borromean property, detected by (4.1) (4.2) or (4.4) (4.6) 3 borders giving ribbon graphs (4.7) 3 surfaces with various degrees of smoothness, for the interior and for the boundary (4.8) 3 dimensions can represent static physical 3-space or dynamical 2-space incorporating time {recall that Newton's planetary orbits are space-time graphs} (4.9) “Curve” in (4.5) can be interpreted as a compact set with “1-dimensional features” as in Gromov's talk.",
"These all relate to the way 2-planes sit in 3-space.",
"They can be usefully compared to (pairs) which lead to curvature, complex numbers and the way lines sit in 2-dimensions.",
"The different aspects of Borromean triples listed above indicate that Borromean triples are not an amusing curiosity for popes or parties but are fundamental in 3-dimensional static phenomena (as is the Möbius band).",
"This explains why stability is so important and how dynamics, which brings in time, will relate to 4-dimensions as in Donaldson 60 These general thoughts, when suitably developed, can be seen to underpin vast areas of natural philosophy and are much studied by scientists of all types, including mathematical ones.",
"But, since the entire universe in all its diversity is too vast to examine in full detail and at all scales, scientists long ago decided to study simple model examples.",
"This was fully understood by Maxwell.",
"As I explained I aim to be Maxwellian and pick simple illustrative examples.",
"That is why the Borromean steel sculptures at the Newton Institute are my inspiration.",
"An example by itself is not enough.",
"We have to see it in a broader context and imagine ways, perhaps several, in which it can be generalized.",
"Then, and only then, does an example become an illustration.",
"So, in this spirit, let me return to the geometry of the Newton Institute sculptures."
],
[
"A Geometric Result about Borromean Rings",
"The geometric data consists of 3 bordered regular $N$ -gons with $N>3$ of width $\\delta $ which are planar or spherical.",
"The figure in section 3 shows the planar case for $N=4$ .",
"The question we ask is (5.1) Can this data be assembled in 3-space ($\\mathbb {R}^3$ or $S^3$ ) by translation and scale changes to form a Borromean triple of $N$ -gons?",
"The trigonometric calculations in Section 3 show that this is possible provided we have the inequality (5.2) $A_N(r,\\delta ) > r\\delta \\sin (2\\pi /N)$ derived from the equality (3.4).",
"The \"error\" is the quadratic term (5.3) $\\delta ^2/2.N \\sin (2\\pi /N)$ This answers the questions raised at the end of section 2.",
"The desired bound $d$ depends on the scale set by the radius $r$ and, for regular $N$ -gons $d=2r$ and we just get the inequality $\\delta /r \\leqslant 2$ .",
"Embodied in planar steel, as at the Newton Institute, formula (5.2) gives the engineering team the precise measurements needed, for any $N$ , and the tolerance or error.",
"As the width tends to zero the tolerance also tends to zero so that, in the (conformal) limit, steel is no longer a suitable material and the Borromean rings have to made of silk thread.",
"For irregular Borromean rings the exact formula (5.3) will be replaced by more general ones, but the leading (topological) term will remain.",
"While (5.2) is elementary trigonometry, because of the symmetry, its natural context is that of Minkowski's mixed areas, or the Gauss scalar curvature.",
"Generalizing (5.2) was therefore standard geometry over a century ago.",
"In more recent times the topology and geometry of Borromean triples has become equally standard though more sophisticated.",
"Different experts develop different machinery and do not always use the same language.",
"For applied mathematicians, engineers or for Olympic pole vaulters, the position of the centre of gravity is crucial.",
"Borromean geometry, based on the stability that comes from topology, may help win gold medals.",
"Maxwell would have understood it all and he would have explained how a few well-chosen examples are all you need.",
"Since he is no longer here in person I have tried to stand in for him, like an actor who stands in for Charles Dickens.",
"3 planets in elliptical orbits may have long term stability, held together by mysterious topology, if the parameters are right.",
"It seems to me more than likely that some versions of the famous 3-body problem will turn out to be Borromean, perhaps in the rings of Saturn, studied by Maxwell or even in the Red Spot of Jupiter.",
".",
"I hope I have whetted the appetite of the reader, and that the Newton Institute will continue to inspire natural philosophers for decades to come.",
"But let me conclude with two final sections extending the scope of the discussion."
],
[
"Force and Centres",
"The Borromean rings at the Newton Institute were our starting point.",
"For artistic reasons they were highly symmetrical with an obvious centre.",
"But, as we start to deform them, their symmetry and centre get lost.",
"How do we track the centre?",
"Newton found the answer by going back to Apollonius and conic sections.",
"The focus (one of a pair) was the appropriate centre and the eccentricity of the ellipse increased as the focus moved away from the centre.",
"What Newton realized, and Apollonius did not, was that the focus was the centre of gravity.",
"In other words, gravitational force changes the geometry, and planetary orbits reflect this changed geometry.",
"On the time scale of Earth-years the gravitational attraction of the sun does not change and planetary orbits are, with modest variations, stable for millennia.",
"Mathematicians, from Laplace onwards, wondered about stability in the “long-term”.",
"Would the solar system survive forever?",
"To an astronomer or an astrophysicist, this was an idealization too far.",
"A time-scale may not matter to the pure mathematician but, if it is not commensurate with physical time-scales, it is irrelevant: the “long-term” will never arrive.",
"Such metaphysical questions acquire more significance in two ways.",
"On the one hand, if gravity were replaced by electro-magnetism, also based on the inverse-square law, planets could be replaced by electrons travelling in orbits at velocities close to the speed of light.",
"This dramatic change of scales brings mathematical time-scales down to physical time-scales and long-term stability becomes important for engineers at CERN, as explained to me many years ago by Jürgen Moser.",
"Another more fundamental change came, a century ago, with Einstein's Theory of General Relativity.",
"The inverse-square law is just the Newtonian linear approximation but non-linear GR has to incorporate Einsteinian corrections, even for the solar system, and the accuracy of GPS requires such corrections.",
"If we also want to understand the magnetic fields that generate solar flares and affect our weather, then both kinds of corrections are necessary.",
"The LIGO triangles designed to detect gravitational waves were testing not just the statics of GR but also its dynamics, which brings in time, so Borromean rings might acquire a cosmic significance.",
"This may not just be “$\\pi $ in the SKY” since respected astronomers have been searching for “cosmic strings” both in theory and in observation.",
"The lessons I draw from this global (or cosmic) perspective are the following (6.1) the importance of forces in geometry (6.2) the key role of the centres of force (6.3) the role of centres in dynamical stability (6.4) the importance of effective criteria for stability (6.5) the understanding of different force scales (6.6) the role of topology in qualitative understanding of statics and dynamics (6.7) the role of geometry in quantitative versions of (6.6) with explicit tolerances.",
"Needless to say most of this is known to most scientists most of the time.",
"But I suspect that not all is known to all scientists all the time.",
"Perhaps it is time for all natural philosophers to get under the Newton Apple Tree and admire the Borromean Rings, just as our Druid ancestors admired the Art and Architecture of Stonehenge.",
"In the next and final section I will descend from Heaven to Earth."
],
[
"Down to Earth",
"The 3 Borromean rings from which we started are the simplest geometric examples in 3-space.",
"They can be generalized in infinitely many ways and we are free to choose the model that fits our needs and our resources.",
"There is no single multi-purpose model which will do everything.",
"I have already indicated that regular polygons can be replaced by convex polygons and that the parameters can all be allowed to vary.",
"For a bordered polygon with width $\\delta $ we started with 3 polygons $C_i$ having the same width, but we can now allow different widths $\\delta _i$ .",
"Ellipses appear when the polygon has an infinite number of sides, so we can end up with 3 ellipses of different widths sitting in 3-space.",
"Topologically they are either Borromean rings or non-Borromean rings, the Papal version of Hamlet's dilemma : To $B$ or not to $B$.",
"Mathematical techniques now exist to answer Hamlet.",
"Shakespeare was an actor as well as a playwright, so he would have embraced the new technology.",
"The new technology discussed at Donaldson 60 would have solved Hamlet's dilemma, depriving the English language of its favourite quotation.",
"All Hamlet had to do was to find out if the 3 rings could balance stably in the earth's gravitational field.",
"The nearby apple tree would have shown him that the small convex set (the tolerance), defined by the widths $\\delta _i$ , should contain the centre of gravity.",
"Fortunately the flat fenland of Cambridge would ensure that the gravitational field had essentially the same strength on both sides of the Institute.",
"Apples could therefore be substituted by steel rings, though the Borromean property would need a high speed camera to capture any entanglement of 3 falling apples, the dynamics of which were famously observed by the occupant of Woolsthorpe manor.",
"The engineers would not need to incorporate the relativistic Einsteinian corrections of section 6, since auroras are rarely seen at latitude $52^\\circ $ North.",
"We thus end up with an effective numerical criterion for the 3 rings to be Borromean or non-Borromean.",
"This describes all the steel sculptures that might be constructed.",
"As pointed out in section 3, Minkowski only dealt with convex bodies, which like spheres, have positive scalar curvature, while the Gauss formula allows regions of negative curvature.",
"But passing to convex hulls swallows up the negative regions.",
"There are many ways of understanding this difference.",
"(7.1) A general Morse function $f$ on a closed surface will have maxima, minima and saddle points.",
"(7.2) The Hessian of $f$ at critical points distinguishes between the 3 cases in (7.1).",
"It defines local geometries with scalar curvature positive at the maxima and minima, but negative at the saddle points.",
"(7.3) In the regions of positive curvature the Minkowski theory agrees with Gauss.",
"(7.4) The Gauss integral formula is about averages and Euler's topological invariant gives the dominant term.",
"(7.5) An average smooths out local irregularities, and gives good results when the averaging process is tailored to the nature of the irregularities.",
"(7.6) In all branches of science local irregularities occur and have to be averaged.",
"Atoms in a molecule, planets in the solar system or galaxies in the cosmos are all “irregularities” in this sense.",
"(7.7) Analysts have developed very refined techniques of smoothing irregularities which lead to stochastic differential equations.",
"(7.8) Geometers have developed techniques of stability descending from Minkowski and used in Mirror Symmetry.",
"(7.9) Physicists have techniques called renormalization which provide powerful tools in quantum field theory.",
"(7.10) Number theorists since Euler and Riemann have used similar ideas to study the “irregular” distribution of primes, with the Riemann Hypothesis as the ultimate goal.",
"References There is a vast literature on geometry and dynamics, much too extensive to list here.",
"The great classic is of course Maxwell's Smith Prize Essay on Saturn's Rings.",
"For more recent literature I can at least point to some of my own papers which are relevant in various ways.",
"M.F.Atiyah, Convexity and commuting Hamiltonians, Bull.",
"Lond.",
"Math.",
"Soc.",
"14 (1981), 1–15.",
"–, The Non-Existent Complex 6-Sphere, arXiv:1610.09366v2 [math.DG] (2016) –, Geometric Models of Helium, arXiv:1703.02532v1 [physics.gen-ph] (2017) –, Groups of Odd Order and Galois Theory, to appear.",
"– and J.Berndt, Projective planes, Severi varieties and spheres, Surveys in Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, International Press, Somerville, MA (2003).",
"–, R.Bott and L.Gårding, Lacunas for hyperbolic differential operators with constant coefficients I., Acta Math.",
"124 (1970), 109–189.",
"– and G.Segal, Twisted K-theory and Cohomology, in Inspired by S.S. Chern: A Memorial Volume in Honor of a Great Mathematician (Editor: P. A. Griffiths), Nankai Tracts in Mathematics Vol.",
"11, World Scientific Publishing Co Inc. (2007).",
"J.C.Maxwell, On the stability of the motion of Saturn's rings, An essay which obtained the Adams Prize in 1856 in the University of Cambridge, Macmillan (1859) Available online at https://archive.org/details/onstabilityofmot00maxw"
]
] | 1709.01539 |
[
[
"Sequence Prediction with Neural Segmental Models"
],
[
"Abstract Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.",
"Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.",
"However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.",
"In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.",
"We also show that instead of including more features to obtain better accuracy, segmental cascades can be used to speed up training and decoding.",
"Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.",
"In the first stage, a frame classifier is trained with manual alignments, and then in the second stage, segmental models are trained with manual alignments and the out- puts of the frame classifier.",
"However, obtaining manual alignments are time-consuming and expensive.",
"We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.",
"We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification.",
"We present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.",
"Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks."
],
[
">=latex SEQUENCE PREDICTION WITH NEURAL SEGMENTAL MODELS BY HAO TANG A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science at the TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO Chicago, Illinois September, 2017 Thesis Committee: Karen Livescu (Thesis Advisor) Kevin Gimpel David McAllester Eric Fosler-Lussier James Glass left=0in,right=0in,top=0in,bottom=0in Figure: NO_CAPTION Sequence Prediction with Neural Segmental Models by Hao Tang Segments that span contiguous parts of inputs, such as phonemes in speech, named entities in sentences, actions in videos, occur frequently in sequence prediction problems.",
"Segmental models, a class of models that explicitly hypothesizes segments, have allowed the exploration of rich segment features for sequence prediction.",
"However, segmental models suffer from slow decoding, hampering the use of computationally expensive features.",
"In this thesis, we introduce discriminative segmental cascades, a multi-pass inference framework that allows us to improve accuracy by adding higher-order features and neural segmental features while maintaining efficiency.",
"Segmental models, similarly to conventional speech recognizers, are typically trained in multiple stages.",
"In the first stage, a frame classifier is trained, and in the second stage, segmental models are trained with the outputs of the frame classifier.",
"Both training stages require manual alignments, and obtaining manual alignments are time-consuming and expensive.",
"We explore end-to-end training for segmental models with various loss functions, and show how end-to-end training with marginal log loss can eliminate the need for detailed manual alignments.",
"We draw the connections between the marginal log loss and a popular end-to-end training approach called connectionist temporal classification, and present a unifying framework for various end-to-end graph search-based models, such as hidden Markov models, connectionist temporal classification, and segmental models.",
"Finally, we discuss possible extensions of segmental models to large-vocabulary sequence prediction tasks.",
"Thesis Supervisor: Karen Livescu Title: Associate Professor First, I would like to thank my advisor, Karen Livescu.",
"Karen has always been very patient, letting me make mistakes after mistakes, mistakes that she can foresee, mistakes that I have made many times already, mistakes that have no easy fix, so that I can learn from them.",
"She taught me when to explore novel ideas and when to persist and complete tedious tasks.",
"I have learned to think critically and argue constructively, while at the same time to be honest and critical to my own work.",
"There are countless other things I have learned from her including phonetics, spectrogram reading, and English word usage, to name a few.",
"I can not express how much I appreciate her guidance and mentorship.",
"I am really fortunate to be her first PhD student.",
"Next, I want to thank my committee members, Kevin Gimpel, David McAllester, Eric Fosler-Lussier, and Jim Glass.",
"David's comments and questions always encourage me to view my work in a broader context and in the most general form possible.",
"I have also greatly benefited from his mathematical rigor through his talks, his lectures, and interactions with him, and have developed the habit of asking whether a mathematical expression is type checked or not.",
"I am grateful to Eric for his insightful comments.",
"I have benefited a lot from conversations we had in workshops and conferences.",
"I enjoyed his humorous and joyful attitude, and him pushing me towards the finish line harder than my advisor does.",
"I thank Jim for keeping the speech science aspect of my research in mind.",
"My research is heavily influenced by his prior work, and it is very special to work on something that he has been working on for decades.",
"I am deeply grateful to Kevin.",
"He is like my second advisor, caring and helpful in many ways.",
"I have benefited from his expertise in natural language processing (NLP), and discovered many similar approaches developed in parallel in both the NLP and the speech community.",
"Besides the technical content, he taught me to be humble yet to be bold when necessary.",
"I have also enjoyed his humorous comments on my writing before stressful deadlines.",
"It has been a great pleasure to have his company during my PhD.",
"I was fortunate to join Mark Hasegawa-Johnson's team in the second Jelinek summer workshop.",
"I have benefited from Mark's extensive knowledge in both speech science and machine learning.",
"It is fair to say that I always learn something new when talking to him.",
"I also thank him for his guidance after the workshop.",
"It has been a great pleasure to work with the people on the team.",
"In particular, I have benefited from interactions with Chunxi Liu, Preethi Jyothi, Amit Das, Vimal Manohar, Paul Hager, Tyler Kekona, and Rose Sloan.",
"People I met during the workshop, including Yi Luan, Yangfeng Ji, Lingpeng Kong, Guoguo Chen, and Trang Tran, also made the overall experience fun and pleasant.",
"In 2013, I worked as an intern with Shinji Watanabe at the Mitsubishi Electric Research Laboratories (MERL).",
"I have benefited from the interactions with him and other people at MERL, including John Hershey, Jonathan Le Roux, and Tim Marks.",
"In particular, I thank Shinji for taking care of me when I had Bell's palsy during the internship.",
"The fellow interns at MERL, including Niao He and Lingling Tao, also made the internship experience fun and rewarding.",
"My first research experience related to speech and language processing was acquired from Lin-Shan Lee's guidance in National Taiwan University back in 2009.",
"Lin-Shan, even with his always busy schedule, has been very caring and helpful.",
"It would not have been possible for me to pursue a PhD without his help and encouragement.",
"I thank the people in Toyota Technological Institute at Chicago (TTIC) for making my PhD journey wonderful.",
"In particular, I have benefited from Julia Chuzhoy's and Madhur Tulsiani's lectures, acquiring the necessary vocabulary to talk to theory people.",
"I thank Nati Srebro for grilling me during my qualifying exam, forcing me to always keep the precise language and mathematical terms in mind.",
"I enjoyed the interactions with Greg Shakhnarovich and his humor.",
"I thank Joseph Keshet for his guidance and mentorship during my first few years of PhD.",
"I have benefited interactions and collaboration with Liang Lu.",
"I thank the fellow students, including Behnam Tavakoli, Somaye Hashemifar, Taehwan Kim, Shubham Toshniwal, Shane Settle, Qingming Tang, Bowen Shi, Lifu Tu, Hai Wang, Renyu Zhang, Jian Yao, Jianzhu Ma, Xing Xu, Avleen Bijral, Payman Yadollahpour, Feng Zhao, Zhiyong Wang, Karthik Sridharan, and Peng Jian, for the interactions, cookie breaks, random interruptions, and random questions.",
"I will never regret choosing a cubicle that invites so many interruptions.",
"I am grateful to Weiran Wang.",
"It is fair to say that much of my work would not be possible without his help.",
"I have also greatly benefited from interactions with past post-docs, including Raman Arora and Herman Kamper.",
"I thank visiting students, Taiki Kawano and Takayuki Yamabe, from Toyota Technological Institute in Japan, for the fun time we had in Chicago and in Tokyo.",
"Thanks to Chrissy Novak, Adam Bohlander, and the rest of the staff for making my PhD life easy and smooth.",
"Lastly, I thank the president of TTIC, Sadaoki Furui, for his guidance, and for pushing me to exercise and to practice my tennis skills with him.",
"Thanks to my family in Taiwan for their support over the years.",
"Thanks to Hsin-Yu, for everything.",
"And thanks to my parents, whom I cannot possibly thank in words.",
"Segmental models, the subject of this thesis, are a family of models that predict sequences of segments.",
"Segmental models are designed for a wide range of sequence prediction tasks, such as named entity recognition [118], speech recognition [104], [35], and action recognition [122], [55].",
"Examples of these tasks are shown in Figure REF .",
"The input of a sequence prediction task is commonly represented as a sequence of real-valued vectors, and the output is a sequence of discrete labels.",
"The goal is to find a function that maps the input vectors to the output labels.",
"Every output label in the output sequence, such as a named entity, a phoneme, or an action, typically has a corresponding chunk of contiguous input vectors, called a segment.",
"Segments come in varying lengths.",
"As a consequence, the lengths of the output sequences typically do not match the lengths of the input sequences.",
"Figure: Examples of sequence prediction tasks.The first example is speech recognition, where the input is a sequence of acousticsignals and the output is a sequence of words.",
"The second example is American SignLanguage fingerspelling recognition, where the input is a sequence of images andthe output is a sequence of letters.",
"The third example is named entity recognition,where the input is a sequence of words and the output is the same sequence of wordswith named entities in parentheses.",
"In the example of speech recognition,the gray arrow connects the word “else” to its corresponding contiguous partin the input sequence.The predominant approach to solving these tasks is to break the varying-length segments into pieces so that the lengths of the input sequences match the lengths of the output sequences.",
"These smaller pieces are then assembled back into varying-length segments with an additional post-processing step.",
"Each individual element in the input sequence is referred to as a frame.",
"Models, such as hidden Markov models [112], [62], linear-chain conditional random fields [43], [99], and recurrent neural networks [40], that map input sequences to output sequences of the same lengths, are called frame-based models [104], [35].",
"To include a wider context, it is common to use windows of frames instead of individual frames for prediction.",
"Nevertheless, the number of windowed frames is still the same as the number of output labels.",
"Because frame-based models are conceptually simple and computationally cheap, they have enjoyed much success and received much attention in the past few decades.",
"Figure: Frame-based approaches and segment-based approaches for sequence prediction.Model accuracies for prediction tasks are heavily influenced by the set of features used to make prediction, where a feature is a real-value function of the input that correlates well with a label or a subset of labels.",
"Breaking up varying-length segments is an extremely successful heuristic, but it also makes extracting certain features from segments difficult.",
"For example, computing durations is difficult for frame-based models.",
"Variants of frame-based models, such as variable-duration HMMs [28], are proposed in order to overcome this difficulty, but it is impractical to construct a model for every new type of feature.",
"In contrast, segment-based approaches do not assume anything about the segments, so the users are free to use arbitrary information about the segments during prediction.",
"For example, given a segment with a precise start time and end time, computing its duration is trivial.",
"For speech recognition, energy distribution at the left and right boundaries are useful features.",
"For named entity recognition, we can check whether a phrase (segment) is in the dictionary.",
"The flexibility makes segment-based models appealing, and many promising results have been reported in the past [147], [104], [35].",
"A comparison of the two approaches is shown in Figure REF .",
"More formally, a segment is a tuple of a start time, an end time, and possibly a label.",
"At a high level, segmental models make predictions by first creating a search space consisting of connected segments, or paths.",
"Segments are given weights based on the start time, end time, and label.",
"The weight values reflect how well the labels match the input.",
"Finding the sequence of segments that best matches the input is equivalent to finding the maximum-weight path.",
"The prediction process is also referred to as decoding.",
"Note that the segment weights can be computed by any function as long as it only makes use of the start time, the end time, the label, and the input sequence.",
"An example is shown in Figure REF .",
"Figure: An example of a segmental model.",
"A search space is built based on theinput frames.",
"Each edge (segment) has a start time, an end time, and a label,which the weight function can make use of.",
"Once the weights of edgesare computed, decoding is the problem of finding the maximum-weight path.Despite their success and attractive properties, segmental models still fall behind frame-based models on many tasks, such as speech recognition [148].",
"There are many possible reasons behind this.",
"The choice of features extracted from the segments might play a role.",
"However, complex features are typically computationally expensive.",
"The large number of segments we need to consider prohibits the use of computationally expensive segment features.",
"The central goal of this thesis is to improve the efficiency of segmental models, allowing us to explore more computationally expensive features and to improve the performance of segmental models.",
"This thesis studies segmental models along three directions: segment features, inference, and learning.",
"We introduce discriminative segmental cascades to speed up inference, allowing us to explore computationally expensive segment representations, such as ones computed with neural networks.",
"We study how segmental models perform when they are trained with various loss functions.",
"Different loss functions have different training requirements, some of them allowing us to train segmental models without manual alignments.",
"We also study whether it is possible to train segmental models end to end from random initialization.",
"We focus on two tasks, phonetic recognition and American Sign Language fingerspelling recognition.",
"In this section, we briefly describe automatic speech recognition (ASR), a task for which segmental models have been studied extensively.",
"We describe a few features that were shown to be useful for disambiguating phonemes, the basic sound units that distinguish words.",
"We then describe why it is hard to incorporate these features in frame-based models, and some of the past efforts to overcome this difficulty.",
"In brief, human speech production is the process of mapping a sequence of discrete linguistic units, such as words or phonemes, into waveforms, and automatic speech recognition is about inverting this process, mapping speech waveforms to sequences of discrete linguistic units.",
"On one side of the process, we have speech waveforms, commonly represented as sequences of real-valued vectors called frames.",
"On the other side of the process, we have the discrete units, namely segments, representing phonemes or words.",
"A waveform can be represented as a sequence of real-valued vectors in multiple ways.",
"One common representation is the amount of sinusoids at different frequencies appearing in the signal.",
"A waveform is first converted into small overlapping chunks.",
"Typically the size of the chunk is 25ms, and a chunk is created every 10ms.",
"Each chunk of waveform is represented as a linear combination of different sinusoids at different frequencies.",
"In particular, a frame in this representation is the vector of coefficients of the linear combination.",
"This representation is known as the spectrogram.",
"Examples of spectrograms are shown in Figure REF .",
"Given a sequence of frames, the goal of ASR is to predict what is being said in the waveform, in the form of a sequence of discrete linguistic units, with words and phonemes being the two most common discrete units.",
"Phonemes are defined as the sound units that distinguish words [76], and a sequence of phonemes can be converted into words by looking up the pronunciations of words in a lexicon.",
"In most settings, the set of phonemes and the lexicon are assumed to be given.",
"A standard evaluation metric for ASR is the Levenshtein distance between the predicted label sequence and the ground-truth sequence.",
"Levenshstein distance measures the minimum number of edits that needs to be performed to transform the predicted label sequence to the ground-truth sequence; hence it is also called the edit distance.",
"Formally, the edit distance of two label sequences $y$ and $\\hat{y}$ is defined as $\\text{edit}(\\hat{y}, y) = \\frac{I + D + S}{|y|},$ where $|y|$ is the length of $y$ , $I$ , $D$ , and $S$ are the number of insertion, deletion, and substitution, respectively.",
"ASR is a difficult task due to the wide range of variabilities in the process of speech production.",
"Even in the most controlled setting, the same isolated word can hardly be pronounced the same way twice by the same speaker.",
"Speech production is a highly context-dependent process.",
"Phonemes are pronounced differently when the neighboring phonemes are different [76].",
"Similarly, words are pronounced differently in different contexts.",
"Different speakers pronounce words and phonemes differently due to the differences in their speech organs [80].",
"Different speaking styles also affect pronunciations.",
"For example, in conversational speech, words are seldom pronounced in the canonical way presented in the lexicon due to the casual speaking style [83].",
"All of the above variabilities make ASR difficult.",
"It is even more difficult when the speech signals are degraded by noise [54].",
"Since the goal of speech recognition is to predict words based on their pronunciations, much work has been dedicated to finding features, sometimes referred to as acoustic cues, that identify phonemes and differentiate phonemes.",
"See [47] and citations therein.",
"Some features are correlated with how humans identify phonemes [93].",
"In general, there are many such features that are useful for speech recognition.",
"Each feature alone is probably not enough to identify a phoneme, but a combination of features might be able to [47].",
"Below we describe two features as examples.",
"Duration is one of the features that differentiate phonemes.",
"For example, the duration of the long vowel /iy/The phonemes are written in ARPAbet.",
"in the word “seat” is typically longer than short vowels /ih/ in the word “sit” [106].",
"In this case, duration serves as a good feature to differentiate /iy/ from /ih/.",
"Unvoiced fricatives, such as /f/, /s/, and /sh/, typically have longer duration than voiced fricatives, such /v/, /z/, and /zh/ [136].",
"Durations have also been shown to help improve recognition results [5].",
"Perhaps the most salient feature for distinguishing phonemes by looking at the spectrogram is the distribution of energy [76].",
"For example, fricatives, such as /s/ and /f/, have evenly spread energy across frequency.",
"Stops, such as /t/ and /p/, have sudden sharp bursts of energy.",
"Bands of high energy in spectrograms, commonly referred to as formants, are another type of energy patterns useful for identifying vowels [76].",
"The first and second formants (counting from low frequencies) are commonly used to differentiate vowels [53].",
"Formants are also useful for speech recognition [57].",
"Integrating arbitrary segment features into frame-based models has always been considered a difficult task.",
"Integrating even just the duration requires nontrivial modification to the models [81].",
"For example, we can expand the label set with durations, but this approach only works for discrete features and generates unnecessarily large search spaces.",
"Integrating these segment features is also one of the driving forces behind the use and development of segmental models [147].",
"While hidden Markov models assume that frame labels follow a Markov process, hidden semi-Markov models assume that segments follow a Markov process.",
"Semi-Markov processes have been used to incorporate duration in frame-based models [116].",
"Hidden semi-Markov models have been applied to speech recognition with the same reason in mind [81], [117].",
"The segment-based speech recognizer SUMMIT, is designed with the goal of integrating arbitrary segment features [147].",
"Segmental models have also been applied to other sequence prediction tasks, such as named entity recognition [118], American Sign Language fingerspelling recognition [71], and action recognition [122], [55].",
"For named entity recognition, the input is a sequence of words, and the output is a sequence of named entity tags, such as person name, organization name, and location name.",
"For American Sign Language fingerspelling recognition, the input sequence is a sequence of video frames, and the output is a sequence of letters.",
"For action unit detection, the input sequence is a sequence of video frames, and the output is a sequence of actions, such as walk, jump, sit, and stand.",
"In general, these tasks follow a left-to-right (or right-to-left) order in both the input sequence and the output sequence.",
"A task is said to be linear or monotonic if the task has a left-to-right (or right-to-left) order.",
"One sequence prediction task that is not monotonic is translation [19].",
"Depending on the source and the target languages, the order of words in the source language can be different from the order in the target language.",
"In this thesis, we only focus on segmental models for monotonic sequence prediction tasks.",
"We have seen why segmental models are favored over frame-based models when incorporating complex segment features is needed.",
"However, the flexibility of segmental models comes with a price.",
"Enumerating segments of different lengths and of different labels can be time-consuming.",
"Consider an input sequence of length 300, and a label set size of 50.",
"The number of possible segments of length up to 30 is around $30 \\times 50 \\times 300 = 450000$ .",
"If we make only a single decision at every time point as for frame-based models, we only need to consider $50 \\times 300 = 15000$ different decisions.",
"The hypothesis space for segmental models is considerably larger, and as a consequence learning and inference for segmental models are significantly slower than for frame-based models.",
"Many researchers have attacked the problem of large hypothesis spaces either implicitly or explicitly.",
"The most common approach is to use frame-based models to generate a set of high-confidence segments, and then use segmental models to rescore the segments with segment features [35], [149].",
"[102] explicitly train a model for pruning segments, while [139] reuse the computation of features whenever possible.",
"The first approach needs to have two separate models and the second approach depends on the exact form of the features.",
"We would like to design segmental models that do not depend on other external models, and can handle large hypothesis spaces in a feature-oblivious manner.",
"Designing and implementing state-of-the-art frame-based models requires a significant amount of engineering effort.",
"Phonemes are modeled with three-state hidden Markov models [119].",
"Since phonemes are influenced heavily by nearby phonemes, phoneme labels that include the previous and the next phonemes, referred to as triphones, are introduced [119].",
"The large number of triphones makes estimating their probabilities difficult, so triphone parameters are shared using with decision trees [143].",
"These design decisions in model structure makes engineering difficult.",
"In contrast to frame-based models, the engineering effort in segmental models is put into designing segment features, while the model structure remains the same.",
"In sum, we aim to design segmental models that perform well on sequence prediction tasks, have a clean and modular mathematical definition, are efficient to train and to decode, and do not depend on other models.",
"The concept of segmental models was not explicitly defined before the late 1980s, and little work has had segmental models as the primary focus.",
"[141] were one of the early attempts in the 1970s to build speech recognizers based on segment features.",
"In the early 1980s, [23] developed a system for recognizing isolated English letters based on segment features.",
"Isolated digits as segments were considered in [75].",
"[147] introduced SUMMIT, a segment-based speech recognizer that can handle arbitrary segment features.",
"While the SUMMIT system was later formulated into a probabilistic framework [34], others were not probabilistic, and have very few parameters to be estimated.",
"The probabilistic view of segmental models can also be traced back to the 1980s.",
"As mentioned earlier, [116] introduced semi-Markov processes and the follow-up [117] introduced hidden semi-Markov models to the speech recognition community.",
"These two studies were mainly motivated by the need to include duration in frame-based models.",
"[103] formalized segmental models as a probabilistic framework to include arbitrary features, but the authors only used frame-based probabilities with one additional duration feature.",
"[115] and [29] further proposed segmental hidden Markov models, but the general form stayed mostly the same.",
"Segmental models proposed in these studies were generative, and were mostly restricted to frame-based Gaussian distributions.",
"This line of work has been summarized in [104].",
"Into the 2000s, studies of segmental models shifted from generative to discriminative.",
"[118] proposed semi-Markov conditional random fields (CRF), the first type of discriminative segmental models.",
"Semi-Markov CRFs require the use of manual alignments to train the models, making them less applicable to speech recognition.",
"[149] proposed to use a different training loss for training semi-Markov CRFs, marginalizing over all possible segmentations.",
"This approach alleviated the need for manual alignments to train segmental models.",
"Later, [150] used segmental models as second-pass rescoring models for word recognition, showcasing the flexibility of segmental models for incorporating a wide variety of features.",
"A segmental model is said to be a first-pass model if it exhaustively searches over the entire hypothesis space.",
"Before [148], all of the segmental models for speech recognition were not first-pass.",
"The early version of SUMMIT [147] used a heuristic approach to estimate phoneme boundaries, instead of searching over the entire search space.",
"A more recent version of SUMMIT [34] avoided searching exhaustively by using a frame-based model to generate high confidence hypotheses.",
"Semi-Markov CRFs are first-pass segmental models, but have only been applied to natural language tasks, where sequences are typically short and label set size is small.",
"[148] showed that it is feasible to use segmental models as first-pass speech recognizers.",
"Since then, studies of segmental models have moved on to first-pass recognition.",
"At the same time, advances in neural networks have allowed us to learn generic feature representations, so many studies have been devoted to finding better segment representations to replace hand-crafted features.",
"[49] improved upon [148] by using a 3-layer multilayer perceptron to produce phoneme probabilities.",
"[1] further improved the results by using a deep convolutional neural network, and it was also the first to train segmental models and neural networks end to end.",
"Along the same line, [85] proposed end-to-end segmental models with segment representations computed with long short-term memory networks.",
"In this thesis, we make the following contributions to the development of segmental models.",
"We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.",
"We develop improved understanding of training approaches and training requirements for segmental models.",
"We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.",
"We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.",
"We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.",
"Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.",
"We introduce discriminative segmental cascades, allowing us to improve sequence prediction performance using rich features while maintaining efficiency.",
"We develop improved understanding of training approaches and training requirements for segmental models.",
"We compare segmental models trained end to end and ones trained in multiple stages, and compare segmental models trained with and without manual alignments Along the way, we present segmental models in a general, modular fashion.",
"We present a unifying framework that encompasses many end-to-end models, such as hidden Markov models, connectionist temporal classification, as special cases.",
"We use phonetic recognition and American Sign Language fingerspelling recognition as test beds for comparing segmental models and frame-based models.",
"Segmental models are able to outperform frame-based models in many settings for both sequence prediction tasks.",
"In Chapter 2, we review finite-state transducers (FST), an important tool for describing the hypothesis spaces of segmental models.",
"For the second part of Chapter 2, we review the essential components in speech recognizers, such as language models, lexicons, and hidden Markov models, represented with FSTs.",
"In Chapter 3, we formally define segmental models.",
"The definition is modular and covers many existing segmental models as special cases.",
"In Chapter 4, we introduce discriminative segmental cascades, which allow us to explore various segment representations while maintaining efficiency.",
"In Chapter 5, we study segmental models trained in different training conditions, comparing end-to-end training and multi-stage training.",
"In Chapter 6, we review several variants of segmental models, and describe how they relate to our definition of segmental models.",
"In Chapter 7, we propose a unified framework encompassing many end-to-end frame-based models and segmental models.",
"Finally, in Chapter 8, we discuss possible ways to extend segmental models.",
"This chapter describes finite-state transducers (FST), a useful tool for representing and manipulating the search space of a sequence prediction task.",
"This chapter also includes a complete example of a standard speech recognizer based on hidden Markov models represented as FSTs.",
"See [96] for a comprehensive review of FSTs and their algorithms, and [97] for their applications to speech recognition.",
"We define FSTs based on multigraphs (graphs that allow multiple edges between any two vertices) instead of regular graphs.",
"A multigraph $G$ is a tuple $(V, E, \\mathrm {tail}, \\mathrm {head})$ , where $V$ is a set of vertices, $E$ is a set of edges, $\\mathrm {tail}: E \\rightarrow V$ is a function that associates an edge to its tail vertex, and $\\mathrm {head}: E \\rightarrow V$ is a function that associates an edge to its head vertex.",
"Note that $E$ is not a subset of $V \\times V$ but a general set, because we allow many edges to have the same tail and head, and a pair of vertices is not enough to uniquely identify an edge.",
"An example of a multigraph is shown in Figure REF .",
"We say that $e_1 \\in E$ and $e_2 \\in E$ are connected if $\\mathrm {head}(e_1) = \\mathrm {tail}(e_2)$ .",
"In addition, we define a path of length $n$ as a sequence of connected edges $(e_1, e_2, \\dots , e_n)$ where $\\mathrm {head}(e_i) = \\mathrm {tail}(e_{i+1})$ for $i = 1, \\dots , n$ .",
"A sub-path is a sequence of connected edges within a path.",
"For example, $(e_3, e_4, e_5)$ is a sub-path of $(e_1, e_2, \\dots , e_7, e_8)$ .",
"A finite-state transducer is a tuple $(G, \\Sigma , \\Lambda , I, F, i, o, w)$ , where $G$ is a multigraph, $\\Sigma $ is a set of input symbols, $\\Lambda $ is a set of output symbols, $I \\subseteq V$ is a set of initial vertices, $F \\subseteq V$ is a set of final vertices, $i: E \\rightarrow \\Sigma $ is a function that associates an edge to its input symbol, $o: E \\rightarrow \\Lambda $ is a function that associates an edge to its output symbol, and $w: E \\rightarrow \\mathbb {R}$ is a function that puts weights on edges.",
"An example is shown in Figure REF .",
"Figure: A multigraph, where V={0,1,2,3}V = \\lbrace 0, 1, 2, 3\\rbrace and E={0,1,2,3,4}E = \\lbrace 0, 1, 2, 3, 4\\rbrace .The functions of tail and head are shown in the table on the right.Figure: An FST based on the multigraph in Figure ,with Σ={𝚊,𝚋,𝚌,𝚍,𝚎}\\Sigma = \\lbrace \\texttt {a}, \\texttt {b}, \\texttt {c}, \\texttt {d}, \\texttt {e}\\rbrace andΛ={𝙰,𝙱,𝙲,𝙳,𝙴}\\Lambda = \\lbrace \\texttt {A}, \\texttt {B}, \\texttt {C}, \\texttt {D}, \\texttt {E}\\rbrace .The initial vertex is shown in bold, and the final vertexis shown with a doubled circle, i.e., I={0}I = \\lbrace 0\\rbrace and F={3}F = \\lbrace 3\\rbrace .We use σ\\sigma :λ\\lambda /w to denote an edgewith input symbol σ\\sigma , output symbol λ\\lambda , and weight ww.The number in parentheses is an identifier of an edge.An FST can be seen as a function that maps strings to strings, where each path in the graph defines an input-output pair.",
"Specifically, a path $(e_1, e_2, \\dots , e_n)$ associates the input string $i(e_1)i(e_2)\\cdots i(e_n)$ with the output string $o(e_1)o(e_2)\\cdots o(e_n)$ .",
"For example, the FST shown in Figure REF defines the mapping $\\lbrace (\\texttt {ad}, \\texttt {AD}), (\\texttt {bd}, \\texttt {BD}),(\\texttt {ce}, \\texttt {CE})\\rbrace $ .",
"There is a special symbol called the empty symbol, denoted $\\epsilon $ .",
"Any string concatenated with an empty symbol is itself, i.e., $\\sigma _1\\sigma _2\\cdots \\sigma _n\\epsilon = \\epsilon \\sigma _1\\sigma _2\\cdots \\sigma _n= \\sigma _1\\sigma _2\\cdots \\sigma _n$ .",
"If the input symbol of an edge is the empty symbol, then we can traverse the edge without consuming any input.",
"If the output symbol of an edge is the empty symbol, then we do not produce any output when traversing the edge.",
"As an example, the FST in Figure REF defines the mapping $\\lbrace (\\texttt {ad}, \\texttt {AD}), (\\texttt {a}, \\texttt {AF}),(\\texttt {bd}, \\texttt {BD}), (\\texttt {b}, \\texttt {BF}), (\\texttt {ce}, \\texttt {CE})\\rbrace $ .",
"Figure: An FST with empty symbols.",
"The edgewith the empty symbol is highlighted in red.We assume there is one unique start vertex and one unique end vertex, i.e., $|I| = |F| = 1$ .",
"If an FST has more than one initial vertices, we can always create an additional vertex and connect all initial vertices to the additional vertex with empty symbols (and possibly zero weights) without affecting the string mapping of the FST.",
"For convenience, we also define $\\text{in}(v) = \\lbrace e \\in E : \\mathrm {head}(e) = v\\rbrace $ and $\\text{out}(v) = \\lbrace e \\in E: \\mathrm {tail}(e) = v\\rbrace $ .",
"We have reviewed FSTs as graphs and string functions.",
"The third view of FSTs is as state machines.",
"Take the FST in Figure REF for example.",
"To get the output AD from the input ad, we first feed the character a and traverse from vertex 0 to vertex 1, and then feed the character d and traverse from vertex 1 to vertex 3.",
"Similarly, to get BF from b, we first feed the character b and traverse from vertex 0 to vertex 1.",
"Due to the empty symbol $\\epsilon $ , we get F and traverse from vertex 1 to vertex 3 without feeding any character.",
"The view of state machines is especially useful when we talk about FST compositions.",
"Other than just manipulating strings, we are also interested in the weights on the edges, especially when paths are considered.",
"We introduce two operators $\\otimes $ and $\\oplus $ , where $\\otimes $ is for combining edge weights and $\\oplus $ is for combining path weights.",
"For example, by traversing the path $(0, 3)$ that produces AD in Figure REF , we collect the weight 0.1 from edge 0 and 0.4 from edge 3, and the weight of the path $(0, 3)$ is defined as $w(0) \\otimes w(3) = 0.1 \\otimes 0.4$ .",
"Suppose in general $S$ is the set of weights.",
"It is desirable to have certain properties, such as commutativity and associativity, for the two operators.",
"For $a \\in S$ , $b \\in S$ , and $c \\in S$ , we say that an operator $*$ is commutative if $a * b = b * a$ , and that an operator $*$ is associative if $(a * b) * c = a * (b * c)$ .",
"The element $0 \\in S$ is an identity with respect to the operator $*$ if $0 * a = a * 0 = a$ .",
"The element $0 \\in S$ annihilates $S$ with respect to $*$ if $0 * a = a * 0 = 0$ .",
"We say that the operator $*$ distributes over the operator $+$ if $a * (b + c) = a * b + a * c$ and $(a + b) * c = a * c + b * c$ .",
"The tuple $(S, \\oplus , \\otimes )$ is a semiring [95] if $\\oplus $ is associative and commutative with identity $0 \\in S$ , $\\otimes $ is associative with identity $1 \\in S$ , $\\otimes $ distributes over $\\oplus $ , and the element $0 \\in S$ annihilates $S$ with respect to $\\otimes $ .",
"An example of semiring is $(\\mathbb {R}, +, \\times )$ with 0 being the identity of $+$ , and 1 being the identity of $\\times $ .",
"Another example is the tropical semiring $(\\mathbb {R} \\cup \\lbrace \\infty \\rbrace , \\min , +)$ with $\\infty $ being the identity of $\\min $ , and 0 being the identity of $+$ .",
"Yet another example is the log semiring $(\\mathbb {R} \\cup \\lbrace -\\infty \\rbrace , \\text{logadd}, +)$ where $\\text{logadd}(a, b) = \\log (\\exp (a) + \\exp (b))$ , $-\\infty $ is the identity of $\\text{logadd}$ , and 0 is the identity of $+$ .",
"Defining the operators for combining weights in a general way encourages algorithm reuse.",
"As we will see in later chapters, FST algorithms tend to look very similar, and typically the only difference is how the weights are combined.",
"By using general operators, we get a family of algorithms.",
"In other words, we create algorithms almost for free by choosing a proper semiring.",
"Given an FST, we are interested in finding a shortest path from the initial vertex to the final vertex.",
"For example, if the weights are negative log probabilities traversing from one vertex to another, then a shortest path corresponds to one of the most likely paths.",
"We assume there are multiple shortest paths, but finding one of them is enough.",
"We also assume the underlying graph is acyclic, meaning that every path can only traverse a vertex at most once.",
"For acyclic graphs, we have a nice necessary condition for shortest paths—every sub-path within a shortest path is also a shortest path.",
"The argument for the necessary condition is simple.",
"If a sub-path is not a shortest path, then we can always substitute the sub-path with another shorter sub-path to create a shorter path overall.",
"The necessary condition can be viewed as a recursive condition.",
"Let $\\mathcal {P}(u, v)$ be the set of paths from vertex $u$ to vertex $v$ , and let $w(p)$ be a shorthand of $\\sum _{e \\in p} w(e)$ .",
"Suppose we want to compute a shortest path from the vertex $u$ to vertex $v$ .",
"We examine the set of edges $\\text{in}(v) = \\lbrace e \\in E: \\mathrm {head}(e) = v\\rbrace $ leading into vertex $v$ .",
"If an edge $e \\in \\text{in}(v)$ is part of the shortest path, then by the necessary condition every path from $u$ to $\\mathrm {tail}(e)$ should also be a shortest path.",
"In other words, $\\min _{p \\in \\mathcal {P}(u, v)} w(p)& = \\min _{e \\in \\text{in}(v)} \\min _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))}[ w(e) + w(p^{\\prime }) ] \\\\& = \\min _{e \\in \\text{in}(v)}\\Bigg [ w(e) + \\min _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))} w(p^{\\prime }) \\Bigg ]$ The recursive condition can be computed efficiently with dynamic programming if we store the shortest distance from vertex $u$ to every other vertex.",
"Specifically, let $d(v)$ be the shortest distance from $u$ to $v$ , i.e., $d(v) = \\min _{p \\in \\mathcal {P}(u, v)} w(p).$ By the recursive condition, we have $d(v) = \\min _{e \\in \\text{in}(v)} [ w(e) + d(\\mathrm {tail}(e)) ].",
"$ Before computing the minimization in (REF ), we need to make sure the shortest sub-paths are computed.",
"Formally, to compute $d(v)$ for vertex $v$ , we need to make sure $d(\\mathrm {tail}(e))$ are computed for $e \\in \\text{in}(v)$ .",
"In other words, if there is a path from vertex $w$ to vertex $v$ , then $d(w)$ should be computed before $d(v)$ .",
"We are looking for an order in which $w$ should come before $v$ if there is a path from $w$ to $v$ .",
"An order in which $w$ comes before $v$ if there is a path from $w$ to $v$ is called a topological order.",
"To be precise, a topological order is a function $f: V \\rightarrow \\lbrace 1, 2, \\dots , |V|\\rbrace $ such that $f(w) < f(v)$ if there is a path from $w$ to $v$ .",
"Given an directed acyclic graph, there exist many topological orders.",
"Any one of them would suffice for our dynamic programming.",
"One way to find a topological order is to run depth-first search (DFS) on the reversed graph.",
"The depth-first search algorithm is shown in Algorithm REF .",
"We maintain a stack $S$ .",
"We say that a vertex is traversed if it has been put on $S$ , and we use a set $T$ to track the traversed vertices.",
"When a vertex $v$ is put on stack, we also put a variable indicating whether we have tried to put its neighbors on the stack, where the set of neighbors is defined by $\\text{in}(\\cdot )$ .",
"We say that a vertex is expanded if we have tried to put its neighbors on the stack.",
"It is not hard to see that every vertex is put on the stack twice, once before it is expanded and once after.",
"When a vertex $v$ popped out from the stack is expanded, all vertices that can be traversed from $v$ are also expanded.",
"Since we put a vertex $v$ in $O$ when $v$ and all vertices traversable from $v$ are expanded, the order $O$ is exactly a topological order.",
"Algorithm REF is an instance of DFS specifically for computing a topological order by defining the set of neighbors with $\\text{in}(\\cdot )$ .",
"The algorithm can be modified for other purposes, for example, letting the set of neighbors be $\\text{out}(\\cdot )$ (and changing $\\mathrm {tail}(\\cdot )$ to $\\mathrm {head}(\\cdot )$ accordingly) if we want to search the graph in a forward direction instead of backwards.",
"Due to the use of a stack, DFS prefers to follow one path until none of the vertices can be expanded; hence the name depth-first search.",
"Depth-First Search (DFS) Let $r$ be the vertex we start to search.",
"A list of vertices $O$ is returned in topological order.",
"$S = \\lbrace (r, \\texttt {false})\\rbrace $ , $T = \\lbrace r\\rbrace $ , $O = ()$ $S$ is not empty pop $(v, z)$ from $S$ $z$ append $v$ to $O$ push $(v, \\texttt {true})$ to $S$ $e \\in \\text{in}(v)$ Expand $v$ .",
"$\\mathrm {tail}(e) \\notin T$ push $(\\mathrm {tail}(e), \\texttt {false})$ to $S$ $\\mathrm {tail}(e)$ is traversed.",
"add $v$ to $T$ Shortest-Path Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.",
"The map $d$ contains the shortest distances to $v$ for all vertices.",
"Let $O = (o_1, o_2, \\dots , o_n)$ be a topological order by running DFS starting from $t$ .",
"$i = 1, \\dots , n$ $d(o_i) = \\min _{e \\in \\text{in}(o_i)} [ w(e) + d(\\mathrm {tail}(e)) ]$ $\\pi (o_i) = \\operatornamewithlimits{\\arg \\!\\min }_{e \\in \\text{in}(o_i)} [ w(e) + d(\\mathrm {tail}(e)) ]$ Backtracking Let $s$ be the initial vertex and $t$ be the final vertex.",
"A shortest path $p$ from $s$ to $t$ .",
"$v \\leftarrow t$ $p = ()$ $v \\ne s$ add $\\pi (v)$ to $p$ $v \\leftarrow \\mathrm {tail}(\\pi (v))$ reverse $p$ Generalized Shortest-Distance Algorithm for Directed Acyclic Graphs Let $t$ be the ending vertex of all paths.",
"The map $d$ contains the shortest distances to $v$ for all vertices.",
"Let $O = (o_1, o_2, \\dots , o_n)$ be a topological order by running DFS starting from $t$ .",
"$i = 1, \\dots , n$ $d(o_i) = \\bigoplus _{e \\in \\text{in}(o_i)} [ w(e) \\otimes d(\\mathrm {tail}(e)) ]$ Up to this point, we are interested in finding a path that is shortest.",
"The weight of a path is defined to be the sum of the edge weights, and path weights are combined with the $\\min $ operator.",
"These operators are the ones used in the tropical semiring, and can be generalized to other semirings, where edge weights are combined with $\\otimes $ and path weights are combined with $\\oplus $ .",
"To be precise, the weight of a path $p$ is defined as $w(p) = \\bigotimes _{e \\in p} w(e)$ , and the distance can be written as $d(v) & = \\bigoplus _{p \\in \\mathcal {P}(u, v)} w(p)= \\bigoplus _{p \\in \\mathcal {P}(u, v)} \\bigotimes _{e \\in p} w(e)= \\bigoplus _{e \\in \\text{in}(v)} \\bigoplus _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))}[ w(e) \\otimes w(p^{\\prime }) ] \\\\& = \\bigoplus _{e \\in \\text{in}(v)} \\left[ w(e) \\otimes \\bigoplus _{p^{\\prime } \\in \\mathcal {P}(u, \\mathrm {tail}(e))} w(p^{\\prime }) \\right] \\\\& = \\bigoplus _{e \\in \\text{in}(v)} [w(e) \\otimes d(\\mathrm {tail}(e))].$ The generalized shortest-distance algorithm is shown in Algorithm REF .",
"See [95] for a general treatment of shortest-distance algorithms.",
"The recursive condition (REF ) only gives us the distance to the final vertex.",
"To obtain the path, we can record down the edge that achieves the minimum while computing the distances.",
"Specifically, let $\\pi (v)$ be the edge that achieves the minimum, i.e., $\\pi (v) = \\operatornamewithlimits{\\arg \\!\\min }_{e \\in \\text{in}(v)} [ w(e) + d(\\mathrm {tail}(e)) ].$ We can iteratively collect the optimal edge with $\\pi $ .",
"Since the algorithm goes from the final vertex to the initial vertex, it is commonly called backtracking.",
"The final shortest-path algorithm and the backtracking algorithm are shown in Algorithm REF and REF .",
"Recall that FSTs can be regarded as functions that map strings to strings.",
"It is natural to have multiple FSTs with one FST consuming the outputs of another FST, similar to function composition.",
"Let $T(x, y)$ be the weight of a path in the FST $T$ consuming $x$ as input and producing $y$ as output.",
"For any two FSTs $T_1$ and $T_2$ , the weight of a path with input $x$ and output $y$ in the composed FST $T_1 \\circ T_2$ is defined as $ (T_1 \\circ T_2)(x, y) = \\bigoplus _{z \\in \\mathcal {Z}(x)} T_1(x, z) \\otimes T_2(z, y),$ where $\\mathcal {Z}(x)$ is the set of output strings that can be produced by feeding $x$ to $T_1$ [2].",
"The weight is undefined if $\\mathcal {Z}(x)$ is an empty set or there is no path with input $z$ and output $y$ in $T_2$ .",
"The definition of (REF ) does not provide an explicit structure of the composed FST.",
"In this thesis, we prefer another approach to composing FSTs which we term structured composition [130].",
"Formally, the structured composition (or $\\sigma $ -composition) of two FSTs $T_1 = (G_1, \\Sigma _1, \\Lambda _1, I_1, F_1, i_1, o_1, w_1)$ with $G_1 = (V_1, E_1, \\mathrm {tail}_1, \\mathrm {head}_1)$ and $T_2 = (G_2, \\Sigma _2, \\Lambda _2, I_2, F_2, i_2, o_2,w_2)$ with $G_2 = (V_2, E_2, \\mathrm {tail}_2, \\mathrm {head}_2)$ is defined as $T_1 \\circ _\\sigma T_2 = (G, \\Sigma , \\Lambda , I,F, i, o, w)$ where $G = (V, E, \\mathrm {tail}, \\mathrm {head})$ with the following constraints.",
"$ V & = V_1 \\times V_2 &E & = \\Big \\lbrace \\langle e_1, e_2 \\rangle \\in E_1 \\times E_2 : o_1(e_1) = i_2(e_2) \\Big \\rbrace $ $\\Sigma & = \\Sigma _1 &i(\\langle e_1, e_2 \\rangle ) & = i_1(e_1) \\\\\\Lambda & = \\Lambda _2 &o(\\langle e_1, e_2 \\rangle ) & = o_2(e_2) \\\\I & = I_1 \\times I_2 &\\mathrm {tail}(\\langle e_1, e_2 \\rangle ) & = \\langle \\mathrm {tail}_1(e_1), \\mathrm {tail}_2(e_2) \\rangle \\\\F & = F_1 \\times F_2 &\\mathrm {head}(\\langle e_1, e_2 \\rangle ) & = \\langle \\mathrm {head}_1(e_1), \\mathrm {head}_2(e_2) \\rangle $ The $\\sigma $ -composed graph is defined over pairs of vertices and pairs of edges.",
"Due to the edge constraints (REF ), the outputs from $T_1$ have to match the inputs to $T_2$ .",
"An example is shown in Figure REF .",
"Note that the definition of the weight function is left to the users.",
"One possible definition is to let $w(\\langle e_1, e_2 \\rangle ) = w_1(e_1) \\otimes w_2(e_2).$ As a consequence of (REF ), the outgoing edges and incoming edges of a vertex in the $\\sigma $ -composed FST can be computed locally with $\\text{out}_1(\\cdot )$ and $\\text{out}_2(\\cdot )$ .",
"Specifically, $ \\text{out}(\\langle v_1, v_2 \\rangle )= \\Big \\lbrace \\langle e_1, e_2 \\rangle : e_1 \\in \\text{out}_1(v_1), e_2 \\in \\text{out}_2(v_2),o_1(e_1) = i_2(e_2) \\Big \\rbrace .$ Since computing $\\text{out}(\\langle v_1, v_2 \\rangle )$ only requires access to edges connected to $v_1$ and $v_2$ , traversing the composed FST, for example with DFS (Algorithm REF ), only needs to keep track of the vertex pairs.",
"The edges can be computed on the fly and do not need to be stored in memory.",
"In general, composing FSTs without explicitly storing the entire graph is commonly known as on-the-fly composition or lazy composition.",
"Another way to compute structured composition is to view FSTs as state machines.",
"Take Figure REF for example.",
"We start from vertex 0 in both $T_1$ and $T_2$ .",
"To get to vertex 1 in $T_1$ , we can either produce A or produce B as output.",
"In $T_2$ , to go from vertex 0 to vertex 1, we only have the choice to consume A.",
"Therefore, in the composed FST $T_1 \\circ _\\sigma T_2$ we have the edge from $(0, 0)$ to $(1, 1)$ with input a and output $\\alpha $ .",
"By walking through the vertices one by one, we are essentially performing a search on both FSTs while computing (REF ) on the fly.",
"Figure: An example of structured composition.Vertices in T 1 ∘ σ T 2 T_1 \\circ _\\sigma T_2 are labeled with pairs of vertices from T 1 T_1 and T 2 T_2.We now have all the necessary tools to describe a basic speech recognizer based on FSTs.",
"Modern speech recognizers are based on a series of string function compositions.",
"Following [98], we represent the string functions as FSTs, and the compositions are realized with structured composition.",
"The first FST $U$ takes acoustic frames as inputs and produces a sequence of frame labels.",
"The second FST $H$ takes in frame labels and converts them into phonemes.",
"The third FST $L$ takes phonemes as inputs and produces words as outputs.",
"The fourth FST $G$ takes words as inputs and produces words as outputs.",
"Finally, the four FSTs are $\\sigma $ -composed into $D = U \\circ _\\sigma H \\circ _\\sigma L \\circ _\\sigma G.$ The path weights in $D$ are typically negative log probabilities, so the maximum-weight path is the most likely path based on the probabilities.",
"To find the maximum-weight path, we simply negate the weights and find the shortest path.",
"The tropical semiring is used to combine the negated weights.",
"In sum, to predict a sequence from a sequence of frames, we create the FST $D$ and find the maximum-weight path in $D$ .",
"Below we describe the four FSTs $U$ , $H$ , $L$ and $G$ in detail.",
"The dynamics of speech are commonly modeled by hidden Markov models (HMM).",
"The generative story of a hidden Markov model is as follows.",
"We first generate a state $y_1$ based on a prior distribution and generate a frame $x_1$ given $y_1$ .",
"A state $y_2$ is generated given $y_1$ , and a frame $x_2$ is generated given $y_2$ .",
"In general, for some $t = 2, \\dots , T$ , the state $y_t$ is generated given $y_{t-1}$ , and $x_t$ is generated given $y_t$ .",
"For a sequence of $T$ frames $x_1, \\dots , x_T$ , the probability distribution defined by an HMM is $p(x_{1:T}, y_{1:T}) = p(y_1) \\prod _{t=2}^T p(y_t | y_{t-1}) \\prod _{t=1}^T p(x_t | y_t),$ where $x_{1:T}$ is a shorthand for $x_1, \\dots , x_T$ , $x_t \\in \\mathbb {R}^d$ for some $d \\in \\mathbb {N}$ and $y_t \\in \\lbrace 1, \\dots , S\\rbrace $ for some $S \\in \\mathbb {N}$ for $t=1, \\dots , T$ .",
"An example of a four-frame HMM is shown in Figure REF .",
"Note that Figure REF is not an FST but a graphical model where vertices are random variables and edges represent conditional dependencies.",
"The learnable parameters in an HMM are $p(y_1)$ , $p(y_t | y_{t-1})$ , and $p(x_t | y_t)$ for $t = 1, \\dots , T$ .",
"The probabilities $p(y_t | y_{t-1})$ are commonly known as transition probabilities, and $p(x_t | y_t)$ emission probabilities.",
"Figure: A hidden Markov model for 4 frames x 1 ,x 2 ,x 3 ,x 4 x_1, x_2, x_3, x_4with hidden labels y 1 ,y 2 ,y 3 ,y 4 y_1, y_2, y_3, y_4.Note that this is a graphical model where vertices are random variablesand edges represent conditional dependencies, not an FST.The observed variables are shaded, and the unobserved are not.The transition probabilities $p(y_t | y_{t-1})$ can be represented as a square matrix $A$ of size $S \\times S$ in which $A_{ij} \\in [0, 1]$ is the probability of transitioning from state $i$ to state $j$ satisfying $\\sum _{j=1}^S A_{ij} = 1$ .",
"To forbid the transition from state $i$ to state $j$ , we simply let $A_{ij} = 0$ .",
"It is common to model a phoneme as a 3-state HMM, and parameterize $A$ as $\\begin{pmatrix}a_{11} & a_{12} & 0 \\\\0 & a_{22} & a_{23} \\\\0 & 0 & a_{33}\\end{pmatrix}$ where we are only allowed to either stay in the current state or move to the next state.",
"The allowed transitions can be conveniently represented as an FST, where vertices are states and edges are transitions associated with transition probabilities.",
"An example of the allowed state transitions is shown in Figure REF .",
"Figure: A 3-state FST for the phoneme ih, where loga ij \\log a_{ij} isthe log transition probability from state ii to state jj.To allow transitions from phonemes to phonemes, we connect the FSTs, such as the ones in Figure REF , in parallel.",
"An example is shown in Figure REF .",
"An additional start state and an additional end state are added.",
"A backward edge from the end state to the start state is also added to allow a sequence of phonemes with indefinite length.",
"The prior distribution $p(y_1)$ entering the phoneme is also added from the start state to each of the phoneme FSTs.",
"Note that when entering one of the phoneme FSTs, the edge consumes a phoneme state and produces a phoneme, and the phoneme FSTs only consume phoneme states and do not produce any output.",
"Silences are modeled with five states rather than three, because they are allowed to be longer than other phonemes.",
"This finishes the construction of the FST $H$ that converts phoneme states to phonemes.",
"Figure: An FST HH that converts phoneme states to phonemes.The emission probabilities $p(x_t | y_t)$ in HMM are typically modeled by Gaussian mixture models (GMM).",
"Specifically, $p(x_t | y_t) = \\sum _{i=1}^C \\pi _{i, y_t} p( x_t | \\mu _{i, y_t}, \\sigma ^2_{i, y_t})$ where there are $C$ components for each state in $\\lbrace 1, \\dots , S\\rbrace $ , each component $p(x_t | \\mu _{i, y}, \\sigma ^2_{i, y})$ is a Gaussian distribution with mean $\\mu _{i, y}$ and variance $\\sigma ^2_{i, y}$ , and $\\pi _{i, y}$ is the probability of selecting the $i$ -th component.",
"For a sequence of $T$ frames, the emission probability for each frame can be independently evaluated.",
"Since each state has $C$ Gaussian components, there are $S \\times C$ evaluations for each frame.",
"We can build an FST that has $S \\times C$ edges for every frame.",
"Each edge has a weight computed from its Gaussian component and has a phone state as the output symbol.",
"An example is shown in Figure REF .",
"Note that the FST is constructed by following the generative story of Gaussian mixture models, evaluating a single Gaussian component (selected based on $\\pi $ ) for every frame rather than evaluating the weighted average of all Gaussian components.",
"Figure: An FST UU for 4 frames with GMM probabilities.For example, if each phoneme has 3 states, and eachstate has 64 Gaussian components, thens∈{𝚊𝚊1,𝚊𝚊2,𝚊𝚊3,⋯,𝚣𝚑1,𝚣𝚑2,𝚣𝚑3}s \\in \\lbrace \\texttt {aa1}, \\texttt {aa2}, \\texttt {aa3}, \\dots , \\texttt {zh1}, \\texttt {zh2}, \\texttt {zh3} \\rbrace ,i∈{1,⋯,C=64}i \\in \\lbrace 1, \\dots , C = 64\\rbrace .If there are LL phonemes (i.e., S=3LS = 3L), then there are 3L×643L \\times 64 (i.e., S×C)S \\times C) edges between each pair of vertices.By $\\sigma $ -composing $U$ and $H$ , we construct an FST where each path in the FST is a sample drawn from the generative story and the weight of each path is the corresponding log probability.",
"To better see the connection, we can traverse the FST $U$ and $H$ synchronously as the way we compute $U \\circ _\\sigma H$ .",
"First when we traverse an edge in $U$ , we select one phoneme state and one Gaussian component, collecting the log probability and producing a phoneme state as output.",
"Then the FST $H$ receives the phoneme state and waits for the next phoneme state, collecting transition probabilities and producing a phoneme when necessary.",
"This completes the construction of $U$ and $H$ .",
"The pronunciation dictionary, or lexicon, is a mapping from a word to its pronunciation.",
"To construct an FST representing the lexicon, we create, for each entry in the lexicon, a path consuming a sequence of phonemes and producing a word.",
"Similar to the FST $H$ , we have a start vertex and an end vertex that joins the pronunciation paths, and we also have a backward edge that allows indefinite amount of words to be produced.",
"An example is shown in Figure REF .",
"Note that the weights on the edges can either be 0 or the probability of a chosen pronunciation.",
"The FST also allows a word to have multiple pronunciations.",
"Silences are considered as words and are included in the FST.",
"The size of the FST can be significantly reduced if we maintain a prefix tree of the phonemes instead of having parallel pronunciations for every word.",
"Figure: An FST LL representing a pronunciation dictionary.Language models assign probabilities to word sequences.",
"For any $k \\in \\mathbb {N}$ , a word sequence $w_{1:k} = w_1, \\dots , w_k$ of length $k$ has probability $p(w_{1:k}) = \\prod _{i=1}^k p(w_i | w_{1:i-1}).$ As the sequence gets longer, i.e., as $i$ gets larger, $p(w_i | w_{1:i-1})$ depends on more words, which makes estimating the probabilities difficult.",
"A simple approach to remedy this problem is to introduce the Markov assumption $p(w_i | w_{1:i-1}) = p(w_i | w_{i-n+1:i-1}).$ A language model that gives a probability for the current word based on the previous $n-1$ words is commonly known as an $n$ -gram language model.",
"To construct an FST representing a language model, we create vertices that corresponds to history words $w_{i-n+1:i-1}$ .",
"For each vertex, the outgoing edges produce the next word $w_i$ with the weight $\\log p(w_i | w_{i-n+1:i-1})$ .",
"A path in this FST consumes a sequence of words and has the probability of the word sequence assigned by the language model.",
"There is a start vertex representing the state of having no history words.",
"We assume the utterance starts and ends with silences, so there is only one edge going out from the start state expecting a silence.",
"An example is shown in Figure REF .",
"Back-off language models can also be approximated with FSTs [3].",
"Figure: An FST GG representing a bigram language modelfor the vocabulary {sil, A, cat, runs}.Some edges are dashed to avoid clutter.The output symbols are the same as the input symbols for every edge,hence ignored.",
"The vertex in red is the silence state.In this chapter, we have reviewed the definition of finite-state transducers (FST) and constructed a basic speech recognizer by $\\sigma $ -composing an HMM emission FST $U$ , an HMM transition FST $H$ , a lexicon $L$ , and a language model $G$ .",
"Since $H \\circ _\\sigma L \\circ _\\sigma G$ is shared across all utterances, it is typically $\\sigma $ -composed and saved.",
"Each individual FST and the composed FST can also be determinized and minimized for efficiency, which we do not cover.",
"Interested readers should refer to [98] for further details.",
"The problem of prediction in general can be considered as a search problem.",
"Given an input $x$ , we first construct a set of possible outputs $\\mathcal {Y}(x)$ , referred to as the search space.",
"For every output hypothesis $y$ in the search space $\\mathcal {Y}(x)$ , we measure how well the output matches the input, assigning a weight to each pair $(x, y)$ .",
"Prediction can be considered as finding the hypothesis $\\hat{y}$ such that the weight of $(x, \\hat{y})$ is larger than any other pairs.",
"Extending the above paradigm, the problem of sequence prediction, such as speech recognition as we have seen in Chapter REF , can be considered as a search problem.",
"Given an input sequence, the search space in this case is a set of sequences of connected segments.",
"The number of such sequences is exponential in the number of possible segments.",
"Fortunately, we do not need to store exponentially many such sequences, and can represent the search space compactly as an finite-state transducer (FST).",
"Consider an FST with vertices corresponding to time points, and edges corresponding to segments.",
"A path in the FST corresponds to a sequence of connected segments, and the set of all paths defined by the FST is the search space.",
"The FST is weighted, and the weights assigned to the edges are based on how well the segments match the input.",
"Prediction can be considered as finding the maximum-weight path, because the maximum-weight path, by definition, is a path that best matches the input.",
"In this chapter, we formally define segmental models, including search spaces constructed from sequences of input vectors, weight functions that measure how well a segment matches the acoustic signals, and loss functions used for training segmental models.",
"Let $\\mathcal {X}$ be the input space, the set of all sequences of real-valued vectors, e.g., log mel filter bank features or mel frequency cepstral coefficients (MFCCs).",
"Specifically, for a sequence of $T$ vectors $x = (x_1, \\dots , x_T) \\in \\mathcal {X}$ , each $x_t \\in \\mathbb {R}^d$ is a $d$ -dimensional vector, also referred to as a frame, for $t \\in \\lbrace 1, \\dots , T\\rbrace $ .",
"Let $\\mathcal {Y}$ be the output space, the set of all label sequences, where each label in a label sequence comes from a label set $L$ , e.g., a phoneme set in the case of phoneme recognition.",
"Given any $T$ frames, a segmentation of length $K$ is a sequence of time points $((1 = s_1, t_1), \\dots , (s_K, t_K = T))$ , where $s_k \\le t_k$ and $t_k + 1 = s_{k+1}$ for $k \\in \\lbrace 2, \\dots , K\\rbrace $ .",
"A segment (typically denoted $e$ in later sections) is a tuple $(\\ell , s, t)$ where $\\ell \\in L$ is its label, $s$ is the start time, and $t$ is the end time.",
"A segmental model is a tuple $(\\Theta , w)$ where $\\Theta $ is a set of parameters, and $w: \\mathcal {X} \\times E \\rightarrow \\mathbb {R}$ is a weight function parameterized by $\\Theta $ and $E$ is the set of all segment tuples $(\\ell , s, t)$ .",
"A sequence of segments forms a path.",
"Specifically, a path of length $K$ is a sequence of segments $(e_1, \\dots , e_K)$ , where $e_k \\in E$ for $k \\in \\lbrace 1, \\dots , K\\rbrace $ .",
"Let $\\mathcal {P}$ be the set of all paths.",
"For any path $p$ , we overload $w$ such that $w(x, p) = \\sum _{e \\in p} w(x, e)$ .",
"We will also abbreviate $w(x, e)$ and $w(x, p)$ as $w(e)$ and $w(p)$ respectively when the context is clear.",
"The concrete form of the weight function will be defined in later sections.",
"Given an input $x \\in \\mathcal {X}$ , segmental models aim to solve sequence prediction by reducing it to finding the maximum-weight path $ \\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(x, p).$ The set of paths $\\mathcal {P}$ , also referred to as the search space, can be compactly represented as an FST.",
"Once we have the search space FST, we can simply invoke Algorithm REF to find the maximum-weight path.",
"We will describe how the search space FST is constructed in the next section.",
"A segmental model can be trained by finding a set of parameters that minimizes a loss function.",
"We emphasize that the model definition is not tied to any loss function, allowing us to study the behavior of segmental models under different loss functions.",
"We define various loss functions for training segmental models in Section REF , and discuss how the properties of the losses affect the training requirement.",
"To represent the set of paths $\\mathcal {P}$ as an FST, we place a vertex at every time point and connect vertices based on the set of segments.",
"Suppose we have $T$ frames.",
"The set of segments $E$ is an exhaustive enumeration of tuples $(\\ell , s, t)$ for all $\\ell \\in L$ and $1 \\le s \\le t \\le T$ .",
"We create a set of vertices $V = \\lbrace v_0, v_1, \\dots , v_T\\rbrace $ and a time function $\\tau : V \\rightarrow \\mathbb {N}$ such that $\\tau (v_t) = t$ for $t \\in \\lbrace 0, 1, \\dots , T\\rbrace $ .",
"For every segment $(\\ell , s, t) \\in E$ , we create an edge $e$ such that $i(e) = o(e) = \\ell $ , $\\mathrm {tail}(e) = v_{s-1}$ , and $\\mathrm {head}(e) = v_t$ .",
"In other words, for any $e \\in E$ , the corresponding segment $(\\ell , s, t) = (o(e), \\tau (\\mathrm {tail}(e)), \\tau (\\mathrm {head}(e)))$ .",
"As a result, we will use $w(e)$ and $w((\\ell , s, t))$ interchangeably.",
"We set $\\Sigma = \\Lambda = L$ , $I = \\lbrace v_0\\rbrace $ , and $F = \\lbrace v_T\\rbrace $ to complete the construction of the FST given any $T$ frames.",
"The number of segments in the graph is $O(|L|T^2)$ .",
"To reduce the size of the search space, a maximum duration constraint is typically imposed while creating the search space.",
"Specifically, we only create segments $(\\ell , s, t)$ with $t - s + 1 \\le D$ , for some maximum duration $D$ .",
"Adding such constraint makes the number of segments $O(|L|TD)$ .",
"An example of a search space is shown in Figure REF .",
"Figure: An example search space for a five-frame input sequencewith a label set LL of size three and maximum segment duration DD of two frames,i.e., T=5T = 5, |L|=3|L| = 3, and D=3D = 3.The three edges between any two nodes are associated with the three labels.Here we detail two types of weight functions based on prior work by ourselves and others [49], [130], [1], [85].",
"We will compare the two weight functions and in Chapter REF .",
"The term feature function is often used in the literature to denote the function $\\phi : \\mathcal {X} \\times E \\rightarrow \\mathbb {R}^m$ for some $m$ , when the weight function $w(x, e)$ is of the form $\\theta ^\\top \\phi (x, e)$ for some parameter vector $\\theta \\in \\mathbb {R}^m$ .",
"In general, the weight function need not be a dot product, but can be any (sub-)differentiable real-valued function.",
"A weight function is typically a composition of two steps: first the input vectors are converted into an intermediate representation, and second the intermediate representation is converted into segment weights.",
"The function in the first step that converts input vectors to an intermediate representation are called an encoder.",
"Specifically, an encoder is a function that takes in $T$ frames $x_1, \\dots , x_T$ and outputs $\\tilde{T}$ feature vectors $h_1, \\dots , h_{\\tilde{T}}$ .",
"The number of output vectors $\\tilde{T}$ can be different from the number of input vectors $T$ depending on the encoder architecture.",
"We defer the actual implementation of encoders in the experimental sections.",
"Here we define weight functions based on $h_1, \\dots , h_{\\tilde{T}}$ .",
"We use $\\Theta _\\text{enc}$ to denote the parameters for the encoder, and let $\\Theta _\\text{dec}$ be the remaining parameters in the weight function.",
"Note that $\\Theta = \\Theta _\\text{enc} \\cup \\Theta _\\text{dec}$ .",
"The following weight function, termed FC weight, is based on a frame classifier and is similar to weight functions used in a variety of prior work [49], [130], [1].",
"The frame classifier takes in the output $h_1, \\dots , h_{\\tilde{T}}$ from the encoder, and produces a sequence of log probability vectors over the labels $z_i = \\text{logsoftmax}(W h_i + b)$ where $z_i \\in \\mathbb {R}^{|L|}$ and $W$ and $b$ are the parameters, for $i \\in \\lbrace 1, \\dots \\tilde{T}\\rbrace $ .",
"Based on these posterior vectors, we define several functions that summarize the posteriors over a segment: The average of transformed log probabilities $w_{\\text{avg}}((\\ell , s, t)) = \\frac{1}{t - s + 1} \\sum _{i=s}^{t} (u_{i})_{\\ell },$ where $u_i = W_\\text{avg} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ .",
"A sample of transformed log probabilities $w_{\\text{spl-}j}((\\ell , s, t)) = (u_{j})_{\\ell }$ at time $j \\in \\lbrace (t - s)/6, (t - s)/2, 5(t - s)/6\\rbrace $ , where $u_i = W_\\text{spl} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ .",
"The average of transformed log probabilities around the left boundary (start) of the segment $w_{\\text{left}-k}((\\ell , s, t)) = (u_{i-k})_{\\ell }$ and around the right boundary (end) $w_{\\text{right}-k}((\\ell , s, t)) = (u^{\\prime }_{i+k})_{\\ell }$ where $u_i = W_\\text{left} z_i$ and $u^{\\prime }_i = W_\\text{right} z_i$ for $i \\in \\lbrace 1, \\dots , \\tilde{T}\\rbrace $ , and $k = 1, 2, 3$ .",
"The label-dependent duration weight $w_{\\text{dur}}((\\ell , s, t)) = d_{\\ell , t - s}.$ A label-dependent bias $w_{\\text{bias}}((\\ell , s, t)) = b^{\\prime }_{\\ell }.$ The final FC weight function is the sum of all the above weight functions.",
"When the FC weight function is used, $\\Theta _\\text{dec}$ is $\\lbrace W_\\text{avg}, W_\\text{spl},W_\\text{left}, W_\\text{right}, d, b^{\\prime }\\rbrace $ .",
"Note that $\\lbrace W, b\\rbrace $ are considered parameters of the encoders.",
"The MLP weight function based on a multi-layer perceptron (MLP) is inspired by [130], [74], [85].",
"Two hidden layers $z^{(1)}_{\\ell , s, t} & = \\text{ReLU}(W_1 [ h_s; h_t; c_\\ell ; d_{\\lfloor \\log _{1.6}(t-s)\\rfloor }] + b_1) \\\\z^{(2)}_{\\ell , s, t} & = \\tanh (W_2 z^{(1)}_{\\ell , s, t} + b_2)$ are computed directly from the outputs $h_1, \\dots , h_{\\tilde{T}}$ of the encoder before computing the final weight, where $c_\\ell $ is a label embedding vector for the label $\\ell $ , $d_k$ is a duration embedding vector for the duration $k$ in log scaleWe use 5 different duration embedding vectors for our experiments.",
"The base 1.6 is chosen such that $k \\in \\lbrace 0, \\dots , 4\\rbrace $ ., with $k = \\lfloor \\log _{1.6}(t-s+1)\\rfloor $ , and $\\text{ReLU}(x) = \\max (x, 0)$ .",
"The final weight for the segment is defined as $w((\\ell , s, t)) & = \\theta ^\\top z^{(2)}_{\\ell , s, t}.$ When the MLP weight function is used, $\\Theta _\\text{dec}$ is $\\lbrace W_1, b_1, W_2, b_2, \\theta \\rbrace $ .",
"Although the MLP weight function is conceptually simple, it is more expensive to compute than the FC weight function.",
"In [74], [85], an LSTM is created for each segment consuming the outputs of the encoder, followed by an MLP taking the output vectors of the per segment LSTM at the segment boundary.",
"In order to reduce the computation, we discard the per segment LSTM, and use the vectors produced by the encoder at the segment boundary as input to the MLP.",
"Recall that a path $p = ((\\ell _1, s_1, t_1), \\dots , (\\ell _K, s_K, t_K))$ consists of a label sequence $y = (\\ell _1, \\dots , \\ell _K)$ and a segmentation $z = ((s_1, t_1), \\dots , (s_K, t_K))$ .",
"In the following, we will use $(y, z)$ and $p$ interchangeably.",
"We will also denote the space of all segmentations $\\mathcal {Z}$ .",
"Training a segmental model aims to find a set of parameters $\\Theta $ that minimizes the expected task loss, for example, the expected edit distance $\\mathbb {E}_{(x, y) \\sim \\mathcal {D}}[\\mathrm {edit}(h_\\Theta (x), y)]$ where $h$ is the inference algorithm (Algorithm REF ) parameterized with $\\Theta $ , $\\mathrm {edit}$ computes the edit distance between two sequences, and the expectation is taken over samples $(x, y) \\in \\mathcal {X} \\times \\mathcal {Y}$ drawn from a distribution $\\mathcal {D}$ .",
"The expectation can be decomposed into two steps $\\mathbb {E}_{x \\sim \\mathcal {D}(x)} \\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathrm {edit}(h_\\Theta (x), y)],$ first sampling $x$ and then sampling $y$ .",
"To obtain a good discriminator $h$ , it suffices to optimize the inner expectation $\\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathrm {edit}(h_\\Theta (x), y)],$ for any $x$ drawn from $\\mathcal {D}(x)$ .",
"However, the edit distance, due to its discrete nature, is difficult to optimize, so instead we minimize the expected loss $ \\mathbb {E}_{y \\sim \\mathcal {D}(y|x)}[\\mathcal {L}(\\Theta ; x, y)],$ where $\\mathcal {L}$ is a surrogate loss function.",
"If segmentations are considered in the loss function, then we can minimize $ \\mathbb {E}_{(y, z) \\sim \\mathcal {D}^{\\prime }(y, z | x)}[\\mathcal {L}(\\Theta ; x, y, z)],$ where $\\mathcal {D}^{\\prime }$ is a conditional distribution over $\\mathcal {X} \\times \\mathcal {Y} \\times \\mathcal {Z}$ .",
"Since the distribution $\\mathcal {D}$ is unknown, we use a tranining set $S = \\lbrace (x_1, y_1), \\dots , (x_n, y_n)\\rbrace $ of size $n$ to approximate the expectation and instead minimize $\\frac{1}{n} \\sum _{i=1}^n \\mathcal {L}(\\Theta ; x_i, y_i).$ If segmentations are considered, we use a tranining set $S = \\lbrace (x_1, y_1, z_1), \\dots , (x_n, y_n, z_n)\\rbrace $ of size $n$ to approximate $\\mathcal {D}^{\\prime }$ and minimize $\\frac{1}{n} \\sum _{i=1}^n \\mathcal {L}(\\Theta ; x_i, y_i, z_i).$ The connection between the surrogate loss $\\mathcal {L}$ and the edit distance depends on the choice of loss.",
"We will optimize the loss functions with first-order methods, such as stochastic gradient descent.",
"When listing the function definitions along with reasons for using them, we will also list the (sub-)gradients with respect to the weight $w(e)$ for some edge $e$ .",
"We assume the weight function $w$ is (sub-)differentiable and the (sub-)gradients with respect to the parameters can be obtained with back-propagation.",
"Given an utterance $x$ and a ground truth path $p = (y, z)$ , the hinge loss is defined as $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)= \\max _{p^{\\prime } \\in \\mathcal {P}} \\left[ \\mathrm {cost}(p^{\\prime }, p) - w(p) + w(p^{\\prime }) \\right]$ where $\\mathrm {cost}$ is a user-defined cost function.",
"The connection between the hinge loss and the task loss is through the cost function.",
"Suppose $\\hat{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(p)$ is the maximum-weight path found by Algorithm REF .",
"The cost of the inferred path $\\hat{p}$ against the ground truth $p$ can be upper-bounded by the hinge loss: $\\mathrm {cost}(\\hat{p}, p) \\le \\mathrm {cost}(\\hat{p}, p) - w(p) + w(\\hat{p})\\le \\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p).$ When the cost function is the edit distance, minimizing the hinge loss is minimizing an upper bound on the edit distance of the predicted sequence.",
"The hinge loss itself is difficult to optimize when the cost function is the edit distance.",
"In practice, the cost function is assumed to be decomposable $\\mathrm {cost}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}(e^{\\prime }, p)$ to allow efficient dynamic programming.",
"When the cost is decomposable, the hinge loss can be written as $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)& = \\max _{p^{\\prime } \\in \\mathcal {P}} \\left[ \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}(e^{\\prime }, p)- \\sum _{e \\in p} w(e) + \\sum _{e^{\\prime } \\in p^{\\prime }} w(e^{\\prime }) \\right] \\\\& = \\max _{p^{\\prime } \\in \\mathcal {P}} \\sum _{e^{\\prime } \\in p^{\\prime }} \\left[ \\mathrm {cost}(e^{\\prime }, p)+ w(e^{\\prime }) \\right] - \\sum _{e \\in p} w(e),$ and the $\\max $ operator in the first term can be solved with Algorithm REF by adding the costs to the weights for all segments.",
"A subgradient of the hinge loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)}{\\partial w(e)}= -{1}_{e \\in p} + {1}_{e \\in \\tilde{p}}$ where $\\tilde{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p^{\\prime } \\in \\mathcal {P}} [\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })],$ which is the path that maximizes the first term in the hinge loss, and can be obtained with Algorithm REF with the cost added.",
"Linear models are called support vector machines (SVM) when trained with the hinge loss, and are referred to as structured SVMs when used for solving structured prediction problems, e.g., sequence prediction in our case.",
"A hinge loss with cost zero is also called perceptron loss [77].",
"Segmental models trained with the hinge loss have been studied in [146], [129], [130].",
"Segmental models can be treated as probabilistic models by defining probability distributions on the set of all paths $\\mathcal {P}$ .",
"Specifically, the probability of a path $p = (y, z)$ is defined as $P(y, z | x) = P(p | x) = \\frac{1}{Z(x)} \\exp (w(x, p))$ where $Z(x) = \\sum _{p^{\\prime } \\in \\mathcal {P}} \\exp (w(x, p^{\\prime }))$ is the partition function.",
"Given an input $x$ and a ground truth path $p$ , the log loss is defined as $\\mathcal {L}_{\\text{log}}(\\Theta ; x, p) = -\\log P(p | x).$ Minimizing the log loss is equivalent to maximizing the conditional likelihood.",
"In addition, the conditional likelihood can be written as $P(y, z | x) & = \\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) = (y, z)}] \\\\& = 1 - \\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}].$ Therefore, maximizing the conditional likelihood is equivalent to minimizing the expected zero-one loss $\\mathbb {E}_{(y^{\\prime }, z^{\\prime }) \\sim P(y^{\\prime }, z^{\\prime } | x)}[{1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}],$ where $P(y, z | x)$ is used to approximate $\\mathcal {D}^{\\prime }(y, z | x)$ and the zero-one loss ${1}_{(y^{\\prime }, z^{\\prime }) \\ne (y, z)}$ is used.",
"The use of the log loss is justified because the expectation above can be seen as an approximation of (REF ).",
"Segmental models trained with log loss have been referred to as semi-Markov CRFs [118].",
"Minimizing log loss is equivalent to maximizing mutual information, and is commonly referred to as the MMI criterion [10].",
"Since the weight for the ground truth path $p$ can be efficiently computed, we are left with the problem of computing the partition function $Z(x)$ .",
"The partition function can also be computed efficiently with the following dynamic programming algorithm.",
"Let $\\mathcal {P}(u, v)$ be the set of paths that start at vertex $u$ and end at vertex $v$ .",
"For any vertex $v$ , define the forward marginal as $\\alpha (v) = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp (w(p^{\\prime })).$ By expanding the edges ending at $v$ , we have $\\alpha (v) &= \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp \\left(\\sum _{e \\in p^{\\prime }} w(e)\\right) \\\\&= \\log \\sum _{e \\in \\text{in}(v)} \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, \\mathrm {tail}(e))}\\exp \\left(w(e) + \\sum _{e^{\\prime } \\in p^{\\prime }} w(e^{\\prime })\\right) \\\\&= \\log \\sum _{e \\in \\text{in}(v)} \\exp \\left(w(e) + \\alpha (\\mathrm {tail}(e))\\right)$ Similarly, the backward marginal at $v$ is defined as $\\beta (v) = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v, v_T)} \\exp (w(p^{\\prime })),$ and has similar recursive structure.",
"The complete algorithm is shown in Algorithm REF .",
"Once all entries in $\\alpha $ and $\\beta $ are computed, the log partition function is $\\log Z(x) = \\alpha (v_T) = \\beta (v_0).$ Note that we store all of the entries in log space to maintain numerical stability.",
"The gradient of the log loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{log}}(\\Theta ; x, p)}{\\partial w(e)}& = -{1}_{e \\in p} + \\frac{1}{Z(x)}\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime })) \\\\& = -{1}_{e \\in p} + \\exp \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e)+ \\beta (\\mathrm {head}(e)) - \\log Z(x) \\Big ],$ which can also be efficiently computed once the marginals are computed.",
"Note that the form of Algorithm REF is very similar to that of Algorithm REF .",
"Recall that to get the standard shortest-path algorithm (Algorithm REF ), we simply use the tropical semiring when running Algorithm REF .",
"If the tropical semiring is replaced by the log semiring, we arrive at Algorithm REF , demonstrating the generality of FST algorithms and semirings.",
"Computing forward and backward marginals $\\alpha (v_0) = 0$ $\\beta (v_T) = 0$ $\\text{logadd}(a, b) = \\log (\\exp (a) + \\exp (b))$ $v = v_0, v_1, \\dots , v_T$ $\\alpha (v) = \\text{logadd}_{e \\in \\text{in}(v)} \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e) \\Big ]$ $v = v_T, v_{T-1}, \\dots , v_0$ $\\beta (v) = \\text{logadd}_{e \\in \\text{out}(v)} \\Big [ \\beta (\\mathrm {head}(e)) + w(e) \\Big ]$ Log loss can be modified to include a cost function.",
"Speficically, we define $P^+_{p}(p^{\\prime } | x) = \\frac{1}{Z(x, p)} \\exp (w(x, p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p))$ where $p$ is the ground-truth path and $Z(x, p) = \\sum _{p^{\\prime \\prime } \\in \\mathcal {P}} \\exp (w(x, p^{\\prime \\prime }) + \\mathrm {cost}(p^{\\prime \\prime }, p)).$ Unlike $P$ , the distribution $P^+$ considers both the weights and the costs.",
"The modified log loss, also known as the boosted MMI criterion [109], is defined as $\\mathcal {L}_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P^+_{p}(p | x).$ Given an input $x$ and a label sequence $y$ , the marginal log loss is defined as $\\mathcal {L}_{\\text{mll}}(\\Theta ; x, y) = -\\log P(y | x) = -\\log \\sum _{z \\in \\mathcal {Z}} P(y, z | x)$ where the segmentation is marginalized compared to log loss; hence the name.",
"Following the same argument as for the log loss, the marginal distribution can be written as $P(y | x) = 1 - \\mathbb {E}_{y^{\\prime } \\sim P(y^{\\prime } | x)}[{1}_{y \\ne y^{\\prime }}],$ and maximizing the marginal distribution is equivalent to minimizing the expected zero-one loss $\\mathbb {E}_{y^{\\prime } \\sim P(y^{\\prime } | x)}[{1}_{y \\ne y^{\\prime }}],$ where $P(y | x)$ is used to approximate $\\mathcal {D}(y | x)$ .",
"Note that the zero-one loss ${1}_{y \\ne y^{\\prime }}$ only depends on the label sequence.",
"While the log loss has a connection to (REF ), the marginal log loss justifies its use by directly approximating (REF ) with the above expected zero-one loss.",
"Note that both the hinge loss and the log loss depend on the ground-truth path, or more specifically, the ground-truth segmentation.",
"The marginal log loss marginalizes over the segmentations, so it only depends on the ground-truth label sequence and not the segmentation.",
"The lack of reliance on the ground-truth segmentation makes the marginal log loss attractive for tasks such as speech recognition, because collecting ground-truth segmentations for phonemes or words is time-consuming and/or expensive.",
"In addition, the boundaries of phonemes and words tend to be ambiguous, so it can be preferable to leave the decision to the model.",
"Segmental models trained with the marginal log loss have been referred to as segmental CRFs [149].",
"To compute the marginal log loss, we can rewrite it as $\\mathcal {L}_{\\text{mll}}(\\Theta ; x, y) & = -\\log \\sum _{z \\in \\mathcal {Z}} P(y, z | x) \\\\& = -\\log \\sum _{z \\in \\mathcal {Z}} \\exp (w(x, (y, z))) + \\log Z(x) \\\\& = -\\underbrace{\\log \\sum _{p^{\\prime }: \\Gamma (p^{\\prime }) = y} \\exp (w(x, p^{\\prime }))}_{\\log Z(x, y)} + \\log Z(x)$ where $\\Gamma $ extracts the label sequence from a path, i.e., for $p^{\\prime } = (y^{\\prime }, z^{\\prime })$ , $\\Gamma (p^{\\prime }) = y^{\\prime }$ .",
"Since the partition function can be efficiently computed from Algorithm REF , we only need to compute $\\log Z(x, y)$ .",
"Since the term $\\log Z(x, y)$ is identical to $\\log Z(x)$ except that it involves a constrained search space considering all paths with the same label sequence $y$ , the strategy is to construct the constrained search space with an FST and run Algorithm REF on the FST.",
"Let $F$ be a chain FST that represents $y$ , with edges $\\lbrace e_1, \\dots , e_{|y|}\\rbrace $ , where $i(e_k) = o(e_k) = y_k$ for all $k \\in \\lbrace 1, \\dots , |y|\\rbrace $ .",
"Let $G$ be the search space consisting of all paths in $\\mathcal {P}$ .",
"The term $\\log Z(x, y)$ can be efficiently computed by running Algorithm REF on the $\\sigma $ -composition of $G$ and $F$ , i.e., $G \\circ _\\sigma F$ .",
"Note that after $\\sigma $ -composing $G$ and $F$ , we only allow paths in $G$ that can produce output sequences that $F$ accepts.",
"Since $F$ only accepts $y$ , the paths in $G \\circ _\\sigma F$ are all the paths in $G$ that produce $y$ as desired.",
"Let the forward and backward marginals computed on $G \\circ _\\sigma F$ be $\\alpha ^{\\prime }$ and $\\beta ^{\\prime }$ .",
"We have $\\log Z(x, y) = \\alpha ^{\\prime }(v_T) = \\beta ^{\\prime }(v_0)$ .",
"The gradient of the marginal log loss with respect to $w(e)$ is $\\frac{\\partial \\mathcal {L}_{\\text{mll}}(\\Theta ; x, y)}{\\partial w(e)}& = -\\frac{1}{Z(x, y)} \\sum _{\\begin{array}{c}p^{\\prime } \\ni e\\\\ \\Gamma (p^{\\prime }) = y\\end{array}} \\exp (w(p^{\\prime }))+ \\frac{1}{Z(x)}\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime })) \\\\& = - \\exp \\Big [ \\alpha ^{\\prime }(\\mathrm {tail}(e)) + w(e) + \\beta ^{\\prime }(\\mathrm {head}(e)) - \\log Z(x, y) \\Big ] \\\\& \\qquad {} + \\exp \\Big [ \\alpha (\\mathrm {tail}(e)) + w(e) + \\beta (\\mathrm {head}(e)) - \\log Z(x) \\Big ].$ and can be efficiently computed once all of the marginals are computed.",
"The empirical Bayes risk (EBR) [125] is defined as $\\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p) = \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)].$ The intuition behind EBR is that if $P$ is a good approximation for $\\mathcal {D}^{\\prime }$ , then EBR is a good approximation for $\\mathbb {E}_{p^{\\prime } \\sim \\mathcal {D}^{\\prime }(p^{\\prime } | x)}[\\mathrm {cost}(p^{\\prime }, p)].$ The goal is to find $P$ that achieves a low expected cost.",
"Empirical Bayes risk is also referred to as minimum phone error (MPE) when the cost is based on phone errors, and is referred to as minimum word error (MWE) when the cost is based on word errors respectively [108].",
"A boosted version of EBR can be obtained by replacing $P$ with $P^+$ [89].",
"Since $Z(x)$ can be computed through the forward and backward marginals $\\alpha $ and $\\beta $ , we are left with $\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)$ .",
"Similar to the forward and backward marginals, we have $\\alpha ^{\\prime \\prime }(v) & = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, v)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{in}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p))\\sum _{p^{\\prime } \\in \\mathcal {P}(v_0, \\mathrm {tail}(e))} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{in}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p) + \\alpha ^{\\prime \\prime }(\\mathrm {tail}(e))) \\\\\\beta ^{\\prime \\prime }(v) & = \\log \\sum _{p^{\\prime } \\in \\mathcal {P}(v, v_T)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{out}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p))\\sum _{p^{\\prime } \\in \\mathcal {P}(\\mathrm {head}(e), v_T)} \\exp (w(p^{\\prime }) + \\log \\mathrm {cost}(p^{\\prime }, p)) \\\\& = \\log \\sum _{e^{\\prime } \\in \\text{out}(v)} \\exp (w(e^{\\prime }) + \\log \\mathrm {cost}(e^{\\prime }, p) + \\beta ^{\\prime \\prime }(\\mathrm {head}(e)))$ where $\\mathrm {cost}$ is assumed to be non-negative, and the marginals are stored in log space to maintain numerical stability.",
"Given the cost-augmented marginals, we have $\\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)] = \\exp (\\alpha ^{\\prime \\prime }(v_T) - \\log Z(x))= \\exp (\\beta ^{\\prime \\prime }(v_0) - \\log Z(x)).$ The gradient of EBR with respect to $w(e)$ of some edge $e$ is $\\frac{\\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p)}{w(e)}& = \\frac{\\sum _{p^{\\prime } \\ni e} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)}{Z(x)}- \\frac{\\left[\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }))\\mathrm {cost}(p^{\\prime }, p)\\right]\\left[\\sum _{p^{\\prime } \\ni e}\\exp (w(p^{\\prime }))\\right]}{(Z(x))^2} \\\\& = \\exp (\\alpha ^{\\prime \\prime }(\\mathrm {tail}(e)) + w(e) + \\log \\mathrm {cost}(e, p) + \\beta ^{\\prime \\prime }(\\mathrm {head}(e)) - \\log Z(x)) \\\\& \\quad {} - \\mathcal {L}_{\\text{ebr}}(\\Theta ; x, p) [\\exp (\\alpha (\\mathrm {tail}(e)) + w(e) + \\beta (\\mathrm {head}(e)) - \\log Z(x))]$ It is also interesting to note that, in general, $\\frac{\\partial }{\\partial w(e)} \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[f(p^{\\prime })]&= \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ni e} f(p^{\\prime })]- \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ni e}]\\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[f(p^{\\prime })] \\\\&= \\text{Cov}(\\mathbb {1}_{p^{\\prime } \\ni e}, f(p^{\\prime })),$ for any function $f: \\mathcal {P} \\rightarrow \\mathbb {R}$ .",
"The ramp loss [24] is defined as $\\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p) = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - \\max _{p^{\\prime \\prime }}w(p^{\\prime \\prime })].$ It is easy to see that $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p)& = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - w(p)] \\\\& \\ge \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) - \\max _{p^{\\prime \\prime }}w(p^{\\prime \\prime })]= \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p).$ In addition, for $\\hat{p} = \\operatornamewithlimits{\\arg \\!\\max }_{p^{\\prime }} w(p^{\\prime })$ , $\\mathrm {cost}(\\hat{p}, p) &= \\mathrm {cost}(\\hat{p}, p) + w(\\hat{p}) - w(\\hat{p}) \\\\& \\le \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })] - w(\\hat{p}) \\\\& = \\max _{p^{\\prime }}[\\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime }) ] - \\max _{p^{\\prime }} w(p^{\\prime })= \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p).$ Therefore, we have $\\mathcal {L}_{\\text{hinge}}(\\Theta ; x, p) \\ge \\mathcal {L}_{\\text{ramp}}(\\Theta ; x, p)\\ge \\mathrm {cost}(\\hat{p}, p).$ In other words, ramp loss is also an upper bound on the cost of the predicted sequence, but is tighter than hinge loss.",
"However, optimizing the ramp loss is more complicated.",
"Ramp loss can be minimized with the concave-convex procedure (CCCP) [145].",
"See [33] for a detailed implementation of CCCP for solving ramp loss.",
"Log loss augmented with a cost function, or boosted MMI, can be seen as a soft version of hinge loss.",
"By the definition of boosted MMI, where $\\mathcal {L}_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\log \\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p)),$ we can see that the second term acts as a soft-max instead of a max function.",
"To see this, note that $\\log (e^{x_1} + e^{x_2} + \\dots + e^{x_n})& = \\log e^{x_{\\text{max}}} (e^{x_1 - x_{\\text{max}}} + \\dots + e^{x_n - x_{\\text{max}}}) \\\\& = x_{\\text{max}} + \\log (e^{x_1 - x_{\\text{max}}} + \\dots + e^{x_n - x_{\\text{max}}}),$ where $x_{\\text{max}} = \\max _i x_i$ .",
"The second term in (REF ) is small when the difference between $x_{\\text{max}}$ and other $x_i$ 's is large.",
"In fact, we can introduce an addition factor $\\epsilon $ to control this effect.",
"Specifically, $\\epsilon \\log (e^{x_1/\\epsilon } + e^{x_2/\\epsilon } + \\dots + e^{x_n\\epsilon })& = x_{\\text{max}} + \\epsilon \\log (e^{(x_1 - x_{\\text{max})/\\epsilon }} + \\dots + e^{(x_n - x_{\\text{max}})/\\epsilon }).$ When $\\epsilon \\rightarrow 0$ , the second term vanishes and (REF ) acts exactly like a max function.",
"We can add the additional $\\epsilon $ factor to the cost-augmented log loss and get $\\mathcal {L}^{\\epsilon }_{\\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\epsilon \\log \\sum _{p^{\\prime }} \\exp \\left(\\frac{1}{\\epsilon }(w(p^{\\prime }) + \\mathrm {cost}(p^{\\prime }, p))\\right).$ As $\\epsilon \\rightarrow 0$ , $\\mathcal {L}^{\\epsilon }_{\\text{bMMI}} \\rightarrow \\mathcal {L}_{\\text{hinge}}$ .",
"This connection was first presented in [52].",
"[32] independently proposed softmax-margin, the equivalence of cost-augmented log loss.",
"As we have noted when we introduced log loss, $P(p | x) = \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } = p}]= 1 - \\mathbb {E}_{p^{\\prime } \\sim P(p^{\\prime }|x)}[\\mathbb {1}_{p^{\\prime } \\ne p}].$ Therefore, minimizing log loss is equivalent to minimizing EBR with $\\mathrm {cost}(p^{\\prime }, p) = \\mathbb {1}_{p^{\\prime } \\ne p}$ .",
"Another connection between cost-augmented log loss and EBR is the following.",
"Let $\\mathcal {L}_{\\lambda , \\text{bMMI}}(\\Theta ; x, p) = -\\log P(p | x)= -w(p) + \\log \\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p)),$ where we introduce a scaling factor $\\lambda $ for the cost function in cost-augmented log loss.",
"If we take the derivative of cost-augmented log loss with respect to $\\lambda $ , we have $\\frac{\\partial }{\\partial \\lambda } \\mathcal {L}_{\\lambda , \\text{bMMI}}&= \\frac{\\sum _{p^{\\prime }} [\\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p)) \\mathrm {cost}(p^{\\prime }, p)]}{\\sum _{p^{\\prime }} \\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p))} \\\\&= \\mathbb {E}_{p^{\\prime } \\sim P^+_\\lambda (p^{\\prime }|x)}[\\mathrm {cost}(p^{\\prime }, p)],$ where $P^+_{\\lambda }(p^{\\prime }|x) = \\frac{\\exp (w(p^{\\prime }) + \\lambda \\mathrm {cost}(p^{\\prime }, p))}{\\sum _{p^{\\prime \\prime }} \\exp (w(p^{\\prime \\prime }) + \\lambda \\mathrm {cost}(p^{\\prime \\prime }, p))}$ Therefore, the derivative of cost-augmented log loss with respect to $\\lambda $ is cost-augmented EBR, or sometimes referred to as boosted MPE.",
"This connection was first presented in [90].",
"In this sectionThese results were published in [129], we describe experiments comparing segmental models trained with various loss functions and two cost functions.",
"Instead of using the entire search space, we follow [149], [150] and use a baseline recognizer to generate sparse search spaces, commonly known as lattices.",
"Segmental models are then applied on these lattices.",
"The weight function used in these experiments is a linear function $w(x, e) = \\theta ^\\top \\phi (x, e)$ where $\\theta $ is a parameter vector and $\\phi (x, e)$ is a feature vector for some edge $e$ .",
"The feature function $\\phi $ is task-dependent, and is described in detail later.",
"Hinge loss and ramp loss are sensitive to the scale of the cost, so the scale of the cost function is tuned.",
"Specifically, we introduce the parameter $\\mu $ in the hinge loss $\\mathcal {L}_{\\text{hinge}} = \\max _{p^{\\prime }} [\\mu \\cdot \\mathrm {cost}(p^{\\prime }, p) - w(p) + w(p^{\\prime })],$ and parameters $\\mu _1$ and $\\mu _2$ for the ramp loss $\\mathcal {L}_{\\text{ramp}} = \\max _{p^{\\prime }} [\\mu _1 \\cdot \\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })]- \\max _{p^{\\prime }}[\\mu _2 \\cdot \\mathrm {cost}(p^{\\prime }, p) + w(p^{\\prime })].$ Note the additional cost term in the above ramp loss.",
"This version of ramp loss is a generalization of the standard ramp loss [20], [33].",
"One cost function was proposed in [108] in the context of MPE/MWE training, and we refer to it as MPE cost.",
"The cost of an edge is the duration of the non-overlapping part of a matching ground-truth edge that gives the lowest error, where the error is one if the label is correct and 0.5 otherwise.",
"Formally, for any hypothesized edge $e^{\\prime }$ , we define $\\mathrm {cost}_{\\text{MPE}}(e^{\\prime }, p) =1 - \\max _{e \\in p} \\left[ \\mathbb {1}_{o(e) = o(e^{\\prime })}\\frac{|e \\cap e^{\\prime }|}{|e|}+ \\frac{1}{2} \\mathbb {1}_{o(e) \\ne o(e^{\\prime })}\\frac{|e \\cap e^{\\prime }|}{|e|} \\right],$ and $\\mathrm {cost}_{\\text{MPE}}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}_{\\text{MPE}}(e^{\\prime }, p),$ where $|e|$ denotes the length of segment $e$ .",
"The MPE cost only penalizes false negatives and does not account for false positives.",
"Therefore, we propose an alternative that we refer to as the overlap cost: $\\mathrm {cost}_{\\text{overlap}}(e^{\\prime }, p) = 1 - \\mathbb {1}_{o(e) = o(\\tilde{e})}\\frac{|e \\cap \\tilde{e}|}{|e \\cup \\tilde{e}|},$ where $\\tilde{e} = \\operatornamewithlimits{\\arg \\!\\max }_{e \\in p} |e^{\\prime } \\cap e|$ .",
"This cost function finds the most overlapping edge in the ground truth and considers any part of the union of the two edges that is not overlapping to be in error.",
"The cost for the whole path is again $\\mathrm {cost}_{\\text{overlap}}(p^{\\prime }, p) = \\sum _{e^{\\prime } \\in p^{\\prime }} \\mathrm {cost}_{\\text{overlap}}(e^{\\prime }, p).$ We study the various losses and cost functions on two tasks.",
"One is a standard speech recognition task, namely TIMIT [31] phonetic recognition.",
"The second is a sign language recognition task from video, in particular recognition of fingerspelled letter sequences in American Sign Language (ASL).",
"Both are tasks on which there is prior work using semi-Markov CRFs [148], [71], and both are small enough (in terms of data set size and decoding search space) to run many empirical comparisons in a reasonable amount of time.",
"For the ASL task, we use the data and experimental setup of [71]: We obtain baseline lattices using their tandem HMM-based system, and we use the same set of segmental feature functions.",
"We use forced alignments of the ground-truth transcriptions for training.",
"We train all models from all-zero weights and optimize with Rprop [114] for 20 epochs.",
"We use $L_1$ and $L_2$ regularization, with parameters tuned over the grid $\\lbrace 0,10^{-6}, 10^{-5}, \\dots , 0.1, 1\\rbrace ^2$ .",
"For hinge and ramp loss, we use the standard forms without tuning the cost weights (i.e., $\\mu =1$ , $\\mu _1 = 0$ , and $\\mu _2 = 1$ ).",
"Table: Letter error rates (%) for a baseline tandem HMM and segmental models trained with various lossesand cost functions on the ASL data set.Table: Phone error rates (%) for a baseline HMM and segmental models trained with various lossesand cost functions on the TIMIT data set.For TIMIT, we use the standard 3696-utterance training set and 192-utterance core test set, plus a random 192 utterances from the full test set (excluding the core test set) as a development set.",
"We collapse the 61 phones in the phone set to 48 for training, and further collapse them to 39 for evaluation [79].",
"We use lattices generated by a baseline monophone HMM system with 39-dimensional MFCCs.",
"The resulting lattices have an average density (average number of hypothesized edges per ground truth edge) of 60.1.",
"The oracle phone error rate is 6.3% for the development set and 7.0% for the core test set.",
"We use oracle paths (paths with minimum phone error) from the lattices as ground truth for training.",
"We implement segmental models with various feature functions.",
"The base features are the acoustic and language model score from the baseline recognizer, and a bias (a feature that is always one).",
"We also include a set of features based on spectro-temporal receptive fields implemented as follows.",
"We begin with 40-dimensional log mel filter bank features.",
"For each segment, we divide it evenly into thirds in both time and frequency, resulting in nine patches for each segment.",
"For each patch, we have a $3 \\times 13$ receptive field of all ones, and convolve it with the patch.",
"The resulting $3 \\times 13 \\times 9$ numbers are lexicalized to form the final features for the segmental model.",
"Specifically, for any feature vector $v$ , we refer to $v \\otimes \\mathbf {1}_\\ell $ as the lexicalized feature vector, where $\\mathbf {1}_\\ell $ is a one-hot vector for the label $\\ell $ .",
"We optimize the loss functions using AdaGrad [27], using step size 0.1 for 10 epochs.",
"$L_1$ and $L_2$ regularization parameters are tuned over the grid $\\lbrace 0, 0.001, 0.1, 1\\rbrace \\times \\lbrace 0, 0.1, 1, 100\\rbrace $ .",
"The results for ASL recognition, averaged over four signers, are shown in Table REF .",
"The evaluation metric is the letter error rate, which is the percentage of letters that are substituted, inserted, or deleted.",
"The results for TIMIT are shown in Table REF .",
"We observe three consistent conclusions: Segmental models perform significantly better than the baseline HMM.",
"Across losses, overlap cost is better than MPE cost.",
"Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.",
"Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.",
"We suspect a warm start might be able to remedy this.",
"Segmental models perform significantly better than the baseline HMM.",
"Across losses, overlap cost is better than MPE cost.",
"Hinge loss with overlap cost is the best performer, but this is only by a small margin, and log loss is very competitive even without using an explicit cost function.",
"Non-convex losses (ramp and EBR) are difficult to optimize and therefore achieve inconsistent results.",
"We suspect a warm start might be able to remedy this.",
"For the ASL task, we tuned on a development set the cost weights for ramp loss over the grid $\\lbrace -100,-10,-1,-0.1,-0.01\\rbrace \\times \\lbrace 0.01,0.1,1,10,100\\rbrace $ , using overlap cost.",
"The test result of tuned ramp slightly improves over hinge loss, confirming that if ramp is tuned carefully, it is able to outperform hinge.",
"However, even though tuned ramp loss achieves very good results, considering the time spent tuning $\\mu _1$ and $\\mu _2$ , we still favor hinge and log loss.",
"We also conducted experiments to determine how the results are affected by different levels of noise in the feature functions, using simulated phone detector-based features.",
"Similarly to [150], we define a detection event as a (time, phone label) pair, and a feature function that is an indicator of whether a phone detection event occurs in the time span of the edge.",
"If we set a high weight for the phone event that occurs in an edge with the same phone label, then we can exactly recover the oracle path.",
"This allows us to conduct a series of simulated experiments with different amounts of noise added to the oracle phone events, or gold events.",
"For all experiments below, we use the same TIMIT setting except that we only use the acoustic and language model score with the simulated phone detector features, with no regularizer and one epoch of AdaGrad.",
"The ramp cost weights are set to $\\mu _1=0, \\mu _2=1$ .",
"The cost weight for hinge is tuned over $\\lbrace 0.01, 0.1, 1, 10, 100\\rbrace $ and results are only shown for the best-performing value.",
"The first set of experiments randomly perturbs the correct phone label of each event to an incorrect label with the corruption probability shown on the $x$ -axis; the event times are not perturbed.",
"The second set of experiments perturbs the time for each event but not the label.",
"We add Gaussian noise with mean set to the time at which the event occurs and with several standard deviations shown on the $x$ -axis.",
"For the third and fourth set of experiments, we randomly include an edge in the lattice as a false positive event, or randomly delete an event from the gold events.",
"The conclusions are consistent with our previous observation, namely, that hinge is the consistent winner but only by a very small margin, that log loss is very competitive, that non-convex losses are hard to optimize, and that overlap cost is better than MPE cost.",
"As a byproduct, we note that we could achieve 17.7% given a phone detector with any of the following characteristics: up to 50% phone error rate but perfect time information, up to 5-frame time perturbations (in standard deviation) but perfect labels, 1.8 false positives per gold edge, or 20% false negatives.",
"Figure: Lattice rescoring with various noise added to a perfectphone detector.From Left to Right: Perturbing phone labels.",
"Perturbing time.Adding false positive events.",
"Adding false negative events.In this chapter, we have described a formal framework for discriminative segmental models.",
"The definition is not tied to any weight function or any loss function, encompassing a large family of segmental models.",
"We have described two weight functions, one based on frame classifiers, and one based on segmental recurrent neural networks.",
"We have also described various loss functions, and have drawn connections between surrogate losses and the true loss that we want to minimize.",
"We have shown how (sub-)gradients of loss functions can be efficiently computed with FST algorithms.",
"The flexibility to use different loss functions allows us to train segmental models in different training settings.",
"Preliminary results have shown that segmental models outperform HMMs by a large margin and that overlap cost is better than MPE cost across loss functions.",
"In terms of loss functions, non-convex losses are more difficult to optimize.",
"Hinge loss is the best performer while log loss is very competitive.",
"Our final task loss is the edit distance, but none of the surrogate losses can be optimized when the cost function is the edit distance.",
"There are two difficulties for optimizing the edit distance: the edit distance is discrete, and it involves a minimization over all alignments to the ground truth.",
"Using a cost function that considers all alignments to the ground truth without the minimization has been explored in [51], [137], [121].",
"Other approaches for directly minimizing task losses [69], [67] are also suitable for minimizing the edit distance.",
"As we have seen in Chapter REF , search spaces defined by segmental models are dense and redundant, in the sense that many segments that have the same label differ only by a few frames (or even just a single frame) and that many segments with the same start time and end time differ only in the labels.",
"In other words, most of the random segments do not look like the ground truths.",
"Decoding in multiple passes is an approach that exploits dense search spaces.",
"Since the search spaces are dense, a simple weight function is typically enough to significantly reduce the size of search spaces.",
"After the first pass, we may use more complicated weight functions to make predictions or to further reduce the search spaces.",
"Reducing the size of search spaces is commonly known as pruning.",
"Pruning has to be done carefully, because it might hurt the final performance if the ground truth segments are pruned.",
"To measure how pruning affects the search spaces and the quality of predictions, we measure the lowest achievable task loss, the oracle error rate, of the search spaces after pruning.",
"A simple experiment on the phonetic recognition task shows that the oracle error rate, i.e., the best edit distance we can achieve, on average stays at zero even if we remove 50% of segments at random.",
"This result shows that the search spaces are indeed dense and redundant.",
"Since the pruned search space is smaller than the original space, inference and learning using the pruned search space are faster in the second pass.",
"As a result, we can afford to use computationally expensive weight functions for better prediction in the second pass.",
"In general, with the same weight function, multi-pass systems have the potential to decode faster than single-pass systems.",
"In other words, under the same decoding time budget, multi-pass systems can use computationally more expensive weight functions than single-pass systems to obtain better performance.",
"Since pruning plays an important role, in the following sections, we present and compare several pruning strategies and their consequences.",
"We then develop a multi-pass system, trading between the size of the search space and the computational complexity of weight functions.",
"In this section, we describe several pruning approaches to reduce search spaces.",
"Note that pruning is a form of inference, and we assume a pruning approach has access to search spaces defined by an FST, and a weight function $w : E \\rightarrow \\mathbb {R}$ parameterized by $\\Theta _{\\text{prn}}$ .",
"We discuss the choice of $w$ and how $\\Theta _{\\text{prn}}$ is obtained in the experimental sections.",
"A very naive approach, referred to as greedy pruning, is to prune the edges branching out from a vertex based on their edge weights.",
"Specifically, the set of edges branching out from a vertex $v$ is pruned based on a threshold $\\tau $ .",
"Let $S_v = \\lbrace e \\in \\text{out}(v): w(e) \\ge \\tau \\rbrace $ be the set of vertices that survive pruning.",
"We collect the edges $\\bigcup _{v \\in V} S_v$ to form the pruned graph.",
"There are two nice properties about this approach.",
"First, the pruning procedure is embarrassingly parallelizable.",
"Second, if we let $\\tau = \\lambda \\min _{e \\in \\text{out}(v)} w(e)+ (1 - \\lambda ) \\max _{e \\in \\text{out}(v)} w(e)$ where $0 \\le \\lambda \\le 1$ is a parameter that controls the amount of pruning, the graph is guaranteed to be connected.",
"To see this, note that there is at least one edge surviving for every vertex.",
"In other words, from the start vertex, there is at least one edge for us to traverse for every vertex, and we eventually arrive at the final vertex.",
"Beam pruning, or more generally beam search [84], is a widely used search and pruning method.",
"The motivation behind beam search is to constrain a search algorithm, such as the shortest-path algorithm (Algorithm REF ), with a fixed memory budget.",
"Due to the memory budget, we cannot afford to remember all the paths we have traversed, so we prune the paths that are less likely to have high weights.",
"Specifically, the beam search algorithm performs the following steps while visiting vertices in topological order.",
"We keep an approximate shortest distance $\\hat{d}(u)$ to every vertex $u$ .",
"Suppose vertex $v$ is the current vertex being visited.",
"A set of surviving edges $S_v = \\lbrace e \\in \\text{in}(v): \\hat{d}(\\mathrm {tail}(e)) \\ge \\tau \\rbrace $ is computed based on a threshold $\\tau $ , and then the approximate shortest distance for $v$ is updated by setting $\\hat{d}(v) = \\max _{e \\in S_v} w(e) + \\hat{d}(\\mathrm {tail}(e)).$ Note that this equation is identical to the shortest-path recursion, except that the set of incoming edges $\\text{in}(v)$ is pruned before updating.",
"To reconstruct the graph after beam pruning, we simply collect the surviving edges $\\bigcup _{v \\in V} S_v$ .",
"We can also let $\\tau = \\lambda \\min _{e \\in \\text{in}(v)} \\hat{d}(\\mathrm {tail}(e))+ (1 - \\lambda ) \\max _{e \\in \\text{in}(v)} \\hat{d}(\\mathrm {tail}(e))$ where $0 \\le \\lambda \\le 1$ is a parameter that controls the amount of pruning.",
"Note that beam pruning is very similar to greedy pruning.",
"For greedy pruning, edges are pruned based on edge weights, while for beam pruning, edges are pruned based on estimates of shortest distances.",
"It is obvious that if an edge is pruned, paths going through the edge are no longer in the search space.",
"Whenever we prune an edge, we discard the maximum-weight path that passes through the edge.",
"Obviously if the maximum-weight path passing through the edge survives pruning, the edge survives pruning.",
"The max-marginal for an edge is defined as the weight of the maximum-weight path passing through the edge.",
"Max-marginal pruning was first proposed by [124] and later rediscovered by [142].",
"Formally, the max-marginal of an edge $e$ is defined as $\\gamma (e) = \\max _{p \\ni e} w(p),$ i.e., the weight of the maximum-weight path that passes through $e$ .",
"The pruning procedure is simple: choose a threshold $\\tau $ and discard edge $e$ if $\\gamma (e) < \\tau $ .",
"One way to set the threshold [142] is $\\tau _\\lambda = \\lambda \\max _{e \\in E} \\gamma (e) + (1 - \\lambda ) \\frac{1}{|E|}\\sum _{e \\in E} \\gamma (e),$ where $E$ is the set of segments and $0 \\le \\lambda \\le 1$ .",
"There are two nice properties about max-marginal pruning with the above threshold: There is at least one path surviving; all paths with weights higher than $\\tau _\\lambda $ survive the pruning.",
"The first property is true because paths that achieve the maximum weight always survive.",
"The second property can be proved by contradiction.",
"Suppose path $p$ has weight $s > \\tau _\\lambda $ but is pruned.",
"Then there exists an edge $e$ in $p$ such that $\\gamma (e) < \\tau _\\lambda $ .",
"On the other hand, since $p$ passes through $e$ , $\\gamma (e)$ is at least $s$ , i.e., $\\gamma (e) > s > \\tau _\\lambda $ , a contradiction.",
"Note that we can only guarantee that paths with weights larger than $\\tau _\\lambda $ survive pruning.",
"It does not imply that all paths that survive pruning have weights larger than $\\tau _\\lambda $ .",
"To calculate $\\gamma (e)$ for each $e \\in E$ , note that $\\gamma (e) = \\max _{p \\ni e} w(p)= \\Big [ \\max _{p_1 \\in \\mathcal {P}(s, \\mathrm {tail}(e))} w(p_1) \\Big ]+ w(e)+ \\Big [ \\max _{p_2 \\in \\mathcal {P}(\\mathrm {head}(e), t)} w(p_2) \\Big ],$ where $\\mathcal {P}(u, v)$ is the set of paths that start at vertex $u$ and end at vertex $v$ .",
"We can construct an algorithm like the shortest-path algorithm to calculate the first and third terms as follows.",
"$d_1(v) & = \\max _{p_1 \\in \\mathcal {P}(s, v)} w(p_1)= \\max _{e \\in \\text{in}(v)} [ d_1(\\mathrm {tail}(e)) + w(e) ] \\\\d_2(v) & = \\max _{p_2 \\in \\mathcal {P}(v, t)} w(p_2)= \\max _{e \\in \\text{out}(v)} [ d_2(\\mathrm {head}(e)) + w(e) ]$ As shown above, the runtime of max-marginal pruning is the same as that of the shortest-path algorithm (Algorithm REF ).",
"Max-marginal pruning can also be applied to vertices.",
"Let $\\gamma (v) = \\max _{p \\ni v} w(p)$ be the max-marginal of the vertex $v$ .",
"We can also obtain a similar recursion $\\gamma (v) = \\max _{p \\ni v} w(p)= \\Big [ \\max _{p_1 \\in \\mathcal {P}(s, v)} w(p_1) \\Big ]+ \\Big [ \\max _{p_2 \\in \\mathcal {P}(v, t)} w(p_2) \\Big ]= d_1(v) + d_2(v),$ and set a threshold based on the convex combination of $\\max _{v \\in V} \\gamma (v)$ and $\\frac{1}{|V|} \\sum _{v \\in V} \\gamma (v)$ .",
"Similar guarantees also hold for max-marginal vertex pruning.",
"Recall that the runtime for finding the shortest path is $O(|E|C)$ where $E$ is the set of segments in the search space and $C$ is the computational cost of evaluating the weight function on a single segment.",
"Suppose we have a fixed time budget.",
"Since the pruned search space is smaller than the original space, we can afford to use a more computationally expensive weight function on the pruned space for better prediction.",
"Since additional pruning can be applied to the search space, we end up with a system having multiple rounds of pruning with increasingly expensive weight functions.",
"Prediction is then done on the pruned space from the final round.",
"The above approach is also commonly referred to as rescoring, because hypotheses are given new scores (weights) after pruning.",
"Models are also named after the pass in which they are used.",
"For example, a first-pass model is one that searches over the entire search space; a second-pass model is a model that rescores the pruned search space produced by a first-pass model.",
"Based on rescoring, [142] proposed structured prediction cascades, a type of multi-pass system with multiple rounds of pruning.",
"The complete search space is denoted $H_1$ .",
"At round $k$ , we have hypothesis space $H_k$ , train a pruning model $\\Theta _k$ on $H_k$ , and prune $H_k$ to get $H_k^{\\prime }$ .",
"The pruned space $H_k^{\\prime }$ is expanded to $H_{k+1}$ by considering more context for every segment, such as label $n$ -grams.",
"The process can be repeated for any number of rounds.",
"Inspired by [142], we propose discriminative segmental cascades, a multi-pass system consisting of segmental models with increasingly expensive weight functions.",
"In our cascades, all search spaces are represented as FSTs.",
"We denote the complete search space $H_1$ .",
"At the $k$ -th pass, based on the search space $H_k$ , we train a segmental model $\\Theta _k$ , and use it to prune the search space to get $H_k^{\\prime }$ .",
"In our framework, we can decide whether to expand the search space to include more context.",
"Since the search spaces are represented as FSTs, we propose a new FST operation, termed self-expansion, to efficiently consider neighboring segments for every segment.",
"If we decide to self-expand, then $H_{k+1} = {H_k^{\\prime }}^\\uparrow $ , where ${H_k^{\\prime }}^\\uparrow $ is the self-expansion of $H_k^{\\prime }$ .",
"Otherwise, $H_{k+1} = H_k^{\\prime }$ .",
"Suppose we decide to self-expand.",
"Then the weight function for the subsequent pass $w_{k+1}$ can make use of a wider context than the weight function of the previous pass.",
"Including a wider context is one possible approach to increase the expressiveness of the weight function.",
"Other approaches include using a deeper or more complex neural network, or extracting more sophisticated features such as tracking formants or pitches, both of which can be done without self-expansion.",
"The process can be repeated many times, forming a segmental cascade.",
"An example of a segmental cascade is shown in Figure REF .",
"Segmental models in each of these passes can be trained with the losses described in Section REF .",
"[142] proposed to train the pruning models with the filtering loss, measuring how well the ground truths are retained after pruning.",
"We decide to train the segmental models in the cascade with losses in Section REF that directly measure the final performance, making sure that we have a model for prediction after each pass.",
"This approach also makes it easier to monitor when to stop stacking the cascade.",
"Figure: An example segmental cascade.",
"The search space H 1 H_1is the complete space.",
"The search space H 1 ' H_1^{\\prime }is pruned from H 1 H_1.",
"The search space H 2 H_2is the self-expansion of H 1 ' H_1^{\\prime }.The additional vertex and edge added to H 1 ' H_1^{\\prime } beforeself-expansion are shown in dashed lines.Given a search space represented as an FST $H$ , we can expand the context of a segment by considering its neighbors.",
"Suppose we want to incorporate the previous segment given the current segment.",
"We create a new FST $H^\\uparrow $ that remembers and keeps track of the previous segment (edge) traversed in $H$ .",
"Vertices in $H^\\uparrow $ correspond to the possible previous segments in $H$ .",
"Edges in $H^\\uparrow $ are of the form $\\langle e_1, e_2 \\rangle $ , where $e_1$ is the edge just traversed and $e_2$ is the edge to be traversed.",
"We refer to $H^\\uparrow $ as the self-expansion of $H$ .",
"Formally, let $H = ((V, E, \\mathrm {tail}, \\mathrm {head}), \\Sigma , \\Lambda , I, F, i, o)$ .",
"The self-expansion of $H$ is defined as $H^\\uparrow = ((V^\\uparrow , E^\\uparrow , \\mathrm {tail}^\\uparrow , \\mathrm {head}^\\uparrow ),\\Sigma ^\\uparrow , \\Lambda ^\\uparrow , I^\\uparrow , F^\\uparrow , i^\\uparrow , o^\\uparrow )$ where $V^\\uparrow & = E \\cup \\lbrace e_0 \\rbrace & E^\\uparrow & = \\lbrace \\langle e_1, e_2 \\rangle : \\mathrm {head}(e_1) = \\mathrm {tail}(e_2) \\rbrace $ $I^\\uparrow & = \\lbrace e_0 \\rbrace & \\mathrm {tail}^\\uparrow (\\langle e_1, e_2 \\rangle ) & = e_1 \\\\F^\\uparrow & = \\lbrace e \\in E : \\mathrm {head}(e) \\in F \\rbrace & \\mathrm {head}^\\uparrow (\\langle e_1, e_2 \\rangle ) & = e_2 \\\\\\Sigma ^\\uparrow & = \\Sigma & i^\\uparrow (\\langle e_1, e_2 \\rangle ) & = i(e_2) \\\\\\Lambda ^\\uparrow & = \\Lambda & o^\\uparrow (\\langle e_1, e_2 \\rangle ) & = o(e_2)$ A vertex $v_0$ and an $e_0$ is created in $H$ before self-expansion, where $i(e_0) = o(e_0) = \\epsilon $ , $\\mathrm {tail}(e_0) = v_0$ , and $\\mathrm {head}(e_0) = v_1$ assuming we only have a single start state $v_1$ , i.e., $I = \\lbrace v_1\\rbrace $ .",
"The edge $e_0$ is created as a sentinel, corresponding to a vertex in $H^\\uparrow $ for not having traversed any edge.",
"The weight function $w^\\uparrow : E^\\uparrow \\rightarrow \\mathbb {R}$ is task-dependent, so we do not define it explicitly here.",
"However, since $w^\\uparrow $ is defined on $E^\\uparrow $ , the weight function after taking a edge $\\langle e_1, e_2 \\rangle \\in E^\\uparrow $ can make use of all the information about $e_1$ and $e_2$ to compute the weight, including the start times, end times, and labels.",
"We also observe that $\\text{out}^\\uparrow (e) & = \\lbrace \\langle e, e^{\\prime } \\rangle : e^{\\prime } \\in \\text{out}(\\mathrm {head}(e)) \\rbrace \\\\\\text{in}^\\uparrow (e) & = \\lbrace \\langle e^{\\prime }, e \\rangle : e^{\\prime } \\in \\text{in}(\\mathrm {tail}(e)) \\rbrace $ for $e \\in V^\\uparrow = E$ .",
"Therefore, similarly to $\\sigma $ -composition, self-expansion can be computed lazily (on the fly) while traversing $H$ .",
"We conduct experiments on the TIMIT data set in the same setting as in Section REF , except that here we follow [42] and use 40-dimensional log mel filter bank outputs instead of 39-dimensional MFCCs.",
"The data set is phonetically transcribed, so we have the option of training the encoders with frame-wise cross entropy based on the ground-truth frame labels.",
"In the following experiments, we explore convolutional neural network (CNN) encoders.",
"The CNN encoders are inspired by [123] in that they are deep convolutional neural networks with small $3 \\times 3$ and $5 \\times 5$ filters.",
"For frame classification, the input to the network is a window of 15 frames of 40-dimensional log mel filter outputs centered on the current frame.",
"The network has five convolutional layers, with 64 filters of size $5 \\times 5$ for the first layer and 128, 128, 256, and 256 filters of size $3 \\times 3$ for layers two to five respectively, each of which is followed by a rectified linear unit (ReLU) activation [101], with max pooling layers after the first and the third ReLU layers.",
"The output of the final ReLU layer is concatenated with a window of 15 frames of 39-dimensional MFCCs centered on the current frame, and the resulting vector is passed through three fully connected ReLU layers with 4096 units each.",
"The network is trained with SGD for 35 epochs with step size 0.001, momentum 0.95, weight decay 0.005, and a mini-batch size of 100 frame predictions.",
"Dropout [128] are applied to the concatenation layer with rate 20% and to the fully connected layers with rate 50%.",
"This classifier was tuned on the development set and achieves a 22.3% frame error rate (after collapsing to 39 phone labels) on the development set and 23.0% on the test set.",
"After confirming that the CNN encoder is able to perform well on frame classification, we use the CNN encoder to generate frame posterior probabilities, and train segmental models with the FC weight function and a maximum duration of 30 frames.",
"Hinge loss is optimized with AdaGrad [27] with step size 0.01 and a mini-batch size of 1 utterance for 70 epochs.",
"No explicit regularizer is used except early stopping on the development set.",
"We use a scaled overlap cost for our experiments: $\\mathrm {cost}(p^{\\prime }, p) &= \\sum _{e^{\\prime } \\in p} \\mathrm {cost}(e^{\\prime }, p) \\\\\\mathrm {cost}(e^{\\prime }, p) &= |e^{\\prime } \\cup \\tilde{e}| - |e^{\\prime } \\cap \\tilde{e}| {1}_{o(e^{\\prime }) = o(\\tilde{e})}$ where $p^{\\prime }$ is the hypothesized path, $p$ is the ground-truth path, $|e^{\\prime } \\cup e|$ denotes the union of the two intervals defined by $e$ and $e^{\\prime }$ , $|e^{\\prime } \\cap e|$ denotes the intersection of the two intervals, $o(e)$ is the output label of $e$ , and $\\tilde{e} = \\operatornamewithlimits{\\arg \\!\\max }_{e \\in p} |e^{\\prime } \\cap e|$ .",
"In words, $\\tilde{e}$ is the edge that overlaps the most with $e^{\\prime }$ .",
"The cost is non-negative and is only zero if $e^{\\prime } = e$ , and it can be seen as an estimate of the number of incorrectly predicted frames.",
"The results are shown in Table REF .",
"Our first-pass segmental model is on par with [1], where a CNN encoder is also used, and it is also on par with a baseline HMM-DNN hybrid system that we trained with Kaldi using the standard recipe for TIMIT [110].",
"Table: Phoneme error rates (%) on TIMITcomparing a HMM-DNN hybrid system todiscriminative segmental models.To construct a second-pass segmental model, we first evaluate pruning approaches based on oracle error rates and lattice densities.",
"The oracle error rate is the lowest achievable phone error rate of a given lattice.",
"The lattice density is defined as the total number of edges divided by the number of ground-truth edges, i.e., the number of hypothesized edges per ground-truth edge.",
"In general, a dense lattice has a higher chance to contain the ground-truth path than a sparse lattice.",
"In fact, the entire search space is a dense lattice that always contains the ground-truth path.",
"A good pruner is expected to produce sparse lattices while maintaining a low oracle error rate.",
"We compare max-marginal pruning with $\\lambda \\in \\lbrace 0.5, 0.6, 0.7, 0.8\\rbrace $ , greedy pruning with $\\lambda \\in \\lbrace 0.95, 0.9, 0.875, 0.85, 0.825\\rbrace $ , random pruning with probability $\\lbrace 0.92, 0.94, 0.96, 0.98\\rbrace $ , and beam pruning with $\\lambda =0.9$ .",
"The results are shown in Figure REF .",
"For reference, the average density of the complete spaces is 11587.67.",
"We observe that search spaces of segmental models are robust to missing edges—the oracle error rate is only 1.7% when we randomly drop 92% of the edges.",
"We also observe that max-marginal pruning produces significantly sparser lattices than greedy pruning and random pruning while maintaining low oracle error rates.",
"Beam pruning performs the worst.",
"Since greedy pruning works well by just comparing edges with the same tail vertex (i.e., with the same start time), we suspect the bad performance of beam pruning is due to comparing distances at different time points.",
"Figure: Left: Oracle error rates vs lattice densities for various pruningapproaches.",
"Right: A re-scaled version of the left figure focusing ongreedy pruning and max-marginal pruning.Based on the above pruning comparison, we decide to use the lattices generated by max-marginal pruning with $\\lambda = 0.8$ .",
"Let $H_1$ be the entire search space, and $H_1^{\\prime }$ be the pruned search space.",
"To train a segmental model with a wider segmental context, we first self-expand $H_1^{\\prime }$ into $H_2 = {H_1^{\\prime }}^\\uparrow $ and introduce the following weight functions.",
"Note that the weight functions can now depend on two segments instead of one.",
"Instead of re-learning all of the weight functions in the first-pass model, we reuse them in the second pass by having $w_{\\text{lat}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= \\alpha \\cdot w((\\ell _2, s_2, t_2))$ from the first level of the cascade, where $\\alpha $ is a learnable parameter.",
"We use the weight $w_{\\text{LM}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= \\beta \\cdot \\log p(\\ell _2 | \\ell _1)$ to include the log probability of the bigram $\\ell _1 \\ell _2$ , where $\\beta $ is a learnable parameter.",
"We define $w_{\\text{left}-k}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )& = u_{s_2 - k, \\ell _1, \\ell _2} \\\\w_{\\text{right}-k}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )& = u_{t_2 + k, \\ell _1, \\ell _2}^{\\prime }$ similarly to the first-order boundary weights, where $u_{t, \\ell , \\ell ^{\\prime }} = \\theta _{\\ell , \\ell ^{\\prime }}^\\top h_t$ and $u_{t, \\ell , \\ell ^{\\prime }}^{\\prime } = {\\theta ^{\\prime }_{\\ell , \\ell ^{\\prime }}}^\\top h_t$ for $t = 1, \\dots , \\tilde{T}$ and some learnable parameters $\\theta , \\theta ^{\\prime }$ .",
"Since the search space is sparse, we can afford to compute segment features based on a neural network for every segment.",
"We choose a similar CNN architecture to the one for frame classification.",
"We pre-train the CNN with a segment classification task.",
"Here the features at the input layer are the log mel filter outputs from a 15-frame window around the segment's central frame.",
"Instead of concatenation with 15-frame MFCCs, we concatenate with a segmental feature vector consisting of the average MFCCs of three sub-segments in the ratio of 3-4-3, plus two four-frame averages at both boundaries and length indicators for length 0 to 20 (similar to the segmental feature vectors of [46], [22]).",
"This CNN is trained on the ground-truth segments in the training set.",
"Finally, we build an ensemble of such networks with different random seeds and a majority vote.",
"This ensemble classifier has a 15.0% segment classification error on the test set.",
"A comparison of our CNN to other segment classifiers is shown in Table REF .",
"Table: TIMIT segment classification error rates (%).Directly running the CNN segment classifier for every edge in the lattice is, however, still too time-consuming.",
"We instead compress the best-performing (single) CNN into a shallow fully connected network with one hidden layer of 512 ReLUs by training it to predict the log probability outputs of the deep network, as proposed by [8].",
"We then use the log probability outputs of the shallow network.",
"We refer to the result as segment NN weights.",
"Let $w_{\\text{seg}}(\\langle (\\ell _1, s_1, t_1), (\\ell _2, s_2, t_2) \\rangle )= (u_{s_2, t_2})_{\\ell _2}$ where $u_{s, t} = \\theta ^\\top z_{s, t}$ , $z_{s, t}$ is a log probability vector of the labels obtained by passing the segment $x_s, \\dots , x_t$ to the CNN segment classifier, and $\\theta $ is some learnable parameter vector.",
"To add more context information, we use the same CNN architecture and training setup to learn a bi-phone frame classifier, but with an added 256-unit bottleneck linear layer before the softmax.",
"Each frame is labeled with its segment label and one additional label from a neighboring segment.",
"If the current frame is in the first half of the segment, the additional label is the previous phone; if it is in the second half, then the additional label is the next phone.",
"The learned bottleneck layer outputs are used to define weights (although they do not correspond to log probabilities) with averaging and sampling as for the uni-phone case.",
"We refer to the resulting weights as bi-phone NN bottleneck weights.",
"We use the sum of the lattice weight, the bigram LM weight, second-order boundary weights, segment NN weights, bi-phone NN bottleneck weights, length indicators, and lexicalized bias as our final weight function for the second level of the cascade.",
"Hinge loss is minimized with AdaGrad for 20 epochs with step size 0.01.",
"Again, no explicit regularizer is used except early stopping based on the PER on the development set.",
"Results with these additional weights are shown in Table REF .",
"Adding the second-order weights, bigram LM, and the above NN weights gives a 1.8% absolute improvement over our best first-pass system, demonstrating the value of including such expensive features with wider context.",
"Table: Phoneme error rates (%)with discriminative segmental cascades.The search space for the first pass is denoted H 1 H_1,and the search space for the second pass H 1 ' ↑ {H_1^{\\prime }}^\\uparrow is denoted H 2 H_2, where H 1 H_1 is pruned to get H 1 ' H_1^{\\prime }.We have shown that segmental cascades can be used to incorporate computationally expensive weight functions while maintaining efficiency by shifting the computationally expensive weights to later passes.",
"In this section, instead of incorporating additional weights, we shift weights to later passes to improve decoding speed.",
"Due to their recent success [42], we also explore long short-term memory (LSTM) networks [56] as encoders instead of CNN encoders.",
"In this section, we describe long short-term memory (LSTM) networks.",
"We start by defining a single-layer LSTM.",
"Suppose the input vectors are $x_1, \\dots , x_T$ .",
"First we apply a linear transformation to the inputs to get $x_t^{\\prime } = \\begin{pmatrix} W_{xu} \\\\ W_{xi} \\\\ W_{xf} \\\\ W_{xo} \\end{pmatrix} x_t,$ which can be done for $t=1, \\dots , T$ in parallel.",
"Next, assume $h_0 = c_0 = 0$ .",
"Suppose we have already computed $h_t$ and $c_t$ , and we want to compute $h_{t+1}$ and $c_{t+1}$ .",
"We apply a linear transformation to get the candidates $u^{\\prime }$ , $i^{\\prime }$ , $f^{\\prime }$ , and $o^{\\prime }$ for the cell update vectors $u$ and the gates $i$ , $f$ and $o$ .",
"$\\begin{pmatrix}u^{\\prime } \\\\ i^{\\prime } \\\\ f^{\\prime } \\\\ o^{\\prime }\\end{pmatrix}= x_t^{\\prime } + \\begin{pmatrix} W_{hu} \\\\ W_{hi} \\\\ W_{hf} \\\\ W_{ho} \\end{pmatrix} h_t+ \\begin{pmatrix} 0 \\\\ W_{ci} \\\\ W_{cf} \\\\ 0 \\end{pmatrix} c_t.$ The update vectors $u$ , the input gate $i$ , and the forget gate $f$ are then computed from the candidates by applying nonlinear transformations.",
"$u & = \\tanh (u^{\\prime }) \\\\i & = \\text{logistic}(i^{\\prime }) \\\\f & = \\text{logistic}(f^{\\prime })$ The new cell vector $c_{t+1}$ is computed with the update vector gated by the input gate and the previous cell.",
"$c_{t+1} & = i \\odot u + f \\odot c_t$ Finally, the output gate $o$ is computed from the updated cell $c_{t+1}$ , and the new hidden vector $h_{t+1}$ is updated with the cell gated by the output gate.",
"$o & = \\text{logistic}(o^{\\prime } + W_{co} c_{t+1}) \\\\h_{t+1} & = o \\odot \\tanh (c_{t+1})$ The final results after running an LSTM on $x_1, \\dots , x_T$ are $h_1, \\dots , h_T$ and $c_1, \\dots , c_T$ .",
"We use $h_{1:T} = \\text{LSTM}(x_{1:T})$ to denote the process above.",
"Since LSTMs have an assigned direction, it is natural to run two LSTMs in opposite directions and combine the hidden vectors.",
"Formally, $h^f_{1:T} & = \\text{LSTM}(x_{1:T}) \\\\h^b_{T:1} & = \\text{LSTM}(x_{T:1}) \\\\h_t & = W_f h^f_t + W_b h^b_t$ We use $h_{1:T} = \\text{BiLSTM}(x_{1:T})$ as a shorthand for the bidirectional LSTMs.",
"Note that the parameters of the two LSTMs are not shared.",
"Another way to increase the complexity of the acoustic encoder is to stack them on top of each other.",
"For example, if we stack 3 layers of bidirectional LSTMs, we get $h^{(1)}_{1:T} &= \\text{BLSTM}(x_{1:T}) \\\\h^{(2)}_{1:T} &= \\text{BLSTM}(h^{(1)}_{1:T}) \\\\h^{(3)}_{1:T} &= \\text{BLSTM}(h^{(2)}_{1:T})$ We use $h_{1:T} = \\text{DBLSTM}^3(x_{1:T})$ to denote the 3-layer bidirectional LSTMs (where “D” refers to “deep”).",
"Again the parameters of the LSTMs in each layer and each direction are not shared.",
"As in Section REF , we first explore LSTMs on a frame classification task, Specifically, the log probability vector of a frame at time $t$ is $z_t = \\text{logsoftmax}(W h_t + b)$ where $h_t$ is the hidden vector at time $t$ produced by an LSTM, and $W$ and $b$ are the parameters.",
"We then explore segmental cascades with weight functions based on log probabilities produced by LSTMs.",
"Experiments are conducted on the TIMIT data set.",
"We compute 41-dimensional log mel filter bank outputs including energy, and concatenate with the delta's and double delta's to form the final 123-dimensional acoustic feature vectors.",
"Instead of using a 192-utterance development set, we follow the Kaldi recipe [110] and use 400 utterances from the complete test set (disjoint from the core test set) as the validation set.",
"We report numbers on the core test set, the same test set as in Section REF .",
"We build a frame classifier by stacking three layers of bidirectional LSTMs.",
"The cell and hidden vectors have 256 units.",
"We train the frame classifier with frame-wise cross entropy and optimize with AdaGrad [27] with mini-batch size of 1 utterance and step size 0.01 for 30 epochs.",
"We choose the best-performing model on the development set (early stopping).",
"Following [138], [92], we consider dropping half of the frames for any given utterance in order to save time on feeding forward.",
"Specifically, we only use $x_2, x_4, \\dots , x_{T - 2}, x_T$ to feed forward through the deep LSTM and generate $z_2, z_4, \\dots ,z_{T - 2}, z_T$ (assuming $T$ is even, without loss of generality).",
"We then copy each even-indexed output to its previous frame, i.e., $z_{i-1} = z_i$ for $i = 2, 4, \\dots , T-2, T$ .",
"During training, the cross entropy is calculated over all frames and propagated back.",
"Specifically, let $E_i$ be the cross entropy at frame $i$ , and $E = \\sum _{i=1}^T E_i$ .",
"The gradient of $z_i$ is the sum of the gradients from the current frame and the copied frame.",
"Dropping even-indexed frames is similar to dropping odd-indexed frames, except the outputs are copied from $z_i$ to $z_{i+1}$ for $i = 1, 3, \\dots , T-1$ .",
"When training LSTMs with subsampling, we alternate between dropping even- and odd-numbered frames every other epoch.",
"The results are shown in Figure REF .",
"Figure: Development set frame error rates vs. number of epochs (left)and vs. training hours (right).We observe that with subsampling the model converges more slowly than without, in terms of number of epochs.",
"However, by the end of epoch 30, there is almost no loss in frame error rates when we drop half of the frames.",
"Considering the more important measure of training time rather than number of epochs, the LSTMs with frame subsampling converge twice as fast as those without subsampling.",
"For the remaining experiments, we use the log posteriors at each frame of the subsampled LSTM outputs as the inputs to the segmental models.",
"Our baseline system, denoted $R$ , is a first-pass segmental model with the FC weight function (Section REF ).",
"The baseline system is trained by optimizing hinge loss with AdaGrad using mini-batch size 1 utterance, step size 0.1, and early stopping for 50 epochs.",
"We decide to shift the FC weight function to the second pass, and train a segmental model, denoted $A_1$ , with just the label posterior and a label-independent bias.",
"Specifically, the label posterior weight is defined as $w((\\ell , s, t)) = \\sum _{i=s}^t (z_i)_\\ell ,$ where $z_i = \\text{logsoftmax}(W h_i + b)$ is the log probability vector at time $i$ based on the vector $h_i$ produced by the LSTM.",
"This reduces the number of features from 24528 to two.",
"We use hinge loss optimized with AdaGrad with mini-batch size 1 and step size 1.",
"Since we only have two features, learning converges very quickly, in only three epochs.",
"We take the model from the third epoch to produce lattices for subsequent passes in the cascade.",
"Lattices are generated with max-marginal pruning with $\\lambda = 0.85$ .",
"The resulting lattices have an average oracle error rates of 1.4% and average density of 213.02.",
"We train a second-pass model, denoted $A_2$ , with the FC weight function except that we add a “lattice score” feature corresponding to the segment score given by the two-feature system.",
"Hinge loss is optimized with AdaGrad with mini-batch size 1, step size 0.1, and early stopping for 20 epochs.",
"The learning curve comparing $R$ and $A_1$ followed by $A_2$ is shown in Figure REF .",
"We observe that the learning time per epoch of the two-feature system $A_1$ is only one-third of the baseline system $R$ .",
"We also observe that training of $A_2$ converges faster than training $R$ , despite the fact that they use almost identical feature functions.",
"The baseline system achieves the best result at epoch 49.",
"In contrast, the two-pass system is done before the baseline even finishes the third epoch.",
"Figure: Learning curve of the proposed two-pass system compared withthe baseline system.The time gap between the first pass and the second pass is thetime spent on pruning.Following Section REF , given the first-pass baseline, we apply max-marginal pruning to produce lattices for the second-pass baseline with $\\lambda =0.8$ .",
"The second-pass baseline features are the lattice score from the first-pass baseline, a bigram language model score, first-order length indicators, and a bias.",
"Hinge loss is optimized with AdaGrad with mini batch size 1, step size 0.01, and early stopping for 20 epochs.",
"For the proposed system, we produce lattices with max-marginal pruning and $\\lambda =0.3$ for the third-pass system.",
"We use the same set of features and hyper-parameters as the second-pass baseline for the third pass.",
"Phone error rates of all passes are shown in Table REF .",
"First, if we compare the one-pass baseline with the proposed two-pass system, our system is close to the baseline.",
"Second, we observe a healthy improvement by just adding the bigram language model score to the second-pass baseline.",
"The improvement for our third-pass system is small but brings our final performance to within 0.4% of the baseline second pass.",
"Table: Phone error rates (%) of proposed and baseline systems.Next we report on the speedups in training and decoding obtained with our proposed approach.",
"Table REF shows the real-time factors for decoding with the baseline and proposed systems.",
"In terms of decoding time alone, we achieve a factor of 2.4 speedup compared to the baseline.",
"If the time of feeding forward LSTMs is included, then our proposed system is two times faster than the baseline.",
"Table: Real-time factors for decoding.Table REF shows the times needed to train a system to get to the performance in Table REF .",
"The speedup mostly comes from the fast convergence of the first pass.",
"In terms of training the segmental models alone, we achieve an 18.0-fold speedup.",
"If the time to train the LSTMs is included, then we obtain a 3.4-fold speedup compared to the baseline.",
"To summarize some of the above results: With a combination of the first-pass two-feature system and edge pruning, we prune 95% of the segments in the first-pass hypothesis space, leading to significant speedup in both decoding and training.",
"The feed-forward time for our LSTMs is halved through frame subsampling.",
"In the end, with a single four-core CPU, we achieve 0.31 times real-time decoding including feeding forward, which is 2.2 times faster than the baseline, and 32.5 hours in total to obtain our final model including LSTM training, which is 3.4 times faster than the baseline.",
"Excluding the LSTMs, the segmental model decoding alone is 2.4 times faster than the baseline, and training the segmental models alone is 18 times faster than the baseline.",
"Table: Hours for training the system.In this chapter, we have described discriminative segmental cascades, a multi-pass system for segmental models.",
"Max-marginal pruning lies at the heart of segmental cascades, producing sparse lattices while having low oracle error rates.",
"After pruning, it becomes feasible to incorporate computationally expensive weight functions, such as higher-order LMs, and weight functions based on neural networks.",
"We demonstrate that incorporating these weights significantly improves the performance of segmental models.",
"We also show that decoding and training speed can be significantly improved with segmental cascades by shifting weights to later passes without sacrificing accuracy.",
"Though we have shown improvement with segmental cascades, how the density and structure of search spaces affect the training of segmental models is still not fully known.",
"Task-dependent priors, such as phoneme duration, are not fully explored and can potentially be integrated into the pruning procedure.",
"The fact that beam search works well for frame-based models but fails miserably for segmental models is also worth investigating.",
"Converting other more sophisticated search algorithms, such as $\\text{A}^*$ search [68], into pruning algorithms is a possibility that might show us signs as to why beam search fails for segmental models.",
"In the previous chapter, we have seen how a sequence recognizer can be trained in multiple stages: a frame classifier is first trained with frame-wise cross entropy, and a segmental model is trained with a sequence loss, such as the ones defined in Section REF , based on the classifier's outputs.",
"In other words, the frame classifier is trained independently from the segmental model, and is held fixed when the segmental model is being trained.",
"A natural question to ask is whether we can further improve the sequence loss by updating the frame classifier while holding the segmental model fixed.",
"A more general question would be whether we can improve the sequence loss by updating the frame classifier and the segmental model simultaneously.",
"Optimizing all of the parameters simultaneously against a single loss function is commonly referred to as end-to-end training.",
"Since the loss function with respect to all the parameters in a neural network is non-convex, end-to-end training might be more difficult than training in multiple stages.",
"On the other hand, since end-to-end training is jointly optimizing all parameters, it might achieve a lower loss, leading to better performance.",
"In this chapter, we formally define these training settings, including multi-stage training and end-to-end training, Different training approaches and losses have different training requirements.",
"We discuss the pros and cons of these training approaches, and conduct experiments to see empirically how well they compare to each other.",
"Recall that the set of parameters $\\Theta $ consists of $\\Theta _\\text{enc}$ and $\\Theta _\\text{dec}$ , where $\\Theta _\\text{enc}$ is the set of parameters in the encoder and $\\Theta _\\text{dec}$ are the rest of the parameters in the weight function.",
"The training approach we have seen in the previous chapter is referred to as multi-stage training.",
"Specifically, the encoder is trained in the first stage with an encoder-specific loss $\\mathcal {L}_\\text{enc}(\\Theta _\\text{enc}),$ for example, the frame-wise cross entropy.",
"Let $\\hat{\\Theta }_\\text{enc} = \\operatornamewithlimits{\\arg \\!\\min }_{\\Theta _\\text{enc}}\\mathcal {L}_\\text{enc}(\\Theta _\\text{enc})$ .",
"In the second stage, we solve the optimization problem $ \\min _{\\Theta _\\text{dec}} \\mathcal {L}_\\text{seq}(\\hat{\\Theta }_\\text{enc}, \\Theta _\\text{dec}),$ where $\\mathcal {L}$ is a sequence loss, such as hinge loss, log loss, or marginal log loss, defined in Section REF .",
"Note that in the second stage the parameters of the encoder are held fixed.",
"Typically, after the first stage, we can feed the inputs in the entire data set forward, and reuse the vectors while optimizing the sequence loss to save computation.",
"Another benefit of optimizing (REF ) with $\\hat{\\Theta }_\\text{enc}$ fixed is that the loss function $\\mathcal {L}$ might be convex in $\\Theta _\\text{dec}$ , for example, when hinge loss or log loss is used and the weight function is linear in $\\Theta _{\\text{dec}}$ .",
"The characteristics of convex functions are well-understood [13], and many algorithms, other than first-order methods, have been developed for optimizing convex functions [12].",
"In contrast to multi-stage training, we refer to solving $ \\min _{\\Theta _\\text{enc}, \\Theta _\\text{dec}}\\mathcal {L}_\\text{seq}(\\Theta _\\text{enc}, \\Theta _\\text{dec})$ as end-to-end training, where all parameters $\\Theta _\\text{enc}$ and $\\Theta _\\text{dec}$ are optimized jointly.",
"Since all parameters are optimized jointly, the loss function is in general non-convex.",
"Though end-to-end training might be harder to solve, it might lead to a lower loss value.",
"Typically, the solution found by multi-stage training is a reasonably good solution for the sequence loss.",
"As a result, a safe strategy, which we refer to as end-to-end fine-tuning, may be to use the solution from multi-stage training as an initialization for solving (REF ).",
"One major difference between multi-stage training and end-to-end training is their resource requirements.",
"Multi-stage training requires extra labels for optimizing the encoder loss $\\mathcal {L}_\\text{enc}$ , while end-to-end training requires only whatever the sequence loss $\\mathcal {L}_\\text{seq}$ requires.",
"For experiments, we use the same phoneme recognition setting on the TIMIT data set as in Section REF .",
"In the sections below, we first compare the performance of CNNs and LSTMs, and then compare two training settings for training segmental models, multi-stage training followed by end-to-end fine-tuning and end-to-end training from random initialization.",
"We also explore various weight functions paired with various loss functions in the context of end-to-end training from random initialization.",
"Recall that the multi-stage training approach demonstrated in Section REF consists of two stages: first training a frame classifier, and second training a segmental model based on the output probabilities of the frame classifier.",
"Instead of a CNN encoder, a 3-layer bidirectional LSTM with hidden vector size of 250 is trained for frame classification.",
"Parameters are initialized according to [36].",
"Vanilla SGD is used to minimize frame-wise cross entropy for 20 epochs with step size 0.1 and a mini-batch size of 1 utterance.",
"Gradients are clipped to norm 5 if the norm is above 5.",
"The best model (measured by frame error rates on the development set) is chosen among the 20 epochs, and trained for another 20 epochs with step size 0.75 decayed by 0.75 after every epoch.",
"For phoneme recognition, we use segmental models with the FC weight function, the same setting as in Section REF .",
"Hinge loss with the overlap cost are minimized with RMSProp [100] for 20 epochs with step size ${10}^{-4}$ and decay 0.9, The frame error rates (FER) and phoneme error rates (PER) compared to the CNN in Section REF are shown in Table REF .",
"The LSTM performs better than the CNN for both tasks.",
"We suspect that the LSTM's superior performance is due to the training approaches: the LSTM is trained on entire utterances, while the CNN is trained on batches of 15-frame windows.",
"This effect was also observed in [58].",
"For the rest of the experiments, we use 3-layer LSTMs as our encoders.",
"Table: Frame error rates (FER) and phoneme error rates (PER)on the development setcomparing the CNN and the LSTM encoder.Next, we compare segmental models trained with different losses, including hinge loss, log loss, and marginal log loss.",
"We use the FC weight function and a maximum segment duration of 30 frames.",
"All losses are minimized with RMSProp for 20 epochs with step size ${10}^{-4}$ and decay 0.9.",
"Results are shown in Table REF .",
"No explicit regularizers are used except early stopping on the development set.",
"All losses achieve similar results with log loss slightly behind.",
"Note that although marginal log loss does not require manual alignments, the frame-wise cross entropy used to train the frame classifier still does.",
"After two-stage training, we use the trained segmental models and frame classifiers as an initialization for end-to-end training.",
"Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 decayed by 0.75 after each epoch.",
"Gradients are clipped with norm 5.",
"Dropout is added to the LSTMs with rate 0.2, and no other regularizers are used except early stopping on the development set.",
"Results are shown in Table REF .",
"End-to-end fine-tuning is able to improve the error rates for all losses with marginal log loss being the best performer.",
"Table: Phoneme error rates (%) comparing different losseswith two-stage training (2s) followed by end-to-end fine-tuning (ft).After end-to-end fine-tuning, the encoders are not constrained to perform well on frame-wise cross entropy.",
"We evaluate the encoders on the frame classification task to see how much the encoders deviate from the initialization, and whether the intermediate representations trained for frame classification are still preserved.",
"Results are shown in Table REF .",
"The encoders fine-tuned with hinge loss and log loss deviate less compared to the one fine-tuned with marginal log loss.",
"We suspect this is because hinge loss and log loss use the ground-truth alignments in training while marginal log loss does not.",
"Table: Frame error rates (%) of LSTM encodersafter end-to-end fine-tuning with different losses .After observing the success of end-to-end fine-tuning, we conduct experiments to see whether it is possible to train segmental models end to end from random initialization.",
"We use the FC weight function and a maximum duration of 30 frames, the same setting as in the multi-stage experiments.",
"We initialize the parameters according to [36].",
"Each loss is optimized with vanilla SGD for 20 epochs with step size 0.1 and gradient clipping with norm 5.",
"The best performing model, chosen in the first 20 epochs, is trained for another 20 epochs with vanilla SGD step size 0.75 decayed by 0.75 after each epoch.",
"Dropout of rate 0.2 is used throughout the training process.",
"No other regularizers are used except early stopping on the development set.",
"Results are shown in Table REF .",
"There is no significant difference in terms of performance between models trained end to end from random initialization and ones trained in multiple stages followed by fine-tuning.",
"This shows that it is possible to train segmental models end to end from random initialization, and that models trained in multiple stages can serve as a good initialization for end-to-end training.",
"Table: Phoneme error rates (%) on the development setcomparing multi-stage trainingfollowed by end-to-end fine-tuning (2s+ft) and end-to-end trainingfrom random initialization (e2e).Next, we consider various weight functions for end-to-end training from random initialization.",
"Note that the initialization scheme in [36] is not suitable the for log-soft-max operation used after the LSTMs for producing log probabilities, because the variances before and after the log-soft-max operation are very different.",
"Based on this intuition, we define a new weight function, called FC bottleneck (FCB) weight function, removing the log-soft-max layer when using the FC weight function.",
"More precisely, suppose $h_1, \\dots , h_T$ is the sequence of vectors produced by an LSTM.",
"Whereas the FC weight function uses the average, samples, and boundaries of $z_i = \\text{logsoftmax}(W h_i + b)$ for $i = 1, \\dots , T$ , the FCB weight function uses the average, samples, and boundaries of $h_i$ 's directly.",
"The learning curves of segmental models trained end to end with FC weight function and the FCB weight function are shown in Figure REF .",
"We observe significantly faster convergence with the FCB weight function than with the FC weight function.",
"Figure: Learning curves of segmental models trained end to endwith the FC weight function and FCB weight functionwith marginal log loss.We do not compare the MLP weight function to the FC and FCB weight functions in the current setting, because it is too time-consuming to train segmental models with the MLP weight function on the entire search space.",
"To train segmental models with the MLP weight function, we follow [85] and reduce the time resolution by a factor of four using pyramid LSTMs.",
"Specifically, for an input sequence $x_{1:T}$ , $h^{(1)}_{1:T} &= \\text{BLSTM}(x_{1:T}) \\\\z^{(1)}_{1:\\lfloor T/2 \\rfloor } &= \\text{subsample}(h^{(1)}_{1:T}) \\\\h^{(2)}_{1:\\lfloor T/2 \\rfloor } &= \\text{BLSTM}(z^{(1)}_{1:\\lfloor T/2 \\rfloor }) \\\\z^{(2)}_{1:\\lfloor T/4 \\rfloor } &= \\text{subsample}(h^{(2)}_{1:\\lfloor T/2 \\rfloor }) \\\\h^{(3)}_{1:\\lfloor T/4 \\rfloor } &= \\text{BLSTM}(z^{(2)}_{1:\\lfloor T/4 \\rfloor })$ where $h_2, h_4, \\dots , h_{2(k-1)}, h_{2k} = \\text{subsample}(h_1, h_2, \\dots , h_{2k})$ if we drop a frame for every two frames.",
"The final output sequence $h^{(3)}$ is then used to compute edge weights.",
"An encoder with several layers of LSTMs interleaved with subsampling is also known as a pyramid LSTM and has been used in other prior work, e.g., [16].",
"In addition, we set the maximum duration to 8 frames, effectively $4 \\times 8 = 32$ frames on the original time scale.",
"The reduced time resolution and maximum duration result in search spaces 16 times smaller than the original ones, making it feasible to compute the MLP weight function on all edges.",
"With 3-layer pyramid LSTMs, we compare segmental models trained with the three weight functions and three losses.",
"All models are trained with the same step size scheduling as in the previous experiments.",
"Results with and without pyramid LSTMs are both shown in Table REF .",
"First, we observe that using pyramid LSTMs leads to a degradation in performance for segmental models with the FC and FCB weight functions.",
"However, using pyramid LSTMs has less impact on segmental models trained with marginal log loss compared to the ones trained with hinge loss and log loss.",
"We suspect this is because hinge loss and log loss rely on the ground-truth alignments for training while marginal log loss does not.",
"Second, we observe that while segmental models with the FC and FCB weight functions are able to achieve low hinge loss values, the ones with the MLP weight function completely fail to do so.",
"Finally, the best result on the development set obtained by using the FC weight function is slightly worse than using the MLP weight function.",
"Table: Phoneme error rates (%) comparing different lossesfor end-to-end training from random initialization.Segmental models trained with marginal log loss do not require ground-truth alignments during training, making annotating data sets easier, cheaper, and less time-consuming.",
"However, it is harder to diagnose errors made by models trained end to end because the intermediate representations are not interpretable.",
"Models trained in multiple stages are easier to diagnose by looking at the performance of proxy tasks, such as frame classification in our case.",
"We compare the time to train segmental models end to end.",
"Times are measured on a 3GHz 4-core CPU.",
"Multithreading is only used for matrix operations; all FST algorithms are implemented single-threaded.",
"Results are shown in Table REF .",
"Reducing the time resolution with pyramid LSTMs significantly improves the runtime for all losses.",
"The MLP weight function is about 2.5 times slower than the FC weight function.",
"Table: Average number of minutes for one epoch of end-to-end training on TIMIT.We are interested in how well segmental models recover the phone boundaries in the end-to-end setting when manual alignments are not used during training.",
"The task, commonly known as phonetic segmentation, is to align the phonetic transcription to the acoustic frames.",
"Unlike sequence prediction, the phonetic transcription is known.",
"We measure the error rates of the boundaries under some tolerance level.",
"Specifically, suppose the predicted boundary is at time $\\hat{b}$ and the ground truth is at time $b$ .",
"The predicted boundary $\\hat{b}$ is considered correct if the absolute distance $t = |\\hat{b} - b| \\le \\tau $ for some $\\tau $ .",
"The error rates under various thresholds for the segmental model with the FC weight function trained with marginal log loss are shown in Table REF .",
"Models proposed in [70], [113], [144], shown in row 1–3 in Table REF , are specifically trained for phonetic segmentation.",
"Though the alignment results are behind models trained specifically to align, the segmental model trained with marginal log loss is not supervised with any ground-truth alignments, and its performance is sufficient for many tasks.",
"Table: Boundary error rates (%) of phonetic segmentation at different tolerance levelson the TIMIT core test set (except the last row)for segmental models trained end to end with marginal log losscompared to past results.We analyze the errors made by the segmental model.",
"The top 30 most errorful boundary types sorted by error rates are shown in Table REF .",
"We observe that most of the boundaries in Table REF involve silences, vowels, and semi-vowels.",
"The ones involving silences often appears at the start or the end of utterances.",
"Boundaries between two vowels and between a semi-vowel and a vowel are known to be ambiguous [120], [135].",
"We see few alignment errors between consonants and vowels.",
"The top 30 most errorful boundary types sorted by error counts are shown in Table REF .",
"We are also interested in boundary types that have hight error counts but not necessary have high error rates.",
"Compared to Table REF , we see a very different pattern.",
"Most of the errors in Table REF involve silences (including voiced and unvoiced closures).",
"For $0\\text{ms}$ and $10\\text{ms}$ , the segmental model errs at the boundaries between closures and the bursts of stop consonants, such as /b/, /d/, /g/, /p/, /t/, and /g/.",
"It also errs at boundaries involving liquids, such as /r/ and /l/, across all tolerance levels.",
"Similarly to semi-vowels, the boundaries of liquids are also known to be ambiguous [120], [135].",
"The errors for higher tolerance levels, such as 30ms and 40ms, mostly involve silences that appear at the start or end of utterances.",
"We notice that 10.5% of the silences in the training set are longer than 30 frames.",
"We suspect the 30-frame maximum duration constraint is too restrictive for silences.",
"Some phoneme boundaries are inherently ambiguous.",
"In other words, there are cases where segments do not have clear start times and end times.",
"We argue that segmental models are still well-motivated and suitable for these tasks, such as phonetic recognition.",
"First, the search space are stochastic when the path weights are interpreted as probabilities.",
"In fact, lattices, when first proposed by [120], were used to account for phoneme boundary ambiguity.",
"Training segmental models with marginal log loss also takes the ambiguity into account by marginalizing over all segmentations.",
"For decoding, ideally we want to find the label sequence that has the maximum marginal weight $\\operatornamewithlimits{\\arg \\!\\max }_{y} \\sum _{p \\in \\Gamma (y)} w(p), $ where $\\Gamma (y)$ is the set of paths with the label sequence $y$ .",
"However, the above is not tractable and we typically approximate it by finding the most probable path $\\operatornamewithlimits{\\arg \\!\\max }_{p \\in \\mathcal {P}} w(p).$ Another approach to approximate (REF ) is to use beam search.",
"In summary, segmental models are capable of handling ambiguous segment boundaries if the training and decoding take the ambiguity into account.",
"We have compared different training approaches for segmental models, including multi-stage training and end-to-end training.",
"End-to-end training improves over multi-stage training, and end-to-end training from random initialization is on par with end-to-end fine-tuning.",
"Multi-stage training is useful for diagnosing end-to-end training, because every model found by multi-stage training is a valid model for end-to-end training objectives.",
"We have shown that segmental models can be trained with marginal log loss from random initialization, without requiring manual alignments for training.",
"As a byproduct, segmental models trained with marginal log loss perform reasonably well on phonetic segmentation.",
"Segmentation errors made by end-to-end segmental models are aligned with the studies in acoustic phonetics.",
"Table: Top 30 boundaries and its preceding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error rates ofthe segmental model trained with marginal log loss.Table: Top 30 boundaries and its preceeding and following phonemes(with at least five occurrences) in the training setsorted according to alignment error counts ofthe segmental model trained with marginal log loss.Automatic speech recognition (ASR) has been posed as a graph search problem since the 1970s [61].",
"A search space, represented as a graph, is first constructed; edges in the graph are assigned weights based on the input and the edges; to predict, we simply run the shortest-path algorithm on the graph.",
"The graph search paradigm has been popularized by the use of hidden Markov models (HMM) [112], [62], where the shortest-path algorithm is known as the Viterbi algorithm.",
"In the probabilistic setting, finding the shortest path can be seen as finding the maximum a posteriori solution, justifying the graph search paradigm.",
"The probabilistic view has spawned many algorithms, such as segmental k-means [63] and expectation maximization [116], for estimating parameters in the weight function.",
"Segmental models follow the same graph search paradigm while having a different type of search spaces and different weight functions.",
"Many variants of segmental models were proposed in the past, such as stochastic segment models [103], [104], semi-Markov HMMs [117], segmental HMMs [115], [29], semi-Markov conditional random fields (CRF) [118], and segmental CRFs [149].",
"However, the type of search spaces and the shortest-path algorithm stay the same.",
"In this chapter, we review these variants from the graph search point of view.",
"The idea of using segmental features for speech recognition can be traced back to the 1970s [141].",
"The definition of segmental models was not explicit.",
"Most of the studies still followed the graph search paradigm, though the graph structures were typically constructed from a lexicon with a small vocabulary, and the weights on the edges were typically estimated with heuristics or a small amount of data [75], [23].",
"In the following sections, we categorize variants of segmental models as either generative or discriminative.",
"This categorization also aligns well with their rough chronological order.",
"Hidden semi-Markov Models are arguably the first segmental models applied to speech recognition [81], [117].",
"Given a sequence of observations $x = (x_1, \\dots , x_T)$ of length $T$ , let $y = (\\ell _1, \\dots , \\ell _K)$ be the label sequence and $z = ((s_1, t_1), \\dots ,(s_K, t_K))$ be the segmentation, where $\\ell _1, \\dots , \\ell _K \\in L$ for some discrete label set $L$ , and $s_1 = 1$ , $t_k = T$ , $s_k \\le t_k$ , $t_{k-1} + 1 = s_k$ , for $k = 2, \\dots , n$ .",
"Let $e_k = (\\ell _k, s_k, t_k)$ for $k = 1, \\dots , K$ .",
"The probability of $x_{1:T}$ defined by hidden semi-Markov models is $p(x, y, z) = p(x_{1:T}, e_{1:K})= p(e_1) \\prod _{k=2}^K p(e_k | e_{k-1}) \\prod _{k=1}^K p(x_{s_k:t_k} | e_k).$ The generative story is straightforward: the segments are generated one by one, following a Markov chain, and each segment $e = (\\ell , s, t)$ generates $t - s + 1$ observations.",
"Hidden Markov models are special cases of hidden semi-Markov models with the constraints $K = T$ and $s_k = t_k$ for $k = 1, \\dots , K$ .",
"There are many ways to define and parameterize $p(e_k | e_{k-1})$ and $p(x_{s_k:t_k} | e_k)$ .",
"The most common assumptions are $p(e_k | e_{k-1}) & = p(\\ell _k | \\ell _{k-1}) \\\\p(x_{s_k:t_k} | e_k) & = p(x_{s_k:t_k} | t-s+1, \\ell _k) p(t-s+1 | \\ell _k)$ where $p(\\ell _k | \\ell _{k-1})$ is commonly known as the transition probability, $p(x_{s_k:t_k} | t-s+1, \\ell _k)$ the emission probability, and $p(t-s+1 | \\ell _k)$ the duration probability.",
"The observations within a segment are typically assumed to be independent, i.e., $p(x_{s_k:t_k} | t-s+1, \\ell _k) =\\prod _{j=s_k}^{t_k} \\mathcal {N}(x_j; \\mu _{\\ell _k}, \\sigma _{\\ell _k}^2),$ where $\\mu _\\ell $ and $\\sigma _\\ell ^2$ are the mean and variance of the Gaussian distribution for the label $\\ell $ .",
"The single Gaussian case can be easily extended to a mixture of Gaussians.",
"Continuously variable duration HMMs [81], stochastic segment models [103] and segmental HMMs [115], [29] are all hidden semi-Markov models with the above assumptions.",
"The difference among them lies in how the emission probability $p(x_{s_k:t_k} | t-s+1, \\ell _k)$ is defined.",
"The subtle differences are summarized in [30], [104].",
"The duration probability is typically defined by a Poisson distribution [116] or Gamma distribution [81].",
"See [104] and the citations therein for other options.",
"Decoding with hidden semi-Markov models is done by solving $\\operatornamewithlimits{\\arg \\!\\max }_{y, z} p(y, z | x)& = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(y, z | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log \\frac{p(x, y, z)}{p(x)} \\\\& = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(x, y, z),$ where $y$ is the label sequence and $z$ is the segmentation, and is equivalent to finding the maximum-weight path with $w((\\ell , s, t)) = \\log p(t-s+1 | \\ell )+ \\sum _{j=s}^t \\log \\mathcal {N}(x_j; \\mu _\\ell , \\sigma _\\ell ^2).$ The term $p(\\ell ^{\\prime } | \\ell )$ is ignored here for simplicity, but can be included once the search space is self-expanded (Section REF ).",
"Training hidden semi-Markov models can be done by maximizing the likelihood of the training set.",
"The likelihood can be maximized using gradient-based methods or expectation maximization (EM) [116].",
"See [30] for a detailed explanation of the EM algorithm for estimating the parameters of hidden semi-Markov models.",
"Originally motivated by using rich segmental features [147], the SUMMIT system was defined as a generative model [34].",
"However, [34] introduced the notion of anti-phones and later [18] introduced near-misses, training the system with a discriminative touch.",
"It held the state-of-the-art result for speaker-independent phoneme recognition on TIMIT [45] until the rise of deep neural networks [94].",
"Since maximum mutual information was introduced as a training criterion for HMMs [10], ASR studies have gradually shifted from generative to discriminative modeling.",
"Parallel to the development of segmental models in the ASR community, [118] proposed semi-Markov CRFs for named entity recognition.",
"Using the same notation as in the previous section, the probability of a label sequence $y = (\\ell _1, \\dots , \\ell _n)$ and a segmentation $z = ((s_1, t_1), \\dots , (s_K, t_K))$ given an observation sequence $x = (x_1, \\dots , x_T)$ is defined as $p(y, z | x) = \\frac{1}{Z(x)} \\exp \\left( \\sum _{k=1}^K \\theta ^\\top \\phi (x, e_k) \\right)$ where $Z(x) = \\sum _{y^{\\prime }, z^{\\prime }} \\exp (\\sum _{e \\in (y^{\\prime }, z^{\\prime })} \\theta ^\\top \\phi (x, e))$ and $e \\in (y^{\\prime }, z^{\\prime })$ is a shorthand for enumerating $(\\ell _1, s_1, t_1),\\dots ,(\\ell _{K^{\\prime }}, s_{K^{\\prime }}, t_{K^{\\prime }})$ for $y^{\\prime } = (\\ell _1, \\dots , \\ell _{K^{\\prime }})$ and $z^{\\prime } = ((s_1, t_1), \\dots , (s_{K^{\\prime }}, y_{K^{\\prime }}))$ of length $K^{\\prime }$ .",
"The function $\\phi $ , called the feature function, extracts feature vectors that are intended to correlate well with $y$ and $z$ .",
"The parameter vector $\\theta $ can be learned by maximizing the likelihood of the training set.",
"Decoding with semi-Markov CRFs is done by solving $\\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\log p(y, z | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y, z} \\sum _{e \\in (y, z)} \\theta ^\\top \\phi (x, e),$ and is equivalent to finding the maximum-weight path if we let $w(x, e) = \\theta ^\\top \\phi (x, e)$ .",
"Similarly, training is done by maximizing the conditional likelihood of the data, and is equivalent to minimizing the negative log likelihood, or log loss.",
"Heavily influenced by [118], [149] proposed segmental CRFs.",
"The difference between segmental CRFs and semi-Markov CRFs lies in the training loss.",
"Instead of optimizing the conditional likelihood $p(y, z | x)$ , segmental CRFs optimize the marginal likelihood $p(y | x) = \\sum _{z} p(y, z | x).$ The connection between the marginal likelihood and marginal log loss is clear once we take the log of the probability distribution.",
"As we defined in Chapter REF , we distinguish between segmental models that consider the entire search space and ones that do not.",
"The former are called first-pass segmental models.",
"Whether a segmental model is first-pass or not is independent of its definition.",
"For example, semi-Markov CRFs were used as first-pass segmental models in [118]; segmental CRFs were first used as a second-pass model in [149], and were later used as a first-pass model in [148].",
"In Table REF , we provide a set of highlights of results in the development of segmental models on the TIMIT data set.",
"[148] was the first to explore discriminative segmental models that search over sequences and segmentations exhaustively, and did not use neural networks.",
"[49] first used (shallow) neural network-based frame classifiers to define weight functions, and later extended the idea to deep neural networks in [48].",
"[1] were the first to use deep convolutional neural networks for the weight functions, and were the first to train segmental models end to end.",
"We have compared different losses and training strategies for segmental models, first in a rescoring framework [129] and then in first-pass segmental models [131].",
"We also introduced segment-level classifiers and segmental cascades for incorporating them (and other expensive features) into segmental weight functions [130].",
"[85] introduced an LSTM-based weight function for every segment, and were also the first to use pyramid LSTMs to speed up inference for segmental models.",
"Table: TIMIT PERs (%) for various segmental modelscompared with HMMs and the current state of the art.The acoustic features are speaker-independent (spk indep) orspeaker-adapted with mean and variance normalization (mvn)or maximum likelihood linear regression (fMLLR) .Some results were obtained with MFCCs and some with log filter bank features.In this chapter, we have reviewed variants of segmental models categorized as either generative or discriminative in rough chronological order.",
"We review hidden semi-Markov models, a broad class of generative segmental models subsuming stochastic segment models and segmental HMMs.",
"We then review semi-Markov CRFs, the discriminative counterpart of hidden semi-Markov models.",
"We have established connections between these special cases and the general segmental models defined in Chapter REF .",
"Beyond segmental models, this chapter reviews other modern models trained end to end within the graph search paradigm.",
"We review connectionist temporal classification (CTC) [41], a popular approach for training frame-based LSTMs end to end.",
"Drawing on the connection between CTC and marginal log loss, we propose a framework, consisting of search spaces (represented as FSTs), weight functions, and training losses, that can encompass many end-to-end models, such as hidden Markov models trained with lattice-free maximum mutual information [111], and LSTMs trained with CTC, as special cases.",
"We compare end-to-end segmental models and end-to-end frame-based models, including one-state HMMs and two-state HMMs with LSTM encoders, and LSTMs trained with CTC.",
"Having these end-to-end models within the unified framework allows us to see the effect of each component while holding the other components fixed.",
"In this section, we review connectionist temporal classification (CTC) [41].",
"While being conceptually simple, LSTMs trained with CTC were the state of art for phonetic recognition in 2013 [42], and have achieved competitive results on large-vocabular speech recognition [92], [88], [151].",
"Consider a sequence of acoustic vectors $x_1, \\dots , x_T$ and its corresponding labels $y_1, \\dots , y_n$ , where $y_i \\in L$ for $i = 1, \\dots , n$ and some label set $L$ .",
"We assume $n < T$ because a phoneme is typically more than a frame long.",
"Suppose there exists a function $\\mathcal {P}$ that maps $y_1, \\dots , y_n$ to a path $a_1, \\dots , a_T$ , where $a_t \\in L^{\\prime }$ for some other label set $L^{\\prime }$ .",
"Minimizing $p(a_{1:T} | x_{1:T})$ can be done by simply minimizing the frame-wise cross entropy $p(a_{1:T} | x_{1:T}) = \\prod _{t=1}^T p(a_t | x_t).$ [41] proposed a mapping $\\mathcal {P}$ as follows.",
"Given a sequence $a_1, \\dots , a_T$ where $a_t \\in L \\cup \\lbrace \\varnothing \\rbrace $ for $t = 1, \\dots , T$ and $\\varnothing $ is the blank symbol.",
"Let $\\mathcal {B}$ be a function that first replaces duplicate contiguous labels into a single label, and second removes all the blank symbols.",
"For example, $\\text{\\texttt {k} \\texttt {ae} \\texttt {t}} = \\mathcal {B}(\\text{$\\varnothing $ \\texttt {k} \\texttt {k} $\\varnothing $ \\texttt {ae} \\texttt {ae} \\texttt {t}\\texttt {t} $\\varnothing $}).$ [41] used $\\mathcal {P}(y) = \\mathcal {B}^{-1}(y) = \\lbrace a = (a_1, \\dots , a_T) :a_t \\in L \\cup \\lbrace \\varnothing \\rbrace , \\mathcal {B}(a) = y\\rbrace $ to map a label sequence to a sequence of the same length as the input sequence.",
"However, this function $\\mathcal {P}$ returns a set of sequences, so the objective is modified to $p(y_{1:n} | x_{1:T}) = \\sum _{a_{1:T} \\in \\mathcal {B}^{-1}(y_{1:n})} p(a_{1:T} | x_{1:T})= \\sum _{a_{1:T} \\in \\mathcal {B}^{-1}(y_{1:n})} \\prod _{t=1}^T p(a_t | x_t),$ marginalizing over all the possible paths.",
"Given an input sequence $x$ , predicting a label sequence is done by finding $\\operatornamewithlimits{\\arg \\!\\max }_{y} p(y | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y} \\sum _{a \\in \\mathcal {B}^{-1}(y)} p(a | x).$ However, there are currently no algorithms that can solve the above efficiently, so it is typically approximated by finding the greedy best path $\\operatornamewithlimits{\\arg \\!\\max }_{y} p(y | x) = \\operatornamewithlimits{\\arg \\!\\max }_{y} \\max _{a \\in \\mathcal {B}^{-1}(y)} p(a | x)= \\mathcal {B}(\\operatornamewithlimits{\\arg \\!\\max }_{a} p(a|x)).$ In other words, we simply find the label that achieves the highest probability at every time point and remove the duplicates and blank symbols to produce the final decoded sequence.",
"As a result of (REF ), the search space of CTC has an edge for every label in the label set (including the blank label) at every time step.",
"Specifically, the search space $G$ includes the edges $\\lbrace e_{\\ell , t}: \\ell \\in L, t \\in \\lbrace 1, \\dots , T\\rbrace \\rbrace $ with $v_{t-1} = \\mathrm {tail}(e_{\\ell , t})$ and $v_t = \\mathrm {head}(e_{\\ell , t})$ .",
"An example is shown in Figure REF .",
"The weight of an edge $e_{\\ell , t}$ is the log probability of label $\\ell $ at time $t$ .",
"By construction, the decision made at every time point is independent of the decision at other time points conditioned on $x_{1:T}$ .",
"In addition, since the probabilities at every time point sum to one, the partition function $Z(x)$ of the search space, i.e., the sum of all the path probabilities, is always 1.",
"Recall that the marginal log loss is defined as $\\mathcal {L}_{\\text{mll}} = -\\log \\sum _{p \\in \\mathcal {P}_{G \\circ _\\sigma F_y}} \\exp (w(p))+ \\log \\sum _{p \\in \\mathcal {P}_{G}} \\exp (w(p))$ where $\\mathcal {P}_K$ is the set of paths in the FST $K$ and $F_y$ is the constraint FST constructed from the ground truth label sequence $y$ .",
"Since $Z(x) = 1$ for CTC, the second term is zero.",
"By the definition of $\\mathcal {B}$ , we construct the constraint FST $F_y$ such that it consists of the sequences of one or more labels with zero or more blanks in between labels.",
"For example, for the label sequence “k ae t,” the constraint FST is the regular expression $\\varnothing ^*\\texttt {k}^+\\varnothing ^*\\texttt {ae}^+\\varnothing ^*\\texttt {t}^+\\varnothing ^*$ .",
"We have $\\mathcal {B}^{-1}(y) = G \\circ _\\sigma F_y$ .",
"In other words, the sum of path probabilities in $G \\circ _\\sigma F_y$ exactly matches the CTC objective.",
"Because the first term matches the CTC objective and the second term is zero, marginal log loss becomes the CTC objective for this type of search spaces and the log probability weight function.",
"Figure: An example of the CTC search space for a five-frame utterancewith a label set of size three (plus one blank).Most mainstream end-to-end speech recognition models can be broadly categorized as either frame-based models or encoder-decoder models.",
"Frame-based LSTMs trained with CTC, HMMs, and some newer approaches like the auto-segmentation criterion (ASG) [25] fall under the first category, because these models emit one symbol for every frame.",
"Falling under the second category, encoder-decoder models proposed by [21], [9], [16] generate labels one at a time while conditioning on the input and the labels generated in the past, without an explicit alignment between labels and frames.",
"Since frame-based models follow the same graph search framework as segmental models, we will focus on discussing the connection between these and segmental models.",
"Recall that marginal log loss requires a search space $G$ and a constraint FST $F$ to limit the search space to ground-truth labels.",
"To compute marginal log loss, we first compute the marginals on $G$ for computing the partition function $Z(x)$ , and then compute the marginals on the $\\sigma $ -composed FST $G \\circ _\\sigma F$ for computing $Z(x, y)$ .",
"CTC, HMMs whtn trained with lattice-free MMI [111], and ASG can all be seen as special cases of this framework.",
"Comparing CTC to HMMs, the search space is different depending on the HMM topology.",
"For example, two-state HMMs are used in [111].",
"Since the transition probabilities and emission probabilities are all locally normalized, the partition function $Z(x)$ is always 1.",
"The constraint FST representing the ground-truth labels consists simply of sequences of repeating labels.",
"For example, for the label sequence “k ae t,” the constraint FST for one-state HMMs is the regular expression $\\texttt {k}^+\\texttt {ae}^+\\texttt {t}^+$ .",
"For two-state HMMs, the constraint FST is the regular expression $\\texttt {k}_1\\texttt {k}_2^*\\texttt {ae}_1\\texttt {ae}_2^*\\texttt {t}_1\\texttt {t}_2^*$ .",
"With the above construction, marginal log loss applied to HMMs is equivalent to lattice-free MMI [111].",
"For ASG, the search space is equivalent to that of one-state HMMs.",
"Instead of assuming conditional independence as in CTC, ASG includes transition probabilities between states.",
"The constraint FST is identical to that of HMMs, with repeated ground-truth labels.",
"However, in ASG the weights on the edges are not locally normalized, so the partition function $Z(x)$ is not always 1 and has to be computed.",
"With the above search space construction, marginal log loss becomes ASG.",
"Another approach similar to CTC proposed in [38] is called RNN transducers.",
"The search space of an RNN transducer is the set of alignments from the speech signal to all possible label sequences, so the search space grows exponentially in the number of labels.",
"The weight function of a path in this approach relies on an RNN, and is not decomposable as a sum of weights of the edges.",
"RNN transducers are trained with marginal log loss.",
"By the independence assumption imposed in [38], the partition function $Z(x)$ is still 1, so we do not need to marginalize over the exponentially large space.",
"During decoding, however, we still have to search over the exponentially large space with, for example, beam search.",
"In view of this framework, even when using the same loss function, i.e., marginal log loss, segmental models and frame-based models differ in their search space, weight functions, and how the search space is constrained by the ground-truth labels during training.",
"A summary of special cases is shown in Table REF .",
"Table: An example instantiation of the components used in marginal log losswith the ground-truth sequence “k ae t”and TT input framesfor segmental models and various end-to-end models.The search space L T L^T consists of sequences of TT labels.The label sets L 1 L_1 and L 2 L_2 contain labels in LL with subscript 1 and 2 respectively.The search space is denoted GG and the constraint FST is denoted F y F_y in the table.We compare various frame-based models trained end to end on the same phonetic recognition task in the same setting as in Section REF .",
"We use 3-layer 256-unit biderectional LSTMs paired with CTC, one-state HMM, and two-state HMMs.",
"We do not use transition probabilities for two-state HMMs.",
"We optimize marginal log loss with the corresponding weight functions and the corresponding constraint FSTs for CTC, one-state HMM, and two-state HMMs.",
"We run vanilla SGD with step size 0.1 and gradient clipping of norm 5 for 20 epochs.",
"Starting from the best performing model of the first 20 epochs, we run vanilla SGD for another 20 epochs with step size 0.75 decayed by 0.75 after each epoch.",
"The dropout rate is 0.2, and no other explicit regularizer is used except early stopping on the development set.",
"The same experiments are repeated with pyramid LSTMs.",
"Results comparing with segmental models are shown in Table REF .",
"When standard LSTMs are used, we fail to minimize the training loss for one-state HMMs and two-state HMMs The only difference between CTC and one-state HMMs is the use of blank symbols, suggesting that blank symbols play an important role in end-to-end frame-based models.",
"In particular, one-state HMMs and two-state HMMs explicitly marginalize over segmentations in training and are thus sensitive to time, while LSTMs trained with CTC are not.",
"Besides, the tail states in $L_2$ can be seen as label-dependent blank symbols.",
"Since two-state HMMs fail to achieve a low training loss, having label-dependent blanks is not helpful in this case.",
"When pyramid LSTMs are used, all variants of frame-based models achieve low PERs on the development set and improve over ones with standard LSTMs.",
"Since the time resolution is reduced by four with pyramid LSTMs, we hypothesize that using the pyramid LSTMs introduces a bias similar to a minimum duration constraint, which helps end-to-end training for frame-based models.",
"The number of minutes per epoch for training segmental models and LSTMs trained with CTC is shown in Table REF , and a scatter plot showing number of minutes per epoch vs.",
"PER is shown in Figure REF .",
"Pyramid LSTMs trained with CTC are the best performer while also being the most efficient to train.",
"The real-time factors for decoding are shown in Table REF .",
"Due to smaller and simpler search spaces, frame-based LSTMs trained with CTC decode faster than segmental models.",
"While frame-based LSTMs train and decode faster, segmental models are more flexible, so they might be improved with additional feature functions.",
"Explicitly hypothesizing segments can serve as a prior, so segmental models might perform better in low-resource settings.",
"Table: Phoneme error rates (%) for end-to-end training fromrandom initialization comparing CTCand segmental models trained with marginal log loss.Table: Average number of minutes for one epoch of end-to-end training on TIMIT.Table: Real-time factors for decoding comparing CTC and segmental models.Figure: Training minutes per epoch vs.",
"PER (%) for segmental modelsand LSTMs trained with CTC.For American Sign Language (ASL) fingerspelling experiments, we follow [72] and compare end-to-end models in both signer-dependent setting and signer-independent setting.",
"The input is a sequence of 128-dimensional histograms of oriented gradients (HoG) feature vectors computed over $128 \\times 128$ hand shape images.",
"The output is a sequence of English letters, and the label set consists of the 26 English characters plus two labels for initial and final non-signing regions.",
"The evaluation metric is the edit distance between the predicted letter sequence and the ground-truth letter sequence.",
"For the signer-dependent setting, the data for each signer is split into ten folds, in which eight are for training, one is for development, and one is for testing.",
"We report the letter error rate averaged over the 10 experiments.",
"We train LSTMs with CTC and segmental models with the FCB weight function.",
"Both models are trained with marginal log loss from random initialization.",
"The encoder is a one-layer 128-unit bidirectional LSTM.",
"For CTC, we run vanilla SGD with step sizes in $\\lbrace 0.1, 0.05, 0.025\\rbrace $ and gradient clipping with norm 5 for 200 epochs.",
"Dropout is used with a rate of 0.2, and no other explicit regularizer is used except early stopping.",
"Step sizes and early stopping are tuned on the development set for each individual experiment.",
"For segmental models, we run vanilla SGD with step size 0.1 for 75 epochs.",
"The maximum segment length is 75 frames.",
"Other hyperparameters, such as dropout and gradient clipping, are the same as for CTC.",
"Results are shown in Table REF .",
"The frame-based LSTMs trained with CTC are better than Tandem HMMs, but are behind the segmental models.",
"The segmental models trained end to end are on par with the two-stage system in [72], but are behind the discriminative segmental cascades (DSC).",
"Table: Letter error rates (%) for signer-dependent models averagedover ten folds.For the signer-independent setting, we train on three signers and use the last signer for development and testing.",
"Specifically, the data of the last signer is split into ten folds, and we use two folds for development and the rest of the eight folds for testing.",
"Instead of dividing the accumulated error counts by the accumulated label sequence lengths, letter error rates are averaged over the folds.",
"We compare LSTMs trained with CTC and segmental models with FCB weight function.",
"The loss function and training procedure stay the same except that we only run 40 epochs of vanilla SGD for segmental models.",
"Results are shown in Table REF .",
"In this setting, CTC and segmental models trained end to end perform significantly better than Tandem HMMs and segmental models trained with the FC weight function in two stages.",
"To compare signer-independent and signer-dependent settings, we average the letter error rates over the same eight folds.",
"Results are shown in Table REF .",
"Segmental models perform significantly better than CTC in both settings.",
"Table: Letter error rates (%) for signer-independent modelsaveraged over eight folds.Table: Letter error rates (%) for signer-dependent and signer-independent models averagedover the same eight folds.We have discussed how other end-to-end frame-based models, such as CTC, HMMs trained with lattice-free MMI, ASG, and RNN transducers are all trained with marginal log loss.",
"The differences among them lie in the search spaces and the weight functions.",
"Drawing these connections allows us to generalize search spaces, loss functions, and weight functions to a broad class of models.",
"From the results of comparing CTC to one-state HMMs and two-state HMMs, we have found that the blank symbol seems to play an important role in training LSTMs end to end.",
"Using pyramid LSTMs improves both the performance and the runtime of decoding and training for CTC, one-state HMMs, and two-state HMMs, but not for segmental models.",
"We have also shown that segmental models with regular LSTMs are better than regular LSTMs trained with CTC on both phonetic recognition and ASL fingerspelling recognition.",
"In this thesis, we have made the following contributions in advancing the study of discriminative segmental models.",
"We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.",
"We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.",
"We obtain improved performance over most earlier work while greatly improving efficiency.",
"We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.",
"We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.",
"We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.",
"Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.",
"We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.",
"Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.",
"We have proposed discriminative segmental cascades for incorporating rich computationally expensive features while maintaining efficiency.",
"We use max-marginal pruning to reduce the size of search spaces, generating sparse lattices while having low oracle error rates.",
"We obtain improved performance over most earlier work while greatly improving efficiency.",
"We have explored the space of losses, multi-stage training, and end-to-end training for segmental models with various losses and weight functions.",
"We have shown that segmental models trained with multi-stage training can serve as a good initialization for end-to-end training.",
"We have presented a unified framework including many end-to-end models, such as hidden Markov models, connectionist temporal classification, and segmental models, as special cases.",
"Drawing this connection allows us to design general search spaces, loss functions, and weight functions applicable to models in a broad class.",
"We achieve competitive results on phonetic recognition and ASL fingerspelling recognition with a segmental model trained end to end.",
"Earlier work (not reported in this thesis) by [72] has obtained the best reported results to date using our discriminative segmental cascades on signer-dependent fingerspelling recognition.",
"In this chapter, we discuss some potential future work extending segmental models to large-vocabulary tasks and to unsupervised settings.",
"Another potential direction for future work is even richer feature functions.",
"Since first-pass segmental models are computationally demanding, it is very slow to train and decode segmental models with label sets of size in the order of 10,000.",
"This poses a challenge for tasks with large label sets, such as word recognition.",
"[150], [88] bypassed this difficulty by using a baseline HMM recognizer to generate word lattices, and used segmental models to rescore these lattices, exploring various word-level features.",
"However, the performance is constrained by the quality of the baseline recognizer and the quality of the lattices.",
"First-pass segmental models have previously been successfully applied to word recognition [35], [50].",
"This previous work treats first-pass segmental models as a drop-in replacement for HMM phoneme recognizers, because both models serve as functions that map acoustic features to phoneme strings.",
"The phoneme recognizers are then composed with a lexicon and a language model to form a word recognizer, as described in Section REF .",
"This is also an option for extending our work to word recognition.",
"Recent work has explored models that directly predict characters, avoiding the need for a lexicon [39], [92] but still allowing for improved performance when constraining the search space with a lexicon (through FST composition) [92].",
"Segmental models can also be used to predict characters by changing the label set, but might be hampered by the poor alignment of characters to acoustics.",
"Syllables might be a better option than characters.",
"Instead of using intermediate discrete representations, such as phonemes or characters, recent advances in computing power have made it feasible to directly predict words [87], [11], [127], [7].",
"In this case, rather than using a pronunciation dictionary, only a list of words is needed for decoding.",
"Segmental models can also be used to directly predict words by using the list of words as the label set.",
"This approach is worth exploring further, although efficiency issues make it nontrivial to train such models [7].",
"We have presented segmental models for supervised sequence prediction in this thesis.",
"Our framework can be extended to unsupervised sequence prediction.",
"The goal in this setting is typically grouping segments of varying length into clusters.",
"We define unsupervised sequence prediction as finding a function that maps an input sequence $x_1, \\dots , x_T$ into a sequence of segments $(\\ell _1, s_1, t_1), \\dots , (\\ell _n, s_n, t_n)$ where $\\ell _i \\in L$ for $i=1, \\dots , n$ and some label set $L$ , and $s_1 = 1$ , $t_n = T+1$ , $s_i < t_i$ , $t_{i-1} = s_i$ for $i = 1, \\dots , n$ , given a data set $S$ of sequences without labels.",
"Since we do not have labels for the data samples, the goal in general is to design a loss function $\\mathcal {L}(\\Theta ; x)$ that only depends on the input sequence $x$ , or even more generally a loss function $\\mathcal {L}(\\Theta ; S)$ that depends on the data set $S$ .",
"The label set $L$ is typically predefined to be just a set of identifiers, and the correspondence between the input sequence and the identifier sequence is learned from a data set.",
"As a result, a label in $L$ does not have a predefined meaning unless the user assigns a post hoc meaning or the loss function enforces one.",
"Generative segmental models, such as hidden semi-Markov models, can be naturally extended to the unsupervised setting by maximizing the marginal likelihood of the data.",
"For example, Bayesian segmental models have been applied to small-vocabulary [64] and larger-vocabulary [65] word discovery.",
"Viterbi-style training for such models has also been explored [66].",
"Recently, [134] proposed unsupervised neural HMMs, which can be easily extended to hidden semi-Markov models.",
"In fact, [26] proposed a neural version of hidden semi-Markov models similar to [134].",
"All the above approaches aim to maximize the marginal likelihood of the data.",
"Besides generative models, approaches for training log-linear models in an unsupervised fashion, notably contrastive estimation [126], [107], noise-contrastive estimation [44], and auto-encoders [4], have also been extensively studied.",
"These approaches are designed for discriminative models, and can be readily applied to our segmental models.",
"A segmental model trained in an unsupervised fashion can be used for segment clustering.",
"One major application of segment clustering is for lexical unit discovery.",
"There is a long history of lexical unit discovery from acoustic signals [14], sometimes called spoken term discovery [105], [60].",
"Much of the recent work relies on dynamic time warping (DTW) [59], [15] for spoken term discovery, with [78] being out of the few exceptions who took a nonparametric Bayesian approach.",
"DTW has also been found to help lexical discovery in an HMM system [140].",
"Other related tasks that we do not cover here include unsupervised word segmentation [37], unsupervised part-of-speech tagging [91], and unsupervised dependency parsing [73].",
"Though segmental models were not used in the above studies, many ideas can be borrowed and help advance the study of segmental models in unsupervised settings.",
"The assumption that input sequences can be decomposed into a sequence of segments serves as a strong inductive bias, and it is particularly useful in unsupervised settings when little is assumed about the data.",
"Segmental models are more flexible than frame-based models when additional assumptions, such as the form of the feature functions, are needed.",
"Therefore, we believe segmental models have the potential to perform well in these unsupervised settings."
]
] | 1709.01572 |
[
[
"Weak solutions for a thermoelectric problem with power-type boundary\n effects"
],
[
"Abstract This paper deals with thermoelectric problems including the Peltier and Seebeck effects.",
"The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects.",
"To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3).",
"They are essentially approximated solutions based on the Rothe method.",
"It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero.",
"The proof of the existence of time discretized solutions relies on fixed point and compactness arguments.",
"In this study, we establish quantitative estimates to clarify the smallness conditions."
],
[
"Introduction",
"The study of the heat equation with constant coefficients is a simplification from both mathematical and engineering points of view.",
"From the real world point of view, constant coefficients are not appropriate because the density and the thermal conductivity both depend on the temperature itself, and often also on the spatial variable.",
"The concern of discontinuous leading coefficient is being a long matter of study in the mathematical literature, as long as the works [22], [25].",
"The complete concern is achieved by the doubly quasilinear parabolic equation [1], [3], [5].",
"It is well known that the determination of estimates is the crucial key in the theory of partial differential equations (PDE), which involve the so-called universal bounds.",
"With their abstract form, these bounds are only qualitative and they do not have any practical use on the real world applications.",
"In their majority, if the proof of estimates should be remade step by step, the expression of the qualitative bounds would be truly cumbersome, or even impossible if the contradiction argument is applied.",
"Also regularity estimates have being a subject of study in the last decades [11], [12], [15], but these ones only occur by admitting data smoothness.",
"With this in mind, our main objective is to find quantitative estimates, i.e.",
"their involved constants have an explicit expression, that are useful on the real applications.",
"In particular, the quantitative estimates clarify the smallness conditions on the data when a fixed point argument is used.",
"For the two-dimensional space situation, a first attempt on the finding smallness conditions that assure the existence and regularity results for some thermoelectric problems is presented in [9], [10], where some domain dependent constants were kept abstract.",
"Indeed, the central existence result of a weak solution for a class of elliptic systems on divergence form, which is only 2D valid, is provided under some higher regularity ($W^{1,p}$ regularity, with $p > 2$ ).",
"The Gehring-type higher integrability technique makes the smallness conditions quite bizarre.",
"Here, we establish more elegant smallness conditions and they are extended to the $n$ -dimensional space situation, by finding weak solutions.",
"The present model also extends the thermal effects, of the previous works [9], [10], to the unsteady state.",
"Existence of solutions for parabolic-elliptic systems with nonlinear no-flux boundary conditions is not a new idea if taking constant coefficients into account [4].",
"Application of elliptic PDE system in divergence form with Dirichlet boundary conditions in doubly-connected domain of the plane are given in [7] to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour effects.",
"In [8], the authors deal with a traditional RLC circuit in which a thermistor has been inserted, representing the microwave heating process with temperature-induced modulations on the electric field.",
"In particular, the existence of a solution to a coupled system of three differential equations (an ODE, an elliptic equation and a nonlinear parabolic PDE) and appropriate initial and boundary conditions is proved.",
"A one-dimensional thermal analysis for the performance of thermoelectric cooler is conducted in [13] under the influence of the Thomson effect, the Joule heating, the Fourier heat conduction, and the radiation and convection heat transfer.",
"Simulation studies have been performed to investigate the thermal balance affected by anode shorting in an aluminum reduction cell [6].",
"The method of discretization in time, whose basic idea (coming from the implicit Euler formula) was investigated by Rothe, is a very well-known effective technique for both theoretical and numerical analysis, [14], [17], [24] and [21], [26], respectively (see also the pioneering work [1] of Alt and Luckhaus).",
"This paper is organized as follows.",
"The thermoelectric (TE) model is introduced in Section .",
"After discussing the physical model, the main result with respect to this model is formulated with a detailed description of the relevant constants.",
"In Section , one abstract model related to the problem under consideration is introduced to simplify the proofs of the existence results of time-discretized solutions (Section ), and their corresponding steady-state solutions (Section ).",
"Indeed, the analysis of the problem is structured via two different approaches to exemplify alternative assumptions on the data smallness, namely the existence results of time-discretized solutions (Subsections REF and REF ), and their corresponding steady-state solutions (Subsections REF and REF )."
],
[
"The thermoelectric model",
"Let $[0, T] \\subset {\\mathbb {R}}$ be the time interval with $ T >0$ being an arbitrary (but preassigned) time.",
"Let $\\Omega $ be a bounded domain (that is, connected open set) in $\\mathbb {R}^n$ ($n\\ge 2$ ).",
"Its boundary is constituted by two disjoint open $(n-1)$ -dimensional sets $\\partial \\Omega =\\overline{\\Gamma _\\mathrm {N}}\\cup \\overline{\\Gamma }$ .",
"We consider $\\Gamma _{\\rm N}$ over which the Neumann boundary condition is taken into account, and $\\Gamma $ over which the radiative effects may occur.",
"Each one, $\\Gamma _{\\rm N}$ and $\\Gamma $ , may be alternatively of zero $(n-1)$ -Lebesgue measure.",
"Set $Q_T=\\Omega \\times ]0,T[$ and $\\Sigma _T=\\Gamma \\times ]0,T[$ .",
"The electrical current density $\\bf j$ and the energy flux density ${\\bf J}={\\bf q}+\\phi {\\bf j}$ , with $\\bf q$ being the heat flux vector, are given by the constitutive relations (see [9] and the references therein) ${\\bf q}&= &- k (\\cdot ,\\theta ) \\nabla \\theta -\\Pi (\\cdot ,\\theta ) \\sigma (\\cdot ,\\theta ) \\nabla \\phi ;\\\\{\\bf j}&=& -\\alpha _{\\rm S}(\\cdot ,\\theta ) \\sigma (\\cdot ,\\theta ) \\nabla \\theta -\\sigma (\\cdot ,\\theta ) \\nabla \\phi .$ Here, $\\theta $ denotes the absolute temperature, $\\phi $ is the electric potential, $\\alpha _{\\rm S}$ represents the Seebeck coefficient, and the Peltier coefficient $\\Pi (\\theta )=\\theta \\alpha _{\\rm s}(\\theta )$ is due to the first Kelvin relation.",
"The electrical conductivity $\\sigma $ , and the thermal conductivity $k=k_{\\rm T}+\\Pi \\alpha _{\\rm s}\\sigma $ , with $k_T$ denotes the purely conductive contribution, are, respectively, the known positive coefficients of Ohm and Fourier laws.",
"The Seebeck coefficient $\\alpha _{\\rm S}$ has a constant sign corresponding to the Hall effect.",
"With positive sign ($\\alpha _{\\rm S}>0$ ), there are as examples: the alkali metals Li, Rb and Cs [2], and the noble metals Ag and Au [2] or [20].",
"With negative sign ($\\alpha _{\\rm S}<0$ ), there are as examples: the alkali metals Na and K [20], the transition metals Fe and Ni [2], and the semiconductor Pb [2].",
"We refer to [10], and the references therein, for more examples and their increase and decrease behaviors.",
"Although heat generation starts instantaneously when the current begins to flow, it takes time before the heat transfer process is initiated to allow the transient conditions to disappear.",
"Thus, the electrical current density $\\bf j$ and the energy flux density ${\\bf J}$ satisfy $&&\\left\\lbrace \\begin{array}{ll}\\nabla \\cdot {\\bf j}=0 &\\mbox{ in }\\Omega \\\\-{\\bf j}\\cdot {\\bf n}=g &\\mbox{ on }\\Gamma _{\\rm N}\\\\{\\bf j}\\cdot {\\bf n}=0 &\\mbox{ on }\\Gamma \\end{array}\\right.\\\\ &&\\left\\lbrace \\begin{array}{ll}\\rho (\\cdot ,\\theta ) c_\\mathrm {v}(\\cdot , \\theta )\\partial _t \\theta -\\nabla \\cdot {\\bf J}=0 &\\mbox{ in } Q_T\\\\{\\bf J}\\cdot {\\bf n}=0 &\\mbox{ on }\\Gamma _{\\rm N}\\times ]0,T[\\\\-{\\bf J}\\cdot {\\bf n}=\\gamma (\\cdot ,\\theta ) |\\theta |^{\\ell -2}\\theta -h &\\mbox{ on }\\Sigma _T,\\end{array}\\right.$ for $\\ell \\ge 2$ .",
"Here, $\\rho $ denotes the density, $c_\\mathrm {v}$ denotes the heat capacity (at constant volume), $\\bf n$ is the unit outward normal to the boundary $\\partial \\Omega $ , and $g$ denotes the surface current source, The boundary operators, $\\gamma $ and $h$ , are temperature dependent functions that express, respectively, the radiative convection depending on the wavelength, and the external heat sources.",
"For $\\ell =5$ , the Stefan-Boltzmann radiation law says that $\\gamma (T)=\\sigma _{\\rm SB}\\epsilon (T)$ and $h(T)=\\sigma _{\\rm SB} \\alpha (T) \\theta _\\mathrm {e}^{\\ell -1}$ , where $\\sigma _{\\rm SB}=\\, 5.67\\times 10^{-8}$ W m$^{-2} $ K$^{-4}$ is the Stefan-Boltzmann constant for blackbodies, and $ \\theta _\\mathrm {e}$ denotes an external temperature.",
"The parameters, the emissivity $\\epsilon $ and the absorptivity $\\alpha $ , both depend on the space variable and the temperature function $\\theta $ .",
"If $\\ell =2$ , the boundary condition corresponds to the Newton law of cooling with heat transfer coefficient $\\gamma =h/ \\theta _\\mathrm {e}^{\\ell -1}$ .",
"In the framework of Sobolev and Lebesgue functional spaces, we use the following spaces of test functions: $V &=& \\left\\lbrace \\begin{array}{l}V(\\Omega ) = \\left\\lbrace v\\in H^{1}(\\Omega ):\\ \\int _{\\Omega } v \\mathrm {dx}=0 \\right\\rbrace \\\\V(\\partial \\Omega ) = \\left\\lbrace v\\in H^{1}(\\Omega ):\\ \\int _{\\partial \\Omega } v \\mathrm {ds}=0 \\right\\rbrace \\end{array}\\right.",
"\\\\V_{\\ell }(\\Omega ) &=&\\left\\lbrace v\\in H^{1}(\\Omega ) :\\ v|_{\\Gamma }\\in L^{\\ell }(\\Gamma ) \\right\\rbrace ;\\\\V_{\\ell }(Q_T) &=&\\left\\lbrace v\\in L^2(0,T; H^{1}(\\Omega )) :\\ v|_{\\Sigma _T }\\in L^{\\ell }(\\Sigma _T) \\right\\rbrace ,$ with their usual norms, $\\ell >1$ .",
"Notice that $V_{\\ell }(\\Omega )\\equiv H^{1}(\\Omega ) $ if $\\ell \\le 2_*$ , where $ 2_*$ is the critical trace exponent, i.e.",
"$2_*=2(n-1)/(n-2)$ if $n>2$ and $2_*>1$ is arbitrary if $n=2$ .",
"In the presence of the previous considerations, the temperature-potential pair does not be expectable to be regular nor even bounded.",
"The thermoelectric problem is formulated as follows.",
"(TE) Find the temperature-potential pair $(\\theta ,\\phi )$ such that if it verifies the variational problem: $\\int _0^T\\langle \\rho (\\cdot , \\theta ) c_\\mathrm {v}(\\cdot , \\theta )\\partial _t\\theta , v\\rangle \\mathrm {dt} +\\int _{Q_T} k (\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} +\\nonumber \\\\ +\\int _{Q_T} \\sigma (\\cdot ,\\theta )\\left(T_\\mathcal {M}(\\phi )\\alpha _{\\rm S}(\\cdot ,\\theta )\\nabla \\theta +(\\Pi (\\cdot ,\\theta )+T_\\mathcal {M}(\\phi ))\\nabla \\phi \\right) \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\nonumber \\\\+\\int _{\\Sigma _T} \\gamma (\\cdot ,\\theta )|\\theta |^{\\ell -2} \\theta v\\mathrm {ds} \\mathrm {dt}= \\int _{\\Sigma _T} h(\\cdot ,\\theta ) v\\mathrm {ds} \\mathrm {dt}; \\\\\\int _\\Omega \\sigma (\\cdot ,\\theta )\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+\\int _\\Omega \\sigma (\\cdot ,\\theta ) \\alpha _{\\rm S}(\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\ \\mbox{ a.e.",
"in } ]0,T[ , $ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ , where $\\langle \\cdot ,\\cdot \\rangle $ accounts for the duality product, and $T_\\mathcal {M}$ is the $\\mathcal {M}$ -truncation function defined by $T_\\mathcal {M}(z)=\\max (-\\mathcal {M},\\min (\\mathcal {M},z))$ .",
"We assume the following.",
"(H1) The density and the heat capacity $\\rho , c_\\mathrm {v}: \\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions, i.e.",
"measurable with respect to $x\\in \\Omega $ and continuous with respect to $e\\in \\mathbb {R}$ .",
"Furthermore, they verify $\\exists b^\\#,b_\\#>0: \\quad b_\\#\\le \\rho (x,e)c_\\mathrm {v}(x,e)\\le b^\\# ,\\quad \\mbox{for a.e. }",
"x\\in \\Omega ,\\quad \\forall e \\in \\mathbb {R} .",
"$ (H2) The thermal and electrical conductivities $k,\\sigma :\\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions.",
"Furthermore, they verify $\\exists k^\\#,k_\\#>0:&&k_\\#\\le k(x,e)\\le k^\\#;\\\\\\exists \\sigma ^\\#, \\sigma _\\#>0:&&\\sigma _\\#\\le \\sigma (x,e)\\le \\sigma ^\\#\\quad \\mbox{for a.e. }",
"x\\in \\Omega ,\\quad \\forall e \\in \\mathbb {R}.",
"$ (H3) The Seebeck and Peltier coefficients $\\alpha _{\\rm S},\\Pi :\\Omega \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ are Carathéodory functions such that $\\exists \\alpha ^\\#>0: &&|\\alpha _{\\rm S}(x,e)|\\le \\alpha ^\\#; \\\\\\exists \\Pi ^\\#>0: &&|\\Pi (x,e)|\\le \\Pi ^\\#,\\quad \\mbox{for a.e. }",
"x\\in \\Omega ,\\quad \\forall e\\in \\mathbb {R}.$ (H4) The boundary function $h$ belongs to $L^{\\ell ^{\\prime }}(\\Sigma _T)$ .",
"(H5) The boundary function $g$ belongs to $ L^{2}(\\Gamma _{\\rm N})$ .",
"(H6) The boundary operator $\\gamma $ is a Carathéodory function from $\\Sigma _T\\times \\mathbb {R}$ into $\\mathbb {R}$ such that $\\exists \\gamma _\\#, \\gamma ^\\#>0:\\quad \\gamma _\\#\\le \\gamma (x,t,e)\\le \\gamma ^\\#; \\quad \\mbox{for a.e. }",
"(x,t)\\in \\Sigma _T,\\quad \\forall e\\in \\mathbb {R}.",
"$ Moreover, $\\gamma $ is strongly monotone: $\\left(\\gamma ( u )| u |^{\\ell -2} u- \\gamma ( v )| v |^{\\ell -2} v \\right) ( u- v)\\ge \\gamma _\\# |u - v|^\\ell .$ Let us state our main existence theorem.",
"Theorem 2.1 Let (H1)-(H6) be fulfilled.",
"The thermoelectric problem (TE) admits a solution $(\\theta ,\\phi )\\in V_\\ell (Q_T)\\times L^2(0,T; V)$ , for $\\mathcal {M}$ being such that $\\mathcal {M}\\alpha ^\\#\\sigma ^\\#< k_\\#,$ and one of the following hypothesis is assured: there holds $4( k_\\# - \\mathcal {M}\\alpha ^\\#\\sigma ^\\# ) \\sigma _\\# > (\\sigma ^\\#)^2 ( \\Pi ^\\#+\\mathcal {M} +\\alpha ^\\#)^2 ;$ there holds $4( k_\\# - \\mathcal {M}\\alpha ^\\#\\sigma ^\\# ) >\\sigma ^\\# ( \\Pi ^\\#+\\mathcal {M} +\\alpha ^\\#)^2 ;$ there holds $k_\\#> \\sigma ^\\# \\alpha ^\\# ( 2 \\Pi ^\\#+ 3 \\mathcal {M}) .$"
],
[
"Existence of approximated solutions",
"The thermoelectric problem provides the abstract initial boundary value problem $b(\\theta )\\partial _t \\theta -\\nabla \\cdot \\left( a ( \\theta ,\\phi )\\nabla \\theta \\right)= \\nabla \\cdot ( \\sigma ( \\theta ) F (\\theta ,\\phi )\\nabla \\phi ) && \\\\-\\nabla \\cdot (\\sigma ( \\theta )\\nabla \\phi )= \\nabla \\cdot \\left( \\sigma ( \\theta ) \\alpha _\\mathrm {S} (\\theta ) \\nabla \\theta \\right) && \\mbox{ in }Q_T ; \\\\\\left( a ( \\theta ,\\phi )\\nabla \\theta + \\sigma ( \\theta ) F ( \\theta ,\\phi )\\nabla \\phi \\right)\\cdot \\mathbf {n} =\\left(h - \\gamma (\\theta ) |\\theta |^{\\ell -2} \\theta \\right)\\chi _{\\Gamma } && \\\\\\left( \\sigma ( \\theta )\\nabla \\phi + \\sigma ( \\theta ) \\alpha _\\mathrm {S} (\\theta )\\nabla \\theta \\right)\\cdot {\\bf n} =g \\chi _{\\Gamma _\\mathrm {N}}&&\\mbox{ on } \\partial \\Omega \\times ]0,T[ .",
"$ This abstract problem is formulated in the form that the coefficients are correlated with the leading coefficient $\\sigma $ .",
"We emphasize that this interrelation must be clear.",
"Let us assume the hypothesis set.",
"(H) The operators $ a,F $ and $b,\\sigma , \\alpha _\\mathrm {S}$ are Carathéodory functions from $\\Omega \\times \\mathbb {R}^2$ and $\\Omega \\times \\mathbb {R}$ , respectively, into $\\mathbb {R}$ , which enjoy the following properties.",
"There exist positive constants $F^\\#, a_\\#,a^\\#, b_\\#,b^\\#$ such that $| F(x,e,d )|\\le F^\\#; && \\\\a_\\#\\le a(x,e,d )\\le a^\\#; && \\\\b_\\#\\le b (x,e )\\le b^\\# &&\\mbox{for a.e. }",
"x\\in \\Omega ,\\quad \\forall e,d \\in \\mathbb {R} , $ and $\\sigma _\\#,\\sigma ^\\#, \\alpha ^\\#$ verifying (), (REF ), respectively.",
"Definition 3.1 We say that $(\\theta ,\\phi )$ is a weak solution to (REF )-() if it solves the variational problem $\\int _0^T\\langle b(\\cdot , \\theta )\\partial _t\\theta , v\\rangle \\mathrm {dt} +\\int _{Q_T} a (\\cdot ,\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma (\\cdot ,\\theta )|\\theta |^{\\ell -2} \\theta v\\mathrm {ds} \\mathrm {dt}=\\nonumber \\\\ = - \\int _{Q_T}\\sigma ( \\theta ) F (\\cdot ,\\theta ,\\phi )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\int _{\\Sigma _T} h(\\cdot ,\\theta ) v\\mathrm {ds} \\mathrm {dt}; \\\\\\int _\\Omega \\sigma (\\cdot ,\\theta )\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\cdot , \\theta ) \\alpha _\\mathrm {S} (\\cdot ,\\theta )\\nabla \\theta \\cdot \\nabla w\\mathrm {dx} =\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\mbox{ a.e.",
"in } ]0,T[ ,$ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ .",
"We define an auxiliary operator.",
"Denote by $B$ the operator from $H^1(\\Omega )$ into $L^2(\\Omega )$ defined by $B(v)=\\int _0^v b(\\cdot ,z)\\mathrm {dz},$ for all $v\\in H^1(\\Omega )$ .",
"Different approaches in the finding of solutions according to Definition REF provide different smallness conditions (REF ), (REF ) or (REF ).",
"We emphasize that the difference between these smallness conditions has its importance in the real-world applications.",
"Theorem 3.1 Let (H) and (H4)-(H6) be fulfilled.",
"If there exists $\\varepsilon >0$ such that one the following relation holds, that is, either $a_\\# >\\varepsilon \\sigma ^\\#(F^\\#+\\alpha ^\\#)/2\\quad \\mbox{ and } \\quad \\varepsilon \\sigma _\\# >\\sigma ^\\#(F^\\#+\\alpha ^\\#)/2,$ or $a_\\# >\\varepsilon \\sqrt{\\sigma ^\\#}(F^\\#+\\alpha ^\\#)/2 \\quad \\mbox{ and } \\quad \\varepsilon >\\sqrt{\\sigma ^\\#}(F^\\#+\\alpha ^\\#)/2, $ then the variational problem (REF )-() admits a sequence of approximate solutions $\\lbrace (\\theta _M,\\phi _M)\\rbrace _{M\\in \\mathbb {N}}$ in the sense established in Section REF .",
"The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx} =\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _{\\Gamma } h_{m} v \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (\\theta ^m)\\nabla \\phi ^m\\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\theta ^m) \\alpha _\\mathrm {S}(\\theta ^m)\\nabla \\theta ^{m} \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ where $\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\in \\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).",
"We call $\\phi ^m$ the corresponding solution to the time independent temperature $\\theta ^{m}$ .",
"Theorem 3.2 Let (H) and (H4)-(H6) be fulfilled.",
"If there holds $a_\\#>2\\sigma ^\\# \\alpha ^\\# F^\\# ,$ then the variational problem (REF )-() admits a sequence of approximate solutions $\\lbrace (\\theta _M,\\phi _M)\\rbrace _{M\\in \\mathbb {N}}$ in the sense established in Section REF .",
"The proof of Theorem REF relies on the limit solution to the recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx} =\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _\\Gamma h_{m} v \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (\\theta ^{m-1})\\nabla \\phi ^m\\cdot \\nabla w\\mathrm {dx}= - \\int _\\Omega \\sigma (\\theta ^{m-1}) \\alpha _\\mathrm {S}(\\theta ^{m-1})\\nabla \\theta ^{m-1} \\cdot \\nabla w \\mathrm {dx} +\\nonumber \\\\ + \\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ where $\\tau $ is the so called time step, $B$ is defined in (REF ), $m\\in \\mathbb {N}$ and $h_m$ is conveniently chosen in Section (the time discretization technique).",
"We call $\\phi ^m$ the corresponding solution to the time independent temperature $\\theta ^{m-1}$ ."
],
[
"Steady-state solvability",
"In this section, we prove the existence of solutions to the recurrent sequence of time-discretized problems (REF )-() and (REF )-() in Subsections REF and REF , respectively.",
"Since $m\\in \\mathbb {N}$ is fixed and $\\theta ^{m-1}\\in V_{\\ell }(\\Omega )$ is given, for the sake of simplicity, we set $f= B(\\theta ^{m-1})$ and $H=h_m$ , and we omit the index to the unknown pair, i.e.",
"we simply write $(\\theta ,\\phi )$ .",
"Denoting by $K_{2}$ the continuity constant of the trace embedding $H^{1}(\\Omega )\\hookrightarrow L^2(\\Gamma )$ , with $2_*=2(n-1)/(n-2)$ if $n>2$ , and any $2_*>2$ if $n=2$ , and by $P_2$ the Poincaré constant correspondent to the space exponent 2, the constant $K_2(P_2+1)$ obeys $\\Vert v\\Vert _{2,\\Gamma }\\le K_2 \\left( \\Vert v\\Vert _{2,\\Omega }+ \\Vert \\nabla v\\Vert _{2,\\Omega }\\right)\\le K_2(P_2+1) \\Vert \\nabla v\\Vert _{2,\\Omega },\\quad \\forall v\\in H^1(\\Omega ).$ Let us introduce [1], [16] $\\Psi (s) := B(s)s-\\int _0^s B(r)\\mathrm {dr} = \\int _0^s(B(s) - B(r) )\\mathrm {dr}.$ We state the main properties of the auxiliary operators $B$ and $\\Psi $ , the ones that we will use later.",
"For completeness sake, we sketch the proof of the property (REF ).",
"Lemma 4.1 There holds $\\int _\\Omega (B(u)-B(v) ) u \\mathrm {dx} \\ge \\int _\\Omega \\Psi (u)\\mathrm {dx} - \\int _\\Omega \\Psi (v)\\mathrm {dx} .$ In particular, if the assumption () is fulfilled then there holds $\\int _\\Omega \\Psi (u)\\mathrm {dx}\\le \\int _\\Omega B(u) u \\mathrm {dx} \\le b^\\# \\Vert u\\Vert _{2,\\Omega }^2 .$ Under the assumption () the operator $B$ verifies $(B(u)-B(v),u-v)\\ge b_\\# \\Vert u-v\\Vert _{2,\\Omega }^2 .$ Let us write the decomposition $(B(u)-B(v) ) u = B(u)u-B(v) v - B(v)(u-v).$ Thanks to the mean value theorem for definite integrals, there exists $c$ between $u$ and $v$ such that $\\int _{v} ^{u} B(r)\\mathrm {dr}= B(c)(u-v).$ Since $-B$ is a decreasing function, we obtain $\\int _\\Omega (B(u)-B(v) ) u \\mathrm {dx} \\ge \\int _\\Omega (B(u)u-B(v) v )\\mathrm {dx} -\\int _\\Omega \\int _{v} ^{u} B(r)\\mathrm {dr} \\mathrm {dx},$ which concludes the proof by definition of $\\Psi $ .",
"Finally, we recall the following remarkable lemma [1].",
"Lemma 4.2 Suppose $u_m$ weakly converge to $u$ in $L^p(0,T;W^{1,p}(\\Omega ))$ , $p>1$ , with the estimates $\\int _\\Omega \\Psi (u_m(t)) \\mathrm {dx}\\le C\\quad \\mbox{for } 0<t<T,$ and for $z>0$ $\\int _0^{T-z}\\int _\\Omega (B(u_m(t+z))-B(u_m(t)) ) (u_m(t+z)-u_m(t)) \\mathrm {dx} \\mathrm {dt} \\le Cz ,$ with $C$ being positive constants.",
"Then, $B(u_m) \\rightarrow B(u)$ in $L^1(Q_T)$ and $\\Psi (u_m) \\rightarrow \\Psi (u) $ almost everywhere in $Q_T$ ."
],
[
"Fixed point argument (solvability to (",
"Let $\\ell \\ge 2$ , and define an operator $\\mathcal {T}$ from $\\mathbf {V}_{\\ell } =V_\\ell (\\Omega )\\times V$ into itself such that $(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ is the unique solution of Proposition REF .",
"Proposition 4.1 Let $\\mathbf {u} =(u_1,u_2)\\in \\mathbf {V}_{\\ell }$ , and $u=u_1$ .",
"Then, there exists a unique solution $(\\theta ,\\phi )\\in \\mathbf {V}_{\\ell }$ to the Neumann-power type elliptic problem $\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta v\\mathrm {dx}+\\int _\\Omega a( \\mathbf {u} )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}+\\int _\\Omega \\sigma (u) F( \\mathbf {u}) \\nabla \\phi \\cdot \\nabla v\\mathrm {dx} +\\nonumber \\\\+\\int _\\Gamma \\gamma (u) |\\theta |^{\\ell -2} \\theta v \\mathrm {ds} =\\frac{1}{\\tau }\\int _\\Omega fv \\mathrm {dx} +\\int _\\Gamma Hv \\mathrm {ds} ; \\\\\\int _\\Omega \\sigma (u)\\nabla \\phi \\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (u) \\alpha _\\mathrm {S}(u) \\nabla \\theta \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ and $w\\in V$ .",
"In addition, the following estimate $\\frac{b_\\#}{2\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2 + (L_{1})_\\#\\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2 +\\frac{(L_{2})_\\# }{2}\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\frac{ 1 }{ 2\\tau b_\\# }\\Vert f\\Vert _{2,\\Omega }^2+ \\nonumber \\\\ +\\frac{1}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}} \\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+\\frac{(K_2)^2(P_2+1)^2}{2 (L_{2})_\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N} } ^2 := \\mathcal {R}( \\Vert f\\Vert _{2,\\Omega }^2 ,\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} ) \\quad $ holds true, if provided by one the following definition $&&\\left\\lbrace \\begin{array}l(L_{1})_\\# = a_\\#- \\varepsilon \\sigma ^\\# \\left( F^\\# + \\alpha ^\\#\\right) /2\\\\(L_{2})_\\# = \\sigma _\\# - \\sigma ^\\# \\left( F^\\# + \\alpha ^\\#\\right) /(2\\varepsilon )\\end{array}\\right.",
"\\\\ &&\\left\\lbrace \\begin{array}l(L_{1})_\\# = a_\\#- \\varepsilon \\sqrt{\\sigma ^\\# }( F^\\#+\\alpha ^\\#) /2\\\\(L_{2})_\\# = \\sigma _\\#\\left( 1- \\sqrt{\\sigma ^\\# }( F^\\#+\\alpha ^\\#) /(2\\varepsilon ) \\right)\\end{array}\\right.",
".$ The existence of a solution to the variational system (REF )-() relies on the direct application of the Browder-Minty Theorem [18].",
"Indeed, the form $\\mathcal {F}: \\mathbf {V}_{\\ell }\\rightarrow \\mathbb {R}$ defined by $\\mathcal {F}(v,w)=\\frac{1}{\\tau }\\int _\\Omega fv\\mathrm {dx}+\\int _\\Gamma Hv \\mathrm {ds} +\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds}$ is continuous and linear, and the form $\\mathcal {L}: \\mathbf {V}_{\\ell }\\times \\mathbf {V}_{\\ell }\\rightarrow \\mathbb {R}$ defined by $\\mathcal {L} \\left( (\\theta ,\\phi ), (v,w)\\right)=\\frac{1}{\\tau }\\int _\\Omega b(u) \\theta v\\mathrm {dx}+\\int _\\Omega \\left(\\mathsf {L}( \\mathbf {u} )\\nabla \\left[\\begin{array}{c}\\theta \\\\\\phi \\end{array} \\right]\\right)\\cdot \\nabla \\left[\\begin{array}{c} v\\\\w\\end{array} \\right]\\mathrm {dx},$ is continuous and bilinear, with $\\mathsf {L}$ being the $(2\\times 2)$ -matrix $\\mathsf {L} ( \\mathbf {u} )=\\left[\\begin{array}{cc}a( \\mathbf {u} )& \\sigma (u) F( \\mathbf {u} ) \\\\\\sigma (u) \\alpha _\\mathrm {S} (u)& \\sigma (u)\\end{array}\\right] .$ Moreover, $\\mathcal {L}$ is coercive: $\\sum _{i,j=1}^{ 2}\\sum _{l=1}^n \\left( L_{i,j} ( \\mathbf {u})\\xi _{j,l}\\right)\\xi _{l,i}\\ge (L_{1})_\\#|\\xi _1|^2+ (L_{2})_\\#|\\xi _2|^2,$ with $(L_{1})_\\#$ and $(L_{2})_\\#$ being the positive constants defined in (REF ) or (), taking the assumptions (REF ) and (REF ) into account.",
"The difference of the definitions is consequence of the different application of the Young inequality $2AB\\le \\varepsilon A^2+B^2/\\varepsilon $ ($\\varepsilon ,A,B>0$ ), see Remark REF .",
"Namely, with $A=| \\xi _{1}| $ and $B=| \\xi _{2}|$ , for (REF ).",
"That is, $ \\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right) \\le \\sigma ^\\# \\left( F^\\#+\\alpha ^\\# \\right)\\left( \\frac{\\varepsilon }{2}A^2+\\frac{1}{2\\varepsilon } B^2\\right).$ $A=| \\xi _{1}| $ and $B=\\sqrt{\\sigma (u)}| \\xi _{2}| $ , for ().",
"That is, $ \\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right)\\le \\sqrt{\\sigma ^\\# }\\left( F^\\#+\\alpha ^\\# \\right)\\left( \\frac{\\varepsilon }{2}A^2+\\frac{ 1}{2\\varepsilon }B^2\\right).$ Finally, observing that the function $e\\in \\mathbb {R}\\mapsto \\gamma (u)|e|^{\\ell -2}e $ is monotonically increasing, we conclude the existence of the required solution.",
"In order to obtain (REF ), we take $v=\\theta $ and $w=\\phi $ as test functions in (REF ) and (), respectively.",
"Summing the obtained relations, and applying (), (REF ), the coercivity (REF ) of $\\mathsf {L}$ , and the Hölder inequality, we find $\\frac{ b_\\# }{\\tau } \\Vert \\theta \\Vert _{2,\\Omega }^2+ (L_{1})_\\# \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+(L_{2})_\\#\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\gamma _\\# \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\nonumber \\\\ \\le \\frac{ 1 }{\\tau } \\Vert f\\Vert _{2,\\Omega }\\Vert \\theta \\Vert _{2,\\Omega }+\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }\\Vert \\theta \\Vert _{\\ell ,\\Gamma }+\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}\\Vert \\phi \\Vert _{2,\\Gamma _\\mathrm {N}}.$ We successively apply (REF ) and the Young inequality to obtain $\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma } \\Vert \\theta \\Vert _{\\ell ,\\Gamma }+\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N} }\\Vert \\phi \\Vert _{2,\\Gamma _\\mathrm {N}} \\le \\nonumber \\\\\\le \\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} }\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+\\frac{\\gamma _\\#}{\\ell }\\Vert \\theta \\Vert _{\\ell ,\\Gamma }^\\ell +\\frac{K_2^2(P_2+1)^2}{ 2(L_{2})_\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2+ \\frac{(L_{2})_\\#}{2}\\Vert \\nabla \\phi \\Vert _{2,\\Omega } ^2.",
"$ Inserting (REF ) into (REF ), we deduce (REF ).",
"Remark 4.1 Even $\\varepsilon >0$ may be an arbitrary (but fixed) number, we may differently define $(L_1)_\\#$ and $(L_2)_\\#$ .",
"Indeed, the Young inequality $2AB\\le \\varepsilon A^2+B^2/\\varepsilon $ ($\\varepsilon ,A,B>0$ ) may be applied to obtain $\\sum _{l=1}^n \\left(\\sigma (u) F( \\mathbf {u} ) \\xi _{2,l}\\xi _{l,1}+\\sigma (u) \\alpha _\\mathrm {S} (u) \\xi _{1,l}\\xi _{l,2} \\right) \\le \\sigma ^\\# \\left( F^\\#\\left( \\frac{\\varepsilon _1}{2}| \\xi _{1}|^2+\\frac{1}{2\\varepsilon _1} | \\xi _{2}|^2\\right)+\\right.",
"\\\\ \\left.",
"+\\alpha ^\\#\\left( \\frac{\\varepsilon _2}{2}| \\xi _{1}|^2+\\frac{1}{2\\varepsilon _2} | \\xi _{2}|^2\\right) \\right).$ Next, let us determine whose radius make possible that the operator $\\mathcal {T}$ maps a closed ball into itself.",
"Proposition 4.2 For $R=\\max \\lbrace R_1,R_2\\rbrace $ with $R_1$ and $R_2$ being defined in (REF ) and (REF ), respectively, the operator $\\mathcal {T}$ verifies $\\mathcal {T}(K)\\subset K$ , with $K=\\left\\lbrace (v,w)\\in \\mathbf {V}_\\ell : \\ \\Vert \\nabla w\\Vert _{2,\\Omega }+\\Vert \\nabla v\\Vert _{2,\\Omega }+ \\Vert v\\Vert _{\\ell ,\\Gamma } \\le R\\right\\rbrace .$ Let $\\mathbf {u}\\in \\mathbf {V}_{\\ell }$ , $u=u_1$ and $(\\theta ,\\phi )$ be the unique solution of Proposition REF , i.e.",
"$(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ .",
"In order to prove that $(\\theta ,\\phi )\\in K$ we consider two different cases: (1) if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le 1$ ; and (2) if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } >1$ , if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le 1$ , then there holds $\\Vert \\nabla \\phi \\Vert _{2,\\Omega }+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }+ \\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le \\sqrt{2}\\left(\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+ \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2\\right)^{1/2}+1,$ by applying the elementary inequality $(a+b)^2\\le 2(a^2+b^2)$ for every $a,b\\ge 0$ .",
"By using (REF ), we may take $R_1=\\left(\\frac{2\\mathcal {R}}{\\min \\left\\lbrace (L_{1})_\\#, (L_{2})_\\#/2\\right\\rbrace } \\right)^{1/2}+1 .",
"$ if $\\Vert \\theta \\Vert _{\\ell ,\\Gamma } > 1$ , then using $\\ell \\ge 2$ there holds $\\Vert \\nabla \\phi \\Vert _{2,\\Omega }+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }+ \\Vert \\theta \\Vert _{\\ell ,\\Gamma } \\le \\sqrt{2}\\left( 2(\\Vert \\nabla \\phi \\Vert _{2,\\Omega }^2+\\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2) + \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\right)^{1/2},$ by applying the elementary inequality $(a+b)^2\\le 2(a^2+b^2)$ for every $a,b\\ge 0$ .",
"By using (REF ), we may take $R_2^2=\\left(\\frac{2 }{\\min \\left\\lbrace (L_{1})_\\#, (L_{2})_\\#/2\\right\\rbrace } +\\frac{\\ell ^{\\prime }}{\\gamma _\\#}\\right)\\mathcal {R} .",
"$ Then, the proof is complete by taking $R$ such that is the maximum of $R_1$ and $R_2$ defined in (REF ) and (REF ), respectively.",
"Proposition 4.3 The operator $\\mathcal {T}$ is continuous.",
"Let $\\lbrace \\mathbf {u}^m\\rbrace _{m\\in \\mathbb {N}}$ be a sequence such that weakly converges to $\\mathbf {u}=(u,u_2)$ in $ \\mathbf {V}_\\ell $ , and $(\\theta _m,\\phi _m)=\\mathcal {T}(\\mathbf {u}^m)$ for each $m\\in \\mathbb {N}$ .",
"Proposition REF guarantees that $(\\theta _m,\\phi _m)$ solves, for each $m\\in \\mathbb {N}$ , the variational system (REF )$_m$ -()$_m$ , with $\\mathbf {u}$ replaced by $\\mathbf {u}^m$ .",
"The uniform boundedness ensured by Proposition REF guarantees the existence of a limit $(\\theta ,\\phi )\\in \\mathbf {V}_\\ell $ , for at least a subsequence of $(\\theta _m,\\phi _m)$ still denoted by $(\\theta _m,\\phi _m)$ , such that $\\theta _m\\rightharpoonup \\theta \\quad \\mbox{in } V_\\ell (\\Omega )\\quad \\mbox{and}\\quad \\phi _m\\rightharpoonup \\phi \\quad \\mbox{in } V \\quad (\\mbox{as } m\\rightarrow +\\infty ) .$ The Rellich-Kondrachov theorem guarantees the strong convergences $u^m\\rightarrow u &\\mbox{ and }& u_2^m\\rightarrow u_2 \\quad \\mbox{in } L^2(\\Omega ); \\\\\\theta _m\\rightarrow \\theta &\\mbox{ and }& \\phi _m\\rightarrow \\phi \\quad \\mbox{in } L^2(\\Omega ); \\\\u^m\\rightarrow u &\\mbox{ and }& \\theta _m\\rightarrow \\theta \\quad \\mbox{in } L^2(\\Gamma ).$ To show that $(\\theta ,\\phi )=\\mathcal {T}(\\mathbf {u})$ , it remains to pass to the limit in the system (REF )$_m$ -()$_m$ as $m$ tends to infinity.",
"Applying the Krasnoselski theorem to the Nemytskii operators $b$ , $a$ , $\\sigma $ , we have $b(u^m) v\\rightarrow b(u) v &\\mbox{ in }& L^2(\\Omega );\\\\a(\\mathbf {u}^m) \\nabla v\\rightarrow a(\\mathbf {u}) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(\\Omega ); \\\\\\sigma (u^m)\\nabla v\\rightarrow \\sigma ( u)\\nabla v &\\mbox{ in }& \\mathbf {L}^2(\\Omega ),$ for all $v\\in H^1(\\Omega )$ , making use of the Lebesgue dominated convergence theorem and the assumptions () and ()-().",
"Also the terms $\\sigma (u^m)F(\\mathbf {u}^m) \\nabla v$ and $ \\sigma (u^m) \\alpha _\\mathrm {S} (u^m) \\nabla v$ pass to the limit making recourse to the assumptions (REF ) and (REF ), respectively.",
"Similarly, the boundary term $\\gamma (u^m)v$ converges to $\\gamma (u) v$ in $L^{\\ell ^{\\prime }}(\\Gamma )$ , for all $v\\in L^{\\ell ^{\\prime }}(\\Gamma )$ , due to (REF ).",
"Observe that $ \\theta _m$ strongly converges to $\\theta $ in $L^p(\\Gamma )$ , for all $1<p<\\ell $ .",
"Then, the nonlinear boundary term $\\gamma (u^m) |\\theta ^m|^{\\ell -2} \\theta ^m $ weakly passes to the limit as $m$ tends to infinity to $\\gamma (u)\\Lambda $ in $L^{\\ell ^{\\prime }}(\\Gamma )$ .",
"Therefore, the variational system (REF )$_m$ -()$_m$ as $m$ tends to infinity to conclude that $\\phi $ is the required limit solution, i.e.",
"it solves the limit equality (), while $\\theta $ verifies $\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta v\\mathrm {dx}+\\int _\\Omega a( \\mathbf {u} )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}+\\int _\\Omega \\sigma (u) F( \\mathbf {u}) \\nabla \\phi \\cdot \\nabla v\\mathrm {dx} +\\nonumber \\\\+\\int _\\Gamma \\gamma (u) \\Lambda v \\mathrm {ds} =\\frac{1}{\\tau }\\int _\\Omega fv \\mathrm {dx} +\\int _\\Gamma Hv \\mathrm {ds}.$ It remains to identify the limit $\\Lambda $ by using the Minty trick as follows.",
"The argument is slightly different from the classical one (see [18]).",
"Making recourse to the the lower bound (REF ) of $\\gamma $ and the monotone property of the function $v\\mapsto |v|^{\\ell -2}v$ , we have $0\\le \\gamma _\\# 2^{2-\\ell }|\\theta _m -v|^\\ell \\le \\gamma (u^m) \\left(|\\theta _{m}|^{\\ell -2} \\theta _{m} -|v|^{\\ell -2}v\\right) (\\theta _{m} -v).$ Thanks to the coercivity coefficients (REF ) or (), the monotonicity property of the boundary term, and the Hölder and Young inequalities, let us consider $\\int _{\\Omega }a(\\mathbf {u}^m)|\\nabla (\\theta _m- v)|^2\\mathrm {dx} +\\int _{\\Omega }\\sigma (u^m)|\\nabla (\\phi _m-\\phi )|^2\\mathrm {dx}+ \\nonumber \\\\ +\\int _{\\Omega } \\sigma (u^m) F (\\mathbf {u}^m)\\nabla (\\phi _m -\\phi ) \\cdot \\nabla ( \\theta _m-v)\\mathrm {dx} + \\nonumber \\\\ +\\int _{\\Omega } \\sigma (u^m) \\alpha _\\mathrm {S} (u^m)\\nabla (\\theta _m-v) \\cdot \\nabla (\\phi _m -\\phi )\\mathrm {dx} + \\nonumber \\\\+\\int _{\\Gamma } \\gamma (u^m) \\left(|\\theta _{m}|^{\\ell -2} \\theta _{m} -|v|^{\\ell -2}v\\right) (\\theta _{m} -v)\\mathrm {ds}\\ge \\nonumber \\\\ \\ge (L_1)_\\# \\int _{\\Omega }|\\nabla (\\theta _m-v) |^2\\mathrm {dx} +(L_2)_\\#\\int _{\\Omega }|\\nabla (\\phi _m -\\phi ) |^2\\mathrm {dx}\\ge 0.",
"$ Let us define $\\mathcal {J}_m:= \\int _{\\Omega }\\left( a(\\mathbf {u}^m) |\\nabla \\theta _m|^2 +\\sigma (u^m) F (\\mathbf {u}^m) \\nabla \\phi _m\\cdot \\nabla \\theta _m\\right)\\mathrm {dx} + \\\\ +\\int _{\\Omega }\\left(\\sigma (u^m) |\\nabla \\phi _m|^2 + \\sigma (u^m) \\alpha _\\mathrm {S} (u^m) \\nabla \\theta _m\\cdot \\nabla \\phi _m\\right)\\mathrm {dx} + \\\\+\\int _{\\Gamma }\\gamma (u^m) |\\theta _m|^\\ell \\mathrm {ds}.$ On the one hand, we deduce $\\lim _{m\\rightarrow \\infty }\\mathcal {J}_m\\ge \\int _{\\Gamma }\\gamma (u)\\Lambda v\\mathrm {ds} +\\int _{\\Gamma }\\gamma (u)|v|^{\\ell -2} v(\\theta -v)\\mathrm {ds}+ \\\\ +\\int _{\\Omega }a(\\mathbf {u})\\nabla \\theta \\cdot \\nabla v\\mathrm {dx} +\\int _{\\Omega }a(\\mathbf {u})\\nabla v\\cdot \\nabla (\\theta - v)\\mathrm {dx} + \\\\ +\\int _{\\Omega }\\sigma (u)|\\nabla \\phi |^2\\mathrm {dx} +\\int _{\\Omega } \\sigma (u) F (\\mathbf {u})\\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx} +\\int _{\\Omega } \\sigma (u) \\alpha _\\mathrm {S} (u)\\nabla \\theta \\cdot \\nabla \\phi \\mathrm {dx}.$ On the other hand, taking in (REF )$_m$ the test function $v=\\theta ^m$ , in (REF ) the test function $v=\\theta $ , in ()$_m$ the test function $w=\\phi _m$ , and in () the test function $w=\\phi $ , we deduce $\\lim _{m\\rightarrow \\infty }\\mathcal {J}_m =\\frac{1}{\\tau }\\int _\\Omega f\\theta \\mathrm {dx} +\\int _\\Gamma H\\theta \\mathrm {ds} -\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta \\mathrm {dx} + \\int _{\\Gamma _\\mathrm {N}} g \\theta \\mathrm {ds}=\\\\=\\int _\\Omega a( \\mathbf {u} ) |\\nabla \\theta |^2\\mathrm {dx}+\\int _\\Omega \\sigma (u) F( \\mathbf {u}) \\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx}+\\int _\\Gamma \\gamma (u) \\Lambda \\theta \\mathrm {ds} + \\\\ +\\int _\\Omega \\sigma (u)|\\nabla \\phi |^2\\mathrm {dx}+ \\int _\\Omega \\sigma (u ) \\alpha _\\mathrm {S} (u)\\nabla \\theta \\cdot \\nabla \\phi \\mathrm {dx} .$ Gathering the above two relations, we find $\\int _{\\Omega }a(\\mathbf {u})|\\nabla (\\theta - v)|^2\\mathrm {dx} +\\int _{\\Gamma }\\gamma (u)(\\Lambda -|v|^{\\ell -2}v) (\\theta -v) \\mathrm {ds} \\ge 0.$ We continue the argument by taking $v=\\theta -\\delta \\varphi $ , with $\\varphi \\in \\mathcal {D}(\\Gamma )$ .",
"After dividing by $\\delta >0$ , and finally letting $\\delta \\rightarrow 0^+$ we arrive to $\\int _{\\Gamma }\\gamma (u)(\\Lambda - |\\theta |^{\\ell -2}\\theta )\\varphi \\mathrm {ds} \\ge 0, \\quad \\forall \\varphi \\in \\mathcal {D}(\\Gamma ),$ which implies that $\\Lambda = |\\theta |^{\\ell -2}\\theta $ .",
"Thus, we are in the condition of concluding that $(\\theta ,\\phi )$ is the required limit solution, i.e.",
"it solves the limit system (REF )-().",
"Thanks to Propositions REF , REF and REF , there exists at least one fixed point of $\\mathcal {T}$ , that is $(\\theta ,\\phi )=\\mathcal {T}(\\theta ,\\phi )$ , which concludes the solvability to (REF )-()."
],
[
"Fixed point argument (solvability to (",
"Let $\\ell \\ge 2$ , and define an operator $\\mathcal {T}$ from $V_{\\ell }(\\Omega )$ into itself such that $\\theta =\\mathcal {T}(u)$ is the unique solution of Proposition REF .",
"Denote by the well defined continuous operator such that $\\mathcal {F}( u)=\\phi $ .",
"The existence of a unique weak auxiliary solution $\\phi $ to the variational equality () is standard and it can be stated as follows.",
"Proposition 4.4 Let $u\\in H^1(\\Omega )$ .",
"Under the assumptions (), (REF ) and (H5), the Neumann problem $\\int _\\Omega \\sigma (u) \\nabla \\phi \\cdot \\nabla w\\mathrm {dx}=\\int _\\Omega \\sigma (u) \\alpha _\\mathrm {S} (u)\\nabla u\\cdot \\nabla w \\mathrm {dx}+\\int _{\\Gamma _\\mathrm {N}} gw \\mathrm {ds},\\qquad \\forall w\\in V,$ admits a unique solution $\\phi \\in V$ .",
"Moreover, the estimate $\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega }\\le \\sqrt{\\sigma ^\\#}\\alpha ^\\# \\Vert \\nabla u\\Vert _{2,\\Omega }+\\frac{K_2 (P_2+1) }{\\sqrt{\\sigma _\\#}}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}$ holds true.",
"Let us establish the quantitative estimate (REF ).",
"We take $w=\\phi $ as a test function in (REF ), and we compute by applying the Hölder inequality and (REF ) $\\Vert \\sqrt{\\sigma (u)}\\nabla \\phi \\Vert _{2,\\Omega } ^2 \\le \\left( \\alpha ^\\#\\Vert \\sqrt{\\sigma (u)}\\nabla u\\Vert _{2,\\Omega }+\\frac{ K_2(P_2+1)}{\\sqrt{\\sigma _\\#}} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}\\right)\\Vert \\sqrt{\\sigma (u)}\\nabla \\phi \\Vert _{2,\\Omega } .$ Then, (REF ) arises.",
"Proposition 4.5 Let $\\mathbf {u} =(u,\\phi ) \\in \\left( H^1(\\Omega ) \\right)^2$ .",
"Under the assumptions (), (REF ) and (REF )-(), there exists a unique solution $\\theta \\in V_\\ell (\\Omega )$ to the power type elliptic problem $\\frac{1}{\\tau }\\int _\\Omega b(u)\\theta v\\mathrm {dx}+\\int _\\Omega a (\\mathbf {u} )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}+\\int _\\Omega \\sigma (u) F ( \\mathbf {u} ) \\nabla \\phi \\cdot \\nabla v\\mathrm {dx} +\\nonumber \\\\+\\int _\\Gamma \\gamma (u) |\\theta |^{\\ell -2}\\theta v \\mathrm {ds} =\\frac{1}{\\tau }\\int _\\Omega fv \\mathrm {dx} +\\int _\\Gamma Hv \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ .",
"If $\\phi \\in V$ satisfies (REF ), then the following estimate $\\frac{b_\\#}{2\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2 +\\frac{ a_\\#}{2} \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\frac{ 1 }{ 2\\tau b_\\# }\\Vert f\\Vert _{2,\\Omega }^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}}\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }}+ \\nonumber \\\\ +\\frac{(F^\\#)^2\\sigma ^\\#}{ a_\\#}\\left( \\sigma ^\\#(\\alpha ^\\# )^2\\Vert \\nabla u \\Vert _{2,\\Omega }^2 +\\frac{K_2 ^2(P_2+1)^2 }{\\sigma _\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2\\right) .",
"$ holds true.",
"Taking $v=\\theta $ as a test function in (REF ), and applying (), (), (REF ), (REF ), and the Hölder inequality, we find $\\frac{b_\\#}{\\tau }\\Vert \\theta \\Vert _{2,\\Omega }^2+ a_\\# \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2+\\gamma _\\# \\Vert \\theta \\Vert _{\\ell ,\\Gamma } ^\\ell \\le \\\\ \\le \\frac{ 1 }{\\tau } \\Vert f\\Vert _{2,\\Omega }\\Vert \\theta \\Vert _{2,\\Omega }+F^\\#\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega } \\Vert \\sqrt{\\sigma (u)} \\nabla \\theta \\Vert _{2,\\Omega }+\\Vert H\\Vert _{\\ell ^{\\prime },\\Gamma }\\Vert \\theta \\Vert _{\\ell ,\\Gamma } .$ Applying (REF ) and the Young inequality, we compute $\\Vert \\sqrt{\\sigma (u)} \\nabla \\phi \\Vert _{2,\\Omega }\\Vert \\sqrt{\\sigma (u)} \\nabla \\theta \\Vert _{2,\\Omega }\\le \\frac{a_\\#}{2} \\Vert \\nabla \\theta \\Vert _{2,\\Omega }^2 + \\\\+\\frac{\\sigma ^\\# }{a_\\#} \\left(\\sigma ^\\# (\\alpha ^\\#)^2 \\Vert \\nabla u\\Vert _{2,\\Omega } ^2+\\frac{K_2^2(P_2+1) ^2}{\\sigma _\\#}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 \\right).$ Then, arguing as in (REF ), we deduce (REF ).",
"Next, let us determine whose radius make possible that the operator $\\mathcal {T}$ maps a closed ball into itself.",
"Proposition 4.6 Let (REF ) be fulfilled.",
"For $\\tau \\le a_\\# / b_\\#$ and $R>0$ being defined as in (REF ), the operator $\\mathcal {T}$ verifies $\\mathcal {T}(\\overline{B_R})\\subset \\overline{B_R}$ , with $B_R$ denoting the open ball of $H^1(\\Omega )$ with radius $R$ .",
"Let $u\\in H^1(\\Omega )$ and $\\theta =\\mathcal {T}(u)$ be the unique solution according to Proposition REF .",
"Considering (REF ) and $\\min \\left\\lbrace b_\\#/\\tau ,a_\\#\\right\\rbrace =a_\\#$ , the proof is complete by defining $R$ such that $R\\left(\\sqrt{ a_\\# }-2\\frac{F^\\# \\sigma ^\\#\\alpha ^\\# }{ \\sqrt{a_\\#}} \\right) &=&\\left(\\frac{ 1 }{\\tau b_\\# }\\Vert f \\Vert _{2,\\Omega }^2 +\\frac{2}{\\ell ^{\\prime }\\gamma _\\#^{1/(\\ell -1)}} \\Vert H \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} \\right)^{1/2}+\\nonumber \\\\&&+ 2F^\\# K_2(P_2+1) \\sqrt{\\frac{\\sigma ^\\#}{ a_\\#\\sigma _\\#}}\\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}} , \\quad $ taking the assumption (REF ) be account.",
"Proposition 4.7 Let $\\lbrace u_m\\rbrace _{m\\in \\mathbb {N}}$ be a sequence such that weakly converges to $u$ in $ H^1(\\Omega )$ , then the solution $(\\theta _m,\\phi _m)$ according to Propositions REF and REF weakly converges in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ , and its limit is a solution according to Propositions REF and REF Let $(\\theta _m,\\phi _m)$ be the solution according to Propositions REF and REF and corresponding to $u_m$ for each $m\\in \\mathbb {N}$ .",
"The estimates (REF ) and (REF ) guarantee that the sequence $(\\theta _m,\\phi _m)$ is uniformly bounded in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ .",
"Thus, we can extract a subsequence of $(\\theta _m,\\phi _m)$ still denoted by $(\\theta _m,\\phi _m)$ , weakly convergent to $(\\theta ,\\phi )$ in $V_\\ell (\\Omega )\\times H^1(\\Omega )$ .",
"Similar arguments in the proof of Proposition REF the weak limit $(\\theta ,\\phi )$ solves the variational system consisting of (REF ) and (REF ), which concludes the proof of Proposition REF .",
"Thanks to Propositions REF , REF , REF and REF , there exists at least one fixed point of $\\mathcal {T}:u\\mapsto (u, \\mathcal {F}(u))\\mapsto \\theta ,$ that is $\\theta =\\mathcal {T}(\\theta )$ and $\\phi =\\mathcal {F}(\\theta )$ , which concludes the solvability to (REF )-()."
],
[
"Time discretization technique",
"In this section, we apply the method of discretization in time [17], [23], [24].",
"We decompose the time interval $I=[0,T]$ into $M$ subintervals $I_{m,M}$ of size $\\tau $ such that $M=T/\\tau \\in \\mathbb {N}$ , i.e.",
"$I_{m,M}=[(m-1)T/M,mT/M]$ for $m\\in \\lbrace 1,\\cdot \\cdot \\cdot ,M\\rbrace $ .",
"We set $t_{m,M}=mT/M$ .",
"Thus, the problem (REF ) is approximated by the following recurrent sequence of time-discretized problems $\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^m) v\\mathrm {dx}+\\int _\\Omega a (\\theta ^m,\\phi ^m )\\nabla \\theta ^{m}\\cdot \\nabla v\\mathrm {dx}+\\int _\\Gamma \\gamma (\\theta ^m)|\\theta ^m|^{\\ell -2} \\theta ^{m} v \\mathrm {ds} +\\nonumber \\\\ +\\int _\\Omega \\sigma (\\theta ^m) F( \\theta ^m,\\phi ^m ) \\nabla \\phi ^m \\cdot \\nabla v\\mathrm {dx}=\\frac{1}{\\tau }\\int _\\Omega B (\\theta ^{m-1} )v\\mathrm {dx}+\\int _\\Gamma h(t_{m,M}) v \\mathrm {ds} ,$ for all $v\\in V_{\\ell }(\\Omega )$ , and the problem () is approximated by either () or () for all $w\\in V$ , corresponding to the two different approaches.",
"The existence of weak solutions pair $(\\theta ^m,\\phi ^m) \\in V_{\\ell }(\\Omega )\\times V$ to the above systems of elliptic problems is established in Section with $H =h(t_{m,M})$ .",
"Since $\\theta ^0\\in L^2(\\Omega )$ is known, we determine $(\\theta ^{1},\\phi ^1)$ as the unique solution of the Neumann-power type elliptic problems (REF )-() or (REF )-(), and we inductively proceed.",
"Denote by $\\lbrace {\\theta }_M\\rbrace _{M\\in \\mathbb {N}}$ , $\\lbrace {\\phi }_M\\rbrace _{M\\in \\mathbb {N}}$ and $\\lbrace Z_M\\rbrace _{M\\in \\mathbb {N}}$ the sequences of the (piecewise constant in time) functions, $ {\\theta }_M : [0,T] \\rightarrow V_\\ell (\\Omega )$ , $ {\\phi }_M : ]0,T] \\rightarrow V$ and $ Z_M:[0,T]\\rightarrow L^2(\\Omega )$ , defined by, respectively, a.e.",
"in $\\Omega $ ${\\theta }_M (t) &:=& \\left\\lbrace \\begin{array}{ll}\\theta ^{0}&\\mbox{ for }t=0\\\\\\theta ^{m}&\\mbox{ for } t\\in ]t_{m-1,M},t_{m,M}]\\end{array}\\right.",
"\\\\{\\phi }_M (t) &:=& \\phi ^m \\quad \\mbox{ for all }t\\in ]t_{m-1,M},t_{m,M} ],$ in accordance with one of the two variational formulations () and (), while $Z_M (t):=\\left\\lbrace \\begin{array}{ll}B(\\theta ^{0} ) &\\mbox{ for }t=0\\\\Z^{m}&\\mbox{ for } t\\in ]t_{m-1,M},t_{m,M}]\\end{array}\\right.",
"\\mbox{ in }\\Omega ,$ with the discrete derivative with respect to $t$ at the time $t=t_{m,M}$ : $Z^{m}:=\\frac{B(\\theta ^{m})-B(\\theta ^{m-1}) }{\\tau }.$ While $\\theta _M$ is the Rothe function obtained from $\\theta ^m$ by piecewise constant interpolation with respect to time $t$ , the Rothe function, obtained from $\\theta ^m$ by piecewise linear interpolation with respect to time $t$ , $\\Theta _M$ is $\\Theta _M(\\cdot ,t)=\\theta ^{m-1}+(t-t_{m-1,M}) \\frac{\\theta ^m- \\theta ^{m-1}}{ \\tau }.$ For our purposes, we introduce the following definition.",
"Definition 5.1 We say that $\\lbrace \\widetilde{B}_M= \\widetilde{B}(\\theta _M)\\rbrace _{M\\in \\mathbb {N}}$ is the Rothe sequence (affine on each time interval) if $\\widetilde{B}(\\cdot ,\\theta _M (t)) = B(\\cdot ,\\theta ^{m-1}) + \\frac{ t-t_{m-1,M}}{\\tau }\\left( B(\\cdot ,\\theta ^m) - B(\\cdot ,\\theta ^{m-1}) \\right)$ in $\\Omega $ , for all $t\\in I_{m, M}$ , for all $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ .",
"Denoting $h_M(t)=h(t_{m,M})$ for $t\\in ]t_{m-1,M},t_{m,M}]$ and $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ , the triple $ ({\\theta }_M ,\\phi _M,{Z}_M )$ solve $\\int _0^T\\int _\\Omega {Z}_M v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} a ( {\\theta }_M , {\\phi }_M )\\nabla {\\theta }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma ( {\\theta }_M )| {\\theta }_M |^{\\ell -2} {\\theta }_M v \\mathrm {ds}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\sigma ( \\theta _M ) F( \\theta _M , \\phi _M ) \\nabla {\\phi }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}=\\int _{\\Sigma _T} h_M v \\mathrm {ds}\\mathrm {dt} .\\qquad $"
],
[
"Proof of Theorem ",
"Here, $ {\\phi }_M $ solves $\\int _\\Omega \\sigma (\\theta _M)\\nabla \\phi _M\\cdot \\nabla w\\mathrm {dx}+ \\int _\\Omega \\sigma (\\theta _M) \\alpha _\\mathrm {S}(\\theta _M)\\nabla \\theta _M \\cdot \\nabla w \\mathrm {dx}=\\int _{\\Gamma _\\mathrm {N}} g w \\mathrm {ds} ,$ for $w\\in V$ and a.e.",
"in $]0,T[$ .",
"We begin by establishing the uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ .",
"Proposition 5.1 Let $ {\\theta }_M $ and $ {\\phi }_M $ be the (piecewise constant in time) functions defined in (REF )-().",
"Then the following estimate holds: $\\max _{1\\le m\\le M} \\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}+( L_1)_\\# \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ^2+\\frac{(L_{2})_\\# }{2}\\Vert \\nabla \\phi _M\\Vert _{2,Q_T} ^2 +\\nonumber \\\\+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } ^\\ell \\le b^\\# \\Vert \\theta ^0\\Vert _{2,\\Omega } ^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} } \\Vert h \\Vert _{\\ell ^{\\prime },\\Sigma _T }^{\\ell ^{\\prime }} +T \\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 .$ Let $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ be arbitrary.",
"Choosing $v=\\theta ^m\\in V_{\\ell }(\\Omega )$ and $w=\\phi ^m \\in V$ as test functions in (REF )-(), we sum the obtained relations, and arguing as in (REF )-(REF ), we have $\\frac{1}{\\tau }\\int _\\Omega (B(\\theta ^m)-B(\\theta ^{m-1}) )\\theta ^m\\mathrm {dx} +( L_1)_\\# \\Vert \\nabla \\theta ^{m} \\Vert _{2,\\Omega }^2 +\\frac{(L_{2})_\\#}{2} \\Vert \\nabla \\phi ^{m} \\Vert _{2,\\Omega }^2 +\\nonumber \\\\+\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds}\\le \\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} )+\\frac{1}{\\ell }\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds}, $ with $\\mathcal {R}$ being the increasing continuous function defined in (REF ).",
"By (REF ), we have $\\sum _{i=1}^m\\int _\\Omega (B(\\theta ^i)-B(\\theta ^{i-1}) )\\theta ^i\\mathrm {dx} \\ge \\int _\\Omega (\\Psi (\\theta ^m) - \\Psi (\\theta ^{0}) )\\mathrm {dx} .$ Therefore, summing over $i=1,\\cdots , m$ into (REF ), multiplying by $\\tau $ , and inserting the previous inequality, we obtain $\\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}& +&\\tau \\sum _{i=1}^m \\left( ( L_1)_\\# \\Vert \\nabla \\theta ^{i} \\Vert _{2,\\Omega }^2+\\frac{1}{\\ell ^{\\prime }}\\int _\\Gamma \\gamma (\\theta ^{i})|\\theta ^{i}|^{\\ell } \\mathrm {ds} +\\frac{(L_{2})_\\#}{2} \\Vert \\nabla \\phi ^{i} \\Vert _{2,\\Omega }^2 \\right) \\nonumber \\\\ &\\le &\\int _\\Omega \\Psi ( \\theta ^0 )\\mathrm {dx} +\\tau \\sum _{i=1}^m\\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} ) .",
"$ Therefore, we find the uniform estimate (REF ) by taking the maximum over $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ in the previous estimate and applying Lemma REF provided by ().",
"A direct application of Proposition REF ensures the following proposition.",
"Proposition 5.2 There exist $\\theta ,\\phi : Q_T\\rightarrow \\mathbb {R}$ and subsequences of $(\\theta _M,\\phi _M)$ , still labelled by $(\\theta _M,\\phi _M)$ , such that ${\\theta }_M \\rightharpoonup \\theta &\\mbox{ in }& V_\\ell (Q_T); \\\\\\phi _M \\rightharpoonup \\phi & \\mbox{ in }& L^2(0,T;V),$ as $M$ tends to infinity.",
"Moreover, there exists $Z: Q_T\\rightarrow \\mathbb {R}$ such that $\\partial _t\\widetilde{B}(\\theta _M)\\rightharpoonup Z \\quad \\mbox{in } L^{\\ell ^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime }),$ as $M$ tends to infinity.",
"Considering the uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ that are established in Proposition REF , we extract subsequences, still denoted by $ {\\theta }_M $ and $ {\\phi }_M $ , weakly convergent in $ V_\\ell (Q_T )$ and $ L^2(0,T;V)$ , respectively, to $\\theta $ and $\\phi $ .",
"Let $p=\\max \\lbrace \\ell ,2\\rbrace =\\ell $ .",
"Note that $L^p(0,T;V_\\ell (\\Omega ))\\hookrightarrow V_\\ell (Q_T)$ .",
"By definition of norm, we find $\\Vert \\partial _t\\widetilde{B}(\\theta _M)\\Vert _{L^{p^{\\prime }}(0,T;(V_\\ell (\\Omega ))^{\\prime })} = \\sum _{m=1}^M\\int _{(m-1)\\tau }^{m\\tau }\\sup _{v\\in L^p(0,T;V_\\ell (\\Omega )) \\atop \\Vert v\\Vert \\le 1} \\langle Z^m,v\\rangle \\mathrm {dt}\\le C,$ with $C>0$ being a constant independent on $M$ , by estimating in (REF ) the term involving $Z^m$ by means of the rest terms using the uniform estimates established in (REF ).",
"Hence, we can extract a subsequence, still denoted by $\\partial _t\\widetilde{B}(\\theta _M)$ , weakly convergent to $Z$ in $L^{p^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime })$ .",
"In the following proposition, we state the strong convergence of $B(\\theta _M)$ and of $\\theta _M$ .",
"Proposition 5.3 Under ()-(REF ) and (REF )-(), the solution $\\theta ^m$ of (REF ) satisfies $b_\\#\\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{2,\\Omega } ^2\\le \\int _\\Omega (B(\\theta ^{m })-B(\\theta ^{m-1 })) (\\theta ^{m }-\\theta ^{m-1 }) \\mathrm {dx} \\le \\nonumber \\\\\\le C\\left(\\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{\\ell ,\\Gamma } \\tau ^{1/\\ell } + \\Vert \\theta ^{m }-\\theta ^{m-1 }\\Vert _{2,\\Omega }\\sqrt{\\tau } \\right),$ with $C$ being a positive constant.",
"Moreover, for a subsequence, there hold $B( {\\theta }_M) \\rightarrow B(\\theta ) &\\mbox{ in }& L^1(Q_T); \\\\\\theta _M \\rightarrow \\theta & \\mbox{ a.e.",
"in }& Q_T,$ as $M$ tends to infinity.",
"Let $k\\in \\mathbb {N}$ .",
"Let us sum up (REF ) for $m=j+1,\\cdots ,j+k$ and multiply by $\\tau $ , obtaining $\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) v \\mathrm {dx} \\le \\mathcal {I }_\\Gamma ^j +\\mathcal {I}_\\Omega ^j ,$ where $\\mathcal {I}_\\Gamma ^j&:=& \\tau \\sum _{m=j+1}^{j+k} \\int _\\Gamma | (\\gamma (\\theta ^m)|\\theta ^{m}|^{\\ell -2} \\theta ^{m} -h(t_{m,M}) )v| \\mathrm {dx} \\\\&\\le & \\int _{j\\tau }^{(j+k)\\tau }\\Vert \\gamma (\\theta _M)|\\theta _M|^{\\ell -2} \\theta _M -h \\Vert _{\\ell ^{\\prime },\\Gamma } \\Vert v \\Vert _{\\ell ,\\Gamma }\\mathrm {dt} ; \\\\\\mathcal {I}_\\Omega ^j &:=& \\tau \\sum _{m=j+1}^{j+k}\\int _\\Omega |( a ( \\theta ^m,\\phi ^m )\\nabla \\theta ^{m} +F ( \\theta ^m,\\phi ^m) \\nabla \\phi ^m )v| \\mathrm {dx} \\\\&\\le & \\int _{j\\tau }^{(j+k)\\tau }\\Vert a ( \\theta _M,\\phi _M )\\nabla \\theta _M +F ( \\theta _M,\\phi _M) \\nabla \\phi _M \\Vert _{2,\\Omega }\\Vert v\\Vert _{2,\\Omega } \\mathrm {dt} .$ Here, we used the Hölder inequality and the definition of $ \\theta _M$ and of $\\phi _M$ .",
"Let us compute $\\mathcal {I }_\\Gamma ^j $ and $\\mathcal {I}_\\Omega ^j$ by applying the estimate (REF ).",
"Using the assumption (REF ) and after the Hölder inequality, we deduce $\\mathcal {I}_\\Gamma ^j\\le \\Vert v \\Vert _{\\ell ,\\Gamma } \\int _{j\\tau }^{(j+k)\\tau }\\left(\\gamma ^\\#\\Vert \\theta _M\\Vert _{\\ell ,\\Gamma }^{\\ell -1}+\\Vert h \\Vert _{\\ell ^{\\prime },\\Gamma } \\right)\\mathrm {dt}\\le \\Vert v \\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell }.$ Using the assumptions (), (REF ) and (), and after the Hölder inequality, we deduce $\\mathcal {I}_\\Omega ^j \\le \\Vert v\\Vert _{2,\\Omega } \\int _{j\\tau }^{(j+k)\\tau }\\left(a^\\#\\Vert \\nabla \\theta _M\\Vert _{2,\\Omega } + \\sigma ^\\#F ^\\#\\Vert \\nabla \\phi _M \\Vert _{2,\\Omega } \\right) \\mathrm {dt} \\le \\Vert v\\Vert _{2,\\Omega }C\\sqrt{k\\tau }.$ Hence, we find $\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) v \\mathrm {dx} \\le \\Vert v \\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell } + \\Vert v\\Vert _{2,\\Omega }C\\sqrt{k\\tau } .$ In particular, the estimate (REF ) follows by taking $j=m-1$ , $k=1$ and $v=\\theta ^{m }-\\theta ^{m-1 }$ , and applying Lemma REF .",
"To prove the convergences, we will apply Lemma REF .",
"Considering the weak convergence of $\\theta _M$ established in Proposition REF and the estimate (REF ), in order to apply Lemma REF it remains to prove that the condition (REF ) is fulfilled.",
"Let $0<z<T$ be arbitrary.",
"Since the objective is to find convergences, it suffices to take $M>T/z$ , which means $\\tau <z$ .",
"Thus, there exists $k\\in \\mathbb {N}$ such that $k\\tau <z\\le (k+1)\\tau $ .",
"Moreover, we may choose $M>k+1$ deducing $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\\\le \\sum _{j=1 }^{M-k} \\int _{(j-1)\\tau }^{(j+k)\\tau }\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) (\\theta ^{j+k }-\\theta ^{j })\\mathrm {dx} .$ Taking $v=\\theta ^{j+k }-\\theta ^j$ in (REF ) and then summing up for $j=1,\\cdots , M-k$ , we find $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\\\le (k+\\tau )\\sum _{j=1 }^{M-k}\\int _\\Omega (B(\\theta ^{j+k })-B(\\theta ^{j })) (\\theta ^{j+k }-\\theta ^{j })\\mathrm {dx} \\le \\\\ \\le \\sum _{j=1 }^{M-k} \\int _{(j-1)\\tau }^{j\\tau +k }\\left( \\Vert \\theta ^{j+k }-\\theta ^{j }\\Vert _{\\ell ,\\Gamma } C (k\\tau )^{1/\\ell } +\\Vert \\theta ^{j+k }-\\theta ^{j }\\Vert _{2,\\Omega }C\\sqrt{k\\tau } \\right) \\mathrm {dt} .$ Arguing as in (REF ) and (REF ), we conclude $\\int _0^{T-z}\\int _\\Omega (B(\\theta _M(t+z))-B(\\theta _M(t))) (\\theta _M(t+z)-\\theta _M(t))\\mathrm {dx} \\mathrm {dt}\\le \\\\ \\le C\\left( (k\\tau )^{1/\\ell }(k\\tau +\\tau )^{ 1/\\ell ^{\\prime }} + (k\\tau )^{1/2} (k\\tau +\\tau )^{1/2}\\right)=C\\left( 2^{ 1/\\ell ^{\\prime }}+ 2^{ 1/2}\\right) z.$ Thus, all hypothesis of Lemma REF are fulfilled.",
"Therefore, Lemma REF assures that $B(\\theta _M)$ strongly converges to $B(\\theta )$ in $L^1(Q_T)$ .",
"Consequently, up to a subsequence, $B(\\theta _M)$ converges to $B(\\theta )$ a.e.",
"in $Q_T$ .",
"Since $B$ is strictly monotone, $\\theta _M$ converges to $\\theta $ a.e.",
"in $Q_T$ (see, for instance, [19]).",
"Now we are able to identify the limit $Z$ .",
"Proposition 5.4 The limit $Z$ satisfies $Z=\\partial _t (B(\\theta )) \\quad \\mbox{ in } L^{\\ell ^{\\prime } }(0,T;(V_\\ell (\\Omega ))^{\\prime }).$ For a fixed $t$ , there exists $m\\in \\lbrace 1,\\cdots ,M\\rbrace $ such that $t\\in ]t_{m-1,M},t_{m,M}]$ .",
"From definition REF we have $\\int ^t_0 {Z}_M(\\varsigma )\\mathrm {d\\varsigma }=\\sum _{j=1}^{m-1}\\int _{(j-1)\\tau }^{j\\tau }\\frac{B(\\theta ^{j})(\\varsigma )-B(\\theta ^{j-1})(\\varsigma )}{ \\tau }\\mathrm {d\\varsigma } + \\\\+\\int _{(m-1)\\tau }^{t}\\frac{ B(\\theta ^m) (\\varsigma )- B(\\theta ^{m-1})(\\varsigma ) }{\\tau }\\mathrm {d\\varsigma } =\\\\=B(\\theta ^{m-1})- B(\\theta ^{0}) + \\frac{ t-(m-1)\\tau }{\\tau }\\left( B(\\theta ^m) - B(\\theta ^{m-1}) \\right) = \\widetilde{B}(\\theta _M(t)) - B(\\theta ^{0})$ in $\\Omega $ .",
"By the Riesz theorem, the bounded linear functional $v\\in L^2(\\Omega )\\mapsto \\int ^t_0 ({Z}_M(\\varsigma ),v)\\mathrm {d\\varsigma }$ is (uniquely) representable by the element $\\widetilde{B}(\\theta _M(t)) - B(\\theta ^{0}) $ from $L^2(\\Omega ).$ Using the corresponding definitions we compute $\\int _0^T\\Vert \\widetilde{B}(\\theta _M(t)) - B(\\theta _M) \\Vert _{2,\\Omega }^2\\mathrm {dt} =\\sum _{m=1}^M \\Vert Z^m \\Vert _{2,\\Omega }^2\\int _{(m-1)\\tau }^{m\\tau } (t-m\\tau )^2\\mathrm {dt}= \\\\=\\frac{\\tau ^3}{3}\\sum _{m=1}^M \\Vert Z^m \\Vert _{2,\\Omega }^2 =\\frac{\\tau }{3}\\sum _{m=1}^M\\Vert B(\\theta ^m) - B(\\theta ^{m-1}) \\Vert _{2,\\Omega }^2\\le \\\\\\le (b^\\#)^2 \\frac{\\tau }{3}\\sum _{m=1}^M\\Vert \\theta ^m - \\theta ^{m-1} \\Vert _{2,\\Omega }^2.$ Applying (REF ) and Proposition REF we have $\\int _0^T\\Vert \\widetilde{B}(\\theta _M(t)) - B(\\theta _M) \\Vert _{2,\\Omega }^2\\mathrm {dt}\\le C\\left( \\tau ^{1/\\ell +1/\\ell ^{\\prime }} +\\tau ^{1/2+1/2} \\right) = C\\tau ,$ and consequently $\\widetilde{B}(\\theta _M)$ converges to $ B(\\theta ) $ .",
"By the uniqueness of limit, we deduce $\\int ^t_0 Z(\\varsigma )\\mathrm {d\\varsigma }=B(\\theta ) - B(\\theta ^{0}),$ which concludes the proof.",
"We emphasize that the above convergences are sufficient to identify the limit $\\phi $ as stated in the following proposition, but they are not sufficient to identify the temperature $\\theta $ as a solution, because on the one hand the apparent nonlinearity of the coefficients destroy the weak convergence, on the other hand, the weak-weak convergence does not imply weak convergence.",
"Corollary 5.1 Let $(\\theta ,\\phi )$ be in accordance with Propositions REF and REF , then they verify ().",
"Let $(\\theta _M,\\phi _M)$ solve (REF )-(REF ).",
"Applying Propositions REF and REF , and the Krasnoselski theorem to the Nemytskii operators $\\sigma $ and $\\alpha _\\mathrm {S}$ , we have $\\sigma ( \\theta _M )\\nabla \\phi _M\\rightharpoonup \\sigma ( \\theta )\\nabla \\phi &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\alpha _\\mathrm {S}( \\theta _M ) \\nabla \\theta _M\\rightharpoonup \\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla \\theta &\\mbox{ in }& \\mathbf {L}^2(Q_T) \\quad \\mbox{ as } M\\rightarrow +\\infty .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\theta ,\\phi )$ verifies ()."
],
[
"Proof of Theorem ",
"This proof follows mutatis mutandis the structure of the proof of Theorem REF (cf.",
"Subsection REF ).",
"We only sketch its main steps.",
"The uniform estimates to $ {\\theta }_M $ and $ {\\phi }_M $ are as follows.",
"The quantitative estimate (REF ) reads $\\int _\\Omega \\Psi ( \\theta ^m )\\mathrm {dx}+ a_\\# \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ^2+\\frac{\\gamma _\\#}{\\ell ^{\\prime }} \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } ^\\ell \\le \\nonumber \\\\\\le b^\\# \\Vert \\theta ^0\\Vert _{2,\\Omega } ^2 +\\frac{1}{\\ell ^{\\prime }\\gamma _{\\#}^{1/(\\ell -1)} } \\Vert h \\Vert _{\\ell ^{\\prime },\\Sigma _T }^{\\ell ^{\\prime }} +T \\frac{K_2^2(P_2+1)^2}{2 (L_{2})_\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}}^2 ,$ by using the argument to estimate (REF ).",
"Namely, (REF ) reads $\\frac{1}{\\tau }\\int _\\Omega (B(\\theta ^m)-B(\\theta ^{m-1}) )\\theta ^m\\mathrm {dx} +a_\\# \\Vert \\nabla \\theta ^{m} \\Vert _{2,\\Omega }^2+\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds} \\le \\\\\\le \\mathcal {R}(0, \\Vert h(t_{m,M}) \\Vert _{\\ell ^{\\prime },\\Gamma }^{\\ell ^{\\prime }} )+\\frac{1}{\\ell }\\int _\\Gamma \\gamma (\\theta ^{m})|\\theta ^{m}|^{\\ell } \\mathrm {ds},$ where $\\mathcal {R}$ is the increasing continuous function defined in (REF ), with $(L_2)_\\#=a_\\#\\sigma _\\#/ (2 F^\\#\\sigma ^\\#)$ .",
"In addition, summing the quantitative estimate $\\sqrt{\\sigma _\\#} \\Vert \\nabla \\phi ^{m} \\Vert _{2,\\Omega } \\le \\sqrt{\\sigma _\\#}a^\\# \\Vert \\nabla \\theta ^{m-1} \\Vert _{2,\\Omega }+\\frac{K_2(P_2+1)}{ \\sqrt{\\sigma _\\#} } \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}},$ it results in $\\Vert \\nabla \\phi _M\\Vert _{2,Q_T} \\le a^\\# \\Vert \\nabla \\theta _M \\Vert _{2,Q_T}+ T\\frac{K_2(P_2+1)}{ \\sigma _\\#} \\Vert g\\Vert _{2,\\Gamma _\\mathrm {N}} .$ For subsequences of $ {\\theta }_M $ and $ {\\phi }_M $ , the weak convergences hold according to Propositions REF and REF , which guarantee the required result."
],
[
"Existence of solutions to the TE problem",
"The objective is the passage to the limit in the abstract boundary value problems introduced in Section as the time step goes to zero ($M\\rightarrow +\\infty $ ), with the coefficients being defined by $b(\\cdot , v) &=& \\rho (\\cdot ,v)c_\\mathrm {v}(\\cdot , v);\\\\a(\\cdot , v,w)&=&k(\\cdot , v) + T_\\mathcal {M}(w)\\alpha _\\mathrm {S}(\\cdot , v)\\sigma (\\cdot , v);\\\\F(\\cdot , v,w)&=&\\Pi (\\cdot , v)+T_\\mathcal {M}(w),$ where $T_\\mathcal {M}$ is the $\\mathcal {M}$ -truncation function defined by $T_\\mathcal {M}(z)=\\max (-\\mathcal {M},\\min (\\mathcal {M},z))$ .",
"By the definition of truncated functions, we choose $a_\\# &=& k_\\#-\\mathcal {M}\\alpha ^\\#\\sigma ^\\#;\\\\F^\\# &=& \\Pi ^\\#+\\mathcal {M}.$ taking (REF ) into account.",
"Under these choices, the assumptions (REF ), (REF ) and (REF ) imply (REF ), (REF ) and (REF ), respectively.",
"Let us foccus the present proof in accordance with the approximated solutions that are established in Theorem REF .",
"Analogous argument is valid for the approximated solutions that are established in Theorem REF .",
"Let us redefine the electrical current density as $\\mathbf {j}( \\theta , \\phi )= \\sigma ( \\theta )\\nabla \\phi +\\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla \\theta .$ Analogously for $\\mathbf {j}_M= \\mathbf {j}(\\theta _M ,\\phi _M)$ or simply $\\mathbf {j}_M$ and $\\mathbf {j}$ whenever the meaning is not ambiguous.",
"Let $ (\\theta _M, {\\phi }_M )$ solve (REF )-(REF ), which may be rewritten as $\\int _0^T\\int _\\Omega {Z}_M v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} k ( {\\theta }_M )\\nabla {\\theta }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\gamma ( {\\theta }_M )| {\\theta }_M |^{\\ell -2} {\\theta }_M v \\mathrm {ds}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\sigma ( \\theta _M ) \\Pi ( \\theta _M ) \\nabla {\\phi }_M \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\phi _M \\mathbf {j}( \\theta _M , \\phi _M ) \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} =\\nonumber \\\\=\\int _{\\Sigma _T} h_M v \\mathrm {ds}\\mathrm {dt} ;\\qquad \\\\\\int _\\Omega \\mathbf {j}( \\theta _M , \\phi _M)\\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds}, \\qquad $ for every $v\\in V_{\\ell }(Q_T)$ and $w\\in V$ .",
"There exist $\\theta ,\\phi : Q_T\\rightarrow \\mathbb {R}$ and subsequences of $(\\theta _M,\\phi _M)$ , still labelled by $(\\theta _M,\\phi _M)$ , weakly convergent in accordance with Proposition REF .",
"By Corollary REF , $(\\theta ,\\phi )$ verify the electric equality $\\\\\\int _\\Omega \\mathbf {j}( \\theta , \\phi )\\cdot \\nabla w\\mathrm {dx}=\\int _{\\Gamma _{\\rm N}}g w\\mathrm {ds},$ for every $w\\in V$ .",
"We emphasize that the weak convergence of $\\mathbf {j}_M$ to $ \\mathbf {j}$ in $ \\mathbf {L}^2(Q_T)$ , taking (REF )-() into account, is not sufficient to pass to the limit the term $\\phi _M \\mathbf {j}( \\theta _M , \\phi _M ) $ .",
"Moreover, the non smoothness of the coefficients destroy the possibility of obtaining strong convergences of $\\nabla \\theta _M$ and of $\\phi _M$ .",
"Thanks to Proposition REF , we have a.e.",
"pointwise convergence for a subsequence of $\\theta _M$ , which we still denote by $\\theta _M$ .",
"Considering the assumptions (REF )-(), the Nemytskii operators are continuous due to the Krasnoselski theorem, and applying the Lebesgue dominated convergence theorem, we obtain $k( \\theta _M )\\nabla v\\rightarrow k(\\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\alpha _\\mathrm {S}( \\theta _M ) \\nabla v\\rightarrow \\sigma ( \\theta ) \\alpha _\\mathrm {S}( \\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) \\Pi ( \\theta _M ) \\nabla v\\rightarrow \\sigma ( \\theta ) \\Pi ( \\theta ) \\nabla v &\\mbox{ in }& \\mathbf {L}^2(Q_T).$ Applying (REF ) to the following estimates $\\Vert T_\\mathcal {M}(\\phi _M) \\nabla \\theta _M\\Vert _{2,Q_T}\\le \\mathcal {M} \\Vert \\nabla \\theta _M\\Vert _{2,Q_T} ; \\\\\\Vert \\sigma ( \\theta _M ) T_\\mathcal {M}( \\phi _M ) \\nabla \\phi _M\\Vert _{2,Q_T}\\le \\sigma ^\\# \\mathcal {M} \\Vert \\nabla \\phi _M\\Vert _{2,Q_T} ;\\\\\\Vert \\gamma ( \\theta _M ) | \\theta _M|^{\\ell -2} \\theta _M\\Vert _{\\ell ^{\\prime },\\Sigma _T } \\le \\gamma ^\\# \\Vert \\theta _M\\Vert _{\\ell ,\\Sigma _T } .$ there exist $\\Lambda _1 , \\Lambda _2 \\in \\mathbf {L}^2(Q_T)$ and $\\Lambda _3\\in L^{\\ell ^{\\prime }}(\\Sigma _T ) $ such that $T_\\mathcal {M}( \\phi _M )\\nabla \\theta _M\\rightharpoonup \\Lambda _1 &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\sigma ( \\theta _M ) T_\\mathcal {M}( \\phi _M ) \\nabla \\phi _M\\rightharpoonup \\Lambda _2 &\\mbox{ in }& \\mathbf {L}^2(Q_T); \\\\\\gamma ( \\theta _M ) | \\theta _M|^{\\ell -2} \\theta _M\\rightharpoonup \\Lambda _3 &\\mbox{ in }& L^{\\ell ^{\\prime }}(\\Sigma _T ) .$ Thus, we may pass to the limit in (REF ) as $M$ tends to infinity, concluding that $(\\theta ,\\phi )$ verifies $\\int _0^T\\int _\\Omega Z v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} k (\\theta )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+ \\nonumber \\\\ +\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\sigma ( \\theta ) \\alpha _\\mathrm {S}(\\theta ) \\Lambda _1 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} +\\nonumber \\\\+\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 v \\mathrm {ds}\\mathrm {dt}=\\int _{\\Sigma _T} h v \\mathrm {ds}\\mathrm {dt} ,\\quad \\forall v \\in V_\\ell (Q_T) .",
"$ To identify the temperature $\\theta $ as a solution, we need to identify $\\Lambda _1 , \\Lambda _2 \\in \\mathbf {L}^2(Q_T)$ and $\\Lambda _3\\in L^{\\ell ^{\\prime }}(\\Sigma _T ) $ .",
"To prove that $\\Lambda _1= T_\\mathcal {M}( \\phi )\\nabla \\theta $ , let us consider the Green formula $\\int _{Q_T} T_\\mathcal {M}( \\phi _M )\\nabla \\theta _M \\cdot \\mathbf {v}\\mathrm {dx}\\mathrm {dt}=-\\int _{Q_T[|\\phi _M|<\\mathcal {M}]} \\theta _M \\nabla \\phi _M \\cdot \\mathbf {v}\\mathrm {dx}\\mathrm {dt},$ for every $\\mathbf {v}\\in \\mathbf {L}^p(0,T; \\mathbf {W}^{1,p}(\\Omega ))$ such that $\\nabla \\cdot \\mathbf {v}=0$ in $Q_T$ and $\\mathbf {v}\\cdot \\mathbf {n}=0$ in $\\partial \\Omega \\times ]0,T [$ .",
"Next we choose the exponent $p>1$ to ensure the meaning of the involved terms.",
"By $\\theta _M\\in L^\\infty (0,T; L^2(\\Omega ))\\cap L^2(0,T; H^1(\\Omega ))$ and $ H^1(\\Omega )\\hookrightarrow L^{2^*}(\\Omega )$ with $2^*$ being the critical Sobolev exponent, i.e.",
"$2^*=2n/(n-2)$ if $n>2$ and any $2^*>1$ if $n=2$ , making recourse to the interpolation with exponents being $\\frac{\\beta }{2}= \\frac{1-\\beta }{2}+\\frac{\\beta }{2^*}=\\frac{1}{q}, \\qquad (0<\\beta <1),$ then $\\theta _M$ converges to $\\theta $ in $L^q(Q_T)$ for every $q<2(n+2)/n$ .",
"In particular, we take $p>n+2$ such that $\\frac{1}{2}=\\frac{1}{p}+\\frac{1}{q} >\\frac{1}{p}+\\frac{n}{2(n+2)}.$ Consequently, we have that $\\theta _M\\mathbf {v}$ converges to $\\theta \\mathbf {v}$ in $\\mathbf {L}^2 (Q_T)$ .",
"Therefore, the uniqueness of the weak limit implies that $\\Lambda _1= T_\\mathcal {M}( \\phi )\\nabla \\theta $ .",
"In particular, we find $\\int _{Q_T} a(\\theta _M,\\phi _M)\\nabla \\theta _M\\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}\\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} \\int _{Q_T} a(\\theta ,\\phi ) \\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt},$ and consequently (REF ) reads $\\int _0^T\\int _\\Omega Z v\\mathrm {dx}\\mathrm {dt} +\\int _{Q_T} a(\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt}+ \\nonumber \\\\+\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 v \\mathrm {ds}\\mathrm {dt}=\\int _{\\Sigma _T} h v \\mathrm {ds}\\mathrm {dt} ,\\quad \\forall v \\in V_\\ell (Q_T) .",
"\\qquad $ Now, we are in the conditions to identify the limits $\\Lambda _2$ and $\\Lambda _3$ by making recourse to the Minty argument as follows.",
"We rephrase (REF ) as $\\mathcal {J}_M -\\mathcal {J}_1^M-\\mathcal {J}_2^M -\\mathcal {J}_3^M -\\mathcal {J}_4^M -\\mathcal {J}_5^M + \\\\+\\int _{\\Sigma _T} \\left(\\gamma (\\theta _M) |\\theta _{M}|^{\\ell -2} \\theta _{M} -\\gamma (v) |v|^{\\ell -2}v\\right)(\\theta _{M} -v)\\mathrm {ds}\\mathrm {dt} \\ge \\\\ \\ge (L_1)_\\# \\int _{Q_T}|\\nabla (\\theta _M-v) |^2\\mathrm {dx} \\mathrm {dt}+(L_2)_\\#\\int _{Q_T}|\\nabla (\\phi _M - \\phi ) |^2\\mathrm {dx}\\mathrm {dt}\\ge 0,$ where $\\mathcal {J}_M &:=& \\int _{Q_T}\\left( a(\\theta _M,\\phi _M) |\\nabla \\theta _M|^2 +\\sigma (\\theta _M) F (\\theta _M, \\phi _M) \\nabla \\phi _M\\cdot \\nabla \\theta _M\\right)\\mathrm {dx}\\mathrm {dt} + \\\\ &&+\\int _{Q_T}\\left(\\sigma (\\theta _M) |\\nabla \\phi _M|^2 + \\sigma (\\theta _M) \\alpha _\\mathrm {S} (\\theta _M) \\nabla \\theta _M\\cdot \\nabla \\phi _M\\right)\\mathrm {dx} \\mathrm {dt}+ \\\\ &&+\\int _{\\Sigma _T}\\gamma (\\theta _M) |\\theta _M|^\\ell \\mathrm {ds}\\mathrm {dt}; \\\\\\mathcal {J}_1^M &:=& 2\\int _{Q_T} a(\\theta _M,\\phi _M)\\nabla \\theta _M\\cdot \\nabla v\\mathrm {dx}\\mathrm {dt}-\\int _{Q_T} a(\\theta _M,\\phi _M)|\\nabla v|^2\\mathrm {dx}\\mathrm {dt}; \\\\\\mathcal {J}_2^M &:=& 2\\int _{Q_T}\\sigma (\\theta _M)\\nabla \\phi _M\\cdot \\nabla \\phi \\mathrm {dx}\\mathrm {dt} -\\int _{Q_T}\\sigma (\\theta _M) |\\nabla \\phi |^2\\mathrm {dx}\\mathrm {dt} ;\\\\\\mathcal {J}_3^M &:=&\\int _{Q_T} \\sigma (\\theta _M) \\Pi (\\theta _M) \\Big (\\nabla \\phi _M \\cdot \\nabla v +\\nabla \\phi \\cdot \\nabla ( \\theta _M-v) \\Big ) \\mathrm {dx} \\mathrm {dt} ; \\\\\\mathcal {J}_4^M &:=&\\int _{Q_T} \\sigma (\\theta _M) T_\\mathcal {M} (\\phi _M) \\Big (\\nabla \\phi _M \\cdot \\nabla v +\\nabla \\phi \\cdot \\nabla ( \\theta _M-v) \\Big ) \\mathrm {dx} \\mathrm {dt} ; \\\\\\mathcal {J}_5^M &:=&\\int _{Q_T} \\sigma (\\theta _M) \\alpha _\\mathrm {S} (\\theta _M)\\Big ( \\nabla \\theta _M \\cdot \\nabla \\phi +\\nabla v\\cdot \\nabla (\\phi _M - \\phi ) \\Big ) \\mathrm {dx} \\mathrm {dt}.$ Considering the convergences $\\mathcal {J}_1^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} & 2\\int _{Q_T} a(\\theta ,\\phi )\\nabla \\theta \\cdot \\nabla v\\mathrm {dx}\\mathrm {dt} -\\int _{Q_T} a(\\theta ,\\phi ) |\\nabla v|^2\\mathrm {dx} \\mathrm {dt}:= \\mathcal {J}_1;\\\\\\mathcal {J}_2^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) |\\nabla \\phi |^2\\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_2; \\\\\\mathcal {J}_3^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) \\Pi (\\theta ) \\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_3;\\\\\\mathcal {J}_4^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T} \\Lambda _2 \\cdot \\nabla v\\mathrm {dx} \\mathrm {dt} +\\int _{Q_T}\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\cdot \\nabla ( \\theta -v)\\mathrm {dx} \\mathrm {dt} := \\mathcal {J}_4;\\\\\\mathcal {J}_5^M & \\mathrel {\\mathop {\\hspace{0.0pt}\\longrightarrow }\\limits _{M\\rightarrow \\infty }} &\\int _{Q_T}\\sigma (\\theta ) \\alpha _\\mathrm {S} (\\theta )\\nabla \\theta \\cdot \\nabla \\phi \\mathrm {dx}\\mathrm {dt}:= \\mathcal {J}_5,$ we deduce $\\lim _{ M\\rightarrow \\infty }\\mathcal {J}_M\\ge \\int _{\\Sigma _T}\\Lambda _3 v\\mathrm {ds} \\mathrm {dt}+\\int _{\\Sigma _T}\\gamma (v)|v|^{\\ell -2} v(\\theta -v)\\mathrm {ds}\\mathrm {dt}+ \\\\ + \\mathcal {J}_1+\\mathcal {J}_2+ \\mathcal {J}_3 +\\mathcal {J}_4 +\\mathcal {J}_5.$ We continue the Minty argument by taking in (REF ) and (), respectively, the test function $v=\\theta _M$ and the test function $w=\\phi _M$ , in (REF ) the test function $v=\\theta $ , and in (REF ) the test function $w=\\phi $ , we deduce $\\lim _{M\\rightarrow \\infty }\\mathcal {J}_M = -\\int _{Q_T} Z\\theta \\mathrm {dx} \\mathrm {dt}+\\int _{\\Sigma _T} h\\theta \\mathrm {ds} \\mathrm {dt}+\\int _0^T \\int _{\\Gamma _\\mathrm {N}} g \\theta \\mathrm {ds}\\mathrm {dt}=\\\\=\\int _{Q_T} a( \\theta ,\\phi ) |\\nabla \\theta |^2\\mathrm {dx}\\mathrm {dt}+\\int _{Q_T} \\sigma (\\theta ) \\Pi (\\theta )\\nabla \\phi \\cdot \\nabla \\theta \\mathrm {dx} \\mathrm {dt} + \\\\ +\\int _{Q_T} \\Lambda _2\\cdot \\nabla \\theta \\mathrm {dx}\\mathrm {dt}+\\int _{\\Sigma _T} \\Lambda _3 \\theta \\mathrm {ds}\\mathrm {dt}+\\int _{Q_T} \\mathbf {j}( \\theta , \\phi ) \\cdot \\nabla \\phi \\mathrm {dx} \\mathrm {dt}.$ Gathering the above two relations, we find $\\int _{Q_T} a( \\theta ,\\phi )|\\nabla (\\theta - v)|^2\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\Big (\\Lambda _2 -\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\Big )\\cdot \\nabla ( \\theta -v)\\mathrm {dx} \\mathrm {dt} + \\nonumber \\\\+\\int _{\\Sigma _T} \\left(\\Lambda _3 -\\gamma (v)|v|^{\\ell -2}v \\right) (\\theta -v) \\mathrm {ds}\\mathrm {dt} \\ge 0.$ Next, taking $v=\\theta -\\delta \\varphi $ , with $\\varphi \\in \\mathcal {D}(Q_T)$ , and after dividing by $\\delta >0$ , we arrive to $\\delta \\int _{Q_T} a( \\theta ,\\phi )|\\nabla \\varphi |^2\\mathrm {dx} \\mathrm {dt}+\\int _{Q_T} \\Big (\\Lambda _2 -\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi \\Big )\\cdot \\nabla \\varphi \\mathrm {dx} \\mathrm {dt} \\ge 0.$ Finally letting $\\delta \\rightarrow 0^+$ then we obtain $\\Lambda _2 =\\sigma (\\theta )T_\\mathcal {M}(\\phi )\\nabla \\phi $ .",
"We conclude the Minty argument by taking $v=\\theta -\\delta \\varphi $ , with $\\varphi \\in \\mathcal {D}(\\Sigma _T)$ , in (REF ).",
"After dividing by $\\delta >0$ , and finally letting $\\delta \\rightarrow 0^+$ we arrive to $\\int _{\\Sigma _T}(\\Lambda _3 -\\gamma (\\theta ) |\\theta |^{\\ell -2}\\theta )\\varphi \\mathrm {ds}\\mathrm {dt} \\ge 0, \\quad \\forall \\varphi \\in \\mathcal {D}(\\Sigma _T),$ which implies that $\\Lambda _3 =\\gamma (\\theta )|\\theta |^{\\ell -2}\\theta $ .",
"Therefore, the weak formulation (REF ) yields concluding the proof of Theorem REF ."
]
] | 1709.01871 |
[
[
"Nehari's theorem for convex domain Hankel and Toeplitz operators in\n several variables"
],
[
"Abstract We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables.",
"A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows.",
"Let $\\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\\Xi$, consider the Hankel operator $$\\Gamma_f (g)(x)=\\int_{\\Xi} f(x+y) g(y) \\, dy, \\quad x \\in\\Xi.$$ Then $\\Gamma_f$ extends to a bounded operator on $L^2(\\Xi)$ if and only if there is a bounded function $b$ on $\\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2\\Xi$.",
"This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix.",
"In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks."
],
[
"Introduction",
"For an open connected set $\\Xi \\subset {\\mathbb {R}}^d$ , $d \\ge 1$ , let $\\Omega = \\Xi + \\Xi = \\lbrace x + y \\, : \\, x\\in \\Xi , \\, y\\in \\Xi \\rbrace ,$ and consider a distribution $f$ defined on $\\Omega $ .",
"The associated general domain Hankel operator $\\Gamma _f = \\Gamma _{f, \\Xi }$ is the (densely defined) operator $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ , given by $\\Gamma _f (g)(x)=\\int _{\\Xi } f(x+y) g(y) \\, dy, \\quad x \\in \\Xi ,$ where $dy$ is the Lebesgue measure on ${\\mathbb {R}}^d$ .",
"The case $\\Xi = {\\mathbb {R}}_+ = (0, \\infty )$ for $d=1$ corresponds to the class of usual Hankel operators; when represented in the appropriate basis of $L^2({\\mathbb {R}}_+)$ , the operator $\\Gamma _{f, {\\mathbb {R}}_+}$ is realized as an infinite Hankel matrix $\\lbrace a_{n+m}\\rbrace _{n,m =0}^\\infty $ [31].",
"Nehari's theorem characterizes the bounded Hankel operators $\\Gamma _f \\colon L^2({\\mathbb {R}}_+) \\rightarrow L^2({\\mathbb {R}}_+)$ .",
"For a function $g$ on ${\\mathbb {R}}^d$ , we let $\\hat{g} = \\mathcal {F}g$ denote its Fourier transform, $\\hat{g}(\\xi ) = \\mathcal {F}g(\\xi ) = \\int _{\\mathbb {R}^d} g(x) e^{-2\\pi i x \\cdot \\xi } \\, dx, \\quad \\xi \\in \\mathbb {R}^d.$ Theorem (Nehari [25]) Suppose that $f$ is a distribution in ${\\mathbb {R}}_+$ , $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}_+)$ .",
"Then $\\Gamma _f \\colon L^2(\\mathbb {R}_+) \\rightarrow L^2(\\mathbb {R}_+)$ is bounded if and only if there exists a function $b\\in L^\\infty ({\\mathbb {R}})$ such that $\\hat{b}|_{{\\mathbb {R}}_+}=f$ .",
"Moreover, it is possible to choose $b$ so that $ \\Vert \\Gamma _f\\Vert = \\Vert b\\Vert _{L^\\infty }.$ Nehari's theorem is canonical in operator theory.",
"The two most common proofs proceed either by factorization in the single variable Hardy space or by making use of the commutant lifting theorem.",
"For $d > 1$ , the operators $\\Gamma _{f, {\\mathbb {R}}_+^d}$ , $\\Xi = {\\mathbb {R}}_+^d$ , correspond to (small) Hankel operators on the product domain multi-variable Hardy space $H^2_d$ .",
"In this case, the analogue of Nehari's theorem remains true, apart from (REF ), but it is significantly more difficult to prove.",
"It was established by Ferguson and Lacey ($d=2$ ) and Lacey and Terwilleger ($d > 2$ ) [18], [23].",
"A precise statement is given in Theorem REF .",
"The main purpose of this article is to prove Nehari's theorem when $\\Xi \\subset {\\mathbb {R}}^d$ is a simple convex polytope.",
"t1altTheorem REF Let $\\Xi $ be a simple convex polytope, and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ where $\\Omega =2\\Xi $ .",
"Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ When $d = 1$ , the only open connected sets $\\Xi \\subset {\\mathbb {R}}$ are the intervals $\\Xi = I$ .",
"In this case, Theorem REF is due to Rochberg [35], who called the corresponding operators $\\Gamma _{f, I}$ Hankel/Toeplitz operators on the Paley–Wiener space.",
"They have also been called Wiener–Hopf operators on a finite interval [30].",
"These operators have inspired a wealth of theory in the single variable setting – see Section REF , where we shall interpret Theorem REF in the context of Paley–Wiener spaces.",
"Even for $d=1$ , our proof of Theorem REF appears to be new.",
"However, in several variables our proof relies on the Nehari theorem of Ferguson–Lacey–Terwilleger, and can therefore not be used to give a new proof of their results.",
"We shall also consider general domain Toeplitz operators $\\Theta _f=\\Theta _{f, \\Xi } \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ .",
"In this context, $f$ is a distribution defined on $\\Omega = \\Xi - \\Xi $ , and $\\Theta _f$ is densely defined via $\\Theta _f(g)(x) = \\int _{\\Xi } f(x-y) g(y) \\, dy, \\quad x \\in \\Xi .$ If $\\Xi $ after a translation is invariant under the reflection $x \\mapsto -x$ , then the classes of Hankel operators $\\Gamma _{f, \\Xi }$ and Toeplitz operators $\\Theta _{\\widetilde{f}, \\Xi }$ are essentially the same, and Theorem REF immediately yields a boundedness result.",
"This reasoning is applicable to the cube $\\Xi = (0,1)^d$ , for example.",
"t2Corollary REF Let $\\Xi $ be a simple convex polytope such that for some $z \\in {\\mathbb {R}}^d$ it holds that $\\Xi +z = -\\Xi - z$ .",
"Let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =\\Xi - \\Xi = 2\\Xi + 2z$ .",
"Then $\\Theta _f$ is bounded if and only if there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .",
"There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Theta _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ On the other hand, when $\\Xi $ is a proper convex unbounded set, containing an open cone say, it is clear that the boundedness characterizations of $\\Theta _{f, \\Xi }$ and $\\Gamma _{f,\\Xi }$ may be completely different; plainly explained by the fact that $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ in the Toeplitz case, while $\\Omega = \\Xi + \\Xi = 2\\Xi \\subsetneq {\\mathbb {R}}^d$ for Hankel operators.",
"In this setting, identifying the boundedness of $\\Theta _f$ carries none of the subtleties of Nehari-type theorems.",
"In Theorem REF we obtain the expected boundedness result for a class of “cone-like” domains $\\Xi $ .",
"Rather than giving a precise statement here, let us record the following corollary of Theorem REF .",
"cor:exCorollary REF Let $\\Xi \\subset {\\mathbb {R}}^d$ be any open connected domain such that $(1,\\infty )^d \\subset \\Xi \\subset (0,\\infty )^d,$ and let $f \\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ .",
"Then $\\Theta _{f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if $f$ is a tempered distribution and $\\Vert \\hat{f}\\Vert _{L^\\infty ({\\mathbb {R}}^d)} < \\infty $ , and in this case $\\Vert \\Theta _f\\Vert = \\Vert \\hat{f} \\Vert _{L^\\infty }.$ In the final part of the paper we shall give an application of Theorem REF to matrix completion theory, essentially obtained by discretizing Corollary REF when $\\Xi $ is a cube.",
"To avoid introducing further notation, we shall only state the result in words for now.",
"Recall that a Toeplitz matrix is one whose diagonals are constant.",
"An $N \\times N$ $d$ -multilevel block Toeplitz matrix is an $N \\times N$ Toeplitz matrix whose entries are $N\\times N$ $(d-1)$ -multilevel block Toeplitz matrices.",
"Here $N$ could be finite or infinite.",
"A 1-multilevel block Toeplitz matrix is simply an ordinary Toeplitz matrix.",
"A 2-multilevel block Toeplitz matrix is what is usually considered a block Toeplitz matrix where each block itself is Toeplitz.",
"thm:blocktoepTheorem REF Every finite $N \\times N$ $d$ -multilevel block Toeplitz matrix can be extended to an infinite $d$ -multilevel block Toeplitz matrix bounded on $\\ell ^2$ , with a constant which only depends on the dimension $d$ .",
"For scalar Toeplitz matrices ($d=1$ ) this result is well-known [5], [26], [36], [37], although not as firmly cemented in the literature as the Nehari theorem itself; see [28] for a proof based on Parrot's lemma and a discussion of the result's history.",
"For $d=1$ , the converse deduction of Theorem REF starting from Theorem REF can be found in [13].",
"The paper is laid out as follows.",
"In Section we will give a more formal background and introduce necessary notation.",
"We will also discuss the relationship between $\\Gamma _{f, \\Xi }$ , Paley–Wiener spaces, and co-invariant subspaces of the Hardy spaces.",
"In Section we will prove approximation results for distribution symbols with respect to Hankel and Toeplitz operators, allowing us to reduce to smooth symbols.",
"Section briefly outlines what we need to know about convex sets and polytopes.",
"In Section we prove Theorem REF , our Nehari theorem for Hankel operators.",
"We also indicate how the proof extends to certain unbounded polyhedral domains.",
"In Section our main result on Toeplitz operators is shown, Theorem REF .",
"Finally, Section gives the proof of Theorem REF ."
],
[
"Hankel operators on multi-variable Hardy spaces",
"Let us begin by placing Hankel operators $\\Gamma _f$ into the context of classical Hankel operators on Hardy spaces.",
"As before, for $g \\in L^2(\\mathbb {R}^d)$ , let $\\hat{g} = \\mathcal {F}g$ denote its Fourier transform, $\\hat{g}(\\xi ) = \\mathcal {F}g(\\xi ) = \\int _{\\mathbb {R}^d} g(x) e^{-2\\pi i x \\cdot \\xi } \\, dx, \\quad \\xi \\in \\mathbb {R}^d.$ For the inverse transform we write $\\mathcal {F}^{-1}(g)=\\check{g}$ .",
"The product domain Hardy space $H^2_d$ is the proper subspace of $L^2(\\mathbb {R}^d)$ of functions whose Fourier transforms are supported in the cone $\\overline{{\\mathbb {R}}_+^d}$ , ${\\mathbb {R}}_+ = (0,\\infty )$ , $ H^2_d = \\left\\lbrace G \\in L^2(\\mathbb {R}^d) \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{\\mathbb {R}_+^d}\\right\\rbrace .$ We let $P_d \\colon L^2(\\mathbb {R}^d) \\rightarrow H^2_d$ denote the orthogonal projection, and let $J \\colon L^2(\\mathbb {R}^d) \\rightarrow L^2(\\mathbb {R}^d)$ be the involution defined by $JG(x) = G(-x)$ , $x \\in \\mathbb {R}$ .",
"Consider $\\Gamma _f = \\Gamma _{f, \\Xi }$ for $\\Xi = \\mathbb {R}_+^d$ with $f \\in L^2(\\mathbb {R}_+^d)$ .",
"For a dense set of $g, h \\in L^2(\\mathbb {R}_+^d)$ we have that $ \\langle \\Gamma _f g, h \\rangle _{L^2(\\mathbb {R}_+^d)} = \\langle \\check{f} J\\check{g}, \\check{h}\\rangle _{H^2_d}.$ It follows that the (possibly unbounded) operator $\\Gamma _f \\colon L^2({\\mathbb {R}}_+^d) \\rightarrow L^2({\\mathbb {R}}_+^d)$ is unitarily equivalent to the small Hankel operator $Z_{\\check{f}} \\colon H^2_d \\rightarrow H^2_d$ , $Z_{\\check{f}} G = P_d(\\check{f} \\cdot JG).$ Note that any $b$ such that $\\hat{b}|_{\\mathbb {R}_+^d} = f$ generates the same Hankel operator as $\\check{f}$ , $Z_b = Z_{\\check{f}}$ .",
"To justify the above computation easily we assumed that $f \\in L^2({\\mathbb {R}}_+^d)$ .",
"An approximation argument is needed to consider general symbols $f$ , which may only be distributions in ${\\mathbb {R}}_+^d$ .",
"We provide this later in Proposition REF .",
"We can then read off the boundedness of $\\Gamma _f$ from the boundedness of the corresponding Hankel operator on $H^2_d$ .",
"When $d=1$ and $\\Xi =\\Omega ={\\mathbb {R}}_+$ , the analogue of Theorem REF is exactly the classical Nehari theorem.",
"In higher dimensions the corresponding theorem is due to Ferguson–Lacey–Terwilleger [18], [23].",
"In our notation, their results read as follows.",
"Theorem 2.1 Suppose $\\Xi =\\Omega ={\\mathbb {R}}_+^d$ and that $f$ is a distribution in ${\\mathbb {R}}^d_+$ , $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d_+)$ .",
"Then $\\Gamma _f \\colon L^2(\\mathbb {R}^d_+) \\rightarrow L^2(\\mathbb {R}^d_+)$ is bounded if and only if there exists a function $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}_+^d}=f$ .",
"Moreover, there exists a constant $c > 0$ , depending on $d$ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty }\\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ For $d > 1$ it is not possible to take $c=1$ in (REF ), see for example [29]."
],
[
"Hankel operators on bounded domains",
"We now discuss bounded domains $\\Xi $ , the setting of our main result.",
"The only convex bounded domains in ${\\mathbb {R}}$ are the intervals $I\\subset {\\mathbb {R}}$ .",
"Translations, dilations, and reflections carry the operator $\\Theta _{f,I}$ onto $\\Gamma _{\\tilde{f},J}$ , where $J \\subset \\mathbb {R}$ is any other interval and $\\tilde{f}$ arises from transforming $f$ appropriately.",
"In one variable it thus suffices to consider operators $\\Gamma _{f,(0,1)}$ where $\\Xi = (0,1)$ .",
"Rochberg [35] called these operators Hankel operators on the Paley-Wiener space and proved Theorem REF in the one-dimensional case.",
"In the same article [35], it is posed as an open problem to characterize the bounded Hankel operators $\\Gamma _{f,\\Xi }$ when $\\Xi $ is a disc in $\\mathbb {R}^2$ .",
"We are not able to settle this question, but Theorem REF does provide the answer when $\\Xi = (0,1)^d$ is a cube in ${\\mathbb {R}}^d$ .",
"As we will see, the Hankel operators $\\Gamma _{f, (0,1)^d}$ constitute a natural generalization of the Hankel operators on the Paley-Wiener space.",
"On a technical level, the reason that we are able to prove Theorem REF when $\\Xi $ is a simple convex polytope, but not when $\\Xi $ is a ball, is that we rely on Theorem REF .",
"In applying Theorem REF to our situation, the corners of the boundary of $\\Xi $ are actually of help rather than hindrance.",
"We consider the case of a ball to be an interesting open problem for which we do not dare to make a firm conjecture.",
"In view of Fefferman's disproof of the disc conjecture [17], Nehari theorems might turn out to be quite different for balls and polytopes."
],
[
"Toeplitz operators",
"When $d=1$ and $\\Xi ={\\mathbb {R}}_+$ , $\\Omega = {\\mathbb {R}}$ , the operators $\\Theta _f$ are known as Wiener-Hopf operators [11].",
"Analogously with Hankel operators, these can be shown to be unitarily equivalent to Toeplitz matrix operators on $\\ell ^2({\\mathbb {N}})$ .",
"In this case the boundedness characterization is easy to both state and prove, $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }.$ In Theorem REF we extend (REF ) to Toeplitz operators $\\Theta _{f, \\Xi }$ for a class of “cone-like” domains $\\Xi \\subset {\\mathbb {R}}^d$ , for which $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ ."
],
[
"Truncated correlation operators",
"For open connected sets $\\Xi , \\Upsilon \\subset {\\mathbb {R}}^d$ it is also convenient to introduce the more general “truncated correlation operators” $\\Psi _{f, \\Upsilon , \\Xi } \\colon L^2(\\Upsilon )\\rightarrow L^2(\\Xi )$ , defined by $\\Psi _{f}(g)({x})=\\int _{\\Upsilon }f({x}+{y}) g({y}) ~d{y},\\quad {x}\\in \\Xi ,$ where $f$ lives on $\\Omega =\\Xi +\\Upsilon $ .",
"This class of operators includes both general domain Hankel and Toeplitz operators, by letting $\\Upsilon = \\Xi $ and $\\Upsilon = -\\Xi $ , respectively.",
"For our purposes, general truncated correlation operators will only appear in intermediate steps toward proving the main results, but they also carry independent interest.",
"They were introduced in [1], where their finite rank structure was investigated.",
"In [2] it was shown that they have a fundamental connection with frequency estimation on general domains, motivating the practical need for understanding such operators not only on domains of simple geometrical structure.",
"In [3] it is explained how one may infer certain results for the integral operators $\\Psi _f$ from their discretized matrix counterparts.",
"We warn the reader that in naming the operators $\\Gamma _f$ , $\\Theta _f$ , and $\\Psi _f$ we have slightly departed from previous work, reserving the term (general domain) Hankel operator for truncated correlation operators of the form $\\Psi _{f,\\Xi ,\\Xi }$ ."
],
[
"Hankel operators on multi-variable Paley–Wiener spaces",
"Another viewpoint is offered through co-invariant subspaces of the Hardy spaces $H^2_d$ .",
"For a domain $\\Xi \\subset {\\mathbb {R}}^d$ , let $\\operatorname{PW}_\\Xi $ denote the subspace of $L^2({\\mathbb {R}}^d)$ of functions with Fourier transforms supported in $\\Xi $ , $\\operatorname{PW}_\\Xi = \\lbrace G \\in L^2({\\mathbb {R}}^d) \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{\\Xi } \\rbrace .$ In the classical case $\\Xi = (0,1) \\subset {\\mathbb {R}}$ , note that $\\operatorname{PW}_{(0,1)} = H^2_1 \\ominus \\lbrace G \\in H^2_1 \\, : \\, \\operatorname{supp}\\hat{G} \\subset [1,\\infty )\\rbrace = H^2_1 \\ominus \\theta H^2_1,$ where $\\theta (x) = e^{i2\\pi x}, \\quad x \\in \\mathbb {R}.$ Hence $\\operatorname{PW}_{(0,1)}$ is the ortho-complement (in $H^2_1$ ) of $\\theta H^2_1$ , the shift-invariant subspace of $H^2_1$ with inner factor $\\theta $ .",
"This space is usually denoted $K_\\theta $ , $\\operatorname{PW}_{(0,1)} = K_\\theta := (\\theta H^2_1)^\\perp .$ By a calculation similar to (REF ) we see that $\\Gamma _{f, (0,1)}$ is unitarily equivalent to the compression of the Hankel operator $Z_{\\check{f}}$ to $\\operatorname{PW}_{(0,1)}$ , $\\Gamma _{f, (0,1)} \\simeq P_{\\operatorname{PW}_{(0,1)}} Z_{\\check{f}}|_{\\operatorname{PW}_{(0,1)}},$ where $P_{\\operatorname{PW}_{(0,1)}} \\colon H^2_1 \\rightarrow \\operatorname{PW}_{(0,1)}$ denotes the orthogonal projection onto $\\operatorname{PW}_{(0,1)}$ .",
"Such truncated Toeplitz and Hankel operators are now very well studied on general $K_\\theta $ -spaces [6], [7], [9], [10], [14], [20], [27], [30], [36].",
"In the case of the cube $\\Xi = (0,1)^d \\subset {\\mathbb {R}}^d$ , $d > 1$ , the Hankel operator $\\Gamma _{f,\\Xi }$ may, just as for $d=1$ , be understood as the compression of a Hankel operator to a co-invariant subspace of $H^2_d$ .",
"Namely, $\\operatorname{PW}_{(0,1)^d} = \\lbrace G \\in H_d^2 \\, : \\, \\operatorname{supp}\\hat{G} \\subset [0,1]^d\\rbrace = \\lbrace G \\in H_d^2 \\, : \\, \\operatorname{supp}\\hat{G} \\subset \\overline{{\\mathbb {R}}_+^d} \\setminus (0,1)^d\\rbrace ^\\perp .$ If $G \\in H_d^2 \\cap L^\\infty ({\\mathbb {R}}^d)$ , it is clear that $G\\operatorname{PW}_{(0,1)^d}^\\perp \\subset \\operatorname{PW}_{(0,1)^d}^\\perp $ , since $\\mathcal {F}(GH)(\\xi ) = \\int _{{\\mathbb {R}}_+^d} \\hat{G}(y) \\hat{H}(\\xi -y) \\, dy = 0, \\quad H \\in \\operatorname{PW}_{(0,1)^d}^\\perp , \\; \\xi \\in [0,1]^d.$ Hence $\\operatorname{PW}_{(0,1)^d}^\\perp \\subset H^2_d$ is an invariant subspace (under multiplication by bounded holomorphic functions), and as before we have that $\\Gamma _{f, (0,1)^d} \\simeq P_{\\operatorname{PW}_{(0,1)^d}} Z_{\\check{f}}|_{\\operatorname{PW}_{(0,1)^d}},$ where $P_{\\operatorname{PW}_{(0,1)^d}} \\colon H^2_d \\rightarrow \\operatorname{PW}_{(0,1)^d}$ denotes the orthogonal projection onto $\\operatorname{PW}_{(0,1)^d}$ .",
"Finally, let us briefly discuss the viewpoint of weak factorization.",
"The Hardy space $H^1_d$ is defined as the closure of $\\mathcal {F}^{-1}(C_c^\\infty ({\\mathbb {R}}_+^d))$ in $L^1({\\mathbb {R}}^d)$ .",
"Similarly, we define $\\operatorname{PW}^1_\\Xi $ as the closure of $\\mathcal {F}^{-1}(C_c^\\infty (\\Xi ))$ in $L^1({\\mathbb {R}}^d)$ .",
"As is well known, see for example [24], Theorem REF is equivalent to the fact that $H^1_d$ is the projective tensor product of two copies of $H^2_d$ , $ H^1_d = H^2_d \\odot H^2_d,$ with equivalence of norms.",
"Here the projective tensor product norm on $X \\odot X$ , $X$ a Banach space of functions, is given by $\\Vert G\\Vert _{X \\odot X} = \\inf \\left\\lbrace \\sum _j \\Vert G_j\\Vert _{X} \\Vert H_j\\Vert _{X} \\, : \\, G = \\sum _j G_j H_j, \\; G_j, H_j \\in X \\right\\rbrace ,$ $X \\odot X$ being defined as the completion of finite sums $\\sum _j G_j H_j$ in this norm.",
"The reason that Theorem REF is equivalent to (REF ) is the following: by (REF ), $\\Gamma _{f, {\\mathbb {R}}_+^d}$ is bounded if and only if $|\\langle \\check{f}, GH \\rangle _{H^2_d} | \\le C\\Vert G\\Vert _{H^2_d} \\Vert H\\Vert _{H^2_d},$ which means precisely that $\\check{f}$ induces a bounded functional on $H^2_d \\odot H^2_d$ , $\\check{f} \\in (H^2_d \\odot H^2_d)^*$ .",
"On the other hand, the existence of $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}_+^d} = f|_{{\\mathbb {R}}_+^d}$ , so that $\\langle \\check{f}, GH \\rangle _{H^2_d} = \\langle b, GH \\rangle _{H^2_d}$ , $G,H \\in H^2_d$ , means, by the Hahn–Banach theorem, precisely that $\\check{f} \\in (H_d^1)^*$ .",
"Theorem REF yields a similar weak factorization theorem for Paley–Wiener spaces.",
"We postpone the proof to Section , mentioning only that corresponding weak factorization for $K_\\theta $ -spaces plays an important role in [6] and [9].",
"cor:factCorollary REF Let $\\Xi $ be a simple convex polytope, and let $\\Omega = 2\\Xi $ .",
"Then $PW_{\\Omega }^1 = \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ The norms of these Banach spaces are equivalent."
],
[
"Brief historical overview",
"Z. Nehari published his famous theorem in 1957 [25], inspiring the search for analogous statements in other contexts; positive results are themselves often referred to as Nehari theorems.",
"The most natural inquiries are perhaps those related to Hankel operators on Hardy spaces of several variables.",
"Nehari's theorem for the Hardy space of the unit ball was proven by Coifman, Rochberg and Weiss in 1976 [15], but this setting is rather different from the one considered in this paper.",
"For the product domain Hardy space $H^2_d$ , Hankel operators can be defined by either projecting on $H^2_d$ or on the larger space $L^2({\\mathbb {R}}^d) \\ominus H^2_d$ .",
"The first option leads to the “small” Hankel operators considered in Section REF , while the second type of operator is commonly referred to as a “big” Hankel operator.",
"In the notation of Section REF , a small Hankel operator is an operator $\\Psi _{f,{\\mathbb {R}}_+^d,{\\mathbb {R}}_+^d}=\\Gamma _{f,{\\mathbb {R}}_+^d}$ , whereas big Hankel operators are of the form $\\Psi _{f,{\\mathbb {R}}_+^d,{\\mathbb {R}}^d\\setminus \\overline{{\\mathbb {R}}_+^d}}$ .",
"When transferred to operators on the Hardy space of the polydisc, small Hankel operators correspond, in the standard basis, to infinite matrices with a certain block Hankel structure (cf.",
"Section ).",
"The big Hankel operators were extensively studied by Cotlar and Sadosky.",
"In particular, boundedness of the big Hankel operators was characterized in terms of certain $\\operatorname{BMO}$ type estimates in [16].",
"Small Hankel operators were investigated by Janson and Peetre [22] in 1988.",
"They introduced “generalized Hankel and Toeplitz operators” as particular cases of a more general class of pseudo-differential operators called paracommutators.",
"In their terminology, an operator of the form $\\Psi _{f,\\Xi ,\\Upsilon }$ is a generalized Hankel operator if $\\Xi $ and $\\Upsilon $ are open cones and $\\overline{\\Xi } \\cap (-\\overline{\\Upsilon })=\\lbrace 0\\rbrace $ , whereas it is called Toeplitz if $\\Xi \\cap (-\\Upsilon ) \\ne \\emptyset .$ Hence the general domain Hankel operators $\\Gamma _{f,\\Xi }$ are generalized Hankel operators a lá Janson–Peetre whenever $\\Xi $ is a cone with mild restrictions, while $\\Theta _{f,\\Xi }$ is a generalized Toeplitz operator a lá Janson–Peetre for every open cone $\\Xi $ .",
"In the Toeplitz case, a full boundedness characterization is given in [22].",
"In the Hankel case, only sufficient conditions for boundedness and Schatten class membership are provided, in terms of $\\operatorname{BMO}$ and Besov spaces, respectively.",
"As previously mentioned, R. Rochberg considered Hankel operators for bounded domains in 1987 [35], studying the case of a finite interval in one dimension.",
"Furthermore, he posed as an open problem to understand the case when $\\Xi \\subset {\\mathbb {R}}^2$ is a disc.",
"In this latter setting, L. Peng [32] characterized when $\\Gamma _{f, \\Xi }$ belongs to the Schatten class $S_p$ , for $1 \\le p \\le 2$ , in terms of certain Besov spaces adapted to the disc.",
"L. Peng also carried out a similar study [33] for the case of the multidimensional cube, $\\Xi = (-1,1)^d$ , describing membership in $S_p$ for all $p$ , $0 < p < \\infty $ , as well as giving a sufficient condition for boundedness.",
"Since then it seems that the field did not see progress until the results of Ferguson–Lacey–Terwilleger [18], [23] settled the issue of boundedness of small Hankel operators."
],
[
"Distribution symbols",
"Let $\\Xi ,\\Upsilon \\subset {\\mathbb {R}}^d$ be any open connected sets and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ be a distribution on $\\Omega $ , $\\Omega =\\Xi +\\Upsilon $ .",
"We follow the notation of [21] in our use of distributions.",
"We then define the truncated correlation operator $\\Psi _f$ as an operator $\\Psi _{f, \\Upsilon , \\Xi }:C^\\infty _c(\\Upsilon )\\rightarrow C^\\infty (\\Xi )$ by the formula $\\Psi _f(\\varphi )(x)=({f,T_x\\varphi }), \\quad x \\in \\Xi ,$ where $(f,\\varphi )$ denotes the action of $f$ on $\\varphi $We reserve the notation $\\langle f,\\varphi \\rangle $ for scalar products which are anti-linear in the second entry.",
"and $T_x\\varphi (\\cdot )=\\varphi (\\cdot -x).$ Since $T_x\\varphi $ is compactly supported in $\\Omega $ for $x \\in \\Xi $ , it follows that $\\Psi _f(\\varphi )$ this is well-defined and smooth in $\\Xi $ (see e.g.",
"[21]).",
"Since $C^\\infty _c(\\Upsilon )$ is dense in $L^2(\\Upsilon )$ , $\\Psi _f$ gives rise to a densely defined operator on the latter space which extends to a bounded operator $\\Psi _f \\colon L^2(\\Upsilon ) \\rightarrow L^2(\\Xi )$ if and only if $\\Vert \\Psi _f\\Vert =\\sup \\left\\lbrace \\frac{\\Vert \\Psi _f(\\varphi )\\Vert _{L^2(\\Xi )}}{\\Vert \\varphi \\Vert _{L^2(\\Upsilon )}}:~\\varphi \\in C_c^\\infty (\\Upsilon ), ~\\varphi \\ne 0\\right\\rbrace <\\infty .$ It is clear that $\\Psi _f(\\varphi )(x) = \\int f(x+y)\\varphi (y) \\, dy$ whenever $f \\in L^1_{\\textrm {loc}}(\\Omega )$ .",
"By slight abuse of notation, we write the action of $\\Psi _f$ in this way even when $f$ is not locally integrable.",
"The central question in this paper is the following: for which domains $\\Upsilon $ and $\\Xi $ is the boundedness of $\\Psi _f$ equivalent to the existence of a function $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega }=f$ ?",
"Some care must be taken in interpreting this question.",
"For example, the prototypical example of a bounded Hankel operator is the Carleman operator $\\Gamma _{1/x, {\\mathbb {R}}_+} = \\Psi _{1/x, {\\mathbb {R}}_+, {\\mathbb {R}}_+}.$ The symbol $f(x) = \\frac{1}{x}\\chi _{{\\mathbb {R}}_+}(x)$ is in this case not a tempered distribution on $\\mathbb {R}$ (so the meaning of $\\check{f}$ is unclear) – it is, however, the restriction of the tempered distribution $\\operatorname{p.v.",
"}\\frac{1}{x}$ to ${\\mathbb {R}}_+$ .",
"An example with a delta function makes it clear that it is not necessary for $f$ to be locally integrable in $\\Omega $ either.",
"We first record the answer to our question in the trivial direction.",
"Proposition 3.1 Consider any connected open domains $\\Xi ,$ $\\Upsilon \\subset {\\mathbb {R}}^d$ , with associated domain $\\Omega =\\Upsilon +\\Xi $ .",
"Let $b\\in L^\\infty ({\\mathbb {R}}^d)$ be given and suppose $f=\\hat{b}|_{\\Omega }$ .",
"Then $\\Psi _f \\colon L^2(\\Upsilon ) \\rightarrow L^2(\\Xi )$ is bounded and $\\Vert \\Psi _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ For $\\varphi \\in C^\\infty _c(\\Upsilon )$ we have that $ \\Psi _{f}(\\varphi )={\\mathcal {F}}M_b J{\\mathcal {F}}^{-1} \\varphi |_{\\Xi },$ where $M_b$ is the operator of multiplication by $b$ .",
"The statement is obvious from here.",
"Next we establish two technical results on the approximation of distribution symbols by smooth compactly supported functions, Propositions REF and REF .",
"They will help us to overcome the technical issues mentioned earlier, in particular allowing us to deduce Theorem REF from the corresponding statements in [18], [23].",
"Given open connected domains $\\Xi ,$ $\\Upsilon \\subset {\\mathbb {R}}^d$ , let $\\left(\\Upsilon _n\\right)_{n=1}^\\infty $ be an increasing sequence of connected open subdomains $\\Upsilon _n \\subset \\Upsilon $ such that $\\operatorname{dist}(\\Upsilon _n,\\partial \\Upsilon )>1/n, \\quad \\cup _{n=1}^\\infty \\Upsilon _n=\\Upsilon .$ Note that $\\Omega _n = \\Upsilon _n + \\Xi $ is also increasing and satisfies $\\operatorname{dist}(\\Omega _n,\\partial \\Omega )>1/n, \\quad \\cup _{n=1}^\\infty \\Omega _n = \\Omega .$ Let $\\psi \\in C^\\infty _c({\\mathbb {R}}^d)$ be a fixed non-negative function with compact support in the ball $B(0,1/2)$ such that $\\int _{{\\mathbb {R}}^d} \\psi (x) \\, dx =1$ .",
"For $n\\ge 1$ let $\\psi _n(x)={n^d}\\psi (nx),$ so that $(\\psi _n)_{n=1}^\\infty $ is an approximation of the identity.",
"Since $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ and $\\operatorname{supp}\\psi _n \\subset B(0,1/2n)$ , the convolution $f*\\psi _n$ is well-defined as a function in $C^\\infty (\\Omega _{2n})$ .",
"Let $\\rho _n$ be a smooth cut-off function which is 1 in a neighborhood of $\\overline{\\Omega _n}$ but zero in a neighborhood of $\\Omega _{2n}^c$ , and note that $\\rho _n(f*\\psi _n)$ then naturally defines a function in $C^\\infty ({\\mathbb {R}}^n)$ .",
"Finally, for a non-negative function $\\eta \\in C^\\infty _c({\\mathbb {R}}^d)$ with $\\Vert \\eta \\Vert _{L^2} = 1$ , let $\\omega = \\eta * \\tilde{\\eta }$ , where $\\tilde{\\eta }(x) = \\eta (-x)$ .",
"Then $\\omega \\in C^\\infty _c({\\mathbb {R}}^d)$ and $\\omega (0)= \\Vert \\hat{\\omega }\\Vert _{L^1} = 1.$ Let $\\omega _n(x)=\\omega (x/n)$ .",
"We introduce $f_n=\\omega _n\\rho _n (f*\\psi _n)$ as an approximant of $f$ , where the role of $\\omega _n$ is to enforce compact support in case $\\Omega $ is unbounded.",
"By construction, $f_n \\in C_c^\\infty (\\Omega )$ and it is straightforward to check that $f_n \\rightarrow f$ in ${\\mathcal {D}}^{\\prime }(\\Omega )$ .",
"As for $\\Psi _{f_n, \\Upsilon _n, \\Xi }$ , we have the following result.",
"Proposition 3.2 Let $\\Xi ,$ $\\Upsilon $ be connected open domains, $\\Omega =\\Upsilon +\\Xi $ , and suppose $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ .",
"For $n \\ge 1$ , let $\\Omega _n=\\Upsilon _n+\\Xi $ and $f_n$ be constructed as above.",
"Then $\\Vert \\Psi _{f_n,\\Upsilon _n,\\Xi }\\Vert \\le \\Vert \\Psi _{f,\\Upsilon ,\\Xi }\\Vert .$ First note that $\\omega _n(x)=\\int _{{\\mathbb {R}}^d} n^d \\hat{\\omega }(n \\xi )e^{2\\pi i x \\cdot \\xi } \\, d\\xi ,$ the integrand on the right having $L^1$ -norm equal to $\\Vert \\hat{\\omega }\\Vert _{L^1({\\mathbb {R}}^d)}$ .",
"Letting $g_n=\\rho _n (f*\\psi _n)$ , we have for $\\varphi \\in C^\\infty _c(\\Upsilon _n)$ and $x \\in \\Xi $ that $\\Psi _{f_n}(\\varphi )(x)=\\int _{\\Upsilon _n} \\int _{{\\mathbb {R}}^d} n^d\\hat{\\omega }(n\\xi )e^{2\\pi i (x+y)\\cdot \\xi }\\,d\\xi \\, g_n(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}^d} n^d \\hat{\\omega }(n \\xi ){e^{2\\pi i \\xi \\cdot x}}\\Psi _{g_n}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y\\cdot \\xi }\\varphi (y)$ .",
"Since $\\Vert \\varphi _\\xi \\Vert _{L^2} = \\Vert \\varphi \\Vert _{L^2}$ it follows by the triangle inequality (for $L^2$ -valued Bochner integrals) that $\\Vert \\Psi _{f_n, \\Upsilon _n, \\Xi }\\Vert \\le \\Vert \\hat{\\omega }\\Vert _{L^1}\\Vert \\Psi _{g_n, \\Upsilon _n, \\Xi }\\Vert = \\Vert \\Psi _{g_n, \\Upsilon _n, \\Xi }\\Vert .$ This reduces our task to proving that the operators $\\Psi _{g_n, \\Upsilon _n, \\Xi } = \\Psi _{\\rho _n (f*\\psi _n), \\Upsilon _n, \\Xi } = \\Psi _{f*\\psi _n, \\Upsilon _n, \\Xi }$ are uniformly bounded in $n$ .",
"We have for $\\varphi \\in C^\\infty _c(\\Upsilon _n)$ and $x \\in \\Xi $ that $\\Psi _{f*\\psi _n}(\\varphi )(x) &=\\int _{{\\mathbb {R}}^d} \\int _{{\\mathbb {R}}^d} f((x+y)-z) \\psi _n(z) \\, dz~\\varphi (y) \\, dy \\\\&= \\int _{{\\mathbb {R}}^d} f(x+z) \\int _{{\\mathbb {R}}^d} \\psi _n(y-z)\\varphi (y)\\, dy \\,dz=\\Psi _f(\\widetilde{\\psi }_n * \\varphi )(x),$ where $\\widetilde{\\psi }_n(x) = \\psi _n(-x)$ .",
"Since $\\Vert \\widetilde{\\psi }_n * \\varphi \\Vert _{L^2(\\Upsilon )} \\le \\Vert \\psi _n\\Vert _{L^1} \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)} = \\Vert \\psi \\Vert _{L^1} \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)} = \\Vert \\varphi \\Vert _{L^2(\\Upsilon _n)},$ this completes the proof.",
"Suppose that $\\Gamma _{f, {\\mathbb {R}}_+^d} = \\Psi _{f, \\Xi , \\Upsilon }$ is bounded, where $\\Xi = \\Upsilon = {\\mathbb {R}}_+^d$ .",
"In this case, we let $\\Upsilon _n = (2/n, \\infty )^d$ .",
"By Proposition REF we then have that $\\Vert \\Gamma _{f_n, \\Upsilon _n}\\Vert \\le \\Vert \\Psi _{f_n, \\Upsilon _n,\\Xi }\\Vert \\le \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert , \\quad n \\ge 1.$ Since $\\Upsilon _n = z_n + {\\mathbb {R}}_+^d$ , $z_n = (2/n, \\ldots , 2/n)$ , we have that $\\Gamma _{f_n, \\Upsilon _n}(g)(x) = \\Gamma _{\\tilde{f}_n, {\\mathbb {R}}_+^d} (\\tilde{g})(x-z_n),$ where $\\tilde{f}_n(x) = f_n(x+2z_n)$ and $\\tilde{g}_n(x) = g(x+z_n)$ .",
"Since $\\tilde{f}_n \\in L^2({\\mathbb {R}}_+^d)$ , the computation that lead to (REF ) is justified, and we conclude from [18], [23] that there is $b_n \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}_n|_{2\\Upsilon _n} = f_n|_{2\\Upsilon _n}, \\quad \\Vert b_n\\Vert _{L^\\infty } \\le C \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert .$ By Alaoglu's theorem it follows that there is a weak-star convergent subsequence $(b_{n_k})_{k=1}^\\infty $ with limit $b \\in L^\\infty $ having norm less than $C \\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert $ .",
"It remains to prove that $f=\\hat{b}|_{{\\mathbb {R}}_+^d}$ , i.e.",
"that $(f,\\varphi )=(b,\\hat{\\varphi })$ holds for all $\\varphi \\in C^\\infty _c({\\mathbb {R}}_+^d)$ .",
"However, this is clear from the construction; since $\\hat{\\varphi }\\in L^1$ we have that $(b,\\hat{\\varphi })=\\lim _{k\\rightarrow \\infty }(b_{n_k},\\hat{\\varphi })=\\lim _{k\\rightarrow \\infty }(f_{n_k},{\\varphi }) =(f,{\\varphi }).",
"$ In Section we will consider Toeplitz operators $\\Theta _{f, \\Xi }$ for which $\\Omega = \\Xi - \\Xi = {\\mathbb {R}}^d$ .",
"In this case $f * \\psi _n$ is a smooth function defined in all of ${\\mathbb {R}}^d$ , and there is no need to multiply with $\\rho _n$ or to introduce the subdomains $\\Upsilon _n$ .",
"In this case we simply let $f_n = \\omega _n (f * \\psi _n).$ Clearly, $f_n \\rightarrow f$ in ${\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ and we have, with the exact same proof as for Proposition REF , the following approximation result.",
"Proposition 3.3 Let $\\Xi ,$ $\\Upsilon $ be connected open domains for which $\\Omega =\\Upsilon +\\Xi = {\\mathbb {R}}^d$ , and suppose $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ .",
"For $n \\ge 1$ , let $f_n$ be constructed as above.",
"Then $\\Vert \\Psi _{f_n,\\Upsilon ,\\Xi }\\Vert \\le \\Vert \\Psi _{f,\\Upsilon ,\\Xi }\\Vert .$"
],
[
"On convex sets and polytopes",
"We recall some basic properties of convex sets.",
"Given an unbounded convex set $\\Omega \\subset {\\mathbb {R}}^d$ which is either open or closed, its characteristic cone, also known as its recession cone, is the closed set $\\operatorname{cc}_{\\Omega }=\\lbrace x\\in {\\mathbb {R}}^d \\, : \\, \\Omega +x{\\mathbb {R}}_+\\subset \\Omega \\rbrace .$ The support function $h_{\\Omega }:{\\mathbb {R}}^d\\rightarrow (-\\infty ,\\infty ]$ is defined by $h_{\\Omega }({\\theta })=\\sup _{{x}\\in \\Omega } {x}\\cdot {\\theta }.$ We refer to [21] for the basic properties of $h_\\Omega $ .",
"The barrier cone of $\\Omega $ is the set $ \\operatorname{bc}_{\\Omega }=\\lbrace \\theta \\in {\\mathbb {R}}^d \\, : \\, h_{\\Omega }({\\theta })<\\infty \\rbrace .$ The characteristic cone $\\operatorname{cc}_{\\Omega }$ coincides with the polar cone of the barrier cone $\\operatorname{bc}_{\\Omega }$ , that is, $\\operatorname{cc}_{\\Omega } = \\lbrace x \\in {\\mathbb {R}}^d \\, : \\, x \\cdot y \\le 0, \\; \\: \\forall y\\in \\operatorname{bc}_{\\Omega }\\rbrace .$ To give a complete reference for this claim, first note that for closed convex sets $\\Omega $ , $\\operatorname{cc}_{\\Omega }$ coincides with the asymptotic cone of $\\Omega $ , giving (REF ) by [4].",
"When $\\Omega $ instead is open and convex we have that $\\Omega $ is equal to its relative interior $\\operatorname{ri}(\\Omega )$ , and since $\\operatorname{cc}_{\\operatorname{ri}(\\Omega )} = \\operatorname{cc}_{\\overline{\\Omega }}$ [8], it follows that $\\operatorname{cc}_{\\Omega } = \\operatorname{cc}_{\\overline{\\Omega }}$ in this case.",
"We next recall some standard terminology and facts of polytopes, referring to for example [12].",
"By an open halfspace in ${\\mathbb {R}}^d$ we mean a set $H_\\nu ^r=\\lbrace x \\in {\\mathbb {R}}^d \\, : \\, x\\cdot \\nu >r\\rbrace ,$ where $\\nu \\in {\\mathbb {R}}^d$ is a non-zero vector and $r\\in {\\mathbb {R}}$ .",
"A closed half-space is the closure of such a set.",
"A finite intersection of half-spaces is called a polyhedral set.",
"A convex polytope is a bounded polyhedral set.",
"A closed convex polytope is the convex hull of a finite set of points.",
"The minimal set of such points coincides with the extreme points of the polytope, that is, its vertices.",
"If the minimal number of defining hyperspaces of a convex polytope is $d+1$ (equivalently, if it has precisely $d+1$ vertices), the polytope is called a simplex.",
"For a non-closed polytope we define its vertices (and its edges and facets) as those of its closure.",
"The boundary of a polytope set is made up of a finite amount of facets (i.e.",
"$d-1$ dimensional faces), see Corollary 7.4 and Theorem 8.1 of [12].",
"For a polytope $\\Pi $ with vertex $x_j$ , we denote by $\\partial _{\\operatorname{far},x_j}\\Pi $ the part of its boundary made up of all facets not containing $x_j$ .",
"A vertex of a polytope will be called simple if it is contained in precisely $d$ of its edges.",
"We say that a polytope is simple if all of its vertices are simple, which coincides with the standard terminology.",
"Equivalently, this means that each vertex is contained in precisely $d$ of its facets (cf.",
"[12]).",
"By an affine linear transformation we mean a map of the form $A(x)=x_0+L(x)$ where $L$ is a linear map, and we call $x_0$ the origin of such a map.",
"The following simple lemma gives a third characterization of simple vertices.",
"Lemma 4.1 Let $\\lbrace x_j\\rbrace _{j=1}^J$ be the vertices of a closed polytope $\\Pi $ .",
"Then the vertex $x_j$ is simple if and only if it is the origin of an invertible affine transformation $A_j$ such that $\\Pi $ locally coincides with $A_j(\\overline{{\\mathbb {R}}_+^d})$ around $x_j$ , i.e.",
"for any neighborhood $U$ of $x_j$ such that $\\overline{U} \\cap \\partial _{\\operatorname{far}, x_j}\\Pi = \\emptyset $ we have that $A_j^{-1}(\\Pi \\cap U) = \\overline{{\\mathbb {R}}_+^d} \\cap A_j^{-1}(U).$ By compactness it is easy to construct a partition of unity adapted to the vertices of $\\Pi $ .",
"Lemma 4.2 Given a polytope $\\Pi $ with vertices $\\lbrace x_j\\rbrace _{j=1}^J$ there exist functions $\\lbrace \\mu _j\\rbrace _{j=1}^J$ such that $\\mu _j \\in C_c^\\infty ({\\mathbb {R}}^d)$ , $\\sum _{j=1}^J \\mu _j(x) = 1$ for $x \\in \\overline{\\Pi }$ , and $\\operatorname{supp}\\mu _j\\cap \\partial _{\\operatorname{far}, x_j}\\Pi =\\emptyset $ ."
],
[
"General domain Hankel operators",
"We now consider general domain Hankel operators $\\Gamma _{f,\\Xi }$ for convex domains $\\Xi $ .",
"Observe that in this case $\\Omega =\\Xi +\\Xi = 2\\Xi $ .",
"We begin with a proposition that links the bounded Hankel operators with weak factorization.",
"Proposition 5.1 Let $\\Xi $ be an open convex domain.",
"Then $X = \\left\\lbrace \\Gamma _{f, \\Xi } \\, : \\, \\Vert \\Gamma _{f, \\Xi } \\Vert < \\infty \\right\\rbrace $ is a closed subspace of the space of bounded linear operators on $L^2(\\Xi )$ .",
"As a Banach space, it is isometrically isomorphic to the dual space $(\\operatorname{PW}_\\Xi \\odot \\operatorname{PW}_\\Xi )^*$ .",
"More precisely, bounded functionals $\\mu $ on the projective tensor product correspond to distributions $f$ on $\\Omega = 2\\Xi $ , $( f, g ) = \\mu (\\mathcal {F}^{-1}g), \\quad g \\in C_c^\\infty (\\Omega ),$ for which $\\Vert \\Gamma _{f, \\Xi }\\Vert = \\Vert \\mu \\Vert .$ The main fact to be proved is that $\\mathcal {F}^{-1}(C_c^\\infty (\\Omega )) \\subset \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ Since $C_c^\\infty (\\Xi )$ is dense in $L^2(\\Xi )$ , it then follows that $\\mathcal {F}^{-1}(C_c^\\infty (\\Omega ))$ is dense in the product $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ .",
"We will actually show a little more than the claim.",
"Namely, every $g \\in C_c^\\infty (\\Omega )$ can be written $g = \\sum _{k} g_k * h_k, \\quad g_k, h_k \\in L^2(\\Xi ),$ in such a way that the corresponding map $g \\mapsto \\sum _k \\Vert g_k\\Vert _{L^2(\\Xi )} \\Vert h_k\\Vert _{L^2(\\Xi )}$ is continuous from $C_c^\\infty (\\Omega )$ , equipped with the usual test function topology, to $\\mathbb {R}$ .",
"By employing a partition of unity in which each member is compactly supported in a cube, it is sufficient to prove the claim when $\\Xi = (0, 1/2)^d$ .",
"For this we employ Fourier series.",
"Let $\\lambda (t) = 1/2 - |t-1/2|$ , $t \\in [0,1]$ , and let $\\Lambda (x) = \\prod _{i=1}^d \\lambda (x_i), \\quad x \\in (0,1)^d.$ Note that $\\lambda $ is in the Wiener algebra $A([0,1])$ , the space of functions on $[0,1]$ with absolutely convergent Fourier series, equipped with pointwise multiplication.",
"Therefore $\\Lambda $ is in the Wiener algebra $A([0,1]^d)$ , since $\\Lambda $ is a tensor power of $\\lambda $ .",
"Since $g \\in C_c^\\infty ((0,1)^d)$ and $\\Lambda $ is non-zero on compact subsets of $(0,1)^d$ it follows by Wiener's lemma [19] that $g/\\Lambda \\in A([0,1]^d)$ (to apply Wiener's lemma, first modify $\\Lambda $ to be nonzero outside the support of $g$ ).",
"Expanding $g/\\Lambda $ in a Fourier series, $(g/\\Lambda )(x) = \\sum _{k \\in {\\mathbb {Z}}^d} a_k e^{i2\\pi k \\cdot x}, \\quad \\sum _{k \\in {\\mathbb {Z}}^d} |a_k| < \\infty , \\quad x \\in [0,1]^d,$ let $h_k(x) = e^{i2\\pi k \\cdot x} \\chi _{(0,1/2)^d}(x)$ , $g_k = a_k h_k$ .",
"Then a computation shows that $(g_k * h_k)(x) = a_k e^{i2\\pi k \\cdot x} \\Lambda (x), \\quad x \\in (0,1)^d,$ so that $g = \\sum _{k \\in {\\mathbb {Z}}^d} g_k * h_k, \\quad \\sum _{k \\in {\\mathbb {Z}}^d} \\Vert g_k\\Vert _{L^2((0,1/2)^d)} \\Vert h_k\\Vert _{L^2((0,1/2)^d)} < \\infty .$ An inspection of the argument shows that the $g \\mapsto g/\\Lambda $ is continuous from $C_c^\\infty ((0,1)^d)$ to $A([0,1]^d)$ , and therefore $g \\mapsto \\sum _k \\Vert g_k\\Vert _{L^2((0,1/2)^d)} \\Vert h_k\\Vert _{L^2((0,1/2)^d)}$ is continuous on $C_c^\\infty ((0,1)^d)$ as promised.",
"Suppose now that $\\mu \\in (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ .",
"We have just demonstrated that $( f, g ) = \\mu (\\mathcal {F}^{-1}g)$ , $g \\in C_c^\\infty (\\Omega )$ , defines a distribution on $\\Omega $ .",
"Hence we may consider the Hankel operator $\\Gamma _{f, \\Xi }$ .",
"For $g, h \\in C_c^\\infty (\\Xi )$ we have that $ \\langle \\Gamma _{f}g, h \\rangle _{L^2(\\Xi )} = ( f, g * \\bar{h} ) = \\mu (\\mathcal {F}^{-1}g \\cdot \\mathcal {F}^{-1}\\bar{h}).$ Since $\\mu $ is a bounded functional on $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ we conclude that $|\\langle \\Gamma _{f}g, h \\rangle _{L^2(\\Xi )}| \\le \\Vert \\mu \\Vert \\Vert \\mathcal {F}^{-1} g\\Vert _{\\operatorname{PW}_\\Xi } \\Vert \\mathcal {F}^{-1}\\bar{h}\\Vert _{\\operatorname{PW}_\\Xi } = \\Vert \\mu \\Vert \\Vert g\\Vert _{L^2(\\Xi )} \\Vert h\\Vert _{L^2(\\Xi )},$ that is, $\\Gamma _{f, \\Xi }$ is bounded, and in fact $\\Vert \\Gamma _f\\Vert = \\Vert \\mu \\Vert $ .",
"Conversely, if $f$ is a distribution on $\\Omega $ such that $\\Gamma _{f, \\Xi }$ is bounded, it is clear that $f$ induces a bounded functional $\\mu $ on $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ by (REF ).",
"This proves that $X$ is isometrically isomorphic to the Banach space $(\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ , which also entails that $X$ is closed, completing the proof.",
"In the remainder of this section we assume that $\\Xi $ is a convex polytope.",
"Next we prove Theorem REF under the additional assumption that $f$ is supported around one simple vertex of $\\Omega $ .",
"Proposition 5.2 Let $\\Xi \\subset {\\mathbb {R}}^d$ be an open convex polytope, $x$ a simple vertex of $\\Omega =2\\Xi $ , and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ be such that $\\operatorname{supp}f \\cap \\partial _{\\operatorname{far},x} \\Omega = \\emptyset $ .",
"If $\\Gamma _f$ is bounded as an operator on $L^2(\\Xi )$ , then there exists a $b\\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ As in Lemma REF , let $A$ be an affine transformation with origin $x$ such that $A({\\mathbb {R}}_+^d)$ coincides with $\\Xi $ in a neighborhood of $x$ .",
"It is straightforward to verify that it suffices to prove the proposition for $\\Gamma _{f\\circ A, A^{-1}(\\Xi )}$ .",
"Since $A^{-1}(\\Xi )$ is also a convex polytope, we may hence assume that $x=0$ and that $\\Omega $ coincides with ${\\mathbb {R}}_+^d$ in a neighborhood $U$ of $\\operatorname{supp}f$ , $\\overline{U} \\cap \\partial _{far,0} \\Omega = \\emptyset $ .",
"In particular, since $\\Omega $ is a convex polytope, we have that $\\Omega \\subset \\mathbb {R}_+^d$ .",
"Since $\\operatorname{supp}f \\subset U \\cap \\overline{\\Omega }$ and $\\overline{U} \\cap \\partial _{far,0} \\Omega = \\emptyset $ , we can extend $f$ to a distribution on all of ${\\mathbb {R}}_+^d$ by letting it be zero outside $\\Omega $ .",
"Our strategy is to show that the operator $\\Gamma _{f,{\\mathbb {R}}_+^d}$ is bounded and to then apply Theorem REF .",
"For $n\\in {\\mathbb {N}}^d$ let $C_n$ denote the cube $(n_1,n_1+1)\\times \\ldots \\times (n_d,n_d+1)$ .",
"For a set $X \\subset {\\mathbb {R}}_+^d$ , let $P_X \\colon L^2({\\mathbb {R}}_+^d) \\rightarrow L^2({\\mathbb {R}}_+^d)$ denote the orthogonal projection of $L^2({\\mathbb {R}}_+^d)$ onto $L^2(X)$ , and let $r>0$ be such that $2\\sqrt{d}r<\\operatorname{dist}(U \\cap \\overline{\\Omega },\\partial _{far,0}\\Omega ).$ By considering test functions $g \\in C_c^\\infty ({\\mathbb {R}}_+^d)$ such that $\\operatorname{supp}g \\cap \\overline{rC_m} \\subset rC_m$ for every $m$ , we give meaning to the equality $\\Gamma _{f,{\\mathbb {R}}_+^d}=\\left(\\sum _{n\\in {\\mathbb {N}}^d}P_{rC_n}\\right)\\Gamma _{f,{\\mathbb {R}}_+^d} \\left(\\sum _{m\\in {\\mathbb {N}}^d}P_{rC_m}\\right)=\\sum _{m,n\\in {\\mathbb {N}}^d}P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m},$ a term $P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m}$ being non-zero only if $(rC_m+rC_n)\\cap \\operatorname{supp}f \\ne \\emptyset .$ Hence there are only finitely many non-zero terms in the decomposition.",
"Since $\\Vert P_{rC_n}\\Gamma _{f,{\\mathbb {R}}_+^d} P_{rC_m}\\Vert =\\Vert \\Psi _{f,rC_m,rC_n}\\Vert ,$ recalling the definition of $\\Psi _f$ from Section , it therefore suffices to prove that $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert $ is bounded whenever (REF ) holds.",
"If $rC_m,rC_n\\subset \\Xi $ there is nothing to prove since $\\Gamma _{f,\\Xi }$ is bounded by hypothesis.",
"For the other terms, note that (REF ), $\\operatorname{supp}f \\subset U \\cap \\overline{\\Omega }$ , and the choice of $r$ implies that $rC_m+rC_n\\subset \\Omega ,$ since $2\\sqrt{d} r$ is the diameter of $rC_m+rC_n$ .",
"For any $z\\in {\\mathbb {R}}^d$ , $x \\in rC_n$ , and $g \\in C_c^\\infty (rC_m)$ we have that $\\Psi _{f,rC_m,rC_n}(g)(x)=\\int _{rC_m} f(x+y)g(y) \\, dy=\\int _{rC_m+z} f(x+(y-z))g(y-z)dy,$ and hence $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert =\\Vert \\Psi _{f,rC_m+z,rC_n-z}\\Vert .$ In particular, for $z=r(n-m)/2$ we obtain that $\\Vert \\Psi _{f,rC_m,rC_n}\\Vert =\\Vert \\Psi _{f,rC_{\\frac{m+n}{2}},rC_{\\frac{m+n}{2}}}\\Vert .$ However, $2rC_{\\frac{m+n}{2}}=rC_m+rC_n$ so by (REF ) we conclude that $rC_{\\frac{m+n}{2}}\\subset \\Xi $ .",
"The desired boundedness now follows as it did in the first case considered.",
"We have just demonstrated that $\\Vert \\Gamma _{f, {\\mathbb {R}}_+^d}\\Vert < \\infty $ .",
"By Theorem REF there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{{\\mathbb {R}}^d_+} = f$ .",
"This in particular implies that $\\hat{b}|_{\\Omega } = f$ when we return to the initial interpretation of $f$ as a distribution on $\\Omega $ .",
"Theorem 5.3 Let $\\Xi $ be a simple convex polytope, and let $f\\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =2\\Xi $ .",
"Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Gamma _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ Assume that $\\Gamma _f$ is bounded.",
"Let $\\lbrace x_j\\rbrace _{j=1}^J$ be the vertices of $\\Omega $ , and let $\\lbrace \\mu _j\\rbrace _{j=1}^J$ be partition of unity as in Lemma REF .",
"For $\\varphi \\in C_c^\\infty (\\Xi )$ and $x \\in \\Xi $ we have that $\\Gamma _{\\mu _j f}(\\varphi )(x)=\\int _{\\Xi } \\int _{{\\mathbb {R}}^d} \\hat{\\mu }_j(\\xi )e^{2\\pi i (x+y)\\cdot \\xi }\\,d\\xi \\, f(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}^d} \\hat{\\mu }_j(\\xi ){e^{2\\pi i \\xi \\cdot x}}\\Gamma _{f}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y\\cdot \\xi }\\varphi (y)$ .",
"Hence, $\\Gamma _{\\mu _j f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded, $\\Vert \\Gamma _{\\mu _j f}\\Vert \\le \\Vert \\hat{\\mu }_j\\Vert _{L^1} \\Vert \\Gamma _f\\Vert .$ Therefore, by Proposition REF there are functions $b_j \\in L^\\infty $ such that $\\mu _jf=\\hat{b}_j|_\\Omega $ .",
"Thus $f=\\hat{b}|_\\Omega $ , where $b = \\sum _{j=1}^J b_j \\in L^\\infty $ .",
"Conversely, if $f=\\hat{b}|_\\Omega $ , where $b \\in L^\\infty $ , then $\\Gamma _f$ is bounded by Proposition REF .",
"The constant $c$ now arises from abstract reasoning.",
"Consider the Banach space $X = \\left\\lbrace \\Gamma _{f, \\Xi } \\, : \\, \\Vert \\Gamma _{f, \\Xi } \\Vert < \\infty \\right\\rbrace $ of Proposition REF .",
"We have just shown that $b \\mapsto \\Gamma _{\\hat{b} |_{\\Omega }, \\Xi }$ is a map of $L^\\infty $ onto $X$ .",
"The open mapping theorem hence guarantees the existence of $c$ .",
"We immediately obtain the corresponding result for Toeplitz operators, when $\\Xi $ is a simple convex polytope which, possibly after a translation, is symmetric under $x \\mapsto -x$ .",
"Corollary 5.4 Let $\\Xi $ be a simple convex polytope such that for some $z \\in {\\mathbb {R}}^d$ it holds that $\\Xi +z = -\\Xi - z$ .",
"Let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega =\\Xi - \\Xi = 2\\Xi + 2z$ .",
"Then $\\Theta _f$ is bounded if and only if there exists a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .",
"There exists a constant $c > 0$ , depending on $\\Xi $ , such that $b$ can be chosen to satisfy $c\\Vert b\\Vert _{L^\\infty } \\le \\Vert \\Theta _f\\Vert \\le \\Vert b\\Vert _{L^\\infty }.$ In this case $\\Theta _{f} g = \\Gamma _{\\tilde{f}} \\tilde{g}$ , where $\\tilde{f}(x) = f(x + 2z)$ , $x \\in 2\\Xi $ , and $\\tilde{g}(x) = g(-x-2z)$ , $x \\in \\Xi $ .",
"Hence the result follows from Theorem REF .",
"We also deduce the weak factorization result for $\\operatorname{PW}_\\Omega ^1$ , see Section REF .",
"Corollary 5.5 Let $\\Xi $ be a simple convex polytope, and let $\\Omega = 2\\Xi $ .",
"Then $PW_{\\Omega }^1 = \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ The norms of these Banach spaces are equivalent.",
"By Cauchy-Schwarz, the inclusion $I \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow PW_{\\Omega }^1$ is bounded.",
"Since $I$ has dense range by Proposition REF , the adjoint $I^* \\colon (\\operatorname{PW}^1_\\Omega )^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ has empty kernel.",
"Suppose $\\mu \\in (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ .",
"Note that $CG(x) = \\overline{G(-x)}$ defines an anti-linear isometric involution $C \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }$ .",
"This induces an anti-linear isometric involution $D \\colon (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ , $D\\mu (G) = \\overline{\\mu (CG)}, \\quad G \\in \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi }.$ According to Proposition REF , $( f, g ) = \\mu (\\mathcal {F}^{-1}g)$ , $g \\in C_c^\\infty (\\Omega )$ , defines a distribution on $\\Omega $ such that $\\Vert \\Gamma _{f,\\Xi }\\Vert = \\Vert \\mu \\Vert $ .",
"By Theorem REF , there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b}|_{\\Omega } = f$ .",
"Since $\\operatorname{PW}_\\Omega ^1 \\subset L^1({\\mathbb {R}}^d)$ , we can interpret $b$ as an element of $(\\operatorname{PW}_\\Omega ^1)^*$ , $b(G) = \\langle G, b \\rangle _{L^2({\\mathbb {R}}^d)}$ .",
"Then, recalling that $JG(x) = G(-x)$ , we have that $DI^* b (G) = (b, JG) = ( f, \\mathcal {F}^{-1} JG ) = ( f, \\mathcal {F} G ) = \\mu (G), \\quad G \\in \\mathcal {F}^{-1}(C_c^\\infty (\\Omega )),$ that is, $DI^*b = \\mu $ , or $I^*b = D\\mu $ .",
"Since $D$ is an involution, it follows that $I^*$ is onto.",
"In other words, $I^* \\colon (\\operatorname{PW}^1_\\Omega )^* \\rightarrow (\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi })^*$ is a Banach space isomorphism, and therefore the inclusion $I \\colon \\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } \\rightarrow \\operatorname{PW}^1_\\Omega $ is as well.",
"Hence, $\\operatorname{PW}_{\\Xi }\\odot \\operatorname{PW}_{\\Xi } = \\operatorname{PW}^1_\\Omega ,$ and the norms of these two Banach spaces are equivalent, by the open mapping theorem.",
"The method used to prove Theorem REF extends to many unbounded polyhedral sets.",
"Instead of pursuing a general statement, let us consider the example of a strip in ${\\mathbb {R}}^2$ , $ \\Xi = {\\mathbb {R}}_+ \\times (0,1).$ This is an interesting addition to Theorem REF , since $\\Xi $ does not have a simple vertex at infinity.",
"In fact, $\\partial \\Xi $ may be considered to have a cusp point there.",
"Proposition 5.6 Let $\\Xi $ be the strip defined in (REF ), and let $f \\in {\\mathcal {D}}^{\\prime }(\\Omega )$ , $\\Omega = 2\\Xi $ .",
"Then $\\Gamma _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if there is a function $b \\in L^\\infty ({\\mathbb {R}}^d)$ such that $\\hat{b} |_{\\Omega }=f.$ Let $\\nu _1, \\nu _2 \\in C_c^\\infty ({\\mathbb {R}})$ be functions such that $\\nu _1(t) + \\nu _2(t) = 1$ for $t \\in [0,2]$ , $\\nu _1$ vanishes in a neighborhood of 2, and $\\nu _2$ vanishes in a neighborhood of 0.",
"Let $\\mu _j(x) = \\nu _j(x_2), \\quad j=1,2, \\; x = (x_1, x_2) \\in \\Omega .$ Then for $\\varphi \\in C_c^\\infty (\\Xi )$ and $x = (x_1, x_2) \\in \\Xi $ we have that $\\Gamma _{\\mu _j f}(\\varphi )(x)=\\int _{\\Xi } \\int _{{\\mathbb {R}}} \\hat{\\nu }_j(\\xi )e^{2\\pi i (x_2+y_2) \\xi }\\,d\\xi \\, f(x+y)\\varphi (y) \\, dy= \\int _{\\mathbb {R}} \\hat{\\nu }_j(\\xi ){e^{2\\pi i x_2\\xi }}\\Gamma _{f}(\\varphi _\\xi )(x) \\, d\\xi ,$ where $\\varphi _\\xi (y) = e^{2\\pi i y_2 \\xi }\\varphi (y)$ , $y = (y_1, y_2) \\in \\Xi $ .",
"Hence, as before we see that $ \\Vert \\Gamma _{\\mu _j f, \\, \\Xi }\\Vert \\le \\Vert \\hat{\\nu }_j\\Vert _{L^1} \\Vert \\Gamma _{f, \\, \\Xi } \\Vert , \\quad j=1,2.$ As in Proposition REF and Theorem REF it is sufficient to see that $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ and $\\Gamma _{\\mu _2 f} \\colon L^2({\\mathbb {R}}_+ \\times (-\\infty , 1) ) \\rightarrow L^2({\\mathbb {R}}_+ \\times (-\\infty , 1))$ define bounded operators, and by symmetry it is sufficient to consider the first of the two.",
"For $n \\in {\\mathbb {N}}$ , let $S_n$ denote the strip ${\\mathbb {R}}_+ \\times (n, n+1)$ , and let $r > 0$ be such that $2r < \\operatorname{dist}([0,2] \\cap \\operatorname{supp}\\nu _1, 2).$ We decompose $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ according to strips instead of cubes, $\\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} = \\sum _{m,n \\in {\\mathbb {N}}} P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m}.$ There are only a finite number of non-zero terms in this decomposition, and for any such term we by our choice of $r$ that $ rS_m + rS_n \\subset \\Omega .$ For $n,m$ corresponding to a non-zero term, we have that $\\Vert P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m} \\Vert = \\Vert \\Psi _{\\mu _1f, rS_m, rS_n}\\Vert = \\Vert \\Psi _{\\mu _1f, rS_m + z, rS_n - z}\\Vert = \\Vert \\Psi _{\\mu _1f, rS_{\\frac{m+n}{2}}, rS_{\\frac{m+n}{2}}}\\Vert ,$ where $z = (0, r(n-m)/2)$ .",
"Since $rS_{\\frac{m+n}{2}} \\subset \\Xi $ by (REF ) and $\\Gamma _{\\mu _1 f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded by (REF ), we conclude that each non-zero term $P_{rS_n} \\Gamma _{\\mu _1 f, \\, {\\mathbb {R}}_+^2} P_{r S_m}$ is bounded.",
"Hence $\\Gamma _{\\mu _1 f} \\colon L^2({\\mathbb {R}}_+^2) \\rightarrow L^2({\\mathbb {R}}_+^2)$ is bounded, finishing the proof."
],
[
"General domain Toeplitz operators",
"In this section we consider general domain Toeplitz operators on open convex domains $\\tilde{\\Xi }\\subset {\\mathbb {R}}^d$ such that both $\\operatorname{cc}_{\\tilde{\\Xi }}$ and $\\operatorname{bc}_{\\tilde{\\Xi }}$ have non-empty interior (as in the classical case $\\tilde{\\Xi }={\\mathbb {R}}_+$ ).",
"This forces $\\tilde{\\Xi }$ to be unbounded and, as we shall soon see, it also entails that $\\tilde{\\Omega }=\\tilde{\\Xi }-\\tilde{\\Xi }= {\\mathbb {R}}^d$ .",
"We shall also consider more general open connected sets $\\Xi $ such that there are points $x_0$ and $x_1$ for which $x_1+\\tilde{\\Xi }\\subset \\Xi \\subset x_0+\\tilde{\\Xi },$ and prove that $\\Vert \\Theta _{f, \\Xi }\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ under this hypothesis.",
"This allows for domains $\\Xi $ with very irregular boundaries, in sharp contrast to Theorem REF .",
"The corresponding class of operators $\\Theta _{f,\\Xi }$ partially extends the class of generalized Toeplitz operators considered in [22], see Section REF .",
"The next theorem can also be recovered by verifying the hypotheses of and keeping track of the constants in the proof of [22].",
"However, for completeness we prefer to give our own concrete proof.",
"Theorem 6.1 Let $\\Xi $ be a set as above.",
"Then $\\Xi -\\Xi ={\\mathbb {R}}^d$ and, for $f\\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ , we have that $\\Theta _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if $f\\in {\\mathcal {F}}^{-1}(L^\\infty )$ .",
"Moreover, $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ .",
"Fix $z\\in {\\mathbb {R}}^d$ and set $|z|=R$ .",
"Pick a vector $e\\in \\operatorname{int}(\\operatorname{cc}_{{\\tilde{\\Xi }}})$ with distance greater than $R$ to the complement of $\\operatorname{cc}_{\\tilde{\\Xi }}$ , which is possible since $\\operatorname{cc}_{{\\tilde{\\Xi }}}$ is a cone with non-empty interior.",
"Then $e+z\\in \\operatorname{cc}_{{\\tilde{\\Xi }}}$ , so for any $x \\in \\tilde{\\Xi }$ we have that $x_1+x+e+z\\in x_1+\\tilde{\\Xi }\\subset \\Xi $ .",
"Similarly, $x_1+x+e\\in \\Xi $ .",
"Since $z$ is the difference of these two vectors, the first claim follows.",
"Suppose that we have proven the theorem for all $f \\in C_c^\\infty ({\\mathbb {R}}^d)$ .",
"If $f$ is a general symbol for which $\\Theta _f$ is bounded, consider the sequence of functions $f_n \\in C_c^\\infty ({\\mathbb {R}}^d)$ from Proposition REF .",
"Then $\\hat{f}_n$ has, by Alaoglu's theorem, a subsequence $\\hat{f}_{n_k}$ which converges weak-star in $L^\\infty $ to some element $g$ .",
"Since $f_n$ converges to $f$ in distribution, it must be that $g = \\hat{f}$ .",
"Hence $f\\in {\\mathcal {F}}^{-1}(L^\\infty )$ and, by Propositions REF and REF , we have that $\\Vert \\hat{f} \\Vert _{L^\\infty } \\le \\varlimsup _{k \\rightarrow \\infty } \\Vert \\hat{f}_{n_k} \\Vert _{L^\\infty } = \\varlimsup _{k \\rightarrow \\infty } \\Vert \\Theta _{f_{n_k}}\\Vert \\le \\Vert \\Theta _{f}\\Vert \\le \\Vert \\hat{f} \\Vert _{L^\\infty }.$ This proves the theorem for general symbols.",
"Hence we assume that $f \\in C_c^\\infty ({\\mathbb {R}}^d)$ .",
"Fix $\\xi \\in {\\mathbb {R}}^d$ , pick any vector $\\nu $ in $\\operatorname{int}(\\operatorname{bc}_{\\tilde{\\Xi }})$ , and consider for $\\varepsilon > 0$ the function $E_{\\varepsilon }(x)=e^{\\varepsilon x\\cdot \\nu +2\\pi i x\\cdot \\xi }\\chi _{\\Xi }(x), \\quad x \\in \\Xi .$ By [1] this function is in $L^2(x_0+\\tilde{\\Xi })$ ,The set $\\operatorname{bc}_{\\tilde{\\Xi }}$ was denoted $\\Theta $ in [1].",
"and hence $E_\\varepsilon \\in L^2(\\Xi )$ .",
"We use $E_\\varepsilon $ as a test function: $\\Vert \\Theta _f\\Vert \\ge \\left|\\frac{\\langle \\Theta _f E_{\\varepsilon },E_{\\varepsilon }\\rangle }{\\Vert E_\\varepsilon \\Vert ^2}\\right| &= \\left|\\frac{1}{\\Vert E_\\varepsilon \\Vert ^2}\\int \\int f(x-y)e^{\\varepsilon (x+y)\\cdot \\nu }e^{2\\pi i (y-x)\\cdot \\xi }\\chi _{\\Xi }(y)\\chi _{\\Xi }(x) \\, dy \\, dx\\right| \\\\&=\\left|\\frac{1}{\\Vert E_\\varepsilon \\Vert ^2}\\int f(z)e^{-2\\pi i z\\cdot \\xi }\\int e^{\\varepsilon (z+2y) \\cdot \\nu }\\chi _{\\Xi }(z+y)\\chi _{\\Xi }(y) \\, dy \\,dz\\right|$ Hence it follows that $\\Vert \\Theta _f\\Vert \\ge |\\hat{f}(\\xi )|$ upon showing that $\\lim _{\\varepsilon \\rightarrow 0^+} \\frac{e^{\\varepsilon z \\cdot \\nu }}{\\Vert E_\\varepsilon \\Vert ^2} \\int e^{2\\varepsilon y \\cdot \\nu }\\chi _{\\Xi }(z+y)\\chi _{\\Xi }(y) \\, dy = 1$ uniformly on compacts in $z$ .",
"Since $\\xi $ is arbitrary this establishes that $\\Vert \\Theta _f\\Vert \\ge \\Vert \\hat{f}\\Vert _{L^\\infty }$ and by Proposition REF we then conclude that $\\Vert \\Theta _f\\Vert =\\Vert \\hat{f}\\Vert _{L^\\infty }$ .",
"Fix $R>0$ and suppose that $z \\in {\\mathbb {R}}^d$ with $|z|<R$ .",
"Again, pick a vector $e\\in \\operatorname{int}(\\operatorname{cc}_{{\\tilde{\\Xi }}})$ with distance greater than $R$ to the complement of $\\operatorname{cc}_{{\\tilde{\\Xi }}}$ .",
"Then $e+z\\in \\operatorname{cc}_{{\\tilde{\\Xi }}}$ , and therefore $-z+\\Xi \\supset -z+(x_1+\\tilde{\\Xi })\\supset -z+x_1+(e+z)+\\tilde{\\Xi }\\supset x_1+e-x_0+x_0+\\tilde{\\Xi }\\supset x_1+e-x_0+\\Xi .$ With $x_2=x_1+e-x_0$ we have just shown that $x_2+\\Xi \\subset -z+\\Xi $ .",
"It also holds that $x_2+\\Xi \\subset \\Xi $ , by the last inclusion in the above chain and the fact that $x_1+e+\\tilde{\\Xi }\\subset x_1+\\tilde{\\Xi }\\subset \\Xi $ .",
"This gives us that $ \\chi _{\\Xi }(y-x_2)=\\chi _{\\Xi }(y)\\chi _{\\Xi }(y-x_2)\\le \\chi _{\\Xi }(y)\\chi _{\\Xi }(y+z)\\le \\chi _{\\Xi }(y),$ and hence that $e^{\\varepsilon 2x_2 \\cdot \\nu }\\Vert E_\\varepsilon \\Vert ^2 =\\int e^{2\\varepsilon y \\cdot \\nu }\\chi _{\\Xi }(y-x_2) \\, dy\\le \\int e^{2\\varepsilon y \\cdot \\nu }\\chi _{\\Xi }(y+z) \\chi _{\\Xi }(y) \\, dy\\le \\int e^{2\\varepsilon y \\cdot \\nu }\\chi _{\\Xi }(y)\\,dy =\\Vert E_\\varepsilon \\Vert ^2.$ The desired equality (REF ) is now immediate, completing the proof.",
"Corollary 6.2 Let $\\Xi \\subset {\\mathbb {R}}^d$ be any open connected domain such that $(1,\\infty )^d \\subset \\Xi \\subset (0,\\infty )^d,$ and let $f \\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ .",
"Then $\\Theta _{f} \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded if and only if $f$ is a tempered distribution and $\\Vert \\hat{f}\\Vert _{L^\\infty ({\\mathbb {R}}^d)} < \\infty $ , and in this case $\\Vert \\Theta _f\\Vert = \\Vert \\hat{f} \\Vert _{L^\\infty }.$"
],
[
"Bounded extension of multi-level block Toeplitz/Hankel-matrices",
"In this section we interpret Corollary REF , when $\\Xi $ is a $d$ -dimensional cube, as a result on the possibility of extending finite multi-level block Toeplitz matrices to infinite multi-level block Toeplitz matrices which are bounded as operators on $\\ell ^2$ .",
"In view of the equivalence between Toeplitz and Hankel operators on the cube (cf.",
"the proof of Corollary REF ), and a similar equivalence for finite Hankel and Toeplitz matrices, we could equally well make the analogous statement for multi-level block Hankel matrices.",
"We present only the Toeplitz-case.",
"Such matrices appear in various applications, for example in multi-dimensional frequency estimation.",
"Note in particular that Pisarenko's famous method for one-dimensional frequency estimation [34], which relies on the classical Carathéodory-Fejér theorem, was recently extended to the multi-variable case [38] (see also [3]).",
"When $d=1$ our statement reduces to a well-known theorem on extending finite (ordinary) Toeplitz matrices, appearing previously for example in [5] and [26].",
"To describe it, recall that a finite $N\\times N$ Toeplitz-matrix is characterized by its constant diagonals, whose values we denote by $a = (a_{-N+1},\\ldots a_{N-1})$ .",
"As an operator $T_a$ on $\\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace )$ , its action is given by $T_a(v)(m)=\\sum _{n=0}^{N-1}a_{m-n}v_n, \\quad v \\in \\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace ), \\; m \\in \\lbrace 0,\\ldots ,N-1\\rbrace .$ We can also consider the case when $N = \\infty $ , the definitions extending in the obvious way.",
"The completion result then states that it is always possible to extend $a$ to a bi-infinite sequence $\\tilde{a}$ such that the corresponding Toeplitz operator $T_{\\tilde{a}}\\colon \\ell ^2({\\mathbb {N}}) \\rightarrow \\ell ^2({\\mathbb {N}})$ satisfies $\\Vert T_{\\tilde{a}}\\Vert \\le 3\\Vert T_a\\Vert .$ It is an open problem whether the constant 3 is the best possible in this inequality.",
"A discussion offering different approaches to the optimal constant can be found in [9].",
"See also [36].",
"When $d>1,$ each multi-sequence $a=(a_n)_{n\\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d},$ generates a multi-level block Toeplitz matrix $T_a$ .",
"As an operator on $\\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace ^d)$ it is given by the formula $T_a(v)(m)=\\sum _{n\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}a_{m-n}v_n, \\quad v \\in \\ell ^2(\\lbrace 0,\\ldots ,N-1\\rbrace ^d), \\; m \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d.$ To understand this matrix, consider the $d$ -level block Toeplitz matrix $T_a$ as an ordinary $N \\times N$ -Toeplitz matrix with entries which are $(d-1)$ -level block Toeplitz matrices, $T_a = \\lbrace A_{i-j}\\rbrace _{i,j \\in \\lbrace 0,\\ldots ,N-1\\rbrace }, \\quad A_i = \\lbrace a_{(i,m-n)}\\rbrace _{m,n \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^{d-1}}.$ For instance, a multi-level block Toeplitz matrix for $d=2$ is an $N\\times N$ Toeplitz matrix whose entries are $N \\times N$ Toeplitz matrices.",
"Again, we allow for the possibility that $N=\\infty $ .",
"We now provide the multi-level block Toeplitz matrix analogue of the Toeplitz matrix completion theorem.",
"Theorem 7.1 There exists a constant $C_d > 0$ such that any finite multi-sequence $a$ can be extended to an infinite multi-sequence $\\tilde{a}$ on ${\\mathbb {Z}}^d$ for which $T_{\\tilde{a}} \\colon \\ell ^2({\\mathbb {N}}^d) \\rightarrow \\ell ^2({\\mathbb {N}}^d)$ is bounded with norm $\\Vert T_{\\tilde{a}}\\Vert \\le C_d\\Vert T_a\\Vert .$ Let $f=\\sum _{n\\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d} a_n\\delta _n,$ where $\\delta _n$ is the Dirac delta function at $n$ , $\\delta _n(\\varphi )=\\varphi (n), \\quad \\varphi \\in C_c^\\infty ({\\mathbb {R}}^d).$ Set $\\Xi =(0,N)^d$ and consider $\\Theta _f=\\Theta _{f,\\Xi }$ .",
"Given $g\\in C^\\infty _c(\\Xi )$ , a short calculation shows that $\\Theta _f(g)(x)=\\sum _{n\\in {\\mathbb {Z}}^d\\cap (x-\\Xi )} a_n g(x-n), \\quad x \\in (0, N)^d.$ With $x=m+r$ , where $m \\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d$ and $r\\in [0,1)^d$ , this can be rewritten $\\Theta _f(g)(m+r)=\\sum _{k\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d} a_{m-k}g(r+k).$ In other words, with $g_r=\\lbrace g(r+n)\\rbrace _{n\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}$ , we have that $\\Theta _f(g)(m+r)=T_a(g_r)(m).$ Hence $\\sum _{m\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}|\\Theta _f(g)(m+r)|^2=\\Vert T_a(g_r)\\Vert ^2\\le \\Vert T_a\\Vert ^2\\Vert g_r\\Vert ^2=\\Vert T_a\\Vert ^2\\sum _{m\\in \\lbrace 0,\\ldots ,N-1\\rbrace ^d}|g(m+r)|^2.$ Integrating both sides over $r \\in (0,1)^d$ gives us that $\\Vert \\Theta _f(g)\\Vert ^2\\le \\Vert T_a\\Vert ^2\\Vert g\\Vert ^2$ .",
"In other words, $\\Theta _f \\colon L^2(\\Xi ) \\rightarrow L^2(\\Xi )$ is bounded and $\\Vert \\Theta _f\\Vert \\le \\Vert T_a\\Vert .$ Noting that the constant $c$ in Corollary REF is invariant under homotheties, we find that there exists a distribution $\\tilde{f} = \\hat{b} \\in {\\mathcal {D}}^{\\prime }({\\mathbb {R}}^d)$ , coinciding with $f$ on $(-N,N)^d$ , such that $\\Vert \\Theta _{\\tilde{f},{\\mathbb {R}}^d}\\Vert \\le C_d\\Vert T_a\\Vert ,$ where $C_d$ only depends on the dimension $d$ .",
"Of course, $\\Theta _{\\tilde{f},{\\mathbb {R}}^d} \\colon L^2({\\mathbb {R}}^d) \\rightarrow L^2({\\mathbb {R}}^d)$ is nothing but the operator of convolution with $\\tilde{f}$ .",
"Now pick any function $\\varphi \\in C^\\infty _c((-1/2,1/2)^d)$ with $\\int |\\varphi |^2 dx=1$ and consider the isometry $I \\colon \\ell ^2({\\mathbb {N}}^d)\\rightarrow L^2({\\mathbb {R}}^d)$ given by $I v(x)=\\sum _{n\\in {\\mathbb {N}}^d}v_n \\varphi (x-n), \\quad v \\in \\ell ^2({\\mathbb {N}}^d), \\; x \\in {\\mathbb {R}}^d.$ Then $I^* g(n) = \\int _{{\\mathbb {R}}^d} g(x) \\overline{\\varphi (x-n)} \\, dx, \\quad g \\in L^2({\\mathbb {R}}^d), \\; n \\in {\\mathbb {N}}^d.$ It follows that $I^*\\Theta _{\\tilde{f}} I v (m) = \\sum _{n \\in {\\mathbb {N}}^d} \\tilde{a}_{m-n} v_n, \\quad v \\in \\ell ^2({\\mathbb {N}}^d), \\; m \\in {\\mathbb {N}}^d,$ where $\\tilde{a}_n = \\int _{{\\mathbb {R}}^d} \\int _{{\\mathbb {R}}^d} f(x-y + n) \\varphi (y) \\overline{\\varphi (x)} \\, dy \\, dx, \\quad n \\in {\\mathbb {Z}}^d.$ That is, $I^*\\Theta _{\\tilde{f}}I = T_{\\tilde{a}}$ .",
"It is clear by construction that $\\tilde{a}$ is an extension of $a$ , $\\tilde{a}_n = a_n \\int _{{\\mathbb {R}}^d} |\\varphi (y)|^2 \\, dy = a_n, \\quad n \\in \\lbrace -N+1,\\ldots ,N-1\\rbrace ^d.$ This finishes the proof."
]
] | 1709.01843 |
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